Energy–time uncertainty relations and time operators

It is proved that for any Hamiltonian in a separable Hilbert space, having a non-empty absolutely continuous spectrum, there exists a time operator densely defined in the subspace of absolutely continuous vectors. This result is obtained by using the Carbó-Dorca parameterized vector spaces and the spectral representation theorem for self-adjoint operators in Hilbert spaces. The restriction of the Hamiltonian to the absolutely continuous subspace and its time operator are incompatible. These results bring a completely new light on the energy–time uncertainty relations. The possible physical interpretations and related facts are also examined.     

Computing the Fukui function from ab initio quantum chemistry: approaches based on the extended Koopmans’ theorem

The extended Koopmans’ theorem is related to Fukui function, which measures the change in electron density that accompanies electron attachment and removal. Two approaches are used, one based on the extended Koopmans’ theorem differential equation and the other based directly on the expression of the ionized wave function from the extended Koopmans’ theorem. It is observed that the Fukui function for electron removal can be modeled as the square of the first Dyson orbital, plus corrections. The possibility of useful generalizations to the extended Koopmans’ theorem is considered; some of these extensions give approximations, or even exact expressions, for the Fukui function for electron attachment.     

Alternatives to Bazley’s special choice for eigenvalue lower bounds

Bazley’s special choice operator is a lesser operator to a positive perturbation of a self-adjoint semi-bounded operator that possesses an exactly soluble base eigenvalue problem. It allows the construction of an exactly soluble intermediate problem that gives eigenvalues not less than the base problem and not greater than the perturbed problem so that lower bounds to the eigenvalues of the perturbed operator are produced. This paper considers alternate derivations of Bazley’s special choice which lead to two alternate methods to determine eigenvalue lower bounds. One is simpler, but gives poorer bounds; the other is more difficult, but sometimes yields superior bounds. Lower bounds to the particle in a box model with a linear perturbation and lower bounds to the helium atom are calculated using the two methods introduced and are compared to those given with Bazley’s special choice.      

Two alternative derivations of Bridgman's theorem

One of the fundamental results of Dimensional Analysis is the so‐called Bridgman' s theorem. This theorem states that the only functions that may have dimensional arguments are products of powers of the base quantities of a given system of units. In this work, Bridgman's theorem is discussed and rederived in two different ways, one not involving calculus, and a second one based on a Taylor series expansion analysis. This revised version was published online in July 2006 with corrections to the Cover Date.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

Improvement of Weyl’s Inequality

Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized.     

Addition Theorem and Matrix Elements of Radiative Multipole Operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of the μth component of the angular momentum operator, L_μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator.     

On some properties of general linear operators

Some elementary properties of a general linear operator, which is defined on a Hilbert space but is not necessarily self-adjoint, are briefly reviewed: the classical canonical form, the bi-orthogonality theorem, the bivariational principle, the spectral resolution of the resolvent, and the properties of exponential forms. © 1993 John Wiley & Sons, Inc.     

Analytical evaluation for two-center nuclear attraction integrals over slater type orbitals by using Fourier transform method

In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s.     

A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces

Bearing in mind the insight into the Hohenberg–Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb’s proof as well as an extension of this theorem to finite subspaces. Contribution to the Serafin Fraga Memorial Issue.     

Calculation of intensity in the resonance Raman and two-photon absorption spectra of polyatomic molecules

A direct method for calculating the resonance Raman and two-photon absorption spectra of polyatomic molecules is described in detail The method is based on the adiabatic model and uses Herzberg-Teller’s approximation. Relations ruling out direct summation over vibrational quantum numbers of excited electronic states and representing the matrix elements of the Green function of a multidimensional oscillator as functions of vibration frequencies and Dushinsky transformation parameters are derived. The relations are convenient for constructing algorithms. Translated from Zhumal Struktumoi Khimii, Vol. 38, No. 2, pp. 248–255, March–April, 1997.     

Recent trends in trace element determination and speciation using inductively coupled plasma mass spectrometry

During the past decade, inductively coupled plasma mass spectrometry (ICPMS) has evolved from a delicate research tool, intended for the well-trained scientist only, into a more robust and well-established analytical technique for trace and ultra-trace element determination, with a few thousand of instruments used worldwide. Despite this immense success, it should be realized that in its ’standard configuration’– i.e. equipped with a pneumatic nebulizer for sample introduction and with a quadrupole filter – ICPMS also shows a number of important limitations and disadvantages: (i) the occurrence of spectral interferences may hamper accurate trace element determination, (ii) solid samples have to be taken into solution prior to analysis and (iii) no information on the ‘chemical form’ in which an element appears can be obtained. Self-evidently, efforts have been and still are made to overcome the aforementioned limitations to the largest possible extent. The application of a double focusing sector field mass spectrometer in ICPMS instrumentation offers a higher mass resolution, such that spectral overlap can be avoided to an important extent. Additionally, in a sector field instrument, photons are efficiently eliminated from the ion beam, resulting in very low background intensities, making it also very well-suited for extreme trace analysis. Also the combination of the ICP as an ion source and a quadrupole filter operated in a so-called ‘alternate’ stability region, an ion trap or a Fourier transform ion cyclotron resonance mass spectrometer allows high(er) mass resolution to be obtained. With modern quadrupole-based instruments, important types of spectral interferences can be avoided by working under ‘cool plasma’ conditions or by applying a collision cell. The use of electrothermal vaporization (ETV) or especially laser ablation (LA) for sample introduction permits direct analysis of solid samples with sufficient accuracy for many purposes. The application range of LA-ICPMS has become very wide and the introduction of UV lasers has led to an improved spatial resolution. Solid sampling ETV-ICPMS on the other hand can be used for some specific applications only, but accurate calibration is more straightforward than with LA-ICPMS. Limited multi-element capabilities, resulting from the transient signals observed with ETV or single shot LA, can be avoided by the use of a time-of-flight (TOF) ICPMS instrument. Finally, when combined with a powerful chromatographic separation technique, an ICP-mass spectrometer can be used as a highly sensitive, element-specific multi-element detector in elemental speciation studies. Especially liquid (HPLC-ICPMS) and – to a lesser extent – gas (GC-ICPMS) chromatography have already been widely used in combination with ICPMS. In speciation work, sample preparation is often observed to be troublesome and this aspect is presently receiving considerable attention. For GC-ICPMS, new sample pretreatment approaches, such as headspace solid phase microextraction (headspace SPME) and the purge-and-trap technique have been introduced. Also supercritical fluid chromatography (SFC) and capillary electrophoresis (CE) show potential to be of use in combination with ICPMS, but so far the application ranges of SFC-ICPMS and CE-ICPMS are rather limited. It is the aim of the present paper to concisely discuss the aforementioned recent ’trends’ in ICPMS, using selected real-life applications reported in the literature. Received: 30 November 1998 / Revised: 22 March 1999 / Accepted: 24 March 1999     

Some current problems in quantum chemistry as described by their key words and the International Journal of Quantum Chemistry

A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.     

Heavy-particle resonance phase shifts: an improved amplitude–phase formula

An improved amplitude–phase formula suitable for non-relativistic heavy-particle resonance phase shifts is derived. The present formula makes use of two amplitude functions instead of one for a central potential; an inner amplitude which is non-oscillatory in the well region of the effective potential, and an outer amplitude function which is non-oscillatory far away from the origin of the effective potential. The low-energy limit is discussed in connection with Levinson’s theorem. Numerical computations at resonance energies and graphical illustrations are presented. Numerical comparisons with an existing single-amplitude formula are made.     

Matrix elements between spin-bonded functions in a hole-particle formalism

Closed formulas are given for Hamiltonian matrix elements between spin-bonded functions in a holeparticle formalism. The derivation is based on Wick's theorem and the use of Jucys diagrams. The final formulas are only slightly more complicated than those for bonded functions in a particle formalism.     

The influence of the support functions on the quality of enhanced multivariance product representation

This paper presents recently developed Enhanced Multivariance Product Representation (EMPR) method for multivariate functions. EMPR disintegrates a multivariate function to components which are respectively constant, univariate, bivariate and so on in ascending multivariance. Although the EMPR method has the same philosophy with the High Dimensional Model Representation (HDMR) method, it has been proposed to get better quality than HDMR’s with the help of the support functions. For this purpose, we investigate the EMPR truncation qualities with respect to the selection of the support functions. The obtained results and a number of numerical implementations to show the efficiency of the method are also given in this paper.     

The geometry and spectra of hyperbolic manifolds

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceH ⁿ by discrete subgroups Г, i.e. ℳ =H ⁿ/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles. Research supported in part by NSF grant DMS93-07438