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6286cbd1d555504c5fa060fd
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Consider the following short time propagation, from t to t + dt, of the reduced density operator ρ (Eq. 8) of a K ≥ 2 level system evolving under the dynamics of the quantum subsystem Hamiltonian Ĥ = ĤQ + ĤQC (R) and a single non-zero Lindblad jump operator L = |0⟩⟨1|. For a small dt, the time evolution can be approximated as
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6286cbd1d555504c5fa060fd
| 24 |
due to the matrix elements of Eq. 24 that involve states {|j⟩}. The equation of motion for the coefficients {c a,ξ (t)} governed by the evolution of ĤQ + ĤQC (or e L Ĥ dt in Eq. 27) can be computed based on the simple MFE approach in Eq. 15. The rest of the derivation will be focused on updating the coefficients to match these jump operator dynamics, separate from any Hamiltonian dynamics (as Eq. 27 allows us to do).
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6286cbd1d555504c5fa060fd
| 25 |
In order to achieve the same Lindblad dynamics from a mixed quantum-classical method such as MFE, the coefficients c 1 and c 0 (and {c j }) must be evolved in a way such that their time evolution corresponds to equations 28a through 29d. The density matrix elements of equations 28a through 29d cannot, however, be directly replaced with the usual products of the coefficients c 1 and c 0 (and {c j }). This is because this density matrix, in general, represents a mixed quantum state due to the fact that Lindblad dynamics describe an interaction with an external environment. The density matrix of a mixed quantum state cannot, in general, be represented by the density matrix of a single pure state as c 0 |ψ 0 ⟩ + c 1 |ψ 1 ⟩. Thus, alternative approaches must be used to perform the time evolution of c 1 and c 0 with a combined result that satisfies Eq. 28a through Eq. 29d.
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6286cbd1d555504c5fa060fd
| 26 |
To this end, we take advantage of the existing multiple trajectories that are already present in mixed quantumclassical or semiclassical methods. We propose to evolve multiple trajectories and use the trajectory average of the electronic coefficients of these trajectories to compute the reduced density matrix elements at each time step, with the estimator of the reduced density matrix elements evaluated as
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6286cbd1d555504c5fa060fd
| 27 |
In addition, we introduce random variation of the time evolved coefficients as (31) where {η a,ξ } are random complex variables with a certain probability distribution yet to be determined, such that the expectation value of the estimator of the time evolved reduced density matrix elements will exactly correspond to Eqs. 28a-29d. Further, {θ a,ξ } are the phases and {χ a,ξ } are the magnitudes of the complex variables.
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6286cbd1d555504c5fa060fd
| 28 |
and all of the decoherence dynamics in Eqs. 29a-29c will be governed by the changes in c 1,ξ and c 0,ξ . Ultimately, Eq. 39 only makes the choice that no arbitrary global phase is added to all coefficients since any other modification to the c j,ξ coefficients, besides adding a global phase, would incorrectly alter some of the reduced density matrix elements involving the {|j⟩} states.
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6286cbd1d555504c5fa060fd
| 29 |
While there are several possible choices for constructing c 1,ξ (t + dt) and c 0,ξ (t + dt) that agree with Eqs. 37a-41b, it is important to consider how these choices affect the Hamiltonian dynamics (evolved by e L Ĥ dt ) that occur in conjunction with the Lindblad dynamics. Certain choices of c 1,ξ (t+dt) and c 0,ξ (t+dt) may cause the Hamiltonian dynamics to diverge from the results of the exact Liouvillian dynamics, for example, when they do not conserve the trace of the density matrix within a trajectory. To avoid this, we add on an additional constraint to the coefficient evolution that
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6286cbd1d555504c5fa060fd
| 30 |
such that Eq. 53 (and thus Eq. 37c) is satisfied. Equations 49 and 54 govern how the (potentially random) variables θ ξ and χ 1,ξ must be selected in order to obey equations 37a through 42. To avoid considering joint probability distributions of dependent random variables, we make the assumption that
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6286cbd1d555504c5fa060fd
| 31 |
The right hand side of Eq. 59 is purely real and is always within the range 0 to 1 for Γdt ≥ 0. Thus, there exist distributions of θ ξ that satisfy the above equation because the expectation value of e iθ ξ can have a magnitude within the range 0 to 1. To reduce large jumps in these phase variables, we assume that
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6286cbd1d555504c5fa060fd
| 32 |
This means that choosing ∆θ ξ according to Eq. 61 as the bounds of the uniform probability distribution of θ ξ will satisfy the expectation value relations in all previous equations. Since Eq. 61 is a transcendental equation, no general closed form exists for the solutions to ∆θ ξ . To find ∆θ ξ , a numerical interpolation function of the first positive solution of sin(x) = ax can be pre-computed for the values 0 ≤ a ≤ 1 and then used during the simulation. Up to this point, the only unspecified variables are the individual phases θ 0,ξ and θ 1,ξ . To determine these variables, Eq. 38a and Eq. 38b can be used. Using the conditions outlined in Eq. 39 and Eq. 58, the left hand side of Eq. 38a can be rewritten as
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6286cbd1d555504c5fa060fd
| 33 |
This means that using the information of how state |1⟩ decoheres with other states |j⟩ / ∈ {|0⟩, |1⟩} allows us to determine how the random phase θ ξ should be partitioned among θ 0,ξ and θ 1,ξ . Eq. 64 suggests that there is no need to add additional random phase to coefficients c 1,ξ (t) if one wants to correctly describe the decoherence dynamics between state |1⟩ and |j⟩ (for j ̸ = 0, 1) governed by Lindblad dynamics (Eq. 24). The information in Eq. 63 was omitted when developing the Ehrenfest+R method. Nevertheless, the same choice was empirically discovered through the use of numerical simulations of Ehrenfest+R. Using a similar analysis of the above procedure for the left-hand side of Eq. 38b, together with Eq. 45 and Eq. 66, it follows that
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6286cbd1d555504c5fa060fd
| 34 |
where T (dt) is a formal transition operator that propagates the coefficients as designed, e iφ 0 ξ = c 0,ξ (t)/|c 0,ξ (t)| is the phase factor of c 0,ξ (t), and the random phase θ ξ is sampled based on a uniform distribution P (θ ξ ) expressed as
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6286cbd1d555504c5fa060fd
| 35 |
Thus, we explicitly use the trajectory average to converge to the expectation value defined in Eq. 33. This is one of the main theoretical results of the paper. Note that when c 0,ξ (t) = 0 and c 1,ξ (t) ̸ = 0, the random angle θ ξ added to the ground state is sampled from -π to π due to Eq. 71 which completely randomizes the phase of the ground state, which renders the undetermined phase factor e iφ 0 ξ irrelevant. These effective Lindblad updates of the electronic coefficients described in Eqs. 69a-69c can be combined with the Ehrenfest part of the electronic coefficients update described in Eq. 15 and Eq. 16 to describe the mixed quantum-classical dynamics subject to a Lindblad type decay. We refer to this approach as the L-MFE method for the remainder of the paper. The coefficient propagation of the L-MFE method can be expressed as
|
6286cbd1d555504c5fa060fd
| 36 |
where c(t) is the vector of the quantum coefficients, T describes the coefficients update due to the jump operator using Eqs. 69a-69c, and e -i ℏ ( ĤQ+ ĤQC)dt describes the unitary evolution of the coefficients due the MFE dynamics described in Eq. 15. Note that ĤQC (R) depends on the nuclear DOFs R, where R(t) evolves based on the force in Eq. 16. Here, we use a symmetrical Trotter decomposition in Eq. 72 to reduce error due to a finite time step dt, and we put the Ehrenfest propagation in the middle because it is computational expensive (which scales as ∼ K 2 ) compared to the Lindblad decay part (denoted by T ) which only requires the update of two coefficients c 1,ξ (t) and c 0,ξ (t).
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6286cbd1d555504c5fa060fd
| 37 |
The L-MFE algorithm outlined in Eq. 72 will, in principle, generate different results compared to Eq. 21 where the Lindblad dynamics are propagated deterministically in the Liouville space. This is because the L-MFE method adds random phases to state |0⟩ (for the L = |0⟩⟨1| jump operator) through T (see Eq. 69b) for each individual trajectory. Further, these random phases in c 0,ξ (t) will also influence the magnitude of populations through coupling term V 01 (R ξ )+ε 01 . Thus, L-MFE adds different phases onto ρ 01,ξ (as well as on ρ 10,ξ ) and generates different populations ρ 00,ξ and ρ 11,ξ for each individual trajectory. The L L [ρ] term in Eq. 21, on the other hand, causes deterministic changes for the density matrix elements of all trajectories. Thus, the random phases in L-MFE will influence the nuclear forces in Eq. 19 and these different nuclear forces for each trajectory further influences the nuclear motion for R, and eventually back influence the electronic dynamics through e -i ℏ ( ĤQ+ ĤQC(R))dt in Eq. 72. Future research will focus on investigating the difference between Eq. 21 and Eq. 72. Nevertheless, numerical results obtained for the model systems (Eqs. 76a-76h and Eq. 77) produce visually identical results using either Eq. 21 or Eq. 72.
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6286cbd1d555504c5fa060fd
| 38 |
We want to emphasize the connections and differences between the current L-MFE method and previous approaches that try to accomplish Lindblad dynamics through stochastic wavefunction approaches, such as Monte Carlo wave-function methods and the Ehren-fest+R approach. The Monte Carlo wavefunction method expresses the stochastic time evolution of the system's wavefunction as follows d|ψ
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6286cbd1d555504c5fa060fd
| 39 |
This term performs a Poisson jump process that captures the projective action of the environment on the system that randomly collapses the system wavefunction into a pure system eigenstate. The trajectory average of Eq. 73 with a large number of realizations (jump trajectories) generates results identical to ρS (t) described by the Lindblad master equation in Eq. 9. The equivalence of these two descriptions can be viewed as the correspondence between the stochastic quantum state diffusion equation (Eq. 73) and the deterministic Wigner Fokker-Planck equation (which can be written in the Lindblad master equation form of Eq. 9). The L-MFE method, on the other hand, does not collapse trajectories onto a single system state, which avoids large changes in the magnitudes of the coefficients. Further, the change in magnitude of the coefficients is deterministic, whereas the magnitude of the coefficients in the Monte Carlo wavefunction method is stochastic. Further, the phases of the coefficients in L-MFE only slightly vary between time steps with no large jumps. When explicitly considering ĤC + ĤQC , the stochastic Schrödinger equation approach can encounter numerical instabilities due to large changes of the electronic coefficients, whereas the L-MFE approach provides a stable numerical integration of the equation of motion due to the small changes of phase in the coefficients. The L-MFE approach, in addition, can be easily combined with any mixed quantumclassical or semiclassical approach.
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6286cbd1d555504c5fa060fd
| 40 |
The Ehrenfest+R method 10 was originally developed to simulate the electronic quantum subsystem coupled to the classical electromagnetic field in order to accurately describes spontaneous emission processes. It effectively captures Lindblad dynamics with a deterministic change of the magnitude of the quantum coefficients, and stochastic changes of the phases. While the L-MFE method is similar to (and largely inspired by) the Ehren-fest+R method, there is a key difference between Ehren-fest+R and L-MFE in the off-diagonal reduced density matrix element decay procedure. To facilitate the theoretical comparison, we briefly summarize the Ehren-fest+R method 10 in Appendix A.
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6286cbd1d555504c5fa060fd
| 41 |
The Ehrenfest+R approach was derived under the condition that Γ ≪ E, where Γ is the decay rate of the jump operator L and E is the energy difference between state |0⟩ and |1⟩ (see Appendix A). This condition is thus the regime where the Ehrenfest+R approach can be applied with guaranteed accuracy. The L-MFE method, on the other hand, does not have any restrictions on the parameter regimes where it is applicable.
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6286cbd1d555504c5fa060fd
| 42 |
The "+R decay" procedure (see Eq. A24 through Eq. A28) in Ehrenfest+R was designed to correct the decay of the diagonal density matrix elements and offdiagonal elements independently, with the goal that the combined Ehrenfest dynamics and +R decay dynamics will match the Lindblad decay of the density matrix elements. However, the procedure to adjust the diagonals of the density matrix by changing the magnitudes of the corresponding coefficients also changes the off-diagonal density matrix elements, causing unintended deviations in the dynamics. The off-diagonal relaxation decay rate in Ehrenfest+R, which is proposed to be (see Eq. 53 in Ref. 10)
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6286cbd1d555504c5fa060fd
| 43 |
for |c 0,ξ (t)c 1,ξ (t)| ̸ = 0. The detailed derivation of this expression is provided in Appendix B. The L-MFE method already accounts for the effect that modifying the magnitudes of the coefficients has on the off-diagonals of the density matrix, as shown in Eq. 59. This is another main theoretical result of the current paper.
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6286cbd1d555504c5fa060fd
| 44 |
Additionally, in the Ehrenfest+R approach, it was numerically found that adding random phase only to c 0,ξ (t + dt) gave the most accurate results; 11 however, these choices lacked a rigorous theoretical reason for why this should be the case and it was speculated that this is because spontaneous emission from states |1⟩ to |0⟩ should not affect the coherence of state |1⟩ with other states {|j⟩}. In contrast, the analysis in this paper shows that the mathematical reason for this choice of random phase lies in the fact that the excited state |1⟩ decoheres with every other state of the quantum subsystem while state |0⟩ only decoheres with state |1⟩, when L = |0⟩⟨1|. This fact is derived from applying the Lindbladian L L (with L = |0⟩⟨1|) to the entire reduced density matrix of K states instead of only the reduced density matrix of |0⟩ and |1⟩. While state |1⟩ decoheres with every other state, the decoherence with states {|j⟩} is entirely captured by the reduction of the magnitude of c 1,ξ (t + dt) by e -Γdt/2 ; thus, the phase of state |1⟩ does not need to provide any additional decoherence. In contrast, state |0⟩ does not decohere with states {|j⟩}, but the increase of the magnitude of c 0,ξ (t + dt) causes an increase in coherence with states {|j⟩}. The role of the random phase applied to state |0⟩, aside from adjusting the coherence with state |1⟩, is to add decoherence to cancel out the increase in coherence with states {|j⟩} due to the increase of the magnitude of c 0,ξ (t + dt).
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6286cbd1d555504c5fa060fd
| 45 |
Lastly, when ĤQC + ĤC = 0 (no nuclear DOFs present in the system Hamiltonian), the L-MFE approach in Eq. 72 (or equivalently Eq. 21) provides identical results as those obtained by solving Eq. 12, regardless of the choice of dt (as long as it is small enough to provide a stable integration of -i ℏ [ ĤQ , ρ]). This is because in the L-MFE approach, the decay dynamics are designed to exactly match the analytical time evolution of the reduced density matrix elements. Consequentially, if the Hamiltonian is 0 and only jump operator dynamics are present, the choice of dt for L-MFE could be arbitrarily large and still give the correct dynamics. On the other hand, under the same condition when ĤQC + ĤC = 0, the Ehrenfest+R approach is only accurate up to first order in dt (see Eq. A28 and Eq. B7).
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6286cbd1d555504c5fa060fd
| 46 |
Simple Model Systems. To assess the accuracy of the L-MFE method, a variety of models are tested and compared with an exact calculation of the corresponding Lindblad dynamics. All of the following simple models are associated with a single Lindblad jump operator L = |0⟩⟨1| with interaction strength Γ = 0.05 a.u. These models are
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6286cbd1d555504c5fa060fd
| 47 |
Here, |α⟩ ∈ {|g⟩, |e⟩} represents the electronic ground or excited state, which are treated as diabatic states in this model, R represents the reaction coordinate, and TR = P 2 /2M is the nuclear kinetic energy operator associated with R, with nuclear mass M = 550 Da. The detailed expression of E α (R) is provided in Appendix C. The second term in Eq. 77 is the Hamiltonian of the quantized photon mode inside the cavity with the frequency ω c , and â † and â are the photon creation and annihilation operators, respectively. The third term in Eq. 77 describes the molecule-photon coupling through electric-dipole interactions under the dipole gauge, where σ † = |e⟩⟨g| and σ = |g⟩⟨e| are the molecular excitonic creation and annihilation operators, respectively, and the light-matter interaction strength ℏg c is treated as a parameter in this model. The dipole self-energy
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6286cbd1d555504c5fa060fd
| 48 |
The lifetime of the cavity mode is finite, due to the coupling between the cavity mode and the far-field photon modes outside the cavity. The detailed discussions for molecular cavity QED with cavity loss are provided in Appendix D. Here, we use the following Lindblad jump operator to model this process
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6286cbd1d555504c5fa060fd
| 49 |
In this paper, we consider this photo-isomerization model under the Jaynes-Cummings approximation where there is a maximum of only one excitation (the singleexcited subspace plus the ground state) because the lightmatter coupling strength ℏg c /ω c < 0.1 thus the results do not significantly change when including states with multiple excitations. In this case, the possible Fock states are just |n⟩ ∈ {|0⟩, |1⟩}. Thus, the system Hamiltonian ĤS can be written as ĤS = TR + E g (R)|g, 0⟩⟨g, 0| + E e (R)|e, 0⟩⟨e, 0| (82) + E g (R) + ℏω c |g, 1⟩⟨g, 1| + ℏg c |e, 0⟩⟨g, 1| + |g, 1⟩⟨e, 0| , and the system Lindblad jump operator can be written as
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6286cbd1d555504c5fa060fd
| 50 |
is a Gaussian wavepacket centered around R 0 = -0.7 a.u. with variance 1/2M ω R , mass M = 550 Da, and frequency ω R = 132.4 cm -1 . The temperature is set to be T = 0 K. For the L-MFE simulation, the corresponding initial condition is ρS (0
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6286cbd1d555504c5fa060fd
| 51 |
Computational Details. The L-MFE method is implemented by using a symmetrical Trotter decomposition in Eq. 72 to reduce time step error. The unitary dynamics e -i ℏ ( ĤQ+ ĤQC)dt are propagated using the 4th order Runge Kutta (RK4) algorithm, while the Lindblad decay dynamics T are propagated using the coefficient modifications described in Eqs. 69a to 71. When classical nuclear DOFs are present, the velocity Verlet algorithm is used to propagate the nuclear DOFs, with mean-field force given by Eq. 16, alongside the quantum subsystem. For all model calculations, the electronic time step used is dt E = 0.05 a.u. For the photo-isomerization system that contains nuclear DOF, the nuclear time step was dt N = 6 a.u. and the time step used in the Lindblad decay T was dt N /4 (thus the unitary propagation was performed 60 times in between Lindblad decay propagations). A total of 24, 000 trajectories were used to ensure fully converged results, although using only 1000 trajectories already provides a mostly converged result (as shown in Fig. ).
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| 52 |
The numerical results obtained from L-MFE were benchmarked against the original Lindblad dynamics in Eq. 21, which are referred to as "exact" results (of performing Lindblad dynamics) in this paper. These numerical results are obtained by using the QuTiP library with the mesolve function, where model Hamiltonian, Lindblad jump operator, and initial wavefunction are entered as arguments to the function. Similar numerical simulations has been recently performed to investigate molecular cavity QED processes as well. For Models 1-8, ĤC + ĤQC = 0 (no nuclear DOF), and the only inputs in the simulation are the matrix elements of ĤQ and the decay rate Γ. For the photo-isomerization coupled to the cavity model, ĤS is described by Eq. 82. The matrix elements of the Hamiltonian in Eq. 82 as well as the jump operator LS in Eq. 83 are evaluated using the basis {|α, n⟩ ⊗ |χ ν ⟩}, where |χ ν ⟩ is the discrete variable representation (DVR) basis for the nuclear DOF R. For the DVR basis |χ ν ⟩, a total of 175 grid points are used in the range of R ∈ [-1.25 a.u., 1.25 a.u.].
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6286cbd1d555504c5fa060fd
| 53 |
Fig. presents the population dynamics of Model 1 (Eq. 76a), obtained from the L-MFE approach (dots) as well as exact Lindblad dynamics (solid lines). In this case, the only dynamics present are the Lindblad exponential decay from state |1⟩ to state |0⟩. In Fig. , the diagonal populations of both states |1⟩ and |0⟩ exhibit the expected exponential decay/growth at a rate of Γ = 0.05, while the coherence between the two states stays at 0. This is expected because, in the absence of any Hamiltonian dynamics, the coherences between states |1⟩ and |0⟩ should only monotonically decrease to 0 from the initial time coherence ρ 01 (0). Fig. presents the population dynamics of Model 2 (Eq. 76b), which has an initial condition of a superposition of state |1⟩ and |0⟩. Similar to Model 1, only Lindblad exponential decay/growth is present. The diagonal populations match the Lindblad dynamics, and the coherence shows the expected Γ/2 decay rate from Lindblad dynamics.
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| 54 |
We note that because ĤQ = 0 in Models 1 & 2, the L-MFE method in Eq. 72 becomes c(t + dt) = T (dt) • c(t), where the dynamics are completely dictated by the Lindblad decay process governed by L L. The dynamics are insensitive to the choice of dt, which means that one can choose an arbitrarily large dt and obtain identical results. This is not the case for Ehrenfest+R approach, where the "+R" dynamics will be sensitive to the choice of dt even when ĤQ = 0.
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| 55 |
Fig. present the incorrect dynamics when the random phases θ ξ (in Eq. 69b) are intentionally ignored by setting them to be zero. One can see that the diagonal populations of states |1⟩ and |0⟩ still show the correct exponential decay/growth, but the coherences take a large departure from the expected coherences because there is no random phase present to correct the artificial coherence that is produced when the magnitudes of the coefficients are changed.
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| 56 |
Fig. presents the population dynamics of Model 3 (Eq. 76c). This model contains Hamiltonian-induced coherences (through the ∆ term in Eq. 76c) which must be properly incorporated with the Lindblad decay dynamics. In Fig. , the diagonal populations match the exact Lindblad dynamics result with both correct oscillation magnitudes and correct longtime populations. The oscillations of the imaginary parts (dark and light orange lines) of the off-diagonal coherences also exactly agree with Lindblad dynamics. Fig. presents the population dynamics of Model 4 (Eq. 76d), which contains both electronic coupling as well as an energy level difference between the two states, which will further impact the dynamics. Again, both the diagonal populations and the off-diagonal coherences match the exact Lindblad dynamics.
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| 57 |
Fig. present the incorrect dynamics when the random phases θ ξ (in Eq. 69b) are intentionally ignored by setting them to be zero. In Fig. , the longtime diagonal populations appear correct, but the diagonal populations oscillate with a much larger magnitude than Lindblad dynamics. This is caused by the fact that the coherences are larger than they should be because there is no random phase to reduce the size of the coherences. Consequentially, the larger magnitude of coherence causes a larger magnitude of population oscillations. In Fig. , not only are the oscillation magnitudes of the diagonal populations too large, but the diagonal populations converge to incorrect longtime populations. In fact, the L-MFE method with no random phase shows the excited state |1⟩ with a larger longtime population, while Lindblad dynamics show that the ground state |0⟩ should have the larger longtime population. This is again caused by the incorrect coherences without random phase which cause a longtime shift in the diagonal populations. Fig. presents the population dynamics of Model 4 (Eq. 76d) with different numbers of trajectories to examine the convergence of the L-MFE method. The numbers of trajectories used in Fig. are (a) 10, (b) 100, (c) 1,000, and (d) 10,000, respectively. For the result using 10 trajectories, the magnitudes and phases of the oscillations of the L-MFE dynamics do not match the exact ones. However, the longtime populations of the states are approximately correct, in contrast to the no-randomphase case in Fig. where the relative magnitude of the longtime populations are flipped versus the exact populations. For the result using 100 trajectories, the magnitudes of the oscillations are almost correct while there are some deviations at later times. For 1,000 trajectories, the relative error of the L-MFE dynamics versus the exact dynamics is only a few percent, and there is little visual difference between the L-MFE dynamics and the exact Lindblad dynamics. For 10,000 trajectories, the L-MFE dynamics and exact Lindblad dynamics are nearly indistinguishable. These results give a better sense of how many trajectories are required to achieve a desired level of accuracy using the L-MFE method.
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| 58 |
). This model contains the third state |2⟩, which is coupled to state |0⟩. Note that in the L-MFE algorithm, when the jump operator is L = |0⟩⟨1|, state |0⟩ is the state that gains random phases (see Eq. 65). Thus, the preservation of correct dynamics when random phases interact with states {|j⟩} outside of the reduced density matrix of |0⟩ and |1⟩ is tested. In Fig. , the excited state shows exponential decay, while states |0⟩ and |2⟩ oscillate together until they reach a longtime population of 0.5, which is predicted by Lindblad dynamics. In Fig. , the corresponding imaginary parts of coherences Im[ρ 10 ] (orange), Im[ρ 12 ] (green), and Im[ρ 20 ] (red) are presented, and all of these coherences obtained from L-MFE (dotted) match the Lindblad dynamics (solid lines).
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| 59 |
In particular, the oscillation in Im[ρ 20 ] is due to the presence of electronic coupling between state |0⟩ and |2⟩, without further decoherence from L (see Eq. 24), suggesting that the L-MFE method can correctly describe dynamics involving {|j⟩} states even in the presence of random phases in the coefficient of state |0⟩.
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| 60 |
Fig. presents the population dynamics of Model 6 (Eq. 76f), which is a more challenging 3-state Hamiltonian that involves multiple electronic couplings to test the validity of the L-MFE method. The diagonal populations generated from L-MFE (dotted) again match Lindblad dynamics (solid lines). In Fig. , all of the coherences, Im[ρ 10 ] (orange), Im[ρ 12 ] (yellow), and Im[ρ 20 ] (red) match perfectly with Lindblad dynamics. Note that due to the electronic couplings between states |0⟩ and |1⟩, as well as between states |0⟩ and |2⟩, all of the coherences in general will be non-zero at a given point in time. Using the random phase (governed by Eq. 66) for state |0⟩, all of these detailed features are captured, further demonstrating the exact equivalence between the L-MFE method and Lindblad dynamics when no nuclear DOFs are present. Fig. presents the population dynamics of Model 7 (Eq. 76g) to assess the importance of which state the random phase is applied to. Fig. ,c present the population and coherence dynamics, respectively, of the L-MFE method versus exact Lindblad dynamics. The L-MFE method applies the random phase only to state |0⟩ and its dynamics match the exact results. Fig. present the incorrect population and coherence dynamics, respectively, of a modified L-MFE method when the random phase (θ ξ ) is only applied to state |1⟩ instead of state |0⟩, such that c 1,ξ (t + dt) = e iθ ξ e -Γdt/2 c 1,ξ (t). The populations dynamics of this modified L-MFE method show incorrect oscillation magnitudes, most notably that the oscillations between states |1⟩ and |2⟩ are considerably smaller than the oscillations of the exact dynamics. This corresponds to the coherence dynamics of Im[ρ 12 ] that are smaller than the corresponding exact coherence dynamics. This coherence inaccuracy is caused by the over-decoherence of state |1⟩ due to the application of the random phase to state |1⟩. This result highlights the importance of applying the random phase to the correct state.
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| 61 |
FIG. . Dynamics of Model 7 using (a,c) the L-MFE approach and (b,d) a modified L-MFE approach with the random phase only applied to state |1⟩. The solid lines are exact Lindblad dynamics while the dotted lines are the original or modified L-MFE approach. Panels (a) and (b) present the diagonal populations, with ρ00(t) (dark blue), ρ00(t) (cyan), and ρ22(t) (magenta). Panels (c) and (d) present the off-diagonal coherences, with Im
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| 62 |
method for this model). Fig. ,c present the population and coherence dynamics, respectively, obtained from Ehrenfest+R (dots), using the same algorithm as described in Ref. 11, where a concise summary of this approach can be found in Appendix A. The +R decay dynamics are implemented as suggested as the original reference 10 (through Eq. A27a and Eq. A27b) using the original γ R (Eq. A22b). The parameters of Model 8 were carefully chosen such that the excited state energy E ′ = 1 a.u. is significantly larger than the decay rate Γ = 0.05 a.u., such that the Ehrenfest+R method should work correctly in the regime E ′ ≫ Γ, while still showing interesting dynamics on the timescale of the population decay (t = 0 ∼ 100 a.u.). In Fig. , the Ehrenfest+R dynamics are qualitatively similar to the exact Lindblad dynamics, but there are some errors in the fluctuations of the state |1⟩ decay as well as in the magnitudes of the oscillations of states |0⟩ and |2⟩. The cause of these deviations can be seen in Fig. , where the magnitudes of the coherences obtained from Ehrenfest+R are larger than those of Lindblad dynamics, causing incorrect population transfer between diagonal populations. This is because the coherence decay rate γ R (Eq. A22b) in the Ehrenfest+R algorithm is smaller than the correct decay rate γ ′ R (Eq. B11), causing artificially large coherences and thus a larger magnitude of population oscillation.
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| 63 |
) Eh+R' Exact FIG. . Dynamics of Model 8 using (a,c) the original Ehren-fest+R approach (labeled as Eh+R) and (b,d) the Ehrenapproach using the modified γ ′ R from Eq. 75 (labeled as Eh+R ′ ). The modified Ehrenfest+R dynamics (Eh+R ′ ) are identical to those obtained from the L-MFE method for this model. The solid lines are exact Lindblad dynamics while the dotted lines are the original or modified Ehrenfest+R approach. Panels (a) and (b) present the diagonal populations, with ρ00(t) (dark blue), ρ11(t) (cyan), and ρ22(t) (magenta). Panels (c) and (d) present the off-diagonal coherences, with Im[ρ10](t) (orange), Im[ρ12](t) (yellow), and Im[ρ20](t) (red).
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Ehrenfest+R approach that uses the modified γ ′ R . This modified Ehrenfest+R approach produces results identical to those obtained from the L-MFE method for this model. Consequentially, in Fig. , the population dynamics of the modified Ehrenfest+R approach are identical to the Lindblad population dynamics. The corresponding coherence dynamics are provided in Fig. , which again match perfectly with the Lindblad coherence dynamics. This demonstrates that it is important to include the effect that modifying the magnitudes of the coefficients has on the coherences between the states (see analysis in Appendix B). We note that using the modified γ ′ R in the Ehrenfest+R method yields the same expectation values for the populations and coherences as the L-MFE method as Γdt → 0, but begins to lose accuracy for larger Γdt due to the use of a Poisson process for decoherence (through Eq. A28), which is sensitive to the choice of Γdt (Eq. B7).
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Fig. presents the potential energy surfaces (PESs) and population dynamics of the photo-isomerization model coupled to an optical cavity, with the system Hamiltonian described in Eq. 82. Here, we explicitly consider the population decay of the photonic DOF through the jump operator L in Eq. 83, which describes the finite lifetime of the cavity mode due to its coupling to the other non-cavity modes. Fig. presents the diabatic PESs for the |g, 0⟩ state (dark blue), the |e, 0⟩ state (cyan), and the |g, 1⟩ state (magenta). These states are diabatic because their electronic character does not depend on the nuclear coordinate R, thus there is no derivative coupling between these states (see Appendix C for details). Conversely, there is a diabatic coupling between the states |e, 0⟩ and |g, 1⟩ which causes coherent population transfer between these states. Fig. presents the adiabatic PESs for the ground |g, 0⟩ state (dark blue), the lower polariton state (middle line, labeled |LP ⟩), and the upper polariton state (upper line, labeled |U P ⟩). These adiabatic states are eigenstates of the polaritonic Hamiltonian ĤS -TR (where ĤS is given in Eq. 82) thus there is no additional diabatic coupling between any of the states. Further, since the electronic character of the upper and lower polariton states changes as a function of the nuclear coordinate R, there exists derivative coupling between the upper and lower polariton states which allows for coherent population transfer between them. Fig. -f presents the population dynamics of diabatic states |g, 0⟩ (dark blue), |e, 0⟩ (cyan), and |g, 1⟩ (magenta). The results of exact Lindblad dynamics (solid lines) are obtained by solving Eq. 12 using the basis {|α, n⟩ ⊗ |χ ν ⟩}, where |α, n⟩ = |α⟩ ⊗ |n⟩, |α⟩ ∈ {|g⟩, |e⟩} are the diabatic electronic states and |n⟩ are the Fock states of the cavity mode. The L-MFE dynamics (dotted) are obtained by treating the electronic and photonic DOFs as the quantum subsystem, with basis |α, n⟩, and the nuclear DOF as a classical DOF. The details of these numerical simulations are provided in Sec. IV.
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Fig. presents the population dynamics with Γ = 0, i.e., with no Lindblad decay dynamics. Thus, Eq. 12 reduces to the exact dynamics of a closed system with the Hamiltonian in Eq. 82 (which is a molecule-cavity hybrid system), and Eq. 21 reduces to the Ehrenfest dynamics for the same system. The Ehrenfest dynamics provides nearly identical results compared to the exact dynamics in this case because the nuclei are mostly oscillating on a single adiabatic upper polariton surface (see Fig. ). Because there are no interactions between the |g, 0⟩ state and the other states in the model, the |g, 0⟩ state is not populated.
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In Fig. , the Lindblad jump operator interaction strength is set to be Γ = 1 meV, which causes the population of the upper and lower polariton states to decay to the ground state |g, 0⟩ of the molecule cavity hybrid system. While the L-MFE dynamics semi-quantitatively match the exact Lindblad dynamics, there are some noticeable differences in the magnitudes of oscillation of states |e, 0⟩ and |g, 1⟩, and the ground state |g, 0⟩ does not rise as quickly as predicted by Lindblad dynamics. Similarly, in Fig. , when the interaction strength is set to be Γ = 2 meV and Γ = 8 meV, respectively, the L-MFE dynamics show similar errors while still maintaining the semi-quantitatively correct dynamics.
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The discrepancy between the dynamics obtained from the L-MFE method and the exact Lindblad dynamics is due to the inadequacy of MFE as a mixed quantumclassical method. This demonstrates the need to incorporate Lindblad dynamics into more accurate mixed quantum-classical or semiclassical approaches that go beyond the approximations present in the mean-field Ehrenfest approach. Note that in recent investigations of molecular cavity quantum electrodynamics, the photonic population decay is incorporated in a similar fashion as described in Eq. 69a-69c. However, in these early investigations, the random phase e iθ ξ (in Eq. 69b) is not incorporated. We have demonstrated the consequence of missing this random phase in Fig. , where artificial coherences are generated. Future wavefunction based investigations in molecular cavity QED should carefully describe the decay dynamics using the approach outlined in L-MFE.
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In this work, we derived the L-MFE method to incorporate Lindblad jump operator dynamics into the meanfield Ehrenfest (MFE) approach. We took the density matrix equations of motion for Lindblad dynamics and mapped them onto an ensemble of pure state coefficients, using trajectory averages and expectation values of random variables. We then derived the L-MFE method to update the MFE coefficients at each time step which rigorously satisfies Lindblad jump operator dynamics. This established a method that exactly reproduces Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase (on coefficients c 0,ξ (t) in Eq. 69b).
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Compared to the Monte Carlo wavefunction methods which randomly collapse the wavefunction onto single states, the L-MFE approach only adds on random phases to the expansion coefficients, providing a more stable dynamics which can be incorporated with any mixed quantum-classical, semiclassical, or wavepacket based approaches. Compared to the Ehrenfest+R method the L-MFE method uses the same procedure to decay the magnitude of the quantum expansion coefficients, but a different choice of the random phase distribution, such that the exact Lindblad dynamics can be recovered, whereas Ehrenfest+R cannot exactly recover Lindblad dynamics. The derivation procedure of L-MFE also does not assume any relation between the energy gap of the two states versus the decay rate, whereas Ehrenfest+R assumes a particular parameter regime where the energy gap is much larger than the decay rate. Our theoretical analysis further provides valuable insights and mathematical justification for the relationship between the dynamics of the reduced density matrix and the dynamics of the ensemble of pure states. Through these careful analyses, we discovered an easy fix of the Ehrenfest+R method with the correct "+R" decay rate, which can be used to fix Ehrenfest+R dynamics to exactly reproduce Lindblad dynamics. Throughout the theoretical development in this work, we demonstrated the importance of including a carefully chosen random phase on both the coherences as well as the diagonal populations of the dynamics. Using numerical simulations, we demonstrated that the L-MFE method is equivalent to Lindblad dynamics for a variety of complicated dynamical scenarios when nuclear DOFs are not present, including scenarios where previous approaches (such as Ehrenfest+R ) do not match Lindblad dynamics. We further demonstrated that when including nuclei in Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach.
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This work provides a general approach for incorporating the Markovian dynamics of the Lindblad master equation into a scalable, wavefunction-type approach, allowing for the description of the dynamics of a quantum subsystem interacting with an anharmonic classical subsystem (nuclei) through a mixed quantum-classical description, as well as a Markovian environment that can be accurately described by Lindblad dynamics. The current approach can readily be used in the context of describing spontaneous emission due to light-matter interactions, incorporating cavity leaking in polariton chemistry, or being combined with Ehrenfest dynamics or a surface hopping approach to incorporate decoherence corrections. We envision that the current approach provides a general framework for future work to incorporate Lindblad dynamics into other mixed quantum-classical or semiclassical approaches.
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the resulting equations of motion for the quantum subsystem are mathematically identical to those of the Lindblad master equation for a two level uncoupled system with a decay from an excited state to a ground state. Thus, while the Ehrenfest+R method was not originally intended to broadly describe and simulate Lindblad dynamics for generic quantum subsystems, the method should be, in principle, able to do exactly this (at least when the energy gap is much larger than the decay rate). Additionally, Ehrenfest+R was a primary source of inspiration for the L-MFE method derived in this paper, so it is fruitful to examine the quantum subsystem part of the Ehrenfest+R method in order to understand its relation to the L-MFE method and to understand any potential issues it has.
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The Ehrenfest+R method may generally be applied to the spontaneous emission between any two quantum states as long as the energy gap is much larger than the decay rate. For simplicity, a two-level system will be considered to clearly understand the method, as was done in the original paper . Consider a two-level system with the following quantum subsystem Hamiltonian ĤQ and a Lindblad jump-operator L defined as follows
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Throughout the discussion of the Ehrenfest+R approach, we ignore the presence of ĤQC (R) + ĤC (see Eq. 5), although it is possible to generalize it to incorporate these terms. The Lindblad decay superoperator L L (whose effect is given in Eq. 25) corresponding to the decay dynamics can be written in the Liouville space as follows
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While the dynamics described by the Lindblad Liouvillian in Eq. A5 are identical to the quantum subsystem dynamics that the Ehrenfest+R method aims to simulate, the particular implementation of the method was developed by additionally considering the role the electromagnetic field should have in spontaneous emission. The authors Chen et al. derive a Hamiltonian interaction term (not present in Eq. A1) that represents the electric dipole coupling between the excited and ground states of the system. This time-dependent interaction term Ĥint is defined as follows
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Under the condition that Γ ≪ E, the oscillations of the phase of the off-diagonal density matrix elements due to E are much faster than the decay dynamics. This condition is thus the regime where the Ehrenfest+R approach can be applied with guaranteed accuracy. Under this condition, one can approximate the coherence as ρ 01 ≈ |ρ 01 |e iEt , thus Im[ρ 01 ] 2 ≈ |ρ 01 | 2 sin 2 (Et), which can be used to get an approximate expression of Eq. A13.
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For the purposes of analysis, it is convenient to average out the insignificant effects that the E-dependent phase oscillations have on the decay rates present in Eq. A13. Note that in the actual Ehrenfest+R simulation, L Eh (Eq. A13) is explicitly used. Thus, following Chen et al. , we define a moving average
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for a timescale τ such that 2π/E ≪ τ ≪ 1/Γ. The result of performing this moving average will subsequently be called the "time average" of the quantity, although it is important to understand that only the effects that the Edependent rapid oscillations have on the decay rates have been averaged out while any remaining time dependence is still present in the "time averaged" quantity. By using Im[ρ 01 ] 2 ≈ |ρ 01 | 2 sin 2 (Et) and sin 2 (Et) = 1 2 , the time average of Eq. A13 is expressed
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| 79 |
The dynamics governed by L Eh (Eq. A17) do not match the Lindblad dynamics governed by L (Eq. A5) since the decay rates k Eh and γ Eh from Eqs. A18 and A19 are not constant and depend on the density matrix elements, among other reasons. This is not surprising because it is impossible to recover open system Lindblad dynamics through the deterministic dynamics of a Hamiltonian (Eq. A10). Thus, an additional relaxation propagation (the "+R" stage of the dynamics) is introduced, which is designed to correct both the diagonals and offdiagonals of the reduced density matrix elements in the {|0⟩, |1⟩} subspace. This additional decay process (the +R dynamics) is governed by
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The random phase ϕ ∈ [-π, π] is sampled at the beginning of each trajectory and is keep constant throughout the evolution of that trajectory. The purpose of this phase, in the original context of the Ehrenfest+R development, is to add randomness to the energy of the classical electromagnetic field that couples to the molecules since the Ehrenfest+R method increases the energy of the classical electromagnetic field proportional to k R at each time step due to energy conservation. While this is a necessary feature to accurately describe the classical electromagnetic field, it only affects the quantum dynamics by increasing the variance between the trajectories. This decay rate k R can be time averaged to yield k R as
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where L [ρ(t)], L Eh [ρ(t)], and L R [ρ(t)] are defined in Eq. A6, Eq. A20, and Eq. A24, respectively. The dynamics associated with the relaxation component L R [ρ(t)] in Eq. A24 cannot be captured through deterministic dynamics. This is because the off-diagonals must be decohered independent of the diagonals, which requires the use of a random phase that cannot appear in a deterministic method governed solely by a Hamiltonian. Instead of using a modified Hamiltonian in the relaxation stage of the Ehrenfest+R method, the electronic expansion coefficients {c i,ξ } of state i and trajectory ξ at each time step are directly modified to attempt to achieve the correct Lindblad dynamics. These coefficient modifications for states |0⟩ and |1⟩ can be summarized as
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where P (Φ) is the probability distribution of the random phase Φ, δ(Φ) is the Dirac delta distribution, and RN is a random number in the range of RN ∈ [0, 1]. The "+R" relaxation stage of the Ehrenfest+R approach is summarized in Eq. A27a, Eq. A27b, and Eq. A28. This provides an effective Poisson process for incorporating decoherence. Generalizing this approach to multiple state has been proposed and tested as well , where the coefficients {c j,ξ } for |j⟩ / ∈ {|0⟩, |1⟩} are not changed by the Lindbladian time evolution, such that c j,ξ (t + dt) = c j,ξ (t).
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What has not yet been determined from this analysis, however, is whether the proposed relaxation method in Eq. A27a, Eq. A27b, and Eq. A28 corresponds to the effective time averaged Liouvillian in Eq. A24 or not. As will be shown in the next section, the proposed relaxation stage does not correspond to this Liouvillian, and the γ R decay rate must be modified in order to satisfy Eq. A26.
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To understand how these coefficient modifications in Eq. A27a and Eq. A27b affect the overall dynamics, it is best to work backwards to determine the effective Liouvillian L R that generated these relaxation dynamics, and figure out whether Eq. A26 is satisfied. The modified coefficients can be written in the density matrix form using the outer product as
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Due to the presence of a random phase, the expectation value of Eq. B1 is what the trajectory-averaged density matrix will converge towards and will be treated as the result of operating the effective Liouvillian L R onto the density matrix, which can be written as where we have explicitly used the expressions in Eq. A27a-A27b to write down these coefficients. The superoperator e LRdt can be determined from Eq. B2 by identifying the coefficients |c 0 (t)| 2 , c 0 (t)c * 1 (t), c 1 (t)c * 0 (t), and |c 1 (t)| 2 on the left-hand side as the density matrix elements ρ 00 , ρ 01 , ρ 10 , and ρ 11 , respectively, and considering the terms that these density matrix elements multiply with as the rates that come from e LRdt . This superoperator can thus be identified as (B5)
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If the Ehrenfest+R method has been successful, this sum in Eq. B6 should be equivalent to the Lindblad Liouvillian in Eq. A5. The k Eh + k R and -k Eh -k R elements in the right-hand column of Eq. B6 match the elements inside the Lindblad Liouvillian L (Eq. A5), Γ and -Γ, respectively, due to Eq. A22a. The elements in the middle two columns of Eq. B6 require closer inspection. The expectation value inside of α can be evaluated as
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such that the expectation value will be treated as an approximation of the exponential function in order to extract γ R outside of the logarithm. This approximation is present due to the choice of a Poisson rate process in determining the distribution of the random phase in Eq. A28 which results in the decoherence being a first order approximation of an exponential decay. Thus, the chosen distribution for the random phase Φ requires γ R dt ≪ 1 for accuracy. Treating the expectation value as the exponential decay in Eq. B7, the leftmost non-zero element of Eq. B6 can be evaluated as where Eq. A22b has be used to substitute Γ/2 in for γ Eh + γ R . Comparing the effective Liouvillian in Eq. B6 and the actual Lindblad Liouvillian in Eq. A5, the terms k Eh + k R = Γ match. However, the term iE -γ Eh + ln β dt in Eq. B6 fails to match the term iE -Γ 2 in Eq. A5, by the difference of the extra last term in Eq. B8. The presence of this extraneous term is due to the modification of the magnitudes of the coefficients in Eqs. A27a and A27b. The expression inside of the logarithm in Eq. B8 is the ratio of the modified coefficient magnitudes to the original coefficient magnitudes. These magnitude modifications change the coherence between the ground and excited states (as well as all other states in larger dimensional systems) thus their impact must be taken into consideration in the off-diagonal decay procedure in order to match Lindblad dynamics.
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To redeem the Ehrenfest+R approach, one can modify the decay rate γ ′ R to absorb this additional term (the last term in Eq. B8), such that the following equality is explicitly enforced where the expression inside the logarithm in Eq. B10 has been replaced by the ratio of the magnitudes of the modified coefficients to the original coefficients (when explicitly using Eq. A27a and Eq. A27b). Eq. B11 is the correct modification regardless of whether a linear approximation of the exponential function is used when modifying the magnitudes of the coefficients (as has been done in most applications of the Ehrenfest+R method). This is validated numerically in Fig. where panels a and c are the results of the Ehrenfest+R method with the original γ R (and a linearization of the exponential when calculating the modified coefficient magnitudes) while panels b and d are the results when replacing γ R with γ ′ R from Eq. B11.
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The same expression has been used in the development of the stochastic Schrödinger equation (see Eq. 73) that is equivalent to Lindblad dynamics. Thus, when completely ignoring the refiling term Γâρ S â † , one can approximate Lindblad dynamics as the time dependent Schrödinger equation (TDSE) with the complex Hamiltonian Ĥeff , which has been used in several recent works on molecular cavity QED. In the situations where the refilling term is negligible, the dynamics can equivalently be described by using the Schrödinger equation of a wave function evolving with the effective Hamiltonian. However, for the applications considered in current work in Fig. , one cannot ignore the Γâρ S â † term as we do care about the population refilling in the |g, 0⟩ state, as well as the proper decoherence among these states.
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Note that despite the common usage of the Lindblad jump operator LS = â ⊗ Îe ⊗ ÎR (Eq. 81) for describing cavity losses, 14 this jump operator is actually derived by considering the simpler system Hamiltonian ĤS = ℏω c â † â + 1 2 ⊗ Îe ⊗ ÎR without explicit consideration for the matter Hamiltonian or molecule-cavity interactions in ĤQED . Thus, Eq. D3 should be viewed as a phenomenological equation, and the correct Lindblad master equation for cavity QED should be derived starting from the total Hamiltonian ĤT = ĤQED + ĤE + ĤI , where ĤE and ĤI are expressed in Eq. D1 and Eq. D2, respectively. The details of the microscopic derivation of the Lindblad master equation for the Jaynes-Cumming model with cavity losses, as well as for comparison with Eq. D3, can be found in Ref. 70.
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Aquilanols A (1) and B (2) are naturally occurring sesquiterpenoids, isolated by Park et al., in 2017 from the agarwood of Aquilaria malaccensis (Figure ). Their intriguing structure comprises a 7/10 bicyclic skeleton, unprecedented within the humulene-type sesquiterpenoids family. The inherent strain of such medium-sized bicyclic frameworks is further intensified by the presence of a double bond on the junction of the two rings and a trans-disubstituted double bond incorporated into the 10membered ring. Although, preliminary evaluation did not reveal any significant activity when tested against certain Grampositive and Gram-negative bacteria, such strained molecular scaffolds hold high potential for biological activity, owned to strain-releasing-driven reactivity. This expectation is supported by the fact that agarwood is produced by the plant in response to microbials infections. Given the interest of our group in the synthesis of small-molecular weight antibacterial agents, 3 combined with the challenging structural features of aquilanols A and B, we embarked on the synthesis of the abovementioned compounds. Our interest became ever greater on the notice that no total synthesis has been reported so far and that price of agarwood, the natural source of aquilanols, is exceptional high and keeps rising due to emerging depletion of the Aquilaria tree population. On a first thought in designing of our synthetic plan, we wished to probe the viability of the biosynthetic hypothesis suggested by Park and co-workers. According to their proposal, aquilanol A (1) was derived from an intramolecular epoxide opening event on a highly oxidized derivative of humulene (3) (Figure , compound 4). Consequently, our initial objective was oriented toward the assembly of the eleven-membered carbocyclic oxygenated intermediate 4. A blueprint of our approach towards this goal encompassed the construction of polyunsaturated precursor 8, upon which a ring closure metathesis would be applied to formulate the ring (Figure ). Albeit rare, such strategies for the preparation of eleven-membered carbocycles are reported in the literature. Unfortunately, this approach proved fruitless on our systems, despite several variations on the cyclisation substrate we attempted the reaction on. As often the case in terpene synthetic world, we turned our attention to terpene chiral pool to identify easily accessible natural compounds, that could serve as alternative starting materials to formulate the macrocyclic core. Our interest was attracted by (-)-caryophyllene oxide (11), a naturally derived material, available in large quantities and low cost (approx. 0,6 USD/g) and in acceptable purity (>95%). 8 Shenvi have recently demonstrated the high added-value conversion of the latter to much more precious humulene oxide (10), stereoselectively, via a radical retro-cycloisomerization reaction catalyzed by Co(III)salen species (Scheme 1). This transformation perfectly suited to the needs of our plan. A retrosynthetic analysis capitalizing on the latter would encompass only an additional regio-and stereospecific allylic oxidation on the pendent methyl group of the trisubstituted double bond in 10 (Scheme 1). Unfortunately, (-) -caryophyllene oxide, the only enantiomer commercially available, leads to the enantiomer of the natural compound, ent-aquilanol A (ent-1). Even so, we decided to test our hypothesis as it constitutes a rapid and versatile access to the broader family of aquilanols.
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Truly, in our hands commercially available (-)-caryophyllene oxide (11), was converted in good yield and on multigram scale to humulene oxide (10), providing ease access to a valuable intermediate, that is very tricky to be reached otherwise (Scheme 2). Next, we explored the allylic oxidation of 10, using standard protocol employing SeO2/t-BuOOH as the catalytic system. Even though TLC monitoring of the reaction indicated an unexpectedly clean conversion, NMR analysis of the reaction mixture revealed the presence of an inseparable mixture of three isomeric allylic alcohols 12a,b and ent-4. Indeed, Shirahama had previously reported that analogous oxidation on the same substrate leads to mixtures of geometrical isomers of allylic alcohols, one of them appearing in the form of two distinct and stable conformers. Unfortunately, the produced mixture of alcohols was extremely difficult to separate, especially given the unstable nature of them under standard purification conditions. Even worse, the desired (Z)-allylic alcohol ent-4, was by far the minor component of the mixture, indicating that typical SeO2 oxidation was not a productive solution to our problem.
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The above discouraging results prompted us to revise our synthetic plan. We envisioned that combining in one step functionalization of the specific position (pendent methyl group) and retro-cycloisomerization event may serve as a solution to our problem. The new concept is briefly described on Scheme 3. We thought that activation of the exocyclic double bond on (-)caryophyllene oxide (11) with a suitable reagent could trigger a concomitant cationic (or radical) mediated retro-cycloisomerization and functionalization of C-14 (intermediate II). Subsequent functional group transformation would provide the desired monocyclic precursor ent-4. Scheme 3. Revised retrosynthetic analysis combining retrocycloisomerization and functionalization of C-14.
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Gratifyingly, our hypothesis proved correct. After much experimentation we found that treatment of (-)-caryophyllene oxide (11) with NBS in acetone afforded the expected humulene-type bromide 13, with vinyl bromides 14a,b accounting for the rest of the consumed starting material (Scheme 4). NBS proved unique electrophilic bromine source among a variety of similar activators tested. Different halogen electrophiles also turned out to be unproductive. Unlikely, the polarity of all three bromides was very similar rendering purification problematic. To circumvent this adverse event, we proceeded with the hydrolysis of the reaction mixture with the expectation that vinylic bromides 14a,b would be inert to the reaction conditions. Indeed, the anticipated (Z)-allylic alcohol ent-4 was the only alcohol detected in the reaction mixture, while bromides 14a,b were recovered intact after fast purification using neutralized silica gel. The combined yield range of alcohol ent-4 for the two-step sequence is 30-35%, consistently, regardless the scale of the reaction, typically applied on gram scale. Scheme 4. Two-step synthesis of (Z)-alcohol ent-4 from (-)caryophyllene oxide (11).
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To further improve our synthesis, we envisioned direct installation of hydroxyl group, while triggering the retro-cycloisomerization rearrangement. A recent work by Lei describes the visible light mediated anti-Markovnikov hydration of olefins, that is claimed to proceed via a radical intermediate, thus deemed to be ideal for our purposes (Scheme 3; I: X = OH, * = radical). Regretfully, on our substrate, such a protocol proved ineffective, since it led to a rather complicated reaction mixture, the 1 H-NMR spectrum of which did not indicate any ring opening of the bicyclic of caryophyllene oxide. Having ensure a rapid access to the oxidized humulene-type substrate ent-4, we were in position to investigate the plausible biomimetic intramolecular epoxide opening. Apparently, acid catalysis was required to favor the anticipated regioselectivity, that forms a seven membered ring, in contrast to the six membered analogue that is expected to be preferred in a non-catalytic process. Thankfully, treatment of ent-4 with various types of acids (SnCl4, BF3•Et2O, TFA, Sc(OTf)3), all brought about
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the desired transformation (Scheme 5). Among them Sc(OTf)3 gave the cleanest reaction profile, providing access to the enantiomer of natural aquilanol A (1) in 74% yield, presumable due to its ability of concomitant activation of epoxide and complexation of the allylic hydroxyl group, thus bringing in proximity the reacting components. Furthermore, m-CPBA oxidation of ent-1, led exclusively to natural aquilanol B (2) in 78% yield, with the disubstituted double bond Δ being the only reactive site. The spectroscopic and physical data were in excellent agreement with the reported ones by the isolation group. In addition, having ensure a rapid access to the respective (E)allylic alcohol, in the form of two stable conformers 12a,b (see Scheme 2), we were curious whether analogues of the natural product could be derived from. Against our expectations, treatment with various types of acids did not cause an analogous intramolecular epoxide opening, instead provided access to several structurally distinct products, often with admirable chemoselectivity, a fact that renders alcohols 12a,b a valuable common intermediate to a versatile array of highly oxidized carbocyclic motifs (Scheme 6). In particular, treatment of 12a,b with typical Lewis acids such as SnCl4, or TiCl4, afforded exomethylene allylic alcohol 15 as the major product (please see supporting information Scheme S1 for mechanistic interpretation). On the other hand, BF3•OEt2 resulted in the formation of a pinacol type rearrangement product, ketone 16. When alcohols 12a,b were subjected to the action of a Brønsted acid, regardless its strength, a known rearrangement took place, triggered by nucleophilic attack of the nearby disubstituted double bond, to provide cyclopropyl triol 17. 15 Scheme 6. Highly oxidized carbocyclic scaffolds from treatment of (E)-allylic alcohols 12a,b with various acids.
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In summary, we have achieved the first synthesis of the enantiomer of naturally occurring aquilanol A (1) and its further oxidized congener aquilanol B (2), in three chemical steps starting from easily accessible (-)-caryophyllene oxide (11). Our strategy consists of an electrophilic activation of the exocyclic double bond on 11, accompanied by a retro-cycloisomerization event to obtain the eleven-membered monocyclic intermediate bromide 13 and, after hydrolysis, alcohol ent-4. Intramolecular acid catalyzed epoxide opening on the (Z)-allylic alcohol precursor completes the synthesis (overall yield 24%), thus providing a logical basis to support the biosynthetic hypothesis by Park and co-workers. Further selective epoxidation of ent-1 leads to aquilanol B (2) in good yield. Finally, various structurally distinct highly oxidized eleven-membered cyclic scaffolds were reached in a selective manner, all originated from common intermediate (E)-allylic alcohol 12a,b, upon the action of specific acidic reagent.
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As essential platforms for the digitalization of modern research laboratories, Electronic Lab Notebooks (ELNs) assist scientists in data management, collaboration, and documentation of scientific experiments, enhancing the reproducibility, sustainability, transparency, and traceability of experiments. Depending on the researchers' needs, ELNs must support various processes and incorporate tools for working in different disciplines. Due to the complexity of requirements as well as the diverse handling and structuring of data, there are different ELNs utilised by researchers in their respective fields. Generic or multidisciplinary ELNs typically provide researchers with significant freedom in data storage methods and process documentation, being often limited in ensuring well-annotated and standardised data availability.
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There are also ELNs with specialised functionality and a data structure well adapted to each respective discipline. The standards and processes prescribed by these discipline-specific ELNs can effectively accelerate scientific work, increasing the acceptance of user scientists. They can also ensure FAIR (Findable, Accessible, Interoperable, Reusable) data storage to ensure the direct and long-term reusability of these data and comparability with other sources. Selecting the appropriate ELN software is not trivial, in particular as ELNs are often established by the respective research institutions that have to address the needs of significantly different disciplines and interdisciplinary work. Providing added value to such interdisciplinary work and the support of different disciplines can only be achieved through a combination of ELNs which then raises the question of interoperability between ELNs immediately. Obviously, interoperability software solutions need to ensure data transfer from one ELN environment to another with minimal loss.
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Unfortunately, there are currently no standardised protocols for exchanging data between ELNs, creating a large hurdle for data transfer or migration from one system to another . There are various approaches to this issue, assuming that the data structure, contents, and functions of the ELNs cannot be harmonised. Possible paths toward using data in different ELNs could include:
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There are already approaches to solve the ELN interoperability problem being advanced by individual providers or consortia, aiming at long-term harmonisation of research data and metadata. The group "The ELN consortium", involving developers of ELNs from academic institutions, is working towards generating a unified format (*.eln) that can be used by all participating ELNs based on the well-established RO-Crate specification. This approach has a high potential for being successful in offering data exchange solutions in the long run. Also, the ELN consortium improves the FAIRness of data due to continuous extensions of the metadata descriptions in the *.eln file, e.g. by adding further semantic annotations using JSON-LD. However, a file exchange requires high implementation effort for the ELN developers, as the transfer of specific data requires adaptation of the file format for each pair of ELNs, also keeping the format updated to newer ELN versions. Moreover, moving files induces much manual work for the users.
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Beyond exchange file formats, there are initiatives that work on the specification of scientific data. Scimesh 9 defines the representation of scientific results in the form of a knowledge graph including physical specimens to facilitate the exchange of information on particular samples from one ELN to another one. Also Bioschemas offer standardised descriptions that can be used to map the content of ELNs to common schemas and ontologies. Being developed for improving the findability of life science records in the web, the types and properties defined by BioSchemas can serve as reference to describe ELN content in a suitable way.
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At first sight, employing established standards, i.e. schemas or file formats, appears to be particularly sustainable to achieve ELN interoperability, regardless of the interfaces of the respective software. While this approach for ELN software with less structured content offers satisfactory solutions in the medium term, applying it to very discipline-specific ELNs with a high degree of structure becomes a complex undertaking that cannot currently be completed to satisfaction. The main reason for this is the inability of a schema to cover all specifications for the diverse contents of discipline-specific documentation, which consequently cannot be transferred clearly and in detail from one ELN to another by a data exchange format. In order to develop satisfactory solutions for those discipline-specific ELNs, for example, an Application-Programming-Interface-based (API-based) exchange of information should be considered.
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Chemotion is an an open-source ELN developed by the Karlsruhe Institute of Technology (KIT) in Germany in collaboration with other partners of the National Research Data Infrastructure of chemistry (NFDI4Chem). Chemotion is tailored to the needs of researchers in the field of chemistry, providing features to facilitate data management, experiment documentation, the analysis of measurements and analytical data, and collaboration. Chemotion aims to streamline the research process in chemistry laboratories by offering tools for recording experimental data, for drawing and processing chemical structures, the transformation of proprietary files into open file formats, the management of a chemical inventory, and the organisation of research workflows. As an open-source project, Chemotion allows users to customise and extend its functionality according to their specific requirements. Overall, Chemotion is designed to enhance efficiency, reproducibility, and collaboration in chemistry research through its features tailored to the needs of chemists with an experimental focus. It can be used to seamlessly transfer data into the research data repository Chemotion to publish research data.
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Herbie is developed at the Helmholtz-Zentrum Hereon as an open-source and modular ELN to interlink heterogeneous process chains in research. It is ideally suited for labs with fabrication and characterization processes that are standardised or performed repetitively, e.g. for materials sciences. To achieve comparability, quality assurance, and reusability, users enter data in wellstructured, pre-defined webforms for each process or entity type. These webforms are automatically generated from ontologies, defined in OWL (Web Ontology Language) and SHACL shapes (Shapes Constraint Language). Through the integration of ontologies, all inserted data is semantically annotated and the webforms are thoroughly interconnected, reflecting the complete lifecycle of a sample. Thus, all entered information is easily accessible in various contexts, such as in a chronological journal, filtered for the linking entities like projects or equipment, or in a tabular database. Users can create, view, update or delete entries via the graphical web interface or its REST API. File export, file import and the API embed Herbie conveniently into linked-data software ecosystems.
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Chemotion and Herbie entail very complimentary features, both from a frontend and backend perspective, and thus jointly offer to seamlessly document the entire process chain from the molecular design to the engineering of components and plants. Therefore, a data exchange mechanism between these two ELNs would enable synthetic chemists, materials scientists, and engineers to create an interoperable and rich database. By providing an API, both ELNs fulfil the basic requirements for this endeavour.
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Mapping the content of one ELN to another -even when covering only parts of interest in a certain data exchange scenario -can be a complex task, as ELNs may have different data structures that may prevent options of a one-to-one mapping of content. Also, the granularity of descriptions referring to certain ELN entities may be quite different, depending on the focus of the ELN's application and use-cases. As a result, one ELN may have a certain set of information stored in well-defined database tables with values therein most likely correlated to standardised units, whereas the same information is given in textual form in the other ELN. In the worst case, one ELN stores information that is neglected in the other and vice versa.
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These schematic conflicts were solved in work described therein for the ELNs Chemotion and Herbie for the use case "Polymer Membrane Post-Modification". For the given use-case, we considered the relevant content of the selected ELNs which was covered in Chemotion ELN with the ELN models (input masks) for "reaction" and "sample", and in Herbie with the ELN models for "post-modification reaction" and "product". The content of the required ELN models was mapped from one ELN to the other, and conflicts were resolved by different means, e.g. the creation of new data fields (if a lack of information was detected and the need for an adaptation obvious) or converting date-time formats(details in the SI).
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The main component to enable the generic connection of two (or more) ELNs is constituted by an API-based adapter server (ELNdataBridge). ELNdataBridge leverages Python APIs to interact with the underlying data structures of various ELN systems, enabling the mapping of information and thereafter the seamless transfer of information between them. The ELNdataBridge covers therefore two main workflows:
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The User Interface (UI) is a common server-client application that can be used to configure the ELN-to-ELN data exchange. This includes the registration process of ELNs to ELNdataBridge and the user-friendly mapping of the content of two ELNs, driven by a data parser in the backend, without expecting coding skills from the users (Figure ). It is based on Django, a robust Python framework. Due to the specifications of SimpleDomControl, a Python/Javascript framework and Django extension, the architecture of the system was developed strictly according to a Model-View-Controller pattern. As a data storage component a SQL database is available. Django enables a database-independent design, capable of interfacing with various SQL databases. However, by default, it is configured to use a PostgreSQL database.
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The business logic of the synchronisation is provided for each ELN via their respective platform API and enables seamless communication between the system and the ELNs, based on a delegation pattern. The so-called synchronisation delegator is provided as a moderator of communication between the platforms (Figure ). The delegator performs the following tasks in each process step: It finds all entries from both platforms that are mapped for synchronisation in the UI. From the entries found, the delegator identifies the matching pairs of entries between the two platforms involved. Then it determines whether values have changed since the last synchronisation process and therefore need to be transferred to the respective complement. All communication to the actual ELN APIs is delegated by the delegator to a delegate. A delegate must be provided for each participating platform. These delegates must implement a so-called platform API translator. To fulfil the requirements of the interface, the delegate must implement a set of Python functions that handle reading and writing individual entries, requesting all entries of a certain type, as well as listing all types that are available.
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The establishment of the ELN-to-ELN mapping process in ELNdataBridge needs the customization of the planned data exchange using the mapping of the ELNs' content as described before. The UI allows researchers to configure the required aspects of the ELN-to-ELN data exchange and to map and configure the transfer of single values and entry types between different ELNs. In a first step, a connection between the ELNs is established by the generation of a new synchronisation instance. This requires the entering of necessary information such as the name, URL, and token or password assigned per ELN instance and user of the instance (Figure ). In the next steps 2 and 3, the setups of ELN A, e.g. Chemotion, and ELN B, e.g. Herbie, have to be configured (Figure , exemplarily for ELN A). The updating of connection details and the configuration of essential properties are critical steps to ensure a smooth synchronisation process in ELNs. The choice of properties encompasses decisions such as whether to include all existing entries in the synchronisation or only those last updated after a specific timeframe. Additionally, consideration must be given to whether the automatic generation of new entries should be permitted within the ELN environment. Given that projects and activities in Chemotion are structured within collections, synchronisation hinges on choosing one within your user account to establish the initial link between ELNs. In contrast, Herbie lacks a comparable feature; hence, no additional outline element necessitates selection.
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After setting up the general mapping framework with steps 1-3, the Sync Model Manager (Figure ) of ELNdataBridge allows defining the main properties and pairing settings (Figure ) by either the creation of new synchronisation models or the management of existing ones. Each sync model requires a unique identifier, which is a human-readable name, and the names of the two ELN models (input masks) that need to be synchronised (Figure ). Additionally, it's crucial to set the keys for both ELN models. The system uses these keys to identify and match pairs to ensure the accuracy of the synchronisation process.
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A detailed configuration of the sync model for the ELN mapping can be set in forms that guide the user through the required information, supporting a customizable synchronisation process. Most of the fields in the user-friendly form rely on a click-and-select mechanism by dropdowns or a selection from the JSON representation of the objects (input fields) displayed below the form. The sync model was designed to also support complex synchronisation processes by including submodel synchronisation (details see supporting information). That allows, e.g., for the synchronisation of all related data of a selected molecule of a reaction across the different ELNs while other materials used within the same reaction can be ignored.
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A descriptive explanation accompanying each mapping of objects, outlining the functionality of the transfer operator. When setting the actual values in the sync model's mapping form, the only property that requires manual input is the identifier, as in the model pairing form. Also as above, the mapping keys can be set using the JSON representation below the form. Moreover, it is important to set the data type.
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(schedules tasks to run them periodically at fixed intervals in order to automate regular, repetitive jobs) triggered routine enables the synchronisation with Activate Autosync. Synchronisation progress, i.e. the server activity and the history of all processes can be monitored in real time in a list in the main menu (SI Figure ). The changes to each individual object can be tracked via a link in the logs (SI Figure ).
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The actual process of synchronisation is illustrated in a simplified form in Figure . All entries are read one after the other, first from ELN A (Figure (1)), then from ELN B (Figure (3)). For each entry, a check is made for changes of the values configured by the mapping specifications (see Figure ) since the last synchronisation process (Figure (2 & 4)). If so, a corresponding data entry is made. As soon as all entries have been read, the process searches for the availability of complementary entries in the target ELN (Figure (5)). In case of missing entries, an attempt is made to create a new one if the corresponding settings allow it (Figure ). The new values are now transferred to the entries (Figure (6.A & 6.B)) found (or created) according to the mapping specifications (see Figure ). In this step, the transfer operator is used to adapt the data to the target system, the latest updated version of two pairing entries will overwrite all existing values in its partner in cases where the settings define that all or parts of the entries are to be synchronised.
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Subsequently, the chemical reactions of molecular substances with a polyacrylonitrile membrane are recorded in Chemotion, while further performance characterization and the assembly of membrane modules will be documented in Herbie. Hence, the metadata of the chemical experiments must be transferred from Chemotion to Herbie for future reference and dataset assembly, creating a full picture of the entire process chain for the researchers.
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Prior to the conducting the chemical experiment, the existing membrane "PAN$20/013-PAN-EDA-01;" is generated as a "product" in Herbie by noting fabrication details, and automatically created as a "sample" in Chemotion via the ELNdataBridge synchronisation. After the postmodification experiment, the filled-in reaction template in Chemotion (Figure ) contains parameters such as employed weight of the membrane (integer), a description of the reaction conditions (free-text) and status of the reaction (selection from list). Selecting the collection and reaction in Chemotion, followed by synchronisation, creates all data in Herbie. Subsequent changes to the three aforementioned parameters are updated in Herbie via ELNdataBridge (Figure ). Thus, ELNdatabridge enables a seamless, quick and low-effort metadata transfer of chemical reactions between Herbie and Chemotion. This provides scientists and ELN users a simple means to transfer data between the ELN platforms.
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BU72 is a μ opioid of exceptionally high affinity and potency (Figure ) . Its dissociation constant (Ki) for μOR ranges from 0.15 nM in crude brain membranes , to lower values in transfected cell membranes , and as low as 0.01 nM for purified μOR with Gi protein . Very few ligands for any protein exceed this extraordinary affinity, which is considered an effective upper bound on the strength of non-covalent binding .
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Figure : Structures of BU72 and analogs BU72 was the ligand in the first crystal structure of active μOR . As noted there, the electron density exhibited two unexplained features. Firstly, fitting the published structure of BU72 (1a, Figure ) required a near-planar orientation of the phenyl group, an implausibly high-energy conformation that required many extreme deviations from ideal geometry and left unexplained density around the benzylic carbon (Figure ).
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The authors considered the possibility that the ligand was actually imine 2 (Figure ), whose planar sp 2 benzylic carbon would resolve this problem, but this was not detected in mass spectra of the crystallization mixture . In a preprint, I proposed an alternative: a revised structure for BU72 with the phenyl group in the opposite (R) configuration (1b, Figure ) . Revised structure 1b fits in a low-energy conformation, eliminating the geometric outliers and unexplained density around the phenyl group, and yielding superior validation metrics (Figure ) . The similar binding affinities of BU72 and imine 2 provide further support to structure 1b: the equatorial phenyl group in 1b is approximately planar, as in imine 2, which also fits the density in a low-energy conformation, unlike 1a (see Extended Data Fig. in ).
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