Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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The goal of this section is to derive classical <strong>molecular</strong> <strong>dynamics</strong> 12,270,217starting from Schrödinger’s wave equation <strong>and</strong> following the elegant route ofTully 650,651 . To this end, the nuclear <strong>and</strong> electronic contributions to the totalwavefunction Φ({r i }, {R I }; t), which depends on both the nuclear <strong>and</strong> 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 <strong>and</strong> electronic wavefunctions are separately normalized to unityat every instant of time, i.e. 〈χ; t|χ; t〉 = 1 <strong>and</strong> 〈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, . . . <strong>and</strong> I = 1, . . . variables {r i } <strong>and</strong> {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 <strong>dynamics</strong>. Note also thatthis product ansatz (excluding the phase factor) differs from the Born–Oppenheimeransatz 340,350 for separating the fast <strong>and</strong> 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| <strong>and</strong> 〈χ| <strong>and</strong> 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 <strong>and</strong> 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 <strong>and</strong> electronic wavefunctions, respectively).Thus, the single–determinant ansatz Eq. (3) produces, as already anticipated, amean–field description of the coupled nuclear–electronic quantum <strong>dynamics</strong>. Thisis the price to pay for the simplest possible separation of electronic <strong>and</strong> nuclearvariables.6