<—
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Full Variational Molecular Orbital
Method: Application to the
Positron-Molecule Complexes
MASANORI TACHIKAWA,1, 2, * KAZUHIDE MORI, 2
KAZUNARI SUZUKI, 2, 3 KAORU IGUCHI 1, †
1
Department of Chemistry, School of Science and Engineering, Waseda University, Tokyo,
Japan 169-8555
2
Waseda Computational Science Consortium, cro Dr. K. Suzuki, Takachiho University, Tokyo, Japan
168-8508
3
Information Media Center, Takachiho University, Tokyo, Japan 168-8508
Received 5 February 1998; revised 21 April 1998; accepted 28 April 1998
ABSTRACT: Optimal Gaussian-type orbital ŽGTO. basis sets of positron and electron
in positron-molecule complexes are proposed by using the full variational treatment of
molecular orbital ŽFVMO. method. The analytical expression for the energy gradient with
respect to parameters of positronic and electronic GTO such as the orbital exponents, the
orbital centers, and the linear combination of atomic orbital ŽLCAO. coefficients, is
derived. Wave functions obtained by the FVMO method include the effect of electronic or
positronic orbital relaxation explicitly and satisfy the virial and Hellmann᎐Feynman
theorems completely. We have demonstrated the optimization of each orbital exponent in
various positron-atomic and anion systems, and estimated the positron affinity ŽPA. as
the difference between their energies. Our PA obtained with small basis set is in good
agreement with the numerical Hartree᎐Fock result. We have calculated the OHy and
w OHy; eqx species as the positron-molecular system by the FVMO method. This result
shows that the positronic basis set not only becomes more diffuse but also moves toward
the oxygen atom. Moreover, we have applied this method to determine both the nuclear
and electronic wave functions of LiH and LiD molecules simultaneously, and obtained
the isotopic effect directly. 䊚 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 491᎐501, 1998
Key words: positron-molecule complex; positron affinity; full variational molecular
orbital method; nuclear wave function; orbital relaxation
*Research Fellow of the Japan Society for the Promotion of
Science; present address is Rikkyo University, Tokyo, Japan,
171-8501.
†
Deceased.
Correspondence to: K. Suzuki.
International Journal of Quantum Chemistry, Vol. 70, 491᎐501 (1998)
䊚 1998 John Wiley & Sons, Inc.
CCC 0020-7608 / 98 / 030491-11
TACHIKAWA ET AL.
Introduction
A positron Ž eq. is an antiparticle of an electron.
Recently many experimental and theoretical researchers have been interested in systems containing antiparticles w 1᎐3x . When a positron is injected
into some species, a pair annihilation between
positron and electron occurs at the final step of
reactions. In some species, however, a stable system containing a positron is detected experimentally w 4, 5x .
Some theoretical researchers have studied the
possibility of the stable existence of a positron-containing system. The simplest systems PsH and
PsHe, where Ps is a positronium Ža bound state of
a positron and an electron., have been studied by
various methods; variational calculations using the
Hylleraas-type functions w 6᎐8x , the combined
Hylleraas configuration interaction w 9x , and the
complete coupled pair procedures w 10x . The more
electronic PsX systems, where X is a halogen atom,
have so far been studied. Cade and Farazdel w 11x
used the numerical restricted Hartree᎐Fock ŽHF.
method, and Kurtz and Jordan w 12, 13x used the
linear combination of atomic orbital ŽLCAO. HF
calculations. Recently, Saito w 14x calculated these
atomic systems, including the correlation effect,
and compared their energies with quantum Monte
Carlo ŽQMC. calculations by Schrader et al.
w 15, 16x .
Di- or tri-atomic molecular systems containing a
positron have been calculated with LCAO HF calculations by Kurtz and Jordan w 12, 13x , Kao and
Cade w 17x , and we have calculated some positronmolecular systems taking into account the secondorder Møller᎐Plesset ŽMP2. correlation energy w 18,
19x . In order to take account of the correlation
effect, it is important to choose the optimum electronic and the positronic molecular orbitals ŽMOs..
The determination of the MOs is very important
for developing a chemical picture of positronmolecular complexes. In this work we denote the
MOs of the electrons and the positron as i and ,
respectively. An orthogonal basis set is used and
each molecular orbital is expanded by LCAO as,
m1
øi s
s
ž
Ý žÝ
rs1
m2
m2
vs1
492
m1
Ý Ý e ŽŽ S e .
y1 r2
p ŽŽ S p .
/
/
. r Crei ,
y1 r2
. v C vp ,
Ž1.
Ž2.
where e and p are the basis functions of Gaussian-type orbital ŽGTO. for the electrons and the
positron, and S r s ² N r :. Though Slater-type
orbitals ŽSTOs. are a better choice for expressing
the atomic orbital, it is difficult to calculate the
multicentered integrals of molecules. The Cartesian GTO has the form:
Ž x , y, z .
m
l
s Ž x y X . Ž y y Y . Ž z y Z .
2
n
=exp y␣ Ž Ž x y X . q Ž y y Y .
qŽ z y Z .
2
.4,
Ž s e, p . .
2
Ž3.
In Eqs. Ž1. and Ž2. there are three kinds of parameters, such as the LCAO coefficients C e, C p 4 , the
orbital exponents ␣ e, ␣ p 4 , and the orbital centers
R e Ž X e, Y e, Z e ., R p Ž X p , Y p , Z p .4 . Note that in
LCAO HF calculations, only LCAO coefficients
C e, C p 4 are determined by the variational theorem, with the other parameters being held fixed.
Theoretical studies of positron-containing systems have shown that the positron orbital is more
diffuse than the occupied electron orbitals because
of the repulsion between the positron and the
nuclei. Therefore, in the LCAO HF calculation it is
necessary to add diffuse functions to express the
positronic orbital adequately. In previous calculations for such systems, however, the empirical
orbital exponents of the diffuse functions were
used, and the orbital centers were fixed on the
nuclei. Of course these exponents and centers
were not optimal, and therefore the virial and
Hellmann᎐Feynman theorems were not satisfied.
In order to calculate much larger system, it is
required to propose the optimal basis sets for
positronic orbital.
We have proposed the full variational treatment
for molecular orbital ŽFVMO. method w 20x to obtain the optimum parameters of positronic basis
sets. In this method the parameters such as the
orbital exponents ␣ e, ␣ p 4 and the orbital centers
R e, R p 4 are optimized as well as the LCAO coefficients C e, C p 4 of both electronic and positronic
basis sets. Moreover, wave functions obtained by
use of the FVMO method include the effect of
electronic or positronic orbital relaxation explicitly
and satisfy the virial and Hellmann᎐Feynman theorems completely. For efficient optimization, the
analytical energy derivative method has been utilized. Some works have been published about optimization of only orbital exponents w 21x or only
VOL. 70, NO. 3
FULL VARIATIONAL MOLECULAR ORBITAL METHOD
orbital centers w 22x for small molecules Žnot containing positron., however, it does not seem that
the parameters optimized under those conditions
are adequate for larger molecules or other electronic states. In particular such exponents of the
electronic basis set are not optimal for the positronic basis set. To our knowledge, no application
of the analytical derivative method to positroncontaining systems for optimization of both orbital
exponents and centers has been done so far. Therefore, first of all, we have determined the optimal
MOs by the FVMO method.
In the next section we give the analytical formulas of the energy derivatives with respect to electronic and positronic orbital parameters under the
HF approximation. In the third section some computational results for positron-atomic and
positron-molecular systems are shown. Moreover,
we have applied this method to determine both
the nuclear and electronic wave functions directly
and simultaneously. Finally conclusions are given
in the last section.
Theory
We consider a molecular system containing a
positron. The total wave function is assumed to be
⌿ s ⌽e ⭈ ⌽p , where ⌽e is the Slater determinant for
N-electron closed-shell system, and ⌽p is a
positronic spin orbital. We denote the set of spatial
orbitals of electrons and a positron as i 4 Ž i s
1, . . . , Nr2. and , respectively. The HF energy of
this system is given as w 18x :
occ
EHF s 2 Ý h ei i q
i
energy of Eq. Ž4. with respect to these parameters,
the analytical formulas of the HF energy derivatives are required. The first derivative of the HF
energy is given as:
⭸
⭸⍀
occ
E HF s 2 Ý h eŽi i ⍀ . q
i
occ
Ý 2 Ž ii N jj . ⍀
Ž
y Ž ij N ij .
.
Ž⍀.
4
i, j
occ
pŽ ⍀ .
q h
y 2 Ý Ž ii N .
Ž⍀.
.
Ž5.
i
The superscript Ž ⍀ . denotes the differentiation by
the parameter ⍀ as:
h eŽi i ⍀ . s
m1
⭸
Ý Crei) Csie ⭸ ⍀ h er s ,
Ž6.
r, s
Ž ij N kl .
Ž⍀.
Ž ii N .
m1
s
Ž⍀.
Ý
Crei) C sej Ctek) Cue l
r , s, t , u
m1 m 2
s
⭸
⭸⍀
Ž rs N tu . , Ž 7 .
⭸
Ý Ý Crei) Csie Cvp) Cwp ⭸ ⍀ Ž rs N vw . .
r, s v, w
Ž8.
We expanded these formulas into GTOs and performed numerical computations.
We applied the optimization algorithm proposed by Davidon, Fletcher, and Powell w 23x . We
used a primitive basis set of Cartesian Gaussian
functions given in Eq. Ž3. without contraction. The
energy derivatives have been evaluated with respect to the logarithm of the orbital exponents. The
convergence of optimization is judged when the
maximum value of gradient becomes less than
10y6 a.u.
occ
Ý w 2 Ž ii N jj . y Ž ij N ij .x
Results and Discussion
i, j
occ
p
q h
y 2 Ý Ž ii N . ,
Ž4.
i
where h ei i and hp are the one-electronic and onepositronic integrals, Ž ii N jj . and Ž ij N ij . the Coulomb
and the exchange two-electronic integrals, and
Ž ii N . the Coulomb integral between the electron and the positron. If we substitute the mass
p
of proton for positron in Eq. Ž4., h
means the
one-protonic integral.
Each molecular orbital is expanded in the basis
set re 4 or vp 4 by Eqs. Ž1. and Ž2.. Hereafter we
denote the parameters, orbital exponents, and orbital centers of electronic and positronic basis sets
as a whole, by ⍀. In order to optimize the HF
POSITRON-ATOMIC SYSTEM
First we show the results for the Xy and w Xy; eq x
species with various basis sets. In these calculations the orbital centers R ii , R ip 4 of each basis set
are fixed at the center of the X nucleus, and only
the orbital exponents ␣ ie, ␣ ip 4 are optimized. The
positron affinity ŽPA. is also calculated as the
difference between energies of the Xy and w Xy; eq x
systems.
[H y; e + ] System
Let us start with the simplest positron-atomic
complex w Hy; eq x . We have tried to calculate the
Hy and w Hy; eq x systems with only three GTOs. In
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
493
TACHIKAWA ET AL.
this calculation we have used the exponents of
STO-3G primitive basis set w 24x placed on hydrogen nucleus as the initial values of electronic and
positronic exponents, and have optimized these
exponents. Table I summarizes the results of these
systems. The initial orbital exponents of STO-3G
primitive are determined by optimization of the
hydrogen atomic energy. Therefore the optimum
exponents of Hy basis set are much smaller than
the initial ones because of the repulsion between
two electrons. The HF calculation using the optimum basis set lowers the energy by 0.065 a.u.
compared to the initial value, and the virial ratio
with the optimum basis set is very close to 2. This
result indicates that the basis set determined by
optimization of energy of neutral H is not adequate for calculating the energy of anion species.
Next we consider the system containing the
positron, w Hy; eq x , in Table I. No optimum exponents of electronic basis set of w Hy; eq x differ significantly from the initial STO-3G primitives of
hydrogen atom. Namely, the electronic molecular
orbital of this system is similar to the H atom, and
the optimum exponents of electronic basis set for
describing the w Hy; eq x system are very different
from those of Hy. This result indicates that the
electronic wave function of the Hy species is significantly changed by the presence of the positron.
On the other hand, the optimal exponents of the
positronic basis set for the w Hy; eq x system are
much smaller. This is caused by the repulsion
between the positron and the proton, and shows
that the positronic orbital spreads out more than
the electronic orbitals. It is noted that the wave
functions obtained by the FVMO method include
the effect of the orbital relaxation explicitly. The
energy improvement obtained with the optimum
exponents is 0.108 a.u., and the virial ratio is also
very close to 2.
In Table II the total energies and the PAs of
each species obtained using the various basis sets
TABLE II
Optimum energies of H y and [H y; e + ] systems
with each basis set.a
Basis set
3 s (STO-3G
primitive)
3s
5s
10 s
10-STO
H y (a.u.)
[H y; e + ] (a.u.)
PA (eV)
y0.415548
y0.553520
3.754
y0.480220
y0.487306
y0.487924
y0.661085
y0.666551
y0.666944
y0.66695 b
4.922
4.877
4.871
4.86 c
a
Experimental PA is 7.1 " 0.2 in Ref. [5].
Ref. [10].
c
Ref. [14].
b
are shown. Our HF energies of the w Hy; eq x system
with the optimized 3s and 5s GTO basis sets give
99.12 and 99.94% of energy calculated by the 10STO basis set w 10x , respectively. The PA estimated
with the optimum 3s basis set is 4.922 eV, though
one with the STO-3G initial primitive is 3.754 eV.
We see an improvement in the PA of 1.167 eV
upon using the optimum basis set. For further
application, it is required to propose the optimal
basis sets of Hy and w Hy; eq x species with a small
number of basis functions. Moreover, we obtained
4.877 and 4.871 eV with 5s and 10 s optimum basis
sets, respectively, and these values are very close
to the 4.87 eV value obtained with the 10-STO
basis set.
[Li y; e + ] System
In Table III the initial and optimized exponents
of a 6 s basis set for the Liy and w Liy; eq x systems
are shown. Since the initial basis set is the STO-3G
primitive set w 24x , which is determined by the
optimization of the energy of the Li atom, the
optimum exponents of Liy significantly differ from
the initial ones. The optimized exponents of 1 s ,
2 s , and 3 s in Liy are four to six times larger
TABLE I
Optimum exponents of 3 s (STO-3G primitive) basis set for H y and [H y; e + ] systems.
Hy
Type
1s
2s
3s
E HF (a.u.)
Virial ratio
494
Initial
3.42525
0.623914
0.168855
y0.415548
1.5939031
[H y; e + ]
Final
ª
ª
ª
ª
ª
2.38851
0.335631
0.0502995
0.480220
2.0000000
Initial( e y and e + )
3.42525
0.623914
0.168855
y0.553520
1.5377113
ª
ª
ª
ª
ª
Final ( e y )
Final ( e + )
3.22590
0.864208
0.462700
0.0649813
0.0853251
0.0250993
y0.661085
2.0000002
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FULL VARIATIONAL MOLECULAR ORBITAL METHOD
TABLE III
Optimum exponents of 6 s (STO-3G primitive) basis set for Li y and [Li y; e + ] systems.
Li y
Type
1s
2s
3s
4 s
5s
6s
E HF (a.u.)
Virial ratio
[Li y; e + ]
Initial
16.1196
2.93620
.794651
.636290
.147860
.0480887
y7.301830
2.0142956
Final
ª
ª
ª
ª
ª
ª
ª
ª
100.566
15.1464
3.32431
.842991
.0530394
.0110185
y7.414790
2.0000000
than the initial values. These exponents are required to express the character of core electrons of
Liy. On the other hand, the exponents of 5 s and
6 s for valence electrons decrease. These results
indicate that when an electron attaches to a system, the core electron orbitals shrink while valence
electron orbitals become more diffuse. Using this
optimum basis set, the HF energy is lowered by
0.113 a.u. from the initial value, and the virial ratio
with the optimal basis set is very close to 2. On the
other hand, in the w Liy; eq x system the exponents
of 1es , 2e s , and 3es increase, and exponents of 5es
and 6es decrease. The optimum exponents of the
electronic basis set of w Liy; eq x tend to be larger
than those of Liy. In particular, the exponent of 6es
is remarkably large. In contrast, the exponents of
the positronic basis set decrease due to the repulsion by Li nucleus. The energy improvement obtained with the optimum exponents is 0.112 a.u,
and the virial ratio of w Liy; eq x is very close to 2.
In Table IV the total energies and PA are shown.
With the optimized 6 s and 8 s basis sets the energies of w Liy; eq x system are 99.98 and 99.99% of
the result calculated by Partrick and Cade w 25x
using the STO basis set. The improvement of the
PA with the optimized 6 s basis set is 0.021 eV.
TABLE IV
Optimum energies of Li y and [Li y; e + ] systems
with each basis set.
Basis set
6 s (STO-3G
primitive)
6s
8s
STO a
a
Ref. [25].
Li y (a.u.)
[Li y; e + ] (a.u.)
PA (eV)
y7.301830
y7.404681
2.799
y7.414790
y7.426901
y7.428232
y7.516885
y7.528848
y7.529882
2.778
2.774
2.7660
Initial ( e y and e + )
16.1196
2.93620
.794651
.636290
.147860
.0480887
y7.404681
2.0107731
ª
ª
ª
ª
ª
ª
ª
ª
Final ( e y )
Final ( e + )
102.709
3.23169
15.4705
.424169
3.39979
.104258
.864439
.0219853
.0715922
.0112615
.0246763
.00634184
y7.516885
1.9999999
Moreover, we obtained a PA value of 2.774 eV
with the 8 s optimum basis set, while that obtained
by the STO basis set w 25x is 2.766 eV.
[F y; e + ] System
In Table V the exponents of STO-3G Ž6 s3 p .
primitive basis set for Fy and w F -; eq x systems are
shown. Upon optimization the exponents of
1es , 2e s , 3es , 4es , 1e p , and 2e p increase, while the
exponents of 5es , 6es , and 3e p decrease. This behavior is similar to that of Liy shown in Table III.
The HF energy with the optimum basis set is
about 1.0 a.u. lower than the initial value. Next we
show the positronic system of w Fy; eq x . The optimum exponents of the electronic basis set of
w Fy; eq x are also significantly changed from the
initial STO-3G primitive ones, and these exponents
are almost the same as those of Fy. Thee positronic
basis functions of s-type GTOs become diffuse,
while the exponents of the p-type positronic basis
functions are not changed under HF approximation. The energy is lowered about 1.7 a.u. by optimization.
In Table VI total energies and positronic affinity
of each species are shown. With the STO-3G primitive set the PA is calculated to be y14.86 eV. This
shows that the initial STO-3G primitive set is a
very poor basis set to express the positronic attachment. The PA given by the optimum basis set is
5.28 eV. Moreover the PA with 8 s5 p optimum
basis set is 5.06 eV. This energy is very close to
5.01 eV obtained by numerical HF calculation w 14x .
POSITRON-MOLECULAR SYSTEM
We show the results of the OHy and w OHy; eq x
systems as the positron-molecular complex. In this
calculation the distance between the hydrogen and
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
495
TACHIKAWA ET AL.
TABLE V
Optimum exponents of 6 s3 p (STO-3G primitive) basis set for F y and [F y; e + ] systems.
Fy
Type
1s
2s
3s
4 s
5s
6s
1p
2p
3p
E HF (a.u.)
Virial ratio
Initial
166.679
30.3608
8.31682
6.46480
1.50228
0.488589
6.46480
1.50228
0.488589
y98.149374
2.0158110
[F y; e + ]
Final
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
1133.90
170.913
38.3962
10.2971
1.39971
0.387909
8.99334
1.82565
0.355823
y99.146618
2.0000000
oxygen nuclei of OHy is fixed at 1.822 bohrs from
the measurements of Schulz et al. w 26x , and we
placed them at Ž0.0, 0.0, y1.61955. and Ž0.0, 0.0,
0.20245. bohr in three-dimensional space. At first
we optimized only orbital exponents ␣ ie, ␣ ip 4 under the condition of orbital centers R ei , R ip 4 fixed
on each nuclei ŽTable VII., and, second, the orbital
centers and the orbital exponents of the electronic
and positronic basis sets are optimized together
ŽTable VIII..
In Table VII the initial and optimized exponents
of the basis sets for OHy and w OHy; eq x are shown.
We have used the STO-3G primitive set w 24x with
the orbital centers fixed on hydrogen and oxygen
nuclei while the exponents of electronic and
positronic basis sets are optimized. In the electronic results of OHy and w OHy; eq x , the exponents of hydrogen basis functions and oxygen
1 s , 2 s , 3 s , 4 s and especially the p z increased,
while oxygen 3 p x and 3 p y decreased. This is
caused by the fact that the p z are used for the
Initial ( e y and e + )
166.679
30.3608
8.21682
6.46480
1.50228
0.488589
6.46480
1.50228
0.488589
y97.603343
1.9944583
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
Final ( e y )
Final ( e + )
1135.25
12.5063
171.116
1.41051
38.4422
0.502874
10.3097
0.0379389
1.41104
0.0825197
0.393412
0.0181275
9.15730
ᎏ
1.86341
ᎏ
0.367773
ᎏ
y99.340666
2.0000000
bonding pair between hydrogen and oxygen nuclei, while the 3 p x and 3 p y for the lone pairs on
oxygen atom. On the other hand, the exponents of
the positronic basis set in w OHy; eq x become small
and delocalize over the whole molecule. The optimum exponents of positronic basis set on hydrogen nucleus are much larger than those of w Hy; eq x
in Table I. It is interesting that the shift from
atomic exponents of w Hy; eq x to molecular ones
w OHy; eq x shows that the changes in orbital shape
or electron distribution of the original atomic are
caused by the chemical bond formation.
Next we show in Table VIII the result of optimization of both orbital centers and orbital exponents. The orbital centers are expressed as the
displacements from original nuclear positions in
bohr. Schematic illustrations of the initial and the
optimum basis sets of w OHy; eq x system are also
shown in Figure 1 in which only the 3s2 p Ž 4 s ,
5 s , 6 s , 2 p , and 3 p . basis sets of oxygen atom
and the 2 s Ž 2 s and 3 s . of hydrogen atom are
TABLE VI
Optimum energies of F y and [F y; e + ] systems with each basis set.a
Basis set
6 s3 p (STO-3G primitive)
6 s3 p
8 s5 p
[14 s7p ](8 s5 p )b
Numericalc
F y (a.u.)
[F y; e + ] (a.u.)
PA (eV)
y98.149374
y99.146618
y99.433592
y99.4188
y99.459454
y97.603343
y99.340666
y99.619442
y99.6022
y99.643417
y14.858
5.280
5.057
4.99
5.0059
a
Experimental PA is 6.3 " 0.5 in Ref. [4].
Ref. [12].
c
Ref. [25].
b
496
VOL. 70, NO. 3
FULL VARIATIONAL MOLECULAR ORBITAL METHOD
TABLE VII
Optimum exponents of 6 s3 p (STO-3G primitive) basis set for OH y and [OH y; e + ] systems with orbital centers
{R ei , R ip } fixed on each nucleus.
OH y
Type
[OH y; e + ]
Initial
Initial ( e y and e + )
Final
Final ( e y )
Final ( e + )
H
1s
2s
3s
3.42525
0.623914
0.168854
ª
ª
ª
7.31507
.996781
.197687
3.42525
0.623914
0.168854
ª
ª
ª
7.46497
1.03835
0.206552
2.54078
0.300747
0.0765415
O
1s
2s
3s
4 s
5s
6s
1p z
1p x, 1p y
2pz
2 p x, 2 p y
3pz
3 p x, 3 p y
E HF (a.u.)
Virial ratio
130.709
23.8089
6.44361
5.03315
1.16960
0.380389
5.03315
5.03315
1.16960
1.16960
.380389
.380389
y74.413977
2.0158189
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
862.020
129.948
29.1523
7.78716
.850538
.202437
11.1070
6.46925
2.35725
1.29566
.586815
.246843
y75.177075
1.9999948
130.709
23.8089
6.44361
5.03315
1.16960
0.380389
5.03315
5.03315
1.16960
1.16960
.380389
.380389
y74.426856
2.0101077
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
ª
864.819
12.3492
130.366
1.230696
29.2455
0.428886
7.81170
0.0768184
0.873756
0.0369078
0.216166
0.0171953
9.88002
1.07887
6.62172
5.03315
2.07786
0.111166
1.33018
1.16960
0.505416
0.0327949
0.257252
0.380389
y75.361417
2.0052047
TABLE VIII
Optimum centers and exponents of 6 s3 p / 3 s (STO-3G primitive) basis set for OH y and [OH y; e + ] systems.a
OH y
Type
H
1s
2s
3s
O
1s
2s
3s
4 s
5s
6s
1p z
1p x, 1p y
2pz
2 p x, 2 p y
3pz
3 p x, 3 p y
E HF (a.u.)
Virial ratio
a
Center
[OH y; e + ]
Exponent
y1.61955
+0.010799
8.08782
+0.118401
1.17179
+0.749584
0.237009
0.20245
y0.000001
867.952
y0.000057
130.834
+0.000029
29.3485
+0.000143
7.83933
y0.196227
0.922243
+0.214614
0.235636
y0.000838
17.1974
y0.001536
6.39218
y0.014867
3.74343
y0.020748
1.27854
y0.034186
1.00764
y0.131813
0.241988
y75.208073
2.0000000
Center ( e
y)
y1.61955
+0.011881
+0.125454
+0.835971
0.20245
y0.000001
y0.000053
+0.000043
+0.000185
y0.184343
+0.255355
y0.000600
y0.001145
y0.014206
y0.017935
y0.029224
y0.088355
Exponent ( e y)
8.20142
1.19261
0.247801
Center ( e + )
y0.024118
+0.113473
+1.365186
869.014
+0.093443
130.992
+0.098179
29.3839
+0.137154
7.84851
y0.334620
0.935227
+0.824741
0.243199
+1.495749
17.5995
y0.309754
6.57817
0.000000
3.83690
y0.216695
1.32076
0.000000
1.03775
y0.471549
0.254609
0.000000
y75.392926
1.9999987
Exponent ( e + )
4.54649
0.676053
0.170910
3.64601
0.649922
0.177810
0.0490555
0.0230946
0.00862750
1.24373
5.03315
0.168798
1.16960
0.0538371
0.380389
Units in a.u.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
497
TACHIKAWA ET AL.
FIGURE 1. Schematic illustration of the initial and the
optimum basis functions of the [OH y; e + ] system. Only
the 3 s2 p( 4 s , 5 s , 6 s , 2 p , and 3 p ) basis sets of the
oxygen atom and the 2 s ( 2 s and 3 s ) basis set of the
hydrogen atom are shown in arbitrary units. (a) Basis
functions for OH y are shown. (b) Electronic basis
functions of the [OH y; e + ] system are shown. The
electronic basis functions of the hydrogen atom move
toward the oxygen nucleus, while p x and p y basis
functions become more diffuse. (c) Positronic basis
functions of the [OH y; e + ] system are shown. The
positronic orbital not only becomes more diffuse but also
moves toward the oxygen atom.
expressed in arbitrary units. The electronic basis
functions of the hydrogen atom of OHy, particularly hydrogen 3 s , move toward the oxygen nucleus by the orbital center optimization, and the
exponents change little compared with those of
OHy ŽTable VII.. These hydrogen basis functions
are mainly used for expressing the orbital of
this system. On the other hand, the p x and p y
type of basis functions become more diffuse than
the p z basis functions used to express the lone
pairs. As expected from the difference of electronegativity between hydrogen and oxygen, all
basis sets tend to move toward the oxygen atom.
Thus we have predicted that the positronic orbital
moves toward the oxygen nucleus, because of the
attraction between the positron and the electrons
gathering near the oxygen atom. In fact, Figure 1
shows that the positronic orbital not only becomes
more diffusive but also moves toward the oxygen
atom.
In Table IX total energies and PAs of both OHy
and w OHy; eq x systems are shown. Our total energies with only a 6 s3 p basis set are within 0.2 a.u.
deviation of those by Kao and Cade with the
w 5s5 p2 dr3s1 p x basis set for the electrons and the
w 4 s4 pr2 s x basis set for positron w 17x . Of course,
they calculated OHy and w OHy; eq x with the same
basis sets. On the other hand we have optimized
the parameters of the basis sets, that is, our PA
includes the effect of electronic orbital relaxation
explicitly. However, our PA is calculated by HF
approximation, so it is less than the values of MP2
w 27x or QMC w 28x computations.
APPLICATION TO POSITRONIC OR NUCLEAR
WAVE FUNCTIONS
In the previous section the basis sets of electronic and positronic MOs are determined by us-
TABLE IX
Optimum energies of OH y and [OH y; e + ] systems with each basis set and method.
Method
Basis set
E HF
6 s3 p (STO-3G primitive)
6 s3 p
Kao a
MP2 b
QMC c
OH y (a.u.)
[OH y; e + ] (a.u.)
PA (eV)
y74.413977
y75.208073
y75.41117
y75.61845
y74.426856
y75.392926
y75.58764
y75.81439
0.350
5.030
4.80
5.332
5.57 " 0.15
a
[5 s5 p2d / 3 s1p ] for electron and [4 s4 p / 2 s ] for positron [17].
(13 s7p3 d / 6 s3 p ) contracted to [6 s5 p3 d / 4 s3 p ] for electron and positron [27].
c
Ref. [28].
b
498
VOL. 70, NO. 3
FULL VARIATIONAL MOLECULAR ORBITAL METHOD
ing the FVMO method. In the FVMO method it is
possible to determine the basis sets of protonic, or
nuclear, MOs directly, which are not proposed
variationally yet. Table X shows the results of the
FVMO calculation applied to w Liy; eq x , LiH, and
LiD systems. The positron or the proton is treated
as the quantum wave, while the Li nucleus as the
point charge. 6 s basis functions are employed for
electrons and 1 s for the positively charged quantum particles. Since all the exponents and centers
of basis functions are optimized, the virial ratios of
these species are fairly close to 2. Figure 2 shows
the schematic illustration of optimized basis functions for electron and positively charged particle in
arbitrary units.
All the optimized centers of the electronic and
positronic basis functions are situated at the Li
nucleus in the w Liy; eq x system, and the exponent
of the positronic basis function is smaller than any
other ones. The result of LiH molecule is significantly different from that of the w Liy; eq x system,
i.e., the protonic orbital center shifts from Li nucleus toward the position of the point charge,
having a large exponent of 15.97. It is noted that
the center of the most diffuse s function is separated from the Li nucleus by 2.64 bohrs. As illustrated in Figure 2Žc., these functions seem to express the bonding pair between the Li and H
atoms. On the other hand the orbital center of Dq
in the LiD molecule is a little bit shorter than that
of Hq in the LiH molecule, and close to the equilibrium distance calculated by conventional FVMO
method. This cause is due to the anharmonicity.
Since the kinetic energy of Dq is a smaller than
FIGURE 2. Schematic illustration of the optimum basis
functions for electron and positively charged particle.
Only 3 s ( 3es , 4es , and 6es ) electronic basis functions
p
and 1s ( 1s ) positively charged particle are shown in
arbitrary units. (a) Basis functions of the Li y anion. (b)
Basis functions of the [Li y; e + ] system. (c) Basis
functions of the LiH molecule.
TABLE X
Optimum basis function and energies for Li y, [Li y; e + ], LiH, and LiD systems.a
Positively particle
E HF (a.u.)
Virial ratio
Dipole (a.u.)
Electron
1s
2s
3s
4 s
5s
6s
Positively particle
1s
a
Li y
ᎏ
ᎏ
y7.4147900
2.0000000
0.0000
Center Exponent
0.0000
100.566
0.0000
15.1464
0.0000
3.32431
0.0000 0.842991
0.0000 0.0530394
0.0000 0.0110185
[Li y; e + ]
e+
Quantum wave
LiH
H+
Quantum wave
LiD
D+
Quantum wave
LiH
Point Charge
y7.5094065
y7.9119399
y7.9194841
y7.9434074
2.0000000
2.0000000
2.0000000
2.0000000
0.0000
y2.4680
y2.4698
y2.4724
Center Exponent Center Exponent Center Exponent Center Exponent
0.0000
102.473
0.0000 96.6625
0.0000 96.5970
0.0000 96.3989
0.0000
15.4348 y0.0005 14.5586 y0.0005 14.5488 y0.0006 14.5191
0.0000
3.39138 y0.0003 3.19100 y0.0004 3.18881 y0.0005 3.18220
0.0000 0.862130 y0.0285 0.809183 y0.0288 0.808629 y0.0295 0.806931
0.0000 0.0690911
3.1667 0.917424
3.1546 0.953425
3.1159 1.07607
0.0000 0.0229214
2.6357 0.116406
2.6335 0.118731
2.6248 0.126232
0.0000 0.0118690
3.2055
15.9704
3.1907
21.8376
R(Li-H) = 3.1449
Units in a.u.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
499
TACHIKAWA ET AL.
that of Hq, the total energy of LiD is lower than
one of LiH molecule, and the orbital exponent is
larger than that of Hq. In the FVMO method this
isotopic effect is clearly and directly shown.
Conclusion
In this study, the FVMO method is developed
for positron-containing systems by using the analytical energy gradient with respect to parameters,
such as orbital exponents, orbital centers, and
LCAO coefficients. Since the basis set determined
by optimization of the atomic energy is not adequate for calculation of the anion species, we have
also applied the FVMO method to anion species.
The optimum exponents of the positronic basis set
obtained become more diffusive, because the
positronic orbital spreads out more than electronic
orbitals for the repulsion between a positron and a
nucleus. The result of the w OHy; eq x system shows
that the positronic orbital not only becomes more
diffuse but also moves toward the oxygen nucleus
upon optimization. Moreover, we have applied
this method to determine both the nuclear and
electronic wave functions simultaneously and directly. The results of the w Liy; eq x , LiH, and LiD
systems seem to correspond to the change from a
quantum wave to a classical particle. The variational principle requires the largest overlap between exact and trial wave functions to lower the
trial energy. We are very interested in the result
that this quantum mechanical full variational
treatment reproduces the classical chemical description of the LiH molecule.
As is well known, original basis sets, which
were determined by optimization of the atomic
state, are not adequate for calculating anion or
positron-countering species. Of course, wave functions calculated with such basis sets do not satisfy
the virial theorem. In the FVMO method it is
possible to determine the optimal basis sets of
positronic wave functions directly, which are not
proposed variationally yet. The wave functions
obtained by the FVMO method includes the effect
of electronic or positronic orbital relaxation explicitly and satisfy the virial theorem completely. In
conclusion it is useful to describe the positron-containing system by the FVMO method, and we
confirm that this method may be useful for problems involving the nuclear motion. More details of
500
the nuclear wave functions and some applications
will be published.
The effect of the optimization of the energy,
including the correlation energy with respect to
electronic and positronic orbital parameters for
various molecule, is important for describing the
molecular wave function. Such a study is now in
progress, and the results will be published in the
near future.
ACKNOWLEDGMENTS
We thank the staff of Waseda Computational
Science Consortium for many helpful discussions.
This work was carried out as a subject of Research
Fellowships of the Japan Society for the Promotion
of Science for Young Scientists. The present calculations were carried out with Nihon SGI-Cray K.K.
computer system, and the VP2200r10 Super Computer at the Centre for Informatics of Waseda
University.
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hydrogen atom 0.0360, 0.0100 for s orbitals and 0.3600,
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