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Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals. (English) Zbl 1253.81054

Summary: We present a fast algorithm to calculate Coulomb/exchange integrals of prolate spheroidal electronic orbitals, which are the exact solutions of the single-electron, two-center Schrödinger equation for diatomic molecules. Our approach employs Neumann’s expansion of the Coulomb repulsion \(1/| \mathbf x - \mathbf y|\), solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later.
Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions.
Application to the diatomic molecules \(O_{2}\) and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V55 Molecular physics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
81-08 Computational methods for problems pertaining to quantum theory

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References:

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