Lect5-MO-Theory

Molecular Orbital Theory

From the last lecture:
 * Introduced the Schrodinger Equation: HY= EY
 * Solved the Schrodinger Equation for the Hydrogen atom
 * Examined the equations for H2 and found that the 1/r12 term (electron-electron interaction) prohibited an analytical solution
 * Introduced the Hartree approximation - non-interacting electrons

The solutions to the Schrodinger equation for the hydrogen atom led naturally to a set of quantum numbers to describe the atomic orbitals (n,l,m): These are atom orbital solutions to the hydrogen atom and perhaps we often use the combination of atomic orbitals to discuss and understand molecular structures.
 * = n ||= l ||= m ||= description ||
 * = 1 ||= 0 ||= 0 ||= 1s ||
 * = 2 ||= 0 ||= 0 ||= 2s ||
 * = 2 ||= 1 ||= -1,0,1 ||= 2px,2py,2pz ||
 * = 3 ||= 0 ||= 0 ||= 3s ||
 * = 3 ||= 1 ||= -1,0,1 ||= 3p ||
 * = 3 ||= 2 ||= -2,-1,0,1,2 ||= 3d ||

Pauli exclusion principle tells us that no more than two electrons can occupy a single orbital, and that these two electrons in one orbital must differ in electron spin. This leads to the ground state electron configurations for atoms:

H 1s H - 1s 2 He 1s 2 Li 1s 2 ,2s Be 1s 2 ,2s 2 B 1s 2 ,2s 2 ,2p 1 C 1s 2 ,2s 2 ,2p 2 N 1s 2 ,2s 2 ,2p 3 O 1s 2 ,2s 2 ,2p 4 F 1s 2 ,2s 2 ,2p 5

Perhaps we can build molecules by overlapping atomic orbitals

Back To Schrodinger
HYmolecule = EYmolecule

LCAO method assumes that we can represent the wave function for a molecule by the linear combination of atomic orbitals: Ymolecule = X1 + X2 + X3 ... where X is an atomic orbital

So for H2 + we could write: Ymolecule = c1X1 + c2X2 where c's are variables that we want to optimize to get the best Ymolecule which will give the lowest E, closest to the true value.

HY = EY YHY = E YY

E = ∫ YHYdt ∕ ∫ YYdt

Let: Y = (c1X1 + c2X2)

E = ∫ (c1X1+c2X2)H(c1X1+c2X2)dt ⁄ ∫ (c1X1+c2X2)(c1X1+c2X2)dt

(From now on I will leave out the integration symbols)

E = (c1X1Hc1X1 + c1X1Hc2X2 + c2X2Hc1X1 + c2X2Hc2X2) ⁄ ( (c1x1) 2 + 2 c1c2X1X2 +(c2X2) 2 )

if X1HX2 = X2HX1 then Let: H11 = X1HX1 H12 = X1HX2 H22 = X2HX2 S11 = X1X1 S12 = X1X2 S22 = X2X2

E = (c1 2 H11 + 2 c1c2H12 + c2 2 H22) / (c1 2 S11 + 2c1c2 S12 + c2 2 S22)

We want to optimize c1 and c2 to obtain the best wave function and to do this we take the derivative of E with respect to c1 and c2.

∂E/∂c1 = (c1 2 S11 + 2c1cS12 + c2 2 S22)(2c1H11 + 2c2H12) / (c1 2 S11 + 2c1cS12 + c2 2 S22) 2 -

(c1 2 H11 + 2 c1c2H12 + c2 2 H22) (2c1S11 + 2c2S12) / (c1 2 S11 + 2c1cS12 + c2 2 S22) 2 = 0

(2c1H11 + 2c2H12) = [(c1 2 H11 + 2 c1c2H12 + c2 2 H22) / (c1 2 S11 + 2c1cS12 + c2 2 S22) ] (2c1S11 + 2c2 S12)

c1(H11 - ES11) + c2(H12 - ES12) = 0

You get a similar equation when taking the derivative of E with respect to c2 and this leads to a set of simultaneous linear equations the solutions of which correspond to the roots of the secular determinant:

⎢ H11 - ES11 H12 - ES12 ⎢ = 0 ⎢ H21 - ES12 H22 - ES22 ⎢

In the general case where Y = ∑ c n x n you can write the general secular determinant. These determinants have a diagonal of symmetry and n real roots. The results to this part are completely general and makes no assumptions other than that we can represent the wave function for a molecule as the sum of atomic wavefunctions. The same mathematical methods are used at all levels of quantum theory. The differences now come in what approximations we use to solve for the Hii, Hij and Sii and Sij.

Overlap Integrals Sij
Sij = ∫ XiXj dτ

if i = j then Sii = ∫ Xi 2 dτ = 1 because we are using a set of normalized atomic orbitals

if i != j Sij = ∫XiXjdτ = 0 when xi and xj are said to be orthogonal and the atoms are far apart. This term is a measure of the overlap of atomic orbitals and can vary from 0 to 1 depending upon how far apart are the atomic centers. In a zeroth order approximation this simplifies the calculations but at better levels of theory these integrals would be evaluated (ab initio theory) or parameterized (sem-empirical theory).

With these two substitutions we can simplify the secular determinate into:

⎢ H11 - E H12 ⎢ = 0 ⎢ H21 H22 - E ⎢

Coulomb Integrals Hii
Hij = ∫ XiHXj dτ

If i = j then Hii = ∫ XiHXi dτ

In a zeroth order approximation Hii is the coulomb energy of an electron with a wave function xi in the field of atom i and is little affected by nucleii further away. This is a reasonable assumption when the surrounding nucleii have no net electrical charge.

Set Hii = αi where αi is a function of nuclear charge and the type of orbital ( αi is a negative number)

Resonance Integrals Hij
Hij = ∫ XiHXj dτ where i != j

In the zeroth order approximation Hij is the energy of an electron in the fields of atoms i and j with the wave functions xi and xj. It is normally called βij or the resonance integral. It is a function of atomic number, orbital types and the degree of overlap. In the zeroth order approximation βij is set to 0 for atoms that are not bonded to each other.

So if we make the substitutions our secular determinant becomes:

⎢ α1 - E β ⎢ = 0 ⎢ β α2 - E ⎢

For H2 α1 = α2 and

α 2 - 2αE + E 2 - β 2 = 0

E = 2α ± √4α 2 - 4(α 2 - β 2 ) ⁄2 E = α ± β

Thus we find two energy levels for H2 and we can now solve for c1 and c2:

c1(α - E) + c2β = 0 c1β + c2(α - E) = 0

c1/c2 = β/(α - E)

for the E = α + β solution

c1/c2 = 1

if E = α - β c1/c2 = -1

So our wave functions when normalized are:

Y1 = (1/√2) (X1 + X2) and Y2 = (1/√2) (X1 - X2)

These results are perfectly general and can be applied to any molecular system - these techniques are used in ab inition quantum chemistry programs to compute the energies and molecular orbital coefficients for all the electrons in the molecule. Many of the details have been left out to make the presentation clearer and many of the approximations about the size of the resonance and overlap integrals would not hold for better theoretical treatments - but the mathematical techniques would be the same.

Before computers became widely available and faster it was recognized that the LCAO method could be applied to organic molecules if we made the assumption that single and multiple bonds could be treated separately and that the single bonds could be considered as localized and fixed and thus could be ignored. We could apply the same techniques of building a molecular orbital from the sum of atomic orbitals, but now the molecular orbitals would be only for the pi system and the atomic orbitals would be the correspond p type orbitals available for overlap. So for ethylene: Y = c1X1 + c2X2 where X1 is a p orbital on atom 1 and X2 is a p orbital on atom 2.

Likewise for butadiene: Y = c1X1 + c2X2 + c3X3 + c4X4

The secular determinate could be written down as:

⎢ α1-E β 0 0 ⎥ ⎥ β α1-E β 0 ⎥ ⎥0 β α1-E β ⎥ = 0 ⎥0 0 β α1-E ⎥

Which can be solved to give four energy levels:

E1 = α + 1.6180β E2 = α + 0.6180β E3 = α - 0.6180β E4 = α - 1.6180β

where E1 and E2 are the bonding levels and E3 and E4 are antibonding. The mo coefficients for the four orbitals are: Y1 = 0.3717 X1 + 0.6015 X2 + 0.6015 X3 + 0.3717 X4 zero nodes Y2 = 0.6015 X1 + 0.3717 X2 - 0.3717 X3 - 0.6015 X4 one node Y3 = 0.6015 X1 - 0.3717 X2 - 0.3717 X3 + 0.6015 X4 two nodes Y4 = 0.3717 X1 - 0.6015 X2 + 0.6015 X3 - 0.3717 X4 three nodes

Given the wave function you can calculate many properties such as bond orders, charge distributions and free valence. Before fast digital computers became available this type of MO theory was widely used to rationalize experimental results.