Multipole moments from the partition–expansion method

Abstract

Analytical expressions for the atomic multipole moments defined from the partition–expansion method are reported for both Gaussian and Slater basis sets. In case of Gaussian functions, two algorithms are presented and examined. The first one gives expressions in terms of generalized overlap integrals whose master formulas are derived here with the aid of the shift-operator technique. The second uses translation methods, which lead to integrals involving Gaussian and Bessel functions, which are also known. For Slater basis sets, an algorithm based on translation methods is reported. In this algorithm, atomic multipoles are expressed in terms of integrals involving Macdonald functions, which have been solved in previous works. The accuracy of these procedures is tested and their efficiency illustrated with practical applications, including the computation of the full molecular electrostatic potential (not only the long-range) in large systems.

Appendix 1: Generalized overlap integrals for spherical Gaussians

A shift-operator [43] acting on the simplest function of a given type (GTO, STO, …) produces a general function of that type. For a GTO centerd at R _I ≡ (X _I , Y _I , Z _I ) one has:

$$\frac{1}{(2 \xi)^L} \; z_L^M(\nabla_I) \; \hbox{e}^{-\xi\, r_I^2} = z_L^M({\bf r}_I) \; \hbox{e}^{-\xi r_I^2}$$

(37)

where z ^M _L (∇ _I ) is the differential operator resultant from the substitution of the Cartesian coordinates in z ^M _L (X _I , Y _I , Z _I ) by the derivatives with respect to them.

From Eq. (37), it follows:

$$\begin{aligned} \left\langle g_{LM} \vert g^{\prime}_{L^{\prime} M^{\prime}} \right\rangle & \equiv & \int \hbox{d}{\bf r} \; z_L^M({\bf r}_I) \; \hbox{e}^{-\xi r_I^2} \; z_{L^{\prime}}^{M^{\prime}}({\bf r}_J) \; \hbox{e}^{-\xi^{\prime} r_J^2}\\ & = & \frac{1}{(2 \xi)^L} \; \frac{1}{(2 \xi^{\prime})^{L^{\prime}}} \; z_L^M(\nabla_I) \; z_{L^{\prime}}^{M^{\prime}}(\nabla_J) \; \left\langle g \vert g^{\prime} \right\rangle \end{aligned}$$

(38)

where $\langle g \vert g^{\prime} \rangle$ is the integral involving the simplest GTO functions. This well-known integral, that can be derived from the Gaussian product theorem, is:

$$\left\langle g \vert g^{\prime}\right \rangle \equiv \int \hbox{d}{\bf r} \; \hbox{e}^{-\xi \, r_I^2} \; \hbox{e}^{-\xi^{\prime} r_J^2} = \left(\frac{\pi}{\xi + \xi^{\prime}}\right)^{3/2} \; \hbox{e}^{-\xi \xi^{\prime} R^2/(\xi + \xi^{\prime})}$$

(39)

where R = |R _I  − R _J |.

Let f(R) be a function which depends on X _I , Y _I , Z _I , X _J , Y _J , Z _J , only through R, although it may depend on other parameters such as exponents. From a theorem due to Hobson [57], it follows that [52]:

$$z_L^M(\nabla_I) \; z_{L'}^{M'}(\nabla_J) \; f(R) =\,(-1)^L \; \sum_{k=0}^{L_<} {\fancyscript{P}}_k^{LM L'M'}({\bf R}) \; \left(\frac{1}{R} \frac{\partial}{\partial R}\right)^{L+L'-k} \; f(R)$$

(40)

where

$${\fancyscript{P}}_k^{LM L^{\prime}M^{\prime}}({\bf R}) = \frac{2^{k}}{k!} \; \sum_{l=k}^{L_<} \; \frac{l! \; \varGamma(L+L^{\prime}-l+3/2) \; R^{2l-2k}}{(l-k)! \; \varGamma(L+L^{\prime}-l-k+3/2)} \; \sum_m \alpha_{L+L^{\prime}-2l, m}^{LM, L^{\prime}M^{\prime}} \; z_{L+L^{\prime}-2l}^{m}({\bf R})$$

(41)

$\alpha_{L+L^{\prime}-2l, m}^{LM, L^{\prime}M^{\prime}}$ being the coefficients appearing in Eq. (16).

By combining Eqs. (38), (39), and (40), and bearing in mind that $\left(\frac{1}{R} \frac{\partial}{\partial R}\right) = 2 \frac{\partial}{\partial(R^2)},$ the master formula for spherical overlap integrals is obtained:

$$\begin{aligned} \left\langle g_{LM} \vert g^{\prime}_{L^{\prime}M^{\prime}}\right\rangle & = (-1)^{L^{\prime}} \; \frac{\pi^{3/2} \; \xi^{L^{\prime}} \; {\xi^{\prime}}^L}{(\xi + \xi^{\prime})^{L+L^{\prime}+3/2}} \;\hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})}\\ &\quad\times \sum_{k=0}^{L_<} (-1/2)^k \; {\fancyscript{P}}_k^{LM L^{\prime}M^{\prime}}({\bf R}) \; \biggl( \frac{\xi+\xi^{\prime}}{\xi \xi^{\prime}} \biggr)^k \end{aligned}$$

(42)

Taking into account that:

$$r_I^{2n} \; \hbox{e}^{-\xi r_I^2} = \left(-\frac{\partial}{\partial\xi}\right)^n \; \hbox{e}^{-\xi r_I^2}$$

(43)

one has:

$$\begin{aligned} \left\langle g^n_{LM} \vert g^{\prime}_{L^{\prime} M^{\prime}} \right\rangle & = \left(-\frac{\partial}{\partial\xi}\right)^n \; \langle g_{LM} \vert g^{\prime}_{L^{\prime} M^{\prime}} \rangle\\ &= (-1)^{L^{\prime}} \; \pi^{3/2} \; \sum_{k=0}^{L_<} (-1/2)^k \; {\fancyscript{P}}_k^{LM L^{\prime}M^{\prime}}({\bf R}) \; {\xi^{\prime}}^{L-k}\\ &\quad \times \biggl(-\frac{\partial}{\partial\xi} \biggr)^n \; \frac{\xi^{L^{\prime}-k}}{{(\xi + \xi^{\prime})^{L+L^{\prime}+3/2-k}}} \; \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} \end{aligned}$$

(44)

so that the problem is reduced to find:

$$\begin{aligned} \biggl(-\frac{\partial}{\partial\xi} \biggr)^n \; \frac{\xi^{L^{\prime}-k}}{{(\xi + \xi^{\prime})^{L+L^{\prime}+3/2-k}}} \; \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} &= \sum_{p=0}^{n} {n \choose p} \; \biggl[ \biggl( - \frac{\partial}{\partial\xi} \biggr)^{p} \; \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} \biggr]\\ &\quad \times \biggl[\biggl(-\frac{\partial}{\partial\xi} \biggr)^{n-p} \; \frac{\xi^{L^{\prime}-k}}{{(\xi + \xi^{\prime})^{L+L^{\prime}+3/2-k}}} \biggr] \end{aligned}$$

(45)

It is not difficult to prove that:

$$\begin{aligned} \biggl( - \frac{\partial}{\partial\xi} \biggr)^{p} \; \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} &= \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} \; \sum_{i=0}^{p-1} \frac{p! \; (p-1)!}{(p-1-i)! \; (i+1)! \; i!} \; \frac{({\xi^{\prime}}^2 R^2)^{i+1}}{(\xi + \xi^{\prime})^{p+1+i}}\\ & = p! \; \hbox{e}^{-\xi \xi^{\prime} R^2 / (\xi + \xi^{\prime})} \; \frac{{\xi^{\prime}}^2 R^2}{(\xi + \xi^{\prime})^{p+1}} \; _1F_1\biggl( -p+1; 2; -\frac{{\xi^{\prime}}^2 R^2}{\xi + \xi^{\prime}} \biggr)\quad p > 0 \end{aligned}$$

(46)

and that:

$$\begin{aligned} \biggl( - \frac{\partial}{\partial\xi} \biggr)^{n-p} \; \frac{\xi^{L^{\prime}-k}}{{(\xi + \xi^{\prime})^{L+L^{\prime}+3/2-k}}} &= \sum_{i=0}^{{\rm min}(n-p, L^{\prime}-k)} (-1)^i \; \frac{\xi^{L^{\prime}-k-i}}{(\xi + \xi^{\prime})^{n+L+L^{\prime}+3/2-k-p-i}}\\ &\quad \times\frac{(n-p)! \; (L^{\prime}-k)! \; (n+L+L^{\prime}+1/2-k-p-i)!}{(n-p-i)! \; (L^{\prime}-k-i)! \; (L+L^{\prime}+1/2-k)! \; i!}\\ & = \frac{\xi^{L^{\prime}-k}}{(\xi + \xi^{\prime})^{n+L+L^{\prime}+3/2-k-p}} \; \frac{(n+L+L^{\prime}+1/2-k-p)!}{(L+L^{\prime}+1/2-k)!} \;\\ &\quad \times _2F_1\biggl(-n+p,-L^{\prime}+k;-n-L-L^{\prime}-1/2+k+p;\frac{\xi+\xi^{\prime}}{\xi} \biggr) \end{aligned}.$$

(47)

Joining these results, Eq. (18) of the main text follows.