Gaussian-type orbitals
In orbital-based quantum chemistry, we usually expand molecular orbitals (MOs) as linear combinations of atomic orbitals (AOs). These AOs are then called basis functions, and are linear combinations (contractions) of Cartesian Gaussian-type orbitals (GTOs). The individual Cartesian GTOs are named primitives. (Szabo 1989) (T. Helgaker, Jørgensen, and Olsen 2000)
The functional form of such a Cartesian GTO is \[\begin{equation} \require{physics} G_{i,j,k}(\vb{r}; \alpha, \vb{K}) = x_K^i y_K^j z_K^k \exp( -\alpha \thinspace \norm{\vb{r}_K}^2 ) \thinspace , \end{equation}\] with \(i,j,k\) the exponent in the respective Cartesian direction, \(\alpha\) the Gaussian exponent, and \(\vb{R}_K = (K_x, K_y, K_z)\) the center of the GTO. The position of the electron is therefore only used relative to the origin of the GTO, i.e. \[\begin{equation} \vb{r}_K = \vb{r} - \vb{K} \thinspace . \end{equation}\] The value \[\begin{equation} l = i + j + k \end{equation}\] is called the angular momentum of the GTO.
From the form of the Cartesian GTO, we can see that it naturally factors out in its three Cartesian components: \[\begin{equation} G_{ijk}(\vb{r}; \alpha, \vb{K}) = G_i(x; \alpha, K_x) \thinspace G_j(y; \alpha, K_y) \thinspace G_k(z; \alpha, K_z) \thinspace , \end{equation}\] where each individual Cartesian component can be calculated similarly to the \(x\)-component: \[\begin{equation} G_i(x; \alpha, K_x) = x_K^i \exp(-\alpha x_K^2) \thinspace . \end{equation}\]
The Cartesian GTOs admit the following recurrence and derivative relations: \[\begin{equation} x_K G_i(x; \alpha, K_x) = G_{i+1}(x; \alpha, K_x) \end{equation}\] and \[\begin{equation} \pdv{ G_i(x; \alpha, K_x) }{K_x} = - \pdv{ G_i(x; \alpha, K_x) }{x} = 2 \alpha G_{i+1}(x; \alpha, K_x) - i G_{i-1}(x; \alpha, K_x) \thinspace . \end{equation}\] The second derivative of a Cartesian GTO is then given by \[\begin{equation} \pdv[2]{ G_i(x; \alpha, K_x) }{x} = 4 \alpha^2 G_{i+2}(x; \alpha, K_x) - 2 \alpha (2i + 1) G_i(x; \alpha, K_x) + i (i-1) G_{i-2}(x; \alpha, K_x) \thinspace . \end{equation}\]
A Gaussian overlap distribution is the product of two (unnormalized) Cartesian Gaussians. For the \(x\)-component, we may write: \[\begin{align} \Omega_{ij}(x) &= G_i(x; \alpha, K_x) G_j(x; \beta, L_x) \\ &= x_K^i x_L^j C_x^{KL} \exp(- p x_P^2) \thinspace , \end{align}\] where we have used the Gaussian product rule and have introduced the total exponent \(p\): \[\begin{equation} p = \alpha + \beta \thinspace , \end{equation}\] the center of mass \(\vb{P}\): \[\begin{equation} \vb{P} = \frac{ \alpha \vb{K} + \beta \vb{L} }{ \alpha + \beta } \thinspace , \end{equation}\] the reduced exponent \(\mu\): \[\begin{equation} \frac{1}{\mu} = \frac{1}{\alpha} + \frac{1}{\beta} \qquad \qquad \mu = \frac{\alpha \beta}{\alpha + \beta} \thinspace , \end{equation}\] the relative difference \(\boldsymbol{\Delta}_{KL}\) \[\begin{equation} \boldsymbol{\Delta}_{KL} = \vb{K} - \vb{L} \end{equation}\] and the exponential prefactor \(C_x^{KL}\): \[\begin{equation} C_x^{KL} = \exp(- \mu \Delta_{KL, x}^2 ) \thinspace . \end{equation}\]
Normalization
Since the self-overlap of for example the \(x\)-Cartesian component is given by \[\begin{align} \braket{G_i} &= \int_{-\infty}^{+\infty} \dd{x} (x - X)^{2i} \exp(-2\alpha (x-X)^2) \\ &= \frac{(2i-1)!!}{(4\alpha)^i} \sqrt{\frac{\pi}{2\alpha}} \thinspace , \end{align}\] a normalization factor \(N\) can be calculated such that \[\begin{equation} \int \dd{\vb{r}} G_{i,j,k}^2(\vb{r}; \alpha, \vb{R}) = 1 \thinspace , \end{equation}\] where the normalization factor of each component is given by: \[\begin{align} N_{\text{comp}}(\alpha, i) &= \int_{-\infty}^{+\infty} \dd{t} t^{2i} \exp(-2\alpha t^2) \\ &= \qty( \frac{2 \alpha}{\pi} )^{1/4} \qty( \frac{(4 \alpha)^i}{(2i - 1)!!} )^{1/2} \thinspace , \end{align}\] using well-known integrals over Gaussian functions. \(!!\) denotes the double factorial, i.e. \[\begin{equation} n!! = n \cdot (n-2) \cdot (n-4) \cdots \thinspace . \end{equation}\]
The gradient of a Cartesian GTO is straightforward to calculate, and we will provide the derivative in the \(x\)-direction here: \[\begin{equation} \pdv{ % \chi_{i,j,k}(\vb{r}; \alpha, \vb{R}) % }{x} % = i \thinspace \chi_{i-1,j,k}(\vb{r}; \alpha, \vb{R}) % - 2 \alpha \thinspace \chi_{i+1,j,k}(\vb{r}; \alpha, \vb{R}) % \thinspace . \end{equation}\] The other components of the gradient are to be calculated analogously.
A contracted GTO (of length \(L\)) is a linear combination of GTOs with the same Cartesian exponents: \[\begin{equation} \chi^{\text{cGTO}}_{ijk} ( \vb{r}; \vb{d}, \boldsymbol{\alpha}, \vb{R} ) = N_{\text{cGTO}} \sum_a^L \chi_{ijk}(\vb{r}; \alpha_a, \vb{R}) d_a \thinspace , \end{equation}\] in which \(\vb{d}\) collects the contraction coefficients \(\set{d_a}\). The total normalization factor can be calculated as \[\begin{equation} N_{\text{cGTO}} % = \qty( % \sum_{ab}^L % N_a d_a N_b d_b % \prod_{x}^{i,j,k} N_{\text{comp}} \qty( % \frac{ \alpha_a + \alpha_b }{2}, x % ) )^{-1/2} % \thinspace . \end{equation}\]