Ab initio molecular dynamics: Theory and Implementation

More documents

Recommendations

Info

Figure 4. (a) Comparison of the x–component of the force acting on one atom of a model systemobtained from Car–Parrinello (solid line) and well–converged Born–Oppenheimer (dots) moleculardynamics. (b) Enlarged view of the difference between Car–Parrinello and Born–Oppenheimerforces; for further details see text. Adapted from Ref. 467 .the physical variations of the force resolved in Fig. 4(a). These correspond to the“large–amplitude” oscillations of T e visible in Fig. 3 due to the drag of the nucleiexerted on the quasi–adiabatically following electrons having a finite dynamicalmass µ. Note that the inertia of the electrons also dampens artificially the nuclearmotion (typically on a few–percent scale, see Sect. V.C.2 in Ref. 75 for an analysisand a renormalization correction of M I ) but decreases as the fictitious massapproaches the adiabatic limit µ → 0. Superimposed on the gross variation in (b)are again high–frequency bound oscillatory small–amplitude fluctuations like for T e .They lead on physically relevant time scales (i.e. those visible in Fig. 4(a)) to “averagedforces” that are very close to the exact ground–state Born–Oppenheimerforces. This feature is an important ingredient in the derivation of adiabatic dynamics467,411 .In conclusion, the Car–Parrinello force can be said to deviate at most instants oftime from the exact Born–Oppenheimer force. However, this does not disturb thephysical time evolution due to (i) the smallness and boundedness of this differenceand (ii) the intrinsic averaging effect of small–amplitude high–frequency oscillationswithin a few molecular dynamics time steps, i.e. on the sub–femtosecond time scalewhich is irrelevant for nuclear dynamics.2.4.4 How to Control Adiabaticity ?An important question is under which circumstances the adiabatic separation canbe achieved, and how it can be controlled. A simple harmonic analysis of thefrequency spectrum of the orbital classical fields close to the minimum defining theground state yields 467 ( ) 1/2 2(ɛi − ɛ j )ω ij =, (52)µ19
where ɛ j and ɛ i are the eigenvalues of occupied and unoccupied orbitals, respectively;see Eq. (26) in Ref. 467 for the case where both orbitals are occupied ones.It can be seen from Fig. 2 that the harmonic approximation works faithfully ascompared to the exact spectrum; see Ref. 471 and Sect. IV.A in Ref. 472 for a moregeneral analysis of the associated equations of motion. Since this is in particulartrue for the lowest frequency ωemin , the handy analytic estimate for the lowestpossible electronic frequencyω mine∝(Egapµ) 1/2, (53)shows that this frequency increases like the square root of the electronic energydifference E gap between the lowest unoccupied and the highest occupied orbital.On the other hand it increases similarly for a decreasing fictitious mass parameterµ.ω maxnIn order to guarantee the adiabatic separation, the frequency difference ωemin −should be large, see Sect. 3.3 in Ref. 513 for a similar argument. But boththe highest phonon frequency ωnmax and the energy gap E gap are quantities that adictated by the physics of the system. Whence, the only parameter in our handsto control adiabatic separation is the fictitious mass, which is therefore also called“adiabaticity parameter”. However, decreasing µ not only shifts the electronicspectrum upwards on the frequency scale, but also stretches the entire frequencyspectrum according to Eq. (52). This leads to an increase of the maximum frequencyaccording toω maxe∝(Ecutµ) 1/2, (54)where E cut is the largest kinetic energy in an expansion of the wavefunction interms of a plane wave basis set, see Sect. 3.1.3.At this place a limitation to decrease µ arbitrarily kicks in due to the maximumlength of the molecular dynamics time step ∆t max that can be used. The time stepis inversely proportional to the highest frequency in the system, which is ωemax andthus the relation( ) 1/2 µ∆t max ∝(55)E cutgoverns the largest time step that is possible. As a consequence, Car–Parrinellosimulators have to find their way between Scylla and Charybdis and have to makea compromise on the control parameter µ; typical values for large–gap systems areµ = 500–1500 a.u. together with a time step of about 5–10 a.u. (0.12–0.24 fs).Recently, an algorithm was devised that optimizes µ during a particular simulationgiven a fixed accuracy criterion 87 . Note that a poor man’s way to keep the timestep large and still increase µ in order to satisfy adiabaticity is to choose heaviernuclear masses. That depresses the largest phonon or vibrational frequency ωnmaxof the nuclei (at the cost of renormalizing all dynamical quantities in the sense ofclassical isotope effects).20
Page 1 and 2: John von Neumann Institute for Comp
Page 4: 2500Number200015001000CP PRL 1985AI
Page 7 and 8: The goal of this section is to deri
Page 9: ¡the Newtonian equation of motion
Page 13 and 14: Ehrenfest molecular dynamics is cer
Page 15 and 16: the Car-Parrinello approach 108 , s
Page 17: According to the Car-Parrinello equ
Page 22 and 23: Up to this point the entire discuss
Page 24 and 25: parameters are those used to repres
Page 26 and 27: in terms of a linear combination of
Page 28 and 29: structure calculations, see e.g. Re
Page 30 and 31: “unbound electrons” dissolved i
Page 32 and 33: Table 1. Timings in cpu seconds and
Page 34 and 35: stressed that the energy conservati
Page 36 and 37: see e.g. the discussion following E
Page 38 and 39: from a set of one-particle spin orb
Page 40 and 41: is used, which represents exactly a
Page 42 and 43: 2.8.3 Generalized Plane WavesAn ext
Page 44 and 45: disposable parameters that can be o
Page 46 and 47: The index i runs over all states an
Page 48 and 49: f(G) are related by three-dimension
Page 50 and 51: 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 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 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.
show all

Ab initio molecular dynamics: Theory and Implementation

Create successful ePaper yourself

Delete template?

Save as template?