Solid State Physical Calculations

2 Results of Calculations of NLO Properties and Their Discussion. - 5.2.1 Solid State Physical Calculations. - The calculation of static linear polarizabilities zz/N = oll/N for the polymers poly(H2), and the H-bonded systems poly(H20) and poly(LiH) using the above described solid state physical formalism has given much larger values116 than those obtained by extrapolating oligomer values with the aid of different procedures. Further computations for 

Subsequently using124 separate first-order and second-order CHF equations for 

Turning to the results of vibrational polarizabilities131 we have obtained for a poly(HF) chain in a zig-zag geometry in the static case a(elect)zz = —44.88 a.u. while the vibrational contribution was still larger, a(displ)zz = —77.92 a.u. using dementi s double ( basis. 

In the case of the application of the polarization propagator method127 the static and dynamic xzz/N values were increasing with the (H2)x chain length. Their saturation with a 3-21G basis was not reached even after 14 H2 molecules, though their values were close to the values belonging to those of an infinite chain (especially with increasing frequency ). 

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Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

Computational solid-state physics and chemistry are vibrant areas of research. The all-electron methods for high-accuracy electronic stnicture calculations mentioned in section B3.2.3.2 are in active development, and with PAW, an efficient new all-electron method has recently been introduced. Ever more powerfiil computers enable more detailed predictions on systems of increasing size. At the same time, new, more complex materials require methods that are able to describe their large unit cells and diverse atomic make-up. Here, the new orbital-free DFT method may lead the way. More powerful teclmiques are also necessary for the accurate treatment of surfaces and their interaction with atoms and, possibly complex, molecules. Combined with recent progress in embedding theory, these developments make possible increasingly sophisticated predictions of the quantum structural properties of solids and solid surfaces. 

J. Ihm. Total energy calculations in solid state physics. Rep Prog Phys 57 105, 1988. 

D. R. Hartree, Reports on Progress in Physics 11, 113 (1948) this survey is brought up to date in D. R. Hartree, The Calculation of Atomic Structures Wiley and Sons, New York, and Chapman and Hall, London, 1957. See also R. S. Knox, Bibliography of Atomic Wave Functions/ in Solid State Physics (Seitz and Turnbull, eds.), Academic Press, New York, 1957, Yol. 4, p. 413. 

The three-dimensional symmetry is broken at the surface, but if one describes the system by a slab of 3-5 layers of atoms separated by 3-5 layers of vacuum, the periodicity has been reestablished. Adsorbed species are placed in the unit cell, which can exist of 3x3 or 4x4 metal atoms. The entire construction is repeated in three dimensions. By this trick one can again use the computational methods of solid-state physics. The slab must be thick enough that the energies calculated converge and the vertical distance between the slabs must be large enough to prevent interaction. 

Nevertheless, DFT has been shown over the past two decades to be a fairly robust theory that can be implemented with high efficiency which almost always surpasses HF theory in accuracy. Very many chemical and spectroscopic problems have been successfully investigated with DFT. Many trends in experimental data can be successfully explained in a qualitative and often also quantitative way and therefore much insight arises from analyzing DFT results. Due to its favorable price/performance ratio, it dominates present day computational chemistry and it has dominated theoretical solid state physics for a long time even before DFT conquered chemistry. However, there are also known failures of DFT and in particular in spectroscopic applications one should be careful with putting unlimited trust in the results of DFT calculations. 

Density-functional theory, developed 25 years ago (Hohenberg and Kohn, 1964 Kohn and Sham, 1965) has proven very successful for the study of a wide variety of problems in solid state physics (for a review, see Martin, 1985). Interactions (beyond the Hartree potential) between electrons are described with an exchange and correlation potential, which is expressed as a functional of the charge density. For practical purposes, this functional needs to be approximated. The local-density approximation (LDA), in which the exchange and correlation potential at a particular point is only a function of the charge density at that same point, has been extensively tested and found to provide a reliable description of a wide variety of solid-state properties. Choices of numerical cutoff parameters or integration schemes that have to be made at various points in the density-functional calculations are all amenable to explicit covergence tests. 

The use of quantum mechanical calculations of solid properties was initially the province of solid-state physics, and the calculation of electron energy levels in metals and semiconductors is well established. Chemical quantum mechanical... 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

For many problems in solid state physics, the computational efficiency of the computer programs is the result of using a planewave basis set and performing part of the calculation in momentum space through the use of Fast Fourier transforms. A planewave basis set is naturally applicable to systems with translational symmetry and this is the key of the success of... 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

In solid state physics, the sensitivity of the EELS spectrum to the density of unoccupied states, reflected in the near-edge fine structure, makes it possible to study bonding, local coordination and local electronic properties of materials. One recent trend in ATEM is to compare ELNES data quantitatively with the results of band structure calculations. Furthermore, the ELNES data can directly be compared to X-ray absorption near edge structures (XANES) or to data obtained with other spectroscopic techniques. However, TEM offers by far the highest spatial resolution in the study of the densities of states (DOS). 

Our first foray into the realm of numerical convergence takes us away from the comfortable three-dimensional physical space where atom positions are defined and into what is known as reciprocal space. The concepts associated with reciprocal space are fundamental to much of solid-state physics that there are many physicists who can barely fathom the possibility that anyone might find them slightly mysterious. It is not our aim here to give a complete description of these concepts. Several standard solid-state physics texts that cover these topics in great detail are listed at the end of the chapter. Here, we aim to cover what we think are the most critical ideas related to how reciprocal space comes into practical DFT calculations, with particular emphasis on the relationship between these ideas and numerical convergence. 

Every example of a vibration we have introduced so far has dealt with a localized set of atoms, either as a gas-phase molecule or a molecule adsorbed on a surface. Hopefully, you have come to appreciate from the earlier chapters that one of the strengths of plane-wave DFT calculations is that they apply in a natural way to spatially extended materials such as bulk solids. The vibrational states that characterize bulk materials are called phonons. Like the normal modes of localized systems, phonons can be thought of as special solutions to the classical description of a vibrating set of atoms that can be used in linear combinations with other phonons to describe the vibrations resulting from any possible initial state of the atoms. Unlike normal modes in molecules, phonons are spatially delocalized and involve simultaneous vibrations in an infinite collection of atoms with well-defined spatial periodicity. While a molecule s normal modes are defined by a discrete set of vibrations, the phonons of a material are defined by a continuous spectrum of phonons with a continuous range of frequencies. A central quantity of interest when describing phonons is the number of phonons with a specified vibrational frequency, that is, the vibrational density of states. Just as molecular vibrations play a central role in describing molecular structure and properties, the phonon density of states is central to many physical properties of solids. This topic is covered in essentially all textbooks on solid-state physics—some of which are listed at the end of the chapter. 

Seiler and Dunitz point out that the main reason for the widespread acceptance of the simple ionic model in chemistry and solid-state physics is its ease of application and its remarkable success in calculating cohesive energies of many types of crystals (see chapter 9). They conclude that the fact that it is easier to calculate many properties of solids with integral charges than with atomic charge distributions makes the ionic model more convenient, but it does not necessarily make it correct. 

As we have shown in Chapters 2 and 3, under the normal operating conditions of STM, the tunneling current can be calculated from the wavefunctions a few A from the outermost nuclei of the tip and the sample. The wavefunctions at the surfaces of solids, rather than the wavefunctions in the bulk, contribute to the tunneling current. In this chapter, we will discuss the general properties of the wavefunctions at surfaces. This is to fill the gap between standard solid-state physics textbooks such as Kittel (1986) and Ashcroft and Mermin (1985), which have too little information, and monographs as well as journal articles, which are too much to read. For more details, the book of Zangwill (1988) is helpful. 

Calori, C., Combescot, R., Nozieres, P., and Saint-James, D. (1972). A direct calculation of the tunneling current IV. Electron-phonon interaction effects. Solid State Physics 5, 21—42. 

The Green s function G is calculated, in particular, in [3, 4] - analytically for b.c.c. lattice and numerically for s.c. and f.c.c. lattices. It could be calculated as a particular case of the static lattice Green s function widely used in the solid state physics. 

Sometimes the estimation of the electronic structures of polymer chains necessitates the inclusion of long-range interactions and intermolecular interactions in the chemical shift calculations. To do so, it is necessary to use a sophisticated theoretical method which can take account of the characteristics of polymers. In this context, the tight-binding molecular orbital(TB MO) theory from the field of solid state physics is used, in the same sense in which it is employed in the LCAO approximation in molecular quantum chemistry to describe the electronic structures of infinite polymers with a periodical structure -11,36). In a polymer chain with linearly bonded monomer units, the potential energy if an electron varies periodically along the chain. In such a system, the wave function vj/ (k) for electrons at a position r can be obtained from Bloch s theorem as follows(36,37) ... 

The Hamiltonian Eq. (7) provides the basis for the quantum dynamical treatment to be detailed in the following sections, typically involving a parametrization for 20-30 phonon modes. Eq. (7) is formally equivalent to a class of linear vibronic coupling (LVC) Hamiltonians which have been used for the description of excited-state dynamics in molecular systems [66] as well as the Jahn-Teller effect in solid-state physics. In the following, we will elaborate on the general properties of the Hamiltonian Eq. (7) and on quantum dynamical calculations based on this Hamiltonian. 

The DFT concept of calculating the energy of a system from its electron density seems to have arisen in the 1920s with work by Fermi, Dirac, and Thomas. However, this early work was useless for molecular studies, because it predicted molecules to be unstable toward dissociation. Much better for chemical work, but still used mainly for atoms and in solid-state physics, was the Xa method, introduced by Slater in 1951. Nowadays the standard DFT methodology used by chemists is based on the Hohenberg-Kohn theorems and the Kohn-Sham approach... 

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