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Item type:Publication, Access status: without files , Rigorous lower bounds for the ground state energy of molecules by employing necessary N-representability conditions(2020-11-13); ; ; ; Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree–Fock method or numerically more exact for a given basis set by full CI. However, Nakata et al. ( J. Chem. Phys.200111482828292) observed that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the full CI energy. For the LiH, CH–, NH–, OH, OH–, and HF molecules violations within one mhartree were observed. We applied the software VSDP which takes all numerical errors due to floating-point arithmetic operations into consideration. For two test libraries VSDP provides tight rigorous error bounds lower than full CI energies reported with an accuracy of 0.1 to 0.01 mhartree. Only little computation work must be spent in order to compute close rigorous error bounds for the ground state energy.Publicationtype: Journal Article97 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Rigorous results in electronic structure calculationsElectronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree-Fock method. However, Nakata, Nakatsuji, Ehara, Fukuda, Nakata, and Fujisawa observed, that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the Hartree-Fock upper bound. Violations within one mhartree were observed. We present here a method for solving electronic structure problems where all numerical errors are taken into consideration. In particular this method provides rigorous error bounds without violations as mentioned above.Publicationtype: Journal Article47 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Verification of semidefinite optimization problems with application to variational electronic structure calculation(2009); Algorithms for computing lower and upper bounds of global optimal value, as well as enclosures of ε-optimal solutions in semidefinite programming are presented. Suggested verification methods are successfully applied to the electronic structure problem (RDM formulation) to calculate a lower bound of the ground state energy of an N-electron system.Publicationtype: Doctoral Thesis604 282 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Rigorous error bounds for the optimal value in semidefinite programming(Society for Industrial & Applied Mathematics (SIAM), 2007-12-01); ; A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many ill-conditioned and real-life problems.Publicationtype: Journal Article43