Transition Dipole Moment Matrix Elements Research Articles

A Spectroscopic Investigation of the Lowest Electronic States of the I+ 2 Cation as a Candidate for Detecting the Time Variation of Fundamental Constants

The four lowest Ω substates (X2Π3/2,g, X2Π1/2,g, A2Π3/2,u and A2Π1/2,u) of the I+ 2 cation have been studied by high-precision ab initio calculations in comparison with experimental high-resolution absorption spectra. The potential energy curves were calculated using the multi-reference configuration interaction (MRCI) method and Dirac method, respectively. Rovibrational levels of these electronic states were derived by solving the radial Schrödinger rovibrational equation. Molecular constants were obtained in fitting energy levels to a spectroscopic model. Using the fit spectroscopic constants and newly calculated transition dipole moment matrix elements, line strengths of vibronic bands in the A2Π3/2,u- X2Π3/2,g system, as well as Einstein A coefficients for 45 of these bands with ν′ = 11−19 and ν′′ = 1−5, have been derived. The Einstein A coefficients were used to compute radiative lifetimes of the ν′ = 11−19 vibrational levels of the A2Π3/2,u state. Enhancement factors for detecting the variation of the fine-structure constant (α) and the proton-to-electron mass ratio(µ) using transitions between nearly degenerate rovibronic levels of these low-lying states have been calculated.

Riemannian geometry of resonant optical responses

The geometry of quantum states is well-established as a basis for understanding the response of electronic systems to static electromagnetic fields, as exemplified by the theory of the quantum and anomalous Hall effects. However, it has been challenging to relate quantum geometry to resonant optical responses. The main obstacle is that optical transitions involve a pair of states, while existing geometrical properties are defined for a single state. As a result, a concrete geometric understanding of optical responses has so far been limited to two-level systems, where the Hilbert space is completely determined by a single state and its orthogonal complement. Here, we construct a general theory of Riemannian geometry for resonant optical processes by identifying transition dipole moment matrix elements as tangent vectors. This theory applies to arbitrarily high-order responses, suggesting that optical responses can generally be thought of as manifestations of the Riemannian geometry of quantum states. We use our theory to show that third-order photovoltaic Hall effects are related to the Riemann curvature tensor and demonstrate an experimentally accessible regime where they dominate the response.

Line lists for the b1Σ+-X3Σ− and a1Δ-X3Σ− transitions of SO

SO is found in many astronomical sources such as the atmospheres of Io and Venus. In order to create more complete line lists, we fit spectroscopic data on SO from the literature using PGOPHER. The fits covered v = 0 to v = 6 for the X3Σ−state, v = 0–5 for the a1Δ state and v = 0–2 for b1Σ+ state. LeRoy's RKR program was used to produce pointwise potential energy curves. High level ab initio calculations, including spin-orbit coupling, were carried out to obtain the electric transition dipole moment functions for the nominally forbidden b1Σ+-X3Σ−and a1Δ-X3Σ−transitions. The RKR potentials and transition dipole moment points were input into LeRoy's LEVEL program to calculate the transition dipole matrix elements for all possible b-X and a-X bands. For the b1Σ+-X3Σ−transition, the electric and magnetic transition dipole matrix elements were scaled using the experimental values obtained by Setzer et al. [J Mol Spectrosc 1999;198:163–174] for the 0–0 band. The transition dipole moment matrix elements were used in PGOPHER to produce our line lists for all possible bands of the b1Σ+-X3Σ−and a1Δ-X3Σ−transitions.

A new line list for the A[formula omitted] electronic transition of OH

A new line list for the A2Σ+−X2Π electronic transition of OH has been calculated. Line positions have been taken from the literature and refitted with Western’s PGOPHER program. Line intensities were calculated using a new ab initio Transition Dipole Moment Function (TDMF) obtained with Molpro 2012. The new TDMF and the potential functions from LeRoy’s RKR program have been used as input to LeRoy’s LEVEL program in order to calculate Transition Dipole Moment Matrix Elements (TDMMEs). These matrix elements were transformed from Hund’s case (b) to Hund’s case (a) as required for the PGOPHER program using the method of Brooke et al. [JQSRT 2016; 168: 142–157] The line list was calculated with PGOPHER for bands with v′=0−4 in the A2Σ+ state and v″=0−9 for the X2Π state.

Measurement of the Na2 51Σg+→A1Σu+ and 61Σg+→A1Σu+ transition dipole moments using optical-optical double resonance and Autler-Townes spectroscopy.

Accurate knowledge of transition dipole moment matrix elements is crucial since important parameters associated with the interaction of light with matter, such as emission and absorption line intensities, lifetimes, and Einstein coefficients, depend on these matrix elements. We report here an experimental study of the Na2 51Σg+↔A1Σu+ and 61Σg+↔A1Σu+ electronic transition dipole moments and their dependence on internuclear distance. We have measured absolute transition dipole matrix elements for ro-vibrational transitions of the Na2 51Σg+↔A1Σu+ and 61Σg+↔A1Σu+ electronic transitions using Autler-Townes and optical-optical double resonance spectroscopy, and we compare the results to ab initio theoretical values [A. Sanli et al., J. Chem. Phys. 143, 104304 (2015)].

Ab initio study of the low-lying electronic states of the [formula omitted] anion

Six low-lying states (X2Σg+, A2Πu, B2Σu+, 4Σg+, 4Πu and 4Σu+) of C2− anion are studied by highly correlated ab initio calculations. The potential energy curves (PECs) are computed in the internuclear separation from 0.8 to 5.0Å using the complete active space self-consistent field method, and then performing the internally contracted multireference configuration interaction with Davidson correction. Core–valence correlation and scalar relativistic effect are considered through employing the aug-cc-pcV5Z-dk basis set. The spin–orbit coupling of the A2Πu state is also calculated using Breit–Pauli Hamiltonian. Based on the PECs derived from the high-level ab initio calculations, the spectroscopic parameters are obtained by fitting the ro-vibrational levels that are acquired by solving the ro-vibrational Schrödinger equation. These spectroscopic parameters, especially for low-lying doublet states, are in good agreement with the previously calculated and experimental results. Finally, the electronic transition dipole moment matrix elements, Franck–Condon factors, radiative lifetimes and oscillator strengths of A2Πu–X2Σg+andB2Σu+-X2Σg+ are calculated and discussed as well.

Electronic transition dipole moment and radiative lifetime calculations of sodium dimer ion-pair states.

We report here ab initio calculated electronic transition dipole moments for the sodium dimer ion pair states of (1)Σg (+) symmetry. They vary strongly as a function of internuclear distance because of the effect of the Na(+) + Na(-) ion pair potential, which also causes the formation of additional wells and shoulders in the molecular potential energy curves. We also present a computational study of the transition dipole moment matrix elements and lifetimes for these ion-pair states.

Studies of singlet Rydberg series of LiH derived from Li(nl) + H(1s), with n ≤ 6 and l ≤ 4.

The 50 singlet states of LiH composed of 49 Rydberg states and one non-Rydberg ionic state derivable from Li(nl) + H(1s), with n ≤ 6 and l ≤ 4, are studied using the multi-reference configuration interaction method combined with the Stuttgart/Köln group's effective core potential/core polarization potential method. Basis functions that can yield energy levels up to the 6g orbital of Li have been developed, and they are used with a huge number of universal Kaufmann basis functions for Rydberg states. The systematics and regularities of the physical properties such as potential energies, quantum defects, permanent dipole moments, transition dipole moments, and nonadiabatic coupling matrix elements of the Rydberg series are studied. The behaviors of potential energy curves and quantum defect curves are explained using the Fermi approximation. The permanent dipole moments of the Rydberg series reveal that they are determined by the sizes of the Rydberg orbitals, which are proportional to n(2). Interesting mirror relationships of the dipole moments are observed between l-mixed Rydberg series, with the rule Δl = ±1, except for s-d mixing, which is also accompanied by n-mixing. The members of the l-mixed Rydberg series have dipole moments with opposite directions. The first derivatives of the dipole moment curves, which show the charge-transfer component, clearly show not only mirror relationships in terms of direction but also oscillations. The transition dipole moment matrix elements of the Rydberg series are determined by the small-r region, with two consequences. One is that the transition dipole moment matrix elements show n(-3/2) dependence. The other is that the magnitudes of the transition dipole moment matrix elements decrease rapidly as l increases.

Einstein A-values and oscillator strengths of the A2П–X2Σ+ system of CP

Line strengths for bands of the A2П–X2Σ+ transition of CP, including the effect of rotation on the vibrational wavefunctions (the Herman–Wallis effect), have been calculated using Western׳s PGOPHER program and Le Roy׳s LEVEL program. The potential energy functions for the A2П and X2Σ+ state were computed using spectroscopic constants obtained from high resolution spectra. The RKR potentials of the two states, and the electronic transition dipole moments of this transition calculated in a recent ab initio study have been used in Le Roy׳s LEVEL program to produce transition dipole moment matrix elements. The matrix elements have been converted from Hund׳s case (b) to (a), and then used in PGOPHER to generate a line list containing observed and calculated wavenumbers, Einstein A coefficients and f-values for 75 bands with v=0–8 for both states. The Einstein A coefficients have been used to compute radiative lifetimes for v=0–5 in the A2П state. The line list may be useful for computing the molecular opacities of CP needed to simulate the spectra of stellar and substellar sources.

EINSTEIN A COEFFICIENTS AND OSCILLATOR STRENGTHS FOR THE A 2 Π- X 2 Σ + (RED) AND B 2 Σ + -X 2 Σ + (VIOLET) SYSTEMS AND ROVIBRATIONAL TRANSITIONS IN THE X 2 Σ + STATE OF CN

Line strengths have been calculated in the form of Einstein A coefficients and f-values for a large number of bands of the A 2Π-X 2Σ+ and B 2Σ+-X 2Σ+ systems and rovibrational transitions within the X 2Σ+ state of CN using Western's PGOPHER program. The J dependence of the transition dipole moment matrix elements (the Herman-Wallis effect) has been taken into account. Rydberg-Klein-Rees potential energy functions for the A 2Π, B 2Σ+ , and X 2Σ+ states were computed using spectroscopic constants from the A 2Π-X 2Σ+ and B 2Σ+-X 2Σ+ transitions. New electronic transition dipole moment functions for these systems and a dipole moment function for the X 2Σ+ state were generated from high level ab initio calculations and have been used in Le Roy's LEVEL program to produce transition dipole moment matrix elements (including their J dependence) for a large number of vibrational bands. The program PGOPHER was used to calculate Einstein A coefficients, and a line list was generated containing the observed and calculated wavenumbers, Einstein A coefficients and f-values for 290 bands of the A 2Π-X 2Σ+ transition with v' = 0-22, v'' = 0-15, 250 bands of the B 2Σ+-X 2Σ+ transition with v' = 0-15, v'' = 0-15 and 120 bands of the rovibrational transitions within the X 2Σ+ state with v = 0-15. The Einstein A coefficients have been used to compute radiative lifetimes of several vibrational levels of the A 2Π and B 2Σ+ states and the values compared with those available from previous experimental and theoretical studies.

Einstein A coefficients and absolute line intensities for the E 2Π–X 2Σ + transition of CaH

Einstein A coefficients and absolute line intensities have been calculated for the E 2Π–X 2Σ + transition of CaH. Using wavefunctions derived from the Rydberg–Klein–Rees (RKR) method and electronic transition dipole moment functions obtained from high-level ab initio calculations, rotationless transition dipole moment matrix elements have been calculated for all 10 bands involving v′=0,1 of the E 2Π state and v″=0,1,2,3,4 of the X 2Σ state. The rotational line strength factors (Hönl–London factors) are derived for the intermediate coupling case between Hund's case (a) and (b) for the E 2Π–X 2Σ + transition. The computed transition dipole moments and the spectroscopic constants from a recent study [Ram et al., Journal of Molecular Spectroscopy 2011;266:86–91] have been combined to generate line lists containing Einstein A coefficients and absolute line intensities for 10 bands of the E 2Π–X 2Σ + transition of CaH for J-values up to 50.5. The absolute line intensities have been used to determine a rotational temperature of 778±3 °C for the CaH sample in the recent study.

All-optical cw quadruple resonance excitation: A coherently driven five-level molecular system

We report here a continuous-wave, high-resolution, and all-optical quadruple resonance laser excitation experiment. To demonstrate this excitation technique, we have used an inhomogeneously broadened molecular sample (Na-2). We emphasize the enhanced ability afforded by the technique to reach new ranges of internuclear separation as well as access to a large range of molecular electronic states. Using the Autler-Townes split line shape of a rovibronic transition and the E field amplitude of the corresponding coupling laser, we have measured with high precision the absolute value of the molecular electronic transition dipole moment matrix elements, with more flexibility than the double and the triple resonance based excitation schemes used previously. This technique may also be used in a stimulated Raman adiabatic passage configuration for effective transfer of ultracold molecules, formed at long internuclear distance, to the lowest vibrational level of the ground electronic state. (Less)

Photoelectron spectroscopy of the Cl−…H2∕D2 anions: A model beyond the rotationless and Franck–Condon approximations

A model for simulating photoelectron spectra of the triatomic van der Waals complexes containing stable atomic anion and diatomic molecule is proposed and applied to the Cl(-)...H(2) and Cl(-)...D(2) anions. The model assumes adiabatic separation of the electronic and nuclear motions and localization of the photodetachment act at the atomic chromophore. Under these approximations, the electronic transition dipole moment matrix elements are evaluated using the atoms-in-molecule approach and explicit expressions for the rovibrational line strength factors are derived. The energies and intensities of a number of rovibronic photoelectron transitions are calculated for the Cl(-)...H(2) and Cl(-)...D(2) anions within the adiabatic bender model, i.e., with the full separation of the vibrational motions, whereas the simulations of the broad spectral envelopes are performed using the equilibrium conditions, asymmetric line shape function, and two choices of the relative abundances of the para- and ortho-forms of the complex. The simulations reproduce experimental spectra reasonably well allowing for their unambiguous assignment in terms of vibronic transitions fully consistent with the previous time-dependent calculations. Agreement with the previous theoretical works, manifestations of non-Franck-Condon effects, and implications to the assessment of the neutral potential energy surfaces are discussed.

Measurement of the electronic transition dipole moment by Autler-Townes splitting: Comparison of three- and four-level excitation schemes for the Na2AΣu+1−XΣg+1 system

We present a fundamentally new approach for measuring the transition dipole moment of molecular transitions, which combines the benefits of quantum interference effects, such as the Autler-Townes splitting, with the familiar R-centroid approximation. This method is superior to other experimental methods for determining the absolute value of the R-dependent electronic transition dipole moment function mu(e)(R), since it requires only an accurate measurement of the coupling laser electric field amplitude and the determination of the Rabi frequency from an Autler-Townes split fluorescence spectral line. We illustrate this method by measuring the transition dipole moment matrix element for the Na2 A 1Sigma(u)+ (v' = 25, J' = 20e)-X 1Sigma(g)+ (v" = 38, J" = 21e) rovibronic transition and compare our experimental results with our ab initio calculations. We have compared the three-level (cascade) and four-level (extended Lambda) excitation schemes and found that the latter is preferable in this case for two reasons. First, this excitation scheme takes advantage of the fact that the coupling field lower level is outside the thermal population range. As a result vibrational levels with larger wave function amplitudes at the outer turning point of vibration lead to larger transition dipole moment matrix elements and Rabi frequencies than those accessible from the equilibrium internuclear distance of the thermal population distribution. Second, the coupling laser can be "tuned" to different rovibronic transitions in order to determine the internuclear distance dependence of the electronic transition dipole moment function in the region of the R-centroid of each coupling laser transition. Thus the internuclear distance dependence of the transition moment function mu(e)(R) can be determined at several very different values of the R centroid. The measured transition dipole moment matrix element for the Na2 A 1Sigma(u)+ (v' = 25, J' = 20e)-X 1Sigma(g)+ (v" = 38, J" = 21e) transition is 5.5+/-0.2 D compared to our ab initio value of 5.9 D. By using the R-centroid approximation for this transition the corresponding experimental electronic transition dipole moment is 9.72 D at Rc = 4.81 A, in good agreement with our ab initio value of 10.55 D.

Multireference configuration interaction based electronic Floquet states for molecules in an intense radiation field: Theory and application to Li2+

A multireference configuration interaction (CI) method which includes single and double excitations based description of adiabatic Floquet states for the electronic structure of a molecule in an intense laser field is introduced. Using a variant of a recently introduced configuration state function (CSF) based Table-CI methodology, it is shown that the multiple states of several irreducible representations required for a good description of low-lying Floquet states can be obtained using modifications of computational molecular electronic structure techniques. In particular, formulas for all components of the transition dipole moment matrix elements within the CSF-based Table-CI method are derived and presented. Moreover, the flexibility of the recently introduced macroconfiguration description of model and external configuration spaces is shown to lead to multiple potential energy surfaces of sufficiently uniform quality to allow construction of useful Floquet states. The formalism and computer programs developed are demonstrated on Li(2) (+) in a 0.9 x 10(12) W/cm(2) field. In analogy with Na(2) (+), the 1,2 (2)Sigma(g) (+), 1,2 (2)Sigma(u) (+), 1 (2)Pi(g), and 1 (2)Pi(u) states are of relevance, although the pattern of couplings is shown to be more complex. A hitherto unnoticed metastable state, which correlates asymptotically with 2 (2)Sigma(u) (+), is described.

Ab initio zero electron kinetic energy spectroscopy of the ArCl− and KrCl− anions

The theory for calculating the energies and relative intensities of the photodetachment transitions of the weakly bound diatomic anions is implemented for simulations of the zero electron kinetic energy spectra of the ArCl− and KrCl− anions using high-quality ab initio potentials. Its key features are the reduction of the molecular electronic photodetachment transition dipole moment matrix elements to the combination of atomic ones within the atoms-in-molecule model and estimation of the latter by well-developed approaches to the atomic photodetachment processes. The difference in the electronic photodetachment transition dipole moments for distinct electronic states of the neutral, importance of vibrational–rotational coupling, and rotational structure of vibronic bands are analyzed. It is shown that ab initio simulations provide almost the same level of agreement with measured spectra as did the empirical potentials adjusted to reproduce the experimental data. The predictive power of the theory is demonstrated by the discovery of the hidden transitions to the I3/2 state of KrCl which strongy overlap with those to X1/2 state.

Ab initio calculation of vibrational dipole moment matrix elements. I. methods of calculation and diatomic test systems

This paper is the first part of a series devoted to the ab initio calculation of some vibrational properties helpful in the interpretation of high resolution spectroscopy, atmospherical and astrophysical data. An ab initio method of calculation of the vibrational transition energies and dipole moment matrix elements is described and analyzed. The main features of this method are the resolution of both electronic and vibrational problems by analytical variational techniques and the representation of the potential energy and dipole moment hypersurfaces as Taylor's series expansions in normal coordinates up to the fourth order. The dipole moment matrix elements are then evaluated by simple analytical formulae. In this first paper, some of the approximations inherent to the methodology are systematically analyzed on the basis of results obtained for diatomic molecules (BeH, HF, NH and NO) with a comparison to experimental and other theoretical results found in the literature. The dependence on the results of the effects of electrical and mechanical anharmonicities, electron correlation and basis set dependencies are investigated. The applicability of the method to small polyatomic systems is discussed in the second paper of this series, on the basis of test calculations on the water molecule.

Stochastic theory of intramolecular energy transfer in the presence of radiation

A theory for internal energy redistribution in polyatomic molecules perturbed by strong radiation fields is constructed. Use is made of stochastic theory, which assumes that a random phase approximation is valid after appropriate time intervals. This approximation permits the Schrödinger equation to be replaced with a finite-difference master equation for the probabilities of occupying the various quantum levels. The semiclassical theory of the radiation–molecule interaction is employed in this work. A laser line profile is incorporated in the formalism, and this is used to simulate the effects of rotational states and collisions. At each stochastic step the energy changes in the molecules due to radiation are estimated. Thus, energy conservation is explicitly taken into account. Model calculations for SO2, whose transition dipole moment matrix elements and anharmonic force constants have been determined, indicate a complex interplay of anharmonic and radiative coupling. Power density, laser linewidth, and detuning from resonance are observed to play a significant role in energy redistribution.