UC Berkeley

Density functional theory (DFT) guarantees a map between a system's electron density and energy. Although the exact density functional is still unknown to us today, with the development and benchmark of modern density functional approximations (DFA), DFT remains the most prevalent method for electronic structure calculations due to its good balance between computational cost and accuracy. However, besides the accurate energies, another main concern to chemists is how to interpret the DFT calculation results for understanding the behavior of interesting chemical systems. This dissertation tries to bridge the gap between accurate DFT calculations and insightful chemical interpretations through the analysis of the key component of DFT: the electron density. The work begins with the comparison between two popular energy decomposition analysis (EDA) schemes for intermolecular interactions, namely the extended transition-state method with natural orbitals for chemical valence (ETS-NOCV) and the absolutely localized molecular orbital-based method with complementary occupied-virtual pairs (ALMO-COVP). Careful analysis shows a deep connection as well as subtle differences between the two methods, which also enables the generalization of the two methods to connect and analyze any two single-determinant quantum states. A special pair of interesting chemical states are the frozen (FRZ) and polarized (POL) states defined in ALMO-EDA, which together reveals the on-fragment polarization associated with an intermolecular interaction. A redesigned version of the ALMO-COVP analysis is designed to connect these two states by connecting the densities of each individual fragment following the ALMO constraint during the self consistent field for molecular interaction (SCF-MI) process. The resulting polarization analysis yields individual relaxation energies and associated COVP orbitals for each fragment that sum exactly to the polarization energy $\Delta E_\text{POL}$, which gives an exact effective one-body picture of the true many-body polarization process, accompanied by an orbital picture. The next development introduces the occupied-virtual orbitals for chemical valence (OVOCV) to replace or complement the closely related idea of NOCV, which is associated with erroneously large amount of charge transfer. The OVOCVs block-diagonalize the density difference operator into $2\times2$ blocks, which are spanned by one level that is filled in the initial state (the occupied OVOCV) and one that is empty (the virtual OVOCV). By contrast, the NOCVs fully diagonalize the density difference matrix and resulting in orbitals with mixed occupied-virtual character. Use of the OVOCVs makes it much easier to identify the donor and acceptor orbitals. In addition, OVOCVs correct the amount of charge associated with each occupied-virtual orbital pair. The total, summed over all pairs, is exactly the same as the ALMO charge decomposition analysis (ALMO-CTA) results and the independently suggested excitation number. Leveraging the new OVOCV analysis, the focus of the next topic is to try to reconcile the amount of charge transfer defined in real space by charge displacement function (CDF) and in Hilbert space by ALMO-CTA. It is shown that the standard real-space CDF analysis mixes the charge flows due to Pauli repulsion, polarization and charge transfer, which can be separated by employing the intermediate frozen orbital (FRZ) and polarized (POL) states from ALMO-EDA. Using these states reduces the discrepancy between CDF and ALMO-CTA from roughly a factor of ten to a factor of three. In addition, the CDF CT charge is demonstrated to be the net electron flow due to both forward and backward donation, while the ALMO-CTA separates forward and back-donation. In the limit of long distance unidirectional charge transfer, these two methods agree with each other. Beginning from the OVOCV analysis between any two single-determinant states, the next topic is to develop a suitable generalization for the analysis of excited states obtained from orbital-optimized density functional theory (OO-DFT) calculations. An intermediate frozen state that is polarization-free is introduced to cleanly separate the primary excitation from the accompanying orbital relaxation of spectator orbitals. It is thus possible to quantify the orbital relaxation effect and to visualize the relevant orbitals using OVOCVs. Moving forward from the single reference OO-DFT state, it is also possible to generalize the analysis to two-determinant excited states obtained from restricted open-shell Kohn-Sham (ROKS) theory, which is the first step towards future work on analyzing multi-reference states. In summary, this work provides significant contributions to computational chemistry by offering both qualitative insights and quantitative analysis for intermolecular interactions and electronic excitations. The summary given here has concentrated on the formal development, but future value will also come from chemical applications of these methods. A range of chemical examples are given for each method to illustrate the potential for future applications. To make this possible, efficient implementation of all methods have been incorporated in the widely used Q-Chem package.