Single coordinate basis functions

A self-consistent field wavefunction (and thus its energy) can be considered a complicated function of the nuclear coordinates, basis functions and basis function coefficients (and, for a Cl calculation, the coefficients of single determinantal wavefunctions). In order to determine the first, second, etc. derivatives of the energy with respect to the nuclear coordinates [Pulay 1977] it is necessary to consider not only how the energy depends directly on the nuclear coordinates but also whether there is an indirect dependence via other parameters. Indeed, it is only the one-electron part of the Hamiltonian that depends directly upon the nuclear coordinates (H (l), Equation (2.125)), to which is added an intemuclear Coulomb repulsion term. For the other parameters the derivative with respect to the nuclear coordinates is generally determined via the chain rule (for first derivatives). For example, for a generic nuclear coordinate and a generic parameter Xj we can write ... 

A product basis is a basis each of whose functions is a product of functions of a single coordinate. Often, the single coordinate functions are labelled with one index and the basis is a direct product basis, may be written. 

In Dirac notation we may represent a single-particle electronic state as the ket f). Suppose that the single-particle states f) form an orthonormal basis. The projection of i) onto the coordinate representation, r), (where r) is an eigenstate of the position operator, f) gives the single-particle wave function (or... 

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