Atomization of correlated molecular-hydrogen chain: A fully microscopic variational Monte Carlo solution

We discuss electronic properties and their evolution for the linear chain of $H_2$ molecules in the presence of a uniform external force $f$ acting along the chain. The system is described by an extended Hubbard model within a fully microscopic approach. Explicitly, the microscopic parameters describing the intra- and intersite Coulomb interactions are determined together with the hopping integrals, by optimizing simultaneously the system ground state energy and the single-particle wave functions in the correlated state. The many-body wave function is taken in the Jastrow form and the variational Monte Carlo (VMC) method is used in combination with an ab initio approach to determine the energy. Both the effective Bohr radii of the renormalized single-particle wave functions and the many-body wave function parameters are determined for each $f$, which is the only external parameter in the whole analysis. Hence the evolution of the system can be analyzed in detail as a function of the equilibrium intermolecular distance, which in turn is determined for each $f$ value. The transition to the atomic state, including the Peierls distortion stability, can thus be studied in a systematic manner, particularly near the threshold of the dissociation of the molecular into an atomic chain. We also show that interelectronic correlations enhance the Peierls distortion. The computational reliability of the VMC approach is also estimated.

Figure 1

(a) Schematic representation of molecular chain, characterized by lattice parameter $a$ and bond length $b$; the atomic centers of molecule are labeled as $\alpha$ or $\beta$. (b) Hopping terms range extends up to $2a$. Note that only one atom $\alpha$ (blue circle) is marked for the sake of clarity. However, by symmetry the same hopping configuration holds for $\beta$ centers. (c) Same as in (b), but for the intersite ($K$) interactions.

Figure 2

Energy calculation flow chart.

Figure 3

Energy per electron for four different chain configurations, all as a function of variational parameter $\zeta$ for $N=10$. The symbols refer to the data obtained by means of VMC and the solid lines are results of the exact diagonalization (EDABI). The size of the symbols is larger than the estimated statistical error, i.e., $\sim {10}^{-3}$ (Ry) per electron.

Figure 4

Total energy as a function of $b$ for selected $a=1.5a_0,1.7a_0,1.85a_0$, and $2.0a_0$ obtained by means of QVMC for $N=50$. At fixed $a=1.5a_0$ the system approaches ALC solution, i.e., the energy minimum is for $b=a/2$.

Figure 5

Enthalpy as a function of $a$ and $b$ for the selected values of $f$ and for $N=50$. The dashed line indicates ALC. Plots are obtained from the mesh with resolution $\mathrm{\Delta }a=\mathrm{\Delta }b=0.025a_0$ by smoothing it within bilinear interpolation for the sake of clarity.

Figure 6

Equilibrium structural parameters $a$ and $b$ as a function of external force $f$ for $N=50$. Note that we plot $a/2$ instead of $a$ to help the identification of MLC $\to$ ALC transition $a/2\to b$. The critical force value is $f_c\approx 4.64\text{Ry}/a_0$. The inset contains corresponding dependence $\zeta \left(f\right)$. The estimated maximal error for $a$ and $b$ is related to the mesh resolution, i.e., $\pm 0.025a_0$.

Figure 7

Ground-state energy per electron for fixed value of $a=1.7$ as a function of $b$ for the selected $N$. Note the double minimum (marked by the arrows) appearing with the increasing system size.

Figure 8

Relation between $a_{\dim }$ and inverted system size $N$. The line indicates the linear fit.

Figure 9

(a) Critical force $f_c$ versus the inverse system size. Points with the error bars indicate obtained values. Solid line is a polynomial fit (explicitly written at the bottom), whereas dashed line marks the estimated value of $f_c(1/N\to 0)$. The line separates the molecular (MLC) from atomic (ALC) configurations; (b) critical force for the considered chain lengths. Note that for $N<34$ the closed-shell configuration is not fulfilled—their inclusion also prevented conclusive extrapolation and therefore they were disregarded in the final analysis. The lines are a guide for the eye.

Figure 10

Exemplary finite-size scaling of the charge gap $\mathrm{\Delta }$ at $f=3\text{Ry}/a_0$. The solid line represents the linear fit.

Figure 11

Extrapolated charge gap $\mathrm{\Delta }\left(f\right)$ for $a\left(f\right),b\left(f\right)$ referring to $N=50$ (see text). The vertical line separates MLC and ALC (for $N=50$), whereas the dashed line is a guide for the eye.

Figure 12

Charge energy gap as a function of the selected ratios between hoppings.

Figure 13

Density-density correlation functions plotted for the results obtained at $f=4.5\text{Ry}/a_0$ and $N=50$ for both $\alpha$ (a) and $\beta$ (b) sites, respectively.

Figure 14

Schematic representation of the essential results.

Figure 15

Distortion evolution parameter obtained by means of the VMC and with $N=50$ (a) and for the noninteracting electrons (b).