Quantum semi-empirical approach

Needless to say, many extensions of the herein presented methods and different approaches are used to theoretically determine the electronic coupling. Conventional quantum chemistry ranging from fundamental ab initio methods to sophisticated semi-empirical approaches for the calculation of excited state properties is a useful tool to evaluate molecular-wire properties. 

The considerations on the intermolecular interactions can be conveniently reduced to considerations of atom-atom nonbonded interactions. Although these interactions can be treated by nonempir-ical quantum mechanical calculations, empirical and semi-empirical approaches have also proved useful in dealing with them. In the description of the atom-atom nonbonded interactions it is supposed that the van der Waals forces originate from a variety of sources. 

It has been mentioned in the introduction that many authors [4] believed that the model-Hamiltonian Hg = OH would give a better basis for the semi-empirical quantum theory than the derivations starting from e.g. the Hartree-Fock Hamiltonian. One has previously had the dilemma that the parameters in the semi-empirical approach determined fi om selected experiments were usually rather different from those calculated by means of the ab-initio methods. This applied e.g. to Slater s F- and G-integrals in the theory of atomic spectra, to Hiickel s parameters a and P in the theory of conjugated systems, or to the y parameter in the Pariser-Parr-Pople scheme. Careful studies by Karl Freed and his group [9] in Chicago have shown that the discrepancy between the two sets of parameters disappears, if one bases the semi-... 

The rapid growth of ab initio quantum mechanical (QM) simulations of condensed matter means that a comprehensive review of theoretical approaches and applications would in itself occupy a book. In this chapter the emphasis will be placed on QM studies of silicate and oxide systems. The key technologies will be identified and a critique of the possibilities and inadequacies of current theory presented. Although we discuss the technical details of implementation of QM methods, and some fundamental issues with regard to the description of electron interactions, the intention is to provide a general reference for non-experts in this field. Recent work based on semi-empirical approaches to QM simulations will not be reviewed (e.g. LaFemina, 1992 Goniakowski etal., 1993). 

There are a number of quantum theories for treating molecular systems. The first we shall examine, and the one which has been most widely used, is molecular orbital theory. However, alternative approaches have been developed, some of which we shall also describe, albeit briefly. We will be primarily concerned with the ab initio and semi-empirical approaches to quantum mechanics but will also mention techniques such as Hiickel theory and valence bond theory. An alternative approach to quantum mechanics, density functional theory, is considered in Chapter 3. Density functional theory has always enjoyed significant support from the materials science community but is increasingly used for molecular systems. 

In Chapter 2 we worked through the two most commonly used quantum mechanical models for performing calculations on ground-state organic -like molecules, the ab initio and semi-empirical approaches. We also considered some of the properties that can be calculated using these techniques. In this chapter we will consider various advanced features of the ab initio approach and also examine the use of density functional methods. Finally, we will examine the important topic of how quantum mechanics can be used to study the solid state. 

The definition of the gas-phase acidity through reaction (7.3) implies that this quantity is a thermodynamic state function. Thus, one could use quantum chemical approaches to obtain gas-phase acidities from the theoretically computed enthalpies of the species involved. However, two points must be noted before one proceeds A chemical bond is being broken and an anion is being formed. Thus, one may anticipate the need for a proper treatment of electronic correlation effects and also of basis sets flexible enough to allow the description of these effects and also of the diffuse character of the anionic species, what immediately rules out the semi-empirical approaches. Hence, our discussion will only consider ab initio (Hartree-Fock and post-Hartree-Fock) and DFT (density functional theory) calculations. 

The Amsterdam Density Functional package (ADF) is software for first-principles electronic structure calculations (quantum chemistry). ADF is often used in the research areas of catalysis, inorganic and heavy-element chemistry, biochemistry, and various types of spectroscopy. ADF is based on density functional theory (DFT) (see Chapter 2.39), which has dominated quantum chemistry applications since the early 1990s. DFT gives superior accuracy to Hartree-Fock theory and semi-empirical approaches, especially for transition-metal compounds. In contrast to conventional correlated post-Hartree-Fock methods, it enables accurate treatment of systems with several hundreds of atoms (or several thousands with QM/MM)." ... 

In order to extend these methods to make them feasible for the study dynamical chemical processes in biopolymers, simplifying assumptions are necessary. The most obvious choice is the use of semi-empirical techniques within the Hartree Fock, linear combination of atomic orbitals framework. These methods can achieve speedups on the order of 1000 over typical ab initio calculations using split valence basis sets within the Hartree Fock approximation. Often greater accuracy can be achieved as well because of the parameterization inherent in the semi-empirical approaches. One semi-empirical approach which has proven successful in representing many chemically interesting processes is the AMI and MNDO Hartree Fock Self-Consistent Field methods developed and paramerterized by Dewar and coworkers [46]. These methods have recently been implemented in a mixed quantum/ classical methodology for the study of chemical and biochemical processes by Field et al. [47]. 

The interatomic potentials define the force field parameters that contribute to the lattice energy of a relaxed or energy minimized structure. The fundamental question is how reliable is a force field The force field used in evaluating a potential function must be consistent and widely applicable to all similar systems. It must be able to predict the crystal properties as measured experimentally. Two main approaches, namely empirical and semi-empirical, are usually employed in the derivation of potential parameters. Empirical derivations involve a least square fitting routine where parameters are chosen such that the results achieve the best correlation with the observed properties. The semi-empirical approach uses an approximate formulation of the quantum mechanical calculations. An example of such an approximation is the electron gas method [57] which treats the electron density at any point as a uniform electron gas. The following is the analytical description of the potential energy function and interatomic potentials we recommend for use in simulation of zeolites and related system. 

In fact almost all went the way that Mulliken proposed, using the molecular orbital model. In this approach it was possible to formulate the equations in a manner suitable for calculation and to develop consistent approximation schemes that at least allowed semi-empirical calculations to be made. A number of semi-empirical schemes were developed, particularly for aromatic and conjugated systems, which can be regarded as inspired by the initial efforts of Hiickel in 1931 to use molecular orbitals in this area. For such systems the idea of a delocalised electron distribution came immediately out of the calculations, so that there was no need to invoke the bond or the idea of resonance. However the parameterisation schemes within these semi-empirical approaches, were cast in terms of integrals between hybrid orbitals, so that aspect of Pauling s ideas remained alive both in chemistry and in quantum chemistry. 

Ncc,cc the number of adjacent CC bonds, A cc.ch the number of adjacent pairs of CC and CH bonds, and Nch,ch the number of adjacent pairs of CH bonds. The quantities xc, Xcc, Xch, Xcc,cc, Xcc,ch, and ch,ch are formally defined through the corresponding quantum chemical integrals, which in semi-empirical approaches can be viewed as adjustable parameters. 

Elementary rate constants can be estimated also using semi-empirical methods, which are not as accurate as quantum mechanical approaches, being able at the same time to reduce the computational costs of model development. One of such computationally inexpensive semi-empirical approaches appHcable to small molecules, is the bond-order conservation (BOG) or unity bond index—quadratic exponential potential (UBI—QEP) technique of Shustorovich and Sellers. This method ensures thermodynamic consistency... 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

A more practical approach for larger systems is molecular dynamics. In this method, the properties of bonds are determined through a combination of quantum-mechanical simulation and physical experiments, and stored in a database called a (semi-empirical) force field. Then a classical (non-quantum) simulation is done where bonds are modeled as spring-like interactions. Molecular... 

Of course, most of what I have said so far is well known. Nevertheless, I hope to have given these issues a new perspective by adopting an almost perversely rigorous approach in demanding that every aspect of electronic configurations should be strictly deducible from quantum mechanics. Although I am not in a position to propose a better explanation, I do not think that we should be complacent about what the present explanation achieves. As I have tried to argue, in terms of deduction from theoretical principles, the present semi-empirical explanation is not fully adequate. 

Hpp describes the primary system by a quantum-chemical method. The choice is dictated by the system size and the purpose of the calculation. Two approaches of using a finite computer budget are found If an expensive ab-initio or density functional method is used the number of configurations that can be afforded is limited. Hence, the computationally intensive Hamiltonians are mostly used in geometry optimization (molecular mechanics) problems (see, e. g., [66]). The second approach is to use cheaper and less accurate semi-empirical methods. This is the only choice when many conformations are to be evaluated, i. e., when molecular dynamics or Monte Carlo calculations with meaningful statistical sampling are to be performed. The drawback of semi-empirical methods is that they may be inaccurate to the extent that they produce qualitatively incorrect results, so that their applicability to a given problem has to be established first [67]. 

Density functional theory (DFT),32 also a semi-empirical method, is capable of handling medium-sized systems of biological interest, and it is not limited to the second row of the periodic table. DFT has been used in the study of some small protein and peptide surfaces. Nevertheless, it is still limited by computer speed and memory. DFT offers a quantum mechanically based approach from a fundamentally different perspective, using electron density with an accuracy equivalent to post Hartree-Fock theory. The ideas have been around for many years,33 34 but only in the last ten years have numerous studies been published. DFT, compared to ab initio... 

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