Download Electron-pair center-of-mass-motion densities of atoms in position

PHYSICAL REVIEW A
VOLUME 57, NUMBER 6
JUNE 1998
Electron-pair center-of-mass-motion densities of atoms in position and momentum spaces
Toshikatsu Koga and Hisashi Matsuyama
Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan
E. Romera and Jesus S. Dehesa
Facultad de Ciencias, Instituto Carlos I de Fı́sica Teórica y Computacional, Universidad de Granada, Granada E-18071, Spain
~Received 17 November 1997!
Spherically averaged center-of-mass-motion ~extracule! densities d(R) in position space and d̄( P) in momentum space represent probability densities of finding center-of-mass radii u r j 1rk u /2 and u p j 1pk u /2 of any
pair of electrons j and k to be R and P, respectively. Theoretical analysis shows that in the Hartree-Fock
approximation, the individual spin-orbital-pair component of the extracule density has a definite relation with
the corresponding one of the relative-motion ~intracule! density, but the total extracule and intracule densities
are not related in a simple manner. Using the numerical Hartree-Fock method, the extracule densities d(R) and
d̄( P) are constructed and examined systematically for the atoms from He to Xe in their ground state. In
position space, the extracule density d(R) is a monotonically decreasing function for all the atoms. In momentum space, however, the extracule densities d̄( P) are found to be classified into two types according to the
location of a maximum. These different behaviors of the densities d(R) and d̄( P) are studied in detail based
on the contributions of electrons in a pair of atomic subshells and spin orbitals. @S1050-2947~98!01206-2#
PACS number~s!: 31.10.1z, 31.15.Ar
I. INTRODUCTION
The electronic relative-motion ~intracule! I(u) and centerof-mass-motion ~extracule! E(R) densities, introduced by
Coleman @1#, are a couple of electron-pair densities useful to
elucidate the motion of a pair of electrons in atoms and molecules ~see, e.g., @2# for a review!. The intracule density I(u)
and its spherical average h(u) are the probability density
functions for the relative vector r j 2rk and its magnitude
u r j 2rk u of any pair of electrons j and k to be u and u,
respectively. They have been used in several physical and
chemical contexts, particularly in relation to the electron correlation problem ~see references given in Refs. @2–4#!. The
momentum-space counterparts Ī(v) and h̄( v ) have been introduced as well. Accurate Hartree-Fock values of the intracule moments ^ u n & in position space and ^ v n & in momentum
space have been reported @3# recently for the atoms from He
to Xe in their ground states, together with a systematic examination @4# of the spherically averaged intracule densities
h(u) and h̄( v ).
The extracule density E(R) in position space represents
the probability density function for the center-of-mass vector
(r j 1rk )/2 of any pair of electrons j and k to be R, and is
defined by
K(
N21
E ~ R! [
N
(
j51 k5 j11
d @ R2 ~ r j 1rk ! /2#
L
~1a!
for an N-electron system (N>2), where d~r! is the threedimensional Dirac delta function and the angular brackets
stand for the expectation value. The spherical average d(R)
of E(R) is defined by
d~ R ![
1
4p
E
dV R E ~ R! ,
1050-2947/98/57~6!/4212~7!/$15.00
~1b!
57
where (R,V R ) with V R [( u R , f R ) are spherical polar coordinates of the vector R. By definition, the extracule densities
E(R) and d(R) are normalized to the number of electron
pairs,
E
E
dR E ~ R! 54 p
`
0
dR R 2 d ~ R ! 5N ~ N21 ! /2, ~1c!
and contain information about the location of electron pairs.
The momentum-space extracule density Ē(P) and its spherical average d̄( P) are defined analogously by
K(
(
d @ P2 ~ p j 1pk ! /2# ,
~2a!
d̄ ~ P ! [
1
4p
E
~2b!
N21
Ē ~ P! [
E
j51 k5 j11
dP Ē ~ P! 54 p
L
N
E
`
0
dV P Ē ~ P! ,
d P P 2 d̄ ~ P ! 5N ~ N21 ! /2, ~2c!
where P[( P,V P ) and V P [( u P , f P ). The functions Ē(P)
and d̄( P) are probability densities for the center-of-mass
momentum vector (p j 1pk )/2 and its magnitude u p j 1pk u /2
of any pair of electrons j and k to be P and P, respectively.
Evaluation of accurate Hartree-Fock values of the electronic extracule moments ^ R n & and ^ P n & has been performed
@5# very recently for the atoms He through Xe in their ground
states. However, only a few studies are reported on the extracule densities of specific atoms and molecules ~see references given in Ref. @5#!. Moreover, all these studies are
based on basis-set-expansion wave functions, and no accurate and/or systematic results concerning the distribution of
extracule densities are known even for atoms.
4212
© 1998 The American Physical Society
57
ELECTRON-PAIR CENTER-OF-MASS-MOTION . . .
In the present paper, we report accurate Hartree-Fock extracule densities d(R) in position space and d̄( P) in momentum space for a series of 53 atoms from He (Z52) to Xe
(Z554) in their ground states, where Z stands for atomic
number. The numerical Hartree-Fock method is used instead
of the conventional Roothaan-Hartree-Fock or basis-setexpansion method. The next section explains the theoretical
structure of the extracule densities in position and momentum spaces. It will be also clarified that a spin-orbital-pair
component of the extracule density has a definite relation
with the corresponding component of the intracule density in
the Hartree-Fock framework. In Sec. III, the computational
details are described and in Sec. IV the results are presented
and discussed. A systematic analysis shows that in position
space, the extracule density d(R) is a unimodal function with
a maximum at R50 for all the 53 atoms. In momentum
space, on the other hand, the extracule densities d̄( P) are
found to be classified into two types; ~i! 42 atoms have a
unimodal density with a maximum at P50, and ~ii! 11 atoms
have a unimodal density with a local minimum at P50 and
a maximum at P.0. These different features of the densities
d(R) and d̄( P) are discussed in detail based on the contributions of electrons in a pair of atomic subshells and of
atomic orbitals. Hartree atomic units are used throughout this
paper.
A. Extracule density in position space
We consider the case where a Hartree-Fock wave function
of an N-electron system is expressed by a single Slater determinant of N spin orbitals c j (r) h j ( s ). Then, using the
Condon-Slater rule ~see, e.g., @6#!, we can rewrite the spherically averaged extracule density d(R), defined by Eq. ~1b!,
as a sum of contributions d jk (R) from pairs of spin orbitals j
and k:
N21
N
E
`
0
with Y lm (V) being a spherical harmonic and r5(r,V r ), the
explicit forms of D j j,kk and D jk,k j appeared in Eq. ~5a! are
obtained @5# as
min~ 2l j ,2l k !
D j j,kk ~ s ! 5
(
~ 2l11 ! a l ~ l j m j ;l k m k ! W l j j ~ s ! W lkk ~ s ! ,
l50
~7a!
l j 1l k
D jk,k j ~ s ! 5 ~ 21 !
(
l j 1l k
l5 u l j 2l k u
~ 2l11 !
3b l ~ l j m j ;l k m k ! W l jk ~ s ! W lk j ~ s ! ,
~7b!
where the summations run over every other integer between
the specified values, the symbols a k (lm;l 8 m 8 ) and
b k (lm;l 8 m 8 ) are Condon-Shortley parameters @7#, and
W l jk ~ s ! [
E
`
0
dr r 2 j l ~ sr ! R *j ~ r ! R k ~ r ! 5W *
lk j ~ s ! .
~8!
A momentum-space N-electron wave function is given by
the 3N-dimensional Fourier transformation of the corresponding position-space wave function. Then the HartreeFock wave function in momentum space has exactly the
same determinantal structure as that in position space, provided that a position orbital c k (r) is replaced with the corresponding momentum orbital
f k ~ p! 5 ~ 2 p ! 23/2
E
dr exp~ 2ip•r! c k ~ r! .
~9!
Expanding exp(2ip•r) in Eq. ~9! by means of plane waves
@8# and integrating over the angular variable V r , we find the
momentum orbital corresponding to Eq. ~6! to be
ds s 2 j 0 ~ 2Rs ! D jk ~ s ! ,
~4!
f k ~ p! 5 P k ~ p ! Y l k m k ~ V p ! ,
D jk ~ s ! 5D j j,kk ~ s ! 2 d m s j m sk D jk,k j ~ s !
~5a!
is the characteristic function of d jk (R), in which
D i j,kl ~ s ! [ ~ 4 p ! 21
E
~6!
~3!
where j l (x) is the lth-order spherical Bessel function of the
first kind and
f jk ~ s! [
c j ~ r! 5R j ~ r ! Y l j m j ~ V r ! ,
d jk ~ R ! ,
( (
j51 k5 j11
d jk ~ R ! 5 ~ 4/p 2 !
have different spins. If we further restrict ourselves to atomic
systems where the spatial function c j (r) has the form
B. Extracule density in momentum space
II. THEORETICAL STRUCTURE
OF EXTRACULE DENSITY
d~ R !5
4213
E
dV s f *
i j ~ s! f *
kl ~ s ! ,
~5b!
dr exp~ 1is•r! c *j ~ r! c k ~ r! 5 f *
k j ~ 2s ! . ~5c!
In Eq. ~5a!, the Kronecker delta d m s j m sk is unity if the two
spin orbitals j and k have the same spin and zero if j and k
~10!
where p5(p,V p ) and
P k ~ p ! 5 ~ 2i ! l k
A E
2
p
`
0
dr r 2 j l k ~ pr ! R k ~ r ! .
~11!
Consequently, theoretical structures of the spherically averaged extracule density are the same in position and momentum spaces, and all the position-space results explained in
Sec. II A hold in momentum space as well, if we replace
d(R) with d̄( P) and R k (r) with P k (p).
C. Relation between extracule and intracule densities
There are some general relations between the extracule
d jk (R) and intracule h jk (u) density components arising from
two spin orbitals j and k. Comparison of the structure of
4214
KOGA, MATSUYAMA, ROMERA, AND DEHESA
57
extracule densities given in Sec. II A with that @4# of intracule densities shows that the two components of their characteristic functions satisfy
D j j,kk ~ s ! 5H j j,kk ~ s ! ,
D jk,k j ~ s ! 5 ~ 21 ! l j 1l k H k j,k j ~ s ! .
~12!
We immediately obtain
d jk ~ R ! 58h jk ~ 2R ! ,
~13a!
if m s j Þm sk or if m s j 5m sk and l j 1l k 5even, and
d jk ~ R ! 58h jk ~ 2R ! 1 ~ 8/p 2 !
E
`
0
ds s 2 j 0 ~ 2Rs ! H k j,k j ~ s ! ,
~13b!
if m s j 5m sk and l j 1l k 5odd. Thus the intracule and extracule densities are not completely independent as suggested
previously @5,9#. A special case of Eq. ~13a! for m s j 5m sk ,
l j 1l k 5even, and R50 gives the electron-electron counterbalance and coalescence ~or Fermi! holes @10# d jk (0)
5h jk (0)50 simultaneously. The same discussion holds in
momentum space.
If the second term on the right-hand side of Eq. ~13b! is
small or if the spin-orbital-pair contributions in the form of
Eq. ~13b! are small by themselves, we obtain an approximation d(R)>8h(2R) between the total extracule and intracule
densities. Our previous study @4# on the intracule density has
clarified that in position space, the 1s a 1s b spin-orbital pair
gives a predominant contribution to the total density h(u).
Since Eq. ~13a! is valid for the 1s a 1s b pair, we anticipate
that the behavior of the extracule density d(R) will be similar to that of the intracule density h(u) in position space. In
momentum space, however, it has been shown @4# that major
contributions to the intracule density h̄( v ) come from pairs
of valence spin orbitals. Equation ~13b! appears more often
for these spin-orbital pairs, and therefore we expect that in
momentum space the behaviors of the extracule d̄( P) and
intracule h̄( v ) densities may show some differences. These
anticipations will be actually confirmed in Sec. IV.
III. COMPUTATIONAL METHOD
For the neutral atoms He (Z52) through Xe (Z554) in
their experimental ground LS term @11#, we have previously
confirmed @3# that among (2L11)(2S11) degenerate
states, there exists at least one state with specific z component M L and M S values of L and S that results in a single
determinant wave function, where L and S represent the total
orbital and spin angular momentum quantum numbers, respectively.
For these ground LS states of the 53 atoms, the radial
functions R j (r) in position space have been generated by the
numerical Hartree-Fock method based on a modified and enhanced version of the MCHF72 code @12#. Then the products
r 2 R j (r)R k (r) of two radial functions have been numerically
Hankel-transformed to obtain W l jk (s), by using the algorithm of Talman @13#. Following Eqs. ~5a!, ~7a!, and ~7b!, we
obtain the function D jk (s) as the sum of products of two
W l jk (s) with appropriate coefficients. The parameters a k and
b k have been taken from Slater’s book @14#. An additional
FIG. 1. Examples of the position-space extracule density d(R).
All values in hartree atomic units.
Hankel transformation @Eq. ~4!# of the function D jk (s) gives
the spherically averaged extracule density d jk (R).
In momentum space, the position-space radial functions
R j (r), generated by numerical Hartree-Fock calculations,
have been first Hankel transformed to obtain the momentumspace radial functions P j (p) according to Eq. ~11!. Then the
same procedure as in position space has been applied to derive d̄ jk ( P).
IV. RESULTS AND DISCUSSION
Throughout this section, total extracule densities are normalized to unity instead of N(N21)/2 for all the 53 atoms in
order to avoid large numbers and to facilitate mutual comparison. Accordingly, all the spin-orbital-pair components
are normalized to 2/N(N21) and the subshell-pair contributions are obtained as their partial sum.
A. Extracule density in position space
Position-space extracule densities d(R) are exemplified in
Fig. 1 for a few selected atoms in their ground states. All the
extracule densities shown in Fig. 1 are monotonically decreasing functions, when R increases. However, the densities
for the Zn and Mo atoms have a shoulder around R50.05.
Analogous and systematic examinations show that the extracule density d(R) in position space is a unimodal function
with a maximum at R50 for all the 53 atoms, as is @4# the
intracule density h(u). Table I summarizes the maximum
height d max5d(0). The maxima d max increase with increasing
Z, and we find an approximate relation d(0)
>0.2942Z 2.4169/(Z21) for 2<Z<54. We also find
d(0)/h(0)58 exactly for Z52 – 4 as proved in Ref. @9#, but
for the remaining atoms the ratio increases monotonically
from 8.108 for Z55 to 10.946 for Z554. The extracule
density d(R) at R50 is called @9# the electron-electron
counterbalance density and represents the probability density
of finding any two electrons precisely at the opposite posi-
57
ELECTRON-PAIR CENTER-OF-MASS-MOTION . . .
4215
TABLE I. The maximum value d max[d(0) of the position-space
extracule density d(R). Note that d(R) is normalized to unity instead of N(N21)/2. A @ n # means A310n . All values in hartree
atomic units.
Z
Atom
d max
Z
Atom
d max
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
1.5248
2.0914
2.7988
3.5508
4.3673
5.2478
6.2382
7.3095
8.4633
9.6723
1.0933@1#
1.2228@1#
1.3558@1#
1.4923@1#
1.6322@1#
1.7755@1#
1.9221@1#
2.0716@1#
2.2238@1#
2.3737@1#
2.5246@1#
2.6769@1#
2.8274@1#
2.9856@1#
3.1437@1#
3.3034@1#
3.4648@1#
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
3.6246@1#
3.7927@1#
3.9630@1#
4.1359@1#
4.3113@1#
4.4889@1#
4.6688@1#
4.8510@1#
5.0349@1#
5.2207@1#
5.4065@1#
5.5933@1#
5.7799@1#
5.9689@1#
6.1603@1#
6.3505@1#
6.5430@1#
6.7355@1#
6.9315@1#
7.1293@1#
7.3282@1#
7.5286@1#
7.7305@1#
7.9336@1#
8.1381@1#
8.3439@1#
tions with respect to the nucleus. Some discussion on the
property of the electron-electron counterbalance density
d(0) is found in Refs. @9,10,15#.
A decomposition of the extracule density into subshellpair contributions shows that for all the atoms, the five largest contributions to d(R) come from the inner subshell pairs
1s1s, 1s2s, 1s2p, 2s2p, and 2 p2p, which consist of 1, 4,
12, 12, and 15 spin-orbital-pair contributions, respectively,
when fully occupied. For the Zn atom (Z530), Fig. 2 illustrates these five contributions out of the total 28 subshell-pair
contributions. Clearly, the 1s1s subshell pair is seen to give
a significant contribution for the region R,0.02, and the
1s2p subshell pair is largest for 0.02,R,0.10. These two
contributions are unimodal with a maximum at R50 and
form the monotonically decreasing behavior of the extracule
density d(R). Due to Eq. ~13b!, the contribution of the 1s2 p
subshell pair is slightly different in d(R) and h(u); in a
small R or u region, the pair has a larger density in d(R)
than in h(u).
Figure 3 depicts typical spin-orbital-pair contributions
d jk (R) to d(R) in Zn atom, where there is a total of 435
spin-orbital pairs. Figure 3~a! shows the contributions from
five spin-orbital pairs with different spins; 1s a 1s b ,
1s a 2s b , 1s a 2p 0 b , 2 p 21 a 2p 11 b , and 2s a 2 p 0 b . Because of Eq. ~13a!, all these contributions have essentially
FIG. 2. Major subshell-pair contributions to the position-space
extracule density d(R) of the Zn atom (Z530). All values in hartree atomic units.
FIG. 3. Major spin-orbital-pair contributions to the positionspace extracule density d(R) of the Zn atom (Z530). ~a! Pairs
with different spins. ~b! Pairs with the same spin. All values in
hartree atomic units.
4216
KOGA, MATSUYAMA, ROMERA, AND DEHESA
FIG. 4. Examples of the momentum-space extracule density
d̄( P). All values in hartree atomic units.
the same behavior as that we have already discussed @4# for
the intracule density. We here simply note that the 1s a 1s b
pair gives a unimodal distribution with a maximum at R
50, and its contribution is more than ten times larger than
the other four contributions. Figure 3~b! shows the contributions from four spin-orbital pairs with the same spin;
1s a 2p 0 a , 2s a 2p 0 a , 1s a 2s a , and 2p 21 a 2p 11 a . The
first two contributions are unimodal with a maximum at R
50, while the last two are unimodal with zero at R50 and a
maximum away from R50. The property d jk (0)50 is
known @10# as the electron-electron counterbalance hole and
valid for any contribution from spin orbitals j and k with the
same spin and with the same inversion symmetry. The counterbalance hole implies that these electrons cannot be at the
opposite positions with respect to the inversion center and is
a consequence of the antisymmetry of electronic wave functions as the electron-electron coalescence or Fermi hole.
Since the essential feature of spin-orbital-pair contributions
shown in Fig. 3 for the Zn atom is the same for all the 53
atoms, we conclude that the position-space extracule density
d(R) is a unimodal function with a maximum at R50 due to
the predominant 1s a 1s b spin-orbital-pair or 1s1s subshellpair contribution, and has a behavior similar to the corresponding intracule density h(u) apart from two constant factors appearing in Eq. ~13a!.
B. Extracule density in momentum space
Figure 4 exemplifies the momentum-space extracule density d̄( P) for the same atoms examined in Fig. 1 in position
space. In contrast with a single common modality observed
for the position-space densities, Fig. 4 shows that there are at
least two different types of modality in the momentum-space
extracule densities. The extracule densities d̄( P) of F and Zn
atoms are unimodal functions with a maximum at P50. The
extracule densities of K and Mo atoms are unimodal with a
maximum at P50.2266 and at P50.3018, respectively.
57
A systematic examination of all the 53 atoms concludes
that the momentum-space extracule densities are classified
into two types based on the location of the maximum:
Type A: Unimodal extracule density with a maximum at
P50. The next 42 atoms have densities of this type; group 2
atoms, transition atoms Sc-V, Mn-Ni, Zn, Y, Zr, Tc, Pd, and
Cd, and group 13–18 atoms. All these atoms have two electrons in the outermost s orbital.
Type B: Unimodal extracule density with a local minimum at P50 and a maximum at P.0. The next 11 atoms
have type B densities; group 1 atoms and transition atoms
Cr, Cu, Nb, Mo, Ru, Rh, and Ag. All these atoms have a
single electron in the outermost s orbital.
Table II summarizes the above classification of the
momentum-space extracule density d̄( P) together with the
maximal subshell-pair contribution to it. If we introduce
symbols S, P, and D for the outermost s, p, and d subshells,
respectively, and a symbol S 8 for the second outermost s
subshell, the subshell-pairs appeared in Table II are grouped
into only four types SS, SS 8 , S P, and PD. Thus, it is immediately seen that the momentum-space extracule density
d̄( P) is governed by a few outermost subshell pairs and
depends explicitly on the valence electronic configuration of
an atom, in marked contrast to the position-space extracule
density. In general, two neighboring atomic subshells have
different spatial inversion symmetries to which Eq. ~13b!
applies. As a result, the extracule d̄( P) and intracule h̄( v )
densities in momentum space show different behaviors. For
the intracule density h̄( v ), we have observed @4# three types
of the modality and five types of major subshell-pair contributions.
Table II also tabulates the maximum value d̄ max
[d̄(Pmax) of d̄( P) and its location P max , which characterize
the distribution of the momentum-space extracule density.
When atomic number increases within a period, there is a
trend that d̄ max decreases, although it is subject to some exceptions. A special case of d̄( P) at P50, d̄(0), represents
@15# the electron-electron counterbalance density in momentum space, which is the probability density for a pair of
electrons to have exactly opposite momenta. For Z52 – 4,
the ratio d̄(0)/h̄(0) is 8 exactly, but for Z55 – 54, it irregularly distributes between 8.432 and 17.439, reflecting the difference in the extracule and intracule densities for these atoms in momentum space.
For the Zn atom having type A density, Fig. 5 depicts four
major subshell-pair contributions, 4s4s, 3 p3d, 3d4s, and
3s4s. These subshell pairs consist respectively of 1, 60, 20,
and 4 spin-orbital pairs, when the subshells are fully occupied. The first two subshell-pair contributions are unimodal
with a maximum at P50, while the last two are unimodal
with a maximum away from P50. The modality of d̄( P) of
the Zn atom is attributed to the largest 4s4s or SS contribution. The type A modality of atoms with an outermost s(2)
configuration is explained analogously except that for the
group 13–18 atoms, the S P contribution is larger than the
SS one. Figure 5 also shows that the type B modality appears
if we remove the SS contribution as in the case of atoms with
an outermost s(1) configuration. We note that the type A
modality of d̄( P) is explained based only on the largest
57
ELECTRON-PAIR CENTER-OF-MASS-MOTION . . .
4217
TABLE II. Classification of the momentum-space extracule
density d̄( P) and the maximal subshell-pair contribution to it. Characteristics P max and d̄ max[d̄(Pmax) of the maximum of the
momentum-space extracule density d̄( P) are also given. Note that
d̄( P) is normalized to unity instead of N(N21)/2. A @ n # means A
310n . All values in hartree atomic units.
Z
Atom
Type
Subshell pair
P max
d̄ max
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
A
B
A
A
A
A
A
A
A
B
A
A
A
A
A
A
A
B
A
A
A
A
B
A
A
A
A
B
A
A
A
A
A
A
A
B
A
A
A
B
B
A
B
B
A
B
A
A
A
A
A
A
A
1s1s
1s2s
2s2s
2s2 p
2s2 p
2s2 p
2s2 p
2s2 p
2s2p
2s2p
3s3s
3s3p
3s3p
3s3p
3s3p
3s3p
3s3p
3s3p
4s4s
4s4s
4s4s
4s4s
3p3d
4s4s
4s4s
4s4s
4s4s
3p3d
4s4s
4s4p
4s4p
4s4p
4s4p
4s4p
4s4p
4s4p
5s5s
5s5s
5s5s
4p4d
4p4d
5s5s
4p4d
4p4d
4p4d
4p4d
5s5s
5s5p
5s5p
5s5p
5s5p
5s5p
5s5p
0
0.3085
0
0
0
0
0
0
0
0.3316
0
0
0
0
0
0
0
0.2266
0
0
0
0
0.3528
0
0
0
0
0.4471
0
0
0
0
0
0
0
0.1949
0
0
0
0.2754
0.3018
0
0.3157
0.3299
0
0.3686
0
0
0
0
0
0
0
4.1821@ 21 #
1.8265@ 21 #
1.2023
6.7778@ 21 #
3.5025@ 21 #
1.9400@ 21 #
1.3633@ 21 #
9.3930@ 22 #
6.5336@ 22 #
4.9862@ 22 #
2.0743@ 21 #
2.3074@ 21 #
1.7735@ 21 #
1.3191@ 21 #
1.1795@ 21 #
9.8403@ 22 #
8.0187@ 22 #
6.7492@ 22 #
1.6733@ 21 #
1.3040@ 21 #
1.0537@ 21 #
8.6509@ 22 #
2.7148@ 22 #
6.0616@ 22 #
5.1287@ 22 #
4.3874@ 22 #
3.7798@ 22 #
1.3259@ 22 #
2.8622@ 22 #
4.0322@ 22 #
3.8252@ 22 #
3.4107@ 22 #
3.5322@ 22 #
3.3715@ 22 #
3.1034@ 22 #
2.8894@ 22 #
6.4037@ 22 #
5.2748@ 22 #
4.5232@ 22 #
2.0497@ 22 #
1.8214@ 22 #
3.0766@ 22 #
1.5290@ 22 #
1.3964@ 22 #
1.2893@ 22 #
1.1653@ 22 #
1.8869@ 22 #
2.5326@ 22 #
2.5269@ 22 #
2.3773@ 22 #
2.5394@ 22 #
2.5207@ 22 #
2.4153@ 22 #
FIG. 5. Major subshell-pair contributions to the momentumspace extracule density d̄( P) of the Zn atom (Z530). All values in
hartree atomic units.
FIG. 6. Major spin-orbital-pair contributions to the momentumspace extracule density d̄( P) of the Zn atom (Z530). ~a! Pairs
with different spins. ~b! Pairs with the same spin. All values in
hartree atomic units.
4218
KOGA, MATSUYAMA, ROMERA, AND DEHESA
subshell-pair contribution given in Table II, but the type B
modality comes from secondary subshell-pair contributions
with a maximum at P.0.
Figure 6 shows some spin-orbital-pair contributions
d̄ jk ( P) to d̄( P) for the Zn atom. Figure 6~a! compares the
contributions from five spin-orbital pairs with different spins;
4s a 4s b , 3s a 4s b , 3 p 0 a 3d 0 b , 3d 21 a 3d 11 b , and
3d 0 a 4s b . Due to Eq. ~13a!, the behavior of these contributions is essentially the same as those examined @4# for the
intracule density. The first four spin-orbital pairs have type A
distributions, whereas the last spin-orbital pair has type B
distribution. When 3d orbitals participate, the contribution of
the spin-orbital pair is about 1/100 times smaller than that of
the 4s a 4s b spin-orbital pair. Figure 6~b! shows the contributions from four spin-orbital pairs with the same spin;
3s a 4s a , 3d 0 a 4s a , 3d 21 a 3d 11 a , and 3p 0 a 3d 0 a . As
discussed in the Sec. IV A the spin-orbital-pair contributions
with the same spin and even l j 1l k have zero value at P
50 due to the counterbalance hole in the Hartree-Fock
theory, and hence such spin-orbital pairs always contribute to
type B distributions. The first three pairs belong to this
group. On the other hand, the 3p 0 a 3d 0 a pair with l j 1l k
53 is unimodal with a maximum at P50.
The analysis of spin-orbital-pair contributions leads us to
the conclusion that in momentum space, the type A modality
of the extracule density is mainly due to electrons with different spins in SS and S P subshell pairs. On the other hand,
the type B modality comes mainly from electrons in SD
@1# A. J. Coleman, Int. J. Quantum Chem. S1, 457 ~1967!.
@2# A. J. Thakkar, in Density Matrices and Density Functionals,
edited by R. M. Erdahl and V. H. Smith, Jr. ~Reidel, Dordrecht, 1987!, pp. 553–581.
@3# T. Koga and H. Matsuyama, J. Chem. Phys. 107, 8510 ~1997!.
@4# H. Matsuyama, T. Koga, E. Romera, and J. S. Dehesa, Phys.
Rev. A 57, 1759 ~1998!.
@5# T. Koga and H. Matsuyama, J. Chem. Phys. 108, 3424 ~1998!.
@6# I. N. Levine, Quantum Chemistry, 4th ed. ~Prentice Hall,
Englewood Cliffs, NJ, 1991!, p. 315.
@7# E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra ~Cambridge University Press, London, 1970!, pp. 175–
176.
@8# A. Messiah, Quantum Mechanics ~North-Holland, Amsterdam,
57
subshell pair and from electrons with the same spin in SS 8 ,
P P, and DD subshell pairs.
V. CONCLUSION
Within the Hartree-Fock framework, the theoretical structure of the spherically averaged extracule densities d(R) in
position space and d̄( P) in momentum space has been clarified in relation to the corresponding intracule densities h(u)
and h̄( v ). Using the numerical Hartree-Fock method, we
have systematically constructed these extracule densities for
the atoms He through Xe in their ground state. The contributions of subshell pairs and spin-orbital pairs to the total
extracule density have been analyzed in both spaces. For all
the 53 atoms, the position-space extracule density has been
found to be a unimodal function with a maximum at R50
due to the predominant 1s1s subshell-pair contribution. In
momentum space, however, the extracule densities have
been classified into two types based on their modalities. We
have further found that the different modalities in the
momentum-space extracule densities originate from a few
outermost subshell pairs.
ACKNOWLEDGMENT
This work was supported in part by a Grant-in-Aid for
Scientific Research from the Ministry of Education of Japan.
1961!, Vol. 1, p. 497.
@9# T. Koga and H. Matsuyama, J. Chem. Phys. 107, 10 062
~1997!.
@10# T. Koga, J. Chem. Phys. 108, 2515 ~1998!.
@11# C. E. Moore, Atomic Energy Levels, Natl. Bur. Stand. ~U.S.!
Circ. No. 35 ~U.S. GPO, Washington, DC, 1971!, Vols. 1–3;
C. E. Moore, Ionization Potentials and Ionization Limits Derived from the Analysis of Optical Spectra. Natl. Bur. Stand.
~U.S.! Circ. No. 34 ~U.S. GPO, Washington, DC, 1970!.
@12# C. Froese-Fischer, Comput. Phys. Commun. 4, 107 ~1972!.
@13# J. D. Talman, Comput. Phys. Commun. 30, 93 ~1983!.
@14# J. C. Slater, Quantum Theory of Atomic Structure ~McGrawHill, New York, 1960!, Vol. 1, pp. 488–490.
@15# T. Koga and H. Matsuyama, J. Phys. 30, 5631 ~1997!.