Multipole expansion of integral powers of cosine theta

Legendre polynomials form the basis for multipole expansion of spatially varying functions. The technique allows for decomposition of the function into two separate parts with one depending on the radial coordinates only and the other depending on the angular variables. In this work, the angular function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos ^k \theta$$\end{document}coskθ is expanded in the Legendre polynomial basis and the algorithm for determining the corresponding coefficients of the Legendre polynomials is generated. This expansion together with the algorithm can be generalized to any case in which a dot product of any two vectors appears. Two alternative multipole expansions for the electron–electron Coulomb repulsion term are obtained. It is shown that the conventional multipole expansion of the Coulomb repulsion term is a special case for one of the expansions generated in this work.

The function x k , where x = cos θ and k is an integer, is generated in any power series expansion involving a dot product of any two vectors with θ as an angle between them. The Taylor expansion of the plane wave e i q·r (where q is a wave vector of length q and r is a position vector of length r) and Coulomb repulsion term in many-body systems are two classic examples where the exponential term x k is present.
The multipole expansion is a powerful mathematical tool useful in decomposing a function whose arguments are three-dimensional spatial coordinates into radial and angular parts. This simplifies the solution of physical problems by reducing the triple integrals into a one-dimensional integral of the radial part and the two angular integrals. The angular integrals are solved using angular momentum algebra 1,2 . The multipole expansion involves expressing a function as a linear combination of Legendre polynomials, or the related spherical harmonics, with the orders of expansion in this case being the orders of the poles in the multipole expansion 3 . Many special functions, such as ordinary and spherical Bessel functions [3][4][5] , arise naturally as the radial part of a function and spherical harmonics as the angular part whenever a function is separated by the multipole expansion series.
In this work, x k is expanded as a function of the Legendre polynomials. The pattern formed by a sequence of coefficients of the first few Legendre polynomials is analyzed and consequently a generalization equivalent to literature values 3,6 is derived. The generalization is used for the multipole expansion of the plane wave e i q·r and electron-electron interaction term 1 | r i − r j | . Two equivalent multipole expansion series for the electron-electron interaction are obtained. The conventional multipole expansion of the electron-electron interaction is found to be a special case for one of the expansion series.

Theory
The Legendre polynomials P l (cos θ) of order l, with l a non-negative integer, are smooth functions defined in the region −1 ≤ cos θ ≤ +1 . They are usually expressed as a power series, with real number coefficients c k and where x = cos θ . The summation in Eq. (1), runs from zero (0) for even values of l while for odd values, it runs from one (1). The first two Legendre polynomials are P 0 (x) = 1 and P 1 (x) = x . Higher-order Legendre polynomials can be generated using the following recurrence relations 3 (1) www.nature.com/scientificreports/ In this work, an expansion is done in a reverse process by expressing x k as in the Legendre polynomial basis and then deriving the functional dependence of the computed real number coefficient a l (k) on orders l and k. Similar to Eq. (1), the summation runs from zero (0) if k is even and from one (1) if k is odd. The first ten (10) Legendre polynomials 3 are expressed in terms of x k , where x = cos θ , in Table 1.
In the reverse process, x k can be expanded by writing it in terms of P k (cos θ) and lower powers of x, and then similarly replacing the lower powers of x with the coresponding functions of Legendre polynomials. This implies that the reverse process has to begin with x 0 , x 1 , up to x k , in the ascending order. Beginning from the first two cases, x 0 = P 0 (x) and x 1 = P 1 (x) , already defined in Table 1, the higher order cases of the reverse process are evaluated recursively as follows: In Eqs. (4) and (5), we have substituted for 1 in the expression for x 2 , and for x in the expression for x 3 , with the corresponding predefined lower order cases, respectively. Following the same recursive process, the next higher order cases x 4 and x 5 are evaluated as and respectively. Likewise, the expansions of x 6 , x 7 , x 8 , and x 9 are obtained as (2b) (3) Scientific Reports | (2020) 10:20126 | https://doi.org/10.1038/s41598-020-77234-4 www.nature.com/scientificreports/ respectively. The computed expansion coefficients for x k , for 0 ≤ k ≤ 9 , with even and odd powers are listed in Table 2.

Results
Our goal in this study is to obtain a multipole expansion of x k , with order k being a non-negative integer and where x = cos θ is generated from a dot product of two vectors. The natural basis functions for the multipole expansion are the Legendre polynomials, or spherical harmonics which have correspondence relation with the Legendre polynomials. In this work, we have chosen the Legendre polynomials as the basis functions. Table 2 shows the coefficients, a l (k) , of the Legendre polynomials, P l (x) , in the basis expansion of x k computed using Eq. (4) up to Eq. (11) for the even and odd values of k and l respectively.
The next task involves forming the sequence of coefficients corresponding to each order of the Legendre polynomial and deductively determining the pattern with the considered cases. Noting that k = l, l + 2, · · · , we express the sequences for a l (k) with the corresponding algebraically deduced parametric dependence on k for each sequence as: Using Eqs. (12)-(19), we then deduced the parametric dependence of each of the sequences of a l (k) on l. The results determined in this study for each sequence are presented in Table 3 for even and odd values of k and l.
Based on observations of the patterns of a l (k) predicted in Table 3, we have derived the generalized pattern (10)  Table 2. Sequence of coefficients, a l (k) , of the Legendre polynomials, P l (x) , computed for the even and odd values of l and k in the basis expansion of x k given by Eq. (4) up to Eq. (11).
x k /a l (k) l = 0 l = 2 l = 4 l = 6 l = 8 x k /a l (k) l = 1 l = 3 l = 5 l = 7 l = 9   Table 3. Parametric dependence of the coefficients a l (k) on k and l for even and odd values of l. a l (k) : Sequence for even k and l a l (k) : Sequence for odd k and l a 0 (k): www.nature.com/scientificreports/ where t = r < /r > , r < = min(r i , r j ) , r > = max(r i , r j ) , and n = −1/2. www.nature.com/scientificreports/ multipole expansion series of the Coulomb repulsion term, to the best of our knowledge, have not been reported in literature. The application of the derived alternative multipole expansion of the Coulomb repulsion term in solving the electron correlation problem in electronic structure theory is a subject of our current research interest.