Showing papers in "Physical Review A in 1994"

Theory of high-harmonic generation by low-frequency laser fields.

Abstract: We present a simple, analytic, and fully quantum theory of high-harmonic generation by low-frequency laser fields. The theory recovers the classical interpretation of Kulander et al. in Proceedings of the SILAP III Works hop, edited by B. Piraux (Plenum, New York, 1993) and Corkum [Phys. Rev. Lett. 71, 1994 (1993)] and clearly explains why the single-atom harmonic-generation spectra fall off at an energy approximately equal to the ionization energy plus about three times the oscillation energy of a free electron in the field. The theory is valid for arbitrary atomic potentials and can be generalized to describe laser fields of arbitrary ellipticity and spectrum. We discuss the role of atomic dipole matrix elements, electron rescattering processes, and of depletion of the ground state. We present the exact quantum-mechanical formula for the harmonic cutoff that differs from the phenomenological law Ip+3.17Up, where Ip is the atomic ionization potential and Up is the ponderomotive energy, due to the account for quantum tunneling and diffusion effects.

Exchange-correlation potential with correct asymptotic behavior

Abstract: In this work we analyze the properties of the exchange-correlation potential in the Kohn-Sham form of density-functional theory, which leads to requirements for approximate potentials. Fulfilment of these requirements is checked for existing gradient-corrected potentials. In order to examine the behavior of approximate potentials over all space we compare these potentials with exact Kohn-Sham potentials calculated from correlated densities using a newly developed iterative procedure. The main failures in the existing gradient-corrected potentials arise in the asymptotic region of the atom where these potentials decay too fast and at the atomic nucleus where the potentials exhibit a Coulomb-like singular behavior. We show that the main errors can be corrected by a simple potential in terms of the density and its gradients leading to considerably improved one-electron energies compared to the local-density approximation. For Be and Ne it is shown that the electron density is improved in the outer region.

Squeezed Atomic States and Projection Noise in Spectroscopy

Abstract: We investigate the properties of angular-momentum states which yield high sensitivity to rotation. We discuss the application of these ``squeezed-spin'' or correlated-particle states to spectroscopy. Transitions in an ensemble of N two-level (or, equivalently, spin-1/2) particles are assumed to be detected by observing changes in the state populations of the particles (population spectroscopy). When the particles' states are detected with 100% efficiency, the fundamental limiting noise is projection noise, the noise associated with the quantum fluctuations in the measured populations. If the particles are first prepared in particular quantum-mechanically correlated states, we find that the signal-to-noise ratio can be improved over the case of initially uncorrelated particles. We have investigated spectroscopy for a particular case of Ramsey's separated oscillatory method where the radiation pulse lengths are short compared to the time between pulses. We introduce a squeezing parameter ${\ensuremath{\xi}}_{\mathit{R}}$ which is the ratio of the statistical uncertainty in the determination of the resonance frequency when using correlated states vs that when using uncorrelated states. More generally, this squeezing parameter quantifies the sensitivity of an angular-momentum state to rotation. Other squeezing parameters which are relevant for use in other contexts can be defined. We discuss certain states which exhibit squeezing parameters ${\ensuremath{\xi}}_{\mathit{R}}$\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}$. We investigate possible experimental schemes for generation of squeezed-spin states which might be applied to the spectroscopy of trapped atomic ions. We find that applying a Jaynes-Cummings--type coupling between the ensemble of two-level systems and a suitably prepared harmonic oscillator results in correlated states with ${\ensuremath{\xi}}_{\mathit{R}}$1.

Binary-encounter-dipole model for electron-impact ionization

Abstract: A theoretical model, which is free of adjustable or fitted parameters, for calculating electron-impact ionization cross sections for atoms and molecules is presented. This model combines the binary-encounter theory with the dipole interaction of the Bethe theory for fast incident electrons. The ratios of the contributions from distant and close collisions and interference between the direct and exchange terms are determined by using the asymptotic behaviors predicted by the Bethe theory for ionization and for stopping cross sections. Our model prescribes procedures to calculate the singly differential cross section (energy distribution) for each subshell using the binding energy, average kinetic energy, and the differential dipole oscillator strengths for that subshell. Then the singly differential cross section is integrated over the ejected electron energy to obtain the total ionization cross section. The resulting total ionization cross section near the threshold is proportional to the excess energy of the projectile electron. We found that this model yields total ionization cross sections for a variety of atoms and molecules from threshold to several keV which are in good agreement (\ensuremath{\sim}10% or better on average) with known experimental results. The energy distributions also exhibit the expected shapes and magnitudes. We offer a simpler version of the model that can be used when differential oscillator strengths are not known. For the ionization of ions with an open-shell configuration, we found that a minor modification of our theory greatly improves agreement with experiment.

Teleportation of quantum states

Abstract: The recent result of Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)] of teleportation of an unknown quantum state is obtained in the framework of nonlocal measurements proposed by Aharonov and Albert [Phys. Rev. D 21, 3316 (1980); 24, 359 (1981)]. The latter method is generalized to the teleportation of a quantum state of a system with continuous variables.

Spontaneous emission near the edge of a photonic band gap

Abstract: We study spontaneous emission near the edge of a photonic band gap. Instead of a simple exponential decay in the vacuum, spontaneous emission displays an oscillatory behavior. A single photon-atom bound dressed state exhibits a fractional steady-state atomic population on the excited state. For a three-level atom we evaluate the spectral splitting and subnatural linewidth of spontaneous emission. In the presence of N-1 unexcited atoms we show that the collective time scale factor is equal to ${\mathit{N}}^{\mathrm{\ensuremath{\varphi}}}$, where \ensuremath{\varphi}=2/3 for an isotropic band gap and \ensuremath{\varphi}=1 or 2 for anisotropic two-dimensional or three-dimensional band edges, respectively.

Quantum theory of continuous feedback.

Abstract: A general theory of quantum-limited feedback for continuously monitored systems is presented. Two approaches are used, one based on quantum measurement theory and one on Hamiltonian system-bath interactions. The former gives rise to a stochastic non-Markovian evolution equation for the density operator, and the latter a non-Markovian quantum Langevin equation. In the limit that the time delay in the feedback loop is negligible, a simple deterministic Markovian master equation can be derived from either approach. Two special cases of interest are treated: feedback mediated by optical homodyne detection and self-excited quantum point processes.

Exact Kohn-Sham scheme based on perturbation theory

Abstract: An exact formal Kohn-Sham scheme is derived with the help of perturbation theory. Through the introduction of a basis set this Kohn-Sham scheme can be used to perform, in principle, exact Kohn-Sham calculations. As a demonstration, only zeroth- and first-order terms in the underlying perturbation theory are considered. As a result an exact basis set ``exchange-only'' method is obtained. The presented perturbation theory expansions of the exchange-correlation energy and potential may serve as a starting point for the development of new approximate exchange-correlation functionals based on Kohn-Sham orbitals and eigenvalues and may be used to check conventional exchange-correlation functionals. The formal structures of the ab initio and the introduced density-functional treatment of electronic systems are compared.

Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms.

Abstract: We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.

Theory of two-photon entanglement in type-II optical parametric down-conversion.

Abstract: The theory of the two-photon state generated by type-II optical parametric down-conversion is studied with emphasis on the space-time and polarization entanglement of the photons. Several experiments are reviewed that demonstrate various aspects of the quantum nature of this state. The theory of a different type of two-photon interferometer is presented.

From electron densities to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies.

Abstract: By developing our previous method [Phys. Rev. A 46, 2337 (1992); J. Chem. Phys. 98, 543 (1993)], we show how to calculate Kohn-Sham kinetic energies, orbitals, orbital energies, and exchange-correlation potentials, starting from accurate ground-state electron densities. In addition, given correct total energies, we also show how to obtain exchange-correlation energies. The scheme used is based on the Levy constrained-search method for determining the Kohn-Sham kinetic energy. In our preferred implementation, the total electron-electron repulsion is written as a Fermi-Amaldi term plus the rest, thereby assuring the correct long-range behavior of the exchange-correlation potential. Results are given for He, Be, Ne, and Ar. It is demonstrated that the exact exchange-correlation functional cannot be local.

Eigenvectors of two particles’ relative position and total momentum

Abstract: We give the explicit form of the common eigenvectors of the relative position ${\mathit{Q}}_{1}$-${\mathit{Q}}_{2}$ and the total momentum ${\mathit{P}}_{1}$+${\mathit{P}}_{2}$, of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing ${\mathit{Q}}_{1}$-${\mathit{Q}}_{2}$ and ${\mathit{P}}_{1}$+${\mathit{P}}_{2}$, are derived by using the technique of integration within an ordered product of operators.

Dark and bright photovoltaic spatial solitons

Abstract: Dark (bright) planar spatial solitons are predicted for photovoltaic photorefractive materials when the diffraction of an optical beam is exactly compensated by nonlinear self-defocusing (focusing) due to the photovoltaic field and electro-optic effect. These solitons may have steady-state irradiances of microwatts to milliwatts per square centimeter and widths as small as 10 [mu]m in lithium niobate. Optical control is provided by incoherent illumination, and the nonlinear index of a dark soliton may be used to trap a bright soliton by rotating the plane of polarization of the soliton field.

Accurate Exchange-correlation Potentials and Total Energy Components for the Helium Iso-electronic Series

Abstract: Starting from exceedingly accurate many-body wavefunctions, we have constructed essentially exact densities, exchange-correlation potentials and components of the total energy for helium and twoelectron ions. These density functional results are compared to the corresponding quantities obtained from a variety of commonly used approximate density functionals, namely the local density approximation and various generalized gradient approximations in order to test the accuracy of the approximate functionals. Although the generalized gradient approximations yield improved energies compared to the local density approximation, the exchange and correlation potentials (especially the latter) obtained from the generalized gradient approximations are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r4) is found to agree with theoretical predictions. The short range behaviour of the exchangecorrelation potential is very close to quadratic. The prospects for the improved generalized gradient approximations are discussed. Keywords: density functional theory, exchange-correlation potential, generalized gradient approximation

Teleportation of an atomic state between two cavities using nonlocal microwave fields

Abstract: Implementing the ideas of Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], we present an experimentally feasible scheme for the teleportation of an unknown atomic state between two high-Q cavities containing a nonlocal quantum superposition of microwave field states. This experiment provides alternative tests of quantum nonlocality involving high-order atomic correlations.

Quantum noise reduction by radiation pressure.

Abstract: We show that a linear Fabry-P\'erot cavity with an oscillating end mirror can be used for quantum noise reduction. For a high-quality factor of the mechanical oscillator the output quantum fluctuations of the monochromatic light beam can be significantly squeezed at a frequency very close to that of the impinging light. The analysis is performed by taking into account the coupling of the system with the external world.

Quantum-noise reduction using a cavity with a movable mirror

Abstract: Because of radiation pressure, an optical cavity with harmonically bound mirrors has an intensity-dependent length and behaves as an effective Kerr medium. We determine the quantum fluctuations of the field reflected by such a cavity, taking into account the input field fluctuations and the mirror Brownian motion. In the regions of parameter space close to bistability turning points, we obtain a significant quantum-noise reduction effect.

Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium

Abstract: We study the quantized field in a one-dimensional electromagnetic resonant cavity. The cavity contains a linear and lossless dielectric medium with frequency-independent polarizability. The dielectric permittivity is an externally prescribed function of both the space and the time. We also allow one of the cavity's mirrors to move in a given trajectory. Unlike other previous studies on the same system, we formulate an effective Hamiltonian so that the dynamics of the cavity field can be described in the Schr\"odinger picture. The effective Hamiltonian is quadratic in structure, therefore two-photon generation from the vacuum state can occur. We also discuss the case of resonant behavior of the system.

Dispersion coefficients for alkali-metal dimers.

Abstract: Knowledge of the long-range interaction between atoms and molecules is of fundamental importance for low-energy and low-temperature collisions The electronic interaction between the charge distributions of two ground-state alkali-metal atoms can be expanded in inverse powers of R, the internuclear distance The coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations

Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom

Abstract: The position- and momentum-space entropies of the isotropic harmonic oscillator and the hydrogen atom in D dimensions are shown to be related to some entropy integrals which involve classical orthogonal polynomials. These integrals are exactly calculated for Chebyshev polynomials and only in an approximate way for Gegenbauer polynomials. The physical entropies are explicitly obtained in the ground state and in a few low-lying excited states. Finally, the dimensionality dependence of the ground-state entropies of the two above-mentioned quantum-mechanical systems is analyzed (numerically) and the values of the entropies in a large class of excited states of the D-dimensional (D=1,2,3) harmonic oscillator and hydrogen atom are tabulated and discussed.

Photon statistics of a cavity-QED laser: A comment on the laser-phase-transition analogy.

Abstract: We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the ``system size'' and show that the concept of laser threshold is well-defined in the ``thermodynamic'' limit, ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.

High-harmonic generation and correlated two-electron multiphoton ionization with elliptically polarized light.

Abstract: The strong ellipticity dependence of correlated two-electron multiphoton ionization of neon and of the high-harmonic emission from argon (harmonic N=21) and neon (N=41) are reported. These measurements suggest a common underlying mechanism and are quantitatively consistent with a recently developed Keldysh-like model of high harmonic generation which treats the interaction between a newly freed electron and the ion core.

Assessment of Kohn-Sham density-functional orbitals as approximate Dyson orbitals for the calculation of electron-momentum-spectroscopy scattering cross sections.

Abstract: One of the principal advantages of electron momentum spectroscopy (EMS) is that peaks in the binding-energy spectrum can be assigned with greater certainty than in photoelectron spectroscopy, through a comparison of the EMS triple-differential cross sections with the theoretically calculated spherically averaged momentum distributions (MD's) of Dyson orbitals. While the target Hartree-Fock approximation is commonly used to calculate the Dyson orbital MD's for this purpose, a computationally less demanding method would allow the advantages of EMS to be extended to larger molecules. This paper considers the use of Kohn-Sham density-functional theory for this purpose. Although Dyson orbitals are not among the quantities that can be calculated exactly (in the limit of the exact exchange-correlation functional) within the framework of Kohn-Sham density-functional theory, the Kohn-Sham equation can be regarded as an approximate form of Dyson's quasiparticle equation, with a local self-energy. The well known shortcomings of this approach for estimating ionization potentials and band gaps do not a priori imply a corresponding problem with the orbitals. After discussing these formal considerations, we introduce the ``target Kohn-Sham approximation'' as a means of approximating Dyson orbitals by Kohn-Sham orbitals. The quality of this approximation for the calculation of MD's is assessed by comparison with high-quality configuration-interaction calculations, the target Hartree-Fock approximation, and experiment, for several small molecules. The quality of the target Kohn-Sham approximation MD's is found to be comparable to that of the MD's from the target Hartree-Fock approximation, with evident practical implications for EMS.

Wave-field phase singularities: The sign principle

Abstract: Phase singularities (topological charges, dislocations, defects, vortices, etc.), which may be either positive or negative in sign, are found in many different types of wave fields. We show that on every zero crossing of the real or imaginary part of the wave field, adjacent singularities must be of opposite sign. We also show that this ``sign principle,'' which is unaffected by boundaries, leads to the surprising result that for a given set of zero crossings, fixing the sign of any given singularity automatically fixes the signs of all other singularities in the wave field. We show further how the sign of the first singularity created during the evolution of a wave field determines the sign of all subsequent singularities and that this first singularity places additional constraints on the future development of the wave function. We show also that the sign principle constrains how contours of equal phase may thread through the wave field from one singularity to another. We illustrate these various principles using a computer simulation that generates a random Gaussian wave field.

Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms

Abstract: The general theory of quantum angular momentum is used to derive the unique Wigner distribution function for arbitrary angular-momentum states. We give the explicit distribution for atomic angular-momentum Dicke states, coherent states, and squeezed states that correspond to a collection of N two-level atoms. These Wigner functions W(\ensuremath{\theta},cphi) are represented as a pseudoprobability distribution in spherical phase space with spherical coordinates \ensuremath{\theta} and cphi on the surface of a sphere of radius \ensuremath{\Elzxh} \ensuremath{\surd}j(j+1) where j is the total angular-momentum eigenvalue.

Definition of a laser threshold.

Abstract: We propose that the threshold of a laser is more appropriately described by the pump power (or current) needed to bring the mean cavity photon number to unity, rather than the conventional ``definition'' that it is the pump power at which the optical gain equals the cavity loss. In general the two definitions agree to within a factor of 2, but in a class of microcavity lasers with high spontaneous emission coupling efficiency and high absorption loss, the definitions may differ by several orders of magnitude. We show that in this regime the laser undergoes a transition from a linear (amplifier) behavior to a nonlinear (oscillatory) behavior at our proposed threshold pump rate. The photon recycling resulting from the high spontaneous emission coupling efficiency and high absorption may in this case result in lasing without population inversion, and coherent light is generated via ``loss saturation'' instead of gain saturation. This mechanism for lasing without inversion is very different from lasing without inversion using a radiation trapped state.

All-optical versus electro-optical quantum-limited feedback

Abstract: All-optical feedback can be effected by putting the output of a source cavity through a Faraday isolator and into a second cavity which is coupled to the source cavity by a nonlinear crystal If the driven cavity is heavily damped, then it can be adiabatically eliminated and a master equation or quantum Langevin equation derived for the first cavity alone This is done for an input bath in an arbitrary state, and for an arbitrary nonlinear coupling If the intercavity coupling involves only the intensity (or one quadrature) of the driven cavity, then the effect on the source cavity is identical to that which can be obtained from electro-optical feedback using direct (or homodyne) detection If the coupling involves both quadratures, this equivalence no longer holds and a coupling linear in the source amplitude can produce a nonclassical state in the source cavity The analogous electro-optic scheme using heterodyne detection introduces extra noise which prevents the production of nonclassical light Unlike the electro-optical case, the all-optical feedback loop has an output beam (reflected from the second cavity) We show that this may be squeezed, even if the source cavity remains in a classical state

Completeness and orthogonality of quasinormal modes in leaky optical cavities.

Abstract: It is shown that for the scalar analog of electrodynamics in one dimension, the quasinormal modes of a leaky cavity form a complete set inside the cavity, provided the cavity is defined by a discontinuity in the refractive index. This condition is sufficiently general to apply to a number of interesting examples. The quasinormal modes are also orthogonal under a modified definition of the inner product. The completeness and orthogonality hold even though the cavity is not a Hermitian system by itself. These properties allow the discrete quasinormal modes to be used as the basis for dynamics of the scalar wave in the cavity.

Dispersionless, highly superluminal propagation in a medium with a gain doublet

Abstract: In the region between two lines of a doublet of an inverted medium, there exists a point of zero group-velocity dispersion, where highly superluminal effects may be observable without significant gain, loss, distortion, or broadening. The results of this group-velocity analysis hold for sufficiently narrow-band, analytic pulses, and do not constitute a violation of causality, although the group, ``signal,'' and energy velocities as defined by Sommerfeld and Brillouin may all exceed c or even become negative. No sharp disturbance in the pulse (a real signal) could propagate faster than light, but the scheme offers an unusual noiseless amplification scheme for the leading edge of a pulse, both at the classical and at the single-photon level.

Obtaining a gradient-corrected kinetic-energy functional from the perdew-wang exchange functional

Abstract: Lee, Lee, and Parr (LLP) have shown that the kinetic energy can be written in the same form as Becke's exchange energy. This conjecture of LLP has been generalized to another exchange functional, namely, the Perdew-Wang exchange functional. As demonstrated by Lee and Parr, the exchange energy can be written K=\ensuremath{\pi}FFs\ensuremath{\Gamma}(r,s)drds with \ensuremath{\Gamma}(r,s)=\ensuremath{\Vert}\ensuremath{\gamma}(r,s)${\mathrm{\ensuremath{\Vert}}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}/${\mathit{n}}^{2}$(r), where \ensuremath{\Vert}\ensuremath{\gamma}(r,s)${\mathrm{\ensuremath{\Vert}}}^{2}$\ifmmode\bar\else\textasciimacron\fi{} is the spherical average of \ensuremath{\Vert}\ensuremath{\gamma}(r,s)${\mathrm{\ensuremath{\Vert}}}^{2}$. Using the generalization of LLP's conjecture, it is demonstrated that \ensuremath{\Gamma}(r,s)= ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{s}2}$/\ensuremath{\beta}(r)+a[${\mathit{s}}^{4}$/${\mathrm{\ensuremath{\beta}}}_{0}^{2}$(r)]${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{s}2}$/${\mathrm{\ensuremath{\beta}}}_{0}$(r), a=const, ${\mathrm{\ensuremath{\beta}}}_{0}$(r)=5[3${\mathrm{\ensuremath{\pi}}}^{2}$n(r)${]}^{\mathrm{\ensuremath{-}}2/3}$. At zeroth order, \ensuremath{\beta}(r)=${\mathrm{\ensuremath{\beta}}}_{0}$(r), the function \ensuremath{\Gamma}(r,s), gives exactly the modified Gaussian approximation proposed by Lee and Parr.