THE JOURNAL OF CHEMICAL PHYSICS 122, 154307 共2005兲
Excited states of boron isoelectronic series from explicitly
correlated wave functions
F. J. Gálvez and E. Buendía
Departamento de Física Moderna, Facultad de Ciencias, Universidad de Granada,
E-18071 Granada, Spain
A. Sarsa
Departamento de Física, Campus de Rabanales, Edificio C2, Universidad de Córdoba,
E-14071 Córdoba, Spain
共Received 2 December 2004; accepted 19 January 2005; published online 19 April 2005兲
The ground state and some low-lying excited states arising from the 1s22s2p2 configuration of the
boron isoelectronic series are studied starting from explicitly correlated multideterminant wave
functions. One- and two-body densities in position space have been calculated and different
n
ជ 兲典, and 具Rn典, where rជ, rជ , and Rជ stand for
典, 具␦共R
expectation values such as 具␦共rជ兲典, 具rn典, 具␦共rជ12兲典, 具r12
12
the electron-nucleus, interelectronic, and two electron center of mass coordinates, respectively, have
been obtained. The energetic ordering of the excited states and the fulfillment of the Hund’s rules is
analyzed systematically along the isoelectronic series in terms of the electron-electron and
electron-nucleus potential energies. The effects of electronic correlations have been systematically
studied by comparing the correlated results with the corresponding noncorrelated ones. All the
calculations have been done by using the variational Monte Carlo method © 2005 American
Institute of Physics. 关DOI: 10.1063/1.1869468兴
I. INTRODUCTION
Boronlike ions in their ground and first excited states
have been studied by means of most of quantum chemistry
methodologies such as configuration interaction, multiconfiguration Hartree–Fock 共MCHF兲, many-body perturbation
theory, or multiconfigurational Dirac–Fock, see e.g., Refs.
1–5 and references therein. Most of those works were focused on excitation energies, oscillator strengths, and transition rates between a number of excited states. However the
knowledge of other interesting properties such as the oneand two-body electron densities is much more scarce due to
the technical difficulties involved in the calculation.
Nowadays the variational Monte Carlo method has become a powerful tool in quantum chemistry calculations because it allows one to evaluate the expectation value of any
operator between wave functions of any type and therefore
one can work with an explicitly correlated trial wave function rather than expanding it in terms of Slater determinants.
Then one can evaluate not only the energy but also some
other properties such as the one- and two-body electron densities which are very difficult to obtain by using some other
methods and which, especially the latter, are very sensitive to
the effect of electronic correlations. These densities provide
detailed and valuable information on the structure and dynamics of the system. For example, they play a key role in
the understanding and interpretation of some interesting features of the electronic structure of atoms such as the Coulomb hole and the Hund’s rule.6–13 This rule states that 共i兲 if
two states arise from the same configuration, the state having
the highest spin will have the lowest energy, and 共ii兲 among
those states with the same spin value, the most bound one is
that with the highest value of the orbital angular momentum.
0021-9606/2005/122共15兲/154307/15/$22.50
An interpretation for item 共i兲 was given by assuming that
electrons with the same spin tend to keep apart, leading to a
reduction of the Coulomb repulsion energy. However this
argument has been shown to fail. By using accurate wave
functions it has been obtained that the electron-electron repulsion energy is bigger for the state with the highest total
spin, and it is the electron-nucleus energy that is responsible
for the higher binding energy of this state. This was first
found for some excited states of heliumlike systems by using
Hylleraas-type wave functions,9 and then for other atomic
systems by using correlated and uncorrelated wave
functions.14–16 It has been also found that there are some
other multiplets for which the first Hund’s rule does not hold
and some others for which the previous interpretation is not
so direct.16 To explain item 共ii兲 it has been argued that electrons circulating in opposite directions, with a low total angular momentum, will meet frequently leading to a larger
repulsive interaction.17 As we shall see later for some states
of the boron isoelectronic series coming from the same configuration, the explanation may not be as simple as that.
The aim of this work is to study the ground and some
excited states of the boron isoelectronic series. Accurate and
compact explicitly correlated trial wave functions are obtained here for these systems. The energy and some related
properties, such as the electron-nucleus and electron-electron
energies which are important in the discussion of the Hund’s
rules, are reported. The variational ansatz used in this work
has shown to provide good results for the ground and lowlying excited states of four electron atomic systems.16,18,19
We study here the 2 P ground state arising from the 1s22s2 p
configuration as well as the states 4 P, 2D, 2S, and 2 P arising
from the configuration 1s22s2p2. The Hund’s rule is fulfilled
122, 154307-1
© 2005 American Institute of Physics
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154307-2
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
except for the 2 P and 2S terms which are reversed for all of
the systems. On the other hand, and as it is well known, the
position of these states in the spectrum of the ion depends
on the nuclear charge. For C+ and higher ions these
states are the four first excited states, while for B the states
1s22s2p2-2S and -2 P are highly excited. However a relevant
difference between these two states takes place in the spectrum of the boron atom. Thus the 1s22s2p2-2S is the fifth
positive parity 2S state of the spectrum while the 2 P states
with lower energy than 1s22s2p2-2 P have negative parity.
With these considerations we have been able to study all
these states for boron, except for the 1s22s2p2-2S one, with
good precision.
Starting from the wave functions obtained in this work
we have computed, in addition to the energy, different oneand two-body position and momentum properties. One-body
properties in position space can be studied in terms of the
single- particle density, ␳共rជ兲,
␳共rជ兲 = 具⌿兩 兺 ␦关rជ − rជi兴兩⌿典
共1兲
i
which gives the charge distribution around the nucleus.
Two-electron properties can be studied in terms of both
the interelectronic, or intracule, I共rជ12兲, and the center of
ជ 兲, densities20,21 defined as
mass, or extracule, E共R
I共rជ12兲 = 具⌿兩
␦关rជ12 − 共rជi − rជ j兲兴兩⌿典,
兺
i⬎j
ជ 兲 = 具⌿兩 兺 ␦关Rជ − 共rជ + rជ 兲/2兴兩⌿典,
E共R
i
j
共2兲
共3兲
i⬎j
respectively. These two-body functions represent the probability density function for a pair of electrons having a relaជ , respectively.
tive vector rជ12 or a center of mass vector R
Their spherical averages will be denoted by h共r12兲 and d共R兲,
respectively.
The intracule density in position space plays an important role in several physical and chemical problems such as,
for example, the electron correlation problem or the interpretation of the Hund’s rules.6–9 For two- and three-electron
atoms it has been calculated by using highly accurate correlated wave functions,6–13,22–24 leading to an extensive study
of its properties. For heavier atoms the results known are
more scarce. Correlated results have been obtained within a
configuration interaction scheme25 for the atoms of the second and third row, and,26 from energy-derivative twoelectron reduced density matrices for some light atoms and
starting from multiconfiguration Hartree–Fock wave functions for the beryllium atom.27 It has been also obtained for
the atoms helium to neon starting from explicitly correlated
wave functions by means of the Monte Carlo method.18,28,29
Much less work has been done for the extracule density.
For He and Be atoms and by using accurate wave functions,
correlated results have been recently reported18,23,24,27 and
for the atoms helium to neon starting from explicitly correlated wave functions.30
These two distribution functions provide insight into the
spatial arrangement of the electronic charge. They have been
analyzed here to elucidate the differences between the electronic clouds of doublets and quartets as well as the differences between states arising from the same configuration
with the same spin and different orbital angular momentum.
Finally the radial moment of order −1 of the single-particle
and the intracule densities give the electron-nucleus attraction and the electron-electron repulsion energy, respectively.
Because of the virial theorem these radial moments, 具r−1典 and
−1
典, and their behavior along the isoelectronic series play a
具r12
key role in the energetic ordering of the excited states and
henceforth on the interpretation of the Hund’s rule.
The structure of this work is as follows. In Sec. II we
show the wave function used in this work. The results obtained are reported and discussed in Sec. III. The conclusions
and perspectives of this work can be found in Sec. IV.
Atomic units are used throughout.
II. WAVE FUNCTION
The correlated trial wave function ⌿ used in this work is
the product of a symmetric correlation factor F, which includes the dynamic correlation among the electrons, and a
model wave function ⌽, which provides the correct properties of the exact wave function such as the spin and the
angular momentum of the atom, and is antisymmetric in the
electronic coordinates:
⌿ = F⌽.
共4兲
For the correlation factor we use the form of Boys and
Handy31 with the prescription proposed by Schmidt and
Moskowitz.32 We have worked with 17 variational nonlinear
parameters in the correlation factor, which include electronnucleus, electron-electron, and electron-electron-nucleus correlations.
The model wave function has been fixed within the optimized effective potential 共OEP兲 framework, which has been
recently generalized to deal with multiconfigurational wave
functions.33 Within this framework the model wave function
is written as
⌽=
兺k Ck␾k ,
共5兲
where ␾k is each one of the states with the proper values of
the total spin and orbital angular momentum arising from the
configurations selected to describe the state under consideration. In this work the orbitals are built starting from the
single-particle configurations 1s2nl pn⬘l⬘q, with n , n⬘ = 2 , 3;
l , l⬘ = s , p , d; p , q 艌 0 and p + q = 3. To study a given state we
have selected only those configurations which provide an
improvement in the energy greater than the statistical error in
the calculation. The importance of some configurations decreases as the nuclear charge Z increases in the isoelectronic
series while the importance of some other configurations increases with Z. In all the configurations and/or states considered in the present work the 1s2 term appears, and so it will
not be written from now on.
Once the model wave function is built, the total trial
wave function is obtained by multiplying it by the correlation
factor F. The nonlinear parameters of the correlation factor
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154307-3
J. Chem. Phys. 122, 154307 共2005兲
Excited states of boron
and the linear coefficients Ck of the expansion of the model
wave function, Eq. 共5兲, are taken as variational parameters.
This constitutes a correlated basis set expansion of the trial
wave function where the Hamiltonian is diagonalized. This
step involves the solution of a generalized eigenvalue problem, with matrix elements computed by Monte Carlo
method, obtaining a new set of the linear coefficients Ck.
Therefore, in the minimization process, the model wave
function is also partially optimized, and the new set of linear
coefficients Ck modifies the position of the nodes of the wave
function. The optimization of the wave function has been
carried out by minimizing the total energy. The use of the
Monte Carlo method limits the number of configurations
than can be used in the expansion Eq. 共5兲 because of the
computing time.
Starting from the best wave function, different one- and
two-body densities, their values at the origin, and the expectation values 具tn典 where t stands for r, r12, and R, have been
calculated. The value of each density at the origin has been
calculated by using the relations
␳共0兲 = −
1
2␲
兺i
冕
Q⌿2共x兲
1 1 ⳵⌿共x兲
dx
Q ⌿共x兲r2 ⳵r
共6兲
for the single-particle density,34 and
␹共0兲 = −
1
2␲ i⬎j
兺
冕
Q⌿2共x兲
1 1 ⳵⌿共x兲
dx
Q ⌿共x兲t2 ⳵t
共7兲
for both the intracule 共␹ = h , t = r12兲 and the extracule 共␹
= d , t = R兲 densities.28 These expressions allow one to obtain a
local property of the corresponding density in terms of the
wave function evaluated in the whole domain. In order to
calculate all these expectation values we have used 兩⌿兩2 as
−2
distribution function, except for h共0兲 and 具r12
典 for which we
2
2
have used Q兩⌿兩 , with Q = 兺i⬍j1 / rij, which provides more
accurate results for those expectation values.35
III. RESULTS
The inclusion of the correlation factor F in the trial wave
function modifies the weights Ck of the different configurations appearing in the description of a given state in the OEP
framework. In general, the weights depend on both the state
studied and the ion considered.
The ground state of the boron isoelectronic series is a 2 P
term, coming mainly from the 2s22p configuration. The Ck
coefficient for this configuration in the correlated basis set
expansion goes from 0.961 for B to 0.974 for C+, increasing
up to 0.982 for Ne5+. The 2s-2p near degeneracy effect is
taken into account by including the 2p3 configuration in the
expansion. Its weight is nearly constant along the isoelectronic series. Thus, the coefficient in the correlated basis set
expansion goes from 0.19 for B to 0.184 for C+ and decreases slowly up to 0.176 for Ne5+. We have also considered
the 2s23p, the 2s2p3s and the 2s2p3d configurations to describe the ground state, but their importance decreases
quickly with the nuclear charge Z. Thus the global contribution of these three configurations is of about 4% for B and
less than 0.5% for Ne5+.
The first excited state for all of the members of the boron
isoelectronic series is the quartet 2s2p2-4 P. This state presents an important mixing with the 2s2p3p configuration
共from more than 6.5% for B to 1% for Ne5+兲, and a smaller
contribution coming from the 3s2p2 configuration 共from
1.5% for B to less than 0.25% for Ne5+兲.
There are three different doublet states arising from the
2s2p2 configuration, the 2D, the 2 P, and the 2S states. The
2s2p2-2D state presents an important admixture with both the
2s23d-2D and the 2s2p3p-2D states for B 共4% and 8%, respectively兲. However their weight decreases for C+ 共2.5%
and 3.5%, respectively兲 and higher ions 共less than 1%兲. Also
there appears a much less important admixture with the
2p23d-2D and 3s2p2-2D states, whose weight is about 0.9%
and 0.6% for B, and much smaller for higher ions. The first
2
P state with positive parity comes mainly from the configuration 2s2p2 共more than 99% for N2+ and higher ions兲. For
C+ there is an admixture of ⬇2.5% with the 2s2p3p-2 P state
which increases to more than 25% for B. In this atom this
state is above the ionization threshold. The importance of the
2p23d configuration decreases from 1% for B to 0.3% for
Ne5+. Finally, the 2S state comes mainly from the configuration 2s2p2. It is very excited for B 共in fact it is the fifth state
of 2S type in its spectrum兲 and has not been studied in the
present work. For the positive ions it is the first excited state
of 2S type. For C+ there is a contribution greater than 7% and
3% from the configurations 2s23s and 2s2p3p, respectively.
For Ne5+ it is almost a 2s2p2-2S state, whereas for the rest of
ions we have found an admixture of about 2% with the
2s2p3p-2S state, and hardly appreciably with the 2s23s-2S
one.
All the relative weights have been obtained within the
correlated basis set.
A. Energies
In Table I we report the energy E of the states of the
boron isoelectronic series studied here as compared with the
estimated exact values, Eexact, taken from the NIST database
for atomic spectroscopy.36 For the ground state we have
taken as exact values those reported in Ref. 37 which were
obtained by combining experimental data and ab initio calculations. In the estimated exact results we have averaged the
corresponding spin-orbit interaction among the different
states of the multiplet.
By comparing our results with the corresponding exact
and Hartree–Fock37 energies for the ground state of the different ions, one can see that the percentage of correlation
energy recovered with our wave functions goes from 92.8%
for B to 95.3% for Ne5+. The absolute difference between
our results and the exact values is around 0.009 a.u., showing the good performance of the wave functions obtained
here. This difference is similar for the excited 2 P state,
smaller for the 2s2p2-4 P one, and a bit higher for the 2D and
2
S states. For the excited states this difference decreases
quickly for F4+ and Ne5+.
For the sake of comparison, let us mention here some
other variational upper bounds previously reported in the literature. For example, the energy of ground state of the boron
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154307-4
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
TABLE I. Energy of some states of the boron isoelectronic series obtained in this work 共E兲 as compared with those that can be considered as exact 共Eexact兲
taken from Ref. 36. The electron-nucleus attraction energy Ven, the electron-electron repulsion energy Vee, the kinetic energy T, and the virial ratio ␩ are also
given.
E
Eexact
Ven
Vee
T
␩
2s22p 共2 P兲
2s2p4 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
−24.645 02共6兲
−24.515 81共6兲
−24.424 86共5兲
−24.653 93
−24.522 35
−24.435 88
B
−57.039共7兲
−56.648共7兲
−56.398共6兲
7.6877共4兲兲
7.6094共4兲
7.5043共4兲
24.706共7兲
24.522共7兲
24.469共6兲
1.9975共6兲
1.9997共7兲
1.9982共7兲
−24.315 1共1兲
−24.323 49
−56.023共9兲
7.3478共5兲
24.360共8兲
1.9982共8兲
2s22p 共2 P兲
2s2p2 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
−37.421 9共1兲
−37.227 6共1兲
−37.081 0共1兲
−36.980 9共1兲
−36.918 5共1兲
−37.430 96
−37.234 88
−37.089 55
−36.991 30
−36.926 79
C+
−84.929共11兲
−84.547共11兲
−84.250共11兲
−83.957共13兲
−84.001共12兲
10.0814共6兲
10.0825共5兲
10.0564共5兲
9.9542共5兲
10.1167共6兲
37.423共11兲
37.237共11兲
37.112共12兲
37.021共13兲
36.966共13兲
2.0000共7兲
1.9997共7兲
1.9992共7兲
1.9989共8兲
1.9987共8兲
2s2p2 共2 P兲
2s2p2 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
−52.957 6共1兲
−52.699 1共2兲
−52.496 4共1兲
−52.356 3共1兲
−52.292 4共1兲
−52.966 25
−52.705 27
−52.505 94
−52.369 35
−52.301 26
N2+
−118.381共17兲
−117.968共17兲
−117.572共15兲
−117.378共17兲
−117.266共18兲
12.4522共7兲
12.5291共6兲
12.5789共6兲
12.6365共6兲
12.6819共6兲
52.971共17兲
52.740共17兲
52.497共14兲
52.385共17兲
52.291共18兲
1.9997共6兲
1.9992共7兲
2.0000共7兲
1.9995共7兲
2.0000共7兲
−71.255 53
−70.929 11
−70.677 14
−70.506 62
−70.432 46
O3+
−157.220共23兲
−156.811共22兲
−156.395共22兲
−156.207共20兲
−156.043共24兲
14.7995共7兲
14.9497共7兲
15.0626共7兲
15.1615共8兲
15.2093共8兲
71.173共22兲
70.937共21兲
70.663共21兲
70.549共20兲
70.407共24兲
2.0010共6兲
1.9998共6兲
2.0001共6兲
1.9993共6兲
2.0003共7兲
−92.297 07
−91.904 39
−91.600 48
−91.396 89
−91.316 58
F4+
−201.703共22兲
−201.266共27兲
−200.741共25兲
−200.477共25兲
−200.336共27兲
17.1593共8兲
17.3698共8兲
17.5341共8兲
17.6595共8兲
17.7300共8兲
92.255共23兲
91.996共29兲
91.612共26兲
91.427共27兲
91.293共28兲
1.9996共6兲
1.9990共6兲
1.9998共6兲
1.9996共6兲
2.0002共7兲
−116.090 06
−115.630 09
−115.274 46
−115.038 22
−114.951 71
Ne5+
−251.661共30兲
−251.106共36兲
−250.483共38兲
−250.172共27兲
−250.101共32兲
19.5057共9兲
19.7695共9兲
19.9955共9兲
20.1379共10兲
20.2421共10兲
116.074共30兲
115.707共36兲
115.215共38兲
114.999共26兲
114.905共32兲
2.0001共5兲
1.9994共6兲
2.0005共7兲
2.0003共6兲
2.0004共7兲
2s 2p 共 P兲
2s2p2 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
2
2
2s 2p 共 P兲
2s2p2 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
2
2
2s 2p 共 P兲
2s2p2 共4 P兲
2s2p2共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
2
2
−71.246 8共1兲
−70.924 8共1兲
−70.669 1共1兲
−70.496 7共1兲
−70.426 7共1兲
−92.288 4共1兲
−91.900 1共2兲
−91.595 0共2兲
−91.390 2共1兲
−91.313 9共2兲
−116.081 6共1兲
−115.629 7共2兲
−115.272 7共2兲
−115.035 3共2兲
−114.953 7共1兲
atom 关2s22p共2 P兲兴 has been calculated in Ref. 2 by using
MCHF-type expansions with 7096 and 32 456 configurations
obtaining −24.651 009 a.u. and −24.652 725 a.u., respectively. Our best result for this state is −24.645 02共6兲 a.u. and
the exact energy is −24.653 93 a.u. In that work,2 the state
2s2p2共2D兲 was also studied by using the same methodology
obtaining an energy of −24.431 353 a.u. with 9161 configurations and −24.433 330 a.u. with 31 336 configurations. Our
best result is −24.424 86共5兲 a.u. and the exact one is
−24.435 88 a.u. Our goal here was not to improve the accuracy of previous variational works, instead we pursued to
study in a common framework and by using accurate wave
functions several aspects of the structure of the low-lying
excited states of some member of the B isoelectronic series.
In order to get further insight into the energetic ordering
of the states along the isoelectronic series, the electron-
nucleus attraction energy Ven, the electron-electron repulsion
energy Vee, the kinetic energy T, and the virial ratio
␩=−
Ven + Vee
T
have been calculated and reported in Table I. In parentheses
we show the statistical error in the Monte Carlo calculation
of all these quantities. As ␩ = 2 for the exact wave function,
the value of this quantity gives us additional information on
the quality of the variational wave functions obtained here.
In our calculations, this virial ratio is well reproduced.
Because of the virial theorem, the discussion of the energetic ordering of the states of a given ion can be reduced to
the behavior of Ven and Vee with E. We have obtained that
Ven presents a systematic trend with E for all of the ions
while Vee does not. Thus, as E increases, the absolute value
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154307-5
Excited states of boron
of the electron-nucleus interaction is reduced for all of the
ions considered here, except for the state 2s2p2-2S of C+
which has a smaller 兩Ven兩 value than the 2s2p2-2 P one. With
respect to the electron-electron repulsion energy we have
found that, for B, Vee decreases as E increases, whereas for
N2+ and higher Z ions the contrary holds. The ion C+ presents some oscillations on the behavior of Vee with the energy. It is the balance between Ven and Vee which gives rise to
the final ordering of the states and provides a quantitative
foundation of the Hund’s rules.
The electron-nucleus attraction is stronger in the 2s22p
2
- P state than in the 2s2p2-4 P one for all the ions considered
whereas the repulsion is weaker in the ground state only for
the positive ions. Therefore in the charged species both terms
contribute to make the 2s22p-2 P state more bound while in
the neutral atom the electronic repulsion is more intense in
the ground state. However, this is not enough to overcome
the attraction and therefore the ground state is the 2s22p-2 P
one.
The ordering obtained for the states arising from the
2s2p2 configuration deserves an especial study. First, the
quartet state is more bound than the doublet ones according
to the Hund’s rule. This is due to a greater attraction energy
in the former which compensates its greater repulsion energy
for B and C+. For the other ions, both the electron-nucleus
attraction and the electron-electron repulsion contribute simultaneously to make the doublets less bound than the
quartet.
The second Hund’s rule is concerned with the energy
ordering for terms with the same value of the total spin and
different values of the total orbital angular momentum. The
classical interpretation for this rule states that the electronelectron repulsion energy increases as the total orbital angular momentum decreases, giving rise to a higher total energy.
This is the opposite to what happens for the S and P terms.
This bigger electron-electron repulsion energy of the P state,
along with the stronger electron-nucleus attraction energy of
the S state, gives rise to the inversion of these states. More
still, for C+, the repulsion energy, which is much smaller in
the 2S state than in the 2 P one, is the only one responsible for
making the former more bound than the latter. According to
our results, the original explanation only holds for the D
states for Z 艌 7 ions which are the states with the smallest
value of the electron-electron repulsion energy. For these
ions, the 2D state presents the smallest values of the electronelectron repulsion energy and the lowest values of the Ven in
such a way that both terms contribute to lower the energy of
the 2D state with respect to the 2S and the 2 P ones. With
respect to the B atom the repulsion is stronger in the 2D state
than in the 2 P one, contrary to the classical interpretation of
the second Hund’s rule. For the C+ ion the smallest value of
the electron-electron energy is for the S term and therefore
the explanation of the Hund’s rule fails. However this interpretation holds for the P and D terms, where the lowest
orbital angular momentum, L = 1, presents the highest repulsion energy.
Thus, and as a conclusion, the electron-nucleus interaction Ven is a major determinant of the experimental ordering
of the states in the Boron isoelectronic series, and therefore
J. Chem. Phys. 122, 154307 共2005兲
is responsible for making the Hund’s rule to hold or to fail,
with the only exception in C+ previously commented.
Most of the results reported in the literature are for excitation energies, and the ground state energy is not usually
given. Therefore a direct comparison between the energies is
not possible. In order to compare our results with others
obtained by using a different theoretical approach, we give in
Table II the excitation energy E, in cm−1, as compared with
the results of Ref. 3 obtained in a relativistic many-body
calculation 共Ermb兲. The values that can be considered as exact, 共Eexact兲 taken from Ref. 36, are also shown. In general,
the agreement between the different excitation energies is
better for low Z, and the biggest discrepancies are found for
F4+ and Ne5+.
B. Moments and densities
The moments 具tn典, with n = −2 , . . . , 4, and where t stands
for r, r12, and R, of the single-particle density and of both the
intracule and extracule two-body densities, are reported in
Tables III–V, respectively. In these tables we also give the
value of the corresponding density at the origin. The moments of negative order inform us about the behavior of the
corresponding density near its origin, whereas the moments
of positive order of a density distribution give us information
at long distances. We will discuss here the average size of the
systems in terms of the square root of the moment of order 2.
The value of the single-particle density at the origin ␳共0兲
for the ground state is greater than the one of many of the
excited states studied. However the precision in our Monte
Carlo calculation is not sufficient to compare adequately the
excited states among themselves. The same behavior is
found for the moment of order −2. The trend of the expectation value 具r−1典 has been previously discussed as it gives
the electron-nucleus potential energy Ven = −Z具r−1典.
The ground state of B has the smallest size in this atom.
As Z increases, first the state 2s2p2-4 P and finally all the
states from the 2s2p2 configuration become smaller than the
ground state. For these positive ions, the smallest size corresponds to the quartet excited state 4 P, but the difference between states of different spin diminishes as the nuclear
charge increases. This does not imply a stronger electronnucleus attraction due to shielding effect and redistribution
of the electrons as we shall see later.
The results reported in Table III for the different ions
show that the excited state 2S presents a more diffuse electronic charge distribution than any other state. Besides, a
comparison between the excited states with spin S = 1 / 2
shows that the higher the total angular momentum the
smaller is the root mean square radius, i.e., the atomic size
decreases as the total angular momentum increases.
In Table IV we report the values corresponding to the
intracule density h共r12兲. As for the single-particle density, the
highest value of the intracule at the origin corresponds to the
ground state. For the doublet excited states, h共0兲 decreases
from the 2S state to the 2D and then to the 2 P one. The
smallest value of h共0兲 corresponds to the quartet state 4 P for
N2+ and higher ions, but takes similar values to those of the
excited 2 P state for B and C+. The moments of negative
Downloaded 22 Dec 2005 to 150.214.68.115. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
154307-6
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
TABLE II. Excitation energy, in cm−1, of some states of the boron isoelectronic series obtained in this work 共E兲
as compared with those that can be considered as exact 共Eexact兲 taken from Ref. 36 and with those obtained from
a relativistic many-body calculation 共Ermb兲 共Ref. 3兲. In parentheses we show the statistical error in the Monte
Carlo calculation.
Z
2s2p2 共4 P兲
2s2p2 共2D兲
2s2p2 共2S兲
2s2p2 共2 P兲
5
Ermb
E
Eexact
29 699
28 358共23兲
28 877
46 783
48 320共22兲
47 857
6
Ermb
E
Eexact
43 783
42 644共42兲
43 035
73 693
74 819共41兲
74 932
98 361
96 788共40兲
96 494
108 950
110 484共44兲
110 652
7
Ermb
E
Eexact
58 006
56 734共65兲
57 277
100 015
101 222共43兲
101 027
131 688
131 970共43兲
131 003
144 325
145 995共44兲
145 949
8
Ermb
E
Eexact
72 315
70 671共40兲
71 328
126 103
126 791共40兲
126 942
164 788
164 628共42兲
164 367
179 201
179 992共42兲
180 644
9
Ermb
E
Eexact
86 804
85 222共60兲
86 428
152 171
152 184共63兲
152 884
197 866
197 132共42兲
197 565
213 914
213 878共62兲
215 193
10
Ermb
E
Eexact
101 508
99 181共64兲
99 769
178 024
177 533共62兲
179 003
231 081
229 638共64兲
230 851
248 690
247 545共42兲
249 839
−2
−1
order 具r12
典 and 具r12
典 do not follow the same trend as h共0兲,
and have been previously discussed in the case of the moment of order −1 because it coincides with the electronelectron repulsion energy, Vee.
The ground state of B and C+ presents the less extended
interelectronic distribution as compared with the other states.
However this situation changes gradually for the other ions,
for which first the excited states 4 P and 2 P and then all the
others present a smaller root mean square radius than the
ground state. A comparison among the excited states shows
that for B the largest average electron-electron separation
takes place in the 2 P state, and, for the positive ions, in the 2S
one. The 2 P state presents the smallest mean interelectronic
distance for the excited states of the positive ions whereas
the 4 P and the 2D lie in between.
The values corresponding to the extracule density d共R兲
are shown in Table V. Again the highest value of the extracule density at the origin corresponds to the ground state. For
the excited states the greatest value is for the 2S state, then
for the 2D and the 4 P ones, and, finally, the smallest value is
for the 2 P.
With respect to the moments 具Rn典, the greatest value of
−1
具R 典 and the smallest value of 具R2典 in the boron atom take
place in the ground state, while for the rest of the ions this
corresponds to the 2s2p2-4 P excited state. In general, the
behavior is the same for all the ions considered, i.e., the
moments of positive order of the 2s2p2-4 P state are clearly
smaller than those of the other states from the same configuration with spin S = 1 / 2, whereas the contrary holds for the
moments of negative order. This indicates that electrons in
72 475
72 409共31兲
72 543
the state with the highest spin seek for a position opposite
with respect to the nucleus, as compared with electrons in
states with S = 1 / 2.
A deeper insight into the differences on the electronic
distribution of the different states along the isoelectronic series can be obtained by means of the one- and two-electron
distributions. These density functions provide a fully local
information 共spherically averaged兲 on the electron cloud and
can be used to elucidate the differences found for the different systems here considered. In Figs. 1–4 we plot the difference functions ⌬1−2 f = 4␲t2关f 1共t兲 − f 2共t兲兴 where t stands for r,
r12 or R, and f 1 共f 2兲 for either the single-particle or intracule
or extracule densities of the state 1 共2兲 of the ion under study.
For example, ⌬2S-2P␳共r兲 is the difference between the radial
one-body density of the 2S and 2 P states.
In Fig. 1 we compare the ground and the first excited
state. Both of them have a total orbital angular momentum
L = 1. The ground state is a doublet coming mainly from the
configuration 2s22p while the excited state is a quartet arising from the 2s2p2 configuration. In Figs. 1共a兲–1共c兲 we plot
the difference function for the single-particle, the intracule,
and the extracule density, respectively. The results plotted in
Fig. 1共a兲, ⌬4P-2P共gs兲␳共r兲, show that the single-particle density
in the ground state reaches bigger values than in the 4 P one
at short distances from the nucleus, i.e., in the ground state
there exists a smaller shielding that leads to a stronger
electron-nucleus attraction; the 4 P state concentrates its
charge at intermediates distances in such a way that in all
cases the absolute value of Ven is smaller than in the ground
state. At larger distances there appears a second positive
maximum for B and C+ from where the difference function
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154307-7
J. Chem. Phys. 122, 154307 共2005兲
Excited states of boron
TABLE III. Value at the origin and some radial moments of the single-particle density for the different states of the members of the boron isoelectronic series
here studied. In parentheses we give the statistical error in the last figure.
␳共0兲
具r−2典
具r−1典
71.9共2兲
70.3共4兲
70.4共3兲
93.5共2兲
92.2共5兲
92.0共3兲
71.3共3兲
92.7共4兲
具r典
具r2典
具r3典
具r4典
11.408共1兲
11.330共2兲
11.280共1兲
6.7107共4兲
6.7921共4兲
7.1139共5兲
15.280共3兲
15.847共3兲
17.904共4兲
44.46共2兲
47.54共2兲
59.08共2兲
153.76共9兲
169.68共9兲
235.5共2兲
11.205共2兲
7.917共1兲
25.41共1兲
121.2共1兲
762共2兲
4.9275共3兲
4.8994共2兲
5.0260共3兲
5.2968共4兲
5.1055共3兲
7.931共1兲
7.918共1兲
8.472共2兲
10.003共3兲
8.818共1兲
16.036共4兲
16.239共4兲
18.400共6兲
26.02共2兲
19.672共5兲
37.85共1兲
39.17共2兲
47.93共3兲
86.8共1兲
52.47共3兲
3.9171共2兲
3.8557共2兲
3.9176共2兲
3.9690共2兲
3.9605共2兲
4.9205共6兲
4.8002共5兲
4.9993共7兲
5.1769共7兲
5.1367共6兲
7.725共2兲
7.545共2兲
8.100共2兲
8.625共2兲
8.474共2兲
14.099共5兲
13.884共5兲
15.442共7兲
16.984共7兲
16.444共6兲
3.2622共1兲
3.1895共2兲
3.2266共2兲
3.2572共2兲
3.2548共2兲
3.3750共4兲
3.2409共4兲
3.3375共4兲
3.4205共5兲
3.4150共4兲
4.3519共8兲
4.1459共9兲
4.362共1兲
4.553共1兲
4.542共1兲
6.508共2兲
6.193共2兲
6.678共3兲
7.112共3兲
7.086共3兲
2.7930共1兲
2.7209共1兲
2.7475共1兲
2.7701共1兲
2.7646共1兲
2.4533共2兲
2.3359共3兲
2.3943共3兲
2.4453共3兲
2.4349共3兲
2.6813共5兲
2.5210共5兲
2.631共1兲
2.7276共6兲
2.7123共6兲
3.396共1兲
3.177共1兲
3.388共5兲
3.563共1兲
3.544共2兲
2.4447共1兲
2.3766共1兲
2.3947共1兲
2.4150共1兲
2.4054共1兲
1.8679共2兲
1.7703共2兲
1.8051共2兲
1.8447共2兲
1.8281共2兲
1.7732共4兲
1.6552共3兲
1.7123共3兲
1.7796共4兲
1.7537共4兲
1.9479共6兲
1.8034共5兲
1.8967共6兲
2.0142共8兲
1.9693共7兲1
B
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
208共1兲
201.0共7兲
201.4共6兲
202.1共7兲
199.6共5兲
194共1兲
189.0共6兲
188.5共6兲
188.8共7兲
186.9共5兲
16.911共2兲
16.853共3兲
16.796共2兲
16.768共3兲
16.752共3兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
314共1兲
304.2共8兲
305共1兲
306共1兲
305共2兲
255.4共7兲
249.2共6兲
250共1兲
250共1兲
250共2兲
19.652共4兲
19.601共3兲
19.549共3兲
19.526共3兲
19.505共3兲
2s2p2
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
461共3兲
446共3兲
444共4兲
441共2兲
439共2兲
331共2兲
324共2兲
324共3兲
321共1兲
319共1兲
22.411共3兲
22.363共3兲
22.305共3兲
22.275共3兲
22.259共3兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
638共3兲
614共2兲
609共2兲
609共2兲
605共2兲
413共2兲
401共1兲
399共1兲
399共1兲
397共1兲
25.166共3兲
25.110共3兲
25.048共4兲
25.017共2兲
25.010共3兲
C+
2
126.3共4兲
124.6共5兲
124.7共4兲
124.1共4兲
123.8共4兲
137.8共4兲
136.2共5兲
136.0共5兲
135.2共4兲
135.1共4兲
14.154共2兲
14.091共2兲
14.042共2兲
13.993共2兲
14.000共2兲
N2+
2
O3+
F4+
Ne5+
approaches to zero. In Fig. 1共b兲 we plot the results for the
intracule density. The short range behavior of this difference
function can be understood as a Fermi hole due to the fact
that in the quartet the spins of the electrons are parallel. For
boron, this hole is big enough to make the repulsion energy
bigger in the ground than in the first excited state. However
the relative importance of this hole decreases rapidly as the
nuclear charge increases, in such a way that for Z 艌 7 both
the repulsion energy and the attraction energy contribute to
make the 2 P state the ground state of these five-electron systems. Again, for larger interelectronic distances there appears
a second positive maximum for B and C+ from where the
difference function approaches zero. Finally, a complementary information on the spatial distribution of the electrons in
those states is provided by the difference function of the
extracule density 关Fig. 1共c兲兴. The positive region near the
origin indicates that there is a tendency of the electrons in the
4
P state to be at opposite positions with respect to the
nucleus as compared with the 2 P. In doing so, a reduction of
the electron-electron repulsion is favored without raising the
value of the electron-nucleus potential energy.
To study the effect of the spin within the same configuration we compare in Fig. 2 the excited states 4 P and 2 P from
the configuration 2s2p2. In Fig. 2共a兲 we plot the difference
function ⌬2P-4P␳共r兲. For B it is oscillating, nearly flat, for
small values of r. Then it takes negative values, and finally,
there appears a positive region from where it approaches
zero. This behavior makes the electron-nucleus attraction energy stronger in the quartet than in the doublet. For all the
positive ions the structure of the difference function is the
same: two negative regions followed by a region where the
difference is positive. Thus the function vanishes asymptotically with positive values. This structure leads to both a
Downloaded 22 Dec 2005 to 150.214.68.115. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
154307-8
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
TABLE IV. Value at the origin and some radial moments of the intracule density for the different states of the members of the boron isoelectronic series here
studied. In parentheses we give the statistical error in the last figure.
h共0兲
具r−2
12 典
具r−1
12 典
3.557共2兲
3.459共2兲
3.462共2兲
17.47共3兲
17.14共2兲
17.15共3兲
3.452共2兲
16.97共3兲
具r12典
具r212典
具r312典
具r412典
7.6877共4兲
7.6097共4兲
7.5043共4兲
22.328共2兲
22.794共2兲
23.789共2兲
66.09共1兲
69.19共1兲
76.92共2兲
237.26共7兲
254.74共7兲
306.2共1兲
991.8共5兲
1090.7共5兲
1431.8共8兲
7.3478共5兲
26.207共5兲
102.25共5兲
535.7共5兲
3527共6兲
16.217共1兲
16.2735共9兲
16.534共1兲
17.313共2兲
16.291共1兲
34.268共5兲
34.680共4兲
36.252共6兲
41.31共1兲
35.033共5兲
86.94共2兲
88.95共2兲
98.84共3兲
124.94共7兲
91.56共2兲
254.92共9兲
263.99共9兲
303.0共2兲
460.1共5兲
279.8共1兲
12.8071共6兲
12.7260共6兲
12.7614共8兲
12.4285共8兲
12.4979共7兲
21.206共2兲
21.041共2兲
21.319共3兲
21.746共3兲
20.292共3兲
42.009共8兲
41.746共8兲
43.02共1兲
44.89共1兲
39.638共9兲
95.97共3兲
95.66共3兲
100.76共4兲
108.26共4兲
89.80共3兲
10.6177共5兲
10.4814共5兲
10.4499共6兲
10.4687共6兲
10.2083共6兲
14.514共2兲
14.204共2兲
14.205共2兲
14.382共2兲
13.446共2兲
23.693共4兲
23.057共5兲
23.253共5兲
23.954共5兲
21.237共5兲
44.55共1兲
43.19共1兲
44.11共1兲
46.47共2兲
38.81共1兲
9.0590共4兲
8.9123共4兲
8.8609共5兲
8.8703共4兲
8.6314共5兲
10.530共1兲
10.234共1兲
10.173共1兲
10.279共1兲
9.565共1兲
14.597共3兲
14.061共3兲
14.047共5兲
14.410共3兲
12.689共3兲
23.290共6兲
22.280共6兲
22.49共2兲
23.495共7兲
19.478共7兲
7.9092共4兲
7.7662共3兲
7.6989共4兲
7.7128共4兲
7.4842共4兲
8.0080共9兲
7.7536共7兲
7.6576共8兲
7.7516共9兲
7.1658共9兲
9.659共2兲
9.254共1兲
9.146共2兲
9.416共2兲
8.197共2兲
13.402共4兲
12.728共3兲
12.640共4兲
13.295共4兲
10.883共4兲
B
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
11.392共6兲
10.941共6兲
10.950共5兲
11.035共6兲
10.918共6兲
40.14共7兲
39.80共6兲
39.90共5兲
40.45共6兲
40.11共6兲
12.4522共7兲
12.5291共6兲
12.5789共6兲
12.6365共6兲
12.6819共6兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
17.875共4兲
17.086共6兲
17.154共8兲
17.32共1兲
17.096共8兲
54.78共7兲
54.64共5兲
55.09共7兲
55.83共8兲
55.28共6兲
14.7995共7兲
14.9497共7兲
15.0626共7兲
15.1615共8兲
15.2093共8兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
26.47共2兲
25.27共2兲
25.41共1兲
25.73共5兲
25.31共1兲
72.3共2兲
71.9共1兲
72.7共1兲
73.9共2兲
73.2共1兲
17.1593共8兲
17.3698共8兲
17.5341共8兲
17.6595共8兲
17.7300共8兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
37.49共2兲
35.65共3兲
35.93共2兲
36.35共2兲
35.76共2兲
92.0共2兲
91.8共3兲
92.6共1兲
944.0共1兲
93.37共9兲
19.5057共9兲
19.7695共9兲
19.9955共9兲
20.138共1兲
20.242共1兲
C+
2
6.703共4兲
6.489共4兲
6.491共3兲
6.531共4兲
6.471共3兲
27.63共5兲
27.31共4兲
27.37共4兲
27.26共4兲
27.42共4兲
10.0814共6兲
10.0825共5兲
10.0564共5兲
9.9542共5兲
10.1167共6兲
N2+
2
O3+
F4+
Ne5+
lower value of Ven and a smaller average size in the quartet
as compared to the doublet. In Fig. 2共b兲 we plot the difference function ⌬2P-4Ph共r12兲. The behavior of this difference
function is completely different for the B atom as compared
to the positive ions. For B it presents a negative region followed by a positive one in such a way that the repulsion
energy is stronger in the quartet than in the doublet. As the
nuclear charge increases this situation reverses completely,
making the repulsion energy in the doublet greater than in
the quartet. Therefore the Fermi hole is clearly appreciable in
the positive ions but not in the neutral atom. This may be due
to the fact that in the neutral atom the average size of the
doublet is much greater than the average size of the quartet,
making the electrons to be, on average, closer in the state
with the highest spin. In Fig. 2共c兲 we plot the difference
function ⌬2P-4Pd共R兲. The behavior of this difference function
is similar for all the systems considered. It has a negative
region followed by a positive one that means that electrons
in the quartet seek for a opposite position with respect to the
nucleus as compared with the behavior in the doublet. Therefore the role of angular correlations is more important when
the spin of the electrons are parallel than in the other case.
The second Hund’s rule states that for terms coming
from the same configuration and with the same value of the
spin, the higher the orbital angular momentum L, the lower is
the energy. However this is not fulfilled for the members of
the boron isoelectronic series here studied, because in all of
the cases considered the 2S term is more bound than the 2 P
one. In order to study the role of the orbital angular momentum on the electron distribution we compare states with different L values. We compare first the states 2 P and 2D and
second the states 2S and 2 P. In Fig. 3 we show the results for
the comparison between the states 2 P and 2D. In Fig. 3共a兲 we
plot the difference function for the single-particle density.
Downloaded 22 Dec 2005 to 150.214.68.115. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
154307-9
J. Chem. Phys. 122, 154307 共2005兲
Excited states of boron
TABLE V. Value at the origin and some radial moments of the extracule density for the different states of the members of the boron isoelectronic series here
studied. In parentheses we give the statistical error in the last figure.
d共0兲
具R−2典
具R−1典
35.98共1兲
35.18共1兲
35.20共1兲
79.0共2兲
78.4共2兲
77.5共1兲
34.82共1兲
75.9共3兲
具R典
具R2典
具R3典
具R4典
16.4230共9兲
16.3989共8兲
16.0170共9兲
10.3072共8兲
10.4110共3兲
11.039共1兲
14.037共3兲
14.396共3兲
16.577共4兲
23.264共7兲
24.259共8兲
30.86共1兲
45.30共2兲
48.01共2兲
68.19共5兲
15.234共1兲
12.948共2兲
25.26共1兲
66.43共7兲
219.3共4兲
7.5042共5兲
7.4222共4兲
7.7119共5兲
8.2759共8兲
8.1674共5兲
7.296共1兲
7.165共1兲
7.880共1兲
9.678共3兲
8.878共1兲
8.509共2兲
8.318共2兲
9.854共3兲
14.707共9兲
11.762共3兲
11.501共4兲
11.218共4兲
14.578共8兲
27.68共3兲
18.255共8兲
5.9411共3兲
5.8042共3兲
5.9793共4兲
6.0834共4兲
6.2996共4兲
4.5395共6兲
4.3401共3兲
4.6689共7兲
4.9173共7兲
5.2003共7兲
4.1435共9兲
3.8791共8兲
4.404共1兲
4.868共1兲
5.175共1兲
4.371共2兲
4.007共1兲
4.841共2兲
5.668共2兲
5.971共2兲
4.9366共2兲
4.7830共2兲
4.9073共3兲
4.9662共3兲
5.1587共3兲
3.1214共3兲
2.9307共4兲
3.1238共4兲
3.2455共4兲
3.4685共5兲
2.3526共4兲
2.1396共4兲
2.3912共5兲
2.5795共6兲
2.8052共6兲
2.0456共6兲
1.8004共6兲
2.1257共7兲
2.3979共9兲
2.6297共9兲
4.2195共2兲
4.0691共2兲
4.1688共2兲
4.2099共2兲
4.3702共2兲
2.2739共3兲
2.1133共2兲
2.2453共3兲
2.3209共3兲
2.4785共3兲
1.4592共3兲
1.3056共3兲
1.4518共5兲
1.5520共3兲
1.6882共4兲
1.0797共3兲
0.9291共3兲
1.089共1兲
1.2120共4兲
1.3317共5兲
3.6888共2兲
3.5477共2兲
3.6278共2兲
3.6614共2兲
3.7957共2兲
1.7339共2兲
1.6023共2兲
1.6958共2兲
1.7514共2兲
1.8647共2兲
0.9691共2兲
0.8593共2兲
0.9500共2兲
1.0164共2兲
1.0988共2兲
0.6239共2兲
0.5297共2兲
0.6162共2兲
0.6902共2兲
0.7488共2兲
B
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
109.95共5兲
107.75共3兲
107.35共3兲
108.19共5兲
105.46共5兲
179.8共5兲
181.4共3兲
180.1共3兲
181.8共4兲
175.1共8兲
26.538共2兲
27.057共1兲
26.749共1兲
26.762共1兲
25.832共2兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
169.90共7兲
166.63共7兲
166.43共6兲
167.86共9兲
162.98共9兲
245.5共5兲
249.3共8兲
248.8共8兲
250.2共4兲
239.7共5兲
31.503共2兲
32.290共2兲
31.992共2兲
32.086共2兲
30.933共2兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
249.23共9兲
244.2共1兲
243.9共1兲
246.7共1兲
238.8共1兲
321.0共4兲
327.5共4兲
326.4共6兲
330.8共7兲
315.0共6兲
36.491共2兲
37.515共2兲
37.202共2兲
37.351共2兲
36.002共2兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
349.7共1兲
343.1共2兲
343.0共1兲
347.0共1兲
334.9共2兲
409共2兲
420共2兲
415.1共6兲
421.7共6兲
401.1共6兲
41.441共2兲
42.698共2兲
42.394共2兲
42.572共2兲
41.069共2兲
C+
2
65.98共2兲
64.79共4兲
64.62共4兲
64.95共3兲
63.63共2兲
123.9共3兲
124.6共5兲
123.6共3兲
123.3共3兲
119.2共2兲
21.511共1兲
21.757共1兲
21.427共1兲
21.136共1兲
20.682共1兲
N2+
2
O3+
F4+
Ne5+
This function presents a similar behavior for all the ions, i.e.,
two minima and two maxima. The first minimum appears at
short distances in such a way that the shielding of the electrons in the 2D state is smaller than in the 2 P one. For B the
differences are much greater than for the rest of ions, which
may be due to the fact that in this atom the state 2 P lies in the
continuum. In Figs. 3共b兲 and 3共c兲 we plot the difference
function for both the intracule and the extracule density, respectively. Again there exists a different behavior in the intracule density between B and the other ions. For the positive
ions there is a greater probability for smaller interelectronic
distances in the 2 P state than in the 2D one, and the contrary
holds for B. The extracule density shows the same behavior
for all the systems, i.e., a greater probability to find the electrons at opposite positions with respect to the nucleus in the
2
D state than in the 2 P one.
In Fig. 4 we plot the difference functions between the
states 2S and 2 P. We have used this order because 2S should
be less bound than 2 P according the Hund’s rules. The comparison with Fig. 3 will provide a deeper insight on the energetic ordering of the states. The single-particle difference
function is shown in Fig. 4共a兲. This difference function is
positive at short distances for all of the systems studied,
making the 2S state more attractive than the 2 P one, except
for C+ for which the great negative region compensates for
the positive one. Figure 4共b兲 represents the difference function for the intracule density. Although it is positive for small
interelectronic distances, making that the probability of electrons to meet is greater in the 2S state, this is not sufficient to
overhead the magnitude of the negative region, which makes
Vee greater in the 2 P state than in the 2S one, contrary to the
classical explanation of the Hund’s rule. These differences on
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154307-10
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
FIG. 1. Difference of the radial one-body 共a兲, intracule 共b兲, and extracule 共c兲
densities between 4 P 共first excited state兲 and 2 P 共ground state兲 for some
members of the boron isoelectronic series.
FIG. 2. Difference of the radial one-body 共a兲, intracule 共b兲, and extracule 共c兲
densities between the excited states 2 P and 4 P for some members of the
boron isoelectronic series.
the charge distribution make the 2S state more attractive and
less repulsive than the 2 P, leading to a bigger binding energy
in the former.
correlation and ␶rជ = 0 stands for either noncorrelated variables in the statistical sense or for independent variables. For
atomic systems, statistically noncorrelated variables mean
that the position vectors of any pair of particles are, on average, orthogonal, while independent variables mean that the
diagonal term of the two-body density matrix is the product
of the one-body distribution functions.
The angular correlation coefficient is related to quantities that may be obtained from experimental measurements,
such as the diamagnetic susceptibility and the dipole oscillator strength distribution,24 or the x-ray and/or high-energy
electron scattering intensities.39 Similar definitions are done
in momentum space. Here 具pជ 1 · pជ 2典 is related to the specific
mass shift correction to the energy and is sensitive to the
quality of the variational wave function. More general correlation coefficients can be defined38 by using different probe
functions in order to characterize and measure the statistical
C. Angular correlations
A systematic analysis of the angular correlations between the electrons can be done by using the angular correlation factor ␶rជ, introduced by Kutzelnigg, Del Re, and
Berthier,38
2
␶rជ =
具rជi · rជ j典
兺
i⬎j
共N − 1兲
兺i 具r2i 典
.
共8兲
This quantity is bounded in magnitude by unity, −1
艋 ␶rជ 艋 1. ␶rជ = 1 共␶rជ = −1兲 means perfect positive 共negative兲
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154307-11
Excited states of boron
J. Chem. Phys. 122, 154307 共2005兲
FIG. 3. Difference of the radial one-body 共a兲, intracule 共b兲, and extracule 共c兲
densities between the excited states 2 P and 2D for some members of the
boron isoelectronic series.
FIG. 4. Difference of the radial one-body 共a兲, intracule 共b兲, and extracule 共c兲
densities between the excited states 2S and 2 P for some members of the
boron isoelectronic series.
correlation. In particular, some scalar and vectorial functions
have been used to study systematically the statistical correlation and its dependence on the nuclear charge, degree of
excitation, and angular momentum at different regions of the
nuclear charge for some excited states of the helium isoelectronic series.40 Radial and angular statistical correlation coefficients have been also calculated for the ground state of
some members of the beryllium isoelectronic series41,42 focusing on the shell effects and the role of the nuclear charge.
These angular correlation factors in both position and
momentum spaces along with some other two-body properties such as 具rជ1 · rជ2典 = 具兺i⬎jrជi · rជ j典 are reported in Table VI. In
position space, and for all the ions considered, our results
indicate that all the states present negative angular correlation, except the 2s2p2-2 P which has a negative angular correlation for B and a positive value for all the positive ions.
The most negative values correspond to the 4 P state which
may be due to the fact that this is the term with the highest
spin. The value of ␶rជ does not present a systematic behavior
along the isoelectronic series. Thus, for the ground state, it
increases slightly as Z does; for the first excited state it decreases from B to C+, but then it remains practically constant
for the rest of ions. For the doublets 2D and 2S this coefficient does not modify appreciably its value along the isoelectronic series. For the excited state 2 P, ␶rជ increases noticeably;
this means that, for this state electrons tend to be at the same
side of the nucleus as Z increases.
In momentum space the behavior is completely different.
For the ground state, ␶ pជ is positive for B, has a null correlation for C+, and is negative for all the rest of ions. This
coefficient is negative for the rest of states and for all the
ions. An important characteristic is that for all the states and
all the ions, this coefficient decreases monotonically as the
nuclear charge increases.
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154307-12
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
TABLE VI. Several two-body position and momentum properties for for the different states of the members of the boron isoelectronic series studied here. In
parentheses we give the statistical error in the last figure.
具rជ1 · rជ2典
具 ␶ r典
具p2典
具p212典
具P2典
具pជ 1 · pជ 2典
具 ␶ p典
−2.486 5共6兲
−2.902 6共6兲
−2.654 3共8兲
−0.081 37共2兲
−0.091 58共2兲
−0.074 13共2兲
49.41共1兲
49.05共1兲
48.94共1兲
197.09共5兲
196.40共6兲
195.84共5兲
49.55共1兲
48.99共1兲
48.91共1兲
0.279共2兲
−0.109共2兲
−0.047共2兲
0.002 82共2兲
−0.001 11共2兲
−0.000 48共2兲
−0.304共1兲
−0.005 99共2兲
48.72共2兲
194.47共6兲
48.82共2兲
0.203共2兲
0.002 08共2兲
299.38共9兲
299.42共9兲
298.3共1兲
296.9共1兲
296.4共1兲
74.85共2兲
74.09共2兲
73.87共2兲
73.85共3兲
73.75共3兲
0.001共5兲
−0.762共4兲
−0.708共3兲
−0.38共2兲
−0.360共3兲
0.000 006共31兲
−0.005 12共3兲
−0.004 77共2兲
−0.002 6共2兲
−0.002 43共3兲
424.7共1兲
425.4共1兲
423.4共1兲
421.9共1兲
421.0共1兲
105.72共3兲
104.61共3兲
104.14共3兲
104.06共4兲
103.91共4兲
−0.447共4兲
−1.744共5兲
−1.714共4兲
−1.43共1兲
−1.339共5兲
−0.002 11共2兲
−0.008 27共3兲
−0.008 16共2兲
−0.006 82共5兲
−0.006 40共3兲
571.5共2兲
573.6共2兲
571.3共2兲
569.8共2兲
568.6共2兲
141.81共5兲
140.36共4兲
139.82共4兲
139.76共4兲
139.47共5兲
−1.063共5兲
−3.031共7兲
−3.007共6兲
−2.68共1兲
−2.684共7兲
−0.003 74共2兲
−0.010 68共2兲
−0.010 64共2兲
−0.009 51共5兲
−0.009 53共2兲
741.7共2兲
745.2共2兲
742.1共2兲
739.8共2兲
738.8共2兲
183.59共5兲
181.70共6兲
180.92共5兲
180.75共6兲
180.46共6兲
−1.85共1兲
−4.591共8兲
−4.607共7兲
−4.20共2兲
−4.24共1兲
−0.005 01共4兲
−0.012 48共2兲
−0.012 57共2兲
−0.011 49共6兲
−0.011 62共3兲
934.2共2兲
938.6共3兲
934.7共3兲
932.2共2兲
931.9共3兲
230.75共6兲
228.17共7兲
227.18共8兲
226.95共5兲
226.65共7兲
−2.805共8兲
−6.48共1兲
−6.50共1兲
−6.10共1兲
−6.32共2兲
−0.006 04共2兲
−0.014 01共3兲
−0.014 11共2兲
−0.013 26共7兲
−0.013 75共5兲
B
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
2
2s 2p
2s2p2
2s2p2
2s2p2
2s2p2
共 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
−0.762 1共2兲
−0.920 1共2兲
−0.661 0共2兲
−0.519 1共2兲
0.127 2共2兲
−0.077 44共2兲
−0.095 84共2兲
−0.066 10共2兲
−0.050 14共2兲
0.012 38共2兲
105.94共3兲
105.48共3兲
104.99共3兲
104.77共3兲
104.58共4兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
−0.507 1共1兲
−0.620 3共1兲
−0.427 5共1兲
−0.349 9共1兲
0.107 1共2兲
−0.075 12共2兲
−0.095 70共2兲
−0.064 04共2兲
−0.051 14共2兲
0.015 67共2兲
142.35共5兲
141.87共4兲
141.33共4兲
141.10共4兲
140.81共5兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
−0.358 6共1兲
−0.445 2共1兲
−0.298 0共1兲
−0.248 8共1兲
0.087 1共1兲
−0.073 09共2兲
−0.095 30共2兲
−0.062 23共2兲
−0.050 87共2兲
0.017 90共2兲
184.51共4兲
183.99共6兲
183.22共5兲
182.85共5兲
182.58共6兲
2s22p
2s2p2
2s2p2
2s2p2
2s2p2
共2 P兲
共4 P兲
共2D兲
共2S兲
共2 P兲
−0.268 09共7兲
−0.336 13共5兲
−0.218 57共7兲
−0.186 49共7兲
0.073 28共7兲
−0.071 76共2兲
−0.094 93共2兲
−0.060 54共2兲
−0.050 55共2兲
0.020 04共2兲
232.15共6兲
231.41共7兲
230.43共7兲
230.00共5兲
229.81共6兲
C+
2
−1.270 7共3兲
−1.504 6共3兲
−1.183 0共4兲
−0.650 3共4兲
0.119 3共4兲
−0.080 10共2兲
−0.095 02共2兲
−0.069 82共2兲
−0.032 51共3兲
0.006 76共2兲
74.85共2兲
74.47共2兲
74.22共3兲
74.04共3兲
73.93共3兲
N2+
2
O3+
F2+
Ne5+
We also report in Table VI some correlated momentum
expectation values which are directly obtained in the Monte
Carlo calculation. In particular, we show the expectation val2
ues 具p2典 共the double of the kinetic energy兲, 具p12
典, 具P2典, and
具pជ 1 · pជ 2典. These two-body expectation values and 具p2典 are not
linearly independent. For five electron atoms, the following
relationships can be straightforwardly obtained:
2
典,
具p2典 = 21 具P2典 + 81 具p12
2
典.
具pជ 1 · pជ 2典 = 具P2典 − 41 具p12
共9兲
␶ pជ = 0, then the contributions to the total kinetic energy coming from the the center of mass of the electron pair and that
from the relative motion are identical. If the system presents
positive angular correlations in momentum space, i.e., ␶ pជ is
positive, then the largest contribution to the kinetic energy
comes from the center of mass movement of the electrons
共ground state of B兲, whereas the interelectronic movement of
the electrons is more important if ␶ pជ is negative 共all the ions
and states except the ground state of both B and C+兲.
共10兲
The first of these two equations shows how the center of
mass and the interelectronic moments of order 2 contribute
to the kinetic energy, and therefore to the total energy due to
the virial theorem. The second one give us information about
the relative importance of the two contributions to the kinetic
energy. If angular correlations in momentum space are zero,
D. Electronic correlations
In order to study the effects of the electronic correlations, the correlated results are compared with the noncorrelated ones, typically obtained in a Hartree–Fock calculation.
For the systems considered here, this can be done for all of
the states studied except for the 2s2p2-2S state of the boron
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154307-13
Excited states of boron
FIG. 5. Difference between the correlated and uncorrelated radial intracule
共a兲 and extracule 共d兲 densities for the ground state of some members of the
boron isoelectronic series.
atom because this is not the lowest 2S state in the spectrum of
this atom. In this work we have used the optimized effective
potential method, 共see, e.g., Ref. 33兲 to obtain the noncorrelated results. The ground state energy obtained from this
method for the boron atom differs less than 0.3 mhartree
from the Hartree–Fock value.
The correlation energy of the ground state of the boron
isoelectronic series increases from 0.125 hartree 共B兲 to
0.183 hartree 共Ne5+兲. For the excited states this correlation
energy is smaller and nearly constant along the isoelectronic
series. Thus it varies between 0.07 and 0.08 hartree for the
2s2p2-4 P, between 0.12 and 0.13 hartree for both the 2s2p2
-2D and 2S states, and between 0.14 and 0.16 hartree for the
2s2p2-2 P state. The difference between the quartet state and
the doublet ones is due to the Fermi statistical hole that is
present in the quartet state and not in the doublet one, making the effect of electronic correlations to be smaller in the
4
P state. The decrease in the total energy when electronic
correlations are included is due, mainly, to the electronic
repulsion energy, except for the ground state of B and C+ for
which the increment in the attraction energy 共greater in B
than in C+兲 is comparable to the decrease in the repulsion
energy when the correlated wave functions are used. Therefore, and as it could be expected, the main modifications in
the electronic distribution appear in the two-body densities,
i.e., in the intracule and extracule densities.
In Fig. 5 we plot the difference function ⌬f共t兲 defined as
J. Chem. Phys. 122, 154307 共2005兲
FIG. 6. Difference between the correlated and uncorrelated radial intracule
共a兲 and extracule 共d兲 densities for the ground state and some excited states of
the N2+ ion.
⌬f共t兲 = 4␲t2关f c共t兲 − f u共t兲兴,
where f c 共f u兲 stands for the correlated 共uncorrelated兲 density.
This quantity gives the effect of the electronic correlations
on f. In Fig. 5共a兲 we show the results obtained for the intracule density and in Fig. 5共b兲 those for the extracule density
of the ground state of the atoms of the boron isoelectronic
series here studied.
The structure of this difference function for the intracule
density is not as simple as in, for example, the He atom, for
which a negative region with only one minimum is followed
by a positive region with a maximum and then it vanishes
asymptotically. In all of the cases studied here, the negative
region of the intracule difference function presents two
minima and one local maximum in between these minima.
For higher values of r12, the difference function changes
sign, reaches its global maximum, and then tends to zero
monotonically from positive values. The relative importance
of the Coulomb hole increases as the nuclear charge does.
In Fig. 5共b兲 we plot the difference extracule density
function. It is positive at low values of the variable and negative for high R values. This means that electrons seek for
opposite positions with respect to the nucleus when electronic correlations are included in the wave function. In this
way the average interelectronic distance increases, diminishing the repulsion energy without increasing the averaged
electron-nucleus distance, i.e., the absolute value of the attraction energy does not decrease.
A study of the excited states is shown in Fig. 6 for N2+,
which is representative for the rest of the ions. The data of
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154307-14
J. Chem. Phys. 122, 154307 共2005兲
Gálvez, Buendía, and Sarsa
the corresponding ground state are also included to have a
reference to compare with. The difference function for the
intracule function 关Fig. 6共a兲兴 is less important for the 4 P state
because the Fermi statistical correlation, which is also
present in the noncorrelated wave function, keeps the electrons apart, diminishing the effect of electronic correlations.
This Coulomb hole is much more important for the excited
2
P state and takes similar values for the other two doublet
states. The only exception we have found is for C+, for which
the 2S state presents a more important difference function
than for the rest of excited states. In Fig. 6共b兲 we plot the
difference extracule function for the same states. In this case
the doublet states present a positive region followed by a
negative one, which means that electrons in the doublets
seek an opposite position with respect to the nucleus when
electronic correlations are considered. As this effect has been
partially accounted in the quartet state due to the Fermi statistical correlation, the difference function ⌬d共R兲 is less important in this state than in the doublet ones. In fact, the
quartet state presents two small positive regions, following
each one of them by a small negative one. Therefore, in the
quartet state, electrons increase the extracule density at R
= 0, but, on average, it decreases at small R values. This
behavior is different to that found for doublet spin states,
showing the importance of the spin on the correlations.
IV. CONCLUSIONS
The ground state and some low-lying excited states of
the boron atom and some members of its isoelectronic series
have been studied. We have considered the ground state 共2 P兲
and the quartet 共4 P兲 and doublets 共2S , 2 P , 2D兲 excited states
coming from the 2s2p2 configuration. Accurate, explicitly
correlated wave functions constituted by a Jastrow-type correlation factor and a multideterminant expansion have been
obtained for the different states. Starting from those wave
function the total kinetic and potential energies have been
calculated. One- and two-body properties as the singleparticle density, the intracule and extracule densities, and
some of their radial moments are reported here. Several aspects of the electronic structure of these systems, the energetic ordering of the states, and the effect of electronic correlations have been analyzed in terms of these quantities. All
the calculations of this work have been performed by using
the variational Monte Carlo method.
The 2s22p-2 P ground state presents the lowest value of
the electron-nucleus attraction energy as well as the lowest
electron-electron repulsion energy, except for the neutral
atom where the repulsion is smaller in the first excited state,
2s2p2-4 P. An analysis of the single-particle density reveals
that in the ground state it presents a higher concentration of
the electron charge density at short distances from the
nucleus leading to a bigger binding. A comparison of the
intracule density of the ground state and the quartet reveals
the presence of the Fermi hole induced by the different spin
coupling of these two states. In spite of this hole the
electron-electron repulsion energy is smaller in the 2s22p
-2 P ground state than in the first excited state, 2s2p2-4 P,
except for the boron atom.
With respect to the terms coming from the 2s2p2 configuration, the most bound term is, according to the Hund’s
rule, the quartet 共S = 3 / 2兲. This is due to the deepest electronnucleus attraction of this state as compared with the others.
Contrary to the traditional interpretation of this rule, the electronic repulsion is higher in the quartet for B and C+. However this is not enough to overcome the effect of the
electron-nucleus interaction. For the other ions considered
here, both terms of the potential energy contribute in the
same direction to make more bound the quartet states. These
values are consistent with the differences found in the singleparticle density that shows a higher charge concentration in
the quartet at short distances. The Fermi-hole appears clearly
in the positive ions but not in the neutral atom. Finally, the
electrons are more concentrated at opposite positions with
respect to the nucleus in the quartet than in the doublets.
With respect to the energy of the states arising from the same
configuration with the same total spin, the Hund’s rule is
only fulfilled by the L = 2 term. The classical explanation of
this fact, assuming higher repulsion for lower L values, only
holds for Z 艌 7 ions. We have found that the deeper electronnucleus attraction energy is finally responsible for the relative energy of these states. With respect to the S and P states,
the electron repulsion is smaller in the former 共contrary to
the previous hypothesis兲 giving rise to the Hund’s rule not
being fulfilled here. As before these values are consistent
with the differences found in the electronic distribution for
these states along the isoelectronic series.
The angular correlation coefficients show that most of
the states considered here present negative angular correlation in both position and momentum space. The term with
the most negative angular correlation in position state is that
with the highest total spin value. In momentum space, the
angular correlation factor takes more negative values as the
nuclear charge increases.
The effect of the electronic correlations has been systematically analyzed. In general, when the correlations are included the Coulomb hole appears and the electron-electron
repulsion is reduced. The effect of the correlations is smaller,
as it could be expected, for the 2s2p2-4 P state because of the
Fermi hole. The 2s2p2-2 P state presents the most important
Coulomb hole, except for the C+ ion for which the 2S state
has a bigger hole. In general, the correlations tend to favor
opposite positions with respect to the nucleus of the electronpairs, increasing in this way the averaged electron-electron
distance with minor effects on the electron-nucleus attraction
energy. The local effect of the correlations, especially in the
case of the extracule density, presents a marked dependence
on the total spin of the system.
ACKNOWLEDGMENTS
This work was partially supported by the Spanish Dirección General de Investigación Científica y Técnica
共DGICYT兲 under Contract No. PB2002-00200 and by the
Junta de Andalucía.
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