The Methodology Development For The Electronically Excited State Within The Framework Of Time-dependent Density Functional Theory

In this thesis, I present the major works that I have done during my PhD study. My research is focused on developing the effective time-dependent density functional method (TDDFT) to study the excited state properties of the complex molecule, such as the force and the force constant that are given as the first-order and the second-order energy derivatives with respect to the nuclear perturbations, respectively; and the dipole moments and polarization that can be computed as the first-order and second-order energy derivatives with respect to an external electric perturbations. By the de-velopment of the effective methods to obtain the excited state structure and dynamics properties, it can theoretically provide an deep insight on understanding and regulat-ing the photophysical and photochemical processes of the organic molecules, biolog-ical systems, and natural or artificial photosynthesized systems since those processes are important in energy and environmental sciences, and photovoltaic device design, etc. The following three manifolds are concerned:(1) Developing the linear scaling method for TDDFT excited-state force. The developed algorithms provide a compu-tational efficient theoretical tool to study the excited-state properties and dynamics of large molecules.(2) Developing the analytical second derivative of the excited state en-ergy within the framework of TDDFT, which is essential for many purposes, ranging from the evaluation of chemical kinetics and thermodynamic properties such as en-tropies and free energies to the analysis of infrared and Raman spectra.(3) By coupled with the widely used Polarizable Continuum Model (PCM), we develop the analytical approaches for the second derivatives of electronically excited state within the frame-work of Conductor-like PCM (CPCM)-TDDFT to study the excited state properties, such as the electronic excitation, structure relaxation, vibrational frequencies and in-frared intensities of the vibrations of chromophores in solution. All these algorithms are implemented in Q-CHEM quantum chemistry software package. The main results are presented as follows:1. Starting from the equation of motion in the density matrix formulation, we reformulate the analytical gradient of the excited-state energy at the time-dependent density functional theory level in the nonorthogonal Gaussian atom-centered orbital (AO) basis. Analogous to the analytical first derivative in molecular-orbital (MO) ba-sis, a Z-vector equation has been derived with respect to the reduced one-electronic density matrix in AO basis, which provides a potential possibility to realize the linear scaling excited state algorithm by exploiting quantum locality of the density matrix and avoiding the matrix transformation between the AO and MO basis.2. Developing the analytical approaches for the electronic excited-state Hessian in TDDFT with and without TDA approximations. They have been used to study the excited state vibrational frequencies of a series of molecules. The calculated results are compared with those from the finite-difference method, the method based on single excitation Configuration Interaction and the experimental results. The good agreement indicates the high computational accuracy of the analytical approaches. At the same time we compare the CPU times spent to calculate the excited state vibrational frequen-cies by different methods to investigate the computational efficiency of the developed analytical approaches. Furthermore we compare the vibrational frequencies calculated by the TDDFT with and without Tamm-Dancoff approximation. The results demon-strate that the accuracy of TDDFT with and without TDA are fairly comparable to each other for our chosen test set. This manifests that the Tamm-Dancoff approximation to TDDFT does not produce a significant error on the excited-state frequencies. At last this method is used to study the scale factor for the excited state vibrational frequencies calculation.3. Developing the analytical excited-state Hessian method within the framework of time-dependent density functional theory (TDDFT) to couple with the conductor-like polarizable continuum model. It demonstrates the correctness of numerical imple-mentation of this method by the calculation of the excitation energy, optimization of the geometry structure and the calculation of the vibrational frequencies. Then the reso-nant Raman spectrum of4-hydroxybenzylidene-2,3-dimethyl-imidazolinone in ethanol and the excited-state IR spectrum of9-Fluorenone in methanol solution are studied. These tests and applications with our developed methods have shown the correctness and efficiency of these methods. It is expected these methods will provide an efficient tool to study the properties of the excited state.