Dipole Moment Coefficient Research Articles

Polarization of a Diffuse Soft Particle Subjected to an Alternating Current Field

The polarization of a diffuse soft particle submerged in an aqueous electrolyte and subjected to a uniform alternating electric field is theoretically analyzed with the standard electrokinetic model (the Poisson-Nernst-Planck equations). The particle consists of a rigid uncharged core and a charged diffuse polyelectrolytic shell (soft layer) permeable to ions and solvent. Our focus is on the impact of the characteristics of the soft layer including the Donnan potential, the soft layer thickness, and the friction coefficient of the soft layer on the dipole coefficient, characterizing the strength of the polarization. Under the limits of thin double layers and thin polyelectrolytic shells, approximate analytical expressions to evaluate the dipole moment coefficients are derived for high-frequency and low-frequency ranges, respectively. The analytical results are compared and agree favorably with those numerically computed by the standard model. Interestingly, we discover that when the double layer is comparable to the soft layer the dipole moment behaves qualitatively differently at different Donnan potentials. When the Donnan potential is small, the dipole moment decreases as the double layer increases. In contrast, at large Donnan potentials, the dipole moment increases with the increase in the double layer. The distinct responses to Donnan potentials are attributed to the impact of the associated double layer on the charge distribution of mobile ions inside the soft layer. The theoretical model provides a fundamental basis for interpreting the polarization of heterogeneous systems, including environmental or biological colloids or microgel particles.

The polarization of a nanoparticle surrounded by a thick electric double layer

The polarization of a charged, dielectric, nanoparticle enveloped by a thick electric double layer and subjected to a uniform, alternating electric field is studied theoretically with the standard model (the Poisson–Nernst–Planck PNP equations). The dipole coefficient ( f) is calculated as a function of the electric field's frequency and the double layer's thickness ( λ D ) . For a weakly charged particle with a small zeta potential ζ, an approximate, analytic expression for the dipole moment coefficient, accurate within O ( ζ 2 ) , is derived. Two processes contribute to the dipole moment: the ion transport in the electric double layer under the action of the electric field and the particle's electrophoretic motion. As the thickness of the electric double layer increases so does the importance of the latter. In contrast to the case of the thin electric double layer, the particle with the thick double layer exhibits only high-frequency dispersion. The theoretical predictions are compared and favorably agree with experimental data, leading one to conclude that the standard, PNP based-model adequately represents the behavior of nanoparticles subject to electric fields.

Electric dipole moment function of H 79Br

An anharmonic potential function with 11 parameters was used in numerically solving the radial Schrödinger wave equation to obtain potential constants of hydrogen bromide molecule, H 79Br. Vibration–rotation energy states up to the vibrational level v=8 and the corresponding wavefunctions were generated. The dipole moment matrix elements, M 0 v′(m) , were evaluated using the experimentally determined line strengths. The dipole moment coefficients, M 0, M 1,…, M 8, were then calculated from these matrix elements and wavefunctions.

Electric dipole moment function of H 35Cl

The radial Schrödinger wave equation was solved numerically using an anharmonic potential function with 11 parameters to generate wavefunctions and vibration–rotation energy levels of the hydrogen chloride molecule H 35Cl. Measured line strengths available in the literature cover transitions up to v=7. Using the measured values of the line strengths along with the computed wavefunctions, a set of 135 dipole moment matrix elements M 0 v′(m) were calculated. Finally, the dipole moment coefficients, M 0, M 1,…, M 7, were evaluated from the matrix elements.

Electric dipole moment function of 12C 16O

Vibration–rotation energy states and corresponding wave functions of carbon monoxide molecule 12C 16O are generated by numerically solving the radial Schrödinger wave equation using an anharmonic potential function with eight parameters. With these wave functions and the measured line strengths that are available for the first five vibrational levels, 183 dipole moment matrix elements M 0 v′(m) are evaluated. Finally, the dipole moment coefficients, M 0, M 1,…, M 4, are extracted from the matrix elements.

Centrifugal distortions in molecules: An a b i n i t i o approach

An ab initio approach to the theoretical description of centrifugal distortions in molecules is presented. The method is based on the calculation of the electronic energy changes and gradients accompanying rotationally induced distortions. A centrifugal distortion pathway in nuclear coordinate space is defined. Results are presented at the HF/6-31G** level for NH3, CH4, BF3, and SF6, and at the Guillemin–Zener level for H+2. Quartic centrifugal distortion parameters are computed for each molecule, with distortion anisotropies presented in the form of tensor coefficients. Centrifugally induced electric dipole moment coefficients are presented for CH4 and BF3 together with dipole moment changes for NH3. Agreement with experiment and with conventional theoretical descriptions is generally quite satisfactory, particularly for the tensor coefficients representing the cubic anisotropy in the distortions of the spherical tops CH4 and SF6.

Dipole Moment Functions of HF and HI Molecules Using an Anharmonic Potential

Abstract Vibration-rotation wave functions for HF and HI are computed by solving the radial Schroedinger wave equation numerically using an anharmonic potential function with seven adjustable parameters. With these wave functions the matrix elements of [(r - re)/re]n are computed. These are then applied in a least squares fit to experimentally measured values of the electric dipole matrix elements to yield the dipole moment coefficients M0, M1,..., M5 .

Herman-wallis factors for higher overtone bands: Application to HCl

Theoretical expressions for the Herman-Wallis factors in terms of the Dunham potential-energy parameters and dipole-moment coefficients were derived for higher overtone bands by the method of computer algebra. These results were applied to an analysis of the vibration-rotational intensities of HCl, leading to a good fit of the data and a dipole-moment function that is improved and extended by comparison with previous results.

Notizen: The Dipole Moment Function of HF Molecule Using Morse Potential

Abstract The radial Schrödinger wave equation with Morse potential function is solved for HF molecule. The resulting vibration-rotation eigenfunctions are then used to compute the matrix elements of (r - re)n. These are combined with the experimental values of the electric dipole matrix elements to calculate the dipole moment coefficients, M 1 and M 2.

High order vibrational matrix elements for diatomic molecules; dipole moment function and transition moments of CO

The fifth and sixth-order contributions to the vibrational matrix elements are obtained for the transitions v→ v'( v'⩽ v+4) using an eight power internuclear Dunham potential and a quartic power series expansion of the dipole moment function. The results for the dipole moment coefficients M 0 to M 4 of CO and the transition moments R v v' (with v=5, 10, 20) deduced from a calculation including succesively third, fourth, fifth and sixth-order perturbation theory are compared. The validity of these results is discussed. Using a quintic dipole moment function, general expressions for the vibrational matrix elements corresponding to the transitions v→ v'( v'⩽ v+5) are also presented and the influence of these contributions on the calculation of the dipole moment coefficients as well as the hot band transition moments of CO is shown.

Vibration-rotation matrix elements for diatomic molecules; vibration-rotation interaction functions F v′v ( m) for CO

General expressions for the fourth-order vibrational matrix elements are obtained for the transitions v → v′ ( v ⩽ v′ v + 4) using a sixth-power internuclear potential and a quartic dipole moment function. The results for the dipole moment coefficients M 0 to M 4 of CO and for some transition moments R v′ v deduced from a calculation including successively second, third and fourth order perturbation theory are compared. Using the rotational potential function expanded through the cubic term as a perturbation, we have also obtained general expressions for the vibration-rotation matrix elements. The vibration-rotation interaction functions F v′ v ( m) are calculated for the transitions v → v′ ( v = 0, 10, 20 and v ⩽ v′ ⩽ v + 4) of CO and the coefficients C, D, E and G of the quartic polynomials fitting these functions are deduced. Taking account of uncertainties in the matrix elements R 0 0 to R 4 0, an error estimate for the coefficients of F 2 0( m) and F 3 0( m) is given.

Long Range Collision-induced dipoles

A method of estimating the long range dipole moment coefficient, D 7, for two colliding dissimilar S-state atoms from their ground state sum rule values is presented and applied to systems involving H. He and Ne.

A study of the fundamental and first overtone bands of NO in NO-rare gas mixtures at pressures up to 10,000 PSI

The absorption intensities of the 1-0 and 2-0 vibration-rotation bands of NO are determined from the absorption coefficients of NOHe and NOAr gaseous mixtures at high pressures at room temperature. The values obtained are: A 1-0 = 121 ± 6 cm −2 Agt −1 and A 2-0 = 2.17 ± 0.11 cm −2 Agt −1. A theory developed by Tipping is applied to evaluate the dipole moment coefficients unambiguously, including their signs, from the absolute intensity values and the difference between the mean frequency factor and the band origin. The following expansion for the dipole moment function in the ground state of NO is determined: M( x) = −0.166 + 2.54 x − 1.99 x 2 (in Debye). The absorption profiles of both the 1-0 and 2-0 bands in NOAr mixtures show marked changes as gas pressure increases; some of the factors influencing the shapes of the bands are also discussed. The plots of the integrated intensity vs rare gas density are found to be straight lines with positive slopes. This linear increase of the band intensity with density is interpreted as mainly due to the apparent induced absorption.

Computation of electric dipole matrix elements for hydrogen fluoride

The electric dipole matrix elements of hydrogen fluoride have been calculated by numerical integration for transitions involving large quantum numbers υ, J. Overtones have been included through Δυ = 5. Molecular wave functions obtained by numerical integration of the Schrödinger equation were used. The influence of the mechanical motion on the matrix elements has been determined for Morse and Rydberg-Klein-Rees (RKR) potential functions. The influence of the electric dipole-moment function approximations has been investigated by a comparison of matrix elements obtained with approximations having the form of a truncated polynomial and a wave-function expansion. The inaccuracies in the matrix elements caused by uncertainties in the dipole-moment coefficients have been investigated.

The Dipole Moment Function of H79Br Molecule

Abstract The radial Schrödinger wave equation with Morse potential function is solved for H79Br molecule. The resulting vibration-rotation eigenfunctions are then used to compute the matrix elements of (r-re)n . These are combined with the experimental values of the electric dipole matrix elements to calculate the dipole moment coefficients, M1 and M2 .

Intensites absolues et forces d'oscillateur de quelques raies des bandes de vibration-rotation 1-0 et 2-1 du radical OH

Line radiancies ( R L ) have been obtained in the 1-0 and 2-1 vibration-rotation bands of the OH radical using a high resolution infrared grill spectrometer, a blackbody and an oxyacetylene flame. From temperature measurements and calculated values of the OH partial pressures in different regions of the flame, the optical path has been derived. Integrated absorption S v'J' vJ and oscillator strength f v'J' vJ have been ded uced for the four components (lambda and spin doubling) of the P branch lines J = 11 through J = 17 in the 1-0 transition and J = 9 through J = 16 in the 2-1 transition. The values | R 0 1| = 3.1 × 10 -2 D and | R 1 2| = 3.9 × 10 -2 D have been obtained for the vibrational matrix elements of the 1-0 and 2-1 bands. Dipole moment coefficients have been calculated. Results have been compared with those of Heaps and Herzberg.

Vibration-rotation band strengths and dipole moment function of the H79Br molecule

From the data on the line strengths, the band strengths of the 1-0 and 2-0 vibration-rotation bands of H79Br molecule have been respectively calculated to be 10.65+or-0.22 and 0.174+or-0.008 cm-2 atm-1 at 294 K. From the vibrational matrix elements involving the Morse and anharmonic oscillator wave functions, two sets of values for the dipole moment coefficients M1 and M2 have been calculated. They are M1=0.316 debye AA-1, M2=0.518 debye AA-2 or M1=0.344 debye AA-1, M2=-0.096 debye AA-2. These values are compared with the values reduced from the low-resolution data.

Dipole moment matrix elements for the 1-0, 2-0, and 3-0 vibration-rotation bands of diatomic molecules

Explicit expressions for the matrix elements of the dipole moment have been derived for the 1-0, 2-0, and 3-0 vibration-rotation bands of diatomic molecules. The derivation is based on a cubic dipole moment function and on radial wave functions obtained using a quintic power series expansion of the internuclear potential and third order perturbation theory. The results are given in the form of a rotationless matrix element multiplied by a rotational or Herman-Wallis factor. The expressions for the Herman-Wallis factors include the leading contributions from each dipole moment coefficient and have a reasonably simple form even in the case of the overtone bands. The rotationless matrix elements are similar to results obtained by Herman and Schuler except that their higher order terms are purposely omitted. The reason for this is discussed as are other approaches to the rotational problem which have incorporated a linear dipole moment function. It is pointed out that the results of the present theory for the case of CO are in very good agreement with the numerical treatment of CO by Young and Eachus. The actual comparison is given in a subsequent paper.

Line intensities in the 3-0 band of CO and dipole moment matrix elements for the CO molecule

Line intensities in the 3-0 band of CO have been determined by the method of equivalent widths for near Doppler lines. The data were obtained with sample pressures between 4 and 40 Torr and a spectral slit width of 0.04 cm −1. The values found for the band intensity at 273°K and the linear and quadratic Herman-Wallis coefficients were 0.0135 cm −2 atm −1, 0.0118 and 1.8 × 10 −4, respectively. The expressions for the dipole moment matrix elements derived in Part I were then evaluated for the vibrational band intensities used by Young and Eachus in order to test the present theory. Good agreement with the results of Young and Eachus was obtained both for the dipole moment coefficients of CO and for the Herman-Wallis factors for the 2-0 and 3-0 bands. New dipole moment coefficients and Herman-Wallis factors based on revised 1-0, 2-0, and 3-0 band intensities at 273°K of 260, 2.06, and 0.0132 cm −2 atm −1 were then calculated by means of the present theory. The dipole moment expansion was found to be, M( r) = −0.112 + 3.10 ( r - r e ) − 0.31 ( r - r e ) 2 − 2.28 ( r - r e ) 3 where r e is the equilibrium internuclear distance in angstroms and M( r) is in Debyes. A negative sign was assigned to M 0 on the basis of the good agreement which this sign gave between observed and calculated Herman-Wallis factors. It was concluded that the general expressions obtained in Part I are of good accuracy. Previous analytic expressions obtained for the 2-0 and 3-0 bands are not as accurate primarily because they do not include quadratic and cubic dipole moment terms.