The Non-Concentric Case

 

Figure 5.2: The non-concentric case

Proceeding in a similar manner to the concentric case, the well-separated approximation is invoked before developing a general formula for R_ij:
$$\begin{align}R_{ij} &= AB x_{i} + DC y_{j} - DB \\ R_{ij}^{2} &= \vert DB\vert^... ... AB\vert\vert DC\vert\cos\phi x_{i}y_{j}+\vert DC\vert^{2}y_{j}^{2}; \end{align}$$
therefore, the reciprocal of R_ij is
$$\begin{align}\frac{1}{R_{ij}} &= \frac{1}{\vert DB\vert}\left(1-2\frac{\vert AB\... ...a_{i}^{2}+2\alpha_{i}\beta_{j}\cos\phi +\beta_{j}^{2} \right)^{-1/2} \end{align}$$
where
$$\begin{align}\alpha_{i}=\frac{\vert AB\vert}{\vert DB\vert}x_{i} \qquad \qquad \beta_{j}=\frac{\vert DC\vert}{\vert DB\vert}y_{j} \end{align}$$
which leads to:
$$\begin{align}\frac{1}{R_{ij}} &= \frac{1}{\vert DB\vert}\sum_{k=0}^{\infty}\sum_... ...vert}\sum_{k=0}^{\infty}\sum_{k=0}^{\infty} a_{kl}x_{i}^{k}y_{j}^{l} \end{align}$$
with
$$\begin{align}a_{kl} = \left(\frac{\vert AB\vert}{\vert DB\vert}\right)^{k}\left(\frac{\vert DC\vert}{\vert DB\vert}\right)^{l}d_{kl}. \end{align}$$

Like the three-center case, the d_kl are recursively related, but now there are two recurrence relations, one for increasing k, another for l:
 $$\begin{align}d_{kl} &= \left(\frac{2k+2l-1}{k}\right)\cos\theta_{x}d_{(k-1)l}-\l... ...\phi d_{(k-1)(l-1)}\nonumber \\ & \qquad \qquad \qquad +d_{(k-2)l}. \end{align}$$
The d_kl relations are easily converted to relations for the more convenient variable a_kl. For example, equation (5.41) can become
$$\begin{align}\left(\frac{\vert AB\vert}{\vert DB\vert}\right)^{k}&\left(\frac{\v... ...^{k}\left(\frac{\vert DC\vert}{\vert DB\vert}\right)^{l-2}d_{k(l-2)} \end{align}$$
which leads to the recurrence relations
$$\begin{align}a_{kl} &= \left(\frac{2k+2l-1}{k}\right) \frac{AB\cdot DB}{\vert DB... ..._{(k-1)(l-1)}+\frac{\vert AB\vert^{2}}{\vert DB\vert^{2}}a_{(k-2)l}. \end{align}$$

This general formula is more expensive than the concentric case, but it uses no extra shell-pair information. Also (as a nice check) if AB=0 the relation reduces to the concentric case. The above recurrence relations can also be extended to deal with higher powers of R_ij (which are needed for higher momentum (0)^(m)s). The argument below shows that only the coefficients are changed:
$$\begin{align}\left(1-2\alpha_{i}\cos\theta_{x}-2\beta_{j}\cos\theta_{y}+\alpha_{... ...^{\infty}\sum_{l=0}^{\infty} d_{kl}^{(m)}\alpha_{i}^{k}\beta_{j}^{l} \end{align}$$

$$\begin{align}d_{kl}^{(m)} &= \left(\frac{2k+2l+m}{k}\right)\cos\theta_{x}d_{(k-1... ...\frac{2k+m}{l}\right)\cos\phi d_{(k-1)(l-1)}^{(m)}+d_{(k-2)l}^{(m)}. \end{align}$$
For example, the first relation in the R_ij^-3 expansion (m=1) is
$$\begin{align}a_{kl} &= \left(\frac{2k+2l+1}{k}\right)\frac{AB \cdot DB}{\vert DB... ..._{(k-1)(l-1)}+\frac{\vert DC\vert^{2}}{\vert DB\vert^{2}}a_{k(l-2)}. \end{align}$$