To set up the general framework for relativistic explicitly correlated wave function methods, the electron-electron coalescence conditions are derived for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipulations make full use of the internal symmetries of the reduced two-electron Hamiltonians such that the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coalescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those of the DC and DCB Hamiltonians have weak singularities of the type $r_{12}^{\nu}$ with $\nu$ being negative and of $\mathcal{O}(\alpha^2)$. The behaviors of the MDC wave functions are related to the original ones in a simple manner, while the spin-free counterparts are somewhat different due to the complicated electron-electron interaction. The behaviors of the ZORA wave functions depend on the chosen potential in the kinetic energy operator. In the case of the nuclear attraction, the behaviors of the ZORA wave functions are very similar to those of the nonrelativistic ones, just with an additional correction of $\mathcal{O}(\alpha^2)$ to the nonrelativistic cusp condition. However, if the Coulomb interaction is also included, the ZORA wave functions become close to the large-large components of the DC wave functions. Note that such asymptotic expansions of the relativistic wave functions are only valid within an extremely small convergence radius $R_c$ of $\mathcal{O}(\alpha^2)$. Beyond this radius, the behaviors of the relativistic wave functions are still dominated by the nonrelativistic limit, as can be seen in terms of direct perturbation theory (DPT) of relativity. However, as the two limits $\alpha\rightarrow0$ and $r_{12}\rightarrow0$ do not commute, DPT is doomed to fail due to incorrect descriptions of the small-small component $\Psi^{SS}$ of the DC wave function for $r_{12}<R_c$. Another deduction from the possible divergence of $\Psi^{SS}$ at $r_{12}=R_c$ is that the DC Hamiltonian has no bound electronic states, although the last word cannot be said. These findings enrich our understandings of relativistic wave functions. On the practical side, it is shown that, under the no-pair approximation, relativistic explicitly correlated wave function methods can be made completely parallel to the nonrelativistic counterparts, as demonstrated explicitly for MP2-F12. Yet, this can only be achieved by using an extended no-pair projector.
In this work, a closure experiment for tropospheric aerosol is presented. Aerosol size distributions and single scattering albedo from remote sensing data are compared to those measured in-situ. An aerosol pollution event on 4 April 2009 was observed by ground based and airborne lidar and photometer in and around Ny-Alesund, Spitsbergen, as well as by DMPS, nephelometer and particle soot absorption photometer at the nearby Zeppelin Mountain Research Station. The presented measurements were conducted in an area of 40 x 20 km around Ny-Alesund as part of the 2009 Polar Airborne Measurements and Arctic Regional Climate Model Simulation Project (PAMARCMiP). Aerosol mainly in the accumulation mode was found in the lower troposphere, however, enhanced backscattering was observed up to the tropopause altitude. A comparison of meteorological data available at different locations reveals a stable multi-layer-structure of the lower troposphere. It is followed by the retrieval of optical and microphysical aerosol parameters. Extinction values have been derived using two different methods, and it was found that extinction (especially in the UV) derived from Raman lidar data significantly surpasses the extinction derived from photometer AOD profiles. Airborne lidar data shows volume depolarization values to be less than 2.5% between 500 m and 2.5 km altitude, hence, particles in this range can be assumed to be of spherical shape. In-situ particle number concentrations measured at the Zeppelin Mountain Research Station at 474 m altitude peak at about 0.18 mu m diameter, which was also found for the microphysical inversion calculations performed at 850 m and 1500 m altitude. Number concentrations depend on the assumed extinction values, and slightly decrease with altitude as well as the effective particle diameter. A low imaginary part in the derived refractive index suggests weakly absorbing aerosols, which is confirmed by low black carbon concentrations, measured at the Zeppelin Mountain as well as on board the Polar 5 aircraft. (C) 2011 Elsevier Ltd. All rights reserved.
Eight molybdenite samples were selected from the Yaojiagou molybdenum deposit. The Re-Os isotopic model ages ranging from 166. 1 +/- 2. 3Ma to 169. 1 +/- 2. 5Ma, yielded an isochron age of 168.8 +/- 3. 9Ma (MSWD =1. 12), which was interpreted to be the ore-forming age of the deposit. Combined with existing geochronologic data of Yaojiagou granit pluton, we assume that there were multistage of intrusions in Yaojiagou area and the Yaojiagou molybdenum deposit was related to the magma intrusion activities in 168. 8 +/- 3. 9Ma. In combination with metallogenic geological background, we infer that the Yaojiagou molybdenum deposit developed from Early to Middle Jurassic, influenced by magma and fluid function of post-collision between North China Craton and Siberia Craton.
This paper measures the contribution of information productivity to the economic growth by a revised Cobb-Douglas production function which takes capital, labor and information as three elements. The result shows that information productivity contributes the most to the economic growth. Meanwhile, we also compare China with other high income countries and find that information input has stronger multiplier effect in China than high income countries.
This paper studies the risk‐free rate in an overlapping generations economy with bequests. It is shown that the risk‐free rate depends on risk aversion, the elasticity of intertemporal substitution, the share of wealth invested in human wealth, life expectancy, and the preference for bequests. In a standard life‐cycle context, mortality increases the subjective time rate of discount, and thus increases the compensation required to postpone consumption. This latter effect is offset in a bequest‐driven model of the type considered here, leading to much more powerful income effects. In this sense, the model provides a bequest‐motive explanation for the risk‐free rate puzzle put forward by Weil in 1989.