Journal of Modern Physics, 2011, 2, 421-430
doi:10.4236/jmp.2011.25052 Published Online May 2011 (http://www.SciRP.org/journal/jmp)
Copyright © 2011 SciRes. JMP
A Scheme for Calculating Atomic Structures beyond the
Spherical Approximation
Mitiyasu Miyasita1, Katsuhiko Higuchi2, Masahiko Hig uchi3
1Graduate S chool of S cience an d E ngineering, Shinshu Un i versi t y, Ueda, Japan
2Graduate Sch ool of Advanced Science of M at te r , Hiros hi ma University, Higashi-Hiroshima, Japan
3Department of Physics, Faculty of Science, Shinshu University, Matsumoto, Japan
E-mail: miyasita.mitiyasu@gmail.com
Received February 11, 2011; revised April 3, 2011; April 4, 2011
Abstract
We present a scheme for calculating atomic single-particle wave functions and spectra with taking into ac-
count the nonspherical effect explicitly. The actual calculation is also performed for the neutral carbon atom
within the Hartree-Fock-Slater approximation. As compared with the conventional atomic structure of the
spherical approximation, the degenerate energy levels are split partially. The ground state values of the total
orbital and spin angular momenta are estimated to be both about unity, which corresponds to the term 3P in
the LS-multiplet theory. This means that the nonspherical effect may play an essential role in the description
of the magnetization caused by the orbital polarization.
Keywords: Nonspherical Distribution of Electrons, Spherical Approximation, Orbital Polarization, Atomic
Structure, Carbon Atom
1. Introduction
Let us start with revisiting the conventional atomic
structures. We consider the isolated neutral atom with the
atomic number
Z
. Neglecting the relativistic effects,
the Schrödinger Equation for the stationary state is given
by
ˆ
H
E  (1)
with

2
1,
21 2
ˆ
2
ZZ
i
iij
iij
ij
Z
Hr

 



rr
, (2)
where i
r and i
r stand for the position of the ith elec-
tron and its magnitude, respectively, and where the
atomic unit is used. Equation (1) can be numerically
solved only in small atomic systems, but in larger atomic
systems we have to utilize the theories to reduce Equa-
tion (1) into the effective single-particle Equation such as
the Hartree [1], Hartre-Fock [2] and Kohn-Sham [3,4]
Equations, etc. The single-particle Equation is generally
written by



2ii ii
V
 

 rr r
, (3)
where
denotes the up-spin
or down-spin
. In
order to solve Equation (3), we have usually used the
spherical approximation, i.e., the central field approxi-
mation [5]. Under such the approximation, Equation (3)
is separable into two Equations, one of which depends on
the radial variable r and the other on the angular vari-
ables, and
. If the effective potential is spherically
symmetric and local [6], and if the solutions are given by
  
1,
nlmnl lm
prY
r

r, (4)
then two Equations are
 
2
0
22
1
d
dnlnl nl
ll Vrprpr
rr



 


, (5)
 
2
ˆ,1,
lm lm
YllY

l, (6)
where ˆ
l is the operator of the orbital angular momen-
tum, and
,
lm
Y
are the spherical harmonics [7].
Radial wave functions

nl
pr
of Equation (5) are cal-
culated easily by means of the numerical methods such
as the Herman-Skillman method [8,9]. Thus, the conven-
tional atomic structures, where the eigenstates are speci-
fied by the quantum numbers
 
,1,nlnmlm l
and
can be obtained.
Here the question is raised of whether the spherical
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
422
approximation is always appropriate or not. The spheri-
cal approximation is reasonable for atoms having the
outermost shell that is fully or half occupied since their
electron densities are exactly spherical. However, in the
other atoms the electron densities are not necessarily
spherical, so that the conventional picture of the atomic
structures is not rigorous but just an approximation. To
what extent the effect of the nonspherical distribution of
electrons (which is hereafter called the nonspherical ef-
fect) modifies the conventional picture of the atomic
structures seems to be interesting and important. This is
because the electronic structures of molecules and solids
have been frequently considered on the basis of the or-
dinary atomic wave functions and spectra. The typical
examples are the Slater integrals contained in the model
Hamiltonians like the Hubbard model [10,11], and in the
LDA + U method [12,13].
In addition to the above, there exists an obvious flaw
in the spherical approximation. The total orbital angular
momentum becomes necessarily zero in the conventional
atomic structure, because the spherical approximation
coincides with the filling approximation in which elec-
trons are uniformly distributed into each state in the out-
ermost shell [14]. This means that the orbital polarization
never appears in the atomic structures of the spherical
approximation. The orbital polarization is an origin of
the magnetism of solids as well as the spin polarization
[15-20], especially for the 5f-electron systems [15-17].
So far the orbital polarization has been discussed as a
part of the correlation effects [17] or on the basis of the
LS-multiplet theory [21]. In this paper, we shall discuss
the nonspherical effect on the orbital polarization from
the viewpoint of the single-particle picture. It will be
shown in the following sections that the orbital polariza-
tion appears without the correlation effects.
As a first attempt to take into account the nonspherical
effect, Slater has proposed a scheme for expanding the
eigenfunctions of Equation (3) with the spherical har-
monics [9]. However his method is difficult to be per-
formed because an infinite number of simultaneous
equations have to be solved. After Slater’s proposal,
there have been two kinds of approaches to this problem.
One is the variational method where the single-particle
wave function is expanded by using appropriately chosen
basis functions [22,23]. Another is the density functional
scheme containing the effect of the orbital current den-
sity explicitly.[24-31] In this paper, we adopt the former
approach. As the basis functions, eigenfunctions for the
spherical part of the single-particle potential are used and
updated for each iteration of the self-consistent calcula-
tions. They are apparently different from those of the
previous works [22,23].
The aim of this paper is to present the tractable
scheme for calculating the atomic structures beyond the
spherical approximation, and to discuss the nonspherical
effect. Organization of this paper is as follows. In Sec-
tion 2, we present a scheme for dealing with the non-
spherical effect explicitly. In order to check the validity
of the scheme, we apply it to the neutral carbon atom in
Section 3. The calculation procedure is also explained.
The results are shown in Section 4, with a focus on the
differences between the present atomic structures and the
conventional one. The ground state values of the total
orbital and spin angular momenta are also estimated.
Finally concluding remarks are given in Section 5.
2. A Variational Method beyond the
Spherical Approximation
In this section, we present a variational method for cal-
culating atomic structures with taking into account the
nonspherical effect.
Let us consider solving the single-particle Equation (3).
The Hartree-Fock-Slater approximation is utilized [32],
i.e., the effective potential of Equation (3) is given in a
local form
V
r. First, we expand the effective poten-
tial with the spherical harmonics:
  
**
,,
lm lmlmlm
lm
VvrYvrY

 

r, (7)
where
lm
vr
are the radial components and their ex-
plicit forms are given in Appendix. For the convenience
of the subsequent discussion, the operator of the left-
hand side (LHS) of Equation (5) is defined as

22
022
ˆ
d
ˆ:d
H
Vr
rr
 
l, (8)
where let
Vr
be the spherically averaged potential
for Equation (7), which is defined as

 

*
00 00
1()sind d
4π
1.
4π
Vr V
vrvr




r
(9)
In the Expansion 7, the term

00lm corresponds
to the spherical part of the effective potential as shown in
Equation (9), while the other terms correspond to the
nonspherical parts.
Next, in a similar way to Equation (7), we shall ex-
pand the solution of Equation (3) with the set of known
functions. As the known functions, we here adopt ones
given by Equation (4), the radial part of which is the ei-
genfunction for the Hamiltonian 8. Thus, the solution of
Equation (3) is written as
  
,
1,
i
inllmnllm
nll m
CprY
r

 


r. (10)
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
423
Substituting Equations (7) and (10) into Equation (3),
and writing distinctly the spherical and nonspherical
parts of the effective potential, we get
  
 
 
0
(00)
**
ˆ,(),
,,
,,
i
NL NLLL
NL L
i
LL NLNL
NL
i
iNLNL
NL
HCprY vrY
vrY CprY
Cp rY




 



 


(11)
where we use Equations (8) and (9), and abbreviate nl as
N” and lm as “L” for ease of seeing. It can be easily
shown that Equation (11) is reduced to the spherical
Equation including Equation (5), if the second term of
the LHS is neglected. This means that the second term of
the LHS represents the nonspherical effect that has been
disregarded in the conventional spherical approximation.
Here, for simplicity, we shall use the common value of
l for N and L, and suppose that the eigenvalues for
the Hamiltonian 8 is denoted as 0
N
. Multiplying
 
11
**
1,
NL
prY
r
on both sides of Equation (11) and
integrating over the whole space, we have
 
1
11
11
00
*
2
d0
NN
LL NNi
NL
i
NLLN NL
O
prVrprrC

 








, (12)
where
 
11
*d,
NN NN
Oprprr

(13)
 
 
1
1
1
*
(00)
***
(00)
()
,,,sindd
,,,sindd.
LL
LLL L
L
LLL L
L
Vr
vrYY Y
vr YYY
 



(14)
Equation (12) is just the generalized eigenvalue prob-
lem. If the matrix elements, 1
N
N
O
and
1
NN
Vr
, and
the energy spectra of the spherical approximation, 0
N
,
are given, then we can obtain the eigenvalues, i
, and
eigenfunctions,

i
N
L
C
. It should be noted that the ei-
genvalues i
are guaranteed to be real since both ma-
trices of Equation (12) are hermitian. The angular inte-
grations in Equation (14) can be analytically calculated
by using the Wigner 3j-symbols. According to the prop-
erties of the Wigner 3j-symbols, matrix elements of
Equation (14) are zero unless 1evenlll
 ,
11
lllll
 and 1
mm m
 [7]. These condi-
tions also determines the upper limit of the summation of
Equation (7).
The eigenfunctions thus obtained yield the new poten-
tials by means of the expressions given in Appendix.
These potentials should coincide with the input ones.
Namely, the self-consistency is required for the poten-
tials. The corresponding basis functions in Equation (10)
are modified for each iteration since the function
nl
pr
is the radial part of solution for the Hamiltonian (8) with
the Potential 9. The iteration is continued until self-con-
sistency for the potentials is achieved.
Let us show the detailed procedure of the self-consis-
tent calculations. The flow chart of self-consistent calcu-
lations is shown in Figure 1. We first give a starting po-
tential in some way, for example via the LDA calculation
within the spherical approximation (Step 1 in Figure 1).
In order to prepare the radial basis functions
nl
pr
, the
spherical parts of the potential are derived. Using these
potentials, atomic structure calculations are performed
(Step2). Then, using the basis functions and correspond-
ing spectra, the generalized eigenvalue problem is solved
(Step3). The resultant eigenfunctions provide the new
potentials (Step 4). Here we check whether the potentials
are converged or not (Step 5). Of course, the checking
should be performed on both convergences for the spheri-
cal and nonspherical parts of potentials. If the conver-
gence is not yet obtained, we return to Step 2 with the
spherical potential calculated from the new potential.
The calculations are repeated until the potentials are con-
verged within some accuracy.
3. Application to the Neutral Carbon Atom
Compared to the previous ones [9,22,23], the present
scheme seems to be more tractable, but as to the effec-
tiveness actual calculations have to be performed. Here
we apply it to the neutral carbon atom.
In the Expansion 10, we choose the common value of
l in both summations for nl and lm . That is to say,
physically meaningful functions are prepared for basis
functions of the expansion. In more detail, we use five
functions having the following quantum numbers:
 
100,200,211,210,21 1nlm
.
Correspondingly, the upper limit of the potential given
by Equation (7) is determined from the properties of the
Wigner 3j symbols, as already mentioned below Equa-
tion (15). This time, the expansion of the potential con-
sists of the following terms:
 
 
()00, 11,10,11 ,22,
21,20,2 1,22
lm 
.
The generalized eigenvalue problem is reduced to
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
424
 
11
11111 111
00
*
,d0
2
nln li
llmmnlnlinllml mnlnlm
nlm
OprVrprrC



 










. (15)
For ease of understanding the matrices of the eigen-value problem, their explicit forms are shown below:



0
10
10,20
00
10,10 20 10
00,1 100,1000,11
00,00
00,00
20,10
00
20,20 20
10 20
00,00
2
2
1021 1021 1021
10 10
10 20
20 10
ii
i
iiii iii
i
i
i
i
i
ii
i
i
iii ii ii
ii
ii
i
ii
O
O
O
O
VVV
V
V
V


 
















0
21
0
00,1 100,1000,11
00,00
21,21
11,0011,0011,1011,11
11,11
10,00
2021 20212021
20 20
211021202121 2121
21 21
21 102
ii
ii iii
i
ii
ii ii
i
i
i
iiiiii
ii
iiiiiii i
ii
ii i
O
VVV
V
VV VV
V
V


 







 




0
21
0
21
21,21
10,0010,1 110,11
10,10
21,21
11,0011,0011,11 11,10
11,11
12021 212121
21 21
211021202121 2121
21 21
ii
ii
ii
ii i
i
ii
iiii
i
ii ii i
ii
iiiii iii
ii
O
O
VV
V
VVVV
V
V


 



  

 




100
200
211
210
21 1
0.
i
i
i
i
i
i
i
i
i
i
C
C
C
C
C

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
(16)
Figure 1. Flow chart of the self-consistent calculations. The detailed procedure is shown in the text.
Solving the above Equations in a self-consistent way,
we can obtain the atomic structure for the neutral carbon
atom. The concrete steps of the calculations are shown in
the flow chart of Figure 1. Here note that there is a pos-
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
425
sibility to exist the multiple self-consistent solutions. In
order to cover such solutions, various kinds of input po-
tentials should be arranged. This time, we prepare the
orbitals of the spherical approximations, i.e. 1s, 2s and
2p orbitals, and take the linear combination of them so as
to construct the input potentials. The 1s and 2s orbitals
are used as they stand, while 2p orbitals are transformed
into the following orbitals:
1
2
3
211
210
21 1
cos sin0
sincos 0
001
cos0 sin
010
sin0cos

























, (17)
where 21121021 1
, and
 

are 2p orbitals of the spheri-
cal approximation. The input potentials are constructed
from the 1s, 2s and two orbitals chosen among three ones
given by Equation (17). The angles and
are
changed by 15°, respectively. A total of 864 kinds of
different potentials are taken as the starting potentials
(

32
360 1518015864C
). As shown in a subsequent
section, six kinds of self-consistent solutions can be ob-
tained correspondingly to the starting potentials.
4. Results and Discussions
In this section, we will give the results of the atomic
structure of the neutral carbon atom. Figure 2 shows the
energy spectra of the present scheme, together with those
of the conventional spherical approximation. It should be
noted that the conventional atomic spectra can be speci-
fied by the quantum numbers nlm
, but in the present
scheme they are specified only by the ordinal numbers
because of lack of the spherical symmetry. For instance,
1s states of the spherical approximation correspond to
the 1st and 2nd states of the present scheme.
The conventional 2p states are split into doubly de-
generate levels and single one due to the nonspherical
effect. There exist two types of splitting. In other words,
two types of the converged self-consistent solutions
(SCS) can be found from the viewpoint of the splitting of
energy levels. One is that the doubly degenerate levels
are higher than the single one, and another is the opposite.
They are denoted as “SCS-A” and “SCS-B”, respectively,
in Figure 2. On the other hand, conventional 1s and 2s
states are little influenced by the nonspherical effect.
This is because the 2p states (5th and 6th states) are di-
rectly influenced by their nonspherical densities of elec-
trons, while the wave functions for 1s and 2s states (from
1st to 4th states) are well localized near the nuclear
where the spherical potential mainly caused by the nu-
clear is dominant.
In order to discuss the ground-state properties in more
detail, we shall investigate the components of the eigen-
functions, i.e. the expansion coefficients of Equation (10).
The SCS-A and SCS-B are classified into two and four
types, respectively, according to the components of the
Figure 2. Energy spectra for the neutral carbon atom. The first column shows the results for the conventional spherical ap-
proximation. The second and third columns are the self-consistent solutions for the present scheme, which are denoted as
“SCS-A” and “SCS-B”, respectively. The up- and down-arrows denote the occupied states, and open circle the unoccupied
states. All values are given in Rydberg Unit.
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
426
wave function. They are denoted as A-1, A-2 and B-1,
B-2, B-3, B-4 respectively. The expansion coefficients
for the 1st, 2nd, 3rd and 4th states are shown in Table 1.
For all of the coverged SCSs, the components are same
as those of the spherical approximation within the accu-
racy of 3
10. These results are consistent with the fact
that the states from 1st to 4th for the present scheme are
in a good agreement with the conventional 1s and 2s
states, respectively (Figure 1). Concerning the 5th and
6th states, there is a large difference between the present
and conventional schemes, which is shown in Table 2. In
the conventional spherical approximation, electrons are
distributed into each shell in an equal weight, so that the
corresponding coefficients are all 23. Meanwhile,
components of each the converged SCS are partial to
some of them. This partiality causes the polarization of
the orbital angular momenta, which is so-called orbital
polarization. In order to verify it, we calculate the ground
state values of the total orbital and spin angular momenta.
The ground state of the Hartree-Fock-Slater approxima-
tion is given by a single Slater determinant that is writ-
ten as

 

  
   
   
11 141 1 111 211
14 444 4
16
15515 525 5
166166266
1
,, 6!
xx




 
 
 
 
  



rrrr
rr
rrr
rrr
(18)
where i
x
denote the coordinates for ith electron in-
cluding spatial coordinate i
r and spin coordinate i
,
and where ()
i
r is the solution of Equation (16), and
where () and ()

are wave functions for up- and
down-spins, respectively. Using Equation (18), the total
orbital angular momentum and its z-component, L and
z
L, are respectively calculated by


2
16 16
ˆ
(, ),1xxxx LLL, (19)

16 16
ˆ
,,
z
z
x
xLx xL, (20)
where
6
1
ˆ
ˆi
i
L
l, and ˆ
i
l is the operator of the orbital
angular momentum for the ith electron. Similarly, the
total spin angular momentum and its z-component, S and
z
S, are respectively given by

2
16 16
ˆ
,,1xxxx SS
S, (21)
16 16
ˆ
,,
z
z
x
xSx xS

, (22)
where
6
1
ˆˆi
i
S
s, and ˆi
s is the operator of the spin an-
gular momentum for the ith electron. The results are
shown in Table 3. In a nonrelativistic many-electron
system, ,, and
z
z
LL SS are the conserved quantities. All
of the coverged SCSs yield 1 and 1LS
within the ac-
curacy of 3
10
. This means that the ground states of the
present scheme correspond to the term 3P that is known
to be the ground state of the LS-multiplet theory. Fur-
thermore, it is noticed that the present scheme obviously
Table 1. The expansion coefficients of Equation (10) for the 1st, 2nd, 3rd and 4th states. Both SCS-A and SCS-B give the
same results, so we don’t lable SCS-A and SCS-B distinctly.
(nlm)
Energy [Ryd.] (100) (200) (21+1) (210) (21-1)
–21.354 1.000 … … … …
–21.206 1.000 … … … …
–1.421 … 1.000 … … …
Spheical approximation
–1.032 … 1.000 … … …
–21.353 1.000 0.000 0.000 0.000 0.000
–21.206 1.000 0.000 0.000 0.000 0.000
–1.421 0.000 1.000 0.000 0.000 0.000
Present results for SCS-A and SCS-B
–1.031 0.000 1.000 0.000 0.000 0.000
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
427
Table 2. The expansion coefficients of Equation (10) for the
5th and 6th states.
(nlm)
Energy
[Ryd.] (100) (200) (21+1) (210)(21-1)
Spheical
approximation –0.782 … … 23 23 23
–0.799 0.000 0.000 0.000 1.0000.000
SCS A-1
–0.773 0.000 0.000 1.000 0.0000.000
–0.799 0.000 0.000 0.000 1.0000.000
SCS A-2
–0.733 0.000 0.000 0.000 0.0001.000
–0.799 0.000 0.000 12 0.000 12
SCS B-1
–0.799 0.000 0.000 12 0.000 12
–0.799 0.000 0.000 1.000 0.0000.000
SCS B-2
–0.799 0.000 0.000 0.000 0.0001.000
–0.799 0.000 0.000 0.000 1.0000.000
SCS C-1
–0.799 0.000 0.000 12 0.000 12
–0.799 0.000 0.000 0.000 1.0000.000
SCS C-2
–0.799 0.000 0.000 12 0.000 12
Table 3. The ground state values of the tptal orbital angular
momentum and itd z-component, L and LZ, are showm in
the 1st and 2nd cplumns. Also, the ground state values of
the total spin angular momentum and its z-component, S
and SZ, are shown in the 3rd and 4th columns.
State L LZ S SZ
SCS A-1 1.000 1.000 1.000 1.000
SCS A-2 1.000 –1.000 1.000 1.000
SCS B-1 1.000 0.000 1.000 1.000
SCS B-2 1.000 0.000 1.000 1.000
SCS B-3 1.000 0.000 1.000 1.000
SCS B-4 1.000 0.000 1.000 1.000
causes the orbital polarization. Since the spherical ap-
proximation never brings it, we may say that the non-
spherical effect is one of the keys to the appearance of
the orbital polarization.
Here note that the SCSs are classified into three kinds
of states, which yield 1,0 and 1
z
L, respectively.
This is not surprising because the term 3P are triply de-
generate with respect to the orbital angular momentum.
These three states, by their nature, should be completely
degenerate and their total energies should be same as
each other. Actually, the total energies of these states
which are also evaluated by taking the expectation values
of the Hamiltonian with respect to Equation (18) coin-
cide with each other.
5. Concluding Remarks
In this paper, we present a scheme for calculating the
atomic structures beyond the spherical approximation
and investigate to what extent the single-particle picture
of atomic systems needs to be modified. It is confirmed
that the orbital polarization can appear only by consider-
ing the nonspherical effect explicitly. Compared to the
conventional atomic structures, we find that the atomic
levels are partially split. The magnitude of splitting for
2p states is about 5%, which is never neglected because
the splitting itself causes the orbital polarization. Also,
such a debacle of the conventional atomic structures
seems to be conceptually important.
Although the present scheme shows the necessity of
modifying the single-particle picture of atomic systems,
we have to consider the following effects that are ne-
glected in the present calculations:
1) enhancement of the expansion basis functions in
Equation (10);
2) treatment of the exchange energy beyond the Har-
tree-Fock-Slater approximation;
3) correlation effects.
Concerning the first effect, we here adopt only 1s, 2s
and 2p orbitals in the expansion of the eigenfunctions.
However we had better take more functions as the basis
functions. Especially for the neutral carbon atom, 3d
orbitals should be added to the expansion basis functions
since the nondiagonal elements between the 2p and 3d
states would not be negligibly small in Equation (12) or
(15). Similarly, the second and third effects seem to be
indispensable for describing the nonspherical effect in
more detail. But anyway, we can say within the knowl-
edge obtained in this paper that the orbital polarization
certainly emerges by taking into account the nonspheri-
cal effect even if the correlation effects are not explicitly
considered. Furthermore, the effect of the nonspherical
distribution of electrons cannot be neglected not only
conceptually but also quantitatively in the study on the
single-particle picture of atomic systems.
6. Acknowledgements
This work was partially supported by Grant-in-Aid for
Scientific Research (No. 19540399) and for Scientific
Research in Priority Areas (No. 17064006) of The Min-
istry of Education, Culture, Sports, Science, and Tech-
nology, Japan.
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
428
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Appendix
Expressions for the Potentials
In this appendix, we present the expressions for the
spherical and nonspherical parts of the effective potential.
The effective potential consists of three terms, i.e., the
nuclear, Hartree and exchange potentials, which are given
by
  
1
1
3
3
23
2d6,
4π
Z
Vr


 

r
rrr
rr (A1)
where the exchange potential is simplified with the aid of
the Slater approximation [32], and where
r and

r denote the electron density and electron density
with
-spin, respectively. Using Equation 10, they are
written as
 
  
occ. 2
2
occ.
,
2
1,,
i
i
i
nll mnllm
inllm
CprY
r



 


rr
(A2)
 



OCC
OCC
.2
spin
2
.
,
2
spin
1(), ,
i
i
i
nll mnllm
inllm
CprY
r




 


rr
(A3)
where the sum of Equation A3 is over only the occupied
states with
-spin. What we need are
1) spherical part of Equation A1, i.e.,
Vr
 
*
00 00
1
4πvrvr

, that appears in Equation 8,
and
2) nonspherical components of Equation A1, i.e.,
kq
vr
, that appear in Equation 12 or 14.
The above 1) is indispensable for deriving the basis
functions
nl
pr
and corresponding spectra 0
nl
,
which are also used in Equation 12 or 14.
Now let us show the explicit forms of 1) and 2) by
considering each term of Equation A1. As for the first
term, we have no problem because it is exactly spherical.
Also, the second term can be easily separated into the
spherical and nonspherical parts by means of the multi-
pole expansion of the Coulomb potential. Concerning the
third term, we have to use an approximation so as to de-
rive one third power of

r. Using the composition
relation for the spherical harmonics,

r is formally
separated into spherical and nonspherical parts as fol-
lows:

SNS
r


rr, (A4)
with


 
 
2
OCC
112 2112 21111
11 2 2112 2
.*
*
,, 22
2
spin
22
22
22
1
12121
8
0
0..,
0
000
m
Sii
nllmnllmnlnl
inllmnllm
rCCprprll
r
ll
ll CC
mm
 
 
 








 
(A5)
and

 
OCC OCC
112 21122
11 2 21122
222
11 11
22
..
2*
,,
2
spin spin
1
2
22
*
0
0
22
22
2
1
() ()
1212121
()()24π
000
NSi i
inllm nllm
iinll mnllm
mll k
nl nlkl lqk
CC
r
llk
prpr
llk
llk
m





 
 


 






 

rr
2
(, )...
kq
YCC
mq




(A6)
M. MIYASITA ET AL.
Copyright © 2011 SciRes. JMP
Since

NS
r is constructed from electrons of the
unfilled outermost shell alone, it is quite smaller than the
spherical part of the electron density. That is to say,
 
SNS
r


r. Using this fact, we can get an ap-
proximate form of the one third power of
r as
follows
  
11 2
33 3
1
3
SSNS
rr

 
rr, (A7)
The first and second terms turn to the spherical and
nonspherical parts of the exchange potential, respectively.
Using these relations, the resultant forms of 1) and 2)
are, respectively, given by
  
 
1122 11221111
11 2211 22
2
occ. *
*
,,
1
1
3
22
22 3
22
22
21
d
0
03
2121216,
0
000 4
ii
nllmnllmnlnl
i nllmnllm
mS
Z
VrCCrp rp r
rr
ll
ll
ll r
mm
 
 
 
 

 
 

 




(A8)
 
 

11 2211 221111
11 2 2112 2
2
11 22122
occ. *
*
,, 1
22
22
22
22
1
2
3*
3,,
2
d
4π2121
1000
21
31
24π
k
ii
kqnllmnllmnlnl
k
inllmnllm
m
Sii
nllmnlm
r
vCCrprpr
r
llk
llk
ll
mmq
k
rCC
r

 
 
 
 

 







r
 
 
OCC
11 1
11 22122
2
.*
( spin)
1
222
22
22
22
1(21)2121 .
000
24π
nl nl
inllmnllm
m
prpr
llk
llk
llk
mmq
 

 









 
(A9)
Here 123
123
lll
mmm



is the Wigner 3j symbol [7], and
Sr
is given by Equation A5.