Summary: diatomic MOs
Thanks to all who responded.
My original question concerned a claim in my inorganic chemistry textbook (and
the many other books that people referred me to) that said that the pz-pz sigma
bonding level lies above the degenerate pi bonding levels in B2, C2 and N2, and
below in O2 and F2.
If the textbook had simply said that this analysis was based on experimental
data my question would have been moot because my interest was not in
experimental results at this point in the course. Indeed, many people directed
me to various sources of "experimental orbital energies" of which the
most convenient turned out to be the NIST webbook. (webbook.nist.gov/chemistry/)
Since my textbook's claim is widely repeated in many books (and since many
people wrote fervently to me on this topic), I think a few more comments are in
1. There is no such thing as an 'experimental orbital energy'. 'Orbital' shapes
and energies are not experimentally measurable.
So if the NIST data are not 'experimental orbital energies', what are they?
The NIST data turn out to be the energies of different electronic states of the
cation produced by ionized of the diatomic molecule. If you look at the NIST
table of X2 CATION energies you will see many states listed for each diatomic
ion, their relative energies, their term symbols ("2_Pi_U"), and other
properties besides. If one assumes that the neutral diatomic and all of its ions
have the same orbitals at the same energies (i.e., Koopman's theorem) one can
use the term symbols and state energies to guess the sequence of orbital
energies in the neutral.
These assumptions may be useful for practicing chemists, but it is important to
realize what a tremendous leap of faith they represent. Orbital energies and
molecular geometries change upon ionization; why can't the sequence of orbitals
change as well? (This would explain the behavior of F2, see below.)
Which brings me to my next comment:
2. Many people advised me NOT to use HF calculations to get the correct sequence
of orbital energies. Specific advice included, "don't use Koopman's
theorem", "HF calculations are too crude to give the correct orbital
energies" and other statements to the same effect.
The advice was well intended, but it was based on the mistaken assumption that I
was trying to fit the ionization measurements, something that I was NOT trying
to do (see below). The advice is also fascinating in another way. Consider:
"don't use Koopman's theorem" - The fact that HF orbital energies do
not reproduce ionization potentials is well-known. I have nothing to add. But, I
do wonder, how does one get from the term symbols in the NIST table to the
orbital energies without using Koopman's theorem? There seems to be a double
standard at work - computational chemists should not use Koopman's theorem, but
experimentalists should use poor Koopman. Isn't it equally flawed for both?
"HF calculations are too crude to give the correct orbital energies" -
What other method would people advise? Orbital energies are not physical
observables. Orbitals do not exist in correlated treatments. What method is
there that is less crude than HF, but still has orbital energies and generates
results that can be anticipated using qualitative MO theory? (DFT orbital
energies may be numerically interesting, but there is no reason to expect that
the qualitative MO theory in my textbook was designed to anticipate DFT orbital
3. Teachers and students beware. The textbook that I am using introduces the
subject of diatomic MO energies in the context of "here is how to mix AO's
and guess MO shapes and energies". I assumed that the book was trying to
teach qualitative mixing techniques that mimic what happens in a more
sophisticated Hartree-Fock calculation. Unfortunately, either I am wrong or the
authors simply did not realize that they were changing the rules of the game in
Given the willingness of textbook authors to use expt'l data to say how orbitals
"mix" I think a different point-of-view must be adopted by students
and teachers. They must view the "MO theory" in the textbook as merely
a conceptual framework in which one mixes AOs and MOs to arrive at an
experimental result. Yes, we mix AOs just like the computer does during an HF
calculation, but we adjust our mixing empirically to fit experimental data
(forget all that perturbation theory and secular determinant stuff).
The situation is entirely analogous to the way organic chemists use
"resonance theory". One starts with a solid foundation (VB-CI theory),
simplifies it so almost anyone can use it, and then drops all theoretical
pretensions and fits the remaining parameters ("major/minor resonance
contributors") to get a desired result. Good-bye theoretical foundation.
Now I will know better than to believe my book when it tells me about orbital
energies, overlaps, and mixings - it is probably talking about empirically
derived quantities, and not calculated ones.
4. I took another look at the calculation of HF orbital energies. Instead of
using UHF for open-shell molecules, I used ROHF (Jaguar) and a large basis set:
cc-pvtz(-f)++ (32 functions per atom). The orbital sequences are:
triplet B2: s-s < s-s* < pi_x (1 electron) = pi_y (1 electron) < pz-pz
singlet C2: same as triplet B2 except pi_x and pi_y are filled.
triplet C2: s-s < s-s* < pi_x (1 electron) = pi_y (1 electron) < pz-pz
(filled) < pz-pz* [LUMO]
>>triplet C2 (ROHF) was given a lower energy than single C2 (RHF). Notice
that the pi levels in triplet C2 are half-filled but lower in energy than the
filled pz-pz level.
N2 : s-s < s-s* < pz-pz < pi_x = pi_y < pz-pz* [LUMO]
triplet O2: s-s < s-s* < pz-pz < pi_x = pi_y < pi*_x (1 electron) =
pi*_y (1 electron) < pz-pz* [LUMO]
F2: s-s < s-s* < pi_x = pi_y < pz-pz < pi_x* = pi_y* < pz-pz*
F2 is the most interesting molecule of all(?). Koopman's theorem predicts that
ionization should happen more easily from the sigma z-z level than from the pi
bonding levels, but the experimental ordering of cation states says the
opposite. Because of the high electron density around two F atoms, this is
exactly the kind of case where electron correlation effects are likely to be
substantial. I have no results to support this, but it seems entirely
conceivable that different electronic states of the CATION represent different
orbital sequences, and these sequences may be quite different from those of the
neutral (differential orbital relaxation, geometry changes, and electron
correlation all seem likely here).
I will tell my class that calculations correctly give the order of orbitals in
the neutral F2 molecule, and this order is different from the one in the book
(so don't trust what the book says about when to mix/not mix orbitals!), mainly
because I think calculations tell us about orbitals and experiments don't. I'll
also point out to them that the orbital sequence in F2 is not a good guide to
the sequence of electronic states in F2+. But F2 and various states of F2+ are
all very different molecules, so why should an orbital sequence in one tell me
about the energy of the other?
Department of Chemistry