If computed in its straightforward fashion, the Yukawa-Jastrow, as any other slowly decaying Jastrow correlation factor, leads to a spurious bias in the kinetic energy. Therefore, all contributions that are originating from the periodic images of the unit cell must be taken explicitly into account in order to avoid discontinuities in the derivatives of the WF when the particle distances switch from one closest image to the other. Needless to say that this approach is computationally relatively time consuming, and a more economic strategy is very desirable.

However, before presenting our solution to this effect, let us start by introducing a particular useful test to verify if all correlations are correctly taken into account. To that extent, the expression for the kinetic energy (for simplicity we consider the kinetic contribution of only one particle *j*) is integrated by parts

$$\begin{array}{ccccc}& {\int}_{\mathrm{\Omega}}\text{d}R{\psi}^{*}\left(R\right){\nabla}_{j}^{2}\psi \left(R\right)\hfill & & & \\ & =\sum _{{x}_{\alpha}}{\int}_{\mathrm{\Omega}}\text{d}R{\psi}^{*}\left(R\right)\left(\frac{{\partial}^{2}}{\partial {x}_{\alpha}^{2}}\right)\psi \left(R\right)\hfill & & & \\ & =\sum _{{x}_{\alpha}}{\int}_{\mathrm{\Omega}}\text{d}{R}^{{\overline{x}}_{\alpha}}{\int}_{-{L}_{{x}_{\alpha}}/2}^{+{L}_{{x}_{\alpha}}/2}\text{d}{x}_{\alpha}{\psi}^{*}\left(R\right)\frac{{\partial}^{2}}{\partial {x}_{\alpha}^{2}}\psi \left(R\right)\hfill & & & \\ & =\sum _{{x}_{\alpha}}\{{\int}_{\mathrm{\Omega}}\text{d}{R}^{{\overline{x}}_{\alpha}}{\left[{\psi}^{*}\left(R\right)\frac{\partial}{\partial {x}_{\alpha}}\psi \left(R\right)\right]}_{-{L}_{{x}_{\alpha}}/2}^{+{L}_{{x}_{\alpha}}/2}\hfill & & & \\ & -{\int}_{\mathrm{\Omega}}\text{d}R\left(\frac{\partial}{\partial {x}_{\alpha}}{\psi}^{*}\left(R\right)\right)\left(\frac{\partial}{\partial {x}_{\alpha}}\psi \left(R\right)\right)\},\hfill & & & \end{array}$$(24)

where ${x}_{\alpha}=({x}_{j},{y}_{j},{z}_{j})$, $\text{d}{R}^{{\overline{x}}_{\alpha}}$ is the same as d*R* but excluding the infinitesimal element d*x*_{α}, Ω represents the domain of integration, i.e. the simulation cell, while ${L}_{{x}_{\alpha}}$ is the length of the edge of Ω along the *x*_{α} axis. However, at the presence of periodic boundary conditions, the WF *ψ*(*R*) and also its derivatives are required to be periodic, meaning that they are invariant with respect to particle translations **v** = (*n*_{x}L_{x}, *n*_{y}L_{y}, *n*_{z}L_{z}), where *n*_{x}, *n*_{y}, and *n*_{z} are all integers. From this follows that the term

$${\left[{\psi}^{*}\left(R\right)\frac{\partial}{\partial {x}_{\alpha}}\psi \left(R\right)\right]}_{-{L}_{{x}_{\alpha}}/2}^{+{L}_{{x}_{\alpha}}/2}$$(25)

vanishes, which leads to a modified Jackson-Feenberg (JF) kinetic energy expression^{3}:

$${E}_{\text{JF}}={\mathrm{\hslash}}^{2}\sum _{j=1}^{N}\frac{1}{2{m}_{j}}{\int}_{\mathrm{\Omega}}\text{d}R{\nabla}_{j}{\psi}^{*}\left(R\right)\cdot {\nabla}_{j}\psi \left(R\right).$$(26)

As a consequence, the equivalence of the Pandharipande-Bethe (PB)

$${E}_{\text{PB}}=-{\mathrm{\hslash}}^{2}\sum _{i=1}^{N}\frac{1}{2{m}_{j}}{\int}_{\mathrm{\Omega}}\text{d}R{\psi}^{*}\left(R\right){\nabla}_{j}^{2}\psi \left(R\right)$$(27)

and JF expressions for the kinetic energy is a necessary but not sufficient condition for the required periodic properties of the WF. Thus, in all of our calculations we have computed both expressions and explicitly verified that both are indeed identical, within the corresponding statistical uncertainties.

However, at the presence of additional correlation terms, such as the Jastrow, (25) must be correctly interpreted as the interparticle distances are computed using the closest periodic image. In fact, even though the particle coordinate **r**_{j} is confined to the unit cell of volume *𝒱* = *L*_{x}L_{y}L_{z}, the distance **r**_{ij} between the particles *i* (assumed as fixed) and *j* does not range within $\sqrt{{\left({\mathbf{r}}_{j}-{\mathbf{r}}_{i}\right)}^{2}}$, but always within the box of volume *𝒱* centered on particle *i*. This concept is illustrated in Figure 1. Therefore, it is possible and convenient to fix the origin at the position of the *i*^{th} particle that is considered. As a consequence, in the following, we will set **r**_{i} = 0, so that **r** = (*x*, *y*, *z*) ≡ **r**_{ij} = **r**_{j} and *r* = | **r|** = $\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$.

Figure 1: Integration of **r**_{j} within a box with periodic boundary conditions, from the point of view of particle *i*. The continuous black line represents the simulation box, whereas the dotted black line denotes the effective volume of integration for the distance between the particles *i* and *j*.

Let us demonstrate the JF test by showing that the Yukawa-Jastrow violates it. For that purpose we consider the simple case of only two interacting particles, i.e.

$$J\left(r\right)={e}^{-\frac{A\left(1-{e}^{-Fr}\right)}{r}}.$$(28)

Its first derivative along the *x* axis reads as

$$\begin{array}{ccccc}\frac{\partial J\left(r\right)}{\partial x}\hfill & =\frac{\partial {e}^{-\frac{A\left(1-{e}^{-Fr}\right)}{r}}}{\partial x}\hfill & & & \\ & =\frac{\partial {e}^{-\frac{A\left(1-{e}^{-Fr}\right)}{r}}}{\partial r}\frac{\partial r}{\partial x}\hfill & & & \\ & ={e}^{-\frac{A\left(1-{e}^{-Fr}\right)}{r}}\left(A\frac{1-{e}^{-Fr}}{{r}^{2}}-AF\frac{{e}^{-Fr}}{r}\right)\frac{x}{r}\hfill & & & \\ & =\frac{\partial J\left(r\right)}{\partial r}\frac{x}{r}.\hfill & & & \end{array}$$(29)

It is then apparent that the first derivative is not continuous at *x* = ±*L*/2, which is the border between its two closest periodic images. As a consequence,

$$\underset{\epsilon \to 0}{lim}{\left[J\left(r\right)\frac{\partial}{\partial x}J\left(r\right)\right]}_{-\left({L}_{x}/2\right)+\epsilon}^{+\left({L}_{x}/2\right)-\epsilon}=J\left(r\right)\frac{\partial J\left(r\right)}{\partial r}\frac{{L}_{x}}{r}\ne 0.$$(30)

Therefore, the JF and PB kinetic energies differ as the term in (25) does not vanish. Yet, if $\frac{1}{r}\frac{\partial J}{\partial r}$ is small enough at *x* = $\pm \frac{L}{2}$, the difference is negligible.

An even deeper understanding can be obtained by means of the distribution theory. In fact,

$$\begin{array}{ccccc}\frac{{\partial}^{2}}{\partial {x}^{2}}J\left(r\right)\hfill & =\frac{x}{r}\left(\frac{{\partial}^{2}J\left(r\right)}{\partial {r}^{2}}\frac{x}{r}+\frac{\partial J\left(r\right)}{\partial r}\frac{1}{r}-\frac{\partial J\left(r\right)}{\partial r}\frac{1}{{r}^{2}}\right)\hfill & & & \\ & -\left(\frac{\partial J\left(r\right)}{\partial x}\frac{x}{r}\right)2\delta \left(x-\frac{{L}_{x}}{2}\right),\hfill & & & \end{array}$$(31)

as

$${\int}_{{L}_{x}/2-\epsilon}^{{L}_{x}/2+\epsilon}\frac{\partial}{\partial x}\left(\frac{\partial J\left(r\right)}{\partial x}\right)=-\frac{\partial J\left(r\right)}{\partial x}\frac{{L}_{x}}{r}.$$(32)

In other words, the discontinuity in the first derivative entails a Dirac delta in the second derivative. Obviously, this artifact must be circumvented in order to avoid a bias in the computation of the kinetic energy.

As a discontinuity in the first derivative affects the validity not only of the JF expression but also of the PB one, the kinetic energy contribution provided by the Yukawa-Jastrow is biased. Mathematically, the problem can be eliminated by enforcing a smooth change between the closest periodic images. Physically, all of this originates from the fact that the simulation box is not large enough to “contain” all correlations between the particles.

A straightforward solution to remedy the latter is inspired by the Ewald summation technique [69]. More specifically, the Jastrow is decomposed into a quickly and a slowly decaying part, which are computed separately in real and reciprocal *k*-space, respectively. However, this method requires a summation over the whole momentum space, which is computationally rather demanding.

An alternative approach, which is not only more elegant and simpler but at the same time also more efficient, is due to Attaccalite and Sorella and results from exploiting periodic coordinates (PC) [70]. As the name suggests, the only modification required is to substitute the original coordinates by

$$x\prime =\frac{L}{\pi}\mathrm{sin}\left(\frac{\pi x}{L}\right),$$(33a)

$$y\prime =\frac{L}{\pi}\mathrm{sin}\left(\frac{\pi y}{L}\right),$$(33b)

$$z\prime =\frac{L}{\pi}\mathrm{sin}\left(\frac{\pi z}{L}\right),$$(33c)

and hence evaluate the distances via

$$r\prime =\frac{L}{\pi}\sqrt{{sin}^{2}\left(\frac{\pi x}{L}\right)+{sin}^{2}\left(\frac{\pi y}{L}\right)+{sin}^{2}\left(\frac{\pi z}{L}\right)}.$$(34)

The employment of PC enforces the correct periodicity of the WF. For example, the first derivative

$$\begin{array}{ccccc}\frac{\partial J(r\prime )}{\partial x}\hfill & =\frac{\partial J(r\prime )}{\partial r\prime}\frac{\partial r\prime}{\partial x\prime}\frac{\partial x\prime}{\partial x}\hfill & & & \\ & =\frac{\partial J(r\prime )}{\partial r\prime}\frac{x\prime}{r\prime}\mathrm{cos}\left(\frac{\pi x}{L}\right),\hfill & & & \end{array}$$(35)

is continuous in *x* = ±*L*/2, i.e. on the borders of the simulation box. The same also holds for all higher order derivatives. The consequential modifications of the Yukawa-Jastrow are illustrated in Figure 2.

Figure 2: Comparison between the Yukawa-Jastrow correlation factor for two interacting particles $\mathrm{exp}\left(-\frac{A\left(1-\mathrm{exp}\left(-Fr\right)\right)}{r}\right)$ (black line) and its modified version as obtained by employing PC (dashed line). We have set *y* = *z* = 0, *L* = 10, and *A* = *F* = 1, respectively.

To demonstrate the effectiveness of PC, we have calculated the kinetic energy using the JS-pw trial WF for two different systems, each consisting of 16 hydrogen atoms. The results of the atomic bcc (atm-bcc) and the molecular hcp (mol-hcp) phases of solid hydrogen including the corresponding Wigner-Seitz radii are shown in . As can be extracted by comparing *E*_{kin} with *E*_{JF}, the aforementioned spurious bias can be completely eliminated by the use of PC with an only negligible additional computational cost.

Table 1: The kinetic energies (in Ry) for the atm-bcc and mol-hcp phases of solid hydrogen as obtained with and without PC.

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