4. Overlap matrix elements

The overlap matrix for any two basis functions $\chi_{n \ell m}^{\alpha}$ and $\chi_{n' \ell' m'}^{\beta}$ centred at ${\bf R}_{\alpha}$ and ${\bf R}_{\beta}$ respectively is

$$S_{\alpha \beta} = \int_{\mathrm{all~space}} \mathrm{d}^3 r~\chi_{n \ell m}^{\alpha}... ...bf r} - {\bf R}_{\alpha}) \chi_{n' \ell' m'}^{\beta}({\bf r} - {\bf R}_{\beta})$$

$$= \int_{\mathrm{all~space}} \mathrm{d}^3 r'~\chi_{n \ell m}^{\alpha} ({\bf r}') \chi_{n' \ell' m'}^{\beta} ({\bf r}' - {\bf R}_{\alpha \beta})$$ (10)

where ${\bf R}_{\alpha \beta} = {\bf R}_{\beta} - {\bf R}_{\alpha}$. The dummy variable of integration has been changed in order to highlight the fact that $S_{\alpha \beta}$ is a function of ${\bf R}_{\alpha \beta}$ only. Using a variant of the convolution theorem and the fact that the basis functions are real enables the integral to be rewritten as

$$S_{\alpha \beta} = {1 \over (2 \pi)^3} \int_{\mathrm{all~spa... ...lpha}({\bf k}) {\tilde \chi}_{n' \ell' m'}^{\beta}(-{\bf k}) .$$ (11)

Using equation (9a) we obtain

$$S_{\alpha \beta} = \left( q_{n \ell} r_{\alpha}^2 \right) \l... ...pha}) j_{\ell' - 1}(q_{n' \ell'} r_{\beta}) ~ I_{\alpha \beta}$$ (12)

where $I_{\alpha \beta}$ is the integral

$$I_{\alpha \beta} = {2 \over \pi}~{\mathrm{i}}^{(\ell-\ell')}... ..._{\ell m}(\Omega_{\bf k}) \bar{Y}_{\ell' m'}(\Omega_{\bf k}) .$$ (13)

Introducing differential operators ${\hat D}_{\ell m}$, obtained from $\bar{Y}_{\ell m}$ by making the replacement

$$\left\{ {x \over r},{y \over r},{z \over r} \right\} \longri... ...ha \beta}},{\partial \over \partial z_{\alpha \beta}} \right\}$$

where ${\bf R}_{\alpha \beta} = \left( x_{\alpha \beta} , y_{\alpha \beta} , z_{\alpha \beta} \right)$ in Cartesian coordinates, equation (13) becomes

$$I_{\alpha \beta} = 4 (-1)^{\ell}~ \hat{D}_{\ell m} \hat{D}_{... ...^2 - q_{n \ell}^2 \right) \left( k^2 - q_{n' \ell'}^2 \right)}$$ (14)

where we have used the fact that the integrand is an even function of $k$ for all values of $\ell$ and $\ell'$ to change the limits of the integral. From equation (14) $I_{\alpha \beta}$ no longer appears manifestly symmetric with respect to swapping $\alpha$ and $\beta$. Nonetheless, it still is because under the swap $\{ \alpha , \ell , m \} \leftrightarrow \{ \beta , \ell' , m' \}$, ${\hat D}_{\ell m} \rightarrow (-1)^{\ell'} {\hat D}_{\ell' m'}$ and ${\hat D}_{\ell' m'} \rightarrow (-1)^{\ell} {\hat D}_{\ell m}$.

The three spherical Bessel functions in equation (14) can all be expressed in terms of trigonometric functions and algebraic powers of the argument, using the recursion rules (40,41, Appendix). The product of three trigonometric functions can always be expressed as a sum of four trigonometric functions with different arguments, using well-known identities. The result is to split the integrand up into terms of the following form:

$$\qquad \qquad \frac{\sin k \left( r_{\alpha} \pm r_{\beta} \pm R_{\alpha \beta}... ...eft( k^2 - q_{n' \ell'}^2 \right)}, \qquad p~{\mathrm{always~an~odd~integer}} ,$$ (15)

$$\frac{\cos k \left( r_{\alpha} \pm r_{\beta} \pm R_{\alpha \beta}... ...ft( k^2 - q_{n' \ell'}^2 \right)}, \qquad p~{\mathrm{always~an~even~integer}} .$$

These terms are individually singular and generally possess a pole of order $p$ on the real axis at $k = 0$ and cannot be integrated. However, since we are integrating finite well-behaved functions we know that the total integrand cannot contain any non-integrable singularities. Therefore we can add extra contributions to each term to cancel all the singularities except simple poles, and all these extra terms must cancel when the terms are added together to obtain the whole integrand.

We shall evaluate the integrals using the calculus of residues so that the general integral to be performed is

$$I = \oint_C \mathrm{d}z ~ \frac{\mathrm{e}^{{\mathrm{i}} R z... ...^2 - q_{n \ell}^2 \right) \left( z^2 - q_{n' \ell'}^2 \right)}$$ (16)

where $R = r_{\alpha} \pm r_{\beta} \pm R_{\alpha \beta}$ and the contour $C$ runs along the real $z$-axis from $-\infty$ to $+\infty$, and is closed in either the upper or lower half $z$-plane, depending upon whether $R$ is positive or negative respectively. Adding the extra terms to regularise the integral we obtain the final form of the integral

$$I = \oint_C \mathrm{d}z ~ \frac{\mathrm{e}^{{\mathrm{i}} R z... ... - q_{n \ell}^2 \right) \left( z^2 - q_{n' \ell'}^2 \right)} .$$ (17)

This integrand now has simple poles lying on the contour of integration at $z = 0, \pm q_{n \ell}, \pm q_{n' \ell'}$. The residues of these poles are

$$\frac{({\mathrm{i}} R)^{p-1}}{(p-1)!~q_{n \ell}^2~q_{n' \ell'}^2} , \qquad z = 0 ,$$ (18)

$$\frac{\mathrm{e}^{\pm {\mathrm{i}} q_{n \ell} R} - \sum_{m=0}^{p-... ...n \ell}^2 - q_{n' \ell'}^2 \right) \left( \pm q_{n \ell} \right)^{p+1}}, \qquad z = \pm q_{n \ell} \quad {(\mathrm{similarly~for}}~z = \pm q_{n' \ell'} .)$$

Summing the residues to perform the Cauchy principal value integrals, and taking real or imaginary parts as appropriate, we obtain the following results:

$$\int_{-\infty}^{\infty} \mathrm{d}k \frac{\sin k R + ({\mathrm{ca... ...} {k^p \left( k^2 - q_{n \ell}^2 \right) \left( k^2 - q_{n' \ell'}^2 \right)} =$$

$$\qquad \frac{\pi~{\mathrm{sgn}}R}{q_{n \ell}^2 - q_{n' \ell'}^2} ... ...1}}{(p-1)!~q_{n' \ell'}^2} + \frac{\cos q_{n \ell} R}{q_{n \ell}^{p+1}} \right.$$ (19)

$$\qquad \left. - \frac{\cos q_{n' \ell'} R} {q_{n' \ell'}^{p+1}} -... ...m+1}} - \frac{(-1)^{m \over 2} R^m}{m!~q_{n' \ell'}^{p-m+1}} \right\} \right] ,$$

$$\int_{-\infty}^{\infty} \mathrm{d}k \frac{\cos k R + ({\mathrm{ca... ...} {k^p \left( k^2 - q_{n \ell}^2 \right) \left( k^2 - q_{n' \ell'}^2 \right)} =$$

$$\qquad \frac{\pi~{\mathrm{sgn}}R}{q_{n \ell}^2 - q_{n' \ell'}^2} ... ...1}}{(p-1)!~q_{n' \ell'}^2} - \frac{\sin q_{n \ell} R}{q_{n \ell}^{p+1}} \right.$$ (20)

$$\qquad \left. + \frac{\sin q_{n' \ell'} R} {q_{n' \ell'}^{p+1}} +... ...m+1}} - \frac{(-1)^{m-1 \over 2} R^m}{m!~q_{n' \ell'}^{p-m+1}} \right\} \right]$$

where

$${\mathrm{sgn}}R = \left\{ \begin{array}{ll} -1, \qquad & R < 0 , \\ +1, & R \geq 0 . \end{array} \right.$$ (21)

For the case when $q_{n \ell} = q_{n' \ell'}$, we note that since the integrand in equation (17) must still only have a simple pole at $z = \pm q_{n \ell}$ we obtain a simplified form in this special case by taking the limit $q_{n' \ell'} \rightarrow q_{n \ell}$ of equations (19,20).

$$\int_{-\infty}^{\infty} \mathrm{d}k \frac{\sin k R + ({\mathrm{cancelling~terms}})} {k^p \left( k^2 - q_{n \ell}^2 \right)^2} =$$

$$\qquad \pi~{\mathrm{sgn}}R~ \frac{(-1)^{p-1 \over 2} R^{p-1}}{(p-... ...ell} R}}{2 q_{n \ell}^{p+3}} - \frac{R {\sin q_{n \ell} R}}{2 q_{n \ell}^{p+2}}$$ (22)

$$\qquad + \sum_{m = 0,~{\mathrm{even}}}^{p-3} \frac{(-1)^{m \over 2} (p-m+1) R^m} {2 (m!) q_{n \ell}^{p-m+3}} ,$$

$$\int_{-\infty}^{\infty} \mathrm{d}k \frac{\cos k R + ({\mathrm{cancelling~terms}})} {k^p \left( k^2 - q_{n \ell}^2 \right)^2} =$$

$$\qquad \pi~{\mathrm{sgn}}R~ \frac{(-1)^{p \over 2} R^{p-1}}{(p-1)... ...ell} R}}{2 q_{n \ell}^{p+3}} - \frac{R {\cos q_{n \ell} R}}{2 q_{n \ell}^{p+2}}$$ (23)

$$\qquad + \sum_{m = 1,~{\mathrm{odd}}}^{p-3} \frac{(-1)^{m-1 \over 2} (p-m+1) R^m} {2 (m!) q_{n \ell}^{p-m+3}} .$$

The result for $S_{\alpha \beta}$ is obtained by summing the results in equations (19,20,22, 23) for all the terms in the expansion of the integrand (14) and then operating with the differential operators ${\hat D}_{\ell m}$.

A second special case occurs when ${\bf R}_{\alpha \beta} = 0$, and in this case it is simplest to perform the integral (10) in real-space using the generalised orthogonality relation for spherical Bessel functions (43) when $q_{n \ell} \not= q_{n' \ell'}$.

$$S_{\alpha \beta} = \frac{1}{q_{n \ell}^2 - q_{n' \ell'}^2} \... ... r_{\beta}), & r_{\alpha} \geq r_{\beta} . \end{array} \right.$$ (24)

There is also the case when ${\bf R}_{\alpha \beta} = 0$ and $q_{n \ell} = q_{n' \ell'}$ which is calculated using equation (44, Appendix).

$$S_{\alpha \beta} = {\textstyle{1 \over 2}}\delta_{\ell \ell'... ...ta}), & \qquad r_{\alpha} \geq r_{\beta} . \end{array} \right.$$ (25)

Finally, it is obvious that the overlap matrix element must vanish when the separation of the the sphere centres exceeds the sum of their radii (i.e. $R_{\alpha \beta} > r_{\alpha} + r_{\beta}$) because then there is no region of space where both basis functions are non-zero. However, this is not obvious from the results presented above, but arises because of the change of sign of the residue sums in equations (19,20,22, 23) (denoted by ${\mathrm{sgn}} R$) which occurs when $R_{\alpha \beta} = r_{\alpha} + r_{\beta}$ and results in exact cancellation of all terms.