3.2 Extension to the stochastic case: correlating the noise by Quantum Monte
3.2.4 Algorithm stability with QMC forces
In the following, we will present some calibration runs carried out with the PIOUD algorithm in the VMC framework, which will help us set the proper simulation parameters (γ0,δt) and show the remarkable stability of the quantum dynamics even with noisy VMC forces for large δtand largeP. The results are obtained by performing QMC-PILD short tests (about8.5ps of dynamics) of the Zundel ion in the gas phase at room temperature (300K). Hereafter, the additional PIOUD parameterγBO is set to zero, to avoid overdamping in the BO propagator.
The Langevin thermostat in theiLBOpart plays solely the role of correcting the BO dynamics
0.02 0.025 0.03 0.035 0.04 0.045 0.05
0 0.01 0.02 0.03 0.04 0.05 0.06
Quantum kinetic energy (Ha)
Friction γ0 (atomic units) T = 300 K
<Tvir> CCSD(T)
<Tpri> CCSD(T)
<Tvir> VMC
<Tpri> VMC
292 294 296 298 300 302
0 0.01 0.02 0.03 0.04 0.05 0.06
Average temperature (K)
Friction γ0 (atomic units) T = 300 K
CCSD(T) VMC
0 5 10 15 20 25 30 35 40
0 0.01 0.02 0.03 0.04 0.05 0.06
Potential autocorrelation time τV (a.u.)
Friction γ0 (atomic units) T = 300 K
CCSD(T) VMC
Figure 3.5 – PIOUD evolution of the quantum kinetic energy estimators hTvir/prii(top panel), the temperatureT (middle panel) and the potential autocorrelation timeτV (bottom panel) as a function of the input frictionγ0. Solid lines correspond to the virial estimator of the kinetic energy whereas the primitive estimator curves are dashed. Deterministic forces are represented in black whereas the noisy QMC forces are in red. The time step and the number of quantum replicas are respectively set toδt= 1fs andP = 32.
for the intrinsic VMC noise. As we have already seen, for deterministic forces iLBO reduces to iLp, i.e. it is a simple velocity step.
For each MD step we have evaluated forces and all the energy derivatives with ≈3.3×105 MC samples, much larger than the total number of variational parameters (p≈2.4×103) of the WF. This allows reaching an accuracy of1.5mHa (0.94kcal/mol) in the total energy per VMC energy minimization step. Between two MD steps, five QMC energy minimizations are performed with the Hessian (SRH) algorithm, in order to sample the correct BO surface. As already mentioned in Section 3.2.1, all variational parameters are evolved during the dynam-ics, except for the GTO exponents, which are kept frozen.
The optimal input friction γ0 is chosen to minimize the potential autocorrelation time τV
(Eq. (3.28)), and generate the most efficient phase space sampling during the dynamics. We will give a general protocol to find this optimal value in the case of stochastic VMC forces, where the situation can be more complicated since the FDT is now sensitive to the QMC intrinsic noise. In our initial tests, the time step is set to 1 fs, a large value, which guarantees a quick and effective exploration of the phase space. Moreover γBO = 0 and ∆0 =δt, as we discovered that the most efficient simulation is the one which minimizes the damping in the BO sector. ∆0 is taken as the minimal value which provides a positive definite γBO. In the top panel of Figure3.5, we first observe that in the VMC case, the virial and the primitive ki-netic energy estimators deteriorate when the applied input friction is too large (>0.02×10−3 a.u.). In this situation, the additional QMC noise makes the coupling between the system and the thermostat too large to be fully controlled for such time step values (δt= 1fs). On the other extreme, at very smallγ0, we see that the presence of a QMC-correlated noise tends to flatten the sharp and deep minimum of the potential autocorrelation time obtained with deterministic CCSD(T) forces (bottom panel). This indicates that, contrary to the determin-istic case where there is a clear advantage to set the input friction γ0 to its optimal value, there is more freedom to choose this parameter in VMC. Indeed, the autocorrelation time divergence shown in the CCSD(T) case for small values of γ0 disappears with VMC forces.
This is due to the implicit low-value cutoff in the γ matrix provided by the intrinsic QMC noise, once the QMC-force covariance matrix is converted into an effective friction, according to Eqs.(2.32) and (3.29). In practice however, the value of γ0 cannot be too small either, in order to avoid too cold temperatures shown in the middle panel of Figure3.5. Consequently, we need to take the largestγ0before the increase of the potential autocorrelation timeτV due to the soft-modes overdamping. This will also let us recover an acceptable target temperature (see middle panel). Therefore, γ0 = 1.46×10−3 a.u seems to be a very good compromise be-tween autocorrelation timeτV, effective temperature, and quality of the phase space sampling revealed by the kinetic energy estimators. All subsequent runs will be performed with that value. It is interesting to remark that this is optimal for both deterministic and stochastic forces. Similarly to the tests performed in Subsection 3.1.4 with analytic CCSD(T) forces, we check here the robustness of our novel PIOUD algorithm with respect to the time step δt and to the number of quantum replicas P in the presence of noisy QMC forces. In the upper panel of Figure 3.6, the difference between the virial and the primitive kinetic energy estimators remains very reasonable with increasing values of the time stepδt, even though the time step error is more important in the stochastic case once compared to the deterministic
0.02
Figure 3.6 – PIOUD evolution of the quantum kinetic energy estimators hTvir/prii as a function of the time stepδt(top panel) and the number of quantum replicas P (bottom panel). Colors and symbols are the same as in Figure 3.5. The input friction was set toγ0 = 1.46×10−3 a.u. fs. The default values of the time step and the number of quantum replicas are respectively δt = 1 fs and P= 32.
one. The PIOUD propagator exhibits a smaller difference between these two estimators with QMC forces than the PILE algorithm in the deterministic case. The superior performances of PIOUD will allow us to use large time stepsδt.
Finally, we also check the stability of the PIOUD integration scheme with increasing number of quantum replicas P. As we can see in the bottom panel of the Figure 3.6, the difference between the kinetic and the primitive energy estimators is well controlled up toP = 64beads in the stochastic case. On the contrary, the PILE propagator already exhibits signs of in-stability at this value with deterministic forces (see upper left panel of Figure3.2). This is a further proof of the robustness of the PIOUD integrator, which we thus recommend when one wants to perform a PILD simulation with a large number of beads with any force field, deterministic or not.
Thanks to the technical developments presented in this Chapter, we have paved the way to perform fully quantum simulations of small protonated water clusters, with highly accu-rate ionic forces evaluated within the VMC approach. The very last step before performing such calculations on realistic systems such as the protonated water hexamer, is to check the reliability of our novel methodology on a benchmark system, as we will do in the next Chapter.
The Zundel ion: a benchmark system
Contents
4.1 Zero temperature results . . . . 88 4.1.1 Potential energy landscape . . . . 88 4.1.2 Equilibrium geometries . . . . 90 4.1.3 Accuracy of the Quantum Monte Carlo approach . . . . 95 4.2 Benchmark calculations . . . . 98 4.2.1 Validation of the classical dynamics . . . . 99 4.2.2 Validation of the quantum dynamics . . . 100 4.3 Proton transfer in the Zundel ion . . . . 104
O
ne of the main issueswhich sets back from a complete understanding of proton trans-fer (PT) in water, is related to the very sensitive thermal behavior of PT. The re-quired Potential Energy Surface (PES) precision, of the order of a few tenths of kcal/mol, has been reached only recently by state-of-the-art computational methods beyond DFT, such as MP4[212], Coupled Cluster (CC)[213] or multi-reference configuration interaction (MRCI)[214]methods. They are however characterized by a much poorer scalability with respect to DFT, and therefore they do not allow simulation of sufficiently large molecular clusters. In the pre-vious Chapter, we have set the stage to perform efficient Path Integral Langevin Dynamics (PILD) simulations of such systems with accurate and noisy Quantum Monte Carlo (QMC) forces, exploiting the mild scaling of the VMC approach with the system size.
To validate our methodological developments, we first apply them to H5O+2, namely the Zundel ion[215], widely used as a benchmark system. Indeed, it is the smallest charged water cluster to exhibit a non trivial proton transfer and its reduced size makes a comprehensive and systematic study of the problem easier. More importantly, there is a huge amount of data to compare our results to, because the description of excess proton in water has been widely studied both theoretically and experimentally in the last fifty years. On the one hand, the fast development of spectroscopical instruments allowed to probe experimentally vibrational properties of ionic species and therefore many studies have been published[17,216]on the H5O+2 ion. On the other hand, several accurate theoretical works have appeared on the Zundel ion to study its structure and energetics[110,216–225]. We can cite for instance the extremely accu-rate Potential Energy Surface (PES) geneaccu-rated by Bowman and coworkers from almost exact CCSD(T) calculations[65], the MS-EVB methods[218,219,226,227], or more recently the LEWIS model developed by Herzfeld[228]. Moreover, because of the great importance of the Nuclear Quantum Effects (NQE) in the Zundel cation, the latter is almost always used to test and validate new approaches[229,230].
In this Chapter, we first assess the ability of the QMC approach to accurately describe both the energetics and the geometric properties of the protonated water dimer in Section 4.1.
Later, we perform in Section4.2benchmark calculations at finite temperature to ensure that our QMC-based methodology provides an accurate description of the Zundel ion. To that purpose, we compare our results to classical or quantum Molecular Dynamics simulations carried out with an analytical CCSD(T) force field to ensure the QMC intrinsic noise does not spoil the protonated water dimer dynamics. In Section 4.3, we finally investigate the impact of the temperature and NQE on the PT processes occurring in the Zundel cation.
4.1 Zero temperature results
The protonated water dimer, represented in Figure4.1, is constituted by an excess proton H+ surrounded by two neighboring water molecules. The nature of the minimum energy structure of the H5O+2 ion has been debated in the literature. There are two candidates with competing energies: a C2 symmetric structure (left hand side of Figure 4.1), commonly known as the Zundel configuration, with the proton evenly shared between the two oxygen atoms, and a Cs-Inv one (right hand side of Figure4.1) with the proton slightly closer to one H2O molecule.
Accurate highly correlated studies[65,222–224] have confirmed that the global minimum is C2 symmetric. In this Section, we will explore the zero temperature properties of the protonated water dimer using different computational methods. In particular, we focus on the description of the PES and the geometric properties which are of paramount importance to predict the possible configurations the H5O+2 ion may adopt at finite temperature.
Figure 4.1 – QMC optimized geometries for global C2minimum (left) and for Cs-Inv local minimum (right) of the Zundel ion[72]. Atom labels for the analysis of the Zundel properties are also indicated.