Theoretical Photochemistry WiSe 2017/18

Lecture 2

Irene Burghardt (burghardt@chemie.uni-frankfurt.de)

http://www.theochem.uni-frankfurt.de/teaching/ −→ Theoretical Photochemistry

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Topics

1. Photophysical Processes

3. Wavepackets

5. The Franck-Condon picture of electronic transitions

6. What do we measure experimentally?

2. The Born-Oppenheimer approximation

4. Beyond Born-Oppenheimer – non-adiabatic transitions

7. Conical intersections

8. Examples: Ethene, Protonated Schiff Bases (Retinal), Azobenzene

9. Some electronic structure aspects

10. Dynamics: trajectories or wavefunctions?

11. Wavefunction propagation techniques

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12. Trajectory surface hopping techniques

13. Non-linear optical spectroscopy: calculation of spectroscopic signals

14. Extended systems: Excitons, light-harvesting, etc.

15. Solvent/environmental effects

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Literature

1. P. W. Atkins and R. Friedman, Molecular Quantum Mechanics, 5thEdition, Oxford University Press (2011).

2. D. Tannor, Introduction to Quantum Mechanics: A Time-DependentPerspective, University Science Books (2006).

3. M. Klessinger, J. Michl, Excited States and Photochemistry of OrganicMolecules, VCH-Wiley (1995).

4. I. N. Levine, Quantum Chemistry, 6th Edition, Pearson InternationalEdition (2009).

5. F. Jensen, Introduction to Computational Chemistry, 2nd Edition, Wiley(2007).

6. C. J. Cramer, Essentials of Computational Chemistry – Theories andModels, 2nd Edition, Wiley (2004).

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Starting point: the molecular Hamiltonian

all electrons and nuclei

H = Te + TN + Ve + VN + VeN

= −Ne∑i=1

h2

2me

∇2i −

NN∑a=1

h2

2Ma

∇2a

+e2

4πε0

{Ne∑i=1

Ne∑j>i

1

rij+

NN∑a=1

NN∑b>a

ZaZb

rab−

Ne∑i=1

NN∑a=1

Za

ria

}

ih ∂∂tψ = Hψ Hψ = Eψ

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But we might eventually do classical moleculardynamics (MD) simulations, e.g., for proteins

• Quantum classical transition due to decoherence• Do any quantum effects survive?

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Of course the electrons always need a quantummechanical treatment. . .

. . . but they are usually integrated out so as to yield effective potentialsfor the nuclear motion (Born-Oppenheimer approach)

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Two Steps

1 – Born-Oppenheimer approximation: separate the electronic and nuclearmotions and generate effective potentials for the nuclear motion

2 – follow the dynamics of the nuclei, either using quantum dynamics(time-dependent Schrodinger equation) or else using classical dynamics(Newton’s equations)(∗)

(∗) . . . or else using a varietyof semiclassical and “mixed”quantum-classical approaches

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Step 1. The Born-Oppenheimer Approximation

Max Born Robert Oppenheimer

• Expansion in orders of the mass ratio m/M ∼ 1/18369

Step 2: calculate the dynamics of the nuclei

ih∂Ψ

∂t= (T + V )Ψ

or

q =p

mp = −∇V

In most biological applications, Newton’s equations are used!

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Molecular Potential Energy Surfaces (PES’s)

• simplest picture: bonding/non-bonding combinations of atomic orbitals

• account for the overall symmetry of the wavefunction11

Born-Oppenheimer – basics

Using the idea of an adiabatic separability of the time scales for electronicvs. nuclear motion, separate the total Hamiltonian

HT = Te + Ve + TN + VN + VeN

= TN + Hel

and solve the electronic Schrodinger equation first – disregarding TN :

Helψn(rel|R) = εn(R)ψn(rel|R)

• The eigenvalues εn(R) depend parametrically on the nuclearcoordinate(s) and constitute the Born-Oppenheimer surfaces

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Born-Oppenheimer, cont’d

If we assume that the overall wavefunction (electronic + nuclear) can bewritten as

ΨT (rel, R) = ψn(rel|R)χn(R)

we obtain nuclear motion in terms of the SE (or TDSE) for the nuclearwavefunction χn(R, t) on the nth Born-Oppenheimer surface:

(−h2

2M

∂2

∂R2+ εn(R)

)χn = Eχn ih

∂χn

∂t=(−h2

2M

∂2

∂R2+ εn(R)

)χn

• Thus we have separated the electronic-nuclear problem into two parts:

Helψn(rel|R) = εn(R)ψn(rel|R) ; ihχn =(TN + εn(R)

)χn

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But something isn’t quite right...

• Born-Oppenheimer surfacesfor the I2 molecule

• note that in the BO picture,the nuclei move only on a givenBO surface at a time

• thus, the multiple crossings ofthe I2 potentials are indicativeof a breakdown of the BOapproximation

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What was wrong about our Born-Oppenheimerderivation?

We have neglected that the nuclear kinetic energy operator can have non-zero matrix elements between different electronic wavefunctions (sincethese depend parametrically upon the nuclear coordinates):

〈ψ1(rel|R)|TN |ψ2(rel|R)〉χ2(R) 6= 0

This generates a coupling between different electronic states, and weobtain “non-adiabatic” transitions if the states come energetically closeto each other. A more general wave function ansatz is therefore needed:

ΨT (rel, R) =∑n

ψn(rel|R)χn(R)

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The Group Born-Oppenheimer Approximation

• use the improved ansatz

ΨT (rel, R) =∑n

ψn(rel|R)χn(R)

• where ψn(rel|R) = solutions of the electronic Schrodinger Equation:Helψn(rel|R) = εn(R)ψn(rel|R)

• now integrate over the electronic coordinates and find that the nuclearwavefunctions {χn(R)} are coupled to each other:(−h2

2M

∂2

∂R2+ εn(R)

)χn +

∑m

Λmnχm = Eχn

• where the non-adiabatic couplings are given as

Λmn = −h2

M〈ψm|

∂

∂R|ψn〉

∂

∂R+ 〈ψm|TN |ψn〉

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Non-adiabatic couplings

The non-adiabatic couplings are often re-written as follows:

Λmn = −h2

M〈ψm|

∂

∂R|ψn〉

∂

∂R+ 〈ψm|TN |ψn〉

= −h2

2M

(2Fmn

∂

∂R+Gmn

)

where Fmn = non-adiabatic derivative couplings

where Gmn = non-adiabatic scalar couplings

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Coupled BO surfaces: matrix notation

ih∂

∂t

χ1(R, t)

χ2(R, t)

=

TN + ε1(R) Λ12(R)

Λ21(R) TN + ε2(R)

χ1(R, t)

χ2(R, t)

TN = −h2

2M

∂2

∂R2

Λ12(R) = −h2

M〈ψ1|

∂

∂R|ψ2〉

∂

∂R+ 〈ψ1|TN|ψ2〉

coupling through nuclear kinetic energy!

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Non-adiabatic coupling at an “avoided crossing”

The non-adiabatic coupling becomes very largeat a so-called avoided crossing

This is because the character of the electronicwavefunction changes very rapidly

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Wavepacket motion through an avoided crossing

S1

S0

En

erg

y

Isomerization coordinate

11-cis

rhodopsin

All-trans

photoproduct

Stimulated

emission

800 n

m

1,0

00 n

m 1,0

00 n

m

800 n

m

Photoinduced

absorption

560 n

m

Conical

intersection

2 1

1

a

2

3

1

2

3

1

b

c

d

Pum

p 5

00 n

m

Polli et al., Nature 467, 440 (2010)

• example: isomerisation of retinal20