Direct minimization Hartree Fock

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

The early VB point of view was based solely on the purely covalent HL wave function. In this wave function the electrons are never allowed to approach each other and therefore their electron repulsion is minimized and their Coulomb correlation is at maximum. Thus, while the Hartree-Fock model has no electron correlation, giving equal weight to covalent and ionic structures, the early VB models overestimated the correlation. The true situation is about half-way in-between. In the same way as the Hartree-Fock wave function is improved by Cl, the purely covalent VB function can be improved by admixture of ionic structures as in eq 5, in which the coefficients X and p would be directly optimized in the VB framework. Both improved models thus lead to wave functions that are strictly equivalent and physically correct, even though their initial expressions appear entirely different. This... 

The Patterson or direct method solution will give a number of electron density peaks which can be identified as atoms of certain types. This is still a very crade model of the stracture, which should be optimized by the least squares (LS) refinement in the following way. Spherically symmetrical Hartree-Fock atoms are placed at the positions of the peaks and the coordinates (Section 2.2.2) and displacement parameters (Section 2.2.3) of these atoms are altered so as to minimize the function... 

Field, M. I. (1991). Constrained Optimization of Ab Initio and Semiemperical Hartree-Fock Wave Functions Using Direct Minimization or Simulated Annealing. I. Phvs. Chem. 95 5104-5108. 

In Sections 3.1.5 and 3.1.6 we discuss two important alternative methods to intraorbit optimization. Both are based on the use of local-scaling transformations in order to produce sets of transformed orbitals which are then directly employed in the calculation of the total energy. In the non-variational case, we deal with arbitrary orbitals which are locally-scaled in order to yield the Hartree-Fock one-particle density, which we assume to be known beforehand. In the second method, the final density is optimized by energy minimization. But as in the previous case, locally-scaled transformed orbitals are used in the energy calculation. 

Other examples of optimizing functions that depend quadraticaUy of the parameters include the energy of Hartree-Fock (HF) and configuration interaction (Cl) wave functions. Minimization of the energy with respect to the MO or Cl coefficients leads to a set of linear equations. In the HF case, the Xy coefficients depend on the parameters Ui, and must therefore be solved iteratively. In the Cl case, the number of parameters is typically 10 -10 and a direct solution of the linear equations is therefore prohibitive, and special iterative methods are used instead. The use of iterative techniques for solving the Cl equations is not due to the mathematical nature of the problem, but due to computational efficiency considerations. 

We can now write down the Hartree-Fock one-electron hamiltonian in the D— oo limit. As usual, we will write the hamiltonian as one for the probability amplitude, and will remove the dominant dimension-dependence of the solutions through use of appropriately scaled units. As discussed in the previous section, this means that energies will be in units of 4/(D —1) haxtrees, and distances in units of D(D—l)/6 Bohr radii. The symmetry assumption allows us to equate all electron-nucleus distances, and to constrain the electrons to positions directly above the nuclei, prior to minimization of the hamiltonian. Also, the Hartree-Fock approximation allows dihedral angles to be fixed at 90°. With these scalings and simplifications, the D— oo limit hamiltonian can be written... 

A restricted Hartree-Fock SCF calculation will be performed using Pulay DIPS + Geometric Direct Minimization Optimization ... 

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