Vibrational Overlap Integrals Research Articles

Non-radiative processes and luminescence quenching in Mn4+ doped phosphors

We present the temperature dependence of the 2E→4A2 emission of Na2TiF6:Mn4+. Our results are discussed in the framework of the single configurational coordinate diagram (SCCD) model proposed by Struck and Fonger. We have critically analyzed the SCCD model and calculated the vibrational overlap integrals directly from the Manneback recurrence formulas. We have considered the physical origin of shift of the excited state parabolae in the configurational space as well as possible different slopes of the ground and excited electronic manifolds. We have compared our results with numerous results obtained for Mn4+ in other lattices available in the literature. We have found the high inconsistency between the experimental results and values obtained under the SCCD model. Moreover, we have found strong correlation between the values of experimental activation energies Enr and values of constant describing the probability of nonradiative process (i.e. frequency constant). We have considered interaction with lattice phonons described by power dependence defined by exponent α, related to dimension of phonon space and effective phonon energy ℏωeff. We found α≈7.42 and ℏωeff = 315 cm −1 for Enr < 4000 cm−1 and α≈19.73, and ℏωeff =1130 cm−1 for Enr > 4000 cm−1.

Electronic Predissociation in the Dichloromethane Cation CH2Cl2+ Electronic State 2A1.

The loss of a Cl atom from metastable CH2Cl2+ in the mass-analyzed ion kinetic energy experiment is characterized by a borderline zero kinetic energy release and large kinetic isotope effects on chlorine and hydrogen. Ab initio calculations are employed to assist the interpretation in terms of a nonadiabatic reaction involving electronic predissociation of the electronically excited state 2A1 and two-dimensional reaction dynamics. Strong curvature in the reaction coordinate leads to a bobsled effect that accounts for the low kinetic energy release. The kinetic isotope effects enter via the predissociation rate and are interpreted in terms of vibrational overlap integrals.

Fourier transform emission spectra and deperturbation analysis of the A2Π – X2Σ+ and B2Σ+ – X2Σ+ electronic transitions of ZnH

A discharge-furnace emission source was used to generate the A2Π→X2Σ+ and B2Σ+→X2Σ+ spectra of ZnH radical. High resolution emission spectra were recorded with a Fourier transform spectrometer, and several bands have been assigned for the 64ZnH major isotopologue. The data span the v″=0–6 levels of the X2Σ+ ground state, the v′=0–3 levels of the A2Π state, and the v′=0–2 levels of the B2Σ+ state, extending to high rotational quantum numbers near and above the dissociation asymptote of the ground state. Large local perturbations were observed in the A2Π and B2Σ+ electronic states, and a deperturbation analysis was carried out using a single Hamiltonian matrix that includes 2Π and 2Σ+ matrix elements, as well as off-diagonal elements coupling vibrational levels of the two electronic states. Band constants and Dunham coefficients were obtained for the A2Π and B2Σ+ excited states by least-squares-fitting of all the experimental data. The equilibrium vibrational constants ωe and ωexe have been determined to be 1907.528(4) and 38.674(2)cm−1, respectively, for the A2Π state, and 1021.135(94) and 17.725(80)cm−1, for the B2Σ+ state, and the equilibrium Zn–H distances (re) are 1.511662(2)Å and 2.26805(7)Å for the A2Π and B2Σ+ states, respectively. The RKR potential curves were constructed for the A2Π and B2Σ+ states, and vibrational radial overlap integrals were computed. The off-diagonal matrix elements coupling the electronic wavefunctions of the A2Π and B2Σ+ states, i.e., a+ and b, were determined to be 228±3cm−1 and 0.73±0.01, respectively, for the ZnH molecule.

Rotational analysis and deperturbation of the A2Π → X2Σ+ and B′2Σ+ → X2Σ+ emission spectra of MgD

High resolution Fourier transform emission spectra of MgD radical were analyzed, and the A2Π → X2Σ+ bands with v′ = 2 and v′ = 3 were rotationally assigned. Several local perturbations were observed in the A2Π and B′2Σ+ excited states of 24MgD, and a deperturbation analysis was carried out using an appropriate Hamiltonian matrix containing off-diagonal terms connecting the vibrational levels of the two states. Dunham coefficients and band constants were determined for the A2Π and B′2Σ+ states, along with off-diagonal parameters. The equilibrium vibrational constants ωe and ωexe have been determined to be 1155.040(6) and 16.764(4) cm−1, respectively, for the A2Π state, and 598.108(11) and 6.394(8) cm−1, for the B′2Σ+ state. The equilibrium Mg–D distances were found to be 1.67819(3) Å and 2.59355(2) Å for the A2Π and B′2Σ+ states, respectively. RKR potential curves were constructed for the A2Π and B′2Σ+ states, and vibrational radial overlap integrals were computed for the perturbed levels. The off-diagonal matrix elements coupling the electronic wavefunctions of the A2Π and B′2Σ+ states were determined independently for MgD to be a+ = 18.9 ± 0.2 cm−1 and b = 0.694 ± 0.005, in excellent agreement with those for the MgH isotopologue.

Full dimensional Franck-Condon factors for the acetylene à (1)Au-X̃ (1)Σ(g)(+) transition. II. Vibrational overlap factors for levels involving excitation in ungerade modes.

A full-dimensional Franck-Condon calculation has been applied to the à (1)Au-X̃ 1Σg+ transition in acetylene in the harmonic normal mode basis. Details of the calculation are discussed in Part I of this series. To our knowledge, this is the first full-dimensional Franck-Condon calculation on a tetra-atomic molecule undergoing a linear-to-bent geometry change. In the current work, the vibrational intensity factors for levels involving excitation in ungerade vibrational modes are evaluated. Because the Franck-Condon integral accumulates away from the linear geometry, we have been able to treat the out-of-plane component of trans bend (ν4('')) in the linear X̃ state in the rotational part of the problem, restoring the χ Euler angle and the a-axis Eckart conditions. A consequence of the Eckart conditions is that the out-of-plane component of ν4('') does not participate in the vibrational overlap integral. This affects the structure of the coordinate transformation and the symmetry of the vibrational wavefunctions used in the overlap integral, and results in propensity rules involving the bending modes of the X̃ state that were not previously understood. We explain the origin of some of the unexpected propensities observed in IR-UV laser-induced fluorescence spectra, and we calculate emission intensities from bending levels of the à state into bending levels of the X̃ state, using normal bending mode and local bending mode basis sets. Our calculations also reveal Franck-Condon propensities for the Cartesian components of the cis bend (ν5('')), and we predict that the best Ã-state vibrational levels for populating X̃-state levels with large amplitude bending motion localized in a single C-H bond (the acetylene↔vinylidene isomerization coordinate) involve a high degree of excitation in ν6(') (cis-bend). Mode ν4(') (torsion) populates levels with large amplitude counter-rotational motion of the two hydrogen atoms.

Full dimensional Franck-Condon factors for the acetylene à (1)A(u)-X̃ (1)Σ(g)(+) transition. I. Method for calculating polyatomic linear-bent vibrational intensity factors and evaluation of calculated intensities for the gerade vibrational modes in acetylene.

Franck-Condon vibrational overlap integrals for the à Au1-X̃ 1Σg+ transition in acetylene have been calculated in full dimension in the harmonic normal mode basis. The calculation uses the method of generating functions first developed for polyatomic Franck-Condon factors by Sharp and Rosenstock [J. Chem. Phys. 41(11), 3453-3463 (1964)], and previously applied to acetylene by Watson [J. Mol. Spectrosc. 207(2), 276-284 (2001)] in a reduced-dimension calculation. Because the transition involves a large change in the equilibrium geometry of the electronic states, two different types of corrections to the coordinate transformation are considered to first order: corrections for axis-switching between the Cartesian molecular frames and corrections for the curvilinear nature of the normal modes at large amplitude. The angular factor in the wavefunction for the out-of-plane component of the trans bending mode, ν4(″), is treated as a rotation, which results in an Eckart constraint on the polar coordinates of the bending modes. To simplify the calculation, the other degenerate bending mode, ν5(″), is integrated in the Cartesian basis and later transformed to the constrained polar coordinate basis, restoring the conventional v and l quantum numbers. An updated Ã-state harmonic force field obtained recently in the R. W. Field research group is evaluated. The results for transitions involving the gerade vibrational modes are in qualitative agreement with experiment. Calculated results for transitions involving ungerade modes are presented in Paper II of this series [G. B. Park, J. H. Baraban, and R. W. Field, "Full dimensional Franck-Condon factors for the acetylene à Au1-X̃ 1Σg+ transition. II. Vibrational overlap factors for levels involving excitation in ungerade modes," J. Chem. Phys. 141, 134305 (2014)].

Franck–Condon Factors for Diatomics: Insights and Analysis Using the Fourier Grid Hamiltonian Method

Franck–Condon factors (FCFs) play a crucial role in determining the intensities of the vibrational bands in electronic transitions. In this article, a relatively simple method to calculate the FCFs is illustrated. An algorithm for the Fourier Grid Hamiltonian (FGH) method for computing the vibrational wave functions and the corresponding energy values for different electronic states of diatomics is outlined. For these computations, the Morse potential energy forms are used for constructing and diagonalizing the molecular Hamiltonians. Once the vibrational wave functions for the ground and the excited states are known, the vibrational overlap integrals are calculated by using the formula ∫ψ*v″ψv′ dτn, where dτn is the volume element for nuclear coordinate, v′ and v″ are the vibrational quantum numbers in the ground and exited electronic states, and ψv′ and ψv″ denote the vibrational wave functions. The effects of the changes in the location, height, and the nature of the excited state electronic energy curve (relative to the ground state) on the FCFs have been evaluated. The method is also illustrated for N2 and O2. This method can be effectively used for introducing the FCFs to the students of undergraduate molecular spectroscopy course and also for post-graduate students.

Rotational analysis and deperturbation of the A 2Π → X 2Σ+ and B′ 2Σ+ → X 2Σ+ emission spectra of MgH

Deperturbation analysis of the A(2)Π → X(2)Σ(+) and B(')(2)Σ(+) → X(2)Σ(+) emission spectra of (24)MgH is reported. Spectroscopic data for the v = 0 to 3 levels of the A (2)Π state and the v = 0 to 4 levels of the B'(2)Σ(+) state were fitted together using a single Hamiltonian matrix that includes (2)Π and (2)Σ(+) matrix elements, as well as off-diagonal elements coupling several vibrational levels of the two states. A Dunham-type fit was performed and the resulting Y(l,0) and Y(l,1) coefficients were used to generate Rydberg-Klein-Rees (RKR) potential curves for the A (2)Π and the B'(2)Σ(+) states. Vibrational overlap integrals were computed from the RKR potentials, and the off-diagonal matrix elements coupling the electronic wavefunctions (a(+) and b) were determined. Zero point dissociation energies (D(0)) of the A(2)Π and B'(2)Σ(+) states of (24)MgH were determined to be 12,957.5 ± 0.5 and 10,133.6 ± 0.5 cm(-1), respectively. Using the Y(0,1) coefficients, the equilibrium internuclear distances (r(e)) of the A(2)Π and B'(2)Σ(+) states were determined to be 1.67827(1) Å and 2.59404(4) Å, respectively.

A general analytical expression for the two-dimensional Franck–Condon integral and simulation of the photoelectron spectra of nitrogen dioxide

On the basis of the harmonic oscillator approximation, the two-dimensional four-mode Franck–Condon vibrational overlap integral under mode-mixing effects has been reduced to a product expression of one-dimensional Gaussian integrals with the aid of expanding Hermite polynomials. A more general analytical expression for the calculation of the two-dimensional four-mode Franck–Condon overlap integrals was derived by directly calculating the Gaussian integrals. This analytical expression was applied to investigate the photoelectron spectra of considering mode-mixing and hot band effects. Geometry optimizations and harmonic vibrational frequency calculations were performed on the 2A1 state of NO2 and the 1A1 state of . Franck–Condon analyses and spectral simulations were carried out on the NO2( 2A1)–( 1A1) photodetachment processes. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectra. In addition, the equilibrium geometric parameters of the 1A1 state of were obtained in the spectral simulations: bond length R(ON) = 1.248 ± 0.005 Å and bond angle ∠ (ONO) = 116.8 ± 0.5°.

The role of rotational relaxation in the intersystem crossing between a triplet and a singlet electronic state

AbstractA calculation of the rate of ISC (intersystem crossing) relaxation at 0 K requires a knowledge of the spin–orbit coupling and the vibrational overlap integrals (Franck‐Condon factors) between the T1 and S0 states. At elevated temperatures, the population of the rotational states needs to be taken into account through the overlap of the rotational wavefunctions between the interacting T1/S0 electronic states. Here, we discuss the effect of the ISC singlet–triplet selection rules on the number and magnitude of the rotational overlap factors for the J = 0, 5, and 10 rotational levels of thiophosgene, Cl2CS. Of equal importance is the influence of the spin–spin and spin‐rotation coupling on the rotational fragmentation factors and the ISC survival probability. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Resonance structure in electron–N2 scattering around 11.5 eV: high-resolution measurements, ab initio calculations and line shape analyses

Using two different experimental setups with energy widths of about 6 and 13 meV, we obtained significantly improved results for the energy dependence of angle-differential (10°–180°) elastic and vibrationally inelastic cross sections for electron scattering from N2 molecules in the energy range around the narrow N−2(R2Σ+g) resonance. The energy location and the natural width of this resonance are determined as 11.497(2) eV and 1.3(2) meV, respectively. Ab initio potential energy curves are obtained from CCSD(T) calculations for the neutral N2(X1Σ+g) and N*2(E3Σ+g) states as well as for the N−2(R2Σ+g) resonance state. They corroborate quite accurately the measured resonance energy and provide accurate energy spacings and overlap integrals for the pertinent vibrational states. A detailed analysis of resonance line shapes for selected scattering angles is performed by applying a model for the interference of resonant and nonresonant scattering processes. It provides a link between the resonance width to absolute DCS and describes elastic and vibrational excitation processes on a common basis. Through both their size and sign, vibrational overlap integrals are shown to determine the observed Fano-type line shapes and account for the opposite asymmetries and intensity changes of adjacent vibrational resonance peaks. Fine-tuning of the fits to the observed shapes is achieved by proper parametrization of the nonresonant amplitudes. A highly resolved excitation function for the formation of the metastable N*2(E3Σ+g) level from threshold (near 11.88 eV) to 13.4 eV is also presented.

Studies of intersystem crossing dynamics in acetylene

We report a new ab initio study of the acetylene T3 potential energy surface, which clarifies the nature of its energy minimum, and present computed equilibrium geometries and diabatic frequencies. This information enables the computation of harmonic vibrational overlap integrals of T3 vibrational levels with the S1 3nu3 state. The results of this calculation support the interpretation of two local perturbations of S1 3nu3, revealed in ultraviolet laser-induced fluorescence/surface electron ejection by laser excited metastables spectroscopy and Zeeman anticrossing measurements, respectively, as arising from two rotational submanifolds of a single T3 vibrational state. We present plausible assignments for this state as a guide for future experimental work.

The NaK 1(b)ΠΩ=3 state hyperfine structure and the 1(b)ΠΩ=3∼2(A)Σ+1 spin–orbit interaction

We have measured the hyperfine structure of mutually perturbing rovibrational levels of the 1(b) 3Pi0 and 2(A) 1Sigma+ states of the NaK molecule, using the perturbation-facilitated optical-optical double resonance method with copropagating lasers. The unperturbed 1(b) 3Pi0 levels are split into four hyperfine components by the Fermi contact interaction bFIS. Mixing between the 1(b) 3Pi0 and 2(A) 1Sigma+ levels imparts hyperfine structure to the nominally singlet component of the perturbed levels and reduces the hyperfine splitting of the nominally triplet component. Theoretical analysis relates these observations to the hyperfine splitting that each 1(b) 3Pi0 level would have if it were not perturbed by a 2(A) 1Sigma+ level. Using this analysis, we demonstrate that significant hyperfine splitting arises because the 1(b) 3Pi0 state cannot be described as pure Hund's case (a). We determine bF for the 1(b) 3Pi0 levels and also a more accurate value for the magnitude of the singlet-triplet spin-orbit coupling HSO=[1(b) 3Pi0(vb,J)(H(SO))2(A) 1Sigma+(vA,J). Using the known spectroscopic constants of the 1(b) 3Pi state, we obtain bF=0.009 89+/-0.000 27 cm(-1). The values of (H(SO)) are found to be between 2 and 3 cm(-1), depending on vb, vA, and J. Dividing (H(SO)) by calculated vibrational overlap integrals, and taking account of the 1(b) 3Pi(Omega) rotational mixing, we can determine the magnitude of the electronic part H(el) of H(SO). Our results yield (H(el))=(16.33+/-0.15) cm(-1), consistent with our previous determinations using different techniques.

Picosecond Relaxation of3MLCT Excited States of [Re(Etpy)(CO)3(dmb)]+and [Re(Cl)(CO)3(bpy)] as Revealed by Time-Resolved Resonance Raman, UV−vis, and IR Absorption Spectroscopy

Picosecond dynamics of Re → polypyridine 3MLCT excited states of [Re(Etpy)(CO)3(dmb)]+ and [Re(Cl)(CO)3(bpy)] were investigated by time-resolved UV−vis, resonance Raman, and IR spectroscopy. Raman bands due to NN•- intraligand vibrations of the excited molecules increase with time during the first 15−20 ps after excitation. The time constant of 6 ± 2 ps was estimated for the increase of areas of excited-state Raman bands of [Re(Etpy)(CO)3(dmb)]+. The growth of Raman bands is accompanied by an increase of the near-UV transient absorption band at 375 nm, which corresponds to a ππ* transition of the dmb•- ligand of the 3MLCT excited state [ReII(Etpy)(CO)3(dmb•-)]+. These effects are attributed to structural reorganization during vibrational cooling, during which the electronic dipole moment and/or vibrational overlap integrals increase. IR bands due to CO stretching vibrations and some of the Raman bands undergo dynamical upward shift and narrowing, that occur with time constants between 1 and 11 ps, manifes...

The Franck–Condon principle, two-photon spectroscopy and the [formula omitted] system of benzene

The application of the Franck–Condon principle to two-photon spectra is discussed, and it is shown that both simple and more complex theories suggest that much of the familiar form of the Herzberg–Teller description for one-photon spectra, and the same vibrational overlap integrals, can be applied to the corresponding two-photon spectra. The results are applied to the well-known A ̃ 1B 2u– X ̃ 1A 1g system of benzene. In the course of this it is shown that the experimental data used in the early Franck–Condon work on the one-photon spectrum are unreliable, and as a result it is necessary to question the usual assumption that a simple Franck–Condon analysis of this spectrum leads to a change in the C–C bond length on excitation similar to that found by rotational analyses. Preliminary work suggests that a single Franck–Condon theory may account for the main features of both the one- and two-photon spectra of benzene if the effective transition moments are allowed to increase with the ν 1 normal co-ordinate. In particular it appears that the capacity of the ν 14( b 2 u ) vibration to induce the two-photon spectrum increases as the carbon ring expands. Further experimental and theoretical work to confirm and understand these results is desirable.

Interference effects in the predissociation of the Cs2 C 1Πu and (2)3Πu states through the dissociative (2)3Σu+ state

The excitation spectrum of the Cs2 molecule monitoring the emission from the dissociated Cs(6p2P3/2) atoms was measured by employing a pulsed supersonic molecular beam crossed with a laser beam. Strong predissociation was found in the 16 600–17 200 cm−1 region. Vibrational progressions with broad asymmetric profiles, which were found via the ionization of dissociated Cs(6p2P3/2) atoms by Kim and Yoshihara, were observed. Transitions from the ground state X 1Σg+ to the dissociative state (2)3Σu+ are allowed by the spin-orbit interaction between the (2)3Σu+ and B 1Πu states. The (2)3Σu+, (2)3Πu, and C 1Πu states are mixed by the spin-orbit interaction. By solving the coupled differential equations using the three-channel model, the transition probability to the mixed levels, (2)3Πu(v) and C 1Πu(v′) was calculated. The coupling between the (2)3Σu+ and (2)3Πu states is larger than that between the (2)3Σu+ and C 1Πu states. Accordingly, the linewidth of the (2)3Πu←X 1Σg+ transition is observed to be wider than that of the C 1Πu←X 1Σg+ transition. The sign of the Fano shape index q was shown to coincide with the sign of the vibrational overlap integral between the discrete level and the continuum.

Electronic nonadiabaticity in highly vibrationally excited O2(X 3Σg−): Spin-orbit coupling between X 3Σg− and b 1Σg+

We report full quantum-state-resolved spectra of highly vibrationally excited O2(X 3Σg−,v=26–31). In addition to providing high precision molecular constants for several new vibrational levels, we observe a local spectral perturbation of X 3Σg−(v=28). We present a deperturbation analysis of the observed spectra and assign the perturber to b 1Σg+(v=19). We predict a crossing between the b 1Σg+ and X 3Σg− state at an internuclear separation R=2.45±0.1 Å, somewhat further extended and higher in energy than the outer classical turning point of O2(X 3Σg−,v=28). Using the appropriate vibrational overlap integral, we are able to determine the spin–orbit interaction between these two electronic states, which is 200±20 cm−1 in the vicinity of the crossing. These results suggest that the collision dynamics of highly vibrationally excited O2(X 3Σg−) may involve excited potential surfaces. Furthermore, they imply that present theoretical approaches to the O4 problem, which use a single potential surface, may not be adequate. Possible implications regarding nonequilibrium models of stratospheric ozone formation and the dynamics of the O+O3→2O2 reaction are discussed.

On the theoretical investigation of vibronic spectra of ethylene by ab initio calculations of the Franck–Condon factors

The vibronic spectra of ethylene have been studied using ab initio molecular orbital methods. Geometries of the singlet π–π*, π–3s, and π–3p excited electronic states have been optimized at the CIS and CASSCF levels of theory with the 6-311(2+)G* basis set. Vertical and adiabatic excitation energies, calculated by the multireference configuration interaction (MRCI) and equation-of-motion coupled cluster (EOM-CCSD) methods are in quantitative agreement with experiment. Vibrational frequencies and normal coordinates for the ground and excited states are used for the calculations of vibrational overlap integrals and Franck–Condon factors, taking into account distortion, displacement, and normal mode mixing (up to four modes). Major features of the observed absorption spectrum of ethylene have been interpreted on the basis of the computed Franck–Condon factors. The role of each electronic state in the spectra has been clarified; the π–3s transition corresponds to the distinct intensive peaks in the 57 000–61 000 cm−1 energy region, the less intensive distinct bands in the interval of 62 000–65 000 cm−1 are due to the π–3pσ states and the π–π* peaks constitute the continuum underlying the spectrum. The theoretical vibronic spectrum is in qualitative agreement with the experimental one, except of some details. Possible reasons for the discrepancies between theory and experiment are also discussed.

Franck-Condon factors involving the Morse potential

An analysis of the vibrational overlap integral, which was developed previously for the Morse potential, is presented. It is accurate up to the fifth order. Our calculations of the Franck-Condon factors for the electronic transitions of the Lyman-Birge-Hopfield ( a 1∏ g − X 1 Σ + g ) band system of N 2, the Schumann-Runge ( B 3 Σ - u − X 3 Σ - g ) band system of O 2 and the fourth positive ( A 1∏− X 1 Σ +) band system of CO are in excellent agreement with the results obtained by Nicholls.

Rotational Structure and Perturbations in the B4Π- X4Σ − (1, 0) Band of VO

The (1, 0) band of the B 4Π- X 4Σ − transition of VO has been analyzed rotationally, from Doppler-limited discharge emission and laser excitation spectra. The B 4Π, v = 1 level is heavily perturbed by a vibrational level of the a 2Σ + state and, with the extra complexity caused by the 51V nuclear hyperfine structure, could be analyzed fully only by extensive wavelength-resolved laser-induced fluorescence experiments. Making use of data from the corresponding perturbations in B 4Π, v = 0 and assuming that the vibrational dependence of the spin-orbit matrix element can be represented by a vibrational overlap integral, it has been possible to deduce the vibrational numbering of the perturbing a 2Σ + state without having to use isotopic data. It is found that the a 2Σ +, v = 0 level lies near 10412 cm −1, and that B 4Π, v = 1 is perturbed by a 2Σ +, v = 3. The a 2Σ + state comes from the same electron configuration, (4 sσ) (3 dδ) 2, as the X 4Σ − ground state. The phases of the spin-orbit matrix elements between 4Π and 2Σ + states must be chosen consistently in order to obtain a correct understanding of the rotational perturbations. A consistent set of phases is obtained with 〈 2Σ +, F 2 | H SO | 4Π 1 2 , f〉 = 〈 2Σ +, F 1 | H SO | 4Π 1 2 , e〉 = [formula] 〈 2Σ +, F 2 | H SO | 4Π − 1 2 , f〉 = −〈 2Σ +, F 1 | H SO | 4Π − 1 2 , e〉 = A/2, where A = 〈 2 Σ + || H spin- orbit || 4 Π〉.