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Molecular DFT

Introduction to Molecular Approaches of Density Functional Theory

Jan K. Labanowski (

Ohio Supercomputer Center, 1224 Kinnear Rd, Columbus, OH 43221-1153

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Typical Program Organization for SCF-KS equations
The single geometry SCF cycle or geometry optimization involve following steps:

Start with a density (for the 1st iteration, superposition of atomic densities is typically used).
Establish grid for charge density and exchanger correlation potential
Compute KS matrix (equivalent to the tex2html_wrap_inline2059 matrix in Hartree-Fock method in equation (57)) elements and overlap integrals matrix.
Solve the equations for expansion coefficients to obtain KS orbitals.
Calculate new density tex2html_wrap_inline2339 .
If density or energy changed substantially, go to step 1.
If SCF cycle converged and geometry optimization is not requested, go to step 10.
Calculate derivatives of energy vs. atom coordinates, and update atom coordinates. This may require denser integration grids and recomputing of Coulomb and exchange-correlation potential.
If gradients are still large, or positions of nuclei moved appreciably, go to step 1.
Calculate properties and print results.

Of course, there may be other variants of this method (e.g., when one computes vibrational frequencies from the knowledge of gradients and energies only).

It is quite popular to limit expense of numerical integration during the SCF cycle. It is frequently done by fitting auxiliary functions to charge density and exchange correlation potential. This allows for much faster integral evaluation. These auxiliary fitting functions are usually uncontracted gaussians (though quite different from the atomic basis sets) for which the integrals required for KS matrix can be calculated analytically. Different auxilliary sets are used for fitting charge density and exchange-correlation potential (see e.g., Dunlap & Rösch, 1990). The need for fitting is recently questioned (see e.g., Johnson, 1995) since it scales as tex2html_wrap_inline1665 even for very large systems, however, it is still very popular in DFT codes. The fitting procedures are in general non sparse, while for large molecules many contributions coming from distant portions may be neglected leading to less steep scaling with molecular size.

Early DFT codes were impaired by the lack of analytical gradients. Currently, expressions for first and second derivatives exist (see e.g.: Dunlap & Andzelm, Komornicki & Fitzgerald, 1993) and are implemented in many programs, thus facilitating geometry optimization and vibrational frequency calculations.

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