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• W76642774 abstract "The Kolos-Wolniewicz (KW) wave functions are well known in quantum chemistry. In this work we study a possibility of generalization of KW functions towards greater flexibility and compactness. We report on a new approach to evaluation of integrals which allows numerical integration to be avoided. Some preliminary results illustrating an acceleration in energy convergence are reported. Introduction Nowadays quantum chemistry enables extremely accurate prediction of energy and other properties of small atomic and molecular systems. Such accurate results can serve as reference values for other less sophisticated theoretical models or for verification of experimental output. Wave functions employed in such precise calculations are designed to properly describe correlation of the movement of electrons. Such functions often contain an explicit interelectron distance variable r12 and are called explicitly correlated wave functions. Among them the two-electron James-Coolidge (JC) and Kolos-Wolniewicz wave functions are the most famous as they supplied benchmark results for many decades [1]. In 1933 James and Coolidge [2] applied their wave function to hydrogen molecule. Their ansatz can be written as ( ∑ = + = Ψ K i i i i c 1 ) 1 , 2 ( ) 2 , 1 ( ) 2 , 1 ( ψ ψ ) (1) with basis functions expressed in elliptic coordinates ( ) ( ) . 2 2 , 1 12 2 1 2 1 2 1 i i i i i R r e l k n m i μ ξ ξ α η η ξ ξ ψ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + − (2) In equation (2) ξi , ηi are the coordinates of the i-th electron, r12 and R are interelectron and internuclear distance, respectively. The basis functions differ from each other by a set of integer powers ki, li, mi, ni, and μi whereas the exponential parameter α is common to all the basis functions. Using thirteen such basis functions (K=13) James and Coolidge achieved energy accurate to within 1 millihartree. In 1960s Kolos and Wolniewicz [3] modified the JC function (2) adding a term which accounts for a proper asymptotic behavior of the function ( ) ( ) ( ) 1 2 1 2 1 2 12 1 2 1 2 2 1, 2 1 . i i i i i i i k l s m n k l i r e e R μ αξ αξ βη βη βη βη ψ ξ ξ η η + + − − + − − ⎛ ⎞ = + − ⎜ ⎟ ⎝ ⎠ e (3) where s determines gerade or ungerade symmetry of the state. Simultaneously, they increased the number of exponential parameters of the wave function to four: β β α α and , , . (4) In the new approach described here we increase the flexibility of the wave function by assigning each basis function ψi its own set of four parameters: i i i i and β β α α , , . (5) In such a multiple basis the total number of the nonlinear parameters to be determined variationaly increases to 4K, where K is the number of the basis functions. Additionally, we relax the limitation on the integer powers ki and li and we let them become real nonnegative numbers which fulfill the following condition { } ( ) , 1,2,..., 0 1 i j i j i j K k l k l ∈ + = ∨ + ≥ ∀ . (6) In this extended approach the powers ki and li can be optimized together with parameters (5) and the total number of variational parameters increases to 6K. The wave functions described above are employed as trial functions in variational method of solving the electronic Schrodinger equation Ψ = Ψ E Ĥ with the clamped nuclei Hamiltonian R r r r r r H b a b a 1 1 1 1 1 1 2 2 ˆ 12 2 2 1 1 2 1 + + − − − − ∆ − ∆ − = . (7) During the optimization of the wave function the energy is computed many thousands of times. It is therefore crucial that the evaluation of matrix elements is as fast as possible. In the past, some of the integrals composing the matrix elements were evaluated numerically. In our approach all the needed integrals are expressed in terms of elementary or special functions or in terms of series with a controlled convergence. The most important of them are described shortly below. Elementary integrals The overlap and Hamiltonian matrix elements can be written down in terms of several types of basic integrals. The first type involves the Legendre’s second type function Qn(x) [4] and can be evaluated using an iterative formula n dx x Q x e n dx x Q x e n dx x Q x e n k ax n k ax" @default.
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• W76642774 modified "2023-09-28" @default.
• W76642774 title "A new approach to Kołos-Wolniewicz wave functions" @default.
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