First-Principles Calculations of the Total Energy

As yet, our discussion has focused on the attempt to construct energy functionals in which there is some level of empiricism that mimics to a greater or lesser extent the presumed exact results of a full quantum mechanics calculation. Our reason for shying away from the full quantum mechanical calculations themselves was founded upon the fact that these methods are computationally intensive, and hence must be undertaken with the foresight to insure that they are used only when really necessary. It is the purpose of the present section to sketch the way in which first-principles calculations are constructed. 

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Modem Electron Theory by M. W. Einnis in Electron Theory in Alloy Design edited by D. G. Pettifor and A. H. Cottrell, Institute of Materials, London England, 1992. This is my favorite article on first-principles calculations. Einnis spells out all of the key points involved in carrying out first-principles calculations of the total energy. 

This provides a path for attempting first-principles calculations of the total energies, without the use of the parameters from the Solid State Table. In addition, since it incorporates orbital corrections, it should be more accurate. However, there has not been a serious... 

The elastic constants of iron have been studied experimentally and theoretically at low temperature and high pressure (Mao et al. 1998 Soderlind et al. 1996 Steinle-Neumann et al. 1999 Stixrude and Cohen 1995), but there has not yet been a first principles calculation of the full elastic constant tensor at inner core conditions (see Nye 1985 for a review of elastic constants). Laio et al. (2000) developed a clever hybrid method that combines first principles total energy and force calculations for a limited number of time steps with a semi-empirical potential fit to the first principles results. These authors investigated a number of properties with their ab initio method including... 

Figure 9. Predicted structure of hep iron from particle in a cell method with first principles calculations of the energetics (dark symbols) at densities of (diamonds) 12.52 Mg m (squares) 13.04 Mg m (circles) 13.62 Mg m. Results are compared with earher theoretical predictions based on more approximate ab initio calculations of the total energy (open squares) (Wasserman et al. 1996b) and (inset) with a polybaric set of experimental results at lower pressure fiom (open circles) (Huang et al. 1987) (15-20 GPa) and (squares) (Funamori et al. 1996) (23-35 GPa).

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The first kind of simplification exclusively concerns the size of the basis set used in the linear combination of one center orbitals. Variational principle is still fulfilled by this type of "ab initio SCF calculation, but the number of functions applied is not as large as necessary to come close to the H. F. limit of the total energy. Most calculations of medium-sized structures consisting for example of some hydrogens and a few second row atoms, are characterized by this deficiency. Although these calculations belong to the class of "ab initio" investigations of molecular structure, basis set effects were shown to be important 54> and unfortunately the number of artificial results due to a limited basis is not too small. 

For chemical purposes, substitution of total energy hypersurfaces by those based on the heat of formation seems more reasonable, with the difference given by the zero point energy corrections. However, their calculations from first principles can be rather cumbersome (12) and, moreover, for a given variation of some nuclear coordinates it usually can be assumed that the change in zero point energy is small compared to that of the total energy. On the other hand, se eral semiempirical quantum chemical procedures which are appropriately parametrized often yield satisfactory approximations for molecular heats of formation (10) and, therefore, AH hypersurfaces have become rather common. 

Parameters of GVFF for silicates and aluminosilicates obtained from first-principles calculations were reported by Ermoshin et al. ° ° Having assumed that the dynamics of zeolite lattices can be described in terms of vibrations of the TO4 tetrahedra (T = Si, Al) and shared 03T-0(H)-T03 tetrahedra, the authors calculated the matrix of second derivatives of the total energy in Cartesian coordinates, the Hessian matrix H, for molecular models of such units. The matrix H was then transformed into a matrix of force constants in internal coordinates F... 

The electronic state calculation by discrete variational (DV) Xa molecular orbital method is introduced to demonstrate the usefulness for theoretical analysis of electron and x-ray spectroscopies, as well as electron energy loss spectroscopy. For the evaluation of peak energy. Slater s transition state calculation is very efficient to include the orbital relaxation effect. The effects of spin polarization and of relativity are argued and are shown to be important in some cases. For the estimation of peak intensity, the first-principles calculation of dipole transition probability can easily be performed by the use of DV numerical integration scheme, to provide very good correspondence with experiment. The total density of states (DOS) or partial DOS is also useful for a rough estimation of the peak intensity. In addition, it is necessary lo use the realistic model cluster for the quantitative analysis. The... 

One of the cleanest routes to determining embedding functions is by invoking the universal binding energy relation (UBER) (Rose et al. 1984). On the basis of a vast collection of experience with first-principles calculations of a range of materials, it was found that the total energy as a function of lattice parameter may be fitted to... 

After the computation of 4>hf became routinely feasible in the 1960s [4], much of the work in quantum computational chemistry has been devoted to the computation from first principles of the total energy, E, via the calculation of the correlation energy, Ecorr/ moving from systems with a few electrons to larger ones, with lighter or heavier atoms. In the latter case, relativistic effects make an increasingly important contribution. 

The thermodynamic stabilities for the series of metal borohydrides M(BH4) (M = Li, Na, K, Mg, Ca, Sc, Zr, Hf, Cu, Zn and Al n = -4) have been systematically investigated by first-principles calculations.The heat of formation of M(BH4) AT/boro were estimated from the difference of the total energies between the left- and right-hand sides of Eq. (15.23) ... 

The electronic and transport properties of an amorphous graphitic carbon model constructed by Townsend et al, [112,114] were studied by first-principles calculations in the local-density approximation. Semiempirical density-functional molecular dynamics (DF-MD) was used to simulate the experiments, e.g., neutron diffraction, inelastic neutron scattering, and NMR, to determine the structure of the system in order to achieve a fundamental understanding of structure-related properties on the molecular level of chemical bonding. The total energy of the system... 

In this chapter, we shall now come back to the question how physical observables are associated with proper operator descriptions, which has already been addressed in section 4.3. All preceding chapters dealt with the proper construction of Hamiltonians for the calculation of energies and wave functions of many electron systems. Here, we shall now transfer this knowledge to the construction of relativistic expressions for first-principles calculations of molecular properties for many-electron systems. The basic guideline for this is the fact that all molecular properties can be expressed as total electronic energy derivatives. 

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