Crystal wave function

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

The method of treatment of the shallow traps by expanding the perturbed wave functions in a series of the delocalized pure-crystal wave functions r [Eq. (10)] is also convenient for intensity calculations. Since 0 is the only spectrally active function its... 

However the vibrationally induced transitions are a different case. We must now specify in the crystal wave functions (3.1) the molecular vibrational wave functions or( ) for a molecule in the i-th vibrational quantum level and the r-th electronic state. If the transition is allowed, leading to the zero wave-vector state (3.1), the a are totally symmetrical, and the transition moments get multiplied by Franck-Condon factors (3.5),... 

In addition to the wave functions of uniform amplitude throughout the crystal, wave functions may exist which have appreciable amplitude only in the surface region they represent localized electronic states at metal surfaces. 

In the non-interacting particle model the optical conductivity may be calculated from the band structure and crystal wave functions. 

There is also a lack of theoretical calculation of the optical conductivity from energy bands and crystal wave functions. Computational techniques are now available (WUliams et al., 1972 Janak et al., 1975). It is hoped that progress will be made in this direction in the next few years. 

With the preceeding general statements as a preface we can discuss the particular material included in the present discussion. The quantum mechanical expression for optical rotation of oriented molecules will be discussed first. Oriented molecules are emphasized because most of the previous theoretical discussions have been limited to randomly oriented molecules. The formalism relates the optical rotation to wave functions for the entire molecule or crystal the next step is to obtain useful expressions for these molecular or crystal wave functions. Of course, obtaining useful wave functions is the key to all of molecular quantum mechanics. For optical activity we are interested in large molecules, therefore, usually we will not try to calculate wave functions directly. Instead we will try to relate the optical rotation to other measurable properties of... 

In the Bra-Ket notation, the LCAO ansatz (Eq. (5.1)) for the crystal wave function reads... 

Semiconductor materials are rather unique and exceptional substances (see Semiconductors). The entire semiconductor crystal is one giant covalent molecule. In benzene molecules, the electron wave functions that describe probabiUty density ate spread over the six ting-carbon atoms in a large dye molecule, an electron might be delocalized over a series of rings, but in semiconductors, the electron wave-functions are delocalized, in principle, over an entire macroscopic crystal. Because of the size of these wave functions, no single atom can have much effect on the electron energies, ie, the electronic excitations in semiconductors are delocalized. 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

A roughly equivalent valence-bond theory would result from allowing the 2s electron of each lithium atom to be involved in the formation of a covalent bond with one of the neighbouring atoms. The wave function for the crystal would be... 

Instead of formulating the wave function for a crystal as a sum of functions describing various ways of distributing the electron-pair bonds among the interatomic positions, as was done in the first section of this paper, let us formulate it in terms of two-electron functions describing a single resonating valence bond. A bond between two adjacent atoms ai and cq- may be described by a function < i3-(l, 2) in which 1 and 2 represent two electrons and the function i may have the simple Heitler-London form... 

The resonating-valence-bond theory of the electronic structure of metals is based upon the idea that pairs of electrons, occupying bond positions between adjacent pairs of atoms, are able to carry out unsynchronized or partially unsynchronized resonance through the crystal.4 In the course of the development of the theory a wave function was formulated describing the crystal in terms of two-electron functions in the various bond positions, with use of Bloch factors corresponding to different values of the electron-pair momentum.5 The part of the wave function corresponding to the electron pair was given as... 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

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