Ground State Asymptote Research Articles

Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in $${{\mathbb {R}}}^N$$

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -\Delta_{p} u + \varepsilon u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N}, \end{equation} where $1<p<N $, $p<q<l<+\infty$ and $\varepsilon> 0 $ is a small parameter. For $\varepsilon\rightarrow 0$, we give a characterisation of asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In particular, we show that the behavior of the groundstates is sensitive to whether $q$ is less than, equal to, or greater than the critical Sobolev exponent $p^{*} :=\frac{pN}{N-p}$.

On the first eigenpair of singularly perturbed operators with oscillating coefficients

ABSTRACTThe paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behavior of the first eigenpair (ground state) of this problem. The main tool in deriving the limit \\mbox(effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.

Long-range coupling ofX1Σ+anda3Σ+states of the atom pairK+Cs

The K and Cs atom pair and its $X{\phantom{\rule{0.16em}{0ex}}}^{1}{\ensuremath{\Sigma}}^{+}$ and $a{\phantom{\rule{0.16em}{0ex}}}^{3}{\ensuremath{\Sigma}}^{+}$ electronic molecular states are studied by Fourier-transform spectroscopy with special emphasis on bound states close to the ground state asymptote $(4s)\text{K}+(6s)\text{Cs}$. Significant vibrational mixing due to hyperfine coupling was observed leading to improved potential energy curves for both states. The observed intensity pattern of the hyperfine structure for the last vibrational levels of $X{\phantom{\rule{0.16em}{0ex}}}^{1}{\ensuremath{\Sigma}}^{+}$ and $a{\phantom{\rule{0.16em}{0ex}}}^{3}{\ensuremath{\Sigma}}^{+}$ reveals deep insight into the multicomponent wave function of such levels with its influence for modeling the asymptotic behavior of the excited electronic states involved in the reported observations. Examples of derived properties of cold collisions are given and the importance of the present results for efficient production schemes for ultracold KCs is discussed.

(X)11Σ+state of LiNa studied by Fourier-transform spectroscopy

Laser-induced fluorescence to the electronic ground state $(X){1}^{\phantom{\rule{0.16em}{0ex}}1}{\ensuremath{\Sigma}}^{+}$ of LiNa has been recorded by a Fourier-transform spectrometer. Optical excitations of the LiNa molecules into the $(A)2{}^{\phantom{\rule{0.16em}{0ex}}1}{\ensuremath{\Sigma}}^{+}$ state were performed with different dye lasers. The spectra showed fluorescence progressions up to ${v}^{\ensuremath{'}\ensuremath{'}}=44$ of the singlet ground state covering 99.9$%$ of the potential well. However, searches for fluorescence into the triplet ground state $(a)1{}^{\phantom{\rule{0.16em}{0ex}}3}{\ensuremath{\Sigma}}^{+}$ induced by spin-orbit coupling in the excited system $(b){1}^{\phantom{\rule{0.16em}{0ex}}3}\ensuremath{\Pi}$--$(A)2{}^{\phantom{\rule{0.16em}{0ex}}1}{\ensuremath{\Sigma}}^{+}$ were not successful. The potential energy curve of the singlet ground state was determined by direct potential fits with a data set of about 8200 assigned lines using two distinct analytical potential representations. In addition, the isotopologue ${}^{6}\text{Li}{}^{23}\text{Na}$ was studied for a possible need of Born-Oppenheimer corrections by simultaneous fits of both observed isotopologues from ${}^{6}$Li and ${}^{7}$Li. Finally, new data on Feshbach resonances from cold-atom experiments at Heidelberg were included for improving the potential description around the dissociation energy for both electronic states $(X)1{}^{\phantom{\rule{0.16em}{0ex}}1}{\ensuremath{\Sigma}}^{+}$ and $(a)1{}^{\phantom{\rule{0.16em}{0ex}}3}{\ensuremath{\Sigma}}^{+}$ of the atom-pair ground-state asymptote.

Geometric bonding effects in the X A21, A Σ2u+, and B Π2g states of Li2F

Published ab initio and pseudopotential calculations for the dialkali halide systems suggest that the preferred colinear geometry is for the metal to approach the metal end of the alkali halide. Here, ab initio calculations on the Li2F system reveal that the well depth on the halide side in this radical is much deeper and is a local saddle point associated with the ionic nonlinear global minima. Although many features of the pseudopotential surfaces are confirmed, significant differences are apparent including the existence of a linear excited A Σ2u+ state instead of a triangular one, a considerably deeper global minimum some 50% lower in energy and a close approach between the X A21 and the A Σ2u+ states, with the A Σ2u+ minimum 87 kJ mol−1 below the ground state asymptote. All the results can be rationalised as the avoided crossings between a long range, covalent potential dominant within the LiLiF geometry and an ionic state that forms the global minimum. Calculations on the third A2′ potential indicate that even for Li+LiF collisions at ultracold temperatures the collision dynamics could involve as many as three electronic states.

Theoretical study of the Ru + CH3F reaction

CASSCF-MRMP2 calculations have been performed to analyse the reaction of fluoromethane with a bare ruthenium atom. Potential energy curves emerging from the ground state and the first excited state of the reactants, Ru(5F, 3F;d7s1) + CH3F, were calculated for representative electronic states associated with different approaching modes of the fragments. Whereas no favourable channels correlating with the ground state asymptote were detected for the insertion of the ruthenium atom into the C–H bond of the methyl fluoride molecule, the oxidative addition of the C–F bond of this molecule to the metal atom along the reaction path evolving from the ground state of the fragments leads to the stable inserted product CH3–Ru–F. Although forming less stable products, insertion of ruthenium into the C–H and C–F bonds of the fluorocarbon molecule can occur through electronic states which emerge from the excited triplet state asymptote.

XΣ+1andaΣ+3states of LiCs studied by Fourier-transform spectroscopy

We present the first high-resolution spectroscopic study of LiCs. LiCs is formed in a heat-pipe oven and studied via laser-induced fluorescence Fourier-transform spectroscopy. By exciting molecules through the $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}\text{\ensuremath{-}}B\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Pi}$ and $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}\text{\ensuremath{-}}D\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Pi}$ transitions vibrational levels of the $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ ground state have been observed up to $3\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ below the dissociation limit enabling an accurate construction of the potential. Furthermore, rovibrational levels in the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ triplet ground state have been observed via singlet-triplet mixing in the higher excited states. With the help of coupled channels calculations, accurate singlet and triplet ground-state potentials were derived reaching the atomic ground-state asymptote. These potentials yield an accurate determination of the singlet and triplet ground-state dissociation energies and allow first predictions of cold collision properties of $\mathrm{Li}+\mathrm{Cs}$ pairs.

The coupling of the X1Σ+ and a3Σ+ states of the atom pair Na + Cs and modelling cold collisions

The states X1Σ+ and a3Σ+ correlated to the ground-state asymptote of Na (3s) and Cs (6s) atoms have been experimentally investigated using high resolution Fourier-transform spectroscopy. The hyperfine splitting of the a3Σ+ state levels is partially resolved. Transitions to asymptotic vibrational levels of the a3Σ+ and X1Σ+ states were recorded simultaneously. The joint evaluation of the data of both the a3Σ+ and the X1Σ+ states allows us to determine accurate potential energy curves of both electronic states. Coupled-channels calculations are finally applied for deriving long range dispersion parameters and the exchange contribution of the molecular potentials, yielding a reliable description of cold collisions between Na and Cs atoms in their two different hyperfine ground states. Scattering lengths and Feshbach resonances are predicted for selected quantum states.

Spectroscopic studies of NaCs for the ground state asymptote of Na + Cs pairs

The ground state X\(^1\Sigma^ + \) of NaCs was studied by laser induced fluorescence Fourier-transform spectroscopy. An accurate potential energy curve was derived from more than 5000 transitions. This potential reproduces the experimental observations within their uncertainties of \(\pm 0.003 \) cm-1 and covers about 99.97% of the potential well depth. Few vibrational levels of the shallow state a\(^3\Sigma^ + \) below the atomic ground state asymptote were observed. The identification is mainly done by the observed and quantitatively interpreted molecular hyperfine structure applying atomic parameters of the ground states of Na and Cs. An estimated potential curve for a\(^3\Sigma^ + \) is reported which can be used together with that of X\(^1\Sigma^ + \) for coupled channel calculations of cold collisions between Na and Cs. An example is given.

Potential of the ground state of NaRb

The $X\phantom{\rule{0.3em}{0ex}}^{1}\ensuremath{\sum}^{+}$ state of NaRb was studied by Fourier-transform spectroscopy. An accurate potential energy curve was derived from more than 8800 transitions in isotopomers $^{23}\mathrm{Na}^{85}\mathrm{Rb}$ and $^{23}\mathrm{Na}^{87}\mathrm{Rb}$. This potential reproduces the experimental observations within their uncertainties of $0.003\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}1}--0.007\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$. The outer classical turning point of the last observed energy level $({v}^{\ensuremath{''}}=76,\phantom{\rule{0.2em}{0ex}}{J}^{\ensuremath{''}}=27)$ lies at $\ensuremath{\approx}12.4\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, leading to an energy of $4.5\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$ below the ground-state asymptote.

Autler-Townes splitting in two-color photoassociation of6Li

We report on high-resolution two-color photoassociation spectroscopy in the triplet system of magneto-optically trapped ${}^{6}\mathrm{Li}.$ Photoassociation is induced from the ground-state asymptote to the $v=59$ level of the excited $1{}^{3}{\ensuremath{\Sigma}}_{g}^{+}$ state. This level is coupled to the $v=9$ level of the triplet ground state $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ by the light field of a Raman laser. The absolute transition frequencies are measured with an iodine frequency standard and the binding energy of the ground state is determined to be $24.391\ifmmode\pm\else\textpm\fi{}0.020\mathrm{GHz}.$ Strong coupling of the bound molecular states has been observed as Autler-Townes splitting in the photoassociation signal for different detunings of the Raman laser. From the splitting we determine the spontaneous bound-bound decay rate to be $4.3\ifmmode\times\else\texttimes\fi{}{10}^{5}{\mathrm{s}}^{\ensuremath{-}1}$ and estimate the molecule formation rate. The observed line shapes are in good agreement with the theoretical model.

Ground state asymptotics of a dilute, rotating gas

We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the so-called Gross-Pitaevskii (GP) limit of a dilute gas. Analyzing both the absolute and the bosonic ground state of the system we show, in particular, their different behavior for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.

Solvent-assisted catalysis in the enolization of acetaldehyde radical cation

The keto–enol isomerization of neutral and ionized acetaldehyde, both isolated and catalyzed by solvent molecules, has been studied using the B3LYP and MP2 levels of theory. Single-point calculations at the CCSD(T) level have also been performed. The results show that ionization favors the enolization of acetaldehyde, both thermodynamically and kinetically. Solvent molecules produce an important catalytic effect, especially for the methanol-solvated system, for which the transition state of the isomerization lies below the ground-state asymptote.

Cold atomic collisions studied by molecular spectroscopy

We observe bound states just below the dissociation limit and shape and Feshbach resonances between the ground state hyperfine asymptotes by Raman spectroscopy on a molecular beam of sodium dimers. The rotational selectivity of a two-photon transition gives access to specific states of nuclear motion and thus to cold collision properties of two colliding atoms. Modeling of the collisional resonance structures requires a multichannel treatment of the nuclear dynamics that uses highly accurate $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ potentials. These potentials are constructed from bound levels just below the ground state asymptote measured in this experiment and bound state information available in the literature. We also present simulations of the spectrum between the ground state hyperfine asymptotes. The good agreement shows that accurate potentials obtained from bound state information are able to reproduce scattering properties of two colliding sodium atoms. From this analysis we find for the scattering lengths ${a(f=2)=a}_{1,\ensuremath{-}1}{=52.98(40)a}_{0},$ ${a}_{\mathrm{singlet}}{=19.20(30)a}_{0},$ and ${a}_{\mathrm{triplet}}{=62.51(50)a}_{0},$ where ${1a}_{0}=0.0529177$ nm.

Simple determination of Na2 scattering lengths using observed bound levels at the ground state asymptote

A simple model is developed to determine a scattering length from the experimental values of the energy of the last, least bound, vibrational levels of the ground state, either or ,of an alkali dimer. It is based on an extrapolation of the positions of the few outermost nodes of the bound vibrational wave functions towards the dissociation threshold. It uses the asymptotic part of the molecular potential only. The method is applied to recently measured levels at the asymptote (3s+3s) of Na2. We obtain precise values for the two scattering lengths respectively associated to the dissociation limits F 1 =F 2 =1,F=2 (55.1 1.6 a0) and F 1 =F 2 =1,F=0 (50.0 1.6 a0); the first value corresponds to the a1,-1 scattering length usually considered in cold collisions. The extrapolation procedure is also applied to Li2, using existing experimental data.

A comparison of density functional theory withab initio approaches for systems involving first transition row metals

Density functional theory (DFT), using the B3LYP hybrid functional, is found to give a better description of the geometries and vibrational frequencies of FeL and FeL+ systems that second-order Moller Plesset perturbation theory (MP2). Namely, DFT correctly predicts the shift in the CO vibrational frequency between free CO and the5Σ− state of FeCO and yields a good result for the Fe-C distance in the quartet states of FeCH 4 + . These are properties where the MP2 results are unsatisfactory. Thus DFT appears to be an excellent approach for optimizing the geometries and computing the zero-point energies of systems containing first transition row atoms. Because the DFT approach is biased in favor of the 3d 7 occupation, whereas the more traditional approaches are biased in favor of the 3d 6 occupation, differences are found in the relative ordering of states. It is shown that if the dissociation energy is computed relative to the most appropriate atomic asymptote and corrected to the ground state asymptote using the experimental separations, the DFT results are in good agreement with high levels of theory. The energetics at the DFT level are much superior to MP2 and in most cases they are in good agreement with high levels of theory.

A neutralization—reionization (NR) case study for cationic iron complexes with simple ligands: NR mass spectra of Fe(C 2H 4) + and Fe(CO) +

The cationic iron(I) complexes Fe(C 2H 4) + and Fe(CO) + were examined by neutralization—reionization mass spectrometry (NRMS). Whereas the NR mass spectrum of the carbonyl complex exhibits a signal corresponding to the reionized neutral molecule Fe(CO), for the neutral Fe(C 2H 4) complex no recovery signal is observed; rather, in the course of the experiment the complex dissociates to Fe and C 2H 4. An explanation for this seemingly contradictory behaviour of structurally related metal complexes in a NR process is provided by ab initio MO calculations, in which the geometries of Fe(C 2H 4) + ( 4B 2), Fe(C 2H 4) ( 5B 2), Fe(CO) + ( 4Σ −, Fe(CO) ( 3Σ −, and Fe(CO) ( 5Σ −) have been fully optimized at the QCISD(T) level of theory. From the theoretical results, a neutralization-reionization scheme for organometallic ions MX + emerges which considers the effects caused by curve-crossing from a bound state to a repulsive ground-state asymptote of the neutral building blocks M and X. Thus, even for bound organometallic complexes MX, recovery signals in the NR mass spectrum can only be detected if the internal energy deposited in MX in the vertical electron-transfer reaction MX + → MX is too small to permit this curve-crossing. If dissociation occurs on the time scale of the NR experiments, the spectrum exhibits features of both the metal M and the ligand X, thus revealing the structural properties of the (organic) ligand X bound to the metal ion.

Spin-forbidden decay of the dication HS2+

The lifetimes of the low-lying vibrational levels of the X2Π state of the recently identified dication HS2+ [Miller et al., Int. J. Mass Spectrom. Ion Proc. 100, 505 (1990)] are considered. The stability of this state is attributable to a barrier formed from the avoided crossing of 2Π states asymptotically characterized as H++S+ and H+S2+. As a result of this barrier, the nonrelativistic X2Π potential energy curve supports several quasibound vibrational levels that are long lived with respect to tunneling. However, this is not the principal decay mechanism. We show that the lifetimes of the low-lying vibrational levels, v=0−4, are controlled entirely by the spin–orbit induced perturbation, 14Σ−∼X2Π, and the corresponding allowed crossing of the X2Π potential energy curve by the dissociative 14Σ− potential energy curve which correlates with the ground state asymptote H++S+(4S).

Radiative and nonradiative decay of the NH(ND) A 3Π electronic state: Predissociation induced by the 5Σ− state

Lifetimes for excited ND A 3Πv′=1–3 and NH v′=2 in specified rotation/fine-structure levels were determined by excitation on isolated rotational lines in the A 3Π–X 3Σ−Δv =+1 sequence, using molecules prepared in a pulsed supersonic beam. The observed lifetimes for NH v′=2 levels were found to be significantly shorter than those expected for purely radiative decay, indicative of additional removal of excited-state population by predissociation. The observed fine-structure dependence of the removal rates is consistent with a mechanism in which the A 3Π state is predissociated by spin–orbit coupling to the repulsive 1 5Σ− state which correlates with the ground-state asymptote, N(4S)+H(2S). This mechanism is also expected to be responsible for the previously observed predissociation of high N′ levels in NH v′=0 and 1. By contrast, no significant evidence for predissociation was found for the decay of excited ND v′=1–3 for the low J′ levels investigated. These observations were confirmed with a combination of ab initio electronic structure, and coupled electronic state dynamics, calculations. Using an extended contracted Gaussian basis and large configuration state function expansions(160 000–380 000 terms) potential energy curves for, and spin–orbit induced coupling (using the full microscopic Breit–Pauli interaction) between, the A 3ΠΩ and 1 5Σ−Ω states were determined. These electronic structure data were used as the basis for the determination of the radiationless decay rates using a semiclassical coupled state model. These results were in turn combined with radiative decay rates for the A 3Π→X 3Σ− transition to determine the total decay rates which were found to be in excellent accord with the available experimental observations, thereby confirming the mechanism of the predissociation.

The electronic structure and photodissociation of HCl

Continuous absorption is possible from the ground state of HCl to the repulsive states that also arise from the ground state asymptote. Under conditions where vibrational excitation of the HCl is possible, continuum absorption can occur from an onset near 345 to 100 nm in the ultraviolet. Since the two states X 1Σ+ and A 1Π, that are dipole coupled, correlate to the same asymptote, the transition moment varies rapidly with the internuclear distance. Using all-electron ab initio calculations based on multiconfiguration self-consistent field (MC-SCF) and first-order configuration interaction (FOCI), the energy curves and wave functions have been obtained for the X 1Σ+, a 3Π,A 1Π, b 3Σ+ states, and the first excited states of each symmetry. The electronic structure of the states are analyzed and the energy curves are compared favorably to experiment. Using the calculated A–X transition moment, the A–X absorption cross section has been obtained as a function of the ground state vibrational level. The possible impact of this process on the operation of the XeCl UV laser which uses HCl as a fuel would not be significant since the cross section at 310 nm never exceeds a few times 10−19 cm2.