Discrete variational linear combinations

We used the discrete variational (DV-Xa) method which uses a linear combination of atomic orbitals (LCAO) expansion of molecular orbitals to calculate electronic states of silicate clusters. In this... 

A sensitivity function describes the functional relationship between the change in an integral parameter caused by a fractional change in some input parameter, when the latter is expressed as a function of independent variables. For most applications a linear functional relationship is desirable. Perturbation theory formulations provide such a linear relationship. A sensitivity function can be defined for any integral parameter it can correspond to variations in any of the input parameters and it can be expressed in terms of any of the independent variables. Thus, the total number of sensitivity functions for a given system can be very large, and can be expressed in terms of different combinations of the independent variables. When the input parameter has discrete variations only, we shall refer to the sensitivity functions as sensitivity coefficients. 

Implementations have been realized using Gaussian functions (GTO s) ([38, 39] and Slater-type orbitals (STO s) [5, 40, 41], and numerical basis sets [42, 43, 44]. The auxiliary basis may be avoided by the use of a purely numerical representation of the potential on a grid (usually called DVM - Discrete Variational Method [45, 5]), by certain approximations for the potential (Multiple Scattering concept within the so-called mufl5n-tin approximation - [46]), the linear combination of muffin-tin orbitals [47, 3], and in connection with the pseudopotential concept the application of plane-wave basis expansions - see, e.g.. Ref. [112]. 

The same authors go beyond the well-known linear relationship between OCV and SOC (Figure 9.9) by describing the variation of SOC from a linear combination of the intermediate electrical measurements on the battery, and the previous SOC value, which in the discrete time domain of sampling period h can be written as... 

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