A Simple Correction for Semi-Empirical Electronic Structure Methods

From the time of their introduction, there has been debate and uncertainty about the role of the atomic orbital basis in semiempirical methods. Almost invariably, an atomic orbital basis is specified (through a set of fixed parameters that might be optimized) and this basis is used to construct an overlap metric matrix which is then used to construct the Hamiltonian matrix. However, the electronic Schrodinger equation which is solved usually proceeds to assume the basis set was orthogonal (i.e., the overlap metric matrix is set to an identity in the solution of the electronic Schordinger equation). This led early on to the assertion that the parameterized Hamiltonian corresponded to a Lowdin-orthogonalized basis (as atomic-like as possible while being orthonormal). Workers have repeatedly tried to improve the behavior of semiempirical methods by explicit or implicit orthogonalization, with varying degrees of success. We take a radically different viewpoint in this work, rejecting the assertion that one has any hope to determine the basis which corresponds to the parameterized Hamiltonian. Free of the chains of this common assertion, we can exploit the uncertainty in the basis to regularize fitting of the semiempirical parameters. Specifically, we show that the accuracy of a semiempirical method can be improved by demanding that it is stationary (or as much so as possible) with respect to general linear basis transformations. This correction replicates the invariance of ab initio results with respect to such transformations and sets the stage for new strategies to fitting semiempirical Hamiltonians.