Chapter 7 Semiempirical Methods 7.1 Introduction Semiempirical methods modify Hartree-Fock (HF) calculations by introducing functions with empirical parameters. These parameters are adjusted with experimen- tal conclusions to improve the quality of computation. The real cost of computation is due to the two-electron integrals in the Hamiltonian that has been simplified in this method. Semiempirical methods are based on three approximation schemes. 1. The elimination of the core electrons from the calculation. Inner electrons do not contribute towards chemical activity, which makes it pos- sible to remove the core electron functions from the Hamiltonian calculation. Normally, the entire core (the nucleus and core electrons) of atoms is replaced by a parameterized function. This has the effect of drastically reducing the com- plexity of the calculation without a major impact on the accuracy. 2. The use of the minimum number of basis sets. In this approximation, while introducing the functions of valence electrons, only the minimum required number of basis sets will be used. This technique also reduces the complexity of computation to a large extent. 3. The reduction of the number of two-electron integrals. This approximation is introduced on the basis of experimentation rather than chemical grounds. The majority of the work in ab initio calculations is in the evaluation of the two electron integrals (Coulomb and exchange). All modern semiempirical methods are based on the modified neglect of differential over- lap (MNDO) approach. In this method, parameters are assigned for different atomic types and are fitted to reproduce properties such as heats of formation, geometrical variables, dipole moments, and first ionization energies. The pa- rameterization was carried out separately for classes of compounds like hydro- carbons, CHO systems, CHN systems, and so on. The latest versions of the MNDO method are referred to as AM1 and PM3. Another method to reduce the two-electron integral is the zero differential overlap (ZDO) approximation, which neglects all products of basis functions depending on the same electron coordinates when located on different atoms. K. I. Ramachandran et al., Computational Chemistry and Molecular Modeling 139 DOI: 10.1007/978-3-540-77304-7, ©Springer 2008 140 7 Semiempirical Methods It means that all the products of atomic orbital functions χuχv are set to be zero, and the overlap integral Suv = δuv (where δuv is the Kronecker delta, i.e., δuv = 0if u = v and δuv = 1ifu = v.). At the ZDO approximation, all three- and four-centered integrals vanish. This reduces the overlap matrix into a unit matrix. One-electron integrals involving three centers (two from the basis functions and one from the op- erator) are set to zero. All three- and four-center two-electron integrals, which are by far the most numerous of the two-electron integrals, are neglected. Parameteri- zation is done to compensate the approximations. Hence, all the remaining integrals are replaced by proper parameters obtained by experimentation. 7.2 The Neglect of Differential Overlap Method The neglect of differential overlap (NDO) method was first introduced by John Pople, and it is now the basis of most successful semiempirical methods. The method involves the modification of the HF equation, FA = SAε, by approximating the overlap matrix S as unit matrix. This allows us to replace the HF secular equa- tion |H − ES| = 0 with a simpler equation |H − E| = 0. We shall see some common techniques used to make the computation possible. 7.3 The Complete Neglect of Differential Overlap Method In the complete neglect of differential overlap (CNDO) method, all integrals involv- ing different atomic orbitals, χu, are ignored. Thus, the overlap matrix becomes the unit matrix, S = 1. The parameterization and implementation scheme of the CNDO method was also proposed by Pople. 7.4 The Modified Neglect of the Diatomic Overlap Method The modified neglect of the diatomic overlap (MNDO) method (by Michael De- war and Walter Thiel, 1977) is the oldest NDDO-based model that parameterizes one-center two-electron integrals based on spectroscopic data for isolated atoms, and evaluates other two-electron integrals using the idea of multipole-multipole in- teractions from classical electrostatics. A classical MNDO model uses only s and p orbital basis sets, while more recent MNDO/d adds d-orbitals that are especially important for the description of hypervalent sulphur species and transition metals. MNDO has a number of known deficiencies, such as the inability to describe the hydrogen bond due to a strong intermolecular repulsion. The MNDO method is characterized by a generally poor reliability in predicting heats of formation. For 7.6 The Parametric Method 3 Model 141 example, highly substituted stereoisomers are predicted to be too unstable, com- pared to linear isomers due to the overestimation of repulsion in sterically crowded systems. Existing semiempirical models differ by further approximations that are made when evaluating one- and two-electron integrals and by the parameterization phi- losophy. While INDO added all one-center two-electron integrals to the CNDO/2 for- malism, NDDO adds all two-center integrals for repulsion between a charge dis- tribution on one center and a charge distribution on another center. Otherwise, the zero-differential overlap approximation is used. 7.5 The Austin Model 1 Method The Austin Model 1 (AM1) method, developed by M. J. S. Dewar and coworkers, takes a similar approach to MNDO in approximating two-electron integrals, but uses a modified expression for nuclear-nuclear core repulsion. The modified expression results in non-physical attractive forces that mimic van der Waals interactions. The modification also necessitated a re-parameterization of the model, which was car- ried out with a particular emphasis on dipole moments, ionization potentials, and geometries of molecules. While this allows for some description of the hydrogen bond, other deficiencies, such as systematic over-estimates of basicities, remained unsolved. Also, the lowest energy geometry for the water dimer is predicted incor- rectly by the AM1 model. On the other hand, AM1 nicely improves some properties, such as heats of formation, over MNDO. 7.6 The Parametric Method 3 Model The Parametric Method 3 (PM3) model, developed by James Stewart, uses a Hamil- tonian that is very similar to the AM1 Hamiltonian, but the parameterization strat- egy is different. While AM1 was parameterized largely based on a small number of atomic data, PM3 is parameterized to reproduce a large number of molecular properties. In some sense, chemistry gave way to statistics with the PM3 model. A different parameterization, and a slightly different treatment of nuclear repul- sion allows PM3 to treat hydrogen bonds rather well, but it amplifies non-physical hydrogen-hydrogen attractions in other cases. This results in serious problems while analyzing intermolecular interactions (methane is predicted to be a strongly-bound dimer) or conformations of flexible molecules (OH is strongly attracted to CH3 in 1-pentanol). The accuracy of thermochemical predictions with PM3 is slightly bet- ter than that of AM1. The PM3 model has been widely used for the rapid estimation of molecular properties and has been recently extended to include many elements, including some transition metals. 142 7 Semiempirical Methods 7.7 The Pairwize Distance Directed Gaussian Method The pairwise distance directed Gaussian (PDDG/PM3) method, developed by William Jorgensen and coworkers, overcomes some of the deficiencies of the earlier NDDO-based methods by using a functional group-specific modification of the core repulsion function. The nPDDG/PM3 modification provides a good description of the van der Waals attraction between atoms, and the PDDG/PM3 model appears to be suitable for calculations of intermolecular complexes. Furthermore, careful re-parameterization has made the PDDG/PM3 model very accurate for estimation of heats of formation. However, some limitations common to NDDO methods re- main in the PDDG/PM3 model: the conformational energies are unreliable, most activation barriers are significantly overestimated, and the description of radicals is erratic. So far, only C, N, O, H, S, P, Si, and halogens have been parameterized for PDDG/PM3 [1]. 7.8 The Zero Differential Overlap Approximation Method The zero-differential overlap (ZDO) method is an approximation that is used to sim- plify the many electrons by ignoring two-electron repulsion integrals. If the molec- φ χA ular orbitals j are expanded in terms of N basis functions, s as: b φ = χA j ∑ csi s (7.1) s=1 where A is the atom the basis function is centered on, and csi are the coefficients. The two-electron repulsion integrals are then defined as: |λσ = χA( )χB( ) 1 χC( )χD( ) τ τ sv s 1 v 1 λ 2 σ 2 d d (7.2) r12 The zero-differential overlap approximation ignores integrals that contain the χA( )χB( ) = product s 1 v 1 where s v. This transforms the equation to: sv|λσ = δsvδλσ ss|λλ (7.3) δsv is the Kronecker delta with δsv = 0ifs = v and δsv = 1ifs = v.Theto- tal number of such integrals is reduced to N(N + 1)/2 (approximately N2/2) from [N(N + 1)/2][N(N + 1)/2 + 1]/2 (approximately N4/8) where N is the number of orbitals. 7.9 The Hamiltonian in the Semiempirical Method 143 7.9 The Hamiltonian in the Semiempirical Method The (CNDO) method and the intermediate neglect of differential overlap (INDO) method are SCF methods solving the Roothaan equations iteratively, with approx- imations for the integrals in the Fock matrix. Only valence electrons are mainly considered in these methods. The Hamiltonian is: n(val) n(val) ˆ = −1∇2 + ( ) + 1 Hval ∑ i V i ∑ ∑ (7.4) i=1 2 i=1 j>1 rij which can be simplified as: n(val) n(val) ˆ = ˆ core( )+ 1 Hval ∑ Hval i ∑ ∑ (7.5) i=1 i=1 j>1 rij where 1 Hˆ core(i)= − ∇2 +V(i) (7.6) val 2 i Here n(val) stands for the number of valence electrons in the system, V(i) is the potential energy of valence electron i in the field of nuclei and the core electrons, ˆ core( ) ˆ Hval i is the one-electron part of Hval.