up e.g. in Refs. 132,37,596,597,428,429,333 .A final observation concerning parallel supercomputers might be useful, see alsoSect. 3.9. It is evident from the Lagrangian Eq. (319) <strong>and</strong> the resulting equationsof motion (e.g. Eqs. (331)–(337)) that most of the numerical workload comes fromcalculating the ab <strong>initio</strong> forces on the nuclei. Given a fixed path configurationEq. (316) the P underlying electronic structure problems are independent fromeach other <strong>and</strong> can be solved without communication on P nodes of a distributedmemory machine. Communication is only necessary to send the final result, essentiallythe forces, to a special node that computes the quantum kinetic contributionto the energy <strong>and</strong> integrates finally the equations of motions. It is even conceivableto distribute this task on different supercomputers, i.e. “meta–computing” iswithin reach for such calculations. Thus, the algorithm is “embarrassingly parallel”provided that the memory per node is sufficient to solve the complete Kohn–Shamproblem at a given time slice. If this is not the case the electronic structure calculationitself has to be parallelized on another hierarchical level as outlined inSect. 3.9.4.4.3 <strong>Ab</strong> Initio Path Centroids: DynamicsInitially the <strong>molecular</strong> <strong>dynamics</strong> approach to path integral simulations was inventedmerely as a trick in order to sample configuration space similar to the Monte Carlomethod. This perception changed recently with the introduction of the so–called“centroid <strong>molecular</strong> <strong>dynamics</strong>” technique 102 , see Refs. 103,104,105,665,505,506,507 forbackground information. In a nutshell it is found that the time evolution of thecenters of mass or centroidsR c I(t) = 1 PP∑s ′ =1R (s′ )I(t) (320)of the closed Feynman paths that represent the quantum nuclei contains quasiclassicalinformation about the true quantum <strong>dynamics</strong>. The centroid <strong>molecular</strong><strong>dynamics</strong> approach can be shown to be exact for harmonic potentials <strong>and</strong> to havethe correct classical limit. The path centroids move in an effective potential whichis generated by all the other modes of the paths at the given temperature. Thiseffective potential thus includes the effects of quantum fluctuations on the (quasiclassical)time evolution of the centroid degrees of freedom. Roughly speakingthe trajectory of the path centroids can be regarded as a classical trajectory of thesystem, which is approximately “renormalized” due to quantum effects.The original centroid <strong>molecular</strong> <strong>dynamics</strong> technique 102,103,104,105,665 relies onthe use of model potentials as the st<strong>and</strong>ard time–independent path integral simulations.This limitation was overcome independently in Refs. 469,411 by combiningab <strong>initio</strong> path integrals with centroid <strong>molecular</strong> <strong>dynamics</strong>. The resulting technique,ab <strong>initio</strong> centroid <strong>molecular</strong> <strong>dynamics</strong> can be considered as a quasiclassical generalizationof st<strong>and</strong>ard ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong>. At the same time, it preservesthe virtues of the ab <strong>initio</strong> path integral technique 395,399,644,404 to generate exacttime–independent quantum equilibrium averages.113
Here, the so–called adiabatic formulation 105,390,106 of ab <strong>initio</strong> centroid <strong>molecular</strong><strong>dynamics</strong> 411 is discussed. In close analogy to ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> 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 <strong>and</strong> accelerating the <strong>dynamics</strong> ofall non–centroid modes artificially by assigning appropriate fictitious masses. Atthe same time, the fictitious electron <strong>dynamics</strong> à la Car–Parrinello is kept in orderto calculate efficiently the ab <strong>initio</strong> 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 <strong>and</strong> 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 <strong>and</strong> A s,P+1 = A s1 <strong>and</strong> 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
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John von Neumann Institute for Comp
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2500Number200015001000CP PRL 1985AI
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The goal of this section is to deri
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¡the Newtonian equation of motion
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Ehrenfest molecular dynamics is cer
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the Car-Parrinello approach 108 , s
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According to the Car-Parrinello equ
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Figure 4. (a) Comparison of the x-c
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Up to this point the entire discuss
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parameters are those used to repres
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in terms of a linear combination of
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structure calculations, see e.g. Re
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“unbound electrons” dissolved i
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Table 1. Timings in cpu seconds and
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stressed that the energy conservati
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see e.g. the discussion following E
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from a set of one-particle spin orb
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is used, which represents exactly a
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2.8.3 Generalized Plane WavesAn ext
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disposable parameters that can be o
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The index i runs over all states an
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f(G) are related by three-dimension
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where j l are spherical Bessel func
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andE self = ∑ I1√2πZ 2 IR c I.
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¢££¤¤¢¢¢n tot (G)inv FTn to
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correlation energyΩ ∑E xc = ε x
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3.4 Total Energy, Gradients, and St
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3.4.3 Gradient for Nuclear Position
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The local part of the pseudopotenti
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- Page 130 and 131: 57. M. Bernasconi, G. L. Chiarotti,
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- Page 134 and 135: 175. E. Ermakova, J. Solca, H. Hube
- Page 136 and 137: 244. H. Goldstein, Klassische Mecha
- Page 138 and 139: 313. T. Ikeda, M. Sprik, K. Terakur
- Page 140 and 141: 384. N. A. Marks, D. R. McKenzie, B
- Page 142 and 143: 442. S. Nosé and M. L. Klein, Mol.
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- Page 146 and 147: 562. F. Shimojo, K. Hoshino, and Y.
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- Page 150: 682. R. M. Wentzcovitch, Phys. Rev.