The magnetic rotation spectrum of formaldehyde: singlet–triplet perturbations in the 2141, 2143, 2243, and 2341 levels of the à lA2 state of H2CO

1983 ◽  
Vol 61 (1) ◽  
pp. 6-14 ◽  
Author(s):  
C. M. L. Kerr ◽  
D. C. Moule ◽  
D. A. Ramsay
Keyword(s):  
Magnetic Activity ◽  
Triplet States ◽  
Magnetic Rotation ◽  
Matrix Elements ◽  
Molecular Constants ◽  
Absorption Bands ◽  
Vibrational Assignments ◽  
The Matrix ◽  
Upper Level ◽  
Rotation Spectrum

The absorption spectra and magnetic rotation spectra of the [Formula: see text],[Formula: see text],[Formula: see text], and [Formula: see text] bands of the [Formula: see text] system of formaldehyde have been studied at high resolution. The earlier analyses of the absorption bands have been refined and improved molecular constants obtained. Several singlet–singlet perturbations have been noted and possible mechanisms discussed.The magnetic activity falls into two classes. Three of the bands show limited singlet–singlet activity with characteristics similar to those found earlier for the [Formula: see text] and [Formula: see text] bands. All the bands show singlet–triplet activity which is characterized by the appearance of rR, rP, pR, and PP transitions associated with a common upper level. Tables of perturbed excited-state singlet levels are given. No appreciable shifts of the perturbed levels have been found indicating that the matrix elements are small. For the [Formula: see text] band two perturbation mechanisms are proposed and the vibrational symmetries and energies of the perturbing triplet states are determined. The energies are too high to permit unambiguous vibrational assignments.

The magnetic rotation spectrum of formaldehyde: singlet–triplet perturbations in the 41 and 43 levels of the Ã1A2 state of H2CO

1979 ◽  
Vol 57 (8) ◽  
pp. 1224-1232 ◽  
Author(s):  
D. A. Ramsay ◽  
S. M. Till
Keyword(s):  
Absorption Spectrum ◽  
High Resolution ◽  
Rotational Level ◽  
Magnetic Rotation ◽  
Matrix Elements ◽  
Coriolis Interaction ◽  
Near Ultraviolet ◽  
The Matrix ◽  
Rotation Spectrum

A survey of the magnetic rotation spectrum (MRS) of formaldehyde in the near ultraviolet is reported. The [Formula: see text] and [Formula: see text] bands of the [Formula: see text] system are also studied under high resolution. The rR and pP branches of these bands are found to be active and it is noted that the intensities appear to increase more rapidly with increasing Ka in the MRS than in the absorption spectrum. The 43 and 61 levels are in Coriolis interaction at higher Ka values, and some rR branches of the [Formula: see text] band are identified.Several strong lines in the MRS are shown to be associated with rR, rP, pR, and pP lines in the absorption spectrum and to reveal the presence of singlet–triplet perturbations. Only one perturbed rotational level is found in the [Formula: see text] band, but for the [Formula: see text] band several rotational levels are found to exhibit such perturbations, e.g. 190,19, 191,19, 171,16, 102,8, 102,9, 112,9, 173,14, 173,15, 183,15, and 183,16. These perturbations are produced by ΔJ = ΔN = ΔKa = 0 interactions with the 415161 level of the ã3A2 state and by ΔJ = ΔN = 0. ΔKa = ± 1 interactions with the 5161 level. The matrix elements are of the order of 0.01–0.10 cm−1.

ON THE SPIN–ORBIT INTERACTION IN DIATOMIC MOLECULES. I

1958 ◽  
Vol 36 (3) ◽  
pp. 309-328 ◽  
Author(s):  
I. Kovács
Keyword(s):  
Diatomic Molecules ◽  
Electron System ◽  
Orbit Interaction ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Molecular Constants ◽  
Spin Orbit Interaction ◽  
The Matrix

The purpose of the present paper is to give further details of an investigation of the spin–orbit interaction in diatomic molecules. The first part of the paper deals with perturbations between states of odd multiplicity in a two-electron system. With the aid of the matrix elements which have been obtained, a comparison of the perturbations in the various components of a given state and in some cases a comparison of perturbations in two different states is possible. Also for certain cases it has been possible to establish relationships between the matrix elements which give the magnitude of the perturbations and experimentally measurable molecular constants of the perturbed states.

The Zeeman Effect in Triplet States of Rotating Asymmetric Molecules

1975 ◽  
Vol 53 (19) ◽  
pp. 1853-1860 ◽  
Author(s):  
J. C. D. Brand ◽  
C. di Lauro ◽  
D. S. Liu
Keyword(s):  
Magnetic Field ◽  
Field Theory ◽  
Energy Levels ◽  
Zeeman Effect ◽  
Homogeneous Magnetic Field ◽  
Triplet States ◽  
Matrix Elements ◽  
The Matrix ◽  
Asymmetric Rotor ◽  
Magnetic Tuning

Intermediate field theory is used to obtain the matrix elements which determine the action of a homogeneous magnetic field on the energy levels of triplet states of asymmetric rotor molecules. Applications of these formulas are discussed (i) in relation to the Zeeman effect on the rotational fine structure of triplet–singlet transitions, where conditions are identified under which individual lines remain unbroadened by the field, and (ii) in connection with the magnetic tuning of singlet–triplet resonance.

The magnetic rotation spectrum of thioformaldehyde: singlet–triplet perturbations in the Ã1A2 state

1982 ◽  
Vol 60 (9) ◽  
pp. 1212-1222 ◽  
Author(s):  
D. J. Clouthier ◽  
D. C. Moule ◽  
D. A. Ramsay ◽  
F. W. Birss
Keyword(s):  
High Resolution ◽  
Vibrational Level ◽  
Interaction Matrix ◽  
Magnetic Rotation ◽  
Matrix Elements ◽  
Selection Rules ◽  
Rotation Spectrum

Singlet–triplet perturbations in the [Formula: see text] and [Formula: see text] bands of the [Formula: see text], system of H2CS have been studied by high resolution absorption and magnetic rotation spectroscopy. Extensive perturbations in the [Formula: see text] band are associated with corresponding activity in the magnetic rotation spectrum. The shifts of the levels are consistent with interaction matrix elements up to ~ 1 cm−1. The selection rules indicate that the zeroth vibrational level of the 1A2 state is perturbed by the 4361 level of the 3A2 state. A number of perturbations are also observed in the [Formula: see text] band.

The Band System of Propynal: Rotational Analysis of the 0–0 Band Near 4145 Å

1973 ◽  
Vol 51 (17) ◽  
pp. 1810-1814 ◽  
Author(s):  
F. W. Birss ◽  
Ronald Y. Dong ◽  
D. A. Ramsay
Keyword(s):  
High Resolution ◽  
Band System ◽  
Transition Moment ◽  
Magnetic Rotation ◽  
Rotational Analysis ◽  
Molecular Constants ◽  
Rotation Spectrum

The 0–0 band of the [Formula: see text] system of propynal near 4145 Å has been photographed under high resolution and a rotational analysis carried out. The principal molecular constants for the ã3A″ state are (in cm−1):[Formula: see text]Altogether, 1237 lines have been assigned to transitions with ΔKa = 0. The dominant transition moment involves mixing of the ã3A″ state with higher 1A′ states.A simple magnetic rotation spectrum has been obtained and the assignments of the lines discussed.

Absorption Spectrum of the NO Molecule. X. The 3d Rydberg Complex, its Vibrational Structure, Spin–Orbit Coupling, and Interactions with Non-Rydberg States

1971 ◽  
Vol 49 (18) ◽  
pp. 2350-2365 ◽  
Author(s):  
E. Miescher
Keyword(s):  
Vacuum Ultraviolet ◽  
Rydberg States ◽  
Vibrational Structure ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Absorption Bands ◽  
Unusual Structure ◽  
Vibrational Levels ◽  
The Matrix

The vibrational levels of the 3dσ,π Rydberg complexes of the NO molecule are studied. They give rise to absorption bands 3d–X in the vacuum ultraviolet, most of which are severely overlapped by other bands, and, in addition, to Rydberg–Rydberg emission bands in the visible (3d–3s) and in the infrared (3d–3p). Both types of spectra have been photographed at high resolution, the absorption spectra for three different isotopes.Interactions with non-Rydberg levels and predissociations are observed. A detailed analysis, including spin–orbit effects, is given of the crossing between the 3dσ,π(ν = 1) and B2Π(ν = 23) rotational levels. Off-diagonal matrix elements are determined and precise vibrational and rotational constants are calculated. Largely responsible for the unusual structure of the bands is the matrix element [Formula: see text] which is shown to be independent of ν.

ON THE SPIN–ORBIT INTERACTION IN DIATOMIC MOLECULES. II

1958 ◽  
Vol 36 (3) ◽  
pp. 329-351 ◽  
Author(s):  
I. Kovács
Keyword(s):  
Diatomic Molecules ◽  
Electron System ◽  
Orbit Interaction ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Molecular Constants ◽  
Spin Orbit Interaction ◽  
The Matrix

This paper deals with perturbations between states of even multiplicity in a three-electron system. With the aid of the matrix elements which have been obtained, a comparison of the perturbations in the various components of a given state and in some cases a comparison of perturbations in two different states is possible. Also for certain cases it has been possible to establish relationships between the matrix elements which give the magnitude of the perturbations and experimentally measurable molecular constants of the perturbed states.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

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