Theoretical Chemistry Accounts

, Volume 129, Issue 3, pp 631–650

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

Authors

  • Stefan Knecht
    • Institute de Chimie de StrasbourgCNRS et Université de Strasbourg, Laboratoire de Chimie Quantique
  • Samuel Fux
    • ETH Zurich, Laboratorium für Physikalische Chemie
  • Robert van Meer
    • Amsterdam Center for Multiscale ModelingVU University Amsterdam
  • Lucas Visscher
    • Amsterdam Center for Multiscale ModelingVU University Amsterdam
  • Markus Reiher
    • ETH Zurich, Laboratorium für Physikalische Chemie
    • Laboratoire de Chimie et Physique Quantique (CNRS UMR 5626), IRSAMCUniversité Paul Sabatier
Regular Article

DOI: 10.1007/s00214-011-0911-2

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

Abstract

The electrostatic contribution to the Mössbauer isomer shift of mercury for the series HgFn (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 HgFn (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 57Fe 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 HgFn (n = 1, 2, 4). Projection analysis shows the expected reduction of the 6s1/2 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 6s1/2 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 HgF4 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 HgF2 and square-planar HgF4 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 re = 2.007 Å and ωe = 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 [13], 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 [49] 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.

Each of these energies can be calculated as the electrostatic interaction between electronic and nuclear charge distributions in the particular states. However, usually some well-investigated approximations are made in order to reduce the energies to descriptors like the contact density. First of all, one assumes that the change in electronic charge distribution for the ground and excited nuclear states, which are characterized by 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),
$$ \Updelta E_{\rm IS} = \frac{2\pi}{5} Z \left[n^A(0) - n^S(0) \right] \left( R^{\star^2} - R_0^2 \right) , $$
(1)
where n(0) denotes the contact electron density at the atomic nucleus of absorber A and source S, while \(R^\star\) and R0 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 [1114] 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 [1518] 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 [1923] 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 [2631] (see Refs. [3234] 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 HgF4 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 HgF4. 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

The calculation of the Mössbauer isomer shift via calculated contact densities requires a number of approximations [44]. In the traditional approach [4547], the energy shift associated with going from a point nucleus to a nucleus of finite size is expressed in terms of perturbation theory and evaluated separately for the ground and excited nuclear state. This is hardly a viable approach in a relativistic framework due to the weak singularity of the electron density at the nucleus. We therefore follow the approach of Filatov [19, 22] in which the isomer shift is calculated as an energy derivative. We start with the clamped-nucleus approximation in which a nucleus provides an electrostatic potential that depends on the nuclear charge density ρn as
$$ \phi_n(\user2{r}_{\user2{e}};R)= \int {\rm d}^3 r_n \, \frac{ \rho_n(\user2{r}_{\user2{n}};R)}{r_{ne}}, $$
(2)
where R is some model-specific radial size parameter that characterizes the extension of ρn [11, 13, 48]. The electrostatic electron–nucleus interaction is given by
$$ E^{el}\left(R\right)=\int \rho_e (\user2{r}_{\user2{e}})\phi_n(\user2{r}_{\user2{e}};R){\rm d}^3 r_e , $$
(3)
and is a function of the nuclear radius R. Following Filatov [19, 22], we calculate the associated shift in frequency of the emitted or absorbed photon as an energy derivative
$$ \Updelta E_{\gamma}\,=\,\left.\frac{\partial E^{el}}{\partial R}\right|_{R=R_0}\Updelta R =\left[ \int \rho_e (\user2{r}_{\user2{e}})\frac{\partial\phi_n(\user2{r}_{\user2{e}})}{\partial R}\hbox{d}^3 r_e +\int \frac{\partial\rho_e (\user2{r}_{\user2{e}})}{\partial R} \phi_n(\user2{r}_{\user2{e}}){\rm d}^3r_e \right]_{R=R_0} \Updelta R $$
(4)
in which R0 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 R0. 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.
The next step is to take a specific model for the charge density of the nucleus and establish an explicit expression for \(\Updelta E_{\gamma}\). A detailed comparison of various models is given by Andrae [11, 13], and we refer to this work for illustrative plots of the various nuclear potentials. Relevant for the present discussion are the Gaussian model that is used to obtain relativistic electron densities, and the homogeneously charged sphere model. The potential of the latter one is
$$ \phi^H_n (\user2{r}) = \left\{\begin{array}{ll} \frac{Z}{2R}( 3- \frac {r^2}{R^2});&r \le R\\ \frac{Z}{r} ;&r > R \\ \end{array}\right. $$
(5)
The corresponding derivative is
$$ \frac{\partial\phi^H_n (\user2{r})}{\partial R}= \left\{ \begin{array}{ll} -\frac{3Z}{2R^2}(1- \frac{r^2}{R^2});&r \le R\\ 0 ;&r > R \\ \end{array}\right. $$
(6)
If we assume that the variation of the electron density inside the nuclear volume is minimal, we may replace the function \(\rho_e(\user2{r}_{\user2{e}})\) in Eq. 4 by a constant effective density \(\bar\rho_e\). We can then integrate over the nuclear volume to obtain a simple expression for the energy difference
$$ \Updelta E_{\gamma} = -\frac{4\pi}{5} Z \bar\rho_e R_0 \Updelta R $$
(7)
In the Gaussian model, the nuclear charge distribution is given as
$$ \rho_n^G\left(\user2{r}\right)=\rho_0\exp\left[-r^2/R_G^2\right];\quad\rho_0=\frac{Z} {\left(\pi^{1/2}R_G\right)^3} $$
(8)
and the corresponding nuclear potential is expressed in terms of the error function
$$ \phi_n^G(\user2{r})=\frac{Z}{r}{\rm erf}(r/R_G). $$
(9)
The nuclear radius parameter RG can be related to the radius parameter R of the homogeneous sphere model by comparing radial second momenta [11, 13]
$$\langle r^2 \rangle=\frac{3}{2}R^2_G=\frac{3}{5}R^2;\quad\Rightarrow\quad R_G=\sqrt{\frac{2}{5}}\,R $$
(10)
Taking the derivative with respect to the nuclear radius parameter RG, we obtain a particularly simple expression
$$ \frac{\partial\phi^G_n (\user2{r})}{\partial R}=-2\pi R_G\left(\frac{\partial R_G}{\partial R}\right)\rho^G_n=-\frac{4\pi}{5} R\rho^G_n $$
(11)
which, following the same procedure as for the homogeneous sphere, leads to an identical expression for the energy shift, that is Eq. 7. This relation can therefore also be used to retrieve the effective density by varying the radius parameter of the Gaussian model.
The effective density \(\bar\rho_e\) is often approximated by the contact density, ρe(0). The full sequence of approximations then reads
$$ \begin{array}{lllll} \Updelta E_{\gamma}&\approx&\left.\frac{\partial E^{el}}{\partial R}\right|_{R=R_0}\Updelta R &\approx&\left[\int \rho_e (\user2{r}_{\user2{e}})\frac{\partial\phi_n(\user2{r}_{\user2{e}})}{\partial R}\hbox{d}^3 r_e\right]_{R=R_0} \Updelta R\\ &\approx&\left[\bar\rho_e\int \frac{\partial\phi_n(\user2{r}_{\user2{e}})} {\partial R}\hbox{d}^3 r_e\right]_{R=R_0} \Updelta R &\approx&\rho_e(0)\left[\int \frac{\partial\phi_n(\user2{r}_{\user2{e}})} {\partial R}\hbox{d}^3 r_e\right]_{R=R_0} \Updelta R \end{array} $$
(12)
As pointed out by Fricke and Waber [49], in the relativistic case, the last approximation is likely to be the most severe one and should in principle lead to an overestimation of energy differences as ρe (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 ρe (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

Since the electron density is an observable, an operator can be assigned to it, which yields the particle distribution at the point \(\user2{r}=(x,\,y,\,z)\) in space. By multiplying the electron density with the negative elementary charge, one obtains the charge density \(\rho_e(\user2{r}) = -en(\user2{r})\). The density operator \(\hat{\rho}_{\user2{r}}\) then reads as
$$ \hat{\rho}_{\user2{r}} = -e\sum_{i=1}^{N} \delta^{(3)}(\user2{r}-\user2{r}_i) $$
(13)
with \(\delta^{(3)}(\user2{r}-\user2{r}_i)\) being the three-dimensional Dirac delta distribution. The density operator allows one to express the electron density as an expectation value. The electronic charge density for a four-component wave function described by a single Slater determinant is accordingly given by
$$ \rho_e^{\rm 4c}(\user2{r}) = \left\langle {\Uppsi^{\rm 4c}(\{\user2{r}_i\})}\right|\hat{\rho}_{\user2{r}}\left|{\Uppsi^{\rm 4c}(\{\user2{r}_i\})}\right\rangle = -e \sum_{i}^{N}\Uppsi_{i}^{{\rm 4c}\dag}(\user2{r})\cdot\Uppsi_{i}^{\rm 4c}(\user2{r}), $$
(14)
with \(\Uppsi_{i}^{\rm 4c}\) denoting a four-component orbital.
At the two-component relativistic level, the situation is much more complicated. The relevant Hamiltonian is obtained by carrying out an exact or approximate block diagonalization U of the parent four-component Dirac Hamiltonian H4c and then retaining the block describing the positive energy spectrum only, that is, the two-component Hamiltonian is given by
$$ H^{2c}=\left[U^{\dagger}H^{4c}U\right]_{++}. $$
(15)
Four-component property operators \(\Upomega^{4c}\) must be subject to the same decoupling transformation as the Hamiltonian, that is
$$ \Upomega^{2c}=\left[U^{\dagger}\Upomega^{4c}U\right]_{++}. $$
(16)
Use of the approximate expression \(\Upomega^{2c}\approx \left[\Upomega^{4c}\right]_{LL}\) leads to picture change errors (PCE) [5052] that may be larger than the relativistic effects [53]. This observation also holds true for the density operator, which implies that at the two-component level the electron density cannot simply be resolved into a sum of the squared absolute values of the orbitals at the position \(\user2{r}\) [17, 5456], that is,
$$ \rho^{\rm 2c}(\user2{r})\ne -e \sum_{i}^{N}\Uppsi_{i}^{{\rm 2c}\dag}(\user2{r})\cdot\Uppsi_{i}^{\rm 2c}(\user2{r}). $$
(17)

2.3 Projection analysis

For Hartree–Fock and DFT, it is natural to decompose the total electron density in terms of partial densities from the occupied molecular orbitals. Both the contact density and the averaged density defined in Sect. 2.1 are extremely local quantities and thus well-suited for analysis by means of the projection analysis introduced in Ref. [57]. In this method, the (occupied) molecular orbitals are expanded in pre-calculated orbitals of the constituent atoms
$$ \vert \Uppsi_i \rangle = \sum_{Aj} \vert \Uppsi_j^A \rangle c_{ji}^A + \vert \Uppsi_i^{\rm pol} \rangle , $$
(18)
where the index 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
$$ \langle \Uppsi \vert \hat{\Upomega} \vert \Uppsi \rangle = \underbrace{ \sum_A \sum_{ijk} \langle \Uppsi_i^A \vert \hat{\Upomega} \vert \Uppsi_j^A \rangle c_{ik}^{A\star} c_{jk}^A }_{\rm intra-atomic} + \underbrace{ \sum_{A \neq B} \sum_{ijk} \langle \Uppsi_i^A \vert \hat{\Upomega} \vert \Uppsi_j^B \rangle c_{ik}^{A\star} c_{jk}^B }_{\rm interatomic} + (\hbox{pol}) , $$
(19)
which can be conveniently divided into three classes: (1) the 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 ij. 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].
In the present study, we investigate the mercury contact density in various molecular species and calculate the shift \(\Updelta n(0)\) in the contact density with respect to the free atom. We may anticipate that the total expectation value will be dominated by intra-atomic contributions from the same center. Fixing our coordinate system at the mercury center, we write this contribution as
$$ \Updelta n_{\rm Hg}^{intra}= \sum_{pq}\left\langle {\Uppsi_p^{\rm Hg}{^\dagger}}\left|\delta^{(3)}(\user2{r})\right|{\Uppsi_q^{\rm Hg}}\right\rangle\Updelta D^{intra}_{pq} $$
(20)
where the difference between the intra-atomic block of the density matrix of the molecular species under study and the isolated atom is given by
$$ \Updelta D^{intra}_{pq}=D^{molecule;intra}_{pq}-D^{{\rm Hg};intra}_{pq};\quad D^{intra}_{pq}=\sum_i c_{pi}^{{\rm Hg}\star} c_{qi}^{\rm Hg}. $$
(21)
This expression can be analyzed further by inserting the explicit form of four-component relativistic atomic orbitals
$$ \Uppsi= \left[\begin{array}{l} R^L(r)\chi_{\kappa,m_j}\left(\theta,\phi\right)\\ iR^S(r)\chi_{-\kappa,m_j}\left(\theta,\phi\right) \\ \end{array}\right] $$
(22)
where \(\chi_{\kappa,m_j}\) are two-component angular functions.
The atomic matrix elements in Eq. 20 can now be expressed as
$$ \begin{aligned} \left\langle{\Uppsi_p^{\rm Hg}{^\dagger}}\right|\delta^{(3)}(\user2{r})\left|{\Uppsi_q^{\rm Hg}}\right\rangle&= \left\{\left\langle {R^L_p}\right|\frac{\delta(r)}{4\pi r^2}\left|{R^L_q}\right\rangle_r\left\langle {\chi_{\kappa_p,m_p}}\right.\left| {\chi_{\kappa_q,m_q}}\right\rangle_{(\theta,\phi)}\right.\\ &\quad\left.+\,\left\langle {R^S_p}\right|\frac{\delta(r)}{4\pi r^2}\left|{R^S_q}\right\rangle_r\left\langle {\chi_{-\kappa_p,m_p}}\right.\left| {\chi_{-\kappa_q,m_q}}\right\rangle_{(\theta,\phi)} \right\} \end{aligned} $$
(23)
where subscripts 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
$$ \Updelta n_{\rm Hg}^{intra}=\frac{1}{4\pi}\sum\limits_{\mathop{pq}\limits_{\kappa_p=\kappa_q,m_p=m_q}} \left[R^L_p(0)R^L_q(0)+R^S_p(0)R^S_q(0)\right]\Updelta D^{intra}_{pq} $$
(24)
To proceed, we now consider small r solutions of the Dirac equation for hydrogenic atoms [14, 15, 59]. We expand the large- and small-component radial functions as
$$ \begin{aligned} R^L&=r^{\gamma-1}\left(p_0+p_1r+p_2r^2\ldots\right)\\ R^S&=r^{\gamma-1}\left(q_0+q_1r+q_2r^2\ldots\right) \end{aligned} $$
(25)
For a point nucleus \(\gamma=\sqrt{\kappa^2-Z^2/c^2}>\left|\kappa\right|\) such that there is a weak singularity for \(\left|\kappa\right|=1\). For an extended nucleus, as employed in this work, we have \(\gamma=\left|\kappa\right|\). For κ < 0, we have \(q_0=p_1=0\), which implies that for s1/2 orbitals RL(0) = p0 and RS(0) = 0, where p0 is determined from normalization. For κ > 0, we have \(p_0=q_1=0\), which implies that for p1/2 orbitals RL(0) = 0 and RS(0) = q0, where q0 is likewise determined from normalization. This result can be compared to the non-relativistic case where the radial function is expanded as
$$ R(r)=r^l\left(a_0+a_1r+a_2r^2+\cdots\right) $$
(26)
for both a point and extended nucleus (a1 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 s1/2 and p1/2 orbitals are non-zero at the origin.

3 Computational details

3.1 Effective densities

In order to assess errors induced by the replacement of the effective density \(\bar\rho_e\) by the contact density ρe(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
$$ \bar\rho_e=-\left[\frac{3}{4\pi Z\langle r^2\rangle^{1/2}}\frac{\partial E^{el}}{\partial \langle r^2\rangle^{1/2}}\right]_{R=R_0} $$
(27)
and was calculated numerically by taking step sizes significantly larger than the physical change in nuclear radius in the Mössbauer transition to avoid numerical inaccuracies. By checking linearity of the results, we found that a five-point finite difference scheme
$$ \left.\frac{\partial E^{el}}{\partial \langle r^2\rangle^{1/2}}\right|_{R=R_0}= \frac{E(-2h) - 8E(-h) + 8E(+h) - E(+2h)}{12h}, $$
(28)
with a step size of \(h=5\cdot 10^{-7} a_0\) is sufficient to get numerically stable derivatives. Each energy value was calculated as an expectation value where the perturbation operator is the modification of the nuclear potential with respect to the default nuclear size. This allows us to decompose the effective density into contributions from individual molecular orbitals. As discussed in Sect. 2.3, only s1/2- and p1/2-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 T1 diagnostics [62] which indicates the importance of single excitations in the CCSD approximation. For all closed-shell systems, the T1 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.

An efficient and sophisticated approach to an analytic evaluation of the contact density at the CC level calls for, e.g., a general-order response theory framework [64] which is at present not available in our CC implementation. We have therefore chosen to rather pursue a finite-field strategy which has proven successful in the calculation for various valence and core properties, see e.g., Refs. [6567]. In this approach, the contact density at a given nucleus is modeled by adding the density operator \(\hat{\rho}_{\user2{r}}\) multiplied with a varying perturbation strength λ (“field strength”) to the unperturbed Hamiltonian \(\hat{H}^{0}\):
$$ \hat{H}^{0} + \hat{H}^{\prime} = \hat{H}^{0} + \lambda \cdot \hat{\rho}_{\user2{r}}. $$
(29)
The total electronic energy E becomes a function of the field strength λ, which allows to obtain the expectation value of the contact density from the first derivative of E(λ) with respect to λ
$$ \rho (0) = \left.\frac{d E({\lambda})}{d {\lambda}}\right|_{\lambda = 0}. $$
(30)
Comprehensive test studies on the Hg atom revealed an optimal field strength of \({\lambda}^{opt} = 10^{-8}\) which not only avoided contamination from higher-order contributions but also ensured numerically stable results independent from the level of relativity included in the Hamiltonian. Moreover, it turned out to be particularly useful to treat HF and correlation contributions to the total contact density separately, similar to the procedure followed in finite-field nuclear quadrupole moment calculations [68]. In so doing, we add to the HF expectation value of ρ(0) (Eq. 13) the expectation value ρcorr(0) obtained from a numerical differentiation of the MP2/CCSD/CCSD(T) electron correlation energy Ecorr
$$ \rho^{total} (0) = \rho^{HF}(0) + \rho^{corr} (0) $$
(31)
It proved further advantageous that Ecorr 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
$$ \rho^{corr} (0) = \frac{-{E}_{corr}^{-3} + 9{E}_{corr}^{-2} - 45{E}_{corr}^{-1} + 45{E}_{corr}^{+1} - 9{E}_{corr}^{+2} +{E}_{corr}^{+3}} {60{\lambda}^{opt}}, $$
(32)
where Ecorrn 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

As outlined in the previous section, the scope of the study is twofold: besides the assessment of various electron correlation approaches, we also address successive approximations to the four-component DC Hamiltonian and the influence of spin-orbit coupling (SOC) in the evaluation of the contact density. In particular, we compare four-component relativistic results to:
  • 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 [8591];

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

The eXact two-Component (X2C) calculations have been carried out including two-electron spin-same-orbit corrections provided by the AMFI [93, 94] code. The four-component spin-orbit free (sfDC) as well as the Lévy-Leblond Hamiltonian are defined by a genuine parametrization of the DC Hamiltonian utilizing a quaternion modified Dirac equation [95]. In particular, no perturbation parameters are required for the separation into non-relativistic, scalar-relativistic and spin-dependent terms, respectively, and the Hamiltonians are thus in this sense exact. All these Hamiltonian schemes including the four-component DC Hamiltonian are available in the dirac10 program package [96], while the DKH Hamiltonians and properties are implemented in the molcas 7.3 software suite [97]. For the latter calculations, we employ the notation DKH (\(n,\,m\)) [17] where n and m refer to the order of DKH transformation in the wave function and density operator, respectively, that is
$$ \begin{aligned} \rho^{\rm DKH(n,m)}(\user2{r}) &= \left\langle {{\user2{U}}^{(n)}\Uppsi(\{\user2{r}_i\}) \left|\user2{U}^{(m)}\hat{\rho}_{\user2{r}} \user2{U}^{(m)\dag}\right| {\user2{U}}^{(n)}\Uppsi(\{\user2{r}\})}\right\rangle \\ &= -e \left\langle{\Uppsi}^{{\rm DKH},n}(\{\user2{r}_i\})\left| \sum_{k=1}^{m}\delta_{\epsilon,k}(\user2{r})\right|{\Uppsi}^{{\rm DKH},n}(\{\user2{r}_i\})\right\rangle. \end{aligned} $$
(33)

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 4DC-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 \(\mathsf{TZ}\,(\mathsf{QZ})\,30s24p15d10f\) (34s30p19d12f) SCF set was augmented by 1f4g1h (1f7g4h1i) 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 [104106] (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 [32s25p15d11f4g1h] ([36s31p19d13f7g4h1i]) 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 [27s23p16d12f4g2h] 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 HgFn (n = 1, 2, 4). The first (“[v]”) space comprises the valence 5d6s shell of mercury and the F 2s2p valence electrons. The second (“[cv]”) space is a superset of the [v] space where the core 5s4f5p 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

Table 1 compiles the geometries for the series of molecular mercury fluoride species HgFn (n = 1, 2, 4) for which contact densities have been computed. The geometries for the linear HgF2 and square-planar HgF4 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, 108111] 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 re 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 re 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 (0h,1p) Fock-space sector, thus arriving at the ground state of HgF:
$$ \hbox{HgF}^{+}(0h,0p) \rightarrow \hbox{HgF} (0h,1p) $$
(34)
Table 1

Spectroscopic constants (re and ωe) 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 5s5p4f5d, F 2s2p)

Molecule

Method

Basis sets (Hg/F)

re (Hg-F) [Å]

ωe [cm−1]

HgF

4c-FS-CCSD

\(\mathsf{TZ+2s1p}\)

ATZa

2.012

509.2b

HgF

4c-FS-CCSD

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

AQZ

2.007

513.5

HgF

CCSD(T)c

ECP

ATZ

2.028

480.6

HgF

SO-PBE0c

ECP

ATZ

2.036

457.2

HgF

SO-M06-Lc

ECP

ATZ

2.085

422.8

HgF

NESC/B3LYPd

\(\mathsf{DZ}^{\rm e}\)

ATZ

2.080

HgF

B3LYPf

ECP

AQZ

2.076

414.7

HgF

QCISDf

ECP

6-311G(2df,2dp)

2.019

493.2

HgF

MP2g

ECP

6-311+G*

2.045

444.4

HgF2

CCSD(T)

ECP

AQZ

1.914

HgF4

CCSD(T)

ECP

AQZ

1.885

The energy truncation threshold for active virtual spinors was set at 95 and 107 hartree, respectively. In addition, geometries taken from Ref. [41] are reported for the HgFn (n = 2, 4) compounds at which the present contact density calculations have been performed

aATZ = aug-cc-pVTZ; AQZ = aug-cc-pVQZ

bExperimental value [113]: 490.8 cm−1

cRef. [108]

dRef. [110]

eA contracted Dyall DZ basis was used

fRef. [37]

gRef. [111]

The active (0h,1p) sector space comprised the Hg 6s shell only. Trial studies at the \(\mathsf{TZ+2s1p}/\hbox{ATZ}\) basis set level with enlarged active spaces for the (0h,1p) sector did not reveal any significance of in particular the Hg 6p 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 re. 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 re (see also Table 1) is of same order of magnitude but of opposite sign. Our best theoretical estimate for re 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 re. 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.

We also consider the elimination reaction of difluoride from the tetrafluoride mercury compound in the gas phase.
$$ \hbox{HgF}_{4}\rightarrow \hbox{HgF}_2 + \hbox{F}_2 $$
(35)
Among the rich thermochemistry of the mercury fluoride compounds HgF,n (n = 1, 2, 4) reaction (Eq. 35) has received particular attention in earlier theoretical and lately also experimental works [37, 3941, 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 HgF4 which among the series of group 12 tetrafluorides MFn (M = Zn, Cd, Hg) exhibits an endothermic F2 elimination [3840, 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 HgF4 and HgF2. Calculations on F2 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 HgF4. 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.
Table 2

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 (+73)

CCSD

\(\mathsf{TZ+2s1p}\)

+ 1 (+ 7)

CCSD(T)

\(\mathsf{TZ+2s1p}\)

+31 (+37)

CCSD(T)a

\(\mathsf{TZ+2s1p}\)

+31 (+37)

CCSD(T)/DCG

\(\mathsf{TZ+2s1p}\)

+29 (+35)

CCSD(T)/sfDC

\(\mathsf{TZ+2s1p}\)

+20 (+26)

B3LYP/sfDC

\(\mathsf{TZ+2s1p}\)

+41 (+48)

B3LYP

\(\mathsf{TZ+2s1p}\)

+52 (+59)

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

+54 (+61)

B3LYPa

\(\mathsf{TZ+2s1p}\)

+54 (+61)

PBE0

\(\mathsf{TZ+2s1p}\)

+60 (+67)

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

+62 (+69)

CAMB3LYP

\(\mathsf{TZ+2s1p}\)

+52 (+59)

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

+52 (+60)

Previous work

\(+36.3^{\rm b},\,+35.5^{\rm c},\, +9.5^{\rm c}\)

+27.4d

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

Values in italics do not contain zero-point vibrational energy corrections. Calculations at the four-component spin-orbit free level are indicated with the abbreviation sfDC. Values marked with the acronym DCG were obtained based on the Dirac-Coulomb-Gaunt Hamiltonian using a molecular mean-field approximation

aCalculated at their respective geometries optimized at the

four-component \(\hbox{DFT/B3LYP}/\mathsf{TZ+2s1p}\) level:

HgF4: 1.905 Å; HgF2: 1.928 Å; F2: 1.397 Å

bRef. [39]: B3LYP value

cRef. [109]: B3LYP and CCSD(T) values

dRef. [41]: CCSD(T) value

eRef. [108]: SO-M06 and CCSD(T) values

In accordance with earlier predictions, we find the tetrafluoride compound of mercury to be thermodynamically stable with respect to a spontaneous elimination of F2 in the gas phase at 0 Kelvin. Turning to Table 2, we find that our four-component DFT data suggest a thermodynamically stable HgF4 on the order of 50–60 kJ mol−1 varying with the chosen density functional. Summarizing our DFT results the F2 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 HgF4 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

Before we start with the calibration and interpretation of contact densities, we investigate the differences between the effective densities evaluated via Eq. 27 and the values obtained by directly evaluating the contact density. Table 3 lists the various contributions to the Hg atom densities for \(\hbox{HF}/\mathsf{QZ+2s1p}\) calculations. The results confirm that only the s1/2 and p1/2 orbitals contribute to the contact density, while the effective density also has small contributions from the p3/2 orbitals. These p3/2 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 d3/2 orbitals (not shown). For other types of orbitals, the contributions are significantly smaller and not discernible from numerical noise. More importantly, the s1/2 and p1/2 contributions to the contact density are significantly higher than for the effective density. Interestingly, though, the deviation is quite systematically 10%.
Table 3

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

For the contact density, we give absolute values; for the effective density, we give the difference between the effective density and the contact density. All values are in atomic units a0−3

Mössbauer spectroscopy is not concerned with absolute energies, but rather with energy shifts caused by the different chemical environment. Table 4 lists the contact and effective density 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.
Table 4

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

 

Contact density

Effective density

HgF

−114.54

−103.05

HgF2

−127.85

−115.01

HgF4

−98.09

−88.22

All values are in atomic units a0−3

4.3 Calibration of contact densities

We have compiled in Table 5 the mercury contact densities calculated in this work using the \(\mathsf{TZ+2s1p}/\hbox{ATZ}\) basis set combination (corresponding \(\mathsf{QZ+2s1p}/\hbox{AQZ}\) values are listed in Table B in the supporting information). For the Hg atom, we give absolute values, whereas for the mercury fluorides HgFn (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.
Table 5

Calculated mercury contact densities

Method

Hamiltonian

Hg

HgF

HgF2

HgF4

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

For the Hg atom, we give absolute values, and for the mercury fluorides HgFn (n = 1, 2, 4), we give values relative to the atom. All values are in atomic units a0−3. The acronym DC(G) refers to the Dirac-Coulomb(-Gaunt) Hamiltonian and sfDC to its spin-orbit free form. LL is the non-relativistic four-component Levy-Leblond Hamiltonian. DKH(10,8) corresponds to the 10-th order scalar-relativistic DKH Hamiltonian and 8-th order density operator. For explanation of other notation, see text (all results for Dyall’s decontracted TZ+2s1p)

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 HgF2, whereas an error of −5% is observed for HgF4. 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 HgF4, we observe errors on the order of a few percent, whereas the error is quite small for HgF2. 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% (HgF2). 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 [120125]. However, computational studies [126128] 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 6p orbitals since their role in bonding has been discussed in the literature [111].

The electron configuration of the mercury atom in the studied molecules, obtained by projection analysis at the HF level, is given in Table 6. In mercury monofluoride, the singly occupied HOMO is dominated (74.2 %) by the Hg 6s1/2 orbital, with some contributions (8.5%) from the fluorine 2p-orbitals. The HOMO-1 orbital has some contributions from Hg 5d which are, however, suppressed upon Pipek–Mezey localization [58, 129] of the closed-shell occupied orbitals, giving an orbital dominated by F 2p (78.1%), 2s (7.1%) and Hg 6s1/2 (12.0%). We find only minor contributions from Hg 6p, 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 6p in the model space did not lead to any measurable alteration in the optimized equilibrium structure. In HgF2 and HgF4, we likewise observe a very limited contribution from the Hg 6p orbitals. Along the HgFn (n = 1, 2, 4) series, we observe an increasingly positive charge on the mercury atom, reaching +2.47 for HgF4, 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 6p 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.
Table 6

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

 

HgF

HgF2

HgF4

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

Table 7

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

HgF2

HgF4

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)

We now turn to a projection analysis of the contact density at the HF level. The results are collected in Table 8. We expect the fluorine ligands to pull density away from the mercury atom, and we indeed observe that the contact density is reduced for all the molecular species with respect to the ground state atom. However, referring now to Table 5, we conclude that the decrease is not monotonous with respect to the number of fluorine ligands. This is in contrast to what is observed with all DFT functionals employed in this work, as well as MP2[cv]. The trend, however, agrees with the results obtained at the MP2[v] and coupled-cluster level. Turning now back to Table 8, we see that the (relative) contact density is indeed a very local quantity. The interatomic contribution as well as the intra-atomic contribution from the fluorine(s) are negligible. The polarization contribution, on the other hand, grows steadily along the series HgFn (n = 1, 2, 4), reaching −72% for HgF4. This contribution can certainly not be ignored and we shall return to it in the following.
Table 8

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

 

HgF

HgF2

HgF4

Intra-atomic contribution

  Hg

−120.33

−161.99

−168.41

    pm

−117.32

−178.30

−185.46

      core

10.05

23.85

49.14

      6s1/2

−127.37

−202.15

−234.60

      6p1/2

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 entries ’pm’ and ’hybrid’ refer to principal moments and hybridization contributions, respectively. All numbers are in atomic units a0−3

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 s1/2 and p1/2 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 s1/2 (p1/2) 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 s1/2 over the p1/2 one is on the order of ten than rather than 1/c2, 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 6p1/2 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 6s1/2 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 6s1/2 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 s1/2 orbitals in the molecule, indicated by the growing importance of the hybridization and polarization contributions, as the number of fluorine ligands increase. Indeed, for HgF4 we find that successively adding the mercury \(7s_{1/2},\,8s_{1/2}\) and 9s1/2 orbitals reduces the polarization contribution from 70.84 to 52.07, 10.98 and \(-2.89\,a_0^{-3}\), respectively.

The corresponding projection analysis at the DFT level is given in Table 9, and the most important contributions at the HF and LDA level are contrasted in Fig. 1. Comparing first Tables 6 and 7, we see that the 6s1/2 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 6s1/2 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 Eh at the HF, LDA and B3LYP levels, respectively. DFT, however, yields a more compact 6s1/2 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 6s1/2 contributions from HF and DFT when going from HgF2 and HgF4, as seen in Fig. 1. We also note from Tables 6 and 7 that whereas the 5d populations are rather similar at the HF and DFT level for HgF and HgF2, there is a more significant drop at the HF level when going to HgF4. Due to larger effective mercury atomic charge at the HF level, one would expect stronger polarization of the 6s1/2 orbital at the HF than the DFT level. However, this will be counterbalanced by the more compact nature and thus lower polarizability of the 6s1/2 orbital at the DFT level, as well as that there is less 6s1/2 population to polarize at the HF level, due to lower occupation. In practice, we find that for both HgF and HgF2 the combined LDA contribution of polarization and hybridization is about \(18\,a_0^{-3}\) larger than for HF, whereas this difference basically vanishes for HgF4. 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 HgFn (n = 1, 2, 4), but not for HF, the latter in agreement with our benchmark CC results.
Table 9

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

 

HgF

HgF2

HgF4

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)

      6s1/2

−112.38 (−105.96)

−196.73 (−192.17)

−254.41 (−256.65)

      6p1/2

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)

The entries ’pm’ and ’hybrid’ refer to principal moments and hybridization contributions, respectively. All numbers are in atomic units a0−3

Table 10

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

 

HF

LDA

BP86

BLYP

B3LYP

CAMB3LYP

PBE

PBE0

1s1/2

1951309.48

−0.10

0.37

0.36

0.25

0.29

0.32

0.24

2s1/2

294992.89

−0.05

0.44

0.43

0.30

0.35

0.38

0.29

3s1/2

67814.67

−0.16

0.36

0.34

0.24

0.28

0.31

0.24

4s1/2

17035.91

1.94

2.17

2.20

1.74

1.79

2.11

1.58

5s1/2

3265.25

9.54

8.96

9.13

7.30

7.32

8.74

6.46

6s1/2

276.31

17.34

16.41

14.69

12.84

13.15

15.89

13.45

2p1/2

21934.06

1.03

1.53

1.52

1.18

1.23

1.45

1.08

3p1/2

5659.07

0.89

1.38

1.38

1.06

1.11

1.30

0.97

4p1/2

1403.46

2.88

3.04

3.09

2.45

2.51

2.96

2.20

5p1/2

238.01

10.68

9.94

10.14

8.10

8.02

9.66

7.11

For HF, we give absolute values in atomic units; for DFT functionals, we give deviations with respect to HF in %

https://static-content.springer.com/image/art%3A10.1007%2Fs00214-011-0911-2/MediaObjects/214_2011_911_Fig1_HTML.gif
Fig. 1

Dominant contributions (in atomic units a0−3) to projection analysis of relative contact densities at the HF and LDA level. pol polarization, hyb hybridization

An interesting final point from Tables 8 and 9 is that the contribution from the core s1/2-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
$$ D_{pp}^{intra} = 1+\varepsilon^2+O\left(\varepsilon^4\right) $$
(36)
where \(\varepsilon\) is the norm of the overlap in question.

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., 57Fe, have been proposed and successfully applied. Entering the domain of heavy element chemistry, we therefore here focused on the series of mercury fluorides HgFn (n = 1, 2, 4) where Hg exhibits two Mössbauer active isotopes, 199Hg and 201Hg.

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 re and harmonic frequency ωe, 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 6p shell to the bonding pattern in the monofluoride radical [111].

Gross population analysis along the HgFn series reveals an increasing positive charge on the Hg central atom yielding a maximum of +2.47 in HgF4 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 HgF4 into HgF2 and F2 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 57Fe 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 6s1/2 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 HgF4, these latter contributions are quite similar at the HF and DFT level, but the more compact 6s1/2 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 197Au compounds for which a growing number of experimental Mössbauer IS data is available (see Refs. [132136] and references therein). Aiming at a comprehensive assessment, one could furthermore take into account complexes of Mössbauer active isotopes such as 127I or 129,131Xe for which it would be interesting to study the p1/2 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 T1 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 HgFn (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 HgF4 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.

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