One-Electron Integrals

OpenQuantum evaluates one-electron integrals using the Obara–Saika recurrence relations, which provide an efficient and numerically stable way to compute integrals for arbitrary angular momentum.

Overlap Integrals

$$S_{\mu \nu }=\int {\chi }_{\mu }(\mathbf{r}){\chi }_{\nu }(\mathbf{r})d\mathbf{r}$$

The overlap matrix $\mathbf{S}$ is used throughout the calculation for orthogonalization, density matrix formation, and energy evaluation.

Kinetic Energy Integrals

$$T_{\mu \nu }=\int {\chi }_{\mu }(\mathbf{r})\left(-\frac{1}{2}{\nabla }^2\right){\chi }_{\nu }(\mathbf{r})d\mathbf{r}$$

The kinetic energy operator in atomic units is $-\frac{1}{2}{\nabla }^2$. These integrals contribute to the core Hamiltonian $H_{\mu \nu }=T_{\mu \nu }+V_{\mu \nu }$.

Nuclear Attraction Integrals

$$V_{\mu \nu }=\int {\chi }_{\mu }(\mathbf{r})\sum _A\left(-\frac{Z_A}{|\mathbf{r}-{\mathbf{R}}_A|}\right){\chi }_{\nu }(\mathbf{r})d\mathbf{r}$$

The nuclear attraction potential is a sum over all nuclei $A$ with charge $Z_A$ at position ${\mathbf{R}}_A$. These integrals are computed efficiently using the Obara–Saika scheme with Rys quadrature for the $1/r$ operator.

Obara–Saika Recurrence

The Obara–Saika method generates higher angular momentum integrals from lower ones using recurrence relations. For a Gaussian product ${\chi }_a{\chi }_b$, the recurrence steps up angular momentum in each Cartesian direction:

$$\begin{array}{rl}(i+1)S_{i+1,j,k}& =R_{A,i}{S}_{i,j,k}+i\alpha S_{i-1,j,k}\\ & \text{etc. for }j,k\end{array}$$

where $R_A$ is the vector from the Gaussian center to nucleus $A$, and $\alpha$ is the Gaussian exponent.

This recurrence is numerically stable and avoids the catastrophic cancellation issues of earlier methods.