Cite this: DOI: 10.1039/c0xx00000x
Relativity and the mercury battery
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This journal is © The Royal Society of Chemistry 2011
Physical Chemistry Chemical Physics, 2011, [vol], 00–00 | 1
Patryk Zaleski-Ejgierda and Pekka Pyykköa
Received (in XXX, XXX) Xth XXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
Comparative, fully relativistic (FR), scalar relativistic (SR)
and non-relativistic (NR) DFT calculations attribute about
30% of the mercury-battery voltage to relativity. The
obtained percentage is smaller than for the lead-acid battery,
but not negligible.
Relativistic effects can strongly influence the chemical properties
of heavier elements. A striking example was found in the recent
study of the cell reaction of the common lead-acid battery.1 We
could well reproduce its standard voltage of 2.107 V by the
calculations on the three solids involved. Furthermore, over 1.7 V
or 80% of this electromotoric force (EMF), EO, arise from
relativistic effects. As stated, ‘cars start due to relativity’.
How about other batteries? An obvious example, containing a
heavy element, is the previously common mercury battery2,
whose two-electron cell reaction3 is:
Zn(s) + HgO(s) → ZnO(s) + Hg(l),
Here, Hg is a liquid, while ZnO, Zn and HgO are solids.4
Qualitatively, we could estimate the magnitude of the relativistic
effects by claiming that Cd is non-relativistic (NR) mercury. The
experimental data3, 7 then gives:
Zn(s) + CdO(s) → ZnO(s) + Cd(s),
Assuming for a moment that Cd ≈ Hg(NR) and CdO ≈ HgO(NR)
and based on the experimental values of the EMF of reactions (1)
and (2) one can estimate that up to +0.88 V or 65% of EO(1)
comes from relativity5. Obviously Cd is not Hg, either relativistic
or not, and there are other factors that may affect the final
relativistic contributions to EMF.6
Let us come back on the reaction (1). Indeed, the experimental
cell thermodynamics are well known7, but here we approach them
from an ab-initio point of view. Our goal is to investigate the
relativistic effects on the EO of the cell, which we derive from the
cell free energy of the reaction (1), ∆GO, using
∆GO = – n·F·EO,
EO = – ∆GO / n·F.
EO = +1.35 V. (1)
EO = +0.47 V. (2)
Here n=2 is the number of electrons transferred and F is the
Earlier calculations were reported on the MO (M=Zn, Hg) oxides
by Glanss et al.8 and Biering et al.9 and on the metals M by
Gaston et al.10, 11, 12, Wedig et al.13 and by Moriarty.14 Gaston et
al. find that the cohesion of solid Hg requires both three-body
correlation and relativistic effects.11 For the cohesive energy
using the incremental method see also the work of B. Paulus et
al.15 Biering et al. found that at the non-relativistic level HgO
spontaneously relaxes to form a rock salt structure, thus already
the very structure of HgO originates from the inclusion of
relativity; in this particular case, from the scalar effects.9
Here, we consider the electronic relativistic effects through the
means of density functional theory (DFT) coupled with the
zeroth-order regular approximation (ZORA). We neglect
structural changes due to relativity, thus capturing the dynamic
electronic effects of relativity only.
Methodology: The prediction of formation energies in a
quantitative manner from ab-initio calculations requires, in
Table 1. Comparison of the experimental and calculated
electromotoric force, EO, of the mercury-battery reaction (1).
Level of relativity: NR – Non-relativistic, SR – Scalar relativistic,
FR – Fully-relativistic.
NR SR FR (FR-NR) (SR-NR) (FR-SR)
--- +1.04 +1.04 --- --- +0.00
--- +1.15 +1.13 --- --- -0.02
BAND LDA +0.70 +1.10 +1.10 +0.40 +0.40 +0.00
PBEsol +0.66 +1.04 +1.05 +0.38 +0.38 +0.00
PBEsol-D +0.93 +1.31 +1.31 +0.38 +0.38 +0.00
Exp.  --- --- +1.346b
Exp.  --- ---
a Structures relaxed b Standard conditions c Standard conditions
but with Hg heat of fusion included
2 | Physical Chemistry Chemical Physics, 2011, [vol], 00–00
This journal is © The Royal Society of Chemistry 2011
Table 2. Heats of formation, ∆H, calculated with respect to
spherically symmetrical atoms at different levels of relativity
ZnO(s) Zn(s) Hg(s) HgO(s)
NR -1.84 -9.87 -2.39 -8.45
+0.10 +0.25 +0.50 +1.42
a f.u. = formula unit
addition to having an accurate underlying theory, also good
convergence of all technical parameters including a sufficiently
large basis set. It is therefore meaningful to approach the problem
with several independent methods, and see if they converge on
the result. In our case, we used a linear combination of local
orbitals, with and without a frozen-core approximation, using the
BAND program16, and a plane-wave based program VASP.17, 18,
19 With BAND we employed the following exchange-correlation
functionals: SVWN,20 PBE,21 PBEsol,22 and also PBE-D and
PBEsol-D alternatives, where the two last ones include dispersion
corrections23, 24. For VASP we used the PBE functional only.
We investigate the relativistic effects using the BAND program
by means of the zeroth-order regular approximation (ZORA, see
Ref.  and references therein). We consider three cases: non-
relativistic (NR) with no ZORA operators, scalar relativistic (SR)
including ZORA but without the spin-orbit coupling part, and the
fully relativistic (FR) case with complete ZORA where first-order
spin-orbit effects are also taken into account. To ensure high-
accuracy results, the convergence of the calculations was checked
with respect to all crucial numerical parameters including the
number of k-points, the basis-set quality, and the size of the
In the BAND calculations we apply Slater-type triple-zeta basis
sets augmented with two polarization functions, taken from the
BAND basis-set repository. The frozen-core approximation is
applied to reduce the size of the variational basis set. The use of
frozen core, as implemented in BAND16, is preferable over
pseudo-potentials because it essentially allows for all-electron
calculations. The frozen-core orbitals are taken from high-
accuracy calculations with extensive Slater-type orbital basis sets.
For oxygen, we use all-electron basis sets. For zinc we include up
to 3p and for mercury up to 4f orbitals in the core.
The calculations are performed for experimental crystal structures
(given as supplementary material), allowing no structural
relaxations. Thus we capture the dynamic electronic effects of
relativity (meaning Dirac versus Schrödinger). While in our
BAND calculations the lattice constants and atomic positions
were kept fixed, we did consider relaxation effects in our VASP
calculations. At scalar and fully relativistic levels we found
relaxation effects to have minor effect on the EO (see Table 1).
Because solids are easier to handle theoretically and because we
perform our calculation at 0K, we use solid rather than liquid Hg
throughout the calculations.26 The thermal effects on the reaction
energies are small and are neglected27 as are the zero-point
vibrational contributions to the energy. If the battery freezes at
low temperatures, it is due to the kinetics and not due to
significantly different ∆G(1) at 0K. In the actual calculation of
the EMF of reaction (1) we thus effectively use ∆GO ≈ ∆G(0K) =
Discussion: In case of the lead battery we had a single, dominant
source for the relativistic effect, viz. the relativistic stabilization
of the Pb 6s shell, which made the Pb(IV)O2 reactant a high-
energy species. An unexpected, smaller contribution came from a
relativistic stabilization of the PbSO4 product of the discharge
How about the present reaction (1)? We see from Table 2 that the
same relativistic 6s stabilization raises the energy of the HgO
reactant. The density of states (DOS) of HgO shows significant
hybridization between the Hg 6s and 5d shells with the O 2p
shell.8, 9 This effect is expected to lower the relativistic
stabilization of HgO. Indeed, in our DFT model the stabilization
of HgO is calculated to be smaller than that of PbO2 by about half
of that amount. In the case of mercury, the 6s shell is full, and
the binding energy is also diminished. Thus, HgO goes up but
also Hg goes up. Similar effects, though smaller, are observed for
Zn and ZnO (see Table 2). This effectively diminishes the total
relativistic effect on EO(1).
Because valence-shell relativistic effects roughly scale, down a
column28, as Z2, the ratio of the analogous effects on Zn and Hg,
and their oxides, is expected to be
Indeed, the ratio of the relativistic shifts, ∆(FR-NR), is for
Zn(s)/Hg(s), +0.10/+0.50 = 0.20 and for ZnO(s)/HgO(s)
+0.25/+1.42 = 0.18.
The dominant uncertainty in our model arises mainly from the
metals, whose cohesive properties already contain substantial
dispersion. These effects have been recently invoked to explain
the large c/a ratios of Zn and Cd9, 29 and the rhombohedral
structure of Hg8
already be seen in the tetrahedral Mn clusters, M=Zn, Cd; n = 35,
56.30 The correlation effects are also of importance. Paulus and
Rosciszewski pointed out that Hg is not bound at the Hartree-
Fock level of theory, and thus all the binding comes from electron
Methodologically one source of uncertainty is the DFT itself
which is inherently unable to reliably reproduce the dispersion
interactions. By replacing the pure DFT, PBE and PBEsol
. Interestingly, the signs for large c/a ratio can
. 14 . 0
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Physical Chemistry Chemical Physics, 2011, [vol], 00–00 | 3
functionals with their dispersion-corrected analogues the
reproduction of the EMF is significantly improved. The best
agreement with experiment is obtained by applying the PBEsol-D
functional. Compared to the experimental cell voltage of +1.35 V
our relativistic best estimate yields +1.31 V.
Our key result is that about +0.38 V or 29% of the calculated
+1.31 V of the mercury battery’s EMF arise from relativistic
effects. Like in the case of the lead-acid battery1, the largest
relativistic contribution comes from the destabilization of a
reactant heavy-metal oxide, HgO(s), where 6s electrons are
formally removed. Scalar relativistic effects dominate. The small
spin-orbit effects come from Hg(s) and HgO(s) but cancel for the
We belong to the Finnish Centre of Excellence in Computational
Molecular Science (CMS). PZE is grateful to its current Head,
Professor Lauri Halonen (University of Helsinki, Laboratory for
Physical Chemistry), for generous economic support.
Notes and references
a Department of Chemistry, University of Helsinki, POB 55 (A. I. Virtasen
aukio 1), FI-00014 Helsinki, Finland. Fax: +358 9 19150169; Tel: +358
9 19150171; E-mails: firstname.lastname@example.org, email@example.com
† Electronic Supplementary Information (ESI) available: [Structures used
to simulate Hg(s), HgO(s),
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2 The mercury battery has been first patented by C. L. Clarke in 1884
(US Patent 298175). It became widely used in the 1940s
when Samuel Ruben developed a balanced mercury cell used for
military applications. The batteries become highly popular during
the World War II due to their large storage capacity, long shelf life
(up to 10 years) and stable voltage output. After the WW II it has
been widely applied in small electronic devices including:
pacemakers, hearing aids, calculators, radios and other portable
electronic apparatus. The battery lost its importance once the ban
on mercury-containing products has been enforced in the 1990s.
3 Standard conditions: 293.15 K, 101.325 kPa.
4 Actually the zinc is in form of an amalgam. We neglect the
5 Here, we assume the following: Cd(FR,s) ≈ Hg(NR,s) and
CdO(FR,s) ≈ HgO(NR,s). Then the difference ∆E = EO(1) - EO(2)
= +0.88 V corresponds to the relativistic contribution of the total
EMF of the reaction (1).
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p(1) = –1.23 J·K-1·mol-1 and
p(1) = –0.003 eV. We note that although this is only a