Variational Coulomb Fitting

Discussion of aspects of variational Coulomb fitting that are specific to periodic systems are in Sect. 1.4. The particular variational Coulomb scheme we use was invented by colleagues at the University of Florida [31, 32], though it was anticipated by Whitten [33] in a paper that none of us knew about until many years later. Subsequently Mintmire et al. generalized it to periodic systems [25-27,34], after which it was evolved by us, by our coUeauges at Technische Univer-sitat Munchen [35]), and elsewhere (e.g., the work of Dunlap [30] and of Salahub... 

Variational Coulomb fitting for periodic systems has evolved quite a bit since the original work, yet the details and woiking equations have never been published systematically. That omission is remedied in this and the next Subsections. 

Variational Coulomb Fitting with Charge Neutrality - Extended Systems... 

Because of the charge neutrality requirements for extended systems discussed above, literal transcription of variational Coulomb fitting for finite systems [by replacing p(r) everywhere by e(r)] clearly will not work. The solution is to do the charge fitting with neutralized fimctions. Here it is useful to allow explicitly for multiple atoms per cell. 

This disparity between extended systems and molecules motivates the approach used in GTOFF. As summarized in the introduction (recall Eq. (9)), GTOFF does exchange-correlation fitting to reference quantities evaluated with the fitted spin densities that are the output of the variational Coulomb fitting procedure, a procedure called fit-to-fit . In that procedure, rather than use Eq. (62), we approximate further ... 

Because variational Coulomb fitting with charge neutrality involves a combination of electron-electron repulsion and nuclear-electron attraction Coulomb integrals, we consider the following prototypical combination of those integrals... 

Do so, we use the formalism of the variational density fitting method [55, 56] where the Coulomb self-interaction energy of the error is minimized ... 

Eq. (22) shows that the Coulomb fitting variation principle is a boimd fixim below, so improving the fitting basis raises the total energy. This behavior is the reverse of what h pens to the total energy as the orbital basis is augmented (because of the Rayleigh-Ritz variational principle). The difference can seem counter-intuitive to new users of the method. 

A. Koster, Hermite gaussian auxiliary functions for the variational fitting of the Coulomb potential in density functional methods. J. Chem. Phys. 118, 9943-9951 (2003)... 

J.W. Mintmire, B.I. Dunlap, Fitting the Coulomb potential variationally in linear-combination-of-atomic-orbitals density-functional calculations. Phys. Rev. A 25, 88-95 (1982)... 

A related approach is due to Snijders, Baerends and Ros [71], who also use the discrete variational method for determining the wave-functions. However the Coulomb potential is handled more exactly by fitting the molecular charge... 

The purported N3 dependence of KS methods refers to procedures which reduce the integral evaluation work by fitting the computationally intensive terms in auxiliary basis sets. There are a number of different approaches which are used (and we shall not attempt to cover them all), but these are all more or less variations on a linear least-squares theme. The earliest work along these lines [21, 42], done in the context of Xa calculations, involved the replacement of the density in the Coulomb potential by a model... 

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

Here, the ERI notation for the four-center integrals is extended to three-center and two-center ERIs. Inserting this inequality into the DFT energy expression (Equation 10.7) then yields the DFT energy npj., with the variational fitting of the Coulomb potential... 

As this equation shows, the variational fitting of the Coulomb potential eliminates the A bottleneck of the ERl calculation and moves the computationally most demanding task to the numerical integration of the exchange-correlation energy and potential. 

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