Wavefunction Bloch waves

At crystalline surfaces, there are three types of wavefunctions as shown in Fig. 4.1. (1) The Bloch states are terminated by the surface, which become evanescent into the vacuum but remain periodic inside the bulk. (2) New states created at the surfaces in the energy gaps of bulk states, which decay both into the vacuum and into the bulk, the so-called surface states. (3) Bloch states in the bulk can combine with surface states to form surface re.sonances, which have a large amplitude near the surface and a small amplitude in the bulk as a Bloch wave. 

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

The localised single-exdton wavefunctions are not eigenfunctions of the crystal Hamiltonian. A wavefunction appropriate to the crystal symmetry and the periodic potential can be found using the Bloch-wave ansatz. From the localised basis functions <1>, one obtains the delocalised wavefunction... 

It is useful to describe the form of localization that occurs in a homogeneously disordered material in contrast to the models described in Section 15.2 and Section 15.3. The 3D models described below assume that the materials are isotropic i.e., the materials should be electrically the same in all directions. In a perfect crystal with periodic potentials, the wavefunctions form Bloch waves that are delocalized over the entire solid [40]. In systems with disorder, impurities and defects introduce substantial scattering of the electron wavefiinction, which may lead to localization. Anderson demonstrated [88] that electronic wavefunctions can be localized if the random component of the disorder potential is large... 

Bloch Waves delocalized electronic wavefunctions which have the form = Ui (f) exp (ik r), where ui (r) is a function with the periodicity of the lattice unit cell and exp (ik f) is a wave of wavelength A = iTi/k. 

Localized States electronic states which are not extended over the entire solid as Bloch waves are localized states. The spatial dependence of the wavefunctions of a localized state is usually assumed to vary as i/f(f) exp( — r — ro / ), decaying exponentially in a characteristic length the localization length, away from fb. Charge transport by electrons in these states is due to hopping. 

The cross-sections for itinerant electrons, as, e.g., electrons in broad bands, are evaluated by taking into account that the electrons in the initial as well as in the final state may be represented by Bloch-wavefunctions P = u,t(/ ) exp(i R) (see Chap. A). In these wavefunctions atomic information is contained in the amplitude factor Uj (i ), whereas the wave part exp (i R) is characterized by the wavenumber k of the propagating wave (proportional to the momentum of the electron). 

The unit cell group description of the normal modes of vibration within a unit cell, many of which are degenerate, given above is adequate for the interpretation of IR or Raman spectra. The complete interpretation of vibronic spectra or neutron inelastic scattering data requires a more generalized type of analysis that can handle 30N (N=number of unit cells) normal modes of the crystal. The vibrations, resulting from interactions between different unit cells, correspond to running lattice waves, in which the motions of the elementary unit cells may not be in phase, if ky O. Vibrational wavefunctions of the crystal at vector position (r+t ) are described by Bloch wavefunctions of the form [102]... 

The wavefunction, according to Bloch s theorem, is one which contains a wave-Uke portion [the exponential term in Eq. (A35)] and a periodic cell piortion [the / (r) term in Eq. (A35) and (A36)]. The wavefunction is expanded so as to take on the same periodicity of the... 

The simpler PW methods are the most popular in the Kohn-Sham periodic-systems calculations. Plane waves are an orthonormal complete set any function belonging to the class of continuous normalizable functions can be expanded with arbitrary precision in such a basis set. Using the Bloch theorem the single-electron wavefunction can be written as a product of a wave-like part and a cell-periodic part ifih = exp(ifer)Mj(r) (see Chap. 3). Due to its periodicity in a direct lattice Uj(r) can be expanded as a set of plane waves M<(r) = C[Pg.281]

Third, careful comparison of Eqs. (15-15) and (15-4) shows that they are not exactly the same. Equation (15-15) instructs us to find a periodic function Uj (p) and multiply it by exp(z j p) at every point in p. Think of the sine or cosine related to the exponential and imagine what this means as we multiply it times a 2p j on some carbon. Say the cosine is increasing in value as it sweeps clockwise past the carbon nucleus at 2 00 on a clock face. This produces a product of cosine and 2p j that is unbalanced—smaller toward 1 00 than toward 3 00, because the cosine wave modulates Ujip) everywhere. But Eq. (15-4) is different. It instructs us to take the value of the cosine at 2 00 and simply multiply the 2p r AO on that atom by that number. The 2p AO is not caused to become unbalanced. Only its size in the MO is determined by the cosine. Equation (15 ) is called a Bloch sum. Such sums are approximations to Bloch functions, but any errors inherent in this form are likely to be quite small if the basis functions and unit cell are sensibly chosen. (Using Bloch sums is similar in spirit to the familiar procedure of approximating a molecular wavefunction as a linear combination of basis functions.)... 

Equation (15-17) tells us how to produce wavefunctions. We simply take exp(/fa) times our basis set, with k taking on all values between —nja and n ja, and pick off values at discrete points corresponding to carbon atom positions. Since it is more convenient to work with the real forms of solutions, we in effect choose a pair of k values (e.g., —7r/4a and 7r/4a) so that we can generate a pair of trigonometric coefficient waves [cos(7Tx/4a) and sin(7rx/4a)]. Thejr MOs for regular polyacetylene produced from Bloch sums are shown in Fig. 15-8 for selected values of k. These can be used to illustrate some important points ... 

These functions are shown graphically in Figure 6.4. Applying translational symmetry constraints gives rise to what a chemist would refer to as symmetry adapted linear combinations of the basis functions, Xb and Xab. In the language of electronic band structure, these wavefunctions are called Bloch functions. The Bloch functions, %, are periodic waves delocalize throughout the crystal and can be mathematically expressed as follows ... 

Prove the Bloch statement for the many-body wavefunction, Eq. (5.57), and relate the value of the wave-vector K to the single-particle wave-vectors. 

---