Energy Correctors for Accurate Prediction of Molecular Energies Jorge M. Seminario,* Martha G. Maffei, Luis A. Agapito, and Pablo F. Salazar Department of Chemical Engineering and Department of Electrical and Computer Engineering, Texas A&M UniVersity, College Station, Texas 77843-3122 ReceiVed: September 26, 2005; In Final Form: NoVember 21, 2005 Energy correctors are introduced for the calculation of molecular energies of compounds containing first row atoms (Li-F) to modify ab initio molecular orbital calculations of energies to better reproduce experimental results. Four additive correctors are introduced to compensate for the differences in the treatment of molecules with different spin multiplicities and multiplicative correctors are also calculated for the electronic and zero- point vibrational energies. These correctors, individually and collectively yield striking improvements in the atomization energies for several ab initio methods. We use as training set the first row subset of molecules from the G1 basis of molecules; when the correctors are applied to other molecules not included in the training set, selected from the G3 basis, similar improvements in the atomization energies are obtained. The special case of the B3PW91/cc-pVTZ yields an average error of 1.2 kcal/mol, which is already within a chemical accuracy and comparable to the Gaussian-n theories accuracy. The very inexpensive B3PW91/6-31G** yields an average error of 2.1 kcal/mol using the correctors. Methods considered unsuitable for energetics such as HF and LSDA yield corrected energies comparable to those obtained with the best highly correlated methods. I. Introduction The Gaussian-n (n ) 1-3) compound methods yield excellent accuracy for molecular energies, mostly within the range considered as chemical accuracy, that is, 1-2 kcal/mol. This accuracy is needed as practical applications for the design of new materials and processes require of extremely good energet- ics. As quantum chemistry methods extend their application to the analysis, design and simulation of nanosized systems (nanotechnology), a size region that is extremely difficult to be approached experimentally, the need for precise calculations is of paramount importance for the development of such new field avoiding trial-and-error experimentation. Gaussian-n and other related methods root their success on precise methods requiring computational resources that can only be practically applied to very small systems. In practice, using these methods for molecules larger than benzene becomes prohibited for an installation composed of a few modern workstations. The Gaussian-1 (G1) method introduced by Pople et al. 1,2 gave an atomization energy accuracy better than 2 kcal/mol for a set of molecules containing only first-row elements (G1 set) and an accuracy better than 3 kcal/mol for the second row molecules. 2 The G2 method developed by the same team, Curtiss et al., 3 yielded an accuracy of 1.2 kcal/mol for an extended set of 99 cases named the G2 set; this set includes the molecules in the G1 set plus 24 molecules that contain second-row elements. The G3 method improves on the G2 by including new corrections such as spin-orbit correction for atoms and correction for core correlation. 3 This method also improved the enthalpy of formation error from 1.56 to 0.94 kcal/mol, for the same G2/97 set of molecules. 4 The Gaussian-n theories were developed to take advantage of the fact that relatively precise ab initio calculations contain systematic errors with some additive features. The nature of errors with one level of theory is possibly different from the errors with other levels of theory and might be separately estimated. Also, errors due to the finite nature of the basis sets can be decomposed in contributions with respect to the angular moment of the basis functions. Low angular momentum contributions can be found at lower levels of theory (such as MP2) using large basis sets and high angular momentum contributions can be calculated using higher levels of theory using smaller basis sets. In most of the cases, the contributions are additive. 1,2 Additional corrections in the Gaussian-n methods, among others, include high-level corrections of paired and unpaired electrons using fitted parameters that reproduce the experimental energies. For example, the G1 method uses the Hartree-Fock (HF) energy, which is further corrected with the MP2, MP3, MP4SD(T)Q, and QCISD(T) energies. The zero-point energy for a molecule in this method is obtained from a HF optimization using the 6-31G(d) basis and scaled by the standard 0.8929 for such a level of theory but the geometry to be used for energy calculations is from an MP2 optimization. This MP2 geometry is used further for the MP4 and QCI methods, as no other geometry optimizations are performed for higher levels of theory. The high cost of G1 methods is because the QCI and MP4 methods scale as N 7 . This scaling means, for instance, if a molecule takes 1 day of CPU time, to calculate a double sized molecule takes 2 7 , i.e., 128 days! Even with this strong restriction, these two methods are still far from chemical accuracy if an extremely good basis set is not used. For example the MP4/6-311G(d,p) and QCI/6-311G(d,p) levels of theory yield errors of 15.4 and 16.7 kcal/mol, respectively, still far from chemical accuracy and the MP4/6-311G(2df,p) yield an average error of 8.3 kcal/mol. As expected, levels of theory such as MP4/6-311G(2df,p) or QCI/6-311G(d,p) cannot be used for precise energetics due to the strong errors in energies they yield, needless to say, for lower levels of theory. For instance, the well-used HF/6-31G(d) yields an average error of 87.3 kcal/ mol; although this method formally scales as N 4 , modern computational algorithms have reduce this scaling to ∼N 2 . Since 1060 J. Phys. Chem. A 2006, 110, 1060-1064 10.1021/jp055460z CCC: $33.50 © 2006 American Chemical Society Published on Web 12/23/2005