Generalized Method for Calculating Löwdin Orbitals

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Abstract
A generalized method for numerically calculating Löwdin orbitals is proposed, the method being an application of Frobenius’ theorem in algebra. Löwdin transformation matrix T=S−1⁄2 (S is overlap matrix) is calculated as T=US0−1⁄2 Ut, where S0 is a diagonal matrix whose diagonal elements are eigen values of S, U is a matrix whose column vectors are eigen vectors of S and Ut is the transposed matrix of U. It is proved that the eigen values of the overlap matrix of S are always positive for any choice of linearly independent atomic orbitals and that S−1⁄2, the inverse square-root of S, exists within the range of real numbers. This method is useful for computer calculation by applying Jacobi’s method.