Bending Energy Levels Research Articles

Energy shift and subharmonics induced by nonlinearity in a quantum dot system

The presence of anticrossings induced by coupling between two states causes curvature in energy levels, yielding a nonlinearity in the quantum system. When the system is driven back and forth along the bending energy levels, subharmonic transitions and energy shifts can be observed, which would cause a significant influence as the system is applied to quantum computing. In this paper, we study a longitudinally driven singlet-triplet (ST) system in a double quantum dot (DQD) system, and illustrate the consequences of nonlinearity by driving the system close to the anticrossings. We provide a straightforward theory to quantitatively describe the energy shift and subharmonics caused by nonlinearity, and find good agreement between our theoretical result and the numerical simulation. Our results reveal the existence of nonlinearity in the vicinity of anticrossings and provide a direct way of analytically assessing its impact, which can be applied to other quantum systems without excessive labor.

On the use of nonrigid-molecular symmetry in nuclear motion computations employing a discrete variable representation: A case study of the bending energy levels of CH5.

A discrete-variable-representation-based symmetry adaptation algorithm is presented and implemented in the fourth-age quantum-chemical rotational-vibrational code GENIUSH. The utility of the symmetry-adapted version of GENIUSH is demonstrated by the computation of seven-dimensional bend-only vibrational and rovibrational eigenstates of the highly fluxionally symmetric CH5+ molecular ion, a prototypical astructural system. While the numerical results obtained and the symmetry labels of the computed rovibrational states of CH5+ are of considerable utility by themselves, it must also be noted that the present study confirms that the nearly unconstrained motion of the five hydrogen atoms orbiting around the central carbon atom results in highly complex rotational-vibrational quantum dynamics and renders the understanding of the high-resolution spectra of CH5+ extremely challenging.

Accurate quantum mechanical study of the Renner-Teller effect in the singlet CH2

The Renner-Teller (RT) effect between the two low-lying electronic states of singlet CH(2), ã(1)A(1) and b̃(1)B(1), is studied using the multi-configuration time-dependent Hartree method with complete treatment of the RT terms. The RT terms, which are the matrix elements of the electronic orbital angular momentum operators, are calculated with ab initio methods and fitted to analytical functions. The ro-vibronic energy levels with complete treatment and constant approximation of the RT terms are calculated and compared. The influences of the geometry dependence of the RT terms on the ro-vibronic energy levels are discussed. The differences of the variation trends and influences of the RT terms between CH(2) and NH(2) are explored. In particular, as the molecule bends from linearity, the curve of the RT term (<linear span>ã(1)A(1)|L(z)(2)|ã(1)A(1)(<linear span>) first goes down to reach a minimum and then goes up leading to decreased zero point energy and bending energy levels for the lower state of CH(2) in contrast to the case of NH(2).

Experimental and theoretical investigation of the dispersed fluorescence spectroscopy of HC4S

A high-resolution single vibronic level emission study from the A (2)Pi(32) state of the HC(4)S radical is reported. Ground state density functional theory frequencies have been used to assign ground state vibronic levels involving three stretching modes nu(2), nu(3), and nu(5) in the region of 0-3250 cm(-1), while the frequency of nu(4) remains speculative. Tentative assignments are given for the complicated structures arising from Renner-Teller and spin-orbit interactions within the bending energy levels. From analysis of the dispersed emission spectra, Fermi resonances involving pairs of bands have been identified in the A (2)Pi(32)<--X (2)Pi(32) laser induced fluorescence spectrum.

Equilibrium Structure and HCC Bending Potential Energy Function of 3A‘‘ HCCN

The molecular parameters of the ground triplet state of cyanocarbene, 3A‘‘ HCCN, have been determined in large-scale ab initio calculations using the coupled-cluster method, RCCSD(T), and basis sets of double-through-quintuple-ζ quality. The equilibrium structure of the molecule was found to be planar and bent, with the trans conformation of the HCCN chain and the parameters re(HC) = 1.069 A, re(CC) = 1.328 A, re(CN) = 1.186 A, ∠e(HCC) = 144.9°, and ∠e(CCN) = 175.4°. The potential energy function for the HCC bending motion (v5 mode) was determined to be strongly anharmonic, with the barrier to linearity of 286 cm-1. Influence of various electron-correlation effects on the shape of the HCC bending potential energy function is discussed. The rotation−bending energy levels of the HCCN and DCCN molecules were then calculated using a semirigid-bender Hamiltonian. For both molecules, the predicted patterns of rotational transitions in the excited ν5 states agree favorably with the experimental data.

Six-dimensional variational calculation of the bending energy levels of HF trimer and DF trimer

The bending energy levels of HF and DF trimer were calculated variationally using a six-dimensional kinetic-energy operator derived by constraining the stretch degrees of freedom to their equilibrium values. A basis of direct products of spherical harmonics is used to represent the wave function. The symmetry adapted Lanczos method is used to calculate levels of all symmetries from one sequence of matrix-vector products. Two efficient ways of implementing the symmetry-adapted Lanczos method are presented. To determine the concerted hydrogen bond breaking tunneling splitting in HF and DF trimer one needs to use the D3h point group. By doing D3h calculations, we find that the splitting is negligible for the low-lying levels. If the tunneling is unfeasible, the appropriate group is C3h. Our calculation using the most accurate SO-3+HF3BG potential developed by Quack, Stohner, and Suhm indicates that two gas-phase overtone bending bands of HF trimer should be reassigned, and that the noble gas matrix spectra of the bending fundamental bands are significantly shifted from their gas-phase counterparts.

Ab Initio Study on the Equilibrium Structure and CCN Bending Energy Levels of Cyanofulminate (NCCNO)

The molecular parameters of cyanofulminate, NCCNO, have been determined in large-scale ab initio calculations using the coupled-cluster method, CCSD(T), and basis sets of double- through quadruple-ζ quality. The equilibrium structure of the molecule was found to be linear, with a strongly anharmonic potential energy function of the CCN bending motion (ν7 mode). The rotation−bending energy levels were then calculated using a semirigid-bender Hamiltonian. The vibrational energy levels and effective rotational constants determined for various ν7 states were found to be in good agreement with the experimental data.

Beyond the Potential Energy Surface: Ab initio Corrections to the Born−Oppenheimer Approximation for H2O

It is customary when computing ro−vibrational transitions in molecules to invoke the Born−Oppenheimer separation between nuclear and electronic motion. However, it is known from accurate calculations on H2+ and H2 that the first-order (diagonal adiabatic) and second-order (nonadiabatic) corrections are not negligible and are both important. In the present work, we have made an ab initio implementation of the Bunker and Moss formalism for the nonadiabatic correction and applied it to H2 and H2O. From comparison to accurate calculations for H2, we find that we can obtain good results for the nonadiabatic correction using CI singles to treat the electronically excited states if we scale the results, but we must go beyond the SCF approximation to obtain an accurate diagonal adiabatic correction. For H2O, we find that the first-order correction is more important than the second-order correction for bending energy levels, but the second-order correction is more important than the first-order correction for stre...

The Torsion–Inversion–Bending Energy Levels in theS1(n, π*) Electronic State of Acetaldehyde: A High-Resolution Study of the Bands #7 to #20 in the Jet-Cooled Fluorescence Excitation Spectrum

The band assignments and analyses of the jet-cooled high-resolution laser-induced fluorescence excitation spectrum of acetaldehyde that results from theS1(n, π*) electronic state have been extended to +600 cm−1from the 000system origin. The new assignments start at Band #7 and finish at Band #21. Bands #8 and #9, originally assigned to 1420, have now been assigned to 1530. The assignments of the lower energy bands remain unaltered. The origins of the bands that involve the torsional modes ν15(v= 1 to 4) in combination with the wagging mode ν14(v= 1 and 2) and the ν10(v= 1) were determined by analyses with a rigid rotational Hamiltonian. These origins were fitted to a set of levels that were derived from a torsion–wagging–bending Hamiltonian that employed flexible large amplitude coordinates. The resulting potential surface was found to have barriers to torsion and inversion of 712.5 and 638.6 cm−1, respectively, with minima in the potential hypersurface at θ = 59.9° and α = 33.5° for the torsion and wagging coordinates.

Atom–molecule van der Waals complexes containing open-shell atoms. I. General theory and bending levels

The theory needed to carry out calculations on atom–molecule van der Waals complexes containing open-shell atoms is developed. The discussion concentrates on complexes containing atoms in P states. Several possible expansions of the total wave function are described, and the matrix elements needed to construct the Hamiltonian matrix are set out. Several different angular momentum coupling cases may arise, analogous to Hund’s coupling cases in diatomic molecules. The bending energy levels of Ca–HCl, B–H2, F–H2, Cl–Cl2, and F–N2 are calculated, using simple models of the interaction potentials.

The bending energy levels of C 4H 2

The two low lying bending fundamentals ν 8 and ν 9 of di-acetylene (C 4H 2) near 628 and 220 cm −1, respectively, have been observed under conditions of high resolution. The principal results for these two fundamentals are: ν 8 = 627.89423(10), B 8e = 0.146434670(5), and B 8f = 0.14651508(2) ν 9 = 219.97713(10), B 9e = 0.14672659(1), and B 9f = 0.14693191(2) Several hot bands have also been observed and they have been fitted with effective parameters. Some of the hot bands have also been fitted using a Hamiltonian which accounts for the various l-type doubling interactions.

The electronic spectra of acetaldehyde-h4 and -d4

The [Formula: see text] electronic absorption spectra of acetaldehyde-h4 and -d4 were recorded in the vapour phase at room temperature. The major experimental requirement was a high pressure × path length (650 Torr × 140 m). The vibrational structure of these electronic transitions was interpreted in terms of the torsional modes [Formula: see text] and [Formula: see text] attached to the [Formula: see text] out-of-plane bending mode. The main CH3CHO results agreed with those obtained earlier (Moule and Ng); the values of the [Formula: see text] transition and torsional barrier height were 27240.1 and 590 cm−1 respectively. Three inversion (out-of-plane bending) energy levels in the excited [Formula: see text] state were found and the inversion potential function was determined with a barrier height of 1110 cm−1. The CD3CDO spectrum confirmed the CH3CHO analysis. The values of the [Formula: see text] transition and torsional barrier height were equal to 27270 and 610 cm−1. Keywords: vibronic spectrum, acetaldehyde, molecular structure.

The bending energy levels of C 2H 2

Absorption spectra of C 2H 2 have been recorded between 50 and 1450 cm −1, with a resolution always better than 0.005 cm −1, using two different Fourier transform spectrometers. Analysis of the data provided two sets of results. First, the bending levels with Σ t V t ( t = 4, 5) ≤ 2 were characterized by a coherent set of 34 parameters derived from the simultaneous analysis of 15 bands, performed using a matrix Hamiltonian. The following main parameters were obtained (in cm −1): ω 4 0 = 608.985196(14), ω 5 0 = 729.157564(10); B 0 = 1.17664632(18), α 4 = −1.353535(86) × 10 −3, α 5 = −2.232075(40) × 10 −3; q 4 0 = 5.24858(12) × 10 −3, and q 5 0 = 4.66044(12) × 10 −3, with the errors (1σ) on the last quoted digit. Second, a more complete set of bending levels with Σ t V t ≤ 4, some of which have never previously been reported, and also including V 2 = 1 have been fitted to 80 parameters. This simultaneous fit involved 43 bands and used the same full Hamiltonian matrix. Some perturbations which affect the higher excited levels are discussed.

The dynamics of open-shell Van der Waals complexes

The theory of Van der Waals complexes formed from atoms and open-shell (Σ and Π) diatomic molecules is developed, paying particular attention to the quantum numbers that are conserved in the complex and the angular momentum coupling cases that may be observed. Complexes formed from diatoms in multiplet Σ states may exhibit several different coupling schemes closely analogous to Hund’s coupling cases for diatomic molecules. Complexes formed from diatoms in Π states usually exhibit a coupling scheme in which the (signed) projection P of the diatom angular momentum j onto the intermolecular axis is nearly conserved. Correlation diagrams showing the bending energy levels as a function of potential anisotropy are given for complexes containing diatomic molecules in both Σ and Π states. The transition from free internal rotor quantum numbers to near-rigid bender quantum numbers with increasing anisotropy is investigated. The cases of Ar–OH and Ne–OH are considered as examples.

Use of trigonometric series for solution of the Schrödinger equation for bending vibrations in triatomic molecules

A method is proposed suitable for an ab initio treatment of bending vibrations in triatomic molecules. The potentials and kinetic energy terms are represented by Fourier series and the hamiltonian is diagonalized in a basis of trigonometric functions. The calculation of the bending energy levels and corresponding wavefunctions for nondegenerate and degenerate electronic states, as well as of transition probabilities between various vibronic species is described. The approach is tested on the C3 and NH2 molecules.

The rotational spectrum of the CD2 radical studied by far infrared laser magnetic resonance spectroscopya)

We report the detection of 17 pure rotation transitions in the ground vibronic state of the CD2 radical using far infrared laser magnetic resonance spectroscopy. Fitting the data using an effective rotational Hamiltonian yields values for the three rotational constants, seven centrifugal distortion constants, the three electronic spin-rotation, and two electronic spin–spin parameters. We also fit this data, and CD2 ν2 band data (published separately), using the semirigid bender Hamiltonian and obtain the effective bending potential function for CD2. Combining this with previous CH2 results enables us to predict the rotation bending energy levels of CHD. We also report here the detection of two further rotational transitions in the ν1 excited vibrational state of CH2.

Formaldehyde as a semirigid bender: The rotation-inversion spectrum in its [formula omitted] excited state

The semirigid bender Hamiltonian [Bunker and Landsberg, J. Mol. Spectrosc. 67, 374–385 (1977)] was used to fit the rotation-inversion energy level separations in the A ̃ 1A 2 excited state of formaldehyde. We fix the r 0(CH) bond length and allow the R(CO) bond length and (HĈH) bond angle to vary with the inversion angle ρ. The fit to 64 rotation-inversion energies (with v 4 and J < 4) is significantly better with a standard deviation of 0.199 cm −1 than when the rigid bender [Bunker and Stone, J. Mol. Spectrosc. 41, 310–332 (1972)] is used. The barrier height to planarity is 358 cm −1 and the equilibrium ρ e = 34.7°. The CO bond length is found to decrease by 0.034 from 1.3670 Å and the HĈH angle by about 6 from 122.4° as the molecular configuration changes from planar to pyramidal. The rigid bender model developed earlier by Moule and Rao for formaldehyde [ J. Mol. Spectrosc. 45, 120–141 (1973)] is then used to fit the 32 rotation-(out-of-plane) bending energy levels (with v 4 = 0 and 1) of the X ̃ 1A 1 ground electronic state of H 2CO. For this, a simple potential consisting of quadratic and quartic terms is used and the standard deviation of the fit is 0.148 cm −1.

The geometry and the out-of-plane bending potential function of thioformaldehyde in the [formula omitted] and [formula omitted] electronic states

The semirigid invertor Hamiltonian and computer program that we developed [see Jensen and Bunker, J. Mol. Spectrosc. in press] is used to calculate the rotation and out-of-plane bending energy levels of thioformaldehyde (H 2CS) in the A ̃ 1A 2 and a ̃ 3A 2 electronic states. By fitting to the experimental data it is found that the equilibrium structure of the A ̃ 1A 2 state is planar and that the a ̃ 3A 2 state has a barrier to planarity of only 7 cm −1. This is markedly different from the situation in formaldehyde, which has a much higher barrier in both of the corresponding electronic states. However, in both molecules the triplet state has the higher barrier.

Ab initio SCF-Cl studies on the large amplitude bending motion of the water molecule: Rigid, semirigid, and nonrigid bender models for the Hamiltonian

Self-Consistent Field (SCF) and Configuration Interaction (CI) studies are performed on the bending mode of the water molecule using a double zeta plus polarization basis set. The ab initio points are fitted to a three-parameter double minimum potential consisting of a quadratic plus Lorentzian terms. The vibration-rotation energies are then evaluated using the large amplitude Hamiltonian developed by P. R. Bunker and co-workers at various levels of approximations. It is found that the calculated frequencies improve significantly as one proceeds from approximate H b 00( ρ) to rigid bender H b 0( ρ) [P. R. Bunker and J. M. R. Stone, J. Mol. Spectrosc. 41, 310–332 (1972)] to semirigid bender H b 0( r, ρ) [P. R. Bunker and P. M. Landsberg, J. Mol. Spectrosc. 67, 374–385 (1977)] Hamiltonian. With H b 0( r, ρ), the ab initio calculated bending frequency ν 2 differs from the observed value (1595 cm −1) by 30 cm −1 and the barrier height is 12 229 cm −1. It is also shown that ν 2 and its first four overtones are better calculated by 45–98 cm −1 when the ab initio potential is used directly instead of the three-parameter analytic potential fitted to ab initio data. Finally, rotation bending energy levels are calculated for v 2 ≤ 3 and J ≤ 10 on the basis of a nonrigid bender Hamiltonian of A. R. Hoy and P. R. Bunker [ J. Mol. Spectrosc. 74, 1–8 (1979)], using the ab initio quadratic force field of P. Hennig, W. P. Kraemer, G. H. F. Diercksen, and G. Strey, [ Theor. Chim. Acta 47, 233–248 (1978)]. These results show that the accuracy of calculated force constants and frequencies is critically dependent not only on the size of the basis set but also on the number and spacing of the ab initio points used to derive the force field.