Dyall Hamiltonian Explained

In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:^[1]

\hat{H} ^{\rm} = \hat{H} ^{\rm D} _{i} + \hat{H} ^{\rm D} _{v} + C \hat{H} ^{\rm D} _{i} = \sum ^{\rm core} _{i} \varepsilon _{i} E _{ii} + \sum ^{\rm virt} _{r} \varepsilon _{r} E _{rr} \hat{H} ^{\rm D} _{v} = \sum ^{\rm act} _{ab} h ^{\rm eff} _{ab} E _{ab} + (1)/(2) \sum ^{\rm act} _{abcd} \left\langle ab \left.\right| cd \right\rangle \left(E _{ac} E _{bd} - \delta _{bc} E _{ad} \right) C = 2 \sum ^{\rm core} _{i} h _{ii} + \sum ^{\rm core} _{ij} \left( 2 \left\langle ij \left.\right| ij\right\rangle - \left \langle ij \left.\right| ji\right\rangle \right) - 2 \sum ^{\rm core} _{i} \varepsilon _{i} h ^{\rm eff} _{ab} = 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, \varepsilon _{i} and \varepsilon _{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.

Notes and References

1. Dyall. Kenneth G.. The choice of a zeroth‐order Hamiltonian for second‐order perturbation theory with a complete active space self‐consistent‐field reference function. The Journal of Chemical Physics. March 22, 1995. 102. 12. 4909–4918. 10.1063/1.469539. 1995JChPh.102.4909D.