Ab initio molecular dynamics: Theory and Implementation

〈ψ i |ψ j 〉 = δ ij . The corresponding constraint minimization of the total energy withrespect to the orbitalsmin {〈Ψ 0 |H e |Ψ 0 〉} ∣(35){ψ i}can be cast into Lagrange’s formalism∣{〈ψi|ψ j 〉=δ ij }L = − 〈Ψ 0 |H e | Ψ 0 〉 + ∑ i,jΛ ij (〈ψ i |ψ j 〉 − δ ij ) (36)where Λ ij are the associated Lagrangian multipliers. Unconstrained variation ofthis Lagrangian with respect to the orbitalsδLδψ ⋆ ileads to the well–known Hartree–Fock equations!= 0 (37)H HFe ψ i = ∑ jΛ ij ψ j (38)as derived in standard text books 604,418 ; the diagonal canonical form H HFe ψ i = ɛ i ψ iis obtained after a unitary transformation and H HFe denotes the effective one–particle Hamiltonian, see Sect. 2.7 for more details. The equations of motioncorresponding to Eqs. (32)–(33) read{〈 ∣ 〉}M I ¨R I (t) = −∇ I min Ψ0 ∣H HF∣ e Ψ0 (39){ψ i}0 = −H HFe ψ i + ∑ jΛ ij ψ j (40)for the Hartree–Fock case. A similar set of equations is obtained if Hohenberg–Kohn–Sham density functional theory 458,168 is used, where H HFe has to be replacedby the Kohn–Sham effective one–particle Hamiltonian HeKS , see Sect. 2.7 for moredetails. Instead of diagonalizing the one–particle Hamiltonian an alternative butequivalent approach consists in directly performing the constraint minimizationaccording to Eq. (35) via nonlinear optimization techniques.Early applications of Born–Oppenheimer molecular dynamics were performedin the framework of a semiempirical approximation to the electronic structure problem669,671 . But only a few years later an ab initio approach was implemented withinthe Hartree–Fock approximation 365 . Born–Oppenheimer dynamics started to becomepopular in the early nineties with the availability of more efficient electronicstructure codes in conjunction with sufficient computer power to solve “interestingproblems”, see for instance the compilation of such studies in Table 1 in a recentoverview article 82 .Undoubtedly, the breakthrough of Hohenberg–Kohn–Sham density functionaltheory in the realm of chemistry – which took place around the same time – alsohelped a lot by greatly improving the “price / performance ratio” of the electronicstructure part, see e.g. Refs. 694,590 . A third and possibly the crucial reason thatboosted the field of ab initio molecular dynamics was the pioneering introduction of13