Accuracy, cohesive energies

Ey in an actinide atom varies typically from 3 Ryds for Ac to about 40 Ryds for Am. Even in this approximation very high standards of accuracy are required to obtain good values for the cohesive energy. 

Calculations. In theory, p can be obtained by minimizing the part of the cohesive energy that depends on it. In practice, for large structures, the various contributions cannot be calculated with enough accuracy to give reliable results, and thus such a calculation is far beyond what is presently feasible. 

What is remarkable about this result is that we have found that the vacancy formation energy is with opposite sign equal to the cohesive energy per atom. An assessment of the accuracy of this conclusion is given in table 7.1. We note... 

Correlations for the cohesive energy and the solubility parameter will be presented in Chapter 5, to allow the calculation of these properties at the same level of accuracy as can be attained by group contributions but for much wider classes of polymers. The pitfalls of using solubility parameters in miscibility calculations will also be highlighted in the context of a discussion of the various types of phase diagrams that are observed for blends and mixtures. 

The values of the properties which will be fitted by using equations 2.9 and 2.10 will be selected from available and apparently reliable experimental data whenever there are sufficient amounts of such data. Some important properties of polymers, such as the van der Waals volume (Chapter 3) and the cohesive energy (Chapter 5), are not directly observable. They are inferred indirectly, and often with poor accuracy, from directly observable properties such as molar volume (or equivalently density) and solubility behavior. When experimental data are unavailable or unreliable, the values of the properties to be fitted will be estimated by using group contributions. The predictive power of such correlations developed as direct extensions and generalizations of group contribution techniques will then be demonstrated by using them... 

The availability of the new three-body nonadditive potential made it possible to investigate the structure of the argon crystal at a much increased level of accuracy compared to that possible earlier and resolve an outstanding problem, often called the rare-gas crystal structure paradox . The cohesion energy of the argon crystal computed... 

However, for the second row, which exhibits relatively small nonlocality effects, a good accuracy pseudo-Hamiltonians can be constructed. Reference 47 gives results for several atoms and dimers. New pseudo-Hamiltonian parameterizations for several elements from the first two rows were calculated very recently [49]. Li et al. [50] used a pseudo-Hamiltonian to carry out a DMC calculations on solid silicon, which resulted in excellent agreement with experiment for the cohesive energy. This demonstrated for the first time the feasibility of the DMC calculations on solids other than hydrogen. 

The equilibrium structure to which a material crystallizes under given conditions of temperature and pressure is completely determined by the interaction forces between its molecules and these, in turn, are determined by the electronic structure of those molecules, as we have seen. Unfortunately it is almost impossible to proceed uniquely in this way from molecules to crystals in any but the very simplest solids, because the interaction forces are not known with sufficient accuracy. We must therefore usually content ourselves with observing that the crystal structure which actually occurs is consistent with what is known of the molecular interactions, and with comparing the value which we can calculate for its cohesive energy with that found from experiment. In the case of ice, determination of the crystal structure has itself posed very difficult problems which have only been answered by reference back to the structure and interaction of the water molecules. 

The accuracy of the local density-pseudopotential method is quite adequate for most studies. The structural properties are often replicated to within l-2%. The cohesive energies are not so accurately determined the binding energy is usually overestimated by 5-10%. However, the relative energy differences between solid state structures are more accurately calculated. Structural energy differences as small as —0.01 eV/atom can be reliably predicted. 

The success of simulation is evident The first-principle approach enables one to predict with a high accuracy the strength of interatomic bonding for metals with different electronic structures. The cohesive energy of metals varies from —0.94 eV/atom (0.905 kj/mol) for potassium to —6.82 eV/atom (656.76 kj/mol) for molybdenum and —8.90eV/atom (857.07 kJ/mol) for tungsten. 

Fig. 2. Calculated properties of the 3d and 4d transition metals-cohesive energy, lattice constant, and bulk modulus- compared with experiment (crosses). This represents a milestone in the development of the methods to calculate with sufficient accuracy to find these quantities. (From refs. 25 and 123, figure courtesy of V. MoruzzO... 

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. 

We intentionally do not give here the mathematical details of the approaches allowing implementation of analytical gradients in the calculations of periodic systems (for these details readers are referred to the cited publications and references therein). We note that such an implementation is essentially more complicated than in the case of molecular systems and requires high accmacy in the total-energy calculation. One can find the detailed analysis of the accuracy in gradients calculations on numerical examples in [660,661,663]. The comparison of the numerical and analytical derivatives values can also be found. In what follows we turn to the results obtained for the equilibrium structure and cohesion energy in the crystalline metal oxides. The LCAO calculations discussed were made with the CRYSTAL code and use of HF, KS and hybrid Hamiltonians. 

In an attempt to achieve a formulation concept with both the theoretical content of Winsor R and the straightforward numerical data feature of the HLB, Beerbower and Hill [134,135] introduced the cohesive energy ratio (CER) approach in 1971. However, they were not able to find a way to compute or measure the solubility parameters of some surfactant groups with accuracy, so the calculation was as approximate as the HUB concept. 

---