Molecular orbital theory Hiickel

The diagonal elements of the Hiickel matrix represent the energies of the contributing AOs, which in this case are all a. Each of the bonds (in this case tpi-Wi arid 3- 4) is assigned the overlap energy and all other elements of 

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than 

In 1965, however, the computational resources needed for the full SCF approach were not yet available. Practical MO theories therefore still needed approximations. The main problem is the calculation and storage of the four-center integrals, denoted (fiv I Aa), needed to calculate the electron-electron interactions within the 

SCF approximation. The indices //, v, A, and o denote four atomic orbital centers, so that the number of such orbitals that needs to be calculated increases proportionally scales with ) N, where N is the number of AOs, This was an intractable task in 1965, so Pople, Santry, and Segal introduced the approximation that only integrals in which = v and J. = o (i.e., li)) would be considered and that, further- 

Hiickel molecular orbital (HMO) theory271 deals only with the Jt-electrons of unsaturated systems the o-electrons are considered to be part of a frozen core. That is, we use only the 2pz-AOs (b on the unsaturated carbon atoms Ci, C2,. .., CM,. .., C,. .., CM, which are assumed to be orthonormal (orthogonal and normalized) (Equation 4.2). 

In order to find solutions for the eigenvalue problem 7/ 1// Ibj/, we have to set up the corresponding secular determinant (see Equation 1.16). No attempt is made to calculate the matrix elements of the secular determinant instead, the matrix elements are 

A somewhat simpler form of the determinant is obtained by dividing all elements by /3 and substituting —x for (a — s)//3. Division by a constant is allowed, because the determinant is set equal to zero. We obtain the Huckel determinant (Equation 4.5). All diagonal elements B in the Huckel determinant are equal to — x and the off-diagonal elements B are equal to zero, except for atoms /jl and v that are connected by a o-bond where B v=l. 

General solutions exist for the Huckel determinants of special systems with any number n of carbon atoms, namely for the linear polyenes (Equation 4.7) and for the monocyclic systems (Equation 4.8). 

Equation 4.7 General HMO energies for linear polyenes with n carbon atoms 

In the HMO approximation, the Ti-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total Ti-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not 

The most easily obtained information from such calculations is the relative orderings of the energy levels and the atomic coefficients. Solutions are readily available for a number of frequently encountered delocalized systems, which we will illustrate by referring to some typical examples. Consider, first, linear polyenes of formula C H 2 such as 1,3-butadiene, 1,3,5-hexatriene, and so forth. The energy levels for such compounds are given by the expression 

Carrying out the numerical operations for 1,3,5-hexatriene gives the results shown in Table 1.15. Because the molecule is a six-re-electron system, i, i 2 3 are all doubly 

Coulson and A. Stieitwieser, Jr., Dictionary of n-Electron Calculations, W. H. Freeman, San Francisco, 1965 E. Heilbronner and P. A. Straub, HUchel Molecular Orbitals, Springer-Verlag, Beilin, 1966. 

The total re-electron energy of benzene is 6a+ P, corresponding to a DE of 2jS. Cyclobutadiene is predicted to have a triplet ground state (for a square geometry) and zero 

Carrying out the numerical operations for 1,3,5-hexatriene gives the results shown in Table 1.15. Since the molecule has six-ir-electrons, i/f, 1/2, and if/j are all doubly occupied, giving a total ir-electron energy of 6a + 6.988)8. The general solution for this system is based on the assumption that the electrons are delocalized. If this 

The success of simple HMO theory in dealing with the relative stabilities of cyclic conjugated polyenes is impressive. Simple resonance arguments lead to confusion when one tries to compare the unique stability of benzene with the elusive and unstable nature of cyclobutadiene. (Two apparently analogous resonance structures can be drawn in each case.) This contrast is readily explained by Hiickel s rule, which states that a species composed of a planar monocyclic array of atoms. 

The discussions of stereochemistry and molecular mechanics in Chapters 2 and 3 were based implicitly on localized electron pair bonds. Now we will consider bonding models in which electrons are not restricted to the space between two atoms but are delocalized over a molecular structure. The approach we will consider in greatest detail is Hiickel molecular orbital (HMO) theory, a simple method that nevertheless yields useful insights into structures and properties of organic compounds. Later we will consider some more advanced computational models. 

The fundamental assumption of HMO theory is that we may calculate molecular orbitals through a process known as LCAO the linear combination of atomic orbitals. That is, we use some combination of the wave functions of the atomic orbitals to produce a set of molecular orbitals. In the Huckel method, we combine a set of atomic p orbitals to produce a set of n molecular orbitals. For a set of n parallel p orbitals, the Huckel molecular orbitals have the form shown in equation 4.1. In this equation is the wave function 

Perspectives on Structure and Mechanism in Organic Chemistry, Second Edition By Felix A. Carroll Copyright 2010 John Wiley Sons, Inc. 

4 APPLICATIONS OF MOLECULAR ORBITAL THEORY AND VALENCE BOND THEORY 

Now we use in the Schrodinger equation, = Eij/, and calculate the energies of the HMOs produced. Details of the procedure were given by Roberts, so only an outline is provided here. Multiplying both sides of Htp = Etp by ip (or ip as is appropriate), dividing by ip, and then integrating over all space in both the numerator and denominator gives equation 4.3. 

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K. Yates, Hiickel Molecular Orbital Theory, Academic Press, 1978. 

Hiickel Molecular-Orbital theory is one of the more useful theoretical guides of the pratising chemist. The central idea is that in certain molecules it is... 

Due to its importance in many industrial processes, the prototypical reaction of CO binding to metal surfaces has received much attention. Using Hiickel molecular orbital theory, Blyholder showed that CO bonding at top sites consists of the donation of electrons from the filled CO 5a HOMO to the metal d 2 orbitals with a back-donation of electrons from the metal dxz caddy orbitals to the CO 2n LUMO. Consequently,... 

In the simple Hiickel molecular orbital theory of w-electronic systems, molecular orbitals are derived by combinations of 2p orbitals, one from each bonded atom. All of the exchange integrals for bonded atoms are taken as equal and interactions between non-adjacent atoms are ignored. The energy of each molecular orbital then has the form,... 

Sce Yates Hiickel Molecular Orbital Theory, Academic Press New York, 1978 Coulson O Leary Mallion Hiickel Theory for Organic Chemists. Academic Press New York, 1978 Lowry Richardson Mechanism and Theory in Organic Chemistry. 3rd ed.. Harper and Row New York, 1987. pp. 100-121. 

In this section, we first discuss the bonding in two linear triatomic molecules BeH2 with only a bonds and C02 with both a and n bonds. Then we go on to treat other polyatomic molecules with the hybridization theory. Next we discuss the derivation of a self-consistent set of covalent radii for the atoms. Finally, we study the bonding and reactivity of conjugated polyenes by applying Hiickel molecular orbital theory. 

Hiickel molecular orbital theory for conjugated polyenes... 

In this section, we first study the jt bonding in conjugated polyenes by means of the Hiickel molecular orbital theory. Then we will see how the wavefunctions obtained control the course of the reaction for these molecules. 

Hiickel molecular orbital theory and its application to ethylene and butadiene... 

For an assemblage of two identical molecules spaced d nm apart, the HOMO and LUMO energies split into four levels, each split by 2t eV apart ("dimer splitting") [26] here t is akin to the Hiickel69 resonance integral (i of Section 3.15 Indeed, chemists will remember Eq. (8.6.10) from the simple Hiickel molecular orbital theory for aromatic 7r-electron systems. As the number of molecules N increases, the energy levels become spaced more closely, until they form a quasi-continuous band of bandwidth W, where... 

As it is well known (see, for example, [16,17]), the Hiickel molecular orbital theory is based on a Hamiltonian operator, ff defined by means of the matrix elements... 

Hiickel molecular orbital theory (HMO) atomic orbital basis Xr m-3/2 3... 

Hiickel molecular orbital theory 17 hydrogen-like wavefunction 16 hyperbolic functions 84 hyperfine coupling constant 26 hyper-polarizability 22 hyper-susceptibility 14... 

Although Hiickel molecular orbital theory is not completely consistent in its application to nonalternant systems such as the pentalenyl dianion, it does implicate a certain degree of stabilization for this species. If the transannular bond introduces little or no perturbation, then the pentalenyl dianion is seen to be closely related to the cyclooctatetraene dianion. Katz and his co-workers successfully developed a... 

The hyperfine structure constant thus allows us to probe the electron distribution in radicals. Theoretically calculated values of the spin densities can then be compared with the experimental values obtained from Eq. (9). One of the simplest methods for calculating electron density in an aromatic hydrocarbon is to use Hiickel molecular orbital theory as discussed later. 

Figure 11 Energy level diagram for the HOMO and first two LUMOs of the 60jr-electron system of Ceo, calculated by simple Hiickel molecular orbital theory...

The band structure of a three-dimensional solid, such as a semiconductor crystal, can be obtained in a similar fashion to that of a polyene. Localized molecular orbitals are constructed based on an appropriate set of valence atomic orbitals, and the effects of delocalization are then incorporated into the molecnlar orbital as the number of repeat units in the crystal lattice is increased to infinity. This process is widely known to the chemical conununity as extended Hiickel theory (see Extended Hiickel Molecular Orbital Theory). It is also called tight binding theory by physicists who apply these methods to calcnlate the band structures of semiconducting and metallic solids. 

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