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Recent experimental studies suggest that the host–guest interaction is stronger between C 6 H 6 Mu and the cation site in NaY zeolite than in the case for benzene, leading to trapping of the radical in an essentially frozen geometry over the temperature range 3–322 K. while benzene exhibits fast rotational motion at these temperatures (J. Phys. Chem. B 106 (2002) 6395). A comparison of the benzene–Na + and cyclohexadienyl–Na + model systems using ab initio methods shows that the binding energies between these systems are virtually identical. However, modeling of the benzene and cyclohexadienyl interactions with a partially optimized segment of the zeolite matrix corresponding to the Na + site indicates that electrostatic dipole–dipole interactions between the radical and the zeolite lead to orienting forces (and a barrier to uniaxial reorientation of several tens of kJ mol −1 ) which do not exist in the case of benzene.

The isotropic hyperfine coupling constants A iso for the protons in the cyclohexadienyl radical, C 6 H 7 , are calculated using ab initio molecular orbital methods. Taking all single excitations from a restricted open shell Hartree–Fock configuration, (CIS), the calculated value for A iso at the methylenic protons is 93.0 MHz. The presence of a sodium cation reduces A iso for the methylenic proton of C6H7 on the same side of the ring as Na+ by 18.2 MHz. For the other methylenic proton the presence of Na+ enhances the value of A iso by 30.8 MHz and the C6H7 ring shows significant distortion.

Using the technique of muon spin relaxation (µSR), repolarisation curves are determined following the deposition of positive muons in samples of apoferritin over the temperature range 50 K to 270 K, and ferritin over the temperature range 15 K to 280 K. Evidence is found for a muonium-substituted species common to both apoferritin and ferritin. In apoferritin at 270 K an estimate for A′, the reduced isotropic muon–electron hyperfine coupling constant, is 168±39 MHz. In ferritin at 280 K the value for A′ is 164±37 MHz.

Ab initio molecular orbital calculations at the CISD level have been used to obtain potential energy and hyperfine coupling constant surfaces for the X2A′ electronic state of the HCS radical and its isotopomers DCS and MuCS. Although MuCS is experimentally unknown the calculations indicate that MuCS is likely to be a viable species. A value of 114.1 MHz is predicted for the reduced isotropic hyperfine coupling constant of MuCS in its lowest vibrational level.

The formation of a molecular radical following muon implantation in sulphur S 8 is established experimentally by level crossing resonance. The position of the resonance determines the hyperfine constant as 233 ± 5 MHz. Of various possible assignments evaluated by ab initio molecular orbital calculations, the most promising appears to be the muonic sulfanyl radical, ṠMu.

The [PF6]− salt of E-tetraethyl-l,4-diaminobut-2-ene has been synthesized and the crystal structure determined. The experimental electronic charge distribution has been obtained by fitting a multipole model to low-temperature, high-resolution, single-crystal X-ray diffraction data. A high-level ab initio molecular orbital calculation has also been performed for the cation, which is used as an antifungal agent. Both experimental and theoretical Laplacian distributions display maxima only at the bond critical points. This correlates with a calculated electrostatic potential which is positive at all points.

Zero-point vibrational corrections are calculated by abinitio molecular orbital methods for the energy and the reduced hyperfine tensor of the muonated sulfanyl radical SMu 2Π. Only a small α-isotope effect is found for the reduced isotropic muon–electron hyperfine coupling constant of SMu relative to SH. Interstitial muonium in S8 is also investigated. The results are discussed in the light of recent observations on muonium in elemental sulfur.

The muonium and deuterium isotopomers of the formyl radical HCO 2 A′ are investigated using a variation-perturbation approach for the calculation of vibrational effects in these radicals. After inclusion of anharmonicity MuCO has a calculated zero-point vibrational energy of 74.15 kJ mol -1 compared with a value of 36.11 kJ mol -1 for HCO. The effect is discussed of these zero-point energy contributions to the formation of MuCO. The calculated reduced muon–electron hyperfine coupling constant A µ ′ is 437.75±2.86 MHz for the muonated radical in the ground vibrational level. This represents a calculated α-isotope effect A µ ′ /A p of 1.18, where A p is the proton–electron hyperfine coupling constant in the formyl radical.

Using a variation-perturbation approach, harmonic and anharmonic corrections to some stationary point properties of the ethyl radical are computed for deuterium and muonium isotopomers. The anharmonic corrections in the muonated radical are found to be highly significant. The fundamental frequency for the C—Mu bond is calculated as 7877 cm–1. The effect upon the muon–electron β-hyperfine coupling constant of excitation of the muonated radical to higher vibrational states is investigated.

Quantum-chemical studies of the surface-site geometries for aluminium–oxo (hydroxo) and chloro aluminium–oxo (hydroxo) cluster models derived from the spinel structure of γ-alumina are reported. The ab initio MO calculations which use an SV-321G basis indicate that a stabilisation of the LUMO energy occurs with a narrowing of the energy gap, Egap′, by the substitution of chlorine for oxygen or hydroxy groups in γ-alumina cluster models containing tetrahedral aluminium environments. The studies also show that a hierarchy of potential Bronsted and Lewis character is exhibited with Altet—O—Altet > Altet—O—Aloct > Aloct—O—Aloct′, indicative of a relationship between morpho-electronic factors and Lewis acidity at surface adsorption sites. The Bronsted acid character of the cluster is generated by the location of a near-neighbour tetrahedral aluminium environment. Substitution of chlorine for oxygen into the cluster model enhances the Bronsted acid character for the aluminium–oxo (hydroxo) cluster. The presence of octahedral aluminium–oxo (hydroxo) environments reduces both the Bronsted and Lewis acid character, respectively, for an aluminium–oxo (hydroxo) cluster.

Quantum-chemical studies of the surface-site geometries for aluminium–oxo (hydroxo) and chloro aluminium–oxo (hydroxo) cluster models derived from the spinel structure of γ-alumina are reported. The ab initio MO calculations which use an SV-321G basis indicate that a stabilisation of the LUMO energy occurs with a narrowing of the energy gap, Egap′, by the substitution of chlorine for oxygen or hydroxy groups in γ-alumina cluster models containing tetrahedral aluminium environments. The studies also show that a hierarchy of potential Bronsted and Lewis character is exhibited with Altet—O—Altet > Altet—O—Aloct > Aloct—O—Aloct′, indicative of a relationship between morpho-electronic factors and Lewis acidity at surface adsorption sites. The Bronsted acid character of the cluster is generated by the location of a near-neighbour tetrahedral aluminium environment. Substitution of chlorine for oxygen into the cluster model enhances the Bronsted acid character for the aluminium–oxo (hydroxo) cluster. The presence of octahedral aluminium–oxo (hydroxo) environments reduces both the Bronsted and Lewis acid character, respectively, for an aluminium–oxo (hydroxo) cluster.

The muon-electron β-hyperfine coupling constant is reported for the 2-muoxyprop-2-yl radical formed by positive muon (μ+) implantation in propan-2-one and aqueous solutions of propan-2-one. Using the technique of muon spin rotation (μSR) spectroscopy the coupling constant was measured over a full temperature range in which the samples were in the liquid state. In all cases, the coupling constants were found to be small, and to show a positive temperature dependence. In the aqueous mixtures the presence of the water produces a slight lowering, at all temperatures, in the coupling constant with respect to the coupling in pure propan-2-one. An interpretation is given which relates the radical structure, to the barrier hindering internal rotation and to solvent interactions.

A variation-perturbation approach is applied to compute isotope-dependent zero-point vibrational corrections to the β-hyperfine coupling constant of the ethyl radical. The corrections increase the coupling by ca. 5, 7 and 23% for the deuterium-, protium- and muonium-substituted radicals, respectively. The large vibrational correction to the muon β-hyperfine coupling constant could explain the ‘residual’ isotope effect observed experimentally.

A multipole model of the charge-density distribution in a ring containing C—S and S—S bonds has been obtained using high-resolution X-ray diffraction data from a single crystal of 3,3,6,6-tetramethyl-S-tetrathiane. Analysis of the bond critical points in both experimental and ab initio densities suggests a sulfur–sulfur bond order only slightly greater than unity. For isolated molecules, ab initio geometry optimisations predict a small energy stabilisation of the chair conformer relative to the twist-boat conformer observed in the crystalline state.

The title radicals have been observed in the NaX zeolite using the transverse-field muon spin rotation (TF-µSR) technique and are found to be strongly associated with the zeolite surface.

We report the observation of muonium-substituted radicals formed in 3-methyl-2-butenal. The reduced hyperfine coupling constants (A′µ) at 273 K are 59.96 MHz for addition to the C—C bond and 2.89 MHz for muonium attachment to CO.

Muonium substituted free radicals are observed by muon spin rotation when positive muons are stopped in liquid olefins and dienes. From muon precession the isotropic muon—electron hyperfine coupling constants have been determined for 44 radicals and analyzed to yield the radical structures. Primary, from mono-olefins, and mainly allylic radicals from conjugated dienes. They are formed by addition of the light hydrogen isotope muonium (Mu = μ + the relations between coupling constants and radical structures follow the principles known for the analogous H substituted radicals. Further, the regi to that of H atoms.

The importance of applying a variety of experimental techniques to unravel the nature of solvated electrons is emphasized. Compton profiles are evaluated for these species from a range of models. Some comparisons are made with positron annihilation studies.

Numerical solutions of semicontinuum model potentials for excess electrons localized in condensed phases have been obtained in a range of media. These are compared with previous approximate results and with experiment. Transitions to higher excited states are found to be unlikely to contribute significantly to the width of the observed spectra, which still remains unexplained. It is suggested that more sensitive criteria for the appropriateness of a particular model for the species be adopted. The static polarizability is evaluated here, showing current models to be inadequate.

Finite–difference methods are used in a study of the polarisabilities of the beryllium isoelectronic sequence Li– to Ne6+. Two approximations to the Coupled Perturbed Hartree–Fock scheme are considered for the evaluation of static dipole and quadrupole polarisabilities, it being demonstrated that, provided the appropriate perturbation equations are solved accurately, both yield results which differ only marginally from those obtainable from the fully coupled method. Similarly encouraging calculations of the dynamic polarisability of Be are also recorded. The practicability of accurate polarisability calculations on medium and large systems is briefly discussed in the light of the present results.

Analysis of the muon spin rotation (μSR) data for the muonic ethyl radical CH 2 MuĊH 2 yields β-hyperfine coupling constants in good agreement with the recent level-crossing resonance (LCR) measurements. Other coupling constants are predicted and await confirmation by LCR. The role of hyperconjugation in the ethyl radical is assessed by an ab initio molecular-orbital calculation.

Multipole polarisabilities to l= 5 have been calculated for the helium isoelectronic sequence. Solution of coupled and uncoupled Hartree-Fock equations has been effected by numerical techniques, thus permitting the geometric approximation for atomic polarisabilities to be closely examined. An extension of an early prediction of Pauling to higher multipoles is noted.

Finite-difference methods have been used to obtain the interaction energy to second order for the ground state and 1S2S excited states of two hydrogen atoms at long-range.

Publisher Summary This chapter reviews some applications of numerical procedures to obtain accurate values for atomic properties. The chapter has cited applications in more diverse areas, as one-center expansions for molecules, the calculation of excitation cross sections by the impact parameter method, or a simple solution of continuum and semicontinuum models for excess electrons in liquids. Yet even greater possibilities lie with time-dependent theories. Recently, a number of extensions of the time-dependent Hartree–Fock (TDHF) method have been proposed to handle the general open-shell system. This development probably offers the most promising line of advance in the subject, particularly for the study of photoionization. Even for closed-shell atoms, such as barium, complicated effects are apparent in df/dЄ at high energies, and though there is broad agreement with experiment substantial discrepancies exist between different RPA computations. Possibly only a direct numerical solution of the equations will be sufficiently accurate for successful applications of the TDHF approximation to complex systems.

By fitting of the observed temperature dependence of the β-hyperfine muon–electron interaction to a theoretical model for the muonic radicals CH2MuĊH2, CHDMuĊH2, CHDMuĊD2 and CD2MuĊD2 values for the barrier height V2 restricting internal rotation are calculated. A decomposition of the barrier into pair interactions indicates the isotope effect upon the barrier to be in the sequence VDD2 < VDH2 < VHH2 < VMuD2 < VMuH2. On this basis the barriers to internal rotation for the radicals CH2MuĊD2 and CD2MuĊH2 are predicted to be of the order 3076 and 3186 J mol–1, respectively.

Muonium-substituted free radicals are observed by muon spin rotation when polarised positive muons are stopped in several substituted liquid dienes. There is a preference for the most energetically stable radical to be formed. The selectivity of formation obeys the Evans–Polanyi relation between activation energies and heats of reaction. Two models are discussed for the radical-formation process and upper limits are set on the lifetimes of possible muonic radical precursors: 20 ps for Mu(≡µ+e–) and 2 ns for µ+ in diamagnetic environments.

The analytic method for the calculation of electrostatic molecular potentials is extended to allow for the inclusion of d-type Gaussian functions in the molecular basis set. The method is illustrated with the isopotential maps for the species CO, HCO+ and HOC+ and the sensitivity of such maps to the choice of basis set is highlighted.The family of ions and radicals of the type HXY (X, Y = C, O, N) is discussed with particular regard to the effect of isotopic substitution of protium by muonium. Vibrational frequencies are calculated to provide a measure of the extent of the large-amplitude vibrations anticipated in the muonic species.

The theory for the calculation of the low-lying energy levels of one-electron diatomic molecules is presented with particular regard to the importance of mass dependence as shown by muonic nuclei. A comparison is given of the calculated vibrational levels with those obtained by a parameterisation in terms of the spectroscopic constants allowing for the mass dependence. Transition frequencies of some low-lying vibrational–rotational energy levels are predicted for muonic species.

A distinction is drawn between the structure of the solvated electron, as pertaining to molecular relaxation at a localisation site, and the diffuse nature of the electron.An upper limit to the dipole polarisability for the solvated electron is calculated. For the adiabatic continuum model, with the localisation site being a spherical cavity of radius R, the upper limit to the polarisability for e–am, at R= 3.69 A is 411 A3. The value for e–aq, at R= 1.84 A is 81 A3. These upper limits are within 10 % of the computed dipole polarisabilities. The effect of introducing a semi-continuum shell is noted.Electron momentum distributions and the Compton profile, calculated for e–am and e–aq are reported. The half-width of the Compton profile for e–am within the adiabatic continuum model is 0.196, for e–aq the half-width is 0.30.

Numerical wave functions are presented for a surplus electron in water and ammonia within both the adiabatic and self consistent field formulations of the continuum model.Heats of solution and spectral properties are evaluated, which bring some recent findings into question.

The meaning of the Born–Oppenheimer approximation with respect to muonic species is discussed. The value for D0, the dissociation energy from the lowest rovibrational level, is calculated to be 255.758 kJ mol–1 for H+2, in comparison with an observed value for D0 of ⩽ 255.765 kJ mol–1; for Mu+2 the value for D0 is calculated to be 229.82 kJ mol–1. Expectation values of 〈R〉 are given for non-adiabatic wavefunctions.Nuclear and electronic natural orbitals are illustrated for the molecules T+2, H+2, Mu+2, MuT+ and MuH+.

A new analytic method is presented for the calculation of the electrostatic potential of any molecule. Analytic expressions are given for s- and p-type gaussian functions and the source for expressions involving d-type gaussian functions is noted. The use of Slater functions requires a representation by a gaussian expansion. An extension of the theory is made for contracted gaussian functions. Isopotential maps are shown for several small molecules.

The temperature dependence of the β-hyperfine coupling constant is reported for muonic radicals formed in ethene CH2CH2, [2H1]ethene CH2CHD, [2H3]ethene CHDCD2 and [2H4]ethene CD2CD2. These studies are complemented by e.p.r. observations upon the radical CH2DĊH2.The barrier to internal rotation for the radical CHD2ĊD2 is discussed.