PHYSICAL REVIEW B VOLUME 48, NUMBER 10 1 SEPTEMBER 1993-II Spectral shapes of Lennard-Jones chains A. Cuccoli and V. Tognetti Dipartimento di Fisica dell'Uni Uersita di Firenze, Largo E. Fermi 2, I-50l25 Firenze, Italy A. A. Maradudin Department of Physics, University of California, Irvine, California 9271 7 A. R. McGurn Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008 R. Vaia Istituto di Elettronica Quantistica del Consiglio Ãazionale delle Ricerche, Via Panciatichi 56/30, I 501-27 Firenze, Italy (Received 11 March 1993) We show how the results of classical molecular-dynamics simulations can be used to improve the cal- culation of quantum spectral densities of anharmonic crystals. In particular we show that the spectral density C(k, co) of the displacement correlation function for a quantum chain of N atoms interacting through a nearest-neighbor Lennard-Jones potential, can be calculated accurately in the following way. C(k, co) is expressed in the form of a continued fraction, whose coeKcients are given in terms of its even-frequency moments. The latter, up through the sixth, are calculated from an e8'ective potential that includes the efFects of quantum fluctuations. The continued fraction is then terminated by the use of a termination parameter that is determined from a fit of the same continued fraction to the spectral den- sity calculated by means of a classical molecular-dynamics simulation. with v(r)=4e ' 12 '6 In a recent paper' we faced the problem of evaluating the time-dependent displacement-displacement correla- tion function of a quantum chain of atoms interacting through a nearest-neighbor Lennard- Jones potential. The Hamiltonian used to model the system was +v(Q; — Q;, ) 2m The function go of the complex variable z admits the fol- lowing continued fraction representation: f„(z) = 1 z+&. +if. +i(» ' and the function g„(t), in the time domain, is called the nth memory function. The coefficients 5„are related to the frequency moments defined above. Because in Ref. 1 the quantum moments of the spectral density were evalu- ated up to the sixth one by the effective potential method, ' the explicit expressions for the first three 6's can be used: po(k) C(k, co) =go(k)F(k, to) = Re[go(k, ito)] . (5) The spectral shape, i.e. , the Fourier transform of the correlation function of the displacement u;(t) =Q; (t) — ia, C(k, co)= — ge '"" j'f dt e' '([u;(t) — uj(0)] ), EJ (3) a being the lattice constant, was approached by expan- sions based on the knowledge of its even moments: p, „(k) = f "dco co'"C(k, co), (4) taking into account that the odd moments are vanishing. Indeed, starting from the knowledge of the frequency moments, a reconstruction of the function C(k, to) itself can be attempted by the continued fraction expansion proposed by Mori. ' Let us define P2 P4 P2 1 P6 1 ~ 2 ~ 3 Po P2 Po &2 P2 2 P4 While the 5„vanish for n ) 1 for the harmonic model, the knowledge of the first 5's allows us to reproduce the spectral shape for anharmonic systems if reasonable ap- proximations for the nth memory function (6) are intro- duced. However, as n increases, the moments are more and more determined by the high-frequency part of the spectrum (i.e. , by the short-time behavior of correlations), and therefore the choice of the termination can be a source of arbitrariness, since the latter is generally related with the long-time behavior of the correlations. Unless some insights into the long-time behavior of the dynamic variables of the system could be obtained, the reconstruc- tion of spectral shapes of strongly anharmonic systems can therefore suffer from poor control on the validity of the approximations. 0163-1829/93/48(10)/7015(5)/$06. 00 48 7015 1993 The American Physical Society