Relativistic Calculation of Hyperfine Parameters of Mercury Compounds

Send Orders for Reprints to [email protected] 284 Current Inorganic Chemistry, 2013, 3, 284-290 Relativistic Calculation of Hyperfine Parameters of Mercury Compounds Michael Filatov, Wenli Zou and Dieter Cremer* CATCO, Department of Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA Abstract: The normalized elimination of the small component (NESC) method is a Dirac-exact relativistic method that leads to reliable first order response properties such as contact densities, Mössbauer shifts, electric field gradients, quadrupole coupling constants, or hyperfine structure constants for heavy atoms. In this review, the calculation of these hyperfine parameters with a NESC analytical derivatives formalism is discussed and demonstrated for mercury containing molecules. There is a distinct need for accurate calculated hyperfine parameters because the possibilities of experiment are limited in a case such as mercury. This need can be fulfilled if, beside scalar relativistic effects, the influence of spin-orbit coupling, electron correlation and the finite dimension of the nucleus are accounted for. Keywords: Contact density, dirac-exact relativistic methods, electric field gradient, hyperfine structure constant, Mössbauer shift, normalized elimination of the small component, quadrupole coupling constant. 1. HYPERFINE FIELDS IN ATOMS AND MOLECULES studying realistic molecular models. At the all-electron level of description, these methods provide a first principles The measurement of electron-nuclear hyperfine account of the effects of relativity for the inner core electrons interactions in molecules and solids yields valuable without resorting to the use of an a posteriori restoration of information on the local electronic structure and bonding, the core electron wavefunctions sometimes proposed in the which is obtained on an atomic scale [1, 2]. Experimental context of relativistic effective core potential techniques (see spectroscopic techniques such as Mössbauer spectroscopy Ref. 12 and references cited therein). [3], nuclear forward scattering (NFS) [4], perturbed angular correlation of -rays (PAC) [5], nuclear quadrupole When describing hyperfine fields in atoms, molecules, resonance (NQR) [1, 6] or electron spin resonance (ESR) [7, and solids, one conventionally uses perturbation theory to 8] are capable of providing information on the oxidation and obtain the parameters of these fields in form of expectation spin state of the resonating atom. In addition, information on values of the corresponding quantum mechanical operators its local chemical environment is gained, which makes these [13-16]. Alternatively, the hyperfine parameters can be spectroscopic methods valuable tools to probe chemical expressed as derivatives of the electronic energy with respect bonding. to suitable perturbing fields [9, 17]. The derivatives based formalism is well suited for use in the context of Dirac-exact The combination of electronic structure theory and the computational methods, as in this way the drawbacks of measured spectroscopic data leads to the desirable hyperfine separately transforming the electronic states and the parameters, which are specified within given molecular interaction operators from the Dirac 4-component models [2, 9, 10]. Following this approach, reasonable representation to the Foldy-Wouthuysen 2-component models of the local environment of the resonating atom can representation are avoided [18]. However, the crucial be established. The knowledge of the local hyperfine fields requirement in this case is the availability of analytic energy enables one to refine parameters of the electron-nuclear derivatives for the respective computational method. This hyperfine interactions such as the nuclear electric quadrupole requirement is fulfilled for the NESC method [19, 20]. In moment Q , the fractional nuclear charge radius R / R , or this review, the salient features of the derivatives formalism the nuclear magnetic dipole moment ì, where the results of for the hyperfine parameters will be briefly reviewed and the quantum chemical calculations for the resonating systems application of this formalism in the context of the NESC play an important role [9-11]. The accurate calculation of method will be outlined. these nuclear properties strongly depends on an accurate account of relativity and electron correlation, which requires the use of the most advanced quantum chemical methods. In 2. CONTACT DENSITY AND MÖSSBAUER ISOMER this connection, the application of the Dirac-exact SHIFT computational methods, which provide an accurate account of relativistic effects and are applicable to large molecular The contact electron density is a property that is required systems, is most awarding as they offer an opportunity of in Mössbauer [3] and nuclear forward scattering spectroscopy [4] to interpret the shift of the absorption line *Address correspondence to this author at the CATCO, Department of of the -transition in a nucleus where its embedment in a Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas specific local chemical environment relative to that a sample 75275-0314, USA; Tel (Int.): 214-768-1300; source nucleus plays an important role. The physical origin Email: [email protected] of this shift, known as the isomer shift, lays in the variation 1877-945X/13 $58.00+.00 © 2013 Bentham Science Publishers Relativistic Calculation of Hyperfine Parameters of Mercury Compounds Current Inorganic Chemistry, 2013, Vol. 3, No. 3 285 of the nuclear electric charge radius during the -transition 3. ELECTRIC FIELD GRADIENT AND because this variation leads to a slight alteration of the QUADRUPOLE COUPLING CONSTANT electron-nuclear interaction [14, 15]. It is a sensitive probe A non-spherically symmetric nucleus (i.e. a nucleus with for the local electronic structure and bonding features of the spin I > 1/2) possesses a quadrupole moment QK , which target compound [2]. The isomer shift is expressed in k leads to a splitting of the nuclear energy levels when the terms of the Doppler velocity (in mm/s) needed to achieve a nucleus K is embedded into an electronic environment resonant absorption of the -quantum (having the energy without cubic symmetry [2, 11, 24]. The nuclear quadrupole E ) between the source (s) and absorber (a) nucleus splitting can be accessed by a number of experimental according to eq. (1): techniques including Mössbauer, NFS, PAC, NQR, and c a s microwave spectroscopy. The interaction of the nuclear = ()E E (1) E quadrupole moment with the electric field of the electrons (nuclear quadrupole interaction, NQI) can be described by a(s) where E is the shift of the energy of the -quantum the Hamiltonian [2, 24]: caused by the electron-nuclear interaction in the absorber ˆ NQI K K a(s) H = Qij Vij (5) (source) system. The quantity E is usually expressed with the help of the contact electron density at the absorber i, j (source) nucleus: where the summation runs over the Cartesian (i,j = x,y,z) 2 components of the nuclear quadrupole moment (NQM) K K 4 cZK RK RK a s a s () () tensor Qij and the electric field gradient Vij at the site of (K)= = (K) (2) K 5 E RK the nucleus K. The electric field gradient operator Vij is defined as a second derivative of the potential V(r) due to the where ZK gives the proton number and RK / RK is the electrons and other nuclei as given in eq. (6): nuclear fractional charge radius that reflects the variation of the root-mean-square (RMS) nuclear electric charge radius K 1 2 R during the -transition. As the quantity R / R can Vij = ij V(r) (6) K K K x x 3 not be accurately determined by experiment, the use of i j calculated contact electron densities in connection with The expectation value of the EFG operator over the the experimentally measured isomer shifts remains the most electronic ground state wavefunction is used to characterize reliable method of its determination [2, 16]. The knowledge the magnitude of the NQI. The nuclear quadrupole coupling of the fractional charge radius or the calibration constant constant (NQCC): (K) is a prerequisite for a reliable interpretation of the eQV Mössbauer and NFS spectra of chemical compounds. = cc (MHz) (7) Q h At a fully relativistic level of description, the isomer shift and the contact electron density can be expressed with the characterizes the magnitude of quadrupole splitting in help of the derivatives of the total electronic energy with Mössbauer and NQR spectra. [16, 27] The nuclear respect to the nuclear electric charge radius [9, 17]. When a quadrupole splitting is also characterized by the asymmetry finite size nucleus model is employed in the quantum parameter =(Vaa Vbb )/ Vcc where the EFG chemical calculations, the isomer shift is given by eq. (3): expectation values are presented in a principal axes system and ordered to satisfy | Vaa | | Vbb | | Vcc | . c E a (R) E s (R) (K)= R (3) Using the energy derivatives formalism, the expectation R=RK R=RK K E R R value of the EFG operator (6) is obtained as a derivative of the total electronic energy with respect to the components of where E a(s) (R) is the electronic energy of the absorbing K the NQM tensor Qij given by eq. (8) [22]: (source) system and RK is the experimentally measured RMS nuclear charge radius [17]. From eq. (3), it follows that E(QK ) V K = ij ij K K (8) the contact density can be also expressed via the derivatives Q Qij 0 of the total electronic

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