Relativity in Quantum Chemistry
with applications to group IV hydrides and EuO69-
RIJKSUNIVERSITEIT GRONINGEN
Relativity in Quantum Chemistry
with applications to group IV hydrides and EuO69-
PROEFSCHRIFT
ter verkrijging van het doctoraat in de
Wiskunde en Natuurwetenschappen
aan de Rijksuniversiteit Groningen
op gezag van de
Rector Magnificus Dr. S. K. Kuipers
in het openbaar te verdedigen op
vrijdag 7 februari 1992
des namiddags te 2.45 uur precies
door
Olivier Visser
geboren op 31 maart 1964
te Vreeswijk
Promotor
: Prof. Dr. W. C. Nieuwpoort.
Referent
: Dr. P. J. C. Aerts.
Voorwoord.
In dit voorwoord maak ik van de gelegenheid gebruik een aantal personen voor hun
directe en indirecte bijdragen aan het tot stand komen van dit proefschrift te
bedanken.
Allereerst bedank ik mijn promotor, Wim Nieuwpoort, voor de ruimte die hij mij
tijdens het onderzoek heeft gegeven. Ook de nauwkeurige bestudering van het
manuscript, en de daaruit voortvloeiende discussies, heb ik zeer gewaardeerd en zijn
buitengewoon nuttig geweest. Patrick Aerts bedank ik voor zijn stimulerende
begeleiding en voor de vele waardevolle op- en aanmerkingen met betrekking tot dit
proefschrift.
Mijn speciale dank gaat uit naar Luuk Visscher. Met name de ontwikkeling van de
'general contraction' methode en de implementatie van de Breit correctie hebben
belangrijk bijgedragen aan de uitvoering van dit onderzoek. Bovendien wil ik hem
bedanken voor zijn optreden als paranimf.
Ria Broer bedank ik voor het kritisch lezen van hoofdstuk 5. Jaap Snijders en Prof.
Baerends van de Vrije Universiteit Amsterdam bedank ik voor het nuttige
commentaar dat zij in de loop der jaren bij verschillende gelegenheden gegeven
hebben. Bauke Kooistra bedank ik voor de hulp bij allerlei groepen-theoretische
problemen. Met name zijn hulp bij het gebruik van subgroepketens is nuttig gebleken.
Johan Heijnen bedank ik voor de hulp bij het ontwerpen van de omslag. Bij deze
bedank ik ook Hirzo Merenga, onder andere voor zijn hulp bij allerlei technische
problemen en voor zijn optreden als paranimf.
Bert van Korler en Jaap Hollenberg bedank ik voor hun hulp bij het gebruik van de
verschillende supercomputers via SARA (de Cyber 205, de NEC-SX2 en de CRAY
Y-MP). Ook de bijdrage van het NLR is van groot belang geweest. Bij deze bedank ik
met name Theo Kuilboer en Marco Schoenmaker voor hun hulp bij het gebruik van
de NEC-SX2, waardoor de berekeningen beschreven in hoofdstuk 4 mogelijk werden.
De operateurs van de verschillende computersystemen (bij het rekencentrum van de
RUG, bij SARA en bij het NLR) zijn ook onmisbaar geweest. Hierbij mijn dank.
I want to thank Ken Dyall (NASA, Ames) and Knut Faegri (Oslo University) for the
helpfull meetings and discussions. In particular, the atomic relativistic program
provided by Ken Dyall has shown to be of great practical value.
Wat zeker niet onvermeld mag blijven, zijn de fantastische kinderkampen (van
Humanitas en SMS) waaraan ik als leider heb deelgenomen. Zulke kampen zijn niet
alleen onvergetelijk voor de kinderen, maar ook voor de leiding. Geen enkele andere
vakantie had mij zoveel plezier en (geestelijke) ontspanning kunnen bezorgen. Alle
kids en medeleiding bedank ik voor die prettige tijden, en ik hoop nog vele vakanties
in kinderkampen door te brengen.
Tot slot wil ik een aantal mensen bedanken voor de gezellige jaren in Groningen.
Allereerst mijn ouders, die er op vele manieren voor hebben gezorgd dat ik het er naar
mijn zin had. Dan de verschillende leden van de werkgroep Theoretische Chemie, die
de afgelopen jaren voor een plezierige werksfeer hebben gezorgd. Hans Wubs en
Lianne Koster wil ik bij deze bedanken voor hun veelvuldige aangename gezelschap.
En Tjeerd Wierda weet zelf wel wat hij voor mij heeft betekend. Bedankt.
Acknowledgment.
This work was sponsored by the Netherlands Foundation for Chemical Research
(SON), the Netherlands Foundation for Fundamental Research on Matter (FOM) and
the National Computing Facilities Foundation (NCF) for the use of supercomputer
facilities, with financial support from the Netherlands Organization for the
Advancement of Research (NWO).
Table of contents.
1. Introduction. .................................................................................... 1
1.1.
Aim and organisation.........................................................................1
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
Dirac equation....................................................................................3
Example of Dirac eigenvalue spectrum. ............................................4
Reduction to two-component form. ...................................................5
Nature of the solutions. ......................................................................6
Many electron equation. ....................................................................6
Symmetry. ..........................................................................................8
1.7.1. Dirac double group symmetry. ...........................................8
1.7.2. Time-reversal symmetry. ....................................................9
Other methods..................................................................................10
1.8.1. No-pair approximation. ....................................................11
1.8.
1.8.2. Perturbation theory. ..........................................................11
1.8.3. Numerical integration methods.........................................12
1.8.4. Dirac-Fock-Slater method. ...............................................12
1.8.5. Relativistic effective core potentials. ................................12
1.9. References........................................................................................13
2. Methodology. .................................................................................. 15
2.1.
Basis set expansion. .........................................................................15
2.1.1. Kinetic and atomic balance. ..............................................16
2.1.2. Basis set optimization. ......................................................19
2.2.
2.3.
2.4.
2.5.
Open Shell Hartree-Fock-Dirac-(Roothaan) method. ......................20
CI method. .......................................................................................22
Symmetry: Pitzers theorem and Dacre and Elder list. .....................23
Overview of the MOLFDIR program package. ...............................24
2.5.1. MOLFDIR. .......................................................................24
2.5.2. RELONEL and RELTWEL. .............................................24
2.5.3. MFDSCF. .........................................................................25
2.5.4. ROTRAN. .........................................................................25
2.5.5. GOSCIP. ...........................................................................25
2.6.
References........................................................................................26
3. Non-relativistic valence-only calculations with relativistic core
and relativistic perturbation corrections ......................................... 27
3.1.
3.2.
Introduction......................................................................................27
Description of the method. ..............................................................28
3.2.1. Résumé of the Hartree-Fock-Dirac method. .....................28
3.2.2. Frozen core approach. .......................................................29
3.2.3. Two-component formalism. .............................................29
3.2.4. Non-relativistic two-component valence-only method.....31
3.2.5. Perturbation corrections. ...................................................33
3.3.
3.2.6. Indirect and direct effects. ................................................34
Testcase: the SnH4 molecule. ..........................................................35
3.4.
3.5.
3.3.1. Technical details. ..............................................................35
3.3.2. Results and discussion. .....................................................35
Conclusions. ....................................................................................41
References........................................................................................42
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit)
calculations on CH4, SiH4, GeH4, SnH4 and PbH4 ....................... 45
4.1.
4.2.
Summary. .........................................................................................45
Introduction......................................................................................45
4.3.
Theory. .............................................................................................46
4.3.1. General..............................................................................46
4.3.2. The relativistic open shell SCF equations. .......................48
4.3.3. Basis functions. .................................................................48
4.3.4. Kinetic balance and General contraction. .........................49
4.3.5. COSCI approach. ..............................................................50
Applications. ....................................................................................51
4.4.1. Computational details. ......................................................51
4.4.2. Results and Discussion. ....................................................53
4.4.
4.5.
4.6.
4.4.2.1. Atomic results. ....................................................53
4.4.2.2. Molecular results. ................................................54
4.4.2.2.1. Accuracy. ............................................55
4.4.2.2.2. Hartree-Fock-Dirac results. ................56
4.4.2.2.3. Effect of the Breit interaction. ............56
4.4.2.2.4. Comparison with other methods. ........58
Conclusions. ....................................................................................60
References........................................................................................61
5. Molecular Open Shell CI calculations using the Dirac-Coulomb
hamiltonian. The f6-manifold of an embedded EuO69- cluster ................ 63
5.1.
5.2.
5.3.
5.4.
Abstract. ...........................................................................................63
Introduction......................................................................................64
Theory. .............................................................................................65
5.3.1. General..............................................................................65
5.3.2. Open shell approach. ........................................................67
5.3.3. The relativistic open shell SCF equations. .......................67
5.3.4. The COSCI method. .........................................................70
5.3.5. Basis functions. .................................................................70
5.3.6. Kinetic balance and general contraction. ..........................71
The Ba2GdNbO6:Eu system. ...........................................................72
5.4.1.
5.4.2.
Description of the system. ................................................72
Physical model. .................................................................74
5.5.
Computational details. .....................................................................74
5.5.1. Basis sets...........................................................................74
5.5.2. Madelung potential. ..........................................................76
5.5.3. The EuO69- clusters. .........................................................78
5.6.
Results. ............................................................................................78
5.6.1. The free Eu3+ ion. .............................................................78
5.6.2. The Eu3+ impurity. ...........................................................79
5.6.3. The Spectrum. ...................................................................84
5.7. Conclusions. ....................................................................................86
5.8. References........................................................................................86
6. Summary and conclusions. ............................................................. 89
6.1.
6.2.
6.3.
Summary. .........................................................................................89
Conclusions. ....................................................................................90
References........................................................................................91
Samenvatting. ...................................................................................... 93
Appendix A: ......................................................................................... 97
A.1. General.............................................................................................97
A.2. Basic algorithm. ...............................................................................98
A.3. Permutation symmetry. ....................................................................99
A.3.1. PSym 1:.............................................................................99
A.3.2. PSym 2:...........................................................................100
A.3.3. PSym 3:...........................................................................101
A.4. Spin. ...............................................................................................102
A.5. The algorithm using permutation symmetry and spin. ..................103
A.6. Dirac double group symmetry. ......................................................104
A.7. Speed-ups.......................................................................................106
A.8. Timings. .........................................................................................107
A.9. References......................................................................................107
Appendix B: Sn basis set. .................................................................. 109
Appendix C: H basis set. ................................................................... 111
1. Introduction
1.
Introduction.
1.1. Aim and organisation.
The ab initio study of relativistic effects on the chemical and physical properties of
molecules is a relatively new field of research. Only recently fully relativistic selfconsistent-field (SCF) and configuration interaction (CI) calculations can be
performed for molecular systems at the same level of approximation as is now
customary in the non-relativistic domain. Relativistic effects are of importance for
many chemical properties.
Valence shell properties as the strength and character of chemical bonds, emission
and absorption spectra and magnetic behaviour are strongly influenced by relativity
when atoms with Z > 50 are present. Inner shell properties such as determined by
photo and Auger electron spectroscopy, and by X-ray absorption and emission
measurements, are influenced at even lower Z values depending on the shells
involved. The importance of relativity can be inferred from the observed break-down
of Russell-Saunders coupling in atomic spectra. For example, the ratio of valence
level splittings attributed to, respectively, the spin-orbit interaction and the
electrostatic electron-electron interactions grows from 4-6% for Br (Z=35) and Kr
(Z=36) to 10-14% for I (Z=53) and Xe (Z=54) and to 40% for Po (Z=84) and Rn
(Z=86). One should keep in mind that the spin-orbit interaction, although the most
conspicuous, reveals only part of the relativistic effects. As is well known, when these
effects are formulated as perturbations on the Schrödinger hamiltonian, the massvelocity and Darwin corrections appear at the same order (~1/c2) of perturbation
theory.
Quite a few methods have been devised to deal with relativistic effects. A good
overview is given in a number of review articles [1-6]. Also, in the bibliography by
Pyykkö [7] most methods used to include relativistic effects in chemical systems can
be found. Some of them (perturbation theory, local density methods, the no-pair
1
1. Introduction
approximation and relativistic effective core potentials) are briefly described at the
end of this chapter. Since, among other things, we want to be able to investigate the
effects of various approximations that are made in many methods, we use the allelectron Hartree-Fock-Dirac method (HFD). This method is the relativistic analogue
of the non-relativistic Hartree-Fock method (HF). The HFD method can be based on
the Dirac-Coulomb or Dirac-Coulomb-Breit hamiltonian and can be used as a starting
point for subsequent inclusion of electron correlation in the same manner as is
commonly done in non-relativistic calculations.
The HFD method has been proposed in 1935 by Swirles to study the properties of
atoms [8]. Nowadays several computer programs are available that solve the HFD
equations for many-electron atoms [9, 10]. These programs often include the
possibility to calculate correlation effects by multiconfiguration methods.
Applications of the Hartree-Fock-Dirac method to molecules have been sparse [11,
12, 13], however, and are limited mainly to diatomic molecules with closed shell
electron configurations. Programs that solve the Hartree-Fock-Dirac equations for
molecules of general shape still are few [14-16]. One of the first has been developed
in our laboratory [14, 15].
For many, if not most, systems of interest a closed shell Hartree-Fock-Dirac approach
is not sufficient. Therefore, open shell methods are necessary. In this work we will
demonstrate the feasibility of molecular open shell Hartree-Fock-Dirac calculations
followed by complete open shell configuration interaction calculations. From the
results, we will try to find an answer to the question whether or not relativistic effects
need to be taken into account from the outset. We will also be able to study the effect
of the Breit interaction [17] (a relativistic correction to the two-electron Coulomb
interaction) on bond properties.
The material in this thesis is organized as follows. The first chapter contains a brief
overview of the theory used in this work. At the end of this chapter, a very brief
description of some alternative approaches to treat relativistic effects is given. In
chapter two we describe how we solve the many-electron Dirac-Coulomb-(Breit)
equation using an SCF technique and a basis set expansion approach. The third
chapter presents an analysis of the direct and indirect core and valence contributions
to the relativistic effects in SnH4, calculated using a frozen Hartree-Fock-Dirac core.
The next chapter completes a study initiated by Aerts [14, 18] on bond length
contractions and the effect of the Breit interaction on these contractions for the
molecules CH4, SiH4, GeH4, SnH4 and PbH4, at the Hartree-Fock-Dirac level. In
2
1. Introduction
chapter 5 we study the luminescence spectrum of an Eu3+ impurity in a solid. This
study requires a good relativistic description of the f6-manifold of the EuO69- ion as
well as a proper embedding method to take the effects of the surrounding crystal into
account. The same system has been studied by Van Piggelen (1978) using a nonrelativistic method [19]. In the last chapter the conclusions of the previous chapters
are summarized, and some future developments are indicated. Finally some material
of a rather technical nature (a detailed description of the relativistic four-index
transformation program and the basis sets used for the SnH4 calculations) is collected
in a number of appendices.
1.2. Dirac equation.
In this work, the Dirac equation [20, 21] forms the basic equation that describes the
states of an electron according to both special relativity and quantum mechanics. A
good description of the Dirac equation can be found in Moss [22] and in other
textbooks [23-25].
The time-independent Dirac equation for an electron with mass m and charge -e
moving in an electrostatic potential  reads
c p + mc2 - e  = E
(1)
In this equation, c is the speed of light, and  p is short-hand for
 p = xpx + ypy + zpz
(2)
The 's and  are operators which can be represented by hermitian matrices of
dimension four:
x =
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
0
; y =
0
0
0
i
0
0
-i
0
0
i
0
0
-i
0
0
0
; z =
0
0
1
0
0
0
0
-1
1
0
0
0
0
-1
0
0
;=
1
0
0
0
0
1
0
0
0
0
-1
0
0
0
0
-1
(3)
This representation is called the standard representation. It is often easier to write
these matrices as two-by-two matrices which contain two-by-two matrices as
elements:
x = 0 x ; y = 0 y ; z = 0 z ;  = 1 0
x 0
y 0
z 0
0 -1
3
(4)
1. Introduction
The 's are the Pauli matrices:
x = 0 1 ; y = 0 -i ; z = 1 0
1 0
i 0
0 -1
(5)
The operator on the left-hand side of equation (1) is usually referred to as the Dirac
hamiltonian. When the Dirac operator is represented by a four-by-four matrix, the
Dirac equation is recognized as a set of four coupled first order differential equations.
The solutions  are thus four-component functions:
1
=
2
(6)
3
4
1.3.
Example of Dirac eigenvalue spectrum.
Figure 1: Dirac spectrum for = Z/r.
To get an impression of the eigenvalue
spectrum of the time-independent
Dirac equation, consider an electron in
the Coulomb field of a point charge Z.
The eigenvalue spectrum has the form
sketched in figure 1. For Z=0 the
positive and negative eigenvalues form
a continuum of states, separated by a
gap of 2mc2. A positive or negative
point charge introduces bound states in
mc 2
E
0
the spectrum from respecectively the
top or the bottom of the gap. The
bound states for Z>0 correspond to an
electron bound to a positive nucleus
Z<0
Z=0
Z>0
(as in the hydrogen atom), while the
bound states for Z<0 correspond to a positron bound to a negative nucleus (as in
positronium).
- mc 2
4
1. Introduction
1.4. Reduction to two-component form.
Consider the time-independent Dirac equation for an electron with potential energy V:
c p + mc2 + V  = 
(7)
V = -e
(8)
with
When the expectation values of the kinetic energy c p and the potential energy V
are small, the term mc2 dominates the equation, so the eigenvalues  are
approximately
-mc2 or +mc2. We subtract mc2 from the energy eigenvalues to facilitate comparison
with non-relativistic results.
Equation (7) in bi-spinor form reads:
V
c p
c p
V-
2mc2
L
S
=
L
S
(9)
From equation (9), we can express S in terms of L:
S =  + 2mc2 - V -1 c p L
(10)
Using this equation, we can eliminate S and get:
V + c p  + 2mc2 - V -1 c p L = L
(11)
Equation (10) and (11) together are equivalent to the Dirac equation within the
domain for which the operators involved are properly defined.
We can expand the inverse operator in (11), and in the non-relativistic limit, c,
this expansion reduces to
V + 1  p  p L,NR = NRL,NR
2m
Because of the operator equality
5
(12)
1. Introduction
 p  p = p2
(13)
equation (12) is just the Schrödinger equation (in two components):
p2
+ V L,NR = NRL,NR
2m
(14)
The 'small component' part of the solution S is given by (10) and is seen to be an
order of c-1 smaller than the 'large component' part L. Nevertheless, the small
component is just as important as the large component since the kinetic energy arises
from the coupling between them.
1.5. Nature of the solutions.
Equations (10), (11) and (14) demonstrate that the positive energy solutions, with
eigenvalues near zero, correspond to the non-relativistic states, in the sense that the
upper two components (the 'large components') resemble the non-relativistic electron
solutions for the two possible spin states. The lower two components (the 'small
components') are small compared to the upper two.
We can also study the negative energy solutions by shifting the eigenvalues with
+mc2 followed by elimination of L. It follows that for these solutions, the 'small'
components are large, and the 'large' components are small. It can be shown that these
solutions describe the behaviour of a particle with charge +1 [22, 24]. Therefore,
these states are usually interpreted as positron states.
1.6. Many electron equation.
In order to perform relativistic quantum mechanical calculations on systems of
chemical interest, we search for an equation which describes many electrons in the
field of the (fixed) nuclei. One can imagine a many-electron generalisation of the
Dirac equation formed by a sum of Dirac operators, one for each electron, completed
by a two-electron interaction term representing the Coulomb interaction between
electrons:
• hDi + • gij
i
i = Ei i
(15)
i<j
with
6
1. Introduction
2
hD
i = ci  pi + i mc + Vi
(16)
gij = r1
(17)
and
ij
and i is a function of all electron coordinates. However, equation (15) is not quite
relativistically invariant.
To describe the electromagnetic interaction between two moving particles within the
theory of special relativity, one has to use retarded potentials. It is possible to find an
analogue to the interaction from these potentials that can be used in relativistic
quantum mechanics, the Breit interaction [17, 22, 24]. A more satisfying derivation of
the two-electron interaction term follows from quantum electrodynamics [2, 17, 26].
The dominant term in the interaction in the low-frequency limit is the Breit
interaction:
1 1  i   j +  i rij  j rij
gBr
ij = r ij
2 rij
r3
(18)
ij
The first (and dominant) term in this equation is just the Coulomb interaction.
Equation (18) can be approximated further leading to the Gaunt interaction [1, 2, 27]:
1 i  j
gG
ij = r - r
ij
(19)
ij
The derivation of the Breit interaction is based on perturbation theory, and it is
therefore often stated (but still a matter of discussion) that the Breit term should not
be included in a variational treatment of the Dirac-Coulomb equation (15) [2, 26, 28,
29].
According to Brown and Ravenhall [30], the Dirac-Coulomb equation (15) has no
normalizable solutions unless subsidiary, physical, constraints are added that restrict
the freedom to occupy the sea of negative energy states. However, from quantum
electrodynamics, a many-electron relativistic equation can be derived that has bound
states [26,31-33].
7
1. Introduction
In this derivation, the Dirac field is expanded in a complete set of one-particle
functions. These functions are the negative (positron) and positive (electron) energy
eigen functions of the Dirac hamiltonian with an arbitrary potential (possibly zero).
The definition of an 'electron' thus depends on the potential used. The approximation
that we shall have a fixed number of electrons (the 'no pair' approximation) leads to a
many-electron relativistic equation that resembles the Dirac-Coulomb equation
closely, but differs from it by the appearance of projection operators to the left and
right of the potential energy terms. The consequences of electron-positron pair
creation processes and higher order corrections can subsequently be investigated by
perturbation theory.
It has been shown that the algorithm used in this work to solve the Dirac-Coulomb(Breit) equation (Hartree-Fock-Dirac followed by CI calculations in the positive
energy space) effectively leads to the normalizable solutions of the no-pair equations
[33].
1.7.
Symmetry.
1.7.1.
Dirac double group symmetry.
Use of the symmetry of a molecule not only leads to qualitative insights before doing
any calculation, but also to significant savings of computational resources when
calculations are carried out. In non-relativistic quantum mechanics it is usually
sufficient to consider only symmetry operators that act on the spatial coordinates of
the system (rotations and reflections). For the Dirac equation, however, one needs
symmetry elements which leave the Dirac hamiltonian invariant. These elements are a
subset of the product space of the same spatial operators as used for the nonrelativistic problem, and the spin-operators which operate on the spin-components of
the functions. The set of operators which leave the Dirac hamiltonian invariant form
the Dirac double group. In this section the relation between the usual point group
operators and the Dirac double group operators will be given. The theory of double
groups, the Dirac double group and other symmetry properties of the Dirac
hamiltonian can be found elsewhere [34 and references therein].
8
1. Introduction
If we have a molecular point group
(20)
G = Ri , i=1.. G
of operators Ri that leave the nuclear skeleton of a molecule invariant (|G| is the order
of the group), the corresponding Dirac double group is given by
G*D = Oi Ri , -Oi Ri , i=1.. G
(21)
where Oi is an operator represented by a four-by-four matrix operating on the spincomponents. The operators Oi can be written as
Oi =
Ui
0
0
Ui
(22)
(the minus sign appears if an improper rotation is involved). Ui is a unitary two-bytwo matrix which depends on the spatial operator Ri. We will not need the explicit
form of these operators. Since Ui is a unitary matrix, Oi is also a unitary matrix.
1.7.2.
Time-reversal symmetry.
The Schrödinger as well as the Dirac hamiltonian commute with the time-reversal
operator T. This is an antiunitary operator:
T (a+ b) = a*T + b*T
(23)
TT = 
(24)
and
The effect of the operator T on the wave function, assuming that T commutes with the
hamiltonian and with the operator  t, can be illustrated using the time-dependent
Dirac equation [34]:
(r,t)
ih= hD(r,t)
t
(25)
Operating on this equation with the operator T gives
9
1. Introduction
(r,t)
T(r,t)
Tih= -ih= hDT(r,t)
t
t
(26)
and by changing t in -t we get
T(r,-t)
ih= hDT(r,-t)
t
(27)
It is thus clear that operating with T on a solution of the Dirac equation yields another
solution with the direction of the time coordinate reversed.
It can be shown [24] that the explicit form of the operator T which commutes with the
Dirac hamiltonian is given by
T = -iyE2K
(28)
where K is the complex conjugation operator, and E2 is a two-by-two unit matrix. If
we require that an operator A is invariant under time-reversal, we obtain relations
between the different 'spin' blocks of the matrix representation of the operator, which,
provided that the basis functions  are real, read:
 p  A  q =  p  A  q
*
(29)
and
 p  A  q = -  p  A  q
*
(30)
In (29) and (30) we have introduced the 'spin-functions'  and  which form the basis
of the matrix representation of y in (28).
1.8. Other methods.
In this work we use the Hartree-Fock-Dirac SCF method followed by CI calculations
to solve the Dirac-Coulomb equation for molecules. The method will be described in
some detail in the next chapter. Several other techniques are in use to obtain
information about relativistic effects in chemical systems. In this section some of
these methods are mentioned. Often, combinations of these are used to describe
10
1. Introduction
relativistic effects in molecules. More references can be found in the bibliography by
Pyykkö [7].
1.8.1.
No-pair approximation.
The "no-pair approximation" has already been described in section 1.6. The results
depend on the potential used to generate the positive energy projection operators. The
simplest method (free electron projection operators) leads to simple equations [35,
36], but a potential closer to the physical situation leads to much better results [37,
38]. The resulting equations are rather complicated and therefore sometimes reduced
to a one-component formalism without spin-orbit like contributions. The latter can be
accounted for by perturbation theory.
The Hartree-Fock-Dirac SCF method is equivalent to a no-pair approximation. The
potential used to define the positive energy projection operators is the self-consistent
field potential. According to Mittleman [33], this is the 'best possible' potential within
this approximation.
1.8.2.
Perturbation theory.
The most important relativistic effects can be included approximately using
perturbation theory. The relativistic perturbation corrections can be found in many
textbooks [e.g. 24, 25, 39, 40]. The leading one-electron perturbation terms are the
spin-orbit term, the mass-velocity term and the Darwin term, but in fact an infinite
number of higher order corrections exists. The perturbation terms can be derived from
the Dirac-Coulomb-(Breit) equation by a reduction to a two-component form, either
by eliminating the small component or by a series of unitary transformations (the
Foldy-Wouthuysen method [41]). Since some of the perturbation terms depend on the
potential generated by the other electrons, it is for some properties essential to include
the first-order corrections to the orbitals in an iterative manner [42, 43].
For light systems, first order perturbation theory using the leading relativistic
perturbation terms is expected to give good results. For heavy systems, however,
higher order relativistic perturbation terms cannot be neglected.
11
1. Introduction
1.8.3.
Numerical integration methods.
The direct solution of the many-electron Dirac-Coulomb equation by numerical
integration has proven to be a very successful method for atoms [9, 10]. Several
program packages have been extended to include correlation effects as well as higher
order quantum electrodynamic corrections. As in the non-relativistic case these
numerical calculations are extremely useful as references for basis set methods. For
molecules of general shape numerical techniques are still too expensive to be of
practical use. Therefore applications have been restricted to atoms and small linear
molecules [44, 45].
1.8.4.
Dirac-Fock-Slater method.
The Dirac-Fock-Slater method is based on a local density approximation to the
density functional theory: the exchange-correlation term is replaced by a local
potential. The resulting equations, which look like the Hartree-Fock-Dirac equations
with the exchange part replaced by a local potential [46, 47], are easier to solve and
have been applied to a large number of molecules. However, the use of a local
exchange-correlation potential is an approximation that up to this moment has not
been related in a transparant, exploitable way to some form of explicit many-electron
theory.
1.8.5.
Relativistic effective core potentials.
An easy to use but very approximate manner to treat relativistic effects is based on the
concept of relativistic effective core potentials [34, 48-58 and references therein].
Similar to what is often done in non-relativistic calculations, in these methods the
Hartree-Fock-Dirac equation is replaced by a non-relativistic Hartree-Fock equation
with an additional potential that should take care of all relativistic effects including
the interaction with the frozen relativistic core. This potential is generally a twocomponent potential, but in some methods an average scalar potential is employed
and the effects of the two-component potential (mainly the spin-orbit effect) are
accounted for by perturbation theory.
Although such methods have, for some cases, shown their use in practice, the
theoretical foundation is not very strong. Also, the approximations involved make it
hard to extract reliable and clean information on relativistic effects from the results.
12
1. Introduction
1.9. References.
1. Pyykkö, P., Adv. Quantum Chem. 11 (1978) 353.
2. Grant, I. P., Quiney, H. M., Adv. Atomic and Mol. Phys. 23 (1988) 37.
3. Wilson, S., Grant, I. P., Gyorffy, B. L. (eds), The effects of relativity in atoms,
molecules, and the solid state. Plenum Press, New York (1991).
4. Malli, G. L. (ed.), Relativistic effects in Atoms, Molecules, and Solids. NATO
ASI Series B., Physics v. 87, Plenum Press, New York (1981).
5. Pyykkö, P., Chem. Rev. 88 (1988) 563.
6. Balasubramanian, K., Pitzer, K. S., in: Lawley, K. P. (ed.), Ab initio Methods in
Quantum Chemistry - I, Wiley (1987) 287.
7.
8.
9.
10.
11.
12.
13.
14.
Pyykkö, P., Relativistic Theory of Atoms and Molecules. Springer-Verlag
Berlin (1986).
Swirles, B., Proc. Roy. Soc. London A 152 (1935) 625.
Desclaux, J. P. Int. J. Quantum Chem. 6 (1972) 25.
Dyall, K. G., Grant, I. P., Johnson, C. T., Parpia, F. A., Flummer, E. P.,
Computer Phys. Comm. 55 (1989) 425.
Desclaux, J. P., Pyykkö, P., Chem. Phys. Lett. 29 (1974) 534.
Lee, Y. S., McLean, A. D., J. Chem. Phys. 63 (1982) 735.
McLean, A. D., Lee, Y. S., Stud. Phys. Theor.Chem. 21 (1982) 219.
Aerts, P. J. C., Nieuwpoort, W. C., Int. J. Quantum Chem Symp. 19 (1986) 267.
15. Aerts, P. J. C., Nieuwpoort, W. C., Proceedings of the 6th seminar on
Computational Methods in Quantum Chemistry, Tegernsee, (1984) 13.
16. Dyall, K. G., Taylor, P. R., Faegri, K. Jr., Partridge, H., J. Chem. Phys. 95, 2583
(1991).
17. Breit, G., Phys. Rev. 34 (1929) 553.
18. Aerts, P. J. C., Towards relativistic quantum chemistry. Thesis, Groningen
(1986).
19. Van Piggelen, H. U., Ab initio calculations on the electronic states of 4fn ions
with applications to [EuO6]9-. Thesis, Groningen (1978).
20. Dirac, P. A. M., Proc. Roy. Soc. London A 117 (1928) 610.
21. Dirac, P. A. M., Proc. Roy. Soc. London A 118 (1928) 351.
22. Moss, R. E., Advanced Molecular Quantum Mechanics. Chapman and Hall,
London (1973).
23. Bethe, H. A., Jackiw, R. W., Intermediate Quantum Mechanics. W. A.
Benjamin, Inc, London (1968).
24. Rose, M. E., Relativistic electron theory. Wiley & Sons, New York (1961).
25. Davydov, A. S., Quantum Mechanics. Pergamon Press, Oxford (1965).
26. Buchmüller, W., Phys. Rev. A 18 (1978) 1784.
13
1. Introduction
27. Gaunt, J. A., Proc. Roy. Soc. A 122 (1929) 513.
28. Quiney, H. M., Grant, I. P., Wilson, S. J., J. Phys. B: At. Mol. Opt. Phys.
(1990) L271.
29. Okada, S., Shinada, M., Matsuoka, O., J. Chem. Phys. 93 (1990) 5013.
30. Brown, G. E., Ravenhall, D. G., Proc. R. Soc. London Ser. A 208 (1951) 552.
31. Sucher, J., Int. J. Quantum Chem. 25 (1984) 3.
32. Sucher, J. Phys. Rev. A 22 (1980) 348.
33. Mittleman, M. H., Phys. Rev. A 24 (1981) 1167.
34. Snijders, J. G., Relativity and Pseudopotentials in the Hartree-Fock-Slater
method. Thesis, Amsterdam (1979).
35.
36.
37.
38.
39.
40.
41.
42.
43.
Hess, B. A., Phys. Rev. A 32 (1985) 756.
Hardekopf, G., Sucher, J., Phys. Rev. A 30 (1984) 703.
Hess, B. A., Chandra, P., Physica Scripta 36 (1987) 412.
Hess, B. A., Phys. Rev. A 33 (1986) 3742.
McWeeny, R., Sutcliffe, B. T., Methods of Molecular Quantum Mechanics,
Academic Press, New York (1969).
Blume, M., Watson, R. E., Proc. Roy. Soc. A 270 (1962) 127.
Foldy, L. L., Wouthuysen, S. A., Phys. Rev. 78 (1950) 29.
Snijders, J. G., Baerends, E. J., Ros, P., Mol. Phys. 38 (1979) 1909.
Snijders, J. G., Baerends, E. J., Mol. Phys. 36 (1978) 1789.
44.
45.
46.
47.
48.
49.
Laaksonen, L., Grant, I. P., Chem. Phys. Lett. 109 (1984) 485.
Laaksonen, L., Grant, I. P., Chem. Phys. Lett. 112 (1984) 157.
Liberman, D., Waber, J. T., Cromer, D. T., Phys. Rev. A 137 (1965) 27.
Rosén, A., Ellis, D. J. Chem. Phys. 62 (1975) 3039.
Schwerdtfeger, P., Silberbach, H., Miehlich, B. J. Chem. Phys. 90 (1989) 762.
Christiansen, P. A., Ermler, W. C., Pitzer, K. S., Ann. Rev. Phys. Chem. 36
(1985) 407.
50. Krauss, M., Stevens, W. J., Ann. Rev. Phys. Chem. 35 (1984) 357.
51. Dolg, M., Wedig, U., Stoll, H., Preuss, H., J. Chem. Phys. 86 (1987) 866.
52.
53.
54.
55.
56.
57.
58.
Pyper, N. C., Marketos, P., Mol. Phys. 42 (1981) 1073.
Pyper, N. C., Mol. Phys. 42 (1981) 1059.
Pitzer, K. S. Int. J. Quantum Chem. 25 (1984) 131.
Kahn, L. R., Hay, P. J., Cowan, R. D., J. Chem. Phys. 68 (1978) 2386.
Lee, Y. S., Ermler, W. C., Pitzer, K. S., J. Chem. Phys. 67 (1977) 5861.
Kahn, L. R., Int. J. Quantum Chem. 25 (1984) 149.
Sakai, Y., Miyoshi, E., Klobukowski, M., Huzinaga, S., J. Comp. Chem. 8
(1987) 256.
14
2. Methodology
2.
Methodology.
2.1. Basis set expansion.
The solutions to the Dirac equation can be approximated by the basis set expansion
technique as is commonly done to approximate the solutions of the Schrödinger
equation. In the latter case the solution i is expanded in a basis set  :
i = • p cpi
(1)
p
The expansion coefficients cpi are determined variationally. In most quantum
chemical applications, the primitive basis functions  p are chosen to be gaussian
functions gp [1] or linear combinations thereof,
gp = N xnxpynypznzp e-p(x +y +z ),
2
2
2
nip, p  0
(2)
because the calculation of multi-center two-electron integrals is then relatively easy.
The same argument applies to relativistic quantum chemical applications. The basis
functions are classified according to their l-type, which is determined by the sum of
the powers of the cartesian coordinates l = nxp + nyp + nzp, leading to a set of s
functions (l=0), a set of p functions (l=1), and so on.
To solve the Dirac equation, various kinds of expansions can be considered, due to
the four components that need to be expressed [2-5]. In this work we use an
expansion in two different basis sets. The 'large component set'  Lp is used to expand
the large component functions in, the 'small component set'  Sp is used to expand the
small component functions in:
15
2. Methodology
NL
•
Li =
NS
 Lp cL
pi
p=1
• SpcS
pi
; Si =
NL
• LpcL
pi
p=1
(3)
NS
• SpcS
pi
p=1
p=1
and
i =
Li
(4)
Si
Details on the form of the basis functions can be found in the chapters 4 and 5.
2.1.1.
Kinetic and atomic balance.
If an expansion in an orthonormal basis is used to approximate the solutions to the
Dirac equation, equation (1.9) becomes
c  p
V LL
c  p
SL
V SS
-
LS
2mc2
L L
1
=
S S
SS
L L
S S
(5)
where we have indicated a matrix representation by the use of square brackets. In this
section we use two-component basis functions. The large component basis set (L) is
defined as 1 , 0  Lp , where  Lp is the scalar large component basis set as in
0
1
the previous section. Similarly, the small component basis set (S) is defined as
1 , 0  S . If we express S S in terms of L L:
p
0
1
S S =  + 2mc2 SS - V SS
-1
c  p
SL
L L
(6)
then
V LL + c  p
LS
 + 2mc2 - V SS
-1
c  p
SL
L L=  L L
(7)
In the non-relativistic limit, c, this equation reduces to
LS
SL
L
L
V LL + 1  p
 p
L,NR = NR L,NR
2m
16
(8)
2. Methodology
However, this equation reduces to the Schrödinger equation only if the equality
 p
LS
 p
SL
= p2 LL
(9)
holds. This will depend on the choice of (S) relative to (L). The requirement that (S)
is constructed so that equation (9) holds exactly is called the kinetic balance condition
[6, 7], because p2 is related to the non-relativistic kinetic energy operator p2/2m.
The kinetic balance condition obviously holds when the small component basis set
Si is constructed from the large component basis set Li by
Si = Ni  p Li
(10)
where Ni is a normalization constant. A basis set which is constructed using this
formula is said to contain minimum kinetic balance. It is a condition which ensures a
proper non-relativistic limit, but does not guarantee proper relativistic solutions.
As an illustration, consider the operation px on a scalar function xe-(x2+y2+z2):
px xe-(x 2+y 2+z2) = - ih-

xe-(x 2+y 2+z2) =
x
= - ih- (1 -2x2 ) e-(x 2+y 2+z2)
(11)
We find that the result is a linear combination of two scalar functions with different
l-types (l=0 and l=2).
When the small component basis set constructed using (10) is extended with
additional functions, we get an extended kinetically balanced basis set:
Si =  p Li  Se
(12)
In practice, several kinds of extensions are being used. When  p operating on a
large component basis function yields a linear combination of functions, like in
example (11), and the terms in this combination are used as independent variational
basis functions, the basis is called an unrestricted kinetically balanced basis set.
As an example, consider a large component basis set which contains a p-type gaussian
function. Operation of  p on this function leads to a linear combination of
17
2. Methodology
functions as indicated in (11). When we choose the small component basis set to
contain both the s-type and the d-type cartesian gaussian functions, we have an
unrestricted kinetically balanced basis set.
In most cases, the primitive unrestricted kinetically balanced basis sets we use are
flexible enough to give a good description of the relativistic solutions. However, in
many cases, in particular for molecular calculations, (general) contracted functions are
used instead of primitive gaussians [8,9]:
i = • gpcpi
(13)
p
The contraction coefficients cpi for the molecular basis functions i are usually
determined from previously obtained atomic solutions, or from parts thereof. A small
component basis derived from the contracted large component basis to yield an
unrestricted kinetically balanced basis set, will often lack sufficient flexibility to
describe the wanted Dirac solutions. This can be remedied by choosing a small
component basis which contains also the small component part of the atomic
solutions (still expressed in the primitive basis set) [10]:
Si =  p Li  Se  Sa
(14)
In this equation,  Sa stands for the set of small component parts of the atomic
solutions. We call such a basis set an atomically balanced basis set. Note that this
extension becomes relevant only when contracted basis sets are employed.
To ensure a proper description of the kinetic energy for all positive energy solutions,
we usually employ a contracted basis set which is atomically as well as kinetically
balanced for molecular calculations.
2.1.2.
Basis set optimization.
The primitive basis sets for the large component employed in this work are
constructed from cartesian gaussian functions which have been optimized for nonrelativistic atomic states. The optimization procedure minimizes the total SCF energy
with respect to the exponential parameters of the basis functions.
The primitive basis sets for the small component are constructed from the large
component basis sets to obtain unrestricted kinetically balanced basis sets. These sets
18
2. Methodology
are thus much larger than the large component basis sets, so that the calculation,
storage and handling of two-electron integrals becomes significantly more involved
than in the non-relativistic case.
Figure 2: Example of the construction of a primitive kinetically balanced basis set.
No restrictions on exponents:
g
8
f
10
d
15
p
18
s
Large: 203 scalar functions
8
10
23
28
15
Small: 457 scalar functions
With restrictions on exponents:
g
8
f
10
d
15
p
18
s
Large: 203 scalar functions
8
10
15
18
15
Small: 379 scalar functions
A significant reduction of the small component basis set size can be obtained. Both
l+1 type and l-1 type gaussian basis functions in the large component set generate ltype basis functions in the small component. By using a common set of exponents for
the l+1 and l-1 gaussian basis functions, the l-type small component basis functions
will coincide. This construction is illustrated in figure 2, in which each pattern
corresponds to a set of exponents. The number of scalar functions is the number of
cartesian gaussian functions in the set.
Using this construction, our basis sets for the large component are optimized under
the constraint that the d-(f-) exponential parameters form a subset of the s-(p-)
exponential parameters. The constraint leads to a slightly less flexible basis set. On
the other hand, the number of integrals which need to be calculated, stored and
handled is reduced significanly (about a factor of two for the Pb-atom).
19
2. Methodology
2.2. Open Shell Hartree-Fock-Dirac-(Roothaan) method.
The many-electron equation we want to solve, equation (1.15), reads
• hDi + • gij
i
 i = Ei  i
(15)
i<j
We write the many-electron function i as an (infinite) linear combination of Slater
determinants:
 i = • di
(16)

Each Slater determinant  is an antisymmetrized product of orthonormal, four
component, one-electron functions. This construction ensures that the complete manyelectron wave function is antisymmetric with respect to interchange of electron labels.
In order to approximate the solution we may at first limit the expansion (16) to one
term only. When we optimize the one-electron functions in the determinant
variationally to yield stationary total energy expectation values, using the constraint
that the one-electron functions are orthonormal, we obtain the Hartree-Fock-Dirac
equation:
hD + • Jj - Kj i = i i
(17)
j
The coulomb and exchange operators Jj and Kj in this equation are defined by the
following relations:
Jj q =
*j (2) r1 j (2) d2 q
(18)
*j (2) r1 q(2) d2 j
(19)
Kj q =
12
12
The matrix elements of these operators are two-electron integrals (given in chargecloud notation on the right-hand side of the equations):
p Jj q =
*p (1)*j (2) r1 q(1)j (2) d1 d2 = p q j j
12
20
(20)
2. Methodology
and
p Kj q =
*p (1)*j (2) r1 j (1)q(2) d1 d2 = p j j q
12
(21)
Equation (17) is a one-electron equation and we can solve it using a basis set
expansion technique as described earlier. A slight complication, but a major one in
terms of computational effort, is that it needs to be solved iteratively: the coulomb
and exchange terms depend on the solutions.
Another complication is that the Dirac operator is an unbounded operator with a
continuum of both positive and negative energy states. As a result, the variation
theorem, which states that the expectation value of the non-relativistic total energy for
any trial function is always higher than the exact ground state total energy, no longer
applies. However, the variational method we use determines just a stationary point on
the total energy surface. By selecting the lowest positive energy solutions as the
occupied orbitals to determine the coulomb and exchange operators, we try to find the
solution that has the lowest energy in the positive energy domain. The use of
kinetically balanced basis sets prevents the occurance of 'spurious' positive energy
solutions which lack kinetic energy. A 'variational collapse' does not occur [2, 4, 5, 6,
11].
After solving (17) we have a set of one electron spin-orbitals and we can construct the
one-determinant approximation to the many-electron wave function as
(1..n) = 1 1 (1)2 (2)...n (n)
n!
(22)
Many open shell systems can not be described with a single determinant. In such
cases, we use an average of configuration open shell method to generate a set of
average orbitals. Each of the determinants from the open shell configuration gives rise
to an energy expression, from which we construct the average energy expression.
Next, we determine the optimum spin orbitals by locating the stationary value of the
average energy with respect to orbital changes, under the constraint that the spin
orbitals remain orthonormal. This step is almost identical to the classical open shell
formalism of Roothaan [12]. The average orbitals are then used to construct a many-
21
2. Methodology
electron basis in terms of which the manifold of open shell states is determined by a
small CI calculation.
Details of the open shell formalism can be found in chapter 5.
2.3. CI method.
A better approximation to the many-electron wave function (16) is obtained by
including more determinants in the expansion. A small set suffices if we just want a
description of a complex open shell system, but a much larger set is needed if we
want to account for correlation effects.
The determinants are built from the occupied and unoccupied spin orbitals that result
from a closed or open shell Hartree-Fock-Dirac calculation as described. Although in
principle the inclusion of negative energy spin orbitals could be taken into
consideration, only the positive energy orbitals are used.
The expansion coefficients in (16) are again determined variationally, leading to an
eigenvalue equation that contains the matrix elements of the hamiltonian in terms of
the Slater determinants. For the evaluation of these elements the two-electron
integrals must be expressed in the one-electron basis (the MO basis), so a four-index
transformation is required as usual [13]. This transformation is described in detail in
appendix A.
2.4. Symmetry: Pitzers theorem and Dacre and Elder list.
The number of two-electron integrals that must be calculated and stored can be
reduced by employing the spatial symmetry properties of the integrals. A common
method is that of Dacre and Elder [14-16], extended by Aerts [17], in which only one
integral from a set of symmetry equivalent two-electron integrals needs to be
calculated. Two integrals are called 'symmetry equivalent' when a spatial symmetry
operation of the point group of the molecule (other then the identity) transforms one
into the other. This results in a reduced list of integrals which we call the Dacre and
Elder list. The integrals in this list are scaled by the number of times they transform
into themselves. It can be shown that the contribution of the integrals that are not
calculated can be generated at a later stage by applying all symmetry operations of the
group to the Fock matrix. A restriction on the use of the Dacre and Elder method is
that the Fock operator needs to be totally symmetric.
22
2. Methodology
Alternatively one might want to work in terms of symmetry adapted basis functions or
symmetry orbitals. Then a theorem due to Pitzer can be employed [18, 19] stating that
'Atomic Orbital integrals related by symmetry contribute equally to Symmetry Orbital
integrals whose integrands are totally symmetric'. In practice this again means that we
need to calculate symmetry unique integrals only.
On the SCF level we use the Dacre and Elder list to construct a Fock matrix as usual
and ignore the fact that we have not yet included all contributions. Next, we average
the Fock matrices for the degenerate irreducible representations resulting in a Fock
matrix which is totally symmetric by construction. It can be shown that this procedure
accounts for the contributions from the integrals which are not part of the Dacre and
Elder list. Computationally, this construction is much simpler and cheaper than the
original Dacre and Elder construction.
To construct MO integrals for the subsequent CI calculation (COSCI) from the list of
Dacre and Elder integrals, we perform a standard four-index transformation on the
latter to obtain a list of 'skeleton MO integrals'. By forming linear combinations of
symmetry equivalent integrals we construct integrals which have a totally symmetric
integrand [20], and this again takes into account the contributions from the integrals
which have been left out in the generation of the Dacre and Elder list. This
construction is described in more detail in appendix A.
2.5. Overview of the MOLFDIR program package.
In this section a brief global description of the program system [5, 21] designed to
perform the Hartree-Fock-Dirac and CI calculations is presented. The main purpose is
to show where the various theoretical and methodological developments described
earlier are being used in the program.
2.5.1.
MOLFDIR.
The MOLFDIR (Molecular Fock-Dirac) program reads the input that defines the
molecule, the basis sets and the symmetry to be used. From this information, Dirac
double group symmetry adapted basis functions are constructed by projection along a
group-chain [5]. The use of the group chain leads to symmetry adapted functions
which are not only adapted to the highest possible symmetry, but also to a suitable
abelian subgroup. This feature is very useful when subsequent CI calculations are to
23
2. Methodology
be performed. Also, the representation matrices for the irreducible representations
which are spanned by the basis set are determined.
2.5.2.
RELONEL and RELTWEL.
The RELONEL (relativistic one-electron integrals) program calculates all the oneelectron integrals which are needed ( p, T, VN, overlap) in the symmetry adapted
basis. The nuclei are represented either by point charges or by gaussian charge
distributions [22, 23].
The RELTWEL (relativistic two-electron integrals) program calculates the integrals
over the r-1
12 operator. Only integrals of (LL|LL), (SS|LL) and (SS|SS) type appear
when the Coulomb interaction between the electrons is included. The Gaunt
interaction requires also integrals of the type (LS|LS). The integrals are evaluated over
the scalar basis functions. Only non-zero spatial and permutation symmetry unique
integrals are calculated, i.e. those integrals which do not transform into each other
under spatial or permutation symmetry operators. This results in a Dacre and Elder list
of two-electron integrals scaled by the number of times they transform into
themselves under the spatial group operations.
2.5.3.
MFDSCF.
The Molecular Fock-Dirac Self-Consistent-Field program solves the Hartree-FockDirac-Roothaan equations using information from MOLFDIR, RELONEL and
RELTWEL. The user needs to specify the open or closed shell occupations.
The program constructs the Fock matrix using only the symmetry unique two electron
integrals. After transformation to the symmetry adapted basis the Fock matrix is
symmetrized by averaging the components of the degenerate representations (an
application of Pitzers theorem). The program also assumes time-reversal symmetry
for the system. This leads to relations between the  and  blocks, and between the
 and  blocks, which reduce the required CPU time and core memory by a factor
of two.
2.5.4.
ROTRAN.
The ROTRAN (relativistic orbital transformation) program transforms the list of
Dacre and Elder two-electron integrals to two-electron integrals over Molecular
24
2. Methodology
Orbitals from the MFDSCF program. In this program Pitzer's theorem is applied
again. Since the relativistic MO's are in general complex, some relations used within
non-relativistic four-index transformation programs no longer hold. In appendix A the
algorithm is described in detail.
2.5.5.
GOSCIP.
The GOSCIP (General Orthogonal Small CI program) program constructs the
hamiltonian matrix in the full CI-space using Slater's rules and the MO-integrals
generated by ROTRAN, and diagonalizes it. Abelian symmetry is employed in order
to reduce the computational effort.
25
2. Methodology
2.6.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
19.
References.
Boys, S. F., Proc. Roy. Soc. A 200 (1950) 542.
Kutzelnigg, W., Int. J. Quantum Chem. 25 (1984) 107.
Okada, S., Shinada, M., Matsuoka, O., J. Chem. Phys. 93 (1990) 5013.
Aerts, P. J. C., Nieuwpoort, W. C., Chem. Phys. Lett. 125 (1986) 83.
Aerts, P. J. C., Towards relativistic quantum chemistry. Thesis, Groningen
(1986).
Stanton, R. E., Havriliak, S., J. Chem. Phys. 81 (1984) 1910.
Mohanty, A. K., Clementi, E., J. Chem. Phys. 93 (1990) 1829.
Raffenetti, R. C., Comp. Chem. 8 (1973) 4452.
Ishikawa, Y., Sekino, H., Chem. Phys. Lett., 165 (1990) 237.
Visscher, L., Aerts, P. J. C., Visser, O., in: Wilson, S., Grant, I. P., Gyorffy, B.
L. (eds), The effects of relativity in atoms, molecules and the solid state. Plenum
Press, New York (1991) 197.
Hegarty, D. Theor. Chim Acta 70 (1986) 351.
Roothaan, C. C. J., Rev. Mod. Phys. 32 (1960) 179.
Saunders, V. R., Lenthe, J. H., Mol. Phys. 48 (1983) 923.
Dacre, P. D., Chem. Phys. Lett. 7 (1970) 47.
Elder, M., Int. J. Quantum Chem. 7 (1973) 75.
Schaad, L. J., Wilson, D. J., Hess, B. A. Jr., Chem. Phys. Lett. 105 (1984) 433.
Aerts, P. J. C., Chem. Phys. Lett. 104 (1984) 28.
Pitzer, R. M., J. Chem. Phys. 58 (1973) 3111.
Davidson, E. R., J. Chem. Phys. 62 (1975) 400.
Carsky, P., Hess, B. A. Jr., Schaad, L. J., J. Comp. Chem. 5 (1984) 280.
Aerts, P. J. C., Nieuwpoort, W. C., Int. J. Quantum Chem Symp. 19 (1986) 267.
Visser, O., Aerts, P. J. C., Hegarty, D., Nieuwpoort, W. C., Chem. Phys. Lett.
134 (1987) 34.
20. Dyall, K. G., Faegri, K., Taylor, P. R., in: Wilson, S., Grant, I. P., Gyorffy, B. L.
(eds.), The effects of relativity in Atoms, Molecules and the Solid state. Plenum
Press, New York (1991) 167.
26
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
3.
Non-relativistic valence-only calculations with
relativistic core and relativistic
perturbation corrections.
3.1. Introduction.
The computational effort (both in terms of CPU time and disk space) needed to solve
the Hartree-Fock-Dirac equation, is much larger than the effort needed to solve the
Hartree-Fock equation. Also, the Hartree-Fock-Dirac method is usually applied to
systems which contain one or more heavy atoms. Even using a non-relativistic
method such atoms require much more computational effort due to the large number
of core orbitals.
Hence, for practical reasons a method in which only the valence orbitals are treated
explicitly is of great value. Most of these methods are based on the frozen core
approximation: it is assumed that the all-electron wave function can be split into a
core part and a valence part, and that the core part is not sensitive to the environment.
The core-part is obtained in an atomic (or small molecular) calculation and used as a
frozen core in molecular calculations. Often, additional approximations are
introduced to describe the core-valence interaction and to simplify the form of the
valence functions in the core region [1 - 4]. In such cases, the interaction with the core
states is replaced by some effective potential which is determined from atomic results.
Some of these potentials also incorporate relativistic effects [5 - 7].
In connection with these techniques, it is often stated in literature that relativistic
effects on the one-electron valence states are for a significant part due to changes in
the electrostatic interaction with and the orthogonalization to the relativistically
modified core (indirect core effects) [1, 4, 8 - 12]. Sometimes it is noted that the
indirect core effects as well as the direct relativistic valence effects (effects of the
relativistic operators on the valence orbitals themselves) can be described using an
27
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
effective potential [2] because they originate mainly from the core region [4 - 7, 9, 13
- 16]. Others have studied the contributions of direct and indirect effects using
perturbation theory, one-center expansion methods, pseudopotential methods, or some
combination of these. In general, they show that not only indirect core effects, but also
direct effects and indirect valence effects (due to the interaction with the
relativistically modified valence orbitals) are important [17 - 22]. It has also been
noted that the indirect effects do not contribute to the total energy to first order [23].
Using all-electron Hartree-Fock-Dirac calculations as a reference, we can analyse the
use of a frozen relativistic core in an otherwise non-relativistic calculation. By
extracting the frozen core from a molecular calculation and using it in a nonrelativistic calculation on the same system, we avoid approximations that arise from
the use of atomic cores. Also, we include the frozen core explicitly, that is, we do not
approximate the interaction with the core by some effective potential. As a result, all
differences between the non-relativistic and relativistic calculations are due to the use
of a non-relativistic hamiltonian instead of a relativistic one in the valence-only
calculation. The direct relativistic effects are analysed by first-order perturbation
theory.
In the following sections our approach is described in detail, and results of test
calculations on Sn and SnH4 are presented. More extensive Hartree-Fock-Dirac and
Hartree-Fock calculations on the SnH4 molecule (and other XH4 molecules, X=C, Si,
Ge, Sn and Pb), including comparison with other methods and experiment, are
described in another chapter [25].
3.2.
Description of the method.
3.2.1.
Résumé of the Hartree-Fock-Dirac method.
Consider the Hartree-Fock-Dirac equation (the energy eigenvalues have been shifted
by -mc2:
c  p + (-1)mc2 + V i = ii
(1)
In this equation  and  are the 4 x 4 Dirac matrices in the standard representation,
and V is the effective potential energy due to the nuclei and electrons
28
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
occ.
V = VN +
•
(2)
Jj - Kj
j
The Jj and Kj in this equation are the usual coulomb and exchange operators. The
Breit interaction has not been included. The one-electron four-component spinors
which are solutions of equation (1) are required to be orthonormal:
i|j = ij
(3)
3.2.2.
Frozen core approach.
The frozen core approach is easily implemented both in the non-relativistic and in the
relativistic case: equation (2) is written as
(4)
V = VN + VC + VV
core
VC =
•c
(5)
Jc - Kc
valence
VV =
•
v
(6)
Jv - Kv
In the Hartree-Fock(-Dirac) scheme both VC and VV are determined iteratively; when
the frozen core method is used we have a fixed set of core spin-orbitals, so VC is a
fixed operator and only VV needs to be determined iteratively. The problem of the
orthogonality of the valence spin-orbitals to the core spin-orbitals can be handled
either by explicit orthogonalisation or by inclusion of a shift-operator.
3.2.3.
Two-component formalism.
The frozen core technique applies to the non-relativistic as well as to the relativistic
formalism. However, within the relativistic formalism the valence states are described
by a four-component wave function, so it is still significantly more expensive in terms
of computational resources to perform relativistic calculations than to perform nonrelativistic calculations. To reduce the computational effort further, one can treat the
29
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
valence functions by an effectively two-component SCF-formalism, with additional
perturbation corrections. Formally, the valence functions are still four-component
functions. In practice we determine only the large component part of the fourcomponent function (with the small component part implicitly given by the large
component part).
Consider the reduction of the valence equations to a set of two-component equations.
The Hartree-Fock-Dirac equation for the valence orbitals can be written in bi-spinor
form (introducing a large component basis L and a small component basis S):
VLL
c(
c( p)LS + VLS
p)SL +
VSL
VSS
-
Lv
Sv
2mc2
= v
Lv
Sv
(7)
with
XY
XY
VXY = VXY
N + VC + VV
VXY
C =
(8)
core
•c
(9)
XY
JXY
c - Kc
val
XY
XY
VXY
V = • Jv - Kv
(10)
v
XY
In these equations X, Y L, S; J XY
i is the X, Y - block of the Coulomb operator, K i
is the X, Y - block of the exchange operator, and VXY
N is the X, Y - block of the
nuclear potential.
From equation (7), we get an expression for Sv in terms of Lv . If we assume that
v << 2mc2 for the positive energy states of interest, we can expand the inverse
operator in this expression:
-1
Sv = v + 2mc2 - VSS c( p)SL + VSL Lv
SS
 1 + V - v c( p)SL + VSL Lv
2mc2
4m2 c4
30
(11)
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
Writing T =
p2
we find (neglecting terms of order smaller than c-2) from equations
2m
(11) and (7)
TLL+VLL+ 1 ( p)LS VSS-v ( p)SL+ 1 ( p)LSVSL+VLS( p)SL LvvLv
2mc
4m2c2
(12)
This equation can be rewritten as a eigenvalue equation. We prefer to use equation
(12) directly: formally we are still working with four-component valence functions. In
practice we determine only the function Lv , with the small component part implicitly
given by equation (11).
It should be noted that the wave functions have to be normalized properly: the
complete one-electron functions should obey equation (3). This is of importance since
the potential energy operators V in equation (12) contain the Coulomb and exchange
interaction with the other electrons. Formally, one can ensure proper contributions to
the potential energy by including a normalization factor in the two-electron
interaction between the valence orbitals:
val
VV = • Nv Jv - Kv
(13)
v
with
Nv = Lv Lv + Sv Sv
-1
(14)
3.2.4.
Non-relativistic two-component valence-only method.
Consider a valence-electron-only calculation in the non-relativistic limit (c). Then
equation (12) reduces to
(0) (0)
F(0)(0)
v = v v
(15)
with
(16)
F(0) = TLL + V(0)LL
31
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
V(0)LL is the same operator as VLL but constructed with the zeroth order valence
orbitals (0)
v (and the same set of four-component frozen core spin-orbitals). In the
non-relativistic limit one can write
(0)
v

(0)L
v
(17)
0
(0)LL
so VV
contains in effect the "normal" non-relativistic Coulomb and exchange
operators, while VLL
C contains the interaction with the large and small component of
the frozen cores.
The orthonormality condition (3) leads to an orthogonality condition between the core
spin-orbitals and the zeroth order valence spin-orbitals: it requires the complete fourcomponent valence functions to be orthogonal to the four-component core functions:
(0)
v c = 0
(18)
In effect, this condition avoids a collapse into the core. Since the small component
part of the zeroth order valence functions is zero, the orthogonality condition (18) has
effect on the large-component part only. Technically, the orthogonalization has been
achieved by using a shift-operator: all core-orbitals have been shifted to a large
positive orbital energy.
The normalisation factors (14) in the two-electron interaction are all equal to one in
the non-relativistic limit provided that the large component part of the wavefunction
is normalized to unity. Hence, the zeroth order Fock operator is the normal nonrelativistic Fock operator except for the interaction with the relativistic fourcomponent frozen core.
3.2.5.
Perturbation corrections.
In this section we will describe the corrections for the relativistic effects on the
valence spin-orbitals (which have been left out in the non-relativistic limit). By
comparing (16) and (12) one finds that the perturbation terms (up to order c-2) can be
written as
F(1)=VLL-V(0)LL+ 1 ( p)LS VSS-v ( p)SL+ 1 ( p)LSVSL+VLS( p)SL (19)
2mc
4m2c2
32
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
The normalization operators contained in the two-electron potential energy terms can
be ignored for the first-order perturbation theory if the zeroth order wavefunction is
normalized to unity.
In principle it is possible to approximate VLL - V(0)LL by using equation (11). The
first order corrections to the large component function could be calculated using firstorder (self-consistent) perturbation theory. In the present work, the correction term
VLL - V(0)LL, which yields part of the indirect valence effects (the nuclear as well as
the core contributions cancel), has been left out. It is interesting that the contribution
of this term to the total energy vanishes in first order since the total energy has been
minimized with respect to the orbitals [23, 24, 26].
Thus,
F(1) 
1 ( p)LS VSS -  ( p)SL + 1 ( p)LSVSL + VLS( p)SL (20)
v
2mc
4m2c2
and if the zeroth-order functions are normalized to unity we get
(1)
v =
1 ( p)LSVSS( p)SL (0)- 1 (0) T (0)+ 1 ( p)LSVSL+VLS( p)SL (0)
v
v
v
v
2mc
4m2 c2
2mc2
(21)
were we have used the abbreviation
(0)
(0)
A (0)
v = v A v
(22)
(0)
-4
The error introduced by using (0)
v in the term containing T v is of order c and is
therefore neglected.
Using (8) one gets
(0)
(0)
(0)
1
( p)LSVSS
( p)SL v + ( p)LSVSS
( p)SL v + ( p)LSVSS
( p)SL v
N
C
V
4m2 c2
LS
LS
SL (0)
LS SL
SL (0)
+ 1 ( p)LS VSL
C + VC ( p) v + ( p) VV + VV ( p) v
2mc
(0)
- 1 (0)
v Tv
2
2mc
(23)
(1)
v =
From this point on, the labels LS and SL on  p are not shown for clarity. The third
and fifth term in (23) are two-electron valence contributions to the  p VSS  p and
33
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
 pVSL + VLS p terms; these terms have also been left out since they turn out to
be small.
In this method the zeroth-order valence functions are orthogonal to the core functions.
This is sufficient since we do not calculate the first order corrections to the
wavefunction. Otherwise, one will have to ensure both that the valence wavefunctions
are orthogonalized properly to the core functions, and that the self-consistent effects
on the two-electron interaction are taken into account.
The perturbation correction to the total energy has been calculated as a sum of the
perturbation corrections to the one-electron orbitals. Note that even if we would have
included indirect valence effects by self-consistent perturbation theory [23, 24, 26],
the first order correction to the total energy is still a sum of the same first order oneelectron perturbation corrections to the orbital energies: the indirect valence effects do
not show up in first order in the total energy.
3.2.6.
Indirect and direct effects.
The method includes all indirect core effects (due to the interaction between the
valence spin-orbitals and the four-component core spin-orbitals), all direct core
effects (these have been included in the Hartree-Fock-Dirac calculation used to
determine the core spin-orbitals), and most of the one-electron direct valence effects
(by first-order perturbation theory). Although we have not calculated the two-electron
direct valence effects, we will estimate their magnitude and show that these are
negligible for the systems considered. Thus, for systems for which first order
perturbation theory gives a sufficient approximation of the relativistic effects, the
differences between the valence-only results including perturbation corrections and
the Hartree-Fock-Dirac results are due to indirect valence effects and to higher order
corrections.
3.3.
Testcase: the SnH4 molecule.
3.3.1.
Technical details.
We have applied the relativistic frozen core method with the two-component valenceonly technique with perturbation corrections to the SnH4 molecule. The calculations
34
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
were performed using Td symmetry in the Dirac double group at bond lengths of
3.214 a.u., 3.260 a.u. and r = 3.330 a.u.. The basis sets used for the calculations are
given in the appendices to this chapter. The non-relativistic and valence-only
calculations have been performed using the large-component basis functions only.
All-electron Hartree-Fock-Dirac calculations (FD) were carried out using the
MOLFDIR (Molecular Fock-Dirac) package [27, 28]. For the Sn-atom the 5s1/22
5p1/22 closed shell state was calculated. For each bond length relativistic cores for the
Sn atom were extracted from the results, so the use of a frozen core is no
approximation for this test case. Only the 5s - and 5p - orbitals of Sn and the 1s orbital of each H (if present) were treated as valence states.
Next, for all systems two-component valence-only calculations were carried out using
the relativistic frozen cores (RFC). The valence spin-orbitals were kept orthogonal to
the core spin-orbitals by shifting the core spin-orbitals to high orbital energies.
Several values for these shifts were tried; the valence results were insensitive to these
values. Using these effectively two-component valence orbitals, the perturbation
corrections (PC) (23) have been calculated.
For reference purposes, non-relativistic calculations (NR) have been performed on the
same systems (using again the 5s1/22 5p1/22 closed shell configuration for Sn).
3.3.2.
Results and discussion.
The results of the calculations are collected in a number of tables. In table 1 the total
energies and core energies calculated with the relativistic frozen core method (without
perturbation corrections) are given. The core energy contains the interaction of the
core electrons with the nucleus, the interaction of the core electrons with each other
and the interaction of the core electrons with the hydrogen nuclei. The dependence of
the core energy on the bond length is mainly due to the last term: if a simple
electrostatic correction is included (using a charge of -46 to represent the core
electrons and 4 charges of +1 to represent the hydrogen nuclei), the first three decimal
places of the core energy are constant. This shows that, in this case, the frozen core
approximation would not introduce errors larger than about 0.001 a.u..
Table 1:
Relativistic frozen core results:
total energy (a.u.) and core energy (a.u.) versus bond length (a.u.).
35
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
r = 3.214
r = 3.260
r = 3.330
EtotRFC
-6177.12003
-6177.12241
-6177.12363
Ecore
-6228.99773
-6228.19077
-6227.00549
Table 2:
SnH4 bond length, bond-energy and force constant (a.u.)
FD
3.28
-6177.194
.28
.54
Rmin
Emin
Ebond
k
Table 3:
E1
E2
F
Table 4:
PC
RFC
NR
-6174.921
-.483
-6174.894
-.479
-6174.827
-.457
-6018.181
-.453
-.219
-.221
-.209
-.208
-6177.194
-.699
-6177.175
-.697
-6177.122
-.683
-6020.488
-.679
-.457
-.460
-.454
-.454
-.448
-.454
-.451
-.454
PC
RFC
NR
.027
.004
.094
.026
156.740
.030
-.002
.010
.011
1/2
SnH4 r=3.26
Etot



FD
Differences with Fock-Dirac results (a.u)
Sn atom
Etot
5s
5p
NR
3.31
-6020.489
.31
.58
1/2
SnH4 r=3.26
Etot



RFC
3.32
-6177.124
.30
.59
Valence properties compared (a.u.)
Sn atom
Etot
5s
5p
PC
3.28
-6177.175
.24
.55
E1
E2
F
.01866
.002
.07145
.016
156.70627
.020
-.003
.003
.003
-.006
-.003
-.005
36
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
In tables 2, 3 and 4 the valence properties calculated using the non-relativistic method
(NR), the Hartree-Fock-Dirac method (FD), the relativistic frozen core method (RFC)
and the relativistic frozen core method with perturbation corrections (PC) are
presented. The radial valence densities calculated for the Sn atom are shown in
figures 1 and 2.
Figure 1:
Hartree-Fock-Dirac (FD) and non-relativistic (NR) valence density of Sn atom in arbitrary
units
2.0
r^2 FD
1.6
r^2 NR
1.2
0.8
0.4
0.0
0.0
0.5
1.0
1.5
2.0
r (a.u.)
37
2.5
3.0
3.5
4.0
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
Figure 2:
Non-relativistic (NR) and relativistic frozen core (RFC) valence density of Sn atom in
arbitrary units
2.0
r^2 NR
1.6
r^2 RFC
1.2
0.8
0.4
0.0
0.0
0.5
1.0
1.5
2.0
r (a.u.)
2.5
3.0
3.5
4.0
From these tables and figures it is clear that the use of a relativistic core in an
otherwise non-relativistic calculation (RFC) does not improve the results (in the sense
that they are closer to the Hartree-Fock-Dirac results) with respect to the completely
non-relativistic results. This holds for all results: the bond properties, the valence
orbital energies and the valence charge density. In fact, some properties are slightly
worse, while the charge density calculated with the relativistic frozen core method is
almost the same as the non-relativistic charge density except for some small
deviations in the core region. None of the inward shift of the charge density found
from the Hartree-Fock-Dirac calculations is reproduced by the relativistic frozen core
method.
Inclusion of the perturbation corrections does improve the results significantly. Only
the bond energy and some orbital energies do deviate significantly from the HartreeFock-Dirac results (the corrections to the E2 and F orbital energies in the SnH4
calculations are over-estimated). Since the terms which have been left out in this
calculation are mainly the indirect valence effects, it can be concluded that the
relativistic effects on the valence properties are mainly direct relativistic effects.
38
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
Table 5:
Perturbation corrections (a.u.); all entries include the prefactor as given in formula (23)
sym rep E1
 (zeroth order)
T
VN
VC
VV
.pVN .p
.pVC .p
.p VC + VC .p
 T
|N|2 (normalization constant)
(1) (first order correction)
(0+1)
Sn atom
-.45656
1.54580
-23.50369
20.60009
0.90124
-.02739
.00487
.00041
.00002
.99996
-.02209
-.47865
r = 3.214
-.68765
1.16231
-22.29493
18.55270
1.89227
-.01836
.00329
.00027
.00002
.99997
-.01478
-.70243
SnH4
r = 3.26
-.68276
1.15052
-22.18111
18.47001
1.87782
-.01820
.00326
.00027
.00002
.99997
-.01465
-.69741
r = 3.33
-.67549
1.13454
-22.01926
18.35269
1.85654
-.01800
.00322
.00027
.00002
.99997
-.01449
-.68998
Sn atom
-.20932
.95314
-18.70554
16.70391
0.83917
-.01410
.00269
.00007
.00001
.99997
-.01133
-.22065
r = 3.214
-.45626
.91266
-18.56910
15.36660
1.83358
-.00811
.00186
.00006
.00001
.99998
-.00618
-.46244
SnH4
r = 3.26
-.45396
.88447
-18.36271
15.20639
1.81789
-.00782
.00179
.00005
.00001
.99998
-.00597
-.45993
r = 3.33
-.45036
.84511
-18.06277
14.97266
1.79464
-.00741
.00168
.00005
.00001
.99998
-.00567
-.45603
r = 3.214
-.45357
.98168
-18.64203
15.37413
1.83265
-.00505
.00194
.00009
.00001
.99997
-.00301
-.45658
SnH4
r = 3.26
-.45133
.95147
-18.43229
15.21253
1.81696
-.00484
.00186
.00009
.00001
.99997
-.00288
-.45421
r = 3.33
-.44783
.90921
-18.12734
14.97662
1.79368
-.00455
.00174
.00008
.00001
.99998
-.00272
-.45055
sym rep E2

(zeroth order)
T
VN
VC
VV
.pVN .p
.pVC .p
.p VC + VC .p
 T
|N|2 (normalization constant)
(1) (first order correction)
(0+1)
sym rep F
 (zeroth order)
T
VN
VC
VV
.pVN .p
.pVC .p
.p VC + VC .p
 T
|N|2 (normalization constant)
(1) (first order correction)
(0+1)
Table 5 displays detailed information on the numerical values of the perturbation
corrections. The 'normalization constant' given in the table is given for reference
purposes; it has not been used to calculate the perturbation corrections. Due to the
39
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
very small numerical value it would not show up in the calculated perturbation
corrections anyway. The normalization constant is defined to normalize the function
v :
v = N
L
(0)
v
(24)
 p (0) L

2mc v
with
N2 = 1 +
1 (0) T (0)
v
v
2mc2
-1
(25)
From the data in the table it is clear that the most important correction term, both in
absolute magnitude as in bond length dependence, is the term  p VSS
N  p. The next
SS
important term is the term  p VC  p which is numerically about 20% to 25% of
SS
the leading correction term but with opposite sign. The term  p VSS
C + VC  p has
a very small contribution to the absolute correction and can be neglected as far as the
bond length dependence is concerned. The last term, (0)T, has no significant
contribution.
SS
SS
As has been mentioned, the terms  p VSS
V  p and  p VV + VV  p have been
left out. According to the data in the table, the two-electron interaction with the
valence electrons VV is an order of magnitude smaller than the interaction with the
core electrons VC. We expect that the direct relativistic corrections to these terms
show a similar trend; thus we expect that the terms which have been left out will not
have a significant influence on our conclusions. However, if high accuracy is
important it will be necessary to include at least the term  p VSS
V  p.
40
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
Table 6:
Estimate of Direct / Indirect effects
Indirect core
Direct valence
Indirect valence
Sn atom
5s (a.u.)
5p (a.u.)
-0.004
-0.022
-0.004
-0.001
-0.011
0.002
 E1 (a.u.)
-0.003
-0.015
-0.002
 E2 (a.u.)
0.000
-0.006
0.003
 F (a.u.)
0.002
-0.003
0.006
rmin (a.u.)
0.01
-0.04
0.00
k (a.u.)
0.01
-0.03
-0.01
-0.01
-0.06
0.02
1/2
SnH4 r=3.26 a.u.
Bond properties
Ebond (a.u.)
If we assume that it is sufficient to include the perturbation corrections to first order,
we can estimate the contribution of the direct and indirect relativistic effects to
several properties. The indirect core effect is the difference between the RFC and the
NR results, the direct valence effect is the difference between the PC and the RFC
results and the indirect valence effect is the difference between the FD and the PC
results. The results are presented in table 6.
For all the properties we find that the direct relativistic effect is by far the largest
effect. The indirect core effect is of the same order of magnitude as the indirect
valence effects. We find an indirect valence effect on the bond energy; since the
indirect valence effects do not contribute to the total energy in first order, this
contribution must be due either to higher order effects or to the neglect of some of the
direct valence effects. Since we are not dealing with very heavy atoms, we expect the
latter (especially the term  p VSS
V  p) to be the main reason for the 'indirect
valence' contribution.
3.4. Conclusions.
In this chapter the analysis of a non-relativistic valence-only method using a frozen
relativistic core, combined with first order perturbation theory for the relativistic
effects on the valence electrons, is presented. The method has been applied to Sn and
SnH4, so strictly speaking the conclusions are only valid for systems like these
(especially, the valence electrons are s and p electrons). Also, some care should be
41
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
taken since the basis set errors need not be equal for the different perturbation
corrections.
Our results show that use of a relativistic core (and thus inclusion of all indirect core
effects) does not lead to improved valence results when no other relativistic
corrections are included for the valence electrons. The relativistic corrections to the
valence properties (valence orbital energies, valence charge densities and bond
properties) are mainly direct relativistic valence effects. Nevertheless, the results
collected in table 6 show that the contributions of indirect core and indirect valence
effects should not be neglected.
In all applications of relativistic pseudo-potential techniques it is assumed that
indirect core as well as direct valence effects can be represented by (semi)-local
potentials that are determined from atomic calculations. This assumption should be
investigated more thorougly considering the fact that the results presented here show
large differences between atomic and molecular relativistic effects.
3.5. References.
1. Krauss, M., Stevens, W. J., Ann. Rev. Phys. Chem. 35 (1984) 357.
2. Sakai, Y., Miyoshi, E., Klubokowski, M., Huzinaga, S., J. Comp. Chem. 8
(1987) 226.
3. Dolg, M., Wedig, U., Stoll, H., Preuss, H., J. Chem. Phys. 86 (1987) 866.
4. Christiansen, P. A., Ermler, W. C., Pitzer, K. S., Ann. Rev. Phys. Chem. 36
(1985) 407.
5. Pyper, N. C., Mol. Phys. 39 (1980) 1327.
6. Pyper, N. C., Mol. Phys. 42 (1981) 1059.
7. Pyper, N. C., Marketos, P., Mol. Phys. 42 (1981) 1073.
8.
9.
10.
11.
12.
13.
14.
Desclaux, J. P., Kim, Y. K., J. Phys. B 8 (1975) 1177.
Pitzer, K. S., Int. J. Quantum Chem. 25 (1984) 131.
Victor, G. A., Taylor, W. R., At. data nucl. data tables 28 (1983) 107.
Victor, G. A., Stewart, R. F., Laughlin, C., Astrophys. J. Suppl. 31 (1976) 237.
Das, G., Wahl, A. C., J. Chem. Phys. 64 (1976) 4672.
Kahn, L. R., Int. J. Quantum Chem. 25 (1984) 149.
Schwarz, W. H. E., Wezenbeek, E. M. van, Baerends, E. J., Snijders, J. G., J.
Phys. B: At. Mol. Opt. Phys. 22 (1989) 1515.
15. Lee, Y. S., Ermler, W. C., Pitzer, K. S., J. Chem. Phys. 67 (1977) 5861.
42
3. Non-relativistic valence-only calculations with relativistic core and relativistic
perturbation corrections
16.
17.
18.
19.
20.
21.
22.
23.
24.
Kahn, L. R., Hay, P. J., Cowan, R. D., J. Chem. Phys. 68 (1978) 2386.
Snijders, J. G., Baerends, E. J., Mol. Phys. 36 (1978) 1789.
Snijders, J. G., Baerends, E. J., Ros, P., Mol. Phys. 38 (1979) 1909.
Snijders, J. G., Pyykkö, P., Chem. Phys. Lett. 75 (1980) 5.
Katriel, J., Feller, D., Davidson, E. R., Int. J. Quantum Chem. 26 (1984) 489.
Rose, S. J., Grant, I. P., Pyper, N. C., J. Phys. B 11 (1978) 1171.
Christiansen, P. A., Ermler, W. C., Mol. Phys. 55 (1985) 1109.
Ziegler, T., Snijders, J. G., Baerends, E. J., J. Chem. Phys. 74 (1981) 1271.
Ziegler, T., Snijders, J. G., Baerends, E. J., Chem. Phys. Lett. 75 (1980) 1.
25. Visser, O., Visscher, L., Aerts, P. J. C., Nieuwpoort, W. C., Theoret. Chim. Acta
(accepted for publication).
26. Diercksen, G., McWeeny, R., J. Chem. Phys. 44 (1966) 3554.
27. Aerts, P. J. C., Towards relativistic quantum chemistry. Thesis, Groningen
(1986).
28. Aerts, P. J. C., Nieuwpoort, W. C. Int. J. Quantum Chem. Symp. 19 (1986) 267.
43
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
4. Relativistic all-electron molecular
Hartree-Fock-Dirac-(Breit) calculations
on CH4, SiH4, GeH4, SnH4 and PbH4.
O. Visser, L. Visscher, P.J.C. Aerts and W. C. Nieuwpoort,
Laboratory for Chemical Physics,
State University of Groningen,
Nijenborgh 16,
9747 AG GRONINGEN,
The Netherlands.
E-mail: olivier@rug.nl
fax: 3150634200
(accepted for publication in Theoret. Chim Acta)
4.1.
Summary.
Results and details of molecular Fock-Dirac-(Breit) calculations on CH4, SiH4, GeH4, SnH4
and PbH4 obtained with the MOLFDIR© program package are presented and compared with
other calculations and experimental results. The relativistic ground state energies (including
the Breit interaction) of the atoms C, Si, Ge, Sn and Pb, necessary for reference purposes,
have been calculated using a small relativistic CI. One of our findings is that for the heavier
systems perturbation theory over-estimates the relativistic bond length contraction. The Breit
interaction has only a small effect on the bond lengths.
Key words: Relativistic ab initio calculations — Hartree-Fock-Dirac method — Breit
interaction — tetrahedral hydrides
45
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
4.2. Introduction.
The ab initio study of relativistic effects in molecular systems is a relatively new field
of research. Only in the past decennium various calculations based on relativistic
quantum mechanics have been performed [1 - 4 and references therein]. Most of these
calculations concern atomic systems, a few concern diatomic molecules. Applications
to polyatomic molecular systems have been sparse, except for calculations based on
local density or other more approximate methods.
Approaches to the study of relativistic effects include perturbation theory, relativistic
pseudo-potential techniques, relativistic local density methods and Fock-Dirac calculations. We use the all-electron Hartree-Fock-Dirac model followed by a small
relativistic configuration interaction (CI). This allows the results to serve as reference
for more approximate methods. The method also enables the study of the effects of
the Breit interaction.
In this article we present all-electron Hartree-Fock-Dirac SCF-results of calculations
on the series CH4, SiH4, GeH4, SnH4 and PbH4. Some calculations published
earlier by some of us [4, 5], concerning CH4, SiH4, GeH4, have been improved. For
reference purposes, we also present results of small relativistic CI calculations on the
group IV atoms C through Pb. Among the results are relativistic bond lengths, bond
energies and the effect of the Breit interaction on these properties.
4.3. Theory.
In this section the basic equations underlying our method are presented and the way
we solve them is described. Some details of the MOLFDIR© (Molecular Fock-Dirac)
program package are given (type of basis functions, kinetic and atomic balance, and
general contraction); more details on the open shell and the COSCI (Complete Open
Shell Configuration Interaction) method used,
implementation details can be found elsewhere [5-7].
symmetry
handling,
and
4.3.1.
General.
The time-independent Dirac equation for one-electron molecular systems [8, 9] can be
written as (in atomic units)
c p + mc2 + V  =  
(1)
46
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
The  and  are the four by four matrices defined by
= 0 
 0
and  = 1 0
0 -1
(2)
in which  is the vector of Pauli spin-matrices, V is the potential energy V r; R due
to the nuclei at R, c is the speed of light in vacuum (the value 137.03602 a.u. is used
in the calculations) and m is the rest-mass of the electron. (r) is a four-component
spinor which is conveniently written as a combination of two bi-spinors:
=
L
(3)
S
L and S refer to 'large' and 'small'-component respectively because to first-order
approximation
S  1  pL
2c
(4)
for positive energy states.
One can show from quantum electrodynamics [10, 11] that an approximate relativistic
many-electron equation (the Dirac-Coulomb-Breit equation) is given by
•
i
 i  pi + i mc2 + Vi + • r1 + Bij  = E 
ij
(5)
i<j
The two-electron interaction in this equation consists of the usual Coulomb term and
the Breit term. The latter term, which for chemical systems contributes numerically
only marginally to the total energy compared to the Coulomb term [10], is given by
Bij = - 1
2
i j
 i  rij  j  rij
rij +
r3ij
(6)
The Breit term can be approximated by the Gaunt term [2, 10, 12], which is just twice
the first term in expression (6). In this article we use the Gaunt term only.
47
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
4.3.2.
The relativistic open shell SCF equations.
The relativistic open shell SCF equations can formally be derived in the same way as
their non-relativistic counterparts [13]. The one-electron Schrödinger operators are
replaced by Dirac operators, the interaction terms appear on the diagonal of a fourcomponent matrix and the scalar spin-orbitals become four component spinors.
Fundamentally there is of course a major difference when variational methods are
applied to find approximate solutions. In the search for stationary points in the
parameter space of the energy functional in the case of the Dirac equation, there is no
variational principle that guarantees that one will never find energies lower than the
exact (positive) ground state energy.
The Breit interaction term can -technically- be included in the hamiltonian and thus be
used in the variational process. Alternatively, it's effect can be included afterwards by
perturbation theory. It is still a matter of discussion whether it is legitimate to include
the Breit term in the variational process [14, 15 and references therein].
4.3.3.
Basis functions.
The MOLFDIR program package uses two distinct sets of, usually atom-centered,
scalar (contracted) cartesian gaussian functions: a 'large component' set gLi and a
'small component' set gSi . The required two-electron repulsion integrals are
calculated over the functions belonging to these sets.
From these two scalar sets two new sets of symmetry adapted molecular Dirac spinors
Ll and  Ss are constructed using the Dirac double group symmetry elements. Ll
and  Ss are defined by
 L
l
 Ll =
L
l
0
0
and
 Ss =
0
(7)
 S
s
 S
s
0
L
S
L L
S
S S
L L
S S
L
l = • gi cil , l = • gi cil , s = • gi cis and s = • gi cis
i
i
i
(8)
i
where the coefficients cX
ij (X{L,S}; {,) in equation (8) are determined by
symmetry. Thus, each non-zero component of these basis spinors consists of a linear
48
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
combination of the basis functions from set gLi or set gSi . This structure of the basis
set is the only technical restriction on the choice of the basis spinors.
4.3.4.
Kinetic balance and General contraction.
Kinetic balance, discussed by several authors [16 - 18], is an important means to yield
systematic improvement of the solutions upon basis set extension. This condition
requires that for each four-spinor Ll in the large component basis, the four-spinor
 p Ll is contained in the small component basis set. Since kinetic balance is neither
a necessary nor sufficient condition to produce the best approximate eigensolutions,
we normally extend the small component basis set beyond kinetic balance [17] (while
maintaining the balance rigourously). In practice this means that the small component
basis set contains more basis functions than those minimally required by the kinetic
balance condition.
The small component basis set extension can consist of two types of additions.
The first set of extra functions results from the fact that, where  p operating on a
single gaussian primitive in the large component yields a fixed linear combination of
two gaussians in the small component, we still use these generated gaussians as two
separate primitives instead of that fixed combination. Especially near the origin the
kinetic balance condition is much too restrictive for the small component and this
extension is an easy way to relax that restriction without violating the kinetic balance
itself.
A second set of extra functions in the small component is used in molecular
calculations where atomic solutions are used to contract the large and small
component basis functions. When the contracted small component functions are
derived by kinetic balance from the contracted large component functions, they differ
significantly from the atomic small component solutions in the uncontracted basis set.
Thus, in addition to the kinetically balanced functions the atomic small component
solutions are used as small component basis functions. A basis set which includes
such an extension is called an atomically balanced basis set.
When general contraction [19] is used to reduce the variational space we still require
that the small component basis fulfils atomic and kinetic balance conditions. The contraction reduction is thereby somewhat counteracted by the fact that in some cases up
to four contracted functions for each large component function are necessary: up to
49
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
two functions to ensure kinetic balance, and up to two functions to ensure atomic
balance. Although these small component functions together may become linear
dependent for practical purposes (so that one or more functions may be removed) the
gain by general contraction in the small component basis set is small compared to the
gain in the large component basis set.
Using basis sets which have been constructed to fulfil the kinetic balance condition,
we have not found any positive energy spurious solutions. Also, a lower bound to the
total energy is not guaranteed by the use of such basis sets.
4.3.5.
COSCI approach.
For a number of applications a one-determinant approximation to the many-electron
wavefunction is insufficient, even if electron correlation is not explicitly considered.
For example if one wants to describe the ground state wavefunction of a carbon atom
with its two 2p valence electrons using a one-determinant approximation, the closed
shell would be a 2p1/22 configuration. But the carbon 3P0 ground state is a mixture of
the three px1/2-p2-x3/2 (x = 0, 1, 2) configurations. We handle this problem by
constructing an average open shell energy expression [7] (with contributions from all
possible determinants belonging to the configuration of interest, in this case p2). Next
the set of SCF spin-orbitals that belongs to the average expression is calculated. Using
these 'average' spin-orbitals, the hamiltonian matrix is constructed in the complete
open shell space and diagonalized. This procedure yields the stationary energies of all
possible states from one configuration. Again for carbon, we get the energy for the
3P , 3P , 3P , 1D and 1S states (all described with the same set of orbitals). This
0
1
2
0
0
method will be called the COSCI method (complete open shell configuration
interaction); note that the orbitals are not optimized as in a CASSCF calculation. The
COSCI method can be used within both the non-relativistic and the relativistic
framework.
This example illustrates the need for the COSCI method to calculate the ground state
of the atoms C to Pb. The results are used to calculate the binding energies of the
hydrides.
Another important application of this method is the determination of the energy
eigenvalues of the f-multiplet of rare-earth atoms in molecules or in solid state
systems. Results from such calculations will be published shortly.
50
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
4.4. Applications.
In this work, Hartree-Fock-Dirac calculations have been performed on the
tetrahydrides XH4, with X one of the atoms C, Si, Ge, Sn or Pb. The bond length and
binding energies are calculated for each of these molecules. To show the accuracy of
the contracted basis sets used, some of the calculations have also been performed with
the uncontracted basis set for comparison. The effect of the Breit interaction has also
been studied.
To prepare for the molecular calculations, and for comparison with atomic
calculations, we have performed separate Fock-Dirac calculations on H, C, Si, Ge, Sn
and Pb. For these atomic systems uncontracted basis sets can be used. The optimal
atomic orbitals are used to construct (general) contracted basis sets for the hydride
molecules.
4.4.1.
Table 1:
Computational details.
Summary of basis sets used in the calculations on XH4
(NR: non-relativistic, FD: Fock-Dirac-(Breit)).
XH4
Basis for X
Basis for H
Remarks
CH4
(10s, 5p | 5s, 10p, 5d)
[8s, 4p | 5s, 10p, 5d]
(6s | 6p)
[4s | 6p]
segmented
contraction
SiH4
(12s, 8p | 8s, 12p, 8d)
[9s, 6p | 8s, 12p, 8d]
(6s | 6p)
[3s | 6p]
segmented
contraction
GeH4
(12s, 9p, 5d | 9s, 17 p, 9 d, 5 f)
(6s | 6p)
uncontracted
SnH4
(15s, 11p, 6d | 11s, 17 p, 11 d, 6 f)
[11s, 9p, 5d | 10s, 13p, 10d, 5f]
(4s | 4p)
[3s | 3p]
segmented
contraction
PbH4
(19s, 16p, 10d, 5f | 16s, 19 p, 16 d, 10 f, 5g)
(4s | 4p)
general
NR: [7s, 6p, 3d, 1f]
FD: [7s, 10p, 6d, 2f | 13s, 18p, 16d, 8f, 4g]
[2s | 2p]
contraction
In table 1 information on the basis sets relevant to the calculations on the systems
described in this article are given. We use the notation (ns, mp, ... | n's, m'p, ...) to
indicate the number of primitive Gaussians used in the large and small component
basis sets respectively. We use square brackets to describe the number of contracted
functions.
51
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
Only for the PbH4 molecule the general contraction scheme has been used. The
primitive basis set was constructed from an existing non-relativistic basis set [20]
which has been re-optimized under the constraint that the d-exponents form a subset
of the s-exponents, and the f-exponents form a subset of the p-exponents. Using the
relativistic atomic solution for the Pb-atom this basis was then contracted to a
minimal atomic basis for the large component, with an extra diffuse s and p function
to give flexibility in the valence region for Pb; for H one diffuse s-function has been
added to the large component. The general contracted small-component basis was
next constructed using both the atomic and kinetic balance conditions.
Details (list of primitive Gaussians and details of the segmented contractions) of the
basis sets used for the calculations with X=C, Si and Ge can be found elsewhere [5].
The primitive functions used in the contracted basis set for Sn can also be found
elsewhere [5]; the contraction scheme we have used is available upon request.
The molecular calculations have been performed at the following X-H distances:
CH4 : r = 1.07, 1.08, 1.09, 1.10, 1.11 Å;
SiH4 : r = 1.46, 1.47, 1.48, 1.49, 1.50, 1.51, 1.52 Å;
GeH4 : r = 1.48, 1.54, 1.56, 1.58, 1.64 Å;
SnH4 : r = 1.68, 1.72, 1.73, 1.74, 1.78 Å;
PbH4 : r = 0.988, 1.587, 1.693, 1.728, 1.737, 1.746, 1.799 Å.
The bond lengths and force constants of these molecules have been determined by a
parabolic fit using three points close to the minimum energy.
In most calculations, the Breit contribution to the total energy has been calculated
using first order perturbation theory. For the GeH4 molecule we have also performed
calculations in which the Breit interaction was treated variationally. We have not
included the Breit interaction in the COSCI calculations on the atoms; to get an
estimate of the ground-state energies of the atoms including the Breit interaction we
have used the Breit correction to the average SCF energy.
For the PbH4 problem, two-electron integrals over small component basis functions
with absolute numerical value less than 10-6 have not been used in order to reduce
disk space. We have studied the possible effect of this approximation by setting the
threshold further to 10-5.
52
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
4.4.2.
Results and Discussion.
4.4.2.1.
Atomic results.
The atomic results are collected in figure 1 and table 2. The calculated values for the
splittings within the 3P terms show satisfactory agreement with experiment. For Ge,
Sn and Pb the 3P2 - 3P0 splitting differs from the experimental value by 3-9%, for the
3P - 3P splitting the differences are 10-20%. Comparison with numerical results
1
0
[21] shows that the differences with experiment are mainly due to the finite basis set
approximation. The remaining error is due to the neglect of the Breit interaction and
to the limited description by the COSCI approach.
3
-1
E - E ( P0 ) (cm )
Figure 1:
atomic FD-COSCI results for the 3P1 and 3P2 states relative to the 3P0 states, compared
with experiment [23].
3P1
3P2
3P1 (exp.)
3P2 (exp.)
10 4
104
10 3
103
10 2
102
10 1
101
C
Si
Ge
Sn
Pb
Atom
Table 2:
Ato
m
atomic FD-COSCI results: Total energies, experimental results (from [23]), contributions
of the p1/22 and p3/22 configurations to the ground state CI-vector. The contribution of a
configuration to the ground state CI-vector has been calculated by summing the squares of
the absolute value of the coefficients of all determinants belonging to the given
configuration.
Etot (3P0)
E(3P1) - E(3P0)
(a.u.)
(cm-1)
E(3P2) - E(3P0)
Exp (cm-1)
53
(cm-1)
Exp (cm-1)
p1/22
p3/22
(%)
(%)
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
C
Si
Ge
Sn
Pb
-37.70287
19
16.4
57
43.5
66.8
33.2
-289.47100
81
77.15
239
223.31
67.6
32.4
-2097.26431
507
557.10
1349
1409.90
71.4
28.6
-6174.94537
1357
1691.8
3111
3427.7
77.5
22.4
-20912.66788
6621
7819.35
10367
10650.47
92.1
8.0
If the Russell-Saunders coupling scheme is a good approximation of the
wavefunction, and if the spin-orbit coupling is a small perturbation [22], the splittings
within the 3P-level can be described using the Landé interval rule
EJ - EJ-1 =  J
(9)
We observe a significant breakdown of the Landé interval rule for the heavier atoms.
As usual, this can be explained by the deviation from LS coupling, which is evident
from the contributions of the different configurations to the ground state wave
function, and by the fact that the Landé interval rule neglects all relativistic effects
except spin-orbit coupling.
4.4.2.2.
Table 3:
Molecular results.
Bond length r in Å (NR: non-relativistic, FD: Fock-Dirac, FD+BR: Fock-Dirac with Breit
interaction included as a perturbation; ∆r: difference with NR value).
r (Å)
Molecule
CH4
SiH4
GeH4
SnH4
PbH4
Table 4:
Molecule
CH4
NR
FD
FD+BR
∆r (Å)
FD
1.08327
1.08314
1.08323
-.00013
-.00004
1.48749
1.48660
1.48677
-.00089
-.00072
1.56379
1.55742
1.55793
-.00637
-.00586
1.75423
1.73280
1.73369
-.02143
-.02054
1.80772
1.73536
1.73703
-.07236
-.07069
FD+BR
Binding energy Eb in a.u. (NR: non-relativistic, FD: Fock-Dirac, FD+BR: Fock-Dirac with
Breit interaction included as a perturbation; ∆Eb: difference with NR value).
NR
Eb (a.u.)
FD
-.50423
-.50371
54
FD+BR
∆Eb (a.u.)
FD
FD+BR
-.50361
.00052
.00062
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
SiH4
GeH4
SnH4
PbH4
Table 5:
-.35645
-.35446
-.35436
.00199
.00209
-.32177
-.30814
-.30805
.01363
.01372
-.28108
-.25144
-.25131
.02964
.02977
-.27643
-.18948
-.18867
.08695
.08776
Force constant k in a.u. (NR: non-relativistic, FD: Fock-Dirac, FD+BR: Fock-Dirac with
Breit interaction included as a perturbation; ∆k: difference with NR value).
k (a.u.)
Molecule
CH4
SiH4
GeH4
SnH4
PbH4
4.4.2.2.1.
∆k (a.u.)
NR
FD
FD+BR
FD
FD+BR
1.52
1.52
1.52
-0.00
-0.00
0.82
0.80
0.81
-0.01
-0.01
0.67
0.65
0.66
-0.02
-0.02
0.53
0.53
0.53
0.00
0.00
0.61
0.62
0.61
0.01
-0.01
Accuracy.
The force constants are of course sensitive to the details of the fit used (the selection
of the points taken in the fit). We estimate the absolute error in the force constants to
be of the order of 10-2 a.u.. The number of decimal places given in table 5 is in
accordance with this accuracy.
The non-relativistic calculations on PbH4 with the uncontracted basis set predict a
bond length rmin = 1.80714 Å, a binding energy for the molecule of Eb = -0.28359
a.u. and a force constant k = 0.62 a.u.. If we compare these results with the results
given in table 3 to 5 (which have been obtained using general contracted basis sets)
we find that the use of general contraction introduces a small error as a price for the
loss of variational freedom: for the bond length a difference of 6*10-4 Å is found and
for the binding energy a difference of 7*10-3 a.u. is found (the force constant did not
change significantly).
The results of the Fock-Dirac calculations on PbH4 in which the small-component
two-electron integrals with absolute value less than 10-5 have been ignored (rmin =
1.73535 Å, Eb = -0.18948 a.u., k = 0.62 a.u.) show (by comparison with the data in
the tables) that the error introduced by leaving out those small integrals is not
significant .
55
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
In tables 3 and 4 we have given five decimal places in order to see the differences
introduced by the Breit correction. We expect that the calculated bond length for
PbH4 will be accurate in three decimal places. The binding energies for PbH4 have an
estimated error of the order of 10-2 a.u.. For the other molecules, the results will be
significantly more accurate since the basis sets used to produce those results have not
been as heavily contracted as the basis sets used for the PbH4 calculations. Basis set
truncation errors (and errors from leaving out functions with higher angular
momentum) have not been estimated.
4.4.2.2.2.
Hartree-Fock-Dirac results.
The Hartree-Fock-Dirac results show the well known relative relativistic bond length
contraction; we find relativistic contractions ranging from 0.01% (CH4) to 4.2%
(PbH4). This leads to a bond length of the Pb-H bond in the PbH4 molecule which is
very close to the Sn-H bond length in SnH4.
The Hartree-Fock-Dirac results also show that, going from the lighter to the heavier
systems, the binding energy decreases significantly in absolute value. This results in a
binding energy of PbH4 which is significantly smaller than the binding energy of
SnH4 (the non-relativistic theory predicts that both binding energies are about equal).
We also find a small decrease in the force constants of the Si-H and Ge-H bonds. For
the other molecules the differences between the force constants calculated by the
Hartree-Fock-Dirac method and by the non-relativistic method are not significant.
4.4.2.2.3.
Effect of the Breit interaction.
In figure 2 the bond length found by the relativistic calculations (with and without the
Breit interaction) is given relative to the bond length predicted by non-relativistic
calculations. It can be seen that for PbH4 the Breit interaction leads to a slightly
longer bond length; for the other molecules the absolute effect is small. The data in
table 3 show the general trend that the effect of the Breit interaction cancels a large
part of the other relativistic effects on the bond length for the lighter systems.
56
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
Figure 2:
Relativistic effect on the bond length of the molecules XH4, X = C, Si, Ge, Sn and Pb.
Fock-Dirac
Fock-Dirac-Breit
0.02
Bond length relative to NR (a.u.)
0
-0.02
-0.04
-0.06
-0.08
CH4
SiH4
GeH4
SnH4
PbH4
Mo lecule
Thus, for the heavy systems the effect of the Breit interaction should be taken into
account (although correlation corrections are expected to be much more important),
while for the lighter systems the Breit interaction should be included when relativistic
effects are considered (in CH4, the Breit interaction cancels a large part of the bond
length contraction found neglecting the Breit interaction).
The binding energy and the force constants are also influenced by the Breit
interaction, but in general no significant deviations from the Dirac-Coulomb results
are found.
In figure 3 the total SCF-energy and the Breit correction to this energy as a function of
the bond length are shown for the PbH4-molecule. In the bond length region we have
examined, we find that the Breit interaction gets less positive with increasing bond
length. The Breit interaction thus favours bond length expansion (relative to the bond
length predicted by a relativistic calculation without the Breit interaction). The slope
of the Breit correction versus the bond length is very small (relative to the depth of
the well provided by the total SCF-energy versus the bond distance), so the resulting
bond length expansion is also very small.
57
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
Figure 3:
Total energy and Breit correction for PbH4 as a function of the bond length r.
Etot (+20914, a.u.)
Ebreit (a.u.)
-0.8250
27.6518
-0.8300
27.6516
-0.8350
27.6514
-0.8400
27.6512
-0.8450
27.6510
-0.8500
-0.8550
27.6508
-0.8600
27.6506
2.9
3
3.1
3.2
3.3
3.4
3.5
r (a.u.)
For the GeH4 molecule, we have also performed some calculations treating the Breit
interaction variationally. The bond length and the force constant obtained by these
calculations are identical to the results given in the tables; for the binding energy we
find a difference of 6*10-4 a.u., which is not significant. Thus we conclude that for
lighter systems, as far as bond lengths, force constants and bond energies are
concerned, it makes no difference whether the Breit interaction is treated as a
perturbation or it is treated variationally.
4.4.2.2.4.
Comparison with other methods.
In table 6 some theoretical and experimental results which have been given by other
authors are presented. This list was largely compiled using data given by Pyykkö [1].
In the various results, different methods and basis sets have been used. Although
therefore care must be taken when extracting general trends from these data we can
make some points.
Table 6:
XH4
CH4
CH4
Bond length results from literature.
rNR (Å)
1.099
rRel (Å)
1.083
1.099
∆r (Å)
0.00010
0.000
Method, reference
P-HF Almlof and Faegri [24]
DF-OCE Desclaux and Pyykkö [26]
58
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
CH4
CH4
CH4
SiH4
SiH4
SiH4
SiH4
SiH4
GeH4
GeH4
GeH4
GeH4
GeH4
GeH4
SnH4
SnH4
SnH4
SnH4
SnH4
SnH4
SnH4
PbH4
PbH4
PbH4
PbH4
PbH4
PbH4
PbH4
PbH4
1.0833
1.4875
1.596
1.564
1.804
1.732
1.736
1.754
1.89
1.827
1.806
1.806
1.808
1.145
1.0832
1.086
1.482
1.572
1.586
1.4868
1.481
1.521
1.586
1.603
1.522
1.558
1.527
1.705
1.772
1.783
1.715
1.717
1.734
1.701
1.703
1.782
1.743
1.795
1.732
1.739
1.737
1.754
0.00004
0.00066
0.001
0.0007
0.0070
0.010
0.0059
0.021
0.032
0.017
0.019
0.0205
0.10
0.107
0.084
0.074
0.067
0.0707
X Aguilar-Ancono, Gázquez and Keller [27]
FD+BR This work
Experiment [25]
P-HF Almlof and Faegri [24]
DF-OCE Desclaux and Pyykkö [26]
X Aguilar-Ancono, Gázquez and Keller [27]
FD+BR This work
Experiment (taken from Desclaux and Pyykkö [26])
P-HF Almlof and Faegri [24]
DF-OCE Desclaux and Pyykkö [26]
X Aguilar-Ancono, Gázquez and Keller [27]
PP-MRCI Das and Balasubramanian [31]
FD+BR This work
Experiment (taken from Desclaux and Pyykkö [26])
P-HF Almlof and Faegri [24]
DF-OCE Desclaux and Pyykkö [26]
X Aguilar-Ancono, Gázquez and Keller [27]
PP-HF Pélissier [28]
PP-HF Fernandez, Arriau and Dargelos [29]
FD+BR This work
Experiment (taken from Desclaux and Pyykkö [26])
P-HF Almlof and Faegri [24]
DF-OCE Desclaux and Pyykkö [26]
PP-HF Pélissier [28]
X Aguilar-Ancono, Gázquez and Keller [27]
ARPP Schwerdtfeger, Silberbach, Miehlich [30]
QRPP Schwerdtfeger, Silberbach, Miehlich [30]
FD+BR This work
'Experiment' (from Desclaux and Pyykkö [26])
The 'experimental' value for PbH4 was deduced (following Desclaux and Pyykkö) from
r (PbH4) = r (PbH) + r (SnH4) - r (SnH).
rNR: non-relativistic bond length;
rRel: relativistic bond length;
∆r: difference between rNR and rRel;
P-HF: Perturbation theory;
PP-HF: Pseudopotential Hartree-Fock;
PP-MRCI: Pseudopotential Multireference CI;
X: one-component relativistic method with X local exchange, using a spherical symmetric
one-center approximation;
ARPP: spin-orbit averaged relativistic pseudopotential;
QRPP: spin-orbit coupled pseudopotential (quasi relativistic pseudopotential);
DF-OCE: Dirac-Fock one center expansion;
FD+BR: Fock-Dirac with Breit interaction included as a perturbation.
59
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
First of all, it is evident that the one-component relativistic one-center X method
used by Aguilar-Ancono et al. significantly over-estimates the bond lengths of the
systems considered. The DF-OCE method used by Desclaux and Pyykkö also overestimates bond lengths. This is almost certainly a defect of the one-center expansion
technique. The other data essentially give the same figures for the molecules other
than PbH4. For PbH4, perturbation theory over-estimates the bond length contraction
quite a lot, while the results from the pseudopotential methods are close to our results
but do not show a systematic trend.
After completion of this work we received a report from Dyall, Taylor, Faegri and
Partridge in which they describe results of Dirac-Hartree-Fock calculations on the
series CH4 to PbH4. The main differences between the method described in this work
and the method used by Dyall et al. are the choice of the small component basis sets
and the inclusion of the Breit interaction (Dyall et al. have not included the Breit
interaction). Nevertheless, their results are in close agreement with our results.
The experimental figures in the table show that there is still a significant discrepancy
between the theoretical and experimental numbers. This illustrates the fact that,
except for the lighter elements, relativistic effects are equally, but not more important
than correlation effects. It is interesting to see that for PbH4 the relativistic effects are
much more important than the correlation effects (assuming these effects to be largely
additive and the quoted 'experimental' value to have sufficient significance). This is
particularly well illustrated by the results of Almlof and Faegri [24], who are
generally right on target with their perturbative approach except where the relativistic
effects are becoming dominant, as is the case for Pb and PbH4.
4.5. Conclusions.
In this article we have given results which show the state of the art in ab initio
molecular relativistic quantum chemical calculations using the Fock-Dirac-(Breit)
formalism. The general contraction method introduced in this context, using kinetic
and atomic balance for the small components of molecular basis sets, contributes to
the feasibility of reliable relativistic ab initio methods.
60
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
In our calculations on the hydrides, we have verified that the Breit interaction leads to
small but sometimes significant corrections to the relativistic results both for light and
heavy atoms (bond length expansion for the PbH4). For the lighter systems the
Hartree-Fock-Dirac method without the Breit interaction over-estimates the bond
length contractions (for CH4 by about a factor 3). We have also found that the
relativistic effects are much more important than the correlation effects for the PbH4
molecule and are therefore to be taken into account first.
4.6.
1.
2.
3.
4.
5.
6.
References.
Pyykkö P (1988) Chem. Rev. 88:563.
Pyykkö P (1978) Adv. Quant. Chem. 11:353.
Christiansen PA, Ermler WC, Pitzer KS (1985) Ann. Rev. Phys. Chem. 36:407.
Aerts PJC, Nieuwpoort WC (1986) Int. J. Quantum Chem.: Quant. Chem.
Symp. 19:267.
Aerts PJC (1986) Towards relativistic quantum chemistry. Thesis, Groningen.
Visscher L, Aerts PJC, Visser O (1991) General Contraction in FourComponent Relativistic Hartree-Fock Calculations. In: Grant IP, Gyorffy B,
Wilson S (ed) The effects of relativity in atoms, molecules and the solid state.
Plenum New York .
7. Visser O, Aerts PJC, Visscher L (1991) Open shell relativistic Molecular DiracHartree-Fock SCF-program. In: Grant IP, Gyorffy B, Wilson S (ed) The effects
of relativity in atoms, molecules and the solid state. Plenum New York .
8. Dirac PAM (1928) Proc. Roy. Soc. London A 117:610.
9. Dirac PAM (1928) Proc. Roy. Soc. London A 118:351.
10. Grant IP, Quiney HM (1988) Adv. At. Mol. Phys. 23:37.
11. Breit G (1929) Phys. Rev. 34:553.
12. Gaunt JA (1929) Proc. R. Soc. A 122:513.
13. Roothaan CCJ (1960) Rev. Mod. Phys. 32:179.
14. Quiney HM, Grant IP, Wilson, SJ (1990) J. Phys. B: At. Mol. Opt. Phys.
23:L271.
15. Okada S, Shinada M, Matsuoka O. (1990) J. Chem. Phys. 93:5013.
16. Stanton RE, Havriliak SJ (1984) Chem. Phys. 81:1910.
17. Aerts PJC, Nieuwpoort WC (1986) Chem. Phys. Lett. 125:83.
18. Kutzelnigg W (1984) Int. J. Quantum Chem. 25:107.
19. Raffenetti RC (1973) J. Chem. Phys. 58:4452.
20. Gropen OJ (1987) Comp. Chem. 8:982.
61
4. Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on
CH4, SiH4, GeH4, SnH4 and PbH4
21. Desclaux JP. (1972) Int. J. Quantum Chem. Symp. 6:25.
22. Condon EU, Shortley GH (1963) The theory of atomic spectra. The University
Press Cambridge.
23. Moore, CE (1958) Atomic energy levels. National Bureau of Standards
Washington.
24. Almlof J, Faegri KJr (1986) Theor. Chim. Acta 69:437.
25. Gray DL, Robiette AG (1979) Mol. Phys. 37:1901.
26. Desclaux JP, Pyykkö P (1974) Chem. Phys. Lett. 29:534.
27. Aguilar-Ancono A, Gázquez JL, Keller J Chem. (1983) Phys. Lett. 96:200.
28.
29.
30.
31.
Pélissier M (1980) Thesis, Toulouse .
Fernandez J, Arriau J, Dargelos A, (1985) Chem. Phys. 94:397.
Schwerdtfeger P, Silberbach H, Miehlich B (1989) J. Chem. Phys. 90:762.
Das KK, Balasubramanian K J. (1990) Chem. Phys. 93:5883.
62
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
5.
Molecular Open Shell CI calculations using the
Dirac-Coulomb hamiltonian.
The f6-manifold of an embedded EuO69- cluster.
O. Visser, L. Visscher, P. J. C. Aerts and W. C. Nieuwpoort,
Laboratory for Chemical Physics,
State University of Groningen,
Nijenborgh 16,
9747 AG GRONINGEN,
The Netherlands.
E-mail: olivier@rug.nl
(accepted for publication in J. Chem. Phys.)
5.1.
Abstract.
We present results of all-electron molecular relativistic (Hartree-Fock-Dirac) and nonrelativistic (Hartree-Fock) calculations followed by a Complete Open Shell Configuration
Interaction (COSCI) calculation on an EuO69- cluster in a Ba2GdNbO6 crystal. The results
include the calculated energies of a number of states derived from the f6-manifold and 5D - 7F
luminescence transition wavelengths. The calculations were performed using the MOLFDIR
program package developed in our laboratory. The theory and methods employed in this
package are briefly described.
The physical models used to analyse the Eu3+ impurity states range from a bare Eu3+ ion to
an EuO69- cluster embedded in a Madelung potential representing the rest of the crystal. We
show that it is necessary to use the embedded cluster model to get a reasonable description
of the crystal field splittings of the states arising from the f6-manifold.
63
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Our results indicate that the calculated splittings are very sensitive to the orbitals used. It is
therefore essential that relativistic orbitals be used from the outset.
5.2. Introduction.
Compounds containing lanthanide ions (Z = 57 - 71) form an interesting class of
materials, both from a scientific and a practical point of view. Their optical and
magnetic properties for example are at present technically widely exploited, while
being subjects of continuing experimental and theoretical investigations. These
properties arise from the complex manifold of electronic states arising from the
incompletely filled 4f shells of the lanthanide atoms or ions. Since the classic work of
Becquerel and Bethe [1, 2] it is well known how to describe these states in crystalline
surroundings semi-empirically. In recent years this theory has been revitalized and has
considerably been extended by Thole in connection with the rationalization of X-ray
absorption spectroscopic data [3, 4]. At least for the lower lying electronic states, the
coulombic interaction between the electrons (on the order of eV's) still dominates the
spin-orbit interaction (on the order of tenths of eV's), although not as heavily as in the
case of the 3d metals. In contrast to the latter case, however, the perturbing effects of
the surrounding are much smaller than the spin-orbit interaction and therefore smaller
than relativistic effects in general. This poses a serious problem when one wants to go
beyond a semi-empirical description and use ab initio computational methods, for
example to obtain more quantitative insight into the origin of crystal fields or into the
rates of various radiative or non-radiative decay or excitation transfer processes. The
problem is whether ab initio results based on the non-relativistic Schrödinger
equation with relativistic corrections added on can be relied upon or that a relativistic
approach is needed from the outset to account for the differences between relativistic
and non-relativistic orbitals.
At present the ab initio treatment of relativistic effects in compounds of the kind
considered here is limited to the use of an effective Schrödinger equation for valence
electrons containing relativistic perturbation terms and a relativistic core potential
derived from atomic Fock-Dirac calculations [5 and references therein]. As it is based
on a Schrödinger equation this useful approach allows the inclusion of electron
correlation effects by standard quantum chemical methods. All electronab initio
calculations, however, including relativistic effects as well as correlation effects on
the same theoretical level, have not yet been performed on molecular systems
containing heavy atoms. Some calculations based on relativistic quantum mechanics
64
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
have been performed [6-8 and references therein]. Most of the calculations concern
atoms, and a few deal with diatomic molecules. Applications to molecular systems
containing more than two nuclei have been sparse [9-12], except for calculations
based on a local density approach.
More than a decade ago Van Piggelen [13] has carried out non-relativistic ab initio
calculations, with spin-orbit perturbation corrections, to describe some of the states
from the f-manifold of the Eu3+ impurity in Ba2GdNbO6, using a molecular model
(an EuO69- cluster embedded in the Madelung field of the rest of the crystal). The
impurity site in this compound has cubic (octahedral) symmetry, which was important
to make the calculations feasible. His interpretation of the calculated crystal field
splitting was inconclusive because of the problem mentioned. In this work we use the
same molecular model but carry out fully relativistic calculations by applying the allelectron Hartree-Fock-Dirac method followed by relativistic configuration interaction
(CI), resulting in a complete intermediate coupling description of the f-like manifold
of the lanthanide impurity. One aim of the work is to obtain results that can serve as a
reference for more approximate methods. The method also enables the study of
correlation effects and the effects of the Breit interaction [11].
5.3. Theory.
We use an average of configuration Self Consistent Field method (Hartree-Fock for
non-relativistic calculations, Hartree-Fock-Dirac for relativistic calculations) to
generate a set of one-electron spin orbitals. Using these average orbitals, we perform a
Complete Open Shell Configuration Interaction (COSCI) calculation to get a
description of the open shell manifold. In this section we give the basic equations of
the method and the way these are solved. Some details of the MOLFDIR (Molecular
Fock-Dirac) program package [9, 11, 14, 15] are given, in particular the form of the
basis functions, kinetic and atomic balance and general contraction.
5.3.1.
General.
The time-independent Dirac equation for one-electron molecular systems [16, 17] is
given by
h = 
(1)
65
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
in which h is the one-electron Dirac operator defined by
(2)
h = c p + (-1)mc2 + V
We have shifted the energy scale by -mc2 to facilitate comparison with nonrelativistic energies. The  and  are four by four matrices defined by
= 0 
 0
and  = 1 0
0 -1
(3)
in which  is the collection of Pauli spin-matrices, V is the potential energy V r; R
due to the nuclei at R, c is the speed of light in vacuum (137.03602 a.u. used) and m
is the rest-mass of the electron. (r) is a four-component spinor, conveniently written
in bi-spinor form:
=
L
(4)
S
The first-order approximation to the positive-energy states is
S  1  pL
2mc
(5)
so the lower bi-spinor is conventionally called the 'small component', and the upper
bi-spinor the 'large component'.
One can generalize the one-electron Dirac equation to an approximate relativistic
many-electron equation (the Dirac-Coulomb equation) [18], which is given (in atomic
units) by
•
i
 i  pi + ( i -1)mc2 + Vi + • r1  = E 
i<j
ij
(6)
The two-electron interaction in this equation consists of the usual Coulomb term and
is not relativistically invariant; the leading correction to the two-electron interaction
(the Breit interaction [19]) can be included either variationally or by perturbation
theory [9, 20, 21], but has been left out in this work.
66
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
5.3.2.
Open shell approach.
The molecules or clusters, modelling bulk or surface properties of solids, we want to
study contain heavy atoms and are in general open shell systems. Especially
lanthanides and actinides have an open f-shell, which gives rise to a large number of
energy eigenstates lying close together. A simple open shell SCF method alone is not
sufficient to describe such a manifold. Firstly, significant interaction between the
Russell-Saunders terms of the f-multiplet should be expected. Secondly, since (as is
generally known) the correct eigenfunctions of the Dirac-Coulomb hamiltonian are
neither pure LS-coupled functions nor pure JJ-coupled functions, a many-electron
function in intermediate coupling should be constructed. This can be accomplished by
forming linear combinations of determinants constructed from LS- or JJ-coupled oneelectron spinors. Furthermore, since we are interested in the optical spectrum of the fmanifold, we need to calculate a large number of the energy eigenstates. A separate
SCF calculation for each of these states is cumbersome, and is certainly not a practical
approach.
By using the COSCI (Complete Open Shell Configuration Interaction) approach these
problems can be solved. We start with a SCF calculation using an average energy
expression (defined as the average of the energy expressions for all possible
individual states of the relevant configuration(s) ), resulting in the 'average' total
energy and a set of 'average' orbitals. Next, a CI calculation is performed within the
full-CI space generated from the set of open shell orbitals from the SCF calculation.
This results in a description of the open shell manifold which is not tied to one of the
extreme coupling schemes.
5.3.3.
The relativistic open shell SCF equations.
The relativistic open shell SCF equations are formally derived in the same way as the
non-relativistic equations [22]. The one-electron Schrödinger operators are replaced
by Dirac operators and the scalar orbitals are now four component spinors.
Fundamentally there is of course an important difference in the meaning of the
variational method since it is used to locate a stationary state in the positive energy
domain instead of an absolute minimum. The resulting SCF-equations are called the
Hartree-Fock-Dirac equations.
67
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
The average of configuration energy expression which is used in the MOLFDIR
program package has the following form [14]:
C
C
E = • hk + 1 • Qkl + f
2 kl
k
O
O
C,O
• hm + 12 af• Qmn + • Qkm
m
mn
k,m
(7)
In this equation, k and l label closed shell spinors (which, by definition, are occupied
by one electron); m and n label open shell spinors (with related to them a fractional
occupation number f and a coupling constant a), hk is the diagonal matrix element of
the one-electron Dirac operator (from equation (1)) over spinor k, and Qij  Jij - Kij
in which Jij and Kij are the usual coulomb and exchange integrals. Apart from the use
of four-component spinors instead of orbitals, the energy expression differs from the
energy expression used by Roothaan in the detailed form of the one-electron operators
(Dirac-operators versus one-electron Schrödinger operators) and of the two-electron
operators when the Breit interaction is included.
If we use the average of configuration total energy expression, the coupling constant
and fractional occupation number are given as functions of the number of open shell
electrons n and the number of open shell orbitals d:
f=n
d
a=
(8)
d(n-1)
n(d-1)
(9)
For completeness, we present the working formulas which are obtained when the SCF
equations are solved using a basis set (consisting of either two or four component
spinors) expansion technique. In these equations p, q, r ... label basis spinors p, q,
r, ... and Spq is the overlap between basis spinors p and q, and (pq||rs) is an antisymmetrized two-electron repulsion integral in charge-cloud notation.
 = 1-a
1-f
(10)
|i = •  p cpi
(11)
p
68
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
C
O
*
DC
pq = • cqk cpk
*
DO
pq = f• cqmcpm
(12)
C
QC
pq = • (pq||rs)Dsr
O
QO
pq = • (pq||rs)Dsr
(13)
C O
O C
LC
pq = • SprDrs Qsq + QprDrs Ssq
O O
O O
LO
pq = • SprDrs Qsq + QprDrs Ssq
(14)
FC = h + QC + QO + LO
FO = h + QC + aQO + LC
(15)
m
k
rs
rs
rs
rs
For the closed shell orbitals we get the equation
FC|k = k|k
(16)
and for the open shell orbitals we get the equation
FO|m = m|m
(17)
Equations (16) and (17) need to be solved iteratively. The coupling operators LC and
LO take care of the orthogonality between the closed shell and open shell solutions.
We get the following expression for the total energy:
E = Tr HDDC + 1 Tr QCDC +
2
D
O
Tr H D + 1 aTr QODO +
2
Tr QODC
(18)
5.3.4.
The COSCI method.
In this section, we mention some features of the COSCI method. The many-electron
function space used in the COSCI calculations is the full-CI space generated from a
(small) set of active spin-orbitals which is usually chosen to be the set of open shell
spin-orbitals from the SCF calculation. Since the average energy is proportional to the
trace of the hamiltonian matrix in this many-electron basis, it is invariant for unitary
69
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
transformations of the one-electron basis. In particular, it does not make any
difference if we had started with an LS coupled instead of a JJ coupled basis set.
Since only the open shell space is used to construct the CI matrix, correlation
corrections are hardly accounted for at this stage. Also, the open shell orbitals are not
optimized for individual states. The method is expected, however, to yield a balanced,
global picture of the open shell manifold of states. More accurate descriptions of
specific states can be found subsequently by the application of MRCI or MCSCF
techniques.
We define the set of frozen orbitals to consist of all one-electron spinors which are
present in all determinants spanning the CI-space. The interaction with these frozen
orbitals is taken into account by using an effective one-electron operator h'
F
h' = h + • J f-K f
(19)
f
where Jf is the Coulomb operator with spin-orbital f, Kf is the exchange operator
with spin-orbital f, and the summation runs over the set of frozen spin-orbitals F).
The matrix elements of this operator can be calculated easily using the fact that the
one-electron spinors are solutions of the SCF equations (16) and (17).
Finally, it is clear that while the COSCI method has been introduced here to perform
open shell calculations with a relativistic hamiltonian, it can also be used for
calculations with a non-relativistic hamiltonian.
5.3.5.
Basis functions.
The MOLFDIR package works with two distinct sets of, usually atom centred, scalar
(contracted) cartesian gaussian functions: a large component set gLi and a small
component set gSi . The two-electron repulsion integrals are calculated over the
functions belonging to these sets. From these two scalar sets, two new sets of
symmetry adapted molecular basis spinors Ll and  Ss are constructed using the
Dirac double group symmetry elements. The spinors Ll and  Ss are defined by
70
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
 L
l
 Ll =
L
l
0
0
and
 Ss =
0
(20)
 S
s
 S
s
0
L
S
L L
S
S S
L L
S S
L
l = • gi cil , l = • gi cil , s = • gi cis and s = • gi cis
i
i
i
(21)
i
where the coefficients cX
ij (X{L,S}; {,) in equation (21) are determined by
symmetry. Thus, each non-zero component of these basis spinors consists of a linear
combination of the basis functions from set gLi or set gSi .
5.3.6.
Kinetic balance and general contraction.
As has been discussed by several authors [23-25], it is important to ensure that the
basis set in which the one-electron functions are expanded fulfils the kinetic balance
condition. This condition requires that for each function Ll in the large component
basis, the function  p Ll is contained in the small component basis set. Since
kinetic balance is neither a necessary nor sufficient condition to produce the best
approximate eigensolutions, we use an extended kinetic balance scheme [15]. In
practice this means that the small component basis set contains more basis functions
than those minimally required by the kinetic balance condition.
The first set of extra functions results from our choice to employ two separate
primitive gaussians instead of the fixed linear combination that results from  p
operating on a single gaussian primitive in the large component.
A second set of extra functions is used in molecular calculations where atomic
solutions are used to contract the large and small component basis functions. In
addition to the kinetically balanced functions the atomic small component solutions
are used as small component basis functions. A basis set which includes such an
extension is called an atomically balanced basis set [15].
When general contraction [26] is used to reduce the variational space we still require
that the small component basis fulfils atomic and kinetic balance conditions. The
contraction reduction is thereby counteracted by the fact that up to four contracted
71
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
functions for each large component function are necessary: up to two functions to
ensure kinetic balance, and up to two functions to ensure atomic balance. Although
these small component functions may tend to form a linear dependent set, in which
case one or more functions can be removed, the gain obtained by the use of general
contraction in the small component basis set is small compared to that obtained in the
large component basis set.
5.4.
The Ba2GdNbO6:Eu system.
5.4.1.
Description of the system.
In 1966, Blasse et al. [27, 28] have obtained experimental data (luminescence spectra)
on some compounds which contained an Eu3+ impurity. Due to the approximate
validity of spin and parity selection rules, electric dipole transitions between the 5D
and 7F levels of the Eu3+ impurity are forbidden in first approximation. The spin
selection rule is lifted by the spin-orbit coupling. The magnetic dipole and quadrupole
transitions are only forbidden by a spin selection rule, which is again lifted by the
spin-orbit coupling. To investigate the effect of the symmetry of the surroundings, a
series of compounds was studied: a compound in which the Eu3+ ion occupies a strict
centre of (Oh) symmetry, a compound in which small deviations from this symmetry
occur, and a compound with no symmetry at all. In the last case, both electric and
magnetic dipole transitions can occur, while in the first case only magnetic dipole
transitions are allowed [29, 30].
We have chosen to study a compound with a strict centre of symmetry,
Ba2GdNbO6:Eu. The advantage is that calculations can be performed using Oh
symmetry, which greatly reduces the computational efforts needed (both in terms of
CPU time and of disk space).
72
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Figure 1:
The Ba2GdNbO6 unit cell. In order to get a better view of the structure of the cell, the ions
have not been drawn to scale. Small black spheres: Ba2+, small grey spheres: O2-,
remaining large spheres: alternating Gd3+ and Nb5+. The Eu3+ impurity occupies a Gd3+
site.
The Ba2GdNbO6 crystal (figure 1) has an ordered perovskite structure [31]. The unit
cell of the crystal is a cube divided in 8 smaller cubes. In the centre of the smaller
cubes we find alternating Gd3+ and Nb5+ ions; on the corners of the smaller cubes we
find Ba2+ ions, and at the middle of the sides of the small cubes the O2- ions are
positioned. The Eu3+ impurity occupies one of the Gd3+ sites with symmetry Oh. It is
surrounded by six O2- ions, forming an 'EuO69- cluster', with an Eu-O distance of
4.02 a.u..
The experimental results obtained by Blasse et al. (the energy distribution of the
emission of Eu3+-activated Ga2GdNbO6) show a sharp line at 595 nm (16800 cm-1)
corresponding to the 5D0 7F1 transition. This is an allowed magnetic dipole
transition. Also, at least 6 weak and broad lines are visible in the region 610-630 nm
(15900-16400 cm-1). These lines were assigned to the 5D0 7F2 transitions (the
large number of lines is due to simultaneous vibronic transitions).
5.4.2.
Physical model.
We have performed all-electron average of configuration open shell Hartree-Fock and
Hartree-Fock-Dirac calculations, followed by COSCI calculations (using respectively
73
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
the Schrödinger hamiltonian (NR) and the Dirac-Coulomb hamiltonian (FD)), on part
of the Ba2GdNbO6 crystal together with the impurity. This part of the material, cut
from the complete system, will be referred to as 'the cluster', even when 'the cluster'
consists of one atom only (the impurity atom). The influence of the surrounding
infinite crystal is accounted for by considering the electrostatic effects on the cluster
only. The Madelung potential of the crystal (with the contributions from the atoms
contained in the cluster removed) has been included in the cluster calculations.
Technically, this potential was represented by a set of fitted point charges [32].
We define three different models of the physical system of interest which successively
improve on the quality of the description:
• Eu: a bare Eu3+ ion. This is the crudest model available since none of the effects
of the surroundings are included.
• EuMP: an Eu3+ ion embedded in the Madelung potential. This model is expected to
give more realistic results since the electrostatic effects of the surroundings are
included in the calculation.
• EuO6MP: an EuO69- cluster embedded in the Madelung potential of the rest of the
crystal.
In principle still larger clusters can be defined, and eventually the results from the
calculations are expected to converge to the results from a hypothetical calculation on
a physical crystal containing the impurity. Relativistic calculations using the HartreeFock-Dirac and subsequent COSCI method on systems much larger than the EuO69cluster are not yet technically feasible. To conclude with, we define one last model
system:
• EuO6: a bare EuO69- cluster. This model system is included for comparison.
5.5.
Computational details.
5.5.1.
Basis sets.
The primitive basis set for the Eu3+ ion has been optimized for the 7F state (the
ground state), starting with the Eu3+ basis set from van Piggelen [33]. For technical
reasons (to improve the efficiency), this basis set was reoptimized under the constraint
that the d-exponents be a subset of the s-exponents, and that the f-exponents be a
subset of the p-exponents. This resulted in a very reasonable basis set (table 1) for the
Eu3+ ion: 18s, 15p, 10d and 8f.
74
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Primitive basis sets used for Eu3+ and for O2-.
Table 1:
Eu3+
basis
S
S
S
S
S
S
S D
S D
S D
S D
S D
S D
S D
S D
S D
S D
S
S
P
P
P
P
P
P
P F
P F
P F
P F
P F
P F
P F
P F
P
O2-
4568512.7638569176
684565.0096138082
155804.2488611462
44117.8494269652
14377.8544660754
5174.3829553280
2003.2436229789
820.6948565584
350.2174426569
149.8921545649
68.4361726183
32.0678038909
15.3178276488
7.1275948086
3.2098115099
1.3224328971
.6754795683
.2924156677
49422.1732518000
11709.3555245315
3801.2033528740
1450.4881257395
611.7598011859
275.0558006964
128.7054241105
61.6082879502
29.2377289033
14.3362471454
6.8946387106
3.2020701090
1.4335005095
.5763759499
.2423067468
basis (R. Broer, unpublished):
S 14362.1302
S 2154.8458
S
490.3337
S
138.5686
S
44.8513
S
15.8495
S
5.8637
S
1.4900
S
0.5612
S
0.1813
P
47.5925
P
10.8657
P
3.2595
P
1.0756
P
0.3412
P
0.0927
The total energy of the 7F ground state using this basis set is -10422.00372 a.u. (van
Piggelen: -10422.01687 a.u.; numerical HF: -10422.03788 a.u.). The radial
expectation value for r is given in table 2. From the table it can be seen that the basis
set calculations agree in reasonable detail with the numerical calculations. The basis
set optimized with the constraint on the exponents yields necessarily results that are
slightly less in quality than the results of the basis set optimized without constraints.
Also by a study of the f-spectrum later on it is concluded that the basis set is good
enough for our purposes.
Table 2:
Expectation value of r (in a.u.) for several Eu3+ orbitals.
<r>
NMCHF
ASCF v. Piggelen
ASCF this work
1s
2s
0.02408
0.10253
0.02408
0.10253
0.02408
0.10253
75
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
3s
4s
5s
2p
3p
4p
5p
3d
4d
0.26442
0.59786
1.45285
0.08712
0.25455
0.61337
1.60675
0.22654
0.65312
0.26442
0.59785
1.45270
0.08712
0.25455
0.61338
1.60631
0.22654
0.65308
0.26443
0.59791
1.45299
0.08712
0.25455
0.61341
1.60731
0.22654
0.65295
4f
0.80644
0.80625
0.80483
NMCHF: Numerical Hartree-Fock results;
ASCF v. Piggelen: basis set results, optimized basis set;
ASCF this work: basis set results, basis set optimized under constraints.
Based on average of configuration atomic open shell Hartree-Fock and Hartree-FockDirac calculations on the Eu3+ ion, general contracted basis sets were constructed for
the molecule. The small component basis set was constructed using atomic and
kinetic balance. This procedure resulted in a (6s, 5p, 3d, 2f) non relativistic basis set,
and in a (6s, 9p, 5d, 3f | 10s, 13p, 14d, 8f, 5g) relativistic scalar basis set.
For the oxygen ion we have used a (10s, 6p) primitive basis set (table 1) which has
been optimized (non-relativistically) for the O- ion [34]. Based on Hartree-Fock and
Hartree-Fock-Dirac calculations on the O2- ion general contracted basis sets were
constructed. Again, the small component basis set was constructed using atomic and
kinetic balance. This procedure resulted in a (3s, 2p) non relativistic basis set, and in a
(3s, 3p | 4s, 4p, 5d) relativistic scalar basis set.
5.5.2.
Madelung potential.
The crystal is taken to be purely ionic: Ba22+Gd3+Nb5+O62-. Using these formal
charges, the Madelung potential in the cluster region has been calculated. After
subtracting the cluster contributions, the remaining potential was fitted with a set of
point charges. The fit results for the two different clusters (the Eu3+ ion, and the
EuO69- system) are given in table 3.
76
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Table 3:
Fit to the Madelung potential for several clusters. Position: position of the
fit-charges in units of 4.02 a.u.; # points: number of symmetry equivalent
points for that site. All numbers in a.u..
VEu a
(∆V)max b
(∆V)av c
Madelung
fit for EuO6
fit for Eu
-1.3221273
1.662956 e
3.5 * 10-4
2.1 * 10-7
6.7 * 10-5
-1.3220842
5.0 * 10-5
-2.4 * 10-5
2.5 * 10-5
# points
fit charge for
EuO6 cluster
fit charge for
(V) d
Site
Position
O2Ba2+
Nb5+
(0, 0, 1)
(1, 1, 1)
(0, 0, 2)
6
8
6
1.96266
4.97018
O2Gd3+
Ba2+
(2, 0, 1)
(2, 2, 0)
(3, 1, 1)
24
12
24
-1.91579
1.59002
-0.48009
Eu cluster
-1.99865
1.99609
4.64023
-1.53386
a Potential at Eu3+ site;
b Largest difference between the fitted potential and the exact potential;
c Average difference between the fitted potential and the exact potential;
d Standard deviation in fitted potential;
e -1.3221185 when the contribution of six O2- ions at 4.02 a.u. is included.
For the interpretation of the results it is useful to note that the Madelung potential
used with the Eu3+ ion can be fitted with only six charges on the O2- sites (at a
distance of 4.02 a.u.). The fit charges are -0.865, with a standard deviation in the
potential of 0.04 a.u. [13]. In the embedded ion calculations, however, we have used
the much more accurate fit given in table 3.
77
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
5.5.3.
The EuO69- clusters.
In the Hartree-Fock-Dirac calculations on the EuO69- clusters, using the experimental
geometry, the symmetry unique two-electron integrals which were larger than 10-12
for the (LL|LL) and (LL|SS) integrals, and than 10-8 for the (SS|SS) integrals, were
calculated. This resulted in 2*106 (LL|LL) integrals, 48*106 (LL|SS) integrals and
241*106 (SS|SS) integrals. If no symmetry would have been employed, we would
have obtained 0.1*109 (LL|LL) integrals, 2.2*109 (LL|SS) integrals and 12.7*109
(SS|SS) integrals.
We found that the convergence of the SCF iterations was very slow. Normally, the
computational efforts needed for a single SCF iteration can be reduced by temporarily
using only part of the large-component two-electron integrals, or by temporarily
fixing the contributions of the small component two-electron integrals to the Fock
matrices; in this case we found that both of these methods did seriously degrade the
convergence behaviour.
We also found that selection of the open shell eigenvectors by overlap with the
eigenvectors of the previous iteration did not work very well for the EuO69- systems.
However, selection by character (thus forcing the open shell orbitals to be constructed
from the Eu f-basis spinors) worked much better.
5.6.
Results.
5.6.1.
The free Eu3+ ion.
In order to get an estimate of the accuracy which can be obtained using the given
general contracted basis set together with the COSCI approach, we have performed
some calculations on the Eu3+ ion in the non-relativistic limit:
1.
Numerical calculations using the NMCHF program of Froese-Fischer [35];
these calculations are used as reference.
2.
Atomic basis set expansion calculations with the ASCF [13] program, using
the uncontracted basis set described before.
3.
MOLFDIR-COSCI calculations using the general contracted basis set.
78
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Table 4:
Stat
e
Comparison of numerical (NMCHF), basis set (ASCF) and COSCI total energies relative
to the average total energy (in a.u.).
NMCHF (I)
ASCF (II)
COSCI (III)
7F
-0.35873
-0.35883
-0.35593
-0.00010
0.00290
5L
-0.22542
-0.22583
-0.22473
-0.00042
0.00110
5K
-0.15029
-0.15066
-0.15014
-0.00037
0.00052
3O
-0.13884
-0.13910
-0.13872
-0.00026
0.00038
3N
-0.11107
-0.11147
-0.11123
-0.00041
0.00024
5P
-0.11043
-0.11057
-0.11028
-0.00015
0.00029
5S
-0.04781
-0.04750
-0.04746
0.00031
0.00005
Eav (NMCHF) =
Eav (ASCF) =
Eav (COSCI) =
II - I
III - II
-10421.67915
-10421.64489
-10421.64489
The results are given in table 4. All absolute energies are too high by about 0.034 a.u.,
due to the finite basis set approximation. The relative errors show the accuracy within
the f-manifold, which is in general much better than 10-3 a.u. For the 7F- and 5Lstates, the error due to the use of a set of average orbitals (column III - column II in
table 4) is significantly larger than for the other states. This error, especially in case of
the 5L-state, is partially cancelled by the basis set error (column II - column I in table
4).
We conclude that in the worst case the accuracy of the calculations, as far as splittings
between Russell-Saunders terms are concerned, can be as bad as 3*10-3 a.u. (about
600 cm-1), and that this is mainly due to the average f-shell approach. The accuracy in
the many-electron total energies will therefore not significantly be improved by using
a more flexible basis set. The splittings within Russell-Saunders terms due to
relativistic and ligand field effects are much smaller, and the absolute accuracy of
these splittings is expected to be much better.
5.6.2.
The Eu3+ impurity.
From the average of configuration open shell SCF calculations we have obtained the
f-or f-like orbital energies, which are given in tables 5 and 6. In figure 2 the splitting
of these orbital energies due to relativistic and crystal field effects is pictorially
represented. In the figure the FD orbital energies have been shifted upward by 0.18
79
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
a.u.. The additional shift (tables 5 and 6) in the embedded ion results corresponds
closely to the value of the Madelung potential at the Eu3+ site. To explain the shift
seen in the embedded cluster results also the interaction with the rest of the cluster
needs to be taken into account. We find that this interaction can be fitted (using data
from tables 3, 5 and 6) with six effective charges of -1.5 at the O2- ion positions, or
with six charges of -2.0 at a distance of 5.3 a.u..
Figure 2:
calculated splittings (in a.u.) of the orbital energies of the f and f-like orbitals due to
relativistic and crystal field effects, relative to the average energies.
0.015
f
0.010
e
1u
f
u
7/2
e
2u
0.005
0.000
t
1u
f
t
2u
a
2u
-0.005
-0.010
f
f
u
5/2
e
2u
-0.015
NR
EuO6
Table 5:
Level
a2u
t2u
t1u
average
NR
EuO6MP
NR
EuMP
NR
Eu
FD
Eu
FD
EuMP
FD
EuO6MP
FD
EuO6
Relative and average NR f-electron orbital energies (a.u.).
Eu NR
EuMP NR
EuO6MP NR
EuO6 NR
0.00000
0.00000
0.00000
0.00000
0.00000
0.00210
0.00258
0.00254
0.00000
0.00420
0.00500
0.00733
-1.77887
-0.45802
-1.14273
0.74840
80
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Figure 4:
Density of states from the COSCI results (7F level shown).
Table 6:
Relative and average FD f-electron orbital energies (a.u.).
Level
e2u
fu
e2u
fu
e1u
average
Eu FD
EuMP FD
EuO6MP FD
EuO6 FD
0.00000
0.00000
0.00000
0.00000
0.00000
0.00262
0.00213
0.00346
0.02410
0.02380
0.02380
0.02369
0.02410
0.02616
0.02563
0.02654
0.02410
0.02771
0.02695
0.02896
-1.59526
-0.27429
-0.97088
0.92339
The COSCI results concerning the 7F and the lower of the 5D levels are presented in
tables 7 and 8. In figure 3 the EuO6MP FD COSCI results for the complete f-manifold
are shown. In figure 4 the FD data on the 7F levels is presented in the form of density
of states plots. These plots have been produced by using a convolution of a Gaussian
with the discrete COSCI eigenvalues. The width of the Gaussian was chosen to match
the accuracy of the calculations.
Figure 3:
Density of states from the EuO6MP FD COSCI results.
10
EuO6MP FD
DOS (arb. units)
8
6
4
2
0
0
0.275
0.55
Eto t (a.u.)
81
0.825
1.1
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
10
Eu FD
8
6
DOS (a rb. unit s)
4
2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.015
0.02
0.025
0.03
0.015
0.02
0.025
0.03
0.02
0.025
0.03
Et ot (a. u.)
5
EuMP FD
4
3
DOS (a rb. unit s)
2
1
0
0
0.005
0.01
Et ot (a. u.)
7
EuO6 MP FD
6
5
4
DOS (a rb. unit s)
3
2
1
0
0
0.005
0.01
E tot (a.u. )
6
EuO6 FD
5
4
DOS (a rb. unit s)
3
2
1
0
0
0.005
0.01
0.015
Et ot (a. u.)
82
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
Table 7:
7
5
F
D
Table 8:
NR COSCI energies in a.u. of the lowest states of the f6-manifold with their degeneracies.
Eu NR
.00000
.11478
49
EuMP NR
.00000
21
EuO6MP NR
.00000 21
EuO6 NR
.00000 21
25
.00228
.00452
.11607
21
7
15
.00138
.00317
.11407
21
7
15
.00343
.00510
.11472
21
7
15
.11670
10
.11435
10
.11545
10
FD COSCI energies in a.u. of the lowest states of the f6-manifold with their degeneracies.
Eu FD
.00000
1
EuMP FD
.00000
1
EuO6MP FD
.00000
1
EuO6 FD
.00000
1
.00171
3
.00166
3
.00170
3
.00166
3
.00482
5
.00399
2
.00449
2
.00408
2
F3
.00894
7
.00513
.00851
3
1
.00499
.00871
3
1
.00507
.00854
3
1
F4
.01377
9
.00917
.00935
.01294
3
3
3
.00894
.00902
.01336
3
3
3
.00894
.00921
.01285
3
3
3
F5
.01910
11
.01420
.01466
.01508
.01866
2
3
1
3
.01377
.01398
.01424
.01876
2
3
1
3
.01419
.01430
.01441
.01857
2
3
1
3
F6
.02474
13
.01881
.01921
.02009
.02412
2
3
3
2
.01881
.01916
.01930
.02430
2
3
3
2
.01874
.01880
.01991
.02398
2
3
3
2
D0
5D
1
5D
2
.09370
1
.02425
.02454
.02606
.02622
.02637
.09383
3
1
3
3
1
1
.02433
.02444
.02494
.02503
.02511
.09224
3
1
3
3
1
1
.02411
.02441
.02542
.02559
.02573
.09218
3
1
3
3
1
1
.10121
3
.10132
3
.09972
3
.09961
3
.11275
5
.11263
3
.11115
3
.11084
3
.11316
2
.11125
2
.11128
2
F0
7F
1
7F
2
7
7
7
7
7
5
Going from the free ion to the embedded ion, certain total energy levels (depending
on the symmetry) and the 4f orbital energies are split due to the Madelung field
representing the crystal. When we include the six O2- ions in the cluster together with
the Madelung field of the rest of the crystal, we find that the total energy splittings are
reduced significantly with respect to the embedded ion results (fig. 3, tables 7 and 8).
This behaviour, which is opposite to the behaviour of the transition metal ions [36],
can be explained by the presence of the filled Eu3+ 5s and 5p shells. The
superimposed charge distributions of these shells and the oxygen ions becomes
83
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
slightly deformed. A small amount of charge is shifted from the Eu-O bond region
into the regions between these bonds. This in effect reduces the electrostatic splitting
of the 4f orbitals. Also, due to the localized nature of the 4f orbitals, covalent effects
are not as important as in the case of transition metal ions, where they lead to an
increase of the splitting.
In the non-relativistic case, crystal field theory based on free ion orbitals yields the
same splittings respectively for the many-electron 7F level and the 4f orbital energy.
Our embedded ion results reflect this behaviour rather closely, in contrast to the
embedded cluster results that show substantial deviations. This is due to changes in
the detailed form of the 4f-like spin-orbitals. From this sensitivity it can be concluded
that it is important to use 'relativistic' orbitals, which deviate significantly from nonrelativistic orbitals, from the outset.
The much larger splittings calculated for the bare EuO69- cluster should be compared
with the splittings which would result from a calculation on the ion with O2- ions
represented by six 2- point charges, not including the rest of the crystal. These
splittings can be estimated from our embedded ion results by assuming that the
splittings are linear in the charge of the point charges. The complete Madelung
potential can approximately be fitted by six charges of -0.865 at the O2- positions
[13], so we should compare the splittings calculated using the bare cluster with the
splittings calculated using the embedded ion multiplied by 2.3 (= 2 / 0.865). Now we
find a similar trend as before: inclusion of the six O2- ions reduces the calculated
splittings.
The differences going from embedded ion to embedded cluster are relatively large (of
the same order of magnitude as the crystal field splittings), and one might ask whether
the cluster should be enlarged. The distance from the Eu3+ ion to the next-nearest
neighbours (the Ba2+ ions) is rather large (about 7 a.u.), and inclusion of these ions in
the cluster is therefore not expected to alter the results significantly. It is expected,
however, that the use of point charges deforms the oxygen charge distributions
somewhat.
5.6.3.
The Spectrum.
The dominant luminescence transitions occurring in this system are basically the
atomic 5D0  7F1 and 5D0  7F2 transitions. The first of these is a magnetic dipole
84
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
transition which can be seen clearly in the spectrum. The second transition is an
induced electric dipole transition, allowed by vibronic coupling to phonon modes that
break the inversion symmetry. This transition is split by the reduction of the
symmetry. In table 9 the calculated transition wave numbers are given.
Table 9:
Energies of the fluorescense tranistions (in cm-1)
Transition
Eu FD
EuO6MP
EuMP FD
v. Piggelen
Exp. a
FD
5D
0
5D
0
5D
0
 7F1

20189
20229
19871
20459
16800
(Eg) 19507
 7F2 19507
19718
19467
19259
19149
19738
19599
16150 b
16150 b
7F
2
(T2g)
a Blasse et al [28, 29].
b unweighted average of the six lines observed in this region. The separation between the lines is about
100 cm-1.
Based on the EuO6MP FD results, we expect that the splitting of the 5D0  7F2
transition is of the order of 100 cm-1. The free atom model shows no splitting. When
we introduce the Madelung potential the splitting is somewhat overestimated (251
cm-1). Inclusion of the neighbouring O2- ions in the model (EuO6MP) reduces the
splitting to 110 cm-1. This number is to be compared with the number found by Van
Piggelen (139 cm-1) [13]. Although the calculated splitting is quite consistent with the
experimental data, a more precise comparison cannot be made at present. The
observed vibronic transitions have not been assigned and in fact more and better
resolved data are required to do this [37]. The EuO6MP model, however, should yield
a reasonable description of the effects of the surroundings. We tend therefore to
conclude that a crystal field model based on just the Madelung field overestimates the
splitting.
The spin-orbit splitting (using the weighted average of the Eg and the T2g levels) is
calculated to be 682 cm-1 , 662 cm-1 and 678 cm-1 for respectively Eu, EuMP and
EuO6MP. This is in good agreement with the experimental data from which we
estimate the splitting to be between 600 and 700 cm-1. Van Piggelen's results yield a
somewhat larger value (804 cm-1). Except for the neglect of the Breit interaction we
have not introduced any approximations in the description of the relativistic effects,
so we expect that our results are reliable at this point.
85
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
The calculated splitting between the 5D level and the 7F level is much too large, in all
calculations. For the 5D0  7F1 transition, we have a discrepancy with experiment of
about 3000 cm-1. This large error cannot be attributed to defects of the COSCI or
basis set approach (in section 5.1 we have seen that the worst of these errors is of
order 600 cm-1). This difference is of course to be expected at the present level of
theory since most of the correlation effects have not yet been included.
5.7. Conclusions.
In this paper we have given results of calculations on EuO69- clusters in a Madelung
potential, as a model for an Eu3+ impurity in the Ba2GdNbO6 crystal. The
calculations indicate the state of the art in ab initio relativistic quantum chemistry i. e.
the Hartree-Fock-Dirac and COSCI formalism applied to molecules. The COSCI
method has proven to be extremely useful in obtaining all the individual states of the
open shell manifold.
Relativistic effects, except for the Breit interaction, the effects of the surroundings on
the Eu3+ ion, and part of the f-shell configuration interaction are treated in a
systematic manner. The open shell configuration interaction method employed yields
an ab initio intermediate coupling description of the f6-like manifold.
Our results show that the splittings due to relativistic effects, which are much larger
than the splittings due to the surroundings, can be calculated with an accuracy of
about 10 cm-1. We have also found that both a pure Madelung field and a bare EuO69cluster significantly over-estimate the splittings introduced by the surroundings. It is
clear from our non-relativistic results that the detailed form of the orbitals is
important in the calculation of these splittings. Hence the preferred approach should
be to use relativistic orbitals from the outset.
A significant discrepancy with experiment is found where splittings between RussellSaunders terms are concerned (as in the luminescence spectra). This discrepancy,
which is also present on the atomic level, is due to the neglect of most of the
correlation effects and does not differ much from the discrepancies found in nonrelativistic calculations.
86
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
5.8. References.
1. J. Becquerel, Z. Physik 58, 205 (1929).
2. H. Bethe, Ann. Physik 3, 135 (1929).
3. B. T. Thole, G. van der Laan, J. C. Fuggle, G. A. Sawatzky, R. C. Karnatak and
J. -M. Esteva, Phys. Rev. B 32, 5107 (1985).
4. B. T. Thole, G. van der Laan and G. A. Sawatzky, Phys. Rev. Lett. 55, 2086
(1985).
5. K. Balasubramanian and K. S. Pitzer, in: Ab initio methods in Quantum
Chemistry - I, K. P. Lawley (ed.), Wiley (1987), p. 287.
6.
7.
8.
P. Pyykkö, Chem. Rev. 88, 563 (1988).
P. Pyykkö, Adv. Quant. Chem. 11, 353 (1978).
P. A. Christiansen, W. C. Ermler and K. S. Pitzer, Ann. Rev. Phys. Chem. 36,
407 (1985).
9. P. J. C. Aerts, Towards relativistic quantum chemistry. Thesis, Groningen
(1986).
10. P. J. C. Aerts and W. C. Nieuwpoort, Int. J. Quantum Chem. Symp. 19, 267
(1986).
11. O. Visser, L. Visscher, P. J. C. Aerts and W. C. Nieuwpoort, Theoret. Chim.
Acta. (accepted for publication).
12. K. G. Dyall, P. R. Taylor, K. Faegri, Jr. and H. Partridge, J. Chem. Phys. 95,
2583 (1991)
13. H. U. van Piggelen, Ab initio calculations on the electronic states of 4fn ions
with applications to [EuO6]9-. Thesis, Groningen (1978).
14. O. Visser, P. J. C. Aerts and L. Visscher, in: The effects of relativity in atoms,
molecules and the solid state, ed. I. P. Grant, B. Gyorffy and S. Wilson, New
York (1990).
15. L. Visscher, P. J. C. Aerts and O. Visser, in: The effects of relativity in atoms,
molecules and the solid state, ed. I. P. Grant, B. Gyorffy and S. Wilson, New
16.
17.
18.
19.
20.
21.
22.
York (1990).
P. A. M. Dirac, Proc. Roy. Soc. London A 117, 610 (1928).
P. A. M. Dirac, Proc. Roy. Soc. London A 118, 351 (1928).
I. P. Grant and H. M. Quiney, Adv. At. Mol. Phys. 23, 37 (1988).
G. Breit, Phys. Rev. 34, 553 (1929).
H. M. Quiney, I. P. Grant and S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 23,
L271 (1990).
S. Okada, M. Shinada and O. Matsuoka, J. Chem. Phys. 93, 5013 (1990).
C. C. J. Roothaan, Rev. Mod. Phys. 32, 179 (1960).
87
5. Molecular Open Shell CI calculations using the Dirac-Coulomb hamiltonian:
the f6-manifold of an embedded EuO69- cluster
23.
24.
25.
26.
27.
R. E. Stanton and S. J. Havriliak, Chem. Phys. 81, 1910 (1984).
P. J. C. Aerts and W. C. Nieuwpoort, Chem. Phys. Lett. 125, 83 (1986).
W. Kutzelnigg, Int. J. Quantum Chem. 25, 107 (1984).
R. C. Raffenetti, J. Chem. Phys. 58, 4452 (1973).
G. Blasse, A. Bril and W. C. Nieuwpoort, in: Proceedings of the International
Conference on Luminescence, 1646 (1966).
28. G. Blasse and A. Bril, Philips Technisch Tijdscrift 31, 314 (1970).
29. B. R. Judd, Phys. Rev. 127, 750 (1962).
30. C. S. Ofelt, J. Chem. Phys. 37, 511 (1962).
31. F. Galasso and W. Darby, J. Phys. Chem. 66, 131 (1962).
32. J. Almlöf and U. Wahlgren, Theoret. Chim. Acta 28, 161 (1973).
33. H. U. van Piggelen, W. C. Nieuwpoort and G. A. van der Velde, J. Chem. Phys.
72, 3727 (1980).
34. R. Broer (unpublished), see table 1.
35. C. Froese-Fischer, The Hartree-Fock method for atoms. New York (1977).
36. A. J. H. Wachters and W. C. Nieuwpoort, Int. J. Quantum Chem. 5, 391 (1971).
37. J. P. Morley, T. R. Faulkner and F. S. Richardson, J. Chem. Phys. 77, 1710
(1982).
88
6. Summary and conclusions
6.
Summary and conclusions.
6.1. Summary.
The non-relativistic valence-only method, using a frozen relativistic core combined
with first order perturbation theory for the relativistic effects on the valence electrons,
has been applied to Sn and SnH4. The results show that the relativistic corrections to
the valence properties are mainly direct relativistic valence effects, and that there is a
large difference between the atomic and molecular relativistic effects. In view of these
results, relativistic effects should not be calculated by the inclusion of a relativistic
pseudopotential without further study if accurate results are required.
The feasibility of all-electron open-shell molecular Hartree-Fock-Dirac calculations
on chemically interesting systems has been demonstrated by the calculations reported
in chapter 4 and 5. The general contraction method, using kinetic and atomic balance
for the small components of molecular basis sets, reduces the computational effort
while retaining sufficient accuracy.
The average of configuration open shell method followed by a complete open shell
configuration interaction calculation (the COSCI method) has proven to be useful to
obtain total energies for the states of interest. The calculations on EuO69- clusters
show that the method is extremely useful in obtaining all the individual states of the
open shell manifold. In that case, it yields an ab initio
description of the f6-like manifold.
intermediate coupling
The results of calculations on an Eu3+ impurity in the Ba2GdNbO6 crystal (modelled
by an EuO69- cluster in a Madelung potential) show that the splittings due to
relativistic effects, which are much larger than the splittings due to the surroundings,
can be calculated within about 10 cm-1. We have also found that both a pure
Madelung field and a bare EuO69- cluster significantly over-estimate the splittings
introduced by the surroundings. The detailed form of the orbitals is important in the
89
6. Summary and conclusions
calculation of these splittings, so the preferred approach should be to use relativistic
orbitals from the outset.
In our calculations on the hydrides, we have verified that the Breit interaction leads to
small corrections to the relativistic results both for light and heavy atoms (a small
bond length expansion for PbH4). For the lighter systems the relativistic bond length
contractions are very small. However, in such cases the neglect of the Breit
interaction when studying relativistic effects leads to contractions which are
significantly over-estimated (for CH4 by about a factor 3).
For the PbH4 molecule the relativistic effects are much more important than the
correlation effects.
6.2. Conclusions.
In general, we conclude that it is important to use relativistic orbitals from the outset
when one wants to calculate properties which depend on the details of the orbitals
(like the small crystal field splittings in lanthanide impurities). We have demonstrated
that open-shell Hartree-Fock-Dirac calculations are feasible for some chemically
interesting systems, and that the open shell CI method yields reliable results. It has
also proven to be possible and important for valence properties to include the Breit
interaction. The inclusion of correlation effects remains of course important. But in
the case of heavy systems, like the PbH4 molecule, relativistic effects are at least as
important so that the use of a relativistic method from the outset leads to a better set
of zeroth order orbitals, which can be used as a basis for CI or MBPT calculations.
In the near future, more work remains to be done to reduce the computational effort. It
will probably be possible to improve the SCF convergence significantly. Also,
significant gains can be obtained by further reducing the number of two-electron
integrals. This could be accomplished by reducing the small component basis. It
might be worthwhile to study the use of molecular basis sets which are atomically
balanced only, and to study whether (SS|SS) integrals with an absolute value smaller
than 10-3 might be neglected without degrading the accuracy of the results.
An important extension of the method is the inclusion of correlation effects. This can
be done using techniques similar to those used to handle correlation effects in nonrelativistic calculations, like many-body perturbation theory of configuration
interaction calculations. The main difference is the hamiltonian used, the presence of
90
6. Summary and conclusions
negative energy MO's (which should not be included in the calculation) and the
handling of the spin-symmetry.
According to the work of Sucher [1], the CI calculations based on the positive energy
MO's solves the relativistic problem within the no-pair approximation. It will be
interesting to study corrections to this approximation by perturbation theory.
6.3. References.
1. Sucher, J., Int. J. Quantum Chem. 25 (1984) 3.
91
Samenvatting
Samenvatting.
Binnen de quantumtheoretische chemie wordt getracht met behulp van fundamentele
natuurkundige theorieën chemische eigenschappen te verklaren en te voorspellen.
Enerzijds kan hierdoor een beter inzicht in de fundamenten van de chemie worden
verkregen, anderzijds kan met behulp van de theoretische methoden informatie over
chemische systemen worden verkregen die niet, of nog niet, experimenteel
beschikbaar is.
Er bestaat een aantal natuurkundige theorieën die een rol spelen binnen de
theoretische chemie. Allereerst is de quantummechanica van groot belang. Deze
theorie, die nodig is om het gedrag van deeltjes zoals electronen en kernen te
beschrijven, is rond het begin van deze eeuw ontwikkeld. Dit heeft geleid tot de
'Schrödinger-vergelijking'. Een andere theorie die in het begin van deze eeuw is
ontwikkeld is Einstein's speciale relativiteitstheorie. Deze theorie is nodig voor het
beschrijven van snel bewegende objecten. Waarom is zo'n theorie van belang voor de
chemie? Veel verschijnselen in de chemie worden bepaald door het gedrag van
electronen in atomen en moleculen. Die electronen kunnen in de buurt van zware
kernen door de grote electrische aantrekking een grote snelheid bereiken. In zulke
gevallen is het nodig om ze zowel quantummechanisch als relativistisch te
beschrijven. In principe is dat mogelijk geworden met de ontwikkeling van de
relativistische quantum–mechanica, in 1928 door Dirac geformuleerd, de 'Diracvergelijking'.
Het toepassen van genoemde natuurkundige theorieën is zo op het oog eenvoudig: los
de vergelijkingen op die het gedrag van de electronen in een of ander molecuul
beschrijven. Helaas kunnen zowel de Schrödinger- als de Dirac-vergelijking voor
dergelijke meer-electronen systemen alleen bij benadering worden opgelost. Zelfs dan
is een grote hoeveelheid rekenwerk nodig om een goede benadering voor de
oplossingen te vinden. Door de opkomst van computers is het tegenwoordig mogelijk
om de Schrödinger-vergelijking routinematig met grote nauwkeurigheid op te lossen,
voor niet al te grote moleculen. Het oplossen van de Dirac-vergelijking kost echter
93
Samenvatting
aanzienlijk meer rekenwerk. Naast enkele fundamentele problemen is dat de
voornaamste reden dat pas sinds kort relativistische quantummechanica kan worden
toegepast op chemisch interessante moleculen, waarbij de wisselwerking tussen de
electronen slechts gemiddeld wordt meegenomen. In verschillende groepen wordt
momenteel gewerkt aan methoden waarin ook de van het gemiddelde afwijkende
interacties tussen de electronen goed wordt meegenomen. Deze 'correlatie' effecten
kunnen worden berekend met behulp van een techniek die 'configuratie interactie'
wordt genoemd. In dit proefschrift wordt een deel van dat werk beschreven. Op dit
moment is onze groep de enige ter wereld waar relativistische configuratie interactie
berekeningen, uitgaande van de Dirac-vergelijking voor moleculen, kunnen worden
gedaan.
In dit proefschrift worden resultaten beschreven die verkregen zijn met behulp van ab
initio methoden binnen de benadering dat de kernen niet bewegen (de BornOppenheimer benadering). Het begrip 'ab initio' houdt in dat uitsluitend gebruik wordt
gemaakt van de vergelijkingen die volgen uit de relativistische quantummechanica en
dat de oplossingen daarvan op een systematische manier worden benaderd. Hierdoor
kunnen de fouten door de gebruikte benaderingen willekeurig klein gemaakt worden.
Het wordt dan mogelijk eventuele tekortkomingen in de basisvergelijkingen te
bestuderen door de resultaten te vergelijken met het experiment. Ook kunnen de
uitkomsten worden gebruikt voor de beoordeling van mogelijke vereenvoudigingen
en semi-empirische methoden.
De hier gevolgde weg om relativistische quantumchemische berekeningen uit te
voeren is geïnitieerd door P. J. C. Aerts en heeft eerder geleid tot een Hartree-FockDirac programma voor eenvoudige moleculen met een gesloten-schil
electronenstructuur. Dit proefschrift beschrijft enkele verdere ontwikkelingen,
waarvan de uitbreiding naar open-schil systemen een wezenlijke vernieuwing
betekent.
Allereerst is, gebruik makend van het Hartree-Fock-Dirac programma, een analyse
gemaakt van de relativistische effecten in SnH4. Er is met name gekeken in hoeverre
de chemisch interessante relativistische effecten (zoals de verkorting van de
chemische binding) het gevolg zijn van de 'snelle' binnen-electronen. Dit is gedaan
door die electronen relativistisch te beschrijven, terwijl de valentie-electronen nietrelativistisch worden beschreven. De resultaten bevestigen de voorname rol van
directe relativistische effecten op valentie-electronen in het geval van SnH4. Ondanks
het feit dat een valentie-electron meestal op grote afstand van de Sn kern is, blijken de
94
Samenvatting
correcties ten gevolge van de snelheid van het electron in de buurt van die kern toch
al van belang voor de chemische eigenschappen. De relativistische correcties op de
binnen-electronen blijken minder belangrijk te zijn voor de chemische eigenschappen.
Ten tweede is het programma pakket verder ontwikkeld in die zin dat het efficiënter is
gemaakt, en dat het mogelijk is geworden een relativistische correctie op de electronelectron interactie mee te nemen (de Breit interactie). Dit, samen met de
beschikbaarheid van supercomputers als de NEC-SX2 en de CRAY Y-MP, heeft er
toe geleid dat de relativistische effecten en het effect van de Breit interactie binnen de
serie CH4, SiH4, GeH4, SnH4 en PbH4 kon worden bestudeerd. De resultaten laten
een relativistische contractie van de binding zien (van 0.01% voor CH4 naar 4.2%
voor PbH4). Voor lichte systemen zijn de correcties dus erg klein. Het aardige is dat
ze dan voornamelijk worden bepaald door de Breit interactie. Vergelijking met
experimentele gegevens geeft aan dat voor PbH4 de relativistische correcties groter
zijn dan de correlatie correcties. In zulke gevallen is het logisch om relativistische
orbitals te nemen als uitgangspunt voor de correlatie berekeningen.
Ten derde is het pakket uitgebreid zodat ook berekeningen kunnen worden gedaan aan
ingewikkelde open schil systemen. De methode die hiervoor gekozen is, is gebaseerd
op een 'gemiddelde' open schil Hartree-Fock-Dirac berekening, gevolgd door een
configuratie interactie berekening met een klein aantal (~3000) determinanten (de
COSCI methode). Dit leidt tot een evenwichtige beschrijving van de open schil
toestanden in 'intermediate coupling' (tussen jj- en LS- koppeling). Dezelfde methode
kan in de toekomst worden uitgebreid om correlatie effecten in rekening te brengen.
De methode heeft haar nut bewezen in een aantal atomaire berekeningen, en in
berekeningen aan een EuO69- cluster als model voor een Eu3+ onzuiverheid in een
kristal.
Lanthanide onzuiverheden worden vaak toegepast in verband met hun luminiscentie
eigenschappen, bij voorbeeld in televisie schermen. In dit onderzoek hebben wij
onder andere de golflengte van een aantal luminiscentie overgangen van Eu3+
uitgerekend. De onzuiverheid is gemodelleerd door een klein stukje van het materiaal
(een EuO69-cluster) volledig relativistisch quantummechanisch te beschrijven, en de
rest van het materiaal in rekening te brengen via de electrostatische interactie (de
Madelung potentiaal). De methode geeft een goede beschrijving van de relativistische
effecten en van de berekende kristalveld splitsingen (het resultaat van de invloed van
het kristal op de onzuiverheid).
95
Samenvatting
Met dit werk is aangetoond dat relativistische quantumchemische berekeningen
kunnen worden uitgevoerd, op Hartree-Fock-Dirac niveau met beperkte configuratie
interactie, voor chemisch interessante systemen. Voort dergelijke berekeningen is nog
wel veel rekenkracht, geheugen en schijfruimte nodig. Op grond van opgedane
ervaringen zal de rekenmethode gaandeweg kunnen worden verbeterd.
Supercomputers zullen echter nog lange tijd een vereiste zijn.
Zoals beschreven, is er ook een begin gemaakt met relativistische configuratie
interactie berekeningen. Op dit moment kunnen slechts weinig configuraties worden
meegenomen, zodat voornamelijk een goede beschrijving van de open schil
toestanden verkregen wordt. Aan uitbreiding wordt echter gewerkt zodat in de nabije
toekomst resultaten mogen worden verwacht waarin de effecten van electronen
correlatie goed worden beschreven. Voor deze uitbreidingen is het gebruik van
supercomputers noodzakelijk.
96
Appendix A: ROTRAN: Relativistic Orbital Transformation
Appendix A:
ROTRAN: Relativistic Orbital Transformation.
A.1. General.
The purpose of ROTRAN is to transform the Dacre and Elder list of scalar twoelectron integrals to a new (MO) basis which is a linear combination of the scalar
basis functions with their spin components. Thus, the basis transformation can be
written as

i = • p Qpi  + Qpi 
(1)
p
The  p in this equation are large or small component basis functions
p = p L if p  NL
p = p S if p > NL
(2)
and the p are scalar basis functions (we have NL large component basis functions).

The Qpi and Qpi are the transformation coefficients which, in general, can be
complex. In this appendix we use the following labeling:
• p, q, r, s: AO's (scalar basisfunctions)
• i, j, k, l: (two- or four-component) MO's
The scalar integrals are defined as
97
Appendix A: ROTRAN: Relativistic Orbital Transformation
(pq|rs) = prqs =
*p (1)*r (2)r1 q(1)s (2) dr1 dr2
=
12
(3)
The transformed two-electron integrals can be written as follows:
(lk|ji) = ljki =
=
=
*l (1)*j (2)r1 k (1)i (2) d1 d2
12
(4)
• (pq|rs) Q*plQqkQ*rjQsi
pqrs
The algorithm to implement (4) optimally is well-known in non-relativistic programs
[1 and references therein]. Complications arise from the use of symmetry
(permutation symmetry as well as Dirac double group symmetry) and from the
incorporation of spin. After presenting the basic algorithm, we demonstrate how the
permutation symmetry and the spin are handled. Next, we show how the Dirac double
group symmetry is taken into account. Finally, we indicate some elementary speedups which have been implemented, and we present some timing results.
A.2. Basic algorithm.
In this section, the well known four-index transformation algorithm is given. Using
this algorithm, all two-electron integrals are transformed to a new basis that contains
no spin functions.
First half transformation:
p = 1, N
q = 1, N
(pq|ri) = • (pq|rs) Qsi
s
(pq|ji) = • (pq|ri) Q*rj
r
Second half transformation:
i = 1, NA
j = 1, NA
(pk|ji) = • (pq|ji) Qqk
q
98
Appendix A: ROTRAN: Relativistic Orbital Transformation
(lk|ji) = • (pk|ji) Q*pl
p
Evidently, this algorithm is of order N5 where N is the number of basis functions. For
the first half of the transformation the integrals must be sorted so that all scalar
integrals with a given (pq) can be transformed in one batch. Similarly, the halftransformed integrals need to be sorted so that all integrals with a given (ji) can be
transformed in one batch. These sorts have been implemented using a bucket sort [2].
A.3. Permutation symmetry.
The real scalar integrals are invariant for interchange of the labels in the following
manners:
• (pq|rs) = (rs|pq)
(PSym 1)
• (pq|rs) = (pq|sr)
(PSym 2)
• (pq|rs) = (qp|rs)
(PSym 3)
and all combinations of these. Using these relations, only a subset of the integrals (the
canonical integrals) need to be stored. The contributions of the integrals which have
not been stored can be accounted for during the transformation. In order to do so, we
use a method similar to the method used for non-relativistic four-index
transformations [1]. The differences are due to the fact that the transformation
coefficients in our case can be complex. The three permutation symmetries are
considered in the next sections.
A.3.1.
PSym 1:
Consider the following contributions to (lk|ji):
• to (lk|ji): (pq|rs) Qsi Q*rj Qqk Q*pl
(5)
• to (lk|ji): (rs|pq) Qqi Q*pj Qsk Q*rl
(6)
If PSym1 is used to reduce the number of integrals stored, we have to take into
account the contribution to (lk|ji) from both (pq|rs) (5) and (rs|pq) (6). But since the
contribution in (6) is the same as in (7)
• to (ji|lk): (pq|rs) Qsk Q*rl Qqi Q*pj
(7)
99
Appendix A: ROTRAN: Relativistic Orbital Transformation
we need only to consider the contributions from (pq|rs), if we add the contributions
(5) and (7) together after the transformation. Thus:
• Transform only the lower triangle in the (pq) and (rs) indices to form the
integrals (lk|ji)'
• Construct the MO-integrals by (lk|ji) = (lk|ji)' + (ji|lk)'
Note that we need a correction factor 1/2 for the diagonal elements (pq|pq) since
PSym 1 transforms such integrals into themselves.
A.3.2.
PSym 2:
This symmetry is used for the first half of the transformation:
(pq|rs)  (pq|ji)
We can rewrite this using matrix multiplications:
(pq|ji) = • (pq|rs) Qsi Q*rj =
(8)
rs
= Q+ApqQ
with
(9)
Apq = (pq|rs)
Since the integrals (pq|rs) are real:
A = A+
(10)
A = L + L+
(11)
Write
where L is, for example, the lower triangle of A (include factor 1/2 on diagonal).
Then:
Q+AQ = Q+ (L + L+) Q =
= Q+LQ + (Q+LQ)+
(12)
100
Appendix A: ROTRAN: Relativistic Orbital Transformation
We find that we need only transform L (which can be any set of integrals which is
unique with respect to PSym 2) when we add the hermitian conjugate after the first
half of the transformation.
A.3.3.
PSym 3:
This symmetry is used for the second half of the transformation:
(pq|ji)  (lk|ji)
Again, rewrite this using matrix multiplications:
(lk|ji) = • (pq|ji) Qqk Q*pl =
(13)
rs
= Q+Aji Q
with
(14)
Aji = (pq|ji)
Since the integrals (pq|ji) are complex we only have:
A = AT
(15)
A = L + LT
(16)
Write
where L is, for example, the lower triangle of A (include factor 1/2 on diagonal).
Then:
Q+AQ = Q+ (L + LT) Q =
= Q+ (LQ + LTQ)
(17)
We find that we need only transform L (which can be any set of integrals which is
unique with respect to PSym 3), but we have to correct for this by an additional
matrix multiplication on the right hand side as shown by (17). As far as permutation
symmetry is concerned, this is the only significant deviation from the method
normally used for real four-index transformations [1].
101
Appendix A: ROTRAN: Relativistic Orbital Transformation
A.4. Spin.
When we transform the scalar integral (pq|rs) we have to consider the integrals which
we obtain after inclusion of the spin functions. Only four integrals remain due to spin
integration:  and . Schematically, we get the following
algorithm:

(pq|rs)  (pq|ri)
 (pq|ri)
(pq|ri)  (pq|ji)
(pq|ri)  (pq|ji)
(pq|ji) + (pq|ji) = (pq|ji)
and similarly for the second half of the transformation.
Up to this moment, we have only considered the effect of permutation symmetry on
the scalar integrals. Since we have included the spin functions in all possible
manners, we have obtained a non-canonical list of integrals.
Consider the effect of this for integrals like (pp|pp). When no spin functions are
included, this integral transforms into itself under all permutation symmetries
considered, so we include a factor of 1/8 (as described in the section on permutation
symmetry):
(pp|pp):
factor 1/8 if no spin
When we include the spin functions we get the following integrals:
1
(pp|pp): factor 1/8
2
(pp|pp):
factor 1/4




3
(p p |p p ):
factor 1/4
4
(pp|pp):
factor 1/8
The factors given are the factors needed to enable the treatment of permutation
symmetry (as described in the previous section) if we would transform the canonical
two-electron integrals including spin. It is evident that the second and third of these
integrals transform into each other by permutation symmetry PSym 1, so we should
transform only one of these. We can correct for this by multiplying integrals 2 and 3
by an additional factor 1/2, so that all four integrals require a factor 1/8. This factor is
the same needed for the scalar integrals.
102
Appendix A: ROTRAN: Relativistic Orbital Transformation
It is easy to check (as has been demonstrated for integrals like (pp|pp)) that, when we
include all four possible spin integrals, all two-electron spin integrals require the same
factor as needed for the scalar integrals. Therefore, we can simply incorporate this
factor in the scalar integrals before the transformation.
A.5. The algorithm using permutation symmetry and spin.
We adapt the basic algorithm to incorporate the use of permutation symmetry and the
inclusion of the spin functions. The next algorithm requires only the canonical scalar
integrals:
first half of transformation
p = 1, N
q = 1, p
r = 1, p
s = 1, r (p ≠ r) or 1, q (p = r)
i = 1, NA
(pq|ri) = • (pq|rs) Qsi
s
(pq|ri)
= • (pq|rs) Qsi
s
r = 1, p
i = 1, NA
j = 1, NA
*
(pq|ji)' = • (pq|ri) Qrj + • (pq|ri) Qrj
*
r
i = 1, NA
j = 1, NA
(pq|ji) = (pq|ji)' + (pq|ij)' *
103
r
Appendix A: ROTRAN: Relativistic Orbital Transformation
second half of transformation
i = 1, NA
j = 1, NA
p = 1, N
q = (see summation)
k = 1, NA
(pk|ji)
p
= • (pq|ji)
Qqk
N
+ • (qp|ji) Qqk
q=1
p
q=p
N
q=1
q=p
(pk|ji) = • (pq|ji) Qqk + • (qp|ji) Qqk
p = 1, N
k = 1, NA
l = 1, NA
*
(lk|ji)' = • (pk|ji) Qpl + • (pk|ji) Qpl
*
p
p
ji = 1, NA*NA
lk = 1, NA*NA
(lk|ji) = (lk|ji)' + (ji|lk)'
A.6. Dirac double group symmetry.
In this section we show how we account for the fact that we have transformed a list of
Dacre and Elder integrals (i.e. a list of integrals which are symmetry unique with
respect to the spatial symmetry operations of the point group of the system).
The treatment is based on Pitzer's theorem [3, 4]: 'Atomic Orbital integrals related by
symmetry contribute equally to Symmetry Orbital integrals whose integrands are
totally symmetric'. The use of this theorem is easily demonstrated by a simple oneelectron example given by Pitzer [3]. We can use this theorem since we are
calculating integrals over molecular orbitals which have been adapted to the
symmetry of the system (the Dirac double group of the system). Therefore, these
MO's are also symmetry orbitals.
Since we want to use Pitzer's theorem, we need a totally symmetric integrand. This
can easily be accomplished by the following construction [5, 6]:
104
Appendix A: ROTRAN: Relativistic Orbital Transformation
G*D
G*D
1
(ij|kl)" = 1
g • S(ij|kl) = g • (Si Sj | Sk Sl )
S
S
(18)
In this equation, G*D is the Dirac double-group of the molecule, and S is one of the
Dirac double-group elements. The MO integral has explicitly been symmetrized with
respect to the Dirac double group. Using the expression for the Dirac doublegroup
elements in terms of spatial symmetry operators R of the corresponding point group G
and the unitary spin-operators OR (given in chapter 1), we can write
G*D
1
g • (Si Sj | Sk Sl )
S
G
1
G
1
n
n
n
n
=1
g • • ((-1) ORRi (-1) ORRj | (-1) ORRk (-1) ORRl )
R n=0
+
+
=1
g • • (Ri Rj | ORORRk ORORRl )
R n=0
=
1
G
N symop •
R
(Ri Rj | Rk Rl )
(19)
In the last line, we have introduced Nsymop = |G| (the number of spatial symmetry
operations). It can be seen that the expression reduces (due to the fact that the spin
operators are unitary operators) to a symmetrization using only the single group
spatial operators. Using Pitzer's theorem we find that all scalar integrals which are
unique with respect to the spatial symmetry operations contribute equally to the
symmetrized MO integral, so we use (19) to calculate the MO integrals.
When a scalar two-electron integral transforms M times into itself under the all the
spatial symmetry operations of the group, we have a set of
N symop
M
(20)
integrals which are equivalent by symmetry. The Dacre and Elder list contains only
one integral from such a set, so we have to multiply the contribution from that integral
by the number given in (20). The integral program scales all two-electron integrals
with M-1, so we need to multiply all contributions by Nsymop. As a result, the factor
N -1
symop in (19) cancels.
The effect of the symmetry operations on the MO's can be handled easily using the
representation matrices ai (R):
105
Appendix A: ROTRAN: Relativistic Orbital Transformation
^
Ri = • a ai (R)
(21)
a
We get the following working formula for the symmetrized MO-integral:
(ij|kl)" = •
^
R
•
*
*
(ab|cd) ai (R)bj (R)ck (R)dl (R)
(22)
abcd
A.7. Speed-ups.
Define
vmax (p) = max (|Qpi|), i = 1, NA
(23)
A block of integrals labeled by (pq) will be transformed only if
vmax(p) vmax(q) > threshold
(24)
In (24), threshold is a small number (for example 10-10). In some cases the speed-up
which results from this numerical check is rather large. For example, if the active set
of MO's is are the f-orbitals from a lanthanide atom, large component basis functions
other than the f-functions do not contribute to the MO integrals. By adjusting the
threshold the accuracy of the transformed integrals can be changed.
Another speed-up comes from careful coding of the matrix multiplications. The
program uses either a linked triad or a dot product form, depending on the vector
length and on the machine. The most efficient code is determined experimentally at
run time by the program.
A.8. Timings.
The following table contains the timing results for various parts of the ROTRAN and
GOSCIP program. The MO-set consisted of the 4f-like orbitals only. Note that two
different machines were used. More details and results can be found in chapter 5.
106
Appendix A: ROTRAN: Relativistic Orbital Transformation
CPU time
in seconds
(pq|rs)  (pq|ji)
(pq|ji)  (lk|ji)
Symmetrizing
GOSCIP construction H
diagonalization
Eu3+ uncontracted
(Convex C230)
EuO69(Cray Y-MP)
71354
1853
16
15600
310
2
60
6700
20
730
It is evident that the four-index transformation is much more expensive than the CI
calculation. This is due to the very small active space. For the same reason, the
second half of the transformation is much faster than the first part. The
symmetrization only takes a few seconds and is mainly dependent on the symmetry
used.
A.9. References.
1. Saunders, V. R., Lenthe, J. H., Mol. Phys. 48 (1983) 923.
2. Yoshimine, M., J. Comput. Phys. 11 (1973) 449.
3.
4.
5.
6.
Pitzer, R. M., J. Chem. Phys. 58 (1973) 3111.
Davidson, E. R., J. Chem. Phys. 62 (1975) 400.
Schaad, L. J., Wilson, D. J., Hess, B. A. Jr., Chem. Phys. Lett. 105 (1984) 433.
Häser, M., Almlöf, J., Feyereisen, M. W., Exploiting non-abelian point group
symmetry in direct two-electron integral transformations. University of
Minnesota Supercomputer Institute Research Report UMSI 90/200 (1990).
107
Appendix B: Sn basis set
Appendix B: Sn basis set.
Large component:
(15s, 11p, 6d) [11s, 9p, 5d]
s contraction scheme:
42111111111
683186.43311090
103722.68940840
23874.71588527
6837.24013772
2254.86663921
819.11415233
315.78856860
107.19781653
50.05924377
22.29315251
8.46677214
2.59783661
.80141966
2.04221612
.11929178
1.
3.9964602
14.963422
48.092035
1.
1.8404803
1.
1.
1.
1.
1.
1.
1.
1.
1.
p contraction scheme
131111111
7559.16834501
1774.04643143
573.99658432
217.75745204
90.49389244
39.74794063
15.94515937
7.05846180
2.44044235
.95133126
.11764132
7559.16834501
1774.04643143
573.99658432
217.75745204
90.49389244
39.74794063
15.94515937
7.05846180
2.44044235
.95133126
.11764132
1.
1.
3.8254032
1.
1.
1.
1.
1.
1.
1.
1.
p contraction scheme:
-2 -2 -2 1 1 1 1 1 1 1 1 1 -2
1.
1.
3.8254032
10.2058344
1.
1.
1.
1.
1.
1.
1.
d contraction scheme:
21111
214.79150226
62.66059456
22.29315251
8.46677214
2.59783661
.80141966
Small component
(11s,17p,11d,6f) [10s,13p,10d,5f]
s contraction scheme:
1 -2 1 1 1 1 1 1 1 1
683186.43311090
1.
103722.68940840
3.9964602
23874.71588527
1.
6837.24013772
3.213973
2254.86663921
1.
819.11415233
1.8404803
315.78856860
1.
107.19781653
1.
50.05924377
1.
22.29315251
1.
8.46677214
1.
2.59783661
1.
.80141966
1.
2.04221612
1.
.11929178
1.
214.79150226
1.
62.66059456
5.4723545
d contraction scheme:
1 -2 1 1 1 1 1 1 1 1
1.
5.4723545
1.
1.
1.
1.
7559.16834501
1774.04643143
573.99658432
217.75745204
90.49389244
39.74794063
15.94515937
7.05846180
2.44044235
.95133126
.11764132
109
1.
1.
3.8254032
1.
1.
1.
1.
1.
1.
1.
1.
Appendix B: Sn basis set
f contraction scheme:
-2 1 1 1 1
214.79150226
62.66059456
22.29315251
8.46677214
2.59783661
.80141966
110
1.
5.4723545
1.
1.
1.
1.
Appendix C: H basis set
Appendix C: H basis set.
Large component
(4s) [3s]
s contraction scheme:
211
13.00402168
1.96153313
0.44444 1.
0.12193403
Small component
(4p) [3p]
0.01917
0.1380
1.
p contraction scheme:
-2 1 1
13.00402168
1.96153313
0.44444 1.
0.12193403
111
0.01917
0.1380
1.