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溶液中の溶質分子の振動量子動力学の 計算機シミュレーション ー 状態緩和とコヒーレンスの消失 ー （分子研）　岡崎　進

溶液中の溶質分子の振動量子動力学の 計算機シミュレーション ー 状態緩和とコヒーレンスの消失 ー （分子研）　岡崎　進. プラズマ科学のフロンティア　２００４、土岐. 分子の運動を追跡する. ニュートンの運動方程式 （微分方程式） F i ( t )= m i d 2 r i ( t ) /dt 2 F i : 原子にかかる力 m i : 原子の質量 r i : 原子の位置 t : 時間 自由度の数だけの 連立微分方程式 N = 10 3 ～ 10 6 分子　－　回転 　　　　　　 振動.

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溶液中の溶質分子の振動量子動力学の 計算機シミュレーション ー 状態緩和とコヒーレンスの消失 ー （分子研）　岡崎　進

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  1. 溶液中の溶質分子の振動量子動力学の 計算機シミュレーション ー 状態緩和とコヒーレンスの消失 ー （分子研）　岡崎　進 プラズマ科学のフロンティア　２００４、土岐

  2. 分子の運動を追跡する ニュートンの運動方程式（微分方程式） Fi(t)=mi d 2ri(t)/dt 2 Fi : 原子にかかる力 mi : 原子の質量 ri: 原子の位置 t : 時間 自由度の数だけの 連立微分方程式 N = 103～ 106 分子　－　回転 　　　　　　 振動 高温高圧の水溶液

  3. 両親媒性分子水溶液の構造形成-ナノ分子集合体両親媒性分子水溶液の構造形成-ナノ分子集合体 　大規模MD　平均力、メモリーの解析 低分子水溶液、胆汁酸 　　ミセル、生体膜、グラミシジン 　新しいシミュレーション手法 　　相互作用の粗視化、LD 吉井範行 岩橋建輔 篠田　渉（産総研） 中島俊夫（大分大） 福村裕史（東北大） 猪飼　篤（東工大） 超臨界流体 超臨界水 小松孝之（特別共同利用） AFMを用いたナノ力学 ポリペプチドの延伸 ポリペプチドの機械的１分子操作 量子系の計算機シミュレーション 振動緩和 経路積分影響汎関数理論 量子古典混合系近似 　プロトン移動 量子液体 三上泰治 佐藤昌宏（特別共同利用） 山田篤志 三浦伸一

  4. 水中に生成される球状ミセル　　　　　　石けん水中に生成される球状ミセル　　　　　　石けん 非イオン性ミセル

  5. 膜を横切る水の透過 透過の自由エネルギープロフィール 薬剤吸収へ

  6. イオンチャンネル 抗生物質 情報伝達 DMPC脂質二重層膜中の グラミシジンA

  7. 初期配置 平衡構造 ０％　　　　 ６％　 　２０％　　 　４０％　 　６０％　 　８０％　 １００％ Mechanical Extension of Polyalanine 変角の自由度 水素結合の切断 α-へリックスの崩壊 疎水性相互作用 ゴム弾性 ３％

  8. Supercritrical FluidMiura, Yoshii, and Komatsu ambient water supercritical water

  9. Superfluid Helium Miura and Tanaka

  10. 過剰エネルギーの溶媒への散逸 光 化学反応

  11. hn t = 0 pump hn’ t = t probe First excited state How does the excessive energy dissipate into the solvent? Coherent state How does the coherence between states annihilate in liquid? Vibrational Relaxation experimental background time-resolved spectroscopy e.g. CN－ion in water 　　～ ２０８０ ｃｍ－１ Heilweil and Hochstrasser(1982) Hamm, Lim, and Hochstrasser(1997) T1 = 28 ps at 0.22 M

  12. Collaborators and publications 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 press

  13. Simulation of Quantum Dynamics theoretical background Non-adiabatic transition Equation of motion Classical Quantum It is impossible to solve time-dependent Schrodingerequation for many-body systems such as solutions. An approximation is needed. New method ● Path integral influence functional theory harmonic oscillators bath approximation ● Mixed quantum-classical approximation mean field approximation Traditional Method Classical MD Langevin equation Fermi’s golden rule

  14. Outline of the Talk CN- ion in the aqueous solution 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) collision

  15. Feynman(1948) Feynman and Vernon(1963) 経路積分と影響汎関数 Propagator xf Path Integral Representation xi t tf 0 溶質と溶媒の自由度で表された凝集系のプロパゲータ ↓ 溶媒の自由度について、熱平均をとりながらあらかじめ経路積分を実行してしまう （溶媒に対する積分可能な近似の導入） ↓ 溶質の自由度だけで表された凝集系中の溶質のプロパゲータ ↓ 影響汎関数

  16. Harmonic Oscillators BathFeynman and Vernon(1963) linear coupling solid glass liquidinstantaneous normal mode N=1 solvent solute q x

  17. Multi-phonon Processesnonlinear coupling Nonlinear Couplings with Bath Coordinates system environment system environment system environment single-phonon process two-phonon process three-phonon process w0 = wk w0 = wk + wl w0 = wk + wl + wm

  18. Perturbative Spectral Density Shiga and Okazaki(1998) 1-phonon N Feynman and Vernon(1963) Sum frequency 2-phonon 2N2 Difference frequency 3-phonon 4N3 1-3 cross N2

  19. Time-dependent Transition Probability Time Dependent Probability influence functional Harmonic Oscillator System rigorous path integral Taylor expansion cumulant expansion Survival Probability

  20. Recipe of Calculation 1. classical MD calculation rigid rotor model 2. instantaneous structures 3. normal mode analysis flexible model 4. coupling constants numerical differentiation 5. spectral density 6. survival probability cumulant expansion 7. relaxation time 8. analysis of solvent modes molecular mechanism

  21. MD Calculation classical 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 30 structures every 5 ps from MD

  22. Potential Model Intermolecular interaction +0.52 e +0.52 e H H -1.0 e +0.8 e -0.8 e -1.04 e ・C・NO Ferrario et al. TIP4P Vinter = S S 1/4pe0 qaqb/rab + Aab/rab12– Bab/rab６ Intramolecular interaction C-N stretching 2059cm-1 H2O symmetric stretching 3657cm-1 antisymmetric stretching 3756cm-1 bending 1595cm-1 Vintra = 1/2 MW2x2 Vtotal = Vinter + Vintra

  23. x V(x) x Bond Length Modulation population relaxation VI = 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 + ・・・・ + ・・・・・・・・・・

  24. Single-phonon Spectral Density No resonance ! Fig. 2 Single-phonon spectral density for the instantaneous normal mode.

  25. Two-phonon Spectral Density (a) sum frequency spectrum (b) difference frequency spectrum Strong resonance ! Fig. 3 Two-phonon spectral density for the instantaneous normal mode.

  26. Survival Probability T1 = 7 ps potential function ? experiment 28 ps Hamm et al.(1997) CMD 15 ps Jang et al.(1999) Fig. 4 Survival probability of the first excited state of CN stretching mode.

  27. Distribution of Relaxation Time distribution coming from statistical mechanical uncertainty of the solvent, which may be observed by single particle measurement Fig. Distribution of relaxation time calculated assuming single particle measurement.

  28. Contribution of Modes to the Relaxation TABLE Percentage contributions from combinations of bath modes to the vibrational relaxation based upon two-and three-phonon processes. two-phonon processthree-phonon process coupling quantum classical coupling quantum classical coupling quantum classical TT 0 0 TR 0 0 TB 24 26 TS 0 0 RR 1 1 RB 75 72 RS 0 0 BB 0 0 BS 0 1 SS 0 0 TTT 0 0 RRR14 9 TTR 0 0 RRB 0 0 TTB 28 30 RRS 0 1 TTS 0 0 RBB 0 0 TRR 25 13 RBS 0 0 TRB 33 47 RSS 0 0 TRS 0 0 BBB 0 0 TBB 0 0 BBS 0 0 TBS 0 0 BSS 0 0 TSS 0 0 SSS 0 0 T : translation, R : rotation, B : bending, and S : stretching

  29. Single-molecular or two-molecular process Single-molecular process inverse transformation from normal mode to laboratory coordinate and assignment of the coupling to the molecule Two-molecular process Fig. Relaxation density matrix rR(r, r’).

  30. Water molecules in the first hydration shellin the direction of C-N bond axis water molecules in the first hydration shell H2O Multiplied by Jacobian CN- Na+ R(i)(t) θ C N Great contribution

  31. Quantum Effect of Solvent (sum frequency) Schofield three wk=wl=wm classical solvent h → 0 limit two wk=4wl single standard Fig. Ratio functions for single-, two-, and three-phonon processes and conventional quantum correction at 300 K.

  32. ½ ½ 0 0 • ½ ½ 0 0 • r(0) = 0 0 0 0 0 0 0 Coherent State コヒーレント状態の理論計算 凝集相系の実在系 　量子力学的取り扱い 　影響汎関数理論 初状態 これですべての物理を代表している 実験 S1 short pulse energetically uncertain vibrational state S0 Watanabe, et. al. CPL(2002)

  33. 影響汎関数 密度行列の非対角項の時間発展　　　　　　　ρ０１（ｔ） 影響汎関数のキュムラント展開

  34. 密度行列の非対角項 本研究 frequency shift population relaxation dephasing 比較のため微分 Redfield方程式では 　　　展開の形式が異なる 　　　溶媒の短時間相関近似 　　　第一摂動項Δを無視 Redfield方程式

  35. x V(x) V(x) x x Bond Length and Frequency Modulation VI = 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

  36. コヒーレンスの消失Re r 10(t) ある一つの溶媒和構造 single-particle measurement 初状態 ： x2qk , x2qkql fast oscillation 16 fs ( 2080 cm-1 ) ０ ｃｍ－１　　　　　差周波 relaxation time 5.1 ps

  37. 状態緩和と位相緩和 R1010(t)以外の寄与が無い 永年近似が通用する Population decay time: Pure dephasing time: 1つの分子配置

  38. probe probe 環境の不均一性 homogeneous broadening Single-particlemeasurement inhomogeneous broadening Many-particle measurement 1500-1600 fs 3100-3200 fs 0-100 fs

  39. 環境の不均一性 Single Many log plot single-particle : T2 = 5.1ps many-particle : T2 = 1.7ps single-particle measurement 数倍の dephasing time

  40. に振動を与える項 に比例するシグナル 対角項　　　　　　　　　　　　　　　　　　　　 Re r11(t) 初期条件にコヒーレンスがある場合だけ生き残る quantum beat 吸収実験 四重極子

  41. Mixed Quantum Classical Approximation mean field approximation Total Hamiltonian x: rapid quantum RN: slow classical Coupled Equation of Motion time-dependent Schrodinger equation Hellmann-Feynman force Total energy is conserved.

  42. Eigenfunction Expansion Coupled Differential Equations

  43. Time Evolution of Wavefunction a schematic picture At t = 0, |Y＞ = |１＞ and solvent was in thermal equilibrium. single-molecular measurement pure state t = 0 ps 10 ps S|ci|2 fi (x) 2 20 ps n = 0 30 ps n = 1 |cn|2 40 ps n = 2 50 ps t / ps Fig. 3 Time evolution of the wave function.

  44. Fermi’s golden rulerelaxation time 2080 cm-1 Simulation t = 23 ps t = 30 ps density of state

  45. Coupling as a Function of Time Solid and glass Liquid Gas ？ coupling coupling coupling t t t coupling amplitude phase matching Isolated Binary Collision Model Fermi’s golden rule influence functional thermal averarage （static） collisional? statistics? collisional, delta function apolar solvent? short-ranged force polar solvent? long-ranged force resonant, stationary

  46. Interaction of Ion in Water < i | V(t) | j > | ci |2 V10(t) looks almost random in liquid frequency matching phase matching Im ρ10 resonance t / ps V10(t)= < 1 | V(t) | 0 > dρ11/dt ～ 2/h ・V01・ Im ρ10

  47. Coupling as a Function of Time liquid solid and glass gas coupling coupling coupling t t t t Fermi’s golden rule influence functional thermal averarage （static） random coupling frequency matching phase matching Isolated Binary Collision Model collisional, delta function resonant, stationary How is it in the apolar solvent ? How is it in the supercritical fluid ?

  48. Sibert III and Rey, JCP 116, 237(2002) Relaxation by Collision chloroform short-ranged force repulsive force direction of the mode

  49. Interaction with the Solvent V01 r = 0.870 g/cm3 r = 0.725 g/cm3 r = 0.580 g/cm3 r = 0.435 g/cm3 r = 0. 290 g/cm3 r = 0.145 g/cm3 r = 0.029 g/cm3

  50. Resonance and Collisions with the Solvent collisions resonance r = 0.870 g/cm3 r = 0. 290 g/cm3 r = 0.029 g/cm3

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