Mcmurchie Davidson Scheme

Overlap

Let us frist consider the simple overlap integral

\[ S_{ab} = \langle G_a|G_b \rangle \]

it can be factorized in the three Cartesian directons

\[ S_{ab} = S_{ij}S_{kl}S_{mn} \]

The Obara-Saika recurrence relations for the Cartesian overlap integral over one direction is:

\[\begin{split} \begin{aligned} S_{i+1,j} &= X_{PA}S_{ij} + \frac{1}{2p}(iS_{i-1,j}+jS_{i,j-1})\\ S_{i,j+1} &= X_{PB}S_{ij} + \frac{1}{2p}(iS_{i-1,j}+jS_{i,j-1})\\ \end{aligned} \end{split}\]

With the boundary condtion

\[ S_{0,0} = \sqrt(\frac{\pi}{p}) exp(-\mu X_{AB}^2) \]

Kinetic

Consider the integral of electrons kinetic energy

\[\begin{split} \begin{aligned} T_{a,b} &= \langle G_a \vert -\sum^N_{i=1}\frac{\hbar^2}{2m_i}\boldsymbol{\nabla}_i^2 \vert G_b \rangle\\ T_{a,b} &= T_{ij}S_{kl}S_{mn} + S_{ij}T_{kl}S_{mn} + S_{ij}S_{kl}T_{mn} \end{aligned} \end{split}\]

The Obara-Saika recurrence relations for the Cartesian kinetic integrals over one direction is:

\[\begin{split} \begin{aligned} T_{i+1,j} &= X_{PA}T_{ij} + \frac{1}{2p}(iT_{i-1,j}+jT_{i,j-1}) + \frac{b}{p}(2aS_{i+1,j}-iS_{i-1,j})\\ T_{i,j+1} &= X_{PB}T_{ij} + \frac{1}{2p}(iT_{i-1,j}+jT_{i,j-1})+ \frac{a}{p}(2bS_{i,j+1}-jS_{i,j-1})\\ \end{aligned} \end{split}\]

With the boundary condtion

\[ T_{0,0} = \left[a-2a^2(X_{PA}^2 + \frac{1}{2p})\right] S_{00} \]

Nuclear Attraction

\[\begin{split} \begin{aligned} V_{a,b} &=\langle G_a \vert -\sum^N_{i=1}\sum^M_{\alpha=1} \frac{Z_\alpha e^2}{\textbf{r}_{i\alpha}} \vert G_b \rangle \\ V_{a,b} &= V_{ijklmn}^{000} = \Theta_{ijklmn}^{0} \end{aligned} \end{split}\]

The Obara-Saika recurrence relations for the Cartesian nuclear attraction integrals:

\[\begin{split} \begin{aligned} \Theta_{i+1,j,k,l,m,n}^{N} &= X_{PA}\Theta_{ijkln}^{N} +\frac{1}{2p}(i\Theta_{i-1,j,k,l,m,n}^{N} + j\Theta_{i,j-1,k,l,m,n}^{N})\\ &-X_{PC}\Theta_{ijklmn}^{N+1} -\frac{1}{2p}(i\Theta_{i-1,j,k,l,m,n}^{N+1} + j\Theta_{i,j-1,k,l,m,n}^{N+1})\\ \Theta_{i,j+1,k,l,m,n}^{N} &= X_{PB}\Theta_{ijkln}^{N} +\frac{1}{2p}(i\Theta_{i-1,j,k,l,m,n}^{N} + j\Theta_{i,j-1,k,l,m,n}^{N})\\ &-X_{PC}\Theta_{ijklmn}^{N+1} -\frac{1}{2p}(i\Theta_{i-1,j,k,l,m,n}^{N+1} + j\Theta_{i,j-1,k,l,m,n}^{N+1})\\ \end{aligned} \end{split}\]

With the boundary condtion

\[ \Theta_{000000}^{N} = \frac{2\pi}{p}(-2p)^{-N}K_{ab}^{xyz}R_{000}^{N} = \frac{2\pi}{p} K_{ab}^{xyz}F_N(pR_{PC}^2) \]

Electron Repulsion

\[ V_{abcd} = \langle G_a G_b\vert \sum^N_{i=1}\sum^N_{j>i} \frac{e^2}{\textbf{r}_{ij}}\vert G_c G_d \rangle \]

The source and target integrals:

\[\begin{split} \begin{aligned} \Theta_{0000;0000;0000}^{N} &= \frac{2\pi^{2.5}}{pq\sqrt{p+q}} K_{ab}^{xyz}K_{cd}^{xyz}F_N(\alpha R_{PQ}^2)\\ \Theta_{i_xj_xk_xl_x;i_yj_yk_yl_y;i_zj_zk_zl_z}^{0} &= g_{i_xj_xk_xl_x;i_yj_yk_yl_y;i_zj_zk_zl_z} \end{aligned} \end{split}\]

The Obara-Saika two electron recurrence relation

\[\begin{split} \begin{aligned} \Theta_{i+1,j,k,l}^{N} &= X_{PA}\Theta_{ijkl}^{N} -\frac{\alpha}{p}X_{PQ}\Theta_{i,j,k,l}^{N+1} +\frac{i}{2p}\left(\Theta_{i-1,j,k,l}^{N}-\frac{\alpha}{p}\Theta_{i-1,j,k,l}^{N+1}\right)\\ &+\frac{j}{2p}\left(\Theta_{i,j-1,k,l}^{N}-\frac{\alpha}{p}\Theta_{i,j-1,k,l}^{N+1}\right) -\frac{k}{2(p+q)}\Theta_{i,j,k-1,l}^{N+1} -\frac{l}{2(p+q)}\Theta_{i,j,k,l-1}^{N+1} \end{aligned} \end{split}\]

Using the horizontal recurrence relation, a similar relation may be written down for increments in j, replaceing \(X_{PA}\) with \(X_{PB}\).

\[\begin{split} \begin{aligned} \Theta_{i,j,k+1,l}^{N} &= X_{QC}\Theta_{ijkl}^{N} -\frac{\alpha}{q}X_{PQ}\Theta_{i,j,k,l}^{N+1} +\frac{k}{2q}\left(\Theta_{i,j,k-1,l}^{N}-\frac{\alpha}{q}\Theta_{i,j,k-1,l}^{N+1}\right)\\ &+\frac{l}{2q}\left(\Theta_{i,j,k,l-1}^{N}-\frac{\alpha}{q}\Theta_{i,j,k,l-1}^{N+1}\right) -\frac{i}{2(p+q)}\Theta_{i-1,j,k,l}^{N+1} -\frac{j}{2(p+q)}\Theta_{i,j-1,k,l}^{N+1} \end{aligned} \end{split}\]