Functions Bloch

The periodic nature of crystalline matter can be utilized to construct wavefunctions which reflect the translational synnnetry. Wavefiinctions so constructed are called Bloch functions [1]. These fiinctions greatly simplify the electronic structure problem and are applicable to any periodic system. 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

Now, let us return to our discussion of carrying out an electronic structure calculation for a nanotube using helical symmetry. The one-electron wavefunc-tions can be constructed from a linear combination of Bloch functions linear combination of nuclear-centered functions Xj(r),... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Electronic states (with an appropriate choice of phases of the Bloch functions) near K and K points of two-dimensional (2D) graphite are described by the k p equation ... 

Let us consider lithium as an example. In the usual treatment of this metal a set of molecular orbitals is formulated, each of which is a Bloch function built from the 2s orbitals of the atoms, or, in the more refined cell treatment, from 2s orbitals that are slightly perturbed to satisfy the boundary conditions for the cells. These molecular orbitals correspond to electron energies that constitute a Brillouin zone, and the normal state of the metal is that in which half of the orbitals, the more stable ones, are occupied by two electrons apiece, with opposed spins. 

It was pointed out in my 1949 paper (5) that resonance of electron-pair bonds among the bond positions gives energy bands similar to those obtained in the usual band theory by formation of Bloch functions of the atomic orbitals. There is no incompatibility between the two descriptions, which may be described as complementary. It is accordingly to be expected that the 0.72 metallic orbital per atom would make itself clearly visible in the band-theory calculations for the metals from Co to Ge, Rh to Sn, and Pt to Pb for example, the decrease in the number of bonding electrons from 4 for gray tin to 2.56 for white tin should result from these calculations. So far as I know, however, no such interpretation of the band-theory calculations has been reported. 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

The translational periodicity of the potential is the necessary and sufficient condition for describing the wavefunction as a linear combination of Bloch functions... 

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

This implies that a density built up from such Bloch functions [cf (111.5) and (HI. 14)] is invariant under all such translations [the "little" period] ... 

An important and interesting question is obviously whether for quasicrystals and incommensurately modulated crystals there is anything corresponding to the Bloch functions for crystals. Momentum space may be a better hunting ground in that connection than ordinary space, where we have no lattice. Not only is there no lattice, one cannot even specify the location of each atom yet [8]. 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

A wave function composed in this way from the contributions of single atoms is called a Bloch function [in texts on quantum chemistry you will And this function being formulated with exponential functions exp(inka) instead of the cosine functions, since this facilitates the mathematical treatment]. 

Bloch functions, 25 7, 8 Bloch state, stationary, 34 237, 246 Blow-out phenomenon, 27 82, 84 Bohr magneton, 22 267 number, 27 37 Boltzmann law, 22 280 Bonding energy, BOC-MP, 37 106-107 Bondouard disproportionation reaction, 30 196 Bond percolation, 39 6-8 Bonds activation... 

The AO-basis Bloch functions are, as sum over reciprocal lattice vectors,... 

If we assume that the Bloch functions have periodic boundary conditions ... 

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

Let us consider a wave vector k of an occupied state close to kp and its equivalent k — 2kp as indicated in Fig. 1.32. The Bloch functions of both states will be given by Eq. (1.31) ... 

The existence of surface states is a consequence of the atomic structure of solids. In an infinite and uniform periodic potential, Bloch functions exist, which explains the band structures of different solids (Kittel, 1986). On solid surfaces, surface states exist at energy levels in the gap of the energy band (Tamm, 1932 Shockley, 1939 Heine, 1963). 

The band structure and Bloch functions of metals have been extensively published. In particular, the results are compiled as standard tables. The book Calculated Electronic Properties of Metals by Moruzzi, Janak, and Williams (1978) is still a standard source, and a revised edition is to be published soon. Papaconstantopoulos s Handbook of the Band Structure of Elemental Solids (1986) listed the band structure and related information for 53 elements. In Fig. 4.14, the electronic structure of Pt is reproduced from Papaconstantopoulos s book. Near the Fermi level, the DOS of s and p states are much less than 1%. The d states are listed according to their symmetry properties in the cubic lattice (see Kittel, 1963). Type 2 includes atomic orbitals with basis functions xy, yz, xz], and type e, includes 3z - r-), (x - y ). The DOS from d orbitals comprises 98% of the total DOS at the Fermi level. 

Fig. 5.1. A metal surface with one-dimensional periodicity. The lowest Fourier components of the charge-density distribution are determined by the Bloch functions at the r and the K points in reciprocal space.

The Bloch functions near the K points have a long decay length and contribute to the second term of Eq. (5.10). Following Eq. (5.12), in general, a surface Bloch function at that point has the form ... 

In addition to the term with n = 0, the term with n=-l have the same decay length, and thus have the same magnitude. Also, the Bloch function that generates the symmetric charge density must also be s metric. The lowest-order symmetric Fourier sum of the Bloch function near K is ... 

The function 4>< ( x) has a value I at each lattice point in real space, and 0 at the center of four neighboring lattice points. Similar to the one-dimensional case, the ao(z) term in Eq. (5.29) comes mainly from the Bloch functions near... 

A single plane wave such as Eq. (5.31) is not a good Bloch function because it does not satisfy the symmetry of these points, 2mm (see Appendix E). The appropriately symmetrized Bloch functions are... 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

The problem of the electron charge-density distribution of a surface with hexagonal symmetry has been treated by Liebsch, Harris, and Weinert (1984). Similar to previous cases, the oo(z) term in Eq. (5.41) comes mainly from the Bloch functions near E, whose lowest Fourier component is ... 

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