Total energy techniques

A review of the applications of the pseudopotential method and total energy techniques to the electronic and structural properties of solids is presented. With this approach, it has recently become possible to determine with accuracy crystal structures, lattice constants, bulk moduli, shear moduli, cohesive energies, phonon spectra, solid-solid phase transformations, and other static and dynamical properties of solids. The only inputs to these calculations, which are performed either with plane wave or LCAO bases, are the atomic numbers and masses of the constituent atoms. Calculations have also been carried out to study the atomic and electronic structure of surfaces, chemisorption systems, and interfaces. Results for several selected systems including the covalent semiconductors and insulators and the transition metals are discussed. The review is not exhaustive but focuses on specific prototype systems to illustrate recent progress. 

Payne M C, Teter M P, Allan D C, Arias T A and Joanopoulos J D 1992 Iterative minimization techniques for ab /M/o total energy calculations molecular dynamics and conjugate gradient Rev. Mod. Phys. 64 1045... 

Fig. 11.38 Lag ejfects in ab initio molecular dynamics. (Figure redrawn from Payne MC, M P Teter, D C Allan, R A Arias and D ] Joannopoidos 1992. Iterative Minimisaticm Techniques for Ab Initio Total-Energy Calculations Molecular Dynamics and Conjugate Gradients. Reviews of Modern Physics 64 1045-1097.)...

The electron alfinity (FA) and ionization potential (IP) can be computed as the difference between the total energies for the ground state of a molecule and for the ground state of the appropriate ion. The difference between two calculations such as this is often much more accurate than either of the calculations since systematic errors will cancel. Differences of energies from correlated quantum mechanical techniques give very accurate results, often more accurate than might be obtained by experimental methods. 

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

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... 

Figure 2. Total energies of ordered (LIq structure, squares), random (circles) and segregated (triangles) fee RhsoPdso alloys as a function of the number of neighboring shells included in the local interaction zone. Values obtained by the LSGF-CPA method are shown by filled symbols and full lines. The energies obtained by the reference calculations are shown by a dashed line (LMTO, ordered sample), a dotted line (LMTO-CPA, random sample), and a dot-dashed line (interface Green s function technique, segregated sample).

The D values may be easy or difficult to measure, and they can be estimated by various techniques, but there is no question as to what they mean. With E values the matter is not so simple. For methane, the total energy of conversion from CH4 to C + 4 H (at 0 K) is 393 kcal mol (1644 kJ mol ). " Consequently, E for the C—H bond in methane is 98 kcal mol (411 kJ mol ) at OK. The more usual practice, though, is not to measure the heat of atomization (i.e., the energy necessary to convert a compound to its atoms) directly but to calculate it from the heat of combustion. Such a calculation is shown in Figure 1.11. 

Table 1. shows the total energies obtained using the RHF method for 1. LCAO minimal basis set STO-IG for the sake of comparison with FSGO, 2. FSGO in its symmetric and broken symmetry solutions and, 3. LCAO minimal basis set STO-3G in order to allow a safer comparison with the quality of the subminimal basis used in the FSGO technique. The dissociation curves are given in Figure 1. 

In the Extended Hartree-Foek (EHF) technique, the minimization is performed on the form of the PHF wave function. This type of wave function should produce for each interatomic distance a further lowering of the energy with respect to the RHF, UHF, and PHF total energies. The values of E(PHF-FSGO), and E(EHF-FSGO) for internuclear distances from 1.0 a.u. to 7 a.u. (step 0.5 a.u.) are also given in Table 2. As in the UHF... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

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