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) = Hˆ Ψ(q, r, t) (1) ~∂t If the Hamiltonian is time-independent, formal solution Ψ(t) = exp iHtˆ Ψ(0) (2) − Phase factor Further, if Hˆ is time-independent we can write

−iωi t Ψ(x, t) = Ψi (x)e (3) and ∂ i Ψ(x, t) = ω Ψ (x)e−iωi t (4) ~∂t ~ i i By comparison with the TDSE, Ψi are solutions to the time-independent Schrödinger equation ˆ HΨi = Ei Ψi = ~ωi Ψi (5)

3 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation

Ψi is a Stationary State as expectation values (properties) are time-independent

ˆ ˆ iωi t −iωi t ˆ O = Ψi O Ψi e e = Ψi O Ψi (6) h i h | | i h | | i

If wavefunction is a superposition of stationary states,

X −iωi t χ(x, t) = ci Ψi (x)e (7) i now, ˆ X X ∗ i(ωi −ωj )t O (t) = i~ ci cj Ψi O Ψj e (8) h i − h | | 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 ˆ X 1 2 X X Zae X e X ZaZbe Hel = i + + (9) −2me ∇ − 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 ˆ X 1 2 X X Za X 1 X ZaZb Hel = i + + (10) −2∇ − 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. ˆ Helψi (r; q) = Ei ψi (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 ˆ ˆ ˆ 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)ψ(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 = Hˆ Ψ (14) ∂t | i | i ∂ χ ˆ ˆ i~ ψ | i = TN + Hel χ ψ (15) | i ∂t | i | i Multiply by ψ and integrate over the electronic coordinates h | ∂ i~ χ = ( ψ TN ψ + V (q)) χ (16) ∂t | i h | | i | i using the potential function

ψ H (q) ψ = V (q) (17) h | el | 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 Tˆ = (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 ψ TN ψ = ψ ψ + 2 ψ ψ + (19) h | | i −2M h |∂q2 i h |∂q i∂q ∂q2 And as the derivative operator ∂/∂q is anti-hermitian ∂ ψ ψ = 0 (20) h |∂q i we obtain 1 ∂2 ∂2 ψ TN ψ = ψ ψ + (21) h | | i −2M h |∂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

ψ TN ψ TN (22) h | | i ≈

we obtain the TDSE in the Adiabatic Approximation: ∂ i~ χ = (TN + V (q)) χ (23) ∂t | i | 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 ψ ∂ ψ is included. Most authors call this latter approximation the h | ∂q2 i Born-Huang Approximation.

9 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures Wavepacket Dynamics

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)ψi (r; q) , (24) i where electronic functions are all the solutions to clamped nucleus Hamiltonian Eq. (11). The TDSE is now

X ∂ χi X ˆ ˆ i~ ψi | i = TN + Hel χi ψi (25) | i ∂t | i | i i i

ψj H (q) ψj = Vj (q) (27) h | el | 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 ψj TN ψi = ψj ψi + 2 ψj ψi + (28) h | | i −2M h |∂q2 i h |∂q i∂q ∂q2

which can be written in terms of the scalar and vector derivative couplings, G and F

1 ~ ˆ ψj TN ψi = Gji 2Fji . + TN (29) h | | i −2M − ∇ where

X ∂2 G = ψ ψ (30) ji j ∂q2 i α h | α i

α ∂ Fji = ψj ψi (31) h |∂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 ∇ the TDSE in the adiabatic picture is therefore

∂ X i~ χj = [(TN + Vj ) δji Λji ] χi (33) ∂t | i − | i i

Finally, using the fact that

G = ( F) + F F (34) ∇ · · Eq. (33) can be written: 1 2 ∂χ ( 1 + F) + V χ = i~ , (35) −2M ∇ ∂t

13 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures The Adiabatic Picture

F Assuming M 0 ≈ h i ∂χ Tˆ + 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

ψi Hel ψj = ψi Hel ψj + ψi Hel ψj + ψi Hel ψj (37) ∇h | | i h∇ | | i h |∇ | i h | |∇ i As we are using the adiabatic basis,

ψi Hel ψj = 0 (38) ∇h | | i

14 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Adiabatic / Diabatic Pictures

And as Hψi = Vi ψi , we obtain

ˆ ψi ∇H ψj h | el | i Fij = for i = j . (39) Vj Vi 6 −

Compare this to the Hellman-Feymann expression to the derivative of a potential ψi Hel ψi = ψi Hel ψi (40) ∇h | | i h |∇ | 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 ∇ ∂t

If we rotate the adiabatic electronic basis via a unitary transformation

ψ(d) = S(q)ψ(a) (42)

then (dropping (g) superscript) † 1 2 † † S ( 1 + F) + V SS χ = i~S χ˙ , (43) −2M ∇

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 = S†VS (45) χ(d) = S†χ(a) (46)

1 † 2 S ( 1 + F) S = TN 1 (47) −2M ∇ The last relationship can be shown to be correct if

S = FS (48) ∇ −

Baer Chem. Phys. Lett. (1975) 35: 112 Worth and Cederbaum Ann. Rev. Phys. Chem. (2004) 55: 127

17 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Surface Crossings

For a 2-state system, the diabatic potentials are

W W W = 11 12 (49) W12 W22

Eigenvalues of diabatic potential matrix are adiabatic PES: q 1 1 2 2 V± = (W11 + W22) ∆W + 4W (50) 2 ± 2 12

Surfaces degenerate at q0 if

∆W = W22 W11 = 0 (51) − W12 = 0 (52)

18 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Avoided Crossing Two conditions need to be met for degeneracy. Impossible if only 1 coordninate. Hence in a 1D system states that have non-zero coupling do not meet and form an avoided crossing.

Diabatic Adiabatic

NaI Potential Surfaces

19 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

In multi-dimensional systems, Eqs. (51,52) can be satisﬁed. Diabatic potential matrix elements are all potential like functions so can be represented by a Taylor expansion around a point such as q0

2 X ∂Wij X 1 ∂ Wij Wij = Wij (q0)+ (qα q0)+ (qα q0)(qβ q0)+... ∂qα − 2 ∂qαqβ − − α q=q0 α,β q=q0 (53) Move to q0 where the states 1 and 2 are degenerate. If at this (d) (a) | i | i point we take ψi = ψi then

(d) (d) Wij = ψ Hel ψ = Ei ; i = j (54) h i | | j i = 0 ; i = j (55) 6

20 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

At q0, E1 = E2 = E and so expanding to ﬁrst order:

~ ! E 0 ~κ1 ~r λ ~r W = + ~ · · (56) 0 E λ ~r ~κ2 ~r · ·

where rα = qα q0,α and λ, κi vectors of potential derivatives. As the − diabatic and adiabatic electronic functions are equal at q0, the on-diagonal derivative is the gradient of the potential

∂ (d) (d) ∂ (a) ∂Hel (a) κα,i = ψi Hel ψi = Vi = ψi ψi (57) ∂qα h | | i ∂qα h | ∂qα | i and the off-diagonal derivative is the derivative coupling (See Eq. (37) and use the fact that thje derivative coupling is 0 in the diabatic basis).

∂ (d) (d) (a) ∂Hel (a) λα = ψ1 Hel ψ2 = ψ1 ψ2 (58) ∂qα h | | i h | ∂qα | i

21 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Thus

∆W = (~κ2 ~κ1) ~r = ~g ~r gradient difference − · · ~ ~ W12 = λ ~r = h ~r derivative coupling · · If we move so that ~r is orthogonal to g, h, the states remain degenerate. If we move so that ~r is in the plane g, h, the degeneracy is lifted. The plane deﬁned by g, h is called the branching plane. And the shape of the surfaces{ in the} branching space have the topography of a double cone. This is a conical intersection

22 / 103 Energy

NB need independent ∆W and W12. Hence non-crossing rule in diatomics. Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections The Importance of Symmetry

Symmetry plays an important role as conical intersections often occur at nuclear conﬁgurations with high symmetry. When this happens, the matrix elements

∂Hel ψi Hel ψj and ψi ψj h | | i h | ∂qα | i will be zero if the integrand is not totally symmetric wrt the point group of the molecule. As the H is always totally symmetric,

ψi Hel ψj = 0 if Γi Γj A1 (59) h | | i 6 ⊗ ⊃ ∂Hel ψi ψj = 0 if Γi Γj Γα A1 (60) h | ∂qα | i 6 ⊗ ⊗ ⊃

where A1 represents the totally symmetric irrep

23 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

In all cases

I κα,i = 0 only for Γα A1 6 ⊃ If the electronic states have different symmetries

I λα = 0 only for Γα = Γi Γj 6 ⊗

For example in pyrazine, molecule is D2h, states S1(B3u), S2(B2u) and as B2u B3u = B1g only modes of this symmetry can couple the ⊗ states (there is only 1 of these ν10a). There are four modes with Ag symmetry that have non-zero κ values. The GD and DCP are thus independent and a conical intersection must occur.

24 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

Some point groups (non-Abelian) have degenerate representations. These impose further constraints on relationships between the values of κ and λ.

For example, in many point groups with an E irrep, E e A1 and the doubly degenerate electronic states are coupled by⊗ doubly⊃ degenerate vibrations. This is the E e Jahn-Teller case and ⊗ κ1 = κ2 = λ. − Higher order terms are goverened by similar relationships. For a complete analysis of the JT problem see Viel and Eisfeld JCP (04) 120: 4603. If no symmetry is present a CoIn can form if the modes involved in the ~κ and ~λ happen to be independent. These are called accidental CoIn in contrast to symmetry induced CoIn.

25 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Types of Conical Intersections

2 1 Sloped conical 0 I

-1

-2 intersection 10 5 -4 0 -2 x2 0 -5 x 1 2 -10 κ , κ same sign 4 I 2 1

2 1 Peaked conical 0 I -1 intersection -2 10 5 -4 -2 0 x2 0 -5 x1 2 κ , κ opposite signs 4 I 2 1

26 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

2 1 E e Jahn-Teller 0 I

-1 ⊗ -2 κ = κ = λ 10 I 2 1 5 -10 x2 − -5 0 0 x1 5 -5 10 I Non-Abelian groups with E irrep

2 1 Jahn-Teller 0 I

-1 -2 κ = κ = λ 10 I 2 1 5 -10 x2 − -5 0 0 x1 5 -5 10 I second-order couplings included

27 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

2 1 E b Jahn-Teller 0 I

-1 × -2 κ = κ = λ 10 I 2 1 5 -10 x2 − 6 -5 0 0 x1 5 -5 10 I D2d

2 1 Renner-Teller 0 I -1 Intersection -2 10 5 -10 -5 x2 0 0 x1 5 κ , κ , λ = 0 10 I 2 1

I Second-order Terms highest order

28 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Peaked v Sloped

For dynamics, 2 main classiﬁcations

Peaked Sloped Atchity, Xantheas and Ruedenberg JCP (91) 95: 1862

29 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Non-adiabatic Crossing in NaI

Pump-probe Spectroscopy

I pump pulse inititiates reaction

I probe pulse provides signal related to dynamics

30 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Signature of Conical Intersections Octatetraene Pyrazine

Yamazaki et al Farad. Discuss. (83) 75:395

31 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Benzene: Photophysics Quantum yield Process 253 nm 248 nm 242 nm 237 nm Formation of photoproducts 0.016 0.022 0.024 0.037 Fluorescence 0.18 0.10 0 0 Formation of triplet (ISC) 0.6 0.6 0 0

Absorption Spectrum “Channel 3”

Clara et al Appl. Phys. B (00)71: 431

32 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Photochemistry

Jablonski Diagram

Evolution over Potential Surfaces Conical Intersections = ”Photochemical Funnels”

33 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Other Conical Intersections

Intersections can also occur in higher dimensions:

I 3-states (Matsika and Yarkony JACS (03) 125: 10672)

I 2-states v 3-states (Coe and Matinez JACS (05) 127: 4560)

I 4-states (Assmann, Worth and Gonzalez JCP (12) 137: 22A524)

3-state CoIns have a 5-dimensional branching space etc. And can also be formed in diatomics when

I Light-Induced (Halasz, Vibok and Cederbaum JPCL (15) 6: 348)

34 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections The Adiabatic-Diabatic Transformation matrix

The ADT matrix, S is deﬁned vy Eq. (48) only through a derivative wrt q. To completely deﬁne it specify that the diabatic and adiabatic basis are identical at a particular point, q0. This may be the CoIn minimum (as above) or the FC point, or some other suitable point such as a dissociation assymptote.

1 0 S(q ) = (61) 0 0 1

This is referred to as ﬁxing the global gauge. NB the ADT is a global function S(q)

35 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

For a 2-state CoIn it is possible to get an analytic expression for S. First write the diabatic potential matrix Σ ∆ W12 W = − (62) W12 Σ + ∆

with 1 1 Σ = (W11 + W22) ; ∆ = (W22 W11) (63) 2 2 − Express the unitary ADT matrix in terms of an angle

cos θ sin θ S = (64) sin θ cos θ − Now choose θ to diagonalise W

SWST = V (65)

36 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

2 2 cos θ sin θ W12 + 2 cos θ sin θ∆ = 0 (66) − 2 2 Σ cos θ sin θ ∆ 2 cos θ sin θW12 = V± (67) ± − − where q 2 2 V± = Σ ∆ + W (68) ± 12 are the eigenvalues (see Eq. (50)). These equations can be re-arranged to give W tan 2θ = 12 (69) ∆ 1 1 1 ∆ 2 1 ∆ 2 cos θ = 1 + ; sin θ = 1 (70) √2 d √2 − d with q 1 2 2 d = (V+ V−) = ∆ + W (71) 2 − 12

37 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections

π θ = 4 (W12 > 0, ∆ = 0) 1 1 √1 2 −1 1 q Travel in Branching Space u Around a CoIn. ADT.

θ = 0 (W12 = 0, ∆ = d) 1 0 π θ = 2 (W12 = 0, ∆ = −d) 0 1 0 1 q −1 0 g θ = π (W12 = 0, ∆ = d) 1 0 × −1 0 1

S is undeﬁned at 0 3π θ = 4 (W12 < 0, ∆ = 0) 1 1 − √1 2 −1 1

38 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Geometric Phase

It is known that an adiabatic wavefunction, ψ undergoing transport has a phase factor I ∂ γ(C) = i ψ ψ dq (72) C h |∂q i which means that the wavefunction changes sign on travelling a closed loop around a conical intersection. This is the Berry Phase. Berry, Proc. Roy. Soc. A (84) 392: 45. Note that the sign change is cancelled by the sign change in S. Importance in scattering: Juanes-Marcos et al Science (05) 309: 1227

39 / 103 Non-Adiabatic Dynamics The Time-Dependent Schrödinger Equation Conical Intersections Landau-Zener Model

For a particle on adiabatic surface 2 in the “non-adiabatic region”, the probability of moving to surface 1 is

2 2π F12 P2→1 = exp (73) − ~ q˙ F1 F2 | − |

If P2→1 1 particle changes adiabatic (stays on diabatic) state. −→

40 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Basis-set expansions Propagating wavepacket: The Standard Method Nuclear wavefunction expanded in primitive basis set:

N1 Nf X X (1) (f ) Ψ(q1,... qf , t) = Aj ...j (t) χ (q1) χ (qf ) ··· 1 f j1 ··· jf j1=1 jp=1

Use Dirac-Frenkel Variational Principle: ∂ δΨ H i Ψ = 0 − ∂t to obtain equations of motion for A: ˙ X (1) (f ) (1) (f ) iAj ...j = χ (q1) χ (qf ) H χ (q1) χ (qf ) Al ...l 1 f h j1 ··· jf | | l1 ··· lf i 1 f L ˙ X iAJ = ΦJ H ΦL AL h | | i L Kulander “Time-dependent methods for quantum dynamics”, Elsevier, 1991 Kosloff, Ann. Rev. Phys. Chem. (94) 45: 145 41 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Basis-set expansions The Hamiltonian matrix elements Need to evaluate matrix elements (integrals)

X (1) (f ) (1) (f ) HJL = χ χ H χ χ h j1 ··· jf | | `1 ··· `f i `1,...`f X (1) (f ) (1) (f ) = χ χ T + V χ χ h j1 ··· jf | | `1 ··· `f i `1,...`f

As written an Nf Nf matrix of multi-dimensional integrals! × I. Usually X T = Tκ κ X (κ) (κ) T = χ T χ δ κ κ JL jκ κ `κ J L κ h | | i N N matrices, which by a suitable choice of basis functions can be solved× analytically

42 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Basis-set expansions II. The potential is a local operator V (q). Thus if basis functions are (κ) localised, χ δ(q qj ) then j ≈ − (1) (f ) (1) (f ) VJL = χ χ V χ χ h j1 ··· jf | | `1 ··· `f i

= V (qj1 ... qjf )δJL

To enable I and II, use a DVR basis set in which

Xij = φi xˆ φj h | | i UXUT = x

where φi are an analytically known FBR. And so X χα = Uαi φi i

are the DVR, χα δ(x xα). Various DVRs possible. ∼ − Beck et al Phys. Rep. (2000) 324: 1 43 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Basis-set expansions Integrating the TDSE Full solution is i − Htˆ Ψ(t) = e ~ Ψ(0) Split-operator method.

i i i − Htˆ − Tt − Vt e ~ = e ~ e ~ 6 so divide propagation into short steps and approximate

i i i − V δt − T δt − Vˆ δt Ψ(t + δt) = e 2~ e ~ e 2~ Ψ(t)

Chebyshev Propagation Represent propagator by polynomial expansion: i − Htˆ X e ~ Ψ = anPn(H)Ψ n

where Pn(H) are generated by a recurrence relationship

44 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Basis-set expansions

Great but limited due to exponential increase in computer resources Nf ..... ∼

45 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH The Multiconﬁgurational Time-Dependent Hartree (MCTDH) Method n1 nf f X X Y (κ) Ψ(q ,..., q , t) = ... A (t) ϕ (q , t) 1 f j1...jf jκ κ j1=1 jf =1 κ=1

Variational equations of motion for A and ϕ. ˙ X iAJ = ΦJ H ΦL AL h | | i L −1 iϕ˙ (κ) = 1 P(κ) ρ(κ) H (κ)ϕ(κ) − h i SPFs expanded in primitive grid I non-linear equations of motion N X f ϕ = a χ I Computer memory n + fnN j ij i i=1 Beck et al Phys. Rep. (00) 324:1 Meyer, Gatti and Worth “Multidimensional quantum dynamics”, Wiley-VCH, 2009 46 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH

Where we have single-hole functions,

n1 nf −1 f −1 (f ) X X Y (κ) ψ = ... A ... (t) ϕ (qκ, t) kf j1 jf −1kf jκ j1=1 jf −1=1 κ=1

mean-ﬁeld operators and density matrices (κ) (κ) (κ) (κ) (κ) (κ) and P ∗ κ H ij = ψi H ψj ρij = ψi ψj = J AJκ AJ h i h κ | | κ i h κ | κ i i j time-dependent matrix elements

(1) (f ) (1) (f ) HJL = ΦJ H ΦL = φ . . . φ H φ . . . φ h | | i h j1 jf | | l1 lf i

P (1) (2) I for efﬁciency need product potential V = s cshs hs ...

47 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH TheConstantMeanField Integration Scheme ¯(0) 1 A(0) K- A(τ/2)

1 ρ¯− (0) , ¯(0) 2 ϕ(0) - H ϕ˜(τ/2)

1 ρ¯− (τ/2) , ¯(τ/2) 3 ϕ(0) - H ϕ(τ/2)

1 ρ¯− (τ/2) , ¯(τ/2) 4 ϕ(τ/2) - H ϕ(τ)

¯(τ) 5 A˜ (0) K A(τ/2)

¯(τ) 6 A(τ/2) K- A(τ)

- t t = 0 t = τ/2 t = τ

˙ X ¯ iAJ (t) = KJLAL(t) L  n  1 s (1) (1) (1) (1) X (1) −1 X (1) (1) (1) iϕ˙ (t) = 1 − P h ϕ (t) + ρ¯ H¯ h ϕ (t) j  j rkl r l  k,l=1 jk r=1

......

48 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Note on Treating Non-Adiabatic Systems

H = T 1 + W

To include the electronic degree of freedom use either: Multi-set

ns n1 nf f X X X (s) Y (s,κ) Ψ(q ,..., q , t) = ... A (t) ϕ (qκ, t) 1 f j1...jf jκ s=1 j1=1 jf =1 κ=1

Or Single-Set:

n1 nf ns f X X X Y (κ) Ψ(q1,..., qf , t) = ... Aj ...j ,s(t) ϕ (qκ, t) s 1 f jκ | i j1=1 jf =1 s=1 κ=1 n n n Xs Xs Xs H = s T s + s Wst t | i h | | i h | s=1 s=1 t=1

49 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Combined Mode Particles

Re-write MCTDH ansatz

n1 np p X X Y (κ) Ψ(q ,..., q , t) = ... A (t) ϕ (Q , t) 1 f j1...jp jκ κ j1=1 jp=1 κ=1

A “particle” may contain more than one coordinate, Qi = (qa, qb,..., qw ) e.g.

X X (1) (2) Ψ(q , q , q , t) = A (t) ϕ (q , t) ϕ (q , q , t) 1 2 3 j1j2 j1 1 j2 2 3 j1 j2 X X (1) (2) = A (t) ϕ (Q , t) ϕ (Q , t) j1j2 j1 1 j2 2 j1 j2

50 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Saving in memory Bench Mark: Pyrazine Spectrum wave length [nm] All 1D functions 1.0

0.8 Mem nf + fnN ∼ 0.6

Now combine d modes in each particle. Intensity 0.4 f d p = d particles with grid lengths of N 0.2 ˜ d If n n save memory. 0.0 ≤ 280 220 240 260 280 wave length [nm] Mem n˜p + pnN˜ d ∼

I full 24D QD If d = f , then full-grid used and n˜ = 1 Figure I 650 MB (205 MB good result) Mem Nf 22 ∼ I ca 2 × 10 MB for Result: MCTDH can treat ca 30 modes “standard” Raab et al JCP (99) 110: 936

51 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Multi-Layer MCTDH (ML-MCTDH) Expand a multi-mode SPF in an MCTDH expansion to create layers:

n1 np p X X Y (κ) Ψ(q ,..., q , t) = ... A (t) ϕ (Q , t) Layer 1 1 f j1...jp jκ κ j1=1 jp=1 κ=1

n1 nQ Q (κ) X X κ,jκ Y (ν) ϕ (Qκ, t) = ... B (t) ν (Rν , t) Layer 2 jκ k1...kQ kν k1=1 kQ =1 ν=1

n1 nR R (ν) X X ν,kν Y (ξ) ν (Rν , t) = ... C (t) ξ (Sξ, t) Layer 3 kν l1...lR lξ l1=1 lR =1 ξ=1 ... = ...

Each layer acts as a set of SPFs for the layer above and a set of coeﬁcients for the layer below. Leads to a recursive sets of variational equations of motion:

Wang and Thoss JCP (2003) 119; 1289 52 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH

˙ X iAJ = ΦJ H ΦL AL h | | i L −1 iϕ˙ (κ) = 1 P(κ) ρ(κ) H (κ)ϕ(κ) − h i −1 iν˙ (ν) = 1 P(ν) ρ(ν) H (ν)ν(ν) − h i ... = ...

135 Mode Quantum Dynamics Photo-induced ET. Spin-Boson Model.

Borelli et al Mol. Phys. (2012) 110: 751

53 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Parametrized Spfs (G-MCTDH)

n1 nf p−n p X X Y (κ) Y (κ) Ψ(Q ,..., Q , t) = ... A (t) ϕ g 1 f j1...jp jκ jκ j1=1 jp=1 κ=1 κ=n+1 Replace single-particle functions with Gaussian functions T T gj (Q, t) = exp Q ζj Q + Q ξj + ηj Propagate parameters λ = ζ, ξ, η { }

p nκ ˙ X −1 X X −1 ∂ iAj = Φk H Φl Al iS gk gl AJκ Sjk h | | i − jk h |∂t i l lk κ=1 l=1

p nκ X −1 X X −1 = Hkl Al iS τkl AJκ Sjk − jk l lk κ=1 l=1 iΛ˙ = C−1Y

Burghardt et al JCP (99) 99:2927 ; Burghardt et al JCP (08) 129:174104 54 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH

ADVANTAGES

I Need more GFs than SPFs,

I BUT set of parameters smaller than no. of grid points

I spatially unrestricted

DISADVANTAGES

I Non-orthogonal basis set - numerically difﬁcult

I Efﬁciency requires approximate integral evaluation 0 00 2 LHA V = V (x0) + V (x x0) + V (x x0) − − I convergence on exact result depends on accuraccy of integrals

55 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH General Scheme

G-MCTDH gives general framework for Quantum — semi-classical — classical dynamics. Can also treat open systems using density matrix formalism.

56 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE MCTDH Alternative Ansatz: MCTDH/G Return to original MCTDH equation and variational derivation:

n1 np p X X Y (κ) Ψ(Q ,..., Q , t) = ... A (t) ϕ (Q , t) 1 f j1...jp jκ κ j1=1 jf =1 κ=1 and now using m X ϕr = gα Dαr ; r = 1, n | i | i α=1

obtain EOMs for the AJ , ϕj and λaα as before, but for SPFs ˙ X −1 −1 X X −1 iDγi = S ρ gα (1 P) H jl ϕl + fmi Dγm S ταβDβi γα ij h | − h i | i − γα ljα m αβ P with P = ϕr ϕr . GWPs are TD basis for SPFs. r | ih | First layer of “Multi-layer G-MCTDH” Römer, Ruckenbauer and Burghardt JCP (13) 138: 064106 57 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Gaussian Wavefunctions Grid-based QD Gaussian Wavepackets In limit of only GWP−→ basis functions G-MCTDH becomes the Variational Multi-conﬁgurational GWP Method: vMCG X Ψ(x, t) = AJ gJ (x, t) J

GWPs long-tradition in time-dependent QD.

I Conceptually simple

I Can be related to semi-classical dynamics

I possible to use for direct dynamics

BUT

I numerically unstable

I convergence properties not clear

I limited to rectilinear coordinates 58 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Gaussian Wavefunctions vMCG Equations of Motion

n X1 Ψ(Q1,..., Qf , t) = Aj (t)gj j=1 EOMs are as for G-MCTDH (above). For frozen GWP basis, the EOMS can be written in terms of the centre coordinate and momentum as

plβ 1 X q˙ = + Im C−1 Y˜ lβ m 2ζ lβmα mα β lβ mα ˙ 0 X −1 ˜ plβ = Vlβ + Re ClβmαYmα − mα Most other GWP based methods use basis functions that follow classical trajectories.

59 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Gaussian Wavefunctions Multiple Spawning Martinez and Ben-Nun have developed GWP propagation for non-adiabatic systems. Basic ansatz:

X X (s) (s) Ψ(q, t) = Aj (t)gj (g, t) s j

Variational solution of TDSE for expansion coefﬁcients

p nκ ˙ X −1 X X −1 ∂ iAj = Φk H Φl Al iS gk gl AJκ Sjk h | | i − jk h |∂t i l lk κ=1 l=1

I AJ as vMCG.

I GWPs follow classical trajectories as Heller.

If basis set complete, and integrals exact, full solution.

60 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Gaussian Wavefunctions

Basis set expanded in non-adiabatic region I Identify region

I Propagate GWPs though region

I “Spawn” GWPs on other surface

I Rewind time

I Propagate coupled functions Different criteria to place functions so as to conserve energy etc. Very efﬁcient Martinez et al JPC (96) 100: 7884 Ben-Nun and Martinez JCP (98) 108: 7244

61 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Gaussian Wavefunctions

E.g. Retinal Isomerisation

6 diabatic states all DOFs. 3 runs with 10 starting GWPs

(Ben-Nun and Martinez JPCA (98) 102: 9607) Tunneling can also be simulated but only if barrier position known (Ben-Nun and Martinez JCP (00) 112: 6113)

62 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Trajectory Based Methods Trajectory Surface Hopping Classical trajectory swarm sampling surfaces. Non-adiabatic term as d 2 hop. P2→1 = log c2 . (Tully) − dt | |

Advantages: Computationally simple. Good scaling. Disadvantages: Can converge slowly (ensemble of emsemble). Looses nuclear coherence during conical intersection crossing.

63 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Trajectory Based Methods Ehrenfest Dynamics Assume a single conﬁguration and a TD electronic function X Ψ(q, r, t) = A(t)χ(q, t)ψ(r, t) . (74) i Evaluating [qˆ, H] and [pˆ, H] leads to the Ehrenfest theorem ∂ 1 ∂ ∂V qˆ = pˆ ; pˆ = . (75) ∂t h i m h i ∂t h i −h ∂q i The localised nature of the nuclear functions means that the dynamics reduces to classical equations of motion:

Pi ∂V q˙ i = ; p˙ i = (76) m − ∂q q=qi with the potential averaged over electronic states

X ∗ V = ψ(t) Hel ψ(t) = c (t) ψi Hel ψj cj (t) (77) h | | i i h | | i ij

64 / 103 Non-Adiabatic Dynamics Methods for Solving the TDSE Trajectory Based Methods Coherent Coupled States / Multi-Conﬁgurational Ehrenfest

CCS is from the ﬁeld of semi-classical dynamics. Superposition of trajectory-guided frozen Gaussians that follow Ehrenfest forces:

q˙ = p ∂V p˙ = h− ∂q i hence some quantum effects in GWP propagation. Various tricks to help expand basis set and acheive good results. Shalashilin and Child JCP (01) 115: 5637 Shalashilin JCP (10) 132; 244111

Compared to vMCG: Shalashilin and Burghardt JCP (08) 129: 084104

65 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model A Simple Hamiltonian: The Vibronic Coupling Model X Assume diabatic basis: Ψ(Q, r) = φα(Q)ψα(r; Q) α H(Q) = T(Q) + W(Q) ω ∂2 Tˆ + V 0 = i + Q2 α α 2 ∂Q2

Wαβ = ψα Hel ψβ h | | i 0 X ∂ ∂ Wαβ Vαδαβ + εα + ψα Hel ψβ ψα Hel ψβ Qi + ... ≈ ∂Qi h | | i ∂Qi h | | i i | {z }

κ , λ = 0 if Γα Γi Γβ A1 i i 6 × × ⊇ Köppel et al Adv. Chem. Phys. (1984) 57: 59 Worth et al, Int. Rev. Phys. Chem. (08) 27: 569 66 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model

H H Butatriene photoelectron spectrum CCC C H H 18 modes D2h 2 2 X B2g ; A B2u 2 X ωi ∂ 2 1 0 1 0 H = − 2 + Qi + 2 ∂Q 0 1 0 2 i i 4 (1) ! X κi 0 0 λ + Qi + QA 0 κ(2) λ 0 u i∈Ag i + ...

∂H hψα| |ψβ i B2g × B2u = Au ∂Qi

linear: 5 modes, 16 parameters bilinear: 18 modes, 79 parameters Z ∞ I(ω) dt Ψ(0) Ψ(t) eiωt ∼ −∞ h | i

67 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Butatriene Cation PES X˜ /A˜

CoIn

11 10.5 FC 10 · · · Xmin V [eV] 9.5 TS 9 Amin · 8.5 ·

90 60 30 -2 0 -1 θ (deg) 0 -30 1 2 -60 Q 3 14 4 -90

68 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Butatriene Dynamics

X~ A~ N MCTDH standard

11 MB MB 10 PES 9 2 0.9 16.4 8 Energy [eV] 4 5 10.3 2.5 × 10 0.1 17 18 431.5 1.5 × 10

Density [arb.] 0.0 0 fs

0.1

Density [arb.] 0.0 10 fs

0.1

Density [arb.] 0.0 20 fs

0.1

Density [arb.] 0.0 30 fs

0.1

Density [arb.] 0.0 40 fs 4 2 2 90 60 0 60 0 30 0 -2 30 0 -2 -30 Q14 -30 Q14 θ -60-90 -4 θ -60-90 -4

69 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Diabatisation

The VC Hamiltonian is referred to as diabatisation by ansatz.A similar, but more powerful version using internal coordinates has been developed by Yarkony (Zhu and Yarkony JCP (16) 144: 044104). Other procedures to provide diabatic states rely on using properties that are assumed to be smooth e.g.

I dipoles + quadrupoles (DQ Method Hoyer et al JCP (16) 144: 194101

I orbitals (Boys localisation Subotnik et al JCP (08) 129: 244101

I generalised Mulliken-Hush ET (Cave and Newton CPL (96) 249: 15)

I overlap wrt a reference MRCI wavefunction (Simah JCP (99) 45: 2193) - Molpro

70 / 103 REPORTS

1ps* state can be populated by a vibronically the overall distribution remains centered at rationalized by assuming that the skeletal 12 13 È j1 5 @ induced transition ( , ). H atom TOF spectra TKER 7000 cm ( ). mode(s) that promote the S1 S0 absorption Y measured after excitation to the pyrrole (S1) These observations indicate a remarkable are largely conserved during the pyrrole (S1) state show structure that is attributable to degree of vibrational adiabaticity in the dissoci- product evolution and thus acting simply as

population of selected vibrational levels of the ation of pyrrole (S1) molecules. Recall that the spectators to the N–H bond fission. The mean Non-Adiabatic Dynamics 5 @ ground state pyrrolyl radical ( ). The TKERv 0 0 S1 S0 transition acquires intensity through TKER of products arising via such vibrationally ÈdE peak is discernable in all spectra, enabling vibronic interaction. The photoexcited S1 adiabatic dissociations should be ,the The Hamiltonian D 0 T determination of 0(pyrrolyl–H) 32850 level(s) will thus involve one or more quanta decrease in potential energy on passing from 40 cmj1. Replacing one CH moiety in the five- of the mode(s) promoting the vibronic tran- the barrier region to the dissociation asymptote The Vibronic Couplingmembered Model ring (pyrrole) by N (imidazole) thus sition, with the requirement that the overall (Fig. 1A). dE can be estimated from the dif- has little effect on the N–H bond strength. All combination mode be nonsymmetric. Each ference between the energetic threshold for D other peaks in the TKER spectra are attribut- such level will constitute a lifetime-broadened parent absorption and 0(pyrrolyl–H); the value @ È j1 able to pyrrolyl fragments with one or more vibronic resonance within the overall S1 S0 so derived ( 7000 cm ) matches the mean of l quanta of vibration in nonsymmetric skeletal absorption, and thus be populated with a phot- the fast H atom distribution in all measured modes. Different H þ pyrrolyl (v) product dependent efficiency. The identities of the TKER spectra (5, 14–16). H þ pyrrolyl pro- l 9 dE Heteroaromaticchannels display different Photodissociation recoil anisotropies, product states formed at any given phot,and ducts recoiling with TKER (of which the l 0 and their relative showings vary with phot,but their state-specific recoil anisotropies can be v 0 products are the most extreme example) arise from vibrational to translational energy transfer in the exit channel, e.g. from modes Fig. 1. Structural formu- R involving N–H motion which disappear as N-H las of imidazole, pyrrole, increases. and phenol, together with In contrast to imidazole and pyrrole, the S schematic PE profiles for 1 state of phenol is a bound 1pp* state. The S @ the ground and first excited 1 S origin band shows resolvable rovibronic 1pp*and1ps*statesof 0 (A)pyrroleand(B) phenol, structure, analysis of which has established the 1 ¶ @ 1 ¶ plotted against the NjH A X A transition symmetry, the wave- 17 and OjHstretchcoor- length of the band origin (275.11 nm) ( ), and R R the excited state lifetime (e2ns)(18). Time- dinates, NjH and OjH, respectively [adapted resolved pump-probe studies confirm IC to the from (1)]. S0 state as the dominant nonradiative decay 0 19 pathway for phenol (S1,v 0) molecules ( ). Replacing the OH hydrogen by deuterium causes a È100-fold reduction in the nonradiative decay rate, implying a dominant role for OjH vibrations as acceptor modes in the IC process (20). 1ps 1 µ The higher lying S2( *) state has A symmetry, a much smaller transition moment 21 from the S0 state ( ), and is dissociative with R respect to extending OjH. Its PES thus exhibits CIs with both the S1 and S0 states (Fig. 1B) (1, 22, 23). Given the weakness of the @ S2 S0 transition, we assume that the S2 state is populated by radiationless transfer from the 1pp optically prepared S1( *) state or, at higher energies, from another 1pp* state that domi- nates the absorption at l G 220 nm (24). Other experimental evidence for OjH bond fission after UV excitation of phenol comes from a

Ashfold et al Science (06) 312: 1637 Excitation to ππ∗ states Dissociation after crossing to πσ∗ states l 0 Fig. 2. (A) TOF spectrum of H atoms from photolysis of imidazole at phot arise from photoexcitation of ground state molecules carrying one quantum n 234 nm. (B) TKER spectrum obtained from the data in (A) using Eq. 1 and of N–H out-of-plane bending vibration, 21.(C)Plotshowingvariationof m 0 0 n 0 j1 R 67.07 dalton. Peaks due to population of the v 0and 6 1and2 TKERv 0 0 with photon energy (both in cm ). The intercept gives D 0 T j1 71 / 103 states of imidazolyl are labeled; the weak satellites appearing at higher TKER 0(imidazolyl–H) 33240 40 cm .

1638 16 JUNE 2006 VOL 312 SCIENCE www.sciencemag.org Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Photodissociation of Aniline

Vertical Excitation Energies at FC using aug-cc-pVDZ State EOM-CCSD EOM-CCSD(T) Experimental 0 XA˜ 0.00 0.00 0.00 00 A˜(ππ∗) A 4.77 4.21 4.40a 0 B˜(πσ∗/3s) A 5.02 4.69 4.60b ˜ 0 C(3pz ) A 5.67 5.31 - ˜ 00 D(3py ) A 5.77 5.42 - 0 E˜ (ππ∗) A 5.85 5.42 5.39a ˜ 0 F(3pz ) A 6.38 6.05 - ˜ 00 G(3dyz ) A 6.39 6.04 -

Wang et al JCPA (13) 117: 7298

72 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Model Hamiltonian

H = T + W with W(Q) = W(0) + W(1)(Q) + W(2)(Q) + . ··· Usual Vibronic Coupling model in mass-frequency scaled normal modes except for N—H bonds: take mode combination

Q35 Q36 34 + (0) X ωα ω h i W (Q˜ ) = E + Q˜ 2 + Q˜ 2 + Q˜ 2 + ω−Q˜ Q˜ ii i 2 α 2 35 36 35 36 α=1

73 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Cuts through PES

74 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Cut through PES along N—H Morse oscillators for bound states, avoided crossing for B˜(πσ∗/3s) ( r ) 1 h i2 W (Q˜ ) = ν + ν − (ν − ν )2 + 4 ∆ tanh ρQ˜ , 22 36 2 b d b d 36

2 h ˜ ˜ b i νb = Db exp(−αb(Q36 − Q36,0)) − 1

˜ ˜ d νd = A exp(−αd (Q36 −Q36,0))+Dd .

Topologies EOM-CCSD. Corrected at Q0 using EOM-CCSD(T).

75 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Aniline Spectrum

Calculated Experiment

(a) (b)

220 240 260 280 300 wavelength / nm

(b)

Lower band

5 4.8 4.6 4.4 4.2 4 energy/eV 76 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Electronic Absorption Spectrum of Pyrrole

12.0 12.0

(a) Q2 (b) Q12 10.0 10.0

8.0 8.0

6.0 6.0

Energy (eV) 4.0 Energy (eV) 4.0

2.0 2.0

0.0 0.0 -10 -5 0 5 10 -10 -5 0 5 10

12.0 10.0

10.0 8.0

8.0 6.0

6.0 4.0 Energy (eV) Energy (eV) 4.0

2.0 2.0

0.0 0.0 Symmetry Character CASSCF CASPT2 -1 0 1 2 3 4 5 6 -10 -5 0 5 10 R (a.u.)

A1 0.00 0.00 8.0 ∗ 7.0 (e) φ A2 3s/πσ 4.17 5.06 6.0 ∗ 5.0 B1 3s/πσ 4.87 5.86 4.0 A 3p 4.91 5.87 (eV) Energy 3.0 2 z 2.0 ∗ A ππ 6.47 6.01 1.0 1 0.0 ∗ -1 -0.5 0 0.5 1 B2 ππ 7.83 6.24 φ (radians) B1 3pz 5.67 6.69

I 6-state, 9-mode model. 77 / 103 I Strong vibronic coupling between states. I Contribution from excitation to three states. I Intensity borrowing allows excitation to lower lying Rydberg states.

S.P. Neville & G.A. Worth, JCP, 2014, 140, 034317 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Pyrrole: 6-State 9(10)-Mode Model Ignoring ν2 Including ν2 State populations State populations

1 1 S0 S0 B (πσ*) B (πσ*) 1 πσ 1 πσ B1( *) B1( *) total total

0.8 0.8

0.6 0.6

0.4 0.4 Probability of dissociation Probability of dissociation

0.2 0.2

0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 Time/fs Time/fs “Trapped” Flux Flux “Geometry”

78 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Pyrrole: Time-resolved Photo-electron spectroscopy

D 0

4.65 eV

S (πσ*) 2 t 2 S (πσ*) 1

t 1

5.7eV

S 0

Pump-probe with 5.7 eV Wu et al JCP (15) 142: 074312 79 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Benzene: Time-Resolved Spectra Total Ion Count Photo-electron spectra

1.0 0.5

0.9 0.4

0.3 Pump 243 nm 0.8 0 1.0 2.0 3.0 4.0 Probe: 260, 254, 235 0.7 0.6

0.5

0.4 Integrated photoelectron signal photoelectron Integrated

0.3

0.2 0 0.5 1.0 1.5 2.0 Delay (ps) Figure 3

Decay 220 fs + 12 ps Minns et al PCCP (10) 12: 15607 Period 1.2 ps

80 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Benzene Potential Energy Surface Cuts

12.0 12.0

10.0 10.0

8.0 8.0 ν4 + ν16

6.0 6.0

4.0 4.0

2.0 2.0 ν16a ν6a

0.0 0.0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15

10.0 16.0

9.0 14.0

8.0 12.0 ν4 7.0 ν 10.0 1 6.0

5.0 8.0

4.0 6.0

3.0 4.0 2.0

2.0 1.0 CASPT2 (6,6)/Roos

0.0 0.0 -15 -10 -5 0 5 10 15 -8 -6 -4 -2 0 2 4 6 8

81 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Simulated Absorption Spectra 5-state 8-mode model.

5.8 6 6.2 6.4 6.6 6.8 7

3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6

B2u B1u + E1u Penfold and Worth JCP (09) 131: 064303

82 / 103 Non-Adiabatic Dynamics The Hamiltonian The Vibronic Coupling Model Importance of Triplet States and ISC 3 1 Need SOC between states: Ψ HSO Ψ h | | i −1 At FC point, S1/T2 coupling zero. In general SOC small (< 5 cm )

5-state model (S0, S1, T1, T2)

12.0

10.0 S + T (red) S1 (green) 8.0 S1FC (blue)

6.0

energy [eV] T populations (inset) 4.0

2.0

0.0 -10 -5 0 5 10 Prefulvene Vector

Penfold et al JCP (12) 137: 22A548 83 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Ab Initio Multiple Spawning: AIMS

Efﬁciency of spawning perfect for direct dynamics. Run in adiabatic picture.

I Review: Ben-Nun and Martinez Adv. Chem. Phys. (02) 141: 439

I QM/ MM: PYP / GFP Chromophores in solution Virchup et. al JPCB (09) 113: 3280

I Rydberg states in ethylene Mori et al JPCA (12) 116: 808

I TRPEspec of uracil Hudock at al JPCA (07) 111:8500

84 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Surface Hopping

A number of programs / methods. E.g.

I Newton-X (Barbatti)

I SHARC (Gonzalez)

I CPMD (Tavernelli)

Simple, but difﬁcult ot converge and loss of nuclear coherence.

85 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics

Butatriene Dynamics From A~

X~ A~

4 1.48 2 1.38 Q14 0 PES 1.28 RCC2 -2 1.19

4 1.48

2 1.38 +++++++++++++++ Q14 +++++++++++ 0 0 fs +++ +++++ 1.28 RCC2 +++++++++++++++ -2 1.19 Surface Hopping 4 1.48

2 1.38 +++ +++ + ++ ++ v Q14 + R 0 ++ ++++ 10 fs + 1.28 CC2 ++ + +++ +++ + + ++++++++ -2 ++++++++ ++ + 1.19 +++++++++ + + wavepacket dynamics 4 ++ + + 1.48 + + + +++++ + 2 + + + + + 1.38 + + ++ + + + + Q14 + ++ + + + + + + R 0 + + + + + 20 fs ++ + 1.28 CC2 + + + ++ ++++ + ++ + + +++ -2 ++ + + + + ++++ + 1.19 + + 4 1.48 + + +++ ++ + + +++ 2 ++++ + +++ 1.38 + + +++ + + Q14 + + + 0 + + + + 30 fs + 1.28 RCC2 + + ++ + + + +++++ + + + +++ ++ + + -2 + + ++++ + + 1.19 + +++++ + + + + 4 1.48 + + ++ + + +++ + ++ + ++++ 2 ++ + ++ + + + 1.38 + + + +++++ + + + Q14 + + + +++ + 0 + + + + + 40 fs + 1.28 RCC2 + + + + + ++++ + + + + + ++ + + +++++ ++ ++ -2 + + ++ + +++++ +++++ 1.19 ++++ + + +++ ++++++ + ++ + + + ++++++ + + + + -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 θ θ

86 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Salicylaldimine Test Case: 2D Proton transfer Hamiltonian in normal modes ﬁtted to RHF/3-21G*

2 4 X ωκ ∂ X = + 2 + n H 2 qκ Anq1 2 ∂qκ κ=1,18 n=1 2 2 +B11q1q18 + B22q1 q18 3 3 +B31q1 q18 + B13q1q18

2D Salicylaldehyde Proton Transfer Flux: Full QD v GWPs 0.15 ν1 ν18

0.1

0.05

0 Flux

-0.05

-0.1

-0.15 0 20 40 60 80 100 16 / 32 GWPs Time [fs] 87 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Trajectories with 16 GWPs Classical vMCG

GWP centre coordinate 16 classical GWPs GWP centre coordinate 16 GWPs 4 4 2 2 0 v1 [au] 0

v1 [au] -2 -2 -4 -4 0 20 40 60 80 100 0 20 40 60 80 100 Time [fs] Time [fs]

GWP trajectory in phase-space 16 classical GWPs GWP trajectory in phase-space 16 GWPs 8

8 6

6 4

4 2

2 0 Mom v1 0 -2 Mom v1 -2 -4

-4 -6

-6 -8 -6 -4 -2 0 2 4 6 -8 -6 -4 -2 0 2 4 6 v1 [au] v1 [au] Richings et al Int. Rev. Phys. Chem. (15) 34: 269

88 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics 4D model: Linear Coupling Autocorrelation function: State Populations:

Full QD v 60, 60 GWPs. Full QD v 60, 60 GWPs. 0.4 1

0.35 0.8 0.3

0.25 0.6 0.2 |C(t)| Pop 0.15 0.4

0.1 0.2 0.05

0 0 0 50 100 150 200 0 20 40 60 80 100 120 Time [fs] Time [fs]

QD basis size: 4060 SPFs, 355,000 primitives

89 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics

Pyrazine wavepacket on S1 GWPs: 10,10 MCTDH

Time : 50.000 Time : 50.000

0.51.52.53.5 01234 0.020.040.060.080.120.14 0.1 0 6 6 4 4 2 2 v6a [au] 0 v6a [au] 0 -2 -2 -4 -4 -6 -6

-6 -4 -2 0 2 4 6 -4 -2 0 2 4 v10a [au] v10a [au]

90 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Direct Dynamics

I For integrals gj H gk , Quantum chemistry to second order. h | | i I Gradients and Hessians directly from quantum chemistry. I Store results in a database (energy, gradient, Hessian)

Ideally use adiabatic PES in direct dynamics as they are readily available from quantum chemistry packages.

I States interact via the non-adiabatic coupling terms (NACT) ˆ ψa Hel ψb Fab = h |∇ | i Vb Va −

I NACTs go to inﬁnity at a conical intersection and adiabatic PES become non-differentiable at such points.

Problem for LHA. Avoid these problems by transforming to the diabatic picture. How can we define diabatic states on-the-fly? 91 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Diabatisation by Propagation Adiabatic - Diabatic transformation, S, defined by the differential equation S = FS ∇ − where F is derivative coupling. Exact for complete set of states.

I At each GBF position evaluate F. I Choose S = 1 at the initial point of the propagation. I Solve for S by propagating from the nearest, previously calculated, point.

1 Z −1 1 Z S (q + ∆q) = I + F.∆q I F.∆q S (q) 2 − 2

I PES and gradients transformed by S. I Applicable to any number of states.

Richings and Worth J. Phys. Chem. A (2015) 119: 12457 92 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Butatriene Model

CoIn

11 10.5 FC 10 · · · Xmin V [eV] 9.5 TS 9 Amin · 8.5 ·

90 60 30 -2 0 -1 θ (deg) 0 -30 1 2 -60 Q 3 14 4 -90

I Ground state normal modes from CAS(6,6)/3-21G* (G03).

I 2-mode model: torsion angle and symmetric central C-C stretch. I Dynamics run on ﬁrst-excited ion states.

I DD-vMCG using 25 GWPs with propagated diabatisation.

I Powell updated Hessian

93 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Butatriene Ion: Excited-State Manifold

Adiabatic PES Diabatic PES

20 20

18 18

16 16

14 14

Energy/eV 12 Energy/eV 12

10 10 -6 -6 -2-4 -2-4 2 0 2 0 8 6 4 5Au 8 6 4 5Au CoIns found -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 I 20 14Ag 20 14Ag 18 18 I Smooth 16 16 Diabatic 14 14

Energy/eV 12 Energy/eV 12 PES

10 10 -6 -6 -2-4 -2-4 2 0 2 0 Surfaces 8 6 4 8 6 4 I -6 -4 -2 0 2 4 6 5Au -6 -4 -2 0 2 4 6 5Au 20 14Ag 20 14Ag depend on 18 18

16 16 Nstate

14 14

Energy/eV 12 Energy/eV 12

10 10 -6 -6 -2-4 -2-4 2 0 2 0 8 6 4 8 6 4 -6 -4 -2 0 2 4 6 5Au -6 -4 -2 0 2 4 6 5Au 14Ag 14Ag 94 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Butatriene Ion: Excited State Populations 1.0 2 States 3 States 4 States 0.8

0.6

0.4

0.2 First Excited State Population

0.0 0 20 40 60 80 100 Time/fs

I Initial wavepacket at origin of the coordinate system on A˜-state. I Rapid de-population of the A˜-state as the CoIn is encountered followed by partial re-population and oscillation.

I DD-vMCG results follow full grid-based dynamics.

95 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Formamide Formamide

I Smallest, most stable molecule consisting of HCNO

I Prebiotic Earth

I Found, by spectral molecular survey, on Hale-Bopp[1]

I "Tentatively" found in IR-spectra of interstellar ices[2] [3] I Decomposition pathways studied

I As yet, no excited state studies

[1]D. Bockelée-Morvan, et al, Astron. Astrophys. (2000) 353 1101-1114 [2]S. Raunier, et al., Astron. Astrophys. (2004) 416 165-169 [3]V. S. Nguyen, et al., J. Phys. Chem. A (2013) 117 2543-2555

96 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Formamidic Acid: Electronic Structure

I SA-CAS(10,7)/6-31++G*

I 6 states

State VEE/eV Transition Dipole/au Character S1 6.761 0.234 nπ* S2 7.082 0.010 πσ* S3 7.747 0.535 nπ* S4 10.245 1.482 ππ* S5 10.781 0.351 πσ*

97 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Formamidic Acid: Direct Dynamics Potential Surfaces

Initial Vertical Excitation to bright ∗ S5(ππ ). 60fs 24 GWPs.

14 12 10 8 V [eV]" 6 4 2 Diabatic Populations 0 10 5 0 v10 -10 -5 -5 1 0 5 S0 v6 10 -10 S1 0.8 S2 S3 S4 Trajectory Analysis 0.6 S5 Product No. 0.4 Population O-H break 14 0.2 HN-CO 6 0 N-CH-O 2 0 10 20 30 40 50 Time [fs] NH + CO 1 HNC-H-O 1

98 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Potential Surfaces Cut along O–H bond Cut along O–H OOP

Adiabatic Adiabatic 14 14

12 12

10 10

8 8

V [eV] 6 V [eV] 6

4 4

2 2

0 0 -10 -5 0 5 10 -10 -5 0 5 10 v10 v2 Diabatic Diabatic 14 14

12 12

10 10

8 8

V [eV] 6 V [eV] 6

4 4

2 2

0 0 -10 -5 0 5 10 -10 -5 0 5 10 v10 v2 99 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Ozone Photodissociation

Two dissociation pathways 1 1 1 Theory O3(X A1) −→ O(2p D2) + O2(A ∆g ) 1 3 3 − O3(X A1) −→ O(2p P2) + O2(X Σg )

Chappuis Band: Expt I 400 - 750 nm 1 1 I Coupled 1 A2/1 B1 Detailed QD study. Full valence CASPT2/DZP. Woywod et al JCP (97) 107:7282 Adjusted Flöhtmann et al JCP (97) 107: 7296

100 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics DD-vMCG of Ozone: Spectrum

20 Intensity

10 CASPT2 5 0 12 14 16 18 20 22 24 26 Energy/1000 [cm-1]

Intensity 20

10 CAS 5 0 12 14 16 18 20 22 24 26 Energy/1000 [cm-1] 101 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics DD-vMCG of Ozone: State Populations

0.8

0.6

Population 0.4

0.2 CASPT2 0 0 20 40 60 80 100 Time [fs]

0.8

0.6

Population 0.4

0.2 CAS 0 0 20 40 60 80 100 Time [fs] 102 / 103 Non-Adiabatic Dynamics The Hamiltonian Direct Dynamics Conclusions Variational time-dependent basis sets are a powerful way of obtaining the full solution to the TDSE including non-adiabatic effects.

I MCTDH provides a complete framework.

I ML-MCTDH grid-based for truly large systems - simple PES

I G-MCTDH ﬂexible route to approximate dynamics - any PES but restricted coordinates I G-MCTDH vMCG GWP methods −→ −→ I still complete solution possible

I numerically difﬁcult

I Vibronic Coupling Model good for short-time dynamics

I More ﬂexibility required for complete description of photochemistry

I Direct Dynamics present state-of-the art: DD-vMCG, AIMS, ...

Present bottleneck: Electronic Structure theory! 103 / 103