Molecular Orbital Theory Diatomics

Shortly after the development of VBT, an alternative model, known as MOT, was introduced by the American physicist Robert Mulliken (and others) around 1932. MOT is a delocalized bonding model, where the nuclei in the molecule are held in fixed positions at their equilibrium geometries and the Schrodinger equation is solved for the entire molecule to yield a set of MOs. In practice, it is possible to solve the Schrodinger equation exactly only for one-electron species, such as H2. Whenever more than one electron is involved, the wave equation can only yield approximate solutions because of the e/ectron correlation problem that results from Heisenberg s principle of indeterminacy. If one cannot know precisely the position and momentum of an electron, it is impossible to calculate the force field that this one electron exerts on every other electron in the molecule. As a result of this mathematical limitation, an approximation method must be used to calculate the energies of the MOs. 

The approximation method of choice is the LCAO-MO method, which is an acronym that stands for linear combinations of atomic orbitals to make molecular orbitals. Unlike VBT, however, the linear combinations used to construct MOs derive from the AOs on two or more different nuclei, whereas the linear combinations used to make hybrid orbitals in VBT involved only the valence orbitals on the central atom. Using the variation theorem, the energy of a particle can be determined from the integral in Equation (I0.I2), which is also written in its Dirac (or bra-ket) notation. 

Now let us consider the Hj molecule. The Hamiltonian for Hj is given by Equation (10.13), where M is the mass of the H nucleus, is the mass of an electron, 

Because the nuclei are several orders of magnitude more massive than the electrons and will therefore move more slowly than the electrons, the Born-Oppenheimer approximation states that the nuclear and electronic motions of the molecule can be treated separately. Essentially, we can assume that the nuclei in the molecule are stationary and solve the equation solely for the electronic motion. This causes the nuclear or first term in Equation (10.13) to drop out. 

Because there is more than one electron in the H2 molecule, we cannot solve Equations (10.12) and (10.13) exactly. As a first approximation, let us assume that the wavefunction for H2 is some linear combination of the I s AOs of the two isolated H atoms, as given by Equation (I0.I4), where y/ is the wavefunction of our molecular orbital and (p is the wavefunction for a Is AO. The constants c and C2 are simply weighting factors. These are the adjustable parameters of our trial wavefunction. Because the energies of the two I s AOs for each H atom are identical, c = c- in this example. 

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Lewis structures Valence bond theory Molecular orbital theory diatomics Octet rule... 

To illustrate molecular orbital theory, we apply it to the diatomic molecules of the elements in the first two periods of the periodic table. 

Among the diatomic molecules of the second period elements are three familiar ones, N2,02, and F2. The molecules Li2, B2, and C2 are less common but have been observed and studied in the gas phase. In contrast, the molecules Be2 and Ne2 are either highly unstable or nonexistent. Let us see what molecular orbital theory predicts about the structure and stability of these molecules. We start by considering how the atomic orbitals containing the valence electrons (2s and 2p) are used to form molecular orbitals. 

Hurley, A. C., Proc. Roy. Soc. [London) A216, 424, The molecular orbital theory of chemical valency. XIII. Orbital wave functions for excited states of a homonuclear diatomic molecule."... 

The molecular orbital theory of polyatomic molecules follows the same principles as those outlined for diatomic molecules, but the molecular orbitals spread over all the atoms in the molecule. An electron pair in a bonding orbital helps to bind together the whole molecule, not just an individual pair of atoms. The energies of molecular orbitals in polyatomic molecules can be studied experimentally by using ultraviolet and visible spectroscopy (see Major Technique 2, following this chapter). 

Molecular orbital theory is more complex than the hybrid orbital approach, but the foundations of the model are readily accessible. Though complex, molecular orbital theory opens the door to many fascinating aspects of modem chemistry. In this section, we introduce the molecular orbital approach through diatomic molecules. 

As their names suggest, molecular orbitals can span an entire molecule, while localized bonds cover just two nuclei. Because diatomic molecules contain just two nuclei, the localized view gives the same general result as molecular orbital theoiy. The importance of molecular orbitals and delocalized electrons becomes apparent as we move beyond diatomic molecules in the follow-ing sections of this chapter. Meanwhile, diatomic molecules offer the simplest way to develop the ideas of molecular orbital theory. 

Of these three diatomic moiecuies, only N2 exists under normal conditions. Boron and carbon form soiid networks rather than isolated diatomic molecules. However, molecular orbital theory predicts that B2 and C2 are stable molecules under the right conditions, and in fact both molecules can be generated in the gas phase by vaporizing solid boron or soiid carbon in the form of graphite. 

The limitation of the above analysis to the case of homonuclear diatomic molecules was made by imposing the relation Haa = Hbb> as in this case the two nuclei are identical. More generally, Haa and for heteronuclear diatomic molecules Eq. (134) cannot be simplified (see problem 25). However, the polarity of the bond can be estimated in this case. The reader is referred to specialized texts on molecular orbital theory for a development of this application. 

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

This chapter consists of the application of the symmetry concepts of Chapter 2 to the construction of molecular orbitals for a range of diatomic molecules. The principles of molecular orbital theory are developed in the discussion of the bonding of the simplest molecular species, the one-electron dihydrogen molecule-ion, H2+, and the simplest molecule, the two-electron dihydrogen molecule. Valence bond theory is introduced and compared with molecular orbital theory. The photo-electron spectrum of the dihydrogen molecule is described and interpreted. 

The symmetry concepts of Chapter 2 and those of molecular orbital theory were applied to the construction of molecular orbitals for a range of diatomic molecules. 

Microwave spectrometer, 219-221 Microwave spectroscopy, 130, 219-231 compilations of results of, 231 dipole-moment measurements in, 225 experimental procedures in, 219-221 frequency measurements in, 220 and molecular structure, 221-225 and rotational barriers, 226-228 and vibrational frequencies, 225-226 Mid infrared, 261 MINDO method, 71,76 and force constants, 245 and ionization potentials, 318-319 Minimal basis set, 65 Minor, 14 Modal matrix, 106 Molecular orbitals for diatomics, 58 and group theory, 418-427 for polyatomics, 66... 

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. 

Molecular Orbital Theory Other Diatomic Molecules 281... 

There are a number of different approaches to the description of molecular electronic states. In this section we describe molecular orbital theory, which has been by far the most significant and popular approach to both the qualitative and quantitative description of molecular electronic structure. In subsequent sections we will describe the theory of the correlation of molecular states to the Russell Saunders states of the separated atoms we will also discuss what is known as the united atom approach to the description of molecular electronic states, an approach which is confined to diatomic molecules. 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

Gonzalez-Lafont, A. Truong, T. N. Truhlar, D. G. Direct dynamics calculations with neglect of diatomic differential overlap molecular orbital theory with specific reaction parameters, J. Phys. Chem. 1991, 95,4618-4627. 

The electronic structures of Group lA and IB metal clusters have been determined using two theoretical methods ah initio molecular orbital theory and the semi-empirical diatomics-in-molecules (DIM)... 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

The bonding in the diatomic halogen molecnles can be described in terms of simple Molecular Orbital Theory, as... 

Knowledge of the physical forces that influence the total energy of a system thus reveals the theoretical underpinnings of nearly aU of experimental chemistry. In fact, much of the early activity in chemical bonding theory was the result of attempts to understand the results of molecular spectroscopy experiments. The developers of what came to be called molecular orbital theory, Robert Mulhken (US) and Friedrich Hund (Germany), established a professional and personal relationship based on their conunon interest in the spectra of diatomic molecules especially in the influence of isotope effects. When compared to other theories of the time, a major advantage of their theoretical approach was the ability to directly apply the results to the elucidation of molecular spectra. ... 

Hence it can be stated that the predicted IPs, bonding energies and the bonding characteristics predicted for ThO using the relativistic and the NRL molecular orbital theories differ considerably and that there are very significant relativistic effects due to the participation of the 6d and 6p DFAO s of the Th atom in the bonding of the ThO diatomic. 

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