Second-order Exchange Research Articles

Microscopic calculation of the imaginary Lane isospin potential W1.

Results of a determination of the imaginary Lane potential W/sub 1/ are presented, based on a second-order calculation of the 25 MeV /sup 48/Ca(p,n)/sup 48/Sc 0/sup +/ analog reaction. Inelastic (p,p/sup -1/), (n,n/sup -1/) and charge exchange (p,n/sup -1/) particle-hole intermediate nuclear states are used, and an exact treatment of the continuum and of second-order knockout exchange are included. A complex-energy intermediate-projectile Green's function is used to account for energy averaging of the incident beam. The sign of W/sub 1/ is shown to be positive, in agreement with phenomenological results, but the calculated magnitude is about a factor of 2 weaker than the Becchetti-Greenlees phenomenological potential. The contribution from an intermediate ground-state deuteron projectile is calculated in the zero-range approximation for pickup and stripping and is found to be about (1/3) of the contribution from the sum of all intermediate particle-hole states.

On some integrals arising in mathematical physics

This paper presents in the first section the exact evaluation of three single integrals relating to the dielectric behavior of two-dimensional electron plasmas. In the second section we present a procedure for reducing 3 d-dimensional integrals of the form: ∫∫∫ d q d p d kD(q)α( p+ k+ q)ƒ( p)[1−ƒ( p+ q)]ƒ(k)[1−ƒ( k+ q)] , where the vectors lie in d-dimensional space and ƒ denotes the Fermi function, to tractable form. The second-order exchange integral for a d-dimensional electron gas is taken as an example and is evaluated in closed form as a function of d.

Unified study of the lattice-mechanical properties of copper halide crystals

The copper halide crystals, namely CuI, CuBr, and CuCl, though they have predominantly ionic binding, possess zinc-blende structure, and crystals having this structure have, in general, rather significantly low ionicity. This peculiarity is attendant with a number of unusual properties of this group of solids, and the various attempts to understand them in terms of the existing models show certain problems. Recently we have developed a general energy expression for an assembly of ions occupying an arbitrary configuration from a microscopic analysis of the problem. Inspection of the energy expression shows that within the ${S}^{2}$ approximation (up to and including second-order exchange) the following distortions are allowed: the two types of short-range dipolar deformations apart from the long-range dipolar one, scalar, and quadrupolar deformations. A model based on this energy expression has been employed to make a unified study of the different lattice-mechanical properties of the three copper halides. The specific properties that we have tried to correlate are the cohesive energy, the elastic and the dielectric properties, and the dispersion of phonons with the same set of parameters for each crystal. Finally, the results for the different crystals and the possible sources of the remaining discrepancies are discussed.

Model calculation of the pressure derivative of the low-frequency dielectric constant of alkali halide crystals

The mechanism of polarization of insulators in a static electric field is still not clearly understood. The current situation presents an anomaly. The model which describes the high-frequency dielectric behavior quite well is found to be inadequate for the static case. In fact, with the same set of parameters within any single phenomenological model, it has not been possible to describe satisfactorily both the dielectric constants and their pressure derivatives. In search of a suitable model which can achieve this we have, in the present work, tried to analyze the mechanism of polarization from first principles. The interesting finding of this analysis is that there are two types of short-range polarization effects having entirely different origins: One of which (the second-order exchange interaction) contributes both to the static and high-frequency dielectric properties and the other (the first-order exchange interaction) to the static one only. We have suggested a model based on this analysis. The present calculation of the pressure derivative of the static dielectric constant of alkali halide crystals within a simplified version of this model together with our earlier calculation of the same for the high-frequency dielectric constant provides a fair overall description of the entire dielectric behavior of insulators. In addition, the present investigation also discusses the approximations which yield the well-known models for ionic solids which include polarization.

NMR study of intramolecular and intermolecular exchange processes in the Zn(II) ion-iminodiacetic acid system

Rate constants for intramolecular and intermolecular exchange processes involving the 1:2 complex formed between Zn(II) ion and iminodiacetic acid (IDA) have been determined by complete line-shape fitting of nuclear magnetic resonance (NMR) spectra. Experimental data obtained from spectra recorded at different temperatures on solutions containing metal ion and ligand in the 1:2 concentration ratio enabled the kinetics and activation parameters to be investigated ( ΔH ≠ = 11 ± 2 kcal/mol; ΔS ≠ = −9 ± 6 e.u. at 25°C) for the intramolecular acetate scrambling processes. Intermolecular ligand exchange reactions have been studied at 35°C on solutions containing ligand to metal ion concentration ratios ranging from 2.5 to 4.0 and evidence has been found for the occurrence of second-order exchange processes between the complex and both the unprotonated and monoprotonated ligand anions, whose rate constants have been evaluated (1.9 ± 0.2) × 10 4 M −1 sec −1 and (1.5 ± 0.2) × 10 3 M −1 sec −1, respectively).

Exact second Born exchange amplitude for elastic electron-hydrogen scattering

An exact calculation of the (plane-wave) second Born exchange amplitude for elastic electron-hydrogen scattering at 30 eV is reported. It is shown that exchange coupling to excited states is as important as exchange with the 1s state (at 30 eV) and that in the forward angular region it acts in the opposite direction to 1s exchange (at least in perturbation theory). A closure approximation to the second-order exchange amplitude is compared with the exact numbers. Poor agreement is obtained but it is not clear whether this results from a bad choice of average energy epsilon or is an intrinsic failing of the closure method. If the closure approximation can be made to work then a value of epsilon greater than -1/18 au would seem to be indicated. The Glauber approximation to the exchange amplitude is criticised on the basis that it involves using a closure approximation with epsilon = epsilon 1s=-1/2 au. An exploratory calculation shows that second-order exchange does exert a pronounced effect on the differential cross section.

Exact evaluation of the second-order exchange energy of a two-dimensional electron fluid

A new rigorous method is presented for the evaluation of the second-order exchange energy of a two-dimensional electron fluid. The resulting energy 0.2287 Ry is significant in the correlation energy, in contrast to the three-dimensional case. The present method is applicable to three dimensions.

Two-dimensional electron gas at finite temperature

The internal energy of a two-dimensional electron fluid was evaluated explicitly for low but finite temperature as a function of ${r}_{s}$, taking into consideration the first- and second-order exchange and ring diagrams. It is found that the energy varies almost quadratically with temperature but involves logarithmic terms, one of them being exactly the square of another. From the temperature-dependent energy, the specific-heat effective mass ${m}^{*}$ was determined. In an approximation $\frac{{m}^{*}}{m}=1+0.168{n}^{\ensuremath{-}\frac{1}{2}}\ifmmode\times\else\texttimes\fi{}{10}^{6}$, where $n$ is the density.

Modulated perturbation theory for molecular interactions

Modulated Murrell-Shaw-Musher-Amos perturbation theory is used in second order to calculate the interaction energy between two ground-state hydrogen atoms in the region of intermediate internuclear separations from 0 to 6 bohr. Using a small set of variationally optimized two-electron functions accounting for the majority of induction and for spherical and dipole contributions to dispersion, fairly accurate results are obtained for the interaction over the whole range of distances including the united atom. A marked reduction in the induction energy is observed in short range when second-order exchange is introduced.

Correlation energy of 2-D electrons

The correlation energy and the Fermi momentum of an electron gas in 2-D are evaluated explicitly as functions of density. The ring diagram and first- and second-order exchange contributions are treated. In comparison with the 3-D case, the kinetic energy for the same r s is approximately one-half and the exchange and correlation energies are somewhat larger. The ground state energy plotted against r s shows a minimum at around r s = 1.65 with a minimum value of −0.9858 Ryd. If the third-order ring contribution is added, the curve is shifted upward. The correlation energy is −0.6258 to order e 4. The third-order ringw contribution increases this value almost linearly with r s . The Fermi momentum decreases with r s due to the contribution. Different from the 3-D case, no ln r s term appears in the correlation energy within the approximation.

The influence of iodide ions on the phosphorescence of aromatic hydrocarbons

The phosphorescence decay functions and spectra of ethanol solutions of six aromatic molecules (AM), naphthalcne- hg, and - d 8, phenanthrene- h 10 and - d 10, and chrysene- h 12 and - d 12 were observed in the presence of different KI concentrations at 77 K. Moment analysis of the decay functions shows that the dependence of the AM phosphorescence rate parameter k PT on its distance from a heavy-atom (HA) iodide ion is described by a Dexter-type relation. The HA enhancement of k PT depends on the AM triplet energy E T. The HA-perturbed AM phosphorescence spectrum P(ν) includes not only an intense component, similar to the unperturbed AM spectrum P o(ν), due to the second-order exchange mechanism, but also a structured subspectrum P x(ν) red-shifted by 600–900 cm −1 relative to P o(ν). P x(ν) originates at an asymmetric out-of-plane CH vibrational mode, and it is attributed to a third-order HA perturbation, involving vibronic coupling.

Nuclear magnetic resonance investigations of carbonium ion intermediates. Part IV. Study of the intermediates involved in the reaction between carboxamides and Lewis acids (extended Vilsmeier reaction) by 1H and 13C nuclear magnetic resonance spectroscopy

Structural and dynamic 1H and 13C n.m.r. spectroscopy have been used to study the reaction between N-substituted carboxamides and Lewis acids. The complex kinetics are explained by the formation of new intermediates and selfcondensation compounds whose structure is demonstrated by 13C n.m.r. Hindered rotations around C–N bonds in the starting materials are also measured by 1H and 13C resonance with a view to discussing the nucleophilicity of carboxamides. A second-order exchange reaction is shown to occur in alkyl-substituted iminium salts and activation parameters are computed.

Many-body effects on the magnetic susceptibilities of metals

The paramagnetic and diamagnetic susceptibilities of an electron fluid at low but finite temperatures are determined by taking into consideration the free electron, first- and second-order exchange, and ring diagrams, without using the variational method adopted frequently in the past, but rather using a grand-ensemble method. In this method, the grand partition function is evaluated as a function of the magnetic field. The contribution from the ring diagrams is treated accurately with the finite momentum corrections to order ${q}^{4}$ and with the finite temperature corrections to order ${(\mathrm{kT})}^{2}$ to the eigenvalues. The paramagnetic susceptibility, when compared with the data on alkali metals, is found satisfactory. Both the ${r}_{s}$ and temperature dependences of the para- and diamagnetic susceptibilities are determined. When plotted against ${r}_{s}$, the paramagnetic susceptibility increases over the Pauli susceptibility. The diamagnetic susceptibility is also somewhat increased by interaction, although this depends on ${r}_{s}$. The Coulomb energy and the thermal energy are found to counteract in the manifestation of the magnetism. The energy associated with the external magnetic field is evaluated as a function of ${r}_{s}$. In comparison with the case without interaction, the field energy varies more slowly with ${r}_{s}$. The correlation part of the field energy is also determined.

The temperature and density dependences of the electronic specific heat

The specific heat of an electron fluid is determined for low but nonzero temperature and for high density based on the first- and second-order exchange graphs and the ring diagrams. The specific heat constant γ is found to increase at lower temperatures for r s of order 2 or 3. The specific-heat effective mass determined by experiments should increase accordingly. For very small r s the specific heat is below the ideal-gas value. For r s = 2, the deviation from a straight line of the specific-heat curve plotted against temperature is expected to start at around 2 K and increases at low temperatures. The temperature dependence is characterized by logarithmic terms.

Exchange interactions between Jahn-Teller ions. II

The problem of the Ham reduction of exchange interactions between orbitally degenerate magnetic ions has previously been studied using a first-order theory and singly occupied orbital triplets coupled to Eg local strains. The present paper extends the theory to second order. A number of new considerations enter, and in particular it is shown, by transformation theory, that to a good approximation the Jahn-Teller effects on intermediate excited electronic states can be neglected. It then follows that if the normal exchange parameters in the spin hamiltonian are available, in the absence of the Jahn-Teller effect, as expressions involving matrix elements, the corresponding expressions can readily be written down in the presence of the Jahn-Teller interaction. The Ham reduction is immediately apparent. However, expressions for the exchange parameters are not available, for exchange between orbitally degenerate ions has not been systematically treated previously. It has therefore seemed desirable to make the required extensions, using recently developed many-electron perturbation methods. The second half of the paper is therefore devoted to second-order exchange interactions, in the absence of the Jahn-Teller coupling. The problem is quite thoroughly discussed, beginning with a general hamiltonian. Many excited states are involved and many pairs of operators contribute. Fortunately the theory lends itself to a systematic treatment and reduces a complicated problem to one of manageable proportions. Attempts are made to relate the theory to previous work on exchange but this is not always possible because much of the previous work has used nonorthogonal orbitals and somewhat ill-defined hamiltonians. However, it is possible to recognise that some contributing processes are similar to those previously obtained in treating exchange interactions in nondegenerate ions. The whole topic of exchange in magnetic insulators is in a much tidier state as a result of this investigation.

Thermodynamic Properties of Nonideal Gases. II. The Strongly Ionized Gas

The thermodynamic equilibrium properties of strongly ionized multicomponent gas mixtures are investigated by application of the free-energy minimization method. For high-temperature regions where the Coulomb interaction is the dominant perturbation, the many-body partition function is developed from quantum cluster-expansion theory. The Coulomb free energy is given as the sum of the first- and second-order direct-interaction terms, plus the first three exchange-interaction terms. All five terms are exact in the classical limit, i. e., where Maxwell-Boltzmann statistics apply. The direct terms are correct for weak electron degeneracy and include wave-mechanical effects, while the first-order exchange term is exact for all degrees of degeneracy. The theoretical model is applied to multicomponent mixtures of hydrogen and of helium in the temperature range 50-2000 eV. The combination of the ring term plus higher-order terms significantly extends the region of applicability of the model over a classical electrostatic model. Specifically, electron degeneracy, the short-range cutoff in the ring term, and the three-rung ladder (second-order direct) term all operate to produce much less divergent thermodynamic results at a given density and temperature. First-order exchange is important even at moderate values of the electron-degeneracy parameter. The thermodynamic results indicate that evaluation of the exact quantum-mechanical ring term is essential for wider application of the perturbation-expansion theory, as is the development of a second-order exchange term for arbitrary degeneracy.

Hyperfine Splitting in a Metal of a Localized Moment

Phenomenological equations describing the hyperfine splitting of a localized electron resonance in a metal are studied. The transverse susceptibility χ+(ω) is obtained for the case of nuclear spin I=½, and its poles and residues are displayed. The corresponding eigenvectors are determined, enabling a physical generalization to arbitrary nuclear spin. The validity of the generalization procedure is verified by an examination of the generalized susceptibility. The conditions for the observation of the hyperfine splitting in a dilute magnetic alloy are detailed. Three distinct regions are found to be important, depending on the strength of the exchange coupling. Region (1), corresponding to a very weak exchange coupling, exhibits hyperfine splittings of the same character as in an insulator. The linewidth of each hyperfine component equals Δse+ΔsL, where Δse is the second-order localized-conduction electron exchange relaxation rate, and ΔsL the localized spin lattice relaxation rate. Region (2) corresponds to sufficiently larger values of the exchange coupling for a magnetic resonance bottle-neck to obtain, but for which ωhf/Δse≫1, where ωhf is the electronic hyperfine splitting frequency. In region (2) the hyperfine splitting is the same as in region (1), but the width of each component equals [2I/(2I+1)]· [Δse+ΔsL]. Region (3) corresponds to yet larger values of the exchange coupling, such that ωhf/Δse≪1. In this more strongly bottlenecked region the hyperfine split resonance is narrowed into a single line, and the resonance properties are those one would derive in the complete absence of a hyperfine interaction. Recent successful observations of the hyperfine splitting of a localized moment resonance for rare-earth alloys belong to category (1). Unsuccessful measurements in transition metal alloys indicate their character is appropriate to region (3). It is suggested that lower temperature measurements may shift some of these materials into region (2) (because Δse ∝T), allowing for a direct determination of the magnitude of the exchange coupling constant, even though the resonance system is still in the bottlenecked regime.

The additivity of second-order contributions to intermolecular potentials: calculations on the3Σu+state of H2

Variational calculations have been carried out on the 3Σ u + state of H2 in the region of the van der Waals minimum using a basis of states that introduce dispersion, induction and charge-transfer energies. The basis is constructed from orbitals with optimum exponents and leads to a total energy which is close to the best obtained from James-Coolidge type calculations. From this a meaningful subdivision of the total energy into dispersion, induction, second-order exchange and charge-transfer energies can be made, and the additivity of these energies examined. The sum of the dispersion and induction energies (calculated without exchange) and the charge-transfer energy, is close to the total stabilization energy deduced from the full variational calculation.

Discussion of the He-He interaction energy in the average energy approximation

Using the Unsold average energy method a double perturbation theory expression for the interaction energy through second order is obtained which includes intra-atomic correction terms arising from the use of trial wavefunctions to represent the non-interacting molecules. These formal expressions are applied to the ground state He-He interaction and results for the interaction energy are obtained that compare favourably with recent semiempirical and ab initio calculations. The Unsold calculations are used to investigate approximations that have been introduced into the average energy calculation by other workers, and as a model for discussing the relative importance of second-order charge overlap and exchange effects and the convergence of the multipole treatment of the interaction energy for this typical non-bonded interaction. The results illustrate the importance of knowing as many terms as possible in the R -1 expansion of the energy. Finally a portion of this work required the recalculation of the He-...

Dynamics of Localized Moments in Metals. II. Second-Order Exchange Effects

We have extended our previous microscopic treatment of the dynamic transverse susceptibility for a random array of localized spins in a metal to include terms of second order in the exchange coupling constant $J$. Lattice relaxation of the localized and conduction electrons is included, as before, in such a way as to ensure relaxation to the instantaneous local field. The results, in the limit of equal conduction-electron and localized-spin $g$ values and no lattice damping, reduce to the correct ("bottlenecked") limit. The linewidth for frequencies close to the localized-spin resonance frequency agrees with previous calculations. Bottlenecking of both the longitudinal (frequency-modulation) ($T_{2}^{}{}_{}{}^{\ensuremath{'}}$) and transverse (spin-flip) ($T_{1}^{}{}_{}{}^{\ensuremath{'}}$) parts of the localized-spin resonance linewidth is demonstrated for equal $g$ values and no lattice relaxation. Similarly, the line-width for frequencies close to the conduction-electron resonance frequency exhibits both $T_{2}^{}{}_{}{}^{\ensuremath{'}}$- and $T_{1}^{}{}_{}{}^{\ensuremath{'}}$-type terms, and again bottleneck effects are present. The results are compared with previous macroscopic treatments. It is demonstrated that it is unnecessary to introduce detailed balance conditions per se in the microscopic theory. The relation between the conduction-electron-hole relaxation width and the one-electron width calculated by Overhauser is examined in an appendix.