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A simple O(4,2) approximation of hydrogenic coulomb integrals
Volume 23, number 3
I December 1973
CHEMICAL PHYSltS LETTERS
A SLMPLE 0(4,2)
APPROXIiMATlON
FOR HYDROGENIC
Carl WULFMAN and Sukeyuki Department
INTEGRALS?
COULOMB
KUMEI
of Physics, Unirlersiry of i7re Pacifk. Srock~on, Cali]orrria 95204.
USA
Received 4 September 1973
Repulsion
integrals
for electrons with hydrogenic wavefunctions are erpresscd in terms ol opcralions X 0(4,2). The result is used to obtain approximate expressions for Coulomb quantum numbers.
algebra of the group O(4.2) functions
ofhydrogenic
of the Lie integrals as
Even in the case of hydrogenic wavefunctions, electron repulsion integrals have a very complex dependence upon the quantum numbers of the functions involved. Indeed, no one has given an explicit formulation of this dependence in the general case, in part perhaps, because the result would be’too complicated to be directly useful. However, it has been recognized recently that the quantum mechanics of an n-particle Coulomb system is formulable in terms of operations of the direct sum ofrz 0(4,2) algebras and the corresponding group, acting upon a denumerable basis [l] . This brings new structure and techniques to bear upon the integrals problem. Using the notation of BednCr [2] the two-electron integral involving hydrogenic functions qOj = 9 ,,~mcl(~,~;ln,) is
Cl) ;
Y = n,“b”cr~d/(ZaZbZcz~)“2 SU-bh = ‘a-‘b,
I’
I
5. = zyj)
= exp
- T,(j)
Liea T2(I?1
;
>bj = %?&n,“i) exp
iieb T2(k)]
T, = lb*-4
T2=r-p--i,
2
lAjk-Bjx.I
in terms of the dynamical
eb = h(zb/nb)
;
. variables r and p = -iv:
B=$(LXp-pXL)-r,
A=J(LXp-pXL)+r,
,
en = ln vp,,
rjk = IA(j)--A(k)-B(j)+B(k)l=
The generators have the following expression L=rXp,
,
-’ rS F=S uibk C]dk ’
;
T3 = t(rp*+r)
_
Note that exp (iOrz)f(r)
= eef(eer)
.
Because rf + rz 2 Pj*rk and the expectation
value of the LHS of this expression
is always much less than that
? Investigation supported by a grant from the Research Corporation.
,.
367
Volume 23, number 3
CHEMICAL PHYSICS LETTERS
% Errors
Table I of approximation to Coulomb
% = 100 x {(UbIT,&Ob)expct b
IS
1 December 1973
integrals
- ~u~If,~Iob~,pp~I/~ublr,~Iub~,,,~
2s
2P
3s
3P
3d
4s
4P
4d
4f
-4.5 7.3
-8.2 4.6 2.8 -2.8
-8.8 5.3 a.7
-13.4 2.0 6.9 6.8
-7.6 A.2 4.5 4.6
-11.5 -1.0 8.1 8.9
-15.3 -3.8 6.6 7.9
-12.7 -0.0 2.9 6.1
-5.7 -5.6 -25 4.4
2PQ
3p+
3PO
4P+
4PO
a 3.9
Is 2s 3s 4s b
w
U
2P+ 2PO 3p 3P0 4P+ 4P0
-3.0
0.1 -5.9
-2.2 -0.4 3.2
-0.4 A.0 6.8 0.1
b
2P+
2PO
3P+
3P0
3d-H
-6.9
3d+ 3dO
-5.2 -4.6
-3.3 -6.9 -8.3
-6.8 -5.3 3.9 6.3 5.3
3d-H
-5.3 -8.2 6.3 1.7 9.1 2.0 3d+
3dO
U
0.2 2.3 3.1
of the RHS for hydrogenic and obtain V=
~-‘(abl@+F;)-~‘~-
An inspection
J
-1.3
4.6 0.2 -1.1
eigenstates,
-3.4
-2.9
-6.6
-4.3 -9.6
we expand the operator in (I) as a power series in R = (Z.s*$)@f+$)-’
~_~I+~R+$R~++)Ic~.
of the matrix elements ofd,
(2)
B, and S shows that when rza = nr and “b = nd
(obl(~~+~~)l~d)=~~,~~,Cabl(i;z+~)l~b?
and that when nn + nc, nb #“d; the matrix elements are either zero, small compared to the diagonal elements, of variable sign. We therefore make the general approximation (nbI~(~~+~~)lcd)=f((obl(~~+~)lub))6=,66d
or
.
Inspection of the matrix elements of the generators then shows that the terms linear in R have no diagonal elements, so that through terms in R2 the Coulomb integral is approximated as Vubab = u-‘~ubl~F~l~b~{bbl~+~l~b))-L’*
{I + ~(obl(~~.~~)21ub)((rrbl~~+~lab))-2}
.
Using the matrix elements of the generators of the group, the matrix elements appearing in this expression are easily evaluated algebraically and one fmds ,g - +r,(r,+ 1)) + (n b /z
b j21Bn
;
-
flb(‘b+l))
7
Volume
23, number
Q(nINI) = (ll/z)‘{3!7’
1 December 1973
CHEMICAL PHYSlCS LE’lTERS
3
- 1(1+ I) ]
(I+ 1)2-,J
+ 12-n?
[ qf+l)‘--1
41*-l
1 .
The results are We have evaluated a variety of Coulomb integrals with Z, = Z, = 1, using this approximation. compared with the exact values of Butler et al. [3] in table 1. It is evident that the approximation should be a useful one in studies of systematics, though some caution should be exercised when 1s electrons are involved. We remark that in no case was the expecta:ion value of jR2 greater than 16)s of that of R”, the ratio often being less than lo-*. However, the R* terms are essential for they express the ITI,, !tzb dependence of the integrals. They correctly order the energies of the (r~lm)~(r~‘lm’)~ and (1~2nz)~(n’lm’)* integrals. However, it would be necessary to include
further
terms in the expansion
to properly
order
grals involving electrons with higher angular momenta. Finally, we note that the approximation is equivalent
the (n2n1)2(n’2rtz’)‘-
to making
energies
the two operator
and the energies
of inte.
approximations
and the bperator substitution
(ri-r,,_)2= ((A,--Bi) - (Ak-Bk))2 in eq. (2). References [l] C. Wulfman, in: Group theory and its appliutions. [2] hl. Bed&, Ann. Phys. 75 (1973) 305. (31 P.H. But1er.P.E.H. Minchin and B.C. Wybourne,
Vol. 2, ed. EM. Atomic
Loebl (Academic
Press, New York.
1971) pp. 145-197.
Data 3 (1971) 153.
369
I December 1973
CHEMICAL PHYSltS LETTERS
A SLMPLE 0(4,2)
APPROXIiMATlON
FOR HYDROGENIC
Carl WULFMAN and Sukeyuki Department
INTEGRALS?
COULOMB
KUMEI
of Physics, Unirlersiry of i7re Pacifk. Srock~on, Cali]orrria 95204.
USA
Received 4 September 1973
Repulsion
integrals
for electrons with hydrogenic wavefunctions are erpresscd in terms ol opcralions X 0(4,2). The result is used to obtain approximate expressions for Coulomb quantum numbers.
algebra of the group O(4.2) functions
ofhydrogenic
of the Lie integrals as
Even in the case of hydrogenic wavefunctions, electron repulsion integrals have a very complex dependence upon the quantum numbers of the functions involved. Indeed, no one has given an explicit formulation of this dependence in the general case, in part perhaps, because the result would be’too complicated to be directly useful. However, it has been recognized recently that the quantum mechanics of an n-particle Coulomb system is formulable in terms of operations of the direct sum ofrz 0(4,2) algebras and the corresponding group, acting upon a denumerable basis [l] . This brings new structure and techniques to bear upon the integrals problem. Using the notation of BednCr [2] the two-electron integral involving hydrogenic functions qOj = 9 ,,~mcl(~,~;ln,) is
Cl) ;
Y = n,“b”cr~d/(ZaZbZcz~)“2 SU-bh = ‘a-‘b,
I’
I
5. = zyj)
= exp
- T,(j)
Liea T2(I?1
;
>bj = %?&n,“i) exp
iieb T2(k)]
T, = lb*-4
T2=r-p--i,
2
lAjk-Bjx.I
in terms of the dynamical
eb = h(zb/nb)
;
. variables r and p = -iv:
B=$(LXp-pXL)-r,
A=J(LXp-pXL)+r,
,
en = ln vp,,
rjk = IA(j)--A(k)-B(j)+B(k)l=
The generators have the following expression L=rXp,
,
-’ rS F=S uibk C]dk ’
;
T3 = t(rp*+r)
_
Note that exp (iOrz)f(r)
= eef(eer)
.
Because rf + rz 2 Pj*rk and the expectation
value of the LHS of this expression
is always much less than that
? Investigation supported by a grant from the Research Corporation.
,.
367
Volume 23, number 3
CHEMICAL PHYSICS LETTERS
% Errors
Table I of approximation to Coulomb
% = 100 x {(UbIT,&Ob)expct b
IS
1 December 1973
integrals
- ~u~If,~Iob~,pp~I/~ublr,~Iub~,,,~
2s
2P
3s
3P
3d
4s
4P
4d
4f
-4.5 7.3
-8.2 4.6 2.8 -2.8
-8.8 5.3 a.7
-13.4 2.0 6.9 6.8
-7.6 A.2 4.5 4.6
-11.5 -1.0 8.1 8.9
-15.3 -3.8 6.6 7.9
-12.7 -0.0 2.9 6.1
-5.7 -5.6 -25 4.4
2PQ
3p+
3PO
4P+
4PO
a 3.9
Is 2s 3s 4s b
w
U
2P+ 2PO 3p 3P0 4P+ 4P0
-3.0
0.1 -5.9
-2.2 -0.4 3.2
-0.4 A.0 6.8 0.1
b
2P+
2PO
3P+
3P0
3d-H
-6.9
3d+ 3dO
-5.2 -4.6
-3.3 -6.9 -8.3
-6.8 -5.3 3.9 6.3 5.3
3d-H
-5.3 -8.2 6.3 1.7 9.1 2.0 3d+
3dO
U
0.2 2.3 3.1
of the RHS for hydrogenic and obtain V=
~-‘(abl@+F;)-~‘~-
An inspection
J
-1.3
4.6 0.2 -1.1
eigenstates,
-3.4
-2.9
-6.6
-4.3 -9.6
we expand the operator in (I) as a power series in R = (Z.s*$)@f+$)-’
~_~I+~R+$R~++)Ic~.
of the matrix elements ofd,
(2)
B, and S shows that when rza = nr and “b = nd
(obl(~~+~~)l~d)=~~,~~,Cabl(i;z+~)l~b?
and that when nn + nc, nb #“d; the matrix elements are either zero, small compared to the diagonal elements, of variable sign. We therefore make the general approximation (nbI~(~~+~~)lcd)=f((obl(~~+~)lub))6=,66d
or
.
Inspection of the matrix elements of the generators then shows that the terms linear in R have no diagonal elements, so that through terms in R2 the Coulomb integral is approximated as Vubab = u-‘~ubl~F~l~b~{bbl~+~l~b))-L’*
{I + ~(obl(~~.~~)21ub)((rrbl~~+~lab))-2}
.
Using the matrix elements of the generators of the group, the matrix elements appearing in this expression are easily evaluated algebraically and one fmds ,g - +r,(r,+ 1)) + (n b /z
b j21Bn
;
-
flb(‘b+l))
7
Volume
23, number
Q(nINI) = (ll/z)‘{3!7’
1 December 1973
CHEMICAL PHYSlCS LE’lTERS
3
- 1(1+ I) ]
(I+ 1)2-,J
+ 12-n?
[ qf+l)‘--1
41*-l
1 .
The results are We have evaluated a variety of Coulomb integrals with Z, = Z, = 1, using this approximation. compared with the exact values of Butler et al. [3] in table 1. It is evident that the approximation should be a useful one in studies of systematics, though some caution should be exercised when 1s electrons are involved. We remark that in no case was the expecta:ion value of jR2 greater than 16)s of that of R”, the ratio often being less than lo-*. However, the R* terms are essential for they express the ITI,, !tzb dependence of the integrals. They correctly order the energies of the (r~lm)~(r~‘lm’)~ and (1~2nz)~(n’lm’)* integrals. However, it would be necessary to include
further
terms in the expansion
to properly
order
grals involving electrons with higher angular momenta. Finally, we note that the approximation is equivalent
the (n2n1)2(n’2rtz’)‘-
to making
energies
the two operator
and the energies
of inte.
approximations
and the bperator substitution
(ri-r,,_)2= ((A,--Bi) - (Ak-Bk))2 in eq. (2). References [l] C. Wulfman, in: Group theory and its appliutions. [2] hl. Bed&, Ann. Phys. 75 (1973) 305. (31 P.H. But1er.P.E.H. Minchin and B.C. Wybourne,
Vol. 2, ed. EM. Atomic
Loebl (Academic
Press, New York.
1971) pp. 145-197.
Data 3 (1971) 153.
369