Non-Adiabatic Dynamics Multidimensional Photochemistry and Non-adiabatic Dynamics Graham Worth Dept. of Chemistry, University College London, U.K. 1 / 103 Non-Adiabatic Dynamics I The Time-Dependent Schrödinger Equation I The Born-Oppenheimer Approximation I Adiabatic and Diabatic Pictures I Conical Intersections I Methods for Solving The TDSE I Basis-set expansions (Grids) I MCTDH I Gaussian Wavepackets I Trajectory Based Methods I The Hamiltonian I The Vibronic Coupling Model I Direct Dynamics Köppel, Domcke and Cederbaum Adv. Chem. Phys. (84) 57: 59 Domcke, Yarkony, Köppel “Conical Intersections” World Scientific (04) Klessinger and Michl “Excited-states and Photochemistry”, VCH (94) many reviews 2 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation The Time-Dependent Schrödinger Equation @ i Ψ(q; r; t) = H^ Ψ(q; r; t) (1) ~@t If the Hamiltonian is time-independent, formal solution Ψ(t) = exp iHt^ Ψ(0) (2) − Phase factor Further, if H^ is time-independent we can write −i!i t Ψ(x; t) = Ψi (x)e (3) and @ i Ψ(x; t) = ! Ψ (x)e−i!i t (4) ~@t ~ i i By comparison with the TDSE, Ψi are solutions to the time-independent Schrödinger equation ^ HΨi = Ei Ψi = ~!i Ψi (5) 3 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Ψi is a Stationary State as expectation values (properties) are time-independent ^ ^ i!i t −i!i t ^ O = Ψi O Ψi e e = Ψi O Ψi (6) h i h j j i h j j i If wavefunction is a superposition of stationary states, X −i!i t χ(x; t) = ci Ψi (x)e (7) i now, ^ X X ∗ i(!i −!j )t O (t) = i~ ci cj Ψi O Ψj e (8) h i − h j j i i j An expectation value changes with time and depends on the initial function (ci coefficients). A non-stationary wavefunction is called a WAVEPACKET. 4 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation The Clamped Nucleus Hamiltonian For a given nuclear configuration q, if we clamp the nuclei in place then the electronic Hamiltonian can be written: Ne Ne Nn Ne 2 Nn 2 ^ X 1 2 X X Zae X e X ZaZbe Hel = i + + (9) −2me r − 4π0ria 4π0rij 4π0Rab i=1 i=1 a=1 i;j=1 a;b=1 eKE VeN Vee VNN In atomic units 2 4π0~ Length: 1 Bohr = 0.529 Å a0 = 2 mee −1 e2 Energy: 1 Hartree = 2625.5 kJ mol Eh = 4π0a0 = 27.21 eV Ne Ne Nn Ne Nn ^ X 1 2 X X Za X 1 X ZaZb Hel = i + + (10) −2r − ria rij Rab i=1 i=1 a=1 i;j=1 a;b=1 5 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation The clamped nucleus Hamiltonian can be solved to provide the electronic wavefunctions at a specific nuclear configuration. This is what is done in quantum chemistry. ^ Hel i (r; q) = Ei i (r; q) (11) i.e. the wavefunction is a function of electronic coordinates but depends parametrically on the nuclear coordinates. Return to full TDSE with Hamiltonian ^ ^ ^ H = TN + Hel (12) and write the full wavefunction as a product of nuclear and electronic parts where the electronic function is a solution of the clamped nucleus Hamiltonian: Ψ(q; r; t) = χ(q; t) (r; q) (13) This is an adiabatic separation of variables 6 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures The Nuclear Schrödinger Equation First remove the electronic motion from the TDSE. @ i~ Ψ = H^ Ψ (14) @t j i j i @ χ ^ ^ i~ j i = TN + Hel χ (15) j i @t j i j i Multiply by and integrate over the electronic coordinates h j @ i~ χ = ( TN + V (q)) χ (16) @t j i h j j i j i using the potential function H (q) = V (q) (17) h j el j i 7 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures To analyse KE function, use a simple form of the KEO X 1 @2 T^ = (18) N 2M @q2 α − α α with Mα the nuclear mass. Remembering that the electronic functions depend on the nuclear coordinate, a single term of this KEO can be written 1 @2 @ @ @2 TN = + 2 + (19) h j j i −2M h j@q2 i h j@q i@q @q2 And as the derivative operator @=@q is anti-hermitian @ = 0 (20) h j@q i we obtain 1 @2 @2 TN = + (21) h j j i −2M h j@q2 i @q2 8 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures Ignoring the first term on the RHS, which is (hopefully) small TN TN (22) h j j i ≈ we obtain the TDSE in the Adiabatic Approximation: @ i~ χ = (TN + V (q)) χ (23) @t j i j i This is the basic picture used in chemistry that the nuclear and electronic motions can be separated. The nuclei then move over a potential surface provided by the electrons that do not depend on the electronic motion, but only on their position. This is usually referred to as the Born-Oppenheimer Approximation. Sometimes, however, B-O Approx is used when the scalar term 2 @ is included. Most authors call this latter approximation the h j @q2 i Born-Huang Approximation. 9 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures Wavepacket Dynamics r V [ev] 0 -0.5 -1 -1.5 R -2 -2.5 -3 -3.5 -4 To solve the TDSE require: -4.5 -5 I Potential surfaces 1 1.5 2 Algorithm to propagate 2.5 I 2 3 2.5 3 3.5 r [au] wavepacket 3.5 4 4 R [au] 4.5 5 4.5 5.5 6 5 10 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures TDSE: The Complete Solution A full solution requires a multi-configurational ansatz. Start by using Born representation X Ψ(q; r) = χi (q) i (r; q) ; (24) i where electronic functions are all the solutions to clamped nucleus Hamiltonian Eq. (11). The TDSE is now X @ χi X ^ ^ i~ i j i = TN + Hel χi i (25) j i @t j i j i i i Now multiply by one electronic function j and again integrate h j @ X i~ χj = [ j TN i + Vj (q)δji ] χi (26) @t j i h j j i j i i using the full set of potential functions j H (q) j = Vj (q) (27) h j el j i 11 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures One term of the KE operator in the electronic basis is now 1 @2 @ @ @2 j TN i = j i + 2 j i + (28) h j j i −2M h j@q2 i h j@q i@q @q2 which can be written in terms of the scalar and vector derivative couplings, G and F 1 ~ ^ j TN i = Gji 2Fji : + TN (29) h j j i −2M − r where X @2 G = (30) ji j @q2 i α h j α i α @ Fji = j i (31) h j@qα i 12 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures Defining the non-adiabatic operator 1 ~ Λji = Gji + 2Fji : (32) 2M r the TDSE in the adiabatic picture is therefore @ X i~ χj = [(TN + Vj ) δji Λji ] χi (33) @t j i − j i i Finally, using the fact that G = ( F) + F F (34) r · · Eq. (33) can be written: 1 2 @χ ( 1 + F) + V χ = i~ ; (35) −2M r @t 13 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures The Adiabatic Picture F Assuming M 0 ≈ h i @χ T^ + V χ = i (36) n ~ @t and we recover the adiabatic TDSE where the nuclei move over a single potential energy surface, V , which can be obtained from quantum chemistry calculations. An expression for the derivative coupling in terms of the energy of the states involved can be obtained from i Hel j = i Hel j + i Hel j + i Hel j (37) rh j j i hr j j i h jr j i h j jr i As we are using the adiabatic basis, i Hel j = 0 (38) rh j j i 14 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures And as H i = Vi i , we obtain ^ i rH j h j el j i Fij = for i = j : (39) Vj Vi 6 − Compare this to the Hellman-Feymann expression to the derivative of a potential i Hel i = i Hel i (40) rh j j i h jr j i and we see this is a “coupling force”. The derivative coupling is singular if 2 potential surfaces become degenerate. 15 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures The Diabatic Picture To remove these singularities, we first separate out a group of coupled states from the rest (group B.O. approximation). (g) 1 (g) (g) 2 (g) (g) @χ ( 1 + F ) + V χ = i~ ; (41) −2M r @t If we rotate the adiabatic electronic basis via a unitary transformation (d) = S(q) (a) (42) then (dropping (g) superscript) y 1 2 y y S ( 1 + F) + V SS χ = i~S χ_ ; (43) −2M r 16 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures such that the Hamiltonian can be written @χ(d) [T 1 + W] χ(d) = i ; (44) N ~ @t where all elements of W are potential-like terms W = SyVS (45) χ(d) = Syχ(a) (46) 1 y 2 S ( 1 + F) S = TN 1 (47) −2M r The last relationship can be shown to be correct if S = FS (48) r − Baer Chem.