In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:[1]
![{\displaystyle {\hat {H}}^{\rm {D}}={\hat {H}}_{i}^{\rm {D}}+{\hat {H}}_{v}^{\rm {D}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c35e2cbeeda6975aa965d531788bd635568b76c)
![{\displaystyle {\hat {H}}_{i}^{\rm {D}}=\sum _{i}^{\rm {core}}\varepsilon _{i}E_{ii}+\sum _{r}^{\rm {virt}}\varepsilon _{r}E_{rr}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd87d05f3ddc0f8521d9eeb1068b4c6ce6b4275)
![{\displaystyle {\hat {H}}_{v}^{\rm {D}}=\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c682d55830f1664f5c066fdc57793887d52cbb)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/170d2aca93fbf624db21f26309c2194a08b5b5f7)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15a135163376f2c02c5a6aa2bad860f38d93e056)
where labels
,
,
denote core, active and virtual orbitals (see Complete active space) respectively,
and
are the orbital energies of the involved orbitals, and
operators are the spin-traced operators
. These operators commute with
and
, 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.
References[edit]
- ^ Dyall, Kenneth G. (March 22, 1995). "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. 102 (12): 4909–4918. Bibcode:1995JChPh.102.4909D. doi:10.1063/1.469539.