Path integral molecular dynamics (with surface hopping) Jianfeng Lu 鲁剑锋 Duke University [email protected] Casa Matemática Oaxaca, November 2018 Based on joint works with Zhennan Zhou (BICMR, Peking University) Quantities of interests for nuclear quantum effects • Time propagation: ⟨푢|푒퐴푒|푢⟩ ; • Thermal equilibrium average: tr푒퐴 ; tr 푒 • Dynamical correlation function: tr푒퐴푒퐵푒 , tr 푒 where 퐻 = − Δ + 푉 is a Hamiltonian, 퐴 and 퐵 are observables, 푢 is some wave function, 훽 is inverse temperature and 푡 is time. Difficulty: Curse of dimensionality. = ⋯ ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩ × ⋯ × ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩d푝 d푞 || () = ⋯ ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 × ⋯ || () × ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 d푝 d푞 |푞 − 푞| ∝ ⋯ exp− − 훽 푉(푞)d푞, 2훽 with 훽 = and the convention 푞 = 푞. R. Feynman: Turn this into a sampling problem of a classical system. Classic example of cross-fertilization → Feynman-Kac rep. Lie-Trotter / Strang splitting for 퐻 = − Δ + 푉: 훽 훽 훽 훽 tr 푒 ≈ trexp Δ exp− 푉 ⋯ exp Δ exp− 푉 2푁 푁 2푁 푁 |푞 − 푞| ∝ ⋯ exp− − 훽 푉(푞)d푞, 2훽 with 훽 = and the convention 푞 = 푞. R. Feynman: Turn this into a sampling problem of a classical system. Classic example of cross-fertilization → Feynman-Kac rep. Lie-Trotter / Strang splitting for 퐻 = − Δ + 푉: 훽 훽 훽 훽 tr 푒 ≈ trexp Δ exp− 푉 ⋯ exp Δ exp− 푉 2푁 푁 2푁 푁 = ⋯ ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩ × ⋯ × ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩d푝 d푞 || () = ⋯ ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 × ⋯ || () × ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 d푝 d푞 R. Feynman: Turn this into a sampling problem of a classical system. Classic example of cross-fertilization → Feynman-Kac rep. Lie-Trotter / Strang splitting for 퐻 = − Δ + 푉: 훽 훽 훽 훽 tr 푒 ≈ trexp Δ exp− 푉 ⋯ exp Δ exp− 푉 2푁 푁 2푁 푁 = ⋯ ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩ × ⋯ × ⟨푞|푒 |푝⟩⟨푝|푒 |푞⟩d푝 d푞 || () = ⋯ ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 × ⋯ || () × ⟨푞|푝⟩푒 ⟨푝|푞⟩푒 d푝 d푞 |푞 − 푞| ∝ ⋯ exp− − 훽 푉(푞)d푞, 2훽 with 훽 = and the convention 푞 = 푞. Ring-polymer representation: |푞 − 푞| tr 푒 ∝ ⋯ exp− − 훽 푉(푞)d푞. 2훽 Sampling problem on a high-dimensional energy landscape. • Path-integral Monte Carlo • Path-integral Molecular Dynamics • Extension to dynamical quantities (ring polymer MD, centroid MD, Matsubara dynamics, Liouville dynamics, etc.) Path-integral molecular dynamics (PIMD): Re-introducing momentum |푝| |푞 − 푞| tr 푒 ∝ ⋯ exp−훽 − − 훽 푉(푞)d푝 d푞 2 2훽 =∶ ⋯ 푒(,) d푝 d푞 with |푝 | |푞 − 푞 | 퐻 (푝, 푞) = + + 푉(푞 ). 2 2훽 Figure: Image credit: Tom Markland @Standford Taking 푁 → ∞, we obtain the path-integral representation as a continuum limit of the ring polymers |픮(휏)|̇ tr 푒 ∝ D[픮] exp− + 푉(픮(휏)) d휏 2 =∶ D[픮] exp− ⟨픮, 퐿픮⟩ + 푉(픮) where the path 픮 ∶ [0, 훽] → ℝ takes periodic boundary conditions. Cross-Fertilization: It is quite similar to diffusion bridge sampling except for the boundary condition. Taking 푁 → ∞, we obtain the path-integral representation as a continuum limit of the ring polymers |픮(휏)|̇ tr 푒 ∝ D[픮] exp− + 푉(픮(휏)) d휏 2 =∶ D[픮] exp− ⟨픮, 퐿픮⟩ + 푉(픮) where the path 픮 ∶ [0, 훽] → ℝ takes periodic boundary conditions. For the auxiliary momentum, using kinetic energy with “diagonal mass” does not make sense in the continuum limit. Instead, it is better to use (similar to [Beskos, Pinski, Sanz-Serna, Stuart 2011] for diffusion bridges) tr 푒 = D[(픮, 픭)] exp− ⟨픮, 퐿픮⟩ + ⟨픭, (퐿 ) 픭⟩ + 푉(픮) where 퐿 = 퐿 + 훼Id to avoid the singularity due to the periodic Laplacian. Regularizing the position part as well, we get tr 푒 = D[(픮, 픭)] exp− ⟨픮, 퐿픮⟩ + ⟨픭, (퐿 ) 픭⟩ + 푉(픮) = D[(픮, 픭)] exp− ⟨픮, 퐿 픮⟩ + ⟨픭, (퐿 ) 픭⟩ + 푈 (픮) with 푈(푞) = 푉(푞) − 훼|푞| (assumed to be growing at infinity). The associate (inf. dim.) (underdamped) Langevin dynamics reads d픮 = (퐿)픭d푡; d픭 = −퐿픮d푡 − ∇푈 d푡 − 훾픭d푡 + 2훾퐿 d휔. Langevin dynamics d픮 = (퐿)픭d푡; d픭 = −퐿픮d푡 − ∇푈 d푡 − 훾픭d푡 + 2훾퐿 d휔. The dynamics is rather stiff due to 퐿, causing trouble for sampling. Introducing 픳 = (퐿)픭 and preconditioning, we arrive at d픮 = 픳 d푡; d픳 = −픮 d푡 − (퐿)∇푈 d푡 − 훾픳d푡 + 2훾(퐿) d휔. This leads to a much better sampling scheme when the number of bead is large (in preparation with Zhennan Zhou). Question: Convergence rate? (ongoing project with Yulong Lu and Jonathan Mattingly) In the above, we have assumed that 퐻 = 푇 + 푉, where 푉 is just a single potential surface for the nuclei degree of freedom. This is known as the adiabatic approximation or Born-Oppenheimer approximation. The adiabatic approximation is justified if 퐸(푞) (ground state) as a potential energy surface is well separated from 퐸(푞) (excited state), and hence the transition of electrons to excited states can be neglected. This is however often not the case, for applications like photoexcited dynamics, electron transfer and surface chemistry, where the Born-Oppenheimer approximation falls apart. Figure: Schematic of ultrafast non-adiabatic photoreaction. © Gerhard Stock This thus leads us to the non-adiabatic scenario, for which, under the diabatic representation, the Hamiltonian takes 1 Δ 푉 푉 퐻 = − + 2 Δ 푉 푉 where a two-level system is assumed for simplicity, generalizations to multi-level is possible. Thermal equilibrium sampling for multi-level systems trne(푒 퐴) ⟨퐴⟩ = , trne(푒 ) where trne = tr(ℝ) trℂ . Question: How to extend the PIMD sampling approach to multi-level systems? Previous works on non-adiabatic ring-polymer representation: • Meyer-Miller-Stock-Thoss mapping variable approach, by mapping the discrete electronic degree freedom to continuous variables using uncoupled harmonic oscillators. Then just apply what we did before. In particular, recent works by the groups of Ananth and T. Miller on PIMD based on mapping variable approaches. • Design a ring-polymer representation that keeps the discreteness of the electronic levels. A much less traveled road. Only previous work we are aware of is [Schmidt, Tully 2007], which used Monte Carlo to sample the non-adiabatic ring polymer configuration. We will take this route, as it is more advantageous when combining with the dynamics. Idea: Further extend the phase space to encode the discrete level. 푧= (푝, 푞, ℓ) ∈ ℝ × ℝ × {0, 1} , so now each bead (푝, 푞, ℓ) lives on two copies of the classical phase space, with ℓ ∈ {0, 1} a level index. 2 1 0 -1 -2 -3 1 0.5 0 1 -0.5 0.5 0 -0.5 -1 -1 Thus, phase space 퐿(ℝ) ⊗ ℂ → ℝ × ℝ × {0, 1}. From adiabatic to non-adiabatic ring polymer represenation. Strang / Trotter splitting can be still used: 훽 훽 훽 훽 tr 푒 ≈ trexp− 푇 exp− 푉 ⋯ exp− 푇 exp− 푉 푁 푁 푁 푁 where 푇 and 푉 are matrices of operators: 1 Δ 푉 푉 푇 = − and 푉 = 2 Δ 푉 푉 and continue the process of inserting resolution of identities, but at this time, we will also need to inserting ℓ to track the matrix components. From adiabatic to non-adiabatic ring polymer represenation. Partition function (in a simpler case; ratios of time average estimator needed in general): tr 푒 ≈ 푒(,) d푝 d푞 (,,ℓ) trne 푒 ≈ 푒 d푝 d푞 ℓ∈{,} (,) tr(퐴푒 ) ≈ 푒 푊[퐴](푝, 푞)d푝 d푞 (,,ℓ) trne(퐴푒 )≈ 푒 푊[퐴](푝, 푞, ℓ)d푝 d푞 ℓ∈{,} The expression of 퐻 and 푊[퐴] is complicated; but follows from straightforward calculations. We will skip the details as our main focus today is sampling. Goal: Sample from 휚(푝, 푞, ℓ) ∝ 푒(,,ℓ) on ℝ × ℝ × {0, 1}. The evolution of (푝, 푞) follows the Hamiltonian dynamics with a Langevin thermostat d푞 = ∇퐻(푝, 푞, ℓ)d푡 d푝 = −∇퐻(푝, 푞, ℓ)d푡 − 훾푝 d푡 + 2훾훽 d푊 The evolution of ℓ follows a surface hopping type dynamics as a Markov jump process with (푧 = (푝, 푞)) ℙℓ(푡 + 훿푡) = ℓ ∣ ℓ(푡) = ℓ, 푧(푡) = 푧 = 훿ℓℓ + 휂휆ℓℓ(푧)훿푡 + 표(훿푡), where 푧 = (푝, 푞) and 휂 is an overall hopping intensity scaling parameter. Surface hopping dynamics ℙℓ(푡 + 훿푡) = ℓ ∣ ℓ(푡) = ℓ, 푧(푡) = 푧 = 훿ℓℓ + 휂휆ℓℓ(푧)훿푡 + 표(훿푡). To guarantee that the dynamics samples the desired distribution, we may choose 휆ℓℓ such that the detailed balance condition is satisfied: (,ℓ) (,ℓ ) 휆ℓℓ(푧)푒 = 휆ℓℓ (푧)푒 . For instance, 훽 푝 (푧) = 퐴 exp 퐻 (푧, ℓ) − 퐻 (푧, ℓ ) ℓ ℓ ℓ ℓ 2 with 퐴ℓℓ a symmetric “accessibility matrix” (e.g., only allow one level change of the beads, etc.). PIMD with surface hopping (PIMDSH): d푞 = ∇퐻(푝, 푞, ℓ)d푡 d푝 = −∇퐻(푝, 푞, ℓ)d푡 − 훾푝 d푡 + 2훾훽 d푊 ℙℓ(푡 + 훿푡) = ℓ ∣ ℓ(푡) = ℓ, 푧(푡) = 훿ℓℓ + 휂휆ℓℓ(푧)훿푡 + 표(훿푡). Cross-Fertilization: Anything similar in Bayesian statistics? PIMD with surface hopping (PIMDSH): d푞 = ∇퐻(푝, 푞, ℓ)d푡 d푝 = −∇퐻(푝, 푞, ℓ)d푡 − 훾푝 d푡 + 2훾훽 d푊 ℙℓ(푡 + 훿푡) = ℓ ∣ ℓ(푡) = ℓ, 푧(푡) = 훿ℓℓ + 휂휆ℓℓ(푧)훿푡 + 표(훿푡). Numerical discretization (2nd weak order): • Operator splitting scheme between (푞, 푝) and ℓ; • BAOAB splitting for Langevin [Leimkuhler, Matthews 2013] (see also [Liu, Li, Liu 2016] for BAOAB type scheme for PIMD); d푞 = 0 d푞 = ∇퐻 d푡 d푞 = 0 (B) (A) (O) d푝 = −∇퐻 d푡 d푝 = 0 d푝 = −훾푝 d푡 + 2훾훽 d푊 Numerical tests: two cases in the diabatic representation 1 0.8 15 V 0.6 0 00 ρ 0.4 V 11 0.2 0 10 V -3 -2 -1 0 1 2 3 10/01 x 1 5 0.8 0.6 1 ρ 0.4 0.2 0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x x 20 1 0.8 V 0.6 00 0 15 ρ 0.4 V 11 0.2 10 0 V × 5 -3 -2 -1 0 1 2 3 01/10 x 5 1 0.8 0.6 1 0 ρ 0.4 0.2 0 -5 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x Also consider two observables 푒 0 0 푒 퐴 = (diagonal), 퐴 = (off-diagonal) diag 0 푒 offd 푒 0 Convergence with number of beads for diagonal observable: For 훽 = 1 and 훽 = , and .