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Volume 62, Issue 5, 01 March 1975

First principles derivation of the effective valence shell Hamiltonian for large molecules
View Description Hide DescriptionWe construct an effective Hamiltonian for the valence shell of a large electronic system. The procedure begins by classifying the complete one electron space according to core, valence, and excited orbitals. An N−electron subspaceT_{ N } spanned by Slater determinants with a fixed set of N _{ c } core orbitals and different sets of N _{ v } = (N − N _{ c }) valence orbitals is defined. A canonical transformation on the Coulomb Hamiltonian is used to eliminate the interaction between T_{ N } and its orthogonal complement, ?_{ N }, thereby defining an effective N−electron Hamiltonian. This effective Hamiltonian is expanded in a cluster development of linked one−, two−, three−,⋅⋅⋅body operators in terms of which the conditions of having vanishing matrix elements between T_{ N } and ?_{ N } can be explicitly formulated. Then starting with this effective N−electron Hamiltonian we construct an equivalent Hamiltonian to operate in the space of antisymmetrized products of valence orbitals. To within a constant (times the identity on the valence space) this equivalent Hamiltonian—dubbed the valence shell Hamiltonian—has the same matrix elements between antisymmetrized products of valence orbitals as does the effective N−electron Hamiltonian on T_{ N }. This formalism is then applied to generate an effective pi−electron Hamiltonian for planar conjugated molecules. We conclude by deriving explicit formulas for the transformed dipole transition operators.

Photoionization and ion cyclotron resonance studies of the reaction C_{2}H_{4} ^{+} + C_{2}H_{4} → C_{3}H_{5} ^{+} + CH_{3}
View Description Hide DescriptionHigh pressurephotoionizationmass spectrometry has been employed to study the ion molecule reaction C_{2}H_{4} ^{+} + C_{2}H_{4} → C_{3}H_{5} ^{+} + CH_{3} at wavelengths between 700 and 1180 Å. Measurements have been made of the apparent ionization cross section of both C_{2}H_{4} ^{+} and C_{3}H_{5} ^{+} as a function of photon energy. The threshold energy for the reaction coincides with the ionization threshold of the parent ion at 10.51 eV. The reaction cross section decreases as the internal energy of the ionic reactant increases. Vibrational structure, observed in the C_{2}H_{4} ^{+}photoionization efficiency curve immediately above threshold (10.51−11.7 eV) permits a rough estimate of the change in reaction cross section with vibrational excitation. For the first three observed levels the change in cross section is ? 10%. However, for the fourth and fifth levels the cross section has dropped by approximately 25%. In the energy region between 11.7 and 12.1 eV a slight increase in the C_{2}H_{4} ^{+}photoionization efficiency curve appears which is not reflected in the C_{3}H_{5} ^{+} curve. A large increase in the photoionization efficiency curve of C_{2}H_{4} ^{+} occurs at 12.2 eV, the threshold of the ^{2} B _{3} first excited electronic state of the ion. A dramatic decrease is observed in the reaction probability of C_{2}H_{4} ^{+} formed at energies above 12.2 eV. The cross section for reaction in the excited ^{2} B _{3} state is ? 13% of the cross section for reaction in the lowest vibrational level of the ground electronic state. The observed decrease in reactivity is independent of repeller voltage at ion exit energies below 3 eV. Pressure studies reveal that after one or two collisions, the reactivity of C_{2}H_{4} ^{+} is re−established, suggesting that relaxation of internal excitation is an efficient process. The change in reactivity of C_{2}H_{4} ^{+} with internal energy is compared to the effects of translational excitation as determined using the techniques of ion cyclotron resonance spectroscopy.

Electron reactions and electron transfer reactions catalyzed by micellar systems
View Description Hide DescriptionThe kinetics of the reaction of hydrated electrons with pyrene, pyrene butyric acid, and pyrene sulfonic acid (PSA) have been investigated in aqueous solutions of cetyltrimethylammonium bromide (CTAB). With all three solubilizates the formation of the electron adduct (P^{−}) occurs very rapidly with rate constants ≳10^{11} M ^{−1}⋅sec^{−1}. These abnormally high rate constants are shown to be due to fast trapping of e _{aq} ^{−} in the positive potential field of the micelle and subsequent efficient penetration of electrons into the micellar interior. A similar enhancement was observed for electron transferreactions between CO^{−} _{2} and solutes solubilized in or on the micelle. For example CO^{−} _{2} readily transfers an electron to pyrene sulfonic acid on the surface of the micelle. This reaction does not occur in homogeneous solution but is catalyzed by the positive electrostatic surface potential. Addition of electrolyte drastically reduces the rate of e _{aq} ^{−} and CO_{2} ^{−} with solubilizates. The Debye−Hückel theory of electrolytes was invoked to elucidate the role of the charged micellar interface in facilitating the penetration of the electron into the micelle and promoting the electron transferreaction on the surface. Ion distributions were calculated via solution of a Poisson−Boltzmann equation, generalized to take into account the probable change of the microscopic dielectric constant in the vicinity of the micellar interface.

A molecular orbital investigation of chemisorption. II. Nitrogen on tungsten (100) surface
View Description Hide DescriptionThe relative bonding energies of nitrogen chemisorbed at three symmetric sites on a W(100) surface, represented by finite arrays of tungsten atoms [L. W. Anders. R. S. Hansen, and L. S. Bartell, J. Chem. Phys. 59, 5277 (1973)] were obtained by means of the extended Hückel molecular orbital theory (EHMO). The preferred site for nitrogen chemisorption was found to be the five coordination number (5 CN) site or the fourfold site with a tungsten atom below four tungsten atoms surrounding the nitrogen atom. The 5p orbital repulsive energy, in the case of hydrogen chemisorption, could be adequately approximated by the sum over pairs of empirical exponential repulsive terms; in the case of nitrogen chemisorption, this same method was approximately 10% in error at the equilibrium bond distance, and repulsive energies were therefore obtained from calculations including tungsten 5p orbitals but with smaller arrays.

Molecular and applied modulation effects in electron electron double resonance. IV. Stationary ELDOR of very slowly tumbling spin labels
View Description Hide DescriptionThe investigation of very slowly tumbling spin labels by stationary electron electron double resonance(ELDOR) is discussed. When a Zeeman modulation frequency of 270 Hz was employed, spectra which were independent of modulation frequency but not modulation amplitude resulted. Under such conditions, the ELDOR technique permits characterization of rotational processes with correlation times from 10^{−7} to 10^{−3} sec even though normal electron spin resonance(ESR)spectra are insensitive to motion in these regions, appearing to be superimposable with the ESRpowder pattern of a static random collection of molecules. Quantitative analysis of stationary ELDORspectra is accomplished employing a density matrix treatment that explicitly includes the interaction of the spins with the applied electromagnetic radiation and Zeeman modulation fields. The effect of molecular motion inducing random modulation of the anisotropic hyperfine and electron Zeemaninteractions can be calculated employing either an orthogonal eigenfunction expansion method or a transition rate matrix method. The present communication reports investigation of the rotational diffusion of the spin label probes 2,2,6,6−tetramethyl−4−piperidinol−1−oxyl and 17β−hydroxy−4′,4′−dimethylspiro− [5α−androstane−3,2′−oxazolidin]−3′−oxyl in s e c−butylbenzene. Experimental spectra are compared with computer simulations of spectra carried out for isotropic Brownian (limit of infinitesimal step size) and jump diffusion (arbitrary step size) models. Best fit is obtained for finite step sizes, i.e., 0.15 rad, and significant deviation of microscopic from macroscopic viscosity is observed at the highest viscosities.

Conformational contribution to the entropy of melting. I. Linear chain hydrocarbons
View Description Hide DescriptionAn experimental method is proposed and used to measure the thermal pressure coefficient γ as a function of volume at the melting temperature of crystalline substances. This method was applied to linear hydrocarbons with 11, 13, 15, 17, and 19 carbon atoms; values of the entropy increase due to volume expansion (ΔS _{ v }) and to conformational disorder on melting (ΔS _{ c }) were obtained.

Multimedia dispersion relation for surface electromagnetic waves
View Description Hide DescriptionWe have derived a general, n−media dispersion relation for surface electromagnetic waves propagating on isotropic layers with complex dielectric functions. The equation is presented in a convenient, compact form for ease of application.

Central‐force model for liquid water
View Description Hide DescriptionWe propose a new class of Hamiltonian models of liquid water based on resolution of the monomeric unit into three effective point charges. Interacting through central forces only, the three charges automatically assume the molecular structure. Two important effects are built into this model which have been neglected in similar attempts: intramolecular modes of vibration and the capacity for self−dissociation in the liquid phase. In addition, a large number of microsopic properties of water can be expressed in very simple terms for this representation: The pressure and internal energy, second virial coefficient, high−frequency elastic moduli, and dielectric function are discussed in explicit terms. A convenient algorithm for computing low−order quantum corrections (proportional to h/^{2}) to thermodynamic properties is given as well. To illustrate the general class of central−force models, we provide a concrete realization which has been determined by fitting phenomenological potentials to a nearly linear hydrogen bond of proper energy and dimer configuration. In order to elucidate the microscopic consequences of assuming central−force interactions in water, we have investigated the energy variation of small polymers (dimers and trimers) and the solvated proton near their minimum energy configurations. On a qualitative level, the results of these initial computations provide considerable encouragement for the view that water molecule interactions can be realistically approximated by linear combinations of central forces.

Self‐diffusion in a binary critical fluid
View Description Hide DescriptionThe self−diffusion coefficient of a large tagged particle in a binary critical fluid ’’bath’’ is calculated. The validity of the Stokes−Einstein law in the critical system, and the possibility of determining the critical behavior of η, the shear viscosity, from measurements of D via the Stokes−Einstein law are discussed. It is concluded that η cannot be directly deduced from D if the critical correlation length exceeds the tagged particle radius.

A new generalized expansion for the potential energy curves of diatomic molecules
View Description Hide DescriptionA new generalized expansion for the potential energy curves of diatomic molecules is proposed. It is given by where λ (p) = sgn(p) [1 − (R _{ e }/R)^{ p }], and contains both the Dunham and the Simons−Parr−Finlan (SPF) expansions as special cases corresponding to p = −1 and p = 1, respectively. In order to justify the new expansion, a perturbation theory is developed which yields the Born−Oppenheimer potential as a series identical in form to the new expansion. The perturbation is a purely kinetic−energy perturbation at R _{ e }. Prescriptions are given for obtaining both the expansion coefficients and the o p t i m a l value of p either from perturbation theory or from spectral data. In terms of spectral data p = −a _{1} −1. Applications of the new expansion to CO, HF, and 20 alkali halides indicate that it predicts dissociation energies in much closer agreement with experiment than the SPF expansion while maintaining the same quality of agreement with the RKR curve as the SPF expansion provides. A number of possible extensions are discussed including potential energy surfaces for polyatomic molecules, techniques for analytic continuation of the new expansion, and expansions for properties other than the potential energy.

Longitudinal diffusion of K^{+} ions in He, Ne, Ar, H_{2}, NO, O_{2}, CO_{2}, N_{2}, and CO
View Description Hide DescriptionLongitudinal diffusion coefficients, measured with a drift tube mass spectrometer, are reported for potassiumions in helium, neon, hydrogen, nitric oxide, oxygen, and carbon dioxide for E/N in the range 1−700 Td (E is the electric field strength, N the neutral gas number density, and 1 Td = 10^{−17} V⋅cm^{2}). These results, together with those previously reported for potassiumions in argon, nitrogen, and carbon monoxide, are compared with the theories of Wannier and of Mason, Whealton, and Viehland, which relate mobilities and diffusion coefficients.

Theory of chemically induced dynamic electron polarization. III. Initial triplet polarizations
View Description Hide DescriptionAn analysis of dynamical aspects of the CIDEP mechanism proposed by Wong e t a l. is given. This mechanism is based upon the formation of an excited triplet by intersystem crossing that populates the three triplet levels unequally. The subsequent rotational averaging of the initial population distribution coupled with the orientational effects of the zero field splitting is carefully treated in this work utilizing the stochastic Liouville equation in a manner closely analogous to that recently given for ESR line shapes and relaxation of slow−tumbling triplets. It is shown that the predicted CIDEP polarizations can indeed be very substantial, in agreement with Wong e t a l., but they will depend in general on the relative magnitudes of not only the zero field (D and E) and Zeeman terms (ω_{0}) but also the relevant reaction rates and the rotational tumbling times (τ_{ R }). A useful perturbation expression valid for D ^{2} ≲ (1/2)[ω_{0} ^{2} + τ_{ R } ^{−2}] is obtained which shows these details. Typical complete solutions, obtained numerically, are given for cases when this inequality does not hold.

The dynamics of dissociation of diatomic molecules and mass effect
View Description Hide DescriptionThe effect of the mass of the third body upon the dynamics of dissociation of Br_{2} at 6000°K was studied using 3D classical trajectory calculations. It was found that the dissociation cross section and the impact parameter, being sensitive functions of Br_{2} total energy, are also sensitive functions of the mass of the third body. However, the total equilibrium dissociation rate coefficient (one−way flux) varies but little with the mass of the third body. The latter finding is in qualitative agreement with the variational theory of reaction rates and the available experimental data.

Mechanism and types of explosive behavior in hydrogen–fluorine systems
View Description Hide DescriptionThe experimental data of Kapralova e t a l. [Kinet. Catal. (USSR) 10, 23 (1969)] have been modeled on a computer to show that two chain−branching reactions occur, H_{2}* + F_{2} → HF + H + F (1), HF* + F_{2} → HF + 2F (2). The asterisks indicate a vibrationally excited molecule. Rate coefficients derived for these reactions depend slightly on the computational model. For a model that is adiabatic and in which HF(0) is not absorbed on the wall, 6×10^{4} < k _{1} < 2×10^{5} and 5×10^{6} < k _{2} < 7×10^{6} cm^{3}/mol−s when all vibrationally excited molecules of HF and H_{2} are included in HF* and H_{2}*. For a model that includes flow of heat out of the reacting gas and also diffusion of HF(0) to the wall, the coefficients are 6×10^{4} < k _{1} < 1×10^{5} and 2×10^{6} < k _{2} < 5×10^{6} cm^{3}/mol−s. The computations show that quenched reactions take place during the time of experimental manipulation, so that the stability of H_{2}−F_{2}−O_{2} and H_{2}−F_{2}−O_{2}−M mixtures can be due in large measure to the HF produced during mixing. Double explosions and oscillatory reactions can also occur.

A Mössbauer effect study of ^{57}Fe in transition metal monoborides
View Description Hide DescriptionThe phase characteristics and nuclear hyperfine properties of 45 transition metal monoborides M _{1−x } Fe_{ x } B were studied by Mössbauer effect spectroscopy of ^{57}Fe. Transmission measurements with powders were completed in the temperature range 4.2−1000 K; additional scattering and 18 kOe applied field data were collected at 300 K. Measured^{57} Fe nuclear hyperfine parameters are discussed in terms of atomic bonding, charge transfer, and spontaneous magnetism in the borides. Within the context of the rigid band model, the isomer shift data for all boride phases indicate an electron transfer from boron to the metal 3d band. Similarly, the effective magnetic hyperfine field in ferromagnetic monoborides is proportional to the magnetization and also reveals a possible charge transfer from boron to metal. Quadrupole coupling constant results show a strong ionic contribution to the electric field gradient at the ^{57}Fe site in the monoborides. Measurements on FeB confirm the existence of two distinct structural modifications possessing different magnetic hyperfine fields but the same Curie temperatures. The defect (α) phase of FeB appears to be stabilized by the addition of manganese.

Ion clustering reactions in carbon monoxide
View Description Hide DescriptionIon−molecule reactions in carbon monoxide have been studied in a drift tube ion source of a quadrupole mass filter. Pressures to approximately 0.35 torr and temperatures from 174−280°K were employed. CO^{+} was found to add up to three molecules of CO. The first and second of these appear to be attached irreversibly, probably by true chemical bonds. Small amounts of water present as impurity contribute to the formation of several product ions, the most important of which is at m/e 46, which we interpret to be the formic acid ion. Rate constants of seven reactions have been determined at each temperature. No equilibria were detected.

Temporary negative ions and electron affinities of benzene and N‐heterocyclic molecules: pyridine, pyridazine, pyrimidine, pyrazine, and s‐triazine
View Description Hide DescriptionElectron transmission spectroscopy is used to study shape resonances (temporary negative ions) in benzene and some isolectronic N−heterocyclic molecules (pyridine, diazines, and s−triazine), in the energy range 0−6 eV. The lowest shape resonance in each of these molecules exhibits vibrational structure which is interpreted in all cases as the totally symmetric C−C stretch mode. The ground vibrational level of this lowest shape resonance is accessible by electron impact only in benzene and pyridine. Thus, their electron affinities can be determined from the present experiment (−1.15 eV for C_{6}D_{6} and −0.62 eV for C_{5}H_{5}N). Only excited vibrational levels are accessible in the diazines and s−triazine, indicating that the electron affinities for these molecules have positive values. For benzene, pyridine, and some other aromatic hydrocarbons, we compare the electron affinities established in the gas phase with the polarographic potentials established in the liquid phase and we find a linear relationship. Using this correlation in conjunction with the measured values of the polarographic potentials, we estimate the electron affinities for pyridazine (0.25 eV), pyrimidine (0 eV), pyrazine (0.40 eV) and s−triazine (0.45 eV).

Study of cross sections, oscillator strengths, generalized oscillator strengths, and atomiclike character of the molecular orbitals of formaldehyde for inelastic transitions to valence and Rydberg states
View Description Hide DescriptionOscillator strengths, generalized oscillator strengths, and cross sections are calculated using Hartree−Fock wavefunctions for spin allowed electronic transitions from the 2b _{2}(n) and 1b _{1}(π) molecular orbitals to the n b _{2} (n = 3, 4, and 5), n b _{1} (n = 2 and 3) and n a _{1} (n = 5,⋅⋅⋅,10) Rydberg series, and for the singlet → triplet 2b _{2}) 2b _{1}) transition. The first Born approximation is utilized in the calculation of f (K), and exchange contributions are included in the calculation of the total cross sections with the Born−Ochkur−Rudge modifications. Contour plots illustrating the radial and angular dependence of the orbitals are presented to substantiate the atomiclike character of the n b _{2} and n b _{1} orbitals and to demonstrate the mixture of atomiclike states which contribute to the n a _{1} molecular orbitals. The qualitative shape and trends in the occurence of extrema in the generalized oscillator strengths are discussed for transitions to Rydberg series whose excited molecular orbitals possess a given atomiclike character within the same point group symmetry. The oscillator strengths presented agree well with those determined both by electronic scattering techniques and with ultraviolet measurements in all but the π → π* excitation.

Determination of molecular pair correlation functions and size and shape parameters for diatomic liquids from x‐ray and neutron diffraction data
View Description Hide DescriptionLiquid state structure factors calculated from x−ray and neutron diffraction data for liquids made up of homonuclear molecules including N_{2}, O_{2}, Br_{2}, and Cl_{2} are analyzed. Theoretical calculations of the structure functions were carried out using hard−core diatomic representations of these molecules in conjunction with the blip function calculation of the radial distribution function, extended to deal with nonspherical molecules. The parameters of the hard−core potential were varied to give the best fit between experiment and theory. The results are compared with the intermolecular potential functions used in other studies of these systems. It is concluded that the diatomic Lennard−Jones potential gives satisfactory agreement between theory and a variety of experiments on N_{2} and O_{2}; ’’best’’ values for the parameters of the potential are suggested for these molecules. It is argued that a satisfactory theoretical treatment of the properties of Cl_{2} and Br_{2} should take account of the large quadrupolar interactions as well as the nonspherical shapes of these molecules.

Complex dependence of the positron annihilation rate on methane gas density and temperature
View Description Hide DescriptionAnnihilation rates have been measured for positrons in methane gas at temperatures of 22, 4.5, −28, and −50°C at densities in the range 6.6×10^{−4}−1.1×10^{−1} g/cm^{3}, in 22.5 and −28°C methane gas as a function of applied electric field strength and gas density, and in 22°C methane−argon gas mixtures as a function of argon concentration. A complex dependence of the slow positron annihilation rate on methane density and temperature was observed. The orthopositronium annihilation rate became temperature dependent at high densities. Both the positron−methane and the orthopositronium−methane annihilation rates were found to be independent of argon concentration in the gas mixtures. Electric fields produced a decrease in the slow positron annihilation rates, but had no measurable effect on the orthopositronium rates. The annihilation behavior of slow positrons is discussed in terms of several annihilation models, and empirical analyses of the data in terms of these models are presented. Only the single−stage formation of a positron−methane complex can account for all the observed phenomena.