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Symmetry of Molecular Vibrations
Self Consistent Field Theory
The Potential Energy Surface
and the Hamiltonian
In the Born-Oppenheimer approximation it is assumed that the motion of
the nuclei is so slow that the electrons instantaneously follow them. This
is because mass(proton)2000mass(electron).
Therefore the electronic wavefunction obeys the Schrodinger equation with
the nuclei at rest. This equation is
Atomic units have been used ( ).
n is the number of electrons, N is the number of nuclei A
and ZA the nuclear charges. We see that the positions
enter the hamiltonian. Thus the energy,
depends upon the position of the nuclei.
is called The Potential Energy Surface. The quantum chemist (computational
chemist) attempts to calculate (sections of, pointwise) potential energy
surfaces. No parameters are input into the code. The Schrodinger equation
is impossibly difficult to solve, and approximations are essential.
The Electron Density
The electron density
is the probability of finding an electron at .
From the laws of quantum mechanics it is expressed in terms of the the
square of the orbitals which contain the electrons. Thus
where we have assumed we have two electrons in each of n orbitals,
i.e. we have a 2n electron molecule. Note that by definition
From studying H2, we understand that the density
is `cusped' at the nuclei
and furthermore the shape of the cusp will depend upon the particular nucleus,
the bigger the nuclear charge ZA the more pronounced
the cusp. In mathematical terms therefore a knowledge of the density tells
you (i) where the nuclei are, (ii) what they are and (iii) how many electrons
are in the molecule, from (63). It follows that in principle the density
contains all information, because once
and ZA are known, the hamiltonian (61) is known, and
so the wavefunction is known, and so all is known.
The Energy Expression
Once the orbitals have been determined (and therefore the density) it is
possible to determine the energy
of the molecule. It is given by
These terms may be identified as (i) the kinetic energy of the electrons,
(ii) the attractive energy between nuclei and electrons, (iii) the coulomb
repulsion energy between electrons, (iv) the difficult self-interaction
energy and `electron-avoidance' energy, (v) the nuclear-nuclear repulsion
energy. It is important to note that the Energy is NOT the sum of the orbital
energies, as may be seen from eqns(86) and (87). The sum of the orbital
energies counts twice the coulomb repulsion energy.
Basis sets and the Self Consistent
The orbitals are expressed as linear combinations of basis functions. Basis
functions are the logical extension of atomic orbitals to a more accurate
expansion set. Today basis functions are gaussian functions
situated on different atoms. Thus we write
M is the number of basis functions and
are the basis functions. Small (STO-3G), medium (6-31G*) and
large (TZ2P) basis sets are in common use, giving results increasingly
close to the accurate result.
To find the orbitals, we solve
which is a set of secular equations as we have met before. This time
we have a definition for Fkj, it is
The important point to observe is that there is an expression for
the F matrix, it can be evaluated, and it involves the orbitals which we
are trying to find. Thus the equations must be solved by an iterative procedure.
A typical output is attached, showing the convergence of the energy with
the iterations, the converged orbital energies and the molecular orbital
In this formaldehyde example note that:
(i) orbital 1 is 1sO and is very low in energy
(ii) orbital 2 is 1sC and is again low in energy
(iii) orbital 3 is 2sO. It does not participate in
(iv) orbital 4 is bonding CH2,involving 2sC,
2pzC and 1sA+1sB,
as well as some 2sO to preserve orthogonality
(v) orbital 5 is bonding CH2, b2 symmetry, constructed
from 2pyC and 1sA-1sB.
(vi) orbital 6 is
bonding between 2pzO and 2pzC
(vii) orbital 7 is
bonding between 2pxO and 2pxC
(viii) orbital 8 is nonbonding 2pyO lonepair.
(ix) observe that the sum of the orbital energies is not equal to the
energy of the molecule. There is double counting in the electron-electron
interactions in the former
(x) The molecule is positively charged at the CH2 end, and
negatively charged at O, leading to the dipole moment of 1.54D
(xi) the orbital energies are reasonable approximations to the ionisation
Density Functional Theory
The above are the working equations for the molecular orbitals and energy
of a molecule, using the theory which is today called `Density Functional
Theory' (DFT). The expression for the energy involves the density as anticipated.
The success of the theory depends upon how well the difficult term
can be represented in terms of the density, the above only gives the most
simple approximation to it. It is to be stressed that the hamiltonian F
is the same for all molecules and contains no adjustable parameters. It
is also important to realise that there is no wavefunction in DFT, but
only the density. Nasty things such as determinantal wavefunctions and
exchange integrals do not exist in DFT. In this sense DFT is an ab initio
theory. 90% of ab initio molecular calculations use DFT today, the
remainder use much more difficult theories which attempt to approximate
the electronic wavefunction, which we have seen is a much more complicated
object (a (linear combination of) determinant(s)). As a measure of the
reliability of DFT predictions, typical dissociation energies ( W(AB)-W(A)-W(B))
are calculated to an accuracy of 10 kJ mol-1, molecular bondlengths
to 0.01Å, and vibrational frequencies to 2%. DFT is an example of
the Self Consistent Field (SCF) method, where the orbital equations have
to be iterated until self-consistency is achieved.
There are now many standard quantum chemistry packages which routinely
perform DFT and more sophisticated calculations. Quantum chemistry is now
an accepted predictive tool of chemistry, used by organic, inorganic and
physical chemists. More in the T2 option!
Finally as an example of DFT, let us reexamine the proof of Hund's rules
in the simple case of the 1s2s excited state of Helium. All we concern
ourselves with is the exchange energy which according to DFT is given by
are the alpha and beta spin densities. In the singlet case
and in the triplet case .
Thus the exchange energies in the two cases are
It follows that the triplet is lower in energy than the singlet because
it can easily be proved that .
- Previous: The
Symmetry of Molecular Vibrations