Orbital theories

Molecular orbital theory is a theory of the electronic structure of molecules in terms of molecular orbitals, which may spread over several atoms or the entire molecule. This theory views the electronic structure of molecules to be much like the electronic structure of atoms. Each molecular orbital has a definite energy. To obtain the ground state of a molecule, electrons are put into orbitals of lowest energy, consistent with the Pauli exclusion principle, just as in atoms. 

According to molecular orbital theory, the atomic orbitals involved in bonding actually combine to form new orbitals that are the property of the entire molecule, rather than of tte atoms forming the bonds. These new orbitals are called molecular orbitals. In molecular orbital theory, electrons shared by atoms in a molecrrle 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 atorrtic orbitals, two electrons residing in the same molecular orbital must have opposite spins, as reqrrired by the Parrli exclusion principle. And, like hybrid orbitals, the nrrmber of molectrlar orbitals we get is eqrral to the nrrmber of atomic orbitals we combine. 

Our treatmerrt of molecrrlar orbital theory in this book will be limited to descriptions of bonding in diatorrric molecrrles consisting of elements from the first two periods of the periodic table (H through Ne). 

The description, in MO theory, of the electronic structure of a system is given in terms of molecular orbitals. A molecular orbital, tp, is a function of the spatial coordinates of the electron. The product tp rp, where cp denotes the complex conjugate of p, gives the electronic density distribution. 

An orbital, pt, is said to be normalized when — 1, thebra-ket notation 

When using a spin-free hamiltonian operator, the spin functions are introduced as mulplicative factors, yielding the spin-orbitals. There are two spin-orbitals per orbital. An electronic configuration is defined by the occupancies of the spin-orbitals. Open-shell configurations are those in which not all the orbitals are doubly occupied. 

The total, antisymmetric function for a closed-shell configuration is expressed as a Slater determinant built-up from the spin-orbitals. In the case of open-shell configurations, a linear combination of Slater determinants may be needed in order to obtain a function with the same symmetry and multiplicity characteristics as the state under consideration. 

The best energy attainable within this approximation is the so-called Hartree-Fock (HF) energy. The difference between this energy and the exact eigenvalue for the electronic hamiltonian operator is denoted as correlation energy. Several schemes have been proposed in order to improve this situation. For the systems 

A second approach to bonding in molecules is known as the molecular orbital (MO) theory. The assumption here is that if two nuclei are positioned at an equilibiium distance, and electrons are added, they will go into molcciiiar orbitals that are in many ways analogous to the atomic orbitals discussed in Chapter 2. In the atom there ares, p, d.f. orbitals determined by various sets of quantum numbers and in the molecule we have rr, r, S... . orbitals determined by quantum numbers. We should expect to find the Pauli exclusion principle and Hund s principle of maximiim multiplicity obeyed in these molecular orbitals as well as in the atomic oibilals. 

When we attempt to solve the SchrOdinger equation to obtain the various molecular orbitals, we run into the same problem found earlier for atoms heavier than hydrogen. We are imaUe to solve the Schrdditiger equation exactly and therefore must make some approximations concerning the form of the wave (unctions fior the molecular orbitals. 

Of (he various methods of approximating the correct molecular orbitals, we shall discuss only one the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time (be electrons will be nearer and hence controlled by one or the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one we employed bi constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We 

The one.electron molecular orbitals thus formed consist of a bonding molecular orbital and an aniihonding molecular orbital If we allow a single electron to cxxupy the bonding molecular orbital (as in. for example), the approximate wave 

For a two.electron system such as Hj. the total wave function is the product of the wave functions for each electron  

Still another model to represent the bonding that takes place in covalent compounds is the molecular orbital theory. In the molecular orbital (MO) theory of covalent bonding, atomic orbitals (AOs) on the individual atoms combine to form orbitals that encompass the 

Atomic Orbital of H Molecular Orbital of H2 Atomic Orbital of H 

The bond order for 02 would be (10 — 6)/2 = 2. (Don t forget to count the bonding and antibonding electrons at energy level 1.) 

Quantum Mechanics (QM). The objective of QM is to describe the spatial positions of electrons and nuclei. The most commonly implemented QM method is the molecular orbital (MO) theory, in which electrons are allowed to flow around fixed nuclei (the Bom-Oppenheimer approximation) until the electrons reach a self-consistent field (SCF). The nuclei are then moved, iteratively, until the energy of the system can go no lower. This energy minimization process is called geometry optimization. 

Molecular Mechanics (MM). Molecular mechanics is a non-QM way of computing molecular structures, energies, and some other properties of molecules. MM relies on an empirical force field (EFF), which is a numerical recipe for reproducing a molecule s PES. Because MM treats electrons in an imphcit way, it is a much faster method than QM, which treats electrons exphcidy. A limitation of MM is that bond-making and bond-brealdng processes cannot be modeled (as they can with QM). 

Molecular Dynamics (MD). Energy-minimized structures are motionless and, accordingly, incomplete models of reality. In molecular dynamics, atomic motion is described with Newtonian laws FXt)=miai, where the force Fj exerted on each atom aj is obtained from an EFF. Dynamical properties of molecules can be thus modeled. Because simulation periods are typically in the nanosecond range, only inordinately fast processes can be explored. 

(2001). Molecular Modeling Principles and Applications, 2nd edition. Englewood Cliffe, NJ Prentice-Hall. 

Lipkowitz, Kenneth B., and Boyd, D. B., eds. (1990-2002). Reviews in Computational Chemistry, Vols. 1-21. New York Wiley-VCH. 

If you are pressed for time, then skip this topic. It rarely appears on the AP test, and it is rather complicated. However, because it has appeared on the test and it is mentioned in the required subject matter, it will be discussed in this book. 

The second possibility is that the electron density in the overlapping regions between the nuclei is less than elsewhere. This region now has a higher energy than the overlapping 

Earlier in this chapter, you learned the definition of bond order in the valence bond theory. In molecular orbital theory, the bond order is defined as one-half the difference between the number of electrons in bonding orbitals and the number of electrons in antibonding orbitals. Mathematically, this can be expressed as 

For hydrogen, the bond order is — (2 — 0) = 1. This is no different from what would be predicted in valence bond theory. 

To finish up this section on molecular orbital theory, let s look at the configurations for the elements from lithium to neon. (See Table 7.4.) These should provide you with sufficient examples to see the main principles behind molecular orbital theory. 

let s look at the simplest example, an 11, molecule. In this molecule, two Is orbitals will overlap. The electrons will fill in the als orbital, as seen in the diagram below  

Another way to represent this is to use orbital diagrams that look like atomic orbital diagrams. The same molecule can be shown as follows  

Effect of hybridization on oveHop and bond Molecule Hybridization C—H bond energy (IcJmoM) C—H bond length (pm) 

Hybridization We may make the generalization that the strength of a bond will be roughly propor- 

Molecular Orbital A second approach to bonding in molecules is known as the molecular orbital (MO) 

Cluster s structure Number of metal atoms N 9N CV MG CVE Examples 

Exceptions among electron rich clusters include compounds in which certain d orbitals are empty, while the s and p orbitals are occupied. This is equivalent to the formation of multiple M — M bonds. Only a few such compounds are presently known, for example, [Re4H4(CO)i2] (56 instead of 60e), and [Os3H2(CO)io] 46e instead of 48 ). The skeletal bonds in those compounds may be represented as follows  

Cluster Number of M atoms Location of atoms of a given type Ns Number of nearest neighbors CV MO atomic orbitals  

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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)... 

Roothaan C C J 1951 New developments in moleoular orbital theory Rev. Mod. Phys. 23 69-89... 

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. 

Frori tier Orbital theory supplies an additional asstim piion to ih is calculation. It considers on ly the interactions between the h ighest occupied molecular orbital (HOMO) and the lowest unoccupied rn olecular orbital (I.UMO). These orbitals h ave th e sin a 1 lest energy separation, lead in g to a sin all den oin in a tor in th e Klopinan -.Salem ct uation, fhe Hronticr orbitals are generally diffuse, so the numerator in the equation has large terms. 

Example Anoth er example of Iron tier orbital theory uses the reaction ol phenyl-butadiene with ph en ylethylene. This reaction is a [4-1-21 pericyclic addition to form a six-membered ring. It could proceed with the two phenyl rings close to each other (head to head) or further away front each other (head to tail). 

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. 

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