《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）10-excitedStates

Lasttime1. Explicit solvation models2. Implicit solvation models2

Last time 2 1. Explicit solvation models 2. Implicit solvation models

ContentsExcitedstates1. Absorption and emission model2. Methods for excited states3. Excited state geometry4. SolvatochromismCramer,Chp14ForesmanandFrisch,Chp9

Contents 3 Excited states 1. Absorption and emission model 2. Methods for excited states 3. Excited state geometry 4. Solvatochromism Cramer, Chp 14 Foresman and Frisch, Chp 9

Ground stateand excited state configurations=...2=.·N/2double excitationground statesingly excited configurations

Ground state and excited state configurations ground state singly excited configurations double excitation 4

VerticalabsorptionandemissionElectronicExcitationofx-x-Excited StateexcitedstateGround StateA=absorptionF=fluorescenceEgroundsQuinineAbsorptionandEmissionSpectraWavelength(Nanometers)300350400600450500100StokesFigure444ShiftEmissionEadiabatic80Absorptionaao6040S+S,20S+S2S+SGeneric coordinatehttp://gaussian.com/freg/33.32.812.412.00Wavenumber(cmx10-3)Option,VibronicSpectra:Franck-Condon,Herzberg-TellerandFCHT

Vertical absorption and emission 5 http://gaussian.com/freq/ Option, Vibronic Spectra: Franck-Condon, Herzberg-Teller and FCHT

KoopmanstheoremOrbital energyofan occupied orbital is approximatelyequal to the minus of the ionization potential of thatorbital;IP of Φ, = -S;Can be derived from the Hartree-Fock energy expression,if one assumes that the orbitals do not relax afterionization;In a similar spirit, one can approximate the excitationenergy;E(a)-E(Y)= 8-8,E(Yab)-E(Y)=8. +&, -&, -8)6

Koopmans theorem • Orbital energy of an occupied orbital is approximately equal to the minus of the ionization potential of that orbital; IP of fi = -ei ; • Can be derived from the Hartree-Fock energy expression, if one assumes that the orbitals do not relax after ionization; • In a similar spirit, one can approximate the excitation energy; a b i j ab i j a i a i E E E E e e e e e e  −  = + − −  −  = − ( ) ( ) ( ) ( ) 0 0 6

SingletvstripletstatesPauli principlesaysthatthewavefunctionmustbe4(1,2)=-4(2,1);antisymmetric,Wavefunction iscomposed of spaceandspinparts;If the space part is symmetric, then the spin part must beantisymmetric;- Only one way to do this, singlet spin state· [α(1)β(2)-β(1)α(2)/21/2If the space part is antisymmetric, then the spin partmust be symmetric;- three ways to do this, triplet spin state: α(1)α(2)· [α(1)β(2)+β(1)α(2)/21/2·β(1)β(2)

Singlet vs triplet states • Pauli principle says that the wavefunction must be antisymmetric, (1,2)=-(2,1); • Wavefunction is composed of space and spin parts; • If the space part is symmetric, then the spin part must be antisymmetric; – Only one way to do this, singlet spin state • [a(1)b(2)-b(1)a(2)]/21/2 • If the space part is antisymmetric, then the spin part must be symmetric; – three ways to do this, triplet spin state • a(1)a(2) • [a(1)b(2)+b(1)a(2)]/21/2 • b(1)b(2) 7

SingletTriplet state[α(1)β(2)-β(1)α(2)/21/2[α(1)β(2)+β(1)α(2)]/21/2α(1)α(2)β(1)β(2)8

Singlet . Triplet state . [a(1)b(2)-b(1)a(2)]/21/2 a(1)a(2) [a(1)b(2)+b(1)a(2)]/21/2 b(1)b(2) - + 8

4SCF· Allow orbitals to relax, by doing a Hartree-Fockcalculation on the excited state as well as on the groundstate;.Only works if excited state different symmetry thanground state (otherwise the attempt at calculating theexcited state collapses to the ground state);.OKfor UHF calculationof the lowest triplet (since thenumber of alpha and beta spin electrons is different thanin the ground state);Possible (but tricky) for an excited singlet or triplet if theorbitals differ in symmetry;. In general, need to use configuration interaction;9

ΔSCF • Allow orbitals to relax, by doing a Hartree-Fock calculation on the excited state as well as on the ground state; • Only works if excited state different symmetry than ground state (otherwise the attempt at calculating the excited state collapses to the ground state); • OK for UHF calculation of the lowest triplet (since the number of alpha and beta spin electrons is different than in the ground state); • Possible (but tricky) for an excited singlet or triplet if the orbitals differ in symmetry; • In general, need to use configuration interaction; 9

Configurationinteractionraleyale..Y=Yo+Ety'+Etyg+ZijkijkabciajjabY。 =|Φ ...Φ, I referencedeterminant(Hartree- Fock wavefunction)Ya -=l d. d-d.di.o, I singly excited determinant(excite occupied orbital p, to unoccupied orbital Φ.)Yab =| ..-++*·-,-Φdj+ ·., I doubly excited determinant(d, →pa,d, →p)etc.10

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 ij i a i a i n a i n abc ijk ijkabc abc ijk ab ij ijab ab ij a i ia a i t t t f f f f f f f f f f f f f f f f f f f f f → →  =  =  =  =  +  +  +  + − + − + − +           10

ConfigurationinteractionDetermineCl coefficientsusingthevariational principletabcyabc+..Y=+Zt+Zyg+ Zt+Tjklijkiajjabijkabcminimize E = [y*HYdt / [y*Ydt with respect to t.Full Cl-all possible excitations; (O+V)!/!V!)?determinants; Exact ground state and excited states for a given basis set;- Limited to ca. 14 electrons in 14 orbitals;CiS - include all single excitations;-Simplestapproximationforexcitedstates;- ComparabletoHartree-Fockforthegroundstate;- Applicable to large system, similar in cost to a vibrational frequencycalculation;Cis(D) - include pertubative correction for double excitationsCorrectssomeofthedeficienciesofCiS;11

• Determine CI coefficients using the variational principle • Full CI – all possible excitations; – ((O+V)!/O!V!)2 determinants; – Exact ground state and excited states for a given basis set; – Limited to ca. 14 electrons in 14 orbitals; • CIS – include all single excitations; – Simplest approximation for excited states; – Comparable to Hartree-Fock for the ground state; – Applicable to large system, similar in cost to a vibrational frequency calculation; • CIS(D) – include pertubative correction for double excitations – Corrects some of the deficiencies of CIS; Configuration interaction / with respect to . minimize ˆ * * 0 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 =        =  +  +  +  +      H  11