Accurate, Large-Scale Density Functional Melting of Hg: Relativistic

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Accurate, Large-Scale Density Functional Melting of Hg: Relativistic Effects Decrease Melting Temperature by 160 K Krista G. Steenbergen,* Elke Pahl,* and Peter Schwerdtfeger* Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study and the Institute of Natural and Mathematical Sciences, Massey University Auckland, Private Bag 102904, 0632 Auckland, New Zealand ABSTRACT: Using first-principles calculations and the “interface pinning” method in large-scale density functional molecular dynamics simulations of bulk melting, we prove that mercury is a liquid at room temperature due to relativistic effects. The relativistic model gives a melting temperature of 241 K, in excellent agreement with the experimental temperature of 234 K. The nonrelativistic melting temperature is remarkably high at 402 K.

M

melting temperature requires more than one million computing (core) hours, but yields a wealth of valuable thermodynamic data within each MD trajectory. Analyzing these results, we are able to compare relativistic and nonrelativistic dynamics for additional insight into how relativity affects mercury’s peculiar thermodynamic properties. The simulation of Hg−Hg interactions in clusters and the solid state has a long and complicated history.9,16 The preferred simulation method would be a many-body decomposition for metallic systems using highly accurate relativistic electron correlation methods; however, these methods are computationally infeasible for large-scale melting simulations, which require hundreds of atoms in a unit cell in order to accurately describe thermodynamics. Only recently have first-principles (DFT) melting simulations on bulk materials become computationally viable with the advent of better supercomputers and highly scalable, periodic DFT codes. Although modern density functional theory has several well-known pitfalls for strongly correlated systems,17 certain density functionals have been specifically tailored to describe solid state properties more accurately. In order to select the density functional best-suited to describe bulk mercury properties, we first completed a series of ground state optimizations for five possible crystal structures and a variety of functionals, using the projector-augmented wave (PAW) method18,19 as implemented in the Vienna ab initio simulation package (VASP).20−23 These results are summarized at the end of the Letter, where additional simulations details are also given. Comparing the experimental

ercury is the only elemental metallic liquid at room temperature. It has been suggested that this is due to relativistic effects substantially diminishing the melting temperature (Tm) of mercury compared to the other Group 12 metals.1−5 For almost two decades, quantum chemistry lacked the theoretical tools and computational power to solve the question of mercury’s curiously low Tm while confirming the importance of relativistic effects in many other molecular and solid-state properties of mercury,4,6−11 and in heavy-atom chemistry in general.12 In 2013, results of simulations published by our group gave a first answer to this long-standing question. That work paired parallel tempering Monte Carlo simulation methods with a rather crude model, the Diatomics-inMolecules (DIM) method,13 to describe Hg interactions. Comparing the DIM relativistic and nonrelativistic results, it was found that relativistic effects caused the melting temperature to drop by 105 K.5 However, the limitations of the DIM method are difficult to quantify, e.g., the DIM model leads to a rhombohedral crystal density that is less than half the experimental density. [In the original DIM research5,14 a conversion error in the unit cell calculation occurred, and the correct DIM structural parameters are a = 3.63 Å and α = 70.2° (as compared to the experimental values of a = 3.005 Å and α = 70.53°).15] Corrections to the DIM model are expected to further increase the relativistic-to-nonrelativistic melting temperature difference.5 For these reasons, an entirely independent evaluation of the solid-to-liquid phase transition of mercury is required. Here we present the results of a more refined simulation method of bulk mercury melting. Using density functional theory (DFT) molecular dynamics (MD), we complete bulk melting simulations of mercury, proving that mercury is a liquid at room temperature due to relativistic effects. Each converged © XXXX American Chemical Society

Received: February 13, 2017 Accepted: March 12, 2017 Published: March 12, 2017 1407

DOI: 10.1021/acs.jpclett.7b00354 J. Phys. Chem. Lett. 2017, 8, 1407−1412

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The Journal of Physical Chemistry Letters results with the relativistic calculations for different DFT functionals, we demonstrate that the density functional Perdew−Burke−Ernzerhof (PBE) revised for solids (PBEsol)24 (without dispersion correction) and the local density approximation (LDA) 25 perform best. Both yield the rhombohedral ground state crystal structure with a cohesive energy and density in close agreement with experiment. PBEsol is the best match to the experimental cohesive energy. LDA, which is commonly used for metallic interactions, overbinds as expected, but gives the best match to the experimental density. For this reason, we choose PBEsol and LDA for the subsequent MD melting simulations. While a higher level of theory for the interatomic interactions should increase the accuracy of the thermodynamic model, any bulk melting simulation seeded with a perfect crystal supercell will suffer from superheating, as melting nucleates from surfaces and/or defects. In order to circumvent this issue, Pedersen et al. developed a clever, powerful method called “interface pinning”,26,27 where the solid and liquid phases are simulated within the same supercell, as illustrated in Figure 1. Using an

Table 1. Thermodynamic Quantities of Interest for the PBEsol and LDA Relativistic (rel) and Nonrelativistic Models (nr)a functional

Tm

ρs

ρS

ΔHfus

PBEsol (rel) PBEsol (nr) LDA (rel) LDA (nr) exp. α-Hg

241(13) 402(15) 283(15) 690(20) 23431

13.71 13.24 13.82 13.40 14.1832,33

13.28 12.82 13.64 12.71 13.6933

2.04 5.64 2.96 7.71 2.3031

a Melting temperature (Tm, K), solid density at Tm (ρs, g/cm3), liquid density at Tm (ρS , g/cm3), and estimated latent heat of fusion (ΔHfus, kJ/mol).

thermodynamics matches closely with experimental measurements at the melting temperature. The melting temperature for relativistic LDA is notably higher than the experimental value, although still yielding a liquid at room temperature. Given the higher ΔHfus and the known overbinding of the LDA solid cohesive energy, our results indicate that the melting temperature is very sensitive to the cohesive energy, as one expects. Comparing the lowtemperature liquid RDF to the experimental results (Figure 2b), we also note that the relativistic LDA model fails to capture the correct liquid structure. As melting temperature depends on both solid and liquid energetics, it is clear that LDA cannot adequately model the thermodynamics of mercury melting. The melting temperatures of both nonrelativistic models are significantly higher than their relativistic counterparts. For both PBEsol and LDA, mercury would be a solid at room temperature in the absence of relativistic effects, thus confirming the original DIM results. Relativistic effects decrease the melting temperature by ∼160 K (!) for the PBEsol functional. The nonrelativistic LDA melting temperature is unrealistically high, falling ∼400 K above the relativistic result. However, plots of the melting temperature as a function of the cohesive energy and latent heat of fusion (Figure 3a,b) demonstrates that our results fit well with the linear trend set by all elements in the Periodic Table. [We note with caution that the correlation between ground state (zero-temperature) cohesive energy and melting temperature has serious limitations, as thermodynamic behavior is dictated by many finite temperature variables as clearly seen in Figure 3.] As expected, there are significant deviations for both PBEsol and LDA nonrelativistic liquid RDFs compared to the relativistic simulation and experimental results (Figure 2a,b). Overall, given that relativistic LDA yields a relatively poor match to the experimental thermodynamic quantities, it is reasonable to conclude that the nonrelativistic LDA model is similarly inadequate. Given the excellent performance of relativistic PBEsol, we trust the PBEsol relativistic-to-nonrelativistic melting temperature difference of 160 K. This large, unprecedented change in the melting temperature due to relativistic effects has its origin in the large relativistic 6s contraction known to substantially influence the chemistry of mercury.10,12,36 In summary, first-principles melting simulations confirm that mercury is a liquid at room temperature due to relativistic effects. The relativistic PBEsol melting temperature most closely matches the experimental results, with only a 7 K difference between the modeled and experimental Tm as well as an excellent description of the solid/liquid densities and liquid

Figure 1. Two-phase periodic simulation box, illustrated here for the relativistic simulation with 250 mercury atoms.

external potential to bias (“pin”) the system toward maintaining the solid state, and exploiting the fact that the Gibbs free energy difference between the solid and liquid phase is zero at the melting temperature, a model Tm can be calculated. This method has been successfully used for the melting of bulk silicon, sodium, aluminum, magnesium,27 and gallium,28 as well as to study scale invariance in metals29 and thermodynamic isomorph invariance in crystals.30 Here, we employ this simulation method to model the bulk α-Hg melting temperature, comparing scalar relativistic to nonrelativistic results. [It has been previously demonstrated that, for the melting temperature, spin−orbit effects are negligible.5] For the nonrelativistic calculations, the PAW data set was modified by adjusting the pseudopotential parameters with the speed of light set to infinity. Table 1 summarizes the melting temperatures calculated for each of the four models. The melting temperature of the PBEsol relativistic model is an excellent match to experiment, with an average Tm only 7 K greater than the experimental value. For this model, the solid and liquid densities deviate only slightly from the experimental values: in the solid phase, the modeled bond lengths are (on average) ∼ 0.03 Å longer than the average experimental bond lengths at Tm. In Figure 2a, we compare the simulated low-temperature liquid radial distribution function (RDF) with experimental results, illustrating that the PBEsol relativistic liquid closely matches the experimental liquid structure. Moreover, the PBEsol latent heat of fusion (ΔHfus) deviates only slightly from the experimental value, demonstrating that the PBEsol description of mercury’s 1408

DOI: 10.1021/acs.jpclett.7b00354 J. Phys. Chem. Lett. 2017, 8, 1407−1412

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Figure 2. Simulated radial distribution functions, g(r), of liquid mercury for (a) the PBEsol relativistic model at 240 K (red) and nonrelativistic model at 410 K (blue), and (b) the LDA relativistic model at 290 K (red) and nonrelativistic model at 700 K (blue). We also include experimental g(r) from X-ray diffraction measurements at 300 K (exp.a, solid black line)34 and at 234 K (exp.b, dashed black line).35 For visual clarity, the simulated results have been scaled so that the first peaks match the experimental first peak.

Figure 3. Melting temperature (Tm) plotted as a function of (a) the ground state cohesive energy (Ecoh) and (b) the latent heat of fusion (ΔHfus) for each of the elements in the periodic table. The experimental results for the group-12 elements are highlighted by colored squares: zinc (magenta), cadmium (green), and mercury (red). The simulation results for the PBEsol and LDA relativistic (blue circles) and nonrelativistic (blue triangles) models are also included. The dashed lines are not regression lines, but serve only as visual guides to the overall trend. In panel b, this line disregards the second “outlier trend” created by the p-block elements, as highlighted by the gray wedge.

structure at Tm. The nonrelativistic PBEsol result is ∼160 K greater than the relativistic melting temperature. Interestingly, the PBEsol nonrelativistic Tm is more in line with the group 12 trend of melting temperatures. Future research will include thermodynamic simulations of zinc, cadmium, and superheavy copernicium.

least doubling the relativistic results for most functionals. This same phenomenon was also observed in a study on mercury oxide.10 Interestingly, nonrelativistic mercury crystallizes in a hexagonal close packed (hcp) structure at zero-temperature, following closely the Group 12 trend set by zinc and cadmium. PBEsol and LDA clearly perform best, yielding the rhombohedral ground state crystal structure with cohesive energies and densities in close agreement with experiment. Other functionals perform adequately for cohesive energy and density (such as PBE with Grimme’s D3 dispersion41,42), but fail to capture the α-Hg rhombohedral structure as the ground state. We note that experimentally, the most stable crystal structure below 78 K is the body-centered tetragonal (bct) βHg structure.15,48 For the PBEsol and LDA relativistic models, the bct and rhombohedral crystals are nearly degenerate in energy, but we find the rhombohedral crystal to be slightly more stable. The melting MD simulations of all models were completed using VASP with the Langevin thermostat52,53 and Parrinello− Rahman barostat,54,55 in the NpT or NpzT ensemble (as specified below). Initially, the solid and liquid phases were simulated separately. For the relativistic models, each of the

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COMPUTATIONAL DETAILS Table 2 summarizes the solid state results for a variety of density functionals: PBEsol,24 LDA,25 PBE,37,38 PW9139 and PBE with Grimme’s dispersion corrections PBE+D2,40 as well as the revised dispersion correction with Becke-Johnson damping (PBE+D3bj and PBEsol+D3bj).41,42 Three “opt” nonlocal correlation functionals were also tested: optPBE-vdW, optB88-vdW, and optB86-vdW.43−46 Five different crystal structures were optimized for each functional: rhombohedral (rhom), hexagonal close packed (hcp), body-centered tetragonal (bct), face-centered cubic (fcc), and body-centered cubic (bcc). At the relativistic level, our DFT calculations are in perfect agreement with earlier results.51 The nonrelativistic crystals have significantly higher (more-binding) cohesive energy, at 1409

DOI: 10.1021/acs.jpclett.7b00354 J. Phys. Chem. Lett. 2017, 8, 1407−1412

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The Journal of Physical Chemistry Letters Table 2. Ground State Density (ρ, g/cm3), Crystal Structure, Cohesive Energy (Ecoh, kJ/mol), and Nearest-Neighbor Distance (rNN, Å) for Various Density Functionals at the Nonrelativistic and Relativistic Level of Theorya functional Relativistic PBEsol

LDA

PW91 PBE PBE+D2 PBE+D3 PBEsol+D3 optPBE-vdW optB88-vdW optB86-vdW exp. α-Hg Exp. β-Hg Nonrelativistic PBEsol LDA PW91 PBE PBE+D2 PBE+D3 PBEsol+D3 optPBE-vdW optB88-vdW optB86-vdW

crystal rhomb (70.5°) bct (0.708) bcc hcp (1.717) fcc rhomb (72.5°) bct (0.708) bcc hcp (1.727) fcc hcp (1.528) hcp (1.528) hcp (1.617) bcc rhomb (67.8°) hcp (1.558) bcc bcc rhomb (70.4°)15 bct hcp (1.775) hcp (1.777) hcp (1.904) rhomb (62.6°) rhomb (60.4°) hcp (1.818) hcp (1.715) rhomb (64.2°) hcp (1.839) rhomb (67.1°)

Ecoh

ρ

to those of the solid simulation, and the liquid simulation is also continued in the NpzT ensemble. The NpzT simulations are run for >50 ps for each separate phase in order to obtain equilibrated thermodynamic average quantities: SteinhartNelson Q6 order parameter,56 volume, and energy. The chemical potential difference between the solid and liquid phases (Δμ) is calculated from the two-phase interface pinning simulation. Here, the supercell is created by adjoining the equilibrated solid and liquid phases along the z-axis (matching the x and y-lattice parameters). The relativistic supercells contain 250 mercury atoms, while the nonrelativistic supercells contain 256 atoms. Molecular dynamics simulations are completed for this solid−liquid supercell in the NpzT ensemble at atmospheric pressure and Tsim. Each two-phase melting simulation was run for >30 ps, from which we calculate the average Δμ, which is proportional to the average force required by the external bias field to maintain an approximately equal number of solid and liquid atoms. More details of the method can be found in the original references of Pedersen et al.26,27 Errors in each melting temperature were calculated by dividing the simulation time into statistically independent blocks, calculating the average Δμ for each block. We then compute the mean and standard deviation of these averages, which leads to the average Tm and error bars on Tm, as given in Table 1. Finally, the ΔHfus, as given in Table 1, is estimated by computing the average energy difference between the separate solid and liquid simulations Tsim.

rNN

54.71 53.74 52.68 51.91 51.23 92.82 91.76 90.02 89.25 89.06 17.95 13.99 110.86 51.52 96.97 44.58 51.72 22.58 64.61b ∼ 67.54c

14.17 14.54 14.14 14.40 14.51 14.78 14.96 14.87 15.12 15.27 11.17 10.95 10.33 13.32 15.02 11.51 13.65 8.73 14.4815,47 14.7615,48

3.01 2.85 3.12 3.15 3.20 2.97 2.82 3.08 3.09 3.14 3.41 3.43 3.56 3.19 2.98 3.39 3.19 3.67 2.9915

128.13 160.55 85.87 82.69 174.25 121.67 168.08 101.21 116.17 45.64

13.58 13.99 12.42 12.43 11.41 13.20 14.25 12.43 13.20 10.41

3.17 3.14 3.19 3.30 3.45 3.18 3.16 3.25 3.16 3.41

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Krista G. Steenbergen: 0000-0002-0797-951X Notes

The authors declare no competing financial interest.

a

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Five crystal structures were tested for each functional (structural abbreviations defined in the text). For relativistic LDA and PBEsol, the energies of all tested crystal structures are given. For the remaining functionals, only the lowest energy crystal structure is given. For the rhombohedral crystal structure, the lattice angle α is given in parentheses; for the hcp and bct crystal structures, the lattice a/c ratio is given in parentheses. bAlthough several values have been published for Ecoh of mercury, we have traced the value given by the U.S. NIST-JANAF Thermochemical Tables31 to the original reference by Hultgren et al.,49 verifying that 64.61(±0.05) kJ/mol is the correct quantity. cObtained by a calculation of ΔE from rhombohedral phase.50

ACKNOWLEDGMENTS The authors thank the Marsden Fund of the Royal Society of New Zealand (MAU1409) for financial support. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www. gauss-centre.eu) for funding this project by providing extensive computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (www.lrz.de), and we thank the New Zealand eScience Infrastructure (nesi00260, nesi00265) for computational resources and support.

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separate solid and liquid unit cells consisted of 125 mercury atoms (solid phase of 5 × 5 × 5 rhombohedral supercell), with a 2 × 2 × 2 k-point grid. For the nonrelativistic models, each of the separate solid and liquid supercells consisted of 128 mercury atoms (solid phase of 4 × 4 × 4 hcp supercell), with a 2 × 2 × 1 k-point grid. The solid phase was annealed to the desired simulation temperature (Tsim), while the liquid phase was quenched to the same Tsim. The solid phase is simulated at constant temperature and pressure (NpT) until the unit cell stress on each lattice vector averages to the desired pressure (atmospheric), at which point the x and y-lattice vectors are fixed, and the simulation is continued in the NpzT ensemble. The x and y-lattice vectors in the liquid simulation are matched

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