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- The Limited Expansion of Diatomic Overlap Density Functional Theory (LEDO-DFT): Development and Implementation of Algorithms, Optimization of Auxiliary Orbitals and Benchmark Calculations (2005)
- This work describes the development and implementation of efficient algorithms for the limited expansion of diatomic overlap density functional theory (LEDO-DFT) into the program package TURBOMOLE. Extensive test calculations with this implementation using optimized auxiliary basis sets document the accuracy and efficiency of LEDO-DFT. LEDO-DFT is a novel formalism in the framework of Kohn-Sham DFT (KS-DFT) with the nice property of a formal scaling behaviour as low as O(N^2) for the setup of the secular matrix. Here, N denotes the size of the system. A straightforward implementation of KS-DFT has a formal O(N^4) scaling behaviour. Reduction to O(N^3) is possible if conventional density fitting is applied. For sufficiently large molecular systems, however, it is the asymptotic scaling behaviour together with the associated prefactor that determines the real cost of a calculation. The asymptotic scaling behaviour for the setup of the secular matrix is O(N^2) for KS-DFT, with and without density fitting. Due to the better formal scaling behaviour, however, LEDO-DFT has the potential to further reduce the prefactor associated with the asymptotic O(N^2) scaling of KS-DFT calculations. Density fitting methods are in wide use in electronic structure calculations nowadays. A review of these techniques describes how the LEDO expansion fits into this general framework. Emphasis is put on the distinction of the different norms that can be applied during the density fitting procedure and especially the consequences of using global or local expansions. This allows for a deeper understanding of the advantages and drawbacks of LEDO-DFT as compared to the various other approaches. A proper norm for the optimization of auxiliary basis sets for the LEDO expansion is derived. Possible forms of the auxiliary basis for the LEDO expansion are discussed and the "a priori" elimination, a highly efficient method to deal with near-linear dependences in the LEDO expansion basis, is introduced. A projection technique suggested by Löwdin is presented in the framework of LEDO-DFT, which shall guarantee for the SCF convergence of critical cases. The main features of the algorithms for energy and analytical gradient calculations with LEDO-DFT which been developed and implemented are thoroughly discussed under the aspect of efficiency. Employing the norm presented in this work, a systematic procedure for the optimization of auxiliary orbitals is developed. Optimized auxiliary orbitals for the Ahlrichs SVP basis set for the first three rows of atoms, rare gases excepted, are presented. It is shown that an "a priori" elimination of near-linear dependences in the LEDO expansion basis is feasible and leads to a drastic speedup for the determination of the LEDO expansion coefficients. The accuracy of LEDO-DFT employing the optimized auxiliary orbitals in combination with the "a priori" elimination is demonstrated by structure optimizations and frequency analyses (by numerical differentiation of the analytical gradients) for a test set of 142 small molecules. Total energies, structural parameters, dipole moments and harmonic frequencies are reproduced with very small errors for most molecules of the test set. Application of an extended expansion basis leads to very good results for critical test molecules as well. Test calculations on linear alkanes with a chain length of up to 30 carbon atoms show that the errors in the structure parameters grow only very slowly with increasing system size. It is proven that the Löwdin projection technique can be successfully applied in the framework of LEDO-DFT in order to guarantee for SCF convergence in critical cases without significantly deteriorating the results. It is demonstrated that LEDO-DFT can be expected to yield reliable results for typical applications of DFT like, e.g., the computation of energy differences such as reaction energies and relative stabilities. Results obtained for a variety of more complex molecules as might be of interest in contemporary research are also very encouraging. The computational efficiency of LEDO-DFT is investigated for linear alkanes with a maximum chain length of 30 carbon atoms. A speedup of up to 11.1 for the SCF and 3.4 for the analytical gradients is observed as compared to DFT without further approximations. The timings show that LEDO-DFT is competitive with conventional density fitting (RI-DFT), although the latter is still slightly in advantage. The use of the "a priori" elimination of near-linear dependences of the one-center overlap densities of the LEDO expansion basis for the determination of the expansion coefficients has been essential in achieving these positive results. However, in order to outperform conventional density fitting methods, the cost/accuracy ratio still has to be improved. This is likely to be achieved by substituting the auxiliary orbitals employed in this work with properly optimized auxiliary functions.