Linear integrator evaluation

One of the major selling points of Q-Chem is its use of a continuous fast multipole method (CFMM) for linear scaling DFT calculations. Our tests comparing Gaussian FMM and Q-Chem CFMM indicated some calculations where Gaussian used less CPU time by as much as 6% and other cases where Q-Chem ran faster by as much as 43%. Q-Chem also required more memory to run. Both direct and semidirect integral evaluation routines are available in Q-Chem. 

With very, very large systems, fast-multipole methods analogous to those described in Section 2.4.2 can be used to reduce the scaling of Coulomb integral evaluation to linear... 

The optimization of basis set non-linear parameters, appearing in equation (5.2), constitute one of the main steps in the preliminary work before many center integral evaluation. There will be described only a step by step procedure in order to optimize non-linear parameters of the involved fimctions one by one. 

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 use of Slater-type orbitals as basis functions is currently feasible only for atoms, diatomics and, with effort, linear polyatomic molecules. However, research into more powerful methods continues and the physical attractiveness of these functions is such that the development of effective integral evaluation methods would be a major breakthrough in quantum chemistry. 

B. EVALUATION OF THE LINEAR INTEGRATORS 1. Test Problem Selection... 

During the past three deeades, three main versions of the MCP method have been developed [1,53]. Version I is based on the local approximation. The core-valence Coulomb repulsion is a local interaction and can be satisfactorily approximated by a local potential function. For convenience of the integral evaluation, such a local potential function is chosen to be a linear combination of Gaussian type functions. The core-valence exchange operator is not a local operator. However, in Version I, this non-local interaction is also approximated by the local potential function of Gaussian type. This non-local to local approximation for the exchange operator shares the same concept with Slater s Xa density functional model [69]. Under such an approximation, the one-electron hamiltonian for the valence space in an atom (Eq. 8.5) is rewritten as... 

To facilitate integral evaluation, the atomic orbitals themselves are usually expressed as a linear... 

The SFR method described under ISO 12233 2014 requires the evaluation of luminance intensity to be worked in the linear domain by inverting the opto-electronic conversion function according to ISO 14524. ISO 16505 2015 has not made this requirement mandatory because of the difficulties to make an analytical approach of the integral evaluation of CMS. 

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