The Electronic Hamiltonian

The electronic Hamiltonian in atomic units is:

$$\hat{H}=\sum _i\hat{h}(i)+\sum _{i<j}\frac{1}{r_{ij}}$$

where:

• $\hat{h}(i)=-\frac{1}{2}{\nabla }_i^2-{\sum }_A\frac{Z_A}{r_{iA}}$ is the one-electron operator (kinetic + nuclear attraction)
• $\frac{1}{r_{ij}}$ is the electron–electron repulsion

In the Born–Oppenheimer approximation, the nuclei are fixed and the electronic Hamiltonian acts only on electronic coordinates. The nuclear repulsion energy $E_{\text{nuc}}={\sum }_{A<B}\frac{Z_A{Z}_B}{R_{AB}}$ is a constant for a given geometry and is added to the electronic energy at the end.

Atomic Units

OpenQuantum uses atomic units throughout (Hartree for energy, Bohr for distance, electron mass for mass, $\hslash =1$). Conversion factors:

One-Electron Operator

The one-electron operator consists of kinetic energy and nuclear attraction:

$$\hat{h}(i)=\hat{T}(i)+\hat{V}(i)=-\frac{1}{2}{\nabla }_i^2-\sum _A\frac{Z_A}{r_{iA}}$$

In the LCAO basis, the matrix elements are:

$$h_{\mu \nu }=T_{\mu \nu }+V_{\mu \nu }$$

where $T_{\mu \nu }$ are kinetic energy integrals and $V_{\mu \nu }$ are nuclear attraction integrals, both computed via the Obara–Saika recurrence relations.