Lebedev discrete variable representation

We describe the construction of a Lebedev discrete variable representation (DVR) (Dickinson and Certain 1968 J. Chem. Phys. 49 4209, Lill, Parker and Light 1982 Chem. Phys. Lett. 89 483, Light, Hamilton and Lill 1985 J. Chem. Phys. 82 1400). This DVR is based upon a new algorithm for constructing a generalized DVR in one or more dimensions. The novel part of this algorithm is a contraction of the underlying functional basis that is based upon the given points and weights. The contraction allows us to define a set of basis functions and a set of points, of equal number, for which the transformation between the 'point basis' and the 'function basis' is neither singular nor close to it. We present both a strictly orthogonal and a discretely orthogonal version of the Lebedev DVR, which are based on the octahedrally symmetric Lebedev quadratures for the sphere. The discretely orthogonal DVR is a classic generalized DVR (Light, Hamilton and Lill 1985 J. Chem. Phys. 82 1400) and the strictly orthogonal version is similar to a diagonalization DVR. These DVRs perform well in test calculations. The algorithm is general and should be applicable to nonseparable bases in many dimensions.