Infrared band intensities: a comparative study of the transition moment matrix elements for the fundamental and overtones: Part II. Application to CO,HCl and OH

The various expressions considered in Part I for the transition moment matrix elements of fundamental and first two overtones are applied to carbon monoxide. The coefficients a_ij in the expressions R^io = Σa_ijp_j (where R^io is the transition moment integral for the O → i vibrational transition and p_j is the dipole moment derivative ?^jP/?XXX^j, XXX = (r — r_e)/r_e, r_e is equilibrium bond distance) are reported for i,j = 1, 2, 3. It is found that these coefficients do not vary by more than 5% when compared for the same i,j values in various expressions irrespective of the most exhaustive treatments used in deriving the original expressions. On the basis of the values of the coefficients obtained for CO, generalisations have been suggested on the effects of inclusion of mechanical and electrical anharmonicity to the intensities of fundamental and first two overtones. It is generally observed that the contribution of p'₁, is about 100 fold more than the contribution of p'₂, for R¹⁰. On the other hand the contributions of p'₁, and p, for R²⁰ and R³⁰ are of nearly equal magnitude but opposite in sign. The contribution of p'₁ to R¹⁰ is much higher than its contribution to R²⁰ and R²⁰. The various observations lead us to conclude that, whereas the effect of inclusion of mechanical anharmonicity on the intensity of the fundamental band is negligible, this effect is almost comparable to the effect of inclusion of electrical anharmonicity for the first two overtones. Simple forms of the a_ij expressions are applied to HC1 and OH to demonstrate the effect of variation of molecular constants on the a_ij values. On the basis of the observed trend in the values of these coefficients for CO, HCl and OH general remarks on the effects of hydrogen bonding on IR band intensities are given.     

Hydrodynamic interaction between macromolecules in sedimentation

The sedimentation constant of hard spheres decreases with concentration, due to the effect of backflow and distortion of the hydrodynamic field. For macromolecules at the θ-temperature, this effect of concentration is much smaller than that of equivalent spheres. Pyun and Fixman suggest that intertwinement of macromolecules gives rise to increase of the sedimentation velocity. However, their detailed model of intertwined molecules is unrealistic. An alternative model is proposed by considering a pair of intertwined homologous molecules with N_i and N_j segments respectively, as a single molecule with (N_i + N_j) segments. This model leads to a simple expression for the concentration coefficient in the sedimentation velocity, containing an adaptable parameter, χ_ij, characterising the probability of intertwining. The probability of intertwining is put equal to the probability of intermolecular approach to within a distance of R_int and χ_ij is defined by χ_ij = 2R_int/(R₁ + R_j), with R_i and R_j the friction equivalent molecular radii. From experiments on a number of sharp polystyrene fractions in cyclohexane at T = θ, a value χ_ji = 2.2 is deduced, decreasing with increase of molecular weight. Qualitative agreement is found between experimental results and predictions from the model. Quantitative agreement is not expected in view of uncertainties in the model. In ternary systems, containing molecules of different velocity, probably additional effects operate besides backflow, field distortion and intertwinement.     

Molar excess volumes of ternary mixtures of nonelectrolytes

Molar excess volumes V^E_ijk of methylenebromide i + pyridine j + β-picoline (k, cyclohexane (i) + pyridine (j) + β-picoline(K), benzene(i)+toluene(j)+1,2-dichloroethane(k), benzene(i) + 0-xylene(j) + 1,2-dichloroethane(k) and benzene(i) + p-xylene(j) + 1,2-dichloroethane(k) mixtures have been determined dilatometrically at 298.15 K. The data have been examined in terms of Sanchez and Lacombe theory and the graph-theoretical approach, and it is found that they are described well by the latter. Self- and cross-volume interaction coefficients V_jk, V_jjk and V_jkk, etc., have also been evaluated and the values utilised to study molecular interactions between the jth and kth molecular species in the presence of the ith in these i + j + k mixtures.     

Auswahl von Phasensystemen in der Lösungs-, Adsorptions- und Ionenaustausch-Chromatographie

In contrast to GC selectivity in LC is determined by the composition of both the stationary as well as the mobile phase. Therefore the main problem in LC results in selecting an appropriate phase system for the given separation problem. The selectivity factorα _ijis defined as the ratio of the capacity factors k′ _i k′_jof two solutes, which corresponds to the ratio of their distribution coefficients ^c K _i, ^cK_j. In LLC α _ijis determined by the relative solubility of the solutes in the two immiscible phases, which were prepared from binary or ternary liquid-liquid-systems. Secondary effects on retention are caused by the support. Two variations exist (LLC, Reverse-Phase-LLC) which differ in whether the polar phase is used as stationary or mobile phase, resp. In LSC the same phase variation is possible. Using a polar support and an unpolar solvent α _ijis governed by the relative strength of interactions between the solute molecules and the surface of the support. In Reverse-Phase-LSC, however, using an unpolar support and a polar solvent, these interactions are very weak and α _ijis mainly determined by the solubility of the solutes in the mobile phase. In IEC α _ijdepends on a set of parameters such as the type of ion-exchange matrix, its pore structure and its degree of crosslinking, resp., the type, surface concentration and distribution of functional groups, the type of the eluent ion, its concentration, the ionic strength and pH-value of the eluent, the temperature. Different methods have been developed in order to calculate the distribution coefficients of solutes for a given phase system.     

Atomic distributions in liquid copper-tin alloys

Abstract

Molten copper-tin alloys have been studied by X-ray diffraction, using a focusing theta-theta diffraetometer and Mo-Kα radiation (monochromator in the diffracted beam). Five alloys with 20, 35, 45, 55 and 78 atomic percent Sn, and pure Cu and Sn were measured at temperatures about 20 °C above the liquidus, and at 1100 °C. The total interference functions I(K), where K = 4π sin θ/δ, were obtained from the observed scattered intensities I_a(K) per atom and the theoretical atomic scattering factors. Splitting of the first peak in I(K) has been observed in the Cu-55 at% Sn alloy at the liquidus temperature.

The partial interference functions I_tj(K) at the liquidus temperature and at 1100°C were evaluated (assuming that they are independent of atomic concentration) using the five total I(K) of the alloys. The functions I_ij(K) are in reasonable agreement with those obtained by Enderby, North and Egelstaff from neutron diffraction data of a Cu-45 at % Sn alloy.

The reduced partial distribution functions G_ij(r) = 4πρ₀ r{g_ij(r) ? 1} and the probability functions g_ij(r) = ρ _ij(r)/c_j ρ₀, where ρ _ij(r) is the number of j-type atoms per unit volume at the distance r from an i-type atom, c_j is the atomic fraction of j-type atoms and ρ₀ is the average atomic density, have been evaluated by Fourier transformation of {I_ij(K) ? 1} K.

The electrical resistivities ρ _R of the alloys, calculated with the Faber-Ziman equation using the measured I_ij(K) and Animalu-Heine pseudo-potential elements U_i(K), are in good agreement with the experimental values of Roll and Motz. Assuming that U_i (2k _F) is independent of the values of the Fermi diameter 2k _F of the alloys, the concentration dependence of (3 - X)ρR, where X is the thermoelectric parameter measured by Enderby and Howe, is well reproduced when using the X-ray values of I_ij (2k _F).     

Resistance distance and Laplacian spectrum

The resistance distance r _ij between two vertices v _i and v _j of a (connected, molecular) graph G is equal to the resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any two adjacent points is unity. We show how the matrix elements r _ij can be expressed in terms of the Laplacian eigenvalues and eigenvectors of G. In addition, we determine certain properties of the resistance matrix R=||r _ij ||. AcknowledgementsThis research was supported by the Natural Science Foundation of China and Fujian Province, and by the Ministry of Sciences, Technologies and Development of Serbia, within Project no. 1389. The authors thank Douglas J. Klein (Galveston) for useful comments.     

Simple electrostatic model for the I — A approximation

Employing the approximation which replaces r _ij ^–1 by (r_i+r_j)^–1 we derive the relation 2/4l+5=I–A where is the orbital exponent of a Slater AO. This semiempirical relation appears to agree with experiment for the cases compared.     

Ring signatures for benzenoids with up to seven rings,Part 2: Pericondensed systems

An array of integers p_ij indicate in how many of the K resonance structures one finds 6 > i > 0 π‐electrons in the corresponding ring (which is labeled with capital letters A through a subsequent letter up to G). The index j for the rows indicates the ring type, denoted by capital letters (that is, j = 1 stands for A, j = 2 stands for B, etc.). Row sums lead to π‐electron ring partitions after division by the number K of resonance structures. Column sums lead to a sequence of seven integers (R_i sequence) which after division by K affords the r_i sequence; this is finally converted into the signature of the benzenoid s_i sequence). The analysis of signatures for pericondensed benzenoids with h = 4 through 7 benzenoid rings has revealed interesting regularities and correlations with structure, and has also allowed a confirmation of Clar's theory by investigating five isomeric dibenzopyrenes. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Asymptotic behaviour of molecular bound state wavefunctions

Upper bounds are derived for |r^μ_i|, μ = 1,2, ·, where ? denotes an exact electronic bound state wavefunction of a molecular system in the Born-Oppenheimer approximation, and r_i is the distance of the ith electron from an appropriately chosen point, e.g., the molecular center. It is further shown that ? decays exponentially if r_i → ∞.     

The general solution of the coagulation equation for branched polymers

The coagulation equation with a bilinear kernel, K_ij = [(f − 2) i + 2][(f − 2) j + 2], and arbitrary initial conditions c_k (0), is solved both for finite f, and in the limit f → ∞. Analytic expressions are given for the size distribution before and after gelation and for the gelation times. The correspondence with a generalized version of Flory's probability argument is shown.     

Homogeneous Fermi liquid with ‘artificial’ repulsive inverse square law interparticle potential energy

A good deal is known by now on the so-called jellium model of the homogeneous electron liquid. However, much of the quantitative progress at experimentally realizable densities has come from quantal computer simulation. Therefore, we here consider a homogeneous Fermion liquid with ‘artificial’ repulsive interaction λ/(r_ij )² between Fermions i and j at separation r_ij . We discuss first of all the way the static structure function S(q), essentially the Fourier transform of the pair correlation function, is changed because of non-zero λ from the ‘Fermi hole’ form due entirely to Pauli principle effects between parallel spin Fermions. Unlike jellium with e ²/r_ij repulsive interactions, S(q) is proportional to q at long wavelengths, whereas the plasmon in jellium annulls the q term and S(q) is quadratic in q as q tends to zero. However for λ/(r_ij )² interactions, the coefficient of q appearing in the Fermi hole structure factor, is renormalized by particle repulsions. Then some discussion is given of Fermion quasiparticle lifetimes τ as the Fermi surface is approached. Arguments are presented that τ^?1 is proportional to |E???E _F| as E tends to the Fermi energy. This is already interesting, in fact, in connection with the jellium model and therefore an approximate analytic form of τ is finally derived.     

Kirchhoff index of linear pentagonal chains

The resistance distance r_ij between two vertices v_i and v_j of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed‐form formulae of the Kirchhoff index of liner pentagonal chain P_n in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of P_n is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

A method for selection and use of numerical integration points for molecules with cylindrical symmetry

Boys and Handy [1] have discussed the solution of the bivariational equations with restricted numerical integration. One of the weaknesses of the method was that in the numerical summations over points, some points arose with r _ij= 0 and non-zero weights. This makes the method quite impractical for the Schrodinger Hamiltonian (because of the singularity at r _ij= 0), and it cannot be advantageous for the transcorrelated Hamiltonian C^–1HC because there will be some discontinuous higher derivatives at r _ij=0. Here it is shown how the symmetry of cylindrically symmetric molecules can be used to eliminate such points, without losing any of the advantages of the overall method, such as the convergence of the eigensolutions. It is also shown how the primary numerical integration points (z _i, r_i) may be chosen in any calculation such that each is associated with an equal amount of one-electron density. The choice of the angular coordinates are governed by the removal of the r _ij=0 points and maintaining the natural orthogonality between orbitals of different symmetry types. The method has been programmed and found to be practical, although no new molecular calculations have yet been performed. It is to be hoped that these points will give a basis for new transcorrelated calculations on diatomic molecules.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

Rate parameters for transfer reactions of some polymer radicals: in terms of Swain and Lupton's f and r and the unique positional weighting factors of Williams and Norrington

Rate parameters of the reaction of phenols, deuterated phenols and nitrobenzenes with polyvinylacetate radicals and of aromatic thiols with polymethylmethacrylate radicals have been correlated with Swain and Lupton's substituent constants (F_k and R_k) and Williams and Norrington's unique positional weighting factors (f_j and r_j) by the following equation: $\text{P}{}_{\mathrm{i}}\text{= α}{}_{\mathrm{i}}\text{?}{}_{\mathrm{j}}\text{F}{}_{\mathrm{k}}\text{+ β}{}_{\mathrm{i}}\text{r}{}_{\mathrm{j}}\text{R}{}_{\mathrm{k}}\text{+ e}{}_{\mathrm{i}}\text{+ P}{}^0{}_{\mathrm{i}}$where P_i's are the rate parameters, P⁰_i being that for a standard reference state. The correlations were found to be quite satisfactory. The sign and magnitude, and the ratio of the reaction dependent parameters α_i and β_i throw light on the nature of the transition state and the relative contributions of the mesomeric and inductive effects. The present studies also show that the mesomeric effect of meta-substituents is significantly greater than reported by earlier workers.     

Analysis of the pressure effect on the local composition in a water-alkanol mixture using Kirkwood-Buff integrals

Kirkwood-Buff integrals are calculated from the thermodynamic data for binary mixtures of water with methanol, ethanol, 1-propanol, and 2-propanol at a temperature of 298.15 K in the pressure range from atmospheric to 100 MPa. The values of local compositions Δn _ij are calculated which characterize the excess (or deficit) of molecules i around the central molecule j. It is found that the pressure affects destructively the homoassociation in all mixtures studied. In a series MeOH < EtOH < 2-PrOH < 1-PrOH an excess of molecules around the similar type molecules increases in the local environment and the pressure effect on the local composition is enhanced.     

Isentropic Compressibility Changes of Mixing for 1,3-Dioxolane or 1,4-Dioxane + Water + Propanol Ternary Mixtures

Speed of sound data, u_ijk, of 1,3-dioxolane or 1,4-dioxane(i) + water(j) + propan-1-ol or propan-2-ol(k) ternary mixtures and their sub-binary mixtures, u_ij, of 1,3-dioxolane or 1,4-dioxane(i) + water or propan-1-ol or propan-2-ol(j) and water(i) + propan-1-ol or propan-2-ol(j) mixtures have been measured over the entire composition range at 308.15 K. Isentropic compressibility changes of mixing, (κ_s^E)_ij and (κ_s^E) _ijk, for the binary and ternary mixtures have been determined by employing the observed speeds of sound data and densities (calculated from their molar excess volumes data). The (κ_s^E) _ij and (κ_s^E) _ijk values have also been predicated by the graph theoretical approach and the Flory theory. It has been observed that (κ_s^E) _ij and (κ_s^E) _ijk predicted by the graph theoretical approach compare well with their corresponding experimental values.     

Some considerations on the microwave electrodeless discharge: Plasma diagnostics in argon,helium or nitrogen atmospheres

Plasma diagnostics of several microwave plasmas are determined by making electrical (with double floating probes) and optical measurements in pure Ar, He or N₂ plasmas, and also in Ar plasmas containing various metals, i.e. Cs, Tl or Zn; plasma parameters, such as, electric field (E), electron (j_e) and ion (j_i) current densities, electron density (n_e), electron temperature (T_e) electron conductivity (σ_e), ion density (n_i), electron mean free path (λ_e) electron (μ_e) and ion (μ_i) mobilities and electron [(v_e)_drift] and ion [(v_i)_drift] volocities are either directly measured or calculated. The reversal temperature (T_r) of excited (0.96 eV lower level) thallium atoms is measured, and the steady-state conditions of the plasma are analyzed by the energy balance equation. The experimental measurements indicate that the electric field strength E decreases as the space charge decreases (ionization extent) increases. Although the plasma appears to be under steady-state conditions, it is not under local thermodynamic equilibrium conditions, i.e. T_e >T_r. In addition, the measurements indicate that there is a deficiency of electrons in the plasma (n_e < n_i), probably due to electron affinity processes; and the plasma has a small positive space charge.     

Substituent effects in the reaction of phenols with polyacrylonitrile radicals

Substituent effects in the reactivities of phenols towards polyacrylonitrile radicals have been studied in terms of Swain and Luptons' field (F_k) and resonance (R_k) components of the substituent parameters and the unique positional weighting factors (f_j and r_j) proposed by Williams and Norrington, with the aid of the following equation:

$\text{P}{}_{\mathrm{i}}\text{= x}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{F}{}_{\mathrm{k}}\text{+ ß}{}_{\mathrm{i}}\text{r}{}_{\mathrm{j}}\text{K}{}_{\mathrm{i}}\text{+ e}{}_{\mathrm{i}}\text{+}\text{P}\text{i}\text{o}$

P_i′s are the rate parameters, P_i⁰ being that for a standard reference compound. Two types of rate parameters have been employed—one, suggested by Simonyi, Tüdös and Pospisil (β) and another, suggested by us, (K), which is obtained from a plot of [Monomer]/(rate of polymerisation) vs [Phenol]. The correlations have been found to be quite satisfactory with both β and K. An attempt has been made to ascertain the nature of the transition state from the reaction parameters α_i and β_i; a dipolar transition state is suggested. The anomalous kinetic behaviour of hydroxy phenols has been discussed.     

Excess Isentropic Compressibilities for 1,3-Dioxolane or 1,4-Dioxane <Emphasis Type="Italic">+</Emphasis> Water <Emphasis Type="Italic">+</Emphasis> Formamide or N,N-Dimethylformamide Ternary Mixtures at 308.15 K

Speeds of sound, u_ijk, of 1,3-dioxolane or 1,4-dioxane (i) + water (j) + formamide or dimethylformamide (k) ternary mixtures and of their binary subsystems, u_ij, of 1,3-dioxolane or 1,4-dioxane (i) + formamide or dimethylformamide (j), and water (i) + formamide or dimethylformamide (j) have been measured over the entire composition range at 308.15 K. The experimental data have been used to evaluate the excess isentropic compressibilities of binary (κ_s^E)_ij and ternary (κ_s^E)_ijk mixtures using their densities calculated from molar excess volume data. The Moelwyn-Huggins concept [M. L. Huggins, Polymer 12, 389 (1971)] of interaction between the surfaces of components of a binary mixture has been employed to evaluate the excess isentropic compressibilities (using the concept of connectivity parameter of third degree of a molecule, ³ξ, which in turn depends on its topology) of binary mixtures, and this method has been extended to predict excess compressibilities of ternary mixtures. Values of (κ_s^E)_ij and (κ_s^E)_ijk have also been calculated by the Flory theory. It was observed that (κ_s^E)_ij and (κ_s^E)_ijk predicted by the Moelwyn-Huggins approach compare well with calculated and experimental values.     

Molar excess enthalpies of ternary mixtures of non-electrolytes

Molar excess enthalpies, H^E_ijk(T₁, x_i, x_j), for methylenebromide (i)+pyridine (j)+β-picoline (k); pyridine (i)+β-picoline (j)+cyclohexane (k); benzene (i)+toluene (j)+1,2-dichloroethane (k); benzene (i)+o-xylene (j)+1,2-dichloroethane (k); and benzene (i)+p-xylene (j)+1,2-dichloroethane (k) mixtures have been measured calorimetrically as a function of temperature and composition. The data have been analysed in terms of the Sanchez and Lacombe theory and using an approach employing the “graph theoretical” concept of connectivity parameters to characterize its pure components. It has been observed that the H^E_ijk (T, x_i, x_j) data calculated from the “graph theoretical” approach using ³ξ values based on δ^v considerations (that take into consideration the valency of individual atoms of the molecular graph constituent components) best reproduces the corresponding experimental H^E_ijk data.