Ab initio molecular dynamics: Theory and Implementation

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down the generalization of the Helmholtz free energy of the interacting many–electron case∫F KS (NV T) = Ω KS (µV T) + µ dr n(r) + ∑ Z I Z J|R I − R J |I
in order to compute first the diagonal elements ρ ii (∆τ) of the “high–temperature”Boltzmann operator ρ(∆τ); here ∆τ = β/P and P is the Trotter “time slice” asintroduced in paragraph Ab Initio Path Integrals: Statics. To this end, the kineticand potential energies can be conveniently evaluated in reciprocal and real space, respectively,by using the split–operator / FFT technique 183 . The Kohn–Sham eigenvaluesɛ i are finally obtained from the density matrix via ɛ i = −(1/∆τ) lnρ ii (∆τ).They are used in order to compute the occupation numbers {f i }, the density n(r),the band–structure energy Ω KS , and thus the free energy Eq. (295).In practice a diagonalization / density–mixing scheme is employed in order tocompute the self–consistent density n(r). Grossly speaking a suitably constructedtrial input density n in (see e.g. the Appendix of Ref. 571 for such a method) is usedin order to compute the potential V KS [n in ]. Then the lowest–order approximantto the Boltzmann operator Eq. (300) is diagonalized using an iterative Lanczos–type method. This yields an output density n out and the corresponding free energyF KS [n out ]. Finally, the densities are mixed and the former steps are iterated until astationary solution n scf of F KS [n scf ] is achieved, see Sect. 3.6.4 for some details onsuch methods. Of course the most time–consuming part of the calculation is in theiterative diagonalization. In principle this is not required, and it should be possibleto compute the output density directly from the Fermi–Dirac density matrix evenin a linear scaling scheme 243 , thus circumventing the explicit calculation of theKohn–Sham eigenstates. However, to date efforts in this direction have failed, orgiven methods which are too slow to be useful 9 .As a method, molecular dynamics with the free energy functional is most appropriateto use when the excitation gap is either small, or in cases where thegap might close during a chemical transformation. In the latter case no instabilitiesare encountered with this approach, which is not true for ground–state abinitio molecular dynamics methods. The price to pay is the quite demanding iterativecomputation of well–converged forces. Besides allowing such applicationswith physically relevant excitations this method can also be straightforwardly combinedwith k–point sampling and applied to metals at “zero” temperature. Inthis case, the electronic “temperature” is only used as a smearing parameter ofthe Fermi edge by introducing fractional occupation numbers, which is known toimprove greatly the convergence of these ground–state electronic structure calculations220,232,185,676,680,343,260,344,414,243 .Finite–temperature expressions for the exchange–correlation functional Ω xc areavailable in the literature. However, for most temperatures of interest the correctionsto the ground–state expression are small and it seems justified to use one ofthe various well–established parameterizations of the exchange–correlation energyE xc at zero temperature, see Sect. 2.7.4.3.3 A Single Excited State: S 1 –DynamicsFor large–gap systems with well separated electronic states it might be desirableto single out a particular state in order to allow the nuclei to move on the associatedexcited state potential energy surface. Approaches that rely on fractionaloccupation numbers such as ensemble density functional theories – including the104
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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.
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 102 and 103: 4.3.2 Many Excited States: Free Ene
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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