MODULE 11
Diatomic 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.
MODULE 11
The Hydrogen Molecule
1
A
2
H2 has an additional electron
over its molecule-ion and we
can imagine a scheme such as
shown in Figure.
B
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).
Thus the added electron in H2 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 BornOppenheimer approximation and in atomic units is
1
1
1
1
1
1 1
2
2
ˆ
H   (1   2 ) 



 
2
r1 A r1B r2 A r2 B r12 R
MODULE 11
1 y b (1) y b (2)
y
2! y b  (1) y b  (2)
where yb is the bonding orbital (1s). This can be rearranged to
 1

y  y b (1)y b (2)   (1)  (2)   (2)  (1) 
 2

Since the hamiltonian is independent of the spin terms, we
calculate the energy using only the spatial part of equation.
Thus our molecular wavefunction yMO is given by
y MO 
1
1sA (1)  1sB (1)1sA (2)  1sB (2)
2(1  S )
Where the first term on the RHS is the normalization constant (it is
the square of the normalization constant for y.
MODULE 11
Thus 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
EMO  y MO (1, 2) Hˆ y MO (1, 2)
Where the hamiltonian and the wavefunction are as above
MODULE 11
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 1Sg, which
echoes the term symbols for
atoms (Module 9A).
The superscript 1 is the multiplicity of the state (here the 2
electrons are paired, S = 0). The S (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 L =l1 + l2 to compute the total OAM
In H2, L = 0, hence S.
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 sg orbital, hence the overall parity
is g. Had one occupied a 2su orbital the overall parity would have
been u.
MODULE 11
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
y   1sA  1sB
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 s
orbitals.
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Because they are constructed from a pair of 1s orbitals they are
conventionally labeled s1s, indicating they are s orbitals formed
from 1s AOs, and nothing else.
The positive combination, y, builds up electron density between
the nucleons and is the bonding orbital,
The negative combination, y, excludes electron density from that
region, and is the antibonding orbital, often written as s*1s.
As we saw in Module 10 the bonding orbital has gerade inversion
symmetry, hence it is symbolized as sg1s, and the antibonding
orbital is u, therefore su1s.
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 factors are important in the LCAO procedure.
The two MOs resulting from 2sA+/-2sB are sg2s and su2s, 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.
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The relative energies are in the
sequence
s g 1s  s u 1s  s g 2s  s u 2s
In the hydrogenic ions the 2s and
2p orbitals are degenerate, but
this degeneracy is lifted in manyelectron atoms because of
differences in the nuclear
screening, thus E2s < E2p
In Figure the way in which the
atomic 2p orbitals combine is
indicated.
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 s-type.
MODULE 11
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 p-bonds.
[An easy way of determining whether a MO is s or p 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 s. If what you see looks like a p-AO (two circles
separated by a nodal plane) then you are dealing with a p-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 11
From figure we see that the
energies depend on Z.
Moreover the sg2pz 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 LCAOMO procedure, with aufbau.
Not shown are H2 and He2.
The former we have already considered, it has two electrons of
opposed spins in the sg1s MO and its configuration is (sg1s)2.
MODULE 11
He2 has a configuration (sg1s)2(su1s)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 11
bond 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
(sg1s)2(su1s)2(sg2s)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
MODULE 11
The important thing to note
about these diagrams is
that the sg1s and su1s
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
sg2s MO.
This leads to an alternative
way of writing molecular
electronic configuration for
Li2 as KK(sg2s)2 in which K
represents the filled n = 1
shell of the Li atom.
MODULE 11
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
(pu2px,y in the case of B and pg2px.y in the case of O).
Hund’s rule informs us that we place electrons into degenerate
sets of orbitals one at a time in order to maximize the spin
multiplicity.
Both molecular boron and molecular oxygen have been shown to
have paramagnetic (triplet) ground states.
MODULE 11
In the Table are shown the ground state electron configurations of
the 2nd row homonuclear diatomics
MODULE 11
The tabulated data are
plotted here, showing the
inverse correlation
between bond energy
and bond length.
MODULE 11
Experimental Demonstration of Orbitals
The concepts of atomic and molecular orbitals have been arrived at
through theoretical manipulations, and one might wonder that there
is any reality in the ideas.
In fact, there are experimental ways to support the actual existence
of orbitals.
One is by the use of photoelectron spectrometry.
This is a technique that measures the energy of electrons ejected
from gaseous molecules by vacuum uv/soft x-ray radiation. This
provides the ionization energy of the molecule, which depends on
the MO from which the ionized electron originated.
MODULE 11
The figure shows the PE spectrum of N2 gas.
Peaks occur that can be assigned to all the occupied orbitals
listed for N2 in the Table
MODULE 11
Heteronuclear Diatomic Molecules
Examples are HF, HI, CO, NO, CN-.
Earlier we stated that for LCAO-MOs we need to use AOs that are
close in energy.
In HF and HI the energies of the atomic orbitals are very different.
In CO, NO, CN- the energies are not very different (atoms are
close in terms of Z).
It is useful to look closely at one from each group.
First consider CO.
MODULE 11
The PE spectrum of
CO
Peaks that are
characteristic atomic
1s orbitals on C and
O are visible
The high energies of s1s and s*1s are indicative of these
electrons being closely associated with the respective nuclei and
supports the contention that K electrons play no part in the
bonding process after Z = 3.
Moreover the energies of these orbitals are very similar to the
energies of the constituent O1s and C1s AOs respectively
MODULE 11
The spectrum is consistent with the electronic configuration of CO
being
KK (s 2s)2 (s *2s)2 (p 2 px )2 (p 2 py ) 2 (s 2 pz ) 2
Thus CO has a bond order of 3.
Note that there are no gerade/ungerade subscripts because there
is no inversion symmetry in heteronuclear diatomics.
MODULE 11
In HF, the valence electrons are in different shells.
In F, E2s = -1.477 au, E2p = -0.684 au. In H, E1s = -0.5 au.
Thus it is more favorable for the H1s and F2p orbitals to form the
combination.
Now we have to consider the symmetry of the situation.
Figure shows clearly why
only the 2pz AO on F has the
proper symmetry for bonding
with the H1s AO.
The coupling of H1s and
F2px,y leads to no bond since
the positive and negative
overlaps cancel out.
MODULE 11
The MO for HF will be approximated by
y  c11sH  c2 2 pzF
the bond has cylindrical symmetry and hence is s.
An energy level diagram for HF is
shown (less the 1sF and the 2sF
orbitals).
H has 1 valence electron and F has
7, so the total of 8 occupy the four
orbitals shown according to the
Pauli principle.
The 2sF (not shown), 2pxF, and
2pyF orbitals are non-bonding MOs
(2sF )2 (s b )2 (2 pxF )2 (2 pyF )2
The bond order is 1
MODULE 11
Most of what we have said so far about diatomics has concerned
MOs that have been formed from a single AO from two
contributing atoms.
When we were considering many-electron atoms in Module 10 we
found that we could lower our computed value of Emin by using
linear combinations of orbitals (e.g. STOs) for the trial function and
using the Hartree-Fock SCF procedure to generate the HartreeFock limit.
The molecular case uses similar procedures.
For example we could use a trial function of the form
y  c1 (1s A  1sB )  c2 (2s A  2sB )  c3 (2 pZ , A  2 pZ , B )  ...
and use variation theory to find the relative contributions of the
different LCAOs by determining the variational coefficients, ci.
MODULE 11
One outcome of using the larger MO basis set is that it becomes
no longer possible to identify an MO as arising from a single AO.
Thus our sg1s, etc designations need to be modified.
Thus we find designations such as 1sg, indicating the first (lowest
energy) s orbital, and 1pu, indicating the first (lowest energy) p
orbital, and so on.
These days MOs are determined by Hartree-Fock SCF
computations employing basis sets of a large number of linear
combinations, and the notation above is the one you will find in
the literature.