:Dyall Hamiltonian

{{refimprove|date=December 2009}}

In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:{{cite journal|last1=Dyall|first1=Kenneth G.|title=The choice of a zeroth‐order Hamiltonian for second‐order perturbation theory with a complete active space self‐consistent‐field reference function|journal=The Journal of Chemical Physics|date=March 22, 1995|volume=102|issue=12|pages=4909–4918|doi=10.1063/1.469539|bibcode=1995JChPh.102.4909D}}

:$\backslash hat\{H\}^\{\backslash rm\; D\}\; =\; \backslash hat\{H\}^\{\backslash rm\; D\}\_i\; +\; \backslash hat\{H\}^\{\backslash rm\; D\}\_v\; +\; C$

:$\backslash hat\{H\}^\{\backslash rm\; D\}\_i\; =\; \backslash sum\_\{i\}^\{\backslash rm\; core\}\; \backslash varepsilon\_i\; E\_\{ii\}\; +\; \backslash sum\_r^\{\backslash rm\; virt\}\; \backslash varepsilon\_r\; E\_\{rr\}$

:$\backslash hat\{H\}^\{\backslash rm\; D\}\_v\; =\; \backslash sum\_\{ab\}^\{\backslash rm\; act\}\; h\_\{ab\}^\{\backslash rm\; eff\}\; E\_\{ab\}\; +$

\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)

:$C\; =\; 2\; \backslash sum\_\{i\}^\{\backslash rm\; core\}\; h\_\{ii\}\; +\; \backslash sum\_\{ij\}^\{\backslash rm\; core\}\; \backslash left(\; 2\; \backslash left\backslash langle\; ij\; \backslash left.\backslash right|\; ij\backslash right\backslash rangle\; -\; \backslash left\; \backslash langle\; ij\; \backslash left.\backslash right|\; ji\backslash right\backslash rangle\; \backslash right)\; -\; 2\; \backslash sum\_\{i\}^\{\backslash rm\; core\}\; \backslash varepsilon\_i$

:$h\_\{ab\}^\{\backslash rm\; eff\}\; =\; h\_\{ab\}\; +\; \backslash sum\_j\; \backslash left(\; 2\; \backslash left\backslash langle\; aj\; \backslash left.\backslash right|\; bj\; \backslash right\backslash rangle\; -$

\left\langle aj \left.\right| jb \right\rangle \right)

where labels $i,j,\backslash ldots$, $a,b,\backslash ldots$, $r,s,\backslash ldots$ denote core, active and virtual orbitals (see Complete active space) respectively, $\backslash varepsilon\_i$ and $\backslash varepsilon\_r$ are the orbital energies of the involved orbitals, and $E\_\{mn\}$ operators are the spin-traced operators $a^\{\backslash dagger\}\_\{m\backslash alpha\}a\_\{n\backslash alpha\}\; +\; a^\{\backslash dagger\}\_\{m\backslash beta\}a\_\{n\backslash beta\}$. 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.