Total energy, forces, and stresses

From the point of view of any first principles theory, solids are composed of nuclei and electrons atoms and ions are constmcts that play no primary role. This departure from our usual way of thinking about minerals is essential and has the following important consequences. We may expect our theory to be equally applicable to the entire range of conditions encountered in planets, the entire range of bonding environments 

Density functional theory is a powerful and in principle exact method of solving the quantum mechanical problem that has revolutionized the theoretical study of condensed matter (Hohenberg and Kohn 1964 Kohn and Sham 1965), see Jones and Gunnarsson (1989) for a review. The essence is the proof that the ground state properties of a material are a unique functional of the charge density p(r). This is important theoretically because the charge density, a scalar function of position, is a much simpler object than the total many-body wavefunction of the system. The total energy 

In addition to the essential approximation to the exchange-correlation functional, some first principles calculations make additional assumptions that are asymptotically 

Monte Carlo method. This method was the first to be applied to the study of many-atom condensed matter systems (Metropolis et al. 1953) (see dso chapters by Cygan and 

That is, given sufficient duration, time averages are equivalent to ensemble averages. 

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The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

Let us now find out whether these classical enthalpies may be reproduced by electronic-structure calculations (VASP) on Sn/Zn supercells using ultra-soft pseudopotentials, plane-wave basis sets and the GGA. We therefore have to theoretically determine the total energies of all crystal structure types under consideration (a-Sn, j6-Sn, Zn) as a function of the composition SnxZni x by a variation of the available atomic sites in terms of Sn and Zn occupation, just as for the preceding oxynitrides (CoOi- N ). In the present case, supercells with a total of 16 atoms were generated, and nine different compositions per structure were numerically evaluated. Because this amounts to a significant computational task, the use of pseudopotentials is mandatory, and this also allows the rapid calculation of interatomic forces and stresses for structural... 

It is now possible to explore the high temperature properties of Earth materials from first principles. The combination of efficient first principles methods for computing the total energy, interatomic forces, and stresses, with a variety of statistical mechanical methods including molecular dynamics, Monte Carlo, and approximate treatments such as the cell model promises rapid progress. With continued advances in computational power, and in the development of new theoretical methods, one foresees significant progress in three areas. 

Let me start with the role of computational physics and in particular the computation of electronic structure of solids including all the properties which can readily be calculated from it. This Advanced Studies Institute focuses on what I believe is the most important advance in recent years, namely the calculation of total energy etc. through solution of the SchrBdinger equation so that one has the forces on individual atoms whose equilibrium positions can therefore be found. Sometimes the forces and stresses are found directly and sometimes from differentiating the... 

It is explained in Ref. 12 how the self-consiste LDF calculation yields the total energy, forces on atoms and average macroscopic stress a. Knowledge of forces is of little use for establishing static equilibrium because, in crystals as symmetric as or T, the site-symmetry makes the net force on each atom... 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

The total rate of energy input into the volume element owing to the work done by the body forces and the surface stresses is... 

From eqn (6.30) it is clear that the virial of the electronic forces, which is the electronic potential energy, is totally determined by the stress tensor a and hence by the one-electron density matrix. The atomic statement of the virial theorem provides the basis for the definition of the energy of an atom in a molecule, as is discussed in the sections following Section 6.2.2. 

The starting point of the investigation is the introduction of a scalar microstruc-tural parameter k which contributes to the total energy E of the body under study as pointed out in Refs. [38] and 39]. In Eq. (1) p, s and x are the mass density, the specific internal energy density and the velocity, respectively. The parameter k in the product pk describes microstructural properties and transfers the square of the rate of k to the dimensions of a specific energy density. In addition, the energy supply Ri and the energy flux R2 are also modified in the form of Eqs. (2) and (3), wherein pb is the body force density, pg is the supply of K, and p r is the heat supply. Further quantities are the stress vector t = T n associated to the Cauchy stress tensor T and to the outer normal n, the microstructural flux s = S n and the heat flux qi = —q n. 

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