che-30042 inorganic, physical & solid state chemistry advanced quantum chemistry: lecture 4 rob...

CHE-30042 Inorganic, Physical & Solid State ChemistryAdvanced Quantum Chemistry: lecture 4

Rob JacksonLJ1.16, 01782 733042

[email protected]/robjteaching

@robajackson

che-30042 Advanced QC lecture 4 2

Lecture 4 contents

1. Representing atomic orbitals – basis sets2. Ab initio calculations3. Molecular geometry and geometry optimisation4. Improving SCF calculations – including

Configuration Interaction (CI)5. Alternatives to MO calculations – Density

Functional Theory (DFT)

Representing atomic orbitals

• If we are going to use atomic orbitals in calculations, we need to know how to represent them.

• The usual expression for an atomic orbital (mentioned in che-20028) is:

= exp (-r) Y (, )– Where (r, , ) are coordinates, and (e.g.) an s-orbital

only depends on ‘r’.• This is used for Slater-type orbitals (STOs)


STOs and basis sets – (i)

• In molecular orbital calculations, atomic orbitals are represented by basis sets.– A minimal basis set has one function per

orbital, but accuracy can be improved by having more than one function (‘double zeta’ basis sets, slide 6).

• STOs are functions that have been fitted to numerical wavefunctions, and have a form similar to the one given in slide 3.


STOs and basis sets – (ii)

• In the Gaussian workshop (lecture 6) you will see the effect of using different basis sets on calculation accuracy.– For example, STO-3G means that 3 Gaussian functions

are fitted to the numerical wavefunction, easier mathematically than fitting to the function on slide 3.

– So an ‘s’ function is written as: = exp (-r) = c11 + c22 + c33

Where 1= (21/) exp(-1r2) (2, 3 are similarly defined).


Split valence basis sets – (i)

• Since it is normally the valence orbitals that are involved in bonding, they can be represented by more than one basis function, each of which can in turn be made up of a number of Gaussian functions.– A 3-21G basis set has 3 Gaussian functions per core

orbital, and 2 basis functions per valence orbital, of which 1 is made up of 2 Gaussian functions and 1 is made up of 1 Gaussian function. This is an example of a double zeta basis set.


Split valence basis sets – (ii)

• Other examples of split valence basis sets are:– 4-21G– 4-31G– 6-31G– 6-311G (triple zeta, 3 basis functions per valence

orbital)• In addition, polarisation functions (e.g. d orbitals)

can be added, as in 6-31G*


Plane wave basis sets and pseudopotentials

• Plane wave basis sets use plane wave functions to represent orbitals. By combining enough plane wave functions it is possible to get a good representation of an orbital.

• They are easier to handle mathematically so are better suited to calculations on crystals, where we have to deal with repeating unit cells.

• Pseudopotentials are used to represent the core electrons which are not involved in bonding, so the plane waves only represent the valence electrons.


Ab initio calculations

• The Hückel method, described in lecture 3, avoided the problem of calculating the integrals Hlk and Slk by setting them equal to numerical constants.

• In ‘ab initio’ calculations (meaning ‘from the beginning’), no such drastic assumptions are made, and the calculations are carried out in full.

• The procedure used will now be explained.


Ab initio SCF calculations – (i)

i. Start with a trial molecular orbital as a sum of weighted atomic orbitals (see lec. 2 slide 10).

– e.g. = cAA + cBB (use estimates for cA, cB)

ii. Set up the secular determinant (lec. 2 slides 14-15)

iii. Solve, having calculated all the integrals Hlk & Slk.

iv. Determine orbital energy, and substitute into the secular equations to get coefficients cA, cB.

v. This gives a new orbital, .


Ab initio SCF calculations – (ii)

vi. Set up the secular determinant again (Hlk and Slk have to be recalculated because has changed).

vii. Calculate new values of the energy, and then the weighting coefficients.

viii.Repeat until no change (self consistency).ix. The final result gives the energy and orbital

coefficients.


Molecular geometry and geometry optimisation

• If we are carrying out an ab initio calculation on a molecule, we will need to specify the distance between the atoms, and the molecular geometry.

• The bond distances etc. can be varied, and the energy calculated.

• It is then possible to find the structure corresponding to the lowest energy – geometry optimisation.

• This will be done in the workshop for H2.che-30042 Advanced QC lecture 4 12

Configuration Interaction (CI) calculations

• In lecture 1 when we introduced the SCF method, it was mentioned that the method doesn’t treat electron correlation (repulsion) very well.

• The CI method is one way of correcting this, by including all possible electron configurations in the calculation (i.e. not just the ground state). So for H2, the antibonding orbital could be included in the calculation.– However the method is computationally expensive!


Density Functional Theory (DFT)

• DFT is widely used in place of molecular orbital calculations, especially for solids.

• The main difference with MO methods is that it works with overall electron density rather than by representing each orbital by a function.

• This makes the calculations quicker, but there are still problems in dealing with electron exchange and correlation, and approximations have to be made here.


Lecture summary

• The representation of orbitals has been revised.• Basis sets have been introduced and explained.• The procedure for ab initio SCF calculations has

been explained.• Geometry optimisation has been mentioned – to

be looked at in the workshop.• Density Functional Theory has been introduced.


che-30042 inorganic, physical & solid state chemistry advanced quantum chemistry: lecture 4 rob...

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