Figure 16. Four patterns of spin densities n α t , nβ t , nα m , <strong>and</strong> nβ m corresponding to the two spin–restricted determinants |t〉 <strong>and</strong> |m〉 sketched in Fig. 15, see text for further details. Taken fromRef. 214 .whereas<strong>and</strong>{ 1−2[1 ]2 ∇2 + V H (r) + V ext (r)+ Vxc[n α α m(r), n β m(r)] − 1 } n+12 V ∑xc[n α α t (r), n β t (r)] φ a (r) = Λ aj φ j (r) , (307){ 1−2[1 ]2 ∇2 + V H (r) + V ext (r)j=1+ Vxc β [nα m (r), nβ m (r)] − 1 n+12 V xc α [nα t (r), nβ t}φ (r)] ∑b (r) = Λ bj φ j (r) . (308)are two different equations for the two singly–occupied open–shell orbitals a <strong>and</strong>b, respectively, see Fig. 15. Note that these Kohn–Sham–like equations featurean orbital–dependent exchange–correlation potential where Vxc[n α α m, n β m] =δE xc [n α m, n β m]/δn α m <strong>and</strong> analogues def<strong>initio</strong>ns hold for the β <strong>and</strong> t cases.The set of equations Eq. (306)–(308) could be solved by diagonalization of thecorresponding “restricted open–shell Kohn–Sham Hamiltonian” or alternatively bydirect minimization of the associated total energy functional. The algorithm proposedin Ref. 240 , which allows to properly <strong>and</strong> efficiently minimize such orbital–dependent functionals including the orthonormality constraints, was implementedin the CPMD package 142 . Based on this minimization technique Born–Oppenheimer<strong>molecular</strong> <strong>dynamics</strong> simulations can be performed in the first excited singlet state.j=1107
The alternative Car–Parrinello formulation seems inconvenient because the singly<strong>and</strong> doubly occupied orbitals would have to be constrained not to mix.4.4 Beyond Classical Nuclei4.4.1 IntroductionUp to this point the nuclei were approximated as classical point particles as customarilydone in st<strong>and</strong>ard <strong>molecular</strong> <strong>dynamics</strong>. There are, however, many situationswhere quantum dispersion broadening <strong>and</strong> tunneling effects play an important role<strong>and</strong> cannot be neglected if the simulation aims at being realistic – which is thegeneric goal of ab <strong>initio</strong> simulations. The ab <strong>initio</strong> path integral technique 395 <strong>and</strong>its extension to quasiclassical time evolution 411 is able to cope with such situationsat finite temperatures. It is also implemented in the CPMD package 142 . The centralidea is to quantize the nuclei using Feynman’s path integrals <strong>and</strong> at the same timeto include the electronic degrees of freedom akin to ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> –that is “on–the–fly”. The main ingredients <strong>and</strong> approximations underlying the ab<strong>initio</strong> path integral approach 395,399,644,404 are• the adiabatic separation of electrons <strong>and</strong> nuclei where the electrons are kept intheir ground state without any coupling to electronically excited states (Born–Oppenheimer or “clamped–nuclei” approximation),• using a particular approximate electronic structure theory in order to calculatethe interactions,• approximating the continuous path integral for the nuclei by a finite discretization(Trotter factorization) <strong>and</strong> neglecting the indistinguishability of identicalnuclei (Boltzmann statistics), <strong>and</strong>• using finite supercells with periodic boundary conditions <strong>and</strong> finite samplingtimes (finite–size <strong>and</strong> finite–time effects) as usual.Thus, quantum effects such as zero–point motion <strong>and</strong> tunneling as well as thermalfluctuations are included at some preset temperature without further simplificationsconsisting e.g. in quasiclassical or quasiharmonic approximations, restricting theHilbert space, or in artificially reducing the dimensionality of the problem.4.4.2 <strong>Ab</strong> Initio Path Integrals: StaticsFor the purpose of introducing ab <strong>initio</strong> path integrals 395 it is convenient to startdirectly with Feynman’s formulation of quantum–statistical mechanics in termsof path integrals as opposed to Schrödinger’s formulation in terms of wavefunctionswhich was used in Sect. 2.1 in order to derive ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong>.For a general introduction to path integrals the reader is referred to st<strong>and</strong>ard textbooks187,188,334 , whereas their use in numerical simulations is discussed for instancein Refs. 233,126,542,120,124,646,407 .The derivation of the expressions for ab <strong>initio</strong> path integrals is based on assumingthe non–relativistic st<strong>and</strong>ard Hamiltonian, see Eq. (2). The corresponding canonicalpartition function of a collection of interacting nuclei with positions R = {R I }108
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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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- 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.
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- Page 150: 682. R. M. Wentzcovitch, Phys. Rev.