Hyperfine interactions in muonium-containing radicals

Hyperfine interactions in muonium-containing radicals

ARTICLE IN PRESS Physica B 374–375 (2006) 290–294 www.elsevier.com/locate/physb Hyperfine interactions in muonium-containing radicals Stephanie L. Th...

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Physica B 374–375 (2006) 290–294 www.elsevier.com/locate/physb

Hyperfine interactions in muonium-containing radicals Stephanie L. Thomas, Ian Carmichael Radiation Laboratory, University of Notre Dame, Notre Dame, IN 46556, USA

Abstract We present an overview of techniques designed for the reliable prediction of hyperfine interaction tensors in muonium-bearing radicals from Density Functional Theory (DFT). In general, the isotropic component is the most difficult to determine accurately. Anisotropic terms are often recovered satisfactorily at modest levels of theory. In small systems high-level electronic structure calculations may be used to demonstrate the importance of a number of key components in a successful treatment. The roles of basis set balance, electron correlation effects, and vibrational averaging over modes in which the muon is active are illustrated by calculations on a number of small inorganic and organic species. In larger systems DFT becomes the method of choice. However, the optimal choice of functional is not yet completely clear. Some cautionary tales are given along with some success stories. r 2005 Elsevier B.V. All rights reserved. PACS: 31.15.Ew; 32.10.Fn; 33.15.Pw; 33.20.Tp; 36.10.Dr Keywords: DFT; Muonium; Radicals; Hyperfine

1. Introduction Radiation chemistry seeks to understand, and ultimately control, the transformation of the energy of the impinging radiation into the observable chemical outcomes, beneficial or deleterious. At the Notre Dame Radiation Laboratory, investigators have access to a variety of radiation sources ranging from high-energy photons (g-rays), through pulsed-electron, proton, and other ‘‘heavy’’ ion beams. For obvious reasons, studies have predominantly focused on aqueous systems. The course of electron and proton radiolysis of water is reasonably well understood, at least at room temperature and pressure. However, with the renewed emphasis on nuclear energy to meet growing power needs and the various proposals for ‘‘Generation IV’’ reactors, it has become clear that at high temperatures and pressures much uncertainty remains. It is to be hoped that the unique characteristics of muon studies [1] can help fill this important gap in our understanding of the kinetics of radical reactions under these conditions.

Corresponding author. Fax: +574 631 9735.

E-mail address: [email protected] (I. Carmichael). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.076

While the radiolyzing electrons and protons quickly lose their identity in water, the added muon, with its microsecond lifetime, persists (perhaps as muonium) through the end of the energy-loss stage of the radiation track and well into the track-end chemistry. Any muonium-containing radicals formed are potentially identifiable by their hyperfine splitting and this work offers some insights into the reliable calculation of the relevant magnetic parameters for a number of such radical species. Experimental gasphase measurements on the hyperfine interactions of the hydrogen-containing versions of many of these radicals have been elegantly summarized in the work of Hirota [2,3]. Solution and matrix magnetic properties have been extensively compiled by various authors in the LandoltBornstein volumes on the magnetic properties of free radicals edited by H. Fischer. The accurate calculation of the isotropic component of the hyperfine interaction tensor has proved particularly challenging for traditional molecular-orbital-based quantum chemical techniques. High levels of electron correlation must be included and the basis set must be particularly flexible. Both of these features militate against routine use of this approach in all but the smallest radicals due to extreme calculational expense. We report here results from

ARTICLE IN PRESS S.L. Thomas, I. Carmichael / Physica B 374–375 (2006) 290–294



aiso [14N] MHz

Density Functional Theory (DFT) which has much more modest computational requirements and often provides comparable accuracy. Muonium-containing radicals will possess high-frequency, large-amplitude vibrations. Dynamical averaging will certainly occur and is taken into account in our calculations. Solvent effects might also be important and can be included. We report results for three triatomics, HCO, COH, and HO2 and four tetra-atomics HOCO, HONO(–), HOOO, and H2CN. These species exhibit a range of electronic states and torsional flexibilities. We also note some of our recent work on muonium addition to thiocarbonyls [4,5] and muonium encapsulation in siloxanes [6].




2. Computational details Calculations were performed with localized versions of the Gaussian03 [7] and Dalton 2.0 [8] suites of electronic structure programs. The DFT approach was used throughout. The hybrid B3LYP functional [9] was used exclusively except for the nitrogen atom test case. There results from a range of many-parameter [10–13] and few-parameter [14–17] functionals are compared. Equilibrium structures for the various radicals were obtained by analytic gradient methods and harmonic frequencies from analytic Hessians. Anharmonic corrections were derived from a perturbation theory approach using both cubic and (partial) quartic force constants [18–20]. The higher-order terms were computed by appropriate numerical differentiations of the Hessian. Properties were averaged as described elsewhere [21]. Medium effects were approximated by reaction field methods using the most recent formulation of the polarized cavity model [22]. The correlation-consistent basis sets of Dunning and coworkers [23–25] were employed to allow extrapolation to the basis set limit. For accurate recovery of the Fermicontact term core-valence correlation must be well described. Molecular magnetic interactions in doublet radicals (i.e. those containing one unpaired electron) can be described by a g-tensor, which locates the overall position of the spectrum, and a hyperfine interaction (A-) tensor which describes line splitting due to the presence of magnetic nuclei. The A-tensor can be decomposed into a dipolar component which is usually straightforward to calculate and an isotropic, or Fermi-contact, term (aiso ) which often proves much more troublesome. Typically, for molecules containing ‘‘light’’ atoms it is calculated as the expectation value of the molecular electronic spin density at the nuclear position. This d-function operator places exacting requirements on the quality of the overall wave function and, in particular, the basis sets employed. 3. Results and discussion Fig. 1 displays the convergence with basis set flexibility of the calculated isotropic coupling for the isolated gas-






basis set aug-cc-pCVXZ Fig. 1. Convergence of aiso in the nitrogen atom with basis set for a number of density functionals (see text for references).

phase nitrogen atom for which the experimental value is accurately known [26]. Values from a range of density functionals are included. Only results for the hybrid B3LYP functional converge to experiment and are, in fact, not visibly distinguishable from those obtained by extensive coupled cluster calculations using the same sequence of basis sets [27]. DFT results for other atoms are, however, less convincing [28]. Note that the calculations employ the correlation-consistent core-valence basis sets. Contributions to the overall coupling arise from both the core (1s-) and valence (2s-) shells. The components are similar in magnitude but opposite in sign so must be recovered in a balanced way for accurate results. The unrestricted Hartree–Fock (UHF) method provides an exact (non-relativistic) solution for the isolated hydrogen atom (p21 in a.u.). Fig. 2 compares the UHF convergence to this value with the 3% overshoot found in B3LYP calculations. Presumably this gives an indication of the level of agreement to be anticipated at the basis set limit in vibrationally corrected calculations of aiso (1H) in molecular systems. The addition of further shells of compact s-functions is seen to speed convergence. The formyl radical (HCO) is a s radical with a 2A0 electronic ground state and is well characterized both experimentally [2] and theoretically [29]. The muoniumcontaining version is implicated in muonium spin relaxation in carbon monoxide [30], and has also been discussed theoretically [31]. Isoformyl (COH) has also been treated theoretically [32]. The H–C bond in formyl is weak, we calculate a homolytic bond scission enthalpy (BDE) of 90 kJ mol–1, at the B3LYP/aug-cc-pVTZ level of theory

ARTICLE IN PRESS S.L. Thomas, I. Carmichael / Physica B 374–375 (2006) 290–294


within the harmonic approximation. Isoformyl is computed to be metastable and to lie 85 kJ mol–1 above the H+CO dissociation asymptote. On the other hand, hydroperoxyl (HO2) is a p-radical with a 2A00 bent ground state and a relatively strong H–O bond. We compute a value of 205 kJ mol–1 as above. The vibrationally averaged bond lengths in the various isotopomers of these triatomics are listed in Table 1. Only the lengths to the H-isotope are given, most other structural parameters are not significantly shifted form their computed equilibrium values, although the MuOC bond angle is predicted to be 31 larger than the corresponding parameter in HOC. These changes are also reflected in the extent of anharmonicity computed in the stretching fundamentals, which is displayed in Table 2. Very large anharmonic effects are noted in the muoniumC/O stretches. The characterization of the anharmonic force fields allows the recalibration of the reaction thermochemistry and is particularly significant for muonium-containing radicals. The Mu–C BDE in MuCO is

calculated to be 53 kJ mol–1, when anharmonic corrections are included, while the Mu–O bond strength in MuO2 is predicted to be 153 kJ mol–1. The convergence of the isotropic coupling constants and g-factor in HCO with increasing basis set flexibility computed from DFT is shown in Table 3. These values are obtained at the optimized B3LYP/aug-cc-pVTZ geometry and may be extrapolated to the complete basis set limit [33] to give a g-factor of 2.000296 and a prediction of 384 MHz for the proton coupling in reasonable accord with the gas-phase experiment [34]. Similar extrapolations lead to estimates of a g-factor of 2.0142 and 1H coupling at –25.8 MHz for HO2. On the other hand, the EPR unknown COH radical is predicted at a g of 1.99876 with a proton coupling of 226 MHz. Of course, vibrational averaging will occur, which is expected to be particularly significant for the muoniumcontaining species. Table 4 presents the results of the present anharmonic vibrationally averaged hyperfine couplings for DCO, HCO and MuCO. Only couplings to the

Table 2 Fundamental vibrationsa (X ¼ D,H,Mu)

aiso [1H] MHz











1904(103) 2092(277) 2506(111)

2401(254) 2753(505) 3382(204)

5630(2080) 5005(4495) 8478(1891)


X–C/O stretching frequencies in cm–1 (anharmonicities in parentheses) from B3LYP/aug-cc-pVTZ.

Table 3 Variation in the magnetic propertiesa of HCO with basis set

Fig. 2. Convergence of aiso in the hydrogen atom with basis set (aug-ccpVXZ). Squares are UHF, circles B3LYP, broken lines include compact sfunction.

Table 1 Vibrationally averaged bond lengthsa in XCO, COX, and XOO radicals (X ¼ D,H,Mu)



or0 4D

or0 4H

or0 4Mu

112.3 98.7 97.6

113.8 100.3 98.7

114.3 100.9 99.0

117.8 104.8 101.7

Bond lengths in pm from B3LYP/aug-cc-pVTZ.






aCD aCT aCQ aC5 Extrp.

2.000379 2.000326 2.000309 2.000300 2.000296

358 375 378 384 386

366 378 387 390 392

23 34 35 35 35

a g-factor dimensionless, a in MHz from B3LYP/aug-cc-pCVXZ (X ¼ D,T,Q,5). The basis on H contains one compact s-function.

Table 4 Vibrational effects on ‘‘proton’’ isotropic hyperfine couplinga in HCO, COH and HO2 isotopomers






353 207 25

357 227 25

371 236 26

409 295 28

a in MHz from B3LYP/aug-cc-pCVXZ averaged over indicated isotopomers and ‘‘reduced’’ to 1H values.

ARTICLE IN PRESS S.L. Thomas, I. Carmichael / Physica B 374–375 (2006) 290–294

light nucleus are included in each case and these are ‘‘reduced’’ to equivalent proton values. Smaller, but still significant, changes are seen in the central-atom splitting in HCO and COH, but are not shown here. The distal coupling is unaffected. Assuming vibrational corrections of a similar magnitude to the above extrapolated values, then the final predicted couplings are a(1H) ¼ 404 MHz and a(Mu) ¼ 1416 MHz, for HCO and MuCO, respectively. As mentioned above, dipolar couplings are generally less sensitive to the details of the calculation with Hartree– Fock values being quite similar to those obtained by DFT calculations. They are also less sensitive to vibrational effects as seen in Table 5 which collects values, in the principal axes coordinates, computed at the respective vibrationally averaged coordinates for the D, H and Mucontaining isotopomers of both HCO and HOO. The dipolar couplings and the ‘anharmonic’ structures are both obtained at the B3LYP/aug-cc-pCVTZ level of theory. Similar sets of calculations have been performed for the tetra-atomic species, HOCO, HONO(–), and HOOO, the first of which presumably exists in the muon radiolysis of supercritical CO2 [35]. Hyperfine coupling constant calculations for the 2A0 s -radical HOCO have been presented previously [36,37]. The anti-, syn- isomerization is prevented by a barrier of about 40 kJ mol–1 and the ‘‘branched’’ HCO2 form lies 51 kJ mol–1 above the most stable isomer (anti-). Protonated CO2 also has an antiplanar arrangement; we estimate a gas-phase proton affinity of 530 kJ mol–1. At the same harmonic level, while the BDE of HOCO into OH and CO is 124 kJ mol–1, the scission into H+CO2 only costs 17 kJ mol–1. MuOCO is thus predicted to be metastable, lying 44 kJ mol–1 above Mu+CO2. The inclusion of vibrational anharmonicities reduces this value to 36 kJ mol–1. HONO(–) has been discussed theoretically [38] and the rotational spectrum of HOOO has recently been recorded [39]. The torsional barriers in these 2A00 p-radicals are computed to be less than 4 kJ mol–1 so that large-amplitude torsional motion is to be expected. Vibrational averaging over this type of motion must be treated explicitly. Some relevant energetics and structural parameters, which reveal

Table 5 Variation of reduced ‘‘proton’’ dipolar couplingsa in HCO and HO2 isotopomers





–15.1 –14.8 –12.8 –21.3 –21.2 –20.4

–8.4 –8.3 –7.5 –12.5 –12.5 –12.5

23.5 23.1 20.3 33.8 33.7 32.9

a Principal values in MHz from B3LYP/aug-cc-pCVXZ at corresponding averaged geometries in indicated isotopomers and ‘‘reduced’’ to 1H values.


Table 6 Structural parametersa and relative energiesb in HOYO species










134.1 186.2 154.6

6.8 1.0 1.2

132.5 175.6 150.6

50.6 45.4 252

180 142 139


Central bond lengths r in pm, dihedral angle W in degrees. Energies, E in kJ mol–1, relative to the anti-isomer all from B3LYP/ aug-cc-pVTZ calculations. b

Table 7 Vibrational effects on ‘‘proton’’ isotropic hyperfine couplinga in HOCO, HONO–, HOOO and HNO–2 isotopomers.





4.1 0.3 0.9 37.7

6.3 2.0 2.7 36.6

9.2 3.0 3.8 38.1

a in MHz from B3LYP/aug-cc-pCVXZ ‘‘reduced’’ to 1H values.

a very long central bond in the p-radicals, are collected in Table 6. By way of contrast to HOCO, in HOOO the preferred scission is into OH+O2, only 4 kJ mol–1 endothermic. MuOOO is again predicted to be metastable, although the strength of the Mu–O bond is high at 315 kJ mol–1. Table 7 displays the effect of vibrational averaging on the proton hyperfine coupling in several tetra-atomics. Results for the two p-radicals are not expected to be accurate since the full torsional potential has not been considered. In aqueous media HONO(–) has been shown to be unstable with respect to dissociation into NO and hydroxide [40]. However, the ‘‘branched’’ form, HNO2(–) is apparently produced from the reaction of H atoms with nitrite. It has been detected by time-resolved EPR spectroscopy and the proton coupling successfully calculated from theory. In that work a detailed description of the inversion motion at N was required and the effects of the aqueous environment were model by a polarized cavity approach. A similar anharmonic vibrational treatment to that described for the other semi-rigid radicals above of the hyperfine interactions in H2CN has recently appeared [41]. Our present calculations produce an extrapolated value of 243 MHz coupled with a vibrational correction of 26 MHz, allowing us to predict a value of 269 MHz for the reduced coupling at Mu in HMuCN. We have also recently shown that caution must be employed when interpreting the EPR spectra in radicals produced by muonium addition to carbonyl and thiocarbonyl linkages [4,5].


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