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

Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations Trygve Helgaker,* Sonia Coriani, Poul Jørgensen, Kasper Kristensen, Jeppe Olsen, and Kenneth Ruud Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway Dipartimento di Scienze Chimiche e Farmaceutiche, Universit a degli Studi di Trieste, Via Giorgieri 1, I-34127 Trieste, Italy Lundbeck Center for Theoretical Chemistry, Department of Chemistry, Aarhus University, 8000 Aarhus C, Denmark Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, N-9037 Tromsø, Norway

Recent advances in wave function-based methods of molecular-property calculations.

For the first time, a complete implementation of coupled perturbed density functional theory (CPDFT) for the calculation of NMR spin–spin coupling constants (SSCCs) with pure and hybrid DFT is presented. By applying this method to several hydrides, hydrocarbons, and molecules with multiple bonds, the performance of DFT for the calculation of SSCCs is analyzed in dependence of the XC functional used. The importance of electron correlation effects is demonstrated and it is shown that the hybrid functional B3LYP leads to the best accuracy of calculated SSCCs. Also, CPDFT is compared with sum-over-states (SOS) DFT where it turns out that the former method is superior to the latter because it explicitly considers the dependence of the Kohn–Sham operator on the perturbed orbitals in DFT when calculating SSCCs. The four different coupling mechanisms contributing to the SSCC are discussed in connection with the electronic structure of the molecule.

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Nuclear magnetic resonance spin–spin coupling constants from coupled perturbed density functional theory

The equation‐of‐motion coupled cluster singles and doubles (EOM‐CCSD) method for general second‐order properties is derived providing a quadratic, CI‐like approximation and its linked form from coupled cluster (CC) energy derivative theory. The effects of the quadratic contribution, of the atomic basis set employed, and of electron correlation on NMR spin–spin coupling constant calculations using EOM‐CCSD methods are investigated for a selected set of difficult molecules, notably CH3F, B2H6, CH3CN, C2H4, and CH3NH2. We find that the quadratic contribution is insignificant for the couplings in the molecules considered in this study and in addition the quadratic contribution only slightly depends on the basis set used. Therefore it seems well justified to use the less expensive CI‐like approximation or its linked‐diagram form to evaluate spin–spin coupling constants. The Fermi‐contact contribution shows the largest variation with the change of basis sets. The diamagnetic spin–orbit (DSO) and the spin–dipole...

Electron correlation effects on the theoretical calculation of nuclear magnetic resonance spin–spin coupling constants

The quantum-chemical calculation of NMR indirect spin-spin coupling constants

The normalized elimination of the small component (NESC) theory, recently proposed by Filatov and Cremer [J. Chem. Phys. 122, 064104 (2005)], is extended to include magnetic interactions and applied to the calculation of the nuclear magnetic shielding in HX (X=F,Cl,Br,I) systems. The NESC calculations are performed at the levels of the zeroth-order regular approximation (ZORA) and the second-order regular approximation (SORA). The calculations show that the NESC-ZORA results are very close to the NESC-SORA results, except for the shielding of the I nucleus. Both the NESC-ZORA and NESC-SORA calculations yield very similar results to the previously reported values obtained using the relativistic infinite-order two-component coupled Hartree-Fock method. The difference between NESC-ZORA and NESC-SORA results is significant for the shieldings of iodine.

Relativistic calculation of nuclear magnetic shielding using normalized elimination of the small component.

The relativistic calculation of nuclear magnetic shielding tensors in hydrogen halides is performed using the second-order regular approximation to the normalized elimination of the small component (SORA-NESC) method with the inclusion of the perturbation terms from the metric operator. This computational scheme is denoted as SORA-Met. The SORA-Met calculation yields anisotropies, Δσ=σ∥−σ⊥, for the halogen nuclei in hydrogen halides that are too small. In the NESC theory, the small component of the spinor is combined to the large component via the operator σ⋅πU/2c, in which π=p+A, U is a nonunitary transformation operator, and c≅137.036 a.u. is the velocity of light. The operator U depends on the vector potential A (i.e., the magnetic perturbations in the system) with the leading order c−2 and the magnetic perturbation terms of U contribute to the Hamiltonian and metric operators of the system in the leading order c−4. It is shown that the small Δσ for halogen nuclei found in our previous studies is...

Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. III. Introduction of gauge-including atomic orbitals and a finite-size nuclear model.

A two-component relativistic theory accurately decoupling the positive and negative states of the Dirac Hamiltonian that includes magnetic perturbations is derived. The derived theory eliminates all of the odd terms originating from the nuclear attraction potential V and the first-order odd terms originating from the magnetic vector potential A, which connect the positive states to the negative states. The electronic energy obtained by the decoupling is correct to the third order with respect to A due to the (2n+1) rule. The decoupling is exact for the magnetic shielding calculation. However, the calculation of the diamagnetic property requires both the positive and negative states of the unperturbed (A=0) Hamiltonian. The derived theory is applied to the relativistic calculation of nuclear magnetic shielding tensors of HX (X=F,Cl,Br,I) systems at the Hartree-Fock level. The results indicate that such a substantially exact decoupling calculation well reproduces the four-component Dirac-Hartree-Fock results.

Decoupling of the Dirac equation correct to the third order for the magnetic perturbation.

The approximate SCF-MO calculation of 13C nuclear magnetic shielding constants was performed using the finite perturbation method. The results indicated that the INDO finite perturbation method calculates 13C chemical shifts in poor agreement with experiment.

INDO Calculation of 13C Chemical Shifts by the Finite Perturbation Method

A formulism for four-component Dirac-Hartree-Fock perturbation theory using the restricted magnetic balance (RMB) condition with gauge-including atomic orbitals (GIAOs) is derived at an ab initio level. In this formulation, the zeroth-order Dirac-Hartree-Fock equations are subject only to the usual restricted kinetic balance (RKB) condition, while the external field-dependent RMB condition is introduced in the calculation of the first-order magnetically perturbed orbitals. The magnetic shielding can be divided into a paramagnetic part and a diamagnetic part, as for nonrelativistic (NR) shielding. The present theory is applied to the calculation of hydrogen halide and molecular halogen shieldings.

H. Fukui

Papers

Relativistic calculation of nuclear magnetic shielding using normalized elimination of the small component.

Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. III. Introduction of gauge-including atomic orbitals and a finite-size nuclear model.

Decoupling of the Dirac equation correct to the third order for the magnetic perturbation.

INDO Calculation of 13C Chemical Shifts by the Finite Perturbation Method

Dirac–Hartree–Fock Perturbation Calculation of Magnetic Shielding Using the External Field-Dependent Restricted Magnetic Balance Condition