1Lecture 4. Beyond the Hückel π-electron theory 4.1 Charge densities and bond orders Charge density is an important parameter that is used widely to explain properties of molecules. An electron in an orbital mmrnrcφψ∑= has density distribution 22mmrncφ∑ (neglecting all overlap density terms such as mnφφ). We can thus define the π-electron charge density qm at atom m as follows: 2∑=rrmrmcnq , (4.1) where crm is the coefficient of the basis orbitals φm in the molecular orbital ψr; the sum is over all the π orbitals with nr electrons in each orbital (nr=0, 1, 2). For the ground state of butadiene (see Fig. 3.2, lecture 3) nr=2 for the two occupied bonding orbitals and nr=0 for the antibonding orbitals, thus the charge densities at atoms a and b are 00.1)37.0(2)60.0(22200.1)60.0(2)37.0(222222221222221=+=+==+=+=bbbaaaccqccq (4.2) By symmetry we know that qa=qd and qb=qc. Thus the π-electron density is unity at each atom in butadiene. Examination of the orbitals of benzene will show that the same is true for all the carbon atoms in benzene. To interpret many phenomena in molecules, and it is desirable to estimate the degree of double-bond character in the bond joining two atoms. By analogy with the charge density at an atom m, we can define a π-electron bond order between atoms m and n as follows: rnrrmrmnccnp∑= . (4.3)2 For ethylene the two electrons in the bonding π orbital, whose wavefunction gives a bond order of 1. For butadiene we have ,89.0222211=+=babaabccccp (4.4) .45.0222211=+=cbcbbcccccp (4.5) By symmetry pab=pcd so that according to Hückel theory the outer bonds in butadiene have much more double-bond character than the central bond but less than that in an isolated ethylene molecule. The conventional representation of butadiene as a single valence structure (CH2=CH-CH=CH2) is in close accord with the Hückel picture but fails to show that there is some double-bond character in the central bond. Fig. 4.1 A common representation of π–electron densities in ethylene, butadiene and benzene. 4.2 Introduction of other atoms into Hückel theory In the simple Hückel π-electron theory, it assumes that all the atomic orbitals in an LCAO expansion were similar and had the same energy, so only two parameters, α and β, are required. The simple Hückel π-electron theory provides reasonable description of alternant hydrocarbons that3consist of only carbon and hydrogen atoms. If we wish to treat molecules, such as CH2=CH-CH=O, C6H5N=NC6H5 then we have to deal with atoms other than carbon atoms. One way to treat this situation is to express αX and βCX of these other atoms (X) in terms of the values for benzene (αC and βCC), αX = αC + hX βCC (4.6) and βCX = kCX βCC, (4.7) where hX and kCX are empirical parameters. 4.3 Extended Hückel π-electron theory The Hückel π-electron theory includes only the π orbitals and ignores completely the contribution due to σ orbitals and hydrogen atoms. Attempts have been made to include the σ and hydrogen orbitals have been developed for hydrocarbons and called all-electron theories. The basic features of the theory are that the atomic orbital basis consists of all the valence atomic orbitals and carbon 2s and 2p orbitals. It is more successful than the simple Hückel π-electron theory, but it has many limitations and is not very satisfactory for polar molecules. This is because like the Hückel π-electron theory, the extended Hückel π-electron theory is based on independent electron model, which ignores electron-electron interactions. 4.4 PPP method The Pariser-Parr-Pople (PPP) method is similar to Hückel π-electron theory, except that it includes electron-electron repulsion. We can write HPPP=HHuckle + repulsion. (4.8)44.5 Ab-initio Calculations Ab-initio is a Latin term and means from the beginning or from first-principle. The first ab-initio calculation was probably the one on H2 molecule by Heitler-London in 1927, but it was not attempted to any larger molecules until the development of electronic computers in the 1950s. The extensive production of ab-initio calculations began in 1960s with the widespread availability of programs for Self-Consistent Field (SCF) calculations on polyatomic molecules. Interestingly, the ever-increasing computer power allows us to understand molecular electronics that could ultimately produce even more powerful computers. Hartree-Fock SCF method is one of widely used ab initio methods. To understand the theory, let us start with a two-electron molecule. The two electrons have orbitals (wavefunctions), φa and φb, respectively. The wavefunction of the two electron system, Ψ(1,2), maybe written as Ψ(1,2)= φa(1)φb(2), where numbers 1 and 2 represent the two electrons. This wavefunction is used in the Hartree SCF, but it has a serious problem. Since the electron is a Fermion (spin is a half-integer), it has to obey the Pauli Exclusion Principle, which requires an antisymmetric wavefunction when electrons 1 and 2 exchange positions. Clearly the wavefunction given above is not antisymmetric. A simple way to construct an antisymmetric wavefunction for the many electron system is to use the Slater determinant, which takes the form of [])()()()()()()()(babababa122121221121)2,1(φφφφφφφφ−==Ψ , (4.9) for the two electron system. The prefactor in the expression is required for normalization. It is easy to verify that Ψ(1,2) = - Ψ(2,1) as expected for antisymmetric wavefunction. Also we note that Ψ(1,1) = - Ψ(1,1) = 0, which means the two electrons cannot occupied the same state, a result of the Pauli exclusion principle. The above considerations ignore electron spin and the molecular orbitals φa and φb are functions of space only. It is rather straightforward to include it in the5wavefunction. Electrons can have either spin up(+1/2) or down (-1/2), which can be described by two spin functions, α and β, as follows .1)(,0)(0)(,1)(=↓=↑=↓=↑ββαα (4.10) The molecular orbitals now have both space and spin parts and take the forms of, αφa, βφa, αφb and βφb, which are often referred to as spin-orbitals. If the two-electron system has only one molecular orbital, φa, available, then the wavefunction is )()()()(aaaa2)2(2)2(1)1(1)1(21)2,1(φβφαφβφα =Ψ, (4.11) which reflects the fact that the first electron takes spin up state, the second electron must take spin down, vice versa. The wavefunction given by 4.11 is