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The singular structure of lower vibrational states of the difluorosilylene molecule (up to four total excitation quanta) is studied by expansion of energies of each state into series of the high-order Rayleigh–Schrodinger perturbation theory and analysis of their implicit properties of multiple-valuedness using Pade–Hermite fourth-order approximants. The quartic surface of potential energy in dimensionless normal coordinates is calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of values of multiple-valued approximants reproduces the variational solution with a high accuracy, while other values (beginning with the fourth polyad) in many cases coincide with energies of other states of the same polyad. The Fermi and Darling–Dennison resonances are analyzed based on facts of coincidence of singular complex branch points of the approximants for mutually interacting states inside a unit circle or near it on the complex plane. It is found that a pair of states can have several coinciding branch points of the solutions (in particular, inside the unit circle). A conclusion is made that this approach is an effective method for the determination of the polyad structure of vibrational states. Calculation parameters necessary for the reproducibility of key results are selected. The calculations are carried out by means of a Fortran-based software package using an arithmetic calculation package with a long mantissa of real numbers (200 digits).

The potential functions of internal rotation V(φ) (PFIR) of methacryloyl fluoride (MAF) in the ground (S0) and excited (S1) electronic states were reconstructed using the TORSIO program from experimental energy levels of the torsional transitions in the s-trans- and s-cis isomers. The energy levels were obtained from the analysis of the vibrational structure in the high-resolution gas-phase UV spectrum. The (S0) and (S1) equilibrium geometries of the s-trans- and s-cis rotational isomers of MAF as well as structural parameters for a set of points along the relaxed geometry scans, which were used to find expansion coefficients of kinematic function F(φ) in the Fourier series, were calculated using extended multiconfiguration quasi-degenerate perturbation theory, XMCQDPT2/SA(2)-CASSCF(6,5)/aug-cc-pVTZ. The ground (S0) state calculations were also done at the MP2 level of theory. In (S0) state, parameters Vn., barriers of internal rotation ΔH‡ and ΔH of MAF isomers were calculated using two theories. The calculated values of Vn, ΔH‡, ΔH in (S0) state are in good agreement with those derived from the vibrationally resolved UV spectrum of MAF in the gas phase. Within the one-dimensional model, the experimentally derived barrier heights that hinder internal rotation in MAF were found to be 2100 ± 50 cm−1 and 4950 ± 250 cm−1 in the (S0) and (S1) states, respectively. The parameters Vn and ΔH‡ in (S1) state are determined for first time.

Quantum-mechanical methods of solving the polyatomic vibrational Schrodinger equation need higher quality zero-order approximations than ones originating from the harmonic oscillator (HO). ...

Hexafluoroacetone (HFA, O C(CF3)2) is widely utilized in organic synthesis, chemistry of heat resistant polymers and biochemistry. In this work, infrared (IR) spectra of gaseous HFA were recorded in the range 2000–340 cm−1 with a resolution of 1 cm−1. For the first time, experimental absolute IR intensities of fourteen fundamental bands of HFA were measured and compared with intensities predicted by anharmonic ab initio calculations. Theoretical molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment (DM) components of HFA were calculated using MP2/cc-pVTZ ab initio quantum-mechanical model. In addition, harmonic frequencies were also calculated at MP2/cc-pVQZ level of theory and used for refining the PES. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic IR spectra of HFA. The deviation from observations of predicted IR intensities for ten fundamentals (ν1, ν2, ν8, ν13–ν21) was less than 25% on average. The significant difference between ab initio harmonic intensities calculated with MP2/cc-pVTZ and MP2/cc-pVQZ models for some normal modes indicates that an inaccuracy of the quantum-mechanical model is probably the major source of errors in theoretical intensities. An interpretation of the IR experimental spectra and associated literature data for HFA is accomplished; the ν4, ν7, ν10 and ν15 fundamentals were reassigned. Three predicted Fermi resonances were identified in experimental spectra, ν3/ν5 + ν8, ν17/ν10 + ν19 and ν20/ν9 + ν23. An empirical fit of harmonic frequencies within anharmonic model assisted in a reliable identification of a number of overtones and binary combination tones. The fitted harmonic frequencies were compared with MP2/cc-pVQZ and a simplified complete basis set (CBS) extrapolation. It was found that MP2/cc-pVQZ frequencies were more reliable. The efficiency and convenience of the theoretical model employed and associated techniques for interpretation of complex spectra of HFA are demonstrated.

Linear-chain carbon films with a thickness of order 100 nm were studied by tunneling spectroscopy. The oscillating dependence of the differential conductivity of the investigated structures is established. The results obtained are interpreted using a model of charge-density wave formation in regular structural kinks of carbon chains. The Raman spectra of the films are recorded. The simulated spectra of harmonic oscillations of polyines (–C≡C–) n and cumulenes (=C=) n of carbon films are theoretically compared.

The Raman spectra of kinked carbon chains with a set of end groups composed of hydrogen, silver, and copper atoms are calculated. It is analyzed how end groups, as well as the positions of impurity atoms in the chain, affect shifts characterizing the main peaks of the harmonic vibrations of chemical bonds between carbon atoms in polyyne- and cumulene-like (respectively, (–C≡C–) n and (=C=) n ) chains.

Bis(trifluoromethyl)ketene (BTFMK, (CF 3 ) 2 C C O) is an important chemical compound for the organic synthesis and in polymer chemistry. In this work, IR spectra of gaseous BTFMK were recorded in the range of 2280–400 cm − 1 with a resolution of 1 cm − 1 . The experimental absolute IR intensities of 17 fundamental bands of BTFMK were measured and juxtaposed with ab initio anharmonic ones. The molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment components ( μ ^ α ) were calculated at the MP2/cc-pVTZ level of theory. The predicted frequencies were refined by substituting the harmonic part of PES with the MP2/cc-pVQZ counterparts. The second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic energy levels, IR integral intensities and full experimental (re)-assignments. The deviation of predicted IR intensities from observations for ν 1 − ν 4 , ν 6 , ν 7 , ν 9 , ν 14 , ν 15 , ν 22 was ≤ 12% on average. Three predicted Fermi resonances were experimentally identified: ν 14 / ν 6 + ν 20 , ν 17 / ν 9 + ν 21 and ν 18 / ν 7 + ν 25 .

A new gas-phase infrared (IR) spectrum of acryloyl fluoride (ACRF, CH2 CH CF O) with a resolution of 0.1 cm− 1 in the range 4000–450 cm− 1 was measured. Theoretical ab initio molecular structures, full quartic potential energy surfaces (PES), and cubic surfaces of dipole moments and polarizability tensor components (electro-optical properties, EOP) of the s-trans and s-cis conformers of the ACRF were calculated by the second-order Moller-Plesset electronic perturbation theory with a correlation consistent Dunning triple-ζ basis set. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory was employed for predicting anharmonic IR and Raman scattering (RS) spectra of ACRF. To improve the anharmonic predictions, harmonic frequencies were replaced by their counterparts evaluated with the higher-level CCSD(T)/cc-pVTZ model, to form a “hybrid” PES. The original operator representation of the Hamiltonian is analytically reduced to a quasi-diagonal form, integrated in the harmonic oscillator basis and diagonalized to account for strong resonance couplings. Double canonical transformations of EOP expansions enabled prediction of integral intensities of both fundamental and multi-quanta transitions in IR/RS spectra. Enhanced band shape analysis reinforced the assignments. A thorough interpretation of the new IR experimental spectra and existing matrix-isolation literature data for the mixture of two conformers of ACRF was accomplished, and a number of assignments clarified.

The complete semiexperimental (SE) equilibrium structure of proline (45 degrees of freedom) is determined using the mixed estimation method. The cubic force fields for the parent and eight isotopologues of proline molecule are evaluated at the MP2-FC/cc-pVTZ level in Cartesian coordinates. The accuracy of the SE structure is verified by optimising the structure with the CCSD(T) model and a basis set of quadruple-ζ quality. A significantly more accurate equilibrium structure of proline is obtained when compared to the previous one. It is shown that the employed technique is efficient for the determination of SE equilibrium structures of rather large molecules. A simple transformation of anharmonic force fields between normal coordinate and Cartesian coordinate representations is proposed. The suggested technique allows efficient evaluation of the rotation–vibration interaction constants for a number of isotopologues, once the cubic force field of any species is found either in normal or Cartesian c...

A quantitative study of the vibrational resonance phenomena of the HOF (hypoflorous acid) molecule for over 100 of its lower vibrational energy levels was carried out using resummation techniques of the Rayleigh-Schrodinger perturbation theory (RSPT) divergent series. Anharmonic vibrational states of HOF were calculated by the matrix configuration interaction (VCI) method, the second-order Van Vleck operator perturbation theory (CVPT2), and the high-order RSPT series. CVPT2 predicts a weak Fermi resonance (0,1,–2) and a medium second-order resonance (1,–2,–1). Resummation of the high-order ( λ n , 0 ≤ n ≤ 203 ) divergent RSPT series, using multivalued quartic 40-th degree Pade-Hermite diagonal approximants up to terms O ( λ n + 1 ) for a subset of low-lying states, restored their correct energies. Considering λ as a complex parameter, solution branch points were found as roots of discriminant polynomials. The coincidence of dominant branch points for pairs of states according to the Katz theorem and the condition | λ | = Re ( λ ) 2 + Im ( λ ) 2 ≤ 1 , revealed resonance couplings. The block-diagonal polyad structure of studied states was reconstructed and the polyad quantum number P = ( 8 , 3 , 2 ) form determined as being good for all studied states, up to nearly 14,000 cm − 1 . Advantages of the developed technique for quantitative characterization of resonance and polyad phenomena is demonstrated.

The structural parameters of s-trans- and s-cis-isomers of a methacrolein molecule in the ground (S0) electronic state are determined by means of MP2 method with the cc-pVTZ basis set. Kinematic factor F(φ) is expanded in a Fourier series. The potential function of internal rotation (PFIR) of methacrolein in this state is built using experimental frequencies of transitions of the torsional vibration of both isomers, obtained from an analysis of the vibrational structure of the high-resolution UV spectrum with allowance for the geometry and difference between the energy (ΔH) of the isomers. It is shown that the Vn parameters of the potential function of internal rotation of the molecule, built using the frequencies of the transition of the torsional vibrations of s-trans- and s-cis-isomers of the methacrolein molecule, determined from vibrational structure of the high-resolution UV spectrum and the FTIR spectrum, are close.

Infrared overtone spectra of the ν(C = O) stretching vibration (ν1) of gaseous hexafluoroacetone ((CF3)2C = O, HFA) were recorded in the spectral range of 7450–3300 cm−1 with a resolution of 0.1 cm−1. Experimental absolute IR intensities and vibrational band centers of the overtones 2ν1, 3ν1, 4ν1 of HFA were measured and compared with their ab initio counterparts, calculated by the second-order canonical vibrational perturbation theory (CVPT2). A hybrid potential energy surface (PES) was evaluated using the quantum-mechanical models MP2/cc-pVQZ for harmonic and MP2/cc-pVTZ for anharmonic parts. Cubic surfaces of dipole moment components were determined using the MP2/cc-pVTZ model. The predicted IR intensities for the first and second overtones reproduced the experimental values with a discrepancy of 6% and 9%, respectively. A weak Fermi resonance between the first overtone 2ν1 and combination tone ν1 + ν2 + ν8 was predicted. The appropriate model was employed for simulating the bands studied using: (i) the asymmetric-top vibration-rotational structure of the ν1 mode, (ii) the inhomogeneous band structure due to contributions of the vibrational states populated at the room temperature, and (iii) the homogeneous broadening of each vibration-rotation transition due to the intramolecular vibrational redistribution (IVR). The rotational and vibrational anharmonic constants were taken from the ab initio calculations, whereas the IVR data were obtained from the experimental data of Chekalin et al. (JPCA 2014, 118:955) with a time resolution of ≈100 fs. A good level of agreement of the predicted shapes of bands studied with experiment is achieved.

The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrodinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associated QNS ladder operators are occupation number, creation and annihilation operators K 0 , K + , K − that satisfy the commutation relations [ K 0 , K ± ] = ± K ± and [ K + , K − ] = − 2 K 0 of the Lie algebra su(1, 1). The efficient applications of the corresponding algebraic techniques within the variational or perturbative approximations require reduction of arbitrary multiplicative combinations of the su(1,1) ladder operators to the linear combinations of the normalized products K 0 a K + b K − c . The closed form solution of this problem for a typical complex form of the operator product is derived and proven mathematically. The validity and the numerical stability of the derived expressions are demonstrated by the systematic numerical tests for the product of six-term multiplicative operators.

The vibrational spectra of 2,3,3,3-tetrafluoropropene (2333TFP) are studied in the infrared experimentally and theoretically by the canonical second-order Van Vleck operator perturbation theory (CVPT2) and full quartic potential energy surface (PES). 2333TFP belongs to hydrofluoroolefins (HFOs) considered among the most promising alternatives to the hydrofluorocarbons presently in use in refrigeration. The medium resolution infrared spectra of gaseous 2333TFP were recorded in the range 9500–30 cm − 1 . The integrated IR intensities for all bands falling between 6300–400 cm − 1 were accurately determined. Theoretical descriptions of up to four-quanta anharmonic vibrational states of 2333TFP are performed using the state-of-the-art numerical-analytic implementation of CVPT2. High quality harmonic frequencies, obtained at different coupled-cluster levels of theory up to the CCSD(T*)-F12c/VTZ-F12 model were combined with the full quartic MP2/cc-pVTZ anharmonic force field to form the hybrid PES. All Fermi and Darling-Dennison resonances were detected using a universal criterion and treated with the variational CVPT2/VCI stage by using two-, three-, and four-quanta harmonic oscillator basis sets. General recommendations for choosing such basis sets in future studies are formulated. In addition to all fundamentals and their resonance satellites, the vibrational analysis led to the assignment of virtually all observed first/second overtone and binary/ternary combination bands. Computed dipole moment first derivatives in the principal axis system yielded reliable predictions of band shapes for fundamental transitions in good agreement with observed counterparts. Computed anharmonic IR intensities showed an excellent agreement with observed data in the measured ranges. The fundamental spectroscopic effect of excited resonances is observed experimentally and explained theoretically as a universal mechanism of formation of vibrational polyads. The efficiency of the CVPT2/VCI approach employed is clearly demonstrated and can be considered as a benchmark for modeling vibrational states and interpretation of vibrational spectra of semi-rigid molecules.

Molecular vibrational resonances are manifest by abnormally small energy differences in denominators of anharmonic constants derived within the second-order vibrational perturbation theory (VPT2), ...

The numerical-analytic implementation of the operator version of the canonical Van Vleck second-order vibrational perturbation theory (CVPT2) is employed for a purely ab initio prediction and interpretation of the infrared (IR) and Raman anharmonic spectra of a medium-size molecule of the diketo tautomer of uracil (2,4(1H,3H)-pyrimidinedione), which has high biological importance as one of the four RNA nucleobases. A nonempirical, semidiagonal quartic potential energy surface (PES) expressed in normal coordinates was evaluated at the MP2/cc-pVTZ level of theory. The quality of the PES was improved by replacing the harmonic frequencies with the "best" estimated CCSD(T)-based values taken from the literature. The theoretical method is enhanced by an accurate treatment of multiple Fermi and Darling-Dennison resonances with evaluation of the corresponding resonance constants W and K (CVPT2+WK method). A prediction of the anharmonic frequencies as well as IR and Raman intensities was used for a detailed interpretation of the experimental spectra of uracil. Very good agreement between predicted and observed vibrational frequencies has been achieved (RMSD ∼4.5 cm(-1)). The model employed gave a theoretically robust treatment of the multiple resonances in the 1680-1790 cm(-1) region. Our new analysis gives the most reliable reassignments of IR and Raman spectra of uracil available to date.

The anharmonic vibrational IR and Raman spectra of the bicyclo[1.1.0]butane molecule have been calculated in the range of up to 4000 cm−1 using a numerical and analytical realization of the van Vleck second-order operator perturbation theory. Cubic and quartic force constants in normal coordinates, as well as cubic surfaces of the dipole moment and polarizability, have been found by numerical differentiation of the corresponding first and second derivatives calculated by the MP2/cc-pVTZ quantum-mechanical method. In order to increase the prediction accuracy of vibrational transitions, corresponding harmonic frequencies have been obtained by the CCSD(T)/cc-pVTZ high-precision quantum mechanical method. The anharmonic intensities of the IR and Raman spectra have been calculated using canonical transformations of the operators of the dipole moment and polarizability expanded into a Taylor series around the equilibrium configuration. The assignment of experimental vibrational bands in the IR and Raman spectra has been analyzed. It has been shown that the anharmonic calculation based on the above-described procedure of combining more exact harmonic frequencies with the anharmonic force field obtained with a more economical method makes possible the reliable interpretation of the majority of spectral bands, including Fermi and Darling-Dennison resonances.

Three dihalogenated methane derivatives (CH2F2, CH2FCl, and CH2Cl2) were used as model systems to compare and assess the accuracy of two different approaches for predicting observed fundamental frequencies: canonical operator Van Vleck vibrational perturbation theory (CVPT) and vibrational configuration interaction (VCI). For convenience and consistency, both methods employ the Watson Hamiltonian in rectilinear normal coordinates, expanding the potential energy surface (PES) as a Taylor series about equilibrium and constructing the wavefunction from a harmonic oscillator product basis. At the highest levels of theory considered here, fourth-order CVPT and VCI in a harmonic oscillator basis with up to 10 quanta of vibrational excitation in conjunction with a 4-mode representation sextic force field (SFF-4MR) computed at MP2/cc-pVTZ with replacement CCSD(T)/aug-cc-pVQZ harmonic force constants, the agreement between computed fundamentals is closer to 0.3 cm-1 on average, with a maximum difference of 1.7 cm-1. The major remaining accuracy-limiting factors are the accuracy of the underlying electronic structure model, followed by the incompleteness of the PES expansion. Nonetheless, computed and experimental fundamentals agree to within 5 cm-1, with an average difference of 2 cm-1, confirming the utility and accuracy of both theoretical models. One exception to this rule is the formally IR-inactive but weakly allowed through Coriolis-coupling H-C-H out-of-plane twisting mode of dichloromethane, whose spectrum we therefore revisit and reassign. We also investigate convergence with respect to order of CVPT, VCI excitation level, and order of PES expansion, concluding that premature truncation substantially decreases accuracy, although VCI(6)/SFF-4MR results are still of acceptable accuracy, and some error cancellation is observed with CVPT2 using a quartic force field.

The anharmonic model of vibrations of polyatomic molecule makes it possible, using the secondorder perturbation theory, to interpret vibrational spectra in detail with resonances taken into account and to calculate the band intensities in spectra for fundamental frequencies, overtones, and combination frequencies. For molecules possessing a center of symmetry (to which trans-1,3-butadiene belongs), many vibration modes have zero intensities of absorption in IR spectra due to the mutual exclusion rule. For meaningful analysis of vibrations of such molecules, measurement of Raman scattering (RS) spectra is necessary, as is developing a theoretical model for calculating the anharmonic intensity. Perturbation theory (PT) in the form of contact transformations (CTs) has proven applicable for calculating the anharmonic intensities in RS spectra. The FORTRAN software program ANCO has been developed, which allows the calculation of vibration frequencies and IR-RS intensities of fundamental vibrations, overtones, and combination frequencies on the basis of second-order PT in form of CTs using the polynomial representation of potential energy, dipole moment and polarizability surfaces. The obtained frequencies and modes of anharmonic vibrations are provided, as well as interpretation of the experimental spectrum of the trans-1,3-butadiene molecule. A procedure for calculating the scale factors of the anharmonic force field is proposed. We also show that within the anharmonic vibration model, these factors are close to unity.

The anharmonic frequencies of fundamental vibrations, overtones, and combination vibrations, as well as the intensities of absorption bands in the IR spectrum of uracil, are calculated. The anharmonic quartic force field and the third-order dipole moment surface calculated by the DFT quantum-mechanical method (B3LYP/6-31+G(d,p)) are taken as the initial parameters. The anharmonic frequencies and intensities of vibrations are determined using the second-order perturbation theory in the form of contact transformations. Multiple Fermi resonances and polyads are determined by the diagonalization of a small interaction matrix of vibrations of different types (fundamental, combination, and overtone frequencies). The total experimental IR spectrum of matrix-isolated uracil is interpreted. It is shown that the used method of calculating anharmonic frequencies and intensities can form a basis for anharmonic calculations of vibrations of moderate molecules.

The systems of torsional vibration levels of the trans and cis methacryloyl chloride isomers in the ground (S 0) and excited (S 1) electronic states obtained by analyzing the vibrational structure of the gas-phase UV spectrum were used to reproduce the internal rotation potential functions of the molecule in both electronic states. The kinematic F factor in the S 0 and S 1 electronic states was calculated taking into account the relaxation of geometric parameters depending on the internal rotation angle. The internal rotation potential function parameters in the S 0 state are substantially different from the parameters obtained using the torsional levels of the IR Fourier transform spectrum; at the same time, they are substantiated by quantum-mechanical calculations.

The problem of describing real vibrational spectra of large molecules in terms of perturbation theory is considered. Equations necessary for presenting theoretical anharmonic force fields in various coordinate systems (Cartesian, normal, and internal curvilinear) are discussed. A review of second-order perturbation theory equations necessary for calculating certain spectroscopic values (anharmonicity constants, rotational-vibrational interaction, etc.) is given. A scheme for including resonances based on the construction of the interaction matrix between vibrational transitions of various types is described. This scheme can be used as a basis for anharmonic calculations of vibrations of medium-sized molecules.

Sources of uncertainties that arise in the search for scale factors used to systematically correct theoretical quantum-mechanical force fields in the harmonic approximation are considered. The special features of various approaches to the traditional procedure for scaling according to Pulay are analyzed. It is suggested that scale factors be determined as quotients obtained by dividing the diagonal elements of the approximation to the exact matrix of force constants by their quantum-mechanical analogues. Using this approach obviates the necessity of solving the inverse problem. Perturbation theory is used to show that, when molecular geometric parameters and scale factors are varied jointly, an exact correlation can appear, which substantiates the effective character of force field corrections with the use of scale factors. The procedure for the scaling of generalized quantum-mechanical force fields based on the transition from the normal-coordinate to natural-coordinate representation is considered.

The internal rotation potential function of the acryloyl chloride molecule in the S 0 and S 1 electronic states was reproduced using systems of torsional vibration levels obtained for its trans and cis isomers by analyzing the vibrational structure of the UV spectrum of the molecule. The kinematic factor F in the S 0 ground state was calculated including geometric parameter relaxation as a function of internal rotation angle. The torsional potential parameters in the S 0 state obtained in this work were substantially different from those determined from the infrared Fourier-transform spectrum ignoring the resonance perturbation of the level with v = 3. The form of the internal rotation potential function and the higher stability of the trans isomer (the main isomer) were substantiated by high-level quantum-mechanical calculations.

The applicability of the classic method for calculating the equilibrium structure (r e) of a polyatomic molecule with the use of experimental rotational constants and rotational-vibrational interaction constants (α ) is considered. Direct minimization of the nonlinear functional of the sum of the squares of discrepancies by the Broyden-Fletcher-Goldfarb-Shanno method is suggested. Statistical parameters (confidence intervals and correlation coefficients) can be calculated by the double numerical differentiation of this functional. If the problem is ill-conditioned or some data are lacking, the Tychonoff regularization method can be used. The effectiveness of the suggested approach is demonstrated for the example of two molecules (cyclopropene and 3,3-difluorocyclopropene).

The second-order vibrational Hamiltonian of a semi-rigid polyatomic molecule when resonances are present can be reduced to a quasi-diagonal form using second-order vibrational perturbation theory. Obtaining exact vibrational energy levels requires subsequent numerical diagonalization of the Hamiltonian matrix including the first- and second-order resonance coupling coefficients. While the first-order Fermi resonance constants can be easily calculated, the evaluation of the second-order Darling-Dennison constants requires more complicated algebra for seven individual cases with different numbers of creation-annihilation vibrational quanta. The difficulty in precise evaluation of the Darling-Dennison coefficients is associated with the previously unrecognized interference with simultaneously present Fermi resonances that affect the form of the canonically transformed Hamiltonian. For the first time, we have presented the correct form of the general expression for the evaluation of the Darling-Dennison constants that accounts for the underlying effect of Fermi resonances. The physically meaningful criteria for selecting both Fermi and Darling-Dennison resonances are discussed and illustrated using numerical examples.

Anharmonic vibrational frequencies and intensities (infrared and Raman) of an isolated free-base porphin molecule are predicted from the quantum mechanical (QM) geometry, the "semi-diagonal" quartic force field, and dipole moment and polarizability surfaces. The second-order vibrational perturbation theory plus the numerical diagonalization of the Hamiltonian matrix containing off-diagonal Fermi and Darling−Dennison resonance couplings (VPT2+WK) was used. The QM calculations were carried out with the Becke−Lee−Yang−Parr composite exchange− correlation functional (B3LYP) and with the 6-31+G(d,p) basis set. The harmonic force field for the equilibrium configuration was transformed into nonredundant local symmetry internal coordinates, and normal coordinates were defined. The semi- diagonal quartic rectilinear normal coordinate potential energy surface (PES), as well as the cubic surfaces of dipole moment (p) and polarizability (α) components, needed for the VPT2+WK calculation, were constructed by a five-point finite differentiation of Hessians (for PES) and of the values and first derivatives of p and α. They were obtained at the point of equilibrium and for 432 displaced configurations. This theoretical approach provides very good agreement between the predicted and experimental frequencies and intensities. However, the favorable result can be partly attributed to error cancellation within the B3LYP/6-31+G(d,p) QM model, as observed in earlier studies. Reassignments of some observed bands are proposed.

The critical ignition conditions for a single particle which interacts with a melting medium to form an intermetallic compound on its surface were investigated using the model of a quasistationary melting front. For linear and parabolic drag laws, three characteristic regions, which transform with change of Biot's criterion, are distinguished on the diagram of critical parameters. It is shown that for small values of this criterion, the conditions of self‐heating of the particle are identical to the ignition conditions in a medium with constant temperature. For values of Biot's criterion greater than unity, the critical conditions are substantially affected by the parameters determining the melting kinetics. The characteristic features of self‐heating of a single particle can determine the qualitative pattern of self‐heating of the heterogeneous system as a whole.

The effect of scaling of an ab initio quantum mechanical force field on the frequencies and forms of normal vibrations are studied in terms of first- and second-order perturbation theory. Scaling the force constant matrix according to Pulay using certain assumptions in first-order perturbation theory is equivalent to scaling vibration frequencies and does not modify the form of vibrations. In this case, the second-order corrections to the frequencies and forms of vibrations become zero. The first-order perturbation theory formulas are used to verify the assumptions by calculating the frequencies and matrices of transition to perturbed forms of vibrations of ethane, propane, ethylene, cyclopropene, and isobutene molecules from quantum mechanical force fields found with the 6-31G basis set. It is shown that the vibration frequencies calculated by the formulas of first-order perturbation theory are in good agreement with exact values; the matrix of transition to perturbed eigenvectors is rarefied, with only ≈1% of its elements being markedly nonzero.

In the theory of anharmonic vibrations of a polyatomic molecule, mixing the zero-order vibrational states due to cubic, quartic and higher-order terms in the potential energy expansion leads to the appearance of more-or-less isolated blocks of states (also called polyads), connected through multiple resonances. Such polyads of states can be characterized by a common secondary integer quantum number. This polyad quantum number is defined as a linear combination of the zero-order vibrational quantum numbers, attributed to normal modes, multiplied by non-negative integer polyad coefficients, which are subject to definition for any particular molecule. According to Kellman's method [J. Chem. Phys. 93, 6630 (1990)], the corresponding formalism can be conveniently described using vector algebra. In the present work, a systematic consideration of polyad quantum numbers is given in the framework of the canonical Van Vleck perturbation theory (CVPT) and its numerical-analytic operator implementation for reducing the Hamiltonian to the quasi-diagonal form, earlier developed by the authors. It is shown that CVPT provides a convenient method for the systematic identification of essential resonances and the definition of a polyad quantum number. The method presented is generally suitable for molecules of significant size and complexity, as illustrated by several examples of molecules up to six atoms. The polyad quantum number technique is very useful for assembling comprehensive basis sets for the matrix representation of the Hamiltonian after removal of all non-resonance terms by CVPT. In addition, the classification of anharmonic energy levels according to their polyad quantum numbers provides an additional means for the interpretation of observed vibrational spectra.

Anharmonic vibration frequencies of 1,1-difluoroethylene (11DFE) in the gas phase are predicted by means of the numerical-analytic operator version of the canonical van Vleck perturbation theory in the second and fourth orders (CVPT2 and CVPT4). The full quartic and "semi-diagonal" sextic rectilinear normal coordinate potential energy surfaces, needed for CVPT2 and CVPT4, respectively, were obtained with the MP2/cc-pVTZ quantum-mechanical model. CVPT2 is superior to the traditional second-order vibrational perturbation theory approach (VPT2) because of the uniform general treatment of the Fermi and second-order Darling-Dennison resonances. The fourth-order version, CVPT4, provides a more refined solution and proves convergence of the perturbative treatment. Labeling of the basis functions by polyad numbers breaks down the infinite Hamiltonian matrix into a block-diagonal form. The polyad expression for 11DFE has been determined as P = 14(ν1 + ν7) + 8ν2 + 6(ν3 + ν8) + 4(ν4 + ν9) + 3(ν6 + ν11 + ν12) + 2(ν5 + ν10), where the νi are quantum numbers. The theoretical prediction of anharmonic infrared absorption intensities corroborated an assignment of the majority of observed gas-phase bands up to 3500 cm(-1). The solution was refined by iteratively fitting harmonic frequencies, until predicted fundamental anharmonic frequencies matched the observed values. The average error for about 90 observed frequencies after fitting only fundamental frequencies is ∼1.05 cm(-1). The fitted "semi-experimental" harmonic frequencies agree very well the quantum-mechanical predictions based on the CCSD(T)/cc-pVTZ and CCSD(T)/cc-pVQZ models.

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.

The molecular structure and conformation of nitrobenzene has been reinvestigated by gas-phase electron diffraction (GED), combined analysis of GED and microwave (MW) spectroscopic data, and quantum chemical calculations. The equilibrium r e structure of nitrobenzene was determined by a joint analysis of the GED data and rotational constants taken from the literature. The necessary anharmonic vibrational corrections to the internuclear distances (r e − r a) and to rotational constants (B e (i) − B 0 (i) ) were calculated from the B3LYP/cc-pVTZ quadratic and cubic force fields. A combined analysis of GED and MW data led to following structural parameters (r e) of planar nitrobenzene (the total estimated uncertainties are in parentheses): r(C–C)av = 1.391(3) A, r(C–N) = 1.468(4) A, r(N–O) = 1.223(2) A, r(C–H)av = 1.071(3) A, $${\angle}$$ C2–C1–C6 = 123.5(6)°, $${\angle}$$ C1–C2–C3 = 117.8(3)°, $${\angle}$$ C2–C3–C4 = 120.3(3)°, $${\angle}$$ C3–C4–C5 = 120.5(6)°, $${\angle}$$ C–C–N = 118.2(3)°, $${\angle}$$ C–N–O = 117.9(2)°, $${\angle}$$ O–N–O = 124.2(4)°, $${\angle}$$ (C–C–H)av = 120.6(20)°. These structural parameters reproduce the experimental B 0 (i) values within 0.05 MHz. The experimental results are in good agreement with the theoretical calculations. The barrier height to internal rotation of nitro group, 4.1±1.0 kcal/mol, was estimated from the GED analysis using a dynamic model. The equilibrium structure was also calculated using the experimental rotational constants for nitrobenzene isotopomers and theoretical rotation–vibration interaction constants.

The problem of determining the Eckart molecule-fixed frame for an arbitrary molecule with nuclei displaced from the equilibrium positions is considered. The solution of the problem is formulated by minimizing the sum of mass-weighted squared deviations (MWSD) of the nuclei of a displaced configuration from the nuclei of the equilibrium configuration. A mathematical proof of the equivalence of Eckart conditions and the minimum of MWSD is given. It is shown that the extrema of the sum of MWSD coincide with eigenvalues of a special 4 × 4 symmetric matrix. Its minimal eigenvalue corresponds to the desired solution, and the respective eigenvector can be treated as the quaternion containing the necessary information for rotating the original coordinate system and aligning its axes with the molecule-fixed coordinate system. A detailed scheme for an efficient numerical implementation of the method is provided, and a numerical example is given.

8. J. A. Barltrop and J. D. Coyle, Excited States in Organic Chemistry [Russian transla- tion], Moscow (1978), pp. 67"70. 9. I. S. Ioffe and V. F. Otten, Zh. Organ. Khim., 32, No. 4, 1196-1201 (1962). i0. I. S. Ioffe and V. F Otten, Zh. Organ. Khim., i, No. 2, 343-346 (1965). ii. Yu. Yu. Lur'e, Handbook on Analytical Chemistry [in Russian], Moscow (1979), p. 322. 12. B. Stevens, N. Conneily, and P. Suppan, Spectrochim. Acta, 2-2, No. 12, 2121-2122 (1966). 13. Sharma Ashutash and M. K. Machwe, Curr. Sci. (India), 5_~i, No. 13, 657-659 (1982). 14. M. D. Galanin, A. A. Kut'enkov, V. N. Smorchkov, et al., Opt. Spektrosk., 53, No. 4, 683-689 (1982). 15. V. Ya. Artyukhov, R. T. Kuznetsova, and R. M. Fofonova, Zh. Prikl. Spektrosk., 37, No. 4, 576-580 (1982). 16. F. V. Schaefer (ed.), Dye Lasers, Springer, New York (1977). VIBRATIONAL SPECTRA AND CONFORMATIONAL C~9OSITION OF ETHYLENE GLYCOL DINITRATE IN SOLID PHASES G. A. Beresneva, L. V. Khristenko, UDC 539.194.01 S. V. Krasnoshchekov, and Yu. A. Pentin The authors of [i] studied the vibrational spectra of ethylene glycol dinitrate (EGDN) and showed that in the liquid state this compound consists of a mixture of conformers, while in the crystalline state it exists in the form of two modifications: a metastable one (crystal II), obtained by intense supercooling the liquid, and a stable modification (crystal I) obtained by either slowly cooling the liquid, or by heating the metastable modification to a temperature of -50~ If we assume that in a restrained internal rotation around the C--C and C--O bonds, the "staggered" conformations will be stable, while the --C-~NO= groups are planar, then EGDN can exist in the form of ten different nuclear equilibrium configurations. Figure i shows these possible conformations of EGDN. The authors of [i] suggested that each of the solid modifications is formed by only one of the ten possible conformers, which were identified from the data of rotation isomerism of ethyl nitrate [2] and the results of vibrational calculation carried out in [3], using force constants obtained by the MINDO/2 method. In this present work, the IR spectra of crystals I and II were recorded on a Bruker Fourier-type spectrometer with a higher,resolution than that obtained in [i], including the region below 400 cm -I, in which the IR spectra of crystals have not yet been studied. Figure 2 shows the IR spectra of a vitreous EGDN and two of its crystalline modifications (I and II). In the spectrum of crystal II in the 100-900 cm -I region, a region of skeletal stretch- ing and deformational vibrations, which are most sensitive to rotation isomerism, the number of bands exceeds that required for one nuclear configuration at equilibrium. On crystal II § crystal I transition, the bands at 213, 390, 588, 662, 715, 862, 928, i036,'1237, 1392, 1431, 1460, and 1646 cm -i disappear. Bands at 364, 405, 581, 656, 680, 937, 1044, 1232, 1309, 1396, 1422, 1449, and 1666 cm -~, with a low intensity in the spectrum of crystal II, become markedly more intense, and the number of bands in the spectrum of crystal I in the 100-900 cm -i region corresponds to the number of bands for one Conformer. These facts indicate that crystal II is formed by at least two conformers, and crystal I by one conformer, and the conformer forming crystal I is also included in the composition of crystal II. For a more reliable identification of the conformers forming crystals I and II, we calculated the frequencies and forms of normal vibrations of all the ten possible conformers of EGDN. The calculations were carried out in an independent system of local symmetry co- ordinates, by a standard method [4]. The natural coordinates introduced, which are repre- Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 48, No. 6, pp. 946-952, June, 1988. Original article submitted April 8, 1987. 614 0021-9037/88/4806-0614512.50 9 1988 Plenum Publishing Corporation

The 3,3-dimethyl-3-silathietane molecule has been investigated by gas-phase electron diffraction analysis with consideration of the vibrational effects. The following geometric parameters were obtained (the distances ra, the angles L~, the errors in the form of 3o in parentheses, for ~ 50): S-C 1.853(4), Si-C m 1.870(5) (m stands for methyl), Si-C r 1.916(3) (r stands for ring), C-H 1.086(3) ~, L(C-Si-C) m = 106.9(7), L(C-Si-C) r = 86.1(3), LSi-C-S = 90.5(5), and LC-S-C = 89.5(4) ~ The fourmembered ring is nonplanar; the angle 9 between the CSiC and CSC planes equals 2 0 . 3 ( 2 0 ) ~