- Email: [email protected]
The self-consistent random phase approximation
VohIxJe 11, number’ 4
CHEhfIC!ALPHYSICS LETTERS ,. i
:
.-
‘.
.. ‘.
.’
1 November 1971. ..
:
‘.
,.’ .I,,
.. _.
.’
'~SE~~NSISTENTRANLX)MP~~SEAPPROXIMATION* .. N.
OSkLUND** and M. ,uJWLUS. ‘. I’
An analysis is made of the se&consistent random phase approxima%n (SC RPA) in whi& theground-state function satisfies the annihiIation corxlition. It is shown that the ground stE;tefor a twoefectron system is identical with that obtained by a variational calculation including doublyexcited mnfiiurationsand that the excitation energies are the same as the Tamm-Dancoff values corrected for the ground-state correIation,energy. To Zlustrate the iterative procedure by which the SC RPA solution is ol$ned, an application to Hz is included. i
Recently cansiderabIe interest has been shown in using the r&dom phase approximation (RPA) f&r calculating excitation energies and oscillator strengths of atoms and mole&es [l-8]. It has also been suggested that a’self-consistent fdrm of the RPA may be art improvement over the nornial RPA treatment [9-I I 1. In this communication we demonstrate several important characicristks of the exact self-consistent RPA; they suggest that the method may be more useful for s&dying ground-state correlation than fey excited states. 6olIowing the general formulation of Rowe [I 21, which provides a very clear development of the RPA, we consider the excitation operators 0; and their adjoints 0, defined by .. o;roi=
C3p=o,
In),
de-hole
form,
where the indices r, 8, . . . refer to particle states (virtual
spin orbitals) and cz,P, .._refer to hole states (occupied spin orbitals) and the operators CT and c, are fermion &ation and annihilation operators, respectively. The resutting matrix form of the RPA equations is [ 121 (we assume real orbita% for simplicity)
where A, El, andCUare the matrices
cu
where to> is the ground state af’tht: system aad IfI)iS an excited’state with excitation energy wn; since the spectrum for atoms and molecules is not harmonic,
different operators 0: are required for each excited state n In the RPA, as applied to a closed-shell grour@ state, the excitation operators have the parti. .
i
B
ra,sp = -4
pr,mq
10) f
u ra,s@= @I lu,,J$J
and the sykmetked defined by
double ~-~t~t~r
:
,_
pairs, rcu;‘isM; there s&$ions to eq. (3), which &I&.!~ both po$tive arjd peg&e wh’s of e&al magnitude;it% . .’
‘are k = ZtU
.’
.,‘.
.
:.’
‘;
_;
,’ ‘,
.
.
.‘,
: ,.
,::
‘”
Jf the nutiber of partkfe-f;olt!
: .-
[a;&,c] is
,’
:
,I ,* Supported ia part by a grant f%rn the National Science Fo&&eon_ ‘. : ** Resent address: Departtier& of Chemistry, Uliiversiry of Arkansas, FayettetiIIe,A.rkansas. :. .’ ., “:.( ‘. ..,
._
(4)
I? ;
I .,‘. ‘.
.’ _’ I.
..,
:
‘, .:..
;_
:, ,’
:
Volume 11, number 4
CHEMICALPHYSICSLETTERS
.assumed, as is appropriate for stable solutions, that there are no solutions with ti,, = 0. In the subsequent development we consider only the M solutions of positive defiiite w,, . For a ba.tis of spin .orbitals, which is convenient for the general arguments given below, the equations are not separated into their singlet and triplet components. If the Hartree-Fock ground state I >with energy Em is used as an approximation to 10) in evaluating the matrices in eq. (4), tIie RPA matrix equations in eq..(3) reduce to those of the usua! or normal random phase-approximation (N RPA) 1131 with
where [RY)=DT, I ) is a singlyexcited state and 6;) =0&O&I ) is a doublyexcited state. If the matrix B” is neglected, only the sum over DFa contributes to the excitation operator in eq. (2) and the excitation energies w, are equivalent to those obtained by diagonalizing the hamiltonian matrix over singlyexcited states (Tamm-Dancoff approximation). The matrix BO is usually assumed to introduce the effects of correlation in the ground state [14] . It can be shown by considering *he variational nature of the RPA solution [ 13, IS] that one effect of BO is that all eigenvalues of the N RPA are less than those of the Tamm-Dancoff approximation (TDA); i.e., that ,N RPA < ,TDA (7) n n * Since the annihilation condition on the ground state in eq. (1) is not satisfied for the function I >, there is an inconsistency in using the Hartree-Fock ground state to calculate A. 6, and U [ 16]_ We con-
sidera two-electronsystemand use the result of Sanderson[17] * that the correctedground state which does satisfy the annihilation condition has the
form . IO)=NeSl.
)
(8)
where N is a normalization constant and the operator S is
1 November 1971
The correlation coefficient Crqss isgiven by C=YX-’
)
(10)
where the matrices X and Y consist of f.he vectors in eq.‘(3) arranged in columns. This immediately suggests an iterative scheme since in eq. (4) the matrices A, B, and U, which are expectation vaIues over [O),depend on C. In a self-consistent random phase approximation one can use &I initial guess at C to calcdate
A, 8, and U, solve the WA eigenvalue equation WI, ca1cuIa te a new value of Cfrom eq. (IO),
[es.
and repeat this procedure until the correlation coefficients become self-consistent. Alternatively, as suggested by Sanderson [ 131, it is possible to make use of the fact that U and C commute and combine eqs. (3) and (10) to obtain CA+CBC+B+AC=O,
depends on the assumption of boyn
‘wmmutationrulesfor the operatorsDh,Dm two-electroncase:
ekaqt in the
(11)
which can serve as an implicit equation forC. The ground state 10)obtained by such a self-consistent random phase approximation (SC MA) are the subject of this communication. We show first that the function [@defied by eqs. (8) and (9). which satisfies the self-consistency condition [eq. (l)] for a two-electron system, is the wavefunction that would be obtained by a variational determination of the coefficients C. The exponential in eq. (8) can tik truncated after two terms for this case; i.e.,
lO>=N(I
>+&z ra crr+s19p) ss
If the ground state IO)is variational, the transition formulas (OH
)=NE,
(W=f$> = NECre ss , (13)
are valid with E equal to rhe energyof the vkational (configuration-interaction) wavefirnction. By inscrting eq. (12) into eq. (4) and commuting operators until it is pqssible to use the expressions in eq. (13), we fiid that A, B, and U have the relativeIy sinipIe form
A=N2iAo tCA”&b.E( B=N*{-AOC-CA%2AEC),
* Sanderson’s p&f
the ma-
trices
ftc-?>),
Volume 11, number 4
.wheret\Eisthe conelstion energy E-EHFs
fnserting ‘.. A, 8,2nd Ufrom eq. (14) into eq. (1 l), we fmd in+ mediately that it satisfied identically; i.e., the varia‘. tiona! vafues oft are ihe same as those of the SC RPA. It is instructive also tc write the matrix equations [eq: (3)] in the form (A+B)(X+Y)
=U(X-Y)w;
(A-B)(X-Y)
(15)
,
=U(X+Y)w
(16)
where o is the diagonal matrix of positive eigenvalues 0,. If one substituteseq. (14) into eq. 115) and eq. (16), multiplies eq. (15) on the left by (I -C>-1, eq, (16) on the left by (I I-G)-‘, adds and subtracts the two equations, one obtains the equations (A0 -AEI)(X
-CY) - (X-CY)o
(A0 - ml){ Y -CX) = -( Y-CX)w
,
073 .
(18)
For matrices A0 - AEI and w which are positive definite, eq. (18) requires that eq. (10) be satisfied in agreement with the above argument. From eq. (17), it follows that the excitation energies in the SC RPA are the eigenvalues of A0 -AH; i.e., they are identical to the eigenvalues of A0 corrected by 2 constant equal to’the correlrztion energy introduced into the ground state by the self-consistent ulculation. Thus, we have the inequalities (since AE < 0) ,SC n
RPA
> ,TDA n
> ,N
1
CHEMICALPHYSIB LETTERS
n
RPA
0%
Since the TDA singlet excifation energies have been found to be too large in most applications [ 14,151, the SC RPA are not expected to lead to improved res&s; for triplet strites, the TDA and N RPA values are generally too !ow, so that the SC RPA can be better [9-11,15], To determme the nature of the excited states obtained in the SC RPA by operating with 0; on the ground st2te iO>,we make use of eqs. (lo), (12), 2nd (14) and write
From the ~orma~tion to eq.f3) f12], x+ux-YfUY=
November 1971
conditions for tfie sojutions .,‘:; 7 (20
I,
it is easy. to show (using the fact that U commutes withC) that the matrix!-jUXis,orthogonal. Thus, the excited states of the SC RPA in contrast to the N RPA excited states, form an orthonormaf set. Further, from eqs. (17) 2nd (ZO), the SC RPA excited states are.identi& with those obtained in the.TammDancoff approximation (i.e.,X -.CYare the eigenvectors of A9 or A0 -ASi). Since the solution of the SC RPA provides the configuration interaction result for 2 wavefunction of the form given in eq, (81, it is of interest to determine the rnte of ~onv~rg~~~~of the SC RPA ~aI~u~ation. fn’table 1 we give the coefficients (intermediate normaliwtion) of the five doublyexcited closed shells obtained with the Fraga-Ransil wavefunction for I-l, [18]. The fust row (0th iteration) is the result from the vectors X and Y of the N RPA. The convergence of the coefficients to the configuration-interaction result is seen to be quite rapid. The correI2tion energy AE ~co~urnn 7) evaluated as the expectation value of Hover the ground-state wavefunctiqn obtained at each iteration, converges even faster than the coefficients. In column 8, we list for comparison the correlati,on energy AE’as given by the tran~tion formula, Trace ;s(BnC), which is exact only for the, correct variational wavefunction. The transition value of AE for the N RPA (0th iteration) is equal [15] to that given by the expression derived by McLachIan 2nd Bail f 141,
From the results described in this paper, it appears that neither the ordinary RPA..(N RPA) or its selfconsistent modification (SC RPA) can be used generally to obtain reliable values for excitation energies. If a procedure of the SC RPA type is going to be employed, 2 better representation of the excited states, suzh as that provided by some form of second RPA [ 1%21], may be required to achieve a balance between the ground and excited state correlation correc-’
tions. A possibleutility of the SCRPAitself maybe for the iterative calcul$ion of ground-state wavefunctiers, since if, in the SC RpA, M X Mm&rices occur. for A $6; and &I,the coiresponding confiiuration~in452
‘.:
.
.
.
,’
.
I : . .;
--.‘
:. ,.,. -:.<..-:
-
.
:
,’
:
: ,_
.,
-
.’
,:.
.,
.-
Volume Ii, number
I November 1971
CHEMICAL PHYSICS LB’ITERS
4
Table 1 Convergence of the SC RPA for Fraga-Ran&
Hz
1221
Coefficients of individual configurations 1331 1471 r53r
0 e) 1 2 3 4 S 6 :.
-&10239786 -0.07440917 -0.08036849 -0.0790 1376 -0.07931057 -0.07924570 -0.07925980 -0.07925740 -0.07925674
-0.06465287. -0.05510086 ;_0,05682346 -0.05647899 -0.05655038 -0.0565 3523 -0.0565 3847 -0.0565 3777 3792
-0.03886405 -0.03056764 -0.03228747 -0.03194663 -0.03201720 -0.03200239 -0.03200554 -0.03200501 -0.03200487
-0,017OSSSO
-0.0086 14Of
-0.01173945
-0.00601353
-0.0128’7065 -0.01265136 -0.01269684 -0.01268736 -0.01268937 -0.01268903 -0.01268894
-0.OU647304 -0.00638641 -0.00640398 -0.0%40032~ -0.00640110 -0.CtO640097 -WlW4ao93
ci d)
-0.07925727
-0.0565 3790
-0.03203498
-0.0126 8902
-0.00640096
Iteration
1661
Corteiation aE a) (atz)
energy aEb) (au)
LO.02329954 -O.i?2539&88 -0.02S479L.3 -0.02548255 -0.02S48271 -0.02548272 -0.02548272
-0.53028311 -0.02459811 -0.02566595 -0.02544454 -0.02549084 -0.02548099 -0.02548310
--0.0’25482?2
-0.02548272
a) Calculated as an expectation value. c) Resuits given by N RPA,
b) Catcutated from transition form&a. d) Confipaticn-interaction resrtIt_
teraction calculation
[6] T. Terasaka and T. hfatsushita, C&em. Phys. Letters 4 (1969) 384. [7] P. Jorgensen and J. Linderberg, Intern. 1. Quantum Cltem. 4 (1970) 587. f8] B. Lukman, J. Koikx, E. Zztrajsek and M. Zaucer, Croat. Chem. Acta 42 (1970) 69.
requires &f2 X M2 matrices.
However, the greater complexity of the SC EWAmatrices suggests that further study is needed to evaluate the relative efficiency of the two approaches. Subsequent papers will present a corresponding discussion of the many-electron case, as well as a more detailed description of the various RPS procedures and their applications.
References [ 11 P.L. Altiek and A.E. Gfawold, Phys Rev, 133 (1964) A632. f2] M.A. Ball and AD. McLachlan, Mot Phys. 7 (1964) 501. 131 T.H. Dunning and V. bfc.Koy, J. Chcm. Phys. 47 (1967) 1735. [4] T.H. Dunning and V. McKay, J. Chem. Phys. 48 (1968) 5263. [S] B. Lukman and A. Azman, MoL Phys. 16 (1969) 201.
,’
..
_:
[9] I”. Shibuya and V. McKay, I. Gem. Phps. 53 (1970) 3308. [lOi T. Sfiibuya and V. McKay, J. Ckem. Phys. 54 (1970) 1738. [li] T. Shibuya and V.‘hldSoy, Phys. Rev. A2 (1970) 2208. 1121 DJ. Rowe, Rev. Mad. Phys. 40 (1968) L53. [13] D.f. Thou!ess, Nucl. Phys. 22 (1961) 78. [14] AD. McL;tchfanand MA. Baff, Rev. Mod. Phyr 36 (1964) 844. [ISI N.S. Ostlund and M. Xarplus, to be published. El61 A. Goswami and hf.% Pal, Nuct Phys. 44 (1963) 294. [17] E.A. Sanderson, Phys. Lettots 19 f196Sl 141. 1181 S. Fraga and BJ. broil, J. aem. Phys. 35 (i961) 1967. [19j 3. Sawicki, Phys. Rev. 126 (1962) 2231. [201 T. Tamura and T. Wagawa.NncL Phys. 53 (1964) 33, [21] J. Da Provide&a, Nucf. Phys 6t (f96S) 87,
CHEhfIC!ALPHYSICS LETTERS ,. i
:
.-
‘.
.. ‘.
.’
1 November 1971. ..
:
‘.
,.’ .I,,
.. _.
.’
'~SE~~NSISTENTRANLX)MP~~SEAPPROXIMATION* .. N.
OSkLUND** and M. ,uJWLUS. ‘. I’
An analysis is made of the se&consistent random phase approxima%n (SC RPA) in whi& theground-state function satisfies the annihiIation corxlition. It is shown that the ground stE;tefor a twoefectron system is identical with that obtained by a variational calculation including doublyexcited mnfiiurationsand that the excitation energies are the same as the Tamm-Dancoff values corrected for the ground-state correIation,energy. To Zlustrate the iterative procedure by which the SC RPA solution is ol$ned, an application to Hz is included. i
Recently cansiderabIe interest has been shown in using the r&dom phase approximation (RPA) f&r calculating excitation energies and oscillator strengths of atoms and mole&es [l-8]. It has also been suggested that a’self-consistent fdrm of the RPA may be art improvement over the nornial RPA treatment [9-I I 1. In this communication we demonstrate several important characicristks of the exact self-consistent RPA; they suggest that the method may be more useful for s&dying ground-state correlation than fey excited states. 6olIowing the general formulation of Rowe [I 21, which provides a very clear development of the RPA, we consider the excitation operators 0; and their adjoints 0, defined by .. o;roi=
C3p=o,
In),
de-hole
form,
where the indices r, 8, . . . refer to particle states (virtual
spin orbitals) and cz,P, .._refer to hole states (occupied spin orbitals) and the operators CT and c, are fermion &ation and annihilation operators, respectively. The resutting matrix form of the RPA equations is [ 121 (we assume real orbita% for simplicity)
where A, El, andCUare the matrices
cu
where to> is the ground state af’tht: system aad IfI)iS an excited’state with excitation energy wn; since the spectrum for atoms and molecules is not harmonic,
different operators 0: are required for each excited state n In the RPA, as applied to a closed-shell grour@ state, the excitation operators have the parti. .
i
B
ra,sp = -4
pr,mq
10) f
u ra,s@= @I lu,,J$J
and the sykmetked defined by
double ~-~t~t~r
:
,_
pairs, rcu;‘isM; there s&$ions to eq. (3), which &I&.!~ both po$tive arjd peg&e wh’s of e&al magnitude;it% . .’
‘are k = ZtU
.’
.,‘.
.
:.’
‘;
_;
,’ ‘,
.
.
.‘,
: ,.
,::
‘”
Jf the nutiber of partkfe-f;olt!
: .-
[a;&,c] is
,’
:
,I ,* Supported ia part by a grant f%rn the National Science Fo&&eon_ ‘. : ** Resent address: Departtier& of Chemistry, Uliiversiry of Arkansas, FayettetiIIe,A.rkansas. :. .’ ., “:.( ‘. ..,
._
(4)
I? ;
I .,‘. ‘.
.’ _’ I.
..,
:
‘, .:..
;_
:, ,’
:
Volume 11, number 4
CHEMICALPHYSICSLETTERS
.assumed, as is appropriate for stable solutions, that there are no solutions with ti,, = 0. In the subsequent development we consider only the M solutions of positive defiiite w,, . For a ba.tis of spin .orbitals, which is convenient for the general arguments given below, the equations are not separated into their singlet and triplet components. If the Hartree-Fock ground state I >with energy Em is used as an approximation to 10) in evaluating the matrices in eq. (4), tIie RPA matrix equations in eq..(3) reduce to those of the usua! or normal random phase-approximation (N RPA) 1131 with
where [RY)=DT, I ) is a singlyexcited state and 6;) =0&O&I ) is a doublyexcited state. If the matrix B” is neglected, only the sum over DFa contributes to the excitation operator in eq. (2) and the excitation energies w, are equivalent to those obtained by diagonalizing the hamiltonian matrix over singlyexcited states (Tamm-Dancoff approximation). The matrix BO is usually assumed to introduce the effects of correlation in the ground state [14] . It can be shown by considering *he variational nature of the RPA solution [ 13, IS] that one effect of BO is that all eigenvalues of the N RPA are less than those of the Tamm-Dancoff approximation (TDA); i.e., that ,N RPA < ,TDA (7) n n * Since the annihilation condition on the ground state in eq. (1) is not satisfied for the function I >, there is an inconsistency in using the Hartree-Fock ground state to calculate A. 6, and U [ 16]_ We con-
sidera two-electronsystemand use the result of Sanderson[17] * that the correctedground state which does satisfy the annihilation condition has the
form . IO)=NeSl.
)
(8)
where N is a normalization constant and the operator S is
1 November 1971
The correlation coefficient Crqss isgiven by C=YX-’
)
(10)
where the matrices X and Y consist of f.he vectors in eq.‘(3) arranged in columns. This immediately suggests an iterative scheme since in eq. (4) the matrices A, B, and U, which are expectation vaIues over [O),depend on C. In a self-consistent random phase approximation one can use &I initial guess at C to calcdate
A, 8, and U, solve the WA eigenvalue equation WI, ca1cuIa te a new value of Cfrom eq. (IO),
[es.
and repeat this procedure until the correlation coefficients become self-consistent. Alternatively, as suggested by Sanderson [ 131, it is possible to make use of the fact that U and C commute and combine eqs. (3) and (10) to obtain CA+CBC+B+AC=O,
depends on the assumption of boyn
‘wmmutationrulesfor the operatorsDh,Dm two-electroncase:
ekaqt in the
(11)
which can serve as an implicit equation forC. The ground state 10)obtained by such a self-consistent random phase approximation (SC MA) are the subject of this communication. We show first that the function [@defied by eqs. (8) and (9). which satisfies the self-consistency condition [eq. (l)] for a two-electron system, is the wavefunction that would be obtained by a variational determination of the coefficients C. The exponential in eq. (8) can tik truncated after two terms for this case; i.e.,
lO>=N(I
>+&z ra crr+s19p) ss
If the ground state IO)is variational, the transition formulas (OH
)=NE,
(W=f$> = NECre ss , (13)
are valid with E equal to rhe energyof the vkational (configuration-interaction) wavefirnction. By inscrting eq. (12) into eq. (4) and commuting operators until it is pqssible to use the expressions in eq. (13), we fiid that A, B, and U have the relativeIy sinipIe form
A=N2iAo tCA”&b.E( B=N*{-AOC-CA%2AEC),
* Sanderson’s p&f
the ma-
trices
ftc-?>),
Volume 11, number 4
.wheret\Eisthe conelstion energy E-EHFs
fnserting ‘.. A, 8,2nd Ufrom eq. (14) into eq. (1 l), we fmd in+ mediately that it satisfied identically; i.e., the varia‘. tiona! vafues oft are ihe same as those of the SC RPA. It is instructive also tc write the matrix equations [eq: (3)] in the form (A+B)(X+Y)
=U(X-Y)w;
(A-B)(X-Y)
(15)
,
=U(X+Y)w
(16)
where o is the diagonal matrix of positive eigenvalues 0,. If one substituteseq. (14) into eq. 115) and eq. (16), multiplies eq. (15) on the left by (I -C>-1, eq, (16) on the left by (I I-G)-‘, adds and subtracts the two equations, one obtains the equations (A0 -AEI)(X
-CY) - (X-CY)o
(A0 - ml){ Y -CX) = -( Y-CX)w
,
073 .
(18)
For matrices A0 - AEI and w which are positive definite, eq. (18) requires that eq. (10) be satisfied in agreement with the above argument. From eq. (17), it follows that the excitation energies in the SC RPA are the eigenvalues of A0 -AH; i.e., they are identical to the eigenvalues of A0 corrected by 2 constant equal to’the correlrztion energy introduced into the ground state by the self-consistent ulculation. Thus, we have the inequalities (since AE < 0) ,SC n
RPA
> ,TDA n
> ,N
1
CHEMICALPHYSIB LETTERS
n
RPA
0%
Since the TDA singlet excifation energies have been found to be too large in most applications [ 14,151, the SC RPA are not expected to lead to improved res&s; for triplet strites, the TDA and N RPA values are generally too !ow, so that the SC RPA can be better [9-11,15], To determme the nature of the excited states obtained in the SC RPA by operating with 0; on the ground st2te iO>,we make use of eqs. (lo), (12), 2nd (14) and write
From the ~orma~tion to eq.f3) f12], x+ux-YfUY=
November 1971
conditions for tfie sojutions .,‘:; 7 (20
I,
it is easy. to show (using the fact that U commutes withC) that the matrix!-jUXis,orthogonal. Thus, the excited states of the SC RPA in contrast to the N RPA excited states, form an orthonormaf set. Further, from eqs. (17) 2nd (ZO), the SC RPA excited states are.identi& with those obtained in the.TammDancoff approximation (i.e.,X -.CYare the eigenvectors of A9 or A0 -ASi). Since the solution of the SC RPA provides the configuration interaction result for 2 wavefunction of the form given in eq, (81, it is of interest to determine the rnte of ~onv~rg~~~~of the SC RPA ~aI~u~ation. fn’table 1 we give the coefficients (intermediate normaliwtion) of the five doublyexcited closed shells obtained with the Fraga-Ransil wavefunction for I-l, [18]. The fust row (0th iteration) is the result from the vectors X and Y of the N RPA. The convergence of the coefficients to the configuration-interaction result is seen to be quite rapid. The correI2tion energy AE ~co~urnn 7) evaluated as the expectation value of Hover the ground-state wavefunctiqn obtained at each iteration, converges even faster than the coefficients. In column 8, we list for comparison the correlati,on energy AE’as given by the tran~tion formula, Trace ;s(BnC), which is exact only for the, correct variational wavefunction. The transition value of AE for the N RPA (0th iteration) is equal [15] to that given by the expression derived by McLachIan 2nd Bail f 141,
From the results described in this paper, it appears that neither the ordinary RPA..(N RPA) or its selfconsistent modification (SC RPA) can be used generally to obtain reliable values for excitation energies. If a procedure of the SC RPA type is going to be employed, 2 better representation of the excited states, suzh as that provided by some form of second RPA [ 1%21], may be required to achieve a balance between the ground and excited state correlation correc-’
tions. A possibleutility of the SCRPAitself maybe for the iterative calcul$ion of ground-state wavefunctiers, since if, in the SC RpA, M X Mm&rices occur. for A $6; and &I,the coiresponding confiiuration~in452
‘.:
.
.
.
,’
.
I : . .;
--.‘
:. ,.,. -:.<..-:
-
.
:
,’
:
: ,_
.,
-
.’
,:.
.,
.-
Volume Ii, number
I November 1971
CHEMICAL PHYSICS LB’ITERS
4
Table 1 Convergence of the SC RPA for Fraga-Ran&
Hz
1221
Coefficients of individual configurations 1331 1471 r53r
0 e) 1 2 3 4 S 6 :.
-&10239786 -0.07440917 -0.08036849 -0.0790 1376 -0.07931057 -0.07924570 -0.07925980 -0.07925740 -0.07925674
-0.06465287. -0.05510086 ;_0,05682346 -0.05647899 -0.05655038 -0.0565 3523 -0.0565 3847 -0.0565 3777 3792
-0.03886405 -0.03056764 -0.03228747 -0.03194663 -0.03201720 -0.03200239 -0.03200554 -0.03200501 -0.03200487
-0,017OSSSO
-0.0086 14Of
-0.01173945
-0.00601353
-0.0128’7065 -0.01265136 -0.01269684 -0.01268736 -0.01268937 -0.01268903 -0.01268894
-0.OU647304 -0.00638641 -0.00640398 -0.0%40032~ -0.00640110 -0.CtO640097 -WlW4ao93
ci d)
-0.07925727
-0.0565 3790
-0.03203498
-0.0126 8902
-0.00640096
Iteration
1661
Corteiation aE a) (atz)
energy aEb) (au)
LO.02329954 -O.i?2539&88 -0.02S479L.3 -0.02548255 -0.02S48271 -0.02548272 -0.02548272
-0.53028311 -0.02459811 -0.02566595 -0.02544454 -0.02549084 -0.02548099 -0.02548310
--0.0’25482?2
-0.02548272
a) Calculated as an expectation value. c) Resuits given by N RPA,
b) Catcutated from transition form&a. d) Confipaticn-interaction resrtIt_
teraction calculation
[6] T. Terasaka and T. hfatsushita, C&em. Phys. Letters 4 (1969) 384. [7] P. Jorgensen and J. Linderberg, Intern. 1. Quantum Cltem. 4 (1970) 587. f8] B. Lukman, J. Koikx, E. Zztrajsek and M. Zaucer, Croat. Chem. Acta 42 (1970) 69.
requires &f2 X M2 matrices.
However, the greater complexity of the SC EWAmatrices suggests that further study is needed to evaluate the relative efficiency of the two approaches. Subsequent papers will present a corresponding discussion of the many-electron case, as well as a more detailed description of the various RPS procedures and their applications.
References [ 11 P.L. Altiek and A.E. Gfawold, Phys Rev, 133 (1964) A632. f2] M.A. Ball and AD. McLachlan, Mot Phys. 7 (1964) 501. 131 T.H. Dunning and V. bfc.Koy, J. Chcm. Phys. 47 (1967) 1735. [4] T.H. Dunning and V. McKay, J. Chem. Phys. 48 (1968) 5263. [S] B. Lukman and A. Azman, MoL Phys. 16 (1969) 201.
,’
..
_:
[9] I”. Shibuya and V. McKay, I. Gem. Phps. 53 (1970) 3308. [lOi T. Sfiibuya and V. McKay, J. Ckem. Phys. 54 (1970) 1738. [li] T. Shibuya and V.‘hldSoy, Phys. Rev. A2 (1970) 2208. 1121 DJ. Rowe, Rev. Mad. Phys. 40 (1968) L53. [13] D.f. Thou!ess, Nucl. Phys. 22 (1961) 78. [14] AD. McL;tchfanand MA. Baff, Rev. Mod. Phyr 36 (1964) 844. [ISI N.S. Ostlund and M. Xarplus, to be published. El61 A. Goswami and hf.% Pal, Nuct Phys. 44 (1963) 294. [17] E.A. Sanderson, Phys. Lettots 19 f196Sl 141. 1181 S. Fraga and BJ. broil, J. aem. Phys. 35 (i961) 1967. [19j 3. Sawicki, Phys. Rev. 126 (1962) 2231. [201 T. Tamura and T. Wagawa.NncL Phys. 53 (1964) 33, [21] J. Da Provide&a, Nucf. Phys 6t (f96S) 87,