Quantum Chemistry
- 1. Quantum Mechanics CalculationsNoel M. O’BoyleApr 2010Postgrad course on Comp Chem
- 2. Overview of QM methodsMolecular mechanicsQuantum mechanics(wavefunction)Quantum mechanics(electron density)Including correlationHF (“ab initio”)Semi-empiricalDFTSpeed/AccuracyForcefields
- 3. What can be calculated?Molecular orbitals and their energiesElectron densityMolecular geometryRelative energies of two moleculesNMR shiftsIR and Raman frequencies and normal modesElectronic transitions (UV-Vis absorption spectrum), associated changes in electron density, optical rotationConductivityIonisation potential, electron affinity, heat of formationTransition states, activation energyCharge distributionInteraction energy between two moleculesSolvation energypKaHow accurately can it be calculated?...
- 4. ReferencesEssentials of Computational Chemistry, Christopher CramerIntroduction to Computational Chemistry, Frank JensenMolecular Modelling: Principles and Applications, Andrew LeachComputational Organic Chemistry, Steven Bachrach(http://comporgchem.com/blog/)(coming soon) Molecular Modelling Basics, Jan Jensen (http://molecularmodelingbasics.blogspot.com/)Quantum Mechanics, Tim Clark, Section 7.4 in Cheminformatics– A Textbook, Ed. Gasteiger and Engel
- 5. The WavefunctionThe wavefunction completely describes the properties of a quantum mechanical (QM) systemΨ(r), PsiIt has a value at every point in 3D spaceBy applying various operators to the wavefunction, we can calculate properties of the systemThe Hamiltonian operator (Ĥ) gives the energy of the system
- 6. ĤΨ=EΨ (the Schrödinger equation)
- 7. ρ = |Ψ|2
- 8. “electron density” or “square or the wavefunction”
- 9. A probability density (3D)
- 10. Integrate over a certain volume to find the probability of finding an electron in that volume
- 11. It follows that ∫|Ψ|2dr = N (number of electrons)Credit: OtherDrK (Flickr)
- 12. Solving the Schrodinger equationBorn-Oppenheimer approximationSince electron motion is so rapid compared to nuclear motion, consider the nuclei as fixedThis allows us to simplify the HamilitonianVariational PrincipleThe true energy of a QM system (as given by the Hamilitonian operator) is always less than the energy found if the Hamilitonian is applied to an incorrect wavefunctionTo find the true wavefunction, make a reasonable guess and then keep altering it to minimise the energyHartree-Fock (HF) theoryHF theory neglects electron correlation in multi-electron systemsInstead, we imagine each electron interacting with a static field of all of the other electronsAccording to the variational principle, the lowest energy will can get with HF theory will always be greater than the true energy of the systemThe difference is the correlation energy
- 13. Expressing a vector in terms of a basis(3.5, 1.5)vjiv = 3.5i + 1.5j
- 14. Linear combination of atomic orbitals (LCAO)The LCAO approximation involves expressing (“expanding”) each molecular orbital (ψ) as a sum of “basis set functions” (φx) centered on each atomψφ2φ3φ1HCNLet’s use this parabola for our basis set functions, φx
- 15. Self-consistent field (SCF) procedureBased on the variational principle and the LCAO approach, a set of equations can be derived that allow the calculation of the molecular orbital coefficients (cx on previous slide)Roothaan-Hall equationsThe catch is that terms in the equations are weighed by elements of a density matrix PBut the elements of P can only be computed if molecular orbitals are knownBut finding the molecular orbitals requires solving the Roothaan-Hall equations...An iterative procedure is used to get around thisMake an initial guess of the values of cxUse these to calculate the elements of PSolve the Roothaan-Hall equations to give new values for cxUse these new values to calculate the elements of PIf the new P is not sufficiently similar to the old P, repeat until it convergesSCF not guaranteed to converge, espec. if initial guess is poor
- 16. Basis setsAny set of mathematical functions can be used as a basisHow many functions should we use? Which functions should we use?The larger (i.e. the more components in) the basis set...The better the wavefunction can be describedAnd the closer the energy converges towards the limit of that methodThe slower the calculation – N4 integrals (bottleneck)We would like to use as small a basis set as possible and still describe the wavefunction wellA good solution is to use functions that have shape similar to s, p, d and f orbitals and are centered on each of the atomsSlater-Type Orbitals (STOs)Radial decay follows e-rWe would like to be able to calculate all of the integrals efficientlyGaussian-Type Orbitals (GTOs) are similar to STOs but have a radial term following e-r^2More efficient to calculate in integrals but have the wrong shape so...Replace each STO with a sum of 3 Gaussian-Type Orbitals(GTOs)
- 17. Radial decay of GTO vs STOImage Credit: Essentials of Computational Chemistry, Chris Cramer, Wiley, 2ndEdn.
- 18. How a sum of three GTOs can approximate a STOImage Credit: Essentials of Computational Chemistry, Chris Cramer, Wiley, 2ndEdn.
- 19. STO-3G Basis SetA minimal basis set, i.e. it has one basis function per orbitalExample: for Li-H, there would be 6 basis functions in total1s on H & 1s, 2s, 2px, 2py and 2pz on the LiEach basis function is a fixed sum of 3 Gaussian functions whose coefficients are optimised to match a STOHence the nameA minimal basis set is not sufficient to describe the wavefunctionHowever, it may be useful to do a quick initial geometry optimisation
- 20. Pople’s split-valence basis setsCore orbitals are only weakly affected by binding, whereas valence orbitals can vary widelySo we should enable additional flexibility for representing valence orbitalsSplit-valence basis sets: 3-21G, 6-21G, 4-31G,6-31G, 6-311G“3-21G” implies that each core orbital is represented by single basis function (a sum of 3 GTOs as for STO-3G) but each valence orbital is represented by two basis functions (the first a sum of 2 GTOs, the other a single GTO)In general, molecular orbitals cannot be described just in terms of the atomic orbitals of the atomsE.g. A HF calculation for NH3 with an infinite basis set just consisting of s and p functions predicts that the planar geometry is a minimumPolarisation functions need to be added, corresponding to atomic orbitals of higher angular momentum (e.g. d, f, etc.)6-31G(d) (“6-31G*”), indicates that d orbitals are added to heavy atomsThis basis set is a sort of standard for general purpose calculations6-31G(3d2fg, 2pd) would indicate that that heavy atoms were polarised by 3 functions, 2 f, one g, while hydrogen atoms were polarised by 2 p and one d.Highest energy MOs of anions and highly excited electronic states tend to be very diffuse (tail off very slowly as the distance to the molecules increases)Add diffuse basis functions: 6-31+G(d), 6-311++G(3df,2pd)A single “+” indicates that heavy atoms have been augmented with an additional diffuse s and a set of diffuse p basis functions; another “+” indicates that hydrogen atoms have also been augmented
- 21. Handling open-shell systemsRestricted Hartree-Fock (RHF or just HF)Closed-shell systems, all electrons pairedTwo approaches to handle unpaired electronsRestricted Open-shell HF (ROHF)An approximation that reuses the RHF code but handle the unpaired electron using two paired ½ electronsFails to account for spin polarizationUnrestricted HF (UHF)The SCF is carried out separately for all electrons of one spinCorresponding α and β electrons will have different spatial distributionCalculations take twice as long
- 22. NotationLOT/BSLevel of Theory/Basis setWhere “Level of Theory” simply means the type of calculationE.g. HF/3-21G or UHF/6-31G(d)Compared to energies, geometry is much less sensitive to the theoretical levelSo high-level calculations are often carried out at geometries optimised at a lower level (faster)LOT2/BS2//LOT1/BS1E.g. HF/6-311+G(d)//HF/6-31G
- 23. Overview of QM methodsMolecular mechanicsQuantum mechanics(wavefunction)Quantum mechanics(electron density)Including correlationHF (“ab initio”)Semi-empiricalDFTSpeed/AccuracyForcefields