Calculations feasibility

That said, the remarkable advances in computer hardware have made ab initio calculations feasible for small systems, provided that various technical details are carefiilly treated. A few examples of recent computations... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

Things have moved on since the early papers given above. The development of Mpller-Plesset perturbation theory (Chapter 11) marked a turning point in treatments of electron correlation, and made such calculations feasible for molecules of moderate size. The Mpller-Plesset method is usually implemented up to MP4 but the convergence of the MPn series is sometimes unsatisfactory. The effect... 

In the present section, we give examples of simulated vibrational bands of These bands start in the lowest inversion states 0 and end in the states V2V2 with U2<4. The rovibrational wavefunctions with 7<18 are calculated variationally as described in Ref. [1]. To make these variational calculations feasible the size of the vibrational basis set is reduced relative to that employed for the calculations of vibrational transition moments in Section 4.1. We now require that... 

Modification of Constraint Sets at X°. At the calculated feasible point X°, some of the inactive constraints (i.e. will become active. In order to preserve the determinancy features of the system of equations, some of the previously active constraints (i.e. cp t) will become inactive. Starting at the current optimum X where cgpt is the set of active inequality constraints,... 

Atomic and molecular adsorption at vanadium oxide surfaces have been studied theoretically using both periodic slab and cluster models where so far studies are restricted to the pentoxide, V2O5, as a substrate due to its possible importance in catalytic applications as mentioned before. Further, adsorbate species include in all cases atoms (H [122-123, 126, 136-142], O (see below)) or rather small molecules (O2 (see below), H2O [143-144], NH3 [145-147], NO [146, 148], C2H4 [149], propene (CsHg) [140], toluene (CeHsCHj) [140]) that are of catalytic interest but also small enough to make meaningful calculations feasible. 

APB converts these into <121> shear planes. It would also obviously be desirable to calculate the activation energies for interstitial migration down the APBs. This is at present beyond the scope of our calculations, although developments in the field may soon render such calculations feasible. 

One of the problems that the early applications of the MCSCF method faced was the construction of the wave function. It was necessary to keep it short in order to make the calculations feasible. Thus, one had to decide beforehand which where the most important CSFs to include in the Cl expansion. Even if this is quite simple in a molecule like H2 it quickly becomes ambiguous for larger systems. However, the development of more efficient techniques to solve large Cl problems made another approach possible. Instead of having to choose individual CSFs, one could choose only the orbitals that were involved and then make a full Cl expansion in this (small) orbital space. In 1976 Ruedenberg introduced the orbital reaction space in which a complete Cl expansion was used (in principle). All orbitals were optimized—the Fully Optimized Reaction Space— FORS [21]. 

Although the R-matrix method for scattering was proposed many years ago, its implementation has been quite restricted because of the slow convergence of the unadorned R-matrix to numerically adequate results. For inelastic scattering the use of a simple zero order Hamiltonian with an analytic solution to generate the Buttle correction makes it a viable option for some systems. In particular the increase in memory and speed of large computers make large basis calculations feasible. 

We decided to investigate corrections to the B. O. approximation for a number of diatomic hydrides and deuterides in order to obtain information about electronic isotope effects on the gas phase isotopic exchange reactions (1). Such calculations require a knowledge of electronic wave functions for molecules and the evaluation of a number of integrals. While such calculations would at one time have been very difficult, the availability of a large digital computer makes such calculations feasible now. The results of some of these calculations will be reported in the following sections they are reported in more detail elsewhere (2, 4),... 

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