5.80 Small-Molecule Spectroscopy and Dynamics

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1 MIT OpeCourseWare Small-Molecule Spectroscopy ad Dyamics Fall 2008 For iformatio about citig these materials or our Terms of Use, visit:

2 Lecture # 33 Supplemet Based o a lecture writte by Professor Patric H. Vaccaro. Outlie (i) true Eigestates: A log, hard climb; (ii) the total molecular Hamiltoia ad its Schrödiger Equatio; (iii) the electroic Schrödiger Equatio; (iv) trasformatio of the molecular Schrödiger Equatio; (v) the Adiabatic Approximatio; (vi) Adiabatic correctios; (vii) No-Adiabatic correctios; (viii) the trasitio momet of the à 1 A 2 X 1 A 1 absorptio i H 2 CO: a vibroic couplig model. Image removed due to copyright restrictios. Figure 1: Various routes to approach the exact o-adiabatic wavefuctio. From What Does the Term Vibroic Couplig Mea by T. Azumi ad K. Matsuzai, Photochemistry ad Photobiology 25, (1977). 1

3 Supplemet Time-Idepedet Schrödiger Equatio for a Molecular System H total (r, Q)Ψ t (r, Q) = E t Ψ t (r, Q) where H total (r, Q) = T e (r) + T N (Q) + U(r, Q) + V (Q) r represets electroic coordiates Q represets mass-weighted uclear coordiates describig displacemets from a referece cofiguratio Q 0 T e (r) ħ2 2 2m e r 2 i i T N (Q) ħ2 2 2 Q 2 represets the electroic ietic eergy represets the uclear ietic eergy U(r, Q) represets the Coulombic potetial eergy V (Q) represets the potetial eergy of the uclei PROBLEM: Hamiltoia does ot permit separatio of variables. Therefore, exact solutio is ot possible. Cosider oly the terms depedig o the electroic coordiates (i.e. the so-called Electroic Hamiltoia) H elec (r, Q) = T e (r) + U(r, Q) = T e (r) + U(R, Q 0 ) + ΔU(r, Q) = H elec (r, Q 0 ) + ΔU(r, Q) where Note that: U(r, Q) = U(r, Q 0 ) + ΔU(r, Q) H elec (r, Q 0 ) = T e (r) + U(r, Q 0 ). U(r, Q) = U(r, Q 0 ) + [ U(r, Q) ] Q Q 0 1 [ 2 U(r, Q) ] + Q Q m Q Q m m 0

4 Supplemet Page 3 sets: Cosequetly: [ ] [ ] U(r, Q) 1 2 U(r, Q) ΔU(r, Q) Q + Q Q m +... Q 0 2 Q Q m 0 Defie two types of Electroic Schrödiger Equatios (i) The Dyamical equatio for H elec (r, Q) (ii) The static equatio for H elec (r, Q 0 ),m {the Bor represetatio} H elec (r, Q)ψ i (r, Q) = ǫ i (Q)ψ i (r, Q) [T e (r) + U(r, Q)]ψ i (r, Q) = ǫ i (Q)ψ i (r, Q) dyamical electroic wavefuctios: ψ i (r, Q) Bor Space {the Loguet-Higgis represetatio} H elec (r, Q 0 )ψ 0 i (r, Q 0 ) = ǫ 0 i (Q 0 )ψ 0 i (r, Q 0 ) [T e (r) + U(r, Q 0 )]ψ 0 i (r, Q 0 ) = ǫ 0 i (Q 0 )ψ 0 i (r, Q 0 ) static electroic wavefuctios: ψ 0 i (r, Q 0 ) Loguet-Higgis Space The Eigestates of the Total Hamiltoia ca ow be expaded i either of these two electroic basis (i) The dyamical or Bor Represetatio: (ii) The static or Loguet-Higgis Represetatio: Ψ ψ (r, Q)χ D t (r, Q) = (Q) Ψ ψ 0 (r, Q 0 )χ S t (r, Q) = (Q). Note that ad ψ (r, Q) = l ψl 0 (r, Q 0 )λ l (Q) Ψ = ψ (r, Q)χ D t (r, Q) (Q) [ ] = ψl 0 (r, Q 0 )λ l (Q) χ D (Q) l = ψ 0 (r, Q 0 )χ S (Q). l l lt

5 Supplemet Page 4 where χ S (Q) = λl (Q)χ D (Q) Now recall the Schrödiger Equatio for the total Hamiltoia (i) the dyamical or Bor Represetatio: lt H total (r, Q)Ψ t (r, Q) = E t Ψ t (r, Q) [T e (r) + U(r, Q) + T N (Q) + V (Q) E t ]Ψ t (r, Q) = 0 [H elec (r, Q) + T N (Q) + V (Q) E t ] ψ (r, Q)χ D (Q) = 0 substitute the dyamical electroic Schrödiger Equatio ad the explicit expressio for T N (Q): [ {ǫ (Q) + V (Q) E t } ψ (r, Q)χ D (Q) ħ2 { 2 ψ (r, Q) χ D 2 χ D (Q) ψ (r, Q) χ D (Q) }] (Q) + ψ (r, Q) + 2 = 0 2 Q 2 Q 2 Q Q Multiply from the left by ψ j (r, Q) ad itegrate over the electroic coordiates realizig that: ψ j (r, Q) ψ (r, Q) = δ j. Oe thus obtais a set of coupled differetial equatios: {T N (Q) + V (Q) + ǫ j (Q) + ψ j (r, Q) T N (Q) ψ j (r, Q) E t } χ D jt(q) + { } ψ j (r, Q) T N (Q) ψ (r, Q) ħ 2 ψ j (r, Q) ψ (r, Q) χ D (Q) = 0 Q Q =j Problem still ot solvable. (ii) The static or Loguet-Higgis Represetatio: H total (r, Q)Ψ t (r, Q) = E t Ψ t (r, Q) [T e (r) + U(r, Q) + T N (Q) + V (Q) E t ]Ψ t (r, Q) = 0 [H elec (r, Q 0 ) + ΔU(r, Q) + T N (Q) + V (Q) E t ] ψ 0 (r, Q 0 )χ S (Q) = 0. By similar maipulatios, realizig that 0 ψj (r, Q 0 ) ψ 0 (r, Q 0 ) = δ g 2 ψ 0 (r, Q 0 ) ψ0 (r, Q 0 ) = ψ 0 (r, Q 0 ) j j ψ0 (r, Q 0 ) Q Q 2 = 0 for all j,

6 Supplemet Page 5 oe obtais Adiabatic Approximatios { TN (Q) + V (Q) + ǫ 0 j (Q 0 ) + ψ 0 j (r, Q 0 ) ΔU(r, Q) ψ 0 j (r, Q 0 ) } E t χ S (Q) + ψ 0 (r, Q 0 ) ΔU(r, Q) ψ 0 (r, Q 0 ) χ (Q) = 0 jt j j Defiitio: Adiabatic refers to ay vibroic approximatio scheme i which the wavefuctio is factorized i the form: Ψ AD (r, Q) = ψ(r, X)χ AD (Q) Oe must distiguish betwee several adiabatic schemes: Ay adiabatic scheme is valid oly if the effective potetial surface is well separated from all other potetial surfaces. (i.e. cocept of a potetial surface for the uclear motio has meaig oly if the adiabatic separatio is a sufficietly good approximatio for the descriptio of the molecular state uder cosideratio) Three Commoly Ecoutered Adiabatic schemes I. The Bor-Huag (BH) Adiabatic Approximatio I the set of dyamical differetial equatios, elimiate the couplig terms betwee by assumig Thus, the decoupled equatios become Xjt D (Q) ad X D (Q) where j ψ j (r, Q) T N (Q) ψ (r, Q) = 0 for = j ψ j (r, Q) ψ (r, Q) = 0 for all, j. Q effective potetial {}}{ [T N (Q) + V (Q) + ǫ j (Q) + ψ j ( r, Q) T N (Q) ψ j (r, Q) ]χ BH (Q) = E BH χ BH (Q) jt jt jt which implies Ψ BH jt (r, Q) = ψ j (r, Q)χ BH jt (Q). II. The Bor-Oppeheimer (BO) Adiabatic Approximatio I the set of dyamical differetial equatios, elimiate the couplig terms betwee χ D jt(q) ad χ D (Q) where j

7 Supplemet Page 6 by assumig ψ j (r, Q) T N (Q) ψ (r, Q) = 0 for all j, ψ j (r, Q) ψ (r, Q) = 0 for all j,. Q Thus, the decoupled equatios become effective potetial {}}{ {T V (Q) + ǫ j (Q) }χ BO E BO χ BO N (Q) + jt (Q) = jt jt (Q) which implies Ψ BO jt (r, Q) = ψ j (r, Q)χ BO jt (Q). III. The Crude Adiabatic (CA) Approximatio I the set of static differetial equatios, elimiate the couplig terms betwee χ S jt(q) ad χ S (Q) where j by assumig 0 ψj (r, Q 0 ) ΔU(r, Q) ψ 0 (r, Q 0 ) = 0 for = j. Thus, the decoupled equatios become [ E CA χ CA TN (Q) + V (Q) + ǫ 0 (Q) + ψ 0 (r, Q 0 ) ΔU(r, Q) ψ 0 (r, Q 0 ) ] j j j χ CA jt (Q) = jt jt (Q) which implies Ψ CA (r, Q) = ψ 0 (r, Q 0 )χ CA (Q). jt j jt

8 Supplemet Page 7 Crude Bor-Oppeheimer Bor-Huag Adiabatic Approximatio Adiabatic Approximatio Adiabatic Approximatio Adiabatic Wavefuctio Ψ CA jt (r, Q) =ψj 0 (r, Q)χ CA jt (Q) Ψ BO jt (r, Q) =ψ j (r, Q)χ BO jt (Q) Ψ BH jt (r, Q) =ψ j (r, Q)χ BH jt (Q) Electroic Equatio [T e (r)+u(r, Q 0 )]ψ j 0 (r, Q 0 ) = ɛ j 0 (Q)ψ j 0 (r, Q 0 ) [T e (r)+u(r, Q)]ψ j (r, Q) = ɛ j (Q)ψ j (r, Q) [T e (r)+u(r, Q)]ψ j (r, Q) = ɛ j (Q)ψ j (r, Q) Vibratioal Equatio [T N (Q)+V (Q)+ɛ 0 j (Q) + ψ j 0 (r, Q 0 ) ΔU(r, Q) ψ j 0 (r, Q 0 ) ] χ CA (Q) =E CA χ CA (Q) jt jt jt [T N (Q)+V (Q)+ɛ j (Q)] χ BO (Q) =E BO χ BO (Q) jt jt jt [T N (Q)+V (Q)+ɛ j (Q) + ψ j (r, Q 0 ) T N (Q) ψ j (r, Q)] χ BH (Q) =E BH χ BH (Q) jt jt jt Approximatios Utilized 0 ψj (r, Q 0 ) ΔU(r, Q) ψ 0 (r, Q 0 ) =0for = j ψ j (r, Q) T N (Q) ψ (r, Q) =0 ad ψ j (r, Q) ψ (r, Q) =0 QN ψ j (r, Q) T N (Q) ψ (r, Q) =0for = j ad ψ j (r, Q) ψ (r, Q) =0 QN

9 Supplemet Page 8 Example of Correctios withi the Adiabatic Approximatio Improvemet from the Crude Adiabatic (CA) Approximatio to the Bor-Oppeheimer (BO) Approximatio (Herzberg-Teller vibroic couplig) ψ 0 (r, Q 0 ) ψ(r, Q) The differece i the electroic Hamiltoias comes from the term ΔU(r, Q) where: By perturbatio theory where thus ΔU(r, Q) [ U(r, Q) ] Q + 1 [ 2 U(r, Q) ] Q Q m +... Q 0 2 Q Q m 0,m ψ i (r, Q) ψ 0 i (r, Q 0 ) + A ji (Q)ψ 0 j (r, Q 0 ) j i 0 ψj (r, Q 0 ) ΔU(r, Q) ψ 0 i (r, Q 0 ) A ji (Q) = ; ǫ 0 i (Q 0) ǫ 0 j (Q 0) Ψ BO ir (r, Q) = ψ i (r, Q)χ BO ir (Q) ] [ψ i 0 (r, Q 0) + A χ BO ji (Q)ψ 0 (r, Q j 0 ) ir (Q). Correctios of Adiabatic Schemes to No-Adiabatic Schemes Goal: To express the total o-adiabatic wavefuctios i terms of adiabatic wavefuctios via o-degeerate perturbatio theory: Ψ ir (r, Q) = (r, Q) + c,ir Ψ AD (r, Q) Ψ AD ir ir where { Ψ AD (r, Q) H (r, Q) Ψ AD (r, Q) } c,ir = E AD E AD. ir The perturbatio operator represets the breadow of the adiabatic approximatio: H (r, Q) = H total (r, Q) H AD (r, Q) = H total (r, Q) Ψ AD (r, Q) } E AD { Ψ AD (r, Q) This leads to Bor-Huag (BH) Couplig ad Bor-Oppeheimer (BO) Couplig. The Trasitio Momet of the à 1 A 2 X 1 A 1 Absorptio Trasitio i Formaldehyde The trasitio momet betwee adiabatic wavefuctios is give by { } Mjt;ir AD = Ψ AD jt (r, Q) O(r) Ψ AD ir (r, Q) ( ) χ AD χ AD = jt (Q) ψ j (r, Q) O(r) ψ i (r, Q) ir (Q). ir

10 Supplemet Page 9 To proceed, eed to ow Q-depedece of electroic itegral. Let us apply this, with the Bor-Oppeheimer, Adiabatic represetatio; to the à 1 A 2 X 1 A 1 trasitio of formaldehyde. Lowest Siglet Electroic States i H 2 CO Eergy (ev) State Desigatio State Number 0 1 A A 2 (, π ) B 2 (, σ ) A 1 (π, π ) B 1 (σ, π ) 4 Assume that the electroic eigefuctio of the groud state ca be expressed i terms of a o-mixed crude adiabatic fuctio: Ψ BO 0t (r, Q) = ψ 0 (r, Q)χ BO 0t (Q) ψ 0 0 (r, Q 0 )χ CA (Q). 0t Perform a Herzberg Teller expasio of the wavefuctio for the first excited siglet state: Ψ BO 1r (r, Q) = ψ 1 (r, Q)χ BO 1r (Q) ψ 1 (r, Q) ψ 0 1 (r, Q 0 ) + A j1 (Q)ψ 0 j (r, Q 0 ) j>1 where ψ 0 (r, Q 0 ) ΔU(r, Q) ψ 1 0 (r, Q 0 ) ad Thus Now A j1 (Q) = A j1 (Q) = = j ǫ 0 1 (Q 0) ǫ 0 j (Q 0) ΔU(r, Q) [ U(r, Q) ] Q. Q 0 ] 0 (r, Q 0 ) U(r,Q) ψ1 [ 0 Q (r, Q 0 ) 0 ǫ 0 1 (Q 0) ǫ 0 j (Q 0) ψ j γj 1Q. ψ 1 (r, Q) ψ 1 (r, Q 0 ) + γ j 1Q ψ j (r, Q 0 ) j>1 ad the Bor-Oppeheimer Adiabatic wavefuctio for the first excited siglet state becomes: Q Ψ BO 1r (r, Q) = ψ 1 (r, Q)χ BO 1r (Q) γ ψ 0 ψ 0 χ CA 1 (r, Q 0 ) + j1 Q j (r, Q 0) 1r (Q). j>1

11 Supplemet Page 10 The trasitio momet ow becomes: { } M BO 0t;1r = Ψ BO 0t (r, Q) O(r) Ψ BO 1r (r, Q) = χ CA (Q) ψ 0 0 (r, Q 0 ) O(r) ψ 0 1 (r, Q 0 ) + γ ψ 0 (r, Q 0 ) j1 Q χ CA (Q) j 1r. Simplificatio yields: 0t j>1 (χ M BO = ψ 0 0 (r, Q 0 ) 1 (r, Q CA O(r) ψ 0 0) (Q) χ CA (Q) ) 0t;1r 0t 1r (χ + γj 1 ψ 0 0 (r, Q 0 ) O(r) ψj 0 CA (r, Q 0 ) 0t (Q) Q 1r (Q) ) χ CA. j>1 Now cosider the coefficiets γ j 1 : [ ] ψ 0 U(r,Q) j (r, Q ψ1 0) 0 Q (r, Q 0 ) 0 γ j 1 =. ǫ 0 1 (Q 0) ǫ 0 j (Q 0) Sice the Hamiltoia must be ivariet uder all symmetry operatios, [ U(r, Q) ] must trasform as Q. Q 0 Give that ψ 0 1 (r, Q 0 ) trasforms as A 2, it is easy to fid the appropriate combiatios of Q ad ψ 0 j (r, Q 0 ) such that γ j 1 does ot vaish via symmetry. Three o-zero coefficiets are obtaied: γ 4, γ 5, γ Thus the trasitio momet for H 2 CO ca be writte, more explicitly, as (χ M BO = ψ 0 0 (r, Q 0 ) 1 (r, Q CA O(r) ψ 0 0) (Q) χ CA (Q) ) 0t;1r 0t 1r (χ + γ 4 ψ 0 CA (Q) Q 4 χ CA (Q) ) 21 0(r, Q 0 ) O( 0 r) ψ2 (r, Q 0 ) 0t 1r (χ + γ ψ 5 0 CA 0 (r, Q 0 ) O(r) ψ4(r, 0 Q 0 ) (Q) Q 5 χ CA (Q) ) 41 0t 1r (χ + γ ψ (r, Q 0 ) 4 (r, Q CA 41 O(r) ψ 0 0) 0t (Q) Q 6 χ CA 1r (Q)) = 1 (χ CA A 1 (r, Q 0 ) O(r) 1 A (Q) χ CA (Q) ) 2 (r, Q 0 ) 0t 1r + γ 4 1 (χ CA A 1 (r, Q 0 ) O(r) 1 B (Q) Q 4 χ CA (Q) ) 21 2 (r, Q 0 ) 0t 1r + γ 5 1 (χ CA A 1 (r, Q 0 ) O(r) 1 B (Q) Q 5 χ CA (Q) ) 1 (r, Q 0 ) 41 0t 1r + γ 6 1 (χ A 1 (r, Q 0 ) O( r) 1 CA B 1 (r, Q 0 ) (Q) Q 6 χ CA (Q) ) 41 0t 1r. Note that: µ z = A 1 µ x = B 1 µ y = B 2 m z = A 2 m x = B 2 m y = B 1