Fi (t )= mi d 2ri(t )/dt 2

Fi : mi : ri : t :

N = 103 106

AFM

DMPCA

Mechanical Extension of Polyalanine-

Supercritrical Fluid Miura, Yoshii, and Komatsuambient watersupercritical water

Superfluid Helium Miura and Tanaka

Vibrational Relaxation experimental backgroundtime-resolved spectroscopy e.g. CN ion in water Heilweil and Hochstrasser(1982) Hamm, Lim, and Hochstrasser(1997) T1 = 28 ps at 0.22 M

Collaborators

Dr. M. Shiga (JAERI) Dr. T. Mikami (IMS) T. Terashima (Tokyo. Inst. Tech.) M. Satoh (Tokyo. Inst. Tech., IMS) ACP 118, 191(2001)JCP 109, 3542(1998)JCP 111, 5390(1999)JCP 114, 5663(2001)JCP 115, 9797(2001)JCP 119, 4790(2003)JCP (2004), in pressCollaborators and publications

Simulation of Quantum Dynamics theoretical backgroundIt is impossible to solvetime-dependent Schrodinger equationfor many-body systems such as solutions.An approximation is needed.New methodTraditional Method Classical MD Langevin equation Fermis golden rule Path integral influence functional theory

harmonic oscillators bath approximation

Mixed quantum-classical approximation

mean field approximationNon-adiabatic transition Equation of motion

ClassicalQuantum

Outline of the Talk 1. Framework of the Theory (1) path integral (2) influence functional (3) higher order coupling

2. Energy Relaxation (1) importance of multi-phonon process (2) important combination of normal modes (3) dissipation pathway to the solvent

3. Dynamics of Coherence between States (1) off-diagonal part of the density matrix (2) disappearance of the coherence (3) quantum beat

4. Interaction in Liquid and Supercritical Water (1) resonance (2) collisionCN- ion in the aqueous solution

xixf0tft

Feynman(1948)Feynman and Vernon(1963)PropagatorPath Integral Representation

Harmonic Oscillators Bath Feynman and Vernon(1963)linear couplingsolidglassliquid instantaneous normal modeN = 1solutesolventqx

Multi-phonon Processes nonlinear couplingNonlinear Couplings with Bath Coordinates system environment system environment system environmentsingle-phonon process two-phonon process three-phonon process w0 = wk w0 = wk + wl w0 = wk + wl + wm

Perturbative Spectral Density1-phonon N2-phonon 2N23-phonon 4N31-3 cross N2 Shiga and Okazaki(1998)Sum frequencyDifference frequencyFeynman and Vernon(1963)

Time-dependent Transition ProbabilityHarmonic Oscillator SystemSurvival Probabilityrigorous path integralTaylor expansioncumulant expansionTime Dependent Probabilityinfluence functional

Recipe of Calculation 1. classical MD calculation rigid rotor model2. instantaneous structures3. normal mode analysis flexible model4. coupling constants numerical differentiation5. spectral density6. survival probability cumulant expansion 7. relaxation time 8. analysis of solvent modes molecular mechanism

NVT ensemble N = 256 Na+ + CN- + 254 H2O 0.22 mol/l r = 1 g/cm3 T = 300 K Nose thermostat

Predictor-corrector method Dt = 0.5 fs 300,000 steps 150 ps Ewald

Normal Mode Analysis instantaneous structure quenched structure 2294 + 1 modes MD Calculation30 structures every 5 ps from MDclassical

Intermolecular interaction +0.52 e +0.52 e H H -1.0 e +0.8 e -0.8 e -1.04 e C N O Ferrario et al. TIP4P

Vinter = S S 1/4pe0 qaqb/rab + Aab/rab12 Bab/rab

Intramolecular interaction C-N stretching 2059cm-1 H2O symmetric stretching 3657cm-1 antisymmetric stretching 3756cm-1 bending 1595cm-1 Vintra = 1/2 M W2 x2

Vtotal = Vinter + Vintra

Potential Model

Bond Length Modulation population relaxationVI = SS Vijab(rab)

= V0 + S Ck(1) x qk + SS Ckl(1) x qkql +

+ S Ck(2) x2 qk + SS Ckl(2) x2 qkql +

+ S Ck(3) x3 qk + SS Ckl(3) x3 qkql +

+

Single-phonon Spectral DensityFig. 2 Single-phonon spectral densityfor the instantaneous normal mode. No resonance !

Two-phonon Spectral DensityFig. 3 Two-phonon spectral densityfor the instantaneous normal mode.(a) sum frequency spectrum

(b) difference frequency spectrumStrong resonance !

Survival ProbabilityFig. 4 Survival probability of the firstexcited state of CN stretching mode.T1 = 7 psexperiment 28 psHamm et al.(1997)

CMD 15 ps Jang et al.(1999)potential function ?

Distribution of Relaxation TimeFig. Distribution of relaxation time calculated assuming single particle measurement.distribution coming from statistical mechanical uncertaintyof the solvent,

which may be observedby single particle measurement

TT 0 0TR 0 0TB 24 26TS 0 0RR 1 1RB 75 72RS 0 0BB 0 0BS 0 1SS 0 0

TTT 0 0 RRR 14 9TTR 0 0 RRB 0 0TTB 28 30 RRS 0 1TTS 0 0 RBB 0 0TRR 25 13 RBS 0 0TRB 33 47 RSS 0 0TRS 0 0 BBB 0 0TBB 0 0 BBS 0 0TBS 0 0 BSS 0 0TSS 0 0 SSS 0 0

Contribution of Modes to the Relaxationcoupling quantum classical coupling quantum classical coupling quantum classicaltwo-phonon processthree-phonon processTABLE Percentage contributions from combinations of bath modes to the vibrational relaxation based upon two-and three-phonon processes.T : translation, R : rotation, B : bending, and S : stretching

Single-molecular or two-molecular processFig. Relaxation density matrix rR(r, r).inverse transformation from normal mode to laboratory coordinateandassignment of the coupling to the moleculeSingle-molecular processTwo-molecular process

Water molecules in the first hydration shellin the direction of C-N bond axisR(i)(t)Great contributionCNMultiplied by Jacobianwater moleculesin the first hydration shellNa+CN-H2O

Fig. Ratio functions for single-, two-, and three-phonon processes and conventional quantum correction at 300 K.single twowk=4wl threewk=wl=wmstandardSchofieldQuantum Effect of Solvent (sum frequency)classical solvent

h 0 limit

Coherent Statevibrational stateS1S0energetically uncertainshort pulseWatanabe, et. al. CPL(2002)

RedfieldRedfieldfrequency shiftpopulation relaxation

dephasing

Bond Length and Frequency ModulationVI = SS Vijab(rab)

= V0 + S Ck(1) x qk + SS Ckl(1) x qkql +

+ S Ck(2) x2 qk + SS Ckl(2) x2 qkql +

+ S Ck(3) x3 qk + SS Ckl(3) x3 qkql +

+

x2

Re r 10(t)fast oscillation 16 fs ( 2080 cm-1 )relaxation time 5.1 pssingle-particle measurement x2qk , x2qkql

R1010(t)Population decay time:Pure dephasing time:1

Single-particle measurementMany-particle measurement0-100 fs1500-1600 fs3100-3200 fshomogeneous broadeninginhomogeneous broadening

log plotsingle-particle : T2 = 5.1psmany-particle : T2 = 1.7pssingle-particle measurement dephasing timeSingleMany

Re r11(t)quantum beat

Mixed Quantum Classical Approximation mean field approximation Total HamiltonianCoupled Equation of Motion x : rapid quantum RN : slow classicaltime-dependent Schrodinger equation Hellmann-Feynman forceTotal energy is conserved.

Eigenfunction ExpansionCoupled Differential Equations

Fig. 3 Time evolution of the wavefunction.Time Evolution of Wavefunction a schematic picturet = 0 ps10 ps20 ps30 ps40 ps50 psAt t = 0,

|Y = |

and solvent was in thermal equilibrium.

single-molecular measurementpure stateS |ci|2 fi (x) 2n = 1n = 0n = 2|cn|2t / ps

Fermis golden rule relaxation timedensity of state2080 cm-1t = 30 psSimulation

t = 23 ps

Coupling as a Function of TimetcouplingttcouplingcouplingFermis golden ruleinfluence functional

thermal averaragestatic coupling amplitude phase matchingSolid and glassGasLiquidIsolated Binary Collision Modelcollisional?statistics?resonant, stationarycollisional, delta functionapolar solvent? short-ranged forcepolar solvent? long-ranged force

Interaction of Ion in Water < i | V(t) | j >t / pslooks almost randomin liquid| ci |2d11/dt 2/h V01 Im 10Im 10V10(t) frequency matchingphase matchingV10(t) = < 1 | V(t) | 0 >resonance

Coupling as a Function of TimetcouplingtcouplinggasliquidIsolated Binary Collision Modeltsolid and glassFermis golden ruleinfluence functional

thermal averaragestatic random coupling frequency matching phase matchingresonant, stationarycollisional, delta functionHow is it in the supercritical fluid ?tcouplingHow is it in the apolar solvent ?

Relaxation by Collision chloroformshort-ranged forcerepulsive force

direction of the mode

Interaction with the Solvent V01r = 0.029 g/cm3r = 0.580 g/cm3r = 0.435 g/cm3r = 0. 290 g/cm3r = 0.145 g/cm3r = 0.870 g/cm3r = 0.725 g/cm3

Resonance and Collisions with the Solventr = 0.029 g/cm3r = 0. 290 g/cm3r = 0.870 g/cm3collisionsresonance

Snapshot of Hydrationr = 0.029 g/cm3r = 0. 290 g/cm3r = 0.870 g/cm3density is high locally still high low

Collisions with the Solventr = 0.0029 g cm-3

SummaryMulti-phonon process is essential for the relaxation of CN- ion.

Bending mode of water plays an important role in the two- and three-phonon processes.

Water molecules in the first hydration shell in the direction of C-N bond axis contribute much to the relaxation.

Dynamics of coherence between vibrational states has been formulated.

Essence of single-particle pure dephasing and effect of inhomogeneity of the environment has been demonstrated.

Mixed quantum-classical molecular dynamics in the mean field approximation is being analysed.

R

dominant

3.7