Developing a Model Chemistry for Multiconfiguration Pair-Density Functional Theory to Study Photochemistry and Molecular Interactions A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jie Bao IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF PH.D. OF SCIENCE Donald G. Truhlar January, 2021 © Jie Bao 2021 ALL RIGHTS RESERVED Acknowledgements I would like to acknowledge my advisor, Dr. Donald Truhlar, for being a helpful and reliable advisor. I would like to acknowledge my collaborator, Dr. Laura Gagliardi, for her help in almost all my projects. I would like to acknowledge Dr. Junwei Lucas Bao, Dr. Shaohong Louis Li, Dr. Kamal Sharkas, Dr. Yinan Shu, Dr. Zoltán Varga, Dr. Chad Hoyer, and Dr. Pragya Verma for their help when I started work on theoretical chemistry. I would like to acknowledge Dr. Andrew Sand for his help in teaching me to code in OpenMolcas. I would like to thank Dr. Sijia Dong and Dr. Meagan Oakley for collaborations in two projects. I would like to thank Dr. Chen Zhou, Dr. Matthew Hermes and Thais Scott, for valuable and interesting discussions of some of my projects. At last, I would like to acknowledge Dr. Donald Truhlar, Dr. Laura Gagliardi, Dr. Jason Goodpaster, and Dr. Aaron Massari for serving on my committee for my thesis. i Dedication To my family who give me support. ii Abstract Photochemical reaction, which starts by exciting a system into an electronically excited state, is ubiquitous, for example, in the atmosphere. This has made photochemical reactions a very interesting topic. Multiconfigurational pair-density functional theory (MC-PDFT) is a powerful and efficient method for studying photochemical processes. This method has proved very efficient compared with other wave function methods, such as multi-state complete active space second order perturbation theory (MS-CASPT2), especially for large systems. Successful as MC-PDFT is, there are some limitations that stop MC-PDFT from being applied to studying photochemistry problems. The first limitation is that, like other multireference methods, the performance of MC-PDFT depends on the quality of the reference wave function, which by convention is optimized by an active-space method, such as complete active space self-consistent field (CASSCF). The second limitation is that MC-PDFT is a single-state method that does not include state interaction between reference states. This means that MC-PDFT gives wrong topologies of potential energy surfaces, which are important in studying photochemical reactions. My work is focused on resolving these two limitations. We proposed the ABC scheme and the ABC2 scheme to automatically generate the active space that gives good-quality reference wave functions thus successfully reproducing vertical excitation energies obtained from experiments or high-level calculations. We proposed the extended multi-state PDFT (XMS-PDFT) and compressed-state multi-state PDFT (CMS-PDFT) as two options to introduce state-interaction in pair-density functional theory. Among two methods, XMS-PDFT is more efficient, while CMS-PDFT is more robust. Both methods proved successful in providing correct topologies of potential energy surfaces for a variety of systems. iii Contents Acknowledgements i Dedication ii Abstract iii List of Tables viii List of Figures xii 1 Introduction 1 2 Multiconfiguration pair-density functional theory for doublet excitation ener- gies and excited state geometries: the excited states of CN 4 2.1 Introduction . 4 2.2 Computational Details . 7 2.3 Results and Discussion . 9 2.3.1 An Illustration of the Dominant Electron Configuration for Each State ofCN................................... 9 2.3.2 Multireference Diagnostic for CN . 9 2.3.3 Adiabatic Excitation Energy for Each of the States . 11 2.3.4 Equilibrium Distances . 15 iv 2.4 Conclusions . 19 3 Weak Interactions in Alkaline Earth Metal Dimers by Pair-Density Functional Theory 22 3.1 Introduction . 22 3.2 Computational Details . 24 3.3 Results and Discussion . 26 3.3.1 Multiconfigurational Character . 26 3.3.2 Binding Energies (De) for dimers from Be2 to Ba2 . 27 3.3.3 Equilibrium Internuclear Distances (re) for dimers from Be2 to Ba2 . 28 3.3.4 Larger Active Spaces . 28 3.3.5 Basis Set Dependence . 30 3.3.6 Comparison to KS-DFT . 31 3.3.7 Potential Energy Curves . 32 3.3.8 Performance of Other On-top Functionals . 33 3.3.9 MP2 vs. CASPT2 . 33 3.3.10 Core Correlation . 34 3.3.11 Diffuse Basis Functions and Core Polarization Basis Functions . 34 3.3.12 De and re for Ra2 ............................ 35 3.4 Conclusion . 35 4 Automatic selection of an active space for calculating electronic excitation spectra by MS-CASPT2 or MC-PDFT 36 4.1 Introduction . 36 4.2 ABC Scheme . 38 4.3 Computational Details . 43 4.4 Results and Discussion . 45 v 4.5 Conclusion . 49 5 Automatic Active Space Selection for Calculating Electronic Excitation Ener- gies Based on High-Spin Unrestricted Hartree-Fock Orbitals 56 5.1 Introduction . 56 5.2 ABC2 Sceheme . 59 5.3 Computational Details . 61 5.4 Systems Studied . 63 5.5 Computational Details . 63 5.6 Methods Compared . 66 5.7 Results and Discussion . 66 5.7.1 Doublet Excited States . 66 5.7.2 Singlet Excited States . 68 5.7.3 Comparison to the ABC scheme . 71 5.7.4 Comparison to other guess schemes . 74 5.7.5 Timing . 78 5.7.6 Active spaces in other methods . 79 5.8 Conclusion . 79 6 Extended multi-state Pair-Density Functional Theory 81 6.1 Introduction . 81 6.2 Theory . 83 6.2.1 MC-PDFT . 83 6.2.2 multi-state MC-PDFT . 84 6.2.3 XMS-PDFT . 85 6.3 Computational Details . 88 6.4 Results and Discussion . 89 vi 6.4.1 Lithium fluoride (LiF) . 89 6.4.2 Lithium hydride (LiH) . 90 6.4.3 Phenol (C6H5OH) . 92 6.5 Conclusion . 93 7 Compressed-State Multi-State Pair-Density Functional Theory 94 7.1 Introduction . 94 7.2 Theory . 97 7.2.1 MC-PDFT . 97 7.2.2 MS-PDFT . 98 7.2.3 CMS-PDFT . 99 7.3 Computational Details . 102 7.4 Results and Discussion . 103 7.4.1 LiF . 103 7.4.2 LiH . 104 7.4.3 HNCO . 105 7.4.4 CH3NH2 ................................. 105 7.4.5 Phenol . 108 7.4.6 Spiro . 109 7.4.7 Computational Cost of Using CMS-, XMS- and FMS- intermediate states111 7.5 Conclusion . 112 8 Concluding Remarks 115 References 116 vii List of Tables 2.1 The Dominant Configuration for the Ground State and the Orbitals with a Changed Number of Electrons in the Dominant Configurations for Excited States at re .................................... 8 2.2 Notations for Three Levels of CASSCF and MR Calculations . 8 2.3 M Diagnostic, Dominant Configurations and the Corresponding Weight of the Configuration in the Wave Function for Each State at the Experimental Equilibrium . 10 2.4 Signed Errors of Adiabatic Excitation Energies (in eV), the Mean Unsigned Error (MUE) Over First n Excitation Energies (MUE_n), and the Rank of MUE_n (rank_n)................................. 13 ˝ 2.5 Signed Errors of Equilibrium Distances (r e in A) . 17 2.6 MUE of re over First n States (MUE_n) and the Rank of MUE_n (Rank_n) . 18 3.1 M Diagnostics and Dominant Configurations at the Equilibrium Geometry As Calculated by FV-CASSCF Calculationsa ................... 26 3.2 Reference Values for Dissociation Energies (De, in kcal/mol) and Equilibrium ˝ Distances (re, in A)............................... 28 3.3 Dissociation Energy, Equilibrium Distance, Mean Signed Error (MSE), Mean Unsigned Error (MUE), and Standard Deviation (StdDev) for Calculations with the Large AR Basis Seta ............................ 29 viii 3.4 Dissociation Energy, Equilibrium Distance, Mean Signed Error (MSE), Mean Unsigned Error (MUE), and Standard Deviation (StdDev) for Calculations with the Large AR Basis Seta ............................ 31 3.5 Dissociation Energy, Equilibrium Distance, Mean Signed Error (MSE), Mean Unsigned Error (MUE), and Standard Deviation (StdDev) for Calculations with KS-DFT Using the PBE Exchange–Correlation Functionala . 32 ˝ 4.1 Geometries in Terms of Bond Lengths (A) and Bond Angles (degree) . 44 4.2 Details of the SA-CASSCF Calculations Used as the Starting Points for MR- CISD+Q Calculations . 45 4.3 Details of ABC Active Spaces . 46 4.4 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for CH3 ........................ 47 4.5 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for NH2 ........................ 48 4.6 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for C2H........................ 49 4.7 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for CN . 50 4.8 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for BO . 51 ix 4.9 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with ` Respect to MR-CISD+Q for N2 ......................... 52 4.10 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for CO` ........................ 53 4.11 Signed Errors (eV), Mean Signed Deviation (MSD), and Mean Unsigned Deviation (MUD) of MC-PDFT and MS-CAPST2 Excitation Energies with Respect to MR-CISD+Q for C2H3 ......................