Ab initio molecular dynamics: Theory and Implementation

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and electrons r = {r i } can be written as a path integral∮ ′ ∮ ′Z = DR Dr exp[− 1 ∫ β¡dτ L E({Ṙ I (τ)}, {R I (τ)}; {ṙ i (τ)}, {r i (τ)}) ] (309)where0L E = T(Ṙ) + V (R) + T(ṙ) + V (r) + V (R,r)= ∑ ( ) 212 M dRII + ∑ e 2 Z I Z Jdτ |R I − R J |II
implies that the electronic degrees of freedom at different imaginary times τ and τ ′become completely decoupled. Thus, the influence functional Z[R] has to be local inτ and becomes particularly simple; a discussion of adiabatic corrections in the pathintegral formulation can be found in Ref. 101 . For each τ the influence functionalZ[R] is given by the partition function of the electronic subsystem evaluated forthe respective nuclear configuration R(τ). Assuming that the temperature is smallcompared to the gap in the electronic spectrum only the electronic ground statewith energy E 0 (R(τ)) (obtained from solving Eq. (20) without the internuclearCoulomb repulsion term) is populated. This electronic ground state dominanceleads to the following simple expression[]Z[R] BO = expwhich yields the final result∮ [Z BO = DR exp −∫ β0−dτ∫ β0dτE 0 (R(τ)), (313)(T(Ṙ) + V (R) + E 0 (R)) ] . (314)Here nuclear exchange is neglected by assuming that the nuclei are distinguishableso that they can be treated within Boltzmann statistics, which corresponds to theHartree approximation for the nuclear density matrix. The presentation given herefollows Ref. 399 and alternative derivations were given in Sect. 2.3 of Refs. 124 andin the appendix of Ref. 427 . There, a wavefunction basis instead of the positionbasis as in Eq. (312) was formally used in order to evaluate the influence functionaldue to the electrons.The partition function Eq. (314) together with the Coulomb Hamiltonian Eq. (2)leads after applying the lowest–order Trotter factorization 334 to the following discretizedexpression][P∏ N∏ (MI ) 3/2 ∫PZ BO = limP →∞ 2πβs=1 I=1[ {P∑ ∑ N× exp −βs=1I=11(2 M IωP2dR (s)IR (s)I) 2− R (s+1) 1(I +P E 0 {R I } (s))}] (315)for the path integral with ω 2 P = P/β2 Thus, the continuous parameter τ ∈ [0, β] isdiscretized using P so–called Trotter slices or “time slices” s = 1, . . ., P of “duration”∆τ = β/P. The paths{{R I } (s)} =({R I } (1) ; . . .; {R I } (P))=()R (1)1 , . . .,R(1) N ; . . .;R(P) 1 , . . .,R (P)N(316)have to be closed due to the trace condition, i.e. they are periodic in imaginarytime τ which implies R I (0) ≡ R I (β) and thus R (P+1)I= R (1)I; the internuclearCoulomb repulsion V (R) is now included in the definition of the total electronicenergy E 0 . Note that Eq. (315) is an exact reformulation of Eq. (314) in the limitof an infinitely fine discretization P → ∞ of the paths.110
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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
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 102 and 103: 4.3.2 Many Excited States: Free Ene
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 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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