The Accuracy of Coulomb Fitting in DFT Calculations
Martyn Guest, Paul Sherwood and Huub van Dam
The fitted Coulomb module of the CCP1-developed DFT code promises to significantly reduce the cost of DFT calculations on molecules of intermediate size (up to 1500 basis functions) This can be achieved by choosing a functional without Hartree-Fock exchange and evaluating the Coulomb energy with an auxiliary basis set. The basic idea behind this technology is described by Dunlap et al. [1] and is referred to as "Coulomb fitting", whereby the charge density is fitted to an auxiliary basis,
where the fitting coefficients C can be obtained from:
In this equation Vuv are the electron repulsion integrals in the charge density basis and bvpq are the three centre electron repulsion integrals between the wavefunction basis set and the charge density basis. This technique enables the evaluation of 4-centre 2-electron integrals to be conducted using at most 3-centre 2-electron integrals, thereby reducing the formal scaling of the computational cost from 4th order to 3rd order. This cost is reduced still further by implementing a mechanism to store the 3-centre 2-electron integrals in main memory in a distributed fashion on parallel computers, an approach that is facilitated using a Schwarz inequality to avoid evaluating and storing small integrals. The Schwarz inequality for 3-centre 2-electron integrals is (in analogy to the 4-centre 2-electron integral inequality):
For efficiency only the maximal values of Computational Studies DFT calculations using the HCTH functional due to Handy et al. [2] have been carried out on the twenty transition metal complexes listed in Table 1. In each calculation a TZVP basis set was employed on the transition metal atom, augmented by a set of f-functions [3]; the ccpv-dz basis set due to Dunning and co-workers was used on each ligand atom [4]. Structures for each complex were computed using the explicit coulomb matrix, and three different fitting basis sets from the literature, the DGauss A1 basis [5], the Demon fitting basis [5], and finally the fitted basis sets tabulated by Ahlrichs and co-workers [6]. Table 1 lists each Metal-ligand bond length (Å) calculated using the DFT HCTH functional with both Explicit and the three fitting basis sets.
Table 1. Metal-ligand bond lengths (Å) calculated using the DFT HCTH functional with both Explicit and Fitted Coulomb Treatments (see text). These results show the exceptional accuracy of the Ahlrichs Fitting basis sets; these typically reproduce the explicit Coulomb M-L bond lengths to an accuracy of < 0.001 Å, with the largest deviation found in the Cr-C bond of Cr(C6H6)2 (0.005 Å). In the vast majority of cases the predicted bond lengths are far more accurate, typically reproducing the explicit values within the accuracy of the optimisation. Agreement using the less extensive Demon and A1-dgauss basis sets is less impressive, with typical bond length deviations in the range of 0.01 – 0.02 Å. While not accurate enough for quantitative structural predictions, they should certainly provide a cost-effective alternative in optimisation studies when deployed during the initial stages of a calculation, after which the somewhat more expensive Ahlrichs basis might be deployed. The more extensive nature of the latter does not impact too adversely on the time to solution, since the exchange correlation times typically dwarf the 2e-integral times regardless of fitting basis. References:[1] B.I. Dunlap, J.W.D. Connolly, and J.R. Sabin, On some approximations in applications of Xα theory, Journal of Chemical Physics 71 (1979) 3396-3402. [2] F.A. Hamprecht, A.J. Cohen, D.J. Tozer and N.C. Handy, J. Chem. Phys. 109 (1998) 6264-6271. [3] T.H. Dunning, Jr. J. Chem. Phys. 90 (1989) 1007; D.E. Woon and T.H. Dunning, Jr., J. Chem. Phys. 98 (1993) 1358; D.E. Woon and T.H. Dunning, Jr., J. Chem. Phys. 100 (1994) 2975; A.K. Wilson, D.E. Woon, K.A. Peterson, T.H. Dunning, Jr., J. Chem. Phys., 110 (1999) 7667. [4] A.D. McLean and G.S. Chandler, J. Chem. Phys., 72 (1980) 5639. [5] N. Godbout, D. R. Salahub, J. Andzelm and E. Wimmer, Can. J. Chem. 70, (1992) 560. [6] K. Eichkorn, O. Treutler, H. Ohm, M. Haser and R. Ahlrichs, Chem. Phys. Lett. 240 (1995) 283; K. Eichkorn, F. Weigend, O. Treutler and R. Ahlrichs, Theor. Chim. Acc. 97 (1997) 119.
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for a given set of shells are stored. We have established that for dense clusters of water molecules using a Schwarz inequality screening tolerance of 10-5 yields a stable relative error of 10-4 in the energy. Since screening is applied on a shell basis, the maximal integrals for each shell quartet are stored. Using this screening, and exploiting the aggregate memory of a parallel machine, it is possible to hold a significant fraction of the 3-centre integrals in core. The formal N3 scaling of the inversion rules out the straightforward Dunlap scheme for very large systems. It is particularly valuable for systems of intermediate size, where the cost is dominated by the integral computation and Fock build steps. For dense clusters of water molecules, and the application of a Schwarz inequality screening tolerance of 10-5, the number of integrals evaluated is reduced by about a factor of 5. More details of the performance of this approach are given elsewhere. In this article we consider the accuracy of the fitted Coulomb approach by comparing the structures of a number of transition metal complexes computed using the explicit treatment of the coulomb matrix, with those derived by using a variety of auxiliary fitting basis sets.