Matrix Elements for Explicitly-Correlated Atomic Wave Functions

Abstract

We refer to atomic wave functions that contain the interelectron distances as “explicitly correlated”; we consider here situations in which an explicit correlation factor $r_{ij}$ can occur as a power multiplying an orbital functional form (a Hylleraas function) and/or in an exponent (producing exponential correlation ). Hylleraas functions in which each wave-function term contains at most one linear $r_{ij}$ factor define a method known as Hylleraas-CI . This paper reviews the analytical methods available for evaluating matrix elements involving exponentially-correlated and Hylleraas wave functions; attention is then focused on computation of integrals needed for the kinetic energy. In contrast to orbital-product and exponentially-correlated wave functions, no general formulas have been developed by others to relate the kinetic-energy integrals in Hylleraas-CI (or its recent extension by the Nakatsuji group) to contiguous potential-energy matrix elements. The present paper provides these missing formulas, obtaining them by using relevant properties of vector spherical harmonics. Validity of the formulas is confirmed by comparisons with kinetic-energy integrals obtained in other ways.

It is a pleasure to present this work as part of a tribute to Professor Josef Paldus in celebration of his eightieth birthday.

Appendix. Angular-Momentum Coefficients

The spherical harmonics $Y_l^m(\theta ,\phi )$, alternatively written $Y_l^m(\varOmega )$, can be defined with the sign convention chosen by Condon and Shortley [17] (Condon-Shortley phase) by the Rodrigues formula

$$\begin{aligned} Y^m_l(\varOmega ) = N_{lm}\frac{(-1)^m}{2^l l!}(1-u^2)^{m/2}\frac{d^{l+m}}{du^{l+m}} (u^2-1)^l e^{im\phi }, \end{aligned}$$

(31)

where $u=\cos \theta$ and $N_{lm}$ is the factor

$$\begin{aligned} N_{lm} = \sqrt{\frac{(2l+1)(l-m)!}{4\pi (l+m)!} } \end{aligned}$$

(32)

that makes the $Y_l^m$ orthonormal. With these definitions,

$$\begin{aligned} Y^{m}_l(\varOmega )^* = (-1)^m Y^{-m}_l(\varOmega ). \end{aligned}$$

(33)

A product of spherical harmonics of the same argument $\varOmega$ can be expanded into a sum of harmonics of that argument. The coefficients in that expansion are known as Gaunt coefficients. Unfortunately there is no unanimity as to the definition of the Gaunt coefficient. Choosing the definition of Pinchon and Hoggan [40], we introduce a bracket notation that we hope will become adopted:

$$\begin{aligned} \left[ \begin{array}{ccc}l_1&{}l_2&{}l_3\\ m_1&{}m_2&{}m_3\end{array}\right] = \int Y^{m_1}_{l_1}(\varOmega )Y^{m_2}_{l_2}(\varOmega )Y^{m_3}_{l_3}(\varOmega )\, d\varOmega \,. \end{aligned}$$

(34)

Expansion of the spherical harmonic product $Y^{m_1}_{l_1}Y^{m_2}_{l_2}$ in the orthonormal set $Y_L^M$, carried out by taking scalar products with $(Y_L^M)^*$, leads after use of Eqs. (33) and (34) to

$$\begin{aligned} Y^{m_1}_{l_1}(\varOmega )Y^{m_2}_{l_2}(\varOmega ) = \sum _{LM} (-1)^M \left[ \begin{array}{ccc}l_1&{}l_2&{}L\\ m_1&{}m_ 2&{}-M\end{array}\right] Y^M_L(\varOmega ). \end{aligned}$$

(35)

Because harmonics with upper indices $m_1$ and $m_2$ form a product all of whose terms have the same value of M, Eq. (35) can be simplified by dropping the M summation, setting $M = m_1+m_2$.

The Gaunt coefficients can be written in terms of Wigner 3-j symbols [39]. Using the standard notation for that symbol (an array of l and m values in ordinary parentheses), the Gaunt coefficients as defined here can be written

$$\begin{aligned} \left[ \begin{array}{ccc}l_1&{}l_2&{}l_3\\ m_1&{}m_2&{}m_3\end{array}\right] = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi } } \left( \begin{array}{ccc}l_1&{}l_2&{}l_3\\ m_1&{}m_2&{}m_3\end{array}\right) \left( \begin{array}{ccc}l_1&{}l_2&{}l_3\\ 0&{}0&{}0\end{array}\right) . \end{aligned}$$

(36)

A pair of angular momenta in two independent variables can be coupled to form a quantity of definite resultant angular momentum by forming a linear combination of products of the individual angular momenta; the coefficients in that expansion are called Clebsch-Gordan coefficients. Coupling of the angular-momentum wave functions $\psi _{j_1}^{m_1}(1)$ and $\psi _{j_2}^{m_2}(2)$ with fixed values of $j_1$ and $j_2$ to form the combined function $\varPsi _J^M(1,2)$ is described by

$$\begin{aligned} \varPsi ^M_J(1,2) = \sum _{m_1 m_2} \left\langle \begin{array}{ccc}j_1&{}j_2&{}J\\ m_1&{}m_2&{}M\end{array}\right\rangle \psi ^{m_1}_{j_1}(1) \psi ^{m_2}_{j_2}(2), \end{aligned}$$

(37)

where the array in angle brackets is our (nonstandard) notation for the Clebsch-Gordan coefficient. Here all contributing terms must satisfy $m_1+m_2=M$, so we can actually reduce Eq. (37) to a single sum over, say, $m_2$, with $m_1$ set to $M-m_2$.

The Clebsch-Gordan coefficients can also be written in terms of 3-j symbols:

$$\begin{aligned} \left\langle \begin{array}{ccc}j_1&{}j_2&{}j_3\\ m_1&{}m_2&{}m_3\end{array} \right\rangle = (-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \left( \begin{array}{ccc}j_1&{}j_2&{}j_3\\ m_1&{}m_2&{}-m_3\end{array}\right) . \end{aligned}$$

(38)

Well-documented computer programs exist for the evaluation of the 3-j symbols, making it straightforward to evaluate expressions involving Gaunt or Clebsch-Gordan coefficients.