The rotational structure of three-photon resonances of polyatomic molecules

A general formulation of the selection rules and line strengths of three-photon excitation spectra is presented for molecules of arbitrary symmetry. For symmetric-top molecules the paper extends the discussion by Nieman in the context of rovibronic transitions of NH₃. For asymmetric-top molecules it is shown that the rotational line strengths can be sensitive to quantum-mechanical interference between contributions from the various components of the three-photon vibronic transition tensor. This transition tensor is discussed within a perturbation-theory framework in terms of several models of intermediate-state participation.     

Multiple quantum1H-NMR of ferrocene in isotropic solution

Summary Excitation of multiple quantum coherence in the C₅ symmetrical A₅-spin system of ferrocene was possible by multiexponential relaxation of degenerated NMR transitions. The mechanism of the excitation was analyzed by means of a tensor operator formalism. Symmetry selection rules were postulated for the relaxation pathways.Dedicated to Professor Dr. Karl Schlögl on occasion of his 65th birthday     

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.     

Microwave-optical double resonance in thioformaldehyde: Further studies of high rovibronic levels of the ground state

Microwave-optical double resonance experiments have been carried out on the J_O,J rotational levels of the 4¹ ùA₂ excited state of thioformaldehyde (H₂CS). The number of microwave transitions to high rovibronic levels of the ground state observed in a given frequency range increases in a roughly linear manner with J, supporting the proposition that K_a is not a good quantum number for these high rovibronic levels.     

On the evaluation of the geometrical factors utilized in ligand polarization calculations

The utility of the Ligand polarization model in solving many physical problems in quantum mechanics has been appreciated among scientists during the last years. Problems such as electric dipole strength, vibronic electric dipole strength, optical activity calculations have been carried out within the framework of a dynamic coupling mechanism. Taking advantage of the irreducible tensor method put forward by Griffith in the case of molecular symmetry groups, both the molecular states and relevant operators can be classified in terms of irreducible representations of the molecular group in question, and therefore it is most convenient to express the relevant operators involved in any specific calculation in a symmetry adapted form. As a starting point, we may classify our molecular states and operators in the 0-rotation group and lower symmetry groups may also be studied by using simple correlation properties. Here we aim to deal with d-d and f-f type of transitions, and hence the 2² (electric quadrupole), 2⁴ (electric hexadecapole) and the 2⁶-multipoles are considered in some detail. We have adopted, the octahedral set of functions as given by Griffith to define the 2^itl (l = 2, 4, 6) multipoles and obtain the corresponding geometrical factors for the various irreducible representations.     

Torsional symmetry dependence of S1 dynamics in molecules that undergo methyl internal rotation

The single‐rovibronic‐level fluorescence of “intermediate‐case” molecules that undergo methyl internal rotation is strongly influenced by the torsional symmetry of the lowest excited singlet state (S₁). The most dramatic example of such symmetry dependence comes from our recent finding that the intensities of the e–e transitions in the high‐resolution S₁←S₀ fluorescence excitation spectra of jet‐cooled acetaldehyde become very weak relative to the a–a transitions at higher beam temperatures. In this study, we rationalize this remarkable torsional symmetry dependence of electronic relaxation in acetaldehyde on the basis of internal‐overall rotation coupling that leads to symmetry‐selective increase in the density of states for singlet‐triplet coupling. Related observations by others on aliphatic carbonyls and diazabenzenes are also discussed within the context of the coupling between the internal and overall rotation. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 167–176, 1999     

The dielectric function of LnSF rare-earth fluorosulfides (Ln=La, Ce): experiment and theory

The absorption properties of LaSF and CeSF in the UV-Visible range have been investigated on the basis of first principles density functional theory (DFT) calculations and from electron energy-loss spectroscopy (EELS) measurements. The extinction coefficient k, as well as the refractive index n, determined experimentally from the loss function Im(−1/ε), were compared with the corresponding factors extracted from the calculated dielectric tensor ε₂. The k and n values for the two compounds were expected to be very close to each other, owing to the chemical similarity of La and Ce. However, it was found that the nature of the electronic transitions in LaSF and CeSF strongly influences the k and n values with the result that the refractive index n and the extinction coefficient k are substantially larger for LaSF than for CeSF.     

The Raman spectrum of tetragonal Eu3+CaF2

The Raman spectrum of a tetragonal single crystal of the compound Eu_xCa_1-(3/2)x F₂(where x = 0.002) is reported. The main features of the spectrum are the ⁷F₀ - ⁷F₁ and ⁷F₀ → ⁷F₂ electronic transitions of Eu³⁺ ions located at a site of C₂ symmetry and vibrational transitions which involve movements of the europium ion. The characteristic phonon band of cubic CaF₂ is for the rare-earth doped crystal moved to a higher frequency and the corresponding Raman tensor is that for a crystal of tetragonal symmetry.     

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.     

Propensity rules for orientation by atom impact: II. Many-state description of direct transitions

This paper contains a theoretical analysis of orientation and alignment created in direct, collision-induced transitions among atomic states with arbitrary angular momentum. Using the natural coordinate frame, general propensity rules are derived in the velocity region of maximum transition probability and their range of validity is investigated. The predictions are tested and illustrated by nine-state calculations for Li(n=2, 3) transitions in Li-He collisions.     

Transient excited singlet state absorption in POPOP and dimethyl POPOP

Transient excited singlet state absorption (ESSA) has been measured in POPOP solution in ethanol and dimethyl POPOP solution in toluene at room temperature in the region 4580–6880 Å using a nitrogen laser and nitrogen laser pumped dye laser. Extensive absorption with several submaxima and shoulders, which represent the vibrational structure, has been observed in both molecules in the region covered in the present study. Energy level schemes of the two molecules have been obtained with the help of the ground state absorption and fluorescence spectra recorded for the purpose. The observed structure in the ESSA has been tentatively interpreted to be due to transitions between the different vibrational levels of the lowest excited state S₁ and two other upper singlet electronic states S₃ and S₄ or S_n. The occurrence of transitions from the higher vibrational levels of S₁ in addition to those from its lowest vibrational level could be understood on the basis of the fact that the pump and the probe beams are made to overlap for a period of ∼ 6–7 ns during which they interact simultaneously with the dye molecules. A plausible explanation for the relative variation in the strength of different transitions from S₀ as well as the observed difference (or otherwise) in the strength of these transitions relative to those of S₁ in each molecule is given on the basis of parity/symmetry selection rules. The experimental results obtained and the transition assignments made are being reported for the first time.     

Dual Mechanism for CO(A1Π)-CO(X1 ∑+) Rovibronic Energy Transfer: An OODR-MP1 Spectroscopic Study

The highly versatile OODR-MPI technique is used, mostly without resort to isotope tagging, to study the quantum state specific rovibronic energy transfer between molecules of the same species, specifically CO(A)-CO(XJ. Two types of rovibronic energy transfer mechanisms are found to be prevailing and are differentiated experimentally, i.e., the long range near resonant dipole-dipole interactions and the short range exciplex formation. The first type is characterized, at first unexpectedly, by its compliance to the selection rules of optical dipole transitions, ΔJ = 0, ±1 and parity + ? ?. Numerical computation procedures are presented for the calculation of rovibronic energy transfer cross-sections when the two types of mechanism are operating concurrently. The computed results and their variation with J are in good accord with the experiments.     

CARS spectroscopy of the NaH2 collision complex: the nature of the Na(32 P)H2 exciplex — ab initio calculations and experimental results

CARS (Coherent Anti-Stokes Raman Scattering) has been used to analyze the rovibronic state distribution of H₂ after collision with Na(3² P). New lines, which do not correspond to H₂ lines are observed in the CARS spectrum. The experiments point to the formation of a complex of Na(3² P)H₂ inA ² B ₂ symmetry. Ab initio calculations of theA ² B ₂ potential were performed. On this surface the vibrational spectra of the exciplex are evaluated. The observed lines can be attributed to vibrational transitions in the complex, in which combinational modes are involved. The connection of experimental and theoretical results indicates that a collisionally stabilized exciplex molecule is formed during the quenching process.     

Rotationally resolved laser-induced fluorescence and zeeman quantum beat spectroscopy of the V 1B2 state of jet-cooled CS2

The rotationally resolved laser-induced fluorescence (LIF) excitation spectrum of V system bands (V¹B₂≈X¹Σ¹_g transition) of CS₂ cooled in a supersonic jet has been observed. In a supersonic jet of CS₂/Ar or He mixture, the rotational temperature of CS₂ is reduced to less than 10 K, and thus the LIF excitation spectrum is simplified significantly. Two types of rotational structure are found; one is composed of P and R branch transitions from even J″ levels and the other is of P, Q, and R branch transitions from even as well as J″. The bands with the former rotational structure are assigned to transitions to K′ = O levels of ¹B₂ state, the bands with the latter structure to transitions to K′ = 1 levels from the (O, 1¹, O) level of the electronic ground state, i.e. vibrationally hot bands. This assignment is supported by the further evidence that these hot bands disappear when the supersonic jet includes a third-body gas such as NH₃ which enhances the vibrational relaxation of CS₂. Calculation of transition moments for respective leads to the conclusion that the upper levels of the V system bands are located in the region close to or higher than the potential barrier of the bending vibration of excited CS₂. The radiative lifetime of CS₂ in single rovibronic levels of the ¹B₂ state is in the range of 2–8 μs which is of the same order of magnitude as that calculated from the absorption coefficient. It tends to be longer for higher J levels or for higher vibronic levels. Zeeman quantum beating is observed in the fluorescence decay of excited CS₂ for a number of rovibronic levels under a weak magnetic field, and thus a magnetic moment associated with each rovibronic level can be determined. The g values are around 0.02 and tend to be smaller in higher J levels for some vibronic states. Based on the the observed radiative lifetime and the g value, it is suggested that the ¹B₂ state is perturbed by a spin-rotation interaction with two spin components, A₁ and B₁ of the ³A₂ orbital state besides a strong spin-orbit coupling with the R ³B₂ state.     

General relativistic symmetry of electron spin vorticity

The general relativistic symmetry of quantum electrodynamics (QED) predicts that the spin vorticity of electron contributes to the kinetic momentum of electron. The canonical quantization of QED is performed by using new b-photon, f-electron, and \(f^{c}\)-positron algebras. These algebras work for interacting particles and are useful for nonperturbationally solving the dual Cauchy problems of QED.     

Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

A Basis for Orbital Symmetry Rules

The influence of molecular symmetry on reaction rates is examined with an approach in which reactions are viewed as electronic transitions between states of reacants and products (described, in turn by quasiadiabatic potential surface). The moleculer Hamiltonian is used to derive selection rules for these transitions. The complete Hamilatonian has no useful symmetery. Neglect of non-Born-Oppenheimer and spin-orbit terms (and of other angular momentum coupling terms) leads to an apporixmate Hamiltonian and to selection rules which from the basis of the Woodward-Hoffmann rules. This apporch provides an alternative to the adiabatic potantial surfaces, reaction coordinates, and transition state theory used in more familiar discussions of the Woodward-Hoffmann rules. Further, it provides a particulary clear method for discussing violations of these symmetry rules, and for differentiating concerted and nonconcerted reactions.     

Selection rules in alternant systems

The CI space X_n generated by n electrons moving over 2n spin orbitals is considered. It is shown that the transitions between different eigenstates ψ_i ? X_n of alternant and weakly alternant Hamiltonians are governed by some special selection rules. These selection rules are characteristic to alternant systems, and they do not apply to nonalternant systems. The set of all such selection rules can be easily derived from the splitting theorem. In particular, the selection rules associated with spin independent alternant systems are considered. As an example, the PPP Hamiltonian ?_p describing netural alternant hydrocarbons is treated. In the case of electron dipole transitions between eigenstates ψ_i ? X_n of the Hamiltonian ?_p, the selection rules obtained are in agreement with the selection rules derived previously by Pariser and McLachlan.     

Symmetry and selection rules in the Raman spectra of carbon nanotubes

The symmetry of achiral single-walled (n,0) and (n,n) carbon nanotubes (CNTs) was examined and the frequencies and types of vibrations allowed in the Raman spectra were calculated. The vibrational spectrum was evaluated as the eigenvalues of the dynamical matrix at the -point of the Brillouin zone. The selection rules for the Raman active vibrations were estimated by the values of the matrix elements responsible for the intensities of corresponding vibrational transitions. The (n,n)-CNTs with even and odd n values are characterized by five and six allowed Raman active vibrations, respectively. The number of Raman active vibrations for (n,0)-CNTs is five if n is even and eight if n is odd. Detailed analysis of the results obtained is presented for the (10,10)-CNT as an example.     

Self consistent field perturbation theory for Frenkel and charge transfer states of hydrogen bonded crystals

Comparison is presented of energies, oscillator strengths, and polarization directions calculated using the dipole length and dipole velocity formulas for the lowest 4–7ππ ^* transitions in eight 6π-electron molecules of C _s symmetry using various amounts of CI, from singly excited configurations only (SCI) to complete CI (CCI), and also using the time-dependent Hartree-Fock (TDHF) method. The standard simple PPP approximation was used. For the strongest transitions, SCI and especially TDHF give results in fairly good agreement with CCI. For weaker transitions both SCI and TDHF give similar results, quite different from the CCI solutions. Rational methods for selecting a small number of configurations needed to reproduce correct order of excited states (SECI-1) or a somewhat larger number reproducing correct order, energies, oscillator strengths and polarization directions of transitions (SECI-2) are presented. Both, and particularly the latter, give similar results from both dipole length or dipole velocity formulas when Linderberg's relation is used for matrix elements of the linear momentum operator and in this respect resemble the TDHF and CCI methods for which both formulas necessarily give identical results.