Dyall Hamiltonian

In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:

${\displaystyle {\hat {\mathcal {H}}}^{D}={\hat {\mathcal {H}}}_{i}^{D}+{\hat {\mathcal {H}}}_{v}^{D}+C}$

${\displaystyle {\hat {\mathcal {H}}}_{i}^{D}=\sum _{i}^{\rm {core}}\epsilon _{i}E_{ii}+\sum _{r}^{\rm {virt}}\epsilon _{r}E_{rr}}$

${\displaystyle {\hat {\mathcal {H}}}_{v}^{D}=\sum _{ab}^{\rm {act}}h_{ab}^{\rm {eff}}E_{ab}+{\frac {1}{2}}\sum _{abcd}^{\rm {act}}\left\langle ab\left.\right|cd\right\rangle \left(E_{ac}E_{bd}-\delta _{bc}E_{ad}\right)}$

${\displaystyle C=2\sum _{i}^{\rm {core}}h_{ii}+\sum _{ij}^{\rm {core}}\left(2\left\langle ij\left.\right|ij\right\rangle -\left\langle ij\left.\right|ji\right\rangle \right)-2\sum _{i}^{\rm {core}}\epsilon _{i}}$

${\displaystyle h_{ab}^{\rm {eff}}=h_{ab}+\sum _{j}\left(2\left\langle aj\left.\right|bj\right\rangle -\left\langle aj\left.\right|jb\right\rangle \right)}$

where labels ${\displaystyle i,j,\ldots }$, ${\displaystyle a,b,\ldots }$, ${\displaystyle r,s,\ldots }$ denote core, active and virtual orbitals (see Complete active space) respectively, $\epsilon _{i}$ and $\epsilon_r$ are the orbital energies of the involved orbitals, and ${\displaystyle E_{mn}}$ operators are the spin-traced operators ${\displaystyle a_{m\alpha }^{\dagger }a_{n\alpha }+a_{m\beta }^{\dagger }a_{n\beta }}$. These operators commute with $S^{2}$ and $S_{z}$, therefore the application of these operators on a spin-pure function produces again a spin-pure function.

The Dyall Hamiltonian behaves like the true Hamiltonian inside the CAS space, having the same eigenvalues and eigenvectors of the true Hamiltonian projected onto the CAS space.