Lecture #3: Supplement

Lecture # 33 Supplement Based on a lecture written by Professor Patrick H. Vaccaro. Outline (i) “true” Eigenstates: A long, hard climb; (ii) the “total” molecular Hamiltonian and its Schrödinger Equation; (iii) the electronic Schrödinger Equation; (iv) transformation of the molecular Schrödinger Equation; (v) the Adiabatic Approximation; (vi) Adiabatic corrections; (vii) Non-Adiabatic corrections; e 1 A2 ← X e 1 A1 absorption in H2 CO: a vibronic coupling model. (viii) the transition moment of the A Image removed due to copyright restrictions. Figure 1: Various routes to approach the exact non-adiabatic wavefunction. From “What Does the Term ‘Vibronic Coupling’ Mean” by T. Azumi and K. Matsuzaki, Photochemistry and Photobiology 25, 315-326 (1977). Time-Independent Schrödinger Equation for a Molecular System Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q) where Htotal (r, Q) = Te (r) + TN (Q) + U (r, Q) + V (Q) “r” represents electronic coordinates “Q” represents mass-weighted nuclear coordinates describing displacements from a reference configuration “Q0 ” Te (r) ≈ TN (Q) ≈ −ℏ2 X ∂ 2 2me i ∂ri2 represents the electronic kinetic energy −ℏ2 X ∂ 2 2 n ∂Q2n represents the nuclear kinetic energy U (r, Q) represents the Coulombic potential energy V (Q) represents the potential energy of the nuclei PROBLEM: Hamiltonian does not permit separation of variables. Therefore, exact solution is not possible. Consider only the terms depending on the electronic coordinates (i.e. the so-called Electronic Hamilto­ nian) Helec (r, Q) = Te (r) + U (r, Q) = Te (r) + U (R, Q0 ) + ΔU (r, Q) = Helec (r, Q0 ) + ΔU (r, Q) where U (r, Q) = U (r, Q0 ) + ΔU (r, Q) Helec (r, Q0 ) = Te (r) + U (r, Q0 ). Note that: U (r, Q) = U (r, Q0 ) + X  ∂U (r, Q)  n ∂Qn Qn 0   1 X ∂ 2 U (r, Q) + Qn Qm + . . . 2 nm ∂Qn ∂Qm 0 Consequently: ΔU (r, Q) ≈ X  ∂U (r, Q)  n ∂Qn Qn + 0   1 X ∂ 2 U (r, Q) Qn Qm + . . . 2 n,m ∂Qn ∂Qm 0 Define two types of Electronic Schrödinger Equations (i) The Dynamical equation for Helec (r, Q) {the “Born” representation} Helec (r, Q)ψi (r, Q) = ǫi (Q)ψi (r, Q) [Te (r) + U (r, Q)]ψi (r, Q) = ǫi (Q)ψi (r, Q) dynamical electronic wavefunctions: ψi (r, Q) ⇒ Born Space (ii) The static equation for Helec (r, Q0 ) {the “Longuet-Higgins” representation} Helec (r, Q0 )ψi0 (r, Q0 ) = ǫ0i (Q0 )ψi0 (r, Q0 ) [Te (r) + U (r, Q0 )]ψi0 (r, Q0 ) = ǫ0i (Q0 )ψi0 (r, Q0 ) static electronic wavefunctions: ψi0 (r, Q0 ) ⇒ Longuet-Higgins Space The Eigenstates of the Total Hamiltonian can now be expanded in either of these two electronic basis sets: (i) The dynamical or Born Representation: Ψt (r, Q) = X ψk (r, Q)χD kt (Q) k (ii) The static or Longuet-Higgins Representation: X Ψt (r, Q) = ψk0 (r, Q0 )χSkt (Q). k Note that ψk (r, Q) = X ψℓ0 (r, Q0 )λℓk (Q) ℓ and Ψt (r, Q) = X ψk (r, Q)χD kt (Q) k = " X X k = X ℓ # ψℓ0 (r, Q0 )λℓk (Q) χD kt (Q) ℓ ψℓ0 (r, Q0 )χSℓt (Q). where χSℓt (Q) = X λℓk (Q)χD kt (Q) k Now recall the Schrödinger Equation for the total Hamiltonian (i) the dynamical or Born Representation: Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q) [Te (r) + U (r, Q) + TN (Q) + V (Q) − Et ]Ψt (r, Q) = 0 X [Helec (r, Q) + TN (Q) + V (Q) − Et ] ψk (r, Q)χD kt (Q) = 0 k substitute the dynamical electronic Schrödinger Equation and the explicit expression for TN (Q): " X {ǫk (Q) + V (Q) − Et } ψk (r, Q)χD kt (Q) k  # ∂ 2 χD ∂ψk (r, Q) ∂χD ℏ X ∂ 2 ψk (r, Q) D kt (Q) kt (Q) =0 χkt (Q) + ψk (r, Q) +2 − 2 n ∂Q2n ∂Q2n ∂Qn ∂Qn 2 Multiply from the left by ψj⋆ (r, Q) and integrate over the electronic coordinates realizing that: hψj (r, Q)|ψk (r, Q)i = δjk . One thus obtains a set of coupled differential equations: {TN (Q) + V (Q) + ǫj (Q) + hψj (r, Q)|TN (Q)|ψj (r, Q)i − Et } χD jt (Q) )   X X ∂ ∂ 2 + ψj (r, Q) ψk (r, Q) χD hψj (r, Q)|TN (Q)|ψk (r, Q)i − ℏ kt (Q) = 0 ∂Q ∂Q n n n k6=j ( Problem still not solvable. (ii) The static or Longuet-Higgins Representation: Htotal (r, Q)Ψt (r, Q) = Et Ψt (r, Q) [Te (r) + U (r, Q) + TN (Q) + V (Q) − Et ]Ψt (r, Q) = 0 X [Helec (r, Q0 ) + ΔU (r, Q) + TN (Q) + V (Q) − Et ] ψk0 (r, Q0 )χSkt (Q) = 0. k By similar manipulations, realizing that ψj0 (r, Q0 )|ψk0 (r, Q0 ) = δgk     ∂2 ∂ 0 0 0 0 ψj (r, Q0 ) ψ (r, Q0 ) ψ (r, Q0 ) = ψj (r, Q0 ) ∂Qn k ∂Q2n k =0 for all j, k one obtains  TN (Q) + V (Q) + ǫ0j (Q0 ) + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) − Et X ×χSjt (Q) + ψj0 (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) χkt (Q) = 0 k6=j Adiabatic Approximations Definition: Adiabatic refers to any vibronic approximation scheme in which the wavefunction is factorized in the form: ΨAD (r, Q) = ψ(r, X)χAD (Q) One must distinguish between several adiabatic schemes: Any adiabatic scheme is valid only if the “effective potential surface” is well separated from all other potential surfaces. (i.e. concept of a potential surface for the nuclear motion has meaning only if the adiabatic separation is a sufficiently good approximation for the description of the molecular state under consideration) Three Commonly Encountered Adiabatic schemes I. The Born-Huang (BH) Adiabatic Approximation In the set of dynamical differential equations, eliminate the coupling terms between D D Xjt (Q) and Xkt (Q) where k 6= j by assuming hψj (r, Q)|TN (Q)|ψk (r, Q)i = 0   ∂ ψk (r, Q) = 0 ψj (r, Q) ∂Qn for k = 6 j for all k, j. Thus, the decoupled equations become effective potential z }| { BH BH [TN (Q) + V (Q) + ǫj (Q) + hψj (r, Q)|TN (Q)|ψj (r, Q)i]χBH jt (Q) = Ejt χjt (Q) which implies BH ΨBH jt (r, Q) = ψj (r, Q)χjt (Q). II. The Born-Oppenheimer (BO) Adiabatic Approximation In the set of dynamical differential equations, eliminate the coupling terms between D χD jt (Q) and χkt (Q) where k 6= j by assuming hψj (r, Q)|TN (Q)|ψk (r, Q)i = 0   ∂ ψk (r, Q) = 0 ψj (r, Q) ∂Qn for all j, k for all j, k. Thus, the decoupled equations become effective potential z }| { BO BO {TN (Q) + V (Q) + ǫj (Q) }χBO jt (Q) = Ejt χjt (Q) which implies BO ΨBO jt (r, Q) = ψj (r, Q)χjt (Q). III. The Crude Adiabatic (CA) Approximation In the set of static differential equations, eliminate the coupling terms between χSjt (Q) and χSkt (Q) where j 6= k by assuming ψj0 (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) = 0 for k = 6 j. Thus, the decoupled equations become  TN (Q) + V (Q) + ǫ0j (Q) + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) which implies 0 CA ΨCA jt (r, Q) = ψj (r, Q0 )χjt (Q).  CA CA χCA jt (Q) = Ejt χjt (Q) Crude Adiabatic Approximation Born-Oppenheimer Adiabatic Approximation Born-Huang Adiabatic Approximation Adiabatic Wavefunction 0 CA ΨCA jt (r, Q) = ψj (r, Q)χjt (Q) BO ΨBO jt (r, Q) = ψj (r, Q)χjt (Q) BH ΨBH jt (r, Q) = ψj (r, Q)χjt (Q) Electronic Equation [Te (r) + U (r, Q0 )]ψj0 (r, Q0 ) = 0j (Q)ψj0 (r, Q0 ) [Te (r) + U (r, Q)]ψj (r, Q) = j (Q)ψj (r, Q) [Te (r) + U (r, Q)]ψj (r, Q) = j (Q)ψj (r, Q) Vibrational Equation  [TN (Q) + V (Q) + j (Q)  + ψj0 (r, Q0 )|ΔU (r, Q)|ψj0 (r, Q0 ) ] CA CA ×χCA jt (Q) = Ejt χjt (Q) [TN (Q) + V (Q) + j (Q)] BO BO ×χBO jt (Q) = Ejt χjt (Q) [TN (Q) + V (Q) + j (Q) + ψj (r, Q0 )|TN (Q)|ψj (r, Q)] BH BH ×χBH jt (Q) = Ejt χjt (Q) ψj (r, Q)|TN (Q)|ψk (r, Q) = 0   and     ψj (r, Q)  ∂Q∂ N  ψk (r, Q) = 0 ψj (r, Q)|TN (Q)|ψk (r, Q) = 0 for k = j   and     ψj (r, Q)  ∂Q∂ N  ψk (r, Q) = 0 Approximations Utilized 0  0  ψj (r, Q0 )|ΔU (r, Q)|ψk0 (r, Q0 ) = 0 for k = j Example of Corrections within the Adiabatic Approximation Improvement from the Crude Adiabatic (CA) Approximation to the Born-Oppenheimer (BO) Approximation (Herzberg-Teller vibronic coupling) ψ 0 (r, Q0 ) −→ ψ(r, Q) The difference in the electronic Hamiltonians comes from the term ΔU (r, Q) where:   X  ∂U (r, Q)  1 X ∂ 2 U (r, Q) ΔU (r, Q) ≈ Qn + Qn Qm + . . . ∂Qn 2 n,m ∂Qn ∂Qm 0 0 n By perturbation theory ψi (r, Q) ≈ ψi0 (r, Q0 ) + X Aji (Q)ψj0 (r, Q0 ) j6=i where Aji (Q) = ψj0 (r, Q0 )|ΔU (r, Q)|ψi0 (r, Q0 ) ; ǫ0i (Q0 ) − ǫ0j (Q0 ) thus BO ΨBO ir (r, Q) = ψi (r, Q)χir (Q) h i X ≈ ψi0 (r, Q0 ) + Aji (Q)ψj0 (r, Q0 ) χBO ir (Q). Corrections of Adiabatic Schemes to Non-Adiabatic Schemes Goal: To express the total non-adiabatic wavefunctions in terms of adiabatic wavefunctions via non-degenerate perturbation theory: Ψir (r, Q) = ΨAD ir (r, Q) + X ckt,ir ΨAD kt (r, Q) kt6=ir where  AD Ψkt (r, Q)|H′ (r, Q)|ΨAD ir (r, Q) ckt,ir = . AD AD Eir − Ekt The perturbation operator represents the breakdown of the adiabatic approximation: H′ (r, Q) = Htotal (r, Q) − HAD (r, Q) X  AD AD = Htotal (r, Q) − ΨAD Ψkt (r, Q) kt (r, Q) Ekt kt This leads to Born-Huang (BH) Coupling and Born-Oppenheimer (BO) Coupling. e 1 A2 ← X e 1A1 Absorption Transi­ The Transition Moment of the A tion in Formaldehyde The transition moment between adiabatic wavefunctions is given by n o AD AD b Mjt;ir = ΨAD jt (r, Q)|O(r)|Ψir (r, Q)  D E  AD b ψ (r, Q)| O(r)|ψ (r, Q) = χAD (Q) χ (Q) . j i jt ir To proceed, need to know Q-dependence of electronic integral. e 1 A2 ← X e 1 A1 transition Let us apply this, with the Born-Oppenheimer, Adiabatic representation; to the A of formaldehyde. Lowest Singlet Electronic States in H2 CO Energy (ev) State Designation State Number 1 0 3.50 7.08 7.97 9.45 A1 A2 (n, π ⋆ ) 1 B2 (n, σ ⋆ ) 1 A1 (π, π ⋆ ) 1 B1 (σn, π ⋆ ) 0 1 2 3 4 1 Assume that the electronic eigenfunction of the ground state can be expressed in terms of a non-mixed crude adiabatic function: BO ΨBO 0t (r, Q) = ψ0 (r, Q)χ0t (Q) ≈ ψ00 (r, Q0 )χCA 0t (Q). Perform a “Herzberg–Teller” expansion of the wavefunction for the first excited singlet state: BO ΨBO 1r (r, Q) = ψ1 (r, Q)χ1r (Q) X Aj1 (Q)ψj0 (r, Q0 ) ψ1 (r, Q) ≈ ψ10 (r, Q0 ) + j>1 where Aj1 (Q) = ψj0 (r, Q0 )|ΔU (r, Q)|ψ10 (r, Q0 ) ǫ01 (Q0 ) − ǫ0j (Q0 ) and ΔU (r, Q) ≈ X  ∂U (r, Q)  ∂Qn n Thus Aj1 (Q) = D h i E ∂U(r,Q) 0 ψ10 (r, Q0 ) X ψj (r, Q0 ) ∂Qn 0 ǫ01 (Q0 ) − ǫ0j (Q0 ) n = Qn . 0 X Qn γjn1 Qn . n Now ψ1 (r, Q) ≈ ψ1◦ (r, Q0 ) + XX γjn1 Qn ψj◦ (r, Q0 ) j>1 n and the Born-Oppenheimer Adiabatic wavefunction for the first excited singlet state becomes: BO ΨBO 1r (r, Q) = ψ1 (r, Q)χ1r (Q)   XX n ≈ ψ10 (r, Q0 ) + γj1 Qn ψj0 (r, Q0 ) χCA 1r (Q). j>1 n The transition moment now becomes: n o BO BO b MBO = Ψ (r, Q)| O(r)|Ψ (r, Q) 0t;1r 0t 1r   D E XX n b  = χCA ψ00 (r, Q0 ) O(r) ψ10 (r, Q0 ) + γj1 Qn ψj0 (r, Q0 ) χCA 1r (Q) . 0t (Q) j>1 n Simplification yields: D E
0 0 CA b MBO = ψ (r, Q )| O(r)|ψ (r, Q ) χCA 0 0 0t;1r 0 1 0t (Q)|χ1r (Q) D E XX
0 CA b + γjn1 ψ00 (r, Q0 )|O(r)|ψ χCA j (r, Q0 ) 0t (Q)|Qn |χ1r (Q) . j>1 n Now consider the coefficients γjn1 : γjn1 = D i h E ∂U(r,Q) ψ10 (r, Q0 ) ∂Qn 0 ǫ01 (Q0 ) − ǫ0j (Q0 ) ψj0 (r, Q0 ) . Since the Hamiltonian must be invarient under all symmetry operations,   ∂U (r, Q) must transform as Qn . ∂Qn 0 Given that ψ10 (r, Q0 ) transforms as A2 , it is easy to find the appropriate combinations of Qn and ψj0 (r, Q0 ) such that γjn1 does not vanish via symmetry. Three non-zero coefficients are obtained: 4 γ21 , 5 γ41 , 6 γ41 . Thus the transition moment for H2 CO can be written, more explicitly, as Note that: D E
0 0 CA b MBO χCA 0t;1r = ψ0 (r, Q0 )|O(r)|ψ1 (r, Q0 ) 0t (Q)|χ1r (Q) D E
4 CA b r)|ψ20 (r, Q0 ) χCA + γ21 ψ00 (r, Q0 )|O( 0t (Q)|Q4 |χ1r (Q) D E
CA 5 0 b χCA + γ41 ψ00 (r, Q0 )|O(r)|ψ 0t (Q)|Q5 |χ1r (Q) 4 (r, Q0 ) D E
0 CA 6 b ψ00 (r, Q0 )|O(r)|ψ (r, Q ) χCA + γ41 0 4 0t (Q)|Q6 |χ1r (Q) D E
1 CA b = 1 A1 (r, Q0 )|O(r)| A2 (r, Q0 ) χCA 0t (Q)|χ1r (Q) D E
4 1 1 CA b + γ21 A1 (r, Q0 )|O(r)| B2 (r, Q0 ) χCA 0t (Q)|Q4 |χ1r (Q) D E
5 1 1 CA b + γ41 A1 (r, Q0 )|O(r)| B1 (r, Q0 ) χCA 0t (Q)|Q5 |χ1r (Q) D E
6 1 CA b r)|1 B1 (r, Q0 ) χCA + γ41 A1 (r, Q0 )|O( 0t (Q)|Q6 |χ1r (Q) . µz =⇒ A1 mz =⇒ A2 µx =⇒ B1 mx =⇒ B2 µy =⇒ B2 my =⇒ B1