Numerical solutions calculations using Slater orbitals

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

According to the ab initio molecular orbital theory methodology, atomic orbitals (set of functions, also called basis sets) combine in a way to form molecnlar orbitals that snrronnd the molecule. The molecular orbital theory considers the molecnlar wave function as an antisymmetiized product of orthonormal spatial molecular orbitals. Then they are constructed as a Slater determinant [56], Essentially, the calculations initially use a basis set, atomic wave functions [57, 58], to constract the molecular orbitals. The first and basic ab initio molecular orbital theory approach to solve the Schrodinger equation is the Hartree-Fock (HF) method [59, 60], Almost all the ab initio methodologies have the same basic numerical approach but they differ in mathematical approximations. As it is clear that finding the exact solution for the Schrodinger equation, for a molecular system, is not possible, various approaches and approximations are used to find the reliable to close-to-accurate solutions [61-68]. 

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