Molecular orbital theories

Molecular orbital theory is a semi-empirical method devoted to interpreting the energy-level structure of optical centers where the valence electron cannot be considered as belonging to a specific ion. In our ABe reference center, this would mean that the valence electrons are shared by A and B ions. The approach is based on the calculation of molecular orbitals (MO) of the ABe pseudo-molecule, V mo, from various trial combinations of the individual atomic orbitals, V a and of the A and B ions, respectively. The molecular orbitals V mo of the center ABe are conveniently written in the form 

The molecular orbital theory (MOT) is widely used by chemists. It includes both the covalent and ionic character of chemical bonds, although it does not specifically mention either. MOT treats the electron distribution in molecules in very much the same way that modem atomic theory treats the electron distribution in atoms. First, the positions of atomic nuclei are determined. Then orbitals aroimd nuclei are defined these molecular orbitals (MO s) locate the region in space in which an electron in a given orbital is most likely to be found. Rather than being localized arormd a single atom, these MO s extend over part or all of the molecule. 

The MO that results Irom addition of two s orbitals includes the region in space between the two nuclei it is called a bonding MO, and is of lower energy 

The preceding discussion is a simplified MO approach to bonding, but it illustrates some of the basic ideas and a little of the usefulness of the theory. MOT is very effective in handling both covalent and ionic contributions to the 

In conclusion, note that all three of these theories are, at best, only good approximations. All three can account qualitatively for many features of metal complexes all three are used currently, and one or the other may be most convenient for a given application. The most versatile is MOT. Unfortunately, it is also the most complicated. 

Determine the BAN of the metal in each of the following compounds. Note that several of these metals do not have an BAN equal to the atomic number of a noble gas. 

According to molecular orbital theory, the atomic orbitals involved in bonding acmally combine to form new orbitals that are the property of the entire molecule, rather than of the atoms forming the bonds. These new orbitals are called molecular orbitals. In molecular orbital theory, electrons shared by atoms in a molecule reside in the molecular orbitals. 

Molecular orbitals are like atomic orbitals in several ways they have specific shapes and specific energies, and they can each accommodate a maximum of two electrons. As was the case with atomic orbitals, two electrons residing in the same molecular orbital must have opposite spins, as required by the Pauli exclusion principle. And, like hybrid orbitals, the number of molecular orbitals we get is equal to the number of atomic orbitals we combine. 

Our treatment of molecular orbital theory in this book will be limited to descriptions of bonding in diatomic molecules consisting of elements from the first two periods of the periodic table (H through Ne). 

At its simplest, molecular orbital theory considers the symmetry properties of all of the atomic orbitals of all of the component atoms of a molecule. The basis of the calculation is to combine the (approximate) energies and wave-functions of the appropriate atomic orbitals to obtain the best possible approximations for the energies and wave-functions of 

4) The best source for an overview of Pauling s ideas and contribution to science is L. Pauling, The Nature of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, 1960. 

The sequence of energy levels obtained from a simple molecular orbital analysis of an octahedral complex is presented in Fig. 1-12. The central portion of this diagram, with the t2g and e levels, closely resembles that derived from the crystal field model, although some differences are now apparent. The t2g level is now seen to be non-bonding, whilst the antibonding nature of the e levels (with respect to the metal-ligand interaction) is stressed. If the calculations can be performed to a sufficiently high level that the numerical results can be believed, they provide a complete description of the molecule. Such a description does not possess the benefit of the simplicity of the valence bond model. 

Representations of the molecular orbitals indicated by the alg and eg levels of Fig. 1-12 are shown in Fig. 1-13. They each contain contributions from metal orbitals (the s for the flj, and the dz2 and the dx2 y2 for the e) and from some or all of the ligands. It is difficult to equate these multi-centred orbitals to the more familiar individual metal-ligand interactions and bonds. The majority of chemists prefer to think in terms of localised bonding, rather than in terms of electrons delocalised over the metal and all six ligands. 

Before leaving this brief introduction to molecular orbital theory, it is worth stressing one point. This model constructs a series of new molecular orbitals by the combination of metal and ligand orbitals, and it is fundamental to the scheme that the ligand energy levels and bonding are, and must be, altered upon co-ordination. Whilst the crystal field model probably over-emphasises the ionic contribution to the metal-ligand interaction, the molecular orbital models probably over-emphasise the covalent nature. 

2-2 MOI.ECULAR-ORBITAL THEORY According to molecular-orbital theory, electrons in molecules are in orbitals that may be associated with several nuclei. Molecular orbitals in their simplest approximate form are considered to be linear combinations of atomic orbitals. We assume that when an electron in a molecule is near one particular nucleus, the molecular wave function is approximately an atomic orbital centered at that nucleus. This means that we can form molecular orbitals by simply adding and subtracting appropriate atomic orbitals. The method is usually abbreviated LCAO-MO, which stands for linear combination of atomic 

Presented in this chapter is a verbal and pictorial description of Hartree-Fock MO theory. No equations will be given but reference will be made to appropriate parts of Appendix A where more details may be found. 

The electronic SE focuses on the energy levels of the molecule. By obtaining the lowest energy, one assumes that the associated wave function will yield the electron distribution of the electronic ground state. An alternative theory has come into recent prominance, in which the SE is bypassed and attention focused on the electron density from which many desired properties including energy can derived directly [density functional theory (DFT)]. 

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal- 

This method, called molecular orbital theory, starts with a simple picture of molecules, but it quickly becomes complex in its details. We will provide only an overview and then focus primarily on the application of molecular orbital theory to diatomic molecules. We will begin our overview by comparing molecular orbital theory with the conceptual model we introduced in Chapter 8 for multielectron atoms. 

As discussed in Chapter 8 (see page 350), we imagine that an atom has available to it a set of orbitals (Is, 2s, 2p, etc.) and can be built up electron by electron by placing electrons into these orbitals in a specified order. In a similar manner, with molecular orbital theory, we imagine that each molecule has available to it a set of orbitals, called molecular orbitals (MOs) and can be built up electron by electron by placing electrons into these orbitals in a specified order. Unlike atomic orbitals, which are centered on a single nucleus, molecular orbitals are defined with respect to all the nuclei. 

A stable molecular species has more electrons in bonding orbitals than in antibonding orbitals. For example, if the excess of bonding over antibonding electrons is tzvo, this corresponds to a single covalent bond in Lewis theory. In molecular orbital theory, we say that the bond order is 1. Bond order is one-half the difference between the number (no.) of bonding and antibonding electrons (e ), that is. 

The interaction of two hydrogen atoms according to molecular orbital theory 

Let s use the ideas just outlined to describe some molecular species of the first-period elements, H and He (Fig. 11-23). 

When two atoms approach each other the atomic orbitals will melt together and new so-called molecular orbitals will be formed. In these molecular orbitals the bond electrons of the covalent bond will be hosted. There are two types of molecular orbitals  

Bond orbitals, denoted with the Greek letter a Anti-bond orbitals, denoted with a  

The bond orbitals have lower energy levels compared to the anti-bond orbitals. As for the atomic orbitals these molecular orbitals are each able to host two electrons. In the following example we are going to se how the atomic orbitals of two hydrogen atoms melt together and form two molecular orbitals during the formation of a hydrogen molecule. 

Since the molecular bond orbital (ais) is lower in energy level at the two individual atomic orbitals, the two valence electrons rather prefer so stay in the bond orbital. The energy level of the anti-bond orbital (a is) is higher than that of the atomic orbitals and thus the valence electrons will no be hosted in this orbital. 

So because the total energy can be minimized it is beneficial for the two hydrogen atoms to create a hydrogen molecule. 

An energy diagram showing the relative energy levels of bonding and antibonding molecular orbitals. 

In most situations, valence bond theory will be sufficient for our purposes. However, there will be cases in the upcoming chapters where valence bond theory will be inadequate to describe the observations. In such cases, we will utilize molecular orbital theory, a more sophisticated approach to viewing the nature of bonds. 

Much like valence bond theory, molecular orbital (MO) theory also describes a bond in terms of the constructive interference between two overlapping atomic orbitals. However, MO theory goes one step further and uses mathematics as a tool to explore the consequences of atomic orbital overlap. The mathematical method is called the linear combination of atomic orbitals (LCAO). According to this theory, atomic orbitals are mathematically combined to produce new orbitals, called molecular orbitals. 

It is important to understand the distinction between atomic orbitals and molecular orbitals. Both types of orbitals are used to accommodate electrons, but an atomic orbital is a region of space associated with an individual atom, while a molecular orbital is associated with an entire molecule. 

That is, the molecule is considered to be a single entity held together by many electron clouds, some of which can actually span the entire length of the molecule. These molecular orbitals are filled with electrons in a particular order in much the same way that atomic orbitals are filled. Specifically, electrons first occupy the lowest energy orbitals, with a maximum of two electrons per orbital. In order to visualize what it means for an orbital to be associated with an entire molecule, we will explore two molecules molecular hydrogen (H2) and bromomethane (CH3Br). 

In contrast to VBT, full-blown MOT considers the electrons in molecules to occupy molecular orbitals that are formed by linear combinations (addition and subtraction) of all the atomic orbitals on all the atoms in the structure. In MOT, electrons are not confined to an individual atom plus the bonding region with another atom. Instead, electrons are contained in MOs that are highly delocalized—spread across the entire molecule. MOT does not create discrete and localized bonds between neighboring atoms. An immediate benefit of MOT over VBT is its treatment of conjugated tt systems. We don t need a patch like resonance to explain the structure of a carboxylate anion or of benzene it falls naturally out of the delocalized nature of the MOs. The MO models of simple molecules like ethylene or formaldehyde also lead to bonding concepts that are pervasive in organic chemistry. 

To create group orbitals or delocalized molecular orbitals, we need to understand how to combine atomic orbitals properly. Therefore, the starting point in developing our second model of organic bonding is a set of rules that lead by inspection to group orbitals and molec-ular orbitals. This procedure is called qualitative molecular orbital theory (QMOT). 

Valence bond theory is one of the two quantum mechanical approaches that explain bonding in molecnles. It acconnts, at least qualitatively, for the stability of the covalent bond in terms of overlapping atomic orbitals. Using the concept of hybridization, valence bond theory can explain molecnlar geometries predicted by the VSEPR model. However, the assnmption that electrons in a molecule occupy atomic orbitals of the individual atoms can only be an approximation, because each bonding electron in a molecnle mnst be in an orbital that is characteristic of the molecule as a whole. 

In some cases, valence bond theory cannot satisfactorily account for observed properties of molecnles. Consider the oxygen molecule, whose Lewis structure is 

Magnetic and other properties of molecules are sometimes better explained by another qnantum mechanical approach called molecular orbital (MO) theory. Molecular orbital theory desCTibes covalent bonds in terms of molecular orbitads, which remit from interaction of the atomic orbitals of the bonding atoms and are associated with the entire molecule. The difference between a molecular orbital and an atomic orbital is that an atomic orbital is associated with only one atom. 

One way to acconnt for the fact that an O2 molecule contains two unpaired electrons is to draw the following Lewis structure  

According to MO theory, the overlap of the li orbitals of two hydrogen atoms leads to the formation of two molecular orbitals one bonding molecular orbital and one antibonding molecular orbital. A bonding molecular orbital has lower 

The further development of the hgand field concept takes place in Molecular Orbitals (MO) Theory. As an atomic orbital is a wave fimction describing the spatial probability density for a single electron bound to the nucleus of an atom, a molecular orbital is a wave function, which describes the spatial probabihty density for a single electron bond to the set of nuclei, which constitute the framework of a molecule. 

The MO theory treats molecular bonds as a sharing of electrons between nuclei. Unlike the valence bond theory, which treats the electrons as locahzed balloons of electron density, the MO theory says that the electrons are delocalized. That means that they are spread out over the entire molecule. Now, when two atoms come together, their two atomic orbitals react to form two possible molecular orbitals. 

In this chapter, we discuss the various applications of group theory to chemical problems. These include the description of structure and bonding based on hybridization and molecular orbital theories, selection rules in infrared and Raman spectroscopy, and symmetry of molecular vibrations. As will be seen, even though most of the arguments used are qualitative in nature, meaningful results and conclusions can be obtained. 

As mentioned previously in Chapter 3, when we treat the bonding of a molecule by applying molecular orbital theory, we need to solve the secular determinant 

In molecular orbital theory, the molecular orbitals are expressed as linear combinations of atomic orbitals  

So we have nE s from eq. (7.1.1), and each E value leads to a set of coefficients, or to one molecular orbital. In other words, n atomic orbitals form n molecular orbitals i.e., the number of orbitals is conserved. 

The solution of eq. (7.1.1) is made easier if the secular determinant can be put in block-diagonal form, or block-factored  

Liquid oxygen caught between the poles of a magnet, because the O2 molecules are paramagnetic, having two parallel spins. 

Although we have used the hydrogen molecule to illustrate molecular orbital formation, the concept is equally applicable to other molecules. In the H2 molecule, we consider only the interaction between Is orbitals with more complex molecules, we need to consider additional atomic orbitals as well. Nevertheless, for all s orbitals, the process is the same as for Is orbitals. Thus, the interaetion between two 2s or 3s orbitals can be understood in terms of the molecular orbital energy level diagram and the formation of bonding and antibonding molecular orbitals shown in Fignre 10.22. 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

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. 

We have included in table 6.3 the electronic state nomenclature, the definition of which we now describe. First, we deal with the resultant orbital angular momentum about the intemuclear axis, which is denoted A, and is equal to the sum of the individual 

The vectors he along the intemuclear axis, so that we have a simple algebraic addition. 

For a single it electron, A = k = 1, and the electronic state is called a n state. For two it electrons, the resultant A is equal to 2 or 0, each of these states arising in two different ways. For A = 2 we have a A state, which is two-fold degenerate. For A = 0 the electronic state is a S state although there are two such states arising from the vectorial addition, they are not degenerate but split into a + and a state. We will define the difference between these states in due course. If A = 3, which can arise when cl orbitals are involved, the state is a b state, and so on. The resultant spin is equal to the sum of the individual spins, that is, 

There are still more reasons to believe that rj, as defined in Equation (2.12), is indeed what is meant by chemical hardness. To understand this, it is necessary to see whether the chemical concepts derived by DFT are compatible with molecular orbital (MO) theory.This theory is certainly the most widely used by chemists and is very successful in many areas. It is almost universally applied to explain structure and bonding, visible-UV spectra, chemical reactivity and detailed mechanisms of chemical reactions. 

Hard molecules have a large HOMO-LUMO gap, and soft molecules have a 

A number of papers have appeared showing a correlation between polarizability, a, and softness.Empirically it is found that is a linear function of (/—the softness. This is equivalent to the classical result for spheres of radius R, that charging energies are proportional to 1 /R, whereas polarizability is proportional to R . Calculations of o using DFT, and EN equalization, shows that a is equal to (/- times a factor dependent on the size of the system. 

It would be convenient if we could find I — A) from absorption spectra. But this is not possible, since (/— A) is usually about twice as large as Ai max- For example, H2S has (/- A) = 12.6 eV, determined experimentally whereas max = 6.4 eV. The difference arises from the additional electron-electron repulsion that results from adding an electron to the LUMO, instead of merely exciting it from the HOMO. For filled-shell molecules. 

The visible range is from 12.5 to 25kK (10 cm ). Some free radicals are 

Mulliken s interest in the electronic levels in molecules in the 1920s was stimulated by suggestions that their molecular spectra bore similarities to atomic spectra and definite relationships could be discerned for isosteric molecules. He found that the spectroscopic analogy between isosteric molecules could be extended to atoms with the same number of electrons, and this relatirmship was to lead subsequently to the united atom approach. He and Birge classified the electronic states in diatomic molecules using the same Russell-Saunders classification used previously for atomic states. Himd s theoretical analysis [ 149,162-166] of the nature of electronic states in molecules therefore proved to be timely for Mulliken and led him to publish [167-171] a summary of the theory and provide extra experimental evidence supporting it. In the molecular orbital theory Hund showed how the concept of atomic orbitals and the mathematical procedures developed to define them could 

The best chemical theory of valence covering all types of compounds is generally agreed to 

somewhat mischievously, made the following comment on the relationship between his molecular orbital analysis and the Lewis electron-pair model Now I have a favourite argument that Lewis electron pair bonding is better described by a pair of electrons in a molecular orbital than by the Heitler-London bond. If the chemical bond has any polarity, it is necessary to add an ionic term, that is a Heitler-London plus an ionic term, to represent the bond. That is rather a messy description whereas the molecular orbital- this is not the spectroscopic but the chemical molecular orbital, the delocalized molecular orbital fits very nicely to the 

Lewis concept . He reserved his major criticisms for the Heitler-London, Pauling-Slater (HLPS) approach  

The theories of HLPS might be called electron-pairing theories if Lewis is called an electron-pair theory. It should also be pointed out that the HLPS electron pair differs considerably from Lewis conception of the electron-pair bond, in that the electrons are much less closely associated in this respect it approaches the truth much more closely than does Lewis conception - Pauling and Slater consider a double bond to be merely two ordinary single bonds sticking out from each atom in different directions, and treat the triple bond in a similar way. In this way they do not agree very well with Lewis, nor do they agree with the results obtained from molecular orbital theory. 

2 Consider the following free-electron model for conjugated polyenes  

3 In an extremely crude model of anthracene, C14H10, the n electrons are considered to be confined to a rectangular box of dimensions 4 A x 7 A. Using the appropriate particle-in-a-box energy expression, calculate the wavelength (in A) of the transition from the ground state to the first excited state. 

4 The energies of monocyclic conjugated polyenes can be obtained graphically by means of a Frost-Hiickel cirde. To achieve this, we first draw a circle with a radius of 2 j3 (recalling that 0), then we place the -sided 

Let us take the simple case of benzene as an example. When we put the hexagon inside our circle, we can readdy see that the energies are a +ip a + p (twice) a — P (twice) a — ip. 

Based on the Frost-Hiickel circle for a general monocydic polyene, 

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Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

A superb treatment of applied molecular orbital theory and its application to organic, inorganic and solid state chemistry. Perhaps the best source for appreciating the power of the independent-particle approximation and its remarkable ability to account for qualitative behaviour in chemical systems. 

Salem L 1966 Molecular Orbital Theory of Conjugated Systems (Reading, MA Benjamin)... 

Thiel W 1996 Perspectives on semiempirical molecular orbital theory New Methods in Computationai Quantum Meohanios (Adv. Chem. Phys. XCiti) ed I Prigogine I and S A Rice (New York Wiley) pp 703-57 Earlier texts dealing with semi-empirical methods include ... 

M. J. S. Dewar, The Molecular Orbited Theory of Organic Chemistry, McGraw-Hill, New York, 1969,... 

Simple Approaches to Quantifying Chemical Reactivity 3.4.2.1 Frontier Molecular Orbital Theory... 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

Hehre, W.J. Kadom, 1,. Schleyer, P,v,R, Pople, J..A. Ah Initio Molecular Orbital Theory, John Wiley and Sons, New York, 1986... 

Presell is the basic theory of tjuaiiHim mechanics, particularly, semi-empirical molecular orbital theory. The authors detail and justify the approximations inherent in the semi-empirical Ham illoTi ian s. Includes useful discussion s of th e appiicaliori s of these methods to specific research problems. 

M Li rrell, J. N. IIurgel, A. J. Sem i empirical Seif ron.sisienf field. Molecular Orbital Theory of MoleculesWes In Icrscieri ce. New York. l J7I. 

I h is chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem KHT calcu lation s. 

Fig. 2.7 The addition of a 3d orbital to 2p gives a distorted orbital. (Figure adapted from Hehre WJ, L Radom, p i)R Sdileycr and ] A Hehre 1986. Ab initio Molecular Orbital Theory. New York, Wiley.)...

I nple J A and D L Beveridge, 1970. Approximate Molecular Orbital Theory. New York, McGraw-Hill. Riduirds W G and D L Cooper 1983. Ab initio Molecular Orbital Calculations for Qieniists. 2nd Edition. Oxford, Clarendon Press. 

V. Intermediate Neglect of Differential Overlap. Journal of Chemical Physics 47 2026-2033. pie J A, D P Santry and G A Segal 1965. Approximate Self-Consistent Molecular Orbital Theory. I. 

Invariant Procedures. Journal of Chemical Physics 43 S129-S135. pie J A and G A Segal 1965. Approximate Self-Consistent Molecular Orbital Theory. II. Calculations with Complete Neglect of Differential Overlap. The Journal of Chemical Physics 43 S136-S149. iple J A and G A Segal 1966. Approximate Self-Consistent Molecular Orbital Theory. III. CNDO Results for AB2 and AB3 systems. Journal of Chemical Physics 44 3289-3296. 

In summary, we have made three assumptions 1) the Bom-Oppenheimer approximation, 2) the independent particle assumption governing molecular orbitals, and 3) the assumption of n-Huckel molecular orbital method. 

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