Dipole-quadrupole Terms Research Articles

Ab initio calculations of the forbidden Bragg reflections energy spectra in wurtzites versus temperature

Thermal-motion induced (TMI) scattering is caused by the influence of atomic displacements on electronic states in crystals and strongly depends on temperature. It corresponds to dipole–dipole resonant x-ray scattering, but is usually accompanied by dipole–quadrupole scattering. The phenomenological theory supposes the dipole–quadrupole term to be temperature independent (TI). As a result, the transformation of the energy spectra with temperature observed experimentally in ZnO and GaN corresponds to the interference between the TMI and TI terms. In the present paper the direct confirmation of this theoretical prediction is given. Ab initio molecular dynamics was used to simulate the sets of atomic sites at various temperatures followed by quantum mechanical calculation of resonant Bragg reflection energy spectra. The results of simulation are in excellent coincidence with experimental energy spectra of forbidden reflections and confirm the earlier phenomenological conjecture about the interference between the TI dipole–quadrupole and TMI dipole–dipole contributions to the resonant atomic factor.

Monte carlo calculation of the born-oppenheimer potential between two helium atoms using hylleraas-type electronic wave functions

The fact that Monte Carlo estimates for the expectation value of the Hamiltonian have no standard deviation for exact eigenfunctions of the Hamiltonian has been used to construct electronic wave functions accounting for the total correlation energy of two helium atoms separated by distances larger than 5aB to within 5 × 10−4 Ry. The biased selection Monte Carlo method used is an extension of information sampling, which enables the analytical elimination of all possible singularities in Vψ and also allows the use of the Hartree-Fock atomic wave function to select the points. The trial wave functions are an antisymmetric combination of Hylleraas-type atomic wave functions multiplying an exponential containing dipole-dipole and dipole-quadrupole terms in a form involving 18 parameters. The standard deviations in the energy values were estimated using 200 well-chosen sample points. Best wave functions were found by keeping the points fixed and varying the parameters in the wave functions until the minimum deviations were obtained. The resulting expectation values and standard deviations were then estimated using 6400 points (˜3 min of Amdahl 460 V time). The range of energy values from this step were in agreement with experiment to within their standard deviations of less than 5 × 10−4 Ry. The wave functions for two separation distances were tested much more severely to make sure that the above results were not due to a failure to sample enough odd points. A 10-hr run was made in which the expectation values were evaluated at 6.6 and 9aB separation and also the difference between these values. The runs using 6400 points were found to be valid, and in addition it was discovered that the energy bounds given by these wave functions are in agreement with experiment to within standard deviations of ˜3 × 10−5 Ry and that their difference (1.3 ± 1.4) × 10−5 Ry differs from the experimental 3.2 by about 1.4 standard deviations.

Inclusion of the quadrupole moment when describing polarization. The effect of the dipole‐quadrupole polarizability

It has come to our attention that there was an error in the computer program as presented in this article. The Böttcher notation of the quadrupole moments was used in one place and the Buckingham notation of the quadrupole moments was used in the rest of the computer program. The Buckingham notation should have been used in the entire program. This change will affect Tables 1 through 4. They are presented in their correct form below. , In addition, the numbers in the text on page 6 were incorrectly displayed. Instead of: The response of the molecule to a dipole placed along the y-axis will be well described as a field induced dipole (as an example along the y-axis error using only dipole polarizability is 0.061, compared to 0.063 when the dipole-quadrupole term is included, using LoProp and the VQZP basis set. This can be compared with 0.042 versus 0.026 when the direction of the perturbing dipole is perpendicular to the molecular plane.), and hence it is not important to include the dipole moments induced by the field gradient or the induced quadrupole moment. the text should have read: The response of the molecule to a dipole placed along the y-axis will be well described as a field induced dipole (as an example along the y-axis error using only dipole polarizability is 0.058, compared to 0.065 when the dipole-quadrupole term is included, using LoProp and the VQZP basis set. This can be compared with 0.041 versus 0.021 when the direction of the perturbing dipole is perpendicular to the molecular plane.), and hence it is not important to include the dipole moments induced by the field gradient or the induced quadrupole moment. We would like to extend our apologies for any confusion this error may have caused.

On the relative contribution of the dipole-monopole and dipole-quadrupole corrections to the soft electron emission from atoms in collisions with fast charged projectiles

Soft electron emission in ionization of light atomic targets by fast charged projectiles is considered within the first Born approximation. The dipole-monopole and dipole-quadrupole corrections to the dipole term have been calculated and compared. For ionization of neutral or positively charged atomic targets the dipole-quadrupole term is shown to be the main correction to the dipole term.

Tetrahedral anisotropy of x-ray anomalous scattering.

Diffraction of synchrotron radiation in crystals of potassium chromate and germanium reveals a tetrahedral anisotropy of anomalous (resonant) scattering with respect to polarization and wave vectors that introduces a third-rank tensor into the intensity equations. This tensor arises from a mixed dipole-quadrupole term that vanishes for forward scattering and for centrosymmetric atoms. At resonances 13 eV below the Cr K edge and 7 eV above the Ge K edge large changes of intensities with rotation of azimuth are observed for weak and forbidden Bragg reflections in agreement with the optical model.

Singlet-Singlet Energy Transfer Along the Helix of a Double-Stranded Nucleic Acid at Room Temperature

An irreversible electronic energy trap has been formed in calf thymus DNA by methylating about 75% of its G bases at position N-7. This has allowed us to measure for the first time the efficiency of transfer of energy along the helix of a double-stranded nucleic acid at room temperature. It is found that about one out of every three photons absorbed by the other bases is trapped. We have also simulated the data with a stochastic model that uses the dipole-dipole interaction to calculate the efficiency of transfer. In order to approximate the experimental results, the model requires that: (i) the fluorescence quantum yield of T, C, and G in DNA be about 2 x 10(-3), which is about two orders of magnitude larger than the value of the fluorescence quantum yield reported for DNA; and (ii) the fluorescence quantum yield of A in DNA be negligibly small. Requirement (i) is consistent with energy transfer taking place before a very efficient fluorescence quenching process sets in, which could be formation of excited-state complexes (excimers) that do not fluoresce appreciably. Requirement (ii) implies a very short fluorescence lifetime for A, which is consistent with the reported absence of a significant number of photoproducts formed by A in DNA. The simulations find that, on the average, the excitation energy takes about 1.2 steps to reach the trap; that is to say, bases that are nearest and next nearest neighbors of the trap are, in effect, the only energy donors. Both intra- as well as interstrand energy transfer (the latter only for the C-trap base pair) make significant contributions. The value of the efficiency for pairwise base-base intrastrand transfer is about 60%, whereas those for base-trap intra- and interstand transfer are 90% and 80%, respectively. The corresponding values for the rate constant of transfer are 2 x 10(11), 1 x 10(12), and 4 x 10(11) s-1. Transfer is inefficient when A is the donor or the acceptor. In addition to the dipole-dipole term, the only other significant term in the expansion of the interaction potential is the dipole-quadrupole term which, however, makes only a small contribution to the overall transfer efficiency. The electron exchange interaction appears to be much less efficient than the coulombic interaction.

Van der Waals force between positronium and hydrogenic atoms: Finite-mass corrections.

The Feshbach projection-operator formalism is used to derive the asymptotic effective interaction potential between two atoms. Beyond the usual van der Waals potential, falling like x exp -6, three x exp -8 terms are also obtained: an attractive dipole-quadrupole term (absent in the positronium-positronium case because of symmetry), a repulsive energy-dependent term, and a repulsive mass-dependent but energy-independent one. This last term was not obtained by Manson and Ritchie (1985) using an independent method.

A map of the cosmic background radiation at 3 millimeters

Data from a series of balloon flights covering both the Northern and Southern Hemispheres, measuring the large angular scale anisotropy in the cosmic background radiation at 3.3 mm wavelength are presented. The data cover 85 percent of the sky to a limiting sensitivity of 0.7 mK per 7 deg field of view. The data show a 50-sigma (statistical error only) dipole anisotropy with an amplitude of 3.44 + or - 0.17 mK and a direction of alpha = 11.2 h + or - 0.1 h, and delta = -6.0 deg + or - 1.5 deg. A 90 percent confidence level upper limit of 0.00007 is obtained for the rms quadrupole amplitude. Flights separated by 6 months show the motion of earth around the sun. Galactic contamination is very small, with less than 0.1 mK contribution to the dipole quadrupole terms. A map of the sky has been generated from the data. 12 references.

Comment on ‘‘Dispersion induced dipole moments’’

An earlier result arguing that multipole dispersion interactions between a pair of spherically symmetric atoms would lead to permanent dipoles in each atom directed along the line of centers if the multipole expansion were carried through the dipole quadrupole terms is reinvestigated following a suggestion by Mahanty. The original calculation employed a variational approximation in which the trial function was linear in the terms of a multipole interaction potential and which included only one-dimensional dipole quadrupole interactions along the line of centers. When the three-dimensional dipole quadrupole terms are employed, using an otherwise similar linear trial function, it is found that spherical symmetry causes the dipole moment of each species to vanish in contrast to the previous result. (AIP)

Sudden theories of rotationally inelastic LiH–HCl and LiH–DCl collisions

The impact parameter sudden approximation is applied to rotationally inelastic collisions between two polar molecules. The transition probabilities can be factored into a sum of squares of angular momentum coupling coefficients multiplied by dynamical factors which are related to coefficients in the expansion of the sudden scattering operator. The evaluation of these dynamical factors is discussed for the case of a potential which consists of the dipole–dipole term as well as both dipole–quadrupole terms in the standard multipole expansion. Even at relatively high collision velocities these long-range potentials will necessitate significant adabatic corrections to the pure sudden approximation, which can be determined by the introduction into the sudden action integral of a ’’dephasing frequency’’ proportional to the average rotational energy transfer per single quantum transition. The resulting adiabatically corrected sudden (ACS) approximation [M. H. Alexander and A. E. DePristo, J. Chem. Phys. 83, 1499 (1979)] yields a factorization similar to that obtained in the pure SA. The appropriate dynamical factors can be evaluated by a two-dimensional quadrature. ACS and SA calculations are reported for the collision of a LiH molecule with both HCl and DCl at a collision energy of 0.78 eV. The presence of the dipole–quadrupole terms results in an interesting alternance in some of the jajb→ja′jb′ cross sections, but leads to little change in the total cross section obtained by summing over the final states of the hydrogen halide partner. In a strong resonance effect the magnitude of these summed cross sections for directly coupled dipole transitions (Δja=±1) varies substantially with the initial rotational state of the partner.

Liquid-gas transitions in the spin-1 asymmetrical Ising model

Liquid-gas transitions in an asymmetrical spin-1 Ising model are studied using the Blume-Emery-Griffiths-Mukamel Hamiltonian. Within the Bragg-Williams approximation it is shown that the liquid-gas transition in the symmetrical model may become first order or second order provided the crossed dipole-quadrupole term vanishes, and that no tricritical point can exist. But an asymmetrical model always permits both first-order and second-order liquid-gas transitions, and a tricritical point can exist for some critical value of the crossed dipole-quadrupole coupling.

Collision-induced I(2P1/2) emission in argon and xenon

Long range perturbation theory is used to estimate the magnitude of the dipole transition moment leading to iodine 2P1/2 → 2P3/2 emission during a collision with a rare gas atom. The leading term in the induced transition moment varies as R−4 owing to the dipole–quadrupole term in the interaction potential. The two-body rate coefficient for collision-induced radiative decay of I 2P1/2 is calculated using approximate model potentials for XeI and ArI. The coefficients are estimated to be 2×10−19 cm3 sec−1 and 6×10−19 cm3 sec−1 for I 2P1/2 in 300 °K Ar and Xe, respectively.

Dipole-quadrupole dispersion forces associated with some metastable atoms

An estimate is made of the coefficients C (6) and C (8) of the R -6 and R -8 terms in the interatomic potential between the metastable 3 P states of Hg and of He interacting with sodium. Limited Hartree-Fock wave functions are used for Hg* and Coulomb wave functions for He*. The one-electron wave functions used still show some deficiencies, but in both systems the dipole-quadrupole term is found to contribute roughly 8–10 per cent to the total cross section at thermal energies. The effect of the C (8) term on the phase shifts near the rainbow region is of similar magnitude. The R -8 term should be allowed for in the analysis of metastable scattering experiments.

Viscosity of Polar–Quadrupolar Gas Mixtures

Viscosities of the polar–quadrupolar gas mixtures, CH3Cl–CO2 and SO2–CO2, have been measured in the temperature range from − 35–35°C by an oscillating-disk viscometer. The results show that viscosity is not significantly affected by the dipole–quadrupole term in the unlike interactions. Better combination rules than those now available are necessary for accurate prediction of properties of such gas mixtures.

Semiempirical Calculations of Ionic Polarizabilities and van der Waals Potential Coefficients for the Alkaline-Earth Chalcogenides

Electronic polarizabilities for the ions of the alkaline-earth chalcogenides have been calculated by assuming an additivity law within the family of salts and a Lorentz factor of $\frac{4}{3}\ensuremath{\pi}$, and using experimental refractive indices and interionic distances. It is shown that the additivity assumption is well obeyed, and the Lorentz factor value of $\frac{4}{3}\ensuremath{\pi}$ gives the best fit to the data. Each polarizability obtained has been related to a mean excitation or characteristic energy for the ion, and hence the coefficients of the dipole-dipole and dipole-quadrupole terms of the van der Waals interaction energy derived.

Effect of the short-range attractive terms on the equation of state of caesium bromide

The study of the equation of state of caesium bromide is based on the method adopted by Verma and Dayal for sodium chloride. It was observed that, corresponding to the volume ratio vT/v0 >or= 1.18 for some wave vectors, some of the frequencies obtained by solving the secular determinant were imaginary. As this ratio was smaller than the maximum volume ratio 1.27 for caesium bromide at the melting point, it implied that the usual form of the potential energy function leads to the instability of the lattice before it melts. All other available forms of the potential energy functions were also tried but did not improve the results. It was found that real frequencies could be obtained by augmenting Born and Mayer's potential by a term which is three times larger than the normal van der Waals and dipole-quadrupole terms. The new potential energy function has been used further for evaluating the following four quantities. cohesive energy, melting point and the temperature variations of thermal expansion and compressibility. The agreement with experiment is good for the first and reasonable for the other quantities.

Dipole—Quadrupole and Retardation Effects in Low-Energy Atom—Atom Scattering

Simple approximation formulas are derived for the dipole—quadrupole and retardation effects upon the total elastic scattering cross section for atomic collisions in the thermal energy range. For a long-range potential containing both the dipole—dipole and dipole—quadrupole terms, i.e., V(r) = —(C(6)/r6)[1+(β/r2)], one finds ΔQ/QMM≈⅔π(β/QMM) where QMM is the Massey—Mohr cross section for an inverse sixth-power attraction. For a long-range potential including dipole—dipole and retardation effects, approximated by V(r) = —(C(6)/r6)[a/(a+r)], one obtains ΔQ/QMM≈−(π−32QMM12)/a. Since these deviations are small and opposite in sign, it is concluded that the contributions of the dipole—quadrupole and retardation effects to the total cross sections are not sufficient to account for any significant discrepancy between theoretical and experimental Q values.

The van der Waals Energy of Two Helium Atoms

An expression is derived for the van der Waals energy of two helium atoms, exchange terms being neglected. This expression can be expanded as an asymptotic series in inverse powers of the internuclear distance. The first two terms in this series are compared with the dipole-dipole and dipole-quadrupole terms of previous workers.

The Role of Quadrupole Forces in Van Der Waals Attractions

van der Waals forces between atoms without permanent poles may be described as resulting from the interactions of multipoles associated with quantum transitions of the atoms. When the atoms are far apart, the dipole interaction is the only appreciable one. But at distances of the order of the kinetic theory radius, higher poles, usually neglected, must be considered. This paper examines and evaluates three terms of the series of interactions: the dipole-dipole, the dipole-quadrupole, and the quadrupole-quadrupole term, instead of the customary first term alone. The last appears to be in general small, but the dipole-quadrupole term requires consideration. Its effect is illustrated by plotting the potential energy curve for two helium atoms (a) neglecting the term, and (b) including it. The method outlined permits an easy estimate of dipole and quadrupole forces for all substances for which the wave function of only the normal state is known.