MODULE 11Diatomic Molecules and Beyond

Our next task is to extend the procedures and ideas that we have developed for H2

+ to larger molecules.

The track we follow is the customary one, that is, start with homonuclear diatomics, of which H2 is the paradigm, then consider heteronuclear diatomics, and finally polyatomic

molecules.

The overall procedure is to construct sets of MOs and then put in the electrons according to the Pauli principle analogous to the

aufbau method for atoms

The result is similar because we end up with the electronic configurations of molecules.

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The Hydrogen Molecule

H2 has an additional electron over its molecule-ion and we

can imagine a scheme such as shown in Figure.

The electron configuration follows the same rule as for

the He atom, where the added electron was placed into the lowest energy 1s orbital with

its spin opposed (Pauli).

A B

1 2

Thus the added electron is placed into the bonding orbital defined for the 1sA + 1sB combination.

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The extra electron adds the distances r12, r2B, and r2A.

This adds complexity in that an electron-electron repulsion term appears, as well as two more electron-nucleon potential energy

terms.

The hamiltonian for the hydrogen molecule in the Born-Oppenheimer approximation and in atomic units is

2 21 2

1 1 2 2 12

1 1 1 1 1 1 1ˆ ( )2 A B A B

Hr r r r r R

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where b is the bonding orbital (1). This can be rearranged to

Since the hamiltonian is independent of the spin terms, we calculate the energy using only the spatial part of equation.

Thus our molecular wavefunction is given by

Where the first term on the RHS is the normalization constant (it is the square of the normalization constant for

(1) (2)1(1) (2)2!

b b

b b

1(1) (2) (1) (2) (2) (1)

2b b

11 (1) 1 (1) 1 (2) 1 (2)

2(1 )MO A B A Bs s s sS

MODULE 11Thus our molecular wavefunction is a product of molecular orbitals

both of which are linear combinations of atomic orbitals.

The procedure for constructing molecular wavefunctions is known as the Linear Combination of Atomic Orbitals-Molecular Orbitals,

or LCAO-MO, method.

This is a commonly used procedure for a variety of molecules.

The ground state energy of H2 can then be calculated as before

using

Where the hamiltonian and the wavefunction are as above

ˆ(1,2) (1,2)MO MO MOE H

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The results of the integration are shown in figure.

As for the molecule ion the results lack accuracy, but the form is

clearly shown and the results can be improved using a larger basis

set.

The ground state configuration of H2 is classified as 1g, which

echoes the term symbols for atoms (not yet covered).

The superscript 1 is the multiplicity of the state (here the 2 electrons are paired, S = 0). The (analogous to atomic S)

indicates that the total OAM around the inter-nuclear axis is zero because both electrons are in s-type AOs

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Whereas for atoms we used L = l1+l2, etc to compute the total

OAM, in molecules we use =1 + 2 to compute the total OAM

In H2, = 0, hence .

The subscript g indicates the overall parity of the state and we calculate this from the individual values for the electrons by the

products g x g = g; g x u = u; u x u = g

In H2 both electrons occupy the g orbital, hence the overall parity

is g. Had one occupied a 2u orbital the overall parity would have

been u.

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Homonuclear Diatomic Molecules Beyond Hydrogen

The MOs for homonuclear diatomics beyond H2 are formed from pairs of AOs of many-electron atoms

Therefore we must to use orbitals beyond 1s (higher energy).

Nevertheless the 1s AOs are involved and we can start there and the first pair of MOs is given by

1 1A Bs s

In Module 10 we saw that the two MOs have cylindrical symmetry with respect

to the inter-nuclear axis.

For this reason they are termed orbitals.

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Because they are constructed from a pair of 1s orbitals they are conventionally labeled 1s, indicating they are orbitals formed

from 1s AOs, and nothing else.

The positive combination, builds up electron density between

the nucleons and is the bonding orbital,

The negative combination, excludes electron density from that

region, and is the antibonding orbital, often written as 1s.

As we saw in Module 10 the bonding orbital has gerade inversion symmetry, hence it is symbolized as g1s, and the antibonding

orbital is u, therefore u1s.

We do not include the asterisk in the symbol since the subscript gives the antibonding designation.

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Next consider combinations (sum and diff) of a pair of 2s AOs.

Both AOs have the same energy and the same symmetry and these are factors important in the LCAO procedure.

The two MOs resulting from 2sA+/-2sB are g2s and u2s, one

bonding and the other antibonding.

These MOs have similar shape to those shown in the figure above, except they are larger because the composite AOs are

larger.

By the same token the energy of the 2s-based orbitals are higher than those based on 1s.

The energies can be calculated in an analogous way to that outlined above for the 1s combination.

MODULE 11The relative energies are in the

sequence

In the hydrogenic ions the 2s and 2p orbitals are degenerate, but this degeneracy is lifted in many-electron atoms because of differences in the nuclear screening, thus E2s < E2p

In Figure the way in which the atomic 2p orbitals combine is

indicated.

1 1 2 2g u g us s s s

2pz AOs are oriented along the inter-nuclear axis.

Both u and g combinations of the 2pz orbitals are symmetric

around the axis and are therefore of -type.

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The 2px and 2py combinations overlap “sideways” and the result is

that the four MOs generated do not have cylindrical symmetry.

This change in symmetry leads to the designation of -bonds.

[An easy way of determining whether a MO is or is to “view” the orbital along the inter-nuclear axis. If you see a circle of

electron density centered on the nucleus (like an s-AO), then that orbital is . If what you see looks like a p-AO (two circles

separated by a nodal plane) then you are dealing with a -orbital.]

The negative combination 2pzA-2pzB is the one that yields the

bonding MO; for the other combinations the positive ones are bonding.

The bonding orbital resulting from the 2pz combination transforms

as g on inversion, whereas the opposite is true for the bonding orbitals resulting from the 2px and 2py pair combinations.

MODULE 11From figure we see that the

energies depend on Z.

Moreover the g2pz orbital

changes so much with Z that between N2 and O2 it

switches with the degenerate x and y-pairs.

Also Figure shows the electron occupancy of the

MOs built up by the LCAO-MO procedure, with aufbau.

Not shown are H2 and He2.

The former we have already considered, it has two electrons of opposed spins in the g1s MO and its configuration is (g1s)2.

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He2 has a configuration (g1s)2(u1s)2 in which there are an equal

number of bonding and antibonding electrons.

The two sets cancel such that there is no net bond, according to the simple version of MO theory used here (He2 has been

detected spectrometrically in the gas phase at T~0.001 K).

The molecular ion He2+ is a stable, but reactive, species. Some

properties of hydrogen and helium molecules are in the Table

MODULE 11bond order = (nb-na)/2

where na and nb are the number of electrons in bonding and

antibonding orbitals, respectively.

Note that bond order can be a half-integer.

Diatomic lithium has an electron configuration of (g1s)2(u1s)2(g2s)2

and thus the bond order is one.

Li2 exists in Li vapor and its bond energy is 105 kJmol-1.

Electron density contour maps, generated by computer solutions of the Schrödinger equation are presented in the figure over page

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The important thing to note about these diagrams is that the g1s and u1s

electrons are found close to the nuclei and play virtually no part in the bonding interaction, which is mostly due to the electrons in the g2s MO.

This leads to an alternative way of writing molecular electronic configuration for Li2 as KK(g2s)2 in which K

represents the filled n = 1 shell of the Li atom.

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This becomes even more accurate as Z increases and the 1s electrons are held progressively more tightly than in the Li case.

At this level of approximation only the valence electrons need to be considered for the homonuclear diatomics beyond He2.

Returning to the orbital energy diagram and focusing on B2 and

O2, we see that each molecule has a degenerate pair of orbitals

(u2px,y in the case of B and g2px.y in the case of O).

Hund’s rule informs us that we place electrons into degenerate sets of orbitals one at a timein order to maximize the spin

multiplicity.

Both molecular boron and molecular oxygen have been shown to have paramagnetic (triplet) ground states.

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In the Table are shown the ground state electron configurations of the 2nd row homonuclear diatomics

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