Crystal hamiltonian

The hope of understanding the concept of molecular structure quantum-mechanically would obviously be at its most realistic for the smallest of molecules at the absolute zero of temperature. However, under these conditions completely different pictures emerge for the molecule in, either total isolation, or in a macroscopic sample. In the latter case the molecule appears embedded in a crystal, which is quantum-mechanically described by a crystal hamiltonian with the symmetry of the crystal lattice. The isolated molecule has a spherically symmetrical hamiltonian. The two models can obviously not define the same quantum molecule. 

Let us consider a crystal of N unit cells, each one containing a nuclei. In the expansion of the electronic energy in powers of nuclear displacements around their equilibrium positions at T = 0, the linear term vanishes. It is usual to make the harmonic approximation, keeping only the quadratic terms. Then the crystal hamiltonian is expressed as a function of the momentum P a, of the mass mnt, and of the position rM of each nucleus hoc (n indexes the cell and a the coordinate) ... 

The crystal hamiltonian embodying the above approximations is written in second quantization ... 

The translational symmetry of a (one-dimensional) crystal lattice demands that the potential in the crystal Hamiltonian... 

The boundary conditions are periodic and the number of allowed values of the wave vector k is equal to the number of unit cells in the crystal. These eigenfunctions constitute the basis for the infinite-dimensional Hilbert space of the crystal Hamiltonian and any function with the same boundary conditions can be expressed as a linear combination of functions in this complete set. 

All familiar molecular structures have been identified in the crystalline state. To describe such molecules quantum-mechanically requires specification of a crystal Hamiltonian. This procedure is never attempted in practice. Instead, history is taken for granted by assuming a specific connectivity among nuclei and the crystal environment is assumed to generate well-defined conformational features characteristic of all molecules. Although these decisions may not always be taken consciously, the conventional approach knows no other route from wave equation to molecular conformation. 

The representation of the crystal Hamiltonian in the form (2.4) is still valid, but the explicit expressions for and are more complicated. To... 

Given a state of interest /q) (normalized to 1) and a model crystal Hamiltonian H, we consider the Green s function... 

We assume that the crystal Hamiltonian is expressed in a local basis representation in the form of Eq. (2.4) ... 

The last term describes the coupling of the electron and phonon variables. Its complication is the price for the accuracy of the intersite interaction operator and the crystal Hamiltonian overall. 

B. The Complete Crystal Hamiltonian and the Coupling between Lattice... 

Now that we have expressed the intermolecular potential, it is easy to write down the crystal Hamiltonian. We associate with each point P = n, / of the lattice a molecule with position vector rP = R, + u,. The vector Rp denotes the position of the lattice point P, i.e., R, = R + R, with Rb being the position vector of the origin of the unit cell to which P belongs and R,- the position vector of P relative to this origin. The displacement vector u/> describes the position of the center of mass of the molecule at P relative to the lattice point P. The Hamiltonian then reads... 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

The exact crystal Hamiltonian H is given by Eq. (23) and H0 is of the form given by Eq. (47) the force constants Fp f> are not given by Eq. (48), however, but they are chosen such as to minimize Avar- Neglecting the difference between the exact kinetic energy operators [(25) and (26)] and their harmonic approximations (see Section III,A), one obtains... 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

The meaning of the symbols is explained in Section III.C [Eq. (83)] H is the exact crystal Hamiltonian [Eq. (23)]. This time, however, we choose as the approximate Hamiltonian H0 a sum of single-particle Hamiltonians ... 

Since the O2 molecule carries a triplet spin momentum, the spin Hamiltonian (140) has to be added to the Hamiltonian (23), which contains the kinetic energies and the spin-independent part of the intermolecular O2-O2 potential, in order to obtain the crystal Hamiltonian for solid 02. The spin-independent O2-O2 potential can be partly extracted from the ab initio calculations on the (62)2 dimer by averaging the calculated interac-... 

Jansen and van der Avoird (1985) have also made spin-wave calculations as described earlier. The RPA equations with the effective spin Hamiltonian (140), averaged over the translations and librations, could be solved analytically for any wave vector q. The optical (q = 0) magnon frequencies emerging from these calculations are 6.3 and 20.9 cm-1, in reasonable agreement with the experimental values 6.4 and 27.5 cm-1. This agreement is very satisfactory if we realize that the spin Hamiltonian has been obtained from first principles, with none of its parameters fitted to the magnetic data. We conclude that the RPA model, both for the lattice modes and the spin waves, when based on a complete crystal Hamiltonian from first principles, yields a realistic description of several properties of solid O2 that were not well understood before. 

In crystalline semiconductors, it is relatively easy to understand the formation of gaps in energy states of electrons and hence of the valence and conduction bands using band theory (see Ziman, 1972). Band structure arises as a consequence of the translational periodicity in the crystalline materials. For a typical crystalline material which is a periodic array of atoms in three dimensions, the crystal hamiltonian is represented by a periodic array of potential wells, v(r), and therefore is of the form, 7/crystai = ip l2m) + v(r), where the first term p l2rri) represents the kinetic energy. It imposes the eondition that the electron wave functions, which are solutions to the hamiltonian equation, H V i = E, Y, are of the form... 

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