In ab initio methods such as the Hartree-Fock method the two-electron, multi-center integrals J_ij and K_ij are solved explicitly. In semi-empirical methods these integrals are neglected or parameterized, and only valence shell electrons are considered. The Hamiltonian operator takes the form

where N_v is the total number of valence electrons in the molecule, V(i) is the potential energy of the i^th electron in the field of nuclei and inner-shell electrons, and¹



The semi-empirical methods that are currently included in both the Spartan software and the Gaussian software are the MNDO (modified neglect of diatomic overlap), AM1 (Austin model 1), and PM3 (parametric method number 3) methods. These methods employ Slater-type orbitals (STOs) as basis set functions



and make the following simplifying approximation



where d_zy = 1 if z = y or if zy and the functions f_z and f_y are on the same atom. In all other cases d_zy = 0. Likewise, d_mn = 1 if m = n or if mn and the functions f_m and f_n are on the same atom, and d_zy = 0 in all other cases. The notation (zy|mn) refers to the two-electron interaction integral ^1,2



    The F_yy terms in the secular determinant are^1,3



where the core integral U_yy is



The orbitals f_z and f_y are centered on atom A, and orbitals f_p and f_q are centered on atom B. The second term in eq 5 is an approximation of the integral < f_y | V_B | f_y >. C_B is the core charge on atom B, i.e. atomic number of atom B minus the number of inner-shell electrons, and (yy|s_Bs_B) is a two-electron, two-center interaction integral. The s_B orbital is the valence s orbital on atom B. P_zz and P_pq are called density matrix elements and are defined as



for closed-shell configurations. There are two types of off-diagonal elements F_zy in the secular determinant. The element in which the f_z and f_y orbitals are on the same atom constitutes one type and is labeled . The other type of off-diagonal element has the f_zand f_p orbitals on different atoms and is labeled .





S_zp is the overlap integral < f_z | f_p> and it is solved exactly. The total energy of the molecule, E_total, is the sum of the total valence electronic energy, E_el, and the energy of repulsion between the cores on atoms A and B.



In the MNDO method



where a_A and a_B are parameters and R_AB is the internuclear distance.

    The one-center, two-electron interaction integrals (zz|yy) and (zy|zy) in eqs 5 and 8 are evaluated by a procedure that involves the fitting of the theoretical energies of the atoms to spectroscopic data. The values of these one-center, two-electron interaction integrals and the internuclear distances are used to compute the two-center, two-electron interaction integrals (zy|pq) in eqs 5, 8, and 9. The atomic parameters z ( the orbital parameter, eq2), U_yy, b_z, b_p, a_A, and a_B are evaluated by a nonlinear least-squares optimization procedure. This procedure involves the selection of a number of molecules that contain elements for which these atomic parameters are to be optimized. Only molecules for which the enthalpy of formation, molecular geometry, and dipole moment are experimentally known are chosen. Initial guesses for the parameters are used to calculate the enthalpies of formation, geometric variables, and dipole moments of these molecules. The calculated and experimental values are compared, a new set of values for the parameters are chosen, and the enthalpies of formation, geometric variables, and dipole moments are recalculated. This iterative process is continued until the squares of the weighted differences between the calculated and experimental values of the enthalpies of formation, geometric variables, and dipole moments are minimized.^1,3 The optimized values of the atomic parameters z, U_yy, b_z, b_p, a_A, and a_B for each element are stored in the MNDO software. These values are accessed and used to calculate the F_yy and F_zy terms in the secular determinant each time that a MNDO calculation is performed.

    The AM1 method differs little from the MNDO method. In the AM1 method the values of orbital parameters z for the s and p orbitals are no longer set equal and an addittional expression has been added to the f_AB term in eq 11. Unlike the AM1 method the PM3 method treats the one-center, two-electron interaction integrals (zz|yy) and (zy|zy) as parameters. Also, the procedure that is used to optimize the atomic parameters differs from the procedure described above.^1,2

    1) Levine, Ira N. Quantum Chemistry, 5th ed.; Prentice Hall : Upper Saddle River, 2000; Chapter 16.
    2) Zerner, Michael C. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B. Ed; VCH: New York, 1991; Vol. 2; p 313.
    3) Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc.,1977, 99, 4899.

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