Ab initio molecular dynamics: Theory and Implementation

Here, the so–called adiabatic formulation 105,390,106 of ab initio centroid moleculardynamics 411 is discussed. In close analogy to ab initio molecular dynamics withclassical nuclei also the effective centroid potential is generated “on–the–fly” as thecentroids are propagated. This is achieved by singling out the centroid coordinatesin terms of a normal mode transformation 138 and accelerating the dynamics ofall non–centroid modes artificially by assigning appropriate fictitious masses. Atthe same time, the fictitious electron dynamics à la Car–Parrinello is kept in orderto calculate efficiently the ab initio forces on all modes from the electronic structure.This makes it necessary to maintain two levels of adiabaticity in the courseof simulations, see Sect. 2.1 of Ref. 411 for a theoretical analysis of that issue.The partition function Eq. (315), formulated in the so-called “primitive” pathvariables {R I } (s) , is first transformed 644,646 to a representation in terms of thenormal modes {u I } (s) , which diagonalize the harmonic nearest–neighbor harmoniccoupling 138 . The transformation follows from the Fourier expansion of a cyclicpathR (s)I=P∑s ′ =1a (s′ )Iexp [2πi(s − 1)(s ′ − 1)/P] , (321)where the coefficients {a I } (s) are complex. The normal mode variables {u I } (s) arethen given in terms of the expansion coefficients according tou (1)I= a (1)Iu (P)Iu (2s−2)Iu (2s−1)I= a ((P+2)/2)I= Re (a (s)I )= Im (a (s)I ) . (322)Associated with the normal mode transformation is a set of normal mode frequencies{λ} (s) given by[ ( )] 2π(s − 1)λ (2s−1) = λ (2s−2) = 2P 1 − cos(323)Pwith λ (1) = 0 and λ (P) = 4P. Equation (321) is equivalent to direct diagonalizationof the matrixA ss ′ = 2δ ss ′ − δ s,s′ −1 − δ s,s′ +1 (324)with the path periodicity condition A s0 = A sP and A s,P+1 = A s1 and subsequentuse of the unitary transformation matrix U to transform from the “primitive”variables {R I } (s) to the normal mode variables {u I } (s)R (s)I= √ PP∑s ′ =1u (s)I= 1 √P P ∑s ′ =1U † ss ′u(s′ )IU ss ′R (s′ )I. (325)114