Binding energy

We first start with the exploration of the bonding water dimer PES as a function of the oxygen-oxygen distance OO. The results of our structural optimizations are presented in Figure5.3, where different functionals are tested within the DFT framework (left panel). The QMC dissociation energy curves, obtained by both VMC and LRDMC, are plotted in the right panel for two different WFs.

Let us start with the discussion of the DFT results. As we already noticed in the case of the non-bonding water dimer, we confirm that the LDA functional (black) significantly overestimates the water dimer binding energy. We foundEbind,LDA = 8.74 kcal/mol, in cor-rect agreement with E_bind,LDA = 9.02 kcal/mol, predicted in Ref. 43. The observed minor discrepancy can be explained by the coarse oxygen-oxygen distance grid used in the disso-ciation energy calculations (left panel of Figure 5.3). The GGA and the hybrid functionals such as PBE (dark green) and B3LYP (magenta) provide much more accurate dissociation curves of the bonding water dimer, the results being significantly closer to the reference CCSD(T) calculations. To be more quantitative, we found Ebind,PBE = 5.28 kcal/mol and E_bind,B3LYP= 4.66 kcal/mol, in very close agreement with the tabulated values in literature (5.2 and 4.57 kcal/mol, respectively^[43]). The predicted CCSD(T) binding energy is slightly too large (5.2 against 5.02 kcal/mol in Ref. 43). This can be attributed to the fact that we took the B3LYP geometries for our CCSD(T) energy calculations without relaxing the geometry at the same level of theory.

In the right panel of Figure5.3, we report the binding dimer PES as a function of the oxygen-oxygen distance OO, computed using the two QMC WFs tested in the previous Section, by both VMC and LRDMC. We notice that the QMC approach is able to properly reproduce the dissociation curve of the binding water dimer, giving correct values of the binding energy at any oxygen-oxygen distanceOO. As expected, the LRDMC result is even more accurate that its VMC counterpart, with an overall agreement within ∼ 0.3 kcal/mol with the reference CCSD(T) curve. A more accurate and detailed study of the QMC binding energies will be presented in Table5.1.

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Figure 5.3 – Binding energy (kcal/mol) of the bonding water dimer (H2O)2 as a function of the OO distance. Comparison between different computational methods. Left panel: DFT results. Right panel: QMC results. Structural relaxation is performed at each level of theory, except for the CCSD(T) and the LRDMC approaches. The LRDMC lattice parameter is set to a= 0.125.

In the perspective of simulating larger water clusters such as the protonated water hexamer in Chapter6, we need to tackle the problem of the number of variational parameterspdefining the QMC WF. Indeed, as discussed in Chapter 3, the computational time spent for a single QMC evaluation of the electronic energy is roughly proportional to p. Therefore, we plan to apply also the Geminal embedding scheme to the orbitals in the Jastrow factor. To the best of our knowledge, this is the very first time that such a strategy is employed for the Jastrow part of the QMC WF in water. We applied 3 distinct contractions schemes, referred as O[m]H[n], with the integersm and ndefining the number of employed hybrid orbitals for each atom. For instance, the O[4]H[1] WF is the most simplified one, since it contains only 4 hybrid orbitals defining the oxygen atom in the Jastrow, whereas a single orbital is used for the hydrogen atom. As one can see in the second column of Table 5.1, the Geminal embed-ding scheme, applied to both the Jastrow and the determinantal part of the WF, drastically reduces the number of parametersp describing the QMC WF. Indeed, only1283 parameters are required for the O[6]H[2] QMC WF, against 6303for the fully primitive WF (both in the Jastrow and the determinantal parts). Therefore, the statistical gain on the subsequent QMC calculations is at least of a factor 5 and may be even more important in larger systems such as the protonated water hexamer.

We need however to ensure that this increase of computational efficiency of the QMC calcu-lations is not obtained at the price of a significant deterioration of the QMC WF, losing for instance a significant proportion of the QMC binding energy. To address that issue, we per-formed a series of structural relaxations of the bonding water dimer at the VMC level, using 3 different contraction schemes for the Jastrow orbitals, namely O[4]H[1],O[6]H[1] and O[6]H[2].

The corresponding results are represented in Figure 5.4, where the binding energies obtained for each QMC WF are given in the insert. We highlight that the computed binding energy the O[4]H[1] and O[6]H[1] hybrid WFs are affected by a significant basis set superposition

error (BSSE). Indeed, we applied a correction³ of 0.68 and 0.2 kcal/mol per water molecule to the O[4]H[1] and O[6]H[1] WFs, respectively. At variance with the two aforementioned hybrid WFs, the BSSE of the O[6]H[2] WF is reduced to0.01kcal/mol and can reasonably be neglected. As we can observe in Figure5.4, the O[6]H[2] contraction scheme seems to be the most adapted to properly describe the overall PES of the bonding water dimer at a moderate computational cost. Indeed, a correct agreement (within ∼ 0.3 kcal/mol) is found with the primitive JSD result.

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−1 0

3 4 5 6 7 8

Binding energy (kcal/mol)

d_OO (Å)

Primitive Jastrow O[4]H[1] Jastrow O[6]H[1] Jastrow O[6]H[2] Jastrow CCSD(T)

3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4

Ebind (kcal/mol)

Level of theory

Figure 5.4 – Binding energy (kcal/mol) of the bonding water dimer (H2O)2 as a function of the OOdistance. Comparison between different Jastrow contraction schemes of the QMC WF. Structural relaxation is performed for each employed basis set.

To pursue our discussion on the binding properties of the water dimer, we report in Table 5.1the values of the water dimer, monomer and binding energies for various QMC WFs. The number of parameters p describing the QMC WF is also given in the second column. We first emphasize that the value of the primitive binding energy is presumably not converged, since it represents roughly only86% of the correct value. In the meantime, the simpler WF with hybrid orbitals in the Slater determinant (SD) recovers 85%of the reference CCSD(T) binding energy, as it is slightly less accurate than the primitive JSD. When the geminal em-bedding scheme is applied to both the Jastrow and the SD, the corresponding binding energy is found to be4.05(8)kcal/mol, which is very reasonable (∼78%of accuracy), considering the

3The correction is defined as the energy difference between twice the energy of a single molecule and the water dimer calculated at a very large distance.

reduced number of parameters employed to describe the QMC WF. To go beyond the VMC approach and try to recover a more important fraction of the overall binding energy of the water dimer, we carried out some LRDMC calculations. In LRDMC, the lattice parameter a= 0.125gives unbiased Fixed-Node (FN) energy differences⁴. We will employ this value for all LRDMC calculation carried out in this thesis. The LRDMC binding energy is estimated around4.7−4.8kcal/mol, in agreement with the value provided by Sterpone in Ref.88. The LRDMC value is close to the reference CCSD(T), proving once again its remarkable accuracy.

We also tested another analytic form of the QMC WF, by taking an Antisymmetric Geminal Power matrix (AGP) instead of a SD to describe the fermionic part of the QMC WF. We find that the JAGP WF is, for a given basis set, less accurate that its JSD counterpart to describe the energetics of the bonding water dimer. Indeed, the JAGP binding energy is about ∼0.7 kcal/mol lower than the one predicted with the JSD WF. We have however to investigate the quality of the water dimer geometric parameters, as it will be done in the following Subsec-tion, before claiming that the JSD WF is more accurate than the JAGP one, or vice versa.

Last but not least, when one includes spin terms in the Jastrow factor, the binding en-ergy is significantly improved (by about ∼0.6 kcal/mol) for both JSD and JAGP WF. This significant improvement of the water dimer binding energy suggests that the spin-charge and spin-spin correlations are not so small in water, and should be taken into account. To the best of our knowledge, this is the first time that the possible influence of the spin on the binding interactions between water molecules is discussed so far. This remark is however preliminary and further analysis should be done to confirm the role of spin-spin correlations in water.

Method p E_dimer (Ha) E_monomer (Ha) E_bind (kcal/mol) VMC JSD primitive 6303 −34.50372(8) −17.24823(4) 4.46(8) VMC JSD hybrids det 2089 −34.50337(9) −17.24818(5) 4.40(8) VMC JSD hybrids Jas + det 1283 −34.49876(8) −17.24615(5) 4.05(8) VMC JSD hybrids det + spin 5713 −34.51471(9) −17.25390(4) 4.34(8) VMC JSD hybrids Jas + det + spin 1953 −34.51069(7) −17.25184(4) 4.40(7) VMC JAGP hybrids Jas + det 1283 −34.50689(9) −17.25089(7) 3.21(9) VMC JAGP hybrids Jas + det + spin 1953 −34.51713(6) −17.25551(5) 3.83(8) LRDMC JSD primitive (a= 0.25) 6303 −34.5341(1) −17.26360(8) 4.3(1) LRDMC JSD primitive (a= 0.125) 6303 −34.5333(1) −17.2629(1) 4.7(2) LRDMC JSD hybrids det (a= 0.125) 2089 −34.5336(1) −17.2630(1) 4.78(9)

Table 5.1 – Water dimer, water monomer and binding energies for various QMC WF. The number of parameters pof the WF employed for the calculations is also indicated.

To summarize, we have demonstrated that the QMC binding energies of both the non-bonding and the bonding water dimer are in good agreement with reference CCSD(T) calculations, albeit they are slightly underestimated. Indeed, describing such binding energies at the QMC level is a complex task since a subtle compromise must be found between extreme accuracy

4The corresponding simulation is more time-consuming (by a factor∼2) than if one setsa= 0.25, which leads to partially converged results.

and computational efficiency. Thanks to an original application of the Geminal embedding scheme to the Jastrow orbitals, we have devised a robust QMC WF that properly describes the water dimer energetics (80%of the CCSD(T) binding energy⁵), with a moderate number of parameters. This development allows us to use such a WF to study larger water clus-ters, such as the protonated water hexamer (Chapter 6). However, we have to ensure that this encouraging result is accompanied by an accurate description of the (H₂O)₂ equilibrium geometry.