Structural problems computational approaches

This algorithm alternates between the electronic structure problem and the nuclear motion It turns out that to generate an accurate nuclear trajectory using this decoupled algoritlun th electrons must be fuUy relaxed to the ground state at each iteration, in contrast to Ihe Car-Pairinello approach, where some error is tolerated. This need for very accurate basis se coefficients means that the minimum in the space of the coefficients must be located ver accurately, which can be computationally very expensive. However, conjugate gradient rninimisation is found to be an effective way to find this minimum, especially if informatioi from previous steps is incorporated [Payne et cd. 1992]. This reduces the number of minimi sation steps required to locate accurately the best set of basis set coefficients. 

In this section, the conceptual framework of molecular orbital theory is developed. Applications are presented and problems are given and solved within qualitative and semi-empirical models of electronic structure. Ab Initio approaches to these same matters, whose solutions require the use of digital computers, are treated later in Section 6. Semi-empirical methods, most of which also require access to a computer, are treated in this section and in Appendix F. 

This branch of bioinformatics is concerned with computational approaches to predict and analyse the spatial structure of proteins and nucleic acids. Whereas in many cases the primary sequence uniquely specifies the 3D structure, the specific rules are not well understood, and the protein folding problem remains largely unsolved. Some aspects of protein structure can already be predicted from amino acid content. Secondary structure can be deduced from the primary sequence with statistics or neural networks. When using a multiple sequence alignment, secondary structure can be predicted with an accuracy above 70%. 

One of the major goals of computational chemistry in the field of drug metabolism and pharmacokinetics (DMPK) is the prediction of oral availability. Several computational approaches have been published to predict this important parameter, starting from the molecular structure [1-3]. However, when applied to real case studies, none of these provided totally convincing results and, correspondingly, there is a lack of any general strategy to address this problem. 

The first calculations on a two-electron bond was undertaken by Heitler and London for the H2 molecule and led to what is known as the valence bond approach. While the valence bond approach gained general acceptance in the chemical community, Robert S. Mulliken and others developed the molecular orbital approach for solving the electronic structure problem for molecules. The molecular orbital approach for molecules is the analogue of the atomic orbital approach for atoms. Each electron is subject to the electric field created by the nuclei plus that of the other electrons. Thus, one was led to a Hartree-Fock approach for molecules just as one had been for atoms. The molecular orbitals were written as linear combinations of atomic orbitals (i.e. hydrogen atom type atomic orbitals). The integrals that needed to be calculated presented great difficulty and the computations needed were... 

As presented, the Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set no experimental data or other input is employed. As described in Appendix F, it is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite F, v elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level coulomb interaction integrals that can be computed in an ab initio manner. This approach forms the basis of so-called semi-empirical methods. Appendix F provides the reader with a brief introduction to such approaches to the electronic structure problem and deals in some detail with the well known Htickel and CNDO- level approximations. 

Such an information system must be able to identify molecules on the basis of their structure. Given a molecule, the system must derive a unique code for the molecule, so that the code can be looked up in a table and the properties of the molecule located. It is this coding or cataloging problem which I want to consider here. A number of codes for molecules have been proposed and used e.g. see (l,2,3,k). The existence of many different codes with no single standard suggests the importance and the difficulty of the problem. I shall attempt to explain why the problem is difficult, and to suggest some computer approaches to it. 

Currently the problems involved in calculating the electronic band structures of molecular crystals and other crystalline solids centre around the various ways of solving the Schrodinger equation so as to yield acceptable one-electron solutions for a many-body situation. Fundamentally, one is faced with an appropriate choice of potential and of coping with exchange interactions and electron correlation. The various computational approaches and the many approximations and assumptions that necessarily have to be made are described in detail in the references cited earlier. 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

Proposed shortly after the VB theory, the MO theory became the most popular approach to molecular structure calculations, mainly because this theory is much more amenable than VB to computer implementation. As a consequence, there is a great number of results of MO calculations on many chemical systems. With the improvement of the numerical techniques and of auxiliary interpretative tools by many research groups, together with the wide availability of computer codes, MO theory was soon established as the computational (and for some also the conceptual) approach to the molecular structure problem. Due to its widespread use, MO theory is frequently pushed beyond its conceptual limits. In this section we will briefly outline some aspects of MO theory and highlight its physico-chemical interpretation. 

Owing to recent developments in theoretical and computational methods, the quantum mechanical approach to the polymer electronic structure problem has begun to associate very fruitfully with experimental research in this field. Combination of the methods of molecular quantum theory with the ideas of theoretical solid-state physics has provided a really efficient tool, not only for the interpretation of experimental results, but also for investigation of fine details in the electronic structure which would be only barely accessible in experiments. 

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