Dyall Hamiltonian

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


 * $$\hat{\mathcal{H}}^D = \hat{\mathcal{H}}^D_i + \hat{\mathcal{H}}^D_v + C$$
 * $$\hat{\mathcal{H}}^D_i = \sum_{i}^{\rm core} \epsilon_i E_{ii} + \sum_r^{\rm virt} \epsilon_r E_{rr} $$
 * $$\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, $$\epsilon_i$$ and $$\epsilon_r$$ are the orbital energies of the involved orbitals, and $$E_{mn}$$ operators are the spin-traced operators $$a^{\dagger}_{m\alpha}a_{n\alpha} + a^{\dagger}_{m\beta}a_{n\beta}$$. 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.

Hamiltoniano di Dyall