Electronic Structure for Excited States (multiconfigurational methods) Spiridoula Matsika

Transcription

1 Electronc Structure for Excted States (multconfguratonal methods) Sprdoula Matska

2 Excted Electronc States Theoretcal treatment of excted states s needed for: UV/Vs electronc spectroscopy Photochemstry Photophyscs Electronc structure methods for excted states are more challengng and not at the same stage of advancement as ground state methods Need balanced treatment of more than one states that may be very dfferent n character The problem becomes even more complcated when movng away from the ground state equlbrum geometry

3 Excted states confguratons Ground state Sngly excted conf. Doubly excted conf.

4 Confguratons can be expressed as Slater determnants n terms of molecular orbtals. Snce n the nonrelatvstc case the egenfunctons of the Hamltoan are smultaneous egenfunctons of the spn operator t s useful to use confguraton state functons (CSFs)- spn adapted lnear combnatons of Slater determnants, whch are egenfunctosn of S Snglet CSF Trplet CSF

5 Excted states can have very dfferent character and ths makes ther balanced descrpton even more dffcult. For example excted states can be: Valence states Rydberg states Charge transfer states

6 Rydberg states Hghly excted states where the electron s excted to a dffuse hydrogen-lke orbtal Low lyng Rydberg states may be close to valence states Dffuse bass functons are needed for a proper treatment of Rydberg states, otherwse the states are shfted to much hgher energes Dffuse orbtals need to be ncluded n the actve space or n a restrcted actve space (RAS)

7 Potental Energy Surfaces and Excted States Energy Vert. Abs. Vert. Ems. Adabatc exc.en. For absorpton spectra one s nterested n the Franck Condon (FC) regon. In the smplest case a sngle pont calculaton s used to gve vertcal exctaton energes coordnate

8 Energy When one s nterested n the photochemstry and photophyscs of molecular systems the PES has to be explored not only n the FC regon but also along dstorted geometres. Mnma, transton states, and concal ntersectons need to be found (gradents for excted states are needed) TS CI Reacton coordnate

9 Electronc structure methods for excted states Sngle reference methods ΔSCF, Δ(DFT), Δ(CI), TDDFT EOM-CCSD Mult-reference methods MCSCF CASPT2, MR-MP2 MRCI

10 In the smplest case one can calculate excted state energes as energy dfferences of sngle-reference calculatons. ΔE=E(e.s)-E(g.s.). Ths can be done: For states of dfferent symmetry For states of dfferent multplcty Possbly for states that occupy orbtals of dfferent symmetry

11 Confguraton Interacton Intally n any electronc structure calculaton one solves the HF equatons and obtans MOs and a ground state soluton that does not nclude correlaton The smplest way to nclude dynamcal correlaton and mprove the HF soluton s to use confguraton nteracton. The wavefuncton s constructed as a lnear combnaton of many Slater determnants or confguraton state functons (CSF). CI s a sngle reference method but forms the bass for the multreference methods Ψ CI = N CSF m=1 c m Ψ m

12 Dfferent orbtal spaces n a CI calculaton Frozen Vrtual orbtals Exctatons from occuped to vrtual orbtals Vrtual orbtals Occuped orbtals Frozen orbtals

13 CSFs are created by dstrbutng the electrons n the molecular orbtals obtaned from the HF soluton. Ψ CI = c 0 HF + occ vrt. c r r Ψ + c rs rs j Ψ j +... r occ. vrt. < j r<s The varatonal prncple s used for solvng the Schrodnger equaton H ˆ Ψ = EΨ H ˆ = T ˆ e + V ˆ ee + V ˆ en + V ˆ NN = 1 2 2m e = h ˆ + Z α j> α 1 r j rα + ˆ V NN j> r j α β >α Z α Z β R αβ E var = Ψ * ĤΨdτ * Ψ Ψdτ 0

14 For a lnear tral functon the varatonal prncple leads to solvng the secular equaton for the CI coeffcents or dagonalzng the H matrx Matrx formulaton (N CSF x N CSF CSF ) H 11 E H H 1N H 21 H 22 E... H 2N H 1N H 2N... H NN E = 0 H mn = Ψ m H Ψ n The Hamltonan can be computed and then dagonalzed. Snce the matrces are very bg usually a drect dagonalzaton s used that does not requre storng the whole matrx.

15 Number of snglet CSFs for H2O wth 6-31G(d) bass (m,n): dstrbute m electrons n n orbtals N = m 2 n!(n +1)!! m 2 +1! n m 2! n m 2 +1! Exctaton level # CSFs Today expansons wth bllon of CSFs can be solved

16 Ψ HF 0 0 Ψ HF sngles doubles trples E HF Ψ HF H Ψ j rs CIS 0 Ψ r H Ψ j s dense Ψ k t dense H Ψ j rs sparse Ψ qp rst lm H Ψ jk sparse CISD Ψ HF H Ψ j rs dense Ψ k t H Ψ j rs sparse 0 Ψ qp rst lm H Ψ jk sparse Ψ HF H Ψ r = 0 Brlloun s thm CIS wll gve excted states but wll leave the HF ground state unchanged

17 τ φ φ φ φ φ φ τ ψ ψ ε ε τ ψ ψ τ ψ ψ τ ψ ψ τ ψ ψ d r st pq where b a j b ja d H a a E d H s theorem Brlloun d H E d H d H H E t s t s q p b j a a HF a a a HF j j (1)] (2) (2) (1) [ 1 (2) (1) ) ( ), ( ) ( ) ( ) ' ( = = + + = = = = = t Ht Condon-Slater rules are used to Condon-Slater rules are used to evaluate matrx elements evaluate matrx elements

18 For sngly excted states CIS HF qualty of excted states Overestmates the exctaton energes Can be combned wth sememprcal methods (ZINDO/S)

19 Sze extensvty/consstency Sze-Extensvty: For N ndependent systems the energy scales lnearly E(N)=N*E(1): Sze-Consstency: dssocaton E(A+B) E(A) + E(B) Example: consder H 2 and then two non-nteractng H 2 molecules Ψ CID (H 2 ) = (1 c) 2 Ψ HF + c 2 22 Ψ 1 1 Ψ CID (2H 2 ) = ((1 c) 2 Ψ HF + c 2 22 Ψ 1 1 )((1 c) 2 Ψ HF + c 2 22 Ψ 1 1 ) = (1 c) 4 Ψ HF Ψ HF + c Ψ 1 1 Ψ (1 c) 2 c 2 22 Ψ HF Ψ 1 1 Correctons: Davdson correcton: E cor = (E-E 0 )(1-c 02 )

20 Sngle reference vs. multreference RHF for H 2 : The Hartree-Fock wavefuncton for H 2 s Ψ = 1 2 σ(1)α(1) σ(1)β(1) σ(2)α(2) σ(2)β(2) = σ u * = 1 2 = 1 2 (σ(1)α(1)σ(2)β(2) σ(1)β(1)σ(2)α(2)) = (σ(1)σ(2))(α(1)β(2) β(1)α(2)) σ g The MO s a lnear combnatons of AOs: σ=1s A + 1s B (spn s gnored) Ψ = 1 2 (1s A (1) +1s B (1))(1s A (2) +1s B (2)) = = 1 2 (1s A (1)1s B (2) +1s B (1)1s A (2) +1s A (1)1s A (2) +1s B (1)1s B (2)) Covalent H H Ionc onc H + H + - H - Ths wavefuncton s correct at the mnmum but dssocates nto 50% H + H - and 50% H H

21 CI for H 2 When two confguratons are mxed: σ(1)α(1) σ(1)β(1) Ψ CI = c 1 σ(2)α(2) σ(2)β(2) + c σ * (1)α(1) σ *(1)β(1) 2 σ *(2)α(2) σ *(2)β(2) = Ignore spn c 1 (σ(1)σ(2) + c 2 σ *(1)σ *(2)) Ψ = c 1 (1s A (1) +1s B (1))(1s A (2) +1s B (2)) + c 2 (1s A (1) 1s B (1))(1s A (2) 1s B (2)) = = c 1 (1s A (1)1s A (2) +1s B (1)1s B (2) +1s A (1)1s A (2) +1s B (1)1s B (2)) +c 2 (1s A (1)1s A (2) +1s B (1)1s B (2) 1s A (1)1s A (2) 1s B (1)1s B (2)) σ u * σ g Ψ 1 Ψ 2 The coeffcents c 1 and c 2 determne how the conf. Are mxed n order to get the rght character as the molecule dssocates. At the dssocaton lmt the orbtals σ, σ* are degenerate and c 1 =c 2 Ionc. H + H -

22 Multreference methods Multreference methods are needed for: Near-degeneracy Bond breakng Excted states radcals Nondynamcal correlaton MCSCF Dynamcal correlaton Varatonal: MRCI Based on perturbaton theory: CASPT2, MS-CASPT2, MRMP2 Not wdely spread yet: MRCC, MRCI/DFT

23 Multconfguraton Self- Consstent Feld Theory (MCSCF) CSF: spn adapted lnear combnaton of Slater determnants Ψ MCSCF = CSFs n=1 c n CSF Two optmzatons have to be performed Optmze the MO coeffcents optmze the expanson coeffcents of the CSFs

24 Choose the actve orbtals Depends on the problem and the questons beng asked For a π system all π orbtals should be ncluded f possble If bond breakng nclude bondng, antbondng Check occupaton numbers of orbtals (between ) Tral and error Choose the confguratons obtaned usng these orbtals Complete actve space (CASSCF or CAS): allow all possble confguratons (Full CI wthn the actve space) (m,n): dstrbute m electrons n n orbtals N = m 2 n!(n +1)!! m 2 +1! n m 2! n m 2 +1!.e. (14,12) generates 169,883 CSFs Restrcted actve space (RASSCF): allow n-tuple exctatons from a subset of orbtals (RAS) and only n-tuple exctatons nto an auxllary orgnally empty set (AUX) Generalzed valence bond (GVB)

25 Orbtal spaces n an MCSCF calculatons CASSCF RASSCF Vrtual orbtals CAS Double occuped orbtals Vrtual orbtals AUX:n exctatons n permtted CAS RAS: n exctatons out permtted DOCC orbtals

26 The most mportant queston n multreference methods: Choosng the actve space The choce of the actve space determnes the accuracy of the method. It requres some knowledge of the system and careful testng. For small systems all valence orbtals can be ncluded n the actve space For conjugated systems all π orbtals f possble should be ncluded n the actve space. For heteroatomc rngs the lone pars should be ncluded also. What cas should be chosen for the followng systems? N 2 Ozone Allyl radcal Benzene Uracl

27 State averaged MCSCF All states of nterest must be ncluded n the average When the potental energy surface s calculated, all states of nterest across the coordnate space must be ncluded n the average State-averaged MOs descrbe a partcular state poorer than state-specfc MOs optmzed for that state State-average s needed n order to calculate all states wth smlar accuracy usng a common set of orbtals. Ths s the only choce for near degenerate states, avoded crossngs, concal ntersectons. Provdes common set of orbtals for transton dpoles and oscllator strengths

28 Multreference confguraton nteracton Includes dynamcal correlaton beyond the MCSCF Orbtals from an MCSCF (state-averaged) are used for the subsequent MRCI The states must be descrbed qualtatvely correct at the MCSCF level. For example, f 4 states are of nterest but the 4th state at the MRCI level s the 5th state at the MCSCF level a 5-state average MCSCF s needed

29 MRCI A reference space s needed smlar to the actve space at MCSCF References are created wthn that space Sngle and double exctatons usng each one of these references as a startng pont Ψ MRCI = c Ψ Frozen Vrtual orbtals Vrtual orbtals CAS DOCC orbtals Frozen orbtals

30 CASPT2 Second order perturbaton theory s used to nclude dynamc correlaton Has been used wdely for medum sze conjugated organc systems Errors for exctaton energes ~0.3 ev There are no analytc gradents avalable so t s dffcult to be used for geometry optmzatons and dynamcs

31 Ab nto package MCSCF MRCI COLUMBUS Analytc gradents for MRCI Graphcal Untary Group Approach (GUGA)

32 Integral SCF MCSCF CI Control nput Colnp (nput scrpt)