Ab initio molecular dynamics: Theory and Implementation

Eq. (2.21) of Ref. 644 . This is the desired relation which allows in centroid moleculardynamics the centroid forces to be simply obtained as the average force which actson the lowest-order normal mode Eq. (328). The non–centroid normal modes u (s)I ,s = 2, 3, . . ., P of the paths establish the effective potential in which the centroidmoves.At this stage the equations of motion for adiabatic ab initio centroid moleculardynamics 411 can be obtained from the Euler–Lagrange equations. These equationsof motion readM ′(1)Iü (1)I= − 1 PM ′(s)Iü (s)I,α = −µ¨φ (s)iP∑ ∂E [ {φ i } (s) , {R I } (s)]s=1∂∂u (s)I,α1PP∑s ′ =1−M (s)IωPu 2 (s)∂R (s)I[ (E {φ i } (s′) ,{R II,α − M ′(s)I˙ξ (s)I,α,1 ˙u(s) I,α= − δE [ {φ i } (s) , {R I } (s)]+ ∑ jδφ ⋆(s)iu (1)I)} ] (s, . . .,u (P)′ )I(331), s = 2, . . ., P (332)Λ (s)ij φ(s) j− µ ˙η (s)1˙φ (s)i, s = 1, . . ., P (333)where u (s)I,αdenotes the Cartesian components of a given normal mode vectoru (s)I= (u (s)I,1 , u(s) I,2 , u(s) I,3). In the present scheme, independent Nosé–Hoover chainthermostats 388 of length K are coupled to all non–centroid mode degrees of freedoms = 2, . . ., P[Q n¨ξ(s) I,α,1 =M ′(s)I( ] 2˙uI,α) (s) − kB T[ ( 2Q n¨ξ(s) I,α,k = Q n ˙I,α,k−1) ξ(s) − kB T]− Q n ˙ ξ(s)− Q n ˙ ξ(s)˙ I,α,1ξ (s)I,α,2(334)˙ I,α,kξ (s)I,α,k+1 (1 − δ kK) , k = 2, ..., K(335)and all orbitals at a given imaginary time slice s are thermostatted by one suchthermostat chain of length L[ ]∑ 〈〉Q e 1¨η(s) 1 = 2 µ φ (s) ∣i ∣φ (s)i− Te0 − Q e 1 ˙η(s) 1 ˙η (s)2 (336)i[ ( ) ] 2Q e l ¨η (s)l= Q e l−1 ˙η (s) 1l−1 −β e− Q e l ˙η (s)l˙η (s)l+1 (1 − δ lL) , l = 2, . . ., L ; (337)note that for standard ab initio path integral runs as discussed in the previoussection the centroid mode should be thermostatted as well. The desired fictitiouskinetic energy of the electronic subsystem Te 0 can be determined based on a shortequivalent classical Car–Parrinello run with P = 1 and using again the relation1/β e = 2Te 0 /6N e ′ where N e ′ is the number of orbitals. The mass parameters {Q e l }associated to the orbital thermostats are the same as those defined in Eq. (271),whereas the single mass parameter Q n for the nuclei is determined by the harmonicinteraction and is given by Q n = k B T/ωP 2 = β/P. The characteristic thermostat116