J. Phys. Chem. 1985,89, 2161-2171 lone pairs remains almost unchanged (-0.001 e). The main effect of the changes in u donation and A back-donation is an increase of the bonding forces between metal and ligand. This is also evident from a comparison of the S C F and CI force constants for the stretching vibrations, which are increased by a factor of 6 and 2, respectively (see Table 111). It is a quite common experience that metal-ligand bonds are too long within the Hartree-Fock approximation. This may be traced back to the fact that A bonding is often important in these systems. In such cases the Hartree-Fock model may become inadequate. The reason for this is a competition between (1) the A bonding which has a very short range and (2) the repulsion between the ligand u bonding orbitals and the partly filled d shell at shorter bond distances. The first of these effects is a near degeneracy effect and should be well described within an M C SCF calculation. The second effect has both near degeneracy and dynamic correlation contributions. Both effects will serve to shorten the metal-ligand bond. The difference between the axial and the equatorial bond lengths at the S C F level can be attributed to the different bonding mechanisms (and hence different correlation effects) for the axially and equatorially coordinated ligands. One effect that clearly is of importance in the description of the axial bond is the increase of correlation due to the penetration of the axial carbon u lone pairs into the metal d shell. This correlation effect allows the carbonyl to move further in, thereby increasing the strength of the A bond. It is clear the this second effect is particularly
2161
important for the axial ligands, where the lone pair penetrates into a formally empty d orbital. The bond distance at the S C F level is therefore more reasonable for the equatorial than for the axial bonds. This change in dynamic correlation is difficult to accomplish in a C I calculation with only a few configurations. The CI coefficients in Table V support the above discussion. Another substantial bond shortening effect is due to the charge-transfer excitations of the mixed type to the metal d s and the ligand A* orbitals (e’al’-e’*al’*, e”al’-e’’*al’*, and e’”’e’*”’*) which have rather high coefficients, increasing at shorter metal-ligand distances. The energetic effect of the latter can be estimated by comparison of the differential correlation energies of the two 4-electron CI calculations with the one of the 8-electron C I calculation (see Table 111, type D1, E l , and C1 CCI calculations. Note that double excitations of the type e‘-‘‘* and e”+’* have almost no weight). The coefficients for excitations on the iron atom which give a rearrangement of the d shell have also large coefficients which increase further at larger distances. The inclusion of these excitations serves to elongate the bonds, though only to a minor extent. For a thorough discussion of this particular topic, however, an M C S C F wave function would be more appropriate.
Acknowledgment. We are grateful to the Computer Center of the University of Zurich for a very generous grant of computer time on their IBM 3033 machine. Registry No. Fe(CO),, 13463-40-6;Fe, 7439-89-6.
Can Simple Localized Bond Orbitals and Coupled Cluster Methods Predict Reliable Molecular Energies? William D. Laidig, George D. Purvis, and Rodney J. Bartlett* Quantum Theory Project, University of Florida, Gainesville, Florida 3261 1 (Received: August 8, 1984)
Conventional large-scale configuration interaction or coupled-cluster (CC) calculations, designed to yield accurate structural and energetic properties, usually begin with delocalized molecular orbitals determined with the Hartree-Fock SCF method or with multiconfigurational SCF methods. These orbitals are spread out over the whole molecule and frequently obscure the interpretation of bonding structure and increase the cost of subsequent configuration interaction or coupled-cluster calculations. Calculations on water, methane, acetylene, ethylene, and ethane in double {and double {plus polarization basis sets show that localized bond orbitals, composed of intuitively simple linear combinations of atomic hybrids, can be used effectively in coupled-cluster calculations yielding errors in the total energy (compared to the full CI and near full CI calculations) which are less than 2 kcal/mol. The ramifications for calculations on large molecules are analyzed. Finite-order many-body perturbation theory (MBPT) calculations through fourth order are also reported for these molecules, requiring the first evaluation of non-Hartree-Fock triple excitation diagrams in fourth order. Various factors involved in the selection of localized orbitals are examined including bond polarity, method of hybridization, and localization of the virtual orbitals. The use of localized SCF orbitals in MBPT/CC is also examined via calculations on methane, acetylene, ethylene, and ethane in canonical SCF and equivalent orbital bases.
I. Introduction The use of localized orbitals (Lo’s) in quantum chemistry,’ while still quite extensive in certain areas of semiempirical theory, has declined significantly in the ab-initio domain. While Lo’s are still occasionally used at the self-consistent field (SCF) level, mainly as qualitative aides to understanding the structure, their use in correlated calculations is almost nonexistant. This is due largely to the widespread acceptance of canonical SCF procedures which produce orbitals that are generally delocalized over the entire molecule. Since these orbitals are optimal in the single determinant sense* and are relatively simple and inexpensive to (1) For a review see H. Weinstein, R. Pauncz, and M. Cohen, Adu. At. Mol. Phys., 7 , 91 (1971).
0022-3654/85/2089-2161$01.50/0
compute, they are employed in most correlated calculations. However, as the size of the largest systems that can be studied is extended through advances in computer performance and calculational techniques certain difficulties with delocalized orbitals become apparent. In correlated calculations which require integrals over molecular orbitals (MO’s) the number of nonzero integrals in the delocalized basis grows as n4 with a time proportional to n5 where n is the number of basis functions. Typical correlated calculations such as configuration interaction singles and doubles (CISD), coupled cluster singles and doubles (CCSD), (2) J. E. Lennard-Jones, Proc. R. SOC.London, Ser. A , 198, 1 (1949);G. G.Hall and J. E. Lennard-Jones, Proc. R. SOC.London, Ser. A , 198,166
(1949).
0 1985 American Chemical Society
2162 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
and fourth-order many-body perturbation theory including single, double, and quadruple excitation diagrams (SDQ-MBPT(4)) are even more expensive requiring a time step of n6. If sufficiently localized orbitals are employed instead, the number of M O integrals above a given threshold is substantially decreased. For large systems the number of integrals above the cutoff will only grow as the square of the number of bonds. Another advantage that LO’s have over the canonical S C F orbitals is that the localized basis allows a simple zeroth-order description of the molecule in terms of bonds, antibonds, lone pairs, and core orbitals. Molecular properties such as the energy or dipole moment can then be approximated as a sum over contributions from each bond, and properties of entire molecular fragments might be transferred between different molecules containing this s ~ b s t i t u e n t . In ~ the delocalized basis the global nature of the orbitals precludes the possibility of this type of transferability. Also since the bonds and antibonds are localized in similar regions of space, the correlation in individual atoms or bonds can be studied more or less independently of the correlation elsewhere. This can be quite advantageous when processes such as bond breaking are examined since the m j o r correlation effects will involve only the bonding and antibonding orbitals between the two bonding groups. The decision to use localized over canonical S C F orbitals, then, revolves around whether the Lo’s will yield comparable correlation energies at the same level of theory. This depends a lot on the type of LO employed and the severity of the approximations used to estimate the correlation energy. Various choices for the Lo’s exist and can be divided into two major categories, S C F or non-SCF. LO’s of the S C F type have a major advantage over the non-SCF varieties since, for certain correlated calculations such as CISD and CCSD, the canonical S C F and LO energies are equivalent. One would also expect for calculations where this is not true, such as finite-order perturbation theory, that the SCF-LO energies will be closer to the canonical S C F energies than the corresponding non-SCF-LO energies. While non-SCF-Lo’s can yield quite different energies at low levels of theory it is expected in calculations where recovery of the correlation energy approaches the full CI limit that the energy will become decreasingly sensitive to changes in the orbitals. For instance, CCSD usually recovers from 95% to 98% of the correlation energy when S C F orbitals are employed, and we expect that, if non-SCF-LO’S are used instead, the final energy will be quite similar assuming that the LO’s can qualitatively describe the bonding, etc. Our hope is that this orbital “pseudo-invariance” will allow accurate calculations at the CCSD and maybe at the SDQ-MBPT(4) levels to be performed more quickly using a localized basis. In two previous publication^^^^ we did some preliminary investigation on the use of non-SCF-LO’S in MBPT/CC calculations. These orbitals which are simple linear combinations of hybrid atomic orbitals (AO’s) are discussed in section 111 and yielded single determinant energies which in all cases were 0.2 to 0.3 hartree above the S C F values. In the first paper water and methane were examined near their equilibrium geometries at the CCSD level and the canonical S C F and LO CCSD energies were found to be very close differing by only 1-2 mhartrees. The methane C-H bond was also symmetrically stretched and compressed about its minimum energy value and the two CCSD energy curves were found to be almost parallel. In the second paper finite-order MBPT calculations through SDQ-MBPT(4) and CCSD energies were computed for double { plus polarization (DZP) ethylene. At the SDQ-MBPT(4) and CCSD levels the two energies agreed to 14 and 2 mhartrees, respectively. Overall, then, in these two papers we found that if an extensive correlated calculation such as CCSD is carried out the total energy is not (3) G . G . Hall, Proc. R. SOC.London, Ser. A, 205, 541 (1951); M. J. S. Dewar and R. Petit, J . Chem. SOC.,1617 (1954). (4) W. D. Laidig, G . D. P u M s , and R. J. Bartlett, Int. J. Quantum Chem. Symp., 16, 561 (1982). (5) W. D. Laidig, G . D. Purvis, and R. J. Bartlett, Chem. Phys. Lett., 97, 209 (1983).
Laidig et al. very sensitive to the orbital choice, assuming that these orbitals are at least qualitatively correct. In the present study we again investigate the properties of these simple hybrid LO’s, but expand the study to include acetylene, ethylene, and ethane all employing the same larger than double { (DZ) basis set. Both CCSD and finite-order MBPT calculations through the SDTQ-MBPT(4) level are reported for these molecules along with finite-order MBPT results for methane and water to compliment the earlier published CCSD results. The simple hybrid orbitals themselves are also investigated and the merits of different hybridization schemes, and the use of optimum bond polarity are examined. The use of SCF-Lo’s is probed via MBPT calculations including SDQ-MBPT(4) on methane and the other hydrocarbons listed above employing equivalent orbitals (EO’S), and the order-by-order energies are compared against the comparable canonical S C F and simple hybrid LO results. 11. Overview of MBFT/CC Methods The basic idea behind coupled cluster (CC) theory is that the exact wave functien \k can be expressed as the exponential of the cluster operator T acting on the reference determinant aO69’
= epl+o) (1) The cluster operator is usually partitioned into one-body, two-My, etc. terms such that T = T I T2 ... (2) The components of ? are composed of sums of products of excitation operators and expansion coefficients, and the one- and two-body operators are defined as
+ +
T , = Ct;a+i i,a
(3)
f2 = i
