The Fourier transforms of some exponential-type basis functions and their relevance to multicenter problems

Abstract

We analyze the properties of Slater-type functions (STFs) and B functions with respect to the Fourier transformation which is one of the most important methods for the evaluation of multicenter integrals. Although B functions have a much more complicated structure than STFs, their Fourier transform is probably the simplest of all expontential-type basis functions. Accordingly, in multicenter integrals, B functions seem to have much more attractive properties than STFs. We demonstrate this by analyzing shifting operators which make it possible to increase quantum numbers of the orbitals. Again we find that the shifting operators of B functions can be applied much more easily and yield much more compact results than the corresponding operators for STFs. We also show that the extremely compact convolution and Coulomb integrals of B functions are direct consequences of the simple Fourier transform of B functions.

Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions

“…The Fourier transform of B functions, which is of exceptional simplicity among exponentially decaying functions, is given by [Niukkanen (1984); ; Weniger & Steinborn (1983b)…”

“…The multi-center molecular integrals over B functions can be computed much more easily than the corresponding integrals of other exponentially decaying functions. This can be explained in terms of the Fourier transform of B functions, which is of exceptional simplicity among exponentially decaying functions [Niukkanen (1984); ; Weniger & Steinborn (1983b)]. Moreover, the Fourier transforms of STFs, of hydrogen eigenfunctions, or of other functions based on the generalized Laguerre polynomials can all be expressed as finite linear combinations of Fourier transforms of B functions [Weniger (1985); Weniger & Steinborn (1983b)].…”

“…This can be explained in terms of the Fourier transform of B functions, which is of exceptional simplicity among exponentially decaying functions [Niukkanen (1984); ; Weniger & Steinborn (1983b)]. Moreover, the Fourier transforms of STFs, of hydrogen eigenfunctions, or of other functions based on the generalized Laguerre polynomials can all be expressed as finite linear combinations of Fourier transforms of B functions [Weniger (1985); Weniger & Steinborn (1983b)]. The basis set of B functions is well adapted to the Fourier transform method [Geller (1962); Grotendorst & Steinborn (1988); Prosser & Blanchard (1962); Trivedi & Steinborn (1983)], which allowed analytic expressions to be developed for molecular multi-center integrals over B functions [Grotendorst & Steinborn (1988); Trivedi & Steinborn (1983)].…”

The Spherical Tensor Gradient Operator

“…From the perspective of quantum chemistry, it is probably more interesting that Y m ℓ (∇) can be extremely useful in the context of molecular multicenter integrals of exponentially decaying functions, as shown in articles by Grotendorst and Steinborn [39,40], Niukkanen [70,71,73], Novosadov [75,76,77,78,79,80], Tai [113], and in my own research [109,120,122,126,129,131,133,136].…”

“…Accordingly, it is an obvious idea to try to generate an explicit expression for a multicenter integral over nonscalar functions by differentiating the simpler expression for the corresponding integral over scalar functions (preferably the simplest scalar functions) with respect to scaling parameters and/or nuclear coordinates (compare also [129,Section IV]). The use of generating differential operators does not necessarily produce closed form expression that hold for arbitrary quantum numbers.…”