Inorganic Chemistry Molecular orbital theory

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Molecular orbital theory

We would like a theory of bonding that can be visualized and is at least semiquantitative.We have a picture of atoms with an electronic structure described by orbitals. Why not do the same thing for molecules ?Employ the orbital approximation

Ψ(r1,r2,r3,....) = ψ(r1)ψ(r2)ψ(r3).......

How do we arrive at an approximation to the orbitals ?

The electron density distribution for an electron in the vicinity of a nucleus in a molecule should be similar to that found in the free atom.Use the idea of Linear Combination of Atomic Orbitals (LCAO).What orbitals do we combine ?Start with a minimal basis (just the valence orbitals).

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Bonding in H2

Take two 1s orbitals as the basisGet two MO’s

ψ+ = φ1s(A) + φ1s(B)ψ- = φ1s(A) - φ1s(B)

One electron in BO gives 2.6 eV bond energy but two electrons give only 4.5 eV. Why?

What holds the molecule together?

There is nothing magic about the molecule being bonded– Electrons preferentially spend time between the

two nuclei. They act as electrostatic “glue”

– ψ+2 = [φ1s(A)]2 + [φ1s(B)]2 +

φ1s(A)φ1s(B) + φ1s(B)φ1s(A) » First two terms are electron density on original atoms,

other terms correspond to density between atoms

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Interatomic potentials

Molecular potential energy curve– the equilibrium bond

length corresponds to the minimum energy bond length

– De is the depth of the potential well

UV - PES

How do we know if the energy level diagrams have any meaning ?We can compare to experiments that directly measure the orbital energiesIlluminate a sample with high energy radiation (usually 21.2 eV - in the UV) and measure the kinetic energies of the ejected electrons.– Ek = hν - I

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The PES experiment

The PES spectrum of N2

Note nitrogen atoms have a first ionization energy of 14.5 eV

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Construction of MO diagrams for other diatomics

We need to select a basis set– usually use valence orbitals

We need to categorize the basis orbitalsaccording to their symmetry– only orbitals with the same symmetry have nonzero

overlapFigure out the relative energies of the orbitals– this may require help from spectroscopic data

Classifying orbitals by symmetry

Orbitals in diatomics can be classified according to their rotational symmetry characteristics as σ, π or δ. These classifications are strictly only valid for diatomics, but we also use them to describe bonds between pairs of atoms in polyatomic molecules.

σ orbitals π orbitals δ orbitalsFound in quadruply bonded species such as [Re2Cl8]2-

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MOs in first row diatomics

Ungerade or gerade ?MOs in molecules that are centrosymmetric can be classified as (g) or (u)– Useful for predicting spectroscopic transitions etc.– (g) implies that the wavefunction does not change sign

on inversion through the center of the molecule. (u) means that it does change sign

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Experimental MO energies

Determining electron configurations

The filling rules are essentially the same as those for atoms– Two electrons per orbital– Fill from the lowest energy up– If orbitals are degenerate go for the electron

configuration with the highest spin (Hund’s rule)Consider O2– 1σg

2 2σu2 3σg

2 1πu4 2πg

2

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Hetronuclear diatomics

The contributions to the MO from each of the atoms is unequal– ψ = cAφ(A) + cBφ(B) +.......

The more electronegative atom contributes strongly to the bonding orbitalThe less electronegative atom contributes strongly to the anti-bonding orbital– gives rise to polarity

Orbital mixing

The difference in energy between orbitals on different atoms leads to reduced mixing– The reduced mixing does

not imply weaker bonding

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Hydrogen fluoride

Carbon monoxide

Note that the HOMO and LUMO are largely on C. Important for metal carbonyl formation

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ICl an interhalogen compound

Bond order

B.O. = 1/2 x (No. bonding e - No. anti-bonding e)

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Bond strength / bond length

H3+

This species is postulated as an intermediate in some reactions– It is the simplest triatomic molecule

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Linear H3

1σ = φ1s(A) + 21/2φ1s(B) + φ1s(C)2σ = φ1s(A) - φ1s(C)3σ = φ1s(A) - 21/2φ1s(B) + φ1s(C)

Triangular

a1 = φ1s(A) + φ1s(B) + φ1s(C)φ1s(A) - φ1s(C)φ1s(A) - 2φ1s(B) + φ1s(C)

e {

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Correlation diagram / Walsh diagrams

There is a relationship between the orbitals in the linear and triangular species. This relationship (how the orbital energies evolve on bending from linear to triangular) is shown on a Walsh or correlation diagram.

Three center two electron bonds

The orbitals in H3 are delocalized over the entire moleculeIn H3

+ 2 electrons hold the molecule together– this is an example of a three center two electron

bond

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MO s for Polyatomic chains

Constructing MOs for polyatomic chains

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MOs for rings

Orbitals in more complex molecules

In general, we form MOs from linear combinations of AOs with the correct symmetry propertiesThe energy of the MOs increases as the number of nodes increasesMOs made up from low energy AOs also have low energies

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MOs for NH3

The basis set consists of 3 H1s orbitals and the N 2s and 2p orbitalsThe molecule is known to have three fold symmetryThe N 2s and 2pz orbitals have cylindrical symmetry (also have three fold symmetry)The linear combination H 1s(A) + 1s(B) + 1s(C) has three fold symmetry

a1 MOs for NH3

Combine orbitals/LCAOS with cylindrical symmetry to form MOs

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e MOs for NH3

A combination of N 2px and 2py orbitals and linear combinations of H1s orbitals have e symmetry

Composition of NH3 MOs

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MO diagram + PES for NH3

SF6 and hypervalence

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Diborane and electron deficient compounds

B2H6 is a compound that you can not draw a reasonable Lewis structure for !

Electron deficient as the three atoms forming the B-H-B bridge are held together using only two electrons

XeF2 and electron excess compounds

The electronic structure of XeF2 and it’s stability can be rationalized by invoking a 3 center 4 electron bond