Finit cluster calculation

The crystal orbital approach (see ref. 94 for a review of the recent computational developments in this field) has dominated the electronic structure calculations on polymers for several years. However, the recently published reports on the finite-cluster calculations reveal that the latter methodology has several definite advantages over the traditional approach. Let P(N) be an extensive property of a finite cluster X-(-A-)j -Y, where N is the number of repeating units denoted by A, while X and Y stand for terminal groups. The corresponding intensive properties, p(N) = P(N)/N, are known only for integer values of N. However, provided the polymer in question is not metallic, P(v) can be approximated by a smooth function p(v) of v = 1/N, which in turn can be extrapolated to v = 0 yielding the property of the bulk polymer. 

Unlike their crystal-orbital counterparts, the finite-cluster calculations carried out in conjunction with extrapolations are free from the problems associated with the reciprocal space integration. Their accuracy can be easily monitored by increasing the maximum size of clusters and comparing the values of properties calculated with different extrapolation methods. The termi-... 

Finite-Cluster Calculations VI. A Finite-Size Perturbation Theory for the Hartree-Fock Energy of Linear Oligomers. 

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

After having described molybdenum trioxide, we intend to specify the best finite clusters allowing to represent each of the (010), (001) and (100) faces in order to study surface properties such as energy and electronic distribution. For this purpose, the evolution of the electronic properties will be studied as a function of the cluster size and referred to the results of an EHT - band calculation [12] all calculations have been made with QCPE programs [13,14] and Hoffmann parameters [15],... 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

In the cellular multiple scattering model , finite clusters of atoms are subjected to condensed matter boundary conditions in such a manner that a continuous spectrum is allowed. They are therefore not molecular calculations. An X type of exchange was used to create a local potential and different potentials for up and down spin-states could be constructed. For uranium pnictides and chalcogenides compounds the clusters were of 8 atoms (4 metal, 4 non-metal). The local density of states was calculated directly from the imaginary part of the Green function. The major features of the results are ... 

In addition to the cluster calculations, we report details of recent first-principles calculations based on the density functional formalism. These calculations employ periodic boundary conditions to allow investigation of the entire zeolite lattice, and therefore the use of a plane-wave basis set is applicable. This has a number of advantages, most notably that the absence of atom-centered basis functions results in no basis set superposition error (BSSE) (272), which arises as a result of the finite nature of atom-centered basis sets. Nonlocal, or gradient, corrections are applicable also, just as they are in the cluster calculations. 

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

The main advantage of periodic boundary conditions is the elimination of edge effects or terminal atom problems that occur in the finite cluster approach. The Hamiltonian in slab calculations is generally limited to density functional theory11, an approach that is not always an appropriate choice. 

Since experimental results were available for the high-frequency (" -2080 cm-1) diatomic CN in water (as opposed to CH3C1) (17), with an estimated T value of some 25 ps, an MD study was undertaken by Rey and Hynes (29) to clarify the role of Coulomb forces for VET in this accesible case. The charge distribution of CN in the solvent was modeled by a negative charge on N and a finite dipole located on the C site (30). The equilibrium solvent structure about this ion involved greater solvation number on the N end compared to the C end, a result consistent with some small cluster calculations (31). Since the frequency shift from the vacuum and the anharmonicity in the CN bond are both relatively small (29), the static vibrational aspects of the ion are evidently fairly clean. ... 

As soon as we wish to compare the binding energies of the levels in such a compilation as Table 3 with each other, or indeed with the binding energies of CO in the gas phase we come up against the reference level problem. More important, if a UP-spectrum is calculated based on a finite cluster approach, an experimentally accessible reference level must be located. In the gas phase it is clear that the vacuum level provides the appropriate reference. For an atom or molecule... 

The coupling of a finite cluster with bulk metal material is treated through a Green function s method. First, the density of states (DOS) of the bulk contact is calculated as indicated above. Next, the influence of the DOS of the bulk contact on the broadening and shifting of the discrete energy levels of the molecular orbitals (MO) of the cluster is accounted for via our DFT-Green function approach, as will be explained below. This yields the total DOS of the cluster as affected by the continuum. 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

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