Remarks on the signs of g factors in atomic and molecular Zeeman spectroscopy

This paper draws attention to the advantages that would be obtained by adopting a new convention for the sign of g factors that would make the g factor for electron spin a negative quantity (g ≈ ?2), rather than a positive quantity as generally adopted at present. The editors are aware that the proposal made in this paper concerning the conventional sign of the g factor for electron spin will be seen by some readers as controversial. We have nonetheless agreed to publish this paper in the hope that it will stimulate discussion. The editors would welcome comments on this proposal in the form of short papers, which they will then be happy to consider for publication together at a later date.

Various magnetic moments, associated with rotational, vibrational, nuclear spin, electron orbital and electron spin angular momenta, can contribute to the Zeeman effect in atoms and molecules. They are considered in this paper in the context of the effective Hamiltonian where relativistic and other corrections as well as the effects of mixing with other electronic states are absorbed in appropriate g factors. In spherically symmetric systems, the magnetic dipole moment arising from a specific angular momentum can be written as the product of three factors: the nuclear or Bohr magneton (which is positive), the g factor (which may be positive or negative), and the corresponding angular momentum (which is a vector). A convention is discussed, in which the sign of the g factor is positive when the dipole moment is parallel to its angular momentum and negative when it is antiparallel. This would have the advantage that it could be applied consistently in any situation. Such a choice would require the g factors for the electron orbital and electron spin angular momenta to be negative. This concept can easily be extended to the case of a general molecule where the relation between the dipole and angular momentum vectors has tensorial character.     

The influence of Coriolis-coupling on the partition function of spherical top molecules

Using Schwinger's operator perturbation theory for the partition function, it is shown that Coriolis-coupling can be neglected for spherical top molecules. The results contain the symmetric top as special case. Spherical top molecules of the XY₄-type are taken as numerical examples for a wide range of rotational constants and vibrational frequencies.     

Forbidden vibrational-rotational transitions and Herman-Wallis factors in fundamental IR bands of symmetric-top molecules

Within the theory of coupled schemes of ordering of vibrational-rotational interactions, the operator of the effective dipole moment of single-quantum vibrational transitions is represented in the form of an infinite series in vibrational (normal coordinates and conjugate momenta) or rotational variables (components of the total angular momentum). Mechanisms of activation of infrared-inactive totally symmetric vibrations in molecules of the D _2a , D _3h , C _3h , D _n (n ≥ 3), S ₄, T, T _a , and O symmetries and forbidden vibrational-rotational transitions in IR bands of active vibrations have been studied. The group-theoretic analysis of tensor parameters in higher-order effective dipole moments of single-quantum vibrational transitions in axially symmetric molecules has been performed. The strengths of allowed transitions and forbidden transitions in fundamental and hot IR bands of axially symmetric molecules are calculated with allowance for the Herman-Wallis factors. For effective dipole moments of multiquantum transitions in molecules, models are developed in the form of infinite series in rotational variables and in the form of Padé approximants.     

Effective Hamiltonian for degenerate vibrational states in symmetric top molecules

A mixed matrix-operator form of the effective rotational Hamiltonian has been discussed for the degenerate vibrational states of symmetric top molecules. In this scheme, a rotational contact transformation can be applied to the effective Hamiltonian such that the operators of the “2, +2,” “2, −2,” and “2, −1” l-type interactions as well as the operators of the Δk = ±3 and ±4 interactions are eliminated from the first-order terms of the expansion of the rotational Hamiltonian in terms of the small parameter λ. The results have been used to discuss the correlation between various interaction parameters in the effective rotational Hamiltonian for the doubly degenerate fundamental vibrational levels of semirigid symmetric top molecules. For example, for C_3v or D₃ molecules, the parameter of the “2, −1” interaction is correlated with other parameters and cannot be determined separately by fitting the experimental data (unless there are certain accidental resonances between vibrational-rotational levels).     

The effects of vibration-rotation interaction on the quadrupole hyperfine structure of molecular rotational levels

By using a contact transformation method similar to that commonly employed when determining higher-order corrections to the harmonic oscillator and rigid rotor energy levels of molecules, analogous centrifugal distortion and anharmonic corrections to the nuclear quadrupole coupling energies have been obtained for molecules containing one quadrupolar nucleus. The J, K dependence and v, l dependence of these higher-order corrections to the quadrupole hyperfine energies can be cast in a form which is remarkably similar to the form taken for ordinary vibrational and rotational energy corrections, a result which was not evident from earlier partial treatments of this general problem. Results are obtained here for asymmetric top, symmetric top, spherical top, and linear molecules.     

Calculation of Normal Vibrations of the Tetraazaporphin Molecule and Interpretation of Quasiline Spectra

Quasiline fluorescence and fluorescence-excitation spectra of tetraazaporphin and its N-deuterated derivative in n-octane at 77 K have been investigated, and the frequencies of normal vibrations in electronic states S ₀ and S ₁ have been determined. Calculation of normal vibrations of these molecules has been done and, on its basis, the experimental data are interpreted. It is shown that in the spectra, predominantly the totally symmetric vibrations of type A _g symmetry of the point D _2h symmetry group are active. Some activation of the nontotally symmetric B _1g vibrations in the fluorescence-excitation spectra is explained by the nonadiabatic interaction of the vibrational sublevels of the excited electronic state S ₁ with the purely electronic level S ₂.     

ESR spectra of the OD radical in the 2Π32, J = 32 state, v = 0–5 levels

The paramagnetic resonance absorption spectrum of the OD radical in the excited vibrational levels up to v = 5 of the ground ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{J =}\text{3}\text{2}$ state has been observed at X-band frequencies. The theory of the Zeeman effect of a paramagnetic ²Π state of a light diatomic molecule has been applied to analyze the spectrum. The Λ-doubling frequencies, the molecular g factors, coefficients of the second order Zeeman effect, and the hyperfine interaction constants were determined for each of the vibrational levels. The experimental results agree reasonably well with the values calculated from (a) optical data and (b) ab initio data. The experimental determinations are still much better than theoretical calculations.     

Application of CI expansion techniques to the computation of two-photon absorptivities

Summary We show how conventional linear expansion techniques for both electronic and vibrational wave functions can be used for the computation of matrix elements governing the two-photon absorption probability in molecules. In particular, the²Σ⁺ A←²∏X transition matrix elements of OH and the¹Σ _g ⁺ E,F←¹Σ _g ⁺ X of H₂are computed using 1) the sum over states in the Born-Oppenheimer approximation with inclusion of vibrational wave functions, 2) solution of perturbation theory equations by expansion in the CI basis at fixed nuclei, followed by a kind of vibrational averaging. The results are compared with experiment and discussed. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

Dipole moment function of carbonyl sulphide

Measurements are reported for the infra-red absorption intensity of some overtone and combination bands in carbonyl sulphide. These are interpreted with the aid of an anharmonic force field and some microwave Stark effect measurements on excited vibrational states to give the values for the dipole in the equilibrium configuration, p _e, and the linear and quadratic derivatives for the axial dipole component shown in table 4. These signs are absolute, being tied to the microwave Zeeman effect on isotopically substituted molecules [4].     

Magnetic moments for lithium quartet S states

The fully correlated calculations of the Zeeman g_J factors for the first three quartet S states of lithium are presented, including relativistic and radiative corrections of orders α², α² m/M, and α³. The isotope shifts in g_J are predicted precisely for various isotopes of lithium. Received 4 December 2000 and Received in final form 26 September 2001     

Isotropic Raman spectra of liquid (isotope) mixtures: Intermolecular coupling of vibrational modes

The theory presented in this paper investigates the isotropic Raman spectra of liquid binary mixtures. It is found that the collective vibrational modes of different molecular species can be significantly coupled. This is a consequence of the (nearly) resonant vibrational transfer processes, which give rise to distinct vibrational correlations (i.e. correlations between adjacent molecules). The coupling, however, occurs only with weakly-separated or overlapping bands. The more general results of the theory are applied to isotope mixtures. The spectral information available from relevant dilution experiments is interpreted. In particular it is shown that the spectral properties of the vibrational self-correlation part can be concluded from the observed collective correlation function. The significance of the distinct vibrational correlations with respect to the infrared and depolarized Raman spectra is discussed.     

Molecular g-tensors from analytical response theory and quasi-degenerate perturbation theory in the framework of complete active space self-consistent field method

The molecular g-tensor is an important spectroscopic parameter provided by electron para magnetic resonance (EPR) measurement and often needs to be interpreted using computational methods. Here, we present two new implementations based on the first-order and second-order perturbation theories to calculate the g-tensors within the complete-active space self-consistent field (CASSCF) wave function model. In the first-order method, the quasi-degenerate perturbation theory (QDPT) is employed for constructing relativistic CASSCF states perturbed with the spin–orbit coupling operator, which is described effectively in one-electron form with the flexible nuclear screening spin–orbit approximation introduced recently by us. The second-order method is a newly reported approach built upon the linear response theory which accounts for the perturbation with respect to external magnetic field. It is implemented with the coupled–perturbed CASSCF (CP-CASSCF) approach, which provides an equivalent of untruncated sum-over-states expansion. The comparison of the performances between the first-order and second-order methods is shown for various molecules containing light to heavy elements, highlighting their relative strength and weakness. The formulations of QDPT and CP-CASSCF approaches as well as the derivation of the second-order Douglas–Kroll–Hess picture change of Zeeman operators are given in detail.     

Many-body correlations for Raman and Rayleigh collision induced scattering

In this paper, the spectral shape of the collision induced scattering is calculated in the dipole-induced-dipole approximation for the Raman symmetric vibrational band of optically isotropic molecules. Comparison is made with Rayleigh results. Experimental data for Rayleigh and v ₁ Raman band of CF₄ are discussed.     

Rotational transitions in degenerate vibrational states of C3v symmetric top molecules,with application to CH3C15N

General algebraic expressions for the vibration-rotation energy levels and the associated rotational transitions of C_3v symmetric top molecules are developed. These expressions are presented in a convenient form for analysing the spectra of a molecule with any degree of excitation of a degenerate vibrational mode and have been applied to the spectrum of CH₃C¹⁵N.     

Zeeman splitting factor of the Er3+ ion in a crystal field

Numerical computations are presented on the energy levels of the Er³⁺ ion in crystalline fields of cubic, trigonal, tetragonal and orthorhombic symmetry. Zeeman splitting factors were obtained from the level splitting in an additional magnetic field. For the quartet Γ₈ states in cubic symmetry the Zeeman effect is described by an effective Hamiltonian ℋ= gμ_BBJ+ uμ_BBJ³ with the parametersg andu calculated for mixed fourth- and sixth-order potentials. For the eight doublets in the lower symmetry of an axial trigonal or tetragonal crystal field the principalg tensor components g_∥ and g_⊥ were calculated. The results of such calculations for a ground-state doublet can exactly account for the experimental data obtained on around 70 erbium centers in various crystalline hosts. However, sometimes different sets of parameters give comparably good results. An empirical rule of constant trace g_∥ + 2g_⊥ is supported by the calculations. In contrast to analytical treatments the effect of the crystalline field can be followed over a continuous range of the crystal field parameters. This allows one to establish relations on the relative signs of tensor components. It is found that the measured trace of tensors |g_∥| + 2|g_⊥| is not always equal to their real trace g_∥ + 2g_⊥. In an exploratory calculation a nonaxial center was simulated in an orthorhombic field, with calculation of the three principal values g_x, g_y and g_z. A good agreement is obtained for the recently reportedg values of an erbium center in silicon.     

Electron resonance studies of the renner effect

The gas phase electron resonance spectra of NCO in its ground ²Π_3/2 vibronic state and in two excited vibronic states are described. Theoretical analysis of the spectra yields effective g values for the three states. In additon the ¹⁴N magnetic hyperfine and electric quadrupole coupling constants and the electric dipole moment are determined. The theory of the Renner coupling of electronic and vibrational motion is extended, and shown to account for anomalous contributions to the g values. The theory also shows that these contributions are closely related to the Renner coupling constant.     

Strong quadrupole interaction of light with molecules and its manifestation in surface-enhanced hyper-Raman scattering spectra

The possibility of the giant enhancement of hyper-Raman scattering by molecules adsorbed on rough metal surfaces is demonstrated. The theory is based on the qualitative consideration of electromagnetic field enhancement near some model rough surfaces and individual irregularities, as well as on the quantum-mechanical features of dipole and quadrupole interactions of light with molecules (as in the theory of surface-enhanced Raman scattering), proposed by the author. A consideration of symmetric molecules makes it possible to obtain selection rules for surface-enhanced hyper-Raman scattering (SEHRS) spectra and establish such a regularity as the occurrence of strong forbidden lines (which are due to totally symmetric vibrations); these lines are transformed according to unitary irreducible representation in molecules with the symmetry groups C _nh , D, and higher. An analysis of the data in the literature for trans-1,2-bis (4-pyridyl)ethylene and pyridine molecules shows that their spectra can be explained in terms of the dipole-quadrupole theory of SEHRS. At the same time, the analysis of the SEHRS spectra of pyrazine revealed the presence of strong forbidden bands due to totally symmetric vibrations. This finding substantiated the proposed theory, which makes it possible to interpret the entire spectrum in detail. These results are in good agreement with the general mechanism of the optical effects enhanced by molecules adsorbed on metal surfaces, which was developed by the author.     

Correlation diagrams for quasi-symmetric top molecules with a single rotor

The rotation-bending-internal rotation Hamiltonian for quasi-symmetric top molecules with a single rotor [A. Wierzbicki, J. Koput, and M. Kr?glewski, J. Mol. Spectrosc.99, 102–115 (1983)] is used to calculate the correlation diagrams between the energy levels of the SiH₃NCO-type molecules in the symmetric and asymmetric top limits, and to predict various features of the microwave spectrum through trial calculations. The height of the barrier to linearity was varied from 0 to 6500 cm^?1. For the barriers 0, 31, and 139 cm^?1, the effects of hindered internal rotation are studied and several (Δl = 3, Δk = 0) resonances are found. In some cases almost all the microwave transitions from a given vibrational state can be perturbed.     

Raman band shapes and the dynamics of liquid N2O4

The Raman spectra of the totally symmetric A _g modes, v ₁, v ₂ and v ₃, of the N₂O₄ molecule have been measured in the liquid state at 262, 279 and 297 K. The vibrational and the rotational correlation functions are calculated. The long-time exponential decay of the rotational correlation functions of all the A _g modes reflects an asymptotic diffusional behaviour of molecular reorientation. The rotational relaxation rate is found to increase with increasing temperature. A marked point of inflection from the short time inertial correlation to the long time exponential decay appears at about 0·35 ps for the v ₂ mode. This is an indication of orientational rebound arising from the librational motion in a temporary solvent cage. The isotropic bandwidth increases in the order v ₁ < v ₂ < v ₃, which is also the order of decreasing vibrational frequency. The temperature dependence of the peak frequency and of the bandwidth are also found to increase in the same order. These observations are analysed qualitatively in terms of two models of vibrational dephasing which take into account the effect of vibrational anharmonicity.