Ab initio molecular dynamics: Theory and Implementation

from a set of one–particle spin orbitals {ψ i } required to be mutually orthonormal〈ψ i |ψ j 〉 = δ ij . The corresponding variational minimum of the total electronicenergy H e defined in Eq. (2)E HF [{ψ i }] = ∑ ∫dr ψi [− ⋆ (r) 1 ]2 ∇2 + V ext (r) ψ i (r)i+ 1 ∑∫ ∫dr dr ′ ψi ⋆ (r)ψj ⋆ (r ′ 1)2|r − r ′ | ψ i(r)ψ j (r ′ )ij+ 1 ∑∫ ∫dr dr ′ ψi ⋆ (r)ψj ⋆ (r ′ )2ij1|r − r ′ | ψ j(r)ψ i (r ′ ) (89)yields the lowest energy and the “best” wavefunction within a one–determinantansatz; the external Coulomb potential V ext was already defined in Eq. (78). Carryingout the constraint minimization within this ansatz (see Eq. (36) in Sect. 2.3for a sketch) leads to⎧⎫⎨⎩ −1 2 ∇2 + V ext (r) + ∑ J j (r) − ∑ ⎬K j (r)⎭ ψ i(r) = ∑ Λ ij ψ j (r) (90)jjj{− 1 }2 ∇2 + V HF (r) ψ i (r) = ∑ Λ ij ψ j (r) (91)jHeHF ψ i (r) = ∑ jΛ ij ψ j (r) (92)the Hartree–Fock integro–differential equations. In analogy to the Kohn–Shamequations Eqs. (81)–(83) these are effective one–particle equations that involve aneffective one–particle Hamiltonian HeHF , the (Hartree–) Fock operator. The set ofcanonical orbitalsH HFe ψ i = ɛ i ψ i (93)is obtained similarly to Eq. (85). The Coulomb operator[∫]J j (r) ψ i (r) = dr ′ ψj ⋆ 1(r′ )|r − r ′ | ψ j(r ′ ) ψ i (r) (94)and the exchange operator[∫]K j (r) ψ i (r) = dr ′ ψj ⋆ (r ′ 1)|r − r ′ | ψ i(r ′ ) ψ j (r) (95)are most easily defined via their action on a particular orbital ψ i . It is foundthat upon acting on orbital ψ i (r) the exchange operator for the j–th state “exchanges”ψ j (r ′ ) → ψ i (r ′ ) in the kernel as well as replaces ψ i (r) → ψ j (r) in itsargument, compare to the Coulomb operator. Thus, K is a non–local operator asits action on a function ψ i at point r in space requires the evaluation and thus theknowledge of that function throughout all space by virtue of ∫ dr ′ ψ i (r ′ ) . . . therequired integration. In this sense the exchange operator does not possess a simpleclassical interpretation like the Coulomb operator C, which is the counterpart of37