A differential identity for Green functions

If P is a differential operator with constant coefficients, an identity is derived to calculate the action of e ^P on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a hierarchy of Green functions. The identity is first derived for scalar fields and the standard hierarchy is recovered. Then the case of fermions is considered and the identity is used to calculate the generating function for the Green functions of an electron system in a time-dependent external potential.     

A note on the generalized Rayleigh equation: limit cycles and stability

The aim of the present paper is to give an analytical proof on the existence and stability of the limit cycles in the generalized Rayleigh equation, which models diabetic chemical processes through a constant area duct where the effect of heat addition or rejection is considered, ${\frac{d^{2}x}{dt^{2}}+x = \varepsilon(1-(\frac{dx}{dt}) ^{2n})\,\frac{dx}{dt}}$ where n is a positive integer and ε a small real parameter. The main tool used for it is the averaging theory.     

Binomial Green functions

A straightforward method is developed for deriving analytic Green functions based on the binomial expansion theorem. The utility of the technique is clearly demonstrated by its application to several tight‐binding models. The simplicity of the approach gives rise to the prospect of wide applicability. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 1–4, 2000     

The r-centroid approximation and generalized functions

The r-centroid approximation implies that molecular vibrational wavefunction products should behave similarly to relatively high sequence members of generalised functions.     

Generation of generalized branching diagram spin functions by Schmidt orthogonalization of spin-paired functions

The generalized branching diagram (GBD ) spin representation is defined as the method of sequentially coupling together a number of subsystem spin eigenfunctions using the general rules of angular momentum coupling. It is shown that any GBD representation may also be obtained by Schmidt orthogonalizing a set of cannonical spin–paired (SP ) functions, provided the SP basis is suitably ordered. The ordering procedure used is well suited to computer implementation. This is a generalization of results known in the literature for the Yamanouchi–Kotani and for the Serber spin representations.     

A note on electro-optic effects

Tinoco's second-order electro-optic effect in isotropic fluids is discussed for the case of very high electric fields. The possibility of observing a linear electro-optic effect in oriented molecules of certain symmetry types is also analyzed.     

A note on generalised time-lags

Derivations of an expression for the generalised outgoing ‘adsorption’ time-lag pertaining to transport through a slab membrane made by [Rutherford and Do, Review of time lag permeation technique as a method for characterisation of porous media and membranes, Adsorption 3 (1997) 283] and by [Ash et al., Sorption and surface flow in graphitized carbon membranes. II. Time-lag and blind pore character, Proc. Roy. Soc. London, Ser. A, 304 (1968) 407] are not in agreement. Causes for the discrepancy are identified. Two extensions of the generalised procedure are given with particular attention being paid to ΔL, the time-lag difference.     

A note on atomic density

In atomic systems, electron density has a simple finite expansion in spherical harmonics times radial factors. The difficulties in the calculation of some radial factors are illustrated in the low‐lying states of the carbon atom. Single‐particle methods such as Hartree–Fock and approximate density functional theory cannot ensure the correct expansion of the density in spherical harmonics. Wave‐function methods are appropriate but, as some expansion terms are entirely due to correlation, these methods only will give correct results for high‐quality variational functions. Using full‐configuration integration (CI), all the terms predicted by the theory appear and are not negligible but the convergence of the term due to correlation toward its correct value is uncertain even for very large CI spaces. © 2012 Wiley Periodicals, Inc.     

Green functions in the theory of semiconductor surfaces

Electronic properties of a few technologically important semiconductor surfaces, explored in surface Green function calculations, are presented and briefly discussed in comparison with experimental data from high-resolution surface spectroscopy. The emphasis is on results of first-principles calculations employing the local density approximation or the generalized gradient approximation of density functional theory. The systems addressed comprise of the prototype surfaces of the elemental semiconductors diamond and Si, as well as the group IV compound semiconductor SiC. The examples show that surface Green function calculations, as performed by Maria St licka and Sydney Davison in their early work on the surfaces of model systems, such as linear monoatomic chains or the Kronig–Penney model, can nowadays be applied to efficiently evaluate electronic properties of real surfaces. The results of such ab initio Green function calculations are found to be in very good agreement with experimental data.     

A note on the partitioning technique

A new derivation of the wave operator of the partitioning technique is given. Furthermore this approach is applied to derive wave operators for the inverse hamiltonian and in general for functions of the hamiltonian.     

A note on general core-valence orthogonality

A method of generating a set of basis functions, which satisfy the condition of strong orthogonality, is developed.     

A note on time-dependent variattonal principles

It is demonstrated that the least action time-dependent variational principle suggested by Kerman and Koonin is equivalent to Frenkel's variational principle. A set of equations suitable for the variational treatment of a number of time-dependent problems has been derived in the course of this analysis.     

A note on sorption measuring instruments

Gas sorption phenomena can be used to characterise porous solids and dispersed materials. Usually isotherms of nitrogen and noble gases like He, Ar, Kr are measured at low temperatures (77–90 K). Other gases and vapours like water, CO₂ and benzene are used at near ambient conditions of pressure and temperature. From the amount of gas adsorbed on the (external or internal) surface characteristic quantities like the specific surface area, specific pore volume and pore spectrum of the material are derived by standardised methods. Experimental techniques most often used are the carrier gas, the volumetric/manometric and the gravimetric method. A comprehensible overview of today's available instruments, their advantages and drawbacks is given.This revised version was published online in November 2005 with corrections to the Cover Date.     

A note on equivariant Euler characteristic

We give a new equivariant cohomological characterization of the equivariant Euler characteristic of aG-simplicial set as defined by Brown. This implies in particular that the equivariant Euler characteristic is aG-homotopy invariant.     

Theory of chirality functions,generalized for molecules with chiral ligands

The theory of chirality functions described in a previous publication is generalized to allow for chiral ligands. In the earlier theory, all symmetry operations of the molecular frame could be thought of as permutations of the ligands among the sites; in the present work, improper rotations not only permute the ligands, but convert them into mirror images. The group that generates all isomers from a given ordered molecule belonging to a frame with n sites is now the hyperoctahedral group of order 2ⁿ n! consisting of all possible combinations of permutations and site reflections. The representation theory of these groups is described, and applied to the problem of constructing qualitatively complete chirality functions, and of deciding which ligand partitions, and which isomer mixtures, are chiral. It is found useful to classify chiral representations of the covering group as ligand specific and class specific. The ligand specific representations describe chiral properties which are common to all frames and arise purely from the chirality of the ligands, while the class specific representations describe the chiral properties of the frame. A number of examples are explicitly worked out.