PES Avoided Crossings Conical Intersections Conclusions

Potential energy surfaces and surface crossings

Felix Plasser

Institute for Theoretical Chemistry, University of Vienna

COLUMBUS in ChinaTianjin, October 10–14, 2016

F. Plasser Potential energy surfaces and surface crossings 1 / 29

PES Avoided Crossings Conical Intersections Conclusions

Potential Energy Surfaces

What are potential energy surfaces?

Electronic Schrödinger Equation

H(R) Ψ0(R,x) = E0(R) Ψ0(R,x)

H(R) Ψ1(R,x) = E1(R) Ψ1(R,x)

...

H(R) Ψn(R,x) = En(R) Ψn(R,x)

R Nuclear coordinatesx Electronic coordinates

EI(R) Potential energy with changing nuclear coordinates→ Potential energy (hyper)surface

F. Plasser Potential energy surfaces and surface crossings 4 / 29

PES Avoided Crossings Conical Intersections Conclusions

Potential Energy Surfaces

Special points on the PESI Local minimaI Transition statesI Conical intersections

F. Plasser Potential energy surfaces and surface crossings 5 / 29

PES Avoided Crossings Conical Intersections Conclusions

Potential Energy Surfaces

Dynamcs on the PESI Vertical excitationI Motion on the PESI Transitions between different PES

F. Plasser Potential energy surfaces and surface crossings 6 / 29

PES Avoided Crossings Conical Intersections Conclusions

Avoided Crossing

Potential curveI Selenocroleine- Twist around double bond- T2/T1

I T1 - Two minima:nπ∗ and ππ∗ character

I States cross around 55◦

I T1 and T2 exchange character

y z

x

Se

C

C

C

H

H

HH

Se

C

C

Cθ

1 F. Plasser et al. J. Chem. Theory Comput. 2016, 12, 1207.F. Plasser Potential energy surfaces and surface crossings 8 / 29

PES Avoided Crossings Conical Intersections Conclusions

Avoided Crossing

ZoomI Avoided crossing at 58◦

- Diabatic states (nπ∗, ππ∗)follow straight lines

- Adiabatic states changecharacter

- No crossing

y z

x

Se

C

C

C

H

H

HH

Se

C

C

Cθ

F. Plasser Potential energy surfaces and surface crossings 9 / 29

PES Avoided Crossings Conical Intersections Conclusions

Avoided Crossing

State overlapI Orthogonal states〈Ψ1(R0)|Ψ2(R0)〉 = 0

I State character changes〈Ψ1(R0)|Ψ2(R1)〉 ≈ 1

I Difference quotient⟨Ψ1(R0)

∣∣∣Ψ2(R1)−Ψ2(R0)R1−R0

⟩≈ 1

R1−R0

I Nonadiabatic coupling⟨Ψ1(R0)

∣∣ ∂∂RΨ2(R0)

⟩≈ 1

R1−R0

R0 = 50◦, R1 = 65◦

F. Plasser Potential energy surfaces and surface crossings 10 / 29

PES Avoided Crossings Conical Intersections Conclusions

Mathematical model

Diabatic statesΦn nπ∗ state wavefunctionΦπ ππ∗ state wavefunction

2× 2 Hamiltonian (En(R) c(R)c(R) Eπ(R)

)En = 〈Φn| H |Φn〉 Energy of the nπ∗ stateEπ = 〈Φπ| H |Φπ〉 Energy of the ππ∗ statec = 〈Φn| H |Φπ〉 Diabatic coupling

F. Plasser Potential energy surfaces and surface crossings 11 / 29

PES Avoided Crossings Conical Intersections Conclusions

Mathematical model

Diagonalization(E1 00 E2

)=

(cos η sin η− sin η cos η

)(En cc Eπ

)(cos η − sin ηsin η cos η

)E1 Adiabatic energy of the T1 stateE2 Adiabatic energy of the T2 state

η(R) Diabatic/adiabatic mixing angle

F. Plasser Potential energy surfaces and surface crossings 12 / 29

PES Avoided Crossings Conical Intersections Conclusions

Mathematical model

Diagonalization(E1 00 E2

)=

(cos η sin η− sin η cos η

)(En cc Eπ

)(cos η − sin ηsin η cos η

)Under what conditions do the adiabatic states cross (E1 = E2)?

E1,2 =En + Eπ

2±

√(En − Eπ

2

)2

+ c2

I En(R) = Eπ(R)

I c(R)2 = 0

I Two independent conditions→ Non-crossing rule for a 1-dimensional curve

(same spatial and spin symmetry)→ Conical intersections in multidimensional space

F. Plasser Potential energy surfaces and surface crossings 13 / 29

PES Avoided Crossings Conical Intersections Conclusions

Mathematical model

2× 2 Transformation(Ψ1(R)Ψ2(R)

)=

(cos η(R) sin η(R)− sin η(R) cos η(R)

)(ΦnΦπ

)Ψ1(R) Wavefunction of the adiabatic T1 stateΨ2(R) Wavefunction of the adiabatic T2 stateη(R) Diabatic/adiabatic mixing angle

F. Plasser Potential energy surfaces and surface crossings 14 / 29

PES Avoided Crossings Conical Intersections Conclusions

Mathematical model

2× 2 Transformation(Ψ1(R)Ψ2(R)

)=

(cos η(R) sin η(R)− sin η(R) cos η(R)

)(ΦnΦπ

)

Nonadiabatic coupling

h12 =

⟨Ψ1

∣∣∣∣ ∂∂RΨ2

⟩=

=

⟨Φn cos η + Φπ sin η

∣∣∣∣−Φn cos η∂η

∂R− Φπ sin η

∂η

∂R

⟩

h12 =

⟨Ψ1

∣∣∣∣ ∂∂RΨ2

⟩= − ∂η

∂R

1 F. Plasser, H. Lischka J. Chem. Phys. 2011, 134, 034309.F. Plasser Potential energy surfaces and surface crossings 15 / 29

PES Avoided Crossings Conical Intersections Conclusions

Avoided Crossing

I Nonadiabatic coupling

h12 = − ∂η∂R

I Integrate∫ R1

R0

h12dR = η(R1)− η(R0)

I Full state rotation

η(R0) = 0, η(R1) = π/2

F. Plasser Potential energy surfaces and surface crossings 16 / 29

PES Avoided Crossings Conical Intersections Conclusions

Nonadiabatic Coupling

Nonadiabatic couplingI Indicator of how fast the adiabatic states change their characterI Derivative of the mixing angle

I Vector in coordinate spaceI h = 〈Ψ1|5Ψ2〉

I Physical meaning- Electronic states interact- Transitions between the states→ Nonadiabatic effects

F. Plasser Potential energy surfaces and surface crossings 17 / 29

PES Avoided Crossings Conical Intersections Conclusions

Conical Intersections

Polyatomic molecule

2× 2 Hamiltonian (En(R) c(R)c(R) Eπ(R)

)R Nuclear coordinate vector

Degeneracy if

I En(R) = Eπ(R)

I c(R)2

= 0

I Two coordinates have to be adjusted→ Conical intersection

F. Plasser Potential energy surfaces and surface crossings 19 / 29

PES Avoided Crossings Conical Intersections Conclusions

Conical Intersections

Conical IntersectionI Two-dimensional branching space

Diabatic pictureI Tuning mode:

(En(R)− Eπ(R))→ 0

I Coupling mode: c(R)→ 0

Adiabatic pictureI Gradient difference vector− Pointing to the intersectionI Nonadiabatic coupling− Circling the intersection

F. Plasser Potential energy surfaces and surface crossings 20 / 29

PES Avoided Crossings Conical Intersections Conclusions

Conical Intersections

I F. Plasser Potential energy surfaces and surface crossings 21 / 29

PES Avoided Crossings Conical Intersections Conclusions

Crossing Seams

I 2 degeneracy lifting coordinatesI 3N − 8 coordinates keep the degeneracy→ Intersection spaceI Crossing seamI Hyperline in coordinate spaceI Hyperpoint on the potential energy (hyper-)surface

F. Plasser Potential energy surfaces and surface crossings 22 / 29

PES Avoided Crossings Conical Intersections Conclusions

Crossing Seams

I Crossing seam- Set of structuresI Contains the

Minimum on thecrossing seam (MXS)

F. Plasser Potential energy surfaces and surface crossings 23 / 29

PES Avoided Crossings Conical Intersections Conclusions

Crossing Seams

I Example: ethyleneI Crossing seam over

different types ofstructures

1 Barbatti et al. J. Chem. Phys. 2004, 121, 11614.F. Plasser Potential energy surfaces and surface crossings 24 / 29

PES Avoided Crossings Conical Intersections Conclusions

Crossing Seams

I Minimum on the crossing seam (MXS) orI Minimum energy conical intersection (MECI)

I Likely structure for an electronic transition- Better : dynamics

I Several local minima can exist

F. Plasser Potential energy surfaces and surface crossings 25 / 29

PES Avoided Crossings Conical Intersections Conclusions

Crossing Seams

Cytosine - S1/S0 MXSI Strongly distorted structures- Ground state destabilized- Excited state stabilized (or weaklydestabilized)

F. Plasser Potential energy surfaces and surface crossings 26 / 29

PES Avoided Crossings Conical Intersections Conclusions

Conclusions

Excited statesI Many close-lying potential energy surfacesI Several local minimaI Conical intersection between the surfaces- Branching space + Intersection space- Crossing seams

F. Plasser Potential energy surfaces and surface crossings 28 / 29

PES Avoided Crossings Conical Intersections Conclusions

Conclusions

Nonadiabatic coupling vectorsI h = 〈Ψ1|5Ψ2〉I Related to changes in state charactersI Component of the branching spaceI Drive interstate transitions

F. Plasser Potential energy surfaces and surface crossings 29 / 29