The calculation of vibrational energy levels of polyatomic molecules including anharmonic effect using contact transformation perturbation method

Using contact transformation perturbation method based on the Taylor expansion of the potential energy function in terms of dimensionless normal coordinates up to sixth‐order, the vibrational energy levels in terms of force constants are derived. The contact transformation theory has been applied to simplify the calculation of perturbation effects. To calculate the second‐order vibrational energy correction, the third and fourth‐order terms of potential function have been placed in the first‐order perturbation Hamiltonian and the second‐order Hamiltonian contains hexatic ones. We present expressions which give relations between the fourth‐ and sixth‐order terms in dimensionless normal coordinates of the potential and the anharmonicity coefficients. For illustration, a set of vibrational energies levels of SO₂, and H₂O molecules including anharmonic effects has been calculated. © 2013 Wiley Periodicals, Inc.     

Anharmonic analysis of the vibrational spectra of some cyanides and related molecules of astrophysical importance

A detailed analysis of the vibrational spectra of carbonyl cyanide, diethynyl ketone and acetyl cyanide has been conducted in harmonic and anharmonic approximations. RHF, MP2 and density functional theory (DFT) methods with 6-311++G(2df,2p) basis sets and B3LYP functionals have been employed. Spectroscopic constants such as anharmonicity constants, rotational and centrifugal distortion constants, rotation-vibration coupling constants and Coriolis coupling coefficients have been calculated for each molecule and compared with the experimental data, where available. A close agreement between the calculated and experimental values of the spectroscopic constants has been obtained. Complete assignments have been provided to the fundamental bands, overtones and combination tones of the molecules. Density functional theory based anharmonic frequencies compare well with the experimental frequencies within +/-18 cm(-1) on an average. RHF and MP2 methods, however, give much higher values for the frequencies that need scaling even in the anharmonic approximation.     

The determination of vibrational anharmonicity in molecules from spectroscopic observations

This review article considers the origin of vibrational anharmonicity in molecules, and the effects that vibrational resonances have on the anharmonicity constants which may be extracted from spectroscopic observations. The importance of the effects of Darling—Dennison resonances, which increase with increasing excitation, as well as Fermi resonances, are considered. The local mode approach to X—Y stretching vibrations is dealt with, as a means of simultaneously accounting for Darling—Dennison resonances and of inter-relating normal mode stretching anharmonicity constants, thus reducing the number of parameters to be determined. The inclusion of Fermi resonances, as necessary, into the calculation is next considered, and the joint local mode-normal mode analysis explained.Applications to ethylenic and methyl group molecules are made. The success of the analyses is demonstrated through complete sets of physically representative anharmonicity constants which reproduce vibrational observations into the visible (16 500 cm⁻¹), and which are mutually self-consistent over molecules containing the same functional groups.Extensions of the simple local mode model are considered, as means of achieving anharmonicity parameters which should describe more closely the molecular potential energy surface, and hence the concomitant physical and chemical processes which it controls.     

Theoretical IR,Raman and NMR spectra of 1,2- and 1,3-dimethylenecyclobutane

Equilibrium geometries, rotational constants, harmonic vibrational frequencies, infrared intensities, Raman activities, and ¹H and ¹³C NMR spectra were calculated for 1,2-dimethylenecyclobutane and its less stable isomer 1,3-dimethylenecyclobutane by using MP2, DFT (B3PW91), and RHF theoretical methods involving the 6-311++G⁷ basis set.The properties calculated theoretically have been compared with the experimental values. The internal coordinates defined for both isomers were used in the potential energy distribution (PED) analysis. The theoretical vibrational and NMR spectra form the basis to differentiate particular compounds in reaction mixture.     

Including anharmonicity in the calculation of rate constants. 1. The HCN/HNC isomerization reaction

A method for calculating anharmonic vibrational energy levels in asymmetric top and linear systems that is based on second-order perturbation theory in curvilinear coordinates is extended to the bound generalized normal modes at nonstationary points along a reaction path. Explicit formulas for the anharmonicity coefficients, x(ij), and the constant term, E0, are presented, and the necessary modifications for resonance cases are considered. The method is combined with variational transition state theory with semiclassical multidimensional tunneling approximations to calculate thermal rate constants for the HCN/HNC isomerization reaction. Although the results for this system are not very sensitive to the choice of coordinates, we find that the inclusion of anharmonicity leads to a substantial improvement in the vibrational energy levels. We also present detailed comparisons of rate constants computed with and without anharmonicity, with various approximations for incorporating tunneling along the reaction path, and with a more practical approach to calculating the vibrational partition functions needed for larger systems.     

Approximate molecular orbital anharmonic parameters for the infrared spectrum of H2O

The vibrational spectrum of H2O was calculated at MP2/6-31G(extended) and MP2/6-311G* levels taking into account anharmonicities through a simple approach to second-order perturbation theory in which molecular energy and dipole moment are expanded as Taylor series in normal coordinates with no cross terms, to simplify calculations. The series coefficients are obtained separately for each normal coordinate through polynomial regression of calculated single point property values corresponding to a few distorted molecular geometries. The energy coefficients are used to calculate the harmonic frequencies and the chi(ii) anharmonicity constants, and so the band origins. For the band intensities, second-order perturbation theory equations derived earlier for diatomic molecules are used for each mode. Estimated frequencies have accuracy equivalent to those of previous complete perturbation calculations at the same ab initio levels, being at most 2.6% above the experimental values for the MP2/6-31G(extended) level. The fundamental intensity estimates are equivalent to those for the complete treatments, with the exception of that at MP2/6-31G(extended) level for the bending mode, which is 7% above the experimental value. Estimated overtone intensities by both complete treatments and the simple approach may still differ in magnitude from the experimental values, though to a lesser extent for the formers.     

Efficiency of post-Hartree-Fock force field for the interpretation of vibrational spectra ofN,N-dimethylnitramine

Harmonic force fields for the molecule ofN,N-dimethylnitramine were calculated in the RHF/6-31G^* and MP2/6-31G^** approximations. Scaling of the force fields obtained made it possible to reliably interpret the vibrational spectra of light and perdeuterated compounds reported in the literature. The assignment is confirmed by good reproducibility of experimental isotope shifts upon¹⁵N-amino- and¹⁵N-nitrosubstitution. The frequencies of intramolecular vibrations in far IR and Raman spectra as well as in neutron inelastic scattering spectra for the light and perdeuterated samples of solidN,N-dimethylnitramine were identified using the force field calculated with the inclusion of electron correlation (MP2). Although general structures of the force fields calculated in the RHF and MP2 approximations are similar, considerable differences in the force constants of the NO and NN stretching vibrations and especially in the constants of the NO_str/NO_str and NO_str/NN_str interactions remain even after scaling the force fields.     

Exploring the effect of anharmonicity of molecular vibrations on thermodynamic properties

Thermodynamic properties of selected small and medium size molecules were calculated using harmonic and anharmonic vibrational frequencies. Harmonic vibrational frequencies were obtained by normal mode analysis, whereas anharmonic ones were calculated using the vibrational self-consistent field (VSCF) method. The calculated and available experimental thermodynamic data for zero point energy, enthalpy, entropy, and heat capacity are compared. It is found that the anharmonicity and coupling of molecular vibrations can play a significant role in predicting accurate thermodynamic quantities. Limitations of the current VSCF method for low frequency modes have been partially removed by following normal mode displacements in internal, rather than Cartesian, coordinates.     

CO_2振动激发态的能级指认及统计分布

利用量子力学方法研究了二氧化碳振动高激发态的能级及其统计分布。在 Radau坐标下采用算法稳定,精度高且所需计算机内存较少的Lanczos算法以及势能 优化的离散变量表象方法,获得了20000 cm~(-1)以下的所有振动束缚态能级,并 对这些振动能级进行了指认。此外,还进一步分析了振动高激发态的Fermi共振与 非谐性振动。统计分布表明,二氧化碳的振动光谱在20000 cm~(-1)以下呈规则分 布。     

Harmoniques des vibrations de valence C?H dans les alcanes et les halogénures de t-butyle

The first and second harmonic C? H and C? D valence vibrations of alkanes and t-butyl halides have been measured. A model is proposed for the methyl and methylene groups, which allows the calculation of the anharmonicity constants of every vibrational mode and of the couplings between the modes. The already known large values (?100 to ?150 cm^?1) for the anharmonicity constants that couple the symmetrical with the asymmetrical modes are confirmed. As compared with lower harmonics combination bands become predominant for the higher ones; they allow the application of a diatomic model to estimate higher harmonics and the energy of dissociation.     

Spectra and Structures of Silicon-Containing Compounds. XXIV.* Raman and Infrared Spectra,r 0 Structural Parameters,Vibrational Assignment,Barriers to Internal Rotation,and Ab Initio Calculations of Ethylsilane

The infrared (3200 to 400 cm^–1) and Raman (3200 to 20 cm^–1) spectra of gaseous and solid ethylsilane, CH₃CH₂SiH₃, have been recorded. Additionally, the Raman spectrum of the liquid has been obtained with quantitative depolarization values. The SiH₃ torsional mode has been observed as sum and difference bands with the silicon-hydrogen stretching vibration. Utilizing the torsional fundamental frequency of 132 cm^–1 the threefold periodic barrier of 590 cm^–1 (7.06 kJ/mol) has been obtained. Utilizing the frequencies of the silicon-hydrogen stretches, Si-H bond distances of 1.485 and 1.484 Å have been obtained for the bonds gauche and trans to the methyl group, respectively. Using previously reported rotational constants from seven different isotopomers, the r ₀ parameters have been calculated and are compared to the corresponding r _s parameters. A complete vibrational assignment is proposed that is consistent with the predicted frequencies utilizing the force constants from ab initio MP2/6-31G(d) calculations. Both the infrared intensities as well as the Raman activities and depolarization values have been obtained from the ab initio calculations. Complete equilibrium geometries have been determined by ab initio calculations employing the 6-31G(d), 6-311 + G(d,p), and 6-311+G(2d,2p) basis sets at levels of restricted Hartree–Fock (RHF) and/or Moller–Plesset (MP) to second order. The results are discussed and the theoretical values are compared to the experimental values when appropriate.     

Canonical statistical adiabatic channel model calculations of the H + CH3 → CH4 recombination reaction on an ab initio potential energy surface. The role of the transitional modes

The canonical formalism of the statistical adiabatic channel model is used to calculate limiting high pressure rate constants for the H + CH₃ → CH₄ recombination reaction on a recently reported analytic potential energy surface based on ab initio calculations. An effective adiabatic channel potential which incorporates the G_?? matrix element of the twofold degenerate H₃C? H transitional bending mode, quartic anharmonicity, and state selected mode coupling effects is implemented. The rate constants calculated over the temperature range 200–1000 K are in very good agreement with recent canonical variational transition state theory calculations performed on the same surface. The comparison with experimental results is also discussed.     

Anharmonic C?H and C=O Interactions in Peptide and Sugar

C?H and C=O stretching modes are two among many structural and dynamic probes of proteins and peptides in condensed phases. Anharmonic properties of these two modes in peptide and sugar have been examined using a second-order perturbative vibrational approach. High order force constants were obtained and examined to ˉnd how crucial they are in determining the degree of mode localization and the nature of mode anharmonicity of the two stretching modes. It is found that the C?H mode is highly localized,and its diagonal anharmonicity is mainly determined by the mode itself. However, the C=O mode is largely delocalized, and the diagonal anharmonicity involves contributions from other modes. The o?-diagonal anharmonicity between C?D and C=O modes is found to be negative in deuterated species, di?ering from those of the non-deuterated ones. It is also found that inter-mode interaction between each of the two modes with low-frequency modes contribute signiˉcantly to the o?-diagonal anharmonicity. These low-frequency modes give rise to a network of energy relaxation or intramolecular vibrational energy redistribution pathways which can be used to examine temporal behavior of intramolecular vibration energy °ow, provided a femtosecond broadband two-dimensional infrared spectroscopy is available.     

Ab initio study of molecular structure and internal rotation in methyldicyanophosphine and methyldiisocyanophosphine. The Raman spectrum of methyldicyanophosphine and its interpretation with the use of scaling ofab initio force fields

The equilibrium geometric parameters and structures of transition states of internal rotation for the molecules of methyldicyanophospine MeP(CN)₂ and its isocyano analog MeP(NC)₂ were calculated by the RHF and MP2 methods with the 6–31G^* and 6–31G^** basis sets. At the MP2 level, the total energy of cyanide is −35 kcal mol⁻¹ lower than that of isocyanide and the barriers to internal rotation of methyl group for MeP(CN)₂ and MeP(NC)₂ are 2.2 and 2.7 kcal mol⁻¹, respectively. For both molecules, the one-dimensionalab initio potential functions of internal rotation approximated by a truncated Fourier series were used to determine the frequencies of torsional transitions by solving direct vibrational problems for a non-rigid model. The Raman spectrum of crystalline MeP(CN)₂ was recorded in the range 3500–50 cm⁻¹. The vibrational spectra of this compound were interpreted by scalingab initio force fields calculated by the RHF and MP2 method. The vibrational spectrum of methyldiisocyanophosphine was predicted with the use of the obtained scale factors. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1703–1714, September, 1998.     

Effect of basic set quality and electron correlation on the scale factors of a harmonic force field

The completely optimized structure and harmonic force field of s-trans-buta-1,3-diene are reported at the MP2/6-31G and MP2/6-31G* levels of computation. Sets of empirical scale factors for the calculated force fields are derived and compared with the corresponding values computed at the RHF/4-31G and RHF/6-31G levels. Changes in the scale factors for this series of force fields are discussed. The vibrational frequencies are also reported for thirteen isotopomers of s-trans-buta-1,3-diene using the MP2/6-31G* force field. Some characteristics of the gauche and cis forms of buta-1,3-diene are also given.     

Including anharmonicity in the calculation of rate constants. II. The OH+H2-->H2O+H reaction

A recently developed method for calculating anharmonic vibrational energy levels at nonstationary points along a reaction path that is based on second-order perturbation theory in curvilinear coordinates is combined with variational transition state theory with semiclassical multidimensional tunneling approximations to calculate thermal rate constants for the title reaction. Two different potential energy surfaces were employed for these calculations, an improved version of the author's surface 5 and the WSLFH surface of Wu et al. [J. Chem. Phys. 113, 3150 (2000)]. We present detailed comparisons of rate constants computed for the two surfaces with and without anharmonicity and with various approximations for incorporating tunneling along the reaction path. The results for this system are quite sensitive to the surface employed, the choice of coordinates (curvilinear versus rectilinear), and the inclusion of anharmonicity. A comparison with experiment provides information on the accuracy of these surfaces.     

Normal Coordinate Analysis in Cartesian Space II. Planar Symmetric XYn Molecules

Force constants of planar symmetric XY_n type molecules in the Cartesian space have been derived in terms of the vibrational frequencies. The force constants and normal coordinates of some planar XY₃ molecules with available isotoptic data are worked out. Further study of infrared absorption spectra of square planar XY₄ complex ions will be needed for the present normal coordinate analysis.     

Normal coordinate analysis and thermodynamic functions of niobium and tantalum pentahalides

Normal coordinate treatment of NbX₅ and TaX₅ (X = F, Cl, Br) in the Urey-Bradley force fields was performed using published vibrational frequencies. The analysis was carried out with Wilson's FG formalism and the force constants were evaluated by a computer program based on the least-squares-fit method. Normal coordinates, potential energy distributions and thermodynamic functions were also determined. The results support the reported assignments of the fundamental frequencies of the pentachlorides and the pentabromides, and suggest that the pentafluoride assignments are incorrect.     

Conformational stability, vibrational assignmenents, barriers to internal rotations and ab initio calculations of 2-aminophenol (d 0 and d3)

The Raman (3700-100 cm(-1)) and infrared (4000-400 cm(-1)) spectra of solid 2-aminophenol (2AP) have been recorded. The internal rotation of both OH and NH2 moieties produce ten conformers with either Cs or C1 symmetry. However, the calculated energies as well as the imaginary vibrational frequencies reduce rotational isomerism to five isomers. The molecular geometry has been optimized without any constraints using RHF, MP2 and B3LYP levels of theory at 6-31G(d), 6-311+G(d) and 6-31++G(d,p) basis sets. All calculations predict 1 (cis; OH is directed towards NH2) to be the most stable conformation except RHF/6-31++G(d,p) basis set. The 1 (cis) isomer is found to be more stable than 8 (trans; OH is away from the NH2 moiety and the NH bonds are out-of-plane) by 1.7 kcal/mol (598 cm(-1)) as obtained from MP2/6-31G(d) calculations. Aided by experimental and theoretical vibrational spectra, cis and trans 2AP are coexist in solution but cis isomer is more likely present in the crystalline state. Aided by MP2 and B3LYP frequency calculations, molecular force fields, simulated vibrational spectra utilizing 6-31G(d) basis set as well as normal coordinate analysis, complete vibrational assignments for HOC6H4NH2 and DOC6H4ND2 have been proposed. Furthermore, we carried out potential surface scan, to determine the barriers to internal rotations of NH2 and OH groups. All results are reported herein and compared with similar molecules when appropriate.