First principles calculations results

The scope of the early papers was to use the SK approach to accurately interpolate the results of first principles calculations of the energy bands and densities of states. An important characteristic of these calculations is that the first, second, and third nearest neighbor interactions are treated as independent parameters, which is advantageous for minimizing the rms deviation from the first principles bands. 

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

As expected, the C-C bond lengA widens significantly for the rotational transition state. Here, the agreement between the semi-empirical MNDO results and the first-principles LDF results is remarkable. The discrepancy in the C-H bond length remains, but the trend of a small bond shortening from the ground state to the rotational transition state can be found for both the MNDO and the LDF calculation. 

The Fe(lll)-azide-precursor and the photolysed product were characterized by NIS spectroscopy coupled to detailed DFT calculations [63]. The result of the study provides additional evidence in favor of a low-spin 5=1/2 ground state of the Fe(V)-nitrido complex. Here we show how first-principles calculations assist in quantitative analysis of experimental NIS data for the Fe(lll)-azide complex. 

The greatest limitation of QC methods is computational expense. This expense restricts system sizes to a few hundred atoms at most, and hence, it is not possible to examine highly elaborate systems with walls that are several atomic layers thick separated by several lubricant atoms or molecules. Furthermore, the expense of first-principles calculations imposes significant limitations on the time scales that can be examined in MD simulations, which may lead to shear rates that are orders of magnitude greater than those encountered in experiments. One should be aware of these inherent differences between first-principles simulations and experiments when interpreting calculated results. 

Integrating over the hysteresis loop between the compression and decompression curves in Figure 19 yields the amount of energy dissipated through the reversible bond formation/dissociation process. Unfortunately, it is not possible to determine the contribution of these transitions to the friction of phosphate films because such a calculation would require knowledge of the number of similar instabilities that occur per sliding distance, which is certainly beyond the limits of first-principles calculations. Nonetheless, the results do indicate that pressure- and shear-induced chemical reactions can contribute to the friction of materials. 

The electron density distribution of a known surface structure can be calculated from first-principles. Thus, the He diffraction data can be compared with theoretical results, in particular, to verify different structural models. Hamann (1981) performed first-principles calculations of the charge-density distributions of the GaAs(llO) surface, for both relaxed and unrelaxed configurations. The He diffraction data are in excellent agreement with the calculated charge-density distributions of the relaxed GaAs(llO) surface, and are clearly distinguished from the unrelaxed ones (Hamann, 1981). 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

The problem of first-principles calculations of the electronic structure of solid surface is usually formatted as a problem of slabs, that is, consisting of a few layers of atoms. The translational and two-dimensional point group symmetry further reduce the degrees of freedom. Using modern supercomputers, such first-principles calculations for the electronic structure of solid surfaces have produced remarkably reproducible and accurate results as compared with many experimental measurements, especially angle-resolved photoemission and inverse photoemission. 

As we have mentioned, the results of first-principles calculations are usually presented in the form of charge-density contours. The following are the commonly used forms of presenting results of first-principles calculations. 

Fig. 5.7. Charge-density contour plot of Al(lll) film. Results of a first-principles calculation of a nine-layer Al(lll) film. The contours are in steps of 0.27 electrons per atom. (Reproduced from Wang et al., 1981, with permission.)...

Figure 6.9 is a comparison of the results discussed previously with the first-principles calculation of the AI(lll) surface as well as the experimental results of STM images on Al(lll) by Wintterlin et al. (1989). A very simple model of the Al(lll) surface is used On each surface Al atom, there is an independent Is state near the Fermi level. The charge density contour (i.e., the image with an. r-wave tip state) agrees with the extrapolated corrugation amplitudes of the first-principles calculation (Mednick and Kleinman, 1980 ... 

An example of an Al(lll) surface is shown in Fig. 8.4. From the measured work function, cj)=3.5 eV, we find k = 0.96A", The atomic distance is a = 2.88 A, which also equals the parameter m in the Morse formula. An estimation of the parameter Uq in the Morse formula, or the binding energy per pair of A1 atoms, can be made as follows The evaporation heat of aluminum is 293 kJ/mol, which is 3.0 eV per atom. Aluminum is an fee crystal, where each atom has 12 nearest neighbors. Therefore, the binding energy per pair of A1 atoms is about 0.5 eV. Substitute these numbers into Eq. (8.16) and Eq. (8.17), we find the forces at the T site and the H sites which reproduce the result of first-principle calculations by Ciraci et al. (1990a). 

Use of ESG-RAPT. Experimental data 40 supported by first-principles calculations. See Refs. 33,39 for other results not agreeing with this work. 

Use of isotopic enrichment. Largest Cq 50 ever reported for Mg. Spectra (very broad) recorded with changing RL transmitter offset. Experimental data supported by first-principles calculations. See Ref. 40 for comparison of these experimental data with other results of first-principles calculations. 

Use of isotopic enrichment. See Ref. 40 for comparisons of these experimental data with results of first-principles calculations. 

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