Ab initio methods

You should read Chapter 6 of Goodman's 'Chemical Applications of Molecular Modelling' from section 6.4.4 onwards, for an easy introduction to this. (Before this is what you learned in stage 1.)
MO theory finds approximate solutions to Schroedinger's Equation, to get energies and orbitals of electrons
Energies can be used for:

Geometry optimisation
Reaction energetics
Activation energies for kinetics (difficult)
UV/vis absorption prediction

Orbitals can be used for:

Graphical display (including assessment of where reactants might attack)
Charges
Dipole moments
Electrostatic potentials
NMR shieldings

Ab initio methods are available in many of the more expensive modelling packages, or in Gaussian, which is available on a UCS unix machine
To do an ab initio calculation, you need to select:

a method
a basis set

The notation written in papers, and in the command line to Gaussian, is in the form method/basis, e.g. RHF/3-21G* or B3LYP/6-31G*

Hartree Fock method

Abbreviation HF or RHF
Simplest method, closely related to what we teach in the stage 1 MO course
Good enough for most geometry calculations
May be good enough for reaction energetics

Just as geometry optimisations work by minimising energy with respect to geometry, so ab initio calculations, for a single geometry point, work by minimising the electronic energy by changing the array of coefficients which control how AOs are added together to make each MO

The Variation Principle says that if there are two approximations to a wavefunction (i.e. two different sets of coefficients c), then the better approximation is the one which gives the lower energy
The HF method works by calculating orbitals for one electron in an average field of the rest, as follows:

Start with guess at the electron distribution

At the start of a calculation, this comes from a semiempirical calculation

Use the guess to calculate the average field of the rest of the electrons
Calculate coefficients to produce a set of one-electron MOs and energies
Fill the MOs according to the Aufbau principle and get total electronic energy
Use this set of MOs as the next approximation to the electron distribution
Repeat the process with gradually improving MOs, until the electronic energy converges, i.e. it changes by less than some pre-set limit
This is called self consistent field (SCF) convergence

If we are doing a geometry optimisation, the program moves on to the next geometry point

The AOs move with the atoms
The set of coefficients from the last geometry point can be a starting point for the new SCF calculation, and so on, until the geometry optimisation converges

Because we have one iterative process, the SCF minimisation, inside another iterative process, the geometry minimisation, the whole process takes a lot longer than force-field or semi-empirical methods
The program can save a lot of time if it knows that all electrons are paired, so it can treat them two at a time

This is so for ordinary, diamagnetic organic compounds, for most main-groups compounds, and for organometallic compounds which obey the 18-electron rule
This is called a restricted Hartree Fock treatment (RHF)
Treating compounds with unpaired electrons is much more difficult

Basis sets

The basis set is the set of AOs, which the HF method is going to find the coefficients of, to make MOs
Schroedinger's equation can be solved exactly only for the H atom, so all other AOs are approximations
The H-like orbitals we teach about in stage 1

HF calculations involve calculating hundreds of thousands of integrals in each SCF step, so it is important to use orbitals which are easy to integrate
Natural orbitals are difficult to integrate, but Gaussian functions are easier: the essential difference is that the r is squared

Gaussian primitives are added together in linear combinations to produce basis functions which are approximations to natural orbitals

The designers of basis sets have optimised the set of coefficients d and the set of exponents a used in the gaussian primitives g to give a balanced set of orbitals for a certain set of elements
There is a library of published basis sets on the web, but many of the most useful are built into programs, so we need only give their name in order to use them
The more primitive gaussians there are in a basis function, the better the quality of the approximation, but the longer the calculations will take
If the calculations take too long, e.g. more than a few days of processor time on the computer we have access to, then this limits the size of molecule we can tackle
There is always a trade-off between quality of basis set and practicability
You need to know 'what is the simplest basis set which will answer my questions'
Some programs allow professional computational chemists to design their own basis set, but you must not, intentionally or accidentally, change any of the numbers in a published basis set, otherwise your work will be unpublishable because it could not be repeated
The way gaussian primitives are put together to make a basis function is called a contraction, because a greater number of functions is contracted into a lesser number
Contractions do not have to have a one to one correspondence with natural orbitals, and they usually do not
If we have more basis functions than natural orbitals, then there are more coefficients c to optimise in our SCF calculation, for our particular molecule

This will give better quality MOs and energies, at the expense of a longer-still calculation

The simplest basis set worth using in research is 3-21G
This means that core AOs are contractions of 3 primitive gaussians
Valence shell orbitals are each represented by two separate basis functions: one is a contraction of 2 primitive gaussians, and the other is just a single gaussian function
This is an example of a 'split valence basis set': the valence shell orbitals are split into two components so that their shape can be varied by taking different amounts of each component

You can see this by looking at the provided file '3-21G basis set for H, C and N'. Open this in a separate browser window, so that you can read these notes at the same time.

The first column of numbers with long decimal parts is the exponents a
The next column is the coefficients d
Hydrogen has just a valence shell, so we have two s-type basis functions, one with two primitive gaussians, and the other with one
The three primitives have quite different exponents:

a big exponent gives a function which drops off quickly: this produces the part of the orbital close to the nucleus
small exponents produce the part of the orbital further from the nucleus, in the bonding region

Notice that the one-primitive basis function has the smallest exponent: by adjusting the coefficient of this part in the SCF process, the program can adjust the orbital in the bonding region
Carbon has a core 1s orbital, represented by three primitives
Notice the much bigger exponents than for H, because of the bigger effective nuclear charge felt by core electrons, which makes them very compact
This particular kind of basis set saves computer time by representing both s and p orbitals by gaussian primitives with the same exponents, but using a different set of coefficients d for each usage. The first column of coefficients is for making s orbitals, and the second column for making p orbitals
The program knows it has to generate one s type orbital and three p type orbitals for each of these sp entries
Nitrogen looks similar to carbon, except that the exponents are bigger because the effective nuclear charge is higher

Polarisation basis function

Besides having two (or more) valence shell basis functions for each natural valence shell orbital, to allow the tails of the AOs to be varied to account for the different electronic properties of an element in different molecular environments, you usually need at least one extra basis function to allow the atom's electron cloud to be polarised off-centre from the nucleus
This is called a polarisation function
It is represented by a star after the name of the basis set, e.g. 3-21G*
A polarisation function is one orbital type higher than the highest natural valence orbital, e.g. in general a p type gaussian is added to a H basis set, a d type gaussian to a C basis set, etc.

The more sophisticated basis sets do that

Polarisation functions make little difference to simple basis sets for small atoms like H or C, in which valence shell electrons are not very polarisable, so 3-21G* has polarisation functions only for second row elements such as P or S (or heavier), when d type gaussians are added
You must use a basis set with polarisation functions to model P-O, P-N or S-O p bonding, otherwise you will predict bonds which are too long and weak, and N will come out pyramidal instead of planar, which it normally is when bonded to P or S

Higher ab initio methods

There are three kinds of error in using the Hartree Fock method with practicable-sized basis sets

We would get a lower absolute energy, and better relative energies, if we used a bigger basis set

As basis set size goes up, the energy converges to a theoretical limit
This unreachable limit is the 'Hartree Fock Limit'

When we look at single electrons in an average field of the rest, we take no account of time dependence

The positions of individual electrons at particular instants are correlated
This error in the HF method is called 'Correlation Energy'

We have taken no account of relativity

Electrons move faster near to heavy nuclei, so their masses change
Important for heavier transition metals or heavy main group atoms, e.g. Sn or I
There should be a relativistic correction

Altogether

_True

_{HF practical}

_{basis set error}

_correlation

_relativistic

A basis set error always has to be tolerated: the question is whether to spend calculation time on a bigger basis set, or whether to spend it on reducing the other errors by using a higher method. This kind of question is discussed in Hehre's 'Practical Strategies for Electronic Structure Calculations'
Some of the relativistic error can be removed by using Effective Core Potential (ECP) basis sets for heavy atoms

Core basis functions are left out of the basis set and the effect of the core electrons is represented by parameterised electrostatic potential functions instead
The potentials can include a relativistic correction
Leaving out basis functions saves SCF calculation time, but calculating forces for a geometry optimisation takes longer. These efficiencies balance out for medium weight elements
ECP basis sets are quicker only for the bromine period and below
ECP cannot be used where core electron properties are required, e.g. for NMR shieldings

What to do about correlation energy is the crunch problem

It is particularly important for loosely bound molecules, like transition states
The traditional next method above HF is MP2

MP2 calculates a correlation correction after a HF calculation, but takes much longer
Because it is slow, the MP2 method is only practicable for small molecules
MP2 is often used for single point calculations to get more accurate energies, after geometries have been found at the HF level

Since 1996, Density Functional (DF) methods are being used instead of MP2

DF methods calculate correlation, working directly with electron density instead of with MOs
A DF method replaces HF, rather than being an additional step, as is MP2
DF methods often contain a certain number of preoptimised parameters: these are not changed by the ordinary user
The most popular DF method for organic molecules at present is B3LYP
Since 2000, we are finding that MPW1PW91 is better for inorganic molecules
These (and other) DF methods are built into Gaussian and some PC packages, and are called up by name
DF methods are quicker than MP2, but may be less accurate
For moderate sized molecules, they are always slower than HF, but for large molecules they should be faster, if used with the same sized basis set
DF does not always produce better results than HF, though it usually takes longer: properties such as NMR shieldings may be predicted more poorly, especially for lighter elements
DF methods are usually better than HF for calculating small energy differences
Some journal referees think HF methods are old-fashioned, and require authors to at least try DF methods