Method of increments

For the enthalpies of formation of tetroxides calculated by the method of increments [163,164], see Table 2.14. 

Early studies of the reaction of bromate and iodide found the order non-integral, as though there was interference by the products of reaction. Most subsequent has used the method of incremental ad-... 

Taylor, B. Methodus Incrementorum Directa et Inversa. Proposition 7. [Direct and indirect methods of incrementation.] Peajsniansis Londini (1715) dusker, J. P., Orehowsky, W. Jr., Casciato, C. A., and Carrell, H. L. X-ray crystal analysis of the substrates of aconitase. X. The structure of dipotassium ds-aconitate. Acta Cryst. B28, 419-425 (1972). 

Taylor, B. Methodus Incrementorum Directa et Inversa. Proposition 7. [Direct and indirect methods of incrementation.] Pearsniansis Londini (1715)... 

If there is no opportunity to determine vapour pressures, it is possible to obtain the required data from figures published in the literature by the method of increments. This method, whether carried out graphically or by calculation, gives approximate results. 

A simple procedure to calculate the energies of the intermediates in a complex chemical degradation along the reaction coordinate, using the method of increments by van Krevelen, will be developed. From the energies along the reaction coordinate we will discriminate among various proposed reaction mechanisms. 

Treating a solid implies dealing with essentially infinitely many-electrons. However, in practice, we can correlate only a relatively small number of them. Therefore, we first have to reduce the calculations to a small number of electrons. This can be done by means of the method of increments considered in the next section. 

A method of increments [111, 159,160] is a wavefunction-based ab-initio correlation method for solids. This method is closely related to the ideas of the local ansatz (LA), [5] where local operators acting on the SCF wavefunction are used to admix suitable one- and two-particle excitations to the mean-field HF ground state. The many-electron Hamiltonian is split according to (5.44) and the ground-state Hamiltonian Hscf and the corresponding wavefunction scf = are assumed to be known. 

The method of increments provides a scheme in which the set of equations (5.51) and hence the correlation energy is evaluated in a hierarchical order [159]. 

The method of increments is useful only if the incremental expansion is well convergent, i.e. if increments up to, say, triples are sufficient, and if increments become rapidly small with increasing distance between localized orbitals. These conditions were shown to be well met in the case of different sohds [110,162]. Ideally, the increments should be local entities not sensitive to the surroundings. 

The theory described above has been applied to a great variety of materials, thus demonstrating the feasibility of calculations of that kind. They include the elemental semiconductors [159], III-V [163] and II-VI compounds [164], ionic crystals like MgO [165], CaO [166], NiO [167], alkali halides [168], Ti02 (with a sizeable amount of covalency) [169], rare-gas crystals [170,171], solid mercury [172,173] and the rar earth compound GdN [174] with the 4/ electrons kept within the core. The method of increments allows the CCSD local correlation scheme to be extended from molecules to solids. In most cases the program package CRYSTAL [23] was used for the SCF part including a localization procedure for determining the Wannier functions. 

Method of Increments Valence-band Structure and Bandgap... 

It is shown that for both small and large correlation-energy contributions the method of increments reduces the amount of computations decisively. 

Elena Voloshina and Beate Paulus Introduction Method of increments General formalism... 

Modifications for applying the method of increments for metals Properties of the embedding Technical details... 

The review is organized in the following manner In the first part of the next section we shortly introduce the method of increments in its general form and in the second part we discuss in detail the modifications which are necessary for metals. Further, we discuss necessary properties of the embedding. Technical details are presented in the last part of this chapter. In Sec. 3 we discuss the influence of the embedding on the calculated... 

Quantum-chemical correlation methods, developed for finite systems, can be applied to periodic systems using the method of increments. In this approach, the total energy is written as... 

Fig. 2 (Color online) Schematic overview of the ground-state calculations with the method of increments. Part which is specific for metals is marked by dashed line.

In the next sections we will show how our approach works for the more sophisticated cases of group 12 elements where no binding has been found at the HF level. Applying the method of increments to Zn, Cd, and Hg the calculated ground-state properties are shown to be in excellent agreement with experiment. Moreover, due to the possibility to analyze individual... 

Our calculated values for the bulk modulus are given in Table 7, both for the complete results of the method of increments, and for the case of truncation after two-body terms only. We also excluded the fi -correlation of the three-body increments to give an i-only value for the bulk modulus. We obtained the bulk modulus by fitting a quadratic curve to the points calculated at a = 70.53°, for values of u = 2.94, 2.97, 3.005, and 3.03 A. For the two-body energies only, an additional point at 3.06 A was included. The estimated error bounds of the two-body bulk modulus are therefore somewhat better than for the bulk modulus with three-body increments included. 

The bulk modulus calculated with two-body increments only is 0.132 Mbar, considerably lower than the experimental value, and similar to the result of LDA (in the rhombohedral lattice), 0.187 Mbar. In the hep lattice the LDA bulk modulus does not change much, with a value of 0.190 Mbar. This strong underestimation of the bulk modulus with LDA is in contrast to what would normally be expected for an overbound structure. The two-body increments with only -correlation are still repulsive, as mentioned above. When only the -correlation of the three-body increments is included, the bulk modulus increases to a value of 0.383 Mbar. The final result of the method of increments, with the inclusion of -correlation for... 

Fig. 25 (Color online) The different eontributions to the total binding energy from the method of increments are plotted, for the experimmlal lattiee parameters, and compared to the experimental value. The zero-point energy (ZPI correction is estimated from the Debye temperature. ... 

Thus, applying the method of increments to Zn and Cd one can reproduce accurately the experimental ground-state properties and to determine the nature of binding in these two elemental metals. The next step is the... 

In order to analyze different effects on the anisotropic bonding properties in Zn in more detail we investigate with the method of increments a wider area of the potential energy surface with respect to the hexagonal lattice parameters, PES(a,c). At each point of the PES(a,c) we consistently include the same increments as ordered at the experimental lattice structure, rather than reordering the increments. This topological approach is important because the sign of the contribution of the three-body terms depends on the... 

In conclusion, using the methods of increments we have been able to draw a detailed picture of the hep structure of zinc. Besides the analysis of the PES and its relation to experimental data, mechanisms for the anisotropy are worked out, that lead to a layered structure. We emphasize the necessity of including the filled d shell in the treatment of the electronic correlation. The existence of a zinc modification with a nearly ideal cja ratio is possible. 

An advantage of the method of increments in comparison with other local correlation methods is that the translational symmetry of the system can be used to reduce the effort of the calculation. Translational symmetry is, however, not a prerequisite of the method. Therefore it is possible to extend the method to systems with reduced symmetry like surfaces and molecules on the surface. " Impurities and functional groups in large biological molecules can be described as well. If the description for metals and the one for impurities are combined, the whole field of Kondo physics opens up. But one has to be aware of the fact that for the small energy scales involved in the Kondo effect require highly accurate quantum-chemical methods, and very extended basis sets are necessary to achieve reliable results. 

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