Numerical properties of a new translation formula for exponential-type functions and its application to one-electron multicenter integrals

Abstract

Hitherto known formulas for the translation of exponential-type functions (ETF's) from one center to another (i.e., addition theorems) encounter serious difficulties of one kind or another in practical applications. In contrast, the recently derived new addition theorem of $\ensuremath{\Lambda}$ functions appears to be free of many of those difficulties. The $\ensuremath{\Lambda}$ functions are a special class of ETF's defined by a product of an exponential, a Laguerre function, and a regular solid spherical harmonic: ${\ensuremath{\Lambda}}_{\mathrm{nl}}^{m}(r,\ensuremath{\theta},\ensuremath{\varphi})={2}^{\frac{3}{2}}N(n,l){e}^{\ensuremath{-}r}{L}_{n\ensuremath{-}l\ensuremath{-}1}^{(2l+2)}(2r){(2r)}^{l}{Y}_{l}^{m}(\ensuremath{\theta},\ensuremath{\varphi}).$We have examined the new one-range addition theorem for its applicability to quantum-mechanical multicenter problems from the computational point of view and found it very useful. Test calculations on molecular multicenter one-electron integrals are reported. Under term-by-term integration, accuracy of ${10}^{\ensuremath{-}6}$ could be achieved for all these integrals with an acceptable number of expansion terms.