Molecular Structure Calculations

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

GTOs are widely used in molecular structure calculations, but have the wrong behaviour at the nucleus. We might expect them to give poorer agreement with experiment. Table 18.2 shows a selection of calculations for the H atom. Standard GTO expansions were taken from the literature and left uncontracted. 

The investigator s choice of method (semi-empirical or ab initio) hinges on a number of factors, one of which is simple practicality concerning both time and expense. Semi-empirical methods usually give reasonable molecular structures and thermodynamic values at a fraction of the cost of ab initio calculations. Furthermore, molecular structures calculated by semi-empirical methods are the starting point for more complex ab initio calculations. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The application of theoretical tools for predicting molecular structure, such as ab initio calculations and density functional methods, are discussed in Chapter 6. These tools provide only a first approximation to the molecular structure. There is much room for further development of theoretical molecular structure calculations, but even so such methods have already become a standard part of molecular structure determinations. 

GeH2 13 C and 73 Ge NMR. Chair conformation as determined from spectroscopy, in accordance with molecular structure calculations, similarly to the analogous carbocyclic compound. Introduction of bulky substituents results in a preferred boat conformation. 60... 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

Modern many-body methods have become sufficiently refined that the major source of error in most ab initio calculations of molecular properties is today associated with truncation of one-particle basis sets e.g. [1]- [4]) that is, with the accuracy with which the algebraic approximation is implemented. The importance of generating systematic sequences of basis sets capable of controlling basis set truncation error has been emphasized repeatedly in the literature (see [4] and references therein). The study of the convergence of atomic and molecular structure calculations with respect to basis set extension is highly desirable. It allows examination of the convergence of calculations with respect to basis set size and the estimation of the results that would be obtained from complete basis set calculations. 

C.A. Coulson, Present state of molecular structure calculations, Conference on Molecular Quantum Mechanics, University of Colorado at Boulder, June 21-27, 1960, Reviews of Modem Physics 32 (1960) 170-177. 

At the same time molecular orbital (MO) methods were seeing a rapid development, also because of increased computational ability. These, at least on the surface, appear to provide a simpler approach to molecular structure calculations. Nevertheless, Matsen and Browne[32] made a forceful case for the use of MCVB methods, indicating the difficulties... 

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. 

For the dissociative excitation process (19.32), proceeding via purely dissociative (N A )d>ss states, only the cross-sections for X E (v) —> 63F + transitions are available. The purely dissociative states of H2 having N > 4 are not well known from molecular structure calculations, and they energetically lie far away from the ground X1 Xg state. The transitions to them from X1 Sg should be, therefore, much weaker. They can be nevertheless important in producing two excited atomic products at high collision energies. However, the process (19.32) may become important for AT > 2 initial states, and it should be included in the kinetic scheme of H2 CR models. 

History and Concepts. A complementary approach for molecular structure calculations is available, and it is referred to as the molecular methanics or force field method it is also known as the Westheimer method. In 1946, twenty years after the impressive development of quantum theory, three papers appeared in the literature which applied classifical mechanical concepts to problems of chemical interest. Westheimer investigated the racemization of some optically active biphenyl derivatives. His work demonstrated the potential usefulness of molecular mechanics. The other two papers were attempts to tackle more complex problems. 

Coulson, C.A. (I960). Present State of Molecular Structure Calculations. Rev.Mod.Phys., 32,170. 

As an example of a mature topic, consider Density Functional Theory (DFT). DFT is far from new and can be traced back to the work of John Slater and other solid state physicists in the 1950 s, but it was ignored by chemists despite the famous papers by Hohenberg/ Kohn (1964) and Kohn/ Sham (KS) (1965). The HF-LCAO model dominated molecular structure theory from the 1960 s until the early 1990s and I guess the turning point was the release of the rather primitive KS-LCAO version of GAUSSIAN. DFT never looked back after that point, and it quickly became the standard for molecular structure calculations. So this Volume of the SPR doesn t have a self contained Chapter on DFT because the field is mature. 

The practical realization of multicomponent MBPT rests on the development of efficient algorithms and the associated computer code. In recent work, we have advocated the use literate programming techniques in the development and publication of computer code for molecular structure calculations. We briefly discuss the application of these methods to the multicomponent many-body perturbation expansion. 

A Hierarchal QSAR Molecular Structure Calculator Applied to a Carcinogenic Nitrosamine Data Base... 

In general QSAR models can be constructed from any combination of the four levels of structure calculation. We are currently developing a software package to compute the different classes of molecular descriptors and to construct action models. This software package is being called "A Hierarchal QSAR Molecular Structure Calculator". 

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