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# STO-nG basis sets

## STO-nG basis sets

STO-nG basis sets are the minimal basis sets, where 'n' represents the number of primitive Gaussian functions comprising a single basis set. For minimal basis sets, the core and valence orbitals are represented by same number primitive Gaussian functions $\mathbf \phi_i$. For example, an STO-3G basis set for the 1s orbital of H atom is a linear combination of 3 primitive Gaussian functions. It is easy to calculate the energy of an electron in the 1s orbital of H atom represented by STO-nG basis sets. In the following sections, the structure of the STO-nG minimal basis sets are explained with H atom as an example.

### STO-1G basis set

$\mathbf \psi(1s_H)= \psi_{STO-1G}=c_1\phi_1$, where $\mathbf c_1 = 1$ and $\mathbf \phi_1 = \left (\frac{2\alpha_1}{\pi} \right ) ^{0.75}e^{-\alpha_1 r^2}$. The optimum value of $\mathbf \alpha_1$ is the one which gives the minimum value for the Energy of the 1s electron of H atom. The exponent $\mathbf \alpha_1$ for the STO-1G basis set can be manually derived by equating the derivative of the energy with respect to the exponent to zero.
Thus $\mathbf \alpha_1 = \frac{8 Z^2}{9 \pi} = 0.28294212$ and for the value $\mathbf \alpha_1 = 0.28294212$, the energy of the 1s electron of H atom can be calculated as $\mathbf -0.42441318$ hartree. The expression for the energy of the 1s electron of H atom is a function only of $\mathbf c_1$, $\mathbf \alpha_1$ and other fundamental constants such as $\mathbf \pi$. For convenience, the basis set details can be represented as follows

 STO-1G $\mathbf \alpha$ $\mathbf c$ 0.2829421200D+00 1.0000000000D+00

### STO-2G basis set

In general an STO-nG basis set is a linear combination of 'n' primitive Gaussian functions. The STO-nG basis sets are usually represented by the exponents and the corresponding coefficients. Thus an STO-2G [Ref. 1] basis set which is a linear combination of 2 primitive Gaussian functions can be represented as follows.

 STO-2G $\mathbf \alpha$ $\mathbf c$ 0.1309756377D+01 0.4301284983D+00 0.2331359749D+00 0.6789135305D+00

### Accuracy

The exact energy of the 1s electron of H atom is -0.5 hartree. Following table illustrates the increase in accuracy as the number of primitive Gaussian functions increases in the basis set.

 Basis set Energy [hartree] STO-1G -0.424413182 STO-2G -0.454397402 STO-3G -0.466581850 STO-4G -0.469806464 STO-5G -0.470742918 STO-6G -0.471039054

### References :

[1] http://gnode2.pnl.gov/bse/portal