Densities, operators, and basis sets

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Abstract
Relationships between density matrices and densities or between operators and local potentials are considered for model problems defined by the introduction of basis sets. Some properties depend only on the space spanned by the basis while others depend on a particular choice of basis functions. Linear-dependency conditions play a critical role. In a model problem defined by a basis with all products linearly independent, the effect of any operator can be reproduced by a local potential, but any complete basis must have linearly dependent products. A one-electron density matrix or single-determinant wave function can be determined from the density (or experimental measurements sensitive only to the density) in the model problem defined by a basis with linearly independent products, but not otherwise. A simple example illustrates some of the general results.

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