potential energy surfaces and surface crossings...potential energy surfaces and surface crossings...
TRANSCRIPT
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)
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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
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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