Algebraic approach to thermodynamic properties of diatomic molecules

A simple model extending Lie algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. Local anharmonic effects are described by means of a Morse-like potential and the corresponding anharmonic bosons are associated with the SU(2) algebra. The total number of anharmonic bosons, fixed by the potential shape, is determined for a large number of diatomic molecules. A vibrational high-temperature partition function and the related thermodynamic functions are derived and studied in terms of the parameters of the model. The idea of a critical temperature is introduced in relation to the specific heat. A physical interpretation in terms of a quantum deformation associated with the model is given.     

Sensitivity to anharmonicity of vibrational energy in a diatomic gasdynamic laser

A parameter for evaluating the sensitivity of quantum vibrational energy to anharmonicity in a diatomic gasdynamic laser is defined and calculated by considering the corresponding diatomic molecules as quantum anharmonic oscillators under an interatomic Morse potential. The variation of the above parameter in terms of the vibrational states and in terms of an involved anharmonic coefficient is discussed. In particular, the parameter in question at the classical limit is examined. Both weak and strong anharmonicities are discussed.     

Anharmonicity of NH2 stretching vibrations of aniline

The vibrational levels related to the NH₂, NDH, and ND₂ groups of aniline and its N-deuterated derivatives were calculated on a four-atomic model by the perturbation method. An out-of-plane type deformation coordinate was introduced to reduce the anharmonicity constant for the NH₂ wagging mode. The assumed anharmonic potential contains no higher-order terms except those derived from the Morse function for the individual bonds. The calculated frequencies agree with the observed values satisfactorily and the main features of the level patterns are explained from the simplicity of the potential and the forms of normal coordinates of each isotope. It is also shown that the observed anharmonicity constants x_aa and x_ss are so much affected by the Darling-Dennison resonance that the theoretical relation for the unperturbed constants, ∥x_aa∥ > ∥x_ss∥, is violated.     

Adequate representation of the dipole moment function for diatomic molecules and the matrices of vibration-rotation transition moments

An exponential representation of perturbations is used as a basis of the perturbed Morse oscillator approach which is applied, in a matrix form, for calculating the radial matrix elements for diatomic molecules. An analytic procedure is developed to deduce an exponential-power series expansion for the dipole moment function M(r) from experimental spectral intensities. It is shown that for real anharmonic molecules, the series expansion in powers of e^ar (α being the Morse parameter) is an adequate form for representing transition operators, just as the usual series expansion in powers of internuclear distance r is adequate for the case of a harmonic oscillator, and it is equivalent to a series expansion in vibrational wavefunctions. An exponential-power series expansion is derived as well for a model dipole moment function which has a correct long-range dependence and limit. To exemplify the accuracy and efficiency of the technique proposed, the (40 × 40) matrices of vibration-rotation transition moments 〈vJ|M(r)|v′J′〉(v, v′ = 0, 1, …, 39) have been calculated for the ground state of CO. Typical results of these computations are presented (up to v = 35, J = 100, and v′ ? v = 1–4) to illustrate the dependence of vibration-rotation interaction functions on the vibrational and rotational quantum numbers.     

Vibrational resonance in the Morse oscillator

The occurrence of vibrational resonance is investigated in both classical and quantum mechanical Morse oscillators driven by a biharmonic force. The biharmonic force consists of two forces of widely different frequencies ω and Ω with Ω???ω. In the damped and biharmonically driven classical Morse oscillator, by applying a theoretical approach, an analytical expression is obtained for the response amplitude at the low-frequency ω. Conditions are identified on the parameters for the occurrence of resonance. The system shows only one resonance and moreover at resonance the response amplitude is 1/dω where d is the coefficient of linear damping. When the amplitude of the high-frequency force is varied after resonance the response amplitude does not decay to zero but approaches a nonzero limiting value. It is observed that vibrational resonance occurs when the sinusoidal force is replaced by a square-wave force. The occurrence of resonance and antiresonance of transition probability of quantum mechanical Morse oscillator is also reported in the presence of the biharmonic external field.     

The resonance interaction of vibrational modes in one-dimensional nonlinear lattices

Resonances of vibrational modes were for the first time revealed for the example of the one-dimensional random Morse lattice. The observation of resonances was possible because of lattice deformation, when, at certain relative deformation values, vibrational modes satisfied the conditions of double (m _i ω _i + m _j ω _j = 0) or triple (m _i ω _i + m _j ω _j + m _k ω _k = 0) resonances. Of all the resonances observed, the resonance with the frequency ratio ω₂: ω₁ = 2: 1 was studied in detail. The dependences of mode lifetimes and the degree of energy exchange between them on such parameters as resonance frequency detuning, excitation energy level, etc. were determined. A model of two nonlinearly coupled harmonic oscillators was considered in detail on the assumption of a one-to-one correspondence between oscillators and vibrational modes. A consideration of the model problem of oscillators revealed analytic dependences of the dynamic behavior of vibrational modes on control parameters. Excellent agreement between the numerical results for the Morse lattice and analytic conclusions was obtained. It was shown that, for the Fermi-Pasta-Ulam lattice, the resonance interaction of vibrational modes was controlled by the same rules as with the Morse lattice.     

Theoretical Interpretation of the Vibrational Spectra of Carboxylic-Acid Dimers in the High-Frequency Range

An anharmonic band shift in the vibrational spectra of carboxylic-acid dimers is estimated on the basis of ab initio quantum calculations of anharmonic force constants. The implementation of ab initio quantum calculations taking into account the anharmonic nature of vibrations is connected with the choice of the atomic basis in the framework of a specific quantum method. All these factors together with the exclusion principle for bands in the infrared and Raman scattering spectra allow identification of the position of the bands of valence vibrations of CH bonds in the range of 2500–3500 cm^–1. The results of model calculations give reason to assert that the fundamental vibrations of the carboxylic fragment are the characteristic frequency and vibrational mode and, for OH bonds, also the characteristic intensity. Small doublet splitting and the exclusion principle for frequencies allow identification of the valence vibrations of CH bonds.     

On the Hulburt-Hirschfelder potential function for the Ar2 molecule

The modified Morse potential function known as the Hulburt-Hirschfelder function has been used for the Ar₂ molecule using experimental values for the spectroscopic constants. The consistency of this potential is checked by recalculating the vibrational levels using Cashion's method. The results are satisfactory. The potential may be useful for calculating the anharmonic properties of crystalline argon.     

Simplified analytical expressions for classical turning points of molecular oscillators

Simplified analytical expressions for classical turning ppoints (r_1,2) are derived by expanding the empirical potential function of Dunham and Morse in series forms. These expressions are equivalent to those obtained by Rees when the energy (E) is quadratic in the vibrational quantum number. Our first approximation approach is also found to be equivalent to Jarmain's factor f/r_e.     

H2SiCl2 中 SiH 伸缩振动的泛频光谱

在2000 ～ 9000 波数 、 12000 ～12900 波数的光谱区间记录了室温下H2SiCl2气体分子的振动泛频光谱，所用的仪器分别是高分辨傅立叶变换光谱仪和高灵敏激光腔内吸收光谱仪。用局域模模型和包含达林－丹尼生共振的简正模模型，归属了SiH伸缩振动的基频和泛频，振动量子数的改变△VSiH=1, 2, 3, 4 and 6。通过对实验能级的非线性拟合，得到SiH伸缩振动的谐振频率ωm、非谐性常数χm、键间耦合系数λ、莫尔斯振子参数 De、α 和相互作用力常数 。实验发现，随着振动能量的增加，振动簇（manifold，两个SiH键的伸缩振动量子数m+n=常数 ）中能量最低的两个能级的间距逐渐减小。当△VSiH≥4时，在实验误差范围内这两个最低的振动态能级简并。这种简并的能级结构类似双原子莫尔斯振子，符合Birge-Sponer 关系。双原子莫尔斯振子直接描述了H2SiCl2分子中SiH 的高泛频伸缩振动，表明在高振动情况下振动能量已经集中到单个SiH键上。     

On the density of electronic states in a quantum well of an anharmonic solid under a perpendicular magnetic field

For the first time, given an anharmonic solid as an atomic array of quantum Morse oscillators, the electronic density of states per unit area in an ideal one-dimensional quantum well under a perpendicular, time-independent, uniform magnetic field is calculated. From this result, some aspects related to the classical limit are discussed.     

An algebraic description of anharmonic diatom–diatom inelastic collisions in the semiclassical approximation

An algebraic model to describe inelastic collisions between two anharmonic diatomic molecules in the semiclassical approximation is presented. The interactions for the diatomic systems are modelled in terms of Morse potentials, while an exponential repulsive potential is taken for the interactions between the nearest atoms of the diatomic systems. This problem is treated in the interaction potential framework, where an approximation in terms of the generators of three SU(2) groups is proposed, two corresponding to the Morse oscillators and the other to the interaction. The transition probabilities are given in terms of a sum of the products of three Wigner's d(β) functions corresponding to the three SU(2) groups. As an example the systems N₂?+?N₂ and H₂?+?H₂ are described and compared with exact quantum mechanical calculations.     

Anharmonic potential function of formaldehyde

By using a simple anharmonic potential including 25 higher-order constants, the vibrational frequencies, vibration-rotation interaction constants, and centrifugal distortion constants of formaldehyde were calculated to agree well with the observed values. The harmonic part of the potential gives favorable signs of compliance constants for the C_2v-type dissociations into the methylene radical and into carbon monoxide, as well as for the C_s-type dissociation into the formyl radical. The cubic constants have adequate signs to predict the changes of quadratic constants on the dissociation of the CH or the CO bonds. From the Morse parameter of the CH bond, the heat of dissociation for the process H₂CO → CO + 2H is calculated to be 137 kcal/mole.     

Stretching vibrational overtone and combination states in silicon tetrafluoride

A simple three-parameter model is shown to account for the observed SiF stretching vibrational states of silicon tetrafluoride. A symmetrized anharmonic bond oscillator basis set is used to calculate stretching overtone and combination eigen values, all of which are given up to v₁ + v₃ = 5. The results show that the highest levels of the nν₃ manifold move gradually out of resonances with n quanta of ν₃ as n increases, which indicates that anharmonic resonances between the ν₃ ladder and some other vibrational ladders and (or) multiphoton resonances are needed to explain the observed multiphoton processes.     

Towards an explanation of collisionless multiple-photon laser dissociation of SF6

The rapidity and high degree of molecular vibrational excitation by the absorption of ir laser light in SF₆ and other molecules may be due in large part to the anharmonic splitting of excited vibrational states. Anharmonic splitting of an overtone or combination vibrational level (i) is possible only in molecules with degenerate vibrational states, (ii) can be comparable in magnitude to the net anharmonic shift of the level, (iii) is generally much larger than the rotational shifts which have previously been proposed as an explanation for the dissociation of SF₆. We find that consecutive nearly resonant transitions are possible in SF₆ up to υ₃ = 5 to 10.     

Classical treatment of vibration-rotation intensities for the Morse oscillator

The effect of vibration-rotation interaction on spectral line intensities is treated classically for the Morse oscillator. To describe the distribution of intensity in the R and P branches, one must take into account both the vibrational (radial) and the rotational (perpendicular) dipole moment functions. When this is done, the leading terms for the Fourier coefficients of these functions, as well as the Herman-Wallis factors, agree exactly with the corresponding quantum mechanical matrix elements in the Dunham limit.     

Nuclear quantum effects on the thermal expansion coefficient of hexagonal boron nitride monolayer

This work examines the importance of vibrational delocalization on a basicthermomechanical property of a hexagonal boron nitride monolayer, namely its thermalexpansion coefficient (TEC). Using a recently parametrized bond-order potential of theTersoff type, the TEC was theoretically obtained from the thermal variations of thelattice parameter a(T) calculated using threedifferent methods: (i) the quasiharmonic approximation; (ii) its anharmonic improvementbased on self-consistent phonons; (iii) fully anharmonic Monte Carlo simulations possiblyenhanced within the path-integral framework to account for nuclear quantum effects. Theresults obtained with the three methods are generally consistent with one another and withother recently published data, and indicate that the TEC is negative at least up to ca.700 K, quantum mechanical effects leading to a significant expansion by about 50% relativeto the classical result. Comparison with experimental data on bulk hexagonal BN suggestssignificant differences, which originate from possible inaccuracies in the model that tendto underestimate the lattice parameter itself, and most likely from the 2D nature of themonolayer and the key contribution of out-of-plane modes. The effects of isotopic purityin the natural abundances of boron are found to be insignificant.     

Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng–Fan potential model

With the introduction of a new improved approximation scheme (Pekeris-type approximation) to deal with the centrifugal term, the energy eigenvalues and the wave functions of the Schrödinger equation of the shifted Deng–Fan molecular potential are obtained, via the asymptotic iteration method. Rotational–vibrational energy eigenvalues of some diatomic molecules are presented, these results are in good agreement with other results in the literature. For these selected diatomic molecules, energy eigenvalues obtained are in much better agreement with the results obtained from the rotating Morse potential model for moderate values of rotational and vibrational quantum numbers. Furthermore, thermodynamic properties such as the vibrational mean U, specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues are studied.