To make a quantum mechanical model of the electronic structure of
a molecule, we must solve the Schrödinger equation.
Solving this equation is a very difficult problem and
cannot be done without making approximations. We have covered some of
these approximations in the Semiempirical MO Theory handout.
In this handout we focus on ab initio methods of solving the
equation, in which no integrals are neglected in the course of the
calculation.
The first approximation is known as the BornOppenheimer approximation, in which we take the positions of the nuclei to be fixed so that the internuclear distances are constant. Because nuclei are very heavy in comparison with electrons, to a good approximation we can think of the electrons moving in the field of fixed nuclei. We first choose a geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy.
When more than one electron is present, the Schrödinger
equation is impossible to solve because of the interelectron terms in
the Hamiltonian. Consider, for instance, the Hamiltonian for the
hydrogen molecule in the BornOppenheimer approximation.
The first two
terms are due to the kinetic energy of the electrons. The last six
terms express the potential energy of the system of four particles.
The potential energy term due to the repulsion of the electrons makes
the Schrödinger equation impossible to solve.
To produce a solvable Schrödinger equation we assume that the Hamiltonian is a sum of oneelectron functions, f_{i}, with an approximate potential energy that takes the average interaction of the electrons into account. This leads to a set of oneelectron equations, called the HartreeFock equations, where is a oneelectron wavefunction.
The total wavefunction that is a solution to the total
Schrödinger equation, , is
approximated as the product of the solutions to the oneelectron
equations.
This product must be adjusted to satisfy the Pauli Exclusion
principle, but we won't get into that here. If you are familiar with
determinants, it involves writing the wavefunction as a
determinant.
The question remains about the approximate potential energy in the oneelectron functions that take the average interaction of the electrons into account. What is the form of the functions f_{i} in the HartreeFock equations? The most common way of handling this is to define
where v_{i} is an average potential energy due to
the interaction of one electron with all the other electrons and
nuclei in the molecule. The average potential depends on the
orbitals, , of the other
electrons, which means we must solve the HartreeFock equations
iteratively.
The iterative solution of the HartreeFock equation is as
follows.
1. Guess reasonable oneelectron orbitals (wavefunctions), , and calculate the average
potential energies, v_{i}.
2. Using the variation principle, solve the HartreeFock equations,
to give new oneelectron orbitals, . Use these new orbitals to calculate new and improved
average potential energies, v_{i}.
Because the solution of the HartreeFock equations depends on
the variation principle, the HartreeFock energy should be higher
than the true energy.
3. Repeat the second step until the oneelectron orbitals and
potential energies don't change (are
selfconsistent).
To take the Pauli Principle into account, we must include
electron spin in our wavefunctions. The orbitals that are
calculated by the HartreeFock method actually are spin
orbitals that are a product of a spatial wavefunction and a spin
function.
In a spin orbital, is the spatial
wavefunction describing the probability of finding the electron in
space and or are spin wavefunctions.
For a closed shell system, in which all of the electrons are
paired, during the solution of the selfconsistent field equations,
we can restrict the solution so that the spatial wavefunctions for
paired electrons are the same. This is called a restricted
HartreeFock (RHF) calculation and generally is used
for molecules in which all the electrons are paired. When the
spin functions are removed, we are left with a set of spatial
orbitals, each occupied by two electrons.
An example would
be the restricted HartreeFock solution to the Schrödinger
equation for the hydrogen molecule,
H_{2}. This would lead to two spatial
orbitals, one occupied by the pair of electrons and one unoccupied.
The orbitals holding electrons are called occupied orbitals
and the unoccupied orbitals are called virtual
orbitals.
For open shell systems that contain unpaired electrons, the assumption made in the restricted HartreeFock method obviously won't work. There is more than one way of handling this type of problem. One way is to not constrain pairs of electrons to occupy the same spatial orbital  the unrestricted HartreeFock (UHF) method. In this method there are two sets of spatial orbitals  those with spin up () electrons and those with spin down
() electrons. This leads to two sets of orbitals as pictured at the right and to a lower energy than if the restricted method were used.
For molecular calculations, the HartreeFock SCF equations
still
cannot be solved without one further approximation. To solve the
equations, each SCF orbital,, is
written as a linear combination of atomic orbitals. For
instance, for the H_{2} molecule, the
simplest approximation is to write each spatial SCF orbital as a
combination of 1s atomic orbitals, each centered on one of the
protons.
This reduces the problem to solving for the coefficients,
c_{1} and
c_{2}, since the atomic orbitals do not
change.
The set of atomic orbitals that is chosen to represent the SCF
orbitals is called a basis set. The
{1s_{A},
1s_{B}} basis set shown above is a
minimal basis set  the smallest set of orbitals possible that
describe an SCF orbital. Usually, the quality of a basis set
depends on its size. For instance, a larger basis set, such as
{1s_{A},
1s_{B},
2s_{A},
2s_{B}}would do a better job
approximating the SCF orbital than
{1s_{A},
1s_{B}}.
For manyelectron atoms, we don't know the actual mathematical
functions for the atomic orbitals, so substitutes are used  usually
either Slatertype orbitals (STO) or Gaussiantype orbitals (GTO). We
won't concern ourselves with the exact form of STO and GTO. Suffice
it to say that they are chosen to behave mathematically like the
actual atomic orbitals: stype, ptype, dtype, and ftype,
for instance. A few commonly used basis sets are listed below. The
symbol of the basis set is given in the left column and the
characteristics of the basis set in the center. At the right is the
basis set that would be used to represent methane. For instance, the
STO3G basis set for methane would be
{1s_{H},
1s_{H},
1s_{H},
1s_{H},
1s_{C},
2s_{C},
2p_{xC},
2p_{yC},
2p_{zC}}.
Basis Sets^{1} 
Characteristics 
Basis Set Example (CH_{4})  

STO3G 
A minimal basis set (although not the smallest possible) using three GTOs to approximate each STO. This basis set should only be used for qualitative results on very large systems 
Each H: 1s  
321G 
Inner shell basis functions made of three GTOs. Valence s and porbitals each represented by two basis functions (one made of two GTOs, the other of a single GTO). Use for very large molecules for which 631G is too expensive. 
Each H: 1s, 1s'  
631G(d) 
Inner shell basis functions made of six GTOs. Valence s and p orbitals each represented by two basis functions (one made of three GTOs, the other of a single GTO). Adds six dtype basis functions to nonhydrogen atoms. This is a popular basis set that often is used for medium and large systems. 
Each H: 1s, 2s  
631G(d,p) 
Like 631G(d) except ptype functions also are added for hydrogen atoms. Use when hydrogens are of interest and for final, accurate energy calculations. 
Each H: 1s, 2s, 2p_{x}, 2p_{y},
2p_{z} 
Generally, the larger the basis set the more accurate the calculation (within limits) and the more computer time that is required. As an example, consider the calculation of the bond length of HF using different basis sets, as shown below. ^{1}
Basis Set 
Bond Length (Å) 
 Error (Å)  

631G(d) 
0.93497 
0.017 
631G(d,p) 
0.92099 
0.003 
631+G(d,p) 
0.94208 
0.025 
631++G(d,p) 
0.92643 
0.009 
6311G(d,p) 
0.91312 
0.004 
6311++G(d,p) 
0.91720 
0.000 
Experimental 
0.917 

You might notice that although the large basis set, 6311++G(d,p), predicts the correct answer to within 0.001 Å, several others are correct to within 0.01 Å (well within the criteria of chemical accuracy). Although a larger basis set usually gives better results, you often have diminishing returns as you choose larger sets. A point may be reached beyond which the additional computer time is not worth it.
Even with a very large basis set calculation, HartreeFock results
are not exact because they rely on the independent
electron approximation. HartreeFock SCF Theory
is a good baselevel theory that is reasonably good at computing the
structures and vibrational frequencies of stable molecules and some
transition states^{2}. Electrons are not independent, though.
We say that they are correlated with each other and that the
HartreeFock method neglects electron correlation. This means
that HartreeFock calculations do not do a good job modeling the
energetics of reactions or bond dissociation. There are several ways
of correcting SCF results to take electron correlation into
account.
One method of taking electron correlation into account is
MøllerPlesset manybody perturbation theory, which is
used after a RHF or UHF calculation has been made. It is assumed that
the relationship between the exact and HartreeFock Hamiltonians is
expressed by an additional term, H^{(1)}, so that H =
f_{i} + H^{(1)}.
Calculations based on this assumption lead to corrections that can
improve SCF results. Various levels of perturbation theory can be
applied to the problem. They are called MP2, MP3,
MP4, etc. MP2 calculations are not timeconsuming and usually
give quite accurate geometries and about onehalf of the correlation
energy. Because perturbation theory is not based on the variation
principle, the energy predicted by MP calculations can fall below the
actual energy.
Another important method of correcting for the correlation
energy is configuration interaction (CI). Conceptually we can
think of CI calculations as using the variation principle to combine
various SCF excited states with the SCF ground state, which lowers
its energy. We won't use CI calculations in our exercises at this
level.
When calculating molecular orbitals, you should remember that
molecular orbitals are not real physical quantities. Orbitals are a
mathematical convenience that help us think about bonding and
reactivity, but they are not physical observables. In fact, several
different sets of molecular orbitals can lead to the same energy.
Nevertheless, they are quite useful. We will use ethylene as an
example to illustrate MO concepts.
The basis functions in SCF molecular orbitals are like atomic
orbitals. A RHF/631G(d) calculation on ethylene uses 38 basis
functions (15 for each carbon and 2 for each hydrogen). Since the
molecular orbital wavefunction is expanded in terms of the all the
basis functions,
it might seem that constructing a picture of the orbital would be
difficult. Luckily, most of the coefficients are zero, so the
molecular orbitals are easy to picture. Consider, for instance, the
highest occupied molecular orbital (HOMO) and lowest unoccupied
molecular orbital (LUMO) of ethylene.
HOMO:
The HOMO is a bondingorbital.
LUMO:
The LUMO is an antibonding orbital.
In the last part of the job output from a frequency calculation you will find the predicted vibrational frequencies (cm^{1}) of the normal modes of the molecule. Also supplied are the predicted intensities of the IR and Raman bands corresponding to these normal modes.
1 2 3 B1 B2 A1 Frequencies  1335.5948 1383.4094 1679.4157 4 5 6 A1 A1 B2 Frequencies  2027.8231 3160.8817 3232.9970
Computational results usually have systematic errors. In the case of HartreeFock level calculations, for instance, it is known that calculated frequency values are almost always too high by 10%  12%. To compensate for this systematic error, it is usual to multiply frequencies predicted at the HF/631G(d) level by an empirical factor of 0.893. Similarly, frequencies calculated at the MP2/631G(d) level are scaled by 0.943. ^{1}
The predicted frequencies after applying the 0.893 scale factor are listed below.
1 2 3 B1 B2 A1 Scaled Frequencies  1193 1235 1450 4 5 6 A1 A1 B2 Scaled Frequencies  1811 2822 2887
^{1}J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 199596, p 102.
^{2}J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1996, p 115.
Converse/USCS/Wofford
Physical Chemistry Consortium