Muonium states in the group 6A elements: computational studies
Roderick M. Macrae a,∗
a School
of Mathematics and Sciences, Marian College, 3200 Cold Spring Road, Indianapolis, IN 46222, USA
Abstract
Recent experimental studies on positive muon implantation in silicon, selenium, and tellurium have been interpreted
on the basis that the primary paramagnetic species observed is XMu (X = S, Se, or Te), the muonium-substituted
analog of the appropriate diatomic chalcogen monohydride radical. However, temperature-dependent signal visibility, broadening and hyperfine shift effects remain puzzling. The interplay of degeneracy, spin-orbit coupling, and
vibrational averaging in these species makes them computationally challenging despite their small size. In this work
computational studies are carried out on all hydrogen isotopomers of the series OH, SH, SeH and TeH. Several different methodological approaches are compared, and the effects of wavefunction symmetry, spin-orbit coupling, and
zero-point vibrational corrections on the isotropic and anisotropic components of the hyperfine interaction examined.
Additionally, some models of the Mu site in rhombic sulfur are briefly considered.
Key words: Muonium, Hydrogen, Density Functional Theory, Sulfur, Selenium, Tellurium, Chalcogen, Hyperfine
1. Introduction
Experimental µSR studies on the chalcogens began exactly 50 years ago with the work of Swanson[1], who observed a loss of 95% of the incident
muon polarization on implantation into sulfur, but
it was not until 1997 that Reid[2] made a definitive observation of a paramagnetic state in that material utilizing ALC-µSR. Subsequent observations
on the heavier chalcogens Se and Te followed[3–5],
and in each case some evidence of a paramagnetic
state was uncovered. While the properties of these
states (temperature dependence of hyperfine parameters, relaxation rates, and visibility using various
µSR techniques) remain incompletely clarified, discussion has coalesced around the possible rôle of diatomic “hydride” radicals such as SMu (the muoniated isotopomer of the mercapto radical) and its
Se and Te analogs. S, Se, and Te in the solid state
∗ Tel. +1-317-955-6064, Fax: +1-317-955-6448, email: [email protected]
Preprint submitted to Elsevier
are structurally similar: rhombic sulfur (the most
stable of the allotropes under standard conditions)
is distinctly molecular, consisting of S8 rings; selenium, too, has an Se8 form (α-monoclinic Se), but
it is higher in energy than a trigonal form (t-Se)
consisting of 31 chains; the single form of Te stable under standard conditions is isostructural with
this trigonal form, but shows less evidence of covalent bonding, consistent with its greater metallicity.
Oxygen, by contrast, consists of strongly covalently
bound diatomic molecules both in the gaseous and
in the solid state, and discussions of positive muon
behavior in various forms of oxygen have focused
upon the possible rôle of MuO2 or MuO+
2 ; while
Kempton accounted for the repolarization behavior
observed as a result of muon attachment to O2 on
the surface of silica grains in terms of the neutral
radical[6], Bermejo et al. found that observations in
solid oxygen were best explained if all magnetic interactions are dipolar in nature, implying a state
such as singlet MuO+
2 , an excited state[7]. The results presented here primarily deal with the hyper18 July 2008
fine properties of the diatomic hydride radicals OH,
SH, SeH, and TeH, calculated at a consistent level
of theory, and with zero-point corrections appropriate for the muonium isotopomers included. Some
additional consideration is given to specific possible
muonium states in rhombic sulfur.
tally symmetric Hartree-Fock hamiltonian. (Their
calculations were, however, limited to the UHF and
ROHF-CIS methods, and are consequently subject
to the errors in the treatment of spin polarization
implicit in those methods.) A similar operator is
available in ORCA and was used in the following.
Figure 1 illustrates the effect of fractional occupancy upon the spin density distribution obtained
for SH at r = 134 pm (the experimentally measured
bond length).
2. Computational Details
Calculations on the diatomic hydride species
mainly used the ORCA quantum chemistry program[8], with some supplementary calculations being carried out using Gaussian 03[9]. Very few highquality all-electron basis sets are available for all
chalcogens up to Te; those used were the segmented
basis sets of triple-zeta quality created by Koga
and co-workers[10], which were employed in both
contracted and uncontracted forms: in more detail,
these sets provide a 6s3p2d basis for H, a 10s5p3d2f
basis for O, a 12s9p4d2f basis for S, a 17s14p10d2f
basis for Se, and a 20s17p13d2f basis for Te. Other
calculations use the correlation-consistent basis sets
of Woon and Dunning[11]. Methods used include
standard post-Hartree-Fock treatments such as
CISD, as well as density functional theory methods.
The highest-quality DFT calculations (on the diatomics) use the “double hybrid” B2PLYP method
of Grimme[12], in which the correlation energy incorporates a second-order perturbative component.
Other DFT calculations use the standard B3LYP
approach.
A further complication in the wavefunction of
the diatomics in their X2 Π3/2 ground states is
that it must yield an electron density distribution
which has the same symmetry as the nuclear geometry, in other words axial symmetry. Almost all
previous electronic structure calculations on these
species have however used standard Hartree-Fock
or DFT approaches using a basis set incorporating real (rather than complex) p-orbitals. In such
a case the minimum energy configuration typically
features a SOMO in which the unpaired electron
breaks the symmetry of the hamiltonian and resides in a πx or πy orbital. For this reason earlier
work on SMu obtained a hyperfine tensor which
the anisotropic component was non-axial[13]. The
single exception to this appearing in the literature
seems to be a study by Bendazzoli et al.[14] on OH
and SH, in which the fractional occupation operator
of Slater[15] was generalized to be used in unrestricted molecular wavefunctions, resulting in a to-
Fig. 1. Distribution of alpha (blue) and beta (red) spin density in the SH radical at r = 134 pm (0.005 a.u. contour)
with (a) conventional wavefunction (b) wavefunction enforcing fractional occupancy.
3. Discussion
3.1. Chalcogen monohydrides
No attempt is made here at a comprehensive review of the experimental and theoretical literature
on the diatomic hydrides; such a review will be presented elsewhere in a more complete treatment of
these systems.
Bond-coordinate scans were carried out on the
four diatomic species using the B2PLYP functional
and the appropriate Koga basis, and with fractional
orbital occupancy enforced. The dependence of the
Fermi contact term upon r for each of the four
species is illustrated in Figure 2. The qualitative
behavior is the same in each case: the coupling is
negative over the entire range of r, and has an absolute magnitude which passes through a shallow
minimum at some r < re , then increases monotonically with r over the remainder of the range
studied. In the valence
¯ region
¯ where there is significant orbital overlap, ¯ dAdriso ¯ is small, but the slope
increases sharply in the bond breaking region before the coupling asymptotically tends to −Aiso,H
(the hyperfine constant of the free hydrogen atom)
2
at large r. The asymptote is −Aiso,H rather than
Aiso,H because the state remains overall doublet in
character; the paramagnetic 3 P state of the chalcogen and 2 S state of H couple antiferromagnetically.
then used together with a polynomial expansion of
Aµ (r − re ) in order to calculate the zero-point corrected values 〈A′µ 〉0 and 〈AH 〉0 appearing in Table
1, which also summarizes the experimental Fermi
contact hyperfine terms extracted from µSR results on S, Se, and Te, together with corresponding
values for the analogous XH species.
The lack of a clear trend in the ZP-corrected values on going down the group is likely the result of
a balance between the trend in re on the one hand
and the trend in the position of max (Aiso (r)) on
the other.
0
Aiso(H)/MHz
–400
X
–800
2
3
O
-73[16]
-96.1
-85.2
S ±73 to ±68[2]
-53[17]
-53.7
-47.4
Se ±73 to 67[3,4]
-40[18]
-60.6
-53.5
Te
±103[5]
-52.6 -42.7
Table 1
Fermi contact terms deduced from level crossing resonance
positions, together with values for diatomic XH species, if
known. All values in MHz, reduced to proton values. Experimental data for SH show considerable matrix dependence.
The tabulated value is for the gas phase.
–1200
1
A′µ for XMu Ap for XH 〈A′µ 〉0 〈AH 〉0
4
r/Å
Fig. 2. Fermi contact term vs r for OH (crosses), SH
(squares), SeH (diamonds), and TeH (circles). The curve is
a spline fit for illustrative purposes.
Uncertainty in the experimental 1 H values renders it difficult to judge the quality of these calculations, but in the case of OH (where the experimental uncertainty is least) the agreement is
quite good[24]. Efforts are underway to refine these
calculations in the hope of improving agreement
between theory and experiment, including the inclusion of spin-orbit coupling effects and an approximate treatment of the relativistic nature of the
core electrons in SeH and TeH. Approximate inclusion of second-order spin-orbit coupling effects in
SH by means of the coupled-perturbed Kohn-Sham
approach implemented in ORCA together with a
Douglas-Kroll-Hess relativistic treatment led to
only very minor changes in Aiso . The discrepancy
between experiment and theory is clearly largest for
TeH, where relativistic effects are most significant;
additionally, the fit to the bonding potential was
poorest in this case.
A comparison of density functional methods and
CISD was conducted on SH using Gaussian 03[9]
and the cc-pVTZ basis set[11]; B3LYP yielded re =
135 pm and Aiso = -37 MHz, while CISD yielded re
= 134 pm and Aiso = -52 MHz. The experimental
values are 134 pm and -53 MHz respectively. Superficially this suggests that CISD is superior to DFT
for this calculation, but it should be noted that these
values do not include any zero-point vibrational correction. Additionally, an important point of these
calculations is the need to find a method extensible
to studies of the solid state; this precludes CISD,
which scales poorly with system size and is computationally expensive.
Zero-point corrections are carried out by fitting
coordinate scan total energy data to the analytically solvable four-parameter potential of Wei
Hua[19], a transformed version of the Tietz potential[20]. One branch of this potential is the
Rosen-Morse potential[21], which yields agreement
with experimentally-derived potential curves superior to the more familiar three-parameter potential of Morse[22,23]. Analytical ¯expressions
¯ for the
¯
l¯
diagonal matrix elements 〈m ¯(r − re ) ¯ m〉 were
3.2. Candidate sites in rhombic sulfur
Preliminary calculations on SH@(S8 )2 (see Figure 3) at the B3LYP/cc-pVDZ level in which the
cyclooctasulfur lattice fragment was treated as rigid
led to a structure in which spin density was almost
entirely localized away from the hydrogenic atom
3
yielding Aiso = 0.3 MHz. The geometrical sensitivity of this result is currently under investigation
through orientation dependence and lattice relaxation studies.
5. Acknowledgements
The author thanks KEK for financial support as
a visiting scientist during the writing of this article,
and Prof. S. F. J. Cox for extensive discussions of the
experimental data and advance copies of a review
work in progress.
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Fig. 3. Optimized geometry of SMu trapped between two
S8 rings (treated as rigid, and with atom positions obtained
from the experimental crystal structure for rhombic sulfur).
Alternative interpretations of these systems include a muonium-like doublet state in which Mu attaches to an S8 ring and a structurally similar state
in which the muon does not pick up an electron
but locally excites S8 into a triplet state; preliminary calculations on the model systems Mu@(S8 )2
and [µ@(S8 )2 ]+ at the B3LYP/cc-pVDZ level lead
to Aiso values of 268 MHz and 142 MHz respectively.
With the S8 rings held rigid, the cation triplet state
lies 2.765 eV above the singlet in energy.
4. Conclusions
Density functional theory calculations on the
chalcogen monohydrides using the B2PLYP method
with triple-zeta basis sets and zero-point vibrational
corrections lead to results in reasonable but not exact agreement with the available experimental data.
Current efforts are focused on building up a comprehensive knowledge base on these diatomic systems
sufficient for application to the exotic matrix-bound
molecular radical states likely to be formed on muon
implantation into the solid chalcogens.
4