THE UNIVERSITY OF CHICAGO REDUCED DENSITY MATRICES AND TENSOR FACTORIZATIONS FOR LOW COST QUANTUM CHEMISTRY A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CHEMISTRY BY ERIK P. HOY CHICAGO, ILLINOIS DECEMBER 2015 Copyright ⃝c 2015 by Erik P. Hoy All Rights Reserved To Alice TABLE OF CONTENTS LIST OF FIGURES . vi LIST OF TABLES . viii ABSTRACT . xiv ACKNOWLEDGMENTS . xv 1 INTRODUCTION . 1 1.1 Hartree-Fock . 1 1.2 Electron Correlation . 3 1.3 2-Electron Reduced Density Matrix . 4 1.4 Computational Cost Reduction in Chemistry . 6 1.5 Tensor Factorization . 7 1.6 Chapter Overview . 8 1.7 References . 9 2 THEORY: PARAMETRIC 2-ELECTRON REDUCED DENSITY METHOD (P2- RDM) . 11 2.1 Introduction and Overview . 11 2.2 Parametric 2-RDM . 12 2.2.1 The M Parametrization . 17 2.3 References . 19 3 ISOELECTRONIC ANALOGUE OF OXYWATER: A PARAMETRIC 2-ELECTRON REDUCED-DENSITY-MATRIX STUDY OF AMMONIA OXIDE . 22 3.1 Introduction . 22 3.2 Applications . 23 3.2.1 Computational Methods . 24 3.2.2 Results . 24 3.3 Discussion and Conclusions . 30 3.4 References . 34 4 RELATIVE ENERGIES AND GEOMETRIES OF THE CIS- AND TRANS-HO3 RADICALS FROM THE PARAMETRIC 2-ELECTRON DENSITY MATRIX METHOD 38 4.1 Introduction . 38 4.2 Applications . 41 4.2.1 Computational Methods . 41 4.2.2 Results . 42 4.3 Discussion and Conclusions . 51 4.4 References . 53 iv 5 ENERGIES AND STRUCTURES IN BIRADICAL CHEMISTRY FROM THE PARA- METRIC TWO-ELECTRON REDUCED-DENSITY MATRIX METHOD: APPLI- CATIONS TO THE BENZENE AND CYCLOBUTADIENE BIRADICALS . 60 5.1 Introduction . 60 5.2 Applications . 62 5.2.1 Methodology . 62 5.2.2 Results . 63 5.3 Discussion and Conclusion . 74 5.4 References . 75 6 COMPARISON OF LOW-RANK TENSOR EXPANSIONS FOR THE ACCELER- ATION OF QUANTUM COMPUTATIONS . 79 6.1 Introduction . 79 6.2 Theory . 80 6.2.1 Low-Rank Spectral Expansion . 80 6.2.2 THC Factorization . 81 6.2.3 Theoretical Scaling . 82 6.3 Applications . 83 6.3.1 Computational Methods . 83 6.3.2 Results . 84 6.4 Discussion and Conclusions . 90 6.5 References . 93 7 POSITIVE SEMIDEFINITE TENSOR FACTORIZATIONS OF THE TWO-ELECTRON INTEGRAL MATRIX FOR LOW-SCALING AB INITIO ELECTRONIC STRUC- TURE . 97 7.1 Introduction . 97 7.2 Theory . 99 7.2.1 Cholesky Factorizations . 99 7.2.2 Positive Semidefinite Factorizations . 101 7.3 Applications . 102 7.3.1 Computational Methods . 102 7.3.2 Results . 102 7.4 Discussion and Conclusions . 110 7.5 References . 111 v LIST OF FIGURES 3.1 Isomerization reactions from the parametric 2-RDM(M), (a) the conversion of hydroxylamine to ammonia oxide and (b) the conversion of hydrogen peroxide to oxywater, are compared at the extrapolated basis set limit. The oxywater energies include a harmonic zero-point correction. 31 4.1 Relative energies of the two HO3 isomers and their transition state from the 2- RDM(M), CCSD, and CCSD(T) methods are shown for the (a) cc-pVQZ and (b) aug-cc-pVQZ basis sets. MRCI(Q) calculations in the cc-pVQZ (a) and aug-cc-pVQZ (b) basis sets including harmonic vibrational zero-point energies extrapolated to the basis set limit from Ref. [73]. 49 4.2 Figures show the (a) cis- and (b) trans-HO3 geometries as calculated by the 2-RDM(M) method in the aug-cc-pVTZ basis set. 50 5.1 Molecules (a) cyclobutadiene, (b) the cis-cyclobutadiene biradical, and (c) the trans-cyclobutadiene biradical are shown at the p2-RDM/cc-pVDZ optimized geometries. 64 5.2 Molecules (a) o-benzyne, (b) m-benzyne, and (c) p-benzyne are shown at the p2-RDM/cc-pVDZ optimized geometries. 69 5.3 Visualizations of the Nth and (N + 1)th natural-orbitals of o-, m-, and p-benzyne are shown at the p2-RDM/cc-pVDZ optimized geometries. 72 6.1 Part (a) shows the convergence of the spectral expansion methods to the full-rank result for water in the cc-pVTZ basis set. Part (b) shows the same results for the tensor hypercontraction method. 85 6.2 OH+ dissociation curve in the cc-pVDZ basis set from the p2-RDM method with the (a) spectral expansion and (b) tensor hypercontraction methods. Results are compared to those from the full-rank p2-RDM method. 88 vi 6.3 HF dissociation curve in the cc-pVDZ basis set from the p2-RDM method with the (a) spectral expansion and (b) tensor hypercontraction methods. Results are compared to those from the full-rank p2-RDM method. 88 7.1 The non-zero eigenvalues of the two-electron integral matrix of methane are shown with the integrals generated with (a) no factorization (b) the Cholesky factoriza- tion, and (c) the PSD factorization at the half-bandwidth w = 4. The first 4 eigenvalues, which have a magnitude larger than 1, as well as the zero eigenval- ues are not shown to make the non-zero eigenvalues more visible. The first four eigenvalues also show good agreement between the three methods. 107 7.2 The potential energy curves of (a) hydrogen fluoride HF and (b) hydroxyl radical OH+ from the p2-RDM method with and without PSD two-electron integral factorizations are compared. The curve from the PSD factorization with either w = 1 or w = 4 shows good agreement with the curve from the p2-RDM method without integral factorization. The basis set is cc-pVDZ. 109 vii LIST OF TABLES 2.1 The table below contains the topological factors for equation [12]. The factor no abcd in the table fnv , represents the factor fijkl from above where n0 is the number of occupied spin orbitals shared by i,j and k,l, and nv is the number of virtual orbitals shared by a,b and c,d. This gives a total of nine possible combinations for the topological factor, as shown this table. The D and Q factors given below represent the parametrizations that result from applying the D and Q Cauchy- Schwarz inequalities to the 2-RDM elements [13, 14]. 19 3.1 Correlation energies for hydroxylamine(HA), ammonia oxide(AO), and their tran- sition state(TS) were calculated using the parametric 2-RDM(M) method as well as several coupled cluster methods in aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc- pVQZ basis sets with extrapolation to the basis-set limit (EBSL). 25 3.2 The occupation numbers of the highest occupied natural orbital (HONO) and the lowest unoccupied natural orbital (LUNO) for all six molecules are presented in the aug-cc-pVTZ basis set from the CCSD and 2-RDM(M) methods. 27 3.3 Optimized geometries for hydroxylamine, ammonia oxide, and their transition state are presented from the parametric 2-RDM(M) method and the coupled cluster methods in the aug-cc-pVTZ basis set. The molecular point groups em- ployed were Cs for hydroxylamine and the transition state and C3v for ammonia oxide. 27 viii 3.4 Energy differences of the ammonia-oxide reaction are compared with those of the oxywater reaction. All energy differences were computed from energies at the extrapolated basis-set limit. The oxywater energies are vibrationally corrected by a harmonic zero-point correlation [15], but the ammonia-oxide results are not since such corrections are small relative to the reaction and barrier energies. The chemical formulas (or abbreviations) used for the molecules are NH2OH (hydroxylamine), NH3O (ammonia oxide), HOOH (hydrogen peroxide), H2OO (oxywater), and TS (transition state). 29 3.5 Trends in barrier heights across basis sets are compared for the ammonia-oxide-to- hydroxylamine and the oxywater-to-hydrogen-peroxide reactions. Coupled clus- ter methods are represented by the highest level method CCSD(T), but the trends are the same for both CCSD and CR-CC(T). 30 4.1 The natural-orbital occupation numbers as calculated by the 2-RDM(M) method in the cc-pVQZ and aug-cc-pVQZ basis sets are presented. 42 4.2 The total electronic energies before harmonic zero-point vibrational energies plus 225 Hartrees are given for the cc-pVQZ and aug-cc-pVQZ basis sets. 43 4.3 The energy differences between states before the addition of harmonic zero-point energies are presented for cc-pVQZ and aug-cc-pVQZ basis sets. For comparison, MRCI(Q) cc-pVQZ and aug-cc-pVQZ trans-cis energy differences calculated by Varandas in 2012 are provided [73]. 44 4.4 The energy barriers after the inclusion of harmonic zero-point vibrational energies for cc-pVQZ and aug-cc-pVQZ basis sets are reported in this table. 44 4.5 The harmonic frequencies calculated in the cc-pVDZ basis set for each molecule and method are provided in the table below. Experimental results from Derro et al: for the trans isomer are provided for comparison[36]. 46 ix 4.6 The oxygen bond lengths calculated in the aug-cc-pVTZ basis set are reported. The total correlation energy recovered in the aug-cc-pVQZ basis set for each molecule is given in the rows marked CE (correlation energy) below the relevant parameters. The experimental values for the trans-HO3 boo1 bond length are 1.225* A˚ [74] and 1.235 A˚ [32] while the values for the boo2 bond length are 1.688 A˚ [74] and 1.648 A˚ [32]. *Fixed at the MRCI+Q value in order to estimate the other geometric parameters. 48 5.1 Geometric parameters (angstroms and degrees) of p2-RDM and CCSD optimized cyclobutadiene and its cis- and trans-biradicals.