Quantum Dynamics of Electronically Excited Molecules:Current Developments and Future Challenges

Susanta Mahapatra

School of ChemistryUniversity of Hyderabad, Hyderabad 500046

Email: smsc@uohyd.ernet.in, Tel: +91-40-23134826; URL: http://172.16.56.51/~sm/

Indo-German workshop, September 7-8, 2009, Neuss, Germany

Central Theme

Entanglement electronic and nuclear motion in molecules;ubiquitous, dynamics of excited electronic states of

Polyatomic Molecules

Crossings of electronic potential energy surfacesNuclear motion is quantum mechanical

A central quest in the new vista of chemical dynamics

Electron-Nuclear Coupling f

hγ

i

Noncrossing Rule : von Neumann and Wigner

“Electronic states of a diatomic molecule do not cross, unless permitted by symmetry”

J. von Neumann and E. Wigner, Physik. Z. 30, 467 (1929)E. Teller, J. Phys. Chem. 41, 109 (1937)Landau & Lifshitz, Quantum Mechanics: Nonrelativistic Theory, 1965H. C. Longuet Higgins, Proc. R. Soc. (London) Ser. A 344, 147 (1975)G. Herzberg and H. C. Longuet Higgins, Discuss. Faraday Soc. 35, 77 (1963)C. A. Mead, J. Chem. Phys. 70, 2276 (1979)

It does not apply to Polyatomic systems !!

Diabatic (?)

Equation of a ellipticalDouble – cone

Adiabatic (?)

Surfaces are degenerate when Δ2 = 0 ; H12

2 = 0

This happens when these terms are independent i.e. they are function of different coordinates

Λ = Linear coupling vector δ = Gradient difference vector

Review of basic concepts

Coordinates

X1=X2=0

22

Diatomics : Only one coordinate

• If the two states have same symmetry they do not crossNon crossing rule

• If the states have different symmetry then H12 = 0 ; they can cross

Polyatomics : Degrees of freedom n > 1

• States can always cross in principle, irrespective of symmetry

Near the degeneracy

Degeneracy is lifted to first – order in the space spanned by the vectors δ and λ

Δ = δ·RH12 = λ·R

J. von Neumann and E. Wigner, Physik. Z. 30, 467 (1929);C. A. Mead, J. Chem. Phys. 70, 2276 (1979)

n-2 dimensional intersection space

2 dimensional branching space

x2

x1

Conical Intersection Glancing Intersection

For linear systems

Crucial: paradigm forsignaling ULTRAFASTdecay of excited molecularstates

Intersections of molecular potential energy surfaces

Consequences:Break-down of the Born-Oppenheimer approximation

Nuclear motion no longer remain confined on a (scalar) adiabatic PES

a nonadiabatic situation

Herzberg & Teller, Z. Phys. Chemie B21, 410 (1933);E. Renner, 92, 172 (1934)

Teller, J Phys. Chem. 41, 109 (1937);Herzberg & Longuet-Higgins, Discuss. Faraday Soc. 35, 77 (1963);Carrington, ibid. 53, 27 (1972)

H11 H12

H21 H22Smooth

cusp

Lichten, Phys. Rev. 131, 229 (1963); Smith, Phys. Rev. 179, 111 (1969)

Adiabatic Diabatic

Complexities in Theoretical Treatments:Singular nonadiabatic coupling terms in the adiabatic representation

Quite Tricky!

Derivative coupling vector

Born-Huang term

Adiabatic electronic wavefunction changes signTopological effect / Geometric phase effect

G. Herzberg and H. C. Longuet-Higgins, Discuss. Faraday Soc. 35, 77 (1963)H. C. Longuet-Higgins, Proc. Roy. Soc. London, Ser. A, 344, 147 (1975)M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)C. A. Mead, Rev. Mod. Phys. 64, 51 (1992)

Construction of theoretical models and algorithms

ab initio electronic structure calculations and quantum dynamical simulations

Indo-German Collaborations: Highlights

Traditional wave packet and also MCTDHMatrix diagonalization

1. Conical intersections in H3

2. The Jahn-Teller and pseudo-Jahn-Teller effects in the radical cations of cyclopropane and methyl fluoride

3. Photophysics/chemistry of phenol and pyrrole

4. Photodetachment spectroscopy of NO3-

“All non-linear nuclear configurations are unstable on an orbitally degenerate electronic state”

“Tend to distort in such a way as to remove the electronic degeneracy” ---- H. A. Jahn & E. Teller 1937

Energy levels cross at E0

Four fold axial symmetry reduces to two foldEσ (I) = Eσ ′ (II)Eσ (II) = Eσ ′ (I)

Eσ Eσ ′

Eσ ′Eσ

III

The Jahn-Teller effect

γ

r

R

H H

H

JT Interactions in the Electronic Ground State of H3

DMBE PES: Varandas et al. J. Chem. Phys. 86, 6258 (1987).

Vcmin = 2.74 eV

D3h

Adiabatic to diabatic transformation angle

H + H2 H2 + H

V+

V-

A1 eQx Qy

χ

Qy

Qx

Pseudorotation angle

R

H3 Vibrations:

S. Mahapatra and H. Köppel, J. Chem. Phys. 109, 1721 (1998).A. Thiel and H. Köppel, J. Chem. Phys. 110, 9371 (1999).

Large shift from the vertical transition energy Huge ZPE (~ 0.7 eV) due to the cusp of the JT split PESs

Crossing at E = 3.089 eV (Vc min = 2.74 eV)

wave packet terminates at the immediate neighborhood of the conical intersections of the final electronic state.

~ 1.2 eV

~ 2.95 eV ~ 4.15 eVV- V+

R. Bruckmeier, Ch. Wunderlich, and H. Figger. Phys. Rev. Lett. 72 2550 (1994)Mahapatra and Köppel, Phys. Rev. Lett. 81, 3116 (1998);

J. Chem. Phys. 109, 1721 (1998); Faraday Discuss, 110, 248 (1998)

Theory Time-dependent Wave Packet

Nonradiative decay How fast ?

Time-dependent electronic population ofthe upper adiabatic ( V+ ) sheet

To within 3-6 fs.

Perhaps the fastest femtosecond decay process treated in the literature !!

Mahapatra and Köppel, Phys. Rev. Lett. 81, 3116 (1998)

Dissociative Recombination (DR) of H3

DR mechanism is complex and not yet understood

Various nonadiabatic transitions are anticipated to contribute to the observed DR rate

DR mechanism is important to the interstellar chemistry

Rydberg electronic states are coupled to the ground electronic state in the DR process

Jahn-Teller conical intersections in 3p (E′) and 3d (E″)

The Rydberg excited H3 undergoes various nonadiabatictransitions and dissociates on the repulsive lower adiabaticsheet of its ground electronic state

Study of JT CIs is important in understanding the DR processes

The (E⊗e)-JT CIs in these electronic states influencethe nuclear dynamics in the DR processes

• Static aspects are studied in terms of ab initio quantum chemistry calculations

• Dynamic aspects are studied in terms of vibronic spectra and nonradiative decay behavior of these electronic states

3p (E′)

3d (E″)

Potential energy surfaces PES Contours

SplineRaw

M. Jungen and coworkers, Khim. Fiz., 23(2), 71 (2004)Rao, Mahapatra, Köppel, Jungen, J. Chem. Phys. 123, 134325 (2005)

JT interactions are low in 3d (E′)

JT CIs in 3p (E′)

JT CIs in 3d (E″)

Vibronic Coupling Model :

E10

E20

E30

Harmonic

κN

KÖppel, Domcke and Cederbaum, Adv. Chem. Phys. 57, 59, (1984);

Hamiltonian to be invariant under the symmetry operation of the corresponding point group

Determines the relative signs of the coupling constants

Parameters are obtained by computing potential energy surfaces

Dynamical Observables

Multi-Configuration Time-Dependent Hartree (MCTDH) scheme

Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165, 73 (1990); Meyer, Manthe, Cederbaum, Chem. Phys. 97, 3199 (1992); Beck, Jäckle, Worth, Meyer, Phys. Rep. 324, 1 (2000).

Worth, Beck, Jäckle, Meyer, The MCTDH Package, Version 8.2, 2000, University of Heidelberg, Germany.Mayer, Version 8.3, 2002. See http://www.pci.uni-heidelberg.de/tc/usr/mctdh/

The photoelectron spectrumFermi’s golden rule:

of the autocorrelation function

Nonradiative decay of electronic populations

Jahn-Teller and pseudo-Jahn-Teller effects in cyclopropane radical cationCyclopropane Radical Cation

MP2/cc-PVTZ Expt.

rC-C 1.50378 A° 1.499rC-H 1.07834 A° 1.074

∠CCC 60 59.98∠HCH 115.081 117.08

2.43 eV

2.59 eV

X 2E ′~

A 2E”~

B 2A1′~E

Eqm. Geometry : D3h

+

0.78

Holland et. al. J. Elect. Spect. 125, 2002, 57-68

Γvib =3A1′+A2′+4E′+A1′′ +2A2′′ +3E′′

[E′ Χ E′]† = [E′′ Χ E′′]† = A′1 +E′

[E′ Χ E′′] =A1′′ + A2′′ +E′′

JT Active

Pseudo- JT Active

Condon Active

Mean Spacing≈ 60.0 meV

D3h

hγ

(E⊗e) -JT

Important vibrational modes

Adiabatic Potential Energy cuts of X2E′ - A2E′′ electronic manifold

Symmetric modes Degenerate modes

X2E' band of CP+: LVC Model

0.15 eV ν1

0.19 eV ν2

0.113 eV ν4

0.131 eV ν5

T. S. Venkatesan et. al, J. Phys. Chem. A 108 2256 (2004)

ΔEJT = 0.98 eV

0.80 eV

Expt

Theory (Sym ⊗ JT)

0.78 eV

Vcmin = 10.64 eV

V0- = 9.66 eV

Slonczewski resonances

J.C.Slonczewski, Phys. Rev. 131,1596 (1963)

X2E’ band of CP+

0.81 eV

0.80 eV

Linear vibronic coupling

Dominant progressions due to υ2, υ3, υ4, and υ6 vibrational modes

A2E" band of CP+ (LVC Model)

ΔEJT = 0.638 eV

υ2 190 meV

υ3 396 meV

υ6 187 meV

υ4 122 meV

Irregular spacingis due to multimodeJT effect.

T. S. Venkatesan et. al, J. Phys. Chem. A, 111, 1746 (2007); J. Mol. Struct. 838, 100 (2007).

4 states and 14 modes quadratic JT plus linear PJT model.

Huge impact of PJTcoupling via a1

” and e”

vibrational modes

Vibronic Spectrum of Coupled X2E’-A2E” Manifold

VPJT (min) = 12.12 eV ~ 1.47eV above and ~ 0.62 eV below the minimum of X and A JT conical intersections.

Vibronic Coupling in Methyl Halides

Cation

~ 4.81eV

Mode Frequency/ eV Descriptionν1(A1) 0.1336 C-F Stretchν2(A1) 0.1865 CH3 Bendν3(A1) 0.3822 C-H Stretchν4(E) 0.1503 C-F Bendν5(E) 0.1893 CH3 Deformationν6(E) 0.3950 C-H Stretch

C

F

HH(C3V)

B2EA2A1~~

X2E~ H

17.15617.127

13.318

Γvib = 3A1+3E[E x E] = A1+ E

Condon ActiveJT + PJT Active

Karlsson et al. Phys. Scrip. 16, 225 (1977)

CH3F+

Adiabatic Potential Energy Surface CASSCF/MRCI ; aug-cc-pVTZ with ECP for Fluorine CAS (11,16)

Mahapatra, Vallet, Woywod, Köppel and Domcke, Chem. Phys. 304, 17 (2004).

Vibronic Structure of the X 2E State ∼

Linear Model Quadratic Model

Sym Sym∼ 0.129

0.185ν1ν2(Very weak)

JT0.19

0.15ν5ν4

Sym ⊗ JTν1+ν4+ν5

∼ 0.104

ν1ν2 (Very weak)

γ ≈- 0.024

JTν5+ ν4

Sym ⊗ JT

CH3F+

X 2E Band ∼CH3F+

JT + PJT

Expt

Vibronic Structure of the B 2E State∼

Linear Model Quadratic Model

Sym Sym

JT JT

Sym ⊗ JT Sym ⊗ JT

ν1 + ν3 (weak)

ν5 + ν6 (weak)

ν1 + ν5

γν 1 = 0.055γν 3 = 0.023

γν 6 = 0.0102

η ∼ 10-3 or less

CH3F+

CH3F+

A 2A1 + B 2E Band∼ ∼

JT + PJT

Expt

B2E + A2A1(2 : 1)

∼ ∼

CP, 304, 17 (2004)

Impact of the intermode (a1-e) bilinear coupling terms

Mahapatra, Vallet, Woywod, Koeppel and Domcke, J. Chem. Phys. 123, 231103 (2005)

Schmidt-Kluegmann, Koeppel, Schmatz and Botschwina, Chem. Phys. Lett. 369, 21 (2003)

Photostability & nonadiabatic transitions

HO

Photochemistry of DNA building blocks: chromophores

• Extremely low quantum yield of fluorescence of the strongly UV absorbing excited singlet ππ*state

• Very fast nonradiative processes

Conversion back to the ground state

hγ

Photochemical Funnel

• The photon energy quickly dissipated before more profound chemical rearrangements Photostability

Upper diabaticLower diabatic

Faraday Discuss. 127, 283 (2004)J. Chem. Phys. 122, 224315 (2005)

123, 144307 (2005)

J. Photochem. Photobio. A 190, 177 (2007)

• 1πσ* Optically dark Two curve crossings which become conicalintersections when out-of-plane modes (e.g. CCOH dihedral angle) are considered

• Both CIs have same coupling coordinate CCOH dihedral angle θ (torsion)

• The photo induced dynamics is simulated by time-dependent WP method

Wave packet dynamics

P1

P2

P3

S0

1ππ*1πσ* Diabatic

Upper diabaticLower diabatic

Acknowledgement

Funding:

DST, CSIR-New DelhiVWS and AvH Stiftung, Germany

Computational Facilities:CMSD, University of Hyderabad

Dr. R. PadmanabanDr. T. S. VenkatesanDr. S. GhosalMr. B. Jayachander RaoMr. V. Sivaranjana ReddyMr. T. MondalMr. S. GhantaMr. T. Rajagopala RaoMr. S. Rajagopala ReddyMr. T. Roy

Coworkers: