In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:
- <math>\hat{\mathcal{H}}^D = \hat{\mathcal{H}}^D_i + \hat{\mathcal{H}}^D_v + C</math>
- <math>\hat{\mathcal{H}}^D_i = \sum_{i}^{\rm core} \epsilon_i E_{ii} + \sum_r^{\rm virt} \epsilon_r E_{rr} </math>
- <math>\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)</math>
- <math>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</math>
- <math>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)</math> where labels <math>i,j,\ldots</math>, <math>a,b,\ldots</math>, <math>r,s,\ldots</math> denote core, active and virtual orbitals (see Complete active space) respectively, <math>\epsilon_i</math> and <math>\epsilon_r</math> are the orbital energies of the involved orbitals, and <math>E_{mn}</math> operators are the spin-traced operators <math>a^{\dagger}_{m\alpha}a_{n\alpha} + a^{\dagger}_{m\beta}a_{n\beta}</math>. These operators commute with <math>S^2</math> and <math>S_z</math>, 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.
