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Reduced interaction matrix for doubly-excited configurations and the correlation energy by a partitioning method
Volume 26, number 1
1 May 1974
CHEMICAL PHYSICS LETTERS
REDUCEDINTERA~IONMATRiXFORDOUBLY-EXCITEDCONFIGURATIONS ANDTHECORRELATIONENERGYBYAPARTITIONlNGMETHOD William J. TAYLOR Department of Chemistry,Ohio State Universiry. Columbus, Received
12 February
Ohio 43210.
1974
Analyses of the correlation energy of anN-&ctron system by Nesbet and L&din reduced interaction matrix for doubly-excited configurations and an exact expression latter is separable in a basis diagonalizing this matrix.
1. Introduction Nesbet [I, 21 has pointed out that when the dominant contribution in a configuration interaction (CI) expansion of an atomic or molecular wavefunction is made by the Hartree-Fock (HF) determinant, the correlation energy, EC, can be expressed in terms of the coefficients of doubly-excited configurations in the expansion. Although this result shows that only doubly-excited configurations contribute direcrZy to the correlation. energy, the remaining excited configurations affect the coefficients of the doubly-excited configurations, and hence contribute indirecrly to EC. In the present note we consider the latter effect, and in particular derive a reduced interaction matnk for doubly-excited-configurations which includes the effects of all other excited configurations. For this purpose we use an extension of the partitioning technique for sol&ion of the CI secular equation discussed by Ewdin [3,41. However, the present anaIysis is primarily concerned with the “structure” of the correlation energy, rather than computational methods..
2. Reduced iute&tion matrix and correlation energy-
USA
are extended to obtain a for correlation energy; the
orthonormal Slater determinants, {a”}, may be constructed by addition to a0 of the normalized deter&nants.of al: other ordered subsets of N spin-orbitals from the set {vi}. All such configurations may be classified as to their degree of excitation. defmed as the number of spin-orbitals they contain which are not in the subset of N HF orbitals. Thus, in singlyexcited configurations, q, tpi(l N); in doubly-excited configurations, q,?, vi and 4 (1 S i r> N); etc. (the maximum degree of excitation is clearly N). In the present analysis the complete set of confiiurations, {a,,), is partitioned into three subsets consisting of the HF configuration, @-,, the subset of alI doubly-excited configurations, {G$f}, and-the subset of all other (singly, triply; etc.j excited configurations, {ai, q$, ...}. The coefficients in the CI expansion, *g = -x,cy@“, of the exact (non-relativistic) N-electron ground state wavefunction, @& are provided by the solution of the eigenvalue equation, (H.- El)c= 0,’ associated \?rith the (algebraically) lowest root, Eg, of the secular equation,‘lH --Eil=O. Here cis acoltimn with elements cy, 0 is the nulI column, I is the unit&at& and H has kiemer&~&= <@~w&D,,>, .‘.-. where &-N-i?.the (non-rel&vis&) N-election .. -; .G-: hamiltonian oq&at&. Tie. correlation.energy ii-..: : .. EC =~Eg_.~E& where ..,Ed =.WOIH”~~~~ is’ the HF i- :I -; _ .. _’ -..e.nergy. ..‘. .‘.: -.
Let {qi) be a complete set 6f orthbno,mal oneelectron sp@+bitais having.as its f=stN niemb& the HF~orbitals; the normalized & confiiuratioti i&jhen-. .a0 = (N!)-“* det(dI? ...’, t&).~$.compl<~e Setof _... :. ._ .: .’ ,:. :: Y, .::.. .‘.~ .,_ ‘:.. :. :-__‘_.-, ::. _,,. _: ;_-__;;t_::.~I____-_-L~,: : .-: .:.. .-.‘.-~.-~_.-‘_ ., _I. ___.’.. 29’ -. : :: : -, i:. .: -._,:‘_. _..,’ I _.:.-. .: -. _.: .I._-. .. _.
CHEMICAL PHYSICS LE?TERS
Volume 26, number 1
tions, {a,,), corresponds to the following partitioning of the hamiltonian matrix, ElJ H = “20 .i
”
“IX
“OfI
“22
“,,I
“,,7
”
9
+Ho2c2=0,
“20 co + (“22 -E’22)5
of(S) for c,, and E= Eg, in (2b) yields
“20 =() + (“>2 -Eg
I&c2
=$
(6)
9
where (1)
n II no 1 where the subscript 0 refers to Q, 2 refers to doubiyexcited configurations, and n to all other excited configurations. Thus, the typical element of the block i-l,, is (Qo&&$), and of the bIock H,,, (@$iH_@$). The matrix H, and therefore al o its 3 etc. diagocal blocks, are hermitian, while H20 = Ho2, Since HN contains only one- and two-electron operators all matrix elements connecting !Q, with configurations of triple or higher excitation vanish, and in addition matrix elements connecting a0 and singlyexcited configurations vanish in accord with Brillquin’s theorem. Thus, H,, and HnO are null matrices and the matrix eigenvalue equation separates into a scalar and two matrix equations as follows,
(E, -E)c,,
Substitution
1 May 1974
@a)
+“2nc,,
=02 ,
12b)
=o, ; (24 3 2 =2 •I-(%, -El&c, here c has been similarly partitioned, c = (co, q, c,). Setting E = Eg in (?a) yields EC = cc1 Ho2 cz, which is equivalent to Nesbet’s result,
The explicit form of the matrix element in (3) in terms of two-electron repulsion integrals is (e = 1) +Po LMNlap =(Qlr;;Irs)(ijrr;; IF)_
(4)
As stated in the Introduction, one of the objectives of the present discussion is to show how the coefficients, cr,of the doubly-excited confi&rations (and therefore also cc) are influenced by other excited configurations. For E = Eg thematrix in parentheses in (2cj is positivedefite and therefore nor&ingular, so that
“;?
= Hz2 -
“212(“n,I
-Eg
l,J-*
“,,
(7)
is a reduced interaction mat& for doubly-excited configurations which includes the effects of all other excited configurations_ The matrix in parentheses in (6) can also be assumed to be non-singular*, so that if Es is known (6) yields an explicit expression for the coefficients of doubly-excited configurations** c2 = - @Hi2
- Eg lX2Y1 H,,
(8)
(note also that (5) and (8) yield c,J Finally, substitution of (8) for cl, and Eg for E, in (2a) yields the following exact expression for the correlation energy, % = - “,,(H;,
-Eg
122j-1 H20 3
(9)
with H& given by (7) (see also the appendix). In the preceding analysis it has been assumed the basis sets {I,+} and {QV) are complete, so that *g and Eg represent the exact ground state wavefunction and energy; further, it has been assumed that a0 and E. are the exact HF wavefunction and energy, and EC = Eg - EO the exact correlation energy. In practice, of course, it is not possible to work with complete basis sets. However, all the results obtained in the present article remain vaIid for the case offinite basis sets if appropriate changes in the interpretation of the preceding quantities are made. In particular, @o must be the so-called SCF (self-consistent field) determinant which minimizes the expectation value Eo =: (cPoviN190) in the space spanned by the set (~~1, in which case the anaIog of Brillouin’s theorem is applicable (ref. [4], p_ 284). Simiily, ek,, Eg, and EC become the best approximations to these quantities in * I%Js.(2a) and (6).
with Eg =E,
represent a reduced
eigen-.
due equation for the vector (co. cz), corresponding to the secular equation @o--E) I Hzo
H, (Hi2
-Elzz)
=‘-
By ihe sepmation theorem (ref. (5 1, p. 103), the lowest root of the equ+ion lH$z - El22 I = 0 is not less than Eg, and ii may safely be assumed to be greater than Eg in view.
of the definitions of @o and Eo_ ** The reciprocal matriceiin (5) and (8) may be regard& as : : resolvents. <, .- .. :
-.. _. :
Volume 26, number 1
the space spanned by the set I+” 1. As previously indicated our principal interest in (9) and (7) is the insight they give into the dependence of the correlation energy on doubly-excited configurations, and less directly on other excited configurations. Of course, if Ep = E. + EC on the r.h.s. of these two equations is not known, they provide onIy an implicit (but exact) relation for Ec. However, these equations may be sotved iteratively for EC [4J, and they atso lend themselves to the development of perturbationtype expansions *. For example, if only diagonal matrix elements ofHN for doubly-excited configurations are retained in (7), so that Hi* z diag (Ez) with EF = (~~ifYN~~~>, and also Eg is approximated by E, in (9), one obtains
where the matrix element is given by (4). This is the second-order perturbation theory approximation sug gested by Nesbet [Z, 61.
3. Diagonalization
of the reduced interaction
matrix
A formal simplification of the correlation, (9), may be achieved by diagonalization of the reduced interaction matrix, H;,, defined in (7). Let $2 be a umtary matrix which diagonalizes the hermitian matrix H&, so that UiZ Hi2 U22 = AZ2 = diag
(A,) ,
(11)
where the &‘s are the eigenvalues of HiZ; on premultiplying (11) by lJ22 its columns are seen to be the eigenvectors of Hi?, in the same order as the eigenvalues on the diagonal of A22. On solution of (I I) for H’zz and substitution in (9) one readily obtains Ec = -Hb2 [diag(h,
1 May 1974
CHEMICAL PHYSICS LETTERS
- Eg)-‘1
H A; ,
(12)
* It should be pointed out that in applications it would not bB necessary to evaluate the reciprocal matrices on the r.h.s. of (7) and (9). Thus, if A is a uare non-singular matrix and “t one wishes to calculate X = A- H. where H is a conformable matrix; one may instead solve in unison the_mul$ple sets of simultaneous Iin@ar inhomogeneous equations, AX = H.
where Hbz = Hu, UZ2. Define the following orthonorma1 linear combinations of doubly-excited configurations,
where iYtikr,a is an eIement of UZ2 in the oath column and the row associated with ‘PC in HZ2 (or Hi*). From (4), (12), and (13), the correlation energy can be put In the form EC =-c
f(3+$$f?&)f2/(Xa
-Eg)
,
(14)
Q where
Relation (14) is formally similar to the secondorder Brillouin-Wigner perturbation theory expression obtained by replacing EO in the Rayleigh-Schriidinger formula, (lo), by Eg. However, (14) is exact (to all orders) for a complete basis, and shows that the correlation energy is separable when expressed in terms of the eigenvalues and eigenvectors of the reduced interaction matrix for double excitations; this would appear to provide a rigorous basis for electron-pair theories of correlation energy.
Appendix The reduced interacti n matrix, H;2, can be further s analyzed. Thus, H2,t = Hn2 on the r.h.s. of (7) connects doubly-excited configurations with singly-, triply-, and quadruply-excited configurations (at most). Denote these excited configurations having nonvanishing matrix elements with doublyexcited configurations
by p, and the remaining
type n by 4. Then, inpartitioned (H2p,02q) and (7) becomes
configurations
l-52 = H,,_ - Hzp C(Hnn - Eg I,J1lPP =H2a -.Hzp($.-E8 where
&+Hpr
of
form, Hzn = HP’2 ,
(+”
..
.Volume 26. number 1
CHEMICAL PHYSICS LETTERS
is a reduced interaction matrix for configurations of type p. The same results are obtained if the set {*U I is partitioned into four subsets, namely, cPO, doublyexcited configurations, and configurations of types p and q, and cII and cP are then eliminated successively from the four equations analogous to (2a)-(2c) which are obtained. This was avoided in order not to confuse the main lines of the discussion in the body of this article. Ii is ctear that a hierarchy of such reductions is possible; e.g., the next partitioning would separate excited configurations of degrees 5 and 6 (which interact with type p configurations) from the remaining type 4 configurations. This process would terminate when excitations of the highest even degree less than
1 May 1974
N were reached. From this point of view the present approach may he characterized as one of nested resolvents. However, the reduction of Hi2 in (Al, A2) is probably sufficient for most purposes.
References [l] [2] [3\ I41 [5]
R.K. Nesbet, Phys. Rev. 109 (1958) 1632. R.K. Nesbet. Advan. Chem. Phys. 9 (1965) 321. P.-O. Ltiwdin, J. Chem. Phys. I8 (1951) 1396. P.-O. Liiwdin, Advan. Chem. Phys. 2 (1959) 207. J-H. Wilkinson, The algebraic eigenvaIue problem (Oxford Univ. Press, London. 1965). [6] R.K. Nesbet, Proc. Roy. Sot. A230 (1955) 312; Rev. hlad. Phys. 33 (1961) 28.
1 May 1974
CHEMICAL PHYSICS LETTERS
REDUCEDINTERA~IONMATRiXFORDOUBLY-EXCITEDCONFIGURATIONS ANDTHECORRELATIONENERGYBYAPARTITIONlNGMETHOD William J. TAYLOR Department of Chemistry,Ohio State Universiry. Columbus, Received
12 February
Ohio 43210.
1974
Analyses of the correlation energy of anN-&ctron system by Nesbet and L&din reduced interaction matrix for doubly-excited configurations and an exact expression latter is separable in a basis diagonalizing this matrix.
1. Introduction Nesbet [I, 21 has pointed out that when the dominant contribution in a configuration interaction (CI) expansion of an atomic or molecular wavefunction is made by the Hartree-Fock (HF) determinant, the correlation energy, EC, can be expressed in terms of the coefficients of doubly-excited configurations in the expansion. Although this result shows that only doubly-excited configurations contribute direcrZy to the correlation. energy, the remaining excited configurations affect the coefficients of the doubly-excited configurations, and hence contribute indirecrly to EC. In the present note we consider the latter effect, and in particular derive a reduced interaction matnk for doubly-excited-configurations which includes the effects of all other excited configurations. For this purpose we use an extension of the partitioning technique for sol&ion of the CI secular equation discussed by Ewdin [3,41. However, the present anaIysis is primarily concerned with the “structure” of the correlation energy, rather than computational methods..
2. Reduced iute&tion matrix and correlation energy-
USA
are extended to obtain a for correlation energy; the
orthonormal Slater determinants, {a”}, may be constructed by addition to a0 of the normalized deter&nants.of al: other ordered subsets of N spin-orbitals from the set {vi}. All such configurations may be classified as to their degree of excitation. defmed as the number of spin-orbitals they contain which are not in the subset of N HF orbitals. Thus, in singlyexcited configurations, q, tpi(l N); in doubly-excited configurations, q,?, vi and 4 (1 S i
Let {qi) be a complete set 6f orthbno,mal oneelectron sp@+bitais having.as its f=stN niemb& the HF~orbitals; the normalized & confiiuratioti i&jhen-. .a0 = (N!)-“* det(dI? ...’, t&).~$.compl<~e Setof _... :. ._ .: .’ ,:. :: Y, .::.. .‘.~ .,_ ‘:.. :. :-__‘_.-, ::. _,,. _: ;_-__;;t_::.~I____-_-L~,: : .-: .:.. .-.‘.-~.-~_.-‘_ ., _I. ___.’.. 29’ -. : :: : -, i:. .: -._,:‘_. _..,’ I _.:.-. .: -. _.: .I._-. .. _.
CHEMICAL PHYSICS LE?TERS
Volume 26, number 1
tions, {a,,), corresponds to the following partitioning of the hamiltonian matrix, ElJ H = “20 .i
”
“IX
“OfI
“22
“,,I
“,,7
”
9
+Ho2c2=0,
“20 co + (“22 -E’22)5
of(S) for c,, and E= Eg, in (2b) yields
“20 =() + (“>2 -Eg
I&c2
=$
(6)
9
where (1)
n II no 1 where the subscript 0 refers to Q, 2 refers to doubiyexcited configurations, and n to all other excited configurations. Thus, the typical element of the block i-l,, is (Qo&&$), and of the bIock H,,, (@$iH_@$). The matrix H, and therefore al o its 3 etc. diagocal blocks, are hermitian, while H20 = Ho2, Since HN contains only one- and two-electron operators all matrix elements connecting !Q, with configurations of triple or higher excitation vanish, and in addition matrix elements connecting a0 and singlyexcited configurations vanish in accord with Brillquin’s theorem. Thus, H,, and HnO are null matrices and the matrix eigenvalue equation separates into a scalar and two matrix equations as follows,
(E, -E)c,,
Substitution
1 May 1974
@a)
+“2nc,,
=02 ,
12b)
=o, ; (24 3 2 =2 •I-(%, -El&c, here c has been similarly partitioned, c = (co, q, c,). Setting E = Eg in (?a) yields EC = cc1 Ho2 cz, which is equivalent to Nesbet’s result,
The explicit form of the matrix element in (3) in terms of two-electron repulsion integrals is (e = 1) +Po LMNlap =(Qlr;;Irs)(ijrr;; IF)_
(4)
As stated in the Introduction, one of the objectives of the present discussion is to show how the coefficients, cr,of the doubly-excited confi&rations (and therefore also cc) are influenced by other excited configurations. For E = Eg thematrix in parentheses in (2cj is positivedefite and therefore nor&ingular, so that
“;?
= Hz2 -
“212(“n,I
-Eg
l,J-*
“,,
(7)
is a reduced interaction mat& for doubly-excited configurations which includes the effects of all other excited configurations_ The matrix in parentheses in (6) can also be assumed to be non-singular*, so that if Es is known (6) yields an explicit expression for the coefficients of doubly-excited configurations** c2 = - @Hi2
- Eg lX2Y1 H,,
(8)
(note also that (5) and (8) yield c,J Finally, substitution of (8) for cl, and Eg for E, in (2a) yields the following exact expression for the correlation energy, % = - “,,(H;,
-Eg
122j-1 H20 3
(9)
with H& given by (7) (see also the appendix). In the preceding analysis it has been assumed the basis sets {I,+} and {QV) are complete, so that *g and Eg represent the exact ground state wavefunction and energy; further, it has been assumed that a0 and E. are the exact HF wavefunction and energy, and EC = Eg - EO the exact correlation energy. In practice, of course, it is not possible to work with complete basis sets. However, all the results obtained in the present article remain vaIid for the case offinite basis sets if appropriate changes in the interpretation of the preceding quantities are made. In particular, @o must be the so-called SCF (self-consistent field) determinant which minimizes the expectation value Eo =: (cPoviN190) in the space spanned by the set (~~1, in which case the anaIog of Brillouin’s theorem is applicable (ref. [4], p_ 284). Simiily, ek,, Eg, and EC become the best approximations to these quantities in * I%Js.(2a) and (6).
with Eg =E,
represent a reduced
eigen-.
due equation for the vector (co. cz), corresponding to the secular equation @o--E) I Hzo
H, (Hi2
-Elzz)
=‘-
By ihe sepmation theorem (ref. (5 1, p. 103), the lowest root of the equ+ion lH$z - El22 I = 0 is not less than Eg, and ii may safely be assumed to be greater than Eg in view.
of the definitions of @o and Eo_ ** The reciprocal matriceiin (5) and (8) may be regard& as : : resolvents. <, .- .. :
-.. _. :
Volume 26, number 1
the space spanned by the set I+” 1. As previously indicated our principal interest in (9) and (7) is the insight they give into the dependence of the correlation energy on doubly-excited configurations, and less directly on other excited configurations. Of course, if Ep = E. + EC on the r.h.s. of these two equations is not known, they provide onIy an implicit (but exact) relation for Ec. However, these equations may be sotved iteratively for EC [4J, and they atso lend themselves to the development of perturbationtype expansions *. For example, if only diagonal matrix elements ofHN for doubly-excited configurations are retained in (7), so that Hi* z diag (Ez) with EF = (~~ifYN~~~>, and also Eg is approximated by E, in (9), one obtains
where the matrix element is given by (4). This is the second-order perturbation theory approximation sug gested by Nesbet [Z, 61.
3. Diagonalization
of the reduced interaction
matrix
A formal simplification of the correlation, (9), may be achieved by diagonalization of the reduced interaction matrix, H;,, defined in (7). Let $2 be a umtary matrix which diagonalizes the hermitian matrix H&, so that UiZ Hi2 U22 = AZ2 = diag
(A,) ,
(11)
where the &‘s are the eigenvalues of HiZ; on premultiplying (11) by lJ22 its columns are seen to be the eigenvectors of Hi?, in the same order as the eigenvalues on the diagonal of A22. On solution of (I I) for H’zz and substitution in (9) one readily obtains Ec = -Hb2 [diag(h,
1 May 1974
CHEMICAL PHYSICS LETTERS
- Eg)-‘1
H A; ,
(12)
* It should be pointed out that in applications it would not bB necessary to evaluate the reciprocal matrices on the r.h.s. of (7) and (9). Thus, if A is a uare non-singular matrix and “t one wishes to calculate X = A- H. where H is a conformable matrix; one may instead solve in unison the_mul$ple sets of simultaneous Iin@ar inhomogeneous equations, AX = H.
where Hbz = Hu, UZ2. Define the following orthonorma1 linear combinations of doubly-excited configurations,
where iYtikr,a is an eIement of UZ2 in the oath column and the row associated with ‘PC in HZ2 (or Hi*). From (4), (12), and (13), the correlation energy can be put In the form EC =-c
f(3+$$f?&)f2/(Xa
-Eg)
,
(14)
Q where
Relation (14) is formally similar to the secondorder Brillouin-Wigner perturbation theory expression obtained by replacing EO in the Rayleigh-Schriidinger formula, (lo), by Eg. However, (14) is exact (to all orders) for a complete basis, and shows that the correlation energy is separable when expressed in terms of the eigenvalues and eigenvectors of the reduced interaction matrix for double excitations; this would appear to provide a rigorous basis for electron-pair theories of correlation energy.
Appendix The reduced interacti n matrix, H;2, can be further s analyzed. Thus, H2,t = Hn2 on the r.h.s. of (7) connects doubly-excited configurations with singly-, triply-, and quadruply-excited configurations (at most). Denote these excited configurations having nonvanishing matrix elements with doublyexcited configurations
by p, and the remaining
type n by 4. Then, inpartitioned (H2p,02q) and (7) becomes
configurations
l-52 = H,,_ - Hzp C(Hnn - Eg I,J1lPP =H2a -.Hzp($.-E8 where
&+Hpr
of
form, Hzn = HP’2 ,
(+”
..
.Volume 26. number 1
CHEMICAL PHYSICS LETTERS
is a reduced interaction matrix for configurations of type p. The same results are obtained if the set {*U I is partitioned into four subsets, namely, cPO, doublyexcited configurations, and configurations of types p and q, and cII and cP are then eliminated successively from the four equations analogous to (2a)-(2c) which are obtained. This was avoided in order not to confuse the main lines of the discussion in the body of this article. Ii is ctear that a hierarchy of such reductions is possible; e.g., the next partitioning would separate excited configurations of degrees 5 and 6 (which interact with type p configurations) from the remaining type 4 configurations. This process would terminate when excitations of the highest even degree less than
1 May 1974
N were reached. From this point of view the present approach may he characterized as one of nested resolvents. However, the reduction of Hi2 in (Al, A2) is probably sufficient for most purposes.
References [l] [2] [3\ I41 [5]
R.K. Nesbet, Phys. Rev. 109 (1958) 1632. R.K. Nesbet. Advan. Chem. Phys. 9 (1965) 321. P.-O. Ltiwdin, J. Chem. Phys. I8 (1951) 1396. P.-O. Liiwdin, Advan. Chem. Phys. 2 (1959) 207. J-H. Wilkinson, The algebraic eigenvaIue problem (Oxford Univ. Press, London. 1965). [6] R.K. Nesbet, Proc. Roy. Sot. A230 (1955) 312; Rev. hlad. Phys. 33 (1961) 28.