QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H–Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree–Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree–Fock polarizabilities. The mean...

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Property-optimized gaussian basis sets for molecular response calculations.

Have you ever felt that the very title, Progress in Optics, conjured an image in your mind? Don’t you see a row of handsomely printed books, bearing the editorial stamp of one of the most brilliant members of the optics community, and chronicling the field of optics since the invention of the laser? If so, you are certain to move the bookend to make room for Volume 16, the latest of this series. It contains seven chapters on widely diverging topics, written by well-known authorities in their fields. These are: 1) Laser Selective Photophysics and Photochemistry by V. S. Letokhov, 2) Recent Advances in Phase Profiles (sic) Generation by J. J. Clair and C. I. Abitbol, 3 ) Computer-Generated Holograms: Techniques and Applications by W.-H. Lee, 4) Speckle Interferometry by A. E. Ennos, 5 ) Deformation  Invariant, Space-Variant Optical  Pattern Recognition by D. Casasent and D. Psaltis, 6) Light Emission from High-Current Surface-Spark Discharges by R. E. Beverly, and 7) Semiclassical Radiation  Theory within a  QuantumMechanical Framework by I. R. Senitzkt. The  breadth of topic  matter  spanned  by  these  chapters makes it impossible, for this reviewer at least, to pass judgement on the comprehensiveness, relevance, and completeness of every chapter. With an editorial board as prominent as that of Progress in Optics, however, it seems hardly likely that such comments should be necessary. It should certainly be possible to take the authority of each author as credible. The only remaining judgment to be  made on these chapters is their readability. In short, what are they like to read? The first sentence of the first chapter greets the eye with an obvious typographical error: “The creation of coherent laser light source, that have tunable radiation, opened the . . . .” Two pages later we find: “When two types of atoms or molecules of different isotopic composition ( A and B ) have even one spectral line that does not overlap with others, it is pos-

Progress in optics

A pedagogical overview of intramolecular vibrational redistribution (IVR) phenomena in vibrationally excited molecules is presented. In the interest of focus and simplicity, the topics covered deal primarily with IVR in the ground electronic state, relying on examples from the literature to illustrate key points. The experimental topics discussed attempt to sample systematically three different energy regimes on the full potential surface corresponding to (i) “low”, e.g., moderate- to high-resolution vibrational spectroscopies, (ii) “intermediate”, e.g., stimulated emission pumping and high overtone spectroscopies, and (iii) “high”, e.g., photofragment/predissociation dynamical spectroscopies. The interplay between experiment and theory is highlighted here because it has facilitated enormous advances in the field over the past decade.

Vibrational Energy Flow in Highly Excited Molecules: Role of Intramolecular Vibrational Redistribution

When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

Bose-Einstein condensation

Classical Dynamics Methods for High Energy Vibrational Spectroscopy

Likely candidates for the global potential energy minima of C60(H2O)n clusters with n < or = 21 are found using basin-hopping global optimization. The potential energy surfaces are constructed using the TIP4P intermolecular potential for the water molecules, a Lennard-Jones water-fullerene potential, and a water-fullerene polarization potential, which depends on the first few nonvanishing C60 multipole polarizabilities. This combination produces a rather hydrophobic water-fullerene interaction. As a consequence, the water component of the lowest C60(H2O)n minima is quite closely related to low-lying minima of the corresponding TIP4P (H2O)n clusters. In most cases, the geometrical substructure of the water molecules in the C60(H2O)n global minimum coincides with that of the corresponding free water cluster. Exceptions occur when the interaction with C60 induces a change in geometry. This qualitative picture does not change significantly if we use the TIP3P model for the water-water interaction. Structures such as C60@(H2O)60, in which the water molecules surround the C60 fullerene, correspond to local minima with much higher potential energies. For such a structure to become the global minimum, the magnitude of the water-fullerene interaction must be increased to an unphysical value.

Global potential energy minima of C60(H2O)n clusters.

The effects of confinement on water clusters inside nonmetallic carbon nanotubes with radii ranging between 4 and 7.5 A have been computationally investigated by means of global optimization and finite temperature simulations. The water–water interaction is described by the TIP4P rigid body potential, and a Lennard-Jones potential is used for the water–carbon interaction. Water clusters containing up to 20 molecules are found to form 1D chainlike configurations for the narrow (7, 5) nanotube and 2D ladderlike structures in the (7, 6) tube. In wider tubes, 3D configurations are then formed showing helical motifs, ringlike or closed cage structures, before the most stable structure on flat graphene is eventually found. The same results are obtained by replacing the fully atomistic water–nanotube potential by its continuous approximation [Breton, J.; Gonzalez-Platas, J.; Giradet, C. J. Chem. Phys.1994, 101, 3334], indicating a negligible effect of corrugation. The effects of additional nanotubes were also co...

Confinement Effects on Water Clusters Inside Carbon Nanotubes

Likely candidates for the global potential energy minima of (H2O)n clusters with n ≤ 21 on the (0001) surface of graphite are found using basin-hopping global optimization The potential energy surfaces are constructed using the TIP4P intermolecular potentials for the water molecules (the TIP3P is also explored as a secondary choice), a Lennard-Jones water−graphite potential, and a water−graphite polarization potential that is built from classical electrostatic image methods and takes into account both the perpendicular and the parallel electric polarizations of graphite This potential energy surface produces a rather hydrophobic water−graphite interaction As a consequence, the water component of the lowest graphite−(H2O)n minima is quite closely related to low-lying minima of the corresponding TIP4P (H2O)n clusters In about half of the cases, the geometrical substructure of the water molecules in the graphite−(H2O)n global minimum coincides with that of the corresponding free water cluster Exceptions

Global Potential Energy Minima of (H2O)n Clusters on Graphite

The Dang−Chang many-body polarizable potential has been used to model the interaction between water molecules and a cationic carbonaceous molecule X+, with X = C60 (buckminsterfullerene), C24H12 (c...

J. M. Gomez Llorente

Papers

Classical Dynamics Methods for High Energy Vibrational Spectroscopy

Global potential energy minima of C60(H2O)n clusters.

Confinement Effects on Water Clusters Inside Carbon Nanotubes

Global Potential Energy Minima of (H2O)n Clusters on Graphite

Modeling water clusters on cationic carbonaceous seeds.