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Extended permutation-inversion groups for simultaneous treatment of the rovibronic states of trans-acetylene, cis-acetylene, and vinylidene
Abstract
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.
... Finally, it has been recently observed that barrierproximal states, which can tunnel through the barrier to isomerization, exhibit a tunneling splitting in the form of K-staggering within their rotational manifolds. 13,14 Though group theory allows for the possibility of these staggerings, 40 their sign and magnitude cannot be easily predicted a priori. Understanding these staggerings and gaining a foothold on quantitative estimates of them is essential for continued progress in the study of S 1 C 2 H 2 . ...
... r A multivalued internal coordinate system is employed, requiring the use of the G (8) 4 extended complete nuclear permutation inversion (CNPI) symmetry group. 40 r A constrained, analytic reduced dimension rovibrational kinetic energy operator (KEO) is used, which properly accounts for the frozen CH bond lengths. ...
... The relevant extended CNPI group theory for treating isomerization in S 1 acetylene using the G (8) 4 symmetry group is fully treated by Hougen and Merer in Ref. 40. It can be shown that nearest-neighbor tunneling interactions should lead to an even/odd K-staggering for in-plane isomerization. ...
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 double group theory appropriate for this case [34] involves an extended permutation-inversion group G 12 isomorphic to C 6v , in which C 3 3 corresponds to a limited identity [35], i.e., C 3 3 corresponds to the permutation-inversion group identity, but not to the G 12 identity. Symmetry species can be labeled A 1s , A 2s , E s , A 1d , A 2d , and E d , where the A 1 , A 2 and E part of the labels are basically the same as in G 6 , while the s and d subscripts indicate a character of +1 or À1, respectively for the C 3 3 operation. ...
... The symmetry coordinates S in Eq. (35) are the coordinates used to carry out the diabatic calculations in Section 5, so we focus on them first. Since the formalism of [25] involves taking derivatives of various quantities with respect to c, we write the first equality in Eq. ...
... (19) and (22) of [25]) p c ¼ S tr f c P ¼ f c xy ðS x P y À S y P x Þ ¼ fðd½AðcÞ tr =dcÞ½M½AðcÞg xy ðS x P y À S y P x Þ ð 41Þ which can be shown to be zero as follows. From Eq. (35) and the notional discussion above, the 9 Â 41) is also zero, as was assumed in Section 5. The adiabatic calculations, however, are to be carried out using the normal coordinates Q in Eq. (35). ...
... Assuming the C 2 H 2 molecule remains planar, with the isomerization occurring exclusively through the local bends, the predicted level pattern is that even-K levels go with one of the two tunneling components of a vibrational state and odd-K levels go with the other. 38 No actual splittings will be observed, but instead there will be a staggering of odd-K levels versus even-K levels. At energies close to the barrier, where internal rotation must also be considered, there will be three tunneling components for each vibrational state, resulting in a further staggering of the K = 2, 6, 10,. . . ...
... One of the most interesting results of this work is the observation of an even-odd staggering in the K-structure of the cis-3 1 6 1 level, which indicates that tunneling through the cis-trans isomerization barrier takes place. Both the Discrete Variable Representation (DVR) calculations of Ref. 36 and the group theory considerations of Ref. 38 predict that such a staggering should occur, but it is not immediately obvious why it arises. A qualitative discussion may help to clarify its origin. ...
... Detailed examination of their effects on the various molecule-fixed coordinates 38 shows that the reversal of the CCH bending angles and the C 2 rotation about the a-axis are inextricably linked. To be exact, when these two operations are carried out in sequence, it looks as though nothing has happened to the molecule, or in other words, that the identity operation has been carried out. ...
A systematic analysis of the S(1)-trans (Ã(1)A(u)) state of acetylene, using IR-UV double resonance along with one-photon fluorescence excitation spectra, has allowed assignment of at least part of every single vibrational state or polyad up to a vibrational energy of 4200 cm(-1). Four observed vibrational levels remain unassigned, for which no place can be found in the level structure of the trans-well. The most prominent of these lies at 46 175 cm(-1). Its (13)C isotope shift, exceptionally long radiative lifetime, unexpected rotational selection rules, and lack of significant Zeeman effect, combined with the fact that no other singlet electronic states are expected at this energy, indicate that it is a vibrational level of the S(1)-cis isomer (Ã(1)A(2)). Guided by ab initio calculations [J. H. Baraban, A. R. Beck, A. H. Steeves, J. F. Stanton, and R. W. Field, J. Chem. Phys. 134, 244311 (2011)] of the cis-well vibrational frequencies, the vibrational assignments of these four levels can be established from their vibrational symmetries together with the (13)C isotope shift of the 46 175 cm(-1) level (assigned here as cis-3(1)6(1)). The S(1)-cis zero-point level is deduced to lie near 44 900 cm(-1), and the ν(6) vibrational frequency of the S(1)-cis well is found to be roughly 565 cm(-1); these values are in remarkably good agreement with the results of recent ab initio calculations. The 46 175 cm(-1) vibrational level is found to have a 3.9 cm(-1) staggering of its K-rotational structure as a result of quantum mechanical tunneling through the isomerization barrier. Such tunneling does not give rise to ammonia-type inversion doubling, because the cis and trans isomers are not equivalent; instead the odd-K rotational levels of a given vibrational level are systematically shifted relative to the even-K rotational levels, leading to a staggering of the K-structure. These various observations represent the first definite assignment of an isomer of acetylene that was previously thought to be unobservable, as well as the first high resolution spectroscopic results describing cis-trans isomerization.
... For coupled identical benders, as in ABBA molecules, the states must be representations of the appropriate point group. The permutation-inversion formalism 50,51 has been forwarded as a robust method and a discussion of the symmetry operations used here is available, 32 so here we briefly review only the two operations used in this work. Operationally, we introduce two symmetry operations on the basis |n 1 1 , n 2 2 . ...
... which is identical to the permutation-inversion symmetry operator E*. 51,52 We identify the symmetry of the states on the basis of the eigenvalues of these operators. Note that, since the HamiltonianĤ is rotationally invariant, the total twodimensional angular momentum 12 = 1 + 2 is conserved. ...
We study the bending motion in the tetratomic molecules C2H2 (X̃ (1)Σg (+)), C2H2 (Ã (1)Au) trans-S1, C2H2 (Ã (1)A2) cis-S1, and X̃ (1)A1 H2CO. We show that the algebraic operator expansion method with only linear terms comprised of the basic operators is able to describe the main features of the level energies in these molecules in terms of two (linear) or three (trans-bent, cis-bent, and branched) parameters. By including quadratic terms, the rms deviation in comparison with experiment goes down to typically ∼10 cm(-1) over the entire range of energy 0-6000 cm(-1). We determine the parameters by fitting the available data, and from these parameters we construct the algebraic potential functions. Our results are of particular interest in high-energy regions where spectra are very congested and conventional methods, force-field expansions or Dunham-expansions plus perturbations, are difficult to apply.
... is one such symmetry reduction sub-group where the individual molecules still have some freedom to rotate in and out of the plane in which they are confined. In more recent work, Hougen et al. [37] have extended permutation-inversion group studying large amplitude motion acetylene. The earliest work in treating large amplitude motion in non-rigid molecules goes back to Longuet-Higgins [15]. ...
The theory of Frame transformation relations between the states of Born Oppenheimer and the weak coupling approximations is developed for polyatomic molecules. The symmetry relations are a generalization of the frame transformation relations rived by Harter and Crogman for coupled rotor molecules. A key internal symmetry label (named "soul") is defined so that it remains a constant label for frame transformation relations, and is conserved during vibronic transitions, ionization, and even dissociation provided the nuclear spin-rotation interaction is relatively small. Simplified procedures are given for obtaining selection rules, statistical weights, and matrix elements of multipole operators for common molecules having various point symmetries.
... The L(c) matrix from Eq. (13) also changes the value of f c in Eq. (48) from À1 to +½. The double group theory appropriate for this case [34] involves an extended permutation–inversion group G 12 isomorphic to C 6v , in which C 3 3 corresponds to a limited identity [35], i.e., C 3 3 corresponds to the permutation–inversion group identity, but not to the G 12 identity. Symmetry species can be labeled A 1s , A 2s , E s , A 1d , A 2d , and E d , where the A 1 , A 2 and E part of the labels are basically the same as in G 6 , while the s and d subscripts indicate a character of +1 or À1, respectively for the C 3 3 operation. ...
Stemming from the observation of inverted A/E splittings for the
ν2 and ν9 asymmetric CH-stretching modes in
methanol, there has been much theoretical interest in attempting to
explain the nature of the inversion. We have recently examined the ab
initio normal-mode vibrational displacement vectors along the internal
rotation path for the three C-H stretching vibrations in methanol, both
in the symmetrized and non-symmetrized PAM coordinates. Graphical
representations of the Cartesian atomic normal mode displacement vectors
di(γ) determined by the Gaussian suite of programs for
the three CH stretching motions, ν2(A1),
ν3(A1) and ν9(A2),
along the steepest-descent internal rotation path γ in methanol
(CH_3OH) will be presented and discussed, where A1 and
A2 are notations in permutation-inversion group
G6. These modes are interesting because the symmetry
environment of each C-H bond changes significantly during the internal
rotation, i.e., each of the methyl bonds takes turns passing (twice for
a complete torsional revolution) through the plane of symmetry of the
COH frame of the molecule. We present some simple theoretical models
which can be used to help understand these displacement vectors.
Although this is work in progress, some explanation is already possible
for the rather irregular (avoided-crossing-like) behavior of these
displacement vectors.
Slow electron velocity-map imaging of cryogenically cooled anions (cryo-SEVI) is a high-resolution variant of anion photoelectron spectroscopy that has been applied with considerable success over the years to the study of radicals, size-selected clusters, and transition states for unimolecular and bimolecular reactions. Cryo-SEVI retains the versatility of conventional anion photoelectron spectroscopy while offering sub-meV resolution, thereby enabling the resolution of vibrational structure in the photoelectron spectra of complex anions. This Feature Article describes recent experiments in our laboratory using cryo-SEVI, including a new research direction in which anions are vibrationally pre-excited with an infrared laser pulse prior to photodetachment.
Slow electron velocity-map imaging of cryogenically cooled anions (cryo-SEVI) is a high resolution variant of anion photoelectron spectroscopy that has been applied with considerable success over the years to the study of radicals, size-selected clusters, and transition states for unimolecular and bimolecular reactions. Cryo-SEVI retains the versatility of conventional anion photoelectron spectroscopy while offering sub-meV resolution, thereby enabling the resolution of vibrational structure in the photoelectron spectra of complex anions. This article describes recent experiments in our laboratory using cryo-SEVI, including a new research direction in which anions are vibrationally pre-excited with an infrared laser pulse prior to photodetachment.
3-Ethynylquinoline was subjected to a Sonogashira-Hagihara reaction with methyl 2-, 3-, and 4-bromobenzoates, respectively, and then N-methylated to give 3-[(-(methoxycarbonyl)phenyl)ethynyl]-1-methylquinolinium salts (two X-ray analyses). On saponification of the 3- and 4-substituted benzoates, the mesomeric betaines 3- and 4-[(1-methylquinolinium-3-yl)ethynyl] benzoates were formed. By contrast, the 2-benzoate derivative gave either the corresponding (1-oxo-1H-isochromen-3-yl)quinolinium derivative, or the mesomeric betaine 2-(1-methylquinolinium-3-yl)-1,3-dioxo-2,3-dihydro-1H-inden-2-ide depending on the reaction conditions. A DFT calculation predicts a transoid conformation of the acetylene bond in the intermediate 2-[(1-methylquinolinium-3-yl)ethynyl]benzoate which is due to a strong hydrogen bond between the carboxylate group and 2-H of the quinolinium ring, in addition to a 1,5-interaction between the carboxylate group and the CC triple bond. The bond angles of the transoid CC triple bond were calculated to be 211.6 degrees and -175.1 degrees in vacuo. The corresponding linear triple bond is 50.4 kJ/mol less stable in vacuo according to the calculation, and the N-heterocyclic carbene quinoline-2-ylidene is not formed as a tautomer.
The proton transfer reaction is complicated due to the proton scrambling from the large amplitude motions in the intermediate. In order to understand this reaction, high-resolution spectroscopic studies are necessary for the reactants/products and the intermediate, and the group theoretical analysis is an essential aspect in the prediction and interpretation of these spectra. With five indistinguishable protons, is characterized using the complete nuclear permutation-inversion (CNPI) group. For most of the configurations sampled by the reaction path, the feasible permutations depend on the distance between the and fragments. Subgroups of can be used to describe these feasible permutations. Specifically, we consider two limits of the molecular configurations. The equilibrium structure of , i.e., , can be described using the molecular symmetry group, while the dissociation products, i.e., , require the molecular symmetry group. In the present study, a group theoretical analysis is performed for both limits, providing the symmetries for the nuclear spins and rovibrational wave functions. Also, spectroscopic properties for , particularly rovibrational couplings and electric dipole selection rules, as well as correlations of energy levels between and , are obtained.
Transition state theory is central to our understanding of chemical reaction dynamics. We demonstrate a method for extracting
transition state energies and properties from a characteristic pattern found in frequency-domain spectra of isomerizing systems.
This pattern—a dip in the spacings of certain barrier-proximal vibrational levels—can be understood using the concept of effective
frequency, ωeff. The method is applied to the cis-trans conformational change in the S1 state of C2H2 and the bond-breaking HCN-HNC isomerization. In both cases, the barrier heights derived from spectroscopic data agree extremely
well with previous ab initio calculations. We also show that it is possible to distinguish between vibrational modes that
are actively involved in the isomerization process and those that are passive bystanders.
We report novel experimental strategies that should prove instrumental in extending the vibrational and rotational assignments of the S1 state of acetylene, C2H2, in the region of the cis-trans
isomerization barrier. At present, the assignments are essentially complete up to ∼500 cm−1 below the barrier. Two difficulties arise when the assignments are continued to higher energies. One is that predissociation into C2H + H sets in roughly 1100 cm−1 below the barrier; the resulting quenching of laser-induced fluorescence
(LIF) reduces its value for recording spectra in this region. The other difficulty is that tunneling through the barrier causes a staggering in the K-rotational structure of isomerizing vibrational levels. The assignment of these levels requires data for K values up to at least 3. Given the rotational selection rule K′ − ℓ′′ = ± 1, such data must be obtained via excited vibrational levels of the ground state with ℓ′′ > 0. In this paper, high resolution H-atom resonance-enhanced multiphoton ionization spectra are demonstrated to contain predissociated bands which are almost invisible in LIF
spectra, while preliminary data using a hyperthermal pulsed nozzle show that ℓ′′ = 2 states can be selectively populated in a jet, giving access to K′ = 3 states in IR-UV double resonance.
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 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.
The modeling of coupled vibrational bending degrees of freedom within
the two-dimensional limit of the Vibron Model is presented. The new
aspects that arise in this case, in comparison to the single bender
case, are examined, with special emphasis on the different geometrical
limits embedded in a simple model Hamiltonian and the phase diagram
associated to this two-fluid bosonic model.
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.
We review recent research on the acetylene S(1) state that illustrates how mechanistic rather than phenomenological information about intersystem crossing (ISC) may be obtained directly from frequency-domain spectra. The focus is on the dynamically rich "doorway-mediated" ISC domain that lies between isolated spectroscopic spin-orbit perturbations and statistical-limit interactions between one singlet "bright state" and a quasi-continuum of triplet "dark states". New and improved experimental and data processing techniques permit the statistical-model curtain to be drawn back to reveal mechanistically explicit pathways via one or more identifiable, hence, manipulatable, doorway states, between a user-selected bright state and the undifferentiated bath of dark states.
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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.
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.
The far-infrared torsional band of hydrazine has been studied by Fourier transform spectroscopy with an apodized resolution of 0.011 cm−1. As a result of torsional as well as inversion tunneling, large splittings are observed in this b-type band. About 700 rRK and pPK transitions of 22 subbands with ΔK·K″ from −10 to +11 were assigned. The A-B, B-A, and E-E transitions were assigned for all subbands except for the ΔK·K″ = −2 and −1 subbands, for which only the nondegenerate transitions were observed. A global fitting, which includes all available ground state microwave data, was made using Hougen's group theoretical formalism. Several fitting constants, i.e., B-C, the trans torsional tunneling constant h3vt, and the inversion tunneling constant h5v, were found to exhibit large changes upon torsional excitation. The values of these constants in the torsional fundamental state are: B-C = 184.52(30) MHz, h3vt = −912.0(21) MHz, and h5v = 1994.1(16) MHz, where the numbers in parentheses are 1 σ.
A theoretical formalism is presented for fitting rotational energy levels in isolated (unperturbed) vibrational states of methylamine. This formalism is obtained by recasting and extending theoretical studies in the earlier literature, which were undertaken to help analyze the methylamine microwave spectrum. The present formalism is applicable when both the NH2 umbrella (wagging) motion and the CH3 internal rotation (torsion) motion take place near the high-barrier limit and leads to the usual Fourier sine and cosine series expansions for molecular energy levels. The derivation is separated into two parts, one treating the large-amplitude vibrational problem (the torsional-wagging problem) by itself, the other treating the torsional-wagging-rotational problem. In both treatments, permutation-inversion group and extended group ideas are used to determine the allowed terms in an effective rotational-tunneling Hamiltonian operator and to block diagonalize the matrix representation of this operator for a near-prolate symmetric top. The resulting energy levels and selection rules are discussed, but application of the method in detail to the methylamine spectrum is planned for a later paper.
Results derived previously for the rotational levels of the eight-framework and three-large-amplitude vibrational problem in N2H4, using a tunneling formalism based on a treatment of the vibration-rotation problem as a whole, are rederived here in a much simpler fashion, using a tunneling formalism based on separate treatment of the vibrational and rotational problems. The present formalism is thus much more akin to the usual vibration-rotation formalism, and the origins of the various contributions to the vibration-rotation energy levels can be understood relatively easily. It is convenient here, as earlier, to make extensive use of permutation-inversion and extended-group (double-group) ideas, but it is necessary in the present treatment to consider tunneling between 16 minima in molecular coordinate space, i.e., between a number of minima which is twice the number of nonsuperimposable molecular frameworks that can actually be constructed for N2H4.
The ideas of Hougen and Longuet-Higgins concerning molecular symmetry have been extended to linear molecules by introducing ``extended permutation-inversion groups''. Using such groups it is possible to show that the rovibronic states (rotational levels) of a linear molecule can be classified according to the symmetry species of the point group (D∞h or C∞v). For a molecule with D∞h symmetry the rotational levels can be labeled Sigmag+, Sigmau+, Sigmag-, or Sigmau-, which is equivalent to the usual notation of +s, +a, -a, or -s respectively. For a molecule with C∞v symmetry the rotational levels can be labeled Sigma+ or Sigma-, which is equivalent to the usual notation of + or -, respectively. Although no new symmetry predictions are obtained this does remove an obscurity concerning the symmetry classification of the levels of a linear molecule. Further-more, there is an interesting group-theoretical relationship between the extended permutation-inversion group of a linear molecule (which is isomorphic with D∞h or C∞v) and the corresponding permutation-inversion group (which is isomorphic with C2v or Cs).
A Hamiltonian matrix suitable for fitting rotational energy levels of the hydrazine molecule in its ground vibrational and electronic states was obtained. This matrix is of dimension 16 × 16, where the 16 functions labeling the rows and columns consist of the two members of a near-prolate asymmetric rotor doublet (with given | Ka |) for the eight different, but chemically equivalent, conformers which the molecule can reach by various combinations of -NH2 inversion at either end of the molecule, and internal rotation about the NN bond. The matrix is derived in a phenomenological fashion, by applying group theoretical arguments to a model in which tunneling among the various frameworks is assumed to be very slow compared with the vibrational frequencies. A comprehensive treatment of the large-amplitude vibrational potential surfaces and associated tunneling pathways has not been carried out, nor have quartic (J4) centrifugal distortion effects been considered in a systematic fashion. Preliminary fits indicate that the model developed can be used to fit the hydrazine microwave data in a consistent fashion, and a full treatment of such data has been undertaken.
In this paper we derive an expression for the vibration-rotation Hamiltonian of a triatomic molecule. In the derivation we use a curvilinear bending coordinate and two rectilinear stretching coordinates in such a way that the Hamiltonian obtained is applicable for any triatomic molecule, linear or bent, and allows for large displacements of the bending coordinate (sometimes said to result from the molecule being ``quasi-linear'' but, in fact, of general occurrence). We derive a zeroth-order Hamiltonian to describe the energy levels associated with the bending vibration, and are able to fit the experimental results on HCN and DCN better than if we had used the standard formalism of rectilinear (and small) displacements. We also use the formalism to describe the dependence of the rotational constant B on the bending vibrational quantum number and apply the results to the microwave data on CsOH and CsOD.
Group theoretical treatments found in the literature for other molecules, based on double-groups of appropriate permutation–inversion groups, are modifed slightly and applied to H2O2. This permits a more unified view of many theoretical results derived in various earlier studies of the molecule. Briefly, if no effects due to internal rotation tunneling were observed in the H2O2 spectrum, the molecule could be treated using the C2 point group of its equilibrium geometry. If effects due to internal rotation tunneling through only the trans barrier are observed (as is presently the case), the molecule should be treated using the C2h point group of its trans planar conformation at the top of the tunneling barrier. If effects due to internal rotation through both the trans and cis barriers can be detected, it will be necessary to treat the molecule using a double group of the C2h, point group. Vibrational, torsional, and rotational basis function symmetry species, energy level diagrams, selection rules for electric dipole transitions and perturbations, etc. are presented here for the double group, but these results can easily be converted to those for the C2h point group by dropping all s and d subscripts. An empirically discovered successful fitting procedure for the ground state rotational levels, reported in a treatment of microwave and millimetre wave measurements on H2O2, can be rationalized on the basis of the present theoretical results.
The rovibronic and vibronic energy levels of some nonrigid molecules occur in nearly degenerate sets, whose splitting is due to tunneling between symmetrically equivalent regions of vibrational phase space. When the splitting is negligible, it is convenient to classify the different sets of energy levels according to a symmetry group that is determined by any one of the symmetrically equivalent regions, whereas when the splitting is appreciable, the individual split levels can be classified further according to a larger group that is determined in a similar way by a region containing all these equivalent regions. The former group is a subgroup of the latter.A simple correlation rule is proved, by means of which the symmetry species (in the larger group) of the individual split levels can be obtained from the symmetry species (in the smaller group) of the nearly degenerate set of levels. Some examples of the application of this rule to rovibronic and vibronic species are given.
It is convenient when performing calculations on a vibrating and rotating molecule to define an axis system which is somehow fixed to the molecule. The orientation of the usual molecule-fixed axis system, however, depends not only upon the instantaneous positions of the nuclei, but also upon the equilibrium positions from which the nuclei are regarded as being displaced. Thus, when a molecule of low enough symmetry undergoes an electronic transition accompanied by a change in geometry, it will, in general, be necessary to consider two molecule-fixed axis systems, corresponding to the two different electronic states. This change in axis system from one electronic state to another will be called axis-switching. The two axis systems can be related to each other by the 3 × 3 rotation matrix which brings them into coincidence. The elements of this matrix are functions of the equilibrium geometries of the two electronic states as well as of the instantaneous positions of the atoms in the molecule. Axis-switching leads to departures from the usual expressions for the intensities of rotational lines, the effects of which are most noticeable in near-symmetric tops. The forbidden subbands occurring in the 2 400 Å system of acetylene can be satisfactorily explained by axis-switching. Axis-switching effects may also be present in the spectra of HCN, HSiCl, and HSiBr.
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.
High resolution spectra of H2O2, recorded by means of Fourier transform spectroscopy between 30 and 460 cm−1, have been analyzed leading to the determination of the rotational levels of the torsional states (n,τ) for n=0,1,2,3. In order to reproduce these energy levels, Watson type Hamiltonians have been used and it has been possible to observe a staggering of the levels with n=2 and 3 caused by the cis barrier. The torsional band centers have then been fitted using a torsional Hamiltonian of the form {Bγγ,J2γ} +V(γ) with the potential function V(γ) written as V(γ)=V1 cos 2γ+V2 cos 4γ+V3 cos 6γ+V4 cos 8γ where the torsional coordinate 2γ is the dihedral angle defining the relative position of the two O–H bonds. The potential constants in cm−1 are V1=1036.97±23.1 cm−1, V2=657.53±5.2 cm−1, V3=50.89±3.3 cm−1, V4=2.524±0.83 cm−1 which correspond to barrier heights Vtrans =387.07±0.20 cm−1, Vcis =2562.8±60 cm−1, and to a potential minimum located at 2γ=111.9°±0.4° from the cis configuration. It is also shown that the rotational constants derived from the fit to the experimental rotational levels cannot be reproduced using a model which does not take into account vibrational corrections.
The permutation-inversion group developed by Longuet-Higgins is extended to a classification of the vibronic, torsional, and rotational wavefunctions of a nonrigid X2Y2 molecule by introducing a symmetry operation , which rotates the top half of the molecule by 2π and, accordingly, the molecule-fixed x axis by π. Since the energy levels of linear (D∞h) and bent (C2h, C2h, and C2) forms of X2Y2 are classified according to a set of common symmetry operations of this extended permutation-inversion group, their energy levels can be correlated, including those of nonrigid forms such as a quasilinear system or a free internal rotor. Nuclear spin weights and selection rules are derived.
An attempt is made to distill from the published literature a summary of some new uses of group theory in high resolution gas-phase spectroscopic studies of molecules with large amplitude and/or tunneling motions, paying particular attention to questions like the following: (i) When is a point group sufficient, and when is it not? (ii) What kind of information is easy, and what kind is difficult to extract from a permutation-inversion group treatment? (iii) What seem to be the advantages and disadvantages of various extended groups of the permutation-inversion group? While most spectroscopists would agree that a general group theoretical approach, suitable for application without modification to the majority of floppy molecules, has not yet been synthesized from the particular cases studied in the literature, some feeling for one direction of progress in the field can be obtained from the several examples presented.
The concept of molecular symmetry is extended to molecules such as ethane and hydrazine which can pass from one conformation to another. The symmetry group of such a molecule is the set of (i) all feasible permutations of the positions and spins of identical nuclei and (ii) all feasible permutation-inversions, which simultaneously invert the coordinates of all particles in the centre of mass. According to the representations of this group one can classify not only the spin states and states of motions of the nuclei, but even the electronic states of the molecule. Examples are given to illustrate the use of this concept in determining the statistical weights of individual levels and selection rules for electric dipole transitions between them.
It is shown that the rotational energy levels of a symmetric‐top molecule can be classified according to symmetry species of the full symmetry group of the molecule. Useful selection rules for electric‐dipole transitions between rotational energy levels thus classified are presented. It is shown that the rotational energy levels can be labeled in addition by a convenient quantum number G, which is essentially a generalization of the +l, —l labels. Selection rules for this quantum number are presented, which, together with those for the symmetry species, allow one to determine easily the allowed branches for a transition between any two vibronic states in a symmetric‐top molecule.
The theory of the symmetry properties of the rotational and torsional levels is developed for molecules consisting of two identical XY2 groups connected by a symmetrical linear chain of atoms, with an arbitrary barrier to internal rotation. The levels are classified according to the representations of the double group of the Longuet-Higgins permutation-inversion group, and selection rules for electric dipole transitions are derived. It is shown that the symmetries of the normal coordinates of ethylene-like molecules can be different from those given recently by Papoušek, Sarka, Špirko and Jordanov. The results are applicable to the vibrational spectra of molecules like B2F4 and B2Cl4, which have low barriers to internal rotation in their ground states, and to the vibrational and rotational structure of electronic transitions of C2H4, where the combining electronic states may have very different torsional potential functions.