Skip to main content
UploadLanguage (EN)Support
BusinessMobileSocial MediaMarketingTechnologyArt & PhotosCareerDesignEducationPresentations & Public SpeakingGovernment & NonprofitHealthcareInternetLawLeadership & ManagementAutomotiveEngineeringSoftwareRecruiting & HRRetailSalesServicesScienceSmall Business & EntrepreneurshipFoodEnvironmentEconomy & FinanceData & AnalyticsInvestor RelationsSportsSpiritualNews & PoliticsTravelSelf ImprovementReal EstateEntertainment & HumorHealth & MedicineDevices & HardwareLifestyle
Uploaded bybaoilleach
PPTX, PDF20,335 views

Quantum Chemistry

This document provides an overview of quantum mechanics (QM) calculation methods. It discusses molecular mechanics, wavefunction methods, electron density methods, including correlation, Hartree-Fock theory, semi-empirical methods, density functional theory, and their relative speed and accuracy. Key aspects that can be calculated using these methods are also listed, such as molecular orbitals, electron density, geometry, energies, spectroscopic properties, and more. Basis sets and handling open-shell systems in calculations are also covered.

Education◦
Downloaded 422 times
Quantum Mechanics CalculationsNoel M. O’BoyleApr 2010Postgrad course on Comp Chem
Overview of QM methodsMolecular mechanicsQuantum mechanics(wavefunction)Quantum mechanics(electron density)Including correlationHF (“ab initio”)Semi-empiricalDFTSpeed/AccuracyForcefields
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?...
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
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
ĤΨ=EΨ (the Schrödinger equation)
ρ = |Ψ|2
“electron density” or “square or the wavefunction”
A probability density (3D)
Integrate over a certain volume to find the probability of finding an electron in that volume
It follows that ∫|Ψ|2dr = N (number of electrons)Credit: OtherDrK (Flickr)
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
Expressing a vector in terms of a basis(3.5, 1.5)vjiv = 3.5i + 1.5j
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
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
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)
Radial decay of GTO vs STOImage Credit: Essentials of Computational Chemistry, Chris Cramer, Wiley, 2ndEdn.