Validation of density functional methods for computing structures and energies of mercury(iv) complexes

While quantum chemical predictions have strongly suggested a decade ago the existence of mercury in its oxidation state +iv, no experimental evidence has been found yet.

To enable the search for alternative targets and preparation routes by quantum chemical methods, the present work has validated density functional methods against accurate CCSD(T) results for structures, reaction energies and activation barriers for X2-elimination, and atomization energies for three HgX4 systems (X = F, Cl, H).

Hybrid functionals with ca. 20% Hartree–Fock exchange like B3LYP, B1LYP or MPW1PW91 have provided the best energetics, whereas local or gradient-corrected “pure” functionals overestimate, and the BHandHLYP hybrid functional underestimates the stability of the HgIV state.

Basis sets are suggested that provide a reasonable compromise between accuracy and computational effort in calculations on larger systems.


The possible existence of species with mercury in an oxidation state higher than +ii has been puzzling experimentalists and theoreticians for almost three decades.

An experimental verification of such high-valent mercury complexes is a fascinating target, as it would turn a group 12 element into a true transition metal.

An initial report of an electrochemically generated, spectroscopically characterized short-lived [Hg(iii)(cyclam)][BF4]2 species by Deming et al. in 19761 has never been confirmed.

But it stimulated Jørgensen2,3 to predict the possible existence of HgF4.

Analogous to the 5d8 AuIII oxidation state, a 5d8 HgIV species should be more stable than a HgIII d9 state.

In 1993, we reported the first application of quantum chemical methods to the problem.4,5

Using high-level quasirelativistic pseudopotential QCISD(T) calculations with, at the time, respectable basis sets, the square-planar D4h symmetrical HgF4 was predicted to be thermodynamically stable in the gas phase with respect to the elimination reaction HgF4 → HgF2 + F2 (see Fig. S1 of electronic supplementary information (ESI) for transition state).

Comparison with nonrelativistic pseudopotential results showed that the stability of the higher oxidation state is of relativistic origin.5

Most notably, the results showed that a better description of electron correlation should increase the reaction energy.

This was confirmed five years later by Liu et al6. using larger basis sets and CCSD(T) methods.

HgCl4 was in contrast suggested to be thermodynamically unstable6 with respect to Cl2 elimination.

Seth et al. predicted that the eka-mercury analogue of HgF4, (112)F4, should be even more stable than HgF4 with respect to F2 elimination.7

Recently, Pyykkö et al8. showed computationally that HgH4 and HgH6 are significantly endothermic, but have moderate activation barriers to H2 elimination.

Up to date, none of the discussed high-valent mercury compounds systems has been confirmed experimentally.

The technical difficulties for their synthesis have apparently been too large.

As aggregation energies disfavor HgF4 against HgF2 in the condensed phase, molecular beam or matrix isolation techniques would seem appropriate.

However, the use of aggressive fluorine compounds and of mercury does not make the former route attractive for experimentalists, and the latter route also has not produced evidence for high-valent mercury.9

The quest for Hg(iv) complexes remains thus a major challenge, and we have recently started to consider new synthetic targets and routes, including electrochemical access using chelate or macrocyclic ligands and/or oxidizing matrix environments.10

A problem arising with quantum chemical predictions is that the accurate coupled-cluster methods employed previously in this field are computationally too expensive to be applied to larger complexes.

The only alternative is currently to use density functional theory (DFT) methods.

However, as the accuracy of the various DFT approaches may not be improved systematically towards the exact result, and the most appropriate functional is not immediately obvious, it is necessary to validate DFT methods on smaller models, for which accurate coupled cluster methods may still be used.

In this work we provide such a systematic validation study of various density functionals and basis sets on structures and stability of small HgX4 complexes (X = F, Cl, H; n = 2, 4) against accurate benchmark CCSD(T) calculations.

Computational methods

The benchmark calculations employed the coupled-cluster method with single and double, as well as perturbative triple excitations [CCSD(T) level].

For comparison, CCSD and MP2 calculations are also reported.

All of these ab initio calculations used a quasirelativistic small-core 20-valence-electron (20-VE) pseudopotential11 (effective-core potential, ECP) with a (11s10p9d4f3g)/[9s6p5d3f2g] valence basis set11 for Hg, as well as aug-cc-pVQZ12,13 basis sets for F, Cl, and H. This ECP/basis-set combination will be denoted basis A. Bond lengths were optimized by fitting a fifth-order polynomial to about 7 single-point energy calculations, which were done with the MOLPRO 2000114. program.

DFT calculations used the Gaussian 9815 program and gradient methods.

The following exchange-correlation functionals were scrutinized: The local density approximation in form of the SVWN516 functional, the gradient-corrected BP8617,18 functional, and the hybrid functionals B1LYP,19,20 B3LYP15 (based on the work of Becke),21 MPW1PW91,22 and BHandHLYP.23

Basis-set requirements in the DFT calculations are expected to be somewhat less dramatic than in the post-HF treatments.

Moreover, future applications to larger systems require a reasonable compromise between accuracy and computational effort.

Three different basis-set combinations were compared, denoted B, C, and D (see Table 1).

We will in the following report the computational levels by the notation method/basis.

Unrestricted Kohn–Sham calculations on nonspherical atoms were performed to obtain atomization energies.

Basis-set superposition errors (BSSE) were considered using the counterpoise correction (CP)24 at optimized minimum structures.

Zero-point energy (ZPE) corrections were computed at the B3LYP/C level.

Spin–orbit corrections were not considered in this study.

They have previously been found to be small for the elimination reaction HgF4 → HgF2 + F2,5 and the theory against theory comparison is not affected by them.

Results and discussion

Minimum structures

The HgX4 systems were generally found to have D4h minima at all computational levels.

The calculated Hg–X bond lengths are shown in Table 2.

Using the CCSD(T)/A results as benchmark, CCSD/A underestimates the bond lengths for HgH4 and HgF4.

MP2/A overestimates the distances for HgF4 but underestimates them for HgCl4 and HgH4.

The large and non-systematic differences document the previously discussed5 importance of non-dynamical correlation in these systems, particularly for HgF4 and HgCl4.

The non-iterative triple excitations in CCSD(T) are known to partially recover the non-dynamical correlation.25

The CCSD T1-diagnostics26 at the CCSD(T) minima are 0.017, 0.011, and 0.012 for HgF4, HgCl4, and HgH4, respectively.

This suggests a reasonable quality of the coupled-cluster results.

The largest T1 value for HgF4 is consistent with large variations between MP2, CCSD, and CCSD(T) results.

This suggests to view MP2 and CCSD energies with caution.5

Compared to the CCSD(T)/A benchmark results, all DFT calculations overestimate the Hg-X bond lengths (see Table 2).

This may in part be due to the neglect of dispersion effects27 in present-day functionals, which would further decrease the distances.

Consistent with this, the discrepancies are by far largest for HgCl4.

Among the various functionals, the BHandHLYP and SVWN5 results are closest to the CCSD(T) values for HgF4 and HgCl4.

While this may be attributed to error compensation with the typical overbinding of the local density approximation for the SVWN5 case,28 the large fraction of Hartree–Fock exchange shortens the bond lengths for the BHandHLYP functional.

As might be expected, basis B provides the shortest DFT bond lengths and thus the best agreement with the benchmark CCSD(T)/A results.

This is largely due to the inclusion of f-functions for mercury and in part to the still relatively flexible basis sets for X. Basis set C provides results similar to basis set B for HgF4 and HgH4 at lower computational cost.

The results indicate that the best DFT structure results for HgX4 are obtained using basis sets B or C, and SVWN5 or BHandHLYP functionals.

Reaction energies for X2-elimination

Calculated energies for the elimination reactions HgX4 → HgX2 + X2 (X = F, Cl, H) are provided in Table 3 and in Fig. 1.

CCSD underestimates and MP2 overestimates the elimination energies in all cases.

This appreciable level dependence of the results indicates again a significant influence of non-dynamical correlation.

As has been discussed previously,5 these effects arise mainly for the “true transition-metal” HgIV d8 species, while non-dynamical correlation effects are much smaller for the HgII d10 complexes, where metal d-orbitals are unimportant for the bonding (see atomization energies below and in Table 5).

The resulting lack of error compensation between these effects for the two sides of the reaction is responsible for the appreciable level dependence.

Comparison with previous results5,6 confirms the notion of larger elimination reactions for larger basis sets.

The full CP procedure to correct for BSSE was not possible at CCSD(T)/A level for HgF4 and HgCl4, as the large dimension of the problem combined with the low symmetry of the CP calculation for the halogen atoms exceeded the available computational resources.

Given the close agreement between CCSD and CCSD(T) CP corrections for the hydride, we may assume that the CCSD values provide also a good estimate for the CCSD(T) CP correction in the other two cases.

In general, the CP corrections tend to lower the reaction energies moderately by ca. 10–15 kJ mol−1.

We estimate this to be less than the underestimate of correlation effects due to basis-set incompleteness errors, which are expected to cause an underestimate of the reaction energies.5

We presume therefore, that the CCSD(T)/A values in Table 3 provide still lower bounds to the true reaction energies.

Turning now to DFT methods, the best agreement with the benchmark CCSD(T)/A results is achieved using those hybrid functionals (B1LYP, B3LYP, and MPW1PW91) that exhibit about 20% Hartree–Fock (HF) exchange.

Functionals with higher HF exchange admixture, such as BHandHLYP (50% HF exchange) give too low elimination energies, whereas the gradient-corrected BP86, and in particular the local SVWN5 functional overestimate the elimination energies significantly.

The comparison of basis sets B, C, and D (Table 3 and Fig. 1) indicates a moderate basis-set dependence.

After inclusion of CP corrections, the intermediate basis C results are relatively close to those with the larger basis B (i.e., both provide good agreement with the CCSD(T) data when using a hybrid functional like B3LYP).

However, the CP corrections obtained for basis C are considerably larger than for basis B with HgF4 and HgH4, making basis B the overall more reliable method of choice, provided the size of system to be studied allows the use of the larger basis.

In particular, CP corrections are not easily applicable in all energy calculations (e.g. for intramolecular processes), and thus a basis with an inherently smaller BSSE may be preferable.

Finally, the basis D results for HgF4 and HgCl4 (including a segmented valence basis for Hg, cf. Table 1) exhibit large BSSE and still appreciable deviations from the basis B results after CP correction.

Zero-point energy (ZPE) corrections calculated at the B3LYP/C level are listed in Table 4.

The ZPE corrections lower the elimination energy almost negligibly for the fluoride and chloride and moderately so for the hydride.

Atomization energies

Table 5 provides computed atomization energies (AE) for the X2, HgX2, and HgX4 systems (with CP corrections but without ZPE corrections).

These data allow us to further analyze the contributions from individual species to the elimination reactions in Table 3.

Triple excitations in the coupled cluster calculations increase the AE notably for HgF4, HgCl4, and for F2, consistent with nonnegligible non-dynamical correlation effects (see above).

In the same three cases, the MP2 calculations overestimate the AE appreciably.

In the other cases, the MP2 results are either somewhat above or slightly below the CCSD(T) data.

In the case of F2, Cl2, and H2, experimental data are available for comparison (see footnote a to Table 5).

The CCSD(T)/A results deviate only by a few kJ mol−1 from experiment, indicating an approximate convergence of the halogen (hydrogen) basis sets and of the correlation level in this case.

Any remaining errors in the CCSD(T)/A results for the elimination reactions (Table 3) will thus mostly be due to the description of the metal complexes themselves, probably in particular that of the HgIV species.

While any of the density functionals tested should perform reasonably well for the AE of H2 (the MPW1PW91/B results appear a bit low), the more complicated electronic structure of the dihalogens is reflected in larger variations between the functionals.

Again, hybrid functionals with ca. 20% Hartree–Fock exchange (B1LYP, B3LYP, MPW1PW91) tend to perform best (with an underestimate of ca. 10–30 kJ mol−1), whereas the BHandHLYP functional underestimates the AE of both F2 and Cl2 appreciably, and the gradient-corrected BP86 overestimates the AE for F2 (the local SVWN5 functional overestimates both AEs appreciably).

Similar behavior of the functionals is seen for the tetrahalide complexes: again B1LYP, B3LYP, and MPW1PW91 appear to perform best, BHandHLYP underbinds compared to the CCSD(T)/A benchmark results, whereas BP86 and particularly the local SVWN5 overbind appreciably.

In these cases, deviations for the tetrahalides are much larger than for the dihalides.

This translates into appreciable errors in the energies of the elimination reactions (cf. Table 3).

While the B3LYP/B AE are too high by ca. 70 and 60 kJ mol−1 for HgF4 and HgF2, respectively, these errors compensate largely for the elimination reaction.

Similar comparisons apply to the B1LYP and MPW91PW91 functionals.

In consequence, these types of hybrid functionals (with basis B) reproduce most reliably the CCSD(T)/A thermochemistry.

Transition states for elimination reactions

We have also attempted to calculate the transition states and activation barriers for the concerted elimination of X2 (Table 6).

Full structure optimization at the CCSD(T)/A level exceeded the available computational resources.

In the case of the post-HF methods, we were thus restricted to single-point energy calculations at various DFT-optimized structures.

Full optimizations were, however, attempted for all density functionals.

The transition states located are structurally similar to that computed for the HgH4 case by Pyykkö et al.,8 a planar arrangement with C2v symmetry (see supporting information).

For HgF4, the computed activation barriers appear to be very large and vary over a wide range (Table 6).

Already the variation of the structure with different functionals changes the CCSD(T)/A barrier over a range of more than 100 kJ mol−1.

The DFT barriers are only about half of the CCSD(T) results but still appear unrealistically large relative to the average Hg–F binding energies deducible from the atomization energies in Table 5.

Matters are less dramatic for HgCl4 and HgH4, where the computed barriers range from ca. 60 to 80 kJ mol−1, and from ca. 40 to 50 kJ mol−1, respectively.

In the latter two cases, the DFT results are not very far from the best CCSD(T) values.

Closer inspection of the electronic structure at the F2-elimination transition state for HgF4 suggests very small HOMO-LUMO gaps already for the hybrid functionals (ca. 1–1.7 eV for B3LYP, B1LYP, and MPW1PW91 and ca. 3–3.5 eV for BHandHLYP).

With gradient-corrected and local functionals, no electronically stable Kohn–Sham wavefunction could be obtained.

This suggests appreciable multi-reference character for the transition state, and both approximate DFT and single-reference coupled-cluster theory appear problematic.

Matters are not much better for the HgCl4 case (although here the BP86 and SVWN5 calculations afford small gaps of ca. 0.1–0.2 eV), and it is presently unclear why the computed barriers are less level dependent.

In contrast, the HgH4 case exhibits appreciable HOMO-LUMO gaps at any of the levels employed (ca. 4.5 eV with BP86 and SVWN5, ca. 6 eV with the “regular” hybrid functionals, and ca. 8 eV with BHandHLYP).

It appears that the simpler electronic structure of the HgH4 transition state, and possibly the relatively large Hg–H covalency, make this system an easier case for single-reference methods (T1 diagnostics at CCSD level in this case are only ca. 0.015, compared to values around 0.04–0.06 for HgF4 and HgCl4).

In particular, we think that repulsive effects between the nonbonding electron pairs of the halogen ligands and the 5p semi-core shell on mercury may be responsible29 for the generally larger non-dynamical correlation effects in the tetrahalides compared to the tetrahydride.

The present results for the HgH4 system agree well with the previous study by Pyykkö et al8.


This validation study of various density functionals and basis sets against accurate benchmark CCSD(T) results for structures and energetics of small HgIV complexes provides a basis for our ongoing studies10 on larger target systems of potential interest for experimental studies.

While relatively reliable structures of minima (except for HgCl4) may already be obtained with the local SVWN5 or the hybrid BHandHLYP functionals (due to error compensation), the energetics are better described by hybrid functionals like B1LYP, B3LYP, and MPW1PW91, that incorporate ca. 20% Hartree–Fock exchange.

The choice of basis sets will depend on the size of system to be studied.

Optimizations may employ the moderate-sized, economical basis C, whereas accurate DFT energy calculations may require the larger basis B that exhibits considerably lower BSSE.

A search for transition states and activation barriers for X2-elimination from HgX4 (X = F, Cl, H) indicated considerably larger multi-reference character for X = F, Cl than for the previously studied HgH4 system, possibly due to the presence of nonbonding electron pairs at the halogens.29

At least for X = F, no reliable activation barriers could thus be computed with the available methods.

Unfortunately, multi-configurational approaches would currently also be prohibitively expensive in this case, due to the large active orbital space that would be required.

However, based on the comparison with the other two cases, we expect an appreciable activation barrier for concerted F2-elimination, in view of the multi-reference character of the transition state probably the largest of the three systems studied.

In any case, the calculations confirm clearly the previously noted exothermic character of HgF4 as a gas phase species, in contrast to HgH4 and HgCl4.