The Bra Concentric Case

 

Figure 5.1: The Bra concentric case

For simplicity the special case of ${\bf A}= {\bf B}$ is considered first. As well as being conceptually simpler it also produces computational savings which are useful, so this special case has been coded into the Q-CHEM program.

The first (and most drastic) approximation is to assume that the Bra and Ket are well separated, something equivalent to assuming that all Gaussians on a shell-pair are point charges.

R_ij is generated from the length DB, which is the same for all elements of the contracted shell-pair and therefore only needs to be calculated once per (0)^(m). Note that y_j is the fraction that Q_j is along DC. Computational time can be saved if the formula generates the reciprocal of R_ij, avoiding a costly divide. Now, by simple vector addition:
$$\begin{align}R_{ij} &= DB - DC y_{j} \\ R_{ij}^{2} &= DB \cdot DB - 2 DC \cdot ... ...rt DC\vert\vert DB\vert\cos\theta y_{j} + \vert DC\vert^{2}y_{j}^{2} \end{align}$$
where
$$\begin{align}\cos\theta = \frac{DC \cdot DB}{\vert DC\vert\vert DB\vert}, \end{align}$$
therefore,
$$\begin{align}\frac{1}{R_{ij}} = \frac{1}{\vert DB\vert}\sqrt{ 1 - 2\frac{\vert D... ...}\cos\theta + \frac{\vert DC\vert^{2}}{\vert DB\vert^{2}}y_{j}^{2}}. \end{align}$$
Using a binomial expansion and the Legendre polynomials P_k, the square root is replaced by an infinite series
$$\begin{align}\frac{1}{R_{ij}} &= \frac{1}{\vert DB\vert} \sum_{k=0}^{\infty} P_{... ... \\ &= \frac{1}{\vert DB\vert} \sum_{k=0}^{\infty} a_{0k} y^{k}_{j} \end{align}$$
with
$$\begin{align}a_{0k} = \left(\frac{\vert DC\vert}{\vert DB\vert}\right)^{k}P_{k}(\cos\theta). \end{align}$$

The magnitude of the kth term is bounded by $\left(\dfrac{\vert DC\vert}{\vert DB\vert}\right)^{k}$ because $0 \le y_{j} \le 1$ and $-1 \le P_{k}(\cos\theta) \le 1$. Therefore the series will converge whenever the ratio of |DC| to |DB| is less than one. The coefficients a_0k can also be generated recursively (providing easier computation):
$$\begin{align}P_{k}(\cos\theta) &= \left(\frac{2k-1}{k}\right)\cos\theta P_{k-1}(... ...k-1}{k}\right)\frac{\vert DC\vert^{2}}{\vert DB\vert^{2}}a_{0(k-2)}. \end{align}$$

The power of this group of formulae is that R_ijs can be calculated for the entire (0)^(m) using only one measurement that involves both shell-pairs (the distance DB). The other information required (the distance DC and the fractional lengths y_j) is all within the www-theor shell-pair, and can therefore be constructed before pairing all shell-pairs, thus removing a large fraction of the O(N²) work, at the cost of a little extra O(N) work. The general case is now presented.