Obara–Saika scheme

Obara–Saika scheme is the way to compute various integrals based on recurrence relations regarding angular momentum quantum numbers and derivative orders.

Three-center overlap 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 three-center overlap integral is of the form:

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

The integration [math]\displaystyle{ \int d\boldsymbol{r} }[/math] means [math]\displaystyle{ \int_{-\infty}^\infty dr_\text{x} \int_{-\infty}^\infty dr_\text{y} \int_{-\infty}^\infty dr_\text{z} }[/math].

The three-center overlap integral has the several recurrence relations shown in the sections below. To describe them, the following special integer vectors are introduced:

[math]\displaystyle{ \boldsymbol{0}=(0,0,0) }[/math],

[math]\displaystyle{ \boldsymbol{1}_\mu=(\delta_{\mu\text{x}},\delta_{\mu\text{y}},\delta_{\mu\text{z}}) }[/math],

where [math]\displaystyle{ \delta_{ij} }[/math] is the Kronecker delta, and [math]\displaystyle{ \mu }[/math] is x, y, or z.

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{n}_\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{n}_\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{n}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)=0 }[/math]

Vertical 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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\text{C}\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right) +\frac{\zeta_\text{A}}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}}{n_\text{C}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C}-\boldsymbol{1}_\mu\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{n}_\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{C}}_\mu\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C}-\boldsymbol{1}_\mu\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]

Overlap integral

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

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

Vertical recurrence relation on partial derivative order

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

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

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

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

Initial integral

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

Kinetic energy integral

The kinetic energy integral can be derived from the overlap integrals:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|-\frac{1}{2}\nabla^2\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)=\frac{1}{2}\sum_{\nu=\text{x},\text{y},\text{z}}\frac{\partial}{\partial {R_\text{A}}_\nu}\frac{\partial}{\partial {R_\text{B}}_\nu}\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\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|-\frac{1}{2}\nabla^2\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)+\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|-\frac{1}{2}\nabla^2\middle|\boldsymbol{l}_\text{B}+\boldsymbol{1}_\mu,\boldsymbol{n}_\text{B}\right)=0 }[/math]

Vertical recurrence relation on partial derivative order

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

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

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

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

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

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

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

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

Initial integral

[math]\displaystyle{ \left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|-\frac{1}{2}\nabla^2\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right)=\frac{\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}\left(3-2\frac{\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}\left|\boldsymbol{R}_\text{B}-\boldsymbol{R}_\text{A}\right|^2\right)\left(\boldsymbol{0}_\text{A},\boldsymbol{0}_\text{A}\middle|\boldsymbol{0}_\text{B},\boldsymbol{0}_\text{B}\right) }[/math]

Nuclear attraction integral

The nuclear attraction integral can be derived from the three-center overlap integral:

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

[math]\displaystyle{ -2\frac{\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}{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-2\frac{{\zeta_\text{A}}^2}{\zeta_\text{A}+\zeta_\text{B}}{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{A}+\zeta_\text{B}}{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}}{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{ +2\frac{\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}{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-2\frac{\zeta_\text{A}\zeta_\text{B}}{\zeta_\text{A}+\zeta_\text{B}}{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}}{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}}{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{ +2\zeta_\text{A}{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{ =2\left(\zeta_\text{A}({R_\text{A}}_\mu-{R_\text{C}}_\mu)+\zeta_\text{B}({R_\text{B}}_\mu-{R_\text{C}}_\mu)\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)}_{m+1} }[/math]

[math]\displaystyle{ +2\zeta_\text{A}{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}+{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{ +2\zeta_\text{B}{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}+{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{ -2(\zeta_\text{A}+\zeta_\text{B}){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}}{\zeta_\text{A}+\zeta_\text{B}}({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+\left({R_\text{C}}_\mu-\frac{\zeta_\text{A}{R_\text{A}}_\mu+\zeta_\text{B}{R_\text{B}}_\mu}{\zeta_\text{A}+\zeta_\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)}_{m+1} }[/math]

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

Electron repulsion integral

The electron repulsion integral can be derived from the nested three-center overlap integrals:

[math]\displaystyle{ \left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\middle|\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C},\boldsymbol{l}_\text{D},\boldsymbol{n}_\text{D}\right)=\frac{2}{\sqrt\pi}\int_0^\infty dv~{\left(\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C}\middle|\left.{\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{0}_2,\boldsymbol{0}_2\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)}_{\boldsymbol{r}_1}\right|_{\zeta_2=v^2}\middle|\boldsymbol{l}_\text{D},\boldsymbol{n}_\text{D}\right)}_{\boldsymbol{r}_2} }[/math],

where the subscript 1 denotes the first electron, and the subscript 2 denotes the second electron.

Auxiliary integral

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

[math]\displaystyle{ {\left(\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\middle|\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C},\boldsymbol{l}_\text{D},\boldsymbol{n}_\text{D}\right)\right)}_m=\frac{2}{\sqrt\pi}\int_0^\infty dv~\left(\frac{v^2}{\xi+v^2}\right)^m{\left(\boldsymbol{l}_\text{C},\boldsymbol{n}_\text{C}\middle|\left.{\left(\boldsymbol{l}_\text{A},\boldsymbol{n}_\text{A}\middle|\boldsymbol{0}_2,\boldsymbol{0}_2\middle|\boldsymbol{l}_\text{B},\boldsymbol{n}_\text{B}\right)}_{\boldsymbol{r}_1}\right|_{\zeta_2=v^2}\middle|\boldsymbol{l}_\text{D},\boldsymbol{n}_\text{D}\right)}_{\boldsymbol{r}_2} }[/math],

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

[math]\displaystyle{ \xi=\frac{(\zeta_\text{A}+\zeta_\text{B})(\zeta_\text{C}+\zeta_\text{D})}{\zeta_\text{A}+\zeta_\text{B}+\zeta_\text{C}+\zeta_\text{D}} }[/math].

Apparently,

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

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

Vertical recurrence relation on partial derivative order

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Initial integral

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

Multipole moment integral

The multipole moment integral can be derived from the three-center overlap integral:

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

Vertical recurrence relation on partial derivative order

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

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

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

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

Vertical recurrence relation on multipole moment index

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

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

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

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