Analytic Construction of One-Electron Reduced Density Matrices from Electron Densities within Finite Basis Sets.

We show how to construct analytically all one-electron reduced density matrices (1-RDMs) compatible with a given electron density within a finite basis set, provided that the density is specified as a symmetric quadratic form in terms of the basis functions. Contrary to the current belief, exact linear dependencies in the basis function products assist, rather than hinder, such constructions. By applying the N-representability conditions to the analytically reconstructed 1-RDMs, one can perform a constrained search over physically acceptable 1-RDMs that yield a given finite-basis-set density. The discussion is illustrated with worked-out examples.

Maximization of linear independence of basis function products.

Basis sets consisting of functions that form linearly independent products (LIPs) have remarkable applications in quantum chemistry but are scarce because of mathematical limitations. We show how to linearly transform a given set of basis functions to maximize the linear independence of their products by maximizing the determinant of the appropriate Gram matrix. The proposed method enhances the utility of the LIP basis set technology and clarifies why canonical molecular orbitals form LIPs more readily than atomic orbitals. The same approach can also be used to orthogonalize basis functions themselves, which means that various orthogonalization techniques may be viewed as special cases of a certain nonlinear optimization problem.

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential v~(r) that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials vXC(r) defined on the infinite-dimensional Hilbert space, the potentials v~XC(r) reconstructed from matrices of vXC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between v~XC(r) and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.