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Chemical Concepts from Molecular Orbital Theory

Feng Long Gu1, Jincheng Yu2, and Weitao Yang2

1South China Normal University, School of Environment, Ministry of Education, Key Laboratory of Theoretical Chemistry of Environment, Guangzhou, 510006, People's Republic of China

2Duke University, Department of Chemistry, Box 90346, Durham, NC, 27708-0346, USA

1.1 Introduction

Quantum chemistry provides the theoretical foundations and quantitative explanations of the physical and chemical properties of atoms and molecules in terms of the physical interactions between electrons and nuclei. It is based on physics, combined with various mathematical treatments, and applies the basic principles and methods of quantum mechanics to study chemical problems [1]. Its research scope includes the microscopic study of the electronic structure properties of atoms, molecules, and bulk systems, intermolecular forces, chemical bond theory, and various spectra and chemical reactions.

The history of quantum chemistry can be traced back to 1927 just after the establishment of quantum mechanics. Till the end of the 1950s, three chemical bond theories, i.e. the valence orbital theory (VOT), the molecular orbital theory (MOT), and the coordination field theory (CFT), have been established to study molecular or crystalline systems by using quantum chemistry.

Among these three chemical bond theories, the VOT was developed by Pauling et al. [2-4] on the basis of Heitler and London's work [5] for the molecular structure of hydrogen. The result is much close to the classical atomic valence theory and generally accepted by chemists. The MOT, however, was first proposed by Mulliken and Hund [6-9] in the late 1920s to the early 1930s.

The main idea of Mulliken's work on MOT is that all electrons of atoms contribute to forming molecules, and the electrons in molecules are no longer belonging to a certain atom, but moving across the entire range of a molecular space. The state of motion of electrons in space in molecules can be described by the corresponding molecular orbitals (MOs), i.e. wave function . The main difference between MOs and atomic orbitals (AOs) is that in molecules, electrons move under the action of all nuclei potential fields. An important consequence is that the MOs can be obtained by the linear combination of atomic orbitals (LCAO) in molecules.

Following Mulliken's ideas, the simplest MOT was proposed by Hückel in 1931 [10], so-called Hückel molecular orbital (HMO) method, to successfully treat conjugated molecular systems. The MOT calculation is relatively simple and now widely appreciated by chemists, and it is supported by photoelectron spectroscopy experiments, making it dominant in chemical bond theory.

From the 1960s, the main goal was to further develop the quantum chemical calculation methods, among which the ab initio calculation method, the semi-empirical methods, and other methods have expanded the application scope of quantum chemistry and the calculation accuracy has been gradually improved. Consequently, some accurate results calculated from quantum chemistry were almost exactly the same as the experimental values. The development of computational quantum chemistry has expanded quantitative computing to large molecules and the applications of quantum chemistry into other disciplines become possible.

With the development of MOT and the upgrading of computer facilities, the systems that quantum chemistry can deal with have become larger and larger, and the calculation accuracy has been continuously improved. Quantum chemical computing programs have also become an increasingly important tool for solving chemical problems, and it is expected that more complex chemical problems can be solved in the future. At present, there are many popular program suites, such as the Gaussian series [11], GAMESS [12, 13], and others.

In the beginning of MOT, there seemed to be no direct relation between MOs and the bonds in a chemical formula, because MOs obtained from MOT normally extend over the whole molecule space and are not restricted to the region between two atoms. The difficulty was overcome by using equivalent localized molecular orbitals (LMOs) instead of the delocalized ones. The mathematical definition of equivalent MOs was given only in 1929 by Lennard-Jones [14]. The concept of localization of MOs leads to the connection between MOs and the pictures of chemical bonds. Benefits from LMOs are at least four followings (i) related to the concepts of chemical bonds, useful to isolate functional groups from different molecules; (ii) reducing the efforts for computation; (iii) transferable from one molecule to others within analogical structures; (iv) more suitable for LMOs to treat correlation.

This chapter is organized as follows, the MOT is surveyed together with the localization methods either for OLMOs and NOLMOs.

1.2 Molecular Orbital Theory

In Niels Bohr's atomic model, which is based on principles of quantum physics, electrons circle the atomic nucleus in different shells containing a fixed number of electrons. The assumption was that attractive forces between the atoms in a molecule are the result of atoms sharing electrons to fill the electron shells.

Heitler and London [5] first adopted quantum mechanics to treat hydrogen molecule in 1927, revealing the nature of the chemical bond between two hydrogen atoms, leading the typical Lewis theory to today's modern VOT. The concept of the atomic bonding created by electron sharing was introduced by Lewis in his 1916s fundamental paper [15]. It was elaborated by Langmuir [16] a few years later. Pauling et al. [2-4] introduced the concept of hybrid orbitals to greatly develop VOT and successfully applied it to the structure of diatomic molecules and polyatomic molecules.

VOT coincides with the classical concept of electron bonding familiar to chemists and has been rapidly developed as soon as it appeared. However, the calculation of VOT is more complicated, which makes the later development slow. With the increasing improvement of computing technology, there will be new developments in this theory.

VOT focuses on the contribution of unpaired electrons in the outermost orbital between bonded atoms in the formation of chemical bonds, which can successfully explain the spatial configuration of covalent molecules. However, the inner electrons of the bonding atom were not considered during the actual situation of bonding.

Meanwhile from the mid-1920s, quantum mechanics has been applied to develop sophisticated models for the movement of electrons within a molecule, so-called molecular orbitals (MOs). Under the work of Hund [6,] Mulliken [9,] and John Lennard-Jones [14,] MOT began to arise. Thus, in the beginning, the MOT was called the Hund-Mulliken theory. The concept of the word "orbital" was first proposed by Mulliken in 1932 [9]. The first paper using MOT was published by Lennard-Jones in 1929 [14] to treat MOT in a quantitative way. The LCAO approximation was introduced for constructing MOs to study the electronic structure of oxygen molecule from quantum principles. This convinced chemists that quantum mechanics is so useful, and the success of the MOT today owes much to their great contributions.

MOT is an effective approximation method for dealing with the structure of diatomic molecules and polyatomic molecules and is an important part of chemical bond theory. It differs from VOT, which focuses on understanding chemistry by hybridizing AOs into bonds, while the former focuses on the cognition of MOs. The idea of MOT is that electrons in a molecule move around the entire molecule. MOT pays attention to the integrity of molecules, so it better illustrates the structure of polyatomic molecules. At present, MOT stands on an important position in modern covalent bond theory and is widely accepted and considered a valid and useful theory.

By the 1950s, MOs were thoroughly defined as eigenfunctions of the self-consistent field Hamiltonian operator, marking the development of MOT into a rigorous scientific theory. Hartree-Fock (HF) method is a more rigorous treatment of MOT, and MOs are expanded according to a set of basis of AOs to develop the Hartree-Fock Roothaan (HFR) equation.

Equation (1.1) is so called linear combination of atomic orbitals (LCAO), where was used in the 1930s by Hund, Mulliken [9,] Hückel [10,] and others to construct MOs for polyatomic molecules, also called the LCAO-MO theory.

The Hartree-Fock Roothaan equation (Eq. (1.2)) is a method of ab initio calculation, and the ab initio method is simply to use a "correct" Hamiltonian operator, except for the most basic constants, no longer citing any experimental data, based on the Schrödinger equation, only using single-electron, nonrelativistic, and Born-Oppenheimer approximations.

On this basis, a variety of ab initio quantum chemical calculation methods have been developed. At the same time, MOT has also been applied to a semi-empirical calculation that uses more approximate methods, known as semi-empirical quantum chemical calculations.

MOT, based on HF...