scholarly journals Analytical Evaluation for Calculation of Two-Center Franck–Condon Factor and Matrix Elements

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Hüseyin Koç
Keyword(s):  
Transition Probabilities ◽  
Polyatomic Molecules ◽  
Harmonic Oscillators ◽  
Binomial Coefficients ◽  
Matrix Elements ◽  
Molecular Physics ◽  
Line Intensities ◽  
Vibrational Levels ◽  
Analytical Formulas ◽  
Franck Condon

The Franck–Condon (FC) factor is defined as squares of the Franck–Condon (FC) overlap integral and represents one of the principle fundamental factors of molecular physics. The FC factor is used to determine the transition probabilities in different vibrational levels of the two electronic states and the spectral line intensities of diatomic and polyatomic molecules. In this study, new analytical formulas were derived to calculate Franck–Condon integral (FCI) of harmonic oscillators and matrix elements (xη, e−2cx, and e−cx2) including simple finite summations of binomial coefficients. These formulas are valid for arbitrary values. The results of formulas are in agreement with the results in the literature.

MULTIPOLE MATRIX ELEMENTS ν FOR H-LIKE ATOMS, THEIR ASYMPTOTICS AND APPLICATIONS (AS β=1, n ≤ 4, n' ≤ 10)

2009 ◽  
Vol 23 (08) ◽  
pp. 2041-2067 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Singular Point ◽  
Transition Probabilities ◽  
Nuclear Charge ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Auxiliary Parameter ◽  
Radial Equation ◽  
Effective Nuclear Charge ◽  
Line Intensities

This article deals with the connection between multipole matrix elements <nl|rβ|n'l' >ν for H-like atoms and new properties of Appell's function F2(x,y) to the vicinity of the singular point (1, 1), where ν is the so-called "auxiliary" parameter of Heun-Schrödinger's radial equation, |1 - ν| = o(1), [Formula: see text] is the "effective" nuclear charge. Exact numerical values for the dipole matrix elements, the average oscillator strengths, the transition probabilities and the line intensities, as n ≤ 4 and n' ≤ 10, in the form of regular rational fractions are given (in Tables 1–4), that make more precise the well-known Tables 13–16 by Hans A. Bethe.

EELS white line intensities calculated for the 3d and 4d metals

1989 ◽  
Vol 47 ◽  
pp. 388-389
Author(s):  
C. C. Ahn ◽  
D. H. Pearson ◽  
P. Rez ◽  
B. Fultz
Keyword(s):  
Experimental Data ◽  
Line Intensity ◽  
Transition Probabilities ◽  
Initial Increase ◽  
White Line ◽  
Hartree Fock ◽  
Line Intensities ◽  
Integrated Intensity ◽  
X Ray Absorption ◽  
The Continuum

Previous experimental measurements of the total white line intensities from L2,3 energy loss spectra of 3d transition metals reported a linear dependence of the white line intensity on 3d occupancy. These results are inconsistent, however, with behavior inferred from relativistic one electron Dirac-Fock calculations, which show an initial increase followed by a decrease of total white line intensity across the 3d series. This inconsistency with experimental data is especially puzzling in light of work by Thole, et al., which successfully calculates x-ray absorption spectra of the lanthanide M4,5 white lines by employing a less rigorous Hartree-Fock calculation with relativistic corrections based on the work of Cowan. When restricted to transitions allowed by dipole selection rules, the calculated spectra of the lanthanide M4,5 white lines show a decreasing intensity as a function of Z that was consistent with the available experimental data.Here we report the results of Dirac-Fock calculations of the L2,3 white lines of the 3d and 4d elements, and compare the results to the experimental work of Pearson et al. In a previous study, similar calculations helped to account for the non-statistical behavior of L3/L2 ratios of the 3d metals. We assumed that all metals had a single 4s electron. Because these calculations provide absolute transition probabilities, to compare the calculated white line intensities to the experimental data, we normalized the calculated intensities to the intensity of the continuum above the L3 edges. The continuum intensity was obtained by Hartree-Slater calculations, and the normalization factor for the white line intensities was the integrated intensity in an energy window of fixed width and position above the L3 edge of each element.

On the Theory of Thermally Weighted Electron-Phonon Transition Probabilities

1980 ◽  
Vol 35 (9) ◽  
pp. 902-914
Author(s):  
J. Schupfner
Keyword(s):  
Transition Probabilities ◽  
Algebraic Method ◽  
Calculation Technique ◽  
Radiative Transitions ◽  
Special Cases ◽  
Explicit Formulae ◽  
The Common ◽  
Common Transition ◽  
Franck Condon ◽  
Refined Calculation

Abstract We present a refined calculation method for the phonon part (Franck-Condon Overlaps) of the transition probabilities of electron-phonon radiative and non-radiative transitions in crystals. The evaluation of the thermal averaged Franck-Condon integrals is a purely algebraic method and the transition probabilities we use are derived from first principles and completely atomistic. For the electronic transitions we take into account the frequency shift of the lattice and the change of the phonon normal coordinates. Explicit formulae of the phonon parts are derived and it is shown that the common transition probabilities used in literature are special cases of our functional calculation technique.

THEORETICAL TRANSITION PROBABILITIES FOR THE $\tilde{A}^{2}A_{1} -\tilde{X}^{2}B_{1}$ SYSTEM OF H2O+ AND D2O+ AND RELATED FRANCK–CONDON FACTORS BASED ON GLOBAL POTENTIAL ENERGY SURFACES

2005 ◽  
Vol 04 (01) ◽  
pp. 225-245 ◽  
Author(s):  
IKUO TOKUE ◽  
KATSUYOSHI YAMASAKI ◽  
SATOSHI MINAMINO ◽  
SHINKOH NANBU
Keyword(s):  
Potential Energy ◽  
Photoelectron Spectroscopy ◽  
Potential Energy Surfaces ◽  
Least Squares Method ◽  
Transition Probabilities ◽  
Three Dimensional ◽  
Equilibrium Geometry ◽  
Condon Factors ◽  
Energy Surfaces ◽  
Franck Condon

To elucidate the ionization dynamics, in particular the vibrational distribution, of H 2 O +(Ã) produced by photoionization and the Penning ionization of H 2 O and D 2 O with He *(2 3S) atoms, Franck–Condon factors (FCFs) were given for the [Formula: see text] ionization, and the transition probabilities were presented for the [Formula: see text] emission. The FCFs were obtained by quantum vibrational calculations using the three-dimensional potential energy surfaces (PESs) of [Formula: see text] and [Formula: see text] electronic states. The global PESs were determined by the multi-reference configuration interaction calculations with the Davidson correction and the interpolant moving least squares method combined with the Shepard interpolation. The obtained FCFs exhibit that the [Formula: see text] state primarily populates the vibrational ground state, as its equilibrium geometry is almost equal to that of [Formula: see text], while the bending mode (ν2) is strongly enhanced for the H 2 O +(Ã) state; the maximums in the population of H 2 O + and D 2 O + are approximately v2 = 11–12 and 15–17, respectively. These results are consistent with the distributions observed by photoelectron spectroscopy. Transition probabilities for the [Formula: see text] system of H 2 O + and D 2 O + show that the bending progressions consist primarily of the [Formula: see text] emission, with combination bands from the (1, v′2 = 4–8, 0) level being next most important.

scholarly journals Large Scale Shell Model Calculations of the Negative-Parity States Structure in 24Mg Nucleus

2021 ◽  
Vol 66 (4) ◽  
pp. 293
Author(s):  
A.A. Al-Sammarraie ◽  
F.A. Ahmed ◽  
A.A. Okhunov
Keyword(s):  
Shell Model ◽  
Large Scale ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Energy Levels ◽  
Form Factors ◽  
Model Space ◽  
Negative Parity ◽  
Matrix Elements ◽  
Model Calculations

The negative-parity states of 24Mg nucleus are investigated within the shell model. We are based on the calculations of energy levels, total squared form factors, and transition probability using the p-sd-pf (PSDPF) Hamiltonian in a large model space (0 + 1) hW. The comparison between the experimental and theoretical states showed a good agreement within a truncated model space. The PSDPF-based calculations successfully reproduced the data on the total squared form factors and transition probabilities of the negative-parity states in 24Mg nucleus. These quantities depend on the one-body density matrix elements that are obtained from the PSDPF Hamiltonian. The wave functions of radial one-particle matrix elements calculated with the harmonic-oscillator potential are suitable to predict experimental data by changing the center-of-mass corrections.

Transition dipole matrix elements for 14NH3 from the line intensities of the 2ν2 and ν4 bands

1984 ◽  
Vol 106 (1) ◽  
pp. 38-55 ◽  
Author(s):  
Š. Urban ◽  
D. Papoušek ◽  
V. Malathy Devi ◽  
B. Fridovich ◽  
Romola D'Cunha ◽  
...  
Keyword(s):  
Matrix Elements ◽  
Transition Dipole ◽  
Line Intensities ◽  
Transition Dipole Matrix
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