Dyall Hamiltonian

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

$\hat{\mathcal{H}}^D_v = \sum_{ab}^{\rm act} h_{ab}^{\rm eff} E_{ab} + \frac{1}{2} \sum_{abcd}^{\rm act} \left\langle ab \left.\right| cd \right\rangle \left(E_{ac} E_{bd} - \delta_{bc} E_{ad} \right)$

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

where labels, denote core, active and virtual orbitals (see Complete active space) respectively, and are the orbital energies of the involved orbitals, and operators are the spin-traced operators . These operators commute with and, 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.