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1.
Average angles between linear momenta of an electron in a subshell nl and another electron in a subshell nl are examined for the 102 atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. Congruency in the mathematical structures of the average interelectronic angles in position and momentum spaces leads to the theoretical results that with even |l–l| are exactly equal to 90°, while with odd |l–l| are always larger than 90°. Numerical analyses of 3,275 subshell-pair angles with odd |l–l| in the 102 atoms clarify that deviations of the total average interelectronic angles from 90° are mainly governed by subshell pairs with |n–n|1 and |l–l|=1, in contrast to the position-space results where only subshell pairs with n=n and |l–l|=1 are important.Acknowledgments. We thank Mr. T. Shimazaki for his assistance in the compilation of data. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. 相似文献
2.
Hisashi Matsuyama Toshikatsu Koga Yoshihisa Kawata 《Theoretical chemistry accounts》2006,115(4):287-290
Angular correlation coefficients τ
nl,n^′ l^′ [p] between linear momenta of an electron in a subshell nl and another electron in a subshell n′ l′ are studied for the 102 neutral atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. We theoretically find that electron momenta are negatively
correlated or uncorrelated; τ
nl,n^′ l^′ [p] < 0 when |l − l′|=1, while τ
nl,n^′ l^′ [p]=0 when |l − l′| ≠ 1. Numerical examinations of the atoms show that except for the He–B atoms, negative correlations are largest between
1s and 2p subshells, which have the most diffuse electron distributions in momentum space. 相似文献
3.
The radial electron-pair intracule (relative motion) H(u) and extracule (center-of-mass motion) D(R) densities in position space were known to reveal four types of maxima which are related to the four inner electron shells,
K, L, M, and N, of atoms. The corresponding radial electron-pair intracule Hˉ(v) and extracule Dˉ(P) densities in momentum space are studied for the 102 atoms from He (atomic number Z=2) to Lr (Z=103). The densities Hˉ(v) and Dˉ(P) are found to have either one maximum or two maxima, and the numbers of maxima in Hˉ(v) and Dˉ(P) are the same for 98 atoms. For these atoms, the locations υ
max and P
max and the heights Hˉ
max
and Dˉ
max
of the corresponding maxima satisfy the approximate relations υ
max ≅ 2P
max and Hˉ
max
≅Dˉ
max
/2. On the basis of their Z-dependence, the maxima in Hˉ(v) and Dˉ(P) of the 102 atoms are classified into five types. Shell-pair decompositions of the radial densities show that these maxima
reflect five outer electron shells of atoms.
Received: 24 January 2001 / Accepted: 12 March 2001 / Published online: 13 June 2001 相似文献
4.
Toshikatsu Koga 《Theoretical chemistry accounts》2004,112(5-6):431-434
The accuracies of approximate formulas are examined for several characteristics of the interelectronic angle density A(12), where 12 (012) is the angle subtended by the position vectors r1 and r2 of two electrons. Numerical results for 102 atoms show that simple approximations have sufficient accuracies for the moments with n=1–4, the central moments n with n=2, 4, and the kurtosis, when measured by the absolute and relative errors. For heavy atoms, however, the relative errors for the third central moment 3 and the skewness are large. 相似文献
5.
Electron correlation effects on the electronic structure of atoms were investigated by means of a variety of position and momentum space related properties such as radial one-electron densities and radial electron momentum densities, Compton profiles and radial electron pair distributions. The results were obtained from MR-SDCI wavefunctions utilizing very large basis sets and are discussed in a comparative manner, analysing characteristic features and trends. 相似文献
6.
For the 53 neutral atoms from He to Xe in their ground states, the average distances < u>
n
l
,
n
′
l
′ in position space and < v>
n
l
,
n
′
l
′ in momentum space between an electron in a subshell nl and another electron in a subshell n
′
l
′ are studied, where n and l are the principal and azimuthal quantum numbers of an atomic subshell, respectively. Analysis of 1700 subshell pairs shows
that the electron-pair distances < u>
n
l
,
n
′
l
′ in position space have an empirical but very accurate linear correlation with a one-electron quantity U
n
l
,
n
′
l
′≡L
r
+S
r
2/(3L
r
), where L
r
and S
r
are the larger and smaller of subshell radii < r>
n
l
and < r>
n
′
l
′, respectively. The correlation coefficients are never smaller than 0.999 for the 66 different combinations of two subshells
appearing in the 53 atoms. The same is also true in momentum space, and the electron-pair momentum distances < >
n
l
,
n
′
l
′ have an accurate linear correlation with a one-electron momentum quantity V
n
l
,
n
′
l
′≡L
p
+S
p
2/(3L
p
), where L
p
and S
p
are the larger and smaller of average subshell momenta < p>
n
l
and < p>
n
′
l
′, respectively. Trends in the proportionality constants between < u>
n
l
,
n
′
l
′ and U
n
l
,
n
′
l
′ and between < >
n
l
,
n
′
l
′ and V
n
l
,
n
′
l
′ are discussed based on a hydrogenic model for the subshell radial functions.
Received: 8 April 1998 / Accepted: 6 July 1998 / Published online: 18 September 1998 相似文献
7.
For the 102 atoms from He to Lr in their ground states, the average interelectronic angles <12>
nl, n'l'
between an electron in a subshellnl and another electron in a subshell n'l' are examined, where n and l are the principal and azimuthal quantum numbers, respectively. Theoretical study clarifies that <12>
nl,n'l'
are 90° precisely if l–l' are even, while they are larger than 90° if l–l' are odd. Numerical analysis of 3,275 subshell pairs with odd l–l' of the 102 atoms shows that the increases in the total average interelectronic angles <12> from 90° are attributed predominantly to subshell pairs with n=n' and l–l'=1. 相似文献
8.
Jean-Louis Calais 《Theoretical chemistry accounts》1993,86(1-2):137-147
Summary Fukutome's group theoretical classification scheme for determinants, based on the transformation properties of the Fock-Dirac density matrix under spin rotations and time reversal, has been extended to momentum space. Particular attention is paid to the transformation properties of orbitals and density matrices under inversion in momentum space. 相似文献
9.
The interelectronic angle density A(theta12) is the probability density function that the angle thetaij (0 < or = thetaij < or = pi) subtended by the vectors ri and rj of any two electrons i and j becomes theta12. For equivalent electrons in atoms, it is shown that the density A(theta12) in the Hartree-Fock theory is given by a simple polynomial of cos theta12. Detailed expressions are reported for all LS terms arising from s2, pN (N = 2-6), dN (N = 2-10), and f(N) (N = 2,12) electron configurations. With no modifications, the present results apply as well to the interelectronic angle density A(theta12) in momentum space, where theta12 is the angle between two electron momenta. 相似文献
10.
Summary Symmetric orthogonalisation is favourable to perform in momentum space, as this article will show. We have used a model of a body centered cubic lattice with 1s- and 2s-Slater orbitals centered at each atom site. Computer programs have been written to calculate the eigenvalues of the overlap matrix which play an important role in constructing symmetrically orthogonalised wavefunctions. 相似文献
11.
For 357 subshells of the 53 neutral atoms He through Xe in their ground states, the two-electron intracule (relative motion)
<u
k
>
nl
and extracule (center-of-mass motion) <R
k
>
nl
subshell moments in position space are examined as well as their counterparts <v
k
>
nl
and <P
k
>
nl
in momentum space, where n and l are the principal and azimuthal quantum numbers of the atomic subshell, respectively. It is clarified that between the intracule
and extracule moments the “2
k
-rule” is strictly valid, which means <u
k
>
nl
= 2
k
<R
k
>
nl
and <v
k
>
nl
= 2
k
<P
k
>
nl
for any nl subshell. Theoretical analysis also proves that for a particular case of k = +2, two relations <u
2>
nl
= (N
nl
−1)<r
2>
nl
and <v
2>
nl
= (N
nl
−1)<p
2>
nl
hold exactly, where N
nl
(≥2) is the number of electrons in the subshell nl, and <r
k
>
nl
and <p
k
>
nl
are the familiar one-electron subshell moments in position and momentum spaces, respectively. The latter equality establishes
a new and rigorous relation between the second electron-pair moments in momentum space and the total energy of an atom through
the virial theorem. For k=+1, −1, and −2, the numerical Hartree-Fock results for the 357 subshells show that there are approximate but accurate linear
relations between <u
k
>
nl
and <r
k
>
nl
and between <v
k
>
nl
and <p
k
>
nl
, in which the proportionality constant in each space depends on n,l, and k.
Received: 27 April 1998 / Accepted: 29 May 1998 / Published online: 28 August 1998 相似文献
12.
Toshikatsu Koga 《Theoretical chemistry accounts》2002,107(4):246-249
On the basis of multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule
(center-of-mass motion) properties are reported for the Li atom in momentum space. The present results are more accurate and
consistent than those in the literature.
Received: 10 September 2001 / Accepted: 11 December 2001 / Published online: 22 March 2002 相似文献
13.
The integrated Hellmann-Feynman theorem is used to derive a rigorous relation between the energy and the electron density in momentum space. Choosing the electron mass as a differential parameter, we obtain a formula corresponding to the Wilson-Frost formula in coordinate space. Analysing the mass-dependence of momentum density, we then show that the present formula is equivalent to one of the previous results deduced from the virial theorem. Use of the integral Hellmann-Feynman theorem is also discussed. Several illustrative examples are given for the calculation of energy from momentum density. 相似文献
14.
Summary One-electron density matrices, which are representable in single-centers-orbital basis sets, have been investigated with respect to their reconstruction from densities. The maximum allowed dimension for reconstruction from a combination of position & momentum density dependent properties is only slightly bigger than the dimension in the case of position (or momentum) densities only. Since for a given one-particle basis of dimensionM, the number of one-matrix elements which can be determined is also of orderM only, while the total number of one-matrix elements is of orderM
2, it is in general necessary to introduce severe constraints and restrictions. The accuracy demands on the data and algorithms increase exponentially for linearly increasing size of basis set. 相似文献
15.
Elvira Romera Toshikatsu Koga Juan Carlos Angulo Jesus S. Dehesa 《Journal of mathematical chemistry》2000,28(4):341-351
Modified functions r
–(r) and p
–(p) of the spherically averaged electron densities (r) in position space and (p) in momentum space are found to be convex (i.e., the second derivatives are nonnegative everywhere) for all the 103 ground-state atoms from hydrogen (atomic number Z=1) to lawrencium (Z=103), if the parameters are chosen to be 0.6 and 1.4. The convex property of r
–(r) and p
–(p) is used to derive upper bounds to the density functions (r) and (p) in terms of their radial moments r
s
and p
s
or frequency moments
t
and
t
. In most cases, the present bounds are shown to be more general and more accurate than those reported in the literature. 相似文献
16.
Toshikatsu Koga 《Theoretical chemistry accounts》2000,105(1):96-99
The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized
electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a
r
j
+b
r
k
of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to
the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results.
Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000 相似文献
17.
To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters. 相似文献
18.
A direct application of the Hirshfeld atomic partitioning (HAP) scheme is implemented for molecular electron momentum densities (EMDs). The momentum density contributions of individual atoms in diverse molecular systems are analyzed along with their topographical features and the kinetic energies of the atomic partitions. The proposed p-space HAP-based charge scheme does seem to possess the desirable attributes expected of any atoms in molecules partitioning. In addition to this, the main strength of the p-space HAP is the exact knowledge of the kinetic energy functional and the inherent ease in computing the kinetic energy. The charges derived from HAP in momentum space are found to match chemical intuition and the generally known chemical characteristics such as electronegativity, etc. 相似文献
19.
When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D
<(r) and outer D
>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V
en and the diamagnetic susceptibility χ
d are the sum of the inner and outer contributions. The electron–electron repulsion energy V
ee has an approximate relation with the minus first moment of the outer density D
>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities
D
<(r), D
>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components
to V
en and are also discussed for these atoms. 相似文献
20.
Toshikatsu Koga 《Theoretical chemistry accounts》1989,75(6):417-424
Based on momentum- and position-space analyses of the moment operators for two-electron atoms, it is shown that there exists a family of two-electron wave functions which satisfy a proportionality relation, r/
1
v
/r
12
v
=p/
1
v
/p
12
v
=2–v/2, between the one and two-electron moments in position and momentum spaces, where v is an arbitrary number for which the moments are well-defined. 相似文献