Underactuation

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

${\widehat{\mathcal{\mathscr{H}}}}^D={\widehat{\mathcal{\mathscr{H}}}}_i^D+{\widehat{\mathcal{\mathscr{H}}}}_v^D+C$

${\widehat{\mathcal{\mathscr{H}}}}_i^D=\sum _i^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{ϵ}_i{E}_{ii}+\sum _r^{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}}{ϵ}_r{E}_{rr}$

${\widehat{\mathcal{\mathscr{H}}}}_v^D=\sum _{ab}^{\mathrm{a}\mathrm{c}\mathrm{t}}{h}_{ab}^{\mathrm{e}\mathrm{f}\mathrm{f}}{E}_{ab}+\frac{1}{2}\sum _{abcd}^{\mathrm{a}\mathrm{c}\mathrm{t}}⟨ab|cd⟩(E_{ac}{E}_{bd}-{\delta }_{bc}{E}_{ad})$

$C=2\sum _i^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{h}_{ii}+\sum _{ij}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}(2⟨ij|ij⟩-⟨ij|ji⟩)-2\sum _i^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}{ϵ}_i$

$h_{ab}^{\mathrm{e}\mathrm{f}\mathrm{f}}=h_{ab}+\sum _j(2⟨aj|bj⟩-⟨aj|jb⟩)$

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

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