J. Phys. Chem. 1996, 100, 6069-6075
6069
Compact Variational Wave Functions Incorporating Limited Triple and Quadruple Substitutions C. David Sherrill† and Henry F. Schaefer III* Center for Computational Quantum Chemistry, UniVersity of Georgia, Athens, Georgia 30602 ReceiVed: September 20, 1995; In Final Form: December 7, 1995X
We have investigated a number of configuration interaction (CI) wave functions which incorporate the dominant effects of triple and quadruple substitutions. If we employ natural orbitals from a CI singles and doubles (CISD) procedure, then the orbitals may be conveniently partitioned into subspaces of varying importance, and electron configurations may be classified according to how they occupy each of the subspaces. We employ a CI model space containing all single and double substitutions and also those classes of triply and quadruply substituted configurations deemed most important on the basis of natural orbital populations. This a priori selection scheme recovers a large fraction of the energy of a CI including all triples and quadruples (CISDTQ) at a drastically reduced computational expense, and it represents a more “black-box” alternative to traditional multireference CI approaches. Implications for nonvariational approaches are also discussed.
I. Introduction A majority of chemical applications of configuration interaction (CI) theory employ model spaces consisting of only single and double substitutions, i.e., the CISD method. This truncation of the CI space is motivated by the fact that only singly and doubly substituted configurations contribute to the first-order wave function in perturbation theory and are sufficient to determine the energy through third order. For small molecules, CISD typically recovers about 95% of the basis set correlation energy. For larger molecules, excited states, or molecules far from their equilibrium geometries, the quality of the CISD wave function can become considerably poorer. In such cases (or indeed even in the case of ground states of small molecules near their equilibrium geometries, when extremely accurate results are required), it becomes necessary to account for triple, quadruple, or even higher-order substitutions. Several benchmark studies have investigated the importance of various substitution, or excitation, classes to the full CI wave function.1-4 In a study of BH, H2O, NH3, and HF, Harrison and Handy1 found that the addition of triple substitutions (CISDT) recovers approximately an additional 1% of the basis set correlation energy compared to CISD (total of 95-96%), whereas the addition of all quadruples (CISDTQ) recovers more than 99%. Moreover, these authors found that even when the O-H bonds of H2O are stretched to twice their equilibrium bond length, the CISDTQ energy remains remarkably accurate, recovering 98.6% of the correlation energy. Unfortunately, the CISDTQ method is prohibitively expensive for any but the smallest molecular systems because of the immense number of triple and quadruple substitutions. Such considerations motivate the pursuit of theoretical methods incorporating the dominant effects of triple and quadruple (and perhaps higher) substitutions in a less computationally expensive manner. Many-body perturbation theory (MBPT) and the coupled-cluster (CC) method effectively deal with the “disconnected” quadruple substitutions appearing at fourth order by expressing them as products of doubles. One may also treat a primary N-electron space (e.g., singles and * To whom correspondence should be addressed. † National Science Foundation Graduate Fellow. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6069$12.00/0
doubles) variationally and then treat the larger secondary space (triples and quadruples) perturbatively. Recent work along these lines by Steiner, Wenzel, Wilson, and Wilkins is quite encouraging.5 Another interesting possibility, put forth by Malrieu and co-workers,6 is to modify the Hamiltonian matrix by a socalled “dressing” procedure which corrects for size extensivity errors and includes some effects of triples and quadruples. Among strictly variational approaches, it is certainly possible to reduce the CI space by eliminating those N-electron functions (CSF’s or determinants) which fail to meet some selection test. Likewise, most multireference CI procedures select references according to the weight of each prospective reference function in a preliminary wave function (e.g., a complete-active-space SCF). Although these approaches are designed to yield the most accurate wave function at the least expense, the CI space selection procedure is a tricky business best left in the hands of specialists. An alternative, more “black-box” approach is to select the references for a multireference CI according to excitation class in some subset of orbitals. This is the procedure used in the second-order CI (SOCI),7 which is a multireference CISD (MR-CISD) in which the references are chosen as all possible configurations generated by a full CI in the active space with no spatial or spin symmetry restrictions on the references. Although the SOCI wave function is known to provide highquality results and potential energy surfaces nearly parallel to those from a full CI,8-10 relatively few investigations have explored other types of excitation-class-selected MR-CI’s. One such investigation was reported in 1982 by Saxe, Fox, Schaefer, and Handy;11 in this study, the authors present results for a DZP MR-CISD wave function for ethylene in which the references are chosen as all single and double excitations in the valence space. This wave function can be viewed as a CISDTQ in which no more than two electrons are allowed outside the valence space, and thus we have designated such wave functions CISD[TQ]. In 1992, Grev and Schaefer9 studied the quality of the SOCI and CISD[TQ] wave functions for a number of small molecules; their results indicate that the CISD[TQ] method provides results which are very close to SOCI when a single reference function dominates. A followup study of several other small molecules by Fermann, Sherrill, Crawford, and Schaefer10 reinforced these conclusions. Excitation-classselected MR-CI’s have also been studied by Olsen and co© 1996 American Chemical Society
6070 J. Phys. Chem., Vol. 100, No. 15, 1996 workers for Ne,12,13 Mg,13 H2O,3 and NH3.4 In 1988, Olsen, Roos, Jørgensen, and Jensen presented the restricted active space (RAS) CI,12 whereby the orbital space is divided into three subsets denoted I, II, and III. The CI space is constructed by taking only those configurations which have a minimum of p electrons in RAS I and a maximum of q electrons in RAS III. It is possible to construct the aforementioned SOCI and CISD[TQ] wave functions as RAS CI’s. Because of their special treatment of the valence space, the SOCI and CISD[TQ] wave functions require that the virtual valence orbitals be good correlating orbitals. Grev and Schaefer have shown9 that if CISD natural orbitals (NO’s) are employed, then the SOCI energies based on these orbitals are in excellent agreement with those obtained using complete-active-space (CAS) SCF orbitals. Not only are CINO’s less expensive to obtain than CASSCF orbitals, but they are also better suited to truncation of the one-particle basis set. That is, deletion of the most weakly occupied NO’s leads to a substantially smaller CI space with very little loss in the correlation energy recovered. Such observations led us to propose a modification to the CISD[TQ] method which would result in substantially smaller CI expansions.9,10 It has been known since the 1960 work of Watson on the Be atom14 that the most important quadruple substitutions have much smaller CI coefficients than the most important double substitutions. This finding was qualitatively explained by Sinanogˇlu,15 who showed that the importance of an unlinked quadruple substitution (more properly, a disconnected quadruple) is related to the product of the two double substitutions from which it may be derived. This suggests that a much smaller NO basis can satisfactorily treat the triple and quadruple substitution contributions than is required to treat single and double substitutions. Thus, we split the virtual NO space into two distinct parts. The CISD part of the configuration space includes all of the NO’s; the triple and quadruple substitutions, however, are restricted to the lower portion of the virtual NO space. Since the number of triple and quadruple substitutions is always large compared to the number of singles and doubles, this can drastically reduce the number of configurations included in the CISD[TQ] wave function. Here we evaluate the performance of this split-virtual CISD[TQ] approach compared to our previous CISD[TQ] method9,10 (henceforth referred to as the “full-virtual” CISD[TQ]) and to several other correlation methods for Ne atom and for H2O. We find that split-virtual CISD[TQ] provides a very good approximation to full-virtual CISD[TQ] at a substantially reduced computational expense. II. Methods The CISD[TQ] wave functions are defined by partitioning the orbitals into an occupied subspace and primary, secondary, and tertiary virtual subspaces. Of course, we may also constrain corelike orbitals to remain doubly occupied, and we may delete very high-lying virtual orbitals. In the case of very large nondynamical correlation effects, it is possible to replace the occupied subspace with a CAS subspace. All single and double substitutions are allowed into the CI space, but we select only those triples and quadruples which lie entirely within the primary and secondary orbital subspaces and for which there are no more than two electrons in the seconary space. Figure 1 illustrates this method of orbital partitioning and configuration selection. The effectiveness of this CI selection scheme is based on the exponentially decreasing occupation numbers of the CI natural orbitals.16 Therefore, all CISD[TQ] wave functions are constructed using CISD natural orbitals, and the orbital occupation numbers are used to guide the selection of the orbital subspaces.
Sherrill and Schaefer
Figure 1. Orbital partitioning scheme and subsequent configuration selection in the CISD[TQ] method. If the tertiary virtual subspace is employed, the resulting wave function is denoted “split-virtual” CISD[TQ]; otherwise, it is denoted “full-virtual” CISD[TQ].
In the RAS nomenclature, our occupied subspace is labeled RAS I (with a minimum of N-4 electrons, where N is the number of electrons in the system), our primary subspace is RAS II, and our secondary subspace is RAS III (with a maximum of two electrons). There is no RAS equivalent for our tertiary subspace. Results for these CISD[TQ] wave functions were obtained using a determinant-based CI program recently developed by the authors. This program is capable of performing CI’s truncated at any excitation level (up to full CI) as well as any CI which can be formulated as a RAS CI, such as SOCI and full-virtual CISD[TQ]. Our program is based upon Handy’s alpha and beta string direct-CI formalism17 as presented in papers by Olsen and co-workers12,13 and by Mitrushenkov.18 We have added to our program the ability to employ more elaborate CI spaces, such as the split-virtual CISD[TQ] approach described here, and we are currently in the process of optimizing our code for the IBM RS/6000 workstations available in our laboratory. For comparison purposes, we also report energies determined with the following methods: Hartree-Fock self-consistent-field (SCF), CISD, CISDT, CISDTQ, coupled-cluster with single and double substitutions (CCSD),19-21 and CCSD including a perturbative estimate of connected triple substitutions [CCSD(T)].22,23 These results, in addition to those for the full-virtual CISD[TQ] method, were obtained using the ab initio program package PSI.24 The CISD and full-virtual CISD[TQ] wave functions were determined using the shape-driven Graphical Unitary Group Approach (SD-GUGA),11 while wave functions involving more-than-double substitutions into the external space (i.e,. CISDT and CISDTQ) were obtained using the loop-driven Graphical Unitary Group Approach (LD-GUGA).25,26 All results obtained with the GUGA-CI code were reproduced with the new determinant-based code in order to verify the reliability of the latter. In all correlated procedures, the core orbitals (the Ne 1s orbital and the O 1s-like orbital in H2O) were constrained to remain doubly occupied. The two basis sets used for Ne are Dunning’s correlation-consistent triple-ζ (cc-pVTZ) and quadruple-ζ (ccpVQZ) sets.27 For H2O, we have used the basis set and geometry of Bauschlicher and Taylor in their 1986 benchmark full CI study.28,29 We have also employed the larger cc-pVTZ
Compact Variational Wave Functions
J. Phys. Chem., Vol. 100, No. 15, 1996 6071
TABLE 1: CISD Natural Orbital Occupation Numbers for the Ne Atom with cc-pVTZ and cc-pVQZ Basis Setsa orbital
cc-pVTZ
cc-pVQZ
orbital
cc-pVTZ
cc-pVQZ
1s 2s 2p 3s 3p 3d 4s 4p
2.000 00 1.991 11 1.982 54 0.006 73 0.010 41 0.003 68 0.000 25 0.000 54
2.000 00 1.990 45 1.981 66 0.006 79 0.010 45 0.003 86 0.000 25 0.000 52
4d 4f 5s 5p 5d 5f 5g
0.000 25 0.000 27 0.000 17
0.000 37 0.000 35
