We present the discovery of nine quasars at z ∼ 6 identified in the Sloan Digital Sky Survey (SDSS) imaging data. This completes our survey of z ∼ 6 quasars in the SDSS footprint. Our final sample consists of 52 quasars at 5.7 < z ≤ 6.4, including 29 quasars with z AB ≤ 20 mag selected from 11,240 deg 2 of the SDSS single-epoch imaging survey (the main survey), 10 quasars with 20 ≤ z AB ≤ 20.5 selected from 4223 deg 2 of the SDSS overlap regions (regions with two or more imaging scans), and 13 quasars down to z AB ≈ 22 mag from the 277 deg 2 in Stripe 82. They span a wide luminosity range of −29.0 ≤ M 1450 ≤ −24.5. This well-defined sample is used to derive the quasar luminosity function (QLF) at z ∼ 6. After combining our SDSS sample with two faint (M 1450 ≥ −23 mag) quasars from the literature, we obtain the parameters for a double power-law fit to the QLF. The bright-end slope β of the QLF is well constrained to be β = −2.8 ± 0.2. Due to the small number of low-luminosity quasars, the faint-end slope α and the characteristic magnitude M * 1450 are less well constrained, with α = −1.90The spatial density of luminous quasars, parametrized as ρ(M 1450 < −26, z) = ρ(z = 6) 10 k(z−6) , drops rapidly from z ∼ 5 to 6, with k = −0.72 ± 0.11. Based on our fitted QLF and assuming an IGM clumping factor of C = 3, we find that the observed quasar population cannot provide enough photons to ionize the z ∼ 6 IGM at ∼ 90% confidence. Quasars may still provide a significant fraction of the required photons, although much larger samples of faint quasars are needed for more stringent constraints on the quasar contribution to reionization.

We report initial results from a large Gemini program to observe z 5.7 quasars with GNIRS near-IR spectroscopy. Our sample includes 50 quasars with simultaneous ∼ 0.85 − 2.5 µm spectra covering the rest-frame ultraviolet and major broad emission lines from Ly α to Mg II. We present spectral measurements for these quasars and compare to their lower-redshift counterparts at z = 1.5 − 2.3. We find that when quasar luminosity is matched, there are no significant differences between the rest-UV spectra of z 5.7 quasars and the low-z comparison sample. High-z quasars have similar continuum and emission line properties and occupy the same region in the black hole mass and luminosity space as the comparison sample, accreting at an average Eddington ratio of ∼ 0.3. There is no evidence for super-Eddington accretion or hypermassive (> 10 10 M ) black holes within our sample. We find a mild excess of quasars with weak C IV lines relative to the control sample. Our results, corroborating earlier studies but with better statistics, demonstrate that these high-z quasars are already mature systems of accreting supermassive black holes operating with the same physical mechanisms as those at lower redshifts.

It is shown that in density functional theory (DFT), Koopmans' theorem for a large molecular system can be stated as follows: The ionization energy of the system equals the negative of the highest occupied molecular orbital (HOMO) energy plus the Coulomb electrostatic energy of removing an electron from the system, or equivalently, the ionization energy of an N-electron system is the negative of the arithmetic average of the HOMO energy of this system and the lowest unoccupied molecular orbital (LUMO) energy of the (N - 1)-electron system. Relations between this DFT Koopmans' theorem and its existing counterparts in the literature are discussed. Some of the previous results are generalized and some are simplified. DFT calculation results of a fullerene molecule, a finite single-walled carbon nanotube and a finite boron nitride nanotube are presented, indicating that this Koopmans' theorem approximately holds, even if the orbital relaxation is taken into consideration.