1. Php. Chem. Solids Vol. 56, No. 3/4. pp. 501-505. 1995 Copyright 0 1995Elwier Science Lid Prinwd in Great Britain. All rights reserved 0022.3697/95 S9.M + 0.00 FIRST-PRINCIPLE-CONSTANT PRESSURE MOLECULAR DYNAMICS M. BERNASCONI,? G. L. CHIAROTTI,? P. FOCHER,? S. SCANDOLO,? E. TOSATTIt and M. PARRINELLOS tInternational School for Advanced Studies (SISSA), I-34014 Trieste, Italy fMax-Planck-Institut fur Festkoerperforschung, D-70569 Stuttgart, Germany Abstract-We present a new method for first-principles numerical simulation of solid-solid phase transformation. The method is applied to the study of pressure induced transformations in silicon and carbon. Keywords: A. semiconductors, C. ab initio calculations, D. phase transition. 1. INTRODUCTION 2. METHOD Solid-solid structural phase transitions with increas- ing pressure and/or temperature are ubiquitous in condensed matter. Existing ab-initio theory of these phase transitions has been mostly restricted to com- paring relative energies of known structural phases at different volumes and at zero temperature. A dynam- ical simulation of these transitions under real con- ditions of pressure and temperature, while still retaining the first principles accuracy required for reliable predictions of new structures, is very desir- able. We have devised an ab-initio molecular dynam- ics scheme which allows the simulation of these phenomena with the correct quantum-mechanical description of interatomic forces and internal stress, along with the correct statistical mechanics of ionic degrees of freedom. The method is obtained by efficiently combining the Car-Parrinello [l] method with the Parrinello-Rahman [2] method to include a variable cell shape. Unlike the rigid cell case, within this scheme phase transformations may spon- taneously take place during the simulation with vari- ation of external pressure. Prediction of new phases may thus be obtained without any initial guess on the final stable structure. The method has been demon- strated by simulating the metal-insulator transition in silicon from the diamond to the simple hexagonal structure under high pressure as reported in a recent letter [lo]. In this paper we discuss in depth the method, including a development devised for dealing with the energy cutoff in the electronic plane wave expansion during the simulation. In Section 2 we describe our new simulation scheme and in Section 3 we report our results on the phase transitions in silicon from diamond to simple hexagonal structures, and in carbon from graphite to diamond. Following the idea of Parrinello and Rahman (PR) [2] for the classical MD, we have extended the fixed-cell original Car-Parrinello (CP) [I] scheme, by including the possibility of shape and volume fiuctu- ations of the simulation cell. The basic idea is to consider an extended Lagrangian, where the edges of the simulation cell, a, b, c are additional degrees of freedom, whose trajectories are determined by appro- priate generalized forces. Defining the matrix b = (a, b, c ), the position R of a particle in the cell can be written in terms of the scaled coordinates S as: R = bS. The presence of the electronic wave-func- tions in the CP formulation makes the definition of this fictitious system slightly subtler here than in the classical case. The Kohn-Sham wave-functions $,+(r), defined and normalized on cell h, are not independent fields, since h themselves are now Lagrangian degrees of freedom. We then define a wave-function II/ onto the scaled variable space s = b-‘r (normalized on the unitary cube) t,Gh(r) = l/fi$(h-’ r) = l/$&(s), where Q is the cell volume. The new CP-PR Lagrangian is: 9’ = p C .s ds)$,(s)12 + ;I M,&‘=$) zyxwvutsrqponm I - NII/ ,), P&H + c 4 (s dsl(l :Wll/i 6) - 4, 8, > + f WTr(ti%) - pR, (1) where the integrals are taken on the scaled cell (of unit volume). E[{$,}. {hS,}] is the DFT-LDA [3] zyxwvu 501