Multi-Coefficient Correlation Method for Quantum Chemistry Patton L. Fast, Jose ´ C. Corchado, Marı ´a L. Sa ´ nchez, and Donald G. Truhlar* Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 ReceiVed: January 29, 1999 We present a new method for extrapolating correlated electronic structure calculations based on correlation- consistent polarized double- and triple- basis sets for calculation of molecular energies (atomization energies). 1. Introduction Advances in computational sciences as well as the develop- ment of new algorithms are making it possible to carry out ab initio electronic structure calculations on small systems with errors approaching the accuracy of experimental measurements and with even better accuracy than experiment in a small but rapidly growing number of cases. Nevertheless, we are still far from being able to make reliable quantitative predictions based on ab initio calculations for large and even medium-sized systems. Even though in theory the algorithms are applicable to any system despite its size, the computational resources required for carrying out such calculations for medium- and large-size systems are beyond the scope of available technology, and for large systems we can assume that this situation will continue for a long time. Thus, it is necessary to develop methods that can be applied to medium- and large-sized systems with a reduced computational cost. The two major sources of error in an ab initio calculation of molecular energies are the truncation of the one-electron basis set and the truncation of the number of excitations or configura- tions used for treating correlation energies. In principle, one could calculate the energy of a system of interest using a relatively small basis set and including a reduced number of configurations or excitation operators in the calculation of the correlation energy, and then improve both the basis set and the configuration or excitation space until convergence is achieved. This convergence could be measured as the difference between the approximate calculation and the exact solution to the Schro ¨dinger equation, namely a full configuration interaction (FCI) calculation using an infinite basis set. This combination is called complete configuration interaction (CCI). However, for most systems of interest, the computational cost of either FCI or CCI makes them impossible. One promising way to circumvent this difficulty is to calculate the first few terms in a sequence of improving calculations and use these data to extrapolate to the CCI limit. In the present paper we propose a new set of semiempirical methods designed to do this as accurately as possible. Four recent reviews may be consulted for a summary of available methods for extrapolation. 3-6 The developments reported in the present paper were motivated by two of the previous extrapolation methods, namely the scaling all correlation (SAC) method 5-9 introduced by Gordon and one of the authors and the ab initio infinite basis (IB) method l0,11 discussed recently. We note that the SAC method is itself based on the earlier scaling external correlation (SEC) method 6,12 of Brown and one of the authors, and the IB method is based on earlier work 4,13-15 extrapolating correlation-consistent basis sets. In fact, the systematic convergence 4,13-17 of correlation- consistent basis sets 18,19 is believed to be a key ingredient in the success of the method proposed here. Section 2 presents some useful notation. Section 3 uses this notation to describe all methods considered in this paper, including the new extrapolation method, which we call the multi- coefficient correlation method (MCCM). 2. Notation Throughout this paper we will use the pipe “|” to represent the energy difference either between two one-electron basis sets B1 and B2 or between two many-body levels L1 and L2, e.g., Møller-Plesset second-order perturbation theory and Hartree- Fock theory. The energy difference between two basis sets will be represented as where L is a particular electronic structure method and B1 is smaller than B2. The energy change that occurs upon increasing the treatment of the correlation energy will be represented by where L1 is a lower level of theory than L2 and B is a common basis set. Finally, the change in energy increment due to increasing the level of the treatment of the correlation energy with one basis set as compared to the increment obtained with a smaller basis set will be represented as All new calculations in this paper are based on three correlation-consistent basis sets, 18,19 namely cc-pVDZ, aug′′- cc-pVDZ (we use the double prime notation to denote that diffuse functions have been omitted on hydrogen and that the diffuse subshell corresponding to the highest angular momentum has been omitted for the heavy atoms, i.e., omitting the diffuse d function from the aug′-cc-pVDZ 20 basis set), and cc-pVTZ. Since we restrict ourselves to only three basis sets, no confusion can result from a shorthand notation, and we call these pDZ, pDZ+, and pTZ, respectively. In addition, in some cases we will compare to previous calculations based on Pople-type basis sets such as 6-311G** which are explained elsewhere. 21 The new methods, explained in section 3, will have names like MCSAC-L, MCCM-L, and MCCM-L2;L1. The energies ∆E(L/B2|B1) ≡ E(L/B2) - E(L/B1) (1) ∆E(L2|L1/B) ≡ E(L2/B) - E(L1/B) (2) ∆E(L2|L1/B2|B1) ≡ E(L2/B2) - E(L1/B2) - [E(L2/B1) - E(L1/B1)] (3) 5129 J. Phys. Chem. A 1999, 103, 5129-5136 10.1021/jp9903460 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/15/1999