Chemistry:Dyall Hamiltonian

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

[math]\displaystyle{ \hat{H}^{\rm D} = \hat{H}^{\rm D}_i + \hat{H}^{\rm D}_v + C }[/math]

[math]\displaystyle{ \hat{H}^{\rm D}_i = \sum_{i}^{\rm core} \varepsilon_i E_{ii} + \sum_r^{\rm virt} \varepsilon_r E_{rr} }[/math]

[math]\displaystyle{ \hat{H}^{\rm 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]\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} \varepsilon_i }[/math]

[math]\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) }[/math]

where labels [math]\displaystyle{ i,j,\ldots }[/math], [math]\displaystyle{ a,b,\ldots }[/math], [math]\displaystyle{ r,s,\ldots }[/math] denote core, active and virtual orbitals (see Complete active space) respectively, [math]\displaystyle{ \varepsilon_i }[/math] and [math]\displaystyle{ \varepsilon_r }[/math] are the orbital energies of the involved orbitals, and [math]\displaystyle{ E_{mn} }[/math] operators are the spin-traced operators [math]\displaystyle{ a^{\dagger}_{m\alpha}a_{n\alpha} + a^{\dagger}_{m\beta}a_{n\beta} }[/math]. These operators commute with [math]\displaystyle{ S^2 }[/math] and [math]\displaystyle{ 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.