Introduction to Computational Quantum …

1 Introduction to Computational QuantumChemistry: TheoryDr Andrew GilbertRm 118, Craig Building, RSC3108 Course Lectures 2007 IntroductionHartree Fock TheoryConfiguration InteractionLectures1 IntroductionBackgroundThe wave equationComputing chemistry2 Hartree Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory3 Configuration InteractionThe correlation energyConfiguration expansion of the wavefunctionIntroductionHartree Fock TheoryConfiguration InteractionLectures4 Correlated MethodsConfiguration interactionCoupled-cluster theoryPerturbation theoryComputational Cost5 Basis SetsBasis functionsAdditional types of functions6 Density Functional TheoryDensity functionalsThe Hohenberg Kohn theoremsDFT modelsIntroductionHartree Fock TheoryConfiguration InteractionLectures7 Model ChemistriesModel chemistriesIntroductionHartree Fock TheoryConfiguration

2 InteractionLecture1 IntroductionBackgroundThe wave equationComputing chemistry2 Hartree Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory3 Configuration InteractionThe correlation energyConfiguration expansion of the wavefunctionIntroductionHartree Fock TheoryConfiguration InteractionBackgroundComputational chemistryComputational Chemistrycan be described as chemistryperformed using computers rather than covers a broad range of topics including:CheminformaticsStatistical mechanicsMolecular mechanicsSemi-empirical methodsAb initioquantum chemistryAll these methods, except the last, rely on empiricalinformation (parameters, energy levelsetc.).In this course we will focus on the last of these Fock TheoryConfiguration InteractionBackgroundAb initio Quantum chemistryAb initiomeans from the beginning or from firstprinciples.

3 Ab initioquantum chemistry distinguishes itself from othercomputational methods in that it is based solely onestablished laws of nature: Quantum mechanicsOver the last two decades powerful molecular modellingtools have been developed which are capable of accuratelypredicting structures, energetics, reactivities and otherproperties of developments have come about largely due to:The dramatic increase in computer design of efficient Quantum chemical Fock TheoryConfiguration InteractionBackgroundNobel recognitionThe 1998 Nobel Prize inChemistry was awarded toWalter Kohn for hisdevelopment of the densityfunctional theory andJohn Pople for his developmentof Computational methods inquantum chemistry .IntroductionHartree Fock TheoryConfiguration InteractionBackgroundAdvantagesCalculati ons areeasy to perform, whereas experimentsare often are becomingless costly, whereasexperiments are becoming more can beperformed on any system, even thosethat don t exist, whereas many experiments are limited torelatively stable aresafe, whereas many experiments have anintrinsic danger associated with Fock TheoryConfiguration InteractionBackgroundDisadvantagesCalcul ations aretoo easy to perform, many black-boxprograms are available to the can bevery expensivein terms of the amountof time can be performed on any system,even thosethat don t exist!

4 Computational chemistry isnot a replacementfor experimentalstudies, but plays an important role in enabling chemists to:Explainand rationalise known chemistryExplorenew or unknown chemistryIntroductionHartree Fock TheoryConfiguration InteractionThe wave equationTheoretical modelThe theoretical foundation for Computational chemistry isthe time-independentSchr odinger wave equation: H =E is thewavefunction. It is a function of the positions of allthe fundamental particles (electrons and nuclei) in thesystem. His theHamiltonianoperator. It is the operator associatedwith the observable thetotal energyof the system. It is a scalar (number).The wave equation is a postulate of Quantum Fock TheoryConfiguration InteractionThe wave equationThe HamiltonianThe Hamiltonian, H, is anoperator. It contains all the termsthat contribute to the energy of a system: H= T+ V Tis thekinetic energyoperator: T= Te+ Tn Te= 12 i 2i Tn= 12MA A 2A 2is theLaplaciangiven by: 2= 2 x2+ 2 y2+ 2 z2 IntroductionHartree Fock TheoryConfiguration InteractionThe wave equationThe Hamiltonian Vis the potential energy operator: V= Vnn+ Vne+ Vee Vnnis thenuclear-nuclearrepulsion term: Vnn= A<BZAZBRAB Vneis thenuclear-electronattraction term: Vne= iAZARiA Veeis theelectron-electronrepulsion term.

5 Vee= i<j1rijIntroductionHartree Fock TheoryConfiguration InteractionThe wave equationAtomic unitsAll Quantum chemical calculations use a special system of unitswhich, while not part of the SI, are very natural and greatlysimplify expressions for various length unit is thebohr(a0= 10 11m)The mass unit is theelectron mass(me= 10 31kg)The charge unit is theelectron charge(e= 10 19C)The energy unit is thehartree(Eh= 10 18J)For example, the energy of the H atom is hartree. In morefamiliar units this is 1,313 kJ/molIntroductionHartree Fock TheoryConfiguration InteractionThe wave equationThe hydrogen atomWe will use the nucleus as the centre of our Hamiltonian is then given by: H= Tn+ Te+ Vnn+ Vne+ Vee= 12 2r 1rAnd the wavefunction is simply a function ofr: (r)IntroductionHartree Fock TheoryConfiguration InteractionThe wave equationThe Born-Oppenheimer approximationNuclei are much heavier than electrons (the mass of aproton 2000 times that of an electron) and thereforetravel much more assume the electrons can reactinstantaneouslyto anymotion of the nuclei (think of a fly around a rhinoceros).

6 This assumption allows us tofactorisethe wave equation: (R,r)= n(R) e(r;R)where the ; notation indicates a parametric energy surfaceis a direct consequence ofthe BO Fock TheoryConfiguration InteractionComputing chemistryThe chemical connectionSo far we have focused mainly on obtaining thetotalenergyof our chemical properties can be obtained fromderivativesof the energy with respect to someexternal parameterExamples of external parameters include:Geometric parameters (bond lengths, anglesetc.)External electric field (for example from a solvent or othermolecule in the system)External magnetic field (NMR experiments)1stand 2ndderivatives are commonly available and derivatives are required for some properties, butare expensive (and difficult!) to derivatives must be computed Fock TheoryConfiguration InteractionComputing chemistryComputable propertiesMany molecular properties can be computed, these includeBond energies and reaction energiesStructures of ground-, excited- and transition-statesAtomic charges and electrostatic potentialsVibrational frequencies (IR and Raman)Transition energies and intensities for UV and IR spectraNMR chemical shiftsDipole moments, polarisabilities and hyperpolarisabilitiesReaction pathways and mechanismsIntroductionHartree Fock TheoryConfiguration InteractionComputing chemistryStatement of the problemThe SWE is a second-order linear differential solutions exist for only a small number of systems:The rigid rotorThe harmonic oscillatorA particle in a boxThe hydrogenic ions (H, He+, Li2+.)

7 Approximations must be used:Hartree-Fock theorya wavefunction-based approach thatrelies on the mean-field Functional Theorywhose methods obtain theenergy from the electron density rather than the (morecomplicated) usually Fock TheoryConfiguration InteractionComputing chemistryClassification of methodsPost-HFDFTHF EcSCFH ybrid DFTI ntroductionHartree Fock TheoryConfiguration InteractionLecture1 IntroductionBackgroundThe wave equationComputing chemistry2 Hartree Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory3 Configuration InteractionThe correlation energyConfiguration expansion of the wavefunctionIntroductionHartree Fock TheoryConfiguration InteractionPreviously on 3108 The Schr odinger wave equation H =E TheHamiltonianis made up of energy terms: H= Tn+ Te+ Vnn+ Vne+ VeeTheBorn-Oppenheimerapproximation clamps the nucleiand implies Tn=0 and Vnnis units(bohr, hartree,etc.)

8 Chemical properties are obtained fromderivativesof theenergy with respect toexternal parameters, ( of the energy nuclear coordinates can beused to find transition structures and equilibriumgeometries.).IntroductionHart ree Fock TheoryConfiguration InteractionClassification of methodsPost-HFDFTHF EcSCFH ybrid DFTI ntroductionHartree Fock TheoryConfiguration InteractionThe molecular orbital approximationHartree-Fock theoryHF theory is the simplest wavefunction-based forms the foundation for more elaborate electronicstructure is synonymous with theMolecular Orbital relies on the following approximations:TheBorn-Oppenheimerapprox imationTheindependent electronapproximationThelinear combination of atomic orbitalsapproximationIt does not model thecorrelation energy, by Fock TheoryConfiguration InteractionThe molecular orbital approximationHartree-Fock theoryConsider the H2molecule:The total wavefunction involves 4 coordinates: = (R1,R2,r1,r2)We invoke the Born-Oppenheimer approximation: = n(R1,R2) e(r1,r2).

9 How do we model e(r1,r2)?IntroductionHartree Fock TheoryConfiguration InteractionThe molecular orbital approximationThe Hartree wavefunctionWe assume the wavefunction can be written as aHartreeproduct: (r1,r2)= 1(r1) 2(r2)The individual one-electron wavefunctions, iare calledmolecular form of the wavefunction does not allow forinstantaneous interactionsof the , the electrons feel theaveragedfield of all theother electrons in the Hartree form of the wavefunction is is sometimescalled theindependent electron Fock TheoryConfiguration InteractionThe molecular orbital approximationThe Pauli principleOne of the postulates of Quantum mechanics is that thetotal wavefunction must beantisymmetricwith respect totheinterchange of electron coordinatesThePauli Principleis a consequence of Hartree wavefunction is not antisymmetric.

10 (r2,r1)= 1(r2) 2(r1)6= (r1,r2)We can make the wavefunction antisymmetric by adding allsigned permutations: (r1,r2)=1 2[ 1(r1) 2(r2) 1(r2) 2(r1)]IntroductionHartree Fock TheoryConfiguration InteractionThe molecular orbital approximationThe Hartree-Fock wavefunctionThe antisymmetrized wavefunction is called theHartree-Fock can be written as aSlater determinant: =1 N! 1(r1) 2(r1) N(r1) 1(r2) 2(r2) N(r2).. 1(rN) 2(rN) N(rN) This ensures the electrons areindistinguishableand aretherefore associated with every orbital!A Slater determinant is often written as| 1, 2,.. N IntroductionHartree Fock TheoryConfiguration InteractionThe molecular orbital approximationThe LCAO approximationThe HF wavefunction is an antisymmetric wavefunctionwritten in terms of the one-electron do the MOs look like?