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Configuration mixing within the.3s-3p subshell
Volume
CHEMICAL PHYSICS LETTERS
103. nttmbcr 1
CONFIGURATION Y-HO*
andC.
MIXING WITHIN THE 3s-3p
SUBSHELL
16 December
1983
*
WULFMAN*
Department of Physic.
University of Canterbury. Utrtktchurcb, New Zealand
Received
1983
20 September
.4 recently dlscovcred approximate N-zlectron constant of motion accurately determines confiiuration mixing due to interaction between electrons within the 3s. 3p hydrogenic subshell It is used to establish a close parallekm between this mtxing and that in the 2s. 2p shell. The physical interpretation of the mnstant is outlined.
1. introduction It was discovered some time ago that configuration mixmg in the doubly excited states of twoelectron atoms was governed by an approximate constant of the motion [ 1,2] _It was shown that ‘this constant, and another, were Casimir operators of an O(4) group and that as a consequence the coefficients of mixing within a hydrogenic shell were determined by angular momentum addition coefficients [I]. If Li and Ai are the angular momentum and Runge-tinz vectors of electron i, the constants found are (A’ - AZ)2 + (Ll + L2)2 , It has recently
{(A’
-&)
- (Ll +L”)p
been found that a generalization
_
(1)
of the first of these, viz.,
governs configuration mixing in atoms containing from two to eight electroIls with principal qtiantum number n equal to two. It fmes the confguration mixing among hydrogenic product states lnl, ml ) InZ2m2).-. so accurately that within this basis the overlap of the eigenstates of q with those of the hamiltonian is, to three significant figures, 0999 in all cases [3]. The mixing is l~lown to be stable to improvements in the oneelectron basis functions t3,41. When acting on hydrogenic available the relation [3] {(Ai)2 + (L’)*} Inilimi)
states In/m) the operator
CT can be replaced by several other operators.
= (,i2 - 1) InilinZi) _
One has
(3)
Using it and the identity
L2 =c(Li)2+2j~kti.Lk, i
(4)
one finds that on a basis of fixed n, CT is equivalent to l Research supported by US. Nattonal Scienoz Foundation.Grants ~~E8014165.1~~8211393. * On leave from Department of Phystcs, Jinan University, Guangzhou. China. * On leave from Department of Physics, University of the kxitic, Stockton. California 9511 I, USA.
0 00%2614/83/S (North-Holland
03.00 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
35
Volume
103. number r2 + N(N
CHEMICAL
1
-_2)(?22
-
PHYSICS
LETTERS
16 December
1) - L2 ,
1983
(5)
where IV is the number of electrons and r2= c(Ai)2
-2
i
CAi-Ak
.
(6)
j
As L2 is a constant of motion, and n,N are fried, it follows that r2 is a constant of motion in the basis, provided by antisymmetrized products of 2s and 2p functions, i.e. its matrix representatkn [r2] almost commutes with the matrix representation [H] of the hamiltonian,
on this basis. We note the following:
(i) If v2] commutes with [H] so also does [cr2 ] where c is any constant. (ii) A knowledge of the matrix elements of A’ is sufficient to determine [F2]. (iii) The matrix elements of Ai are [S] MWIA,Inlm)
= ollllA IlnlXlml~llll’m’)
,
(84
where the reduced matrix element has the values 011’llAIlnl)={(n+l+
1)(&i-
1)(1+1)/(21+3)}1/2,
I’=[+
= -{(n + i)(rz- r)r/(2r - 1)}1/2 ,
l’=l-1,
=O
I’flA
(iv) The ratio oft
Le
1,
(8b) (84
I_
WI
I* 2 t 1 and I + 1 --f 1 reduced matrix elements
M + lIIdlln~~~6~1llAllnl+1) = -{(21+ 1)/(21+3)}1/2 ,
(9)
is independent of n. It follows from these observations that the matrix representing r 2, obtained from that of A on a 3s, 3p basis
is 813 times the matrix representing r2 on a 2s, 2p basis. 77~s the hypothesis that the approximate constant of morion r2 (or equiv&ently, c)govems configuration interaction within the 3s. 3p hydrogenic subshell entails the prediction that the wnfi..ration_mixing coemcients within this s&shell have the same vaIues as the corresponding ones in the 2s. 2p shell. The accuracy of this prediction is indicated in table 1 where it is seen that in the 3s, 3p basis the overlap of the eigenstates of q and those of H is, to three &nificant figures, 0987 or better. One naturally wonders whether this result has its origin in a proportionality between the Slater integrals for n = 3 and those of n = 2. Table 2 however indic&es that the ratio #j,~zz;~; : R~~,w,Y;w, varies from 2.08 to 253 so that the similarity between the 3s, 3p manifold of states and the 2s, 2p manifold probably has a more subtle origin. That this is the case will become evident in the next sections.
2. cafculatioIls Using the scaled variables 7 = Zr, etc., of the l/Z expansion we have
(10) and btmQ be L&tbfma 36
anticommuting creation and.annih&tion
operators for the one-particle state with &in pro-
16 December 1983
CHEMICAL PHYSICS LETTERS
Volume 103. number 1
Table 1 Comparison of exact and O(4) configumtlon miung
Exact (approsvnate) mteractmn matrices +_m 200’
3$1-
400+
402+
2385
-3906
2
-390&
1836 (1605)
0 2
5961 ( -39oJz
0
11922
-780
2
-780
11049 (10362)
2 411+
5:1-
(
11563.8 ( -390 11325
0
-390 10690.8 (10003 8) 390
( 390
10452 (9765) )
0
19074 ( 39ofi
39oa
6 0 0+
28611 3904
jection
-390& 5250 (479 1) )
2
2
Q. kt
WapWeu)
Z’e2
0
0
Euct interaction matrices In terms of Slater integrals
18039 (17124) ) 390& 27414 (26271)
the second quantized form of Z-2H
be H = w + Z-’
vi, where
(114
Here the spincontracted -I’m’ elm
_ - c
(I
Using methods
operators
P are defied
by
b,mob”m’o .
set forth in detail by Chacon et al. [4] we reexpress
R as a function
of: the Slater integrals
Table 2 Comparison of hydrogenic Slater integralsa) Integral
n=2
n=3
Rho
R&lsJls#Ins
0.1503906
0.0664062
2.26471
R~PPWnp~S
0.1621091
0.0687934
235647
RitPJlPJwlS
0.0818906 0.1816406
0.0423177 0.0718678
2.07692 252743
0.0878906
0.0359881
2.44221
%vWlnp~np %vv.nPflnp a) All values are in atomic unils,e2/ao
and are taken from the compdation of Butler et al. 161. 37
CHEMICAL
Volume 103. namber 1
PHYSICSLE-ITERS
16 December 1983
I$$, &;fl, the number operator;D, the number operator for 3s occupancy; i2, the second quantized total angular momentum operator; F2, the second quantized version of P2, and c, the quadratic Casimir operator of the U(3) group with generators “ii’ _ \ire find
+,:F~,(3~--i2-~2-~2+2ND-5~55~).
(12)
Theopratorccanbe expressed as [4,7] 5(N-@2
-$(N-E)-23', wheres2 isthesecond tal spin angular momentum operator. Then we can write P as 8= vu + v. + v,,g, where ---~~=~FOOOD(D-~)+~~D(N-~)+;G~~D+~~~D(D+ 1-2Ky-i%~F:,ir@-2N+4). vo=-
r C’ jz si:o1
quantizedto-
’
Vti=~G;,N+ff+,ti(fi-
1)-&F;,[S#(&4)+3~2
+ 12S2].
(13)
Note that v. and vu0 commute with F2 and @but vu does not do so because it is a function of b. For P2 (and hence CT) to be an exact constant of motion, vu must vanish. Rewriting it as ~“=(-;Fo&-&c;,+;~*
-$F:r)B+(;F$-F&
+;I+1
-&&)02
+;F~'l)~~~SsD~+SDD~2+S~~~,
+(F&-fl,
(14)
we see that for v, to vanish when fi has its allowed eigenvalues of 0, 1,2 2nd fi its allowed eigenvalues, 1 through SD, SD,, SND must each vanish. From table 2 we find
8, the three expressions
SD= 24416
X low3 e2/ao,
SDD =
-32552
X 10m3e2/a 09
S ND = 4.1223 X 10-3e2/ao
.
(15)
The
values of these “S integrals” are of the order of l/l0 + l/20 of the average values of the Slater integrals. As can be seen in table 2 this makes the contribution of v,, to the hamiltonian matrix quite small in comparison to the conmbution of vu0 and v. together, i.e. it is the relative smallness of the ‘S integrals” and vu that is immediately responsible for CT being a good apprxiomate constant of motion in the 3s-3p manifold. The same statement is true for the 2s-2p manifold as the reader can verity using table 2 and eqs. (14), which are correct when n=2.
3. Discussion We have seen that if CT were an exact constant of motion for m-shell interactions, in-shell configuration mixing in the 2s, 2p shell and the 3s, 3p subsbell would be identical. In fact they are-very nearly so. This is a consequence of the fact that the dynamical symmetry that would require q to be a constant of motion is a good approximate symmetry within the shells. The smallness of the SD, SDD , SND seen in eqs. (15), which appears there to arise from a collusion between radial and angular integrals in (14) is due to this symmetry. In classical mechanics the generator of the corresponding one-parameter symmetry group is given by {CT -}, the Poisson bracket operator derived from (the classical analog of) q. For a prcof see ref. [S]. The operator of the group, exp@{ CT - )) acts on initial Points p&i, qt) to generate “group orbits”&(a), ~(a)). Th ese orbits either carry the system point along a trajectory or from one trajectoryto%nother. If at P@p,_@) there are no nearby trajectories of the same energy, then the group orbit coincides with a trajectog when {H, CT) T 0. The evolution operator exp(t (H-3) then carries the representative point along the orbit with0 = a(f). _ _ Ia the twoelectron case CT = (At - A2)2, + (L1 + L2)2. Here-(L j + L2)2 is itself a constant of motion and 5 --f (A’ -A2)2 + L(L + 1). Within a hydrogenic shell A’ a r’ ad one can study the correlated motion within SchrXdinger conGguration space.
CHEMICAL PHYSICS LETTERS
Volume 103. number 1 in the classical mechanical cr’) a A’, so that one can only ration space. Moreover in the from the qu~tum-rne~~~c~ nate the position dependence
16 December 1983
system obtain case of system
it is 69, the average value of # over a Kepler orbit that obeys the relation a crude understandmg of the correlation by studying it in classtcal configuthree or more electrons one cannot ehminate the momentum dependence by using the constancy of the angular momen~m. One can however ehmrin both the classical and quanti systems by studymg the momentum correlations
via Fock’s projection onto the hypersphere S3 in 4space [9J. There the classical motions are all locally rotations on the hypersphere that are generated by{CF-f_ In the corresponding qu~tum-rnech~~ problem the group generator CT is also a sum of products of O(4) rotation generators that act on a basis composed of antisymmetrized products of one-particle-functions, products of O(4) spherical harmonics Y,,+&Y,, 8,, 9,) and spin functions of-
Though the method of projection onto the curved space S3 reduces the number of classical variables from 6% to 3iV and is thus most economical for studies of electron correlation, this should not obscure the fact that ttt ordimtry pftase space tile motions iiw?lw irreducible correlations of the positionurrd-momerrtrlnr of every electron witli that of every other efectrott. There is another unusual Feature of the electron correlations discussed here. The operator & acts on 3p functions to produce both 3s and 3d functions so that one might expect C’s to be a constant of motion only if one allowed 3s-3p-3d mixing. It is certainly true that if we had allowed ourselves a 3s, 3p, 3d basis we would have found that the eigenstates of C’$ contained large admixtures of 3d functions. Despite this the marti represent+ tion ofC$ on the 3s, 3p basts, obtained from the matrices of A' on the same bm, almost commutes with the matrix representation ofH on this basis. Mixing in the 3s, 3p, 3d shell will be dealt with m another arxicle. The possibitity that electron correlation in the 3s, 3p subshel! could be related to that in the 2s, 2p shell by group theoretic advents was fust ~vesttgated by Novaro and Freyre [7f at a time when it was believed that (~~Ai)2 + be an approximate constant of motion. Novaro and Freyre wished to obtain a better understanding of the close parallelism between the properties of the 2s, 2p shell and the 3s, 3p subshell. The anaiysis given here was st~ulated by their work, and is directed toward the same end _ (GjU)2
might
References [I ] C. Wulfman, Chem. Phys Letters 23 (1973) 370. [Z] D.R. HerrIck and 0 S~no~lu, Phys Rev. Al I (1975) 97; 0. Siiano~!uand D-R. Herrick, J. Chem. Phys 62 (1975) 886,65 (1976) 830. [3] C. Wulfman.Phys. Rev. Letters 51 (1983) 1159. (4) E. Chaoon.M. hloshinsky, 0. Nomro and C. Wulfman, Phys. Rev. A3 (1971) 166. f.51 L.C. i3iedenham.J. -Math. Phys 2 (1961)433. 161 P-H. Butler, PEH. h¶mchin and B.C.Wyboume, At. Data 3 (1971) 153. [6] 0. Novaro and A. rreyre, hiof. Phys 20 (1971) 861. 181C. W~lfman, Recent advances m group theory and Lheir apphutton IO spectroscopy, York, 1979) pp. 376-378. [9j G Cyorgyr. Nuovo Cmento 53A (1968) 717.
ed. J.C. Denim (Plenum
FHSS,
N~H
39
CHEMICAL PHYSICS LETTERS
103. nttmbcr 1
CONFIGURATION Y-HO*
andC.
MIXING WITHIN THE 3s-3p
SUBSHELL
16 December
1983
*
WULFMAN*
Department of Physic.
University of Canterbury. Utrtktchurcb, New Zealand
Received
1983
20 September
.4 recently dlscovcred approximate N-zlectron constant of motion accurately determines confiiuration mixing due to interaction between electrons within the 3s. 3p hydrogenic subshell It is used to establish a close parallekm between this mtxing and that in the 2s. 2p shell. The physical interpretation of the mnstant is outlined.
1. introduction It was discovered some time ago that configuration mixmg in the doubly excited states of twoelectron atoms was governed by an approximate constant of the motion [ 1,2] _It was shown that ‘this constant, and another, were Casimir operators of an O(4) group and that as a consequence the coefficients of mixing within a hydrogenic shell were determined by angular momentum addition coefficients [I]. If Li and Ai are the angular momentum and Runge-tinz vectors of electron i, the constants found are (A’ - AZ)2 + (Ll + L2)2 , It has recently
{(A’
-&)
- (Ll +L”)p
been found that a generalization
_
(1)
of the first of these, viz.,
governs configuration mixing in atoms containing from two to eight electroIls with principal qtiantum number n equal to two. It fmes the confguration mixing among hydrogenic product states lnl, ml ) InZ2m2).-. so accurately that within this basis the overlap of the eigenstates of q with those of the hamiltonian is, to three significant figures, 0999 in all cases [3]. The mixing is l~lown to be stable to improvements in the oneelectron basis functions t3,41. When acting on hydrogenic available the relation [3] {(Ai)2 + (L’)*} Inilimi)
states In/m) the operator
CT can be replaced by several other operators.
= (,i2 - 1) InilinZi) _
One has
(3)
Using it and the identity
L2 =c(Li)2+2j~kti.Lk, i
(4)
one finds that on a basis of fixed n, CT is equivalent to l Research supported by US. Nattonal Scienoz Foundation.Grants ~~E8014165.1~~8211393. * On leave from Department of Phystcs, Jinan University, Guangzhou. China. * On leave from Department of Physics, University of the kxitic, Stockton. California 9511 I, USA.
0 00%2614/83/S (North-Holland
03.00 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
35
Volume
103. number r2 + N(N
CHEMICAL
1
-_2)(?22
-
PHYSICS
LETTERS
16 December
1) - L2 ,
1983
(5)
where IV is the number of electrons and r2= c(Ai)2
-2
i
CAi-Ak
.
(6)
j
As L2 is a constant of motion, and n,N are fried, it follows that r2 is a constant of motion in the basis, provided by antisymmetrized products of 2s and 2p functions, i.e. its matrix representatkn [r2] almost commutes with the matrix representation [H] of the hamiltonian,
on this basis. We note the following:
(i) If v2] commutes with [H] so also does [cr2 ] where c is any constant. (ii) A knowledge of the matrix elements of A’ is sufficient to determine [F2]. (iii) The matrix elements of Ai are [S] MWIA,Inlm)
= ollllA IlnlXlml~llll’m’)
,
(84
where the reduced matrix element has the values 011’llAIlnl)={(n+l+
1)(&i-
1)(1+1)/(21+3)}1/2,
I’=[+
= -{(n + i)(rz- r)r/(2r - 1)}1/2 ,
l’=l-1,
=O
I’flA
(iv) The ratio oft
Le
1,
(8b) (84
I_
WI
I* 2 t 1 and I + 1 --f 1 reduced matrix elements
M + lIIdlln~~~6~1llAllnl+1) = -{(21+ 1)/(21+3)}1/2 ,
(9)
is independent of n. It follows from these observations that the matrix representing r 2, obtained from that of A on a 3s, 3p basis
is 813 times the matrix representing r2 on a 2s, 2p basis. 77~s the hypothesis that the approximate constant of morion r2 (or equiv&ently, c)govems configuration interaction within the 3s. 3p hydrogenic subshell entails the prediction that the wnfi..ration_mixing coemcients within this s&shell have the same vaIues as the corresponding ones in the 2s. 2p shell. The accuracy of this prediction is indicated in table 1 where it is seen that in the 3s, 3p basis the overlap of the eigenstates of q and those of H is, to three &nificant figures, 0987 or better. One naturally wonders whether this result has its origin in a proportionality between the Slater integrals for n = 3 and those of n = 2. Table 2 however indic&es that the ratio #j,~zz;~; : R~~,w,Y;w, varies from 2.08 to 253 so that the similarity between the 3s, 3p manifold of states and the 2s, 2p manifold probably has a more subtle origin. That this is the case will become evident in the next sections.
2. cafculatioIls Using the scaled variables 7 = Zr, etc., of the l/Z expansion we have
(10) and btmQ be L&tbfma 36
anticommuting creation and.annih&tion
operators for the one-particle state with &in pro-
16 December 1983
CHEMICAL PHYSICS LETTERS
Volume 103. number 1
Table 1 Comparison of exact and O(4) configumtlon miung
Exact (approsvnate) mteractmn matrices +_m 200’
3$1-
400+
402+
2385
-3906
2
-390&
1836 (1605)
0 2
5961 ( -39oJz
0
11922
-780
2
-780
11049 (10362)
2 411+
5:1-
(
11563.8 ( -390 11325
0
-390 10690.8 (10003 8) 390
( 390
10452 (9765) )
0
19074 ( 39ofi
39oa
6 0 0+
28611 3904
jection
-390& 5250 (479 1) )
2
2
Q. kt
WapWeu)
Z’e2
0
0
Euct interaction matrices In terms of Slater integrals
18039 (17124) ) 390& 27414 (26271)
the second quantized form of Z-2H
be H = w + Z-’
vi, where
(114
Here the spincontracted -I’m’ elm
_ - c
(I
Using methods
operators
P are defied
by
b,mob”m’o .
set forth in detail by Chacon et al. [4] we reexpress
R as a function
of: the Slater integrals
Table 2 Comparison of hydrogenic Slater integralsa) Integral
n=2
n=3
Rho
R&lsJls#Ins
0.1503906
0.0664062
2.26471
R~PPWnp~S
0.1621091
0.0687934
235647
RitPJlPJwlS
0.0818906 0.1816406
0.0423177 0.0718678
2.07692 252743
0.0878906
0.0359881
2.44221
%vWlnp~np %vv.nPflnp a) All values are in atomic unils,e2/ao
and are taken from the compdation of Butler et al. 161. 37
CHEMICAL
Volume 103. namber 1
PHYSICSLE-ITERS
16 December 1983
I$$, &;fl, the number operator;D, the number operator for 3s occupancy; i2, the second quantized total angular momentum operator; F2, the second quantized version of P2, and c, the quadratic Casimir operator of the U(3) group with generators “ii’ _ \ire find
+,:F~,(3~--i2-~2-~2+2ND-5~55~).
(12)
Theopratorccanbe expressed as [4,7] 5(N-@2
-$(N-E)-23', wheres2 isthesecond tal spin angular momentum operator. Then we can write P as 8= vu + v. + v,,g, where ---~~=~FOOOD(D-~)+~~D(N-~)+;G~~D+~~~D(D+ 1-2Ky-i%~F:,ir@-2N+4). vo=-
r C’ jz si:o1
quantizedto-
’
Vti=~G;,N+ff+,ti(fi-
1)-&F;,[S#(&4)+3~2
+ 12S2].
(13)
Note that v. and vu0 commute with F2 and @but vu does not do so because it is a function of b. For P2 (and hence CT) to be an exact constant of motion, vu must vanish. Rewriting it as ~“=(-;Fo&-&c;,+;~*
-$F:r)B+(;F$-F&
+;I+1
-&&)02
+;F~'l)~~~SsD~+SDD~2+S~~~,
+(F&-fl,
(14)
we see that for v, to vanish when fi has its allowed eigenvalues of 0, 1,2 2nd fi its allowed eigenvalues, 1 through SD, SD,, SND must each vanish. From table 2 we find
8, the three expressions
SD= 24416
X low3 e2/ao,
SDD =
-32552
X 10m3e2/a 09
S ND = 4.1223 X 10-3e2/ao
.
(15)
The
values of these “S integrals” are of the order of l/l0 + l/20 of the average values of the Slater integrals. As can be seen in table 2 this makes the contribution of v,, to the hamiltonian matrix quite small in comparison to the conmbution of vu0 and v. together, i.e. it is the relative smallness of the ‘S integrals” and vu that is immediately responsible for CT being a good apprxiomate constant of motion in the 3s-3p manifold. The same statement is true for the 2s-2p manifold as the reader can verity using table 2 and eqs. (14), which are correct when n=2.
3. Discussion We have seen that if CT were an exact constant of motion for m-shell interactions, in-shell configuration mixing in the 2s, 2p shell and the 3s, 3p subsbell would be identical. In fact they are-very nearly so. This is a consequence of the fact that the dynamical symmetry that would require q to be a constant of motion is a good approximate symmetry within the shells. The smallness of the SD, SDD , SND seen in eqs. (15), which appears there to arise from a collusion between radial and angular integrals in (14) is due to this symmetry. In classical mechanics the generator of the corresponding one-parameter symmetry group is given by {CT -}, the Poisson bracket operator derived from (the classical analog of) q. For a prcof see ref. [S]. The operator of the group, exp@{ CT - )) acts on initial Points p&i, qt) to generate “group orbits”&(a), ~(a)). Th ese orbits either carry the system point along a trajectory or from one trajectoryto%nother. If at P@p,_@) there are no nearby trajectories of the same energy, then the group orbit coincides with a trajectog when {H, CT) T 0. The evolution operator exp(t (H-3) then carries the representative point along the orbit with0 = a(f). _ _ Ia the twoelectron case CT = (At - A2)2, + (L1 + L2)2. Here-(L j + L2)2 is itself a constant of motion and 5 --f (A’ -A2)2 + L(L + 1). Within a hydrogenic shell A’ a r’ ad one can study the correlated motion within SchrXdinger conGguration space.
CHEMICAL PHYSICS LETTERS
Volume 103. number 1 in the classical mechanical cr’) a A’, so that one can only ration space. Moreover in the from the qu~tum-rne~~~c~ nate the position dependence
16 December 1983
system obtain case of system
it is 69, the average value of # over a Kepler orbit that obeys the relation a crude understandmg of the correlation by studying it in classtcal configuthree or more electrons one cannot ehminate the momentum dependence by using the constancy of the angular momen~m. One can however ehmrin both the classical and quanti systems by studymg the momentum correlations
via Fock’s projection onto the hypersphere S3 in 4space [9J. There the classical motions are all locally rotations on the hypersphere that are generated by{CF-f_ In the corresponding qu~tum-rnech~~ problem the group generator CT is also a sum of products of O(4) rotation generators that act on a basis composed of antisymmetrized products of one-particle-functions, products of O(4) spherical harmonics Y,,+&Y,, 8,, 9,) and spin functions of-
Though the method of projection onto the curved space S3 reduces the number of classical variables from 6% to 3iV and is thus most economical for studies of electron correlation, this should not obscure the fact that ttt ordimtry pftase space tile motions iiw?lw irreducible correlations of the positionurrd-momerrtrlnr of every electron witli that of every other efectrott. There is another unusual Feature of the electron correlations discussed here. The operator & acts on 3p functions to produce both 3s and 3d functions so that one might expect C’s to be a constant of motion only if one allowed 3s-3p-3d mixing. It is certainly true that if we had allowed ourselves a 3s, 3p, 3d basis we would have found that the eigenstates of C’$ contained large admixtures of 3d functions. Despite this the marti represent+ tion ofC$ on the 3s, 3p basts, obtained from the matrices of A' on the same bm, almost commutes with the matrix representation ofH on this basis. Mixing in the 3s, 3p, 3d shell will be dealt with m another arxicle. The possibitity that electron correlation in the 3s, 3p subshel! could be related to that in the 2s, 2p shell by group theoretic advents was fust ~vesttgated by Novaro and Freyre [7f at a time when it was believed that (~~Ai)2 + be an approximate constant of motion. Novaro and Freyre wished to obtain a better understanding of the close parallelism between the properties of the 2s, 2p shell and the 3s, 3p subshell. The anaiysis given here was st~ulated by their work, and is directed toward the same end _ (GjU)2
might
References [I ] C. Wulfman, Chem. Phys Letters 23 (1973) 370. [Z] D.R. HerrIck and 0 S~no~lu, Phys Rev. Al I (1975) 97; 0. Siiano~!uand D-R. Herrick, J. Chem. Phys 62 (1975) 886,65 (1976) 830. [3] C. Wulfman.Phys. Rev. Letters 51 (1983) 1159. (4) E. Chaoon.M. hloshinsky, 0. Nomro and C. Wulfman, Phys. Rev. A3 (1971) 166. f.51 L.C. i3iedenham.J. -Math. Phys 2 (1961)433. 161 P-H. Butler, PEH. h¶mchin and B.C.Wyboume, At. Data 3 (1971) 153. [6] 0. Novaro and A. rreyre, hiof. Phys 20 (1971) 861. 181C. W~lfman, Recent advances m group theory and Lheir apphutton IO spectroscopy, York, 1979) pp. 376-378. [9j G Cyorgyr. Nuovo Cmento 53A (1968) 717.
ed. J.C. Denim (Plenum
FHSS,
N~H
39