In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:
H ^ D = H ^ i D + H ^ v D + C H ^ i D = ∑ i c o r e ϵ i E i i + ∑ r v i r t ϵ r E r r H ^ v D = ∑ a b a c t h a b e f f E a b + 1 2 ∑ 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 ∑ i c o r e h i i + ∑ i j c o r e ( 2 ⟨ i j | i j ⟩ − ⟨ i j | j i ⟩ ) − 2 ∑ i c o r e ϵ i h a b e f f = h a b + ∑ j ( 2 ⟨ a j | b j ⟩ − ⟨ a j | j b ⟩ ) where labels i , j , … , a , b , … , r , s , … 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.
