ECP integral

The unnormalized Cartesian GTO with the partial derivative operators of the center coordinates is defined as follows:

[math]\displaystyle{ \varphi(\boldsymbol{r};\zeta,\boldsymbol{l},\boldsymbol{n},\boldsymbol{R}) = \left(\frac{\partial}{\partial R_\text{x}}\right)^{l_\text{x}} \left(\frac{\partial}{\partial R_\text{y}}\right)^{l_\text{y}} \left(\frac{\partial}{\partial R_\text{z}}\right)^{l_\text{z}} (r_\text{x}-R_\text{x})^{n_\text{x}} (r_\text{y}-R_\text{y})^{n_\text{y}} (r_\text{z}-R_\text{z})^{n_\text{z}} e^{-\zeta|\boldsymbol{r}-\boldsymbol{R}|^2} }[/math],

where

• [math]\displaystyle{ \boldsymbol{r}=(r_\text{x},r_\text{y},r_\text{z}) }[/math]: the coordinates of the electron,
• [math]\displaystyle{ \zeta }[/math]: the orbital exponent; a positive real number,
• [math]\displaystyle{ \boldsymbol{l}=(l_\text{x},l_\text{y},l_\text{z}) }[/math]: the partial derivative orders; nonnegative integers,
• [math]\displaystyle{ \boldsymbol{n}=(n_\text{x},n_\text{y},n_\text{z}) }[/math]: the angular momentum indices; nonnegative integers, and
• [math]\displaystyle{ \boldsymbol{R}=(R_\text{x},R_\text{y},R_\text{z}) }[/math]: the coordinates of the center.

The ECP integral is comprised of the type 1 integrals and the type 2 integrals. The type 1 integral with the partial derivative operators can be written as:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} \int_0^\infty dr_\text{C} \int d\Omega_\text{C}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) |\boldsymbol{r}-\boldsymbol{R}_\text{C}|^n e^{-\zeta_\text{C}|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^2} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}) }[/math],

and the type 2 integral with the partial derivative operators can be written as:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_2(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} \sum_{m=-l}^l \int_0^\infty dr_\text{C} \left( \int d\Omega_\text{C}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) {Y_\text{C}}_{l,m} \right) }[/math]

[math]\displaystyle{ \times |\boldsymbol{r}-\boldsymbol{R}_\text{C}|^n e^{-\zeta_\text{C}|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^2} \left( \int d\Omega_\text{C}~ {Y_\text{C}}_{l,m} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}) \right) }[/math],

where

• A, B, and C denote the centers of the left-side GTO, the right-side GTO, and the potential core respectively,
• [math]\displaystyle{ l }[/math]: the angular momentum of the orbital in the core (do not confuse with [math]\displaystyle{ \boldsymbol{l} }[/math]),
• [math]\displaystyle{ n }[/math]: the power restricted to the values [0, 1, 2] (do not confuse with [math]\displaystyle{ \boldsymbol{n} }[/math]),
• [math]\displaystyle{ \zeta_\text{C} }[/math]: the potential exponent; a positive real number, and
• [math]\displaystyle{ {Y_\text{C}}_{l,m} }[/math]: the real spherical harmonics centered on the potential core.

The integration [math]\displaystyle{ \int_0^\infty dr_\text{C} }[/math] means the radial integral centered on the potential core, and the integration [math]\displaystyle{ \int d\Omega_\text{C} }[/math] means the angular integral centered on the potential core.

Type 1 integral

Using the original method

McMurchie and Davidson showed the method to compute the type 1 integral as described in this section.

The type 1 integral with the partial derivative operators is shown here again:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} \int_0^\infty dr_\text{C} \int d\Omega_\text{C}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) |\boldsymbol{r}-\boldsymbol{R}_\text{C}|^n e^{-\zeta_\text{C}|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^2} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}) }[/math].

Using Obara–Saika scheme

Since the integration on the polar coordinate system [math]\displaystyle{ \int_0^\infty dr_\text{C} \int d\Omega_\text{C} }[/math] can be converted to the integration on the Cartesian coordinate system [math]\displaystyle{ \int_{-\infty}^\infty dr_\text{x} \int_{-\infty}^\infty dr_\text{y} \int_{-\infty}^\infty dr_\text{z}~{r_\text{C}}^{-2} }[/math], the type 1 integral can be rewritten as:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} \int_{-\infty}^\infty dr_\text{x} \int_{-\infty}^\infty dr_\text{y} \int_{-\infty}^\infty dr_\text{z}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) }[/math]

[math]\displaystyle{ \times|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^{n-2} e^{-\zeta_\text{C}|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^2} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}) }[/math].

It can be derived from the three-center overlap integral:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)=\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\frac{1}{|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^{2-n}}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \begin{cases} \displaystyle 2\int_0^\infty dv~v\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} & (n = 0) \\ \displaystyle \frac{2}{\sqrt{\pi}}\int_0^\infty dv~\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} & (n = 1) \\ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) & (n = 2) \end{cases} }[/math].

Therefore, the type 1 integral can be computed using Obara-Saika scheme .

Case of n = 0

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) = 2\int_0^\infty dv~v\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} }[/math]

Auxiliary integral

To describe the recurrence relations, the auxiliary integral is introduced:

[math]\displaystyle{ \left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)_m = 2\int_0^\infty dv~v\left(\frac{v^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}+v^2}\right)^m\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} }[/math],

where [math]\displaystyle{ m }[/math] is a nonnegative integer. Apparently,

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) = {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_0 }[/math].

Recurrence relations

Horizontal recurrence relation on partial derivative order

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_m+{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}+\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m=0 }[/math]

Vertical recurrence relation on partial derivative order #1

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ =\frac{2\zeta_\text{A}\left(\zeta_\text{B}{R_\text{B}}_\mu+\zeta_\text{C}{R_\text{C}}_\mu-(\zeta_\text{B}+\zeta_\text{C}){R_\text{A}}_\mu\right)}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\left((\zeta_\text{A}+\zeta_\text{B}){R_\text{C}}_\mu-\zeta_\text{A}{R_\text{A}}_\mu-\zeta_\text{B}{R_\text{B}}_\mu\right)}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ -\frac{2\zeta_\text{A}(\zeta_\text{B}+\zeta_\text{C})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{2\zeta_\text{A}^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ -\frac{\zeta_\text{B}+\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{2\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_m-\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+\frac{2\zeta_\text{A}(\zeta_\text{A}+\zeta_\text{B})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

Vertical recurrence relation on partial derivative order #2

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}+\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ =\frac{2\zeta_\text{C}\left(\zeta_\text{A}{R_\text{A}}_\mu+\zeta_\text{B}{R_\text{B}}_\mu-(\zeta_\text{A}+\zeta_\text{B}){R_\text{C}}_\mu\right)}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ +\frac{2(\zeta_\text{A}+\zeta_\text{B})\left(\zeta_\text{A}{R_\text{A}}_\mu+\zeta_\text{B}{R_\text{B}}_\mu-(\zeta_\text{A}+\zeta_\text{B}){R_\text{C}}_\mu\right)}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+\frac{2\zeta_\text{A}(\zeta_\text{A}+\zeta_\text{B})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A},\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+\frac{\zeta_\text{A}+\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{B}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_m+\frac{2\zeta_\text{B}(\zeta_\text{A}+\zeta_\text{B})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_m+\frac{\zeta_\text{A}+\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ -\frac{2(\zeta_\text{A}+\zeta_\text{B})\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{2{(\zeta_\text{A}+\zeta_\text{B})}^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

Horizontal recurrence relation on angular momentum index

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}+\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}+\boldsymbol{1}_\mu\right)\right)}_m }[/math]

[math]\displaystyle{ =({R_\text{B}}_\mu-{R_\text{A}}_\mu){\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ -{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

Vertical recurrence relation on angular momentum index

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}+\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ =\frac{\zeta_\text{B}{R_\text{B}}_\mu+\zeta_\text{C}{R_\text{C}}_\mu-(\zeta_\text{B}+\zeta_\text{C}){R_\text{A}}_\mu}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ +\frac{(\zeta_\text{A}+\zeta_\text{B}){R_\text{C}}_\mu-\zeta_\text{A}{R_\text{A}}_\mu-\zeta_\text{B}{R_\text{B}}_\mu}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ -\frac{\zeta_\text{B}+\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{A}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_m-\frac{\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_m-\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{B}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right)\right)}_{m+1} }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_m+\frac{\zeta_\text{A}+\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu{\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_{m+1} }[/math]

Initial integral

[math]\displaystyle{ {\left(\left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ =2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})\left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)G_m\left(\ \frac{{(\zeta_\text{A}+\zeta_\text{B})}^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left|\frac{\zeta_\text{A}\boldsymbol{R}_\text{A}+\zeta_\text{B}\boldsymbol{R}_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}-\boldsymbol{R}_\text{C}\right|^2\right) }[/math]

Here,

[math]\displaystyle{ G_m(u)=\int_0^1 dt~\frac{t}{\sqrt{1-t^2}}t^{2m}e^{-ut^2}=\frac{1}{2}\sqrt{\pi}\frac{\Gamma(m+1)}{\Gamma(m+\frac{3}{2})}{}_1F_1\left(m+1;m+\frac{3}{2};-u\right) }[/math],

where

[math]\displaystyle{ \Gamma(m+1)=m! }[/math],

[math]\displaystyle{ \Gamma(m+\frac{3}{2})=\sqrt{\pi}\prod_{k=0}^m\frac{2k+1}{2} }[/math],

and [math]\displaystyle{ {}_1F_1(a;b;z) }[/math] is the confluent hypergeometric function of the first kind.

Case of n = 1

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) = \frac{2}{\sqrt{\pi}}\int_0^\infty dv~\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} }[/math]

Auxiliary integral

To describe the recurrence relations, the auxiliary integral is introduced:

[math]\displaystyle{ \left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)_m = \frac{2}{\sqrt{\pi}}\int_0^\infty dv~\left(\frac{v^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}+v^2}\right)^m\left.\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right|_{\zeta_\text{C}=\zeta_\text{C}+v^2} }[/math],

where [math]\displaystyle{ m }[/math] is a nonnegative integer. Apparently,

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) = {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)\right)}_0 }[/math].

Recurrence relations

Horizontal recurrence relation on partial derivative order

The same as the case of n = 0.

Vertical recurrence relation on partial derivative order #1

The same as the case of n = 0.

Vertical recurrence relation on partial derivative order #2

The same as the case of n = 0.

Horizontal recurrence relation on angular momentum index

The same as the case of n = 0.

Vertical recurrence relation on angular momentum index

The same as the case of n = 0.

Initial integral

[math]\displaystyle{ {\left(\left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)\right)}_m }[/math]

[math]\displaystyle{ =\frac{2}{\sqrt{\pi}}\sqrt{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)F_m\left(\frac{(\zeta_\text{A}+\zeta_\text{B})^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left|\frac{\zeta_\text{A}\boldsymbol{R}_\text{A}+\zeta_\text{B}\boldsymbol{R}_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}-\boldsymbol{R}_\text{C}\right|^2\right) }[/math]

Here, [math]\displaystyle{ F_m(u) }[/math] is the Boys function:

[math]\displaystyle{ F_m(u)=\int_0^1 dt~t^{2m}e^{-ut^2} }[/math].

Case of n = 2

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_1(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) = \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

Recurrence relations

Horizontal recurrence relation on partial derivative order

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)+\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)+\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}+\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)=0 }[/math]

Vertical recurrence relation on partial derivative order #1

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ =\frac{2\zeta_\text{A}(\zeta_\text{B}({R_\text{B}}_\mu-{R_\text{A}}_\mu)+\zeta_\text{C}({R_\text{C}}_\mu-{R_\text{A}}_\mu))}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ -\frac{2\zeta_\text{A}(\zeta_\text{B}+\zeta_\text{C})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) -\frac{\zeta_\text{B}+\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right) +\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right) }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

Vertical recurrence relation on partial derivative order #2

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}+\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ =\frac{2\zeta_\text{C}(\zeta_\text{B}({R_\text{B}}_\mu-{R_\text{C}}_\mu)+\zeta_\text{A}({R_\text{A}}_\mu-{R_\text{C}}_\mu))}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{A}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{A}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ +\frac{2\zeta_\text{B}\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right) +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right) }[/math]

[math]\displaystyle{ -\frac{2\zeta_\text{C}(\zeta_\text{A}+\zeta_\text{B})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

Horizontal recurrence relation on angular momentum index

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}+\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)- \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}+\boldsymbol{1}_\mu\right) }[/math]

[math]\displaystyle{ = ({R_\text{B}}_\mu-{R_\text{A}}_\mu)\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)- {l_\text{A}}_\mu\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)+ {l_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right) }[/math]

Vertical recurrence relation on angular momentum index

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}+\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ =\frac{\zeta_\text{B}({R_\text{B}}_\mu-{R_\text{A}}_\mu)+\zeta_\text{C}({R_\text{C}}_\mu-{R_\text{A}}_\mu)}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ -\frac{\zeta_\text{B}+\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{A}}_\mu\left(\boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) +\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{A}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right) +\frac{1}{2(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})}{n_\text{B}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu\right) }[/math]

[math]\displaystyle{ +\frac{\zeta_\text{C}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{l_\text{C}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu,\boldsymbol{0}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

Initial integral

[math]\displaystyle{ \left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{C},\boldsymbol{0}_\text{C}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)=\left(\frac{\pi}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}\right)^\frac{3}{2}e^{-\frac{\zeta_\text{A}\zeta_\text{B}|\boldsymbol{R}_\text{A}-\boldsymbol{R}_\text{B}|^2+\zeta_\text{B}\zeta_\text{C}|\boldsymbol{R}_\text{B}-\boldsymbol{R}_\text{C}|^2+\zeta_\text{C}\zeta_\text{A}|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}|^2}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}} }[/math]

Type 2 integral

Using the original method

McMurchie and Davidson showed the method to compute the type 2 integral as described in this section.

The type 2 integral with the partial derivative operators is shown here again:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_2(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} \sum_{m=-l}^l \int_0^\infty dr_\text{C} \left( \int d\Omega_\text{C}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) {Y_\text{C}}_{l,m} \right) }[/math]

[math]\displaystyle{ \times |\boldsymbol{r}-\boldsymbol{R}_\text{C}|^n e^{-\zeta_\text{C}|\boldsymbol{r}-\boldsymbol{R}_\text{C}|^2} \left( \int d\Omega_\text{C}~ {Y_\text{C}}_{l,m} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}) \right) }[/math].

Angular integral

Two angular integrals are included in the type 2 integral. Here, the way to compute the left-side one is described, but the right-side one can be computed in the same manner.

The angular integral can be written as below:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|{Y_\text{C}}_{l,m}\right)_{\Omega_\text{C}}(r_\text{C}) =\int d\Omega_\text{C}~ \varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}) {Y_\text{C}}_{l,m} }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \int_0^\pi d\theta \int_{-\pi}^\pi d\phi~ {(r_\text{x}-{R_\text{A}}_\text{x})}^{{n_\text{A}}_\text{x}} {(r_\text{y}-{R_\text{A}}_\text{y})}^{{n_\text{A}}_\text{y}} {(r_\text{z}-{R_\text{A}}_\text{z})}^{{n_\text{A}}_\text{z}} }[/math]

[math]\displaystyle{ \times e^{-\zeta_\text{A}\left|\boldsymbol{r}-\boldsymbol{R}_\text{A}\right|^2}{Y_\text{C}}_{l,m}(\theta,\phi)\sin{\theta} }[/math].

Applying the relation [math]\displaystyle{ \boldsymbol{r}=\boldsymbol{r}_\text{C}+\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|{Y_\text{C}}_{l,m}\right)_{\Omega_\text{C}}(r_\text{C}) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \int_0^\pi d\theta \int_{-\pi}^\pi d\phi~ {({r_\text{C}}_\text{x}+{R_\text{C}}_\text{x}-{R_\text{A}}_\text{x})}^{{n_\text{A}}_\text{x}} {({r_\text{C}}_\text{y}+{R_\text{C}}_\text{y}-{R_\text{A}}_\text{y})}^{{n_\text{A}}_\text{y}} {({r_\text{C}}_\text{z}+{R_\text{C}}_\text{z}-{R_\text{A}}_\text{z})}^{{n_\text{A}}_\text{z}} }[/math]

[math]\displaystyle{ \times e^{-\zeta_\text{A}\left|\boldsymbol{r}_\text{C}+\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|^2}{Y_\text{C}}_{l,m}(\theta,\phi)\sin{\theta} }[/math].

After several deformations, finally the equations shown below can be obtained.

Firstly, in the case of [math]\displaystyle{ \boldsymbol{R}_\text{A} \neq \boldsymbol{R}_\text{C} }[/math]:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|{Y_\text{C}}_{l,m}\right)_{\Omega_\text{C}}(r_\text{C}) }[/math]

[math]\displaystyle{ = 4\pi \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} e^{-\zeta_\text{A}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|^2} \sum_{k_\text{x}=0}^{{n_\text{A}}_\text{x}} \sum_{k_\text{y}=0}^{{n_\text{A}}_\text{y}} \sum_{k_\text{z}=0}^{{n_\text{A}}_\text{z}} \frac{{n_\text{A}}_\text{x}!}{k_\text{x}!({n_\text{A}}_\text{x}-k_\text{x})!} \frac{{n_\text{A}}_\text{y}!}{k_\text{y}!({n_\text{A}}_\text{y}-k_\text{y})!} \frac{{n_\text{A}}_\text{z}!}{k_\text{z}!({n_\text{A}}_\text{z}-k_\text{z})!} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{A}}_\text{x})^{{n_\text{A}}_\text{x}-k_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{A}}_\text{y})^{{n_\text{A}}_\text{y}-k_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{A}}_\text{z})^{{n_\text{A}}_\text{z}-k_\text{z}} }[/math]

[math]\displaystyle{ \times r_\text{C}^{k_\text{x}+k_\text{y}+k_\text{z}} e^{-\zeta_\text{A}{r_\text{C}}^2} \sum_{i=0}^{l+k_\text{x}+k_\text{y}+k_\text{z}} (-1)^i M_i(2\zeta_\text{A}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|r_\text{C}) \sum_{j=-i}^i Y_{i,j}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}}{\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|}\right) \Omega_{l,m,i,j}^\boldsymbol{k} }[/math],

where [math]\displaystyle{ M_i(z) }[/math] is the modified spherical Bessel function of the first kind, [math]\displaystyle{ \boldsymbol{k}=(k_\text{x},k_\text{y},k_\text{z}) }[/math], and

[math]\displaystyle{ \Omega_{l,m,i,j}^\boldsymbol{k} = \sum_{\begin{gathered}m_\text{x},m_\text{y},m_\text{z}\\m_\text{x}+m_\text{y}+m_\text{z}=l\end{gathered}} \sum_{\begin{gathered}j_\text{x},j_\text{y},j_\text{z}\\j_\text{x}+j_\text{y}+j_\text{z}=i\end{gathered}} s_{m_\text{x},m_\text{y},m_\text{z}}^{l,m} s_{j_\text{x},j_\text{y},j_\text{z}}^{i,j} }[/math]

[math]\displaystyle{ \times \begin{cases} \displaystyle 4\pi\frac{ (k_\text{x}+j_\text{x}+m_\text{x}-1)!! (k_\text{y}+j_\text{y}+m_\text{y}-1)!! (k_\text{z}+j_\text{z}+m_\text{z}-1)!! }{ (k_\text{x}+j_\text{x}+m_\text{x}+k_\text{y}+j_\text{y}+m_\text{y}+k_\text{z}+j_\text{z}+m_\text{z}+1)!! } & \left({\begin{aligned} k_\text{x}+j_\text{x}+m_\text{x}\in\text{even}&~\wedge \\ k_\text{y}+j_\text{y}+m_\text{y}\in\text{even}&~\wedge \\ k_\text{z}+j_\text{z}+m_\text{z}\in\text{even}& \end{aligned}}\right) \\ 0 & \text{othrewise} \end{cases} }[/math].

Here, [math]\displaystyle{ s_{m_\text{x},m_\text{y},m_\text{z}}^{l,m} }[/math] are the polynomial coefficients of the Cartesian form of the real spherical harmonics:

[math]\displaystyle{ Y_{l,m}\left(\frac{\boldsymbol{r}}{|\boldsymbol{r}|}\right) = \sum_{\begin{gathered}m_\text{x},m_\text{y},m_\text{z}\\m_\text{x}+m_\text{y}+m_\text{z}=l\end{gathered}} s_{m_\text{x},m_\text{y},m_\text{z}}^{l,m} \left(\frac{r_\text{x}}{r}\right)^{m_\text{x}} \left(\frac{r_\text{y}}{r}\right)^{m_\text{y}} \left(\frac{r_\text{z}}{r}\right)^{m_\text{z}} }[/math].

Secondly, in the case of [math]\displaystyle{ \boldsymbol{R}_\text{A} = \boldsymbol{R}_\text{C} }[/math]:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|{Y_\text{C}}_{l,m}\right)_{\Omega_\text{C}}(r_\text{C}) }[/math]

[math]\displaystyle{ = 2\sqrt{\pi} \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} {r_\text{C}}^{{n_\text{A}}_\text{x}+{n_\text{A}}_\text{y}+{n_\text{A}}_\text{z}} e^{-\zeta_\text{A}{r_\text{C}}^2} \Omega_{l,m}^{\boldsymbol{n}_\text{A}} }[/math],

where [math]\displaystyle{ \boldsymbol{n}_\text{A}=({n_\text{A}}_\text{x},{n_\text{A}}_\text{y},{n_\text{A}}_\text{z}) }[/math], and

[math]\displaystyle{ \Omega_{l,m}^\boldsymbol{n} = \frac{1}{2\sqrt{\pi}} \sum_{\begin{gathered}m_\text{x},m_\text{y},m_\text{z}\\m_\text{x}+m_\text{y}+m_\text{z}=l\end{gathered}} s_{m_\text{x},m_\text{y},m_\text{z}}^{l,m} }[/math]

[math]\displaystyle{ \times \begin{cases} \displaystyle 4\pi\frac{ (n_\text{x}+m_\text{x}-1)!! (n_\text{y}+m_\text{y}-1)!! (n_\text{z}+m_\text{z}-1)!! }{ (n_\text{x}+m_\text{x}+n_\text{y}+m_\text{y}+n_\text{z}+m_\text{z}+1)!! } & \left({\begin{aligned} n_\text{x}+m_\text{x}\in\text{even}&~\wedge \\ n_\text{y}+m_\text{y}\in\text{even}&~\wedge \\ n_\text{z}+m_\text{z}\in\text{even}& \end{aligned}}\right) \\ 0 & \text{othrewise} \end{cases} }[/math].

Radial integral

Applying the angular integral result to the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}\neq\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}\neq\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_2(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = 16\pi^2 \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{x}}\right)^{{l_\text{B}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{y}}\right)^{{l_\text{B}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{z}}\right)^{{l_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} e^{-\zeta_\text{A}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|^2} e^{-\zeta_\text{B}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}\right|^2} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{A}}_\text{x}=0}^{{n_\text{A}}_\text{x}} \sum_{{k_\text{A}}_\text{y}=0}^{{n_\text{A}}_\text{y}} \sum_{{k_\text{A}}_\text{z}=0}^{{n_\text{A}}_\text{z}} \frac{{n_\text{A}}_\text{x}!}{{k_\text{A}}_\text{x}!({n_\text{A}}_\text{x}-{k_\text{A}}_\text{x})!} \frac{{n_\text{A}}_\text{y}!}{{k_\text{A}}_\text{y}!({n_\text{A}}_\text{y}-{k_\text{A}}_\text{y})!} \frac{{n_\text{A}}_\text{z}!}{{k_\text{A}}_\text{z}!({n_\text{A}}_\text{z}-{k_\text{A}}_\text{z})!} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{B}}_\text{x}=0}^{{n_\text{B}}_\text{x}} \sum_{{k_\text{B}}_\text{y}=0}^{{n_\text{B}}_\text{y}} \sum_{{k_\text{B}}_\text{z}=0}^{{n_\text{B}}_\text{z}} \frac{{n_\text{B}}_\text{x}!}{{k_\text{B}}_\text{x}!({n_\text{B}}_\text{x}-{k_\text{B}}_\text{x})!} \frac{{n_\text{B}}_\text{y}!}{{k_\text{B}}_\text{y}!({n_\text{B}}_\text{y}-{k_\text{B}}_\text{y})!} \frac{{n_\text{B}}_\text{z}!}{{k_\text{B}}_\text{z}!({n_\text{B}}_\text{z}-{k_\text{B}}_\text{z})!} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{A}}_\text{x})^{{n_\text{A}}_\text{x}-{k_\text{A}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{A}}_\text{y})^{{n_\text{A}}_\text{y}-{k_\text{A}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{A}}_\text{z})^{{n_\text{A}}_\text{z}-{k_\text{A}}_\text{z}} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{B}}_\text{x})^{{n_\text{B}}_\text{x}-{k_\text{B}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{B}}_\text{y})^{{n_\text{B}}_\text{y}-{k_\text{B}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{B}}_\text{z})^{{n_\text{B}}_\text{z}-{k_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \sum_{i_\text{A}=0}^{l+{k_\text{A}}_\text{x}+{k_\text{A}}_\text{y}+{k_\text{A}}_\text{z}} \sum_{i_\text{B}=0}^{l+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}} (-1)^{i_\text{A}+i_\text{B}} Q_{i_\text{A},i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},(n-2)+{k_\text{A}}_\text{x}+{k_\text{A}}_\text{y}+{k_\text{A}}_\text{z}+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}}(\boldsymbol{R}_\text{A},\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) }[/math]

[math]\displaystyle{ \times \sum_{m=-l}^l \left[\sum_{j_\text{A}=-i_\text{A}}^{i_\text{A}}Y_{i_\text{A},j_\text{A}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}|}\right)\Omega_{l,m,i_\text{A},j_\text{A}}^{\boldsymbol{k}_\text{A}}\right] \left[\sum_{j_\text{B}=-i_\text{B}}^{i_\text{B}}Y_{i_\text{B},j_\text{B}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|}\right)\Omega_{l,m,i_\text{B},j_\text{B}}^{\boldsymbol{k}_\text{B}}\right] }[/math],

where

[math]\displaystyle{ Q_{i_\text{A},i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},k}(\boldsymbol{R}_\text{A},\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) = \int_0^\infty dr~r^2 M_{i_\text{A}}(2\zeta_\text{A}|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}|r) M_{i_\text{B}}(2\zeta_\text{B}|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|r) r^k e^{-(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})r^2} }[/math].

For the details of the computation, see the ECP type 2 radial quadrature.

Applying the angular integral result to the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}=\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}\neq\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_2(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = 8\pi^\frac{3}{2} \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{x}}\right)^{{l_\text{B}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{y}}\right)^{{l_\text{B}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{z}}\right)^{{l_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} e^{-\zeta_\text{B}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}\right|^2} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{B}}_\text{x}=0}^{{n_\text{B}}_\text{x}} \sum_{{k_\text{B}}_\text{y}=0}^{{n_\text{B}}_\text{y}} \sum_{{k_\text{B}}_\text{z}=0}^{{n_\text{B}}_\text{z}} \frac{{n_\text{B}}_\text{x}!}{{k_\text{B}}_\text{x}!({n_\text{B}}_\text{x}-{k_\text{B}}_\text{x})!} \frac{{n_\text{B}}_\text{y}!}{{k_\text{B}}_\text{y}!({n_\text{B}}_\text{y}-{k_\text{B}}_\text{y})!} \frac{{n_\text{B}}_\text{z}!}{{k_\text{B}}_\text{z}!({n_\text{B}}_\text{z}-{k_\text{B}}_\text{z})!} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{B}}_\text{x})^{{n_\text{B}}_\text{x}-{k_\text{B}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{B}}_\text{y})^{{n_\text{B}}_\text{y}-{k_\text{B}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{B}}_\text{z})^{{n_\text{B}}_\text{z}-{k_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \sum_{i_\text{B}=0}^{l+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}} (-1)^{i_\text{B}} Q_{i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},(n-2)+{n_\text{A}}_\text{x}+{n_\text{A}}_\text{y}+{n_\text{A}}_\text{z}+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}}(\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) }[/math]

[math]\displaystyle{ \times \sum_{m=-l}^l \Omega_{l,m}^{\boldsymbol{n}_\text{A}} \left[\sum_{j_\text{B}=-i_\text{B}}^{i_\text{B}}Y_{i_\text{B},j_\text{B}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|}\right)\Omega_{l,m,i_\text{B},j_\text{B}}^{\boldsymbol{k}_\text{B}}\right] }[/math],

where

[math]\displaystyle{ Q_{i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},k}(\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) = \int_0^\infty dr~r^2 M_{i_\text{B}}(2\zeta_\text{B}|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|r) r^k e^{-(\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C})r^2} }[/math].

For the details of the computation, see the ECP type 2 radial quadrature.

Applying the angular integral result to the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}=\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}=\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\hat{u}_2(\boldsymbol{l}_\text{C})\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) }[/math]

[math]\displaystyle{ = 4\pi \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{x}}\right)^{{l_\text{B}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{y}}\right)^{{l_\text{B}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{z}}\right)^{{l_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} Q^{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C},(n-2)+{n_\text{A}}_\text{x}+{n_\text{A}}_\text{y}+{n_\text{A}}_\text{z}+{n_\text{B}}_\text{x}+{n_\text{B}}_\text{y}+{n_\text{B}}_\text{z}} \sum_{m=-l}^l \Omega_{l,m}^{\boldsymbol{n}_\text{A}} \Omega_{l,m}^{\boldsymbol{n}_\text{B}} }[/math],

where

[math]\displaystyle{ Q^{\zeta,k} = \int_0^\infty dr~ r^{(k+2)} e^{-\zeta r^2} = \frac{(k-1)!!}{(2\zeta)^\frac{k+3}{2}} \times \begin{cases} \sqrt{\frac{\pi}{2}} & (k\in\text{even}) \\ 1 & (k\in\text{odd}) \end{cases} }[/math].

Recurrence relations

To describe the recurrence relations, the auxiliary integral is introduced :

[math]\displaystyle{ \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = \left(\frac{\partial}{\partial{R_\text{A}}_\text{x}}\right)^{{l_\text{A}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{y}}\right)^{{l_\text{A}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{A}}_\text{z}}\right)^{{l_\text{A}}_\text{z}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{x}}\right)^{{l_\text{B}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{y}}\right)^{{l_\text{B}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{B}}_\text{z}}\right)^{{l_\text{B}}_\text{z}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{x}}\right)^{{l_\text{C}}_\text{x}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{y}}\right)^{{l_\text{C}}_\text{y}} \left(\frac{\partial}{\partial{R_\text{C}}_\text{z}}\right)^{{l_\text{C}}_\text{z}} }[/math]

[math]\displaystyle{ \times \sum_{m=-l}^l \int_0^\infty dr_\text{C}~ \left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A} \middle| \left(\frac{{r_\text{C}}_\text{x}}{r_\text{C}}\right)^{{h_\text{A}}_\text{x}} \left(\frac{{r_\text{C}}_\text{y}}{r_\text{C}}\right)^{{h_\text{A}}_\text{y}} \left(\frac{{r_\text{C}}_\text{z}}{r_\text{C}}\right)^{{h_\text{A}}_\text{z}} {Y_\text{C}}_{l,m} \right)_{\Omega_\text{C}}(r_\text{C}) }[/math]

[math]\displaystyle{ \times {r_\text{C}}^{n+{h_\text{A}}_\text{x}+{h_\text{A}}_\text{y}+{h_\text{A}}_\text{z}+{h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z}} e^{-\zeta_\text{C}{r_\text{C}}^2} \left( \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B} \middle| \left(\frac{{r_\text{C}}_\text{x}}{r_\text{C}}\right)^{{h_\text{B}}_\text{x}} \left(\frac{{r_\text{C}}_\text{y}}{r_\text{C}}\right)^{{h_\text{B}}_\text{y}} \left(\frac{{r_\text{C}}_\text{z}}{r_\text{C}}\right)^{{h_\text{B}}_\text{z}} {Y_\text{C}}_{l,m} \right)_{\Omega_\text{C}}(r_\text{C}) }[/math],

where [math]\displaystyle{ \boldsymbol{h}=(h_\text{x},h_\text{y},h_\text{z}) }[/math]. Apparently,

[math]\displaystyle{ \left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B} \right) = \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{0}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{0}_\text{B} \right)\right) }[/math].

Vertical recurrence relation on derivative order #1

[math]\displaystyle{ \left(\left( \boldsymbol{l}_\text{A}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = 2\zeta_\text{A}({R_\text{C}}_\mu-{R_\text{A}}_\mu) \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) +2\zeta_\text{A} \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A}+\boldsymbol{1}_\mu \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ -2\zeta_\text{A}{l_\text{A}}_\mu \left(\left( \boldsymbol{l}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) -{n_\text{A}}_\mu \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ +2\zeta_\text{A}{l_\text{C}}_\mu \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

Vertical recurrence relation on derivative order #2

[math]\displaystyle{ \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}+\boldsymbol{1}_\mu) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = -2\left(\zeta_\text{A}({R_\text{C}}_\mu-{R_\text{A}}_\mu+\zeta_\text{B}({R_\text{C}}_\mu-{R_\text{B}}_\mu)\right) \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ -2\zeta_\text{A} \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A}+\boldsymbol{1}_\mu \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) -2\zeta_\text{B} \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B}+\boldsymbol{1}_\mu \right)\right) }[/math]

[math]\displaystyle{ +{n_\text{A}}_\mu \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}-\boldsymbol{1}_\mu,\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) +{n_\text{B}}_\mu \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}-\boldsymbol{1}_\mu,\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ -2(\zeta_\text{A}+\zeta_\text{B}){l_\text{C}}_\mu \left(\left( \boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{l}_\text{C}-\boldsymbol{1}_\mu) \middle| \boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

Initial integral

In the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}\neq\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}\neq\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\left( \boldsymbol{0}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{0}_\text{C}) \middle| \boldsymbol{0}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = 16\pi^2 e^{-\zeta_\text{A}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}\right|^2} e^{-\zeta_\text{B}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}\right|^2} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{A}}_\text{x}=0}^{{n_\text{A}}_\text{x}} \sum_{{k_\text{A}}_\text{y}=0}^{{n_\text{A}}_\text{y}} \sum_{{k_\text{A}}_\text{z}=0}^{{n_\text{A}}_\text{z}} \frac{{n_\text{A}}_\text{x}!}{{k_\text{A}}_\text{x}!({n_\text{A}}_\text{x}-{k_\text{A}}_\text{x})!} \frac{{n_\text{A}}_\text{y}!}{{k_\text{A}}_\text{y}!({n_\text{A}}_\text{y}-{k_\text{A}}_\text{y})!} \frac{{n_\text{A}}_\text{z}!}{{k_\text{A}}_\text{z}!({n_\text{A}}_\text{z}-{k_\text{A}}_\text{z})!} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{B}}_\text{x}=0}^{{n_\text{B}}_\text{x}} \sum_{{k_\text{B}}_\text{y}=0}^{{n_\text{B}}_\text{y}} \sum_{{k_\text{B}}_\text{z}=0}^{{n_\text{B}}_\text{z}} \frac{{n_\text{B}}_\text{x}!}{{k_\text{B}}_\text{x}!({n_\text{B}}_\text{x}-{k_\text{B}}_\text{x})!} \frac{{n_\text{B}}_\text{y}!}{{k_\text{B}}_\text{y}!({n_\text{B}}_\text{y}-{k_\text{B}}_\text{y})!} \frac{{n_\text{B}}_\text{z}!}{{k_\text{B}}_\text{z}!({n_\text{B}}_\text{z}-{k_\text{B}}_\text{z})!} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{A}}_\text{x})^{{n_\text{A}}_\text{x}-{k_\text{A}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{A}}_\text{y})^{{n_\text{A}}_\text{y}-{k_\text{A}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{A}}_\text{z})^{{n_\text{A}}_\text{z}-{k_\text{A}}_\text{z}} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{B}}_\text{x})^{{n_\text{B}}_\text{x}-{k_\text{B}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{B}}_\text{y})^{{n_\text{B}}_\text{y}-{k_\text{B}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{B}}_\text{z})^{{n_\text{B}}_\text{z}-{k_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \sum_{i_\text{A}=0}^{ \begin{aligned} l+{k_\text{A}}_\text{x}+{k_\text{A}}_\text{y}+{k_\text{A}}_\text{z} \\ +{h_\text{A}}_\text{x}+{h_\text{A}}_\text{y}+{h_\text{A}}_\text{z} \end{aligned} } \sum_{i_\text{B}=0}^{ \begin{aligned} l+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z} \\ +{h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z} \end{aligned} } (-1)^{i_\text{A}+i_\text{B}} Q_{i_\text{A},i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},(n-2)+{k_\text{A}}_\text{x}+{k_\text{A}}_\text{y}+{k_\text{A}}_\text{z}+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}+{h_\text{A}}_\text{x}+{h_\text{A}}_\text{y}+{h_\text{A}}_\text{z}+{h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z}}(\boldsymbol{R}_\text{A},\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) }[/math]

[math]\displaystyle{ \times \sum_{m=-l}^l \left[\sum_{j_\text{A}=-i_\text{A}}^{i_\text{A}}Y_{i_\text{A},j_\text{A}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{A}|}\right)\Omega_{l,m,i_\text{A},j_\text{A}}^{\boldsymbol{k}_\text{A}+\boldsymbol{h}_\text{A}}\right] \left[\sum_{j_\text{B}=-i_\text{B}}^{i_\text{B}}Y_{i_\text{B},j_\text{B}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|}\right)\Omega_{l,m,i_\text{B},j_\text{B}}^{\boldsymbol{k}_\text{B}+\boldsymbol{h}_\text{B}}\right] }[/math].

In the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}=\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}\neq\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\left( \boldsymbol{0}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{0}_\text{C}) \middle| \boldsymbol{0}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = 8\pi^\frac{3}{2} e^{-\zeta_\text{B}\left|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}\right|^2} }[/math]

[math]\displaystyle{ \times \sum_{{k_\text{B}}_\text{x}=0}^{{n_\text{B}}_\text{x}} \sum_{{k_\text{B}}_\text{y}=0}^{{n_\text{B}}_\text{y}} \sum_{{k_\text{B}}_\text{z}=0}^{{n_\text{B}}_\text{z}} \frac{{n_\text{B}}_\text{x}!}{{k_\text{B}}_\text{x}!({n_\text{B}}_\text{x}-{k_\text{B}}_\text{x})!} \frac{{n_\text{B}}_\text{y}!}{{k_\text{B}}_\text{y}!({n_\text{B}}_\text{y}-{k_\text{B}}_\text{y})!} \frac{{n_\text{B}}_\text{z}!}{{k_\text{B}}_\text{z}!({n_\text{B}}_\text{z}-{k_\text{B}}_\text{z})!} }[/math]

[math]\displaystyle{ \times ({R_\text{C}}_\text{x}-{R_\text{B}}_\text{x})^{{n_\text{B}}_\text{x}-{k_\text{B}}_\text{x}} ({R_\text{C}}_\text{y}-{R_\text{B}}_\text{y})^{{n_\text{B}}_\text{y}-{k_\text{B}}_\text{y}} ({R_\text{C}}_\text{z}-{R_\text{B}}_\text{z})^{{n_\text{B}}_\text{z}-{k_\text{B}}_\text{z}} }[/math]

[math]\displaystyle{ \times \sum_{i_\text{B}=0}^{ \begin{aligned} l+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z} \\ {h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z} \end{aligned} } (-1)^{i_\text{B}} Q_{i_\text{B}}^{\zeta_\text{A},\zeta_\text{B},\zeta_\text{C},(n-2)+{n_\text{A}}_\text{x}+{n_\text{A}}_\text{y}+{n_\text{A}}_\text{z}+{k_\text{B}}_\text{x}+{k_\text{B}}_\text{y}+{k_\text{B}}_\text{z}+{h_\text{A}}_\text{x}+{h_\text{A}}_\text{y}+{h_\text{A}}_\text{z}+{h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z}}(\boldsymbol{R}_\text{B},\boldsymbol{R}_\text{C}) }[/math]

[math]\displaystyle{ \times \sum_{m=-l}^l \Omega_{l,m}^{\boldsymbol{n}_\text{A}+\boldsymbol{h}_\text{A}} \left[\sum_{j_\text{B}=-i_\text{B}}^{i_\text{B}}Y_{i_\text{B},j_\text{B}}\left(\frac{\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}}{|\boldsymbol{R}_\text{C}-\boldsymbol{R}_\text{B}|}\right)\Omega_{l,m,i_\text{B},j_\text{B}}^{\boldsymbol{k}_\text{B}+\boldsymbol{h}_\text{B}}\right] }[/math].

In the case of [math]\displaystyle{ \boldsymbol{R}_\text{A}=\boldsymbol{R}_\text{C}~\wedge~\boldsymbol{R}_\text{B}=\boldsymbol{R}_\text{C} }[/math],

[math]\displaystyle{ \left(\left( \boldsymbol{0}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{h}_\text{A} \middle| \hat{u}_2(\boldsymbol{0}_\text{C}) \middle| \boldsymbol{0}_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{h}_\text{B} \right)\right) }[/math]

[math]\displaystyle{ = 4\pi Q^{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C},(n-2)+{n_\text{A}}_\text{x}+{n_\text{A}}_\text{y}+{n_\text{A}}_\text{z}+{n_\text{B}}_\text{x}+{n_\text{B}}_\text{y}+{n_\text{B}}_\text{z}+{h_\text{A}}_\text{x}+{h_\text{A}}_\text{y}+{h_\text{A}}_\text{z}+{h_\text{B}}_\text{x}+{h_\text{B}}_\text{y}+{h_\text{B}}_\text{z}} \sum_{m=-l}^l \Omega_{l,m}^{\boldsymbol{n}_\text{A}+\boldsymbol{h}_\text{A}} \Omega_{l,m}^{\boldsymbol{n}_\text{B}+\boldsymbol{h}_\text{B}} }[/math].