Variational Monte Carlo



N

X

i<j

Φ_J(r_i,r_j)



, (1.47)

with

Φ_J(r_i,r_j) =

Nat

X

a,b N_basis⁰

X

µ,ν

g^a,b_µ,νΨ^J,a_µ (r_i−q_a)Ψ^J,b_ν (r_j−q_b), (1.48)

where Ψ^J,aµ ≡ G^a_µ (see Eq. (1.41)), and G = {g_µ,ν^a,b} is the mathematical equivalent of the AGP matrix Λ for the Jastrow part of the WF. N_basis⁰ is the number of primitive elements constituting the basis set of the Jastrow. The impact of N_basis⁰ and the selected basis set elements on the interaction between water molecules in gas phase will be discussed in this thesis in the Chapter 5. From Eq. (1.48), it is apparent that ΦJ(ri,rj) correlates electrons sitting on different atoms a and b. In this way, interatomic-induced polarization effects are included in the WF. Therefore,J₃is an essential ingredient to include rigorously polarizability and dispersion contribution in the QMC trial WF. Let us also remark that there is a large flexibility^[81] on the way one decides to write the QMC WF in both Jastrow and determi-nantal parts. The degree of sophistication of the WF must be a good compromise between computational efficiency and target accuracy, required for the type of problem we want to solve. Indeed, the WF evaluation is the most demanding task of the QMC approach. In our case, we aim at resolving tiny energy differences (beyond the chemical accuracy) so we have to be very cautious by systematically checking that our QMC WF provides relevant energies and geometries compatible with the physics of PT in water clusters.

1.2.3 Variational Monte Carlo

Variational Monte Carlo (VMC) is one of the simplest QMC approach and has been applied for the first time on fermionic systems in the seminal work of Ceperleyet al. in the late 70s^[82]. Let us consider again the QMC antisymmetric trial WF whose expression have been detailed in the previous pages of this manuscript Ψ_T(r₁, . . . ,r_N). The electronic energy, defined as the quantum expectation value of the Hamiltonian Hˆ given by Eq. (1.3) over the trial WF Ψ_T reads as

hHiˆ =E_VMC =

RdRΨ^∗_T(R) ˆHΨ_T(R)

RdR|Ψ_T(R)|² , (1.49)

where R = (r1, . . . ,rN) represent the coordinates of the QMC walker in the space spanned by electronic configurations and the spin has been omitted for simplicity. The variational principle states that the VMC energy defined in Eq. (1.49) is an upper bound to the true ground state energy of the system: E_VMC ≥E0. Eq. (1.49) can be then rewritten under the following form

EVMC = Z

dR |Ψ_T(R)|² RdR⁰|Ψ_T(R⁰)|²

HΨˆ T(R)

Ψ_T(R) (1.50)

= Z

dRπ(R)e_L(R) =he_Li ≥E₀,

whereh.i denotes an equilibrium average. In the above equation, we demonstrate that it is possible to compute the VMC energyEVMC as a statistical average of the local energy

e_L= HΨˆ _T(R)

Ψ_T(R) (1.51)

thanks to walkers sampling the probability densityπ(R) = ^R_dR^|Ψ0^T|Ψ^(R)|_T(R²⁰)|². This approach is known as the importance sampling since it favors the QMC random walk into the electron configurations space near regions where the amplitude of the QMC trial WF is high (|Ψ_T|²).

Let us notice that if the trial QMC WF Ψ_T is an exact eigenstate of the quantum Hamil-tonian H, the local energyˆ eL is a constant since it is equal to the ground state energy of the system, independently of the configuration explored by walkers. Therefore, the closer the trial WFΨ_T to |Ψ₀i, the smaller the fluctuations of the estimated VMC energyE_VMC. This well-known phenomenon is calledzero variance propertyand is extremely important to ensure the efficiency of any QMC simulation. This implies that a very careful optimization of the WF must be done before estimating the energy and other observables by a VMC calculation.

The probability density π(R) is sampled using the Metropolis-Hastings algorithm first in-troduced by Metropolis in 1953^[83] and then generalized by Hastings 17 years later^[84]. This algorithm, also used in classical MC simulations, enables to sample any unknown probability distribution by generating a memoryless process in the3N-configurational space of electronic positions. Such a random walk in which each new configuration only depends on the previous one is called a Markov chain. In the TurboRVB code, at each step of the Markov process, a new configuration is created by performing a single electron move. The move is then accepted or rejected by application of the Metropolis’ rule^[83]. In a practical way, the MC displacement is calibrated to obtain an optimal acceptance ratio of the MC samples (about60%of accepted moves). The efficiency of the MC sampling can also be increased if the amplitude of the single electron move depends on its distance with the nearest nucleus: if the electron is far from an ion, it moves with a larger step than the one of an electron closer to a nucleus. Moreover, when the system is composed by two distant fragments, as it will be the case in the Chapter 5 where the dissociation of the water dimer will be studied, instantaneous moves from one fragment to the other one are allowed after a given number of MC samples. This avoids to

"freeze" the random walk into the configurational space of electronic positions in a confined region of the space, thus alleviating ergodicity issues.

Finally, the VMC energy E_VMC is straightforwardly computed as we accumulate statistics

of its local estimator, namely the local energy, as defined in Eq. (1.50):

where we recall N_gen is the number of QMC generations. Analogously, any operatorOˆ repre-senting a physical quantity (the oxygen-oxygen distance for instance) can be evaluated as in Eq. (1.52)

T being the local operator. By virtue of the central limit theorem, this estimator of the true expectation value of Oˆ is totally free of biases. The corresponding unbiased estimator of the variance σ²[O_QMC]is given by

σ²[O_QMC] = τC

N_genσ²[O_L] = τC

N_gen(hO²_Li − hO_Li²), (1.54) where σ²[OL] represents the statistical fluctuations of the local measurements of OL. τC is theautocorrelation time of MC iterations, defined as

τC = 1 + 2 is equal to the unit in the ideal case of completely uncorrelated measurements, which is in practice never the case. We thus have to perform block averages (blocking technique) of the Eq. (1.53) not only to obtain a reliable VMC estimation of the desired observable but also to estimate correctly its variance by significantly reducingτ_C in Eq. (1.54).

As already mentioned, the major advantage of the QMC approach over deterministic methods is thus its ability to cope with the high dimensionality of electronic integrals at a reasonable computational cost. Indeed, the central limit theorem states that the intrinsic statistical er-ror σ_QMC² (Eq. (1.54)) on the estimated integral is independent of the dimensionality of the problem. This variance can be simply reduced by increasing the number of MC iterations to reach the desired accuracy, and a compromise must be found between accuracy and savings of computational resources.

To summarize, the guideline to perform accurate VMC calculations on physical and chemical systems, such as water clusters, is extremely simple. The very first step consists in building the WF. During this phase, we specify the mathematical form ofΨ(JSD or JAGP), the basis sets for both the Jastrow and the determinantal parts of the WF, the geometry of the system and the use of pseudopotentials for the heaviest atoms (oxygen in our case)^[80]. Then, we perform a DFT calculation, using the LDA functional, to fill the single or multiple determinants with KS orbitals, which constitute a good starting point for the optimization of the WF. Then, the QMC function is optimized using minimization procedures such as Stochastic Reconfiguration

(SR)^[85] and Stochastic Reconfiguration with Hessian accelerator (SRH)^[86] methods. From the practical point of view, this is the most delicate step in the QMC calculation. Technical details about these minimization techniques are given in AppendixC. Finally, once the WF is fully optimized and as close as possible to the ground state WF, |Ψ₀i, of the system, we can estimate physical quantities by means of VMC statistical averages.

During the 2000s, VMC has been mostly used to study small and neutral water clusters (from the monomer to the hexamer). Looking at Ref.87, we have the confirmation that the VMC energy of both the water monomer and dimer strongly depends on the quality of the WF. Indeed, there is a significant gain of 26 mHa (16.3kcal/mol) in the computed energy for the water monomer when many-body corrections are included in the determinant. Sterpone and his colleagues demonstrated the ability of VMC to reproduce properly the dissociation energy curve of the bonding water dimer with a reasonable, albeit slightly underestimated (about∼0.5kcal/mol), value of the dissociation energyDe. Furthermore, thanks to the great flexibility of the JAGP WF given in Eqs. (1.40) and (1.42), they have been able to estimate the covalent (1.1(2) kcal/mol) and the correlated VdW fluctuations (1.5(2) kcal/mol) contri-butions to the total binding energy^[88].

More recently, Zen and coworkers made an extensive study on the role of the WF ansatz and the basis set size to properly describe the properties of the water molecule^[89]. They have reported the following general hierarchy for the QMC WF JDFT < JSD < JAGP, which clearly indicates that JAGP WF seems to be the more suitable to study water. The accuracy of JAGP-VMC calculations for the dipole are in good agreement with CCSD results, but less accurate than CCSD(T) as expected which suggests VMC is "between" these two approaches for water. Protonated dimer, namely the Zundel ion H₅O⁺₂ has been studied very recently in our research team, which is the starting point of this thesis^[72]. Indeed, it has been demon-strated that obtained geometries for the Zundel complex at the VMC level are in excellent agreement with reference CCSD(T) calculations and the proton static barriers computed by VMC are slightly overestimated (about∼0.7 kcal/mol for d_OO= 2.7 Å) but reasonable.

To conclude, the VMC method is a simple stochastic method which has proven its efficiency to describe a large variety of physical and chemical systems, including neutral or charged water clusters, with an accuracy close to the most sophisticated quantum chemistry calculations.

Therefore, thanks to its milder scaling with the system size, VMC appears to be an ideal candidate to give a correct enough PES to solve the PT problem in liquid water and this point will be largely exploited in this thesis (see Chapters 4 and 6). However, we will see in the following Subsection how to improve the VMC results via another stochastic method, namely theDiffusion Monte Carlo (DMC).