Ab initio molecular dynamics: Theory and Implementation

The goal of this section is to derive classical molecular dynamics 12,270,217starting from Schrödinger’s wave equation and following the elegant route ofTully 650,651 . To this end, the nuclear and electronic contributions to the totalwavefunction Φ({r i }, {R I }; t), which depends on both the nuclear and electroniccoordinates, have to be separated. The simplest possible form is a product ansatz[ ∫ i t]Φ({r i }, {R I }; t) ≈ Ψ({r i }; t) χ({R I }; t) exp dt ′ Ẽ e (t ′ ) , (3)t 0where the nuclear and electronic wavefunctions are separately normalized to unityat every instant of time, i.e. 〈χ; t|χ; t〉 = 1 and 〈Ψ; t|Ψ; t〉 = 1, respectively. Inaddition, a convenient phase factor∫Ẽ e = drdR Ψ ⋆ ({r i }; t) χ ⋆ ({R I }; t) H e Ψ({r i }; t) χ({R I }; t) (4)was introduced at this stage such that the final equations will look nice; ∫ drdRrefers to the integration over all i = 1, . . . and I = 1, . . . variables {r i } and {R I },respectively. It is mentioned in passing that this approximation is called a one–determinant or single–configuration ansatz for the total wavefunction, which at theend must lead to a mean–field description of the coupled dynamics. Note also thatthis product ansatz (excluding the phase factor) differs from the Born–Oppenheimeransatz 340,350 for separating the fast and slow variablesΦ BO ({r i }, {R I }; t) =∞∑˜Ψ k ({r i }, {R I })˜χ k ({R I }; t) (5)k=0even in its one–determinant limit, where only a single electronic state k (evaluatedfor the nuclear configuration {R I }) is included in the expansion.Inserting the separation ansatz Eq. (3) into Eqs. (1)–(2) yields (after multiplyingfrom the left by 〈Ψ| and 〈χ| and imposing energy conservation d 〈H〉 /dt ≡ 0) thefollowing relationsi ∂Ψ∂t = − ∑ ii ∂χ∂t = − ∑ I2{∫∇ 22miΨ +e2{∫∇ 22MIχ +I}dR χ ⋆ ({R I }; t)V n−e ({r i }, {R I })χ({R I }; t) Ψ (6)}dr Ψ ⋆ ({r i }; t)H e ({r i }, {R I })Ψ({r i }; t) χ . (7)This set of coupled equations defines the basis of the time–dependent self–consistentfield (TDSCF) method introduced as early as 1930 by Dirac 162 , see also Ref. 158 .Both electrons and nuclei move quantum–mechanically in time–dependent effectivepotentials (or self–consistently obtained average fields) obtained from appropriateaverages (quantum mechanical expectation values 〈. . . 〉) over the other class ofdegrees of freedom (by using the nuclear and electronic wavefunctions, respectively).Thus, the single–determinant ansatz Eq. (3) produces, as already anticipated, amean–field description of the coupled nuclear–electronic quantum dynamics. Thisis the price to pay for the simplest possible separation of electronic and nuclearvariables.6