frequency of the electronic degrees of freedom ω e should again lie above the frequencyspectrum associated to the fictitious nuclear <strong>dynamics</strong>. These is the methodthat is implemented in the CPMD package 142 .An important issue for adiabatic ab <strong>initio</strong> centroid <strong>molecular</strong> <strong>dynamics</strong> 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),<strong>and</strong> γ is the “centroid adiabaticity parameter”; note that this corrects a misprintof the def<strong>initio</strong>n of M ′(s)Ifor s ≥ 2 in Ref. 411 . By choosing 0 < γ ≪ 1, therequired time–scale separation between the centroid <strong>and</strong> 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 <strong>dynamics</strong> of all other nuclear modess > 1 is fictitious <strong>and</strong> 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 <strong>molecular</strong> <strong>dynamics</strong>.4.4.4 Other ApproachesIt is evident from the outset that the Born–Oppenheimer approach to generatethe ab <strong>initio</strong> forces can be used as well as Car–Parrinello <strong>molecular</strong> <strong>dynamics</strong>in order to generate the ab <strong>initio</strong> forces on the quantum nuclei. This variationwas utilized in a variety of investigations ranging from clusters to <strong>molecular</strong>solids 132,37,596,597,428,429,333 . Closely related to the ab <strong>initio</strong> path integral approachas discussed here is a method that is based on Monte Carlo sampling of the pathintegral 672 . It is similar in spirit <strong>and</strong> in its implementation to Born–Oppenheimer<strong>molecular</strong> <strong>dynamics</strong> sampling as long as only time–averaged static observables arecalculated. A semiempirical (“cndo” <strong>and</strong> “indo”) version of Born–Oppenheimer ab<strong>initio</strong> path integral simulations was also devised 656 <strong>and</strong> applied to study muonatedorganic molecules 656,657 .A non–self–consistent approach to ab <strong>initio</strong> path integral calculations was advocated<strong>and</strong> used in a series of publications devoted to study the interplay of nuclearquantum effects <strong>and</strong> 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 <strong>and</strong> the117
subsequent analysis of its electronic properties is performed using different Hamiltonians.Several attempts to treat also the electrons in the path integral formulation –instead of using wavefunctions as in the ab <strong>initio</strong> path integral family – werepublished 606,119,488,449,450 . These approaches are exact in principle, i.e. non–adiabaticity <strong>and</strong> full electron–phonon coupling is included at finite temperatures.However, they suffer from severe stability problems 121 in the limit of degenerateelectrons, i.e. at very low temperatures compared to the Fermi temperature, whichis the temperature range of interest for typical problems in chemistry <strong>and</strong> materialsscience. Recent progress on computing electronic forces from path integral MonteCarlo simulations was also achieved 708 .More traditional approaches use a wavefunction representation for both theelectrons in the ground state <strong>and</strong> for nuclear density matrix instead of path integrals.The advantage is that real–time evolution is obtained more naturallycompared to path integral simulations. A review of such methods with the emphasisof computing the interactions “on–the–fly” is provided in Ref. 158 . An approximatewavefunction–based quantum <strong>dynamics</strong> method which includes severalexcited states <strong>and</strong> their couplings was also devised <strong>and</strong> used 385,386,387,45 . An alternativeapproach to approximate quantum <strong>dynamics</strong> consists in performing instantonor semiclassical ab <strong>initio</strong> <strong>dynamics</strong> 325,47 . Also the approximate vibrationalself–consistent field approach to nuclear quantum <strong>dynamics</strong> was combined with“on–the–fly” MP2 electronic structure calculations 122 .5 Applications: From Materials Science to Biochemistry5.1 Introduction<strong>Ab</strong> <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> was called a “virtual matter laboratory” 234 , a notionthat is fully justified in view of its relationship to experiments performed in thereal laboratory. Ideally, a system is prepared in some initial state <strong>and</strong> than evolvesaccording to the basic laws of physics – without the need of experimental input.It is clear to every practitioner that this viewpoint is highly idealistic for morethan one reason, but still this philosophy allows one to compute observables withpredictive power <strong>and</strong> also implies a broad range of applicability.It is evident from the number of papers dealing with ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong>,see for instance Fig. 1, that a truly comprehensive survey of applicationscannot be given. Instead, the strategy chosen is to provide the reader with a wealthof references that try cover the full scope of this approach – instead of discussingin depth the physics or chemistry of only a few specific applications. To this endthe selection is based on a general literature search in order to suppress personalpreferences as much as possible. In addition the emphasis lies on recent applicationsthat could not be covered in earlier reviews. This implies that several olderkey reference papers on similar topics are in general missing.118
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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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¢¢¢¢¢i = 1 . . .N b¢c i (G)¢
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¢¢¢¢¢c i (G)123g, E self∆V I
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- Page 128 and 129: AcknowledgmentsOur knowledge on ab
- Page 130 and 131: 57. M. Bernasconi, G. L. Chiarotti,
- Page 132 and 133: Superiore di Studi Avanzati (SISSA)
- 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.
- Page 144 and 145: 502. L. M. Ramaniah, M. Bernasconi,
- Page 146 and 147: 562. F. Shimojo, K. Hoshino, and Y.
- Page 148 and 149: 620. A. Tongraar, K. R. Liedl, and
- Page 150: 682. R. M. Wentzcovitch, Phys. Rev.