## Introduction

Ever since Dirac announced in 1929 that “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble,” scientists have been trying to develop efficient numerical strategies to solve the Schrödinger equation for atoms and molecules which allowed quantum chemistry to become an individual scientific discipline. To have a classical interpretation of quantum mechanics, several researchers have been trying to propose different versions of quantum mechanics in three-dimensional (3D) space. Unlike the wave function which is a 3N dimensional function for an N-electron system, the single-particle density is a 3D function and any quantum theory based on the latter would help developing models, making direct connection with experiments, for example, using X-ray, apart from providing a considerable amount of reduction in computational labor. Modern density functional theory (DFT) and its different variants to tackle excited states and time-dependent (TD) situations have provided proper theoretical justification for all such density-based theories heuristically proposed earlier.

Just like the density functional scheme developed by Hohenberg, Kohn, and Sham was adequate for the ground state, a theory was needed to describe the excited states which are difficult to handle in DFT. The generalization of the Hohenberg-Kohn theorem to the excited states was given by Gunnarsson and Lundqvist1 through the thermodynamic version of spin density formalism. von Barth2 pointed out that the exchange correlation energy in their theory should be symmetry dependent; however, in actual calculation exchange correlational energy functional is approximated by the local density approximation (LDA) with no dependence in symmetry. This method hence failed as it does not reduce to a lone determinant.2 To solve this problem, Xα method was incorporated expressing the multiplet energy as sum of one-determinant energies. But all the above methods could not be extended past the lowest state with a given space and spin symmetry. A DFT method which would account for the general excited state was needed and hence Slater's transition state theory3 was a step forward in this direction.

The success of DFT for stationary systems has inspired its applications toward the TD situations. Bloch4 laid the foundations to a TDDFT in the form of TD version of the Thomas–Fermi (TF) theory. Runge and Gross5 showed that the mapping between the TD density and the TD potential is unique in general conditions. However, Xu and Rajagopal6 criticized this fact and indicated that not the TD potential to TD density but the TD potential to the TD current density mapping is uniquely invertible which was later proved wrong by Dhara and Ghosh.7 Another TDDFT for many electron systems is the quantum fluid DFT (QFDFT) which is an amalgamation of TDDFT (TDDFT) and quantum fluid dynamics (QFD). QFDFT has been successfully applied in studies of ion-atom/molecule collisions,8–10 atom–field interactions,11, 12 and dynamics of chemical reactivity parameters13–16 such as electronegativity, chemical hardness, entropy, and polarizability in chemical reactions.

This review essentially focuses on some representative examples through the study of chaos and nonlinear dynamical systems carried out by our group using the QFD and the QFDFT and also the study of the reactivity dynamics of various systems in free as well as confined environment.