Potential parameters 61 metals

Table A2.3.2 Halide-water, alkali metal cation-water and water-water potential parameters (SPC/E model). In the SPC/E model for water, the charges on H are at 1.000 A from the Lennard-Jones centre at O. The negative charge is at the O site and the HOH angle is 109.47°.

Practical measurements providing data on corrosion risk or cathodic protection are predominantly electrical in nature. In principle they concern the determination of the three principal parameters of electrical technology voltage, current, and resistance. Also the measurement of the potential of metals in soil or in electrolytes is a high-resistance measurement of the voltage between the object and reference electrode and thus does not draw any current (see Table 3-1). 

H = di(Z—iy di are the potential parameters I is the orbital quantum number 3 characterizes the spin direction Z is the nuclear charge). Our experience has show / that such a model potential is convenient to use for calculating physical characteristics of metals with a well know electronic structure. In this case, by fitting the parameters di, one reconstructs the electron spectrum estimated ab initio with is used for further calculations. 

Abstract. The physical nature of nonadditivity in many-particle systems and the methods of calculations of many-body forces are discussed. The special attention is devoted to the electron correlation contributions to many-body forces and their role in the Be r and Li r cluster formation. The procedure is described for founding a model potential for metal clusters with parameters fitted to ab initio energetic surfaces. The proposed potential comprises two-body, three-body, and four body interation energies each one consisting of exchange and dispersion terms. Such kind of ab initio model potentials can be used in the molecular dynamics simulation studies and in the cinalysis of binding in small metal clusters. 

Additional approximations are introduced in order to further simplify the overall calculation, and more importantly to provide a framework for the introduction of empirical parameters. Except for models for transition metals, parameterizations are based on reproducing a wide variety of experimental data, including equilibrium geometries, heats of formation, dipole moments and ionization potentials. Parameters for PM3 for transition metals are based only on reproducing equilibrium geometries. The AMI and PM3 models incorporate essentially the same approximations but differ in their parameterization. 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

This approach proves that a phase diagram can be modeled when the solution microstructure is known (i.e., aggregation number and micellar aggregate number per unit volume) together with an experimental determination of the potential between aggregates. If the variation of the potential versus various parameters (metal salt in the organic phase) can be obtained experimentally, the limits of the phase separation can be reliably correlated with theory. 

As in the MD method, PES for KMC can be derived from first-principles methods or using empirical energy functionals described above. However, the KMC method requires the accurate evaluation of the PES not only near the local minima, but also for transition regions between them. The corresponding empirical potentials are called reactive, since they can be used to calculate parameters of chemical reactions. The development of reactive potentials is quite a difficult problem, since chemical reactions usually include the breaking or formation of new bonds and a reconfiguration of the electronic structure. At present, a few types of reactive empirical potentials can semi-quantitatively reproduce the results of first-principles calculations these are EAM and MEAM potentials for metals and bond-order potentials (Tersoff and Brenner) for covalent semiconductors and organics. 

The importance of solvent parameters such as DN and AN and the advantage of their use over physical-electrostatic parameters was further demonstrated by Mayer et al. [21], who studied correlations between the DN and AN of solvents and redox potentials and their chemical equilibrium and ion pair equilibria. According to the Born theory, redox potentials should depend linearly on the reciprocal of the solvent s dielectric constant. Plotting Em values of a redox such as Cd/Cd2+ versus 1/e of the solvents in which it is measured results in a very scattered picture. In contrast, it has been clearly shown by Mayer et al. [15] that redox potentials of metals (e.g., Zn/Zn2+, Cd/Cd2+, Eu/Eu2+) can be nicely... 

In order to use these equations in an a priori fashion one must have information about the potential parameters, which in turn depend on the potential components Uio, J = 2,4. It is instructive to begin by considering the limiting case of a potential U( r) which is concentrated along the metal-ligand bond axis disregarding the spherical component this can be idealized as... 

There exist several linkages of the SBUs which should be avoided, for example, the short separation of two metal atoms (M Mj), one ligand atom and one bridging atom (Lj Bj), and L Mj in SBUs unlinked. To prevent these undesirable linkages, other potential functions have to be considered, including the repulsive potential between Mj Mj pairs, the attractive potential between L, - - - Mj pairs, and the repulsive potential between Lj Bj pairs. A repulsive potential between Mr Mj pairs prevents SBUs from overlapping with each other. The distance between the Mr Mj pair is limited to 3.4 A for D4R. The Lennard-Jones potential parameters used in assembling the D4R are provided in Table 7.2. 

The concept of canonical bands [1.19] was used by Tettifor in a series of three papers [1.47-49] where he related the superconducting and cohesive properties of the 4d transition metals to the variation of ASA potential parameters as functions of volume and atomic number. By similar means Duthie and Petti for [1.50] established a correlation with the d-band occupation numbers which explained the particular sequence of crystal structures found in the series of rare-earth metals. 

Chapter 9 contains a manual for a series of computer codes based upon the theory presented in the first 8 chapters of the book. With the programmes and the examples given there the user should be able to perform full-scale self-consistent calculations of his own. Finally, the book contains a table of self-consistent potential parameters which together with the LMTO programme will allow the user to reproduce the self-consistent energy bands of 61 metals at normal volume. 

The description of the formation of energy bands contained in (2.28) and illustrated in Fig.2.7 constitutes a scaling principle according to which the unhybridised band structure of any close-packed solid of a given crystal structure may be synthesised from the same canonical bands. Hence, the unhybridised energy bands of all elemental metals with, for instance, fee structure may be obtained from the fee canonical bands shown in Fig.2.4 once their one-electron potentials (or potential parameters) are known. 

As an example of a set of standard parameters, Table 4.1 lists all the potential-dependent information needed to perform an energy-band calculation for (non-magnetic) chromium metal. In the following, chromium is used as an example when we discuss the physical significance of each of the four potential parameters (4.1). At the end of the chapter we derive free-electron potential parameters, give expressions for the volume derivatives of some se-... 

In this chapter we list the four standard potential parameters (4.1) for 61 metals as obtained in self-consistent LMTO calculations using the exchange-correlation potential given by von Barth and Hedin [10.1]. Table 10.1 containing parameters for d-transition metals was prepared by O.K. Andersen and D. Glotzel, whom we wish to thank for permission to quote these results. Table 10.2 was prepared by the author. 

Fig. 2.37 A sketch of the first pseudopotentials (with potential parameters A and k) for two alkali metals as envisaged by Hellmann.

Expressions for the force constant, i.r. absorption frequency, Debye temperature, cohesive energy, and atomization energy of alkali-metal halide crystals have been obtained. Gaussian and modified Gaussian interatomic functions were used as a basis the potential parameters were evaluated, using molecular force constants and interatomic distances. A linear dependence between spectroscopically determined values of crystal ionicity and crystal parameters (e.g. interatomic distances, atomic vibrations) has been observed. Such a correlation permits quantitative prediction of coefficients of thermal expansion and amplitude of thermal vibrations of the atoms. The temperature dependence (295—773 K) of the atomic vibrations for NaF, NaCl, KCl, and KBr has been determined, and molecular dynamics calculations have been performed on Lil and NaCl. Empirical values for free ion polarizabilities of alkali-metal, alkaline-earth-metal, and halide ions have been obtained from static crystal polarizabilities the results for the cations are in agreement with recent experimental and theoretical work. 

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