Ab initio molecular dynamics: Theory and Implementation

frequency of the electronic degrees of freedom ω e should again lie above the frequencyspectrum associated to the fictitious nuclear dynamics. These is the methodthat is implemented in the CPMD package 142 .An important issue for adiabatic ab initio centroid molecular dynamics 411 ishow to establish the time–scale separation of the non–centroid modes compared tothe centroid modes. This is guaranteed if the fictitious normal mode masses M ′(s)Iare taken to beM ′(1)IM ′(s)I= M I= γ M (s)I, s = 2, . . ., P , (338)where M I is the physical nuclear mass, M (s)Iare the normal mode masses Eq. (326),and γ is the “centroid adiabaticity parameter”; note that this corrects a misprintof the definition of M ′(s)Ifor s ≥ 2 in Ref. 411 . By choosing 0 < γ ≪ 1, therequired time–scale separation between the centroid and non–centroid modes can becontrolled so that the motion of the non–centroid modes is artificially accelerated,see Sect. 3 in Ref. 411 for a systematic study of the γ–dependence. Thus, thecentroids with associated physical masses move quasiclassically in real–time in thecentroid effective potential, whereas the fast dynamics of all other nuclear modess > 1 is fictitious and serves only to generate the centroid effective potential “on–the–fly”. In this sense γ (or rather γM I ) is similar to µ, the electronic adiabaticityparameter in Car–Parrinello molecular dynamics.4.4.4 Other ApproachesIt is evident from the outset that the Born–Oppenheimer approach to generatethe ab initio forces can be used as well as Car–Parrinello molecular dynamicsin order to generate the ab initio forces on the quantum nuclei. This variationwas utilized in a variety of investigations ranging from clusters to molecularsolids 132,37,596,597,428,429,333 . Closely related to the ab initio path integral approachas discussed here is a method that is based on Monte Carlo sampling of the pathintegral 672 . It is similar in spirit and in its implementation to Born–Oppenheimermolecular dynamics sampling as long as only time–averaged static observables arecalculated. A semiempirical (“cndo” and “indo”) version of Born–Oppenheimer abinitio path integral simulations was also devised 656 and applied to study muonatedorganic molecules 656,657 .A non–self–consistent approach to ab initio path integral calculations was advocatedand used in a series of publications devoted to study the interplay of nuclearquantum effects and electronic structure in unsaturated hydrocarbons like benzene544,503,81,543,504 . According to this philosophy, an ensemble of nuclear pathconfigurations Eq. (316) is first generated at finite temperature with the aid of aparameterized model potential (or using a tight–binding Hamiltonian 504 ). In a second,independent step electronic structure calculations (using Pariser–Parr–Pople,Hubbard, or Hartree–Fock Hamiltonians) are performed for this fixed ensemble ofdiscretized quantum paths. The crucial difference compared to the self–consistentapproaches presented above is that the creation of the thermal ensemble and the117