New Approaches for Quantum Dissipative Dynamics and Spectroscopy of Condense-Phase Molecular Systems

Department of Chemistry & Center for Quantum �Science and Technology, National Taiwan University

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Physics Division, National Center for Theoretical Sciences

QFort Workshop, NCKU, Tainan

April 16, 2024

Yuan-Chung Cheng*��yuanchung@ntu.edu.tw

Molecular Modeling of Excitonic Dynamics

Research in the Cheng group concerns molecular simulations and quantum dynamic calculations for excitonic phenomena in molecular systems

Generic Model for Condensed-Phase Quantum Dynamics

J

Generic Model for Condensed-Phase Quantum Dynamics

J

Electronic Hamiltonian

Spectral density function:�coarse-grained system-bath �interactions

Strategy for Excitonic System Modeling

J

Exciton Hamiltonian�Quantum chemistry �+ electrostatic �embedding model

Spectral fitting & MD simulations

Spectral density function

Dynamical theory

Conventional Theories for Condensed-Phase Quantum Dynamics

J

Forster theory: weak electronic coupling, incoherent dynamics

Redfield theory: weak system-bath coupling, coherent dynamics

HEOM/QUAPI/…: non-perturbative methods, numerically exact

Conventional Methods Inadequate for Coherent EET in Photosynthesis

Conventional Förster theory or Redfield equation do not adequately describe photosynthetic EET dynamics – new theories suitable for the intermediate regime are needed.

Comparative study with exact calculations based on the reduced hierarchy equation approach on a model chlorophyll dimer (Ishizaki &Fleming, JCP 2009).

Theories for Dissipative Quantum Dynamics

We develop perturbative methods to provide accurate descriptions of quantum dynamics in various parameter regimes:�

• Small-Polaron Quantum Master Equation (SP-QME): based on a small-polaron basis for quantum excitations
• Pure-Dephasing Reference System (PDRS) theory: based on a pure-dephasing electronic basis for excitons and electrons (e.g. coherent modified-Redfield theory)
• Schrodinger-Langevin Equation: based on a separation of frictional effects and fluctuations around the mean – specifically suitable for coupled electronic/nuclear wavepacket dynamics

Simulating Quantum Dynamics & Spectra

• Full quantum dynamics
• Ultrafast & non-Markovian dynamics
• General form of system-bath interactions
• Coherence transfer dynamics
• Dynamics in τ and t periods
• Two-exciton states
• Explicit treatment of laser fields
• Pulse-overlap effects, pulse profiles
• Avoids rotating-wave approximation

🡪 Single consistent theory for signals and dynamics �going beyond response function formalism

http://quantum.ch.ntu.edu.tw/ycclab/QDAS.html

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Theories for Dissipative Quantum Dynamics

We develop perturbative methods to provide accurate descriptions of quantum dynamics in various parameter regimes:�

• Small-Polaron Quantum Master Equation (SP-QME): based on a small-polaron basis for quantum excitations
• Pure-Dephasing Reference System (PDRS) theory: based on a pure-dephasing electronic basis for excitons and electrons (e.g. coherent modified-Redfield theory)
• Schrodinger-Langevin Equation: based on a separation of frictional effects and fluctuations around the mean – specifically suitable for coupled electronic/nuclear wavepacket dynamics

Accurate quantum dynamics via 2^nd-order perturbative quantum master equation

Pure Dephasing Reference System (PDRS)

11

Electronic reference basis

unitary transformation

U

Diagonal H_e

Site basis

PDRS Dynamics via 2^nd-order Perturbations

12

PDRS Quantum Master Equation

Naturally leads to one FT-term caused by excitonic coupling, one mRT-term by electron-phonon coupling, and cross-terms

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(Förster theory)

(modified Redfield theory)

Mix perturbation by an intermediate reference system

Variational Polaron Reference System

• To find an appropriate PDRS basis, we must treat electron-phonon coupling properly
• variational polaron transformation

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Pure Dephasing Variational Polaron Reference System (PDVPRS)

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artifact of variational small polaronic ansatz

PDVPRS vs QUAPI Dynamics

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Dynamics at the Intermediate Regime

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PDVPRS-QME, �CMRT, FT, QUAPI

Theories for Dissipative Quantum Dynamics

We develop perturbative methods to provide accurate descriptions of quantum dynamics in various parameter regimes:�

• Small-Polaron Quantum Master Equation (SP-QME): based on a small-polaron basis for quantum excitations
• Pure-Dephasing Reference System (PDRS) theory: based on a pure-dephasing electronic basis for excitons and electrons (e.g. coherent modified-Redfield theory)
• Schrodinger-Langevin Equation: based on a separation of frictional effects and fluctuations around the mean – specifically suitable for coupled electronic/nuclear wavepacket dynamics

Simulate nonlinear spectra of strongly coupled electronic-vibrational systems

Two-dimensional Spectroscopy: Four-wave Mixing

E_s(τ, Τ, ω_t) ~

spectrometer

sample

local

oscillator (LO)

1

2

3

Rich Information Content in a 2D Spectrum

Theoretical Simulation of 2DES Spectra

The method with explicit treatment on vibrational relaxation dynamics is still underdeveloped

The Quantum Schrodinger-Langevin Equation Approach

An efficient, novel method for coupled vibronic dynamics and 2D spectroscopy in a dissipative environment

Dissipative dynamics + Light-matter interactions

M. T. Wong & Y.-C. Cheng, J. Chem. Phys., 154, 154107 (2021)

pulse overlap, realistic pulse shape, multiple pathways,… automatically included

The Quantum Langevin Equation Approach

An efficient, novel method for coupled vibronic dynamics and 2D spectroscopy in a dissipative environment

M. T. Wong & Y.-C. Cheng, J. Chem. Phys., 154, 154107 (2021)

• Friction operator (quantum Schrodinger-Langevin equation)�����Explicit treatment of vibrational relaxation dynamics in each mode on each individual electronic potential energy surface

Drag force

Random force

Dissipative dynamics + Light-matter interactions

Separates friction and fluctuation, unlike stochastic approaches

The Quantum Langevin Equation Approach

An efficient, novel method for coupled vibronic dynamics and 2D spectroscopy in a dissipative environment

M. T. Wong & Y.-C. Cheng, J. Chem. Phys., 154, 154107 (2021)

• Explicit external fields in dynamical simulations�Phase-matching

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• Photon-echo �signals

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Dissipative dynamics + Light-matter interactions

The Quantum Langevin Equation Approach

An efficient, novel method for coupled vibronic dynamics and 2D spectroscopy in a dissipative environment

M. T. Wong & Y.-C. Cheng, J. Chem. Phys., 154, 154107 (2021)

• Combinations of auxiliary wavefunctions

Dissipative dynamics + Light-matter interactions

Vibrational Relaxation in 2DES Signals

Simple model simulation to demonstrate the capability of the QLE approach

Displaced-oscillator model

Vibrational Relaxation in 2DES Signals

• Separation of ground-state vs. excited-state coherences:�beating in diagonal peaks 🡪 GS�beating in off-diagonal peaks 🡪 ES

Spectroscopy of Tagged Hydronium Ions

X=Ar or N₂

SLE simulated IR Spectra

Ar-tagged:�Fermi resonance

N₂-tagged:�No Fermi resonance

Collaboration with Jer-Lai Kuo & Asuka Fujii

SLE 2DIR Simulations & Fermi Resonance

Ar-tagged

N₂-tagged

Strong coupling leads to square peak pattern for Fermi resonance systems

With Fermi resonance

Without Fermi resonance

Vibronic Effects in Conical-Intersection (CI) Systems

Dissipative CI dynamics with QME based methods

Solvent effects on CI geometry and mixed quantum-classical methods

Malhado et al., J. Chem. Phys., 137, 22A543 (2012)

Chen et al. Faraday Discuss., 194, 61 (2016)

Irregular intensity distribution for CI model (left)

Excited-state absorption due to CI dynamics

Duan et al., J. Phys. Chem. Lett., 7, 382−386 (2016)

Theoretical studies of 2DES spectra for CI systems

Krčmář et al., J. Chem. Phys., 143, 074308 (2015)

Also Cederbaum/Domcke/Dassia Egorova/Fleming/Mexim Gelin/Yarkony/Artur Izmaylov/David Jonas/ …

Decisive evidence for CI dynamics for molecular systems in condensed-phase remains elusive (Berry phase effect).

Probing Vibronic Dynamics via 2DEV Spectroscopy

No vibronic coupling

Franck-Condon coupling

Herzberg–Teller coupling

Fleming and coworkers, J. Chem. Phys. 155, 054201 (2021).

2DEV beat maps

Probing Vibronic Dynamics via 2DEV Spectroscopy

No vibronic coupling

Franck-Condon coupling

Herzberg–Teller coupling

Fleming and coworkers, J. Chem. Phys. 155, 054201 (2021).

2DEV beat maps

Three-state Two-mode Model

Tuning mode 𝜔_𝑡 = 1536 cm⁻¹

Coupling mode 𝜔_𝑐 = 1317 cm⁻¹

(model pyrazine system)

Three-state Two-mode Model

Elucidate CI dynamics and 2D spectra by comparative studies

Two vibronic coupling models

Possible CI Signatures in 2DEV Spectra

2DEV 🡪 electronic excitation (first two pulses), IR probe (last two pulses)

CI

AvC

2DEV Beat Maps?

CI

AvC

Black curves: 2DEV @ T=0

Color map: beat amplitudes

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• Most peaks exhibit significantly beating signals
• CI & AvC show very difficult beat patterns

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We found it difficult to understand these differences!�

How do we interpret these 2DEV results?

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Interpretation of 2DEV Spectra

2DEV 🡪 electronic excitation (first two pulses), IR probe (last two pulses)

Probe IR signals from the eigenstate selected by optical excitation frequency

2DEV probes nonadiabatic nuclear dynamics on the excited-state potential energy surface after optical excitation 🡪 seeking information at a giving excitation frequency to elucidate CI dynamics?

We should focus on IR signals probed at a given visible excitation frequency!

CI vs. AvC Dynamics in 2DEV Spectra

Simulated spectra show distinctive difference for the CI system

Probe IR signals from the eigenstate selected by optical excittaion frequency

CI

AvC

CI vs. AvC Dynamics in 2DEV Spectra

Ground-state IR

Additional signals sensitive to the transitions between eigenstates strongly affected by CI

Not too different from the two peaks in the ground state IR

CI

AvC

CI vs. AvC Dynamics in 2DEV Spectra

Ground-state IR

CI

AvC

FT

Fourier transform along T

2DEV Signal Beating Frequency Map

CI

AvC

2DEV Signal Beating Frequency Map

CI

The signal is from a special coherence transfer pathway that should be unique to CI dynamics

Spectral Dynamics in 2DEV Signals

• Effectively focusing on anharmonic effects in the potential energy surfaces and linear form of the couplings
• No true “degenerate CI point (seam)” in quantized energy levels
• Spectral evolution in T 🡪 detailed vibronic wavepacket dynamics and features of the potential surfaces
• Genuine spectral signitures of geometric phase effects

CI

AvC

FT

Spectral-evolution based analysis

🡪 provide a less ambiguous way to identify characteristic beating frequencies

Concluding Remarks

• A novel quantum Langevin equation approach for dissipative dynamics and 2D spectra simulations with emphasis on vibrational relaxation effects is proposed and developed
• The QLE approach was applied to investigate 2DES spectra of vibrational relaxation dynamics in a displaced-oscillator mode to demonstrate its capability to describe and capture complete vibrational dynamics in the model
• 2DES as well as 2DEV spectra for a model CI system were simulated by the QLE approach to search for spectral signatures of CI dynamics
• We proposed that 2DEV signals probed at a given optical excitation frequencies provide relevant representation of CI dynamics, and showed clearly distinct features in the CI model compared to a avoided crossing model for a representative system

Acknowledgements

Funding: NCTS/NTU/CoS/CQSE

PDVPRS-QME: Pin-Ze Huang (黃品澤)�2DIR: Jun-Hao Yu (尤俊皓)

Thank You!!