《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）11-correlation

LasttimeExcitedstates1. Absorption and emission model2. Methods for excited states3. Excited state geometry4.Solvatochromism

Last time 2 Excited states 1. Absorption and emission model 2. Methods for excited states 3. Excited state geometry 4. Solvatochromism

Contents1. Electron Correlation2. Configuration Interaction3. Size extensivity4. Perturbation methods5. Coupled-cluster methods6. Ab initio solution of the Schrodinger equation7. DFT8. Model chemistriesCramer,Chp7Jensen, Chp 4ForesmanandFrisch,Chp63

Contents 3 1. Electron Correlation 2. Configuration Interaction 3. Size extensivity 4. Perturbation methods 5. Coupled-cluster methods 6. Ab initio solution of the Schrödinger equation 7. DFT 8. Model chemistries Cramer, Chp 7 Jensen, Chp 4 Foresman and Frisch, Chp 6

Electronicenergycomponents. Total electronic energy can be partitioned;E = Et + Ene +E, + Ex +EcEt= kinetic energy of the electrons;ENE=Coulomb attraction energy between electrons and nuclei;E,=CoulombrepulsionenergybetweenelectronsEx= Exchange energy, a correction for the self-repulsions ofelectrons;Ec= Correlation energy between the motions of electrons withdifferent spins;Er, EnE, & E, are largest contributors to E;: Ex> Ec;C

Electronic energy components • Total electronic energy can be partitioned; E = ET + ENE +EJ + EX +EC ET = kinetic energy of the electrons; ENE = Coulomb attraction energy between electrons and nuclei; EJ = Coulomb repulsion energy between electrons EX = Exchange energy, a correction for the self-repulsions of electrons; EC = Correlation energy between the motions of electrons with different spins; • ET , ENE, & EJ are largest contributors to E; • EX > EC ; 4

Electron correlationenergyIn the Hartree-Fock approximation, each electron sees theaverage density of all of the other electrons;Two electronscannot be in the same placeat the sametime;Electrons must move two avoid each other, i.e. theirmotion must be correlated;For a given basis set, the difference between the exactenergy and the Hartree-Fock energy is the correlationenergy; ca 2o kcal/mol correlation energy per electron pair;Types of electron correlation;- Dynamical- Non-dynamical5

Electron correlation energy • In the Hartree-Fock approximation, each electron sees the average density of all of the other electrons; • Two electrons cannot be in the same place at the same time; • Electrons must move two avoid each other, i.e. their motion must be correlated; • For a given basis set, the difference between the exact energy and the Hartree-Fock energy is the correlation energy; • ca 20 kcal/mol correlation energy per electron pair; • Types of electron correlation; – Dynamical – Non-dynamical 5

Static correlationW_=N_(1sA-1sg) (1,2)- (1)师 (2)1=(1s (1)-1s())*(1s (2)-sp(2))(αβ-βα)-ls(1)ls.(2)-1sg(1)ls(2)+Is()s(2)+ 1s.()s (215A1SB（1,2)-(1(2)不(1sA(l)+ls(1)*(Is (2)+Is(2))(αβ-βα)= Is (1)ls (2)+ Isg (1)Isa (2)+ Is (1)Is (2)+ Is (1)sg (2)8ONV+=N,(1SA+1Sp)Covalent configuration lonic configuration6

Static correlation 6 Covalent configuration Ionic configuration

DynamiccorrelationStaticcorrelationisimportantformoleculeswheretheground state is well described only with more than one(nearly-)degenerate determinant;- Hartree-Fock single Slater determinant;MCSCF (and CASSCF)treat the single determinant deficienciesof Hartree-Fock, but they don't do a very good job of handlingdynamic correlation that comes from instantaneous repulsionof electrons at all separations;- In the Hartree-Fock approximation, each electron sees theaverage density of all of the other electrons (mean-fieldapproximation lacks electron dynamics);

Dynamic correlation • Static correlation is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant; – Hartree-Fock single Slater determinant; • MCSCF (and CASSCF) treat the single determinant deficiencies of Hartree-Fock, but they don’t do a very good job of handling dynamic correlation that comes from instantaneous repulsion of electrons at all separations; – In the Hartree-Fock approximation, each electron sees the average density of all of the other electrons (mean-field approximation lacks electron dynamics); 7

GoalsforcorrelatedmethodsWell defined- Applicable to all molecules with no ad-hoc choices;- Can be used to construct model chemistries;Efficient-Notrestrictedtoverysmallsystems;Variational- Upper limittotheexactenergy;Size extensive- E(A+B) = E(A) + E(B)- Needed for properdescription of thermochemistry;.Hierarchy of costvs.accuracy; So that calculations can be systematically improved;8

Goals for correlated methods • Well defined – Applicable to all molecules with no ad-hoc choices; – Can be used to construct model chemistries; • Efficient – Not restricted to very small systems; • Variational – Upper limit to the exact energy; • Size extensive – E(A+B) = E(A) + E(B) – Needed for proper description of thermochemistry; • Hierarchy of cost vs. accuracy; – So that calculations can be systematically improved; 8

Configurationinteractionaeyalo..=o+Zt+Z+ZijkijabijkabciaY。 =|Φ ...Φ,/referencedeterminant(Hartree- Fock wavefunction)Ya -=l d .Φ--d.di+Φn Isingly excited determinant(excite occupied orbital Φ, to unoccupied orbital Φ.)Yab =l d..-d.i...j-pdj*.., I doubly excited determinant(d, →Φa, Φ, →)etc.Ifcarriedouttoall possibleexcitationstoall possibleorbitals,calledafullconfigurationiteration(fullci)model;Thiswavefunctionwouldbeexactwithinagivenbasis;9

Configuration interaction etc. ( , ) | | doubly excited determinant (excite occupied orbital to unoccupied orbital ) | | singly excited determinant (Hartree - Fock wavefunction) | | referencedeterminant 1 1 1 1 1 1 1 1 0 1 0 i a j b i a i j b j n ab i j i a i a i n a i n abc ijk ijkabc abc ijk ab i j ijab ab i j a i i a a i t t t                      → →  =  =  =  =  +  +  +  + − + − + − +           9 If carried out to all possible excitations to all possible orbitals, called a full configuration iteration (full CI) model; This wavefunction would be exact within a given basis;

ConfigurationinteractionDetermine Cl coefficients using the variational principle+-o+y++Z.iaijabijkabcminimize E = [y'HYdt / [?"Ydt with respect to tCiS-include all single excitations- Useful for excited states, but does not change orimprove the ground state;CisD-includeall singleanddoubleexcitations-Mostusefulforcorrelatingthegroundstate;-O2V2 determinants (O=number of occ.orb., V=number of unocc. orb.);CISDT-singles,doublesandtriples-Limitedto small molecules,caO3v3determinants;Full Cl-all possibleexcitations((O+V)!/o!v!)?determinants;Exactforagivenbasisset;limitedtoca.14electronsin14orbitals;10

• Determine CI coefficients using the variational principle • CIS – include all single excitations – Useful for excited states, but does not change or improve the ground state; • CISD – include all single and double excitations – Most useful for correlating the ground state; – O2V 2 determinants (O=number of occ. orb., V=number of unocc. orb.); • CISDT – singles, doubles and triples – Limited to small molecules, ca O3V 3 determinants; • Full CI – all possible excitations – ((O+V)!/O!V!)2 determinants; – Exact for a given basis set; – limited to ca. 14 electrons in 14 orbitals; Configuration interaction E d d t t t t abc ijk ijkabc abc ijk ab i j ijab ab i j a i i a a i / with respect to minimize ˆ * * 0 =        =  +  +  +  +      H  10

Configuration interactionHt=EtVery largeeigenvalue problem, can be solvediteratively; Only linear terms in the Cl coefficients;Upper bound to the exact energy (variational)Applicable to excited states;Gradients simpler than for non-variationalmethods;11

Configuration interaction • Very large eigenvalue problem, can be solved iteratively; • Only linear terms in the CI coefficients; • Upper bound to the exact energy (variational); • Applicable to excited states; • Gradients simpler than for non-variational methods; Ht = E t 11