Comment on a sum rule for vibrational raman optical activity

The expressions for the sum of vibrational Raman optical activity (ROA) intensities indicate that the ROA intensity sum for chiral molecules is non-zero for those with an anisotropic electric dipole polarizability. The non-zero sum depends upon the electric dipole, magnetic dipole and electric quadrupole polarizability components and moments of inertia at equilibrium geometry.     

A sum rule approach to surface-plasmon dispersion in simple metals

We present a sum rule approach to the momentum dispersion of the surface plasmon in alkali metals by using the Lang-Kohn energy density functional. In this approach the dispersion relation is fully determined by the one-body ground state density. The linear slope at smallq is determined by the tail of the electron density outside the jellium surface. A comparison with density-functional calculations of the surface response function is performed.     

A simple sum rule for total radiative decay rates in diatomics

The quantity (ν³μ_e²), which occurs in expressions for the total radiative decay rate for an excited vibronic state of a diatomic, is evaluated as a simple expectation value for the initial state. In test calculations this expectation value agrees with the exact sum over states within ≈0.2%.     

A frequency-weighted sum rule for vibrational Raman optical activity

The sums of frequency-weighted vibrational Raman optical activity (ROA) intensities of isotopic molecules are shown to be related through the elements of the inverse force constant matrix. These relations could be used to determine force constants from experimental ROA intensities.     

An exact sum rule in optical activity

We prove that the lowest order contributions to the high frequency optical activity of any system is non-zero only if there are two or more electrons in the system.     

Differential virial theorem in relation to a sum rule for the exchange-correlation force in density-functional theory

Holas and March [Phys. Rev. A. 51, 2040 (1995)] gave a formally exact theory for the exchange-correlation (xc) force F(xc)(r)= -inverted Deltaupsilon(xc)(r) associated with the xc potential upsilon(xc)(r) of the density-functional theory in terms of low-order density matrices. This is shown in the present study to lead, rather directly, to the determination of a sum rule nF(xc)=0 relating the xc force with the ground-state density n(r). Some connection is also made with an earlier result relating to the external potential by Levy and Perdew [Phys. Rev. A. 32, 2010 (1985)] and with the quite recent study of Joubert [J. Chem. Phys. 119, 1916 (2003)] relating to the separation of the exchange and correlation contributions.     

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 Λⁱ_x. is the square of the Xth harmonic frequency of the ith isotopic species, and σⁱ is a weighting factor for the ith species, taken so that

for all a, where Mⁱ_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 test for the first moment sum rule in the two-dimensional one-component plasma

The recent exact solution of the one-component two-dimensional inhomogencous plasma by Jancovici is shown to be consistent with the sum rule for the first moment of the pair correlation function     

A new theory of determination of vibrational raman intensities in crystal lattices

Quantum-chemical methods have been used for the calculation of the intensities of Raman-active internal and external modes in crystalline oxamide. In a first step the frequencies and eigenvectors of the modes in oxamide are calculated by use of the concept of a flexible vibrating molecule within the intermolecular atom-atom forces. The quantum-mechanical aspects are considered and finally, a semiempirical method is extended to cover the relative Raman intensities of the external and internal modes.     

A sum rule approach to collective spin modes in paramagnetic clusters,cavities and shells

We study spin collective modes in paramagnetic systems using a sum rule approach. We give a simple and analytical expression for the static magnetic polarizability of finite size systems in terms of their radii and of the magnetic susceptibility of a uniform system. A systematics for the mean excitation energy of spin modes in clusters and shells is given. In metal spheres, the hydrodynamical model is solved and compared with the sum rule approach. An estimate for the mean excitation energy of the spin dipole collective state in fullerene is given.     

Group theory approach for raman scattering of triatomic molecules

The symmetry of the rotation-vibration spectra for triatomic molecules is described by means of the group U(5). Group theory approach is adopted to give the transition matrix elements for calculation of the cross sections of vibrational and rotational Raman scattering. The results are in good agreement with the experimental values.     

苯酚分子拉曼光谱的密度泛函理论研究

采用B3LYP杂化泛函,6-31++G(d,p)基组,对苯酚分子的结构进行了优化.通过频率计算,获得了苯酚分子的拉曼光谱,并利用势能函数分布(PED)对拉曼光谱进行了指认.绘制了苯酚分子的静电势分布图,讨论了苯酚分子发生化学反应的位置,同时计算了HOMO-LUMO的能级差.通过TD-DFT计算获得了苯酚分子的吸收光谱和激发态.     

A simple rule for prediction of intensities of vibrational overtone spectra

The intensities of vibrational overtone absorption transitions are described in terms of vibronic coupling of the ground molecular state to excited electronic configurations. Model calculations indicate an important role of nuclear geometry of excited electronic states relative to the ground state in determination of molecular overtone spectra. A simple rule for qualitative predictions of the overtone spectra for diatomic molecules or local bond modes of polyatomic molecules is proposed.     

Chiroptical properties of 2-substituted piperidines : A simple helicity rule

The CD spectra of fifteen 2-substituted piperidines 2 and their N-methylated derivatives 1 have been determined. The sign of the Cotton effect observed for the n→σ^* transition of nitrogen is shown to be correlated, for a given absolute configuration, with the axial or equatorial orientation of the nitrogen lone pair, and to be determined by the screw sense of the helicity between the nitrogen lone pair and the 2-substituent R in 1 and 2: for compounds of the absolute D configuration, a positive helicity gives a positive CD and vice versa.     

Oscillator strength sum rule and the hydrodynamic analogy to quantum mechanics

It is shown that the functional obtained from the hydrodynamic analogy to quantum mechanics identically satisfies the 2^l-pole oscillator strength sum rule if certain conditions are satisfied by the linear trial functions. Further, the eigenfrequencies are the same as those found by the Rayleigh—Ritz variational principle.     

A simple theory of molecular organization in fullerene-containing liquid crystals

Systematic efforts to synthesize fullerene-containing liquid crystals have produced a variety of successful model compounds. We present a simple molecular theory, based on the interconverting shape approach [Vanakaras and Photinos, J. Mater. Chem. 15, 2002 (2005)], that relates the self-organization observed in these systems to their molecular structure. The interactions are modeled by dividing each molecule into a number of submolecular blocks to which specific interactions are assigned. Three types of blocks are introduced, corresponding to fullerene units, mesogenic units, and nonmesogenic linkage units. The blocks are constrained to move on a cubic three-dimensional lattice and molecular flexibility is allowed by retaining a number of representative conformations within the block representation of the molecule. Calculations are presented for a variety of molecular architectures including twin mesogenic branch monoadducts of C60, twin dendromesogenic branch monoadducts, and conical (badminton shuttlecock) multiadducts of C60. The dependence of the phase diagrams on the interaction parameters is explored. In spite of its many simplifications and the minimal molecular modeling used (three types of chemically distinct submolecular blocks with only repulsive interactions), the theory accounts remarkably well for the phase behavior of these systems.     

On the E1 partial wave sum rule and the WKB approximation

The electric dipole (E1) partial wave sum rule, relevant to Coulomb scattering, is derived semiclassically within the WKB theory. Limiting expressions are presented according to the values of the angular momentum l of the colliding particle, the Sommerfeld number and the adiabaticity parameter ξ = ν _? - ν _i. A comparison with the quantal results, of the WKB sum rule and the classical orbital integrals, shows that the WKB approximation is extremely accurate even in the case of light and/or fast particles (ν ? 1). An algorithm is presented for the calculation of the modified Bessel function of the second kind with imaginary argument K _iξ(x) and its derivative dK _iξ(x)/dx.     

A simple theory of liquid—solid phase transitions

A simple phenomenological theory of liquid—solid phase transitions, based on the use of perturbation theory about a hard-sphere fluid, is examined. A temperature dependent hard-sphere diameter is determined which permits the prediction of solid and liquid densities and melting pressures. Calculations presented here for the Lennard-Jones 12-6 fluid show good agreement with the computer simulation results. Provided pair potentials are available, the theory may also be used for other fluids, including liquid metals.     

General theory of vibrational mode mixing and frequency shifts in resonance raman scattering

A general formalism is given for treating vibrational mode mixing, frequency shifts, and atomic equilibrium position shifts under electronic excitation in resonance Raman scattering. The theory is exact for first-order scattering at T = 0 K for all linear and quadratic electron-phonon coupling strengths. Numerical results illustrating mode mixing are presented.