The maximum-entropy approach to the solution of under determined inverse problems is studied in detail in the context of the classical moment problem. In important special cases, such as the Hausdorffmoment problem, we establish necessary and sufficient conditions for the existence of a maximum-entropy solution and examine the convergence of the resulting sequence of approximations. A number of explicit illustrations are presented. In addition to some elementary examples, we analyze the maximum-entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems. We also briefly indicate possible applications to the Lee-Yang theory of Ising models, to the summation of divergent series, and so on. The general conclusion is that maximum entropy provides a valuable approximation scheme, a serious competitor of traditional Pade-like procedures.

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We studied the dynamical properties of Au using our previously developed tight-binding method. Phonon-dispersion and density-of-states curves at T=0 K were determined by computing the dynamical-matrix using a supercell approach. In addition, we performed molecular-dynamics simulations at various temperatures to obtain the temperature dependence of the lattice constant and of the atomic mean-square-displacement, as well as the phonon density-of-states and phonon-dispersion curves at finite temperature. We further tested the transferability of the model to different atomic environments by simulating liquid gold. Whenever possible we compared these results to experimental values.

We present calculations of energetic, electronic, and vibrational properties of silicon using a nonorthogonal tight-binding ͑TB͒ model derived to fit accurately first-principles calculations. Although it was fit only to a few high-symmetry bulk structures, the model can be successfully used to compute the energies and structures of a wide range of configurations. These include phonon frequencies at high-symmetry points, bulk point defects such as vacancies and interstitials, and surface reconstructions. The TB parametrization reproduces experimental measurements and ab initio calculations well, indicating that it describes faithfully the underlying physics of bonding in silicon. We apply this model to the study of finite temperature vibrational properties of crystalline silicon and the electronic structure of amorphous systems that are too large to be practically simulated with ab initio methods.

We present an alternate approach to parametrizing the expression for the total energy of solids within the second-moment approximation ͑SMA͒ of the tight-binding theory. In order to obtain the necessary parameters, we do not use the experimental values of the lattice constant, the elastic constants, and the cohesive energy, but we fit to the total energy obtained from first-principles augmented-plane-wave calculations as a function of volume. In addition, we shift the total-energy graphs uniformly so that at the minimum they give the experimental value of the cohesive energy. We have applied the above methodology to perform molecular-dynamics simulations of the noble metals. For Cu and Ag our results for vacancy formation energies, relaxed surface energies, phonon spectra, and various temperature-dependent quantities are of comparable accuracy to those found by the standard SMA, which is based on fitting to several measured data. However, our approach does not seem to work as well for Au. ͓S0163-1829͑97͒00903-X͔

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In this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients. The perturbation expansion for the ground-state energy E(g) of the octic oscillator defined by H=p2/2+x2/2+gx8 is a series of the form E(g)∼ 1/2 +∑(−1)n+1 Angn. This series is terribly divergent because for large n the perturbation coefficients An grow like (3n)!. This growth is so rapid that the solution to the moment problem is not unique and ordinary Padé summation of the divergent series fails. A completely different kind of procedure based on the principle of maximum entropy for reconstructing the function E(g) from its perturbation coefficients is presented. Very good numerical results are obtained.

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