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Generalized Møller—Plesset perturbation theory applied to general MCSCF reference wave functions
Volume 183, number 5
CHEMICAL PHYSICS LETTERS
6 September 199 I
Generalized Msller-Plesset perturbation theory applied to general MCSCF reference wave functions Robert B. Murphy a and Richard P. Messmer a-b a Department ofPhysics, Universityof Pennsylvania,Philadelphia,PA 19104, USA b General Electric Corporate Research and Development,Schenectady, NY 12301, US.4 Received 28 May 1991: in final form 28 June 1991
We report the application of a generalized Msller-Plesset perturbation theory to multiconfigurational reference wave functions. The theory makes use of a more compact first-order space than earlier investigations and results in a computationally simplified approach. The low-order perturbation theory applied to multiconfigurational reference functions obtains a significant fraction of dynamic correlation energy for a relatively low computational cost. The Importance of using a reference wave function that dissociates properly is discussed. Results are reported for the symmetric dissociation of Hz0 and the energy differcncc between the ‘A, and ‘8, states of CHZ.
1. Introduction We describe a version of generalized Merller-Plesset theory involving a full implementation of MrallerPlesset perturbation theory applied to general multiconfigurational reference wave functions. The method yields a simpler computational approach than previous [ l-31 formulations of the theory. The perturbation theory at low orders obtains a large fraction of the dynamic correlation energy for a relatively small computational cost and converges well over the full potential surface provided the reference wave function contains the essential configurations for dissociation. We have chosen the symmetric dissociation of water and the singlet-triplet energy difference in methylene as initial test applications. It is well established that a correct description of a potential surface including the dissociation limit requires a multiconfigurational (MCSCF) [ 41 wave function. The selection of the configurations that give the dominant correlation effects at equilibrium and allow for proper dissociation is facilitated by the use of a generalized valence bond (GVB) multiconfigurational wave function [ 5 1. For a normal twoelectron covalent bond a perfect-pairing GVB wave function dissociates properly by using two natural orbitals and two configurations to correlate the bond 0009-2614/91/$
pair. A GVB wave function can easily be extended to treat non-perfect-pairing spin couplings by incorporating in the Cl expansion non-perfect-pairing open-shell configurations constructed from the GVB natural orbitals. These non-perfect-pairing configurations are important to treat multiple bonds or dissociation of a molecule into high spin moieties. Such a wave function has been referred to as a GVB-RCI (restricted CI) wave function [ 6 1. A GVB-RCI wave function is constructed by allowing for the three possible distributions of two electrons in each pair of natural orbitals thus giving 3N spatial configurations for N pairs. The difference between a self-consistent GVB-RCI wave function and a complete active space (CASSCF) [ 7,8] wave function defined in the active space of the bond pair natural orbitals is that the CASSCF wave function includes all excitations within and between bond pair orbitals while the GVB-RCI wave function only contains excitations within bond pair orbitals. Although GVB-RCI and CASSCF wave functions give the correct shapes of potential surfaces, these wave functions neglect possibly significant correlation effects. This point is illustrated by comparing large multi-reference Cl or full Cl calculations with CASSCF results [ 41. Since large CI calculations are computationally expensive, several efforts [ l-3,9-
03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
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Volume 183, number 5
141 have been made to apply perturbation theory to MCSCF reference wave functions, One of the most successful and thorough of these theories is the generalized Mraller-Plesset theory of Wolinski et al. [ 1,2]. This theory naturally extends traditional Moller-Plesset theory from the use of a Hartree-Fock reference to use of a MCSCF reference wave function. The theory is size consistent for size consistent reference wave functions. The full implementation of the generalized Msller-Plesset theory applied to two-configuration references was shown to provide excellent estimates of full CI energies when carried out to third order [ 1,2]. An attempt [ 31 was made to make the theory more computationally tractable by an empirical simplification of the reference Hamiltonian with some success.
where orbitals i, j are occupied in the reference, u, v are virtual orbitals, E,; and euivjare single and double excitation operators [4]. The functions { YA} span the spin space having the same total spin and MS as the reference wave function. The 1YA) are mutually orthogonal, orthogonal to all configurations in the reference, and hence to the full 1!Po). The set { YA} spans the entire first-order interacting space [ 171, and has been used in other multicontiguration perturbation theories [ 12- 14 1. By contrast, the first-order space used in refs. [l31 is defined in terms of all external and semi-internal double excitations out of the full reference wave function using a generator state formalism. In that formalism [l-3] an external double excitation is generated by a double excitation to the virtual space orbitals U, v from the full reference wave function:
LE,I%b>
2. Theory The zeroth-order reference wave function MCSCF [ 15]#’ wave function expanded as, IYou,>= TC,l
Y?>,
Iv>=4@,ell~
is a
(1)
where Drn is a spatial orbital product and 8, is a spin eigenfunction. The index I labels each unique antisymmetrized product of an orbital product and a spin eigenfunction, i.e. I labels each configuration state function. We define the first-order correction, Y(i), to the reference in terms of the configurations 1!PA) generated by single and double excitations of the reference configurations I YF> with the constraint that none of the I !PA) are equivalent to any reference configuration. For the sake of comparison to the first-order space defined in refs. [ l-3 ] we will assume in the following discussion that the reference is a CASSCF wave function. For a CASSCF reference the single and double excitations are to all virtual orbitals not occupied in the reference from the configurations I !@) used to define the reference wave function:
(3) *’ The ALIS system of MCSCF programs was used [ 161.
>
(4)
where the notation of refs. [ l-31 is used. A semi-internal excitation involves an excitation within the set of orbitals j, k occupied in the reference and an excitation from reference orbital i to a virtual orbital lA: EM,Ejk I % > .
(5)
The external double excitation E,,E, applied to the full reference (eq. (4) ) will create a linear combination of the l YA) generated by the double excitation to the virtual space orbitals u, u from a set of reference configurations ( Yp ). Similarly, the semiinternal double excitation E,iEjk applied to the full reference wave function (eq. (5)) will create a linear combination of the l YL) generated by the single excitation to the virtual orbital u from a set of reference configurations 1YVp)The coefficients of the linear combinations are determined by the coefticients of the configurations in the reference wave function (eq. ( 1) ) and the type of excitation. The linear combination generated by the semi-internal double excitation (eq. ( 5) ) differs from the linear combination obtained by a single excitation to the virtual space from the full reference wave function: E,,IYo).
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CHEMICAL PHYSICS LETTERS
(6)
In fact the space generated by single excitations of the full reference wave function to the virtual space do not interact through the full Hamiltonian with the
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CHEMICAL PHYSICS LETTERS
reference space by virtue of the extended Brillouin theorem. As a consequence this space is not included in the first-order space of refs. [ l-31. The first-order space used here, consisting of the individual 1!I’:), does not explicitly exclude the space generated (eq. (6) ) by single excitations from the full reference wave -function (which does not interact with the converged MCSCF reference by the Brillouin theorem). The first-order basis generated by the semi-internal and external double excitations of the reference wave function [ l-3] is not orthogonal and an orthogonalization step is necessary to avoid the complications of perturbation calculations in a non-orthogonal basis. The resulting orthogonalized basis can make the evaluation of the first-order correction Vi1 more difficult as discussed below. The use of the individual 1YL ) as the first-order basis avoids the orthogonalization step and simplifies the perturbation calculations. The reference Hamiltonian Ho for Y,, is defined analogously to that in ref. [ 2 1,If,, = P&P,, + PxdFPsd+ ... . POis the projector onto the reference wave function, Psd projects onto the first-order space and higher-order spaces can be similarly defined. The operator F is a sum of one-particle generalized Fock operators [2],f(i). i.e. F= x,f(ij, with matrix elements among spatial orbitals p, q given by, (7)
where h is the bare one-electron Hamiltonian,
6 September 1991
order space of refs. [ l-3] where the orthogonalization step makes (Ho - 1 ED) not as sparse and can create large off-diagonal elements, both factors making the inversion more difficult. An effort was made in ref. [ 31 to simplify (Ho- I&) by keeping only the elements of Ho diagonal in the orbital indices (eq. (7)). The results presented below using the full H,, indicate that the approximation of a H, diagonal in the orbital indices [3] is somewhat successful but not entirely reliable since it can give second-order corrected energies below the full CI energies. Given the first-order C,, the second- and third-order corrections to the reference energy can be evaluated [ 21. The third-order energy correction requires more computational work than the secondorder correction since the full Hamiltonian matrix elements over the first-order space are required as opposed to only Hamiltonian matrix elements between the first-order space and the reference wave function for the second-order energy. The third-order energy correction is typically an order of magnitude smaller than the second-order correction for reference wave functions that dissociate properly. We note that this formulation gives numerical results which are nearly identical to those of the full implementation of generalized Meller-Plesset theory to two-configuration reference wave functions in ref. [ 2 1. The small differences #’ between the two implementations are attributed to different definitions of the first-order space.
and
D, is the first-order density matrix of the reference wave function. The operator F can be derived by ex-
panding the reference wave function in single determinants with M,=O. The coefficients C, (eq. (2 ) ) defining the lirst-order correction Y(’ ) are determined by the set of linear equations [ 3 1,
where the sum over a is over the first-order space, H is the full Hamiltonian, and Eo= ( Y01Ho I!Po) _Thus an inversion of the matrix (Ho- lE,) over the firstorder space is required to find the C,. In the firstorder basis that we have used, (Ho - I&) is a very sparse matrix which can be easily inverted using an iterative scheme [ 18 1.This is in contrast to the first-
3. Results 3.1. H,O A correct reference wave function for the symmetric dissociation of water must include two orbitals per O-H bond and allow for the fact that the oxygen atom will have a )P ground state in the dissociated limit. The reference wave function considered first which satisfies these dissociation conx2
For the case of the two-configuration reference for CH4 in ref. [2] at the equilibrium geometry the second-order corrected energy calculated within the formalism presented here is 0.05 eV lower than that of ref. [ 21.
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straints is a CASSCF wave function defined in the space of the four (a,, a:, bZ, bi ) O-H orbitals henceforth labeled as CASSCF-4. The double-zeta basis set and geometries are those of refs. [ 19,201. The CASSCF-4 energies shown in table 1 from the generalized Meller-Plesset perturbation theory are within 0.25 eV of the full CI results. The second-order corrected energies are within a few millihartrees of those obtained in ref. [ 31 with the same CASSCF4 reference using a Ho which only retained elements diagonal in the orbital indices (GMP2 diag values in table 1). The second reference consists of a self-consistent GVB-RCI [6] wave function which correlates the O-H bonds and dissociates properly by using four natural orbitals to describe the O-H bonds (GVBRCI-4). The GVB-RCI-4 wave function consists of ten configuration state functions while the CASSCF4 wave function above contains twelve. The GVBRCI-4 wave function expanded in terms of symmetry orbitals rather than natural orbitals contains all of the CASSCF-4 configurations, but with constraints on the weights of the configurations. The second- and third-order corrected GVB-RCI-4 energies in table 1 are nearly identical to the correspondTable I Generalized M&z-Plesset
ing CASSCF-4 values. The final MCSCF reference wave function considered is a CASSCF function with an eight-orbital active space consisting of the O-H orbitals (a,, a;, b?, b:), and the lone-pair (a,, a?, b,, by ) orbitals labeled as CASSCF-8. The second-order corrected energies of the CASSCF-8 reference in table 1 are all within 0.1 eV of the full CI values. The second-order corrected results are essentially identical to those obtained in ref. [3] with a Ho diagonal in the orbital indices and the same CASSCF-8 reference. This near agreement with the diagonal Ho results for the CASSCF-8 reference suggests that the diagonal approximation is reasonable if the reference wave function is sufficiently well correlated, however, more tests are needed to affirm this conjecture. The thirdorder CASSCF-8 corrected energies are within 0.05 eV of the full CI energies. A comparison of the CASSCF-4 and GVB-RCI-4 results with the CASSCF-8 results suggests that a reference wave function that correlates all valence electrons combined with the perturbation theory gives very good estimates of full CI energies. The perturbation theory is clearly converging for the reference wave functions above which dissociate
results ‘) for the symmetric dissociation of Hz0 (energies in hartree units)
Method
R,
l.SR,
2.OR,
2.5R,
HF MP2 MP3 GVB-PP GMP2 GMP3 CASSCF-4 GMP2 GMP3 GMP2 diag b, GVB-RCI-4 GMP2 GMP3 CASSCF-8 GMP2 GMP3 GMP2 diag b’ full CI c’
-76.00984 -76.14932 -76.15071 -76.05415 -76.14918 -76.14914 - 76.06289 _ i6.15009 - 76.15287 - 76.1490 - 76.06238 -76.14977 _ 76.15275 -76.13201 - 16.15417 - 76.15691 _ i6.1548 - 76.15787
-75.80353 - 75.99458 - 75.98939 -75.91630 - 76.00586 - 76.00227 -15.92435 - 76.00797 -76.00736 -76.0095 -75.92384 - 76.00740 - 76.0068 I -75.98160 -76.01047 -76.01268 -76.0105 -76.01452
-75.59518 -75.85246 -75.83481 -75.81774 -75.89226 -75.88723 -75.82723 -75.90059 -75.89822 -75.9011 -75.82632 -75.89940 -75.89709 -75.86575 -75.90232 -75.90319 -75.9025 -75.90525
-75.41482
a’ The 0 Is core is correlated in the MP and full CI calculations. b, Results from ref. [ 31 which used only diagonal elements of H,. ‘) Refs. [ 19,201.
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6 September I99 1
- 75.78503
-75.80150 -75.86712 - 75.86807 -75.80104
-75.86689 -75.86713 -75.83164 - 75.86995 - 75.87068
Volume183.number 5
1
[ 12-141 must be done to ascertain which contigurations should be part of the reference and which should be treated perturbatively.
.oo
B 2
0.75
s Y 'Z de
0.50
3.2. CH,
g $
6 September I99 1
CHEMICAL PHYSICS LETTERS
0.25
d 0.00
1.0
15
2.0
0-H Bond Length ( unirsofR, )
Fig. I. Symmetric dissociation of H1O. Moller-Plesset secondand third-order corrected energies for HF, GVB-PP, CASSCF-4 and CASSCF-8 reference wave functions relative to full CI results. (0) HF-M2; (0) HF-MP3: (0) GVB-MP2: (A) GVBMP3; (+) CAWMP2; (n ) CAS4-MP3: ( A ) CASMP2; (0) CASX-MP3.
properly with the third-order correction energies roughly an order of magnitude less than the secondorder energies. This convergent behavior is in contrast to the Hartree-Fock based Msller-Plesset results (table 1) which diverge strongly from the full CI results as the bond length is increased (fig. 1). This divergence results from a lack of the configurations in the reference wave function necessary for dissociation. We have also used a GVB perfect-pairing reference wave function which incorrectly dissociates to an oxygen atom with 314 triplet and l/ 4 singlet coupling. The GVB-PP based Merller-Plesset energies (table 1) also diverge from the full, CI values at large O-H bond lengths due to the lack of the coniigurations which allow for the 3P oxygen atom (fig. 1). These results for the dissociation of water indicate that a prerequisite to the convergence of the perturbation theory over the full potential surface is that the reference wave function must contain all of the dominant configurations necessary for dissociation. The local nature of a GVB-RCI MCSCF expansion and the physical interpretability of the GVB wave function in terms of bond structures [ 21,221 is an advantage that allows for the construction of a reference wave function with the dominant correlation effects accounted for and with the proper dissociation characteristics. Without a physically motivated reference wave function a significant amount of work
As a more stringent test of generalized MerllerPlesset theory we have calculated the second- and third-order energy corrections to the ‘A, and 3B, states of CH2. The double-zeta plus polarization basis set and geometries are from ref. [ 231. The reference wave function consists of a CASSCF defined by the six CH2 valence orbitals, the C-H orbitals (a,, a:, bZ, b:) and the lone-pair (a,, b,) orbitals. The third-order corrected energies in table 2 are within 0.1 eV of the full CI. We note that the perturbation theory of ref. [ 31 which retained only the diagonal elements of Ho finds a second-order corrected energy below the full CI for the 3B, state and the same CASSCF reference (GMP2 diag values in table 2). We did not encounter this problem using the full Hw The third-order energy correction is smaller for the 3B, state than the ‘A, state in accord with the general trend of a larger correlation problem in singlets than higher spin state. The energy gap between the ‘A, and 3B, states is poorly described by the second-order theory. A similar discrepancy at second order was found in refs. [ 3,9 1. The reason for the overly large energy gap at second order seems to be that the singlet is less well described by the second-order correction than the triplet as mentioned. This imbalance puts the singlet too high in energy relative to the triplet at second order. The third-order correction Table 2 Generalized hartree)
Meller-Plesset
results ‘) for CHI
(energies
in
Method
&‘A,)
E(‘B,)
a (kcal/mol)
HF CASSCF GMPZ GMP3 GMPZ diag b1 full CI c)
- 38.88630 - 38.94553 -39.01331 - 39.02322 - 39.02545 - 39.027 18
-38.92795 - 38.96595 - 39.03794 - 39.04338 -39.05117 - 39.04626
26.14 12.82 15.45 12.65 16.13 II.97
a> The C Is core is not correlated. ” Results from ref. [ 31 which used only the diagonal part of Ho. ‘) Ref. [23].
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improves the singlet energy sufficiently to give a gap which is 0.68 kcal/mol within the full CI result.
6
September199 1
References
[ I] K. Wolinski, H.L. Sellers and P. Pulay, Chem. Phys. Letters 4. Conclusion The results above show that generalized MsllerPlesset perturbation theory applied to MCSCF reference wavefunctions which dissociate properly is a computationally viable method for obtaining a large fraction of dynamic correlation energy. The tirst-order space defined here simplifies previous presentations of the theory. Since the generalized MerllerPlesset perturbation theory gives accurate energies over the full potential surface, realistic model potentials for use in dynamics calculations could be constructed using results of the perturbation theory. In future applications of perturbation theory to larger systems, the generalized valence bond approach (in particular a self-consistent GVB-RCI reference wave function) will prove to be invaluable for defining a MCSCF reference wave function which dissociates properly. The localized nature of the natural orbitals comprising the GVB-RCI wave function allows for an implementation of local correlation theory through perturbation theory as has been done for HartreeFock based Merller-Plesset theory [ 241. A successful local correlation theory of this type would be computationally more efficient and provides the possibility of selectively and accurately correlating the most important parts of a system.
Acknowledgement RBM acknowledges the support of an AT&T graduate scholarship and a useful discussion with P. Pulay.
140 (1987) 225.
[ 21 K. Wolinski and P. Pulay, I. Chem. Phys. 90 ( 1989) 3647. [ 31 K. Andersson, P. Malmqvist, B.O. Roos, A.J. Sadlej and K. Wolinski, J. Phys. Chem. 94 (1990) 5483. [4] R. Shepard, Advan. Chem. Phys. 69 (1987) 63. [ 51F.W. Bobrowiz and W.A. Goddard III, in: Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977) p. 79. [6] E.A. Carter and W.A. Goddard III, J. Chem. Phys. 86 (1987) 862. [i’] L.M. Cheung, K.R. Sundberg and K. Ruedenberg, Intern. J. Quantum Chem. 16 (1979) 1103. [8] P. Siegbahn, A. Heiberg, B. Roos and B. Levy, Physica Scripta 21 (1980) 323. [9] J.W. McDouall, K. Peasley and M. Robb, Chem. Phys. Letters 148 ( 1988) 183. [ 101 B.O. Roos, P. Lime, P.E.M. Slegbahn and M.R.A. Blomberg, Chem. Phys. 66 (1981) 197. [ 111 G. Hose, J. Chem. Phys. 84 ( 1986) 4505. [ 121 B. Huron, J.P. Malrieu and P. Rancurel, J. Chem. Phys. 58 (1973) 5745. [ 131 R.J. Harrison, J. Chem. Phys. 94 (1991) 5021. [ 141 J. Novoa, F. Mota and F. Arnau, J. Phys. Chem. 95 (1991) 3096. [ 151 W.I. Salmon and K. Ruedenberg, J. Chem. Phys. 57 ( 1972) 2776. [ 161 K. Ruedenberg, L.M. Cheung and S.T. Elben, Intern. J. Quantum Chem. 16 (1979) 1069. [ 171 A.D. McLean and B. Liu, J. Chem. Phys. 58 ( 1972) 1066. [I81 W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical recipes - the art of scientific computing (Cambridge Univ. Press, London, 1989) p. 71. [ 191 P. Saxe, H.F. Schaefer III and N.C. Handy, Chem. Phys. Letters 79 (1981) 202. [20] R.J. Harrison and N.C. Handy, Chem. Phys. Letters 95 (1983) 386. [21] W.A. Goddard III and L.B. Harding, Ann. Rev. Phys. Chem. 29 (1978) 363. [221 R.P. Messmer, J. Plm. Chem. Sot. 113 ( 199 1) 433. [23] C.W. Bauschlicher Jr. and P.R. Taylor, J. Chem. Phys. 85 (1986) 6510. [24] S. &be and P. Pulay, J. Chem. Phys. 86 (1986) 914.
CHEMICAL PHYSICS LETTERS
6 September 199 I
Generalized Msller-Plesset perturbation theory applied to general MCSCF reference wave functions Robert B. Murphy a and Richard P. Messmer a-b a Department ofPhysics, Universityof Pennsylvania,Philadelphia,PA 19104, USA b General Electric Corporate Research and Development,Schenectady, NY 12301, US.4 Received 28 May 1991: in final form 28 June 1991
We report the application of a generalized Msller-Plesset perturbation theory to multiconfigurational reference wave functions. The theory makes use of a more compact first-order space than earlier investigations and results in a computationally simplified approach. The low-order perturbation theory applied to multiconfigurational reference functions obtains a significant fraction of dynamic correlation energy for a relatively low computational cost. The Importance of using a reference wave function that dissociates properly is discussed. Results are reported for the symmetric dissociation of Hz0 and the energy differcncc between the ‘A, and ‘8, states of CHZ.
1. Introduction We describe a version of generalized Merller-Plesset theory involving a full implementation of MrallerPlesset perturbation theory applied to general multiconfigurational reference wave functions. The method yields a simpler computational approach than previous [ l-31 formulations of the theory. The perturbation theory at low orders obtains a large fraction of the dynamic correlation energy for a relatively small computational cost and converges well over the full potential surface provided the reference wave function contains the essential configurations for dissociation. We have chosen the symmetric dissociation of water and the singlet-triplet energy difference in methylene as initial test applications. It is well established that a correct description of a potential surface including the dissociation limit requires a multiconfigurational (MCSCF) [ 41 wave function. The selection of the configurations that give the dominant correlation effects at equilibrium and allow for proper dissociation is facilitated by the use of a generalized valence bond (GVB) multiconfigurational wave function [ 5 1. For a normal twoelectron covalent bond a perfect-pairing GVB wave function dissociates properly by using two natural orbitals and two configurations to correlate the bond 0009-2614/91/$
pair. A GVB wave function can easily be extended to treat non-perfect-pairing spin couplings by incorporating in the Cl expansion non-perfect-pairing open-shell configurations constructed from the GVB natural orbitals. These non-perfect-pairing configurations are important to treat multiple bonds or dissociation of a molecule into high spin moieties. Such a wave function has been referred to as a GVB-RCI (restricted CI) wave function [ 6 1. A GVB-RCI wave function is constructed by allowing for the three possible distributions of two electrons in each pair of natural orbitals thus giving 3N spatial configurations for N pairs. The difference between a self-consistent GVB-RCI wave function and a complete active space (CASSCF) [ 7,8] wave function defined in the active space of the bond pair natural orbitals is that the CASSCF wave function includes all excitations within and between bond pair orbitals while the GVB-RCI wave function only contains excitations within bond pair orbitals. Although GVB-RCI and CASSCF wave functions give the correct shapes of potential surfaces, these wave functions neglect possibly significant correlation effects. This point is illustrated by comparing large multi-reference Cl or full Cl calculations with CASSCF results [ 41. Since large CI calculations are computationally expensive, several efforts [ l-3,9-
03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
443
Volume 183, number 5
141 have been made to apply perturbation theory to MCSCF reference wave functions, One of the most successful and thorough of these theories is the generalized Mraller-Plesset theory of Wolinski et al. [ 1,2]. This theory naturally extends traditional Moller-Plesset theory from the use of a Hartree-Fock reference to use of a MCSCF reference wave function. The theory is size consistent for size consistent reference wave functions. The full implementation of the generalized Msller-Plesset theory applied to two-configuration references was shown to provide excellent estimates of full CI energies when carried out to third order [ 1,2]. An attempt [ 31 was made to make the theory more computationally tractable by an empirical simplification of the reference Hamiltonian with some success.
where orbitals i, j are occupied in the reference, u, v are virtual orbitals, E,; and euivjare single and double excitation operators [4]. The functions { YA} span the spin space having the same total spin and MS as the reference wave function. The 1YA) are mutually orthogonal, orthogonal to all configurations in the reference, and hence to the full 1!Po). The set { YA} spans the entire first-order interacting space [ 171, and has been used in other multicontiguration perturbation theories [ 12- 14 1. By contrast, the first-order space used in refs. [l31 is defined in terms of all external and semi-internal double excitations out of the full reference wave function using a generator state formalism. In that formalism [l-3] an external double excitation is generated by a double excitation to the virtual space orbitals U, v from the full reference wave function:
LE,I%b>
2. Theory The zeroth-order reference wave function MCSCF [ 15]#’ wave function expanded as, IYou,>= TC,l
Y?>,
Iv>=4@,ell~
is a
(1)
where Drn is a spatial orbital product and 8, is a spin eigenfunction. The index I labels each unique antisymmetrized product of an orbital product and a spin eigenfunction, i.e. I labels each configuration state function. We define the first-order correction, Y(i), to the reference in terms of the configurations 1!PA) generated by single and double excitations of the reference configurations I YF> with the constraint that none of the I !PA) are equivalent to any reference configuration. For the sake of comparison to the first-order space defined in refs. [ l-3 ] we will assume in the following discussion that the reference is a CASSCF wave function. For a CASSCF reference the single and double excitations are to all virtual orbitals not occupied in the reference from the configurations I !@) used to define the reference wave function:
(3) *’ The ALIS system of MCSCF programs was used [ 161.
>
(4)
where the notation of refs. [ l-31 is used. A semi-internal excitation involves an excitation within the set of orbitals j, k occupied in the reference and an excitation from reference orbital i to a virtual orbital lA: EM,Ejk I % > .
(5)
The external double excitation E,,E, applied to the full reference (eq. (4) ) will create a linear combination of the l YA) generated by the double excitation to the virtual space orbitals u, u from a set of reference configurations ( Yp ). Similarly, the semiinternal double excitation E,iEjk applied to the full reference wave function (eq. (5)) will create a linear combination of the l YL) generated by the single excitation to the virtual orbital u from a set of reference configurations 1YVp)The coefficients of the linear combinations are determined by the coefticients of the configurations in the reference wave function (eq. ( 1) ) and the type of excitation. The linear combination generated by the semi-internal double excitation (eq. ( 5) ) differs from the linear combination obtained by a single excitation to the virtual space from the full reference wave function: E,,IYo).
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(6)
In fact the space generated by single excitations of the full reference wave function to the virtual space do not interact through the full Hamiltonian with the
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reference space by virtue of the extended Brillouin theorem. As a consequence this space is not included in the first-order space of refs. [ l-31. The first-order space used here, consisting of the individual 1!I’:), does not explicitly exclude the space generated (eq. (6) ) by single excitations from the full reference wave -function (which does not interact with the converged MCSCF reference by the Brillouin theorem). The first-order basis generated by the semi-internal and external double excitations of the reference wave function [ l-3] is not orthogonal and an orthogonalization step is necessary to avoid the complications of perturbation calculations in a non-orthogonal basis. The resulting orthogonalized basis can make the evaluation of the first-order correction Vi1 more difficult as discussed below. The use of the individual 1YL ) as the first-order basis avoids the orthogonalization step and simplifies the perturbation calculations. The reference Hamiltonian Ho for Y,, is defined analogously to that in ref. [ 2 1,If,, = P&P,, + PxdFPsd+ ... . POis the projector onto the reference wave function, Psd projects onto the first-order space and higher-order spaces can be similarly defined. The operator F is a sum of one-particle generalized Fock operators [2],f(i). i.e. F= x,f(ij, with matrix elements among spatial orbitals p, q given by, (7)
where h is the bare one-electron Hamiltonian,
6 September 1991
order space of refs. [ l-3] where the orthogonalization step makes (Ho - 1 ED) not as sparse and can create large off-diagonal elements, both factors making the inversion more difficult. An effort was made in ref. [ 31 to simplify (Ho- I&) by keeping only the elements of Ho diagonal in the orbital indices (eq. (7)). The results presented below using the full H,, indicate that the approximation of a H, diagonal in the orbital indices [3] is somewhat successful but not entirely reliable since it can give second-order corrected energies below the full CI energies. Given the first-order C,, the second- and third-order corrections to the reference energy can be evaluated [ 21. The third-order energy correction requires more computational work than the secondorder correction since the full Hamiltonian matrix elements over the first-order space are required as opposed to only Hamiltonian matrix elements between the first-order space and the reference wave function for the second-order energy. The third-order energy correction is typically an order of magnitude smaller than the second-order correction for reference wave functions that dissociate properly. We note that this formulation gives numerical results which are nearly identical to those of the full implementation of generalized Meller-Plesset theory to two-configuration reference wave functions in ref. [ 2 1. The small differences #’ between the two implementations are attributed to different definitions of the first-order space.
and
D, is the first-order density matrix of the reference wave function. The operator F can be derived by ex-
panding the reference wave function in single determinants with M,=O. The coefficients C, (eq. (2 ) ) defining the lirst-order correction Y(’ ) are determined by the set of linear equations [ 3 1,
where the sum over a is over the first-order space, H is the full Hamiltonian, and Eo= ( Y01Ho I!Po) _Thus an inversion of the matrix (Ho- lE,) over the firstorder space is required to find the C,. In the firstorder basis that we have used, (Ho - I&) is a very sparse matrix which can be easily inverted using an iterative scheme [ 18 1.This is in contrast to the first-
3. Results 3.1. H,O A correct reference wave function for the symmetric dissociation of water must include two orbitals per O-H bond and allow for the fact that the oxygen atom will have a )P ground state in the dissociated limit. The reference wave function considered first which satisfies these dissociation conx2
For the case of the two-configuration reference for CH4 in ref. [2] at the equilibrium geometry the second-order corrected energy calculated within the formalism presented here is 0.05 eV lower than that of ref. [ 21.
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straints is a CASSCF wave function defined in the space of the four (a,, a:, bZ, bi ) O-H orbitals henceforth labeled as CASSCF-4. The double-zeta basis set and geometries are those of refs. [ 19,201. The CASSCF-4 energies shown in table 1 from the generalized Meller-Plesset perturbation theory are within 0.25 eV of the full CI results. The second-order corrected energies are within a few millihartrees of those obtained in ref. [ 31 with the same CASSCF4 reference using a Ho which only retained elements diagonal in the orbital indices (GMP2 diag values in table 1). The second reference consists of a self-consistent GVB-RCI [6] wave function which correlates the O-H bonds and dissociates properly by using four natural orbitals to describe the O-H bonds (GVBRCI-4). The GVB-RCI-4 wave function consists of ten configuration state functions while the CASSCF4 wave function above contains twelve. The GVBRCI-4 wave function expanded in terms of symmetry orbitals rather than natural orbitals contains all of the CASSCF-4 configurations, but with constraints on the weights of the configurations. The second- and third-order corrected GVB-RCI-4 energies in table 1 are nearly identical to the correspondTable I Generalized M&z-Plesset
ing CASSCF-4 values. The final MCSCF reference wave function considered is a CASSCF function with an eight-orbital active space consisting of the O-H orbitals (a,, a;, b?, b:), and the lone-pair (a,, a?, b,, by ) orbitals labeled as CASSCF-8. The second-order corrected energies of the CASSCF-8 reference in table 1 are all within 0.1 eV of the full CI values. The second-order corrected results are essentially identical to those obtained in ref. [3] with a Ho diagonal in the orbital indices and the same CASSCF-8 reference. This near agreement with the diagonal Ho results for the CASSCF-8 reference suggests that the diagonal approximation is reasonable if the reference wave function is sufficiently well correlated, however, more tests are needed to affirm this conjecture. The thirdorder CASSCF-8 corrected energies are within 0.05 eV of the full CI energies. A comparison of the CASSCF-4 and GVB-RCI-4 results with the CASSCF-8 results suggests that a reference wave function that correlates all valence electrons combined with the perturbation theory gives very good estimates of full CI energies. The perturbation theory is clearly converging for the reference wave functions above which dissociate
results ‘) for the symmetric dissociation of Hz0 (energies in hartree units)
Method
R,
l.SR,
2.OR,
2.5R,
HF MP2 MP3 GVB-PP GMP2 GMP3 CASSCF-4 GMP2 GMP3 GMP2 diag b, GVB-RCI-4 GMP2 GMP3 CASSCF-8 GMP2 GMP3 GMP2 diag b’ full CI c’
-76.00984 -76.14932 -76.15071 -76.05415 -76.14918 -76.14914 - 76.06289 _ i6.15009 - 76.15287 - 76.1490 - 76.06238 -76.14977 _ 76.15275 -76.13201 - 16.15417 - 76.15691 _ i6.1548 - 76.15787
-75.80353 - 75.99458 - 75.98939 -75.91630 - 76.00586 - 76.00227 -15.92435 - 76.00797 -76.00736 -76.0095 -75.92384 - 76.00740 - 76.0068 I -75.98160 -76.01047 -76.01268 -76.0105 -76.01452
-75.59518 -75.85246 -75.83481 -75.81774 -75.89226 -75.88723 -75.82723 -75.90059 -75.89822 -75.9011 -75.82632 -75.89940 -75.89709 -75.86575 -75.90232 -75.90319 -75.9025 -75.90525
-75.41482
a’ The 0 Is core is correlated in the MP and full CI calculations. b, Results from ref. [ 31 which used only diagonal elements of H,. ‘) Refs. [ 19,201.
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6 September I99 1
- 75.78503
-75.80150 -75.86712 - 75.86807 -75.80104
-75.86689 -75.86713 -75.83164 - 75.86995 - 75.87068
Volume183.number 5
1
[ 12-141 must be done to ascertain which contigurations should be part of the reference and which should be treated perturbatively.
.oo
B 2
0.75
s Y 'Z de
0.50
3.2. CH,
g $
6 September I99 1
CHEMICAL PHYSICS LETTERS
0.25
d 0.00
1.0
15
2.0
0-H Bond Length ( unirsofR, )
Fig. I. Symmetric dissociation of H1O. Moller-Plesset secondand third-order corrected energies for HF, GVB-PP, CASSCF-4 and CASSCF-8 reference wave functions relative to full CI results. (0) HF-M2; (0) HF-MP3: (0) GVB-MP2: (A) GVBMP3; (+) CAWMP2; (n ) CAS4-MP3: ( A ) CASMP2; (0) CASX-MP3.
properly with the third-order correction energies roughly an order of magnitude less than the secondorder energies. This convergent behavior is in contrast to the Hartree-Fock based Msller-Plesset results (table 1) which diverge strongly from the full CI results as the bond length is increased (fig. 1). This divergence results from a lack of the configurations in the reference wave function necessary for dissociation. We have also used a GVB perfect-pairing reference wave function which incorrectly dissociates to an oxygen atom with 314 triplet and l/ 4 singlet coupling. The GVB-PP based Merller-Plesset energies (table 1) also diverge from the full, CI values at large O-H bond lengths due to the lack of the coniigurations which allow for the 3P oxygen atom (fig. 1). These results for the dissociation of water indicate that a prerequisite to the convergence of the perturbation theory over the full potential surface is that the reference wave function must contain all of the dominant configurations necessary for dissociation. The local nature of a GVB-RCI MCSCF expansion and the physical interpretability of the GVB wave function in terms of bond structures [ 21,221 is an advantage that allows for the construction of a reference wave function with the dominant correlation effects accounted for and with the proper dissociation characteristics. Without a physically motivated reference wave function a significant amount of work
As a more stringent test of generalized MerllerPlesset theory we have calculated the second- and third-order energy corrections to the ‘A, and 3B, states of CH2. The double-zeta plus polarization basis set and geometries are from ref. [ 231. The reference wave function consists of a CASSCF defined by the six CH2 valence orbitals, the C-H orbitals (a,, a:, bZ, b:) and the lone-pair (a,, b,) orbitals. The third-order corrected energies in table 2 are within 0.1 eV of the full CI. We note that the perturbation theory of ref. [ 31 which retained only the diagonal elements of Ho finds a second-order corrected energy below the full CI for the 3B, state and the same CASSCF reference (GMP2 diag values in table 2). We did not encounter this problem using the full Hw The third-order energy correction is smaller for the 3B, state than the ‘A, state in accord with the general trend of a larger correlation problem in singlets than higher spin state. The energy gap between the ‘A, and 3B, states is poorly described by the second-order theory. A similar discrepancy at second order was found in refs. [ 3,9 1. The reason for the overly large energy gap at second order seems to be that the singlet is less well described by the second-order correction than the triplet as mentioned. This imbalance puts the singlet too high in energy relative to the triplet at second order. The third-order correction Table 2 Generalized hartree)
Meller-Plesset
results ‘) for CHI
(energies
in
Method
&‘A,)
E(‘B,)
a (kcal/mol)
HF CASSCF GMPZ GMP3 GMPZ diag b1 full CI c)
- 38.88630 - 38.94553 -39.01331 - 39.02322 - 39.02545 - 39.027 18
-38.92795 - 38.96595 - 39.03794 - 39.04338 -39.05117 - 39.04626
26.14 12.82 15.45 12.65 16.13 II.97
a> The C Is core is not correlated. ” Results from ref. [ 31 which used only the diagonal part of Ho. ‘) Ref. [23].
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improves the singlet energy sufficiently to give a gap which is 0.68 kcal/mol within the full CI result.
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September199 1
References
[ I] K. Wolinski, H.L. Sellers and P. Pulay, Chem. Phys. Letters 4. Conclusion The results above show that generalized MsllerPlesset perturbation theory applied to MCSCF reference wavefunctions which dissociate properly is a computationally viable method for obtaining a large fraction of dynamic correlation energy. The tirst-order space defined here simplifies previous presentations of the theory. Since the generalized MerllerPlesset perturbation theory gives accurate energies over the full potential surface, realistic model potentials for use in dynamics calculations could be constructed using results of the perturbation theory. In future applications of perturbation theory to larger systems, the generalized valence bond approach (in particular a self-consistent GVB-RCI reference wave function) will prove to be invaluable for defining a MCSCF reference wave function which dissociates properly. The localized nature of the natural orbitals comprising the GVB-RCI wave function allows for an implementation of local correlation theory through perturbation theory as has been done for HartreeFock based Merller-Plesset theory [ 241. A successful local correlation theory of this type would be computationally more efficient and provides the possibility of selectively and accurately correlating the most important parts of a system.
Acknowledgement RBM acknowledges the support of an AT&T graduate scholarship and a useful discussion with P. Pulay.
140 (1987) 225.
[ 21 K. Wolinski and P. Pulay, I. Chem. Phys. 90 ( 1989) 3647. [ 31 K. Andersson, P. Malmqvist, B.O. Roos, A.J. Sadlej and K. Wolinski, J. Phys. Chem. 94 (1990) 5483. [4] R. Shepard, Advan. Chem. Phys. 69 (1987) 63. [ 51F.W. Bobrowiz and W.A. Goddard III, in: Methods of electronic structure theory, ed. H.F. Schaefer III (Plenum Press, New York, 1977) p. 79. [6] E.A. Carter and W.A. Goddard III, J. Chem. Phys. 86 (1987) 862. [i’] L.M. Cheung, K.R. Sundberg and K. Ruedenberg, Intern. J. Quantum Chem. 16 (1979) 1103. [8] P. Siegbahn, A. Heiberg, B. Roos and B. Levy, Physica Scripta 21 (1980) 323. [9] J.W. McDouall, K. Peasley and M. Robb, Chem. Phys. Letters 148 ( 1988) 183. [ 101 B.O. Roos, P. Lime, P.E.M. Slegbahn and M.R.A. Blomberg, Chem. Phys. 66 (1981) 197. [ 111 G. Hose, J. Chem. Phys. 84 ( 1986) 4505. [ 121 B. Huron, J.P. Malrieu and P. Rancurel, J. Chem. Phys. 58 (1973) 5745. [ 131 R.J. Harrison, J. Chem. Phys. 94 (1991) 5021. [ 141 J. Novoa, F. Mota and F. Arnau, J. Phys. Chem. 95 (1991) 3096. [ 151 W.I. Salmon and K. Ruedenberg, J. Chem. Phys. 57 ( 1972) 2776. [ 161 K. Ruedenberg, L.M. Cheung and S.T. Elben, Intern. J. Quantum Chem. 16 (1979) 1069. [ 171 A.D. McLean and B. Liu, J. Chem. Phys. 58 ( 1972) 1066. [I81 W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical recipes - the art of scientific computing (Cambridge Univ. Press, London, 1989) p. 71. [ 191 P. Saxe, H.F. Schaefer III and N.C. Handy, Chem. Phys. Letters 79 (1981) 202. [20] R.J. Harrison and N.C. Handy, Chem. Phys. Letters 95 (1983) 386. [21] W.A. Goddard III and L.B. Harding, Ann. Rev. Phys. Chem. 29 (1978) 363. [221 R.P. Messmer, J. Plm. Chem. Sot. 113 ( 199 1) 433. [23] C.W. Bauschlicher Jr. and P.R. Taylor, J. Chem. Phys. 85 (1986) 6510. [24] S. &be and P. Pulay, J. Chem. Phys. 86 (1986) 914.