The Path Integral approach

We present here the Path Integral (PI) method that will be used in this thesis to take into account NQE. Based on the Feynman PI formulation of quantum mechanics, it has experi-enced a significant interest in the last twenty years. Indeed, this approach enables a rigorous evaluation of the quantum partition functionZ(N_at, T) defined as:

Z(N_at, T) =Tr h

e^−βH i

. (2.50)

Tr[.] represents the trace of the operator inside the squares andH is the Hamiltonian given in Eq. (2.1). One can rewrite Eq. (2.50) as

Z(N_at, T) = Z

d^3Natqhq|e^−βH|qi, (2.51)

wherehq|e^−βH|qiare diagonal matrix elements in the coordinates space. In the case of quan-tum particles, the kinetic energy K and potential energy V operators do not commute, im-plying that we have to apply a Trotter-Suzuki break-up to evaluate these matrix elements.

Indeed, using the same factorization as in Eq. (2.10) appliedP times, one obtains e^−βH =e^−β(K+V) = lim

P→+∞[e^{−βV /2P}e^−βK/Pe^{−βV /2P}]^P. (2.52) To simplify the derivation, we introduce an operatorξ =e^{−βV /2P}e^−βK/Pe^{−βV /2P}. Combining Eqs. (2.51) and (2.52) and insertingP−1times the identity operatorI =R

dx|xihx|between theP factors ofξ, the quantum partition function reads as

Z(N_at, T) = lim

P→+∞

Z

d³q₂. . . d³q_Phq|ξ|q_Pihq_P|ξ|q_P−1i. . .hq₂|ξ|qi. (2.53) The interaction potential V only depends on the atomic coordinates and is consequently diagonal in the coordinate basis. The matrix elements in Eq. (2.53) can be evaluated as

hq_j+1|ξ|q_ji=e^{−βV(qj+1)/2P}hq_j+1|e^−βK/P|q_jie^{−βV(qj)/2P}, (2.54) forj= 0, . . . , P−1. The matrix element implying the kinetic energy operatorK is less trivial to evaluate since one has to work in the momentum basis, as follows

hq_j+1|e^−βK/P|q_ji = where we have used once again the identity operatorIand the knowledge ofhq|pi= 1/√

2π~e^ipq/~. The above equation is simply a gaussian integral and combining it with Eq. (2.54), one obtains

hq_j+1|ξ|q_ji= By inserting the above equation into Eq. (2.53), we finally have the expression of the quantum partition function can interpret the above formula as an exact evaluation of the quantum partition function Z(N_at, T) via a discretized cyclic path which depicts the quantum delocalization of the nu-clei. We can notice that in the high temperature limit (β →0), one recovers the expression of the classical canonical partition function Q(N_at, T) given in Eq. (2.15). Thus, the high temperature limit of the PI formalism is strictly equivalent to the classical limit.

In practice, the number of slicesP is finite and the expression in Eq. (2.57) is approximated

using the quantum-to-classical isomorphism^[168]. This property states that the description of a true quantum system can be replaced by a fictitious classical path in the quantum imaginary time τ = _P^β which reproduces the quantum behavior of the system. This idea is at the basis of the so-called Path Integral Molecular Dynamics (PIMD) and Path Integral Monte Carlo (PIMC) techniques. In this case, the quantum partition function can be evaluated as

Z(N_at, T) = 1

2π~

PZ

d^fΓe^{−τ HP(Γ)}, (2.58) whereH_P(Γ) is the quantum-to-classical isomorphism Hamiltonian, given by

H_P(Γ) = In other words, the quantum nuclei are replaced by fictitious classical ring polymers whose beads are connected to each other by harmonic springs with frequency ω˜P = _β^P

~. The lower the temperature, the more spatially extended the necklace, as intuitively drawn³ in Figure 2.2.

Figure 2.2 – Intuitive representation of the classical ring polymer as a function of the simulation temperatureT.

The PIMD method is a way to calculate quantum mechanical properties using the recipes from classical statistical mechanics with a modified Hamiltonian containing an additional quantum kinetic term (see Eq. (2.59)). Starting from the definition of the quantum average of a generic observableO given by,

it is straightforward to evaluate the potential energy of the system:

hVi ' 1

at) is the bead-dependent potential averaged over all the

3In principle, the number of beadsP remains the same for each temperatureT but is modified here for the sake of representability.

replicas. The quantum kinetic energy can also be evaluated according to hTi ' 1

(2π~)^fZ Z

d^fΓe^{−τ HP(Γ)}T_P(q), (2.62) whereTP(q) can be either chosen to be the centroid primitive estimator⁴,

T_P,pri(q) = 3NatP

or the more often employed centroid virial estimator, T_P,vir(q) = N_at

It is clear from Eq. (2.63) that the primitive estimator is more unstable and less conve-nient to use than the virial because of the possible presence of negative contributions in the quantum kinetic energy. This apparent drawback actually provides a stringent test for the path sampling since one should obtainhTi_vir =hTi_pri for fully converged calculations in the limitδt→0 for any value ofP. In practice this is almost never met exactly and it is partic-ularly difficult to fulfill it for largeP. Indeed, two constraints limit the choice of P. On one side,P must be large enough to recover all quantum properties of the physical system; on the other side, P should not be too large, otherwise the quantum imaginary time τ = _P^β would become too small and, for example, too large fluctuations in the primitive energy evaluation would appear.

PIMD simulations have been widely used to study the impact of NQE in water clusters and liquid water. To the best of our knowledge, the first PI simulation of liquid water and the three first water clusters has been performed in 1985 with an empirical force field^[170]. Later, Marx and coworkers used PIMD simulations to demonstrate that the hydrated proton forms a fluxional defect in the H-bond network, rather than any individual idealized hydration state such as Zundel or Eigen along the water chain. They also established that the quantum tunneling of the proton is negligible and the transition state theory does not apply. Proton diffusion looks however defined by the thermally-induced H-bond breaking and formation of the second solvation shell^[13]. Many other PIMD calculations of charged water clusters or liquid water have been carried out so far,[109,110,171–173] and they all confirm the importance of NQE, even at room temperature to describe PT in such systems. Nevertheless, the precise role of NQE in PT mechanisms is not fully elucidated yet since it can be significantly different from one system to another, and such effects are still under discussions. We refer the inter-ested reader to the recent reviews of Ceriotti^[174] and of Markland^[175]for the latest advances in the theoretical treatment of NQE in water and aqueous systems.

4We refer the reader to the Ref.169for a complete derivation of the primitive energy.

In this thesis, we have the key advantage that our study of NQE will a priori not suffer from a bias coming from the electronic description of the problem thanks to the use of ad-vanced QMC methods. QMC-MD driven simulations will be, as we have previously discussed, incorporated into a LD framework to control the QMC intrinsic noise on the computed ionic forces. Thus, we have to combine together PI and LD methodologies to perform PILD sim-ulations of small water clusters. In the following, we will describe a reference algorithm to perform such calculations.