Molecular force constants

In this study the authors develop simplified equations relating equilibrium fractionations to mass-scaling factors and molecular force constants. Equilibrium isotopic fractionations of heavy elements (Si and Sn) are predicted to be small, based on highly simplified, one-parameter empirical force-field models (bond-stretching only) of Sip4, [SiFJ, SnCl4, and [SnCl,] -. 

From a theoretical perspective, isotope effects are fairly trivially computed. The stationary points on the PES and their electronic energies are independent of atomic mass, as are the molecular force constants. Thus, one simply needs to compute the isotopically dependent zero-point energies and translational, rotational, and vibrational partition functions, and evaluate Eq. (15.33). 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

U(Re) is the value of the electronic energy (including nuclear repulsion) at the equilibrium internuclear separation. ke is called the equilibrium molecular force constant. Making the change of variable (4.15) in (4.11), we have... 

Morino, Y. and Shimanouchi, T., Definition and Symbolism of Molecular Force Constants, Pure Appl. Chem. 50 (1978) 1707-1713. 

The force constant is a measure of the stiffness of a chemical bond. Larger values of k imply sharply curved potential energy functions and are often associated with deeper potential wells and stronger bonds. Molecular force constants are typically in the range of 200-2000 N/m, remarkably, not very different from those for bedsprings. From the solution for the harmonic oscillator, we identify the ground-state vibrational energy, with quantum number u = 0, as... 

There have been a large number of determinations of molecular force constants, mean amplitudes of vibration, bond asymmetry parameters, Coriolis coupling constants (and inertia defects) and centrifugal distortion constants [146,152,259,271,304,581,840,1221,1222,1278,1312, 1416,1448,1449,1549,1550,1575-1578,1587,1618,1671,1682,1806,1807,1858,1931,1961,1984,2021,2045, 2108,2109-2111,2167a], as well as a determination of the atomic potential energy distribution... 

Six vibrational modes are expected for the pseudohalide hydracids for which the normal mode description and energies together with the molecular force constants are shown in Table 6. It is worth noting here that the N-H bond stretching force constant for HN3 is lower than that for HNCO and HNCS. This indicates that the latter are thermodynamically more stable compared with HN3. 

Molecular force constants for TiCl4 have been calculated from i.r. data on the isotopic species TiCU and Ti Cl, in argon matrices, and in the gas phase. 

From the point of view of molecular force constants, diatomic molecules generate a class of the simplest systems. Therefore, Stepanov s idea about the transferability of force constants from this simplest class to a more complex one is very fruitful in the study of polyatomic molecules and emphasizes the essential place of diatomics (Stepanov, 1941 Wilson et al., 1955). 

Moseley (46) has derived a semiempirical relationship between the sound velocity and the orientation function by using the fact that the force constants are different along molecular segments from across. His derivations consider the separate cases of series addition and parallel addition of the molecular force constants. He has shown by comparison with experimental data that the series case appears to be well correlated with orientation while the parallel case leads to physically meaningless results and will not be discussed further. The basic equation for the series case is... 

Expressions for the force constant, i.r. absorption frequency, Debye temperature, cohesive energy, and atomization energy of alkali-metal halide crystals have been obtained. Gaussian and modified Gaussian interatomic functions were used as a basis the potential parameters were evaluated, using molecular force constants and interatomic distances. A linear dependence between spectroscopically determined values of crystal ionicity and crystal parameters (e.g. interatomic distances, atomic vibrations) has been observed. Such a correlation permits quantitative prediction of coefficients of thermal expansion and amplitude of thermal vibrations of the atoms. The temperature dependence (295—773 K) of the atomic vibrations for NaF, NaCl, KCl, and KBr has been determined, and molecular dynamics calculations have been performed on Lil and NaCl. Empirical values for free ion polarizabilities of alkali-metal, alkaline-earth-metal, and halide ions have been obtained from static crystal polarizabilities the results for the cations are in agreement with recent experimental and theoretical work. 

Hie harmonic-oscillator force constant k in Eq. (4.28) is obtained ask = cPV/dx, and the harmonic-oscillator curve essentially coincides with the U R) curve aX.R = Rt, so the molecular force constant is k = d lJ/dR n=n (see also Problem 4.28). Differences in nuclear mass have virtually no effect on the electronic-energy curve U(R), so different isotopic species of the same molecule have essentially the same force constant k. 

The development of the finite orthogonal polynomial method provides the basis for the understanding of the total reduced partition function ratio, ln(s/s )f, of a pair of isotopic molecules and each of the quantum terms, ( /kT) -j, of order j to ln(s/s )f. To this end Bigelelsen, Ishida, and Splndel have undertaken a systematic program for the correlation of ln(s/s )f for Isotopes of a number of elements in a variety of compounds with the molecular force constants in the compounds. 

Lasaga AC, Gibbs GV (1988) Qirantum mechanical potential surfaces and calcitlations on minerals and molecular clusters. I STO-3G and 6-3IG results. Phys Chem Miner 16 29-41 Lasaga AC, Gibbs GV (1990) Ab-initio quantirm mechanical calculations of water-rock interactions adsorption and hydrolysis reactions. Amer J Sci 290 263-295 Lazarev AN, Mirgorodsky AP (1991) Molecular force constants in dynamic model of a-quartz. Phys Chem Miner 18 231-243... 

The development of quantum chemistry and the growth of computing power have led to the fact that modem semiempirical, DFT and ab initio methods of the quantum chemistry can significantly improve the speed and accuracy of calculations of various physical and chemical characteristics of the objects or processes and in many cases allow to achieve precise agreement with the experimental data [5]. This makes it possible to predict the molecular force constants and frequencies of normal vibrations. The second derivative of the total energy of the molecule on the internal... 

Hard sphere elastic cross sections [o.-] have been obtained from averaged molecular force constant as determined from experimental equation of state and transport property data (t6.77). The 2h.2 value for represents the self-collision elastic cross section for Ne. The mixed,values for vs. Hg, Ar and... 

The procedure and pertinent issues one has to consider when determining anharmonic force fields by methods of electronic structure theory may be described as follows. Once the definition of molecular force constants involving selection of an appropriate coordinate system is clear, one may need to identify all unique force (potential) constants to be determined. Then selection of the reference geometry follows, which affects precision of the force field determined and how the theoretical force field can be transformed from one coordinate system (representation) to another. Given that an appropriate basis set and level of electronic structure theory are chosen for the actual computations, the necessary quantum chemical calculations can be performed after one has carefully considered how to obtain the high-order force constants from low-order analytic information without much loss of precision. Last but not least one needs to understand the potential uses and misuses of anharmonic molecular force fields. 

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