Calculation using molecular orbital theory

This is the basic equation used to derive normal coordinate analysis [110] as well as to define the vibrational quantities to be calculated using molecular orbital theory [79,94], The coefficients, g , are the forces acting on the nuclei, which are zero at equilibrium geometiy. This leaves the quadratic terms Vs the first term in the change of potential energy with instantaneous vibrational displacement. The quadratic terms Fy, are conveniently ordered as a matrix which is known as the force field or force matrix. These terms correspond to the derivatives of the potential energy V ... 

This molecular-orbital model of metallic bonding (or band theory, as it is also called) is not so different in some respects from the electron-sea model. In both models Ihe electrons are free to move about in the solid. The molecular-orbital model is more quantitative than the simple electron-sea model, however, so many properties of metals can be accounted for by quantum mechanical calculations using molecular-orbital theory. 

The molar absorption (extinction) coefficient, Einstein coefficient and absorption cross-section are commonly used measures of transition probability. The first three are used for atomic and molecular species in the gas or solution phase, while the latter is commonly used in solid-state studies. These are absolute measures of absorption probability and are ultimately derived from the transition dipole moment, and are therefore all related. They can be measured experimentally firom the absorption spectrum and can, in some cases, be calculated using molecular orbital theory programs. (Note that generally MO calculations will give the oscillator strength for any forbidden transition as zero). 

A unified theoretical explanation using molecular orbital theory has been proposed. Grimme [65] investigated the PFR of phenyl acetate as well as the photo-Claisen rearrangement of allyl phenyl ether and the 3-cleavage of para-substituted phenoxyacetones. A unified description of the three reactions has been invoked according to MNDOC-CI and AMl/AMl-HE calculations. No matter what ex-... 

It is essential to have tools that allow studies of the electronic structure of adsorbates in a molecular orbital picture. In the following, we will demonstrate how we can use X-ray and electron spectroscopies together with Density Functional Theory (DFT) calculations to obtain an understanding of the local electronic structure and chemical bonding of adsorbates on metal surfaces. The goal is to use molecular orbital theory and relate the chemical bond formation to perturbations of the orbital structure of the free molecule. This chapter is complementary to Chapter 4, which... 

The same principles that we have used for the description of diatomic molecules will now be used in a description of the electronic structures of triatomic molecules. However, let it be clear that in using molecular-orbital theory with any hope of success, we first have to know the molecular geometry. Only in very rare cases is it possible from qualitative molecular-orbital considerations to predict the geometry of a given molecule. Usually this can only be discussed after a thorough calculation has been carried out. 

While molecular orbital calculations have been used to calculate the quadratic hyperporaizability, p, (4-5) we believe this to be the first attempt to approach both the molecular and macromolecular aspects of the design of these materials in a consistent manner using molecular orbital theory. 

In the foregoing it has been assumed that the relatively large positive coupling to /3-protons in 7r-radicals such as ethyl occurs as a result of hyperconjugation. This is not necessarily the case, although arguments in favour of the concept are considerable (Symons, 1962). Inevitably, there are several alternative descriptions suitable as a starting point for theoretical calculations. Thus, molecular-orbital theory has been used by some (Bersohn, 1956 Chesnut, 1958), and a... 

Relative Electronegativities of Zeolite Anion and Halides—Comparison of Anion Net Charges Calculated by Molecular Orbital Theory using B3LYP/LANL2DZ with NBO Method ... 

A more quantitative approach, using molecular orbital theory and the known frequencies of the molecular bending modes, to show that the pi bonding term is proportional to the number of vacancies in the t%g metal -orbitals has also been made (70). The populations of the three halogen /(-orbitals, and their variance with bending vibrations (assumed to be harmonic), were calculated by means of molecular orbital... 

Kubicki, J.D., Interpretation of vibrational spectra using molecular orbital theory calculations, in Molecular Modeling Theory Applications in the Geosciences, Cygan, R.T. and Kubicki, J.D., Eds., Geochemical Society of America, Washington, D.C., 2001, p. 459. 

Hoffmann, Shaik, and Hiberty, with additional commentary by Rob-erts ° and by Streitwieser, presented reasons why valence bond theory was initially appealing to organic chemists but was supplanted by molecular orbital theory. Roberts noted, in particular, the difficulty in carrying out valence bond calculations relative to that of using molecular orbital theory. An observation by Hiberty reinforces a point made in Chapter 1. 

Here, n, is the number of atoms of kind i, x fbeir susceptibility per gram-atom, and the sum extends over all atoms in the molecule. In addition, is the number of certain structural elements of kind j, Apj their contribution to the susceptibility, and the sum is over all these contributions. The quantities Xai and /Ipj are called the Pascal constants and the constitutive corrections, respectively. The most frequently used constants Za and constitutive corrections Ap are listed in Tab. 7. In general, the diamagnetic susceptibility of an organic substance may be estimated on the basis of the tabulated values with an accuracy of a few percent. Even though the Pascal scheme was established purely on an empirical basis, it is justified by more recent calculations involving molecular orbital theory [170]. Critical discussions and reviews on the subject are available in the literature [118, 167] and illuminating examples for the calculation of susceptibilities within the Pascal scheme are included in various textbooks [2,8,14]. 

Consider the quantum mechanical description of the hydrogen molecular ion in its simplest version. Let us use molecular orbital theory with the atomic basis set composed of only two Slater Type Orbitals (STO) l a and Isi, centred on the nuclei a and b. The mean value of the Hamiltonian calculated with the bonding (+) and antibonding (-) orbital (see Chapter 8 and Appendix D) reads as... 

Interpretation of Vibrational Spectra Using Molecular Orbital Theory Calculations... 

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