List of integrals

Frequently used integrals are listed below.

The GTO is denoted by [math]\displaystyle{ \varphi }[/math]. The GTO whose center coordinates are [math]\displaystyle{ \boldsymbol{R}_\text{A}=({R_\text{A}}_\text{x},{R_\text{A}}_\text{y},{R_\text{A}}_\text{z}) }[/math] is denoted by [math]\displaystyle{ \varphi_\text{A} }[/math]. The complex conjugate is specified by [math]\displaystyle{ * }[/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], where [math]\displaystyle{ \boldsymbol{r}=(r_\text{x},r_\text{y},r_\text{z}) }[/math] are the coordinates of the electron.

Overlap integral

[math]\displaystyle{ \left(\varphi_\text{A}\middle|\varphi_\text{B}\right)=\int d\boldsymbol{r}~{\varphi_\text{A}}^*(\boldsymbol{r})\varphi_\text{B}(\boldsymbol{r}) }[/math].

Kinetic energy integral

[math]\displaystyle{ \left(\varphi_\text{A}\middle|-\frac{1}{2}\nabla^2\middle|\varphi_\text{B}\right)=-\frac{1}{2}\int d\boldsymbol{r}~{\varphi_\text{A}}^*(\boldsymbol{r})\nabla^2\varphi_\text{B}(\boldsymbol{r}) }[/math].

This can be expressed also as:

[math]\displaystyle{ \left(\varphi_\text{A}\middle|-\frac{1}{2}\nabla^2\middle|\varphi_\text{B}\right)=\frac{1}{2}\int d\boldsymbol{r}\sum_{\nu=\text{x},\text{y},\text{z}}\frac{\partial}{\partial {R_\text{A}}_\nu}{\varphi_\text{A}}^*(\boldsymbol{r})\frac{\partial}{\partial {R_\text{B}}_\nu}\varphi_\text{B}(\boldsymbol{r}) }[/math].

Nuclear attraction integral

[math]\displaystyle{ \left(\varphi_\text{A}\middle|\frac{1}{|\boldsymbol{r}-\boldsymbol{R}_\text{C}|}\middle|\varphi_\text{B}\right)=\int d\boldsymbol{r}~{\varphi_\text{A}}^*(\boldsymbol{r})\frac{1}{|\boldsymbol{r}-\boldsymbol{R}_\text{C}|}\varphi_\text{B}(\boldsymbol{r}) }[/math].

Electron repulsion integral

[math]\displaystyle{ \left(\varphi_\text{A}\varphi_\text{B}\middle|\varphi_\text{C}\varphi_\text{D}\right)=\int d\boldsymbol{r}_1\int d\boldsymbol{r}_2~{\varphi_\text{A}}^*(\boldsymbol{r}_1)\varphi_\text{B}(\boldsymbol{r}_1)\frac{1}{|\boldsymbol{r}_1-\boldsymbol{r}_2|}{\varphi_\text{C}}^*(\boldsymbol{r}_2)\varphi_\text{D}(\boldsymbol{r}_2) }[/math].

Multipole moment integral

[math]\displaystyle{ \left(\varphi_\text{A}\middle|{r_\text{x}}^{m_\text{x}} r_\text{y}^{m_\text{y}} r_\text{z}^{m_\text{z}}\middle|\varphi_\text{B}\right)=\int d\boldsymbol{r}~{\varphi_\text{A}}^*(\boldsymbol{r}){r_\text{x}}^{m_\text{x}} r_\text{y}^{m_\text{y}} r_\text{z}^{m_\text{z}}\varphi_\text{B}(\boldsymbol{r}) }[/math],

where [math]\displaystyle{ ({m_\text{x}},{m_\text{y}},{m_\text{z}}) }[/math] are nonnegative integers.