VOLUME 55, NUMBER 22 PHYSICAL REVIEW LETTERS 25 NOVEMBER 1985

Unified Approach for Molecular Dynamics and Density-Functional Theory

R. Car International School for Advanced Studies, Trieste, Italy and M. Parrinello Dipartimento di Fisica Teorica, Universita di Trieste, Trieste, Italy, and International School for Advanced Studies, Trieste, Italy (Received 5 August 1985)

We present a unified scheme that, by combining molecular dynamics and density-functional theory, profoundly extends the range of both concepts. Our approach extends molecular dynamics beyond the usual pair-potential approximation, thereby making possible the simulation of both co- valently bonded and metallic systems. In addition it permits the application of density-functional theory to much larger systems than previously feasible. The new technique is demonstrated by the calculation of some static and dynamic properties of crystalline silicon within a self-consistent pseu- dopotential framework.

PACS numbers: 71.10.+x, 65.50.+m, 71.45.Gm

Electronic structure calculations based on density- very large and/or disordered systems and to the com- functional (DF) theory' and finite-temperature com- putation of interatomic forces for MD simulations. puter simulations based on molecular dynamics (MD) We wish to present here a new method that is able have greatly contributed to our understanding of to overcome the above difficulties and to achieve the condensed-matter systems. MD calculations are able following results: (i) compute ground-state electronic to predict equilibrium and nonequilibrium properties properties of large andlor disordered systems at the of condensed systems. However, in all practical appli- level of state-of-the-art electronic structure calcula- cations MD calculations have used empirical intera- tions; (ii) perform ah initio MD simulations where the tomic potentials. This approach, while appropriate for only assumptions are the validity of classical mechan- systems like the rare gases, may fail for covalent ics to describe ionic motion and the Born- andi or metallic systems. Furthermore, these calcula- Oppenheimer (BO) approximation to separate nuclear tions convey no information about electronic proper- and electronic coordinates. ties. On the other hand, DF calculations have provid- Following Kohn and Sham3 (KS) we write the elec- ed an accurate, albeit approximate, description of the tron density in terms of occupied single-particle ortho- chemical bond in a large variety of systems, ' but are normal orbitals: n(r) = X,. ~tlt;(r) ~2. A point of the computationally very demanding. This has so far pre- BO potential energy surface is given by the minimum cluded the application of DF schemes to the study of with respect to the Q;(r) of the energy functional, I — E[(p,},(Rt), (ct„}]= X,. d3r tlt (r) [ (h2/2m)Vz]tlt;(r) + U[n(r), {Rt), (o,„}]. Here (Ri) indicate the nuclear coordinates and (n„} are all the possible external constraints imposed on the ing with the size of the problem. Since the whole pro- system, like the volume fl, the strain etc. The e„„, cedure has to be repeat ed for new atomic confi- functional U contains the internuclear Coulomb repul- any guration, the theoretical prediction of the equilibrium sion and the effective electronic potential energy, in- geometries, when these are not known from experi- eluding external nuclear, Hartree, and exchange and still remains an correlation contributions. ment, u nsolved problem in most cases. We adopt a quite different and the In the conventional formulation, minimization of approach regard minimization of the KS functional as a complex optim- the energy functional [Eq. (1)] with respect to the or- ization problem which can be solved the bitals p;, subject to the orthonormality constraint, by applying concept of simulated an nealing, recently introduced by leads to the self-consistent KS equations, i.e., Ktrkpatrtck, Gelatt, and Vecchi. 4 In this approach an objective function O({p)) is minimized relative to the , y, (r) = e;y;(r). (2) 2m ri'+,6n(r) parameters (p), by generation of a succession of (p) s with a Boltzman-type probability distribution The solution of Eq. (2) involves repeated matrix diag- ~ exp( —O((P) )/T) via a Monte Carlo procedure. onalizations with a computational effort rapidly grow- For T 0 the state of lowest O((p) ) is reached un- 1985 The American Physical Society VOLUME 55, NUMBER 22 PHYSICAL REVIEW LETTERS 25 NOVEMBER 1985 less the system is trapped into some metastable state. Vecchi, can be applied efficiently to minimize the KS In our case the objective function is the total-energy functional. This approach, which may be called functional and the variational parameters are the coef- "dynamical simulated annealing, " not only is useful as ficients of the expansion of the KS orbitals in some a minimization procedure but, as we demonstrate convenient basis and possibly the ionic positions here, it allows also the study of finite temperature and/or the (n„}'s. We found that a simulated anneal- properties. ing strategy based on MD, rather than on the Metrop- In our method we consider the parameters (p, ) olis Monte Carlo method of Kirkpatrick, Gelatt, and (Rt), (n„) in the energy-functional [Eq. (1)] to be dependent on time and introduce the Lagrangean

. L = p, r I + + p, E X, 2 „d I+; Xt & MtRt X„2 „n„— [(+;), (Rt), (n„}], (3) where the Q, are subject to the holonomic constraints laxation, and volume and strain relaxation are J{ d'r y, (r, t)yj(r, t) =5,, (4 achieved simultaneously Th.e amount of classical kinetic energy is a measure of the departure of a sys- In the dot indicates time derivative, are the Eq. (3) Mt tem from the self-consistent minimum of its total en- physical ionic masses, and and are arbitrary p, p, , ergy. parameters appropriate units. of It should be stressed that the dynamical simulated The Lagrangean in Eq. generates a dynamics for (3) annealing technique introduced above is a method of the parameters 's, 's, and 's through the (p;) (Rt) {n„) quite general applicability in the context of functional equations of motion: minimization. As such it can be useful in many areas of physics. For instance, it can be applied to the study pP;(r, t) = —oE/o$,'(r, t) + t), (5a) XkA,„P„(r, of classical field theories or to obtain the ground-state ~ ~ IRI = —PRIE, (Sb) energy in Hartree-Fock or configuration interaction schemes. We also observe that, as far as functional p„n„= —, (8E/Bn„), (5c) minimization is concerned, Newtonian dynamics may be conveniently where A;k are Lagrange multipliers introduced in order replaced by Langevin or other types to satisfy the constraints in Eq. (4). The ion dynamics of dynamics. in Eqs. (5) may have a real physical meaning, whereas In order to illustrate how our method works in prac- the dynamics associated with the (p;)'s and the {n„}'s tice, we present results obtained for the ground-state is fictitious and has to be considered only as a tool to electronic structure of Si as follows. We have con- perform the dynamical simulated annealing. Equation sidered a simple cubic supercell containing eight atoms (3) defines a potential energy E and a classical kinetic subject to periodic boundary conditions. We have energy L given by used a local pseudopotentials and a local-density ap- proximation to the exact exchange and correlation ' ' K= X,. —,p, JI d r IP;I'+ Xt ,'MtRt + X„——,p, „n„. functional. The single-particle orbitals for the valence electrons have been expanded in plane waves with an energy cutoff of 8 Ry, which amounts to including 437 The equilibrium value (K) of the classical kinetic en- plane waves at the I point. For simplicity, only the I ergy can be calculated as the temporal average over the point of the Brillouin zone (BZ) of the supercell has trajectories generated by the equations of motion [Eqs. been considered in the evaluation of the energy func- 'o (5)] and related to the temperature of the system by tional. This leads to a total of 16&& 437 complex elec- suitable normalization. By variation of the velocities, tronic variational parameters, since sixteen is the i.e., the {P;}'s, {Rt}'s, and (n„)'s, the temperature of number of doubly occupied KS levels. A simulated the system can be slowly reduced and for T 0 the annealing run is illustrated in Fig. 1. The lattice equilibrium state of minimal E is reached. At equilib- parameter was allowed to vary while the ions were kept rium Q;=0, Eq. (Sa) is identical within a unitary in their perfect diamond arrangement. The total ener- transformation to the KS equation [Eq. (2)], and the gy, the lattice parameter, and the eigenvalues of the eigenvalues of the A matrix coincide with the occupied matrix of the Lagrangean multipliers are plotted as KS eigenvalues. Only when these conditions are satis- functions of the simulation "time. " The initial condi- fied does the Lagrangean in Eq. (3) describe a real tions for the electronic orbitals were fixed by filling physical system whose representative point in config- the lowest available plane-wave states and giving a urational space lies on the BO surface. For large sys- Maxwellian distribution of velocities to the com- tems our scheme is more efficient than standard diago- ponents of the fields. The value of p, was chosen to be 5 nalization techniques. Furthermore, in the present 1 a.u. The mass p, & associated with variation in the approach, diagonalization, self-consistency, ionic re- volume was taken to be 10 5 a.u. The deerlet algo- 2472 VOLUME 55, NUMBER 22 PHYSICAL REVIEW LETTERS 25 NOVEMBER 1985

5x IO I~ sec

- W cn E . b) CO lo.o — -. a). ~ I I 0.025 r25' ~ I- —3.5 0.000 ------.-- - Z Z UJZ ~ 9.5 0 D -0.025- UJ oo :9.0 LLJ 4.I925- ~ X — Ep I- ~ 4.0 ~ 0 4.I935- . = 0 I- - 8.5 -¹ «f ~ 0.03 0.02 0.0I- 04. 1"I 0.5 CO &g) -0.5 LU ~ Q Z Izs 200 400 200 400 cf W 0 t /4& &/h, t z o FIG. 2. From top to bottom, temporal evolution of aver- UJ ~ ~o age atomic displacement along er, potential energy per Lai I— 25

atom, and I and I ~ multiplets for two different MD runs. —0.5 tO The lattice constant was taken to be equal to the experimen- IA lA 0 C4 O 0 tal value of 10.26 a.u. ; At and p, were taken to be 10 and 300 O O 0 0 a.u. The dashed line in l00 200 300 , respectively. the second panel from the top indicates the T=O ground-state energy. The trian- gles indicate a doubly degenerate level. FIG. 1. Evolution of total energy per atom, lattice con- stant, and eigenvalues of the A matrix, during a typical dynamical annealing run. The partial averages of the classi- dynamics generated via Eqs. (5) provides a good cal kinetic energy E during each subsection of the run are representation of the actual dynamics of a physical sys- indicated in the lower part of the picture. For E 0 the tem. eigenvalues of the A matrix tend to the KS eigenvalues. In Fig. 2 we report the results of two different sets The various multiplets are labeled according to the sym- of calculations in which we have performed dynamical metry of the diamond lattice. simulations for our model. In these calculations it was not necessary to reoptimize the electronic variational parameters at each point along the trajectory, since the rithm" with a time step of 0.1 a.u. was used and the thickness of the BO surface never exceeded the value values of the Lagrange multipliers were determined by of 7X 10 6 a.u. per atom, a rather small fraction of the the method of Ryckaert, Ciccotti, and Berendsen. ' potential energy variation. This is perfectly adequate After some initial equilibration the temperature was to represent the interionic forces in this particular case. progressively reduced to very small values. A satisfac- If the thickness of the BO surface were too large, not tory degree of convergence is seen to be achieved after only would the forces be incorrectly estimated but also —200 time steps, when our results agree within nu- they might depend upon the path along which a given merical errors with those of a conventional self- point of the potential surface is consistent calculation for the same model. '3 energy approached. On the other hand, for very small thicknesses the low We can consider now a situation in which the ions, velocities of the electronic variational parameters to which we associate their actual physical masses, are might lead to intolerably long relaxation times. In allowed to move at a given temperature, while the such a case a compromise would be necessary. In Fig. kinetic energy of the electronic variational parameters 2(a) the atoms were initially displaced from their per- remains equal to zero. In this case the electrons are at fect lattice position according to the eigenmode er any time in their ground state and the ions move 25 under the action of BO forces. This can be achieved corresponding to the optical phonon mode at the I either by conveniently reoptimizing the electronic vari- point of the diamond lattice. The system undergoes ational parameters or by realizing a metastable situa- slightly anharmonic oscillations whose frequency is 20 tion in which the kinetic energy associated with the THz, in perfect agreement with the results of a static s remains always very small compared to the typical frozen-phonon calculation for the same model, '3 variations of the potential energy of the system. This showing that the thickness of the BO surface was ade- is equivalent to giving the BO surface a finite thickness quate. In Fig. 2 (a) we also report how the effect of proportional to the temperature associated with the the ionic oscillatory motion is reflected in some elec- s. If this temperature remains very small, the ion tronic properties. Notice that the threefold degenerate 2473 VOLUME 55, NUMBER 22 PHYSICAL REVIEW LETTERS 25 NOVEMBER 1985 topmost I, level splits, in perfect phase with the ion- Nazionale di Struttura della Materia del Consiglio Na- ic motion, into a singlet and a doublet, whereas the zionale delle Richerche and by the Ministero della low-lying I t state is hardly affected by the ionic Pubblica Istruzione. motion. These results are contrasted with those re- ported in Fig. 2(b). Here the ions were first randomly displaced from their equilibrium position and a simu- tSee, for instance, Theory of the Inhomogeneous Electron lated annealing was performed in order to bring the Gas, edited by S. Lundqvist and N. M. March (Plenum, New electrons in the corresponding ground state. The ions York, 1983). were then allowed to move. After some equilibration zA. Rahman, in Correlation Functions and Quasiparticle In the average kinetic energy associated with ionic motion teraetions in Condensed Matter, edited by J. Woods Halley, had a value corresponding to —250 K and the NATO Advanced Study Series Vol. 35 (Plenum, New York, behavior of the system was as illustrated in Fig. 2(b). 1977). 3W. Kohn and L. J. Sham, Phys. Rev. A1133 The projection of the ionic displacement along the er 140, (1965). 25 4S. Kirkpatrick, C. D. Gelatt, Jr. , and M. P. Vecchi, Sci- eigenvector and the electronic properties do not show ence 220, 671 (1983). any apparent correlation. The degeneracy of the 12s, 5For instance, if the %occupied single-particle orbitals are into one-electron eigenstate is completely lifted by thermal expanded M plane waves (M )& N), a standard diago- nalization requires O(M3) re- disorder, while the I state still remains hardly affect- operations, whereas Eq. (5a) t quires both O(NMln M) and O(N M) operations. ed by the ionic motion. P. Carnevali and A. Selloni, private communication. The calculations presented here can all be performed P. J. Rossky, J. D. Doll, and H. L. Friedman, J. Chem. on a VAX-like minicomputer. Access to supercom- Phys. 69, 4628 (1978); C. Pangali, M. Rao, and B. J. Berne, puters can make possible the simulation of larger sys- Chem. Phys. Lett. 55, 413 (1978). tems and more realistic models. Because of the simpli- J. A. Appelbaum and D. R. Hamann, Phys. Rev. B 8, city of Newton's equations the computer code can be 1777 (1973). fully vectorized with not much effort. However the 9J. P. Perdew and A. Zunger, Phys. Rev. 8 23, 5048 main advantages of the present approach lie in its abili- (1981). ty to perform a global minimization of the energy DF ~ The sampling of the BZ and the pseudopotential used in and, more importantly, in offering a convenient and, this paper are not very accurate and should be replaced with better in in principle, exact tool for studying finite temperature ones more realistic calculations. t tL. Verlet, Phys. Rev. 159, 98 (1967). effects and dynamical properties. & J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. We benefited from discussions with A. Baldereschi, Comput. Phys. 23, 327 (1977). P. Carnevali, A. Nobile, S. T. Pantelides, A. Selloni, &3When accurate pseudopotentials are used together with E. Tosatti, and A. R. Williams. Special thanks are due an accurate sampling of the BZ, local-density calculations to S. Baroni for precious suggestions and valuable agree very well with experiment [see M. T. Yin and M. L. help. This work has been supported by the Gruppo Cohen, Phys. Rev. B 26, 3259 (1982)].