This review presents techniques for the computation of multi-electron integrals over Cartesian and solid-harmonic Gaussian-type orbitals as used in standard electronic-structure investigations. The review goes through the basics for oneand two-electron integrals, discuss details of various two-electron integral evaluation schemes, approximative methods, techniques to compute multi-electron inte...

This paper describes some of the results in [5] for a stochastic calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals are defined with explicit expressions for their first two moments. Multiple and iterated integrals of a fractional Browinian motion are defined and various properties of these integrals are given. A square integrab...

An exact analytical method for the calculation of volume of overlapping spheres is presented. In the considered procedure volume of overlapping spheres is expressed as surface integrals over closed regions. Using a natural continuous correspondence between the points of the sphere surface and the points of the plane the surface integrals are computed by the next transformation onto double integ...

The approach used by Ahlrichs [Phys. Chem. Chem. Phys., 2006, 8, 3072] to derive the Obara-Saika recurrence relation (RR) for two-electron integrals over Gaussian basis functions, is used to derive an 18-term RR for six-dimensional integrals in phase space and 8-term RRs for three-dimensional integrals in position or momentum space. The 18-term RR reduces to a 5-term RR in the special cases of ...

A new method for the reduction of one-loop tensor 5-point integrals to related 4-point integrals is proposed. In contrast to the usual Passarino–Veltman reduction and other methods used in the literature, this reduction avoids the occurrence of inverse Gram determinants, which potentially cause severe numerical instabilities in practical calculations. Explicit results for the 5-point tensor coe...

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equati...

All three-loop on-shell QCD Feynman integrals with two masses can be reduced to 27 master integrals. Here we calculate these master integrals, expanded in ε, both exactly in the mass ratio and as series in limiting cases.

The concept of universal integral, recently proposed and axiomatized, encompasses several integrals, including the Choquet, Shilkret and Sugeno integrals. In this paper we present two new axiomatizations of universal integrals on the scale 1⁄20;1 . In the first characterization, we look at universal integrals on the scale 1⁄20;1 as families of aggregation functions F satisfying some desired pro...

We use the fractional integrals to describe fractal solid. We suggest to consider the fractal solid as special (fractional) continuous medium. We replace the fractal solid with fractal mass dimension by some continuous model that is described by fractional integrals. The fractional integrals are considered as approximation of the integrals on fractals. We derive fractional generalization of the...