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Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II
- Part of
- John C. Morrison, Kyle Steffen, Blake Pantoja, Asha Nagaiya, Jacek Kobus, Thomas Ericsson
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- Journal:
- Communications in Computational Physics / Volume 19 / Issue 3 / March 2016
- Published online by Cambridge University Press:
- 16 March 2016, pp. 632-647
- Print publication:
- March 2016
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- Article
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In order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.
Chapter 11 - Policy, Financing and Implementation
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- By Catherine Mitchell, Janet L. Sawin, Govind R. Pokharel, Daniel Kammen, Zhongying Wang, Solomone Fifita, Mark Jaccard, Ole Langniss, Hugo Lucas, Alain Nadai, Ramiro Trujillo Blanco, Eric Usher, Aviel Verbruggen, Rolf Wüstenhagen, Kaoru Yamaguchi, Douglas Arent, Greg Arrowsmith, Morgan Bazilian, Lori Bird, Thomas Boermans, Alex Bowen, Sylvia Breukers, Thomas Bruckner, Sebastian Busch, Elisabeth Clemens, Peter Connor, Felix Creutzig, Peter Droege, Karin Ericsson, Chris Greacen, Renata Grisoli, Erik Haites, Kirsty Hamilton, Jochen Harnisch, Cameron Hepburn, Suzanne Hunt, Matthias Kalkuhl, Heleen de Koninck, Patrick Lamers, Birger Madsen, Gregory Nemet, Lars J. Nilsson, Supachai Panitchpakdi, David Popp, Anis Radzi, Gustav Resch, Sven Schimschar, Kristin Seyboth, Sergio Trindade, Bernhard Truffer, Sarah Truitt, Dan van der Horst, Saskia Vermeylen, Charles Wilson, Ryan Wiser, David de Jager, Antonina Ivanova Boncheva
- Edited by Ottmar Edenhofer, Ramón Pichs-Madruga, Youba Sokona, Kristin Seyboth, Susanne Kadner, Timm Zwickel, Patrick Eickemeier, Gerrit Hansen, Steffen Schlömer, Christoph von Stechow, Patrick Matschoss
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- Book:
- Renewable Energy Sources and Climate Change Mitigation
- Published online:
- 05 December 2011
- Print publication:
- 21 November 2011, pp 865-950
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Summary
Executive Summary
Renewable energy can provide a host of benefits to society. In addition to the reduction of carbon dioxide (CO2) emissions, governments have enacted renewable energy (RE) policies to meet a number of objectives including the creation of local environmental and health benefits; facilitation of energy access, particularly for rural areas; advancement of energy security goals by diversifying the portfolio of energy technologies and resources; and improving social and economic development through potential employment opportunities. Energy access and social and economic development have been the primary drivers in developing countries whereas ensuring a secure energy supply and environmental concerns have been most important in developed countries.
An increasing number and variety of RE policies–motivated by a variety of factors–have driven substantial growth of RE technologies in recent years. Government policies have played a crucial role in accelerating the deployment of RE technologies. At the same time, not all RE policies have proven effective and efficient in rapidly or substantially increasing RE deployment. The focus of policies is broadening from a concentration almost entirely on RE electricity to include RE heating and cooling and transportation.
RE policies have promoted an increase in RE capacity installations by helping to overcome various barriers. Barriers specific to RE policymaking (e.g., a lack of information and awareness), to implementation (e.g., a lack of an educated and trained workforce to match developing RE technologies) and to financing (e.g., market failures) may further impede deployment of RE.
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