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

{\hat{ℋ}}^{D} = {\hat{ℋ}}_{i}^{D} + {\hat{ℋ}}_{v}^{D} + C

{\hat{ℋ}}_{i}^{D} = \sum_{i}^{c o r e} ϵ_{i} E_{i i} + \sum_{r}^{v i r t} ϵ_{r} E_{r r}

{\hat{ℋ}}_{v}^{D} = \sum_{a b}^{a c t} h_{a b}^{e f f} E_{a b} + \frac{1}{2} \sum_{a b c d}^{a c t} ⟨ a b | c d ⟩ (E_{a c} E_{b d} - δ_{b c} E_{a d})

C = 2 \sum_{i}^{c o r e} h_{i i} + \sum_{i j}^{c o r e} (2 ⟨ i j | i j ⟩ - ⟨ i j | j i ⟩) - 2 \sum_{i}^{c o r e} ϵ_{i}

h_{a b}^{e f f} = h_{a b} + \sum_{j} (2 ⟨ a j | b j ⟩ - ⟨ a j | j b ⟩)

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 $E_{m n}$ operators are the spin-traced operators $a_{m α}^{†} a_{n α} + a_{m β}^{†} a_{n β}$ . 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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Underactuation

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