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Hypervirial Theorems and SCF Wavefunctions for Coupled Oscillators
Physics 58 (1981) 65-69 North-HollandPublishingCompany
Comical
H’YPERVIRIAL FOR COUPLED
THEOREMS AND OSCILLATORS
Francisco M. FERN&E2
SCF WAVEFUNCTIONS
and Eduardo A. CASTRO*
Znstifutode Znuestigaciones FikicoquimicasTe&icas y Aplicadas, Secci& Qtlimica Te&ica. La PIam 1900. Argentina Received 1 October 1980 Revised manuscript received 12 January
1981
The SCF formalism is analyzedin connection with hyperviriaf theorems. Coupled oscillator models are chosen in order
to
illustratethe formalresults.
1. introduction For &J a trial wavefunction and H the hamiltonian operator of a given system, Epstein and Hirschfelder [l] demonstrated that for a variation &5 = i W&, with W an hermitian operator, the extremum condition 6E = 0 conducts towards the diagonal hypervirial theorem
the SCF method consists in searching for those optimum functions which make the functional energy E, E(4) = (+lHd),
(3)
an extremum. That is to say s,E=(6~~1Hb)+(~IH~~~)=O,
(4)
where
The SCF method is of a variational nature, so that the corresponding SCF wavefunction will satisfy certain hypervirial relationships. When taken into account such relationships provide us with very useful formulas, as will be shown later. The application of the SCF method to coupled oscillator systems has recently been made by Bowman [Z], and by Tobin and Bowman [S]. If we approximate the eigenfunctions of N as a product of basis functions Qi
Due to the fact that functions & have to satisfy the condition (Z), the variations S&k must have
the form
84, = A,&,
where A,, is an antihermitian operator (A: = -A& Replacing eqs. (5) and (6) in eq. (4) we conclude that 4 fulfills the hypervirial theorems for any antihermitian operator which depends on just one coordinate, i.e., (&l[H. A&#)
subject to the normalisation (&i&)=1; *
i=l,2
condition
,_.., N,
Author to whom correspondence
0301-10l4/81/0000-0000/S02.50
(2) shouid be addressed.
@ North-Holland
(6)
= 0.
(7)
If w(xk) is an arbitrary linear Operator which depends on one coordinate, then ( W- W’)4 .zs well as i( W + W’)+ are compatible variations for 4, so that it follows at once that (&x
WI&) = 0.
Publishing Company
(8)
F.&f_ Femdndrz.
66
EA.
As an immediate consequence, we can assert that eq. (8) will be valid for any linear operator which is expressible as a sum of one-dimensional linear operators.
Castro / Hyperuirial
Such se!ection is class of operator physics systems. (8) adopt a very (Gl[K
2. Hyperviria!
theorems
theorem
wk
We have seen that SCF solutions satisfy the diagonal hypervirial theorems for any unidimensional operator. Now we wish to demonstrate the converse statement: If a product-like function (1) satisfies the diagonal hypervirial theorems for any unidimensionai linear operator, then d is an SCF solution for the system. In order to demonstrate this theorem, we follow Hirschfelder [4]. Assuming 4 reaI and replacing K’(Q) = f(x*)L& (with DC = a/&x, and f(x) an arbitrary function such that fDkQ belongs to the domain of H) in eq. (S), we get
wkw)
supported by the fact that this is typical of several chemical Diagonal hypervirial equations simple form too
= <@kI[Fk,
Wk$#‘k)
= 0;
(14)
= wk (xk )-
The application of this last derive certain relationships useful when managing SCF consecutive replacement of for W, in eq. (14) gives the
formula allows us to which will be very equations_ The f(xk)Dk and f(xn) following equations:
-~(~~:lf)D~~~)k)-(~lcif)~:~li) -(&lf(v:
+h)‘dk)=o,
-f(~liI~~~;)-(C58IfDlc~r~)
(15) = 0.
(16)
When f has an appropriate form, then it is possible to eliminate all the terms Dk and Di from eqs. (1 l), (15) and (16). For j(xk) =xc we get
(9) where
we suppose
.\’ H=X.HP i=*
-2AQe1x:-r
that
+ V(.x,,
-.. .Xx).
(10)
l
Denoting by Fk the SCF operator z i {&&~,,ll’&~&) we deduce from eq. (9) (F;,&)l&
(11)
=&Jr-,
where e,, is a scalar. In this work we are interested in those hamiltonians as (10) with interaction potentials like V(xr,
-.. ,x.X) =c
1 g,jui(xi)ui(xj), following
form
3.
Coupled
oscillator models the hamiltonian
H = -i(~f+~f)+Ir~vT~:+~~,ix~)i~~fxS. The SCF oscillators
for
(17)
In the next section we will see that eq. (17) allows a great simplification when dealing with integro-differential equations (11).
Model I. Let us consider operator [5]
(12)
i j
which leads to the simple SCF operators
(VE -iZllc)&)
-(~klX~(V~+r(li~rPk)=O-
(18)
equations correspond to two harmonic with effective operators
F;=Hyi-ui; (13)
F,=
-~D~+3,v;+2aAr)x;;
A~=(~+:@JI). (20)
FM.
The
eigenenergies
Ferntinder.
EA.
en,= (nz+l/2)(wf+2aAz)“-.
(22)
l/2 ,
A,=(n1tl/2)(rvti2aA1)-
(23)
A, = (n2+1/2)(~v~i2aAz)-‘/‘.
(24)
l?qs. (23) and (24) can be deduced in a very simple fashion by minimizing the total energy [2,3] with respect to A, and A2:
(25)
where b is a parameter satisfies the equation = 0,
to be adjusted
so that &
V-N.
(26)
From the resolution of eq. (26) we find that the optimum value for b is unique for any iV, and it satisfies the relationship b-0.125
R =O.
(27)
For a = 1, b = 0.66236, and thus (I;) = (xf) = 0.37744. This result corresponds to identical values obtained from eqs. (23) and (24). Model II. This model has a hamiitonian operator H = -$(Df+Df:if(W:XT+
and it has been operators are Fl=
,vfxf)
+ rx:,, widely
T,-&ufx:+pA~x,
(28) studied +prx:,
[7-181.
The
WI and the eigenvalues
are
e,,2=(nzif)~CliprA;.
(31)
A1 =(n2i$j~C~‘.
This fact confirms that hypervirial equations (17) are correct. Eqs. (23) and (24) can be solved numerically without further ditiiculties. In order to verify the theorem presented in section 2, we consider the hamiltonian (18) with wf = tvf = 1, and we choose as trial wavefunction for the ground state d(b,~~,~~)=Nexp[-_(xttx~)],
The eigenfunctions of F2 correspond to an harmonic oscillator with an effective frequency
The utiiization of eq. (17) with N = 1 for F, gives for A1 the result
E nrnz=en,+ez2-2aA1A2.
t&xl
67
(21)
1
The problem is solved by using eq. (17) for N = 1, k = 1,2. In such a way, we obtain two relationships which relate At with AZ by the simple expression [6]
b3-0.25
&-orems
are
e,, = (nl;l/2)(rc,ft2aA1)“‘,
(&lr-i, XYDJd)
Castro / Hypemirial
SCF
(29)
(32)
On the contrary, F* (N=O, 1):
we need
- wfAr-pAi-3pr(xl’) 2e,,-
two relationships
for
= 0,
(33)
2~~.‘(xf)-3pA,Ar-5prA~=0.
(34)
Eqs. (33) and (34) allow us to express Ai as a function of A?, so that the numerical calculation of the necessary integrals is reduced to only one. The calculation scheme is as follows: we start the iteration cycle with F”’ = Tzi wf.x~/2 which gives us ezz and A:. Then we construct F\” =T,t-wfsfj2i~A:s~tpr.u:. From numerical integration we get e:,. From A:, ~2, and eqs. (33) and (34) we can express A’: as a function of AZ, so that we need to calculate numerically only one of the couple of integrals. tpA~.x~tprA~, and Then we form Fy’ =Fi” so on. This method is really efficient and useful when dealing with a large number of variables, because the necessary matrix elements to be evaluated are drastically reduced and it has a positive effect on the increase of the accuracy and the speed of the calculation. This model can easily be solved for r=O (Barbanis’ hamiltonian) e,, = (n,+&
-Af/2w;,
(35)
AZ = -pA,/w;. The final iteration between eqs. (32) conducts us to the desired solutions.
(36) and (36)
4. Conchding reinarks The SCF function (1) is eigenfunction operator
of the
(37)
F = fl Fik), which can he considered as a zero-order tonian
hamil-
H=F+H’,
(33)
where _M’ is a perturbation. If R is function of H, then we can write
an
condition
g
f
x (giixix: + giix:xi i g;x’xi’>
i>i
1
(44)
is treated by the SCF method, it leads to a set of SCF operators which are similar To those corresponding to shifted oscillators. It is an easy matter to see that in this case the trial function
(d IS) = 0. According to perturbation expressed as [19]
= 0.
Through a more complicate and tedious procedure, Sinanoglu [19] deduced a similar resu!t for atomic and moiecuIar systems. When the hamiltonian
(39)
with the orthogonality
=
(PI&)
eigen-
R=c$tS,
s
Eq. (43) is satisfied fOi any operator w(xk) whenever F,Z, = en,&,, or 2, = 0. The fust option does not fulfill the condition (S”‘l4) = 0, so that we are led to the second one, which is an orthogonality ccndition stronger than
theory, S can be
Q =n &7i(Qjxi + 6i)
s”‘.
(45)
I
i=*
From previous results presented by Robinson [20] and following our prior discussions in sections l-3, for any operator i[Fk, W(X~,)] with W’(xk) = W(xk), the first order corrections are null, i.e. (+,[[Fk,- W-IS”‘) = 0.
(41)
Due to the fact that the operator W’ is arbitrary, this result can be expressed in the following sentence: the first-order correction to the expectation value of any operator which can be expressed as a so/mof one-coordinate operators vanishes if Hartree wavefunctions are used in its evahiation 1211. Defining the function Zk (xt) as
-Z&k)
(with {@,} bekg the set of the harmonic oscillator eigenfunctions) is a SCF solution when the parameters ai and 6, are VariationaIIy optimized_ This constitutes a positive proof for our recent proposal of using the translation as a manner to introduce variational parameters in trial wavefunctions [22]. Finally, we point out that SCF wavefunction must obey the Hellmann-Feynman theorem, as previously stated [23-261.
References [l] S.T. Epsteinand J.O. Hinchfelder. Phys. Rev. 123 (1961) 1495. [2] J.M. BoHrnan. J. Chem. Phys. 68 (1978) 608. [3]
F.L.
Tobin
and J.M.
Bowman.
Chem.
Phys. 47
(1980)
151.
J.O. Hirschfelder. J. Chem. Phys. 33 (1960) 1462. [S] J.C. Percival and N. Pomphrey. Mol. Phys. 31 (1976) 97. [63 R.A. Harris. J. Chem. Phys. 72 (1980) 1776. [7] S. Nordholm and S.A. Rice, J. Chem. Phys. 61 (1974) 203. [4]
eq. (41) is transformed (wG%ki(Fk-%k)zkj=o-
in (43)
F.&l. Ferndrrdez, E.A. Casmro f Hypeminkl 183 S. Nordholm and S.A. Rice. J. Chem. Phys. 61 (1974) 768. [9] W. Evtes and R.A. Marcus, 3. Chem. Phys. 61 (1974) 43131.
[lo]
N. Pomphrey, J. Phys. B 7 (1974)
1909.
[ 1 I] D.W. Noid and R..4. ~Marcus, I. Chem. Phys. 62 (1975) 2119. [121 S. Nordholm and S.A. Rice, J. Chem. Phys. 62 (1975) 157. 1131 KS. Sorbic, Mol. Phys. 32 (1976) 1577. [14] S. Chapman, B.C. Garrett and W.H. Miller, J. Chem. Phys. 64 (1976) 502. [IS] D.W. Noid. M.L. Koszykowski and R.A. Marcus. J. Chem. Php. 67 (1977) 404. 1161 D.W. Noid and R.A. Marcus. J. Chem. Phys. 67 (1977) 559. [17] KS. Sorbic and NC. Handy, Mot. Phys. 33 (1977) 1319.
iheorems
69
[18] M. Cohen and S. Greita, Chem. Phys. Letters 60 (1979) 445. 1191 0. Sinano%lu. Proc. Roy. Sot. (London) A260 (1961) 379. [20] P.D. Robinson. Proc. Roy. Sot. (London) A283 (1965) 229. [2?] M. Cohen and A. Dalgarno, Prcc. Phys. Sac. 77 (196i) 748. [22] EA. Castro and F-M. Fwx%,des Len. Nuovo Cimento 28 (1980) 484. [23] AC. Hurley. Proc. Roy. Sac. (London) A226 (1954) 179. 1243 .R.E. Stanton. J. Chem. Phys. 36 (19623 1298. [25] E. Yurtsever and J. Hinze, J. Chem. Phys. 71 (1979) 1511. [26] S.T. Epstein, Theoret. Chin Acta (Berlin) 55 (1980) 251.
Comical
H’YPERVIRIAL FOR COUPLED
THEOREMS AND OSCILLATORS
Francisco M. FERN&E2
SCF WAVEFUNCTIONS
and Eduardo A. CASTRO*
Znstifutode Znuestigaciones FikicoquimicasTe&icas y Aplicadas, Secci& Qtlimica Te&ica. La PIam 1900. Argentina Received 1 October 1980 Revised manuscript received 12 January
1981
The SCF formalism is analyzedin connection with hyperviriaf theorems. Coupled oscillator models are chosen in order
to
illustratethe formalresults.
1. introduction For &J a trial wavefunction and H the hamiltonian operator of a given system, Epstein and Hirschfelder [l] demonstrated that for a variation &5 = i W&, with W an hermitian operator, the extremum condition 6E = 0 conducts towards the diagonal hypervirial theorem
the SCF method consists in searching for those optimum functions which make the functional energy E, E(4) = (+lHd),
(3)
an extremum. That is to say s,E=(6~~1Hb)+(~IH~~~)=O,
(4)
where
The SCF method is of a variational nature, so that the corresponding SCF wavefunction will satisfy certain hypervirial relationships. When taken into account such relationships provide us with very useful formulas, as will be shown later. The application of the SCF method to coupled oscillator systems has recently been made by Bowman [Z], and by Tobin and Bowman [S]. If we approximate the eigenfunctions of N as a product of basis functions Qi
Due to the fact that functions & have to satisfy the condition (Z), the variations S&k must have
the form
84, = A,&,
where A,, is an antihermitian operator (A: = -A& Replacing eqs. (5) and (6) in eq. (4) we conclude that 4 fulfills the hypervirial theorems for any antihermitian operator which depends on just one coordinate, i.e., (&l[H. A&#)
subject to the normalisation (&i&)=1; *
i=l,2
condition
,_.., N,
Author to whom correspondence
0301-10l4/81/0000-0000/S02.50
(2) shouid be addressed.
@ North-Holland
(6)
= 0.
(7)
If w(xk) is an arbitrary linear Operator which depends on one coordinate, then ( W- W’)4 .zs well as i( W + W’)+ are compatible variations for 4, so that it follows at once that (&x
WI&) = 0.
Publishing Company
(8)
F.&f_ Femdndrz.
66
EA.
As an immediate consequence, we can assert that eq. (8) will be valid for any linear operator which is expressible as a sum of one-dimensional linear operators.
Castro / Hyperuirial
Such se!ection is class of operator physics systems. (8) adopt a very (Gl[K
2. Hyperviria!
theorems
theorem
wk
We have seen that SCF solutions satisfy the diagonal hypervirial theorems for any unidimensional operator. Now we wish to demonstrate the converse statement: If a product-like function (1) satisfies the diagonal hypervirial theorems for any unidimensionai linear operator, then d is an SCF solution for the system. In order to demonstrate this theorem, we follow Hirschfelder [4]. Assuming 4 reaI and replacing K’(Q) = f(x*)L& (with DC = a/&x, and f(x) an arbitrary function such that fDkQ belongs to the domain of H) in eq. (S), we get
wkw)
supported by the fact that this is typical of several chemical Diagonal hypervirial equations simple form too
= <@kI[Fk,
Wk$#‘k)
= 0;
(14)
= wk (xk )-
The application of this last derive certain relationships useful when managing SCF consecutive replacement of for W, in eq. (14) gives the
formula allows us to which will be very equations_ The f(xk)Dk and f(xn) following equations:
-~(~~:lf)D~~~)k)-(~lcif)~:~li) -(&lf(v:
+h)‘dk)=o,
-f(~liI~~~;)-(C58IfDlc~r~)
(15) = 0.
(16)
When f has an appropriate form, then it is possible to eliminate all the terms Dk and Di from eqs. (1 l), (15) and (16). For j(xk) =xc we get
(9) where
we suppose
.\’ H=X.HP i=*
-2AQe1x:-r
that
+ V(.x,,
-.. .Xx).
(10)
l
Denoting by Fk the SCF operator z i {&&~,,ll’&~&) we deduce from eq. (9) (F;,&)l&
(11)
=&Jr-,
where e,, is a scalar. In this work we are interested in those hamiltonians as (10) with interaction potentials like V(xr,
-.. ,x.X) =c
1 g,jui(xi)ui(xj), following
form
3.
Coupled
oscillator models the hamiltonian
H = -i(~f+~f)+Ir~vT~:+~~,ix~)i~~fxS. The SCF oscillators
for
(17)
In the next section we will see that eq. (17) allows a great simplification when dealing with integro-differential equations (11).
Model I. Let us consider operator [5]
(12)
i j
which leads to the simple SCF operators
(VE -iZllc)&)
-(~klX~(V~+r(li~rPk)=O-
(18)
equations correspond to two harmonic with effective operators
F;=Hyi-ui; (13)
F,=
-~D~+3,v;+2aAr)x;;
A~=(~+:@JI). (20)
FM.
The
eigenenergies
Ferntinder.
EA.
en,= (nz+l/2)(wf+2aAz)“-.
(22)
l/2 ,
A,=(n1tl/2)(rvti2aA1)-
(23)
A, = (n2+1/2)(~v~i2aAz)-‘/‘.
(24)
l?qs. (23) and (24) can be deduced in a very simple fashion by minimizing the total energy [2,3] with respect to A, and A2:
(25)
where b is a parameter satisfies the equation = 0,
to be adjusted
so that &
V-N.
(26)
From the resolution of eq. (26) we find that the optimum value for b is unique for any iV, and it satisfies the relationship b-0.125
R =O.
(27)
For a = 1, b = 0.66236, and thus (I;) = (xf) = 0.37744. This result corresponds to identical values obtained from eqs. (23) and (24). Model II. This model has a hamiitonian operator H = -$(Df+Df:if(W:XT+
and it has been operators are Fl=
,vfxf)
+ rx:,, widely
T,-&ufx:+pA~x,
(28) studied +prx:,
[7-181.
The
WI and the eigenvalues
are
e,,2=(nzif)~CliprA;.
(31)
A1 =(n2i$j~C~‘.
This fact confirms that hypervirial equations (17) are correct. Eqs. (23) and (24) can be solved numerically without further ditiiculties. In order to verify the theorem presented in section 2, we consider the hamiltonian (18) with wf = tvf = 1, and we choose as trial wavefunction for the ground state d(b,~~,~~)=Nexp[-_(xttx~)],
The eigenfunctions of F2 correspond to an harmonic oscillator with an effective frequency
The utiiization of eq. (17) with N = 1 for F, gives for A1 the result
E nrnz=en,+ez2-2aA1A2.
t&xl
67
(21)
1
The problem is solved by using eq. (17) for N = 1, k = 1,2. In such a way, we obtain two relationships which relate At with AZ by the simple expression [6]
b3-0.25
&-orems
are
e,, = (nl;l/2)(rc,ft2aA1)“‘,
(&lr-i, XYDJd)
Castro / Hypemirial
SCF
(29)
(32)
On the contrary, F* (N=O, 1):
we need
- wfAr-pAi-3pr(xl’) 2e,,-
two relationships
for
= 0,
(33)
2~~.‘(xf)-3pA,Ar-5prA~=0.
(34)
Eqs. (33) and (34) allow us to express Ai as a function of A?, so that the numerical calculation of the necessary integrals is reduced to only one. The calculation scheme is as follows: we start the iteration cycle with F”’ = Tzi wf.x~/2 which gives us ezz and A:. Then we construct F\” =T,t-wfsfj2i~A:s~tpr.u:. From numerical integration we get e:,. From A:, ~2, and eqs. (33) and (34) we can express A’: as a function of AZ, so that we need to calculate numerically only one of the couple of integrals. tpA~.x~tprA~, and Then we form Fy’ =Fi” so on. This method is really efficient and useful when dealing with a large number of variables, because the necessary matrix elements to be evaluated are drastically reduced and it has a positive effect on the increase of the accuracy and the speed of the calculation. This model can easily be solved for r=O (Barbanis’ hamiltonian) e,, = (n,+&
-Af/2w;,
(35)
AZ = -pA,/w;. The final iteration between eqs. (32) conducts us to the desired solutions.
(36) and (36)
4. Conchding reinarks The SCF function (1) is eigenfunction operator
of the
(37)
F = fl Fik), which can he considered as a zero-order tonian
hamil-
H=F+H’,
(33)
where _M’ is a perturbation. If R is function of H, then we can write
an
condition
g
f
x (giixix: + giix:xi i g;x’xi’>
i>i
1
(44)
is treated by the SCF method, it leads to a set of SCF operators which are similar To those corresponding to shifted oscillators. It is an easy matter to see that in this case the trial function
(d IS) = 0. According to perturbation expressed as [19]
= 0.
Through a more complicate and tedious procedure, Sinanoglu [19] deduced a similar resu!t for atomic and moiecuIar systems. When the hamiltonian
(39)
with the orthogonality
=
(PI&)
eigen-
R=c$tS,
s
Eq. (43) is satisfied fOi any operator w(xk) whenever F,Z, = en,&,, or 2, = 0. The fust option does not fulfill the condition (S”‘l4) = 0, so that we are led to the second one, which is an orthogonality ccndition stronger than
theory, S can be
Q =n &7i(Qjxi + 6i)
s”‘.
(45)
I
i=*
From previous results presented by Robinson [20] and following our prior discussions in sections l-3, for any operator i[Fk, W(X~,)] with W’(xk) = W(xk), the first order corrections are null, i.e. (+,[[Fk,- W-IS”‘) = 0.
(41)
Due to the fact that the operator W’ is arbitrary, this result can be expressed in the following sentence: the first-order correction to the expectation value of any operator which can be expressed as a so/mof one-coordinate operators vanishes if Hartree wavefunctions are used in its evahiation 1211. Defining the function Zk (xt) as
-Z&k)
(with {@,} bekg the set of the harmonic oscillator eigenfunctions) is a SCF solution when the parameters ai and 6, are VariationaIIy optimized_ This constitutes a positive proof for our recent proposal of using the translation as a manner to introduce variational parameters in trial wavefunctions [22]. Finally, we point out that SCF wavefunction must obey the Hellmann-Feynman theorem, as previously stated [23-261.
References [l] S.T. Epsteinand J.O. Hinchfelder. Phys. Rev. 123 (1961) 1495. [2] J.M. BoHrnan. J. Chem. Phys. 68 (1978) 608. [3]
F.L.
Tobin
and J.M.
Bowman.
Chem.
Phys. 47
(1980)
151.
J.O. Hirschfelder. J. Chem. Phys. 33 (1960) 1462. [S] J.C. Percival and N. Pomphrey. Mol. Phys. 31 (1976) 97. [63 R.A. Harris. J. Chem. Phys. 72 (1980) 1776. [7] S. Nordholm and S.A. Rice, J. Chem. Phys. 61 (1974) 203. [4]
eq. (41) is transformed (wG%ki(Fk-%k)zkj=o-
in (43)
F.&l. Ferndrrdez, E.A. Casmro f Hypeminkl 183 S. Nordholm and S.A. Rice. J. Chem. Phys. 61 (1974) 768. [9] W. Evtes and R.A. Marcus, 3. Chem. Phys. 61 (1974) 43131.
[lo]
N. Pomphrey, J. Phys. B 7 (1974)
1909.
[ 1 I] D.W. Noid and R..4. ~Marcus, I. Chem. Phys. 62 (1975) 2119. [121 S. Nordholm and S.A. Rice, J. Chem. Phys. 62 (1975) 157. 1131 KS. Sorbic, Mol. Phys. 32 (1976) 1577. [14] S. Chapman, B.C. Garrett and W.H. Miller, J. Chem. Phys. 64 (1976) 502. [IS] D.W. Noid. M.L. Koszykowski and R.A. Marcus. J. Chem. Php. 67 (1977) 404. 1161 D.W. Noid and R.A. Marcus. J. Chem. Phys. 67 (1977) 559. [17] KS. Sorbic and NC. Handy, Mot. Phys. 33 (1977) 1319.
iheorems
69
[18] M. Cohen and S. Greita, Chem. Phys. Letters 60 (1979) 445. 1191 0. Sinano%lu. Proc. Roy. Sot. (London) A260 (1961) 379. [20] P.D. Robinson. Proc. Roy. Sot. (London) A283 (1965) 229. [2?] M. Cohen and A. Dalgarno, Prcc. Phys. Sac. 77 (196i) 748. [22] EA. Castro and F-M. Fwx%,des Len. Nuovo Cimento 28 (1980) 484. [23] AC. Hurley. Proc. Roy. Sac. (London) A226 (1954) 179. 1243 .R.E. Stanton. J. Chem. Phys. 36 (19623 1298. [25] E. Yurtsever and J. Hinze, J. Chem. Phys. 71 (1979) 1511. [26] S.T. Epstein, Theoret. Chin Acta (Berlin) 55 (1980) 251.