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Reduced dimension rovibrational variational calculations of the S-1 state of C2H2. I. Methodology and implementation
The Journal of Chemical Physics
Abstract
The bending and torsional degrees of freedom in S1 acetylene, C2H2, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S1 state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.
... Paper I (Ref. 41) contains details of the methodology and implementation, and the reader is directed there for full documentation of the computational methods. Paper II presents our variational results. ...
Reduced dimension variational calculations have been performed for the rovibrational level structure of the S1 state of acetylene. The state exhibits an unusually complicated level structure, for various reasons. First, the potential energy surface has two accessible conformers, trans and cis. The cis conformer lies about 2700 cm(-1) above the trans, and the barrier to cis-trans isomerization lies about 5000 cm(-1) above the trans minimum. The trans vibrations ν4 (torsion) and ν6 (asym. bend) interact very strongly by Darling-Dennison and Coriolis resonances, such that their combination levels and overtones form polyads with unexpected structures. Both conformers exhibit very large x36 cross-anharmonicity since the pathway to isomerization is a combination of ν6 and ν3 (sym. bend). Near the isomerization barrier, the vibrational levels show an even-odd K-staggering of their rotational levels as a result of quantum mechanical tunneling through the barrier. The present calculations address all of these complications, and reproduce the observed K-structures of the bending and C-C stretching levels with good qualitative accuracy. It is expected that they will assist with the assignment of the irregular patterns near the isomerization barrier.
Medium resolution (Δν~ 3 GHz) laser-induced fluorescence (LIF) excitation spectra of a rotationally cold sample of YbOH in the 17300-17950 cm-1 range have been recorded using two-dimensional (excitation and dispersed fluorescence) spectroscopy. High resolution (Δλ~ 0.65 nm) dispersed laser induced fluorescence (DLIF) spectra and radiative decay curves of numerous bands detected in the medium resolution LIF excitation spectra were recorded. The vibronic energy levels of the X²Σ state were predicted using a discrete variable representation approach and compared with observations. The radiative decay curves were analyzed to produce fluorescence lifetimes. DLIF spectra resulting from high resolution (Δν < 10 MHz) LIF excitation of individual low-rotational lines in the A²Π1/2(000)-X²Σ((000), A²Π1/2(100)-X²Σ((000), and [17.73]Ω=0.5-X²Σ((000) bands were also recorded. The DLIF spectra were analyzed to determine branching ratios which were combined with radiative lifetimes to obtain transition dipole moments. The implications for laser cooling and trapping of YbOH are discussed.
In a recent paper, we demonstrated that one-colour (∼220 nm), resonance-enhanced (S1−S0), photodissociation of acetylene generates strong C2 Swan band (d3Πg−a3Πu) and C2 Deslandres-d'Azambuja band (C1Πg−A1Πu) fluorescence, and long-lived (>3 µs) fluorescence from an electronically-excited C2H∗ species. It was not known whether the C2A1Πu and X1Σg+ states are also directly populated in this process. In this work, multiple vibration-rotation transitions between the C2A-state v = 2 and the X-state v = 0 level are examined by time-resolved frequency-modulation (FM) spectroscopy. The photolysis laser wavelength is tuned into resonance at the one-photon level with S1−S0 transitions that populate individual rotational levels of the S1trans-conformer 32, 33, and 34 vibrational states. By comparing the phase of the FM signals from the C2A−X transitions with that from the Rb D1-line absorption transition, we determine that, for all of the probed A−X transitions, the X-state level is more populated than the A-state level. We propose that the acetylene S1 level is excited by the second photon to an acetylene dissociation precursor state, which undergoes sequential C-H bond-breaking to produce the C2X state. The dissociation precursor is assigned as the 11Bg(trans-bent)/11B1(cis-bent) valence state, which correlates to a doubly-excited configuration, (1πu)2(1πg)2, at linear geometry. Based on the rotational distributions of the C2X-state fragments, we believe that at least one of the transition states involved in the photolysis via S134 has a larger CC-H bend-angle for the departing H-atom than that involved in the S132 and 33 photolysis.
One-color (212–220 nm) multi-photon dissociation of acetylene, resonantly enhanced at the one-photon level by the acetylene S1–S0 transition, gives rise to strong photofragment fluorescence signals in the visible and near UV regions. The photofragment fluorescence detection scheme has been applied to obtain rotationally-resolved action-spectra of predissociated S1 acetylene levels near 47,000 cm⁻¹ Jiang et al.(2018) [1]. In this work, fluorescence signals from the photofragments, generated via individual intermediate S1 rovibrational levels, are studied in detail, in both flowing gas cell and supersonic jet conditions. In the flow cell setup (∼1 Torr), dispersed fluorescence (DF) spectra of the photofragments are obtained. For all three S1 levels used in the DF experiment (rotational levels of the trans 34,trans 35, and cis 3161 vibrational states), we observe C2 Swan band (d3Πg-a3Πu) and C2 Deslandres-d’Azambuja band (C1Πg-A1Πu) emissions. In the supersonic jet setup (collision-free), fluorescence time-traces at selected wavelength regions are analyzed. In addition to the two C2 emission bands, we observe a long-lived (>3 μs) visible fluorescence signal, which is likely due to emission from C2H fragments. Based on our analysis of the DF spectra and the fluorescence time-traces obtained in the supersonic jet apparatus, we propose that the long-lived C2H species is generated by one-photon photodissociation of S1 acetylene, and an additional photon photolyzes the C2H fragment into the C2 C and d states.
Rotational analyses are reported for the and bands of the transition of C2H2 near 45,000 cm (+2800 cm relative to T0) from jet-cooled laser-induced fluorescence spectra. While the band is unperturbed and straightforward to assign, the 1 level is strongly perturbed by interactions with the 2 B polyad, where υB′ = υ4′ + υ6′. In order to assign the lines of this band, a population-labelling technique was used, employing an infrared laser to deplete the population in selected ground state rotational levels before probing with the ultraviolet laser. Deperturbation of the 1/2 B interaction leads to the value cm for the fundamental symmetric C–H stretching frequency. Assignments are also reported for the 2 and 12 levels, completing all assignments of levels containing excitation in only the totally symmetric vibrational modes up to +4500 cm. The reassignment of implies that some of currently accepted assignments above 47,000 cm are in error and suggests that the interpretation of some aspects of the near-threshold photodissociation measurements of Mordaunt et al. [J. Chem. Phys. 108, 519 (1998)] may need to be revisited.
The rotation-vibration level structure of ground electronic state HCP is investigated at vibrational energies approaching and exceeding that of the linear CPH saddle point. With respect to energies above the saddle point, we investigate possible spectroscopic manifestations of strong Coriolis interactions between the hindered, bond-breaking internal rotation of the hydrogen about the CP core and the rotation of the molecule in the space-fixed axis system. With respect to energies below the saddle point, we provide new interpretations, from quantum and semiclassical points of view, of previously observed anomalously large B rotational and g 22 energy dependence on the vibrational angular momentum constants for the large-amplitude pure bending states of HCP referred to elsewhere as ''isomerization'' or saddle node states. We also predict similar anomalies in other spectroscopic constants, including the ''centrifugal distortion'' constant D and the ''rotational l-resonance'' parameter q 2 . These changes in the effective spectroscopic rotation-vibration constants are shown to be a direct consequence of the spherical pendulum topology of the HCP bend/internal rotor system, which is associated with a phenomenon called quantum monodromy, defined as the absence of a smoothly valid set of quantum numbers for all states. Our semiempirical model for the HCP bend/internal rotor mode is derived using principles of semiclassical inversion and provides new insights into the breakdown in the ability of rovibrational effective Hamiltonians to model highly vibrationally excited states of HCP. © 2001 American Institute of Physics.
Introduction and history“Pointwise“ representations in one dimensionMultidimensional DVRs and applicationsCaveatsSummary and conclusions
The problem of the vibration rotation spectrum of water vapor is treated by means of the theory of semi-rigid polyatomic molecules developed by Wilson and Howard. The potential energy is expanded as a power series in the normal coordinates and involves three zeroth-order constants, six first-order and six second-order constants. The positions of the band centers are calculated and found to depend upon ten quantities, Xi, Xik, and γ which are functions of the potential constants. A new feature of the treatment is the recognition of a resonance interaction between certain of the overtone bands which arises from the near equality of ν1 and ν3. Eighteen band centers are known experimentally. These serve to determine the Xi, Xik, γ and furnish eight self-consistency checks which are very adequately satisfied. There exists no discrepancy between the Raman and infrared spectra as reported earlier. In order to obtain the geometric displacements corresponding to each normal co-ordinate it is necessary to examine the spectrum of D2O. This not only furnishes the required information but also allows two independent checks upon the theory both of which turn out to be in nearly perfect accord. The interaction between vibration and rotation is considered and the effective moments of inertia are calculated. These are functions of the normal frequencies and of the first-order potential constants. It is shown that Δ=IC−IA−IB depends only upon the normal frequencies and hence may be computed at once. A comparison between the observed and predicted Δ yields a very satisfactory agreement. The analysis of the rotational structure made by Mecke is supplemented by taking account of the rotational stretching. The resulting molecular constants fix the valence angle to be 104°31′ and the O-H distance to be 0.9580A. From the effective moments of inertia the first-order potential constants may be evaluated and these, together with Xik determine the second-order potential constants. It is now possible to compute the interaction constant γ and a comparison with the observed γ again results most satisfactorily.
We calculate second-order vibrational perturbation theory (VPT2) anharmonic force fields for the cis and trans conformers of S1 C2H2, and compare the results to experiment. The vibrational assignments of recently observed levels belonging to the cis well are of particular interest. A refined estimate of the cis origin position (44,870 ± 10 cm−1) is proposed, and preliminary low-energy fits to the global J = K = 0 trans level structure are also described. The performance of perturbation theory in this isomerizing system is examined, and both surprising successes and failures are encountered. We examine these and their causes, and offer practical suggestions for avoiding the pitfalls of applying perturbation theory to systems with large amplitude motions.
The ungerade vibrational levels of the 1Au (S1-trans) state of C2H2 lying in the region 45,800–46,550 cm−1 have been assigned from IR–UV double resonance spectra. The aim has been to classify the complete manifold of S1-trans levels in this region, so as to facilitate the assignment of the bands of S1-cis C2H2. The rotational structure is complicated because of the overlapping of vibrational polyads with different Coriolis and Darling–Dennison parameters, but assignments have been possible with the help of predictions based on the properties of polyads at lower energy. An important result is that the analysis of the (1141, 1161) polyad determines the anharmonicity constants x 14 and x 16, which will be needed to proceed to higher energies. Some regions of impressive complexity occur. Among these is the band given by the 3361, K = 1 state at 45,945 cm−1, where a three-level interaction within the S1 state is confused by triplet perturbations. Several probable S1-cis states have been observed, including cis-62, K = 1; this vibrational level appears to show a K-staggering, of the type that arises when quantum mechanical tunnelling through the barrier to cis-trans isomerization is possible. The total number of identified cis vibrational states is now 6 out of an expected 10 up to the energies discussed in this paper.
The performance of the NASA Ames atomic natural orbital (ANO) basis sets for calculating fundamental vibrational frequencies is examined, using the CCSD(T) treatment of electron correlation and second-order vibrational perturbation theory. Particular attention is paid to the performance of the small, cost-effective truncations ([3s 2p 1d] and [4s 3p 2d 1f] on second-row atoms) known as ANO0 and ANO1, as similarly sized basis sets must necessarily be used for high-level correlation treatment of ‘large’ molecules. It is found that the ANO0 and ANO1 basis sets – particularly the former – outperform comparably sized correlation consistent basis sets for the calculation of vibrational frequencies, suggesting that the ANO0 basis is a useful tool for this area of computational chemistry.
A mathematical procedure to derive in a systematic manner exact quantum
mechanical Hamiltonians for N-particle systems is presented. As an
application, exact quantum mechanical Hamiltonians are obtained for the
motion of three atoms, with the help of various sets of curvilinear
internal coordinates and for various body-fixed frames of reference.
Reduced dimension variational calculations have been performed for the rovibrational level structure of the S1 state of acetylene. The state exhibits an unusually complicated level structure, for various reasons. First, the potential energy surface has two accessible conformers, trans and cis. The cis conformer lies about 2700 cm(-1) above the trans, and the barrier to cis-trans isomerization lies about 5000 cm(-1) above the trans minimum. The trans vibrations ν4 (torsion) and ν6 (asym. bend) interact very strongly by Darling-Dennison and Coriolis resonances, such that their combination levels and overtones form polyads with unexpected structures. Both conformers exhibit very large x36 cross-anharmonicity since the pathway to isomerization is a combination of ν6 and ν3 (sym. bend). Near the isomerization barrier, the vibrational levels show an even-odd K-staggering of their rotational levels as a result of quantum mechanical tunneling through the barrier. The present calculations address all of these complications, and reproduce the observed K-structures of the bending and C-C stretching levels with good qualitative accuracy. It is expected that they will assist with the assignment of the irregular patterns near the isomerization barrier.
The equation-of-motion coupled-cluster method with the full inclusion of the single, double, and triple excitations (EOM-CCSDT) has been formulated and implemented. The proper factorization procedure ensures that the method scales as n8, i.e., in the same manner as the standard CCSDT method for ground states. The method has been tested on the vertical excitation energies of the N2 and CO molecules for several basis sets up to 92 basis functions. The full inclusion of the triple excitations improves the EOM-CCSD results by up to 0.2 eV for considered systems.
There are now a large number of papers in the spectroscopic literature which make use of multiple-valued (frequently double-valued) coordinate systems and the associated multiple-groups of the permutation-inversion group to deal with the symmetry properties of large-amplitude motions in molecules of high symmetry. The use of multiple-valued coordinate systems, and the resultant appearance of more minima on the potential surface than would be found on the surface for a single-valued coordinate system, can lead to conceptual confusion and questions of mathematical legitimacy. In the present paper we demonstrate that treatments using multiple-valued coordinate systems simply represent one scheme for applying the appropriate quantum mechanical boundary conditions to Schrödinger’s partial differential equation defined in a single-valued coordinate system. The demonstration is not general, but rather focuses on the specific example of the S1 electronic state of C2H2, which has local minima only for nonlinear configurations, and on the twofold and eightfold extended permutation–inversion groups recently introduced to simultaneously treat symmetry questions in trans-bent and cis-bent acetylene. Some discussion of the mathematical convenience lost by using a single-valued coordinate system is also presented.
In the first half of this study, rotational and vibrational constants of six Franck-Condon bright vibrational levels of S1 doubly-substituted (13)C acetylene are determined from laser-induced fluorescence spectra and an updated geometry of the trans conformer of S1 acetylene is obtained. In the second half, we determine the quadratic force field of S1 acetylene based on the harmonic frequencies of four isotopologues of acetylene. The effects of both diagonal and off-diagonal xij anharmonicities are removed from the input harmonic frequencies. Results from both experimental and theoretical studies of various isotopologues of acetylene (including those from the first half of this paper) are used to obtain a set of force constants that agrees well with ab initio calculations. Our set of force constants for S1 acetylene is an improvement over previous work by Tobiason et al., which did not include off-diagonal anharmonicities.
In π-electron excitations of unsaturated molecules, the lowest orbitals available to receive the promoted electron, supposing the σ-bond structure to remain unaltered, may interpenetrate so strongly with occupied σ-orbitals neighbouring the centre of unsaturation, that energetically lower promotions can be realised at the cost of a changed σ-bond hybridisation. The result is a gross change of shape on excitation, as is illustrated by a preliminary consideration of the case of acetylene. An unappreciated prevalence of such changes may be the reason why so few upper states of polyatomic molecules have as yet been characterised.
The intensity distributions for the optical transitions S0→S1 and S1→S0 and the electron impact induced transitions S0→Tn
(n=1–3) were calculated using quantum dynamical methods. The problem was analyzed in three degrees of freedom, the CC-bond length and the two CCH-bond angles. The three-dimensional potential energy surfaces needed for such a treatment were calculated at the CASPT2 level employing an ANO-type basis set and an active space that consisted of 10 electrons in 10 orbitals. The complex intensity distribution of the S0↔S1 transition in absorption and emission is reproduced in great detail. Spectral quantization allows an analysis of the modes that dominate vibronic coupling. Most of these modes bear no resemblance to the normal modes of the trans
and cis form of S1. An emission from a higher excited state starting at 6.23 eV could tentatively be assigned to T4. T2 and T3 most likely do not contribute to the first band observed in the electron energy loss (EEL) spectrum. The same is true for the band that starts at 6.6 eV in the EEL spectrum. It results mostly from S1 and possibly in part from T4.
A pulsed-laser double resonance technique provides previously unavailable spectroscopic data on the rovibrational structure of A˜ 1Au acetylene (C2H2). Our assignment and analysis of transitions to the A˜ state ν4’ (torsion) and ν6’ (antisymmetric in-plane bend) vibrational fundamentals uncovers a strong Coriolis interaction between these two nearly degenerate modes and weaker Coriolis interactions between the ν4’/ν6’ pair and remote A˜ state rovibrational levels. We deperturb the direct Coriolis interaction between ν4’ and ν6’ to obtain vibrational frequencies, Coriolis coupling constants and partially deperturbed rotational and centrifugal distortion constants for these previously unobserved fundamentals. Parity selection rules for the A˜←X˜ band permit an unambiguous assignment of the vibrations (ν4’=764.9±0.1 cm−1 and ν6’=768.3±0.2 cm−1). We use these new experimental values to reassign several A˜ state vibrations and to assign previously unidentified A˜ state levels. We also identify two vibrational resonances that seem to be important in determining the rovibrational structure of A˜ 1Au C2H2.
Molecular electronic structure theory has been used to predict the equilibrium geometries and energies of acetylene in its excited singlet electronic states. A double zeta plus polarization basis set of contracted Gaussian functions was used in conjunction with self-consistent field and large scale configuration interaction wave functions. The first excited singlet state of acetylene is the trans 1Au state, in agreement with the experimental studies of King, Ingold, and Innes. This result is particularly interesting because the lowest triplet state of C2H2 is not the 3Au state but rather the cis 3B2 state. The predicted geometry of the ? 1Au state is re(CC)=1.384 A˚, re(CH)=1.096 A˚, thetae(HCC)=121.7 °, in good agreement with available spectroscopic data. The predicted relative energies of the excited singlet states are 5.06 eV (1Au), 5.54 eV (cis 1A2), 6.87 eV (1B2), and 7.29 eV (1Bu). Thus the energetic ordering of the singlet states is Au
The present investigation is a part of the program of a series of studies aiming at a satisfactory analysis of the near ultraviolet spectrum of acetylene. The spectrum of C2H2 at room temperature and 300°C was taken with a Bausch and Lomb large quartz spectrograph. The spectrum extends from λ2430A down to the ultraviolet limit of the instrument. More than 1000 bands and lines have been measured. Most of the bands revealed intensification at elevated temperature. Some of them have been partially resolved into rotational lines with alternating intensities. Seven main progressions all with frequency difference ∼1050 cm—1 have been found. Only the bands of three of the progressions showed no intensification at higher temperature. From the differences between these bands, a frequency 580 cm—1 has been observed. Weaker bands on the red side of stronger ones have been proved through temperature variation to be due to transitions from the excited 2v5″(Eg) level in the ground state. Through a tentative assignment, 2v5′=1050 cm—1 and v4′(Eu)=580 cm—1 and the empirical selection rules, all of the bands in the seven progressions can be satisfactorily explained. Other possible assignments and the nature of the electronic transition have also been briefly discussed.
Recent theoretical developments that facilitate characterization of excited‐state potential energy surfaces are applied to study five stationary points on the S1 surface of C2H2. Although previous calculations have focused on the acetylenic trans and cis forms, the present research predicts that the vinylidene isomer is the global minimum on the potential surface. However, a high activation barrier calculated for rearrangement to the trans isomer suggests that S1 vinylidene is not likely to be formed via photoexcitation of ground‐state acetylene. The trans and cis acetylenic forms of the S1 state are shown to interconvert along an in‐plane reaction coordinate with an activation energy of ∼4000 cm−1, a value which is significantly lower than usually assumed in spectroscopic analyses.
The electronic ground state potential surface of acetylene (HCCH) has a minimum at the linear conformation, but the excited electronic states may have potential minima at a variety of nonlinear equilibrium shapes. This work is concerned with the group theoretical ideas necessary to treat simultaneously the symmetry properties of rovibronic states associated with three different planar acetylene equilibrium configurations, namely trans bent acetylene, cis bent acetylene, and vinylidene (H2CC). We make use of three different kinds of groups: (i) point groups, (ii) permutation-inversion (PI) groups, and (iii) extended PI groups. The PI group is G4 or G8, depending on whether CH bond breaking is impossible (no bent acetylene vinylidene interconversion) or possible. The extended PI groups are G4(2) and G8(2), respectively, when the only large amplitude motions are the CCH bends at each end of the molecule, and G4(8) and G8(8), respectively, when internal rotation is added as a third large amplitude motion. Applied to acetylene, the results indicate that there will be no splittings of the rovibronic levels unless CH bond breaking occurs. Even without bond breaking, however, states of the cis and trans isomers just below their interconversion barrier will show "staggerings" in their K-structures, i.e., a given vibrational level will have three tunneling components at slightly different energies: one component will have levels with K = 4n only (where n is an integer), a second component will have levels with K = 4n + 2 only, and the third will have only odd-K levels. New experimental results for the S1-cis electronic state of acetylene [21] are reviewed, and are found to be consistent with the group theory in so far as comparison is possible.
In molecular vibration-rotation calculations it is frequently desirable to work in a nuclear coordinate system tailored to the nuclear potential energy surface. It is now possible to derive the kinetic energy operator for virtually any coordinate system by using computer algebra programs. We discuss how to choose coordinates for use in practical calculations, in particular variational energy level calculations, with special consideration of the vibration-rotation separation. As an example we present a kinetic energy operator suitable for calculating the rovibrational spectrum of sequentially bonded four-atomic molecules, using valence coordinates. We carry out a rigorous symmetry classification using the MS group, and also consider how the problem of singularities affects the choice of basis set.
The IR−UV double resonance spectroscopy has been applied to observe the rovibronic level system of the ungerade nν3‘ + ν4‘ and nν3‘ + ν6‘(n = 2, 3) vibrational states in the Ã1Ag(S1) state of acetylene which are accessible from the selected rotational level J‘ ‘ of the ν3‘ ‘ state in the X̃1Σu+ state. As was reported by Utz et al. [J. Chem. Phys. 1993, 98, 2742] for the ν4‘ and ν6‘ bands, the nν3‘ + ν4‘ and nν3‘ + ν6‘(n = 2,3) states are found to couple with each other by the a- and b-axis Coriolis interactions. The rotational analysis is performed taking the Coriolis interactions into account to determine the spectroscopic constants including the vibrational term values. The extent of the Coriolis interactions between nν3‘ + ν4‘ and nν3‘ + ν6‘(n = 2,3) is not so significant as that between ν4‘ and ν6‘. This is due to a larger anharmonic coupling of the in-plane trans-bending ν3‘ mode with the in-plane cis-bending ν6‘ mode than with the out-of-plane torsion ν4‘ mode, which causes a larger energy spacing between the pairs of the interacting levels as the ν3‘ quantum number increases. It is also found that most of rotational lines in the 3ν3‘ + ν6‘ band split into two or more peaks due to the S1−T3 interaction, while such rotational line splittings are not found in the 3ν3‘ + ν4‘ band. The present finding that the additional excitation in the out-of-plane torsion (ν4‘) mode suppresses the splittings suggests that the S1−T3 mixing occurs at the planar C2h or C2v geometry rather than at the nonplanar C2 geometry which is distorted along the torsional coordinate from the planar geometry.
This article presents permutation–inversion group-theoretical strategies and recipes aimed at helping a high-resolution molecular spectroscopist use the existing pedagogical literature to carry out their own treatment of the basic symmetry questions in rotating molecules with large-amplitude vibrational motions. Topics addressed include: determination of the feasible permutation-inversion group and its symmetry species and character table; a general equation defining coordinates that can describe translation, overall rotation, large-amplitude vibrations and small-amplitude vibrations for a large class of floppy molecules; and the determination of symmetry species for basis functions and selection rules for operators written in these coordinates. The article is intended to be more advanced than existing pedagogical works, but it still leaves many important topics untreated.
Whereas model Constraints (e.g., frozen bonds, fixed bending or torsonial angles, rigid atomic groups. etc.) are often imposed in calculations of the potential energies of polyatomic molecules by quantum-chemical methods, the derivation of exact expressions for the corresponding kinetic energy operators is generally difficult. This is because of the changes in the metrics of the configuration spaces, which modify the differential operators but not the local operators. An appropriate method for overcoming this difficulty is presented, and examples for four-atom molecules and various coordinate systems are given. The coplanar-atom constraint deserves special attention, because the problem can be formulated either in 2 (i.e., the molecule lies in the space-fixed plane) or in 3 (i.e., the molecule is planar, but the plane, which is body-fixed, rotates with respect to the space-fixed frame). Other types of constraints are also examined.
The potential energy surface of acetylene in its lowest excited single state (S1) is studied by using an ab initio MCSCF method to explore the mechanism of the photodissociation process. Predissociation leading to C2H and H suggested by the LIF spectrum is shown to occur on the single potential surface of the S1 state, which is directly correlated with the excited state of C2H(2Π). The dissociation of the CH bond from trans-bent acetylene in the S1 state is found to proceed via a cis-bent transition state. The calculated energy barrier from trans-bent to cis-bent acetylene is lower than that of the CH cleavage. The present results suggest that dissociation follows trans—cis isomerization.
The efficiency of the theoretical methods to obtain exact expressions of classical and quantum-mechanical Hamiltonians for N-particle systems described in terms of curvilinear coordinates (q), is shown to depend strongly on whether the system is subjected to constraints or not. If it is free, the method based upon the use of the relations q = q(x), where x denotes moving-frame Cartesian coordinates of the particles, is preferable. If it is constrained, the method making use of x = x(q) can become more efficient.
An elaboration and a more complete analysis of Witmer's work on the asymmetrical top treated as a perturbation of the symmetrical one show that one can deduce the rigorous solution of the problem from those of the algebraic equations of degree 2j+1 or less. Without actually solving such equations, we find the terms divisible into even and odd groups just as in the case of the sigma-type doubling in diatomic molecules treated by Kronig, Van Vleck and others. For the case where the asymmetry is slight, an explicit expression for the separation of such similar doublets is obtained. The selection rules, which are rigorous for any degree of asymmetry, consist of the following: (a) Kronig's rule; (b) Δj=0, ±1; Δm=0, ±1; and (c) rule for the quantum number sigma, Δσ=evenforelectricmoment in z direction and Δσ=oddformoment in x−y plane. The effect of the electronic motions on the rotation of a polyatomic molecule as a whole is also briefly discussed.
Recent experimental results permit a detailed normal modes analysis of A˜-state acetylene (C2H2) and its isotopomers (C2HD and C2D2). Using only experimentally determined frequencies and measured or estimated anharmonicities, we determine harmonic frequencies for the 11 directly observed and unambiguously assigned vibrational fundamentals. The normal modes calculation varies force constants to fit the 11 harmonic frequencies and yields a complete set of harmonic frequencies, force constants, and Coriolis coefficients for the three isotopomers. A complete set of fundamental frequencies calculated from the set of harmonic frequencies allows a comparison to and, in some cases, suggests a reassessment of frequencies for tentatively assigned fundamental vibrations.
The previous study [J. Mol. Spectrosc.95, 101-132 (1982)] of the high-resolution absorption spectrum of the A~1Au(C2h)-X~1Sigmag+(D∞h) electronic transition of C2H2 is extended to the region 2190-2050 Å. The vibrational levels nnu'3 (n = 4-6), nu'2 + nnu'3 (n = 3-5), and nu'1 + nnu'3 (n = 2-3) of the A~ state are assigned. These levels and another unidentified level are observed as upper states of vibrational progressions in the known nu''4(pig) vibrational mode of the ground state. Detailed rotational analyses of all these bands are performed. The rotational levels of each of these upper state vibrational levels are perturbed by interactions with unidentified vibronic levels, which probably also belong to the A~ electronic state. As a result, only approximate upper rovibrational constants are obtained.
Rotational analyses are reported for a number of newly-discovered vibrational levels of the S1-trans (Ã1Au) state of C2H2. These levels are combinations where the Franck-Condon active nu2' and nu3' vibrational modes are excited together with the low-lying bending vibrations, nu4' and nu6'. The structures of the bands are complicated by strong a- and b-axis Coriolis coupling, as well as Darling-Dennison resonance for those bands that involve overtones of the bending vibrations. The most interesting result is the strong anharmonicity in the combinations of nu3' (trans bend, ag) and nu6' (in-plane cis bend, bu). This anharmonicity presumably represents the approach of the molecule to the trans-cis isomerization barrier, where ab initio results have predicted the transition state to be half-linear, corresponding to simultaneous excitation of nu3' and nu6'. The anharmonicity also causes difficulty in the least squares fitting of some of the polyads, because the simple model of Coriolis coupling and Darling-Dennison resonance starts to break down. The effective Darling-Dennison parameter, K4466, is found to increase rapidly with excitation of nu3', while many small centrifugal distortion terms have had to be included in the least squares fits in order to reproduce the rotational structure correctly. Fermi resonances become important where the K-structures of different polyads overlap, as happens with the 2131B1 and 31B3 polyads (B = 4 or 6). The aim of this work is to establish the detailed vibrational level structure of the S1-trans state in order to search for possible S1-cis (1A2) levels. This work, along with results from other workers, identifies at least one K sub-level of every single vibrational level expected up to a vibrational energy of 3500 cm-1.
The high-resolution study of the A~1Au(C2h)-X~1Sigmag+(D∞h) absorption system of C2H2 is extended to the region 2050-1930 Å. Bands involving the upper levels nnu'3(n = 7, 8), nu'2 + nnu'3(n = 6-8), and nu'1 + nnu'3(n = 4, 5), and the lower levels mnu''4(m = 0-3) are tentatively identified. The other observed bands have not been assigned. The various bands overlap strongly and many perturbations affect the upper state rovibrational energy levels, preventing rotational analyses of all but a few of the bands. Most of the bands can be characterized by features such as the temperature dependence and the presence or absence of an intensity alternation, and these are reported here. A general discussion of the analysis of the entire band system is also presented.
Parts of the potential surfaces for the ground and excited states (below 11 eV) of acetylene are computed with an ab initio frozen core method. Various planar and non-planar geometrics are investigated. Rydberg states, including the 11Σ+u Vππ* state, are found to be linear and to retain their Rydberg character upon bending. Valence ·(1πu)3(1π*g) states are cis and trans bent if they result chiefly from excitation to the component of the 1π*g MO in the molecular plane. They are further distorted to gauche, non-planer, geometries if they arise from excitation to the out-of-plane component of the 1π*g MO. Walsh diagrams are used to rationalize these trends and the computed results are compared with the available experimental data.
Ab initio SCF MO and CI calculations on acetylene in linear and bent conformations are reported. It is found that acetylene possesses stable trans- and cis-isomers for most of its low-lying electronically excited states. The calculated transition energies are discussed in relation to the observed optical absorption bands and to electron-impact excitation energies.
The electronic energy levels of cis- and trans-bent planar centrosymmetric acetylene molecules, with and r cc = 1.383 Å, have been calculated by the LCAO/ASMO/CI procedure. The lowest energy state that can spectroscopically combine with the linear ground state is found to be of symmetry class Au belonging to the trans-bent molecule and to lie at 4.58 ev above the ground state. This compares favorably with the experimentally observed electronic transition of lowest energy which is of type (1Au−1∑g+) and at 5.24 ev. The lowest-energy allowed transition to a cis-bent state of these dimensions is at 9.39 ev, and hence this may not form part of the unanalyzed system of bands in the 2000–1500 Å region, as has been suggested. However, a transition to a trans-bent state of type (1Bu−1∑g+) is predicted to fall in this region. The energies of other electronic states are discussed in relation to the observed absorption systems of acetylene.
The present and prospective rapidly increasing usefulness of good LCAO-MO calculations on the electronic states of simple molecules in the interpretation of observed excited states is pointed out. As examples, the observed and predicted states of, in particular, the π3π configurations of N2 and CO are compared with those of C2H2 and HCN and with those of CO2 and CS2. Further, the results of LCAO-SCF calculations on CO2 and on O3 are surveyed, and it is shown how these can be helpful in interpreting and understanding the ground and excited states of AB2 (expecially AO2) molecules in general. A new interpretation of the so-called d3П state of CO as a case b πu3πg, 3Δu state is proposed. Tentative interpretations of some of the ultraviolet absorption spectra of C2H2, HCN, and of a number of AB2 molecules are reviewed or suggested, including some discussion of the shapes of excited states. The AB2 discussion is a revision of one given earlier. Finally, following up a suggestion of Coon, it is pointed out that there exists strong evidence for slightly unequal A—O distances in certain excited states of C1O2 and SO2.
Whereas model constraints (namely, internal degrees of freedom either frozen or stepwise adjusted by gradient methods) are often imposed for calculating the potential energies of polyatomic molecules by quantum-chemical methods, the derivation of exact expressions for the corresponding kinetic energy operators is difficult because of the changes of metrics of the configuration spaces, which modify the differential operators but not the multiplicative operators. An appropriate method for overcoming this difficulty has been designed in the case of rigid constraints (e.g., frozen groups) (M. Menou and X. Chapuisat,J. Mol. Spectrosc.159,300–328, 1993). In this article, it is generalized to the case of adiabatic constraints; i.e., the variations of certain internal degrees of freedom are adjusted to those of other degrees of freedom. Exact kinetic energy operators are derived. An example is analyzed.
The electronic ground state and the first valence excited states of acetylene (3Σ+u, 1,3Σ−u and 1,3Δu) are investigated with the aid of ab initio methods applying large and flexible basis sets. Electron correlation has been taken into account on various levels of sophistication, including SCF CI, MC SCF and MC SCF CI methods. Equilibrium geometries are determined for all states mentioned, excitation energies, vertical and 0–0 transitions are reported and the dependence of the computed results on the type of wavefunction is discussed. Harmonic force constants and vibrational frequencies are calculated for all states treated in this paper and compared to experimental values.
A new efficient and fully symmetry‐adapted finite‐basis variational method for calculating the rovibrational eigenstates (J≥0) of any sequentially bonded four‐atom molecule is presented. The exact kinetic‐energy operator VR in valence coordinates is used in a scheme of successive basis‐set contractions. The success of the method is demonstrated with new results for the molecules HCCH, HOOH, and HCNO, respectively linear, nonlinear, and quasilinear. The complexity of VR contributes little to the computational cost, yielding a Hamiltonian matrix whose elements can all be cheaply calculated from products of arbitrarily accurate one‐dimensional integrals; the dominant cost is that of matrix diagonalization. Matrices of up to 6000×6000 are used to obtain low‐lying levels converged to small fractions of 1 cm−1, even for the difficult case of HOOH. The generalization to J≥0 allows the calculation of Π states for HCCH and HCNO and effective rotational constants for HCCH, all usefully converged. For the first time nine‐dimensional rovibrational wave functions for four‐atom systems are calculated without dynamical approximation and with basis sets well completed in all degrees of freedom. For HCCH it is straightforward to obtain hundreds of rovibrational (J=0,1,2) levels converged to better than 1.5 cm−1, opening up the possibility of the systematic refinement of the nuclear potential function N.
The absorption spectrum of acetylene in the region 1970—2470A has been photographed in the fourth and fifth orders of a twenty-one-foot grating spectrograph with absorbing paths of 0.007 to 70 m-atmos.
It has been possible to extend the arguments which led King and Ingold to conclude that in the upper electronic state the molecule is bent and has C2h symmetry. The upper electronic state is Au or, possibly Ag.
The effective moment of inertia Ia of the excited state is small, causing sub-band separations of 50 to 150 cm—1. These sub-bands all have P, Q, and R branches, are well resolved, and have the appearance of bands of diatomic molecules. The following rotational constants of the upper state are derived: A0=12.94, B0=1.1247, C0=1.0297 cm−1. If the C–H distance is assumed to be between 1.070 and 1.090A, one finds rcc=1.388A, [open phi]CCH=120∘. Ground-state combination differences are in agreement with precise values derived from infrared data. Some new lower state B values are obtained.
The O–O vibrational transition is shown to be at 42 197.69 cm—1. The sub-bands form ν4″ progressions and ν3′ progressions. Analysis gives two upper state vibrational frequencies ν3(CCH)=1047.70 cm−1 and ν2(CC)=1389 cm−1. The following lower state vibrational constants are obtained: ω40=608.26, x44=+3.29, g44=+0.17 cm−1. The fundamental frequency ν4″ is the sum of these constants, that is, 611.72 cm—1.
A pulsed‐laser double resonance technique (vibrational overtone excitation combined with laser‐induced fluorescence detection) provides previously unavailable spectroscopic data on the rovibrational structure of 1Au acetylene (C2H2). We collect fluorescence excitation spectra of transitions to vibronic levels lying between 2800 and 4300 cm−1 above the state origin. In this region, we observe only two vibronic levels that are relatively unperturbed, which we assign to the state antisymmetric C–H stretching fundamental vibration ν′5 and its combination with the trans‐bending vibration, ν′3 + ν′5. Parity and symmetry selection rules for the ← band, ab initio predictions for the ν′5 fundamental frequency, and the known frequencies of other state vibrations permit an unambiguous assignment of the vibrations. The fit of ν′5 and ν′3 + ν′5 to a near‐prolate asymmetric top Hamiltonian yields the observed vibrational frequencies (ν′5= 2857.4 ± 0.2 cm−1 and ν′3 + ν′5 = 3894.4 ± 0.1 cm−1) and rotational and centrifugal distortion constants. We deperturb the first‐order a‐axis Coriolis interactions of remote perturbers with ν′5 and ν′3 + ν′5 to obtain partially deperturbed a‐axis rotational constants. We discuss other weak perturbations and present an energy level diagram for the vibrational states with ungerade symmetry in state acetylene.
Valence-excited singlet (S1,S2) and triplet (T1–T4) states of acetylene have been studied by means of extended multireference electron correlation techniques (MR-CISD, MR-CISD+Q, and MR-AQCC). Extrapolations to the basis set limit have been performed. Minima and saddle points have been calculated using a recently developed analytic gradient method for excited states. Planar as well as nonplanar structures have been considered. In particular, the existence of an asymmetric, planar cis-type minimum on the S2 surface has been confirmed conclusively. Moreover, an intersection S1/S2 has been located close to this minimum. This situation will most probably affect the interpretation of the absorption bands attributed to the trans 1 1Bu state. In-plane and out-of-plane saddle points for cis–trans isomerization have been determined and characterized by harmonic vibrational analysis. Several interesting surface crossings for different electronic states (S1/S2, T2/T3, and S1/T3) have been characterized. Implications of the flatness of the T3 surface around linear structures and the location of the S1/T3 crossing seam on the anomalities observed in the ZAC spectrum of the 1Au state are discussed. © 2003 American Institute of Physics.
A general formulation of the vibrational kinetic energy operator expressed in internal bond-angle coordinates is presented. This formulation is based on Podolsky’s expression for the covariant form of the Laplace–Beltrami operator. When a valid set of internal bond-angle coordinates is employed, it is possible to adapt a systematic approach to solve for the Jacobian determinant governing the coordinate transformation from Cartesian coordinates. In the general case of an arbitrary N-atom system, this Jacobian always factorizes to a simple form. This allows one to evaluate all the terms that contribute to ′, the effective potential that arises from transforming the kinetic energy operator to internal coordinates. We discuss restrictions on the choice of internal vibrational coordinates that may be included in a valid set. We then provide tabular information from which the vibrational kinetic energy operator for any molecular system can be constructed directly with no matrix inversion or chain rule manipulation required. © 1999 American Institute of Physics.
A general contraction scheme for Gaussian basis sets is presented. The contraction coefficients are defined by the natural orbitals obtained from an atomic configuration‐interaction calculation. Such atomic natural orbitals provide an excellent basis for molecular electronic structure calculations. Large primitive sets can be contracted to only a few functions without significant loss in either the SCF or correlation energy. Polarization functions can be included using the same approach.
The A~-X~ band system of acetylene with origin near 2400 Å was
photographed in absorption with higher resolution than in previous
studies. A detailed rotational analysis of bands in the
2470-2150-Å region, involving the levels nν''4 (n =
0-4) of the
X~1Σg+(D∞h)
state and nν'3 (n = 0-3) and ν'2 +
nν'3 (n = 0-2) of the
A~1Au(C2h) state, is presented. Some of
the previous assignments are modified. Both the principal subbands with
ΔK = +/- 1 and the satellite subbands with ΔK = 0, +/-2 (due
to axis-switching) and ΔK = +/-3 (due to axis-switching,
asymmetry, and l resonance) are included in a comprehensive
least-squares fit. Rotational levels of all the l components of the
levels of ν''4 are observed, and are consistent with the
theory of l doubling and l resonance of a linear molecule. The
rotational structures of the vibrational levels of the nonlinear A~
state are fitted with asymmetric top rotational constants Av,
Bv, Cv and principal centrifugal constants
DvJ, DvJK,
DvK. A small perturbation of ν'2 +
ν'3, together with the inertial defect
Δ'v, are discussed in terms of the unobserved
vibrational frequencies of the A~ state.
It is shown that the ultraviolet absorption bands reported previously do
not belong to acetylene but to some impurity. In pure acetylene, bands
are observed only below 2400A. These bands are arranged into three
progressions and their interpretation is proposed using information from
the work on the infrared absorption spectrum of acetylene.
Potential curves for trans- and cis-bending and C-C and C-H stretching vibrations in ground and singlet excited states of acetylene of both valence and Rydberg character are calculated using a large-scale MRD-CI approach. The energy surfaces obtained will be used for an interpretation of the U.V. spectrum of acetylene. The computed vertical energies for the lowest two transitions Ã-[Xtilde] and [Btilde]-[Xtilde] are found to agree within 0·2 eV of the corresponding values reported experimentally.
Fr Untersuchungen im Schumanngebiet wurden zwei Spektrographen gebaut, die in ihrer Konstruktion besonders fr Aufnahmen von Absorptionsspektren geeignet sind. Diese beiden Spektrographen dienen zu Untersuchungen der Absorption mehratomiger organischer Molekle und der Bestimmung ihrer Struktur aus spektroskopischen Daten. Zunchst wurden mit dem neueren der beiden Spektrographen (siehe Untersuchungen im Schumanngebiet II) die orientierenden bersichtsaufnahmen ausgefhrt. Dann wurden mit dem Przisionsspektrographen (siehe unten) die betreffenden Banden aufs genaueste aufgenommen. — Ein eingehendes Literaturstudium ber die Struktur mehratomiger organischer Molekle ist bereits durchgefhrt worden und wurde zum Teil im Handbuch der Spektroskopie Kayser-Konen, Bd. VIII, 338–342, 1932 von H. Hese zusammengestellt. Auf diese Literaturbersicht soll in den spteren Arbeiten Bezug genommen werden. — Als weiterer Punkt unseres Arbeitsplanes ist die Schaffung einwandfreier Wellenlngennormalen im Schumanngebiet zu nennen. Die Vorarbeiten dazu sind bereits abgeschlossen und sollen demnchst unter dem Titel: Untersuchungen im Schumanngebiet. III von R. Grfin zu Dohna hier verffentlicht werden.
Valence and low-lying Rydberg states of acetylene (C2H2) are reexamined in the singlet as well as in the triplet manifold. The major goal of this work is a better understanding
of the valence states that contribute to the low-energy electron-energy-loss spectrum recorded under conditions where transitions
to triplet states are enhanced. An appropriate theoretical treatment of these states has to include the low-lying Rydberg
states because of their energetic proximity to some of the valence states. The CASSCF/CASPT2 method provides a suitable framework
for such a task. For some important states the geometry was optimized at the CASPT2 level to allow a comparison with the results
of other highly accurate methods that have been applied to acetylene in the past.
In this paper the current status of the variational method for the determination of the rotational-vibrational energy levels of polyatomic systems is reviewed. Special attention is made for the derivation of the kinetic energy operator in various coordinate systems, and several forms are given. Similarly, analytic forms which are in current use for the potentials are given. The calculation of the Hamiltonian matrix elements (expansion functions, numerical integration grid points and weights) is described in detail, and a description of our programs for this problem is given in section 6.
Ab initio MO calculations at the EOM-CCSD and CASSCF level have been carried out for several low lying electronic states of C2H2. Many minima on the seam of crossing and transition states are located. For photodissociation from the S1 state, a non-adiabatic path involving a S1 → S2 transition followed by a S2 → S0 crossing which produces mainly C2H(X2Σ+) is preferred at low energy to the S1 adiabatic path producing mainly C2H(A2II), in agreement with experimental propensity. A minimum on the seam of crossing between S1 and T3 is found. T3 may be a strong perturbing state in the ZAC spectrum of S1.