Molecular orbitals from atomic orbital wave

A FIGURE 9.A0 Molecular orbitals from atomic orbital wave functions. 

In setting up molecular orbitals from atoms whose wave functions have already been determined, the following principle has been found to be both general and useful If the electron clouds of iwo atoms are likely to overlap when the atoms approach, then the molecular wave function for the combination can he obtained by a linear combination of the individual wave functions. (A linear combination of two quantities is obtained by multiplying each by a constant and adding.) Stated mathematically ... 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

A molecular orbital wave function for the bond between H and Cl in HCl, constructed, assuming that the bond is formed from the Is electron of H atom and a 3p electron of Cl atom ... 

Qa Qb are the charges on atoms A and B, obtained from the Mulliken population analysis [28] of the molecular orbital wave function R is the A-B bond length in A. The Vmid values for the C-NO2 bonds in nitroaromatic molecules are taken to be indicative of the relative instabilities of the respective bonds indeed eq. (2) is closely related to a bond energy expression that has been developed by Fliszar [29,30]. 

Edgecombe, K. E.,Boyd,R. J. (1987). Atomic orbital populations and atomic charges from self-consistent field molecular orbital wave functions. J. Chem. Soc., Faraday Trans. 

An approximate molecular orbital wave function for the LiH molecule was constructed using hybrid orbitals called sp hybrids, which are a linear combination of 5 and p atomic orbitals on the same nucleus. A bonding molecular orbital made from a 2sp hybrid on the Li nucleus and a U orbital on the H nucleus provides an adequate description of the bonding in LiH. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Molecular orbital (MO) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so called because they belong to the entire molecule rather than to an individual atom. Just as an atomic orbital, whether un hybridized or hybridized, describes a region of space around an atom where an electron is likely to be found, so a molecular orbital describes a region of space in a molecule where electrons are most likely to be found. 

Molecular orbital (MO) theory (Section 1.11) A description of covalent bond formation as resulting from a mathematical combination of atomic orbitals (wave functions) to form molecular orbitals. 

To be more exact, every bond is a multi-center bond with contributions of the wave functions of all atoms. However, due to the charge concentration in the region between two atoms and because of the inferior contributions %H2, Xm> and Xh4> the bond can be taken to a good approximation to be a two-center-two-electron bond (2c2e bond) between the atoms C and HI. From the mathematical point of view the hybridization is not necessary for the calculation, and in the usual molecular orbital calculations it is not performed. It is, however, a helpful mathematical trick for adapting the wave functions to a chemist s mental picture. 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

One of the simplest approaches to comprehensive molecular orbital calculations is the extended Hiickel method. This method was developed by Roald Hoffman in the 1960s, and it was applied to hydrocarbon molecules. From the discussion presented in Chapters 2 and 3, we know that one of the first things that has to be done is to choose the atomic wave functions that will be used in the calculations. One of the most widely used types of wave functions is that known as the Slater wave functions (see Section 2.4). In the extended Hiickel method, the molecular wave functions are approximated as... 

From this wave function, one sees how even in the early beginning of molecular quantum mechanics, atomic orbitals were used to construct molecular wave functions. This explains why one of the first AIM definitions relied on atomic orbitals. Nowadays, molecular ab initio calculations are usually carried out using basis sets consisting of basis functions that mimic atomic orbitals. Expanding the electron density in the set of natural orbitals and introducing the basis function expansion leads to [15]... 

Mulliken introduced the term "orbital" distinct from "orbital wave function" in 1932 in the second of fourteen papers carrying the general title, "Electronic Structures of Polyatomic Molecules and Valence." Mulliken defined atomic orbitals (AOs) and molecular orbitals (MOs) as something like the... 

Mulliken, Life, 90. On the "orbital," Mulliken wrote in 1932 "From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. The method followed here will be to describe unshared electrons always in terms of atomic orbitals but to use molecular orbitals for shared electrons." In Robert Mulliken, "Electronic Structures of Polyatomic Molecules and Valence," Physical Review 41 (1932) 4971, on 50. 

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