Transcription of Lecture 2 Hamiltonian operators for molecules
1 1 Lecture 2 Hamiltonian operators for SkylarisCHEM6085: Density Functional TheoryCHEM6085 Density Functional TheoryThe (time-independent) Schr dinger equation is an eigenvalueequationoperator for property AeigenfunctioneigenvalueEnergy operator ( Hamiltonian )wavefunctionEnergy eigenvalue2 CHEM6085 Density Functional TheoryConstructing operators in Quantum MechanicsClassical quantityQuantum operatorQuantum mechanical operators are the same as their corresponding classical mechanical quantitiespositionPotential energy ( energy of attraction of an electron by an atomic nucleus)With one exception!The momentum operator is completely different:3 CHEM6085 Density Functional TheoryBuilding HamiltoniansTheHamiltonianoperator(=tota lenergyoperator)isasumoftwooperators:the kineticenergyoperatorandthepotentialener gyoperatorKineticenergyrequirestakingint oaccountthemomentumoperatorThe potential energy operator is straightforward4 The Hamiltonian becomes:CHEM6085 Density Functional TheoryTheaboveexampleprovidestheexpectat ionvalue(averagevalue)ofthepositionalong thex-axis.
2 Experimentalmeasurementsofphysicalproper tiesareaveragevalues Quantummechanicspostulatesthatwecancalcu latetheresultofanysuchmeasurementby averaging theappropriateoperatorandthewavefunction asfollows:5 CHEM6085 Density Functional TheoryExpectation values of operators6 Force between two charges: Coulomb s LawrEnergy of two chargesCHEM6085 Density Functional Theory7 Coulomb potential energy (or operator )Examples:In one dimensionIn 2 dimensionsIn 3 dimensions Difficult to visualise (would require a 4-dimensional plot!) We live in a 3-dimensional world so this is the potential we useCHEM6085 Density Functional TheoryHamiltonian for hydrogen atomrr-RRelectronnucleusnuclear kinetic energyelectronic kinetic energyelectron-nucleus attractionO (origin of coordinates)8 CHEM6085 Density Functional TheoryAtomic unitsWe will use Atomic Units as they simplify quantum chemistry expressions. : 9 QuantityAtomicUnitValuein SI Energy 2/mea0 (Hartree) x x x x 10-31kgIn SI units:In atomic units:CHEM6085 Density Functional Theory10 Born-Oppenheimer approximationFor a molecule, the wavefunctionis a function of the coordinates of all the electrons and all the nuclei: The Born-Oppenheimer approximation is based on the fact that nuclei have much larger masses than the electrons To a good approximation, one can solve the Schr dinger equation only for the electrons and assume the nuclei are frozen We will use this approximation from now on As a result, our wavefunctionswill be functions only of electronic coordinates:CHEM6085 Density Functional Theory11 Average energyThe energy operator is the Hamiltonian .
3 For a molecular system, under the approximation, this isWe can write this also asFunctionalsof CHEM6085 Density Functional TheoryHamiltonian for helium atomkinetic energy of nucleuskinetic energy of electron 1kinetic energy of electron 2attraction of electron 1 by nucleusattraction of electron 2 by nucleusrepulsion between electrons 1 and 2 r2r1-RRelectron 1nucleuselectron 2r1r2-Rr1-r2O (origin of coordinates) Density Functional Theory13 Sums Extremely useful shorthand notation Allows to condense summations with many terms (5, 10, 100, many millions, infinite!) into one compact expression Single sum example:Double sum example:CHEM6085 Density Functional TheoryHamiltonian operator for water O fortheoxygen(atomicnumberZO=8)nucleus, H1 and H2 (atomicnumbersZH1=1andZH2=1) Quiteacomplicatedexpression!Hamiltonians formoleculesbecomeintractable Fortunately, ,muchmorecompactexpressionsthatapplytoan ymolecule,irrespectiveofitssizeKinetic energy of OKinetic energy of H1 Kinetic energy of H2 Kinetic energy of electron iElectron attraction to OElectron attraction to H1 Electron attraction to H2 Electron-electron repulsionnucleus-nucleus AssumeZ1=ZOZ2=ZH1Z3= Density Functional Theory15 Example: Nuclear attraction potential for one O and two HH2 OOH + HO + H2 How does the Hamiltonian operator differ between these examples?
4 Can you suggest how you may model the reaction OH+H H2 OCHEM6085 Density Functional Theory16 Homework1)Usesummationsymbolstogeneralis etheexampleofthewatermoleculefromthelect urestoanexpressionfortheelectronicHamilt onianoperatorofanymolecule, )Writedownanexpressionfortheexpectationv alueofeachofthetermsoftheaboveHamiltonia n( ,electron-electronrepulsionenergy,etc.)3 )Assumethatchargedparticlessuchaselectro nsandnuclei,insteadofhavingelectrostatic interactionsthatobeyCoulomb sLaw(andincludedintheHamiltonianintermso fCoulombpotentialenergyexpressions)havei nteractionsthatobeyHooke sLaw,whichgivesthepotentialenergyoftwopa rticlesconnectedbyaspringas k(x-x0)2,wherex0istheequilibriumdistance (springisfullyrelaxed) Density Functional Theory17 Name :Surname:5-minute quiz1)Find out what is wrong in the following statement and correct it: The electronic molecular Hamiltonian operator does not contain any terms with the coordinates of the nuclei 2)Write down an expression for the electronic molecular Hamiltonian operator of the HeH+molecule3) Are there any parts of the electronic molecular Hamiltonian operator that will ( by leading to very high energy) prevent two electrons from being at the same point in space and how?
5 (work with a specific example, as the one from the previous question)Date :CHEM6085 Density Functional TheoryPlease note that in all of the questions below and for the rest of the course, whenever we mention electronicmolecular Hamiltonian we assume the molecular Hamiltonian operator after the application of the BO approximatio
