Ab initio molecular dynamics: Theory and Implementation

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4.3.2 Many Excited States: Free Energy FunctionalThe free energy functional approach to excited–state molecular dynamics 5,7 is amean–field approach similar in spirit to Ehrenfest molecular dynamics, see Sect. 2.2.The total wavefunction is first factorized into a nuclear and an electronic wavefunctionEq. (3) followed by taking the classical limit for the nuclear subsystem.Thus, classical nuclei move in the average field as obtained from averaging over allelectronic states Eq. (25). A difference is that according to Ehrenfest moleculardynamics the electrons are propagated in real time and can perform non–adiabatictransitions by virtue of direct coupling terms ∝ d kl between all states Ψ k subjectto energy conservation, see Sect. 2.2 and in particular Eqs. (27)–(29). The averageforce or Ehrenfest force is obtained by weighting the different states k according totheir diagonal density matrix elements (that is ∝ |c k (t)| 2 in Eq. (27)) whereas thecoherent transitions are driven by the off–diagonal contributions (which are ∝ c ⋆ k c l,see Eq. (27)).In the free energy approach 5,7 , the excited states are populated according tothe Fermi–Dirac (finite–temperature equilibrium) distribution which is based onthe assumption that the electrons “equilibrate” more rapidly than the timescaleof the nuclear motion. This means that the set of electronic states evolves at agiven temperature “isothermally” (rather than adiabatically) under the inclusionof incoherent electronic transitions at the nuclei move. Thus, instead of computingthe force acting on the nuclei from the electronic ground–state energy it isobtained from the electronic free energy as defined in the canonical ensemble. Byallowing such electronic transitions to occur the free energy approach transcendsthe usual Born–Oppenheimer approximation. However, the approximation of aninstantaneous equilibration of the electronic subsystem implies that the electronicstructure at a given nuclear configuration {R I } is completely independent from previousconfigurations along a molecular dynamics trajectory. Due to this assumptionthe notion “free energy Born–Oppenheimer approximation” was coined in Ref. 101in a similar context. Certain non–equilibrium situations can also be modeled withinthe free energy approach by starting off with an initial orbital occupation patternthat does not correspond to any temperature in its thermodynamic meaning, seee.g. Refs. 570,572,574 for such applications.The free energy functional as defined in Ref. 5 is introduced most elegantly 7,9by starting the discussion for the special case of non–interacting FermionsH s = − 1 2 ∇2 − ∑ IZ I|R I − r|(286)in a fixed external potential due to a collection of nuclei at positions {R I }. Theassociated grand partition function and its thermodynamic potential (“grand freeenergy”) are given byΞ s (µV T) = det 2 (1 + exp [−β (H s − µ)]) (287)Ω s (µV T) = −k B T lnΞ s (µV T) , (288)where µ is the chemical potential acting on the electrons and the square of the101
determinant stems from considering the spin–unpolarized special case only. Thisreduces to the well–known grand potential expressionΩ s (µV T) = −2k B T ln det (1 + exp [−β (H s − µ)])= −2k B T ∑ ( [ ( )])ln 1 + exp −β ɛ (i)s − µi(289)for non–interacting spin–1/2 Fermions where {ɛ (i)s } are the eigenvalues of a one–particle Hamiltonian such as Eq. (286); here the standard identity lndet M =Tr lnM was invoked for positive definite M.According to thermodynamics the Helmholtz free energy F(NV T) associatedto Eq. (288) can be obtained from an appropriate Legendre transformation of thegrand free energy Ω(µV T)F s (NV T) = Ω s (µV T) + µN + ∑ I
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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
Page 52 and 53: andE self = ∑ I1√2πZ 2 IR c I.
Page 54 and 55: ¢££¤¤¢¢¢n tot (G)inv FTn to
Page 56 and 57: correlation energyΩ ∑E xc = ε x
Page 58 and 59: 3.4 Total Energy, Gradients, and St
Page 60 and 61: 3.4.3 Gradient for Nuclear Position
Page 62 and 63: The local part of the pseudopotenti
Page 64 and 65: ¢¢¢¢¢i = 1 . . .N b¢c i (G)¢
Page 66 and 67: ¢¢¢¢¢c i (G)123g, E self∆V I
Page 68 and 69: where G c is a free parameter that
Page 70 and 71: and a matrix form of the Gram-Schmi
Page 72 and 73: The two sets of equations are coupl
Page 74 and 75: introducing different masses for di
Page 76 and 77: The Lagrange multiplier have to be
Page 78 and 79: Table 3. Relative size of character
Page 80 and 81: of the G vectors, and only real ope
Page 82 and 83: CALL SGEMM(’T’,’N’,M,N b ,2
Page 84 and 85: ing standard communication librarie
Page 86 and 87: over processors. All processors sho
Page 88 and 89: ab = 2 * sdot(2 * N p D ,A(1),1,B(1
Page 90 and 91: ENDCALL ParallelFFT3D("INV",scr1,sc
Page 92 and 93: are the improved load-balancing for
Page 94 and 95: situations where• it is necessary
Page 96 and 97: espectively), but completely analog
Page 98 and 99: 4.2.3 Imposing Pressure: BarostatsK
Page 100 and 101: in the previous section. The isobar
Page 104 and 105: down the generalization of the Helm
Page 106 and 107: free energy functional discussed in
Page 108 and 109: Figure 16. Four patterns of spin de
Page 110 and 111: and electrons r = {r i } can be wri
Page 112 and 113: The effective classical partition f
Page 114 and 115: up e.g. in Refs. 132,37,596,597,428
Page 116 and 117: The eigenvalues of A when multiplie
Page 118 and 119: frequency of the electronic degrees
Page 120 and 121: 5.2 Solids, Polymers, and Materials
Page 122 and 123: the penetration of the oxidation la
Page 124 and 125: in terms of their electronic struct
Page 126 and 127: culations on very accurate global p
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.
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