Parameterization density functional theory

Blomberg, M. R. A., Siegbahn, P. E. M., Svensson, M., 1996, Comparisons of Results From Parameterized Configuration Interaction (PCI-80) and From Hybrid Density Functional Theory With Experiments for First Row Transition Metal Compounds , J. Chem. Phys., 104, 9546. 

An objective function measuring the deviation from the parameterized model and the target response, determined, for example, from density functional theory based methods can be defined as... 

The electronic coupling of donor and acceptor sites, connected via a t-stack, can either be treated by carrying out a calculation on the complete system or by employing a divide-and-conquer (DC) strategy. With the Hartree-Fock (HF) method or a method based on density functional theory (DFT), full treatment of a d-a system is feasible for relatively small systems. Whereas such calculations can be performed for models consisting of up to about ten WCPs, they are essentially inaccessible even for dimers when one attempts to combine them with MD simulations. Semiempirical quantum chemical methods require considerably less effort than HF or DFT methods also, one can afford application to larger models. However, standard semiempirical methods, e.g., AMI or PM3, considerably underestimate the electronic couplings between r-stacked donor and acceptor sites and, therefore, a special parameterization has to be invoked (see below). 

The choice of the exchange correlation functional in the density functional theory (DFT) calculations is not very important, so long as a reasonable double-zeta basis set is used. In general, the parameterized model will not fit the quantum mechanical calculations well enough for improved DFT calculations to actually produce better-fitted parameters. In other words, the differences between the different DFT functionals will usually be small relative to the errors inherent in the potential model. A robust way to fit parameters is to use the downhill simplex method in the parameter space. Having available an initial set of parameters, taken from an analogous ion, facilities the fitting processes. 

A parameterization method of the Hamiltonian for two electronic states which couple via nuclear distortions (vibronic coupling), based on density functional theory (DFT) and Slaters transition state method, is presented and applied to the pseudo-Jahn-Teller coupling problem in molecules with an s2-lone pair. The diagonal and off-diagonal energies of the 2X2 Hamiltonian matrix have been calculated as a function of the symmetry breaking angular distortion modes and r (Td)] of molecules with the coordination number CN = 3... 

Semiempirical methods - based on approximate solutions of the Schrodinger equation with appeal to fitting to experiment (i.e. using parameterization) Density functional theory (DFT) methods - based on approximate solutions of the Schrodinger equation, bypassing the wavefunction that is a central feature of ab initio and semiempirical methods Molecular dynamics methods study molecules in motion. 

Label these statements true or false (1) For each molecular wavefunction there is an electron density function. (2) Since the electron density function has only x, y, z as its variables, DFT necessarily ignores spin. (3) DFT is good for transition metal compounds because it has been specifically parameterized to handle them. (4) In the limit of a sufficiently big basis set, a DFT calculation represents an exact solution of the Schrodinger equation. (5) The use of very big basis sets is essential with DFT. (6) A major problem in density functional theory is the prescription for going from the molecular electron density function to the energy. 

The successor model CM2 (Li et al., 1998) employed a slightly different mapping procedure and was also parameterized for phosphorus. For a training set of 198 compounds, CM2-AM1 and CM2-PM3 yielded rms deviations of 0.25 and 0.23 D, respectively. Inclusion of a secondary set of 13 more complex compounds resulted in rms deviations of 0.30 (CM2-AM1) and 0.23 D (CM2-PM3). CM2 was also parameterized for ab initio and density functional theory (DFT) schemes. 

Another approach to treating the boundary between covalently bonded QM and MM systems is the connection atom method, 125,126 in which, rather than a link atom, a monovalent pseudo-atom is used. This connection atom is parameterized to give the correct behaviour of the partitioned covalent bond, and has been implemented at semi-empirical molecular orbital (AMI and PM3)125 and density functional theory 126 levels of QM theory. It has been suggested that the connection atom approach is more accurate than the standard link atom approach.125... 

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

Theoretical approaches to parameterization have become popular. Theory has not been a great aid to inorganic modeling because of the excessive computational requirements for systems with so many electrons and because the veracity of the results is uncertain. This situation, however, appears to be changing rapidly as higher powered computers become more available and as new methods, such as local density functional theory, progress. 

First principles approaches are important as they avoid many of the pitfalls associated with using parameterized descriptions of the interatomic interactions. Additionally, simulation of chemical reactivity, reactions and reaction kinetics really requires electronic structure calculations [108]. However, such calculations were traditionally limited in applicability to rather simplistic models. Developments in density functional theory are now broadening the scope of what is viable. Car-Parrinello first principles molecular dynamics are now being applied to real zeolite models [109,110], and the combined use of classical and quantum mechanical methods allows quantum chemical methods to be applied to cluster models embedded in a simpler description of the zeoUte cluster environment [105,111]. 

A short introduction to electronic structure theory is included as well as a chapter on Density Functional Theory, which is the underlying method behind all calculations presented in the accompanying papers. Multiple Scattering Theory is also discussed, both in more general terms as well as how it is used in the methods employed to solve the electronic structure problem. One of the methods, the Exact Muffin-Tin Orbital method, is described extensively, with special emphasis on the slope matrix, which energy dependence is investigated together with possible ways to parameterize this dependence. 

Since it s inception in 1967, the Advances in Quantum Chemistry series has attempted to present various aspects of atomic, molecular, and solid state theory at the cutting edge. The contributions have taken various forms, from longer review articles to conference proceedings, and most of them have been well received. In this issue, we continue in this trend, and address some of the fundamental issues in density functional theory (DFT). DFT is extremely popular these days in its applied form It is almost impossible to read an experimental article where explanation of the results does not involve a DFT calculation using one or another commercial program and implementing one or another parameterized potential scheme. However, the success of applied DFT seems to be that it works, rather than the result of critical analysis of the fundamentals of the theory. However, the community also needs to explore the limits of the theory itself, and it is with this that the first four contributions in this issue concern themselves. 

An additional issue in the development of the density functional theory is the parameterization of the trial function for the one-body density. Early applications followed the Kirkwood-Monroe [17,18] idea of using a Fourier expansion [115-117,133]. More recent work has used a Gaussian distribution centered about each lattice site [122]. It is believed that the latter approach removes questions about the influence of truncating the Fourier expansion upon the DFT results, although departures from Gaussian shape in the one-body density can also be important as has been demonstrated in computer simulations [134,135]. 

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