A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry

We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.Comment: 64 pages, 3 figures, tMPH2e.cls style file, doublesp, mathbbol and subeqn package

1989 as a mimetic revolution: Russia and the challenge of post-communism

Various terms have been used to describe the momentous events of 1989, including Jürgen Habermas’s ‘rectifying revolution,’ and my own notion of 1989 as a type of ‘anti-revolution’: repudiating not only what had come before, but also denying the political logic of communist power, as well as the emancipatory potential of revolutionary socialism in its entirety. In the event, while the negative agenda of 1989 has been fulfilled, it failed in the end to transcend the political logic of the systems that collapsed at that time. This paper explores the unfulfilled potential of 1989. Finally, 1989 became more of a counter- rather than an anti-revolution, replicating in an inverted form the practices of the mature state socialist regimes. The paucity of institutional and intellectual innovation arising from 1989 is striking. The dominant motif was ‘returnism,’ the attempt to join an established enterprise rather than transforming it. Thus, 1989 can be seen as mimetic revolution, in the sense that it emulated systems that were not organically developed in the societies in which they were implanted. For Eastern Europe ‘returning’ to Europe appeared natural, but for Russia the civilizational challenge of post-communism was of an entirely different order. There could be no return, and instead of a linear transition outlined by the classic transitological literature, Russia’s post-communism demonstrated that the history of others could not be mechanically transplanted from one society to another

The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity

In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain-gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants . The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics

Beam Energy Dependence of Jet-Quenching Effects in Au plus Au Collisions at root s(NN)=7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV

We report measurements of the nuclear modification factor, $R_{ \mathrm{CP}}$, for charged hadrons as well as identified $\pi^{+(-)}$, $K^{+(-)}$, and $p(\overline{p})$ for Au+Au collision energies of $\sqrt{s_{_{ \mathrm{NN}}}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV. We observe a clear high-$p_{\mathrm{T}}$ net suppression in central collisions at 62.4 GeV for charged hadrons which evolves smoothly to a large net enhancement at lower energies. This trend is driven by the evolution of the pion spectra, but is also very similar for the kaon spectra. While the magnitude of the proton $R_{ \mathrm{CP}}$ at high $p_{\mathrm{T}}$ does depend on collision energy, neither the proton nor the anti-proton $R_{ \mathrm{CP}}$ at high $p_{\mathrm{T}}$ exhibit net suppression at any energy. A study of how the binary collision scaled high-$p_{\mathrm{T}}$ yield evolves with centrality reveals a non-monotonic shape that is consistent with the idea that jet-quenching is increasing faster than the combined phenomena that lead to enhancement.We report measurements of the nuclear modification factor RCP for charged hadrons as well as identified π+(-), K+(-), and p(p¯) for Au+Au collision energies of sNN=7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV. We observe a clear high-pT net suppression in central collisions at 62.4 GeV for charged hadrons which evolves smoothly to a large net enhancement at lower energies. This trend is driven by the evolution of the pion spectra but is also very similar for the kaon spectra. While the magnitude of the proton RCP at high pT does depend on the collision energy, neither the proton nor the antiproton RCP at high pT exhibit net suppression at any energy. A study of how the binary collision-scaled high-pT yield evolves with centrality reveals a nonmonotonic shape that is consistent with the idea that jet quenching is increasing faster than the combined phenomena that lead to enhancement