DFT and Molecular Dynamics

Transcription

1 Chapter 6 DFT and Molecular Dynamcs Let us stck to our prevous assumpton that ons behave as classcal partcles. Ths turns out to be a very good approxmaton, except for the lghtest atoms and for vbratonal modes around the equlbrum postons at low temperature. Also, let us assume the valdty of the Born-Oppenhemer (BO approxmaton and that electrons are always on the BO surface, that s, n the ground state correspondng to ther nstantaneous postons. Under these assumptons the dynamcal behavor of ons can be descrbed by a classcal Lagrangan L = 1 M Ṙ 2 E tot ({R} (6.1 2 where M are the mass of ons. The correspondng equatons of moton: d L dt Ṙ L R = 0, P = L Ṙ (6.2 are nothng but Newton s equatons. It s temptng to use Eq.6.1 as the bass for a molecular dynamcs (MD study. In classcal MD, the forces are generated by an nteratomc potental (often a sum of twobody terms lke Lennard-Jones potentals and the Newton equatons are dscretzed and numercally solved. The dscrete nterval of tme s called tme step. A sequence of atomc coordnates and veloctes s generated startng from a sutable ntal set of coordnates and veloctes. The sequence can be used to calculate thermodynamcal averages. Straghtforward MD wll sample the mcrocanoncal ensemble: constant energy at fxed volume, but t s possble to buld a dynamcs at constant temperature (canoncal ensemble usng a thermostat that smulates a thermal bath, or at constant pressure, by addng a fcttous dynamcs on the volume, and even more complex cases. MD can also be used to fnd the global mnma usng the smulated annealng technque. The confguraton space s sampled at equlbrum, then the knetc energy s gradually removed from the systems that has the possblty (but s not guaranteed to do so to reach the global mnmum. Such procedure s sometmes the only practcal way to fnd the global mnmum for especally hard problems. In mathematcal terms, easy problems are exactly solvable by computer algorthms n polynomal tme, that s, n a number of steps that s a polynomal functon of the dmenson of the problem; hard problems are solved n exponental tme. A problem s NP (nondetermnstc polynomal f ts soluton (f one exsts can be guessed and verfed n polynomal tme. Ths s the knd of problems 34

2 for whch the smulated annealng has been devsed. The determnaton of the structure n clusters s beleved to be a NP-hard problem. 6.1 Classcal Molecular Dynamcs Let us consder the most basc MD : a purely mechancal system of N atoms, enclosed n a volume V (usually wth perodcal boundary condtons, PBC, for a condensed-matter system, havng mechancal energy E = T + E p, where T = 1 2 MṘ2 s the knetc energy of ons, E p = E p ({R} s the nteratomc potental energy. Ths s known as the N V E, or mcrocanoncal, ensemble Dscretzaton of the equaton of moton The numercal soluton (ntegraton of the equatons of motons s generally performed usng the Verlet algorthm. Ths s obtaned from the followng basc and very smple equatons : R (t + δt = R (t + δtv (t + δt2 2M f (t + δt3 6 b (t + O(δt 4 (6.3 R (t δt = R (t δtv (t + δt2 2M f (t δt3 6 b (t + O(δt 4 (6.4 where V = Ṙ are veloctes, f forces actng on on. By summng and subtractng Eqs. (6.3 and (6.4 we get the Verlet algorthm: R (t + δt = 2R (t R (t δt + δt2 M f (t + O(δt 4 (6.5 V (t = 1 2δt [R (t + δt R (t δt] + O(δt 3. (6.6 The veloctes are one step behnd the postons, but they are not used to update the postons. It s possble to recast the Verlet algorthm nto an equvalent form (one gvng exactly the same trajectores n whch both veloctes and postons are updated n the same step. By combnng Eq.(6.3 wth Eq.(6.4 dsplaced n tme by +δt, one fnds V (t + δt = V (t + δt 2M [f (t + f (t + δt] (6.7 R (t + δt = R (t + δtv (t + δt2 2M f (t. (6.8 Note that the update of veloctes requres the forces for the new postons. Ths algorthm s known as Velocty Verlet. Its equvalence to the Verlet algorthm may not seem evdent, but t can be proved qute smply. In spte of hs smplcty, the Verlet algorthm, n any ncarnaton, s effcent and numercally stable. In partcular, t yelds trajectores that conserve to a very good degree of accuracy the energy E. A small loss of energy conservaton, due both to numercal errors and to the dscretzaton, s unavodable, but a systematc drft of the energy s not acceptable. In ths respect Verlet s superor to apparently better (.e. hgher-order schemes. In one of the followng sectons we wll see one reason why ths happen. 35

3 6.1.2 Thermodynamcal averages In the followng we wll use the phase space canoncal varables, collectvely ndcated as R, P, nstead of coordnates and veloctes. Let us consder an observable quantty A = A(R, P. From a practcal pont of vew, the calculaton of thermodynamcal averages n classcal MD s an average over many tme steps: A T = 1 T T A(R(t, P(tdt 1 M A(t n, t n = nδt, t M = Mδt = T. (6.9 0 M n=1 For an ergodc system (that s, one whose trajectores n a suffcently long tme pass arbtrary close to any pont n the phase space, t s beleved that: lm T A T A (6.10 where s the average over the correspondng ensemble: A = ρ(r, PA(R, PdRdP (6.11 where ρ s the probablty of a mcroscopc state. In NV E MD the mcrocanoncal ensemble s sampled: ρ NV E (R, P = g(n δ(h E (6.12 Ω where H s the Hamltonan correspondng to the Lagrangan of Eq.(6.1, E s the mechancal energy (ncludng knetc energy of ons g(n = (h 3N N! 1 for N ndstngushable atoms, and Ω, related to the entropy S by the Boltzmann relaton S = k B log Ω, s the total number of mcroscopc states: Ω = g(n drdpδ(h E. (6.13 The tme step must be as bg as possble n order to sample as much phase space as possble, but at the same tme t must be small enough to allow to follow the moton the ons wth lttle loss of accuracy (whch typcally shows up under the form of non-conservaton of the energy. Typcally δt δt max, where δt max s the perod of the fastest phonon mode: δt max = 1/ω max Verlet algorthm as untary dscretzaton of the Louvllan The tme evoluton of an observable A = A(R, P, t can be wrtten as da dt = ( A A Ṙ + R Ṗ + A = LA + A P t t (6.14 where the operator L s called the Louvllan and s related to the Hamltonan va: LA = [H, A] (6.15 where [...] stands for Posson s brakets. Assumng that A = A(R, P does not depend explctly on the tme, the Louvllan entrely determnes the tme evoluton of A: formally, A(t = e Lt A(t = 0 = U(tA(t = 0. (

4 It can be shown that L s an Hermtan operator and thus U s a untary operator,.e., U = U 1 (unsurprsngly, snce U (t = U( t and tme-reversal symmetry must hold. We can wrte L as L = ( Ṙ + R Ṗ = ( Ṙ + f (6.17 P R P and fnally as a sum of two terms, one actng on coordnates and one on momenta: L = L p + L r, where L p = f, P L r = Ṙ. (6.18 R Untl now, we have just recast the classcal equatons of moton nto a more elegant but not especally useful formalsm. Let us now dscretze the tme evoluton operator, by dvdng t nto N small ntervals δt = t/n, and apply the Trotter approxmaton: e (Lp+Lrt = [e (Lp+Lrδt] N [ N = e Lpδt/2 e Lrδt e Lpδt/2 + O(δt ] 3. (6.19 Remember that L p and L r are operators: the Trotter approxmaton s not trval. Let us apply the operator between square brackets to a pont (R (t, P (t n phase space at tme t. We wll use the known result e a / x f(x = f(x + a (6.20 f a does not depend on x. Snce L p and L r are sums of terms actng on each partcle separately, we can consder ther acton on each partcle ndependently. ( e Lpδt/2 (R, P = R, P + δt 2 f (R (R, P (6.21 e Lrδt ( ( R, P = R + δt P M, P (6.22 ( = R + δt P + δt2 M 2m f (R, P + δt 2 f (R (R, P (6.23 e Lpδt/2 ( ( R, P = R, P + δt 2 f (R (6.24 = ( R + δt M P + δt2 2M f (R, P + δt 2 [f (R + f (R ] (6.25 Notng that f (R = f (t, f (R = f (t+δt, the last expresson s nothng but the velocty Verlet algorthm for (R (t + δt, P (t + δt. In concluson: the Verlet algorthm may be derved by a dscretzaton of the tme evoluton operator that conserves untarty. Such property s crucal for any well-behaved ntegraton algorthm one can thnk of Canoncal ensemble n MD We are often nterested n systems n thermal equlbrum wth a thermal bath at temperature T : the NV T or canoncal ensemble, for whch ρ NV T (R, P = g(n Z e H(R,P/k BT (

5 where Z s the partton functon: Z = g(n drdpe H(R,P/k BT. (6.27 Integraton over P gves for the partton functon of N dentcal atoms: where λ s the thermal wavelength: Z = Z r /(N!λ 3N (6.28 λ = h 2πMkb T (6.29 and Z r s the confguratonal partton functon: Z r = dr 1... dr n e Ep(R/kBT. (6.30 In the canoncal ensemble, the temperature s related to the expectaton value of the knetc energy: N P 2 NV T = 3 2M 2 Nk BT. (6.31 =1 Ths suggests a rather straghtforward algorthm (the velocty rescalng to keep the system at the desred temperature: rescale the veloctes every tme the knetc energy departs from the target value by more than some threshold. Whle effectve to brng the system to the desred temperature, the velocty rescalng does not guarantee that the averages wll be correct, although some varants of t do qute a good job even n ths respect. The correct way to sample the canoncal ensemble s to ntroduce a sutable thermostat that smulates the contact wth a thermal bath at temperature T. A well-known one s the Nosé-Hoover thermostat. Ths ntroduces an addtonal fcttous degree of freedom, producng a dynamcal frcton force havng the effect of heatng ons when the knetc energy s lower than the desred value, coolng them n the opposte case. Specfcally, the equatons of moton become R = f M ζṙ (6.32 ζ = 1 [ N ] M Ṙ 2 3Nk b T (6.33 Q =1 where Q plays the role of thermal mass. The constant of moton for ths system s H = H + Q 2 ζ 2 + 3Nk b T ζ (6.34 but H does not generate the dynamcs (the dynamcs s non-canoncal. It can be shown that such dynamcs samples the canoncal ensemble. Although all thermodynamcal propertes could n prncple be determned from the free energy F, t s not possble to calculate drectly F from a MD smulaton. The free energy (and the partton functon and the entropy as well cannot be smply expressed as a thermodynamcal average (unlke the energy. Specalzed algorthms are needed for free energy calculatons. 38

6 6.1.5 Constant-pressure MD Very often we are nterested n smulatng systems kept at a gven pressure P rather that occupyng a fxed volume V. Constant-pressure MD can be obtaned by addng the volume V or, n a more general case, the cell parameters, to the dynamcal varables. In the smple case of a lqud, one defnes a Lagrangan: L = 1 N 2 M (V 1/3 σ Ep ({V 1/3 σ} W V 2 P V (6.35 =1 where σ = R /V 1/3 are scaled varables, P s the desred external pressure, and W s a (fcttous mass for V. For a sold, we may be nterested n knowng the equlbrum unt cell volume and form under a gven stress state (typcally a constant external hydrostatc pressure rather than workng at fxed cell and calculatng the correspondng stress. In ths case one ntroduces a matrx h, formed by the unt cell vectors a : h = (a 1, a 2, a 3, and defnes scaled varables S as S = h 1 R. The extended Lagrangan becomes L = 1 N M Ṡ GṠ E p ({hs} W Trḣt ḣ P V (6.36 =1 where G = h t h s the metrc tensor. The nterest of varable-cell dynamcs for sold-state systems resde n the possblty to smulate structural phase transtons (under appled pressure but also as a functon of temperature. 6.2 Bblography A Molecular Dynamcs Prmer, by Furo Ercoless, An ntroducton to Classcal Molecular Dynamcs, wth examples and sample codes. 39

7 6.3 Car-Parrnello Molecular Dynamcs Implementatons of MD usng frst-prncple nteratomc potental calculated from DFT, as n Eq.(6.1, are wdely used. All the MD machnery developed for classcal nteratomc potentals can be used. However these mplementaton suffer from a serous drawback. MD s qute senstve to the qualty of forces. If the forces are not the dervatves of the energy wth hgh accuracy, the MD smulaton wll have problems, appearng as a drft of quanttes that should be conserved (lke e.g. the energy from ther values. The error on DFT forces s lnear n the selfconsstency error of the charge densty (whle for the DFT energy t s quadratc. As a consequence, a very good and expensve convergence to self-consstency s requred at every tme step. In 1985 Car and Parrnello (CP proposed a dfferent approach. They ntroduced a Lagrangan for both electronc and onc degrees of freedom: L = µ dr 2 ψ k (r M Ṙ 2 E tot ({R}, {ψ} + ( Λ kl ψ 2 k(rψ l (rdr δ kl k k,l (6.37 whch generates the followng set of equatons of moton: µ ψ k = Hψ k l Λ kl ψ l, M R = E tot R (6.38 where µ s a fcttous electronc mass, and the Lagrange multplers Λ kl enforce orthonormalty constrants. The electronc degrees of freedom are, n the typcal mplementaton, expanson coeffcents of KS orbtals nto PW. The forces actng on them at each tme step are determned by the KS Hamltonan calculated from the current values of ψ k and of R. The sum over orbtals for an nsulatng system of n electrons ncludes n/2 states, assumng that spn polarzaton s neglected (every orbtal s occuped by two electrons. Most CP calculatons are done for aperodc systems or for systems havng a large unt cell (or supercell, so that typcally only the Γ pont (k = 0 s used to sample the Brlloun Zone. Note that the entre Hamltonan operator s not requred: only products Hψ are. The forces actng on ons have the Hellmann-Feynman form: E tot R = k ψ k V R ψ k (6.39 where V s the electron-on nteracton (pseudo-potental. Note however that Hellmann- Feynman theorem holds only on the exact ground state. The relaton of Car-Parrnello forces to Hellmann-Feynman forces s explaned n the next secton. Orthonormalty constrants are exactly mposed to the ψ at each tme step, usng an teratve procedure that explots the fact that the loss of orthonormalty at each tme step s small. The smulaton starts by brngng the electrons to the BO surface (that s, to the ground state at fxed ons and proceeds, usng classcal MD technology, on both electronc and onc degrees of freedom. Wth approprate values of µ and δt, the electrons always reman close to the BO surface, whle the ons follow a trajectory that s close to the trajectory they would follow n the BO approxmaton. 40

8 The Car-Parrnello dynamcs has turned out to be very successful especally n the study of low-symmetry stuatons: surfaces, clusters, lquds, dsordered materals, and for the study of chemcal reactons Why Car-Parrnello works The reasons why the Car-Parrnello dynamcs works so effectvely are qute subtle. The dynamcs for the electrons s purely classcal (and fcttous: t has nothng to do wth real electron dynamcs. As a consequence the energy would tend to equpartton between electronc and onc degrees of freedom, causng an energy transfer from onc to electronc degrees of freedom. Ths does not happen (and must not happen, otherwse the electrons wll leave the BO surface even on long smulaton tmes. If we analyze the dynamcs n terms of oscllators, we fnd that the typcal frequences assocated to the fcttous electron dynamcs are gven by ω el (ɛ ɛ j /µ, f there s a gap n the electronc spectrum. For ons, the oscllator frequences are the typcal phonon frequences. It turns out that, for reasonable values of the gap and of the fcttous electron mass µ, the maxmum phonon frequency s much smaller than the mnmum electron frequency: ωmax ph << ωmn el. The energy transfer from onc to electronc degrees of freedom s as a consequence very small even on long tmes. Ths stuaton generates a fast electron dynamcs that keeps the electrons close to the BO surface and averages out the error on the forces, so that the much slower onc dynamcs turns out to be correct (that s, very close to the BO dynamcs one would obtan from hghly converged selfconsstency. A detaled explanaton s contaned n a 1991 paper by Pastore, Smargass, and Buda. If there s no gap n the electronc spectrum, or f the gap s too small, the above pcture breaks down. It may be needed to add separate thermostats to onc and electronc degrees of freedom n order to prevent the flow of energy from the former to the latter Choce of the parameters The choce of the electronc mass µ must strke a compromse between conservaton of adabatcty (favored by small values of µ, see above and maxmum admssble tme step (that s lmted by the maxmum electronc frequency, so that the heavest µ, the smaller ωmax, el the larger δt max. Typcally µ 200 amu (1 amu=1 electron mass. For large gap systems, such as SO 2 or H 2 O, n whch adabatcty problems are mnor, µ may be ncreased up to amu and even more. Such values of µ correspond to a typcal tmestep of fs. In order to ncrease the tme step, t s customary to ntroduce the so-called mass precondtonng. In a PW bass set, the tme step s lmted by hgh-frequency components wth the largest G vector. These components are domnated by the knetc energy h 2 G 2 /2µ. Snce electronc masses are fcttous, t s advantageous to ntroduce a mass that for hgh-frequency components goes lke µ(g µ(1 + G 2. The correspondng equatons of motons are only slghtly more complex. It should be notced, however, that too heavy electron masses adversely affect the qualty of smulaton va an electron drag effect. The electron moton follow the onc moton wth some delay, thus ntroducng a drag force that appears as f the ons were heaver than ther masses. Ths mass renormalzaton must be taken nto account when 41

9 extractng vbratonal frequences from MD runs. In some cases, ths effect can ntroduce a nonneglgble devaton from the true onc dynamcs. 6.4 Bblography D. Marx and J. Hutter, n Modern Methods of Quantum Chemstry, (John von Neumann Instute for Computng, FZ Jülch, 2000 p All you wanted to know about Car-Parrnello molecular dynamcs G. Pastore, E. Smargass, F. Buda, Phys. Rev. A 44, 6334 (1991. The paper that explans why the Car-Parrnello dynamcs work. 42