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The transient behavior of the phase-error probability density function of the first-order phase-locked loop (PLL) in the presence of additive white Gaussian noise is determined using numerical techniques. In addition, the modulo-2π probability density function and cycle-slipping density functions are found. The algorithm used in these calculations involves numerical integration of a modified form of the Fokker-Planck (F-P)equation. Results are shown for cases involving various signal-to-noise ratios (SNR's and initial conditions.

Sampling techniques provide a practical means of obtaining cross-correlation functions. In this paper, the correlation function is described by sums of the form Z = \begin{equation*}Z = \Sigma^{N}_{j=1}X_{j}Y_{j}\end{equation*}. A general expression is derived for the probability density function of the random variable Z under the condition that Xj and Yj are stationary, jointly Gaussian random processes with nonzero means and unit variances.

An experimentally determined electron density in metallic Be is used to calculate the one-body potential in density functional theory. The scheme, first proposed by Hohenberg and Kohn, is based upon the summation of an infinite subset of terms in a density gradient development for the exchange and correlation energyExc leading to an expression in the formExc[ϱ]=∫drɛxc(r), ɛxc =ɛxc0+∫dr′B(r′, ϱ(r)) · [ϱ(r + 1/2r′) − ϱ (r−1/2r′]2, and a corresponding expression for the one-body potentialVxc(r)=δɛxc/δϱ(r). A discussion is given of the nonlocal part arising from the summation of gradient terms. Attention is focussed on the possible analytic forms of the Fourier transform of the kernel,\(\hat B(q,k_F )\), which can be expressed uniquely in terms of the static dielectric functions e(q). It is emphasized that the singularities of\(\hat B(q)\) are of particular importance, and the results are sensitive to the model which is used. As a consequence of the lack of knowledge of the analytic form of\(\hat B(q)\) arguments are given in favour of a cut-off in this correction term at the Thomas-Fermi screening radius. The form of the first gradient correction in the present scheme can be compared with that in the exchange energy scheme of Hermanet al. However with the inclusion of correlation effects the coefficient of (▽ϱ)2 differs substantially from Herman’s.

We attempt to overcome one of the shortcomings in the Thomas-Fermi and related theories for atoms by connecting a quantum-mechanical electron density for the region near the nucleus to a density given by a statistical model. The joining may be done in a nonarbitrary manner: All parameters are determined. It suffices to use a very crude and easily calculated approximation to the density to start with, so the calculation is quite easy. The resulting electron density is in much better agreement with reality than that from the unmodified theories.

A method of approach to the correlation problem in a uniform electron gas, within the framework of density matrix theory, is discussed. (auth)

The paper contains an investigation of convergence and asymptotic behaviour of the generating functionals on mass shell in the Quantum Field Theory of one real, scalar, selfinteracting field. Also the relations between the Hilbert space fo entire functionals and the various spaces of exponentially bounded entire functionals are treated in some detail. There result various estimates for the functional series and double series representing vectors, operators and their functional derivatives.

We calculate one-electron energy levels in potentials derived from the solution of the atomic Thomas-Fermi-Dirac equation. A local exchange potential is used. Corrections are made to a previous theory, which joins a TFD density to a quantum mechanical electron density near the nucleus, where the Thomas-Fermi-Dirac density has an incorrect singularity. Use of a potential derived from this theory leads to improvement in inner-shell energies. Changes in the potential, suggested in the literature as being reasonable for valence electrons, do not always give the expected improvement. Nor does naively correcting for self-energy lead to improved one-electron energy levels in general.

Recent theories of Singwi et al. for the dielectric response beyond the random-phase approximation are examined in the density-functional formalism. Their approximation and another from the perturbation expansion in the electron-electron interaction are used to calculate the correction to the local-density approximation of the exchange energy in atoms. The latter gives better results.

The density matrix for a many-electron system has been examined with a view toward extending the statistical theory of Thomas and Fermi. The single-particle density matrix is expanded in terms of a power series in $\ensuremath{\hbar}$, according to a procedure developed here for operators consisting of a series of powers of the classical Hamiltonian, the zeroth-order terms corresponding to the Thomas-Fermi theory. Second-order terms in the expansion are identifiable in the expression for the energy: they correspond to exchange energy, the Weizs\"acker inhomogeneity correction for the kinetic energy and, possibly, correlation energy. Their presence follows from the expansion procedure, rather than from an a priori insertion into the theory.While the development here has been concerned mainly with the single-particle density function at the absolute zero of temperature, the procedure has been developed in sufficiently general terms to permit an evaluation to be made of the many-electron density matrix as a function of temperature.The results which have been obtained have been utilized to modify the theory of Fermi and Amaldi which has then been applied to the following atomic states: H ($^{2}S$), He ($^{1}S$), He ($^{3}S$), ${\mathrm{C}}^{++}$ ($^{1}S$), O ($^{1}S$). The disparity between the calculated energy for He ($^{3}S$) and the observed value is less than 6%, while for the remaining cases the disparity does not exceed 2.5%. The radial densities obtained correspond roughly to available quantum densities; they fail, however, to indicate any "shell-structure" nor to display the expected behavior for distances very close to, or very far from the atomic nucleus.

The energy-density functional proposed by Brueckner etal. is used to calculate the average nuclear-binding-energy surface on the $N\ensuremath{-}Z$ plane at $\ensuremath{\beta}$ stability \ifmmode\pm\else\textpm\fi{} about 40 units. The equibinding contours are compared with those predicted by recent liquid-droplet models. Normally, some disagreement begins about 15 units away from the region of known nuclei. The deviations towards lower binding are significant for neutron-rich transuranium (superheavy) nuclei. This would seem to indicate a physically important uncertainty in conventional masslaw extrapolations. The semiempirical dependence on neutron excess cannot be established very well, because of the narrowness of the region of known isotopes. The energy-density functional, on the other hand, incorporates the present knowledge about nuclear matter - including recent neutron-matter results. Our disagreement with other extrapolations, therefore, questions the validity of stability and formation (e.g., $r$-process) calculations based on conventional mass formulas.

In a previous paper, the validity of gradient corrections to the Dirac-Slater exchange energy was discussed within the density functional theory, for a local potential. This local potential, by definition, generated the exact charge density in the (infinite) system considered. The present work considers the full Hartree-Fock theory within the same density functional framework. In particular, using a perturbation expansion of the Dirac density matrix, we obtain the exchange energy as an expansion in the displaced charge Δ H . F . ( r ) = ρ H . F . ( r ) − ρ 0 , , p0 being the average electron density and p H.F. (r) the Hartree-Fock spatially dependent density. In contrast to the local potential theory, a basic integral equation must be solved to express the above relation in explicit form. No exact solution of this equation exists to date, but using the work of Kleinman and Langreth on the vertex function we are able to find an explicit approximation for the exchange energy to second-order in Δ H.F. The difficulties encountered in defining gradient series in a local potential theory are not removed in the full Hartree-Fock theory. However, it is possible to sum sub-series of gradient terms to infinite order to give a useful generalization of Dirac-Slater theory. Although it is known in metals that correlation effects have a major influence on the Hartree-Fock results, the results obtained here may well be useful in atomic and molecular problems with a sufficiently large number of electrons, as well as in non-metallic crystals. The influence of electronic screening in the case of metals is briefly referred to.

A density-functional formulation of magnetic states in metals is set up in which, initially, a spin-dependent external potential is switched on to the metal. The usual Hartree-Fock treatment of strong short-range correlations in an s -band is shown to follow very simply from the theory. Building up this Hartree-Fock theory for the lattice by switching on the interactions one site at a time suggests an approximate way of incorporating correlation effects into the density-functional theory. This is consistent with the philosophy that electron correlation effects are best treated via a picture in which the electron density, and, in magnetic metals, the spin density distributions are thought of as localized round atomic sites. In keeping with this philosophy, a new general magnetic instability condition for the lattice is proposed, which is essentially that for the one-site model and involves the density of states through the band, and not simply at the Fermi level, as with the Hartree-Fock condition. The properties of the Green function associated with this new instability condition are considered. While it is not possible, within the present framework, to resolve completely some ambiguity in higher-order terms of the Green function, nevertheless it is possible to set up a Green function which includes the band and the atomic limits correctly, has required symmetry properties, and exactly conserves states in the band. The density of states in this model does not show simple exchange in the magnetic metal but corresponds to deformation of the magnetic sub-bands by the electron interactions. It seems possible still that the inclusion of the long-range Coulomb correlations may, even in the non-magnetic case, after the basic character of the bands from that of a “cell” model.

Recently, correlation functions have had to be considered where both the reference waveform, which is usually the desired signal, and the input waveform are masked by different samples of additive noise. In this article, we derive the probability density function for the random variable \beta where \begin{equation} \beta = \sum_{i=1}^k (As_{i,x} + N_{i,x})(Bs_{i,y} + N_{i,y}). \end{equation} The s_{i,x} and s_{i,y} are the signal components, and N_{i,x} and N_{i,y} are samples of Gaussian noise. Exact expressions involving Bessel and Whittaker functions are given for several cases. Asymptotic expressions allow W(\beta) to be plotted when these exact expressions cannot be obtained or conveniently evaluated.

An integral equation for the phase density function is solved numerically to give the density of electron states and the average logarithmic decay of the envelope of the wave functions in a simple one-dimensional liquid model. The results are presented graphically to show the effect of varying amounts of disorder.

The response function of a uniform extended system of noninteracting fermions, and thus the energy of a weakly perturbed system of this kind, is computed exactly and in the von Weizsacker approximation; it is thus shown that (a) the original von Weizsacker coefficient leads to an upper bound to the energy which becomes asymptotically exact for short wavelength perturbations and (b) the coefficient reduced by a factor of 9 leads to a lower bound which becomes asymptotically exact for long waves. Generalisations of these results are discussed for the case when the fermions are interacting electrons.

The correlation energy (the exact energy minus the Hartree-Fock energy) of an electron gas with a high and slowly varying density is examined. The term proportional to the square of the density gradient is evaluated by the application of perturbation theory to the external field and of the random-phase (or high-density) approximation to the Coulomb interaction. This term has the form $\ensuremath{\Delta}{E}_{c}[\ensuremath{\rho}]=\ensuremath{\int}{d}^{3}xB(\ensuremath{\rho}(x)){|{\ensuremath{\nabla}}_{\ensuremath{\rho}}(x)|}^{2}$, where $\ensuremath{\rho}(x)$ is the electron density. $B(\ensuremath{\rho})$ is found, by summing the leading divergent diagrams, to be $[8.470\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}+O({\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}\mathrm{ln}\ensuremath{\rho})+O({\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}})]{\ensuremath{\rho}}^{\ensuremath{-}\frac{4}{3}}$ Ry, with the length measured in units of the Bohr radius. The role of the density gradient in the correlation energy problem of atoms is discussed.

From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.

The high resolution proton magnetic resonance spectra of himgravine and himbacine have been obtained in CHCl3 solution at 60 Mc/s. The spectra are consistent with only one of the two possible structures proposed. Other information with regard to relative orientation of groups within the molecule has also been obtained.

Se calculó teóricamente la energía superficial de Ag, Pb, Ge y Si de los sistemas formados por capas metálicas (Ag y Pb) en los semiconductores (Ge y Si), respectivamente para formar los sistemas Ag/Ge y Pb/Si. Para ello, se utilizó el funcional Perdew Wang (PW91) dentro de una aproximación DFT. Para modelar el bulk y las superficies (111) de Ag, Pb, Ge y Si se utilizó el modelo periódico de slab. Los modelos de los semiconductores se construyeron con 12, 11, 10, 9, 8, 7 y 6 capas, los de metales con 1, 2, 3, 4, 5 y 6 capas y para estudiar las monocapas metálicas en los semiconductores se empleó un modelo de 12 capas en total: SC12-nMn (donde SC es Ge o Si; M es Ag y Pb, respectivamente y n es el número de capas). Conforme a los resultados obtenidos, se concluyó que el metal modifica la energía superficial de los semiconductores, y éstos a su vez modifican la energía superficial del metal. Los valores de los sistemas SC12-nMn oscilan en forma similar a la de los semiconductores, aunque con valores intermedios entre el metal y el semiconductor. Estos resultados indican que si el número de monocapas metálicas en un semiconductor se puede controlar, también se puede controlar la energía de superficie del sistema.

A series representation is derived for a class of bivariate probability density functions. The representation uses orthonormal sets generated by the first-order density functions.

The kinetic energy density functional of electrons moving in a common effective one-body potential is calculated in a locally linear potential approximation and given in terms of electron density ρ and y ≡ ∣▽ρ∣ 1 2 ρ −2 3 for all ρ and y . The limits y → 0 and y → ∞ are discussed.

Probability density functions and the corresponding cumulative distribution functions for density and Velocity fluctuations at the surface of a thermionic emitter are discussed. In particular, it is shown that as far as velocity fluctuations are concerned, the appropriate probability density function is given by f(w | r_{c}) = (2 / \pi^{3/2}) \exp (-w^{2}) , where w is the reduced velocity. Some of the misunderstandings encountered in the technical literature are also discussed. It is shown that the mean energy of the emitted electrons is 3/2kT and not 2kT, as is claimed sometimes, the quantity 2kTJ s , where J s , is the saturation current, representing the magnitude of energy flow.

Experimental electric dipole moments are reported for a number of substituted benzaldehydes (o-, m-, and p-XC6H4CHO; X = NO2, CN, F, Br) as solutes in carbon tetrachloride or benzene at 25�C. Analysis of results yields information concerning the preferred conformations of thesemolecules.

The application of the Woodward-Hoffmann rules to thermal, intermolecular two-component cycloaddition reactions generates a set of 'symmetry favoured' pericyclic reactions. The problem of choosing, within this set, the most favoured reaction (i.e. periselectivity) is discussed. It has been found that a second order perturbational molecular orbital (PMO) treatment, employing Huckel eigenvectors and eigenvalues, and which is restricted to include only those terms containing highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals, gives results which are in satisfactory agreement with the available experimental data. The treatment has been extended to cover the effect of substituents on the preferred mode of cycloaddition.

A non-empirical valence-electron method suitable for molecular orbital calculations on large molecules containing heavy atoms is described. The method uses a minimum STO basis for the valence orbitals and accounts for core-valence interactions by means of a simple pseudopotential which affords corrections to the diagonal F matrix elements. All integrals involving one and two centres are included, and these are evaluated theoretically. The method produces electronic energies and electron density properties which are comparable with those of corresponding all-electron ab-initio calculations. In this paper two versions of the method are described, the basic non-parametric method and the parametric method which involves a simple modification for the two-centre integrals involving atoms of different size or electronegativity. Results of calculations on first-row diatomic molecules are described. It is hoped that the method will be suitable for calculations on larger molecules and molecules containing heavier atoms.

There has recently been a revival of interest in theories of liquid mixtures based on analytic equations of state for pure fluids. We have shown that the method used to determine the parameters of the pure-liquid equation of state has a significant effect on the excess thermodynamic properties obtained from such theories.

Densities of binary mixtures of ZnBr2 + KBr, CdI2 + NaI and CdI2 + CsI, have been measured by an Archimedian method over a wide range of temperature. Results have been expressed in terms of simple linear equations in temperature. Molar volumes for the systems have excess values in the ZnBr2 + KBr and CdI2 + CsI systems but not to such an extent in the CdI2+NaI system. The excess molar volumes are related to the possible formation of complex ions in the systems concerned.

M�ssbauer spectra are reported for a series of iron complexes with sulphur-containing Schiff base ligands. These tridentate ligands are based on the Schiff base formed by the condensation of dithiocarbazic acid and pyridine-2-carbaldehyde and form complexes with a metal-to-ligand ratio of 1 : 1 and 1 : 2. The M�ssbauer study confirms that most of these complexes contain iron(11) with examples of both high-and low-spin and one example of a spin crossover. Several iron(111) complexes were also studied. The quadrupole splittings show the anisotropy in bonding expected for metal atoms coordinated to different types of atoms within the same chelate ring. However, for some complexes the splittings are small, showing that low local symmetries need not produce large electric field gradients. In cases where it is applicable, the temperature dependence of the quadrupole splitting has been used to derive a splitting of the t2 and e orbitals produced by the non-cubic symmetry. For compounds of the form Fe(L)X2, where X is a halide and L the tridentate ligand, the Mossbauer results are best interpreted in terms of a five-coordinate structure rather than a six-coordinate bridged dimer.

The crystal structural transformation from aragonite-type KNO3I produces a variety of orientations as shown by polarized light microscopy. Extinction directions are listed and include one which is consistent with the reported orientation for CaCO3 and one consistent with another orientation previously reported for KNO3. Interfaces are produced parallel to the pseudohexagonal basal plane of 11, but the basal planes of the two phases do not necessarily form the interface.

A number of complexes in the series Cu(acac)(X), where X = OMe-, OEt-, OCH2Ph-, N3- and pz- (the pyrazolate anion), have been prepared and their magnetic properties determined down to liquid nitrogen temperatures. Also reported is the complex Cu2(acac)2(OH)(pz). In each case, the magnetism is interpreted in terms of spin coupling within a binuclear unit. All coupling is antiferromagnetic in nature, except for X = N3- which appears to be weakly ferromagnetic.

A study has been made of the bonding of some negative polyvalent iodine ions by means of CNDO/2 molecular orbital calculations. The theoretical equilibrium geometries, harmonic force constants, nuclear quadrupole coupling constants and charge distributions have been calculated. The results tend to confirm the Pimentel theory of bonding, whereby iodine achieves its polyvalence through the delocalization of electrons in the molecular orbital composed of the bonding p-orbitals of the atoms in the ions.

For symmetrical four-electron three-centre bonding units, the standard valence-bond (VB), delocalized molecular orbital (MO), increased-valence (IV) and non-paired spatial orbital (NPSO) representations of the electrons are Diagram O3, NO2- and CF2 with four π-electrons, and N3-, CO2 and NO2+ with eight π-electrons, have respectively one and two four-electron three-centre bonding units for these n-electrons. By means of Pople-Parr-Pariser type approximations, the MO, standard VB, IV and NPSO wave functions for these systems are compared with complete VB (or best configuration interaction) wave functions for the ground states. Similar studies are reported for the n-electrons of N2O. Further demonstration is given for the important result obtained elsewhere that the IV formulae must always have energies which are lower than those of the standard VB formulae, provided that the same technique is used to construct electron-pair bond wave functions. The extra stability arises because IV formulae summarize resonance between the standard VB formulae and long-bond formulae of the type Diagram As has been discussed elsewhere, the latter structure can make appreciable contributions to the complete VB resonance when its atomic formal charges are either zero or small in magnitude.If two-centre bond orbitals are used to construct the wave functions for the one-electron bond(s) and the two-electron bond(s) of IV formulae, then the IV and MO wave functions are almost identical for the symmetrical systems. Further numerical evidence is provided for this near-equivalence.

13C Substituent chemical shifts (scs) data for the C4 and C10 carbon atoms in 1-x-naphthalenes and 9-x-anthracenes respectively do not corroborate the proposal that the meso disposition in anthracene is a suitable model for the detection of the π-inductive or inductomesomeric effect.

From a study of the isotropic proton hyperfine interaction constants for a series of metal dithiocarbamates, and their redox behaviour, it is shown that for iron(111) and manganese(111) dithiocarbamates there is a simple relationship between E1/2 and the NCH2 proton hyperfine interaction constant in the 6Al state. Hence the redox process for these compounds is inferred to take place through the nitrogen atom.

In the conformational analysis of organic molecules by electric birefringence and Rayleigh scattering techniques, the value assigned to the aliphatic C-C bond, γC-C, is of fundamental importance. Thiscommunication discusses the implications of recent changes in the experimental parameters used to derive γC-C.