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    Rigorous lower bounds for the ground state energy of molecules by employing necessary N-representability conditions
    (2020-11-13)
    Chaykin, Denis
    ;
    Jansson, Christian
    ;
    Keil, Frerich
    ;
    Lange, Marko
    ;
    Ohlhus, Kai Torben
    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 Article
      97
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    Rigorous results in electronic structure calculations
    (2016-11)
    Chaykin, Denis
    ;
    Jansson, Christian
    ;
    Keil, Frerich
    ;
    Lange, Marko
    ;
    Ohlhus, Kai Torben
    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. 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 Article
      47
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      41
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    Self-validating methods : some recent developements
    (Shaker Verlag, 1994)
    Jansson, Christian
    ;
    Rump, Siegfried M.
    Publicationtype: Book Part
      38
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    Rigorous solution of linear programming problems with uncertain data
    (Springer, 1991-03-01)
    Jansson, Christian
    ;
    Rump, Siegfried M.
    This note gives a synopsis of new methods for solving linear systems and linear programming problems with uncertain data. All input data can vary between given lower and upper bounds. The methods calculate very sharp and guaranteed error bounds for the solution set of those problems and permit a rigorous sensitivity analysis. For problems with exact input data in general the calculated bounds differ only in the last bit in the mantissa, i.e. they are of maximum accuracy.
    Publicationtype: Journal Article
      33
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      33