Slater functions orbital energy calculations using

Figure 4.12 Calculation of the Is and 2s orbital energies in hydrogen using the sto-3g) basis sets of Table 1.6 and canonical orthonormalization. A better Is energy compared with the result found for Schmidt orthonormalization, since the [sto-3g) Is basis is not improved in that calculation. On the scale of the chart, there appears to be no difference in the approximate functions obtained. However, when the scale is enlarged, a small difference is evident and accounts for the different Is orbital energy calculated. Note, in this calculation both Slater exponents, cells H 4 and H 5, have been allowed to vary in the SOLVER routine.

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

The non-relativistic DV-Xa calculation [21] was performed with the Slater exchange parameter, a = 0.7, for all atoms and with 50,000 DV sampling points, which provided a precision of less than 0.1 eV for valence electron energy eigenvalues. We employed the basis functions of the central nickel atom ls-4p orbitals, while those of the nitrogen and carbon atoms were used ls-2p orbitals. The calculations were carried out self-consistently until the difference in the orbital populations between the initial and final states of the iteration was less... 

E is calculated by summing and integrating the forces on the nuclei as the atoms approach each other due to the superposition of rigid atom density functions using Slater atomic orbitals. Eq is approximated by the change in one-electron molecular orbital energy, A mo> calculated using... 

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