Thompson, Travis Hunter (2020): Integral bounds and rigorous screening algorithms for reduced scaling in explicitly correlated, seminumerical, and nonHermitian quantum chemistry. Dissertation, LMU München: Faculty of Chemistry and Pharmacy 

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Abstract
The properties of chemical systems can be determined computationally by solving the physical equations that govern them. This typically requires the calculation of a very large number of molecular integrals that take various forms depending on the approximations used. Most of these integrals are negligible and avoiding the calculation of negligible integrals can increase computational efficiency immensely. In this work, novel upper bounds and screening algorithms are developed for this purpose. These bounds and algorithms are applicable to a wide range of quantum chemical theories and can be used in combination with the stateoftheart integral approximation methods that are at the heart of today’s most efficient numerical implementations. A Schwarztype bound that captures the decay of fourcenter twoelectron integrals due to the decreased interaction of distant charge distributions is developed and tested in the context of HartreeFock and rangeseparated density functional theory, and on fourcenter integrals over shortrange correlation factors that arise in explicitly correlated theories. An integral partitioning procedure is developed which leads to extremely flexible upper bounds, integral partition bounds (IPBs), for molecular integrals over any number of elec trons, any number of basis function centers, and various combinations of integral operators. The procedure allows for the inexpensive calculation of rigorous extents for charge distri butions within these various contexts. The IPBs are completely separable into twocenter factors, which capture all the sources of asymptotic decay. This allows for the formulation of scalingconsistent screening algorithms, even for the three and fourelectron integrals that arise, e.g., within explicitly correlated MøllerPlesset perturbation theory (MP2F12). The IPBs are used to increase the efficiency and reliability of seminumerical tech niques for calculation of the exchange matrix in HartreeFock and hybrid DFT calcula tions, where real space numerical quadrature is used to approximate electron repulsion integrals. Similarly, a framework for very efficient MP2F12 working equations based on optimal combinations of resolutionoftheindentity, densityfitting, and numerical quadra ture approximations are given. They reduce the fifthorder formal scaling of the MP2F12 method to fourthorder, while drastically reducing the cost of the initially most expensive terms. The resulting equations involve various types of sparse integral tensors which can all be treated using IPBs, and asymptotically linear scaling implementations are possible. Furthermore, a Schwarz bound is developed and implemented for screening the novel fourcenter twoelectron integrals that arise in the treatment of resonance states using the method of complex basis functions within nonHermitian quantum chemistry. This is a crucial step for increasing the efficiency and reach of the method.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  Computational Chemistry, Quantum Chemistry, Integral bounds, integral screening, explicitly correlated methods, F12 Correction, seminumerical integration, nonHermitian quantum chemistry 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 540 Chemistry and allied sciences 
Faculties:  Faculty of Chemistry and Pharmacy 
Language:  English 
Date of oral examination:  27. May 2020 
1. Referee:  Ochsenfeld, Christian 
MD5 Checksum of the PDFfile:  14689264fe187577c0fea9dbe517c416 
Signature of the printed copy:  0001/UMC 27988 
ID Code:  26208 
Deposited On:  14. Jun 2021 14:14 
Last Modified:  16. Jun 2021 08:06 