Frequency Sum Rules Research Articles

A general isotopic sum-rule for frequencies, band intensities, and Coriolis coupling constants

A general isotopic sum-rule for frequencies, band intensities, and Coriolis coupling constants

Vibrational spectra of 1,1-dichloroethanes and βCH stretching interaction force constants

Infrared and Raman spectra are described and assignments are reported for CH 3CDCl 2 and CD 3CHCl 2, together with new i.r. data for CH 3CHCl 2 and CD 2HCHCl 2. Using previously obtained “isolated” CH stretching frequencies, the above data enable the trans and gauche stretch—stretch interaction force constants to be determined: f′ g = ±0.011, f′ t = ∓0.026 mdyn/Å. Similar values, f′ g = +0.016, f′ t = −0.029, may be obtained from the earlier spectra of CHD 2CHD 2 due to Van Riet. The relations between the fundamental deformation bands and their overtones and combinations are examined in order to estimate the accuracy of predictions of v s (= v 3) for the CH 3 and CD 3 groups from the 4 × 4 refinement and frequency sum rule. The data can be interpreted so as to give good agreement on the values of v O s of 2920 cm −1 (CH 3) and 2115 cm −1 (CD 3), provided that there are significance resonance shifts on both 2δ a and 2δ′ s levels. That this is the case seems to be true for the CD 3 group, but is uncertain for the CH 3 one.

Vibrational spectra of bromobenzene-4- d1 and bromobenzene-2,3,5,6- d4 and force field of bromobenzene

Infrared and Raman spectra of bromobenzene-4- d 1 and bromobenzene-2,3,5,6- d 4 have been recorded in the liquid state, and vibrational assignments have been made by referring to isotope effects and Raman depolarization ratios. The normal coordinate analysis has been carried out for the four isotopic bromobenzenes, − d 0, − d 1, − d 4 and − d 5. The valence force field for benzene and toluene due to La Lau and Snyder has been used with a slight change in off-diagonal terms based on the modified frequency sum rule for symmetry species of isotopic molecules. After least squares refinement of the force field, a good fit between the observed and the calculated frequencies has been obtained, and a close correspondence between the force constants of bromobenzene and those of benzene and toluene has been indicated.

Spectra of C2ν deuterium substituted pyrimidines

Pyrimidine, pyrimidine-2- d, pyrimidine-2,5- d 2, pyrimidine-4,6- d 2, pyrimidine-2,4,6- d 3 and pyrimidine- d 4 have been prepared. Isotopic purity and label specificity were established by proton magnetic resonance and mass spectroscopy. The principal fragmentation pathway in the mass spectrometer ion beam is shown to involve a splitting of HCN from the pyrimidine ring produced by loss of the 4 position carbon. Liquid phase Raman and liquid and vapor phase i.r. spectra are reported. Vibrational assignments consistent with frequency sum rule and product rule criteria are given for most of the 144 fundamentals involved. The two A 2 fundamentals of pyrimidine-4,6- d 2, pyrimidine-2,4,6- d 3 and pyrimidine- d 4, and one of the A 2 fundamentals of pyrimidine, pyrimidine-2- d and pyrimidine-2,5- d 2 are identified. Some assignments of earlier authors are altered in order to achieve reasonable frequency sum and product rule agreement.

An inverse sum rule for isotopic frequencies

A rule relating to the square of the inverse harmonic frequencies of a set of isotopically substituted molecules is developed. The rule is ▪ where Λ i x . is the square of the Xth harmonic frequency of the ith isotopic species, and σ i is a weighting factor for the ith species, taken so that ▪ for all a, where M i a is the mass of the ath atom in the ith species. An upper bound for the residue (Res) i can be evaluated from the frequencies of the ith species alone. If the rule is not satisfied, no harmonic force field is compatible with the set of assigned frequencies. The rule is applied to ethylene. The rule is most sensitive to the assignment of the low frequencies. The rule is shown to apply to the vibrations of crystals, including the lattice modes, and if the sum is taken over K = K' ≠ 0 modes, the residue→ 0.

A new sum rule for the kinetic theory of classical fluids

For the collision kernal of the kinetic theory of classical fluids, we have obtained an additional high frequency sum rule. The result is useful since it involves only the pair correlation function.

Calculation of Atomic Force Constants from Electron Densities

An expression is derived which relates molecular electron densities to the atomic force constants obtained from an analysis of the frequency sum rule for polyatomic molecules. Using approximate electron densities computed by Pople and Segal by the CNDO method [J. A. Pople and G. A. Segal, J. Chem. Phys. 43, S136 (1965)], it was found that not only the magnitudes of the atomic force constants for H, C, N, O, and F atoms were accounted for, but also their observed independence of bond multiplicity.

Sum rules and the landau theory of fermi liquids

The Landau theory predicts spectral weight functions for density and spin-density fluctuations which in the long-wavelength, low-frequency limit should be equal to the true spectral weight functions of a system of interacting fermions. This correspondence plus well-known frequency sum rules provides a means to examine the Landau quasiparticle interaction parameters. Two inequalities are derived:F1 ≥G1 and another relating the first three spin-symmetric Landau coefficients to the radial distribution function and the two-particle potential. We find that the third-moment sum rule is not entirely exhausted by Landau-accountable excitations, and therefore it is the highest order sum rule which can still probe the theory. A third-moment sum rule for the spin-spin correlation function is derived and is found to have an unexpected dependence onk, i.e., one that disagrees with the corresponding Landau prediction. We do not examine this result in detail, but merely conjecture that it may indicate a basic weakness in the Landau theory of spin-density fluctuations.

Vibrational Frequency Sum Rules for Distorted Complexes

The calculation of thermodynamic quantities that involve the change of frequencies due to distortion of octahedral complexes can be effectively simplified by calculating the mean frequency change within a vibrational type, rather than the change in each component individually. On the average each stretching mode frequency changes by the same amount, each bending mode frequency changes by the same amount, while the mixed mode frequencies change by amounts that depend on the percentages of stretching and bending motions in the mode.

Use of Field-Sweep Double Resonance in the Unambiguous Analysis of Three-Spin Spectra

A systematic procedure for uniquely determining NMR parameters from a spectrum is applied to the case of three coupled spins, and several examples are treated. The analysis involves three steps: (1) use of frequency sum rules to determine consistent assignments; (2) removal of ambiguities in assignment by field-sweep double resonance; and (3) iterative refinement of arbitrary initial parameters.

Modified Sum Rule for the Vibrational Frequencies of Symmetrical Isotopic Molecules

A general method of applying the modified sum rule for the vibrational frequencies of symmetrical isotopic molecules has been derived. This rule states that the difference in the sum of the frequency parameters between two isotopic molecules A and A*Δσ=Σλ(A)−Σλ(A*)is the same for the appropriate sets I, II, ... of the symmetry species. The sets I, II, ... are selected in accordance with the interactions between the equivalent atoms involved in the isotopic change from A into A*. Various sum-rule relations are obtained for a given pair of isotopic molecules depending upon the interaction terms of the potential function. The proof of the rule is given by using the property of the transformation from the internal coordinates into the symmetry coordinates. The application of the rule is illustrated for ethylene, benzene, and their deuterated homologs.

High resolution nuclear-magnetic-resonance spectra of hydrocarbon groupings

The interpretation of ABC-type spectra is discussed in relation to several series of calculated spectra; some frequency and intensity sum rules are indicated. The hydrogen spectra of some vinyl derivatives, X. CH = CH2 (X = F, Cl, Br, O and C) have been analysed. It is shown that and that a modified correlation holds between the Jgem coupling constants of CH2 groups and the H-C-H angle in a range of molecules. All three coupling constants of the vinyl compounds decrease approximately linearly with the electronegativity of X; these effects are probably transmitted through the sigma bonds. By contrast the difference between the chemical shift of the = CH2 group and that of the XHC = group is approximately linearly dependent on Taft's resonance parameter, σR , suggesting that the pi-electron distribution is principally concerned in this case.

Second-Order Sum Rule for the Vibrations of Isotopic Molecules and the Second Rule of the Mean

A second-order sum rule for the vibrational frequencies of isotopic molecules is derived. The rule relates the second power of the λ's, Σλi2 and Σi≠j λiλj, for isotopic molecules of the type X′Yz—rYr′ and XYz—rYr′ with the second power of the λ's for the symmetrical molecules X′Yz′, XYz′, X′Yz, and XYz. Special consideration is given to the case where X′ and X are in fact Y′ and Y in positions structurally equivalent to the z Y atoms in XYz. In the latter case, if z is greater than 2, the rule may involve isotopic isomers. Just as in the case of the first-order sum rule, the second-order rule can be applied separately to the individual symmetry classes common to all of the molecules. It is shown that the second-order sum rules for a homologous series of isotopic molecules can be found easily by a simple tabulation of the interactions between the equivalent isotopic atoms. Interaction tables are given for the deutero-ethylenes and the deutero-benzenes. The second-order sum rule is used to derive the second rule of the mean for isotopic exchange equilibria. Examples of the latter are given involving the carbon dioxide, ethylene, and benzene molecules.

THE INFRARED AND RAMAN SPECTRA OF cis AND trans DIBROMOETHYLENE, TRIBROMOETHYLENE, AND AN APPLICATION OF THE FREQUENCY SUM RULE

Complete vibrational assignments have been obtained for tribromoethylene and cis dibromoethylene and thermodynamic functions calculated. Ten of the twelve fundamentals of trans dibromoethylene have been assigned and the approximate frequency of the missing pair evaluated by application of the frequency sum rule. The heat capacity of trans dibromoethylene has been calculated.

A Sum Rule for the Vibrational Frequencies of Homologs

It is shown empirically that the zero-point energy in substituted methanes may be treated as a constitutive property when account is taken of the interaction between nonbonded atoms. Thus, the sum of the vibration frequencies (Σν) of the halogenated methanes is found to have the predicted quadratic dependence on the number of substituents. This reduces to a linear dependence in the case of isotopically substituted homologs, compared to the linear dependence for Σν2 found theoretically by Decius and Wilson, and Sverdlov. Both Σν and Σν2 give good agreement between observed and calculated results. Various applications are given for the Σν rule and it is shown further that Σν, Σν2, Σ1/ν, Σ1/ν2 are all approximately constitutive properties of the halogenated methanes.