Dyall Hamiltonian

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

\hat{\mathcal{H}}^D = \hat{\mathcal{H}}^D_i + \hat{\mathcal{H}}^D_v + C
\hat{\mathcal{H}}^D_i = \sum_{i}^{\rm core} \epsilon_i E_{ii} + \sum_r^{\rm virt} \epsilon_r E_{rr}
\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)
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
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 i,j,\ldots, a,b,\ldots, r,s,\ldots denote core, active and virtual orbitals (see Complete active space) respectively, εi and εr are the orbital energies of the involved orbitals, and Emn operators are the spin-traced operators a^{\dagger}_{m\alpha}a_{n\alpha} + a^{\dagger}_{m\beta}a_{n\beta}. These operators commute with S2 and Sz, 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.

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