Gaussian type orbital
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Cartesian GTO[edit | edit source]
The unnormalized Cartesian GTO (Gaussian type orbital) can be written as
- [math]\displaystyle{ \varphi(\boldsymbol{r};\zeta,\boldsymbol{n},\boldsymbol{R})=(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{n}=(n_\text{x},n_\text{y},n_\text{z}) }[/math]: the nonnegative integers, and
- [math]\displaystyle{ \boldsymbol{R}=(R_\text{x},R_\text{y},R_\text{z}) }[/math]: the coordinates of the center.
The angular momentum quantum number [math]\displaystyle{ l }[/math] of the orbital is [math]\displaystyle{ n_\text{x}+n_\text{y}+n_\text{z} }[/math]. The number of Cartesian GTOs for a given [math]\displaystyle{ l }[/math] is [math]\displaystyle{ (l+1)(l+2)/2 }[/math].
| [math]\displaystyle{ l }[/math] | [math]\displaystyle{ n_\text{x} }[/math] | [math]\displaystyle{ n_\text{y} }[/math] | [math]\displaystyle{ n_\text{z} }[/math] | Orbital |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | [math]\displaystyle{ s }[/math] |
| 1 | 1 | 0 | 0 | [math]\displaystyle{ p_x }[/math] |
| 0 | 1 | 0 | [math]\displaystyle{ p_y }[/math] | |
| 0 | 0 | 1 | [math]\displaystyle{ p_z }[/math] | |
| 2 | 2 | 0 | 0 | [math]\displaystyle{ d_{x^2} }[/math] |
| 1 | 1 | 0 | [math]\displaystyle{ d_{xy} }[/math] | |
| 1 | 0 | 1 | [math]\displaystyle{ d_{xz} }[/math] | |
| 0 | 2 | 0 | [math]\displaystyle{ d_{y^2} }[/math] | |
| 0 | 1 | 1 | [math]\displaystyle{ d_{yz} }[/math] | |
| 0 | 0 | 2 | [math]\displaystyle{ d_{z^2} }[/math] | |
| 3 | 3 | 0 | 0 | [math]\displaystyle{ f_{x^3} }[/math] |
| 2 | 1 | 0 | [math]\displaystyle{ f_{x^2y} }[/math] | |
| 2 | 0 | 1 | [math]\displaystyle{ f_{x^2z} }[/math] | |
| 1 | 2 | 0 | [math]\displaystyle{ f_{xy^2} }[/math] | |
| 1 | 1 | 1 | [math]\displaystyle{ f_{xyz} }[/math] | |
| 1 | 0 | 2 | [math]\displaystyle{ f_{xz^2} }[/math] | |
| 0 | 3 | 0 | [math]\displaystyle{ f_{y^3} }[/math] | |
| 0 | 2 | 1 | [math]\displaystyle{ f_{y^2z} }[/math] | |
| 0 | 1 | 2 | [math]\displaystyle{ f_{yz^2} }[/math] | |
| 0 | 0 | 3 | [math]\displaystyle{ f_{z^3} }[/math] | |
| [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] |
The normalized Cartesian GTO is
- [math]\displaystyle{ \Phi(\boldsymbol{r};\zeta,\boldsymbol{n},\boldsymbol{R})=N(\zeta,\boldsymbol{n})\varphi(\boldsymbol{r};\zeta,\boldsymbol{n},\boldsymbol{R}) }[/math],
where
- [math]\displaystyle{ N(\zeta,\boldsymbol{n})=\left(\int_{-\infty}^\infty dr_\text{x} \int_{-\infty}^\infty dr_\text{y} \int_{-\infty}^\infty dr_\text{z}~\left|\varphi(\boldsymbol{r};\zeta,\boldsymbol{n},\boldsymbol{R})\right|^2\right)^{-\frac{1}{2}}=\sqrt{\left(\frac{2\zeta}{\pi}\right)^{\frac{3}{2}}\frac{(4\zeta)^{n_\text{x}+n_\text{y}+n_\text{z}}}{(2n_\text{x}-1)!!(2n_\text{y}-1)!!(2n_\text{z}-1)!!}} }[/math].
Spherical GTO[edit | edit source]
The unnormalized spherical GTO can be written as
- [math]\displaystyle{ \bar{\varphi}_{l,m}(\boldsymbol{r};\zeta,\boldsymbol{R})=Y_{l,m} |\boldsymbol{r}|^l e^{-\zeta|\boldsymbol{r}-\boldsymbol{R}|^2} }[/math],
where [math]\displaystyle{ Y_{l,m} }[/math] is a real spherical harmonics. The number of spherical GTOs for a given [math]\displaystyle{ l }[/math] is [math]\displaystyle{ 2l+1 }[/math]. The unnormalized spherical GTO can be expressed as a linear combination of the unnormalized Cartesian GTOs.
| [math]\displaystyle{ l }[/math] | [math]\displaystyle{ m }[/math] | Orbital | Expression |
|---|---|---|---|
| 0 | 0 | [math]\displaystyle{ s }[/math] | [math]\displaystyle{ \bar{\varphi}_{0,0}=\frac{1}{2}\sqrt{\frac{1}{\pi}}\varphi_{\boldsymbol{n}=(0,0,0)} }[/math] |
| 1 | -1 | [math]\displaystyle{ p_y }[/math] | [math]\displaystyle{ \bar{\varphi}_{1,-1}=\frac{1}{2}\sqrt{\frac{3}{\pi}}\varphi_{\boldsymbol{n}=(0,1,0)} }[/math] |
| 0 | [math]\displaystyle{ p_z }[/math] | [math]\displaystyle{ \bar{\varphi}_{1,0}=\frac{1}{2}\sqrt{\frac{3}{\pi}}\varphi_{\boldsymbol{n}=(0,0,1)} }[/math] | |
| 1 | [math]\displaystyle{ p_x }[/math] | [math]\displaystyle{ \bar{\varphi}_{1,1}=\frac{1}{2}\sqrt{\frac{3}{\pi}}\varphi_{\boldsymbol{n}=(1,0,0)} }[/math] | |
| 2 | -2 | [math]\displaystyle{ d_{xy} }[/math] | [math]\displaystyle{ \bar{\varphi}_{2,-2}=\frac{1}{2}\sqrt{\frac{15}{\pi}}\varphi_{\boldsymbol{n}=(1,1,0)} }[/math] |
| -1 | [math]\displaystyle{ d_{yz} }[/math] | [math]\displaystyle{ \bar{\varphi}_{2,-1}=\frac{1}{2}\sqrt{\frac{15}{\pi}}\varphi_{\boldsymbol{n}=(0,1,1)} }[/math] | |
| 0 | [math]\displaystyle{ d_{z^2} }[/math] | [math]\displaystyle{ \bar{\varphi}_{2,0}=\frac{1}{4}\sqrt{\frac{5}{\pi}}\left(2\varphi_{\boldsymbol{n}=(0,0,2)}-\varphi_{\boldsymbol{n}=(2,0,0)}-\varphi_{\boldsymbol{n}=(0,2,0)}\right) }[/math] | |
| 1 | [math]\displaystyle{ d_{xz} }[/math] | [math]\displaystyle{ \bar{\varphi}_{2,1}=\frac{1}{2}\sqrt{\frac{15}{\pi}}\varphi_{\boldsymbol{n}=(1,0,1)} }[/math] | |
| 2 | [math]\displaystyle{ d_{x^2-y^2} }[/math] | [math]\displaystyle{ \bar{\varphi}_{2,2}=\frac{1}{4}\sqrt{\frac{15}{\pi}}\left(\varphi_{\boldsymbol{n}=(2,0,0)}-\varphi_{\boldsymbol{n}=(0,2,0)}\right) }[/math] | |
| 3 | -3 | [math]\displaystyle{ f_{y(3x^2-y^2)} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,-3}=\frac{1}{4}\sqrt{\frac{35}{2\pi}}\left(3\varphi_{\boldsymbol{n}=(2,1,0)}-\varphi_{\boldsymbol{n}=(0,3,0)}\right) }[/math] |
| -2 | [math]\displaystyle{ f_{xyz} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,-2}=\frac{1}{2}\sqrt{\frac{105}{\pi}}\varphi_{\boldsymbol{n}=(1,1,1)} }[/math] | |
| -1 | [math]\displaystyle{ f_{yz^2} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,-1}=\frac{1}{4}\sqrt{\frac{21}{2\pi}}\left(4\varphi_{\boldsymbol{n}=(0,1,2)}-\varphi_{\boldsymbol{n}=(2,1,0)}-\varphi_{\boldsymbol{n}=(0,3,0)}\right) }[/math] | |
| 0 | [math]\displaystyle{ f_{z^3} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,0}=\frac{1}{4}\sqrt{\frac{7}{\pi}}\left(2\varphi_{\boldsymbol{n}=(0,0,3)}-3\varphi_{\boldsymbol{n}=(2,0,1)}-3\varphi_{\boldsymbol{n}=(0,2,1)}\right) }[/math] | |
| 1 | [math]\displaystyle{ f_{xz^2} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,1}=\frac{1}{4}\sqrt{\frac{21}{2\pi}}\left(4\varphi_{\boldsymbol{n}=(1,0,2)}-\varphi_{\boldsymbol{n}=(3,0,0)}-\varphi_{\boldsymbol{n}=(1,2,0)}\right) }[/math] | |
| 2 | [math]\displaystyle{ f_{z(x^2-y^2)} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,2}=\frac{1}{4}\sqrt{\frac{105}{\pi}}\left(\varphi_{\boldsymbol{n}=(2,0,1)}-\varphi_{\boldsymbol{n}=(0,2,1)}\right) }[/math] | |
| 3 | [math]\displaystyle{ f_{x(x^2-3y^2)} }[/math] | [math]\displaystyle{ \bar{\varphi}_{3,3}=\frac{1}{4}\sqrt{\frac{35}{2\pi}}\left(\varphi_{\boldsymbol{n}=(3,0,0)}-3\varphi_{\boldsymbol{n}=(1,2,0)}\right) }[/math] | |
| [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] | [math]\displaystyle{ \dots }[/math] |
The normalized spherical GTO is
- [math]\displaystyle{ \bar{\Phi}_{l,m}(\boldsymbol{r};\zeta,\boldsymbol{R})=\bar{N}(\zeta,l)\bar{\varphi}_{l,m}(\boldsymbol{r};\zeta,\boldsymbol{R}) }[/math],
where
- [math]\displaystyle{ \bar{N}(\zeta,l)=\sqrt{\left(\frac{2\zeta}{\pi}\right)^{\frac{3}{2}}\frac{(4\zeta)^l}{(2l-1)!!}} }[/math].
Contraction[edit | edit source]
To use GTOs as a basis set in practical quantum chemistry computation, they are almost always contracted using normalized GTOs:
- [math]\displaystyle{ \psi(\boldsymbol{r};\boldsymbol{n},\boldsymbol{R})=\sum_i c_i \Phi(\boldsymbol{r};\zeta_i,\boldsymbol{n},\boldsymbol{R}) }[/math] when Cartesian GTOs are contracted, or
- [math]\displaystyle{ \bar{\psi}_{l,m}(\boldsymbol{r};\boldsymbol{R})=\sum_i c_i \bar{\Phi}_{l,m}(\boldsymbol{r};\zeta_i,\boldsymbol{R}) }[/math] when spherical GTOs are contracted.
Here, [math]\displaystyle{ c_i }[/math] is the contraction coefficient for the [math]\displaystyle{ i }[/math]-th primitive GTO.
Before quantum chemistry computation, the contracted basis function [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \bar{\psi} }[/math] are normalized by scaling the contraction coefficients.