# Mössbauer spectroscopy for heavy elements: a relativistic benchmark study of mercury

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DOI: 10.1007/s00214-011-0911-2

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- Knecht, S., Fux, S., van Meer, R. et al. Theor Chem Acc (2011) 129: 631. doi:10.1007/s00214-011-0911-2

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## Abstract

The electrostatic contribution to the Mössbauer isomer shift of mercury for the series HgF_{n} (*n* = 1, 2, 4) with respect to the neutral atom has been investigated in the framework of four- and two-component relativistic theory. Replacing the integration of the electron density over the nuclear volume by the contact density (that is, the electron density at the nucleus) leads to a 10% overestimation of the isomer shift. The systematic nature of this error suggests that it can be incorporated into a correction factor, thus justifying the use of the contact density for the calculation of the Mössbauer isomer shift. The performance of a large selection of density functionals for the calculation of contact densities has been assessed by comparing with finite-field four-component relativistic coupled-cluster with single and double and perturbative triple excitations [CCSD(T)] calculations. For the absolute contact density of the mercury atom, the Density Functional Theory (DFT) calculations are in error by about 0.5%, a result that must be judged against the observation that the change in contact density along the series HgF_{n} (*n* = 1, 2, 4), relevant for the isomer shift, is on the order of 50 ppm with respect to absolute densities. Contrary to previous studies of the ^{57}Fe isomer shift (F Neese, Inorg Chim Acta 332:181, 2002), for mercury, DFT is not able to reproduce the trends in the isomer shift provided by reference data, in our case CCSD(T) calculations, notably the non-monotonous decrease in the contact density along the series HgF_{n} (*n* = 1, 2, 4). Projection analysis shows the expected reduction of the 6*s*_{1/2} population at the mercury center with an increasing number of ligands, but also brings into light an opposing effect, namely the increasing polarization of the 6*s*_{1/2} orbital due to increasing effective charge of the mercury atom, which explains the non-monotonous behavior of the contact density along the series. The same analysis shows increasing covalent contributions to bonding along the series with the effective charge of the mercury atom reaching a maximum of around +2 for HgF_{4} at the DFT level, far from the formal charge +4 suggested by the oxidation state of this recently observed species. Whereas the geometries for the linear HgF_{2} and square-planar HgF_{4} molecules were taken from previous computational studies, we optimized the equilibrium distance of HgF at the four-component Fock-space CCSD/aug-cc-pVQZ level, giving spectroscopic constants *r*_{e} = 2.007 Å and ω_{e} = 513.5 cm^{−1}.

### Keyword

Mössbauer spectroscopyRelativistic quantum chemistryDensity functional theoryCoupled clusterContact densityMercury compoundsPicture change effects## 1 Introduction

Molecular properties of heavy element compounds are known to be affected by relativistic effects [1–3], especially if they are probed at a heavy atomic nucleus. One such property is the contact density, i.e., the electron (number) density at the center of an atomic nucleus. The contact density can be related to the chemical isomer shift [4–9] that is observed in Mössbauer spectra if the frequency of the γ-radiation absorbed by a nucleus (absorber nucleus) in a solid is not equal to the one emitted by a source nucleus of the same element. Rephrased in terms of relative energy, the energy difference between ground and excited states of the absorbing compound is different from that of the emitting compound. This relative energy shift is usually expressed in terms of the speed of the source relative to the absorber which creates the Doppler shift necessary to bring the emitter and absorber into resonance.

*nuclear*charge distributions of different extension, can be neglected. Then, it is assumed that the electronic charge distribution is approximately constant over the size of the atomic nucleus and hence can be described by a single value, namely by the contact density. For a homogeneously charged, spherical nucleus, one then obtains a simplified expression for the energy difference \(\Updelta E_{\rm IS}\) due to the isomer shift (IS),

*n*(0) denotes the contact electron density at the atomic nucleus of absorber

*A*and source

*S*, while \(R^\star\) and

*R*

_{0}are the nuclear radii in the excited and ground states, respectively.

*Z*is the nuclear charge.

The usual point charge approximation of the nuclear density in the determination of the electronic density should be carefully examined in the calculation of contact densities. In non-relativistic theory, this approximation results in a cusp of the electron density at the position of the nuclei [10]. In relativistic theory, the electron density shows a weak (integrable) singularity at the nucleus. This limiting short-range behavior is, however, an artifact of the simple point charge model and is radically different for the more physical model of an extended nucleus. In the relativistic case, it is already common to introduce an extended-nucleus model [11–14] because this facilitates the use of Gaussian type orbitals (GTOs). GTOs have zero slope at the nucleus, which is consistent with the exact solutions for extended-nuclear models [15–18] in both non-relativistic and relativistic theory. This feature thus makes GTOs a natural expansion set for relativistic orbitals for which one may rely on well-established basis sets augmented by steep functions (as e.g., demonstrated recently for contact densities at iron nuclei [18]). By employing an extended-nucleus model, one may also go beyond the contact density approximation and explicitly calculate the change in nucleus–electron interaction corresponding to the nuclear transition measured in Mössbauer spectroscopy. This approach has been pursued by Filatov and co-workers [19–23] who introduce a difference in radius between the ground- and excited state nucleus as a finite perturbation that can be used to numerically differentiate the electronic energy. An added advantage is that such a finite difference scheme also works for methods for which it is difficult to obtain the relaxed density directly.

A more pragmatic approach, that works well for light elements, is based on the fact that the errors in the density arising from the approximation of the nucleus by a point charge model and by calculating the wave function in a non-relativistic framework, are almost exclusively atomic in nature. This feature makes it possible to derive corrections that scale non-relativistic contact densities, calculated with a GTO basis set and a point nucleus model, towards the true values. Such a useful pragmatic approach to the isomer shift has been suggested by Neese [24, 25] who employed non-relativistic density functional theory to capture valence-shell effects on the contact density, while the difficult-to-capture atomic contributions were absorbed in fit parameters upon parametrization against experimental results.

Although tin and especially iron nuclei are the most prominent Mössbauer nuclei in practice, we focus here on mercury, which is also Mössbauer active but for which there are much less experimental data available [26–31] (see Refs. [32–34] for additional spectroscopic properties of Hg in mercury compounds). Interestingly though, γ-ray fluorescence was first detected in liquid mercury [35, 36]. Mercury compounds feature two advantages: (1) they are prone to relativistic effects so that the reliability of different relativistic Hamiltonians can be thoroughly assessed and (2) they are usually closed-shell molecules so that highly accurate single-reference electron correlation methods can be employed, which allow us to assess different electronic structure methods for such closed-shell species.

In this work, we therefore focus on relativistic calculations of the mercury contact density in the atom as well as in fluorides of mercury. Among the mercury fluorides, which have been studied for various practical reasons [37], the tetrafluoride HgF_{4} has in particular attracted attention in recent years [38, 39] since its theoretical prediction in 1993 [40]. This compound was discovered by matrix spectroscopy [41, 42], but has not yet been obtained in macroscopic yields. However, once this would have been accomplished, a solid sample of this material may be subjected to γ-radiation in a Mössbauer experiment in order to determine the oxidation state of Hg in HgF_{4}. This can then clarify a theoretical prediction by Pyykkö et al. [43] who assigned an extraordinarily high oxidation state of +IV to Hg (considering the fact that the common oxidation state of Hg in chemical compounds is only +II).

This paper is organized as follows: In the Theoretical section, we will first examine the validity of the contact density approximation in relativistic finite nucleus calculations, then discuss the calculation of the contact density in the four-component relativistic formulation, and finally the interpretation of calculated densities in terms of atomic contributions via a projection analysis. We then provide Computational Details about the calculations before embarking on the results in Sect. 4.

## 2 Theoretical considerations

### 2.1 Validity of the contact density approximation

_{n}as

*R*is some model-specific radial size parameter that characterizes the extension of ρ

_{n}[11, 13, 48]. The electrostatic electron–nucleus interaction is given by

*R*. Following Filatov [19, 22], we calculate the associated shift in frequency of the emitted or absorbed photon as an energy derivative

*R*

_{0}is the nuclear ground state radius and \(\Updelta R = R^{\star} - R_0\) the change in radius upon excitation, assumed to be much smaller than

*R*

_{0}. The first of the approximations made to derive the contact density expression is to assume that only the first term of Eq. 4 is relevant for the calculation of energy differences between emitting and absorbing nuclei, that is the change of electron density due to the nuclear excitation is not important in the calculation of the shift. This approximation is discussed by Fricke and Waber [49] who estimated the effect of the other term to be of the order of 0.2 to 0.4% of the isomer shift. We can thus take the difference derivative of the nuclear potential with respect to nuclear radius as a perturbing operator and calculate the energy differences for nuclei in chemically different environments via first-order perturbation theory.

*R*

_{G}can be related to the radius parameter

*R*of the homogeneous sphere model by comparing radial second momenta [11, 13]

*R*

_{G}, we obtain a particularly simple expression

_{e}(0). The full sequence of approximations then reads

_{e}(0) represents a maximum of the charge density. Assuming the intermediate approximations to be minor, we can assess the effect of this approximation by extracting the effective density \(\bar\rho_e\) from the derivative of the energy with respect to nuclear radius and then compare with the calculated contact density ρ

_{e}(0). Results of this procedure for the Gaussian model will be reported in Sect. 4.2

### 2.2 The contact density for two- and four-component wave functions

*U*of the parent four-component Dirac Hamiltonian

*H*

^{4c}and then retaining the block describing the positive energy spectrum only, that is, the two-component Hamiltonian is given by

### 2.3 Projection analysis

*A*labels the individual atoms. In order to avoid overcompleteness and to obtain a meaningful analysis, the expansion is limited to the ground state-occupied orbitals and possibly some virtual orbitals of each center. Whatever part of the molecular orbital \(\vert \Uppsi_i \rangle\) which is not spanned by the reference orbitals is denoted the polarization contribution \(\vert \Uppsi_i^{\rm pol} \rangle \) and is by construction orthogonal to those. Inserting the expansion into the SCF expectation value expression for a general operator \(\hat{\Upomega}\) leads to a series of terms

*intra-atomic contribution*involves only orbitals from a single center. It can be further subdivided into the principal moments involving only diagonal atomic matrix elements \(\langle \Uppsi_i^A \vert \hat{\Upomega} \vert \Uppsi_j^A \rangle\), (i = j) and hybridization contributions for which

*i*≠

*j*. The principal moments contribute to the atomic expectation value, whereas the hybridization contribution arises from mixing of atomic orbitals in the molecular field (2) The

*interatomic contribution*involves two centers, whereas (3) the

*polarization contribution*involves the parts of the molecular orbitals not spanned by the selected atomic orbitals. The usefulness of the analysis is generally reduced if the polarization contribution is important, a feature that can be remedied by including more reference orbitals. However, in a comparative study as the present one, the polarization contribution does carry information, as will be seen in Sect. 4.4. Selecting \(\hat{\Upomega}=1\) allows the formulation of a population analysis similar to the Mulliken one, but without the undesirable strong basis set dependence of the latter [58].

*r*and (θ, ϕ) refer to radial and angular integration, respectively. From the angular integration, we find that non-zero contributions require that \(\kappa_p=\kappa_q\) and \(m_p=m_q\). The intra-atomic contribution to the relative contact density can therefore be written

*r*solutions of the Dirac equation for hydrogenic atoms [14, 15, 59]. We expand the large- and small-component radial functions as

*s*

_{1/2}orbitals

*R*

^{L}(0) =

*p*

_{0}and

*R*

^{S}(0) = 0, where

*p*

_{0}is determined from normalization. For κ > 0, we have \(p_0=q_1=0\), which implies that for

*p*

_{1/2}orbitals

*R*

^{L}(0) = 0 and

*R*

^{S}(0) =

*q*

_{0}, where

*q*

_{0}is likewise determined from normalization. This result can be compared to the non-relativistic case where the radial function is expanded as

*a*

_{1}is zero for an extended nucleus, thus removing the cusp). In the non-relativistic case, we accordingly see that only

*s*-orbitals have non-zero values at the nucleus, whereas in the relativistic case both

*s*

_{1/2}and

*p*

_{1/2}orbitals are non-zero at the origin.

## 3 Computational details

### 3.1 Effective densities

_{e}(0), as in Eq. 12, we have calculated the derivative of the energy with respect to the nuclear radius, using a Gaussian model for the nuclear charge distribution. This derivative was formulated as the derivative of the energy with respect to the radial

*rms*value

*s*

_{1/2}- and

*p*

_{1/2}-type orbitals contribute to the contact density, through their large and small components, respectively. However, other orbitals may contribute to the effective density, as will be shown in Sect. 4.2

### 3.2 Electron correlation methods

Due to the dominant influence of the nuclear potential, the absolute magnitude of the density close to the Hg nucleus is already well described at the Hartree–Fock level. The isomer shift does, however, depend on the changes of density due to the chemical environment and is thus much more sensitive to the valence contributions. An accurate description of the valence electronic structure, which yields the most sensitive contributions to the contact density, requires to properly account for electron–electron correlation [23]. For this purpose, we performed Møller–Plesset second-order perturbation theory (MP2) as well as coupled-cluster (CC) calculations with a full iterative treatment of single and double excitations (CCSD) and including perturbative corrections for triple excitations (CCSD(T)) [60, 61]. Since all atomic and molecular compounds considered in the present study exhibit a closed-shell ground state (except for the monoradical HgF), the use of a single-determinant reference *ansatz* seems thus well justified. This belief is further corroborated by the *T*_{1} diagnostics [62] which indicates the importance of single excitations in the CCSD approximation. For all closed-shell systems, the *T*_{1} diagnostics was less than 0.023. The molecular radical HgF, on the other hand, has a \(^2\Upsigma^+_{{{1}\over {2}}}\) ground state which in general calls for a multi-reference treatment. We therefore applied both the Fock-space coupled-cluster singles and doubles [63] (FS-CCSD) method, a genuine multi-reference CC method, and an open-shell CCSD(T) *ansatz* for the calculation of the ground state contact density. As the relative deviations in the electron densities between the two approaches was less than 10 ppm, we will in the following only refer to the open-shell CCSD(T) data.

^{corr}(0) obtained from a numerical differentiation of the MP2/CCSD/CCSD(T) electron correlation energy E

_{corr}

_{corr}showed a nearly linear dependence on the perturbation strength λ which allowed us to use for the numerical differentiation the well-known central-difference method [69]. Sufficiently high accuracy could then already be achieved with a seven-point stencil

_{corr}

^{n}denotes the electron correlation energy calculated at a perturbation of \(n\cdot{\lambda}^{opt}\).

We also carried out four-component density functional theory (DFT) calculations based on the Dirac–Coulomb (DC) Hamiltonian DFT calculations in order to investigate their performance in the evaluation of contact densities in comparison with wave function methods (WFM) such as HF and CC. For this purpose, an ample set of exchange-correlation functionals has been used, namely LDA (VWN5) [70, 71], BP86 [72, 73], BLYP [72, 74, 75], B3LYP [72, 76, 77], CAMB3LYP [78], PBE [79], and PBE0 [80]. Moreover, it will allow for a detailed examination of internal consistency within the DFT contact densities owing to the approximate nature of the functionals.

As pointed out by Neese [24], the numerical integration scheme must be carefully calibrated to reproduce the electron density in the core region. We therefore employed an ultrafine grid (using the .ULTRAFINE option in dirac) in order to ensure converged results in the exchange-correlation evaluation. The option implies the use of the basis-dependent radial integration scheme of Lindh et al. [81] with a convergence threshold of \(2\times 10^{-15}\) and a 2D Lebedev angular integration correct to angular momentum *L* = 64. The grid size given as (radial points, angular points) thus reads as (237, 54) and (163, 142) for the Hg and F atomic center, respectively, when using the extended triple-\(\zeta\) basis sets (see Sect. 3.4). The convergence of the contact density evaluation with respect to the grid size was validated by means of single-point calculations reverting to the standard grid definition in dirac which yielded identical results within numerical accuracy.

### 3.3 Hamiltonian operator

relativistic densities using the eXact two-Component (X2C) Hamiltonian [82];

four-component spin-orbit free [83, 84] (scalar-relativistic) densities;

one-component scalar-relativistic densities employing the Douglas–Kroll–Hess (DKH) Hamiltonian [85–91];

non-relativistic densities employing the Lévy-Leblond [92] Hamiltonian.

*n*and

*m*refer to the order of DKH transformation in the wave function and density operator, respectively, that is

For reasons of computational efficiency, the molecular mean-field approximation [98] to the four-component DC Hamiltonian was applied in all molecular wave-function-based calculations. In this scheme, the required set of two-electron integrals for the post-HF correlation step are computed in molecular orbital basis by neglecting all integrals of the AO basis that involve the small component. Nonetheless, the approximate integrals are combined with the exact orbital energies available from the HF solutions. The resulting Hamiltonian is denoted \(^4\hbox{DC}^{\ast\ast}\) in Ref. [98]. The relative deviation in the total contact density from the exact ^{4}DC-CCSD(T), with the full set of two-electron integrals, was in all cases tested less than 0.1 ppm, and we therefore consider this as a reliable approach. For the purpose of investigating a potential importance of spin-other-orbit contributions for the evaluation of relative contact densities and reaction energies, we additionally performed molecular mean-field calculations based on the Dirac-Coulomb-Gaunt Hamiltonian. The latter Hamiltonian takes particularly into account both the charge-charge (Coulomb term) and the current–current instantaneous interactions between the electrons in the chosen reference frame [99].

Furthermore, as already discussed above, we have used a finite size Gaussian nuclear model with exponents taken from the reference tables provided by Visscher and Dyall [100]. This approach avoids all singularities of the wave function that arise in point nucleus two- and four-component calculations.

### 3.4 Basis set considerations

Owing to the nature of the contact density as a core property, all calculations were carried out using atom-centered basis sets in their fully uncontracted form. In the relativistic four- and two-component as well as spin-orbit free and non-relativistic case, large-component scalar, Gaussian type orbitals (GTO) were employed. The small-component basis functions, if appropriate, were then generated by the restricted kinetic balance condition [95]. In the calculations performed with dirac triple-\(\zeta\) (\(\mathsf{TZ}\)) and quadruple-\(\zeta\) (\(\mathsf{QZ}\)), basis sets of Dyall [101, 102] were used for Hg. The starting large component \(\mathsf{TZ}\,(\mathsf{QZ})\,30s24p15d10f\) (34*s*30*p*19*d*12*f*) SCF set was augmented by 1*f*4*g*1*h* (1*f*7*g*4*h*1*i*) diffuse functions. Both \(\mathsf{TZ}\) and \(\mathsf{QZ}\) basis sets thus contain the primitives recommended for valence dipole polarization and valence correlation as well as for core-valence correlation. As fluorine basis the augmented correlation-consistent valence triple-\(\zeta\) aug-cc-pVTZ (ATZ) and quadruple-\(\zeta\) aug-cc-pVQZ (AQZ), basis sets [103] were chosen. The basis for the F atom was being left in uncontracted form to allow for valence polarization of the electron density around the neighboring Hg nucleus. In the scalar-relativistic DKH calculations, both the triple-\(\zeta\) basis set of Dyall (see results in Table 5 as well as in Table C of the supporting information) and additionally also all-electron atomic natural orbital (ANO) sets for Hg and F [104–106] (see results in Table D in the supporting information) were employed in a completely decontracted manner.

To particularly ensure basis set saturation at the Hg nucleus with respect to an accurate computation of the contact density may nevertheless require an augmentation with tight *s* and *p* functions [18]. Our calibration studies at the Hartree–Fock level by means of the numerical atomic grasp code [107] revealed a sufficient convergence by supplementing the Dyall \(\mathsf{TZ\,(QZ)}\) basis set in an even-tempered fashion with two more tight \(\{\zeta = 864721150.0, 230133640.0\}\) (\(\{\zeta = 864477130.0, 230139440.0\}\)) *s* functions and one tight \(\{\zeta = 130716620.0\}\) (\(\{\zeta = 194566990.0\}\)) *p* function (see Table A in the supporting information for more details). The final large-component basis thus comprises a [32*s*25*p*15*d*11*f*4*g*1*h*] ([36*s*31*p*19*d*13*f*7*g*4*h*1*i*]) set for Hg (denoted as \(\mathsf{TZ+2s1p (QZ+2s1p)}\) for the following). Similarly, the uncontracted ANO basis set was augmented with two tight \(\{\zeta = 993262470.3, 227944396.6\}\,s\) functions and one tight \(\{\zeta = 91654636.2\}\,p\) function yielding a [27*s*23*p*16*d*12*f*4*g*2*h*] set of primitives (denoted as \(\mathsf{ANO+2s1p}\)).

### 3.5 Choice of active space

In the wave-function-based correlation methods, we chose two active spaces for the Hg atom and the mercury fluoride compounds HgF_{n} (*n* = 1, 2, 4). The first (“[v]”) space comprises the valence 5*d*6*s* shell of mercury and the F 2*s*2*p* valence electrons. The second (“[cv]”) space is a superset of the [v] space where the core 5*s*4*f*5*p* shells of Hg are additionally considered for correlation. The size of the virtual space included in [v] and [cv] was identical and consistently adapted for either basis set combinations \(\mathsf{TZ+2s1p}\,\hbox{(Hg)} + \hbox{ATZ (F)}\) and \(\mathsf{QZ+2s1p}\,\hbox{(Hg)} + \hbox{AQZ (F)}\), respectively. In both cases, the virtual space limit for the [v] and [cv] spaces was tailored to contain all recommended core- and valence correlation as well as valence dipole polarization functions. This corresponds for the TZ set to a threshold of 95 hartree, whereas for the QZ set the cutoff is fixed at 107 hartree. For the Hg atom only, a third active space (“[all]”) is furthermore taken into account. Here, the correlation treatment spans the full space of occupied and virtual orbitals.

## 4 Results and discussion

### 4.1 Molecular structures and energetics

_{n}(

*n*= 1, 2, 4) for which contact densities have been computed. The geometries for the linear HgF

_{2}and square-planar HgF

_{4}molecules were taken from Ref. [41]. In this combined experimental and theoretical work, Wang and co-workers optimized the Hg-F bond length at the CCSD(T) level of theory using a small-core effective core-potential (ECP) combined with an augmented valence-basis set for Hg and an aug-cc-pVQZ one-particle basis for fluorine. They obtained Hg-F distances of 1.914 Å and 1.885 Å, respectively, for the di- and tetrafluoride mercury compound. Their values are in excellent agreement with the most recent two-component spin-orbit (SO) DFT and SO-CASPT2 data by Kim and co-workers [108] who report internuclear Hg-F distances of 1.912 Å and 1.884 Å (SO-PBE0) (1.886 Å; SO-CASPT2) for the two species. Moreover, the structural Hg-F parameters used in this work for both complexes fall within the range of data of 1.91–1.94 Å [37, 39, 108–111] and 1.88–1.89 Å [39, 108, 109], respectively, which is known from literature. The case is, however, different for HgF. Depending on the Hamiltonian, method and quality of basis sets applied the theoretically derived equilibrium bond distances r

_{e}in the monoradical HgF comprise a spread of as large as 2.00–2.17 Å [37, 108, 110, 111] (see also Table 1 for a selection of data). Since furthermore no experimental data for r

_{e}is available, this encouraged us to optimize the Hg-F bond distance at the four-component FS-CCSD level, thus providing a new theoretical reference value. In these benchmark calculations, we used the [cv] correlation space combined with the extensive \(\mathsf{QZ+2s1p}\) basis for Hg and the fluorine AQZ basis set in order to minimize the basis set superposition error. A description of the ground state of the radical diatom HgF by means of the FS-CCSD method requires a closed-shell starting electronic structure. We took the monocation as a point of departure and proceeded from the ground state of the ionic compound to the (0

*h*,1

*p*) Fock-space sector, thus arriving at the ground state of HgF:

Spectroscopic constants (r_{e} and ω_{e}) for the ground state of the radical \(^{202}\hbox{Hg}^{19}\hbox{F}\) calculated at the four-component Fock-space CCSD level correlating 40 electrons (Hg 5*s*5*p*4*f*5*d*, F 2*s*2*p*)

Molecule | Method | Basis sets (Hg/F) |
| ω | |
---|---|---|---|---|---|

HgF | 4c-FS-CCSD | \(\mathsf{TZ+2s1p}\) | ATZ | 2.012 | 509.2 |

HgF | 4c-FS-CCSD | \(\mathsf{QZ+2s1p}\) | AQZ | 2.007 | 513.5 |

HgF | CCSD(T) | ECP | ATZ | 2.028 | 480.6 |

HgF | SO-PBE0 | ECP | ATZ | 2.036 | 457.2 |

HgF | SO-M06-L | ECP | ATZ | 2.085 | 422.8 |

HgF | NESC/B3LYP | \(\mathsf{DZ}^{\rm e}\) | ATZ | 2.080 | – |

HgF | B3LYP | ECP | AQZ | 2.076 | 414.7 |

HgF | QCISD | ECP | 6-311G(2df,2dp) | 2.019 | 493.2 |

HgF | MP2 | ECP | 6-311+G* | 2.045 | 444.4 |

HgF | CCSD(T) | ECP | AQZ | 1.914 | – |

HgF | CCSD(T) | ECP | AQZ | 1.885 | – |

The active (0*h*,1*p*) sector space comprised the Hg 6*s* shell only. Trial studies at the \(\mathsf{TZ+2s1p}/\hbox{ATZ}\) basis set level with enlarged active spaces for the (0*h*,1*p*) sector did not reveal any significance of in particular the Hg 6*p* shell in terms of bonding participation. Moreover, using the same basis set combination, we also estimated the importance of core-valence electron correlation for the equilibrium bond length r_{e}. Extending the valence [v] to the core-valence [cv] correlation space yielded a bond length increase of 0.004 Å. The relative shift of \(-0.005\,\AA\) for \(\Updelta\)(QZ-TZ) in r_{e} (see also Table 1) is of same order of magnitude but of opposite sign. Our best theoretical estimate for r_{e} is thus 2.007 Å using the extended QZ basis sets and the [cv] electron correlation space, and all contact density calculations reported herein for the monoradical were carried out at this internuclear Hg-F distance. Looking at Table 1, it becomes obvious that our FS-CCSD value is located at the lower end of all existing theoretical predictions for r_{e}. Given the extensive basis sets, high level of correlation and \(a\,priori\) inclusion of spin-orbit coupling this benchmark nevertheless ought to be deemed close to the not yet measured equilibrium bond distance of HgF.

As can be seen from Eq. 34, the FS-CC scheme also allows to compute the ionization potential (IP of HgF as a by-product of the geometry optimization. We calculated the IP at the FS-CCSD[cv] correlation level using either the TZ or QZ basis sets. Our data of 232.0 kcal mol^{−1} and 233.2 kcal mol^{−1}, respectively, are in very good agreement with the IP of 235.3 kcal mol^{−1} reported by Cremer and co-workers [110] who derived their value from scalar-relativistic DFT calculations.

_{n}(

*n*= 1, 2, 4) reaction (Eq. 35) has received particular attention in earlier theoretical and lately also experimental works [37, 39–41, 108, 109]. The active interest originated from its importance in answering the question whether or not mercury is a genuine transition metal. Experimental studies in rare-gas matrices at low temperatures [41] recently confirmed the existence of HgF

_{4}which among the series of group 12 tetrafluorides

*M*F

_{n}(

*M*= Zn, Cd, Hg) exhibits an endothermic F

_{2}elimination [38–40, 109, 112] only. Table 2 summarizes in this context our results obtained from single-point energy calculations in a four-component framework at the above discussed reference geometries for HgF

_{4}and HgF

_{2}. Calculations on F

_{2}were carried out at the experimentally determined internuclear distance of \(\hbox{r}_e = 1.41193\,\AA\) [113]. Table 2 furthermore compiles a selection of reaction energies taken from previous scalar-relativistic and spin-orbit studies on the thermodynamical stability of HgF

_{4}. We provide estimations for zero-point vibrational energy corrections to our four-component data by means of scalar-relativistic calculations using the Gaussian09 program [114]. In these vibrational frequency calculations, a small-core effective core-potential (ECP) combined with an augmented valence-basis set for Hg and an aug-cc-pVTZ (aug-cc-pVQZ) basis [103] for F (see Ref. [109] for more details on the Hg ECP/basis set) was used.

Reaction energies (in kJ mol^{−1}) for the elimination reaction \(\hbox{HgF}_{4}\rightarrow\hbox{HgF}_{2}+ \hbox{F}_2\) in the gas phase calculated at different levels of theory within a four-component framework

Method | Basis set | Reaction energy |
---|---|---|

MP2 | \(\mathsf{TZ+2s1p}\) | +67 ( |

CCSD | \(\mathsf{TZ+2s1p}\) | + 1 ( |

CCSD(T) | \(\mathsf{TZ+2s1p}\) | +31 ( |

CCSD(T) | \(\mathsf{TZ+2s1p}\) | +31 ( |

CCSD(T)/DCG | \(\mathsf{TZ+2s1p}\) | +29 ( |

CCSD(T)/sfDC | \(\mathsf{TZ+2s1p}\) | +20 ( |

B3LYP/sfDC | \(\mathsf{TZ+2s1p}\) | +41 ( |

B3LYP | \(\mathsf{TZ+2s1p}\) | +52 ( |

\(\mathsf{QZ+2s1p}\) | +54 ( | |

B3LYP | \(\mathsf{TZ+2s1p}\) | +54 ( |

PBE0 | \(\mathsf{TZ+2s1p}\) | +60 ( |

\(\mathsf{QZ+2s1p}\) | +62 ( | |

CAMB3LYP | \(\mathsf{TZ+2s1p}\) | +52 ( |

\(\mathsf{QZ+2s1p}\) | +52 ( | |

Previous work | – | \(+36.3^{\rm b},\,+35.5^{\rm c},\, +9.5^{\rm c}\) |

+27.4 | ||

\(+41.0^{\rm e},\,+24.3^{\rm e}\) |

In accordance with earlier predictions, we find the tetrafluoride compound of mercury to be thermodynamically stable with respect to a spontaneous elimination of F_{2} in the gas phase at 0 Kelvin. Turning to Table 2, we find that our four-component DFT data suggest a thermodynamically stable HgF_{4} on the order of 50–60 kJ mol^{−1} varying with the chosen density functional. Summarizing our DFT results the F_{2} elimination reaction seems slightly less favorable by about 10–20 kJ mol^{−1} when compared with earlier scalar-relativistic and spin-orbit DFT predictions by Riedel et al. [39], Wang et al. [41] as well as Kim and co-workers [108]. The correlated WFMs results compiled in Table 2, on the other hand, clearly reveal that both post-HF methods, MP2 and CCSD, may not be suitable to claim predictive character concerning the thermochemistry of mercury fluoride compounds. Only the successive inclusion of pertubative triples (CCSD(T); third row in Table 2) in our four-component calculations yields a reaction energy of 31 kJ mol^{−1} which is in very good agreement with the scalar-relativistic CCSD(T) data of 27.4 and 24.3 kJ mol^{−1} reported by Riedel et al. [109] and Kim et al. [108], respectively.

In order to reveal a potential significance of spin-same-orbit and in particular spin-other-orbit (SOO) coupling with regard to our present thermo chemistry data, we performed additional single-point CCSD(T) calculations based on the spin-orbit free (sfDC) and molecular mean-field Dirac-Coulomb-Gaunt Hamiltonian (DCG). Regarding the CCSD(T)/DC reaction energy value of 31 kJ mol^{−1} as our reference point, it can be seen from Table 2 that a consideration of SOO contributions in the evaluation of the thermo stability of HgF_{4} leads to a slight correction of the CCSD(T)/DC value by 2 kJ mol^{−1}. A complete neglect of spin-orbit coupling contributions, on the other hand, gives rise to an underestimation by about −11 kJ mol^{−1}.

We conclude this paragraph by noting that the calculated reaction energies do not show any significant geometry dependence within computational error bars, neither with DFT nor with CCSD(T) (see for example third and fourth row in Table 2). For the purpose of comparison, we thus optimized the geometries of each reaction compound at the four-component DFT/B3LYP using the augmented \(\mathsf{TZ+2s1p}/\hbox{ATZ}\) basis sets for Hg and F, respectively.

### 4.2 Justification of the use of contact densities

_{1/2}and p

_{1/2}orbitals contribute to the contact density, while the effective density also has small contributions from the p

_{3/2}orbitals. These p

_{3/2}contributions to the effective density arise due the fact that these orbitals reach a significant value inside the nuclear volume. We also observe similar contributions from d

_{3/2}orbitals (not shown). For other types of orbitals, the contributions are significantly smaller and not discernible from numerical noise. More importantly, the s

_{1/2}and p

_{1/2}contributions to the contact density are significantly higher than for the effective density. Interestingly, though, the deviation is quite systematically 10%.

Atomic matrix elements \((\hbox{HF}/\mathsf{QZ+2s1p})\)

| Contact density | Effective density |
---|---|---|

1s1/2 | 1951311.50 | −194467.78 |

2s1/2 | 294993.24 | −29548.24 |

3s1/2 | 67814.71 | −6798.36 |

4s1/2 | 17035.79 | −1708.17 |

5s1/2 | 3265.26 | −327.42 |

6s1/2 | 276.32 | −27.71 |

2p1/2 | 21856.04 | −2107.28 |

2p3/2 | 0 | 2 × 0.51 |

3p1/2 | 5638.93 | −544.14 |

3p3/2 | 0 | 2 × 0.14 |

4p1/2 | 1398.44 | −134.96 |

4p3/2 | 0 | 2 × 0.03 |

5p1/2 | 237.17 | −22.89 |

5p3/2 | 0 | 2 × 0.01 |

Total | 2363827.39 | −235685.57 |

*shifts*of the mercury fluorides relative to the mercury atom for \(\hbox{HF}/\mathsf{QZ+2s1p}\) calculations. Also here, we see significant deviations, but systematically on the order of 10%. This feature suggests that a calibration approach similar to that of Neese [24, 25] can be employed, thus justifying the use of contact densities for the determination of Mössbauer isomer shifts for elements as heavy as mercury. However, we will demonstrate in the next section that a DFT scheme, such as proposed by Neese [24, 25], does not even qualitatively reproduce the trends observed in relative contact densities between the fluorides of mercury.

Contact and effective densities differences relative to the Hg atom \((\hbox{HF}/\mathsf{QZ+2s1p})\)

| Contact density | Effective density |
---|---|---|

HgF | −114.54 | −103.05 |

HgF | −127.85 | −115.01 |

HgF | −98.09 | −88.22 |

### 4.3 Calibration of contact densities

_{n}(

*n*= 1, 2, 4), we give values relative to the Hg atom, which seems most pertinent with respect to Mössbauer spectroscopy. The first entry is the four-component relativistic Hartree–Fock value. It is compared to the HF values obtained with neglect of the (

*SS*|

*SS*) class of two-electron Coulomb integrals in building the Fock matrix. As this approximation is usually accompanied with a Simple Coulombic Correction (SCC) [115] to correct for errors made in energy evaluation, we denoted this approach as HF[scc] in the table. It can be seen that this leads to errors of around 0.1 % for the absolute and relative contact densities with respect to the reference HF value and shows that this is probably a viable approximation for the calculation of contact densities even for heavy elements. At the SCF level, the SCC gives significant computational savings and is certainly recommended for the calculation of spectroscopic constants. However, in the present work, we have chosen not to invoke the SCC since our focus was on calibration and since SCC does not yield computational savings at the correlated WFM level, that is, once the four-index transformation to molecular orbital basis has been carried out. We next observe that spin-orbit free HF calculations (the Hamiltonian denoted sfDC in the table) give an error on the order of one percent for the absolute contact density of the Hg atom, but larger errors (up to 5%) for the relative values for the molecules. Such errors may not be acceptable, and it is therefore judicious to include spin-orbit coupling variationally in the calculation of contact densities of heavy atoms. Adding spin-other orbit (SOO) interaction through the Gaunt term (entry DCG) has only a minor effect on absolute and relative contact densities and can be ignored for these systems. What is certainly clear is that the use of a non-relativistic Hamiltonian is meaningless (Lévy-Leblond Hamiltoninan: LL in the table), as it gives orders of magnitude errors for both absolute and, most importantly, relative contact densities. We furthermore see no simple linear relation between relativistic and non-relativistic HF contact densities.

Calculated mercury contact densities

Method | Hamiltonian | Hg | HgF | HgF | HgF |
---|---|---|---|---|---|

HF | DC | 2363929.12 | −114.48 | −127.92 | −98.09 |

HF | DCG | 2354927.26 | −113.83 | −127.16 | −97.22 |

HF[scc] | DC | 2366645.73 | −114.51 | −127.83 | −97.65 |

HF | sfDC | 2342622.43 | −115.59 | −130.69 | −103.52 |

HF | LL | 361818.93 | −11.92 | −13.35 | −6.77 |

HF | X2C | 2358245.44 | −114.06 | −127.40 | −97.43 |

HF | X2C[pce] | 7579179.34 | −712.55 | −432.86 | −350.65 |

HF | DKH(10,8) | 2359971.35 | −107.65 | −131.33 | −103.81 |

CCSD(T)[all] | DC | 2364016.40 | |||

CCSD(T)[cv] | DC | 2363990.74 | −95.11 | −110.46 | −104.16 |

CCSD(T)[cv] | DCG | 2354988.44 | – | −109.70 | −103.20 |

CCSD(T)[v] | DC | 2363952.83 | −95.35 | −110.55 | −101.54 |

CCSD[all] | DC | 2364013.22 | |||

CCSD[cv] | DC | 2363988.25 | −94.33 | −116.59 | −105.16 |

CCSD[cv] | DCG | 2354985.97 | – | −115.79 | −104.18 |

CCSD[v] | DC | 2363952.22 | −94.39 | −115.93 | −102.18 |

MP2[all] | DC | 2364050.05 | |||

MP2[cv] | DC | 2364020.01 | −95.10 | −121.99 | −124.34 |

MP2[v] | DC | 2363963.15 | −96.78 | −118.88 | −115.16 |

LDA | DC | 2362802.35 | −74.38 | −99.03 | −113.69 |

LDA | DKH(10,8) | 2359359.64 | −74.33 | −103.75 | −120.77 |

BP86 | DC | 2373796.03 | −74.35 | −98.52 | −114.00 |

BLYP | DC | 2373687.69 | −72.82 | −95.88 | −111.48 |

B3LYP | DC | 2370863.15 | −85.86 | −105.87 | −113.54 |

CAMB3LYP | DC | 2371811.55 | −95.79 | −113.92 | −116.38 |

PBE | DC | 2372713.57 | −75.11 | −98.74 | −113.42 |

PBE0 | DC | 2370507.83 | −91.30 | −111.07 | −115.62 |

The exact two-Component relativistic Hamiltonian (X2C) [82] performs very well compared to the four-component Dirac Hamiltonian provided the density operator is correctly transformed. The use of the untransformed density operator (denoted XCE[pce] in the table) leads to catastrophic picture change errors. The absolute and relative contact densities as well as the values relative to the atomic density, obtained with the DKH(10,8) Hamiltonian, are somewhat larger than those obtained for the X2C Hamiltonian (which is equivalent to the infinite-order DKH Hamiltonian). Note that even orders of the property-operator transformation approach the infinite-order result from above, whereas odd orders approach it from below [18] (as can be seen in Table D of the supporting information). We also find that in the DKH calculations it is possible to reduce the total number of primitive basis functions by decontracting only the s- and p-shells instead of using fully decontracted basis sets. The deviation for the DKH(10,8) Hartree–Fock contact densities of the mercury species is smaller than 0.1%, when employing the partially decontracted ANO-RCC basis set instead of the fully decontracted one (see Table E of the supporting information).

We next turn to the study of correlation effects in the calculation of contact densities. Comparing our HF value for the Hg atom with the value obtained from CCSD(T) with all electrons correlated (denoted CCSD(T)[all] in the table), we find that the error is extremely small, on the order of 40 pm. While this value is probably not converged with respect to the one-particle basis, the use of uncontracted sets should give a reasonable estimate of the size of core correlation contributions. The small value is perhaps not entirely surprising since we are considering a closed-shell system dominated by a single Slater determinant. However, these errors are on the order of the *relative* contact densities. The correlation errors for these quantities are therefore significant, as seen for the other wave-function-based correlation methods listed in Table 5. For instance, with respect to CCSD(T) calculations including core-valence correlation (denoted CCSD(T)[cv] in the table), HF gives errors around 15% for HgF and HgF_{2}, whereas an error of −5% is observed for HgF_{4}. This clearly shows that electron correlation is mandatory for the calculation of relative contact densities. The effect of core-valence correlation can be seen by comparing our CCSD(T)[cv] results with CCSD(T) values obtained obtained with valence correlation only (denoted CCSD(T)[v] in the table). For HgF and HgF_{4}, we observe errors on the order of a few percent, whereas the error is quite small for HgF_{2}. The effect of the inclusion of perturbative triples can be assessed by comparing CCSD(T)[cv] and CCSD[cv] values in the table. It can be seen that omitting the perturbative triples may lead to errors close to 6% (HgF_{2}). This calls for a study of the effect of the inclusion of iterative triple and higher excitations [116] on calculated relative contact densities. Such calculations are expensive but possible with the recent interfacing [117] of the MRCC code [118] of Kállay and co-workers with dirac. Comparing CCSD(T)[cv] and MP2[cv], we see errors in the relative values on the order of 10–20% which shows that MP2 cannot be recommended for the calculation of relative contact densities for these systems.

When considering the performance of DFT in the calculation of contact densities, two points should be taken into consideration: (1) Most of today’s available approximate density functionals contain parameters that are fitted against experimental data such as atomization energies, electron affinities and reaction barrier heights (see for instance Ref. [119]). To our knowledge, contact densities and Mössbauer data do not enter such training data and so a good performance of semi-empirical density functionals is not guaranteed for these properties. (2) Relativistic DFT generally employs non-relativistic density functionals due to the limited availability of relativistic functionals [120–125]. However, computational studies [126–128] indicate that the effect of such relativistic corrections is negligible for spectroscopic constants. For core properties, here represented by the extreme case of the contact density, the situation is less clear. Turning now to Table 5, we find that all density functionals give errors on the order of 0.5% for the contact density of the mercury atom, with the exception of LDA for which the error is 0.05%, but still larger than the relative contact densities. Considering the performance of the different density functionals in the case of the scalar-relativistic DKH Hamiltonian, the same trends are observed as for the Dirac–Coulomb Hamiltonian, whereas the absolute values are smaller (see Table C in the supporting information). For the molecular shifts, the errors are significant, but less than observed with HF, with GGA functionals performing no better than LDA and hybrid functionals providing only a slight improvement.

### 4.4 Analysis of contact densities

In this section, we will investigate the contact density of the mercury atom in more detail using the projection analysis described in Sect. 2.3. Neese studied the composition of the contact density in terms of molecular orbitals [24]. We believe we can get more detailed information from a decomposition of this expectation value in terms of *atomic* orbitals. We have included the occupied orbitals of the constituent atoms in the analysis. For the mercury atom, we also considered the virtual 6*p* orbitals since their role in bonding has been discussed in the literature [111].

*s*

_{1/2}orbital, with some contributions (8.5%) from the fluorine 2

*p*-orbitals. The HOMO-1 orbital has some contributions from Hg 5

*d*which are, however, suppressed upon Pipek–Mezey localization [58, 129] of the closed-shell occupied orbitals, giving an orbital dominated by F 2

*p*(78.1%), 2

*s*(7.1%) and Hg 6

*s*

_{1/2}(12.0%). We find only minor contributions from Hg 6

*p*, contrary to the analysis of Schwerdtfeger et al. [111], and the bonding corresponds rather to the three-electron two-orbital model discussed by Cremer et al. [110]. These findings corroborate also our conclusions from the FS-CCSD calculations in Sect. 4.1 where the inclusion of the Hg 6

*p*in the model space did not lead to any measurable alteration in the optimized equilibrium structure. In HgF

_{2}and HgF

_{4}, we likewise observe a very limited contribution from the Hg 6

*p*orbitals. Along the HgF

_{n}(

*n*= 1, 2, 4) series, we observe an increasingly positive charge on the mercury atom, reaching +2.47 for HgF

_{4}, still far from a formal charge of +4 suggested by the oxidation state, indicating increasing covalent character of bonding in the series [112]. At the DFT level, the covalent contributions to bonding are further strengthened and the 6

*p*contributions increase slightly, as seen from Table 7. We also note that mercury atomic charges are systematically smaller at the DFT level than at the HF level.

Electron configuration and charge *Q* of mercury (gross populations) in the studied molecules obtained by projection analysis at the HF level

| HgF | HgF | HgF |
---|---|---|---|

5d | 9.93 | 9.74 | 8.98 |

6s | 1.08 | 0.68 | 0.48 |

6p | 0.11 | 0.07 | 0.07 |

Q | 0.88 | 1.51 | 2.47 |

Electron configuration and charge *Q* of mercury (gross populations) in the studied molecules obtained by projection analysis at the DFT level using the B3LYP(LDA) functional

| HgF | HgF | HgF |
---|---|---|---|

5d | 9.91 (9.90) | 9.71 (9.71) | 9.18 (9.22) |

6s | 1.29 (1.36) | 0.89 (0.97) | 0.56 (0.61) |

6p | 0.17 (0.19) | 0.15 (0.20) | 0.20 (0.28) |

Q | 0.63 (0.55) | 1.24 (1.12) | 2.07 (1.89) |

_{n}(

*n*= 1, 2, 4), reaching −72% for HgF

_{4}. This contribution can certainly not be ignored and we shall return to it in the following.

Projection analysis of Hg contact density, relative to the ground state atom, at the HF level

| HgF | HgF | HgF |
---|---|---|---|

Intra-atomic contribution | |||

Hg | −120.33 | −161.99 | −168.41 |

pm | −117.32 | −178.30 | −185.46 |

core | 10.05 | 23.85 | 49.14 |

6s | −127.37 | −202.15 | −234.60 |

6p | 0.06 | 0.01 | 0.01 |

hybrid | −3.01 | 16.32 | 17.06 |

F | 0.00 | 0.00 | 0.00 |

Interatomic contribution | −0.07 | −0.33 | −0.53 |

Polarization contribution | 5.93 | 34.39 | 70.84 |

Total | −114.48 | −127.92 | −98.09 |

The relative Hg contact density is indeed dominated by the intra-atomic contribution from mercury, as one could expect, and can therefore be rationalized in terms of changes in *s*_{1/2} and *p*_{1/2} populations from the free atom to the molecule, as discussed in Sect. 2.3. The non-zero diagonal atomic matrix elements over the density operator at the origin are given in Table 10. At the HF level, we observe an exponential decay \(a\exp(-bn)\) of these elements with respect to the main quantum number *n*. For *s*_{1/2} (*p*_{1/2}) orbitals the fitted parameters are \(a=1.08\times10^7(4.78\times10^5)\) and *b* = 1.69(1.50). We also note that for a given main quantum number *n* the ratio between a diagonal matrix element for the *s*_{1/2} over the *p*_{1/2} one is on the order of ten than rather than 1/*c*^{2}, which one would naively expect. However, one should keep in mind that the pointwise ratio between large and small components goes rather like v/*c* than 1/*c*. [130]. In Table 10 we also give the relative deviation with respect to HF of corresponding atomic matrix elements calculated with the various DFT functionals employed in this study. Interestingly, the relative deviations grow as one goes from core to valence orbitals. This suggests that the observed deviations are not to be attributed to a particular failure of the approximate DFT functionals in the description of the core orbitals.

Returning now to Table 8, we see that the intra-atomic Hg contribution is dominated by the principal moments, although the hybridization contribution grows in importance with an increasing number of fluorine ligands. For the principal moments, we find that the contribution from the 6*p*_{1/2} orbital is negligible for all systems, albeit non-zero, in accordance with the analysis in Sect. 2.3. The relative contact density for the three molecules is dominated by the contribution from the Hg 6*s*_{1/2} orbitals. For this contribution, we observe a monotonic decrease of the contact density relative to the atom which is directly related to the corresponding decrease in the 6*s*_{1/2} population shown in Table 6. We can therefore conclude that the non-monotonic trend observed for the total relative contact density at the HF and CC level is related to the increasing polarization of the mercury *s*_{1/2} orbitals in the molecule, indicated by the growing importance of the hybridization and polarization contributions, as the number of fluorine ligands increase. Indeed, for HgF_{4} we find that successively adding the mercury \(7s_{1/2},\,8s_{1/2}\) and 9*s*_{1/2} orbitals reduces the polarization contribution from 70.84 to 52.07, 10.98 and \(-2.89\,a_0^{-3}\), respectively.

*s*

_{1/2}population is systematically lower at the HF level than at the DFT level, thus leading to higher atomic charges with the former method. This is contrary to what one would expect from looking at the 6

*s*

_{1/2}orbital energy in the neutral atom, which at the TZ+2s1p level is \(-0.328\,\hbox{E}_h,\,-0.261\,\hbox{E}_h\) and −0.274

*E*

_{h}at the HF, LDA and B3LYP levels, respectively. DFT, however, yields a more compact 6

*s*

_{1/2}orbital (the radial

*rms*value is \(4.33\,a_0,\,4.06\,a_0\) and \(4.14\,a_0\) for HF, LDA and B3LYP, respectively) which in turn leads to a smaller polarizability [131] and larger contact density (cf. Table 10). The larger contact density leads to a crossover between the 6

*s*

_{1/2}contributions from HF and DFT when going from HgF

_{2}and HgF

_{4}, as seen in Fig. 1. We also note from Tables 6 and 7 that whereas the 5

*d*populations are rather similar at the HF and DFT level for HgF and HgF

_{2}, there is a more significant drop at the HF level when going to HgF

_{4}. Due to larger effective mercury atomic charge at the HF level, one would expect stronger polarization of the 6

*s*

_{1/2}orbital at the HF than the DFT level. However, this will be counterbalanced by the more compact nature and thus lower polarizability of the 6

*s*

_{1/2}orbital at the DFT level, as well as that there is less 6

*s*

_{1/2}population to polarize at the HF level, due to lower occupation. In practice, we find that for both HgF and HgF

_{2}the combined LDA contribution of polarization and hybridization is about \(18\,a_0^{-3}\) larger than for HF, whereas this difference basically vanishes for HgF

_{4}. A similar picture emerges when comparing HF and B3LYP. In conclusion, we see that there is a delicate balance between polarization and electron withdrawal, which in the case of DFT leads to a monotonous decrease of contact density along the series HgF

_{n}(

*n*= 1, 2, 4), but not for HF, the latter in agreement with our benchmark CC results.

Projection analysis of Hg contact density, relative to the ground state atom, at the DFT level using the B3LYP(LDA) functional

| HgF | HgF | HgF |
---|---|---|---|

Intra-atomic contribution | |||

Hg | −100.69 (−93.96) | −147.93 (−142.40) | −180.05 (−180.52) |

pm | −100.40 (−94.22) | −171.60 (−168.58) | −195.90 (−200.33) |

core | 11.99 (11.74) | 25.13 (23.59) | 58.51 (56.32) |

6s | −112.38 (−105.96) | −196.73 (−192.17) | −254.41 (−256.65) |

6p | 0.21 (0.29) | 0.11 (0.20) | 0.13 (0.25) |

hybrid | −0.29 (0.26) | 23.66 (26.18) | 15.85 (19.81) |

F | 0.00 | 0.00 | 0.00 |

Interatomic contribution | −0.11 (−0.11) | −0.41 (−0.41) | −0.65 (−0.67) |

Polarization contribution | 14.93 (19.69) | 42.46 (43.78) | 67.17 (67.50) |

Total | −85.86 (−74.38) | −105.88 (−99.03) | −113.54 (−113.69) |

Atomic matrix elements \((\hbox{HF}/\mathsf{TZ+2s1p})\)

| HF | LDA | BP86 | BLYP | B3LYP | CAMB3LYP | PBE | PBE0 |
---|---|---|---|---|---|---|---|---|

1s | 1951309.48 | −0.10 | 0.37 | 0.36 | 0.25 | 0.29 | 0.32 | 0.24 |

2s | 294992.89 | −0.05 | 0.44 | 0.43 | 0.30 | 0.35 | 0.38 | 0.29 |

3s | 67814.67 | −0.16 | 0.36 | 0.34 | 0.24 | 0.28 | 0.31 | 0.24 |

4s | 17035.91 | 1.94 | 2.17 | 2.20 | 1.74 | 1.79 | 2.11 | 1.58 |

5s | 3265.25 | 9.54 | 8.96 | 9.13 | 7.30 | 7.32 | 8.74 | 6.46 |

6s | 276.31 | 17.34 | 16.41 | 14.69 | 12.84 | 13.15 | 15.89 | 13.45 |

2p | 21934.06 | 1.03 | 1.53 | 1.52 | 1.18 | 1.23 | 1.45 | 1.08 |

3p | 5659.07 | 0.89 | 1.38 | 1.38 | 1.06 | 1.11 | 1.30 | 0.97 |

4p | 1403.46 | 2.88 | 3.04 | 3.09 | 2.45 | 2.51 | 2.96 | 2.20 |

5p | 238.01 | 10.68 | 9.94 | 10.14 | 8.10 | 8.02 | 9.66 | 7.11 |

*s*

_{1/2}-orbitals to the principal moments is positive. This seemingly counter-intuitive result is due to the small, but generally non-zero overlap between Hg core orbitals and fluorine atomic orbitals. One can easily show that diagonal elements corresponding to Hg core orbitals of the intra-atomic block of the density matrix appearing in Eq. 21 are

## 5 Conclusions and perspectives

The objective of this study has been to evaluate the performance of two- and four-component relativistic *ab initio* wave function methods and density functional theory approaches in the prediction of the Mössbauer isomer shift (IS) for compounds containing heavy elements using a contact density approach. In the recent past, various computational approaches [19, 23, 24] to the calculation of the IS in molecular systems containing lighter elements such as, e.g., ^{57}Fe, have been proposed and successfully applied. Entering the domain of heavy element chemistry, we therefore here focused on the series of mercury fluorides HgF_{n} (*n* = 1, 2, 4) where Hg exhibits two Mössbauer active isotopes, ^{199}Hg and ^{201}Hg.

The geometries for the di- and tetrafluoride compounds have been taken from a recently published scalar-relativistic study [41], whereas for the monoradical HgF we performed a geometry optimization at the relativistic four-component Fock-space CCSD level using augmented basis sets of quadruple-\(\zeta\) quality. Based on our results, we propose new theoretical reference values for the internuclear distance *r*_{e} and harmonic frequency ω_{e}, that are 2.007 Å and 513.5 cm^{−1}, respectively. In addition, we find an excellent agreement with existing theoretical predictions concerning the ionization potential of HgF. Besides, on the basis of our benchmark, we could safely rule out a previously discussed distinct contribution of the Hg 6*p* shell to the bonding pattern in the monofluoride radical [111].

Gross population analysis along the HgF_{n} series reveals an increasing positive charge on the Hg central atom yielding a maximum of +2.47 in HgF_{4} at the HF level. We attribute the significantly lower charge (the formal oxidation state of mercury is +IV) to a considerable proportion of covalent bonding in the square-planar tetrafluoride mercury complex. In addition, we estimate for the latter system the elimination reaction energy at the four-component coupled-cluster level regarding the decomposition of HgF_{4} into HgF_{2} and F_{2} in gas phase. Our results computed at both literature reference geometries and herein at the four-component DFT level optimized structures are in excellent agreement with previous scalar-relativistic studies based on the use of effective core-potentials [108, 109]. It is shown that the neglect of spin-same-orbit contributions may lead to a severe underestimation of the elimination reaction energy whereas spin-other-orbit contributions are of minor importance.

Our calibration study shows that our selection of density functionals gives errors in the absolute contact density for the neutral mercury atom on the order of 0.5% compared to the CCSD(T) reference data. This is not only significantly larger than HF, but about two orders of magnitude larger than the relative density shifts observed in the molecular species with respect to the neutral atom. Contrary to what was found for ^{57}Fe Mössbauer IS by Neese [24], DFT is not able to qualitatively reproduce the non-monotonic decrease of the contact density of the heavier atom mercury that we obtain from our benchmark CCSD(T) calculations and even at the HF level. Projection analysis shows the expected monotonic decrease of the 6*s*_{1/2} contribution to the relative contact density with an increasing number of binding fluorine atoms, but this contribution is opposed by the increasing contributions of polarization and hybridization. For HgF_{4}, these latter contributions are quite similar at the HF and DFT level, but the more compact 6*s*_{1/2} orbital provided by DFT gives a larger contact density which in turn assures a monotonic decrease of the total value.

In order to further investigate the predictive value of approximate DFT functionals in the relativistic domain, it will thus be worthwhile to extent the present study to other heavy element containing systems such as ^{197}Au compounds for which a growing number of experimental Mössbauer IS data is available (see Refs. [132–136] and references therein). Aiming at a comprehensive assessment, one could furthermore take into account complexes of Mössbauer active isotopes such as ^{127}I or ^{129,131}Xe for which it would be interesting to study the *p*_{1/2} contributions to the relative shift. Concerning the performance of the coupled-cluster wave function methods, on the other hand, we observe a partially distinctive effect of the inclusion of perturbative triples on the relative shift of the electron density. Although T_{1} diagnostics at the CCSD level indicate a justified use of a single-reference *ansatz*, we will look in more detail into the effect of including full triples as well as higher excitations on relative shifts of the electron density in a forthcoming publication thereby exploiting the recent interface [117] of the dirac10 program package [96] to a genuine and efficient multi-reference coupled-cluster code [118].

Finally, we have also assessed the application of the contact density approximation for calculating the Mössbauer IS which most notably assumes a constant electronic charge distribution over the finite-sized nucleus. Relying on a Gaussian model of the nuclear charge distribution, the contact density approach yields a systematic overestimation of 10% of the relative shifts along the HgF_{n} (*n* = 1, 2, 4) series when compared to the more sophisticated effective density approach (that is, the integration of the electron density over the nuclear volume). The systematic nature of the observed error, however, allows us to derive a correction factor which facilitates the calculation of Mössbauer ISs within the computationally straightforward contact density *ansatz*.

## Acknowledgments

We dedicate this paper to Pekka Pyykkö, a pioneer of relativistic quantum chemistry. With a unique combination of impressive chemical insight and judicious pragmatism, he has picked many of the bigger berries in the field, but graciously left some for others as well. We would like to thank one of the unknown referee’s for her/his elaborate report and comments which led to the discovery of an initial computational problem in the calculation of reaction energies and contact densities of the HgF_{4} compound. This issue has then been solved for the final version of this paper. S.K. thanks l’Université de Strasbourg (UDS) for a post-doctoral research grant and the supercomputer centers at ETH Zürich as well as UDS for ample computing time. M.R. and S.F. gratefully acknowledge financial support by ETH Zürich (Grant TH-26 07-3) and the Swiss national science foundation SNF (project no. 200020-132542/1). L.V. has been supported by NWO through the VICI programme.