Electronic Structure Models Hückel Model (1933) Basic Assumptions: (a) One orbital per atom contributes to the basis set; all orbitals "equal" (b) The relevant integrals involving the Hamiltonian are α if µ =ν (same site) Hµν = β if µ ν but µ and ν are bonded (nearest neighbors) 0 otherwise (c) The atomic orbitals are assumed normalized and orthogonal: Sµν = 1 if µ = ν 0 if µ ν α and β are parameters; their values will be adjusted to eperimental data of some sort, typically after the calculation is done (or not at all) Consider, as the easy (trivial?) eample, ethylene with the 2p(π) orbitals as the basis set. The secular determinant for two basis functions is H 11 - ES 11 H 12 - ES 12 A
= 0 H 21 - ES 21 H 22 - ES 22 which, in the Hückel model, becomes α - E β β α - E = 0 Multiply out the secular determinant and find the roots: (α - E)2 - β2 = 0 => (α - E)2 = β2 => α - E = ± β or E 2 = α - β α α β E 1 = α + β E 1 = α + β and E 2 = α - β We may then find the two sets of coefficients (one for φ 1, one for φ 2 ) from the secular equations B
α - E β c 1 0 β α - E c 2 0 (α - E) c 1 + β c 2 = 0 β c 1 + (α - E) c 2 = 0 When E = E 1 = α + β, we find by insertion (into either equation) (α - (α + β)) c 1 + β c 2 = 0 => - β c 1 + β c 2 = 0 => - β c 1 = - β c 2 => c 1 = c 2 When E = E 2 = α - β, we find (α - (α - β)) c 1 ' + β c 2 ' = 0 => β c 1 ' + β c 2 ' = 0 => β c 1 ' = - β c 2 ' => c 1 ' = -c 2 ' The coefficients must, of course, be equal in magnitude, since the two carbon atoms are symmetry equivalent. So, φ 1 = c 1 (χ 1 + χ 2 ) φ 2 = c 1 ' (χ 1 - χ 2 ) E1 = α + β E2 = α - β C
The MO's should be properly normalized in order to represent probability distributions. The atomic orbitals χ 1 and χ 2 are assumed to be properly normalized, and orthogonal V χ 1 χ 1 dv = V χ 2 χ 2 dv = 1 V χ 1 χ 2 dv = V χ 2 χ 1 dv = 0 So we find the actual value for c1 or c1' from the normalization condition: V φ 1 φ 1 dv = 1 => V c 1 2 (χ 1 + χ 2 )2 dv = 1 => c 1 2 V χ 1 χ 1 dv + 2c 1 2 V χ 1 χ 2 dv + c 1 2 V χ 2 χ 2 dv = 1 => c 1 2 + 2c 1 2S 12 + c 1 2 = 1 => c 1 = [2(1+ S 12 )]-1/2 Similarly: V φ 2 φ 2 dv = 1 => c 1 ' = [2(1 - S 12 )] -1/2 Within the Hückel-model, S 12 = 0 and we get c 1 = (2)-1/2 and c 1 ' = (2)-1/2. The overlap between the two pi-orbitals in ethylene is in fact near 0.25, and if there were no overlap between the orbitals, there would be no pi-bonding at all. Interatomic overlap is a prerequisite for covalent chemical bonding. Nevertheless, in most semi-empirical methods the overlap D
integrals are neglected in normalization considerations, as a mathematical convenience. Thus, the normalized Hückel π-mo's for ethylene are φ 1 = (2)-1/2 (χ 1 + χ 2 ) φ 2 = (2)-1/2 (χ 1 - χ 2 ) E 1 = α + β E 2 = α - β φ 2 = (2) -1/2 (χ 1 - χ 2 ) E 2 = α - β χ 1 α α χ 2 β E 1 = α + β φ 1 = (2) -1/2 (χ 1 + χ 2 ) E
φ 1 increases the electron density between the atoms relative to the two atoms non-interacting; φ 2 has depleted electron density between the atoms relative to the two atoms non-interacting. φ 1 is a "bonding" orbital, φ 2 is an "antibonding" orbital. Therefore, E1 is less than E2 and β must be a negative quantity. α = energy of an electron in orbital µ in the molecule (negative quantity). α depends on orbital type and nuclear charge. β = energy gained by electron "delocalization" or "resonance" between orbitals µ and ν. β depends on orbital types, nuclear charges, and degree of overlap between the orbitals. Electronic energy: E = 2E 1 = 2(α + β) E(isolated atoms) = α + α = 2α; Energy gained by bonding: 2β F
In fact, most useful results from Hückel calculations are epressed in terms of β. (1) Optical ecitation; Plot HOMO-LUMO energy difference vs. UVmaimum for lowest allowed absorption band for a series of polyenes or rings. E = E(LUMO) - E(HOMO) ν ma (cm -1 ) Optimal value of β = -2.7 ev (2) Ionization potential; IP = -E(HOMO) Optimal value of β = -2.9 ev (3) Electron affinities; EA = E(LUMO) Optimal value of β = -2.4 ev (4) Delocalization energy; defined as E (delocalized) - E (localized), should correlate with heats of formation data. Optimal value of β = -0.69 ev (!!!) G
Conclusion: Different optimal parameters for different properties. The optimal parameters tend to cluster in two groups, one for "spectroscopic" properties and one for "thermodynamic" properties (dynamic vs. static). This observation and idea carries over to more elaborate electronic structure models as well --- a particular parameterization scheme works best for ground state properties whereas a (quite) different set of parameters are more appropriate for spectroscopic (ecited state) properties H
The pi molecular orbitals and energies for butadiene are attached. They are the results of diagonalizing a 44 determinant (χ 1,χ 2,χ 3,χ 4 ) within the Hückel model. Useful indices: charges and bond orders π-charge, i.e. number of pi-electrons per atom q µ = Σ i N i c µi 2 N = # electrons in MO i (2,1, or 0) c µi = coefficient of atom µ in MO i Sum is over all occupied MO's Ethylene: q 1 = 2[(2) -1/2 ]2 = 1.00 = q 2 Butadiene: q 1 = 2 (0.37)2 + 2 (0.60)2 = 1.00 = q 2 = q 3 = q 4 Bond order: The bond order between two (adjacent) atoms µ and ν is p µν = Σ i N i c µi c νi Ni = # electrons in MO i (2,1, or 0) c µi = coefficient on atom µ in MO i c νi = coefficient on atom ν in MO i Note: p µµ = q µ Ethylene: p 12 = 2 (2) -1/2 (2) -1/2 = 1.00 (standard) Butadiene: I
p 12 = 2(0.37)(0.60) + 2(0.60)(0.37) = 0.89 p 23 = 2(0.60)(0.60) + (0.37)(-0.37) = 0.45 p 34 = 2(0.37)(0.60) + 2(0.60)(0.37) = 0.89 p 12 < 1.00 and p 23 > 0.45 => delocalization of pi-electron into the central bond Pi bond-orders between adjacent atoms range from 0.0 to 1.0 (0.66 in benzene, for eample). There is ecellent correlation between pi bond orders and bond lengths. For eample, R(C 1 -C 2 ) = 1.517Å - 0.189 pc1-c2 Å The matri P which has as its elements the charges (diagonal) and bond-orders (off-diagonal) is called the charge - bond order matri. In fancier language, it is called the density matri; in very fancy language, it is called the "first order reduced density matri". J
CALCULATION FOR ETHYLENE HUCKEL MATRIX BY COLUMNS 1 2 1 0.00000 1.00000 2 1.00000 0.00000 0 MO- 1 MO- 2 MO- 0 E/BETA 1.00000-1.00000 C - 1 0.707107-0.707107 C - 2 0.707107 0.707107 ORBITAL OCCUPANCY MO 1--2 MO 2--0 MO HUCKEL ENERGY (UNITS OF BETA) = 2.00000 BOND-ORDER MATRIX BY COLUMNS 0 1 2 1 1.00000 1.00000 2 1.00000 1.00000 CHARGE NET SPIN DENSITY CHARGE DENSITY C 1 1.00000 0.00000 0.00000 C 2 1.00000 0.00000 0.00000 SPIN DENSITY IS FOR HIGHEST MULTIPLICITY CALCULATION FOR BUTADIENE HUCKEL MATRIX BY COLUMNS 0 1 2 3 4 1 0.00000 1.00000 0.00000 0.00000 2 1.00000 0.00000 1.00000 0.00000 K
3 0.00000 1.00000 0.00000 1.00000 4 0.00000 0.00000 1.00000 0.00000 0 MO- 1 MO- 2 MO- 3 MO- 4 MO- 0 E/BETA 1.61803 0.61803-0.61803-1.61803 C - 1 0.371748 0.601501 0.601501 0.371748 C - 2 0.601501 0.371748-0.371748-0.601501 C - 3 0.601501-0.371748-0.371748 0.601501 C - 4 0.371748-0.601501 0.601501-0.371748 ORBITAL OCCUPANCY MO 1--2 MO 2--2 MO 3--0 MO 4--0 MO HUCKEL ENERGY (UNITS OF BETA) = 4.47214 BOND-ORDER MATRIX BY COLUMNS 0 1 2 3 4 1 1.00000 0.89443 0.00000-0.44721 2 0.89443 1.00000 0.44721 0.00000 3 0.00000 0.44721 1.00000 0.89443 4-0.44721 0.00000 0.89443 1.00000 CHARGE NET SPIN DENSITY CHARGE DENSITY C 1 1.00000 0.00000 0.00000 C 2 1.00000 0.00000 0.00000 C 3 1.00000 0.00000 0.00000 C 4 1.00000 0.00000 0.00000 SPIN DENSITY IS FOR HIGHEST MULTIPLICITY L