Electronic Structure Calculations Numerical Approach

For any dynamical simulation, a continuous representation of the PES is mandatory since the potential and the gradients are needed for arbitrary configurations. One can in fact perform ab initio molecular dynamics simulations in which the forces necessary to integrate the classical equations of motion are determined in each step by an electronic structure calculations. There have been few examples for such an approach [35-37], However, in spite of the fact that electronic structure calculations can nowadays be performed very efficiently, still there is a significant numerical effort associated with ab initio calculations. This effort is so large that in the ab initio dynamics simulations addressing molecular adsorption and desorption at surfaces the number of calculated trajectories has been well below 100, a number that is much too low to extract any reliable reaction probabilities. 

Electronic structure calculations - The numerical approach We will focus in this subsection on electronic structure calculations (non-relativistic and relativistic) using numerical techniques, i.e., we do not use an expansion of single-electron functions (orbitals or spinors) in terms of analytic basis functions. Nowadays such numerical calculations are routinely feasible only for atoms and diatomics. Within the non-relativistic approach we have to determine rad.ial functions Pj(r) (z = nl) as parts of single-centre functions... 

Modern computer hardware, innovative numerical techniques, and novel theoretical approaches have dramatically transformed the character of quantum chemical calculations. Not only can properties of small molecules be computed with better accuracy, but also one can now obtain chemically useful information on large and very large systems almost routinely from ab initio electronic structure calculations. In particular, elimination of the need for storage of ERIs disposed of the 10 bottleneck that inhibited earlier methods from being successfully scaled up. The direct techniques are now commonly employed at the SCF level, and they claim an ever-increasing share of correlated calculations. 

As we have seen above chemical bonding in crystals (as well as in molecules) is analyzed in terms of the local properties of the electronic structure, obtained from the one-electron density matrix, written in a localized basis. Since local properties of electronic structure are essential ingredients of a number of theories and models (for example, the numerical values of atomic charges are used in the atom-atom potentials of the shell model), their estimation is of great importance. TVaditionally, the same AO basis is used both in LCAO SCF calculations and in the local properties definition, as was noted above. However, this approach is not always reliable, since the results of the population analyses are often strongly dependent on an inclusion of diffuse orbitals into the basis (useful for the electronic-structure calculations) and on the scheme chosen for the population analysis. 

Because the electronic energy Ee(q) in Eq. [8] and its derivatives must be calculated at each integration step of a classical trajectory, a direct dynamics simulation is usually very computationally intense. A standard numerical integration time step is /St = 10 " s. Thus, if a trajectory is integrated for 10 s, 10" evaluations of Eq. (8) are required for each trajectory. An ensemble for a trajectory simulation may be as small as 100 events, but even with such a small ensemble 10 " electronic structure calculations are required. Because of such computational demands, it is of interest to determine the lowest level of electronic structure theory and smallest basis set that gives an adequate representation for the system under study. In the following parts of this section, semiempirical and ab initio electronic structure theories and mixed electronic structure theory (quantum mechanical) and molecular mechanical (i.e. QM/MM) approaches for performing direct dynamics are surveyed. 

Further evaluations [164,165] have demonstrated the applicability of the fitted system of parameters for calculations of the electronic structure and spectra of numerous complexes of divalent cations using merely the CNDO parametrization for the /-system. In [140,169] the EHCF method is also extended for calculations of the ligands by the INDO, MINDO/3, and SINDO/1 parametrizations. In all calculations the experimental multiplicity (spin) and spatial symmetry of the corresponding ground stales were reproduced correctly. The summit of this approach has been reached in [170] by calculations on the i.s-[Fe(NCS btbipyjo] complex. Its molecular geometry is known both for the HS and LS isomers of the said compound. The calculation... 

The approach that will be used in this text is different, in that the description of electronic structures is greatly simplified to provide a more vivid understanding of the properties numerical estimates of properties will be obtained with calculations that can be carried through by hand rather than machine. We shall concentrate on the physics of the problem. In this context a scmicmpirical determination of matrix elements is appropriate. The first attempt at this (Harrison, 1973c) followed Phillips (1970) in obtaining the principal matrix element Kj from the measured dielectric constant. A second attempt (Harrison and Ciraci,... 

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