The non-additive ligand field

The semi-empirical ligand field is a perturbation operator whose consequences are taken to first order using a basis set ofl functions. Since the basis spans an irreducible representation of the 3-dimensional rotation-inversion groupR _3i it is useful to express the operator as a sum of components of irreducible tensor operators with respect to this group. IfR _3i is reduced with respect to the molecular subgroup the electronic factor of each term in the sum must be totally symmetrical within this group. This choice of operator leads to thecrystal field parameterization without implying an electrostatic model. Alternatively a shift operator withinl space may be chosen as the essential part of the perturbation operator. This leads to theligand field parameterization. Between the two parameterizations there exists a one to one relationship, whose coefficients are proportional to 3l symbols. This relationship is given together with a brief discussion of the reasons for the proposed parameterizations.     

Selfadjoint extensions of a singular differential operator

In this work, firstly in the Hilbert space of vector-functions all selfadjoint extensions of the minimal operator generated by linear singular symmetric differential expression with a selfadjoint operator coefficient A in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.     

Généralisation de l'identité de Germano et application à la modélisation sous-maille

In the large-eddy simulation frame, a dynamic procedure based on Germano's identity is often employed to adjust the subgrid-scale model constant. Applying an n-th degree linear operator to the Germano identity, a generalized dynamic procedure is defined to model this operator applied to the subgrid-scale tensor. This procedure leads to very satisfactory results when applied to the divergence operator in the frame of freely decaying turbulence.     

Matrix elements of multi-electron operator in restricted Hartree-Fock theory

A theorem has been established for the matrix elements of a general t-electron operator between N-dimensional determinantal wave functions arising in the solution of the atomic and molecular multi-electron problem by the restricted Hartree-Fock (RHF) method (1 ≤ t ≤ N). The required matrix elements of this operator are sums of matrix elements over t-dimensional basic determinantal wave functions. The final results are especially useful in the determination of multi-electron properties for atoms and molecules when the Hylleraas approach in RHF theory is employed.     

New approach to single-particle time-correlation function theory: Application to intramolecular dephasing in the condensed phase

A new approach to solve the linewidth expression of infrared line shape based on the Kim-Wilson single-particle time-correlation function theory is presented. A projection-operator technique coupled with an explicit series-expansion method is used to reduce the unusual operator exp[1(1 - P)Lt] in the damping function to the usual operator exp(i L⁰t). This approach is applied to a study of a temperature dependence of the width and shift of the infrared line shape for intramolecular dephasing of a polyatomic molecule in the condensed phase.     

l-splitting of energy levels for almost coulombic central potentials

An operator which for purely Coulomb problems acts as a raising operator for the total angular momentum is employed to provide an expression for the l-splitting due to the non-coulombic part of an arbitrary central potential. Both bound and continuum states are examined. Several useful approximate forms are also obtained and tested.     

The coupling algebra of trigonally subduced crystal field eigenvectors

The general theory of subduction of eigenvectors between infinite groups is used to derive a finite group subduction operator and define the corresponding subduction coefficients. The coupling behaviour of these subduced eigenvectors can then be described in terms of 3 Γ symbols. These symbols, defined only in relation to complex basis sets are all fully real and have all phases fixed by the subduction operator. They differ from V coefficients in two phase relationships and have the advantage, unlike V coefficients, of retaining all the symmetry properties and selection rules of Wigner 3-j symbols. Appropriate label systems which render these properties in terms of simple algebras are given for all quantizing axes available in O _h . The specific set of 3Γ symbols for each quantization is determined by the orientation of the coordinate axes in the Hamiltonian. The four possible orientations for trigonal quantization are examined and the operator chosen which produces eigenvectors with conventional conjugate phases and a fully real set of 3Γ symbols.     

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.     

Calculations of Jahn-Teller coupling constants in the weak-field basis via the angular overlap model: a simple operator equivalent technique

Within the framework of the angular overlap model the matrix elements of the linear Jahn-Teller operator may easily be calculated in the weak-field basis in terms of simple operator equivalents. The method is applied to the calculation of the 〈M_J|?V/?Q|M_j〉 matrix elements for the |LSJMj〉 ground states of t^x species (x = 1) ? 13) in )^* symmetry.     

Reliability of the usual differential collision operator in describing the relaxation of a heavy-particle flow in a light gas

The reliability of the usual (approximate) differential collision operator for heavy particles in a light gas (Rayleigh gas, brownian particles) is examined when the heavy particles, having initially temperature T₀ and flow (or drift) velocity <−<ν− > ₀, relax in a background gas in equilibrium at temperature T. It is found that severe limitations must be imposed on T₀/T and |〈ν〉₀|, in order that the usual approximate collision operator can have a fair reliability at the initial time.     

Selection rules and line-strength factors for multiphoton transitions in gas phase molecular spectroscopy

Multiphoton transitions are discussed in terms applicable to experimental spectroscopy. A simple tensor operator for the equations governing multiphoton transitions in molecules is obtained. Selection rules for these processes are derived by symmetry considerations and procedures for calculation of rotational line-strength factors are given. For the general transition, the rovibronic n-photon selection rules can be determined by the symmetry product of a n-photon operator with the rovibronic wavefunctions. The line-strength factors are found by considering the integral over rotations of the nth rank tensor operators. The line-strength evaluations do not depend on exact knowledge of the vibronic overlap, to within a constant factor.     

Representation of one-dimensional motion in a morse potential by a quadratic hamiltonian

A representation of the algebraic hamiltonian for the anharmonic Morse oscillator as a quadratic form, H = T1ω($\text{1}\text{2}$P² + $\text{1}\text{2}$Q²), where P and Q are operators is derived. The commutator of P and Q is an operator that tends to i (times the identity operator) in the harmonic limit. Coherent states and anharmonic normal modes for a linear triatomic molecule are discussed as potential applications.     

Erratum

Feshbach's projection operator technique as adapted in atomic collision theory is applied to obtain the T-matrix element for indirect dissociative recombination     

Quantum theory of isotropic Raman spectra-changes with gas density

A new formulation of the theory of overlapping line broadening of isotropic Raman scattering is presented in the framework of the impact approximation. This formulation takes into account close or “strong” collisions in the exponential (sudden) approximation. The projection operator formalism is employed to consider both the weak line overlap (resolved isotropic Raman Q-band) and the pressure narrowing limit (unresolved Q-band). The quantum model of j-diffusion is assumed to lead to the matrix factorization of the impact operator. The factorization opens the way for determining all matrix elements of the impact operator using a set of experimental line widths and shifts of the isotropic Raman Q-band. The resulting possibility to determine the rotational energy relaxation time by measuring the line widths of the Q-band spectrum is discussed.     

Application of a spin-adapted coupled-cluster based linear response theory: Direct calculation of π -π* singlet and triplet excitation energies of conjugated systems

A procedure is tested for directly calculating exciation energies for spin-conserving and spin-forbidden transitions using a spin-adapted coupled-cluster based linear response theory. The excited states are generated from the ground state through an excitation operator S, a combination of various nh—np excitations of spin-rank zero and one for singlet and triplet excitations.     

Field shifts in hafnium II

By means of classical interference spectroscopy, using enriched isotope samples, the isotope shift between¹⁷⁸Hf and¹⁸⁰Hf has been measured for 33 transitions in the Hf II spectrum. For the pure Russell-Saunders terms 5d ²6s ⁴ F and² F the parametric analysis yields a field-shift difference of 17(2) mK produced by the second-order interaction of the electrostatic operator and the field-shift operator. Semi-empirical calculations based on the non-relativistic Hartree-Fock method reproduce this value as well as the experimental field shifts if a factor of 1.68(6) is used to scale theab initio electron densities at the nucleus. The corresponding factor for the Hf atom is much smaller. This leads to a re-evaluation of screening ratios for Hf and to a more accurate value of the nuclear parameter λ^178,180 (Hf)=0.072(4) fm².     

Author index to volume 48

The vibrational energy transfer in a polyatomic molecule is described by introducing an irreducible set of operators. Each operator represents a specific interaction induced by the intermolecular potential. We present a graphic method which significantly facilitates the solution if the number of quantum modes (M) is large.     

Exact semiclassical solution for the time evolution of a quantum-mechanical system in a circularly polarized monochromatic driving field

A closed-form exact solution is presented for the time evolution operator of a nonrelativistic hydrogen atom driven by a circularly polarized monochromatic light beam. The solution has the form U(t, 0) = exp(?itF)exp(?itG), where F and G are time independent operators. All temporal effects of the oscillating field are included. Extension to other systems is easily made. A generalization of the rotating wave approximation to multilevel systems is also presented.     

A semiclassical formulation of the time dependent perturbation theory: The N coupled harmonic oscillator model

A new semiclassical formulation of the time dependent perturbation theory is proposed to describe the evolution ofN discrete quantum levels perturbed by time dependent couplings. A formal identity exists between the evolution operator matrix elements of the two following systems: (i) the real molecular system composed of N coupled quantum levels, (ii) a system composed of N harmonic oscillators, specifically prepared to account for a given transition, interacting via a specific potential. By working on the second system rather than the first one a formal expression of the time displacement operator results and an original interaction representation technique with respect to the diagonal and non-diagonal couplings can be derived. This allows one to “dress each transition channel” with slowly varying phase shifts generated by the coupling terms involving the transition itself. An application to the collision system N₂ + CO (OC) illustrates a generalized Mies effect due to a coupling term between quasi-resonant levels.