5.61 F17 Lecture 32: Adiabatic↔Diabatic (Zewail)

5.61 Fall, 2017 Lecture #32 Page 1 Lecture #32: Adiabatic↔Diabatic: Zewail We are interested in wavepacket dynamics. 1. 2. How to create a wavepacket with selected characteristics? What are the basic features of wavepacket dynamics on a simple diatomic molecule potential energy curve with H independent of time? 3. What are the additional features of a wavepacket on a 3N – 6 dimensional potential energy surface? 4. intramode anharmonicity intermode anharmonic interactions curve crossings, surface crossings stationary phase regions - where are they, what can happen as a wavepacket travels through or near one of them? 5. Landau-Zener inter-surface transition probability per crossing 6. How to probe wavepacket evolution? Examples: Zewail: I-CN dissociation NaI dissociation via ionic covalent curve crossing “Mechanism” Diabatic vs. Adiabatic Potential Energy Curves Ab Initio Potentials are “clamped nuclei” = adiabatic exact solution of electronic Schrödinger Equation, ψ is parametrically dependent on R Non-Crossing Rule Potential curves belonging to same electronic symmetry cannot cross because there are no symmetry restrictions on off-diagonal matrix elements between them. If there is a will, molecules will find a way. There will be a non-zero interaction term. 5.61 Fall, 2017 Lecture #32 2 + Page 2 2 + 1 + 1 + large curvature 1 – 1 – small curvature – 2 Rc 2 + 2 + 1 Weak interaction + V+(Rc) – V–(Rc) = 2H12 1 – Rc This is an electronic interaction (usually 1 Strong interaction + 1 ) r12 + 1 – What is missing in the adiabatic representation? Clamped nuclei. Nuclear kinetic energy. – 2 2 must be treated as H(1). ∇ R nuclear kinetic energy (and rotation) Rc – Seems harmless but the electronic Schrödinger Equation gives Weak interaction Rc Strong interaction Rc ψi(qi, Q) – 2 parametric dependence on nuclear coordinates ∂2 ∂ 2 operate on ψi(qi, Q) + ∂Q 2 ∂Q 1 so if we have a weakly avoided crossing 1 V+(Rc) – V–(Rc) = 2H12 + 2 H12 is very small, can we ignore it? YES! – But there is something else that we cannot ignore: matrix elements of ∇ 2R between vibrational levels are enormous! Rc – 2 1 – 5.61 Fall, 2017 Lecture #32 – 2 Page 3 ∂V± is enormous. H(1) is not small. Big trouble. Huge interaction between vibrational levels of the V+ Rc∂Q Rc – and V– states. 2 Weak interaction Strong interaction The actual energy levels are very different from what are generated by H(0). Why is this bad news? The adiabatic representation is a very bad representation for the case of a weakly avoided crossing. It gives useless2clues about how1to assign the observed vibrational levels. 2 1 + + + + Strongly avoided crossing + 1 – 1 – – 2 Rc Rc – Weak interaction Strong interaction – 2 Rc + V+(Rc) – V–(Rc) = 2H12 huge. But H12 is explicitly taken into account in the ab initio computation of V–(R) and V+(R) potentials. We solve for the vibrational levels of V+(R) and V–(R) and we see close relationships between the vibrational levels and H(0). What about ∂ψ ( r; R) – near Rc? Very small because electronic character is changing slowly. ∂R Diabatic Curves do cross Rc 2 1 1 2 How is this possible? We exclude some term in Helectronic which enables the electronic wavefunctions to remain orthogonal. Electronic character does not change near Rc. But there is no such explicit term that we can exclude from H(0) and treat as H(1)! 5.61 Fall, 2017 Lecture #32 Page 4 – Quantum Chemists do not like the diabatic representation even though it is what we like as Chemists. Why? Rc There are ways to construct a diabatic representation from the ab initio adiabatic representation Solid curves are adiabatic V+ V1 V2 Dashed curves are diabatic Adiabatic gives Rc and H12 Make linear approximation for diabatic potential in region of curve crossing. We have an R-dependent 2-level problem. V– DIABATIC ⎛ V (R) H12 (Rc ) 1 †⎜ H(R) = T ⎜ H12 (Rc ) V2 (R) ⎝ ADIABATIC ⎞ ⎛ V (R) 0 ⎟T = ⎜ + ⎟ ⎜ 0 V– (R) ⎠ ⎝ ⎞ ⎟ ⎟ ⎠ We know V±(R) and H12(Rc) This is sufficient to generate V1(R) and V2(R). We do a least squares fit by varying the slope of V1 and V2 at Rc. Once we have V1 and V2 or V+ and V– we can fit the spectrum. H12 large: adiabatic representation is best. H12 small: diabatic representation is best. For polyatomic molecules we have potential surfaces, not curves. Two adiabatic surfaces can have different symmetries. This permits them to cross at the high symmetry point. But as you move away from Rsym, the 2 states interact and the surfaces repel. They look like a pair of cones that are joined at the apex. “Conical Intersection” “Vibronic Coupling” Many things that violate chemical intuition are “explained” by vibronic coupling. 5.61 Fall, 2017 Lecture #32 Page 5 How is this related to Landau-Zener? Transition Probability. P12 = 1− e−2 πγ γ= V122 !v s1 − s2 transition probability 1 → 2 at each passage through Rc. P12 = 0 when γ = 0, P12 = 1 when γ large γ small when v large (P12 small), v is velocity at Rc γ large when s1 ≈ s2, s is slope of diabatic curve at Rc. 1 2 1 Rc Look at examples from the Zewail experiments. 2 Rc 5.61 Fall, 2017 Lecture #32 Page 6 Off-Diagonal Matrix Elements in the Diabatic and Adiabatic Representation In the diabatic representation, the electronic-vibration matrix elements are H ediab = χ vd1 Φde1 H el Φde2 1 ,v1 ;e2 ,v2 r χ vd2 R where right subscripts specify integration over electronic (r) or nuclear (R) coordinates. This matrix element simplifies to the product of two integrals χ vd1 χ vd2 Φde1 H ( Rc ) Φde2 or, more simply, χ vd1 χ vd2 R H1,2 ( Rc ) . The vibrational overlap integral is calculated using the vibrational wavefunctions of electronic states 1 and 2. The electronic matrix element is evaluated at Rc because that is the stationary phase point at which the vibrational integral accumulates to its final value. The matrix elements in the adiabatic representation are the result of a lengthy derivation, given on pages 168-172 of H. Lefebvre-Brion and R. W. Field (see end of Lecture 32 Supplement #2). The term that is neglected in the “clamped nuclei” calculation is ! 2 ⎡ ∂2 2 ∂ ⎤ ! 2 T =− ⎢ 2 + R2 ⎥+ 2 2 µ ⎣ ∂R R ∂R ⎦ 2 µR N (1) The last term is nuclear rotation and we will ignore it. Since the electronic wavefunctions depend parametrically on R, we need to operate with TN on the product of the electronic wavefunction, ψei (r; R) and the vibrational wavefunction, χ vi (R) . After some algebra we get H e1v1;e2 v2 (cm −1 ) = − − 16.8576 ad ∂2 ξ v1 Φad Φad e1 e2 2 µ(amu) ∂R 33.7152 ad ∂ ad ξ v1 Φad Φe e1 µ(amu) ∂R 2 r (Å−2 ) ξ vad2 ( Å−1 ) r R d ξv dR 2 (2) R The r,R right subscripts specify integration over the electron (r) and nuclear coordinates (R). 5.61 Fall, 2017 Lecture #32 Page 7 The vibrational wavefunction ξ vad (R) ≡ R χ vad (R) (3) where ξ(R) and χ(R) are normalized as χ v (R)* R 2 χ v (R) ξ v (R)* ξ v (R)* R R =1 (4a) = 1. (4b) The diabatic curves cross and the 2 × 2 configuration interaction secular equation is Eed1 (R) − E e H12 (R) e H12 (R) Eed2 (R) − E (5) =0 where the diabatic potential energy curves are Eed1 (R) = V1d (R), Eed2 (R) = V2d (R), (6) and we get the adiabatic wavefunctions from the diabatic electronic wavefunctions d d Φad e1 = cosθ (R)Φ e1 − sin θ (R)Φ e2 (7a) d d Φad e2 = sin θ (R)Φ e1 − cosθ (R)Φ e2 . (7b) Note that the mixing angle, θ, is dependent on R. At the crossing point, Rc, between the diabatic potential curves, R = Rc and θ = π/4. The vertical energy difference at Rc between the adiabatic potential curves is e Eead1 ( Rc ) − Eead2 ( Rc ) = 2H12 ( Rc ) (8) and the slopes of the assumed linear diabatic potentials near R = Rc are Eed1 ( R) − Eed2 ( R) = a ( R − Rc ) . Since the diabatic electronic wavefunctions are, by definition, independent of R, (7) operates only on cos θ(R) and sin θ(R), thus (9) ∂ applied to equation ∂R 5.61 Fall, 2017 Lecture #32 Φad e1 Page 8 ∂ ad ⎛ 2 ∂θ 2 ⎞ ∂θ Φe2 = ⎜sin θ + cos θ = ⎟ !########" ∂R ∂R . ∂R ⎝ ⎠ =1 (10) By expanding the determinant in Eq. (5), we obtain the R-dependence of θ near Rc e H12 sin θ cosθ 1 ( Rc ) = − H12e ( Rc ) . = tan 2 θ = a ( R − Rc ) cos2 θ − sin θ 2 2 Eed2 − Eed1 (11) ⎡ −2H e ( R ) ⎤ 1 ⎡ a ( R − R ) ⎤ 1 12 c c ⎥ = cot ⎢ θ (R) = tan−1 ⎢ ⎥. e ⎢⎣ a ( R − Rc ) ⎥⎦ 2 ⎣ −2H12 2 ( Rc ) ⎦ (12) ⎛x⎞ d −c cot −1 ⎜ ⎟ = 2 2 ⎝c⎠ c +x dx (13) Thus Since Φad e1 e aH12 ∂ ad ∂θ ( Rc ) Φe2 = = . 2 ∂R ∂R 4 H e ( R ) + a 2 ( R − R )2 12 c c ( ) (14a) e For convenience, define b = H12 ( Rc ) a Φad e1 ∂ ad Φe ∂R 2 = r b 2 4b + ( R − Rc ) 2 ≡ W e (R) (14b) e It is reasonable to treat H12 as independent of R, because the integral over R accumulates only near Rc. Then Eq. (14b) requires that We(R) is a Lorentzian in R with a full width at half-maximum of 4b = 4H e ( Rc ) a where a is the difference in slope between H ed1 and H ed2 near Rc. Recall that the Landau-Zener formula for the probability of jumping from the V+(R) to the V–(R) adiabatic curve is P+– = 1− e−2 πγ 2 γ= where s1 − s2 = dH ed1 dR − dH ed2 dR ! Vibrational Matrix Elements H e ( Rc ) !v s1 − s2 5.61 Fall, 2017 Lecture #32 Page 9 Our goal is to derive the matrix elements between the vibrational levels of adiabatic potential curves Eead1 (R) and Eead2 (R) that result from ∇ 2R . If it is possible to ignore vibronic interactions of state Φ1ad and Φad 2 with other electronic states, the two terms in Eq. (2) reduce to H e1v1;e2 v2 (cm −1 ) = − 16.8576 ad dW e ad ξ v1 ξ v2 µ(amu) dR − R 33.7152 ad e d ad ξ v1 W (R) ξv µ(amu) dR 2 . R Since both R-integrals accumulate near Rc, W e ( Rc ) = dW e dR 1 a = e 4b 4H12 =0 Rc and the dominant term in the inter-electronic vibrational matrix elements is H e1v1;e2 v2 (cm −1 ) = − 33.7152 ad a d ad ξ v1 ξ e dR v2 µ(amu) 4H12 R where d ad ξ v2 ∝ ξ vad2 ±1 dR because d ( ∝ a − a† ) . dR e Note that when H12 is small (weakly avoided crossing) and the difference in slope of the diabatic potential curves is large (also weakly avoided crossing), the vibrational matrix elements are large. If the adiabatic potential curves are very different in shape, the vibrational matrix elements will not follow any sort of restrictive propensity rule. This means that the interaction matrix elements between the vibrational levels of two adiabatic potential energy curves will be large and involve many simultaneous interactions. An enormously non-diagonal interaction matrix will have to be diagonalized in the e adiabatic representation. At the opposite extreme, if the crossing is strongly avoided, H12 will be large and the difference in slopes near Rc will be very small. All of the off-diagonal vibrational matrix elements will be very small. In the diabatic representation, the opposite behavior of the off-diagonal matrix elements is expected. el For a weakly avoided crossing, H12 is small, the difference in slopes is large, and the only interactions between vibrational levels of diabatic states 1 and 2 that need to be considered will be between pairs of 5.61 Fall, 2017 Lecture #32 el near-degenerate vibrational levels. For a strongly avoided crossing, H12 is enormous and many vibrational overlap integrals will be large. Page 10