positions, necessitating the simultaneous use of quantum mechanics to compute intermolecular forces and classical mechanics to compute the nuclear motions (Car and Parrinello, 1985).

The quantum mechanical description of a bulk material is given by the Schrödinger equation, which is a partial differential equation (PDE) involving on the order of 1023 degrees of freedom. Cohesive energies, of solids, for instance, are obtained from differences in eigenvalues of this equation for the solid and the separated atoms, but the accuracy of solution required is enormous because this cohesive energy represents only on the order of 10-4 of the total eigenvalue. Thus, mathematical approximations are necessary, and this has led to the development of a wide variety of methods, including model Schrödinger equations that contain within them a set of empirical interactions, pseudopotential methods that likewise rely on empirical information or (more recently) on accurate quantum mechanical calculations for atomic systems upon which curves/surfaces are fitted, and the nonempirical density functional and ab initio quantum mechanical methods.

Density functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965; Callaway and March, 1984; see also Interfaces in Polymer Systems in Chapter 3) provides a framework for the calculation of the electronic structure and total energy of any solid-state or molecular system. However, the underlying Hohenberg-Kohn theorem only proves the existence of the density functional, but does not describe its exact form. Thus, various approximate and heuristic density functionals are used in practice. Rather little has been done to compute these density functionals from first principles theory (Levy, 1991; Freed and Levy, 1982), and mathematical assistance would be welcomed (see Chapters 3 and 8) in proving theorems concerning properties of density functionals (Levy, 1991) and in developing new approximation schemes as means for systematically improving available and perforce approximate density functional methods.

When the atomic positions are assumed known, density functional methods reduce the quantum mechanical description to a nonlinear integrodifferential eigenvalue problem for an effective single-particle Hamiltonian operator whose potential energy depends on the operator's own eigenfunctions. The problem is usually solved iteratively until self-consistency is achieved between the eigenfunctions and the potential energy. The development of fast Fourier transform methods has been a key mathematical contribution to improving the efficiency of this process. The effective single-particle Hamiltonian operator's eigenvalues are identified with the electron energy levels and the eigenfunctions with the electron wave functions. The eigenfunctions and eigenvalues are then used to calculate densities of states, optical spectra, carrier life times, and so forth. Most implementations of density functional methods employ what is called the local-density approximation to incorporate exchange and correlation in the effective single-particle potential energy. In recent years, a number of methods, including quantum Monte Carlo approaches (Zhu and Louie, 1991), have been developed to go beyond this approximation, resulting in significant improvement in the calculated electron energy levels. There is growing use of electronic structure calculations based on approximate solution of the Schrödinger equation for a cluster of atoms (see Chapter 8). The main theoretical problem here lies in extrapolating solutions for small clusters to the bulk or in extracting parameters for model Hamiltonian treatments of, for instance, high-temperature superconductivity.

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