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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Eq. (2.21) of Ref. 644 . This is the desired relation which allows in centroid <strong>molecular</strong><strong>dynamics</strong> 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 <strong>initio</strong> centroid <strong>molecular</strong><strong>dynamics</strong> 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)<strong>and</strong> 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 st<strong>and</strong>ard ab <strong>initio</strong> 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 <strong>and</strong> 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 <strong>and</strong> is given by Q n = k B T/ωP 2 = β/P. The characteristic thermostat116