The theory of chemical bond formation in molecules and extended crystals is outlined. We start from the Born–Oppenheimer approximation, which associates the forces experienced by nuclei to the quantum electronic state. The Schrödinger equation for diatomic molecules reveals the formation of stable molecules when electrons are occupying “bonding” molecular orbitals. These are linear combinations of atomic orbitals (LCAO), in which the nuclei “share” electrons that effectively mask the electrostatic repulsion between them. The formation of effective LCAOs relies on compatibility in symmetry and energy of the underlying atomic orbitals. This is ubiquitously found in covalently bounded molecules, including conjugated polyatomic molecules. In the absence of effective LCAO, ionic bonds can be formed by charge transfer between atomic orbitals. In periodic lattices, effective LCAOs result in broad energy bands, which increase electrical conductivity. Conductor-to-insulator transition in response to the type of LCAO in the underlying material is demonstrated for a model system.

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Formulas are derived for the rates of elementary processes in nanoscale systems. Particularly we derive thermal rate constants for charge transfer in a condensed phase environment (Marcus formula), electronic energy transfer between chromophores (Forester resonant energy transfer), and radiation emission/absorption by electronic and vibronic transitions in molecules. All these processes are characterized by changes in the electronic state, strongly coupled to nuclear motions in the nano-system or in its surroundings. The relevant systems are mapped on a generic spin-boson model Hamiltonian, where different meanings are assigned to the model parameters in the different scenarios. In each case, rate constants are derived under appropriate approximations and are identified as different realizations of Fermi’s golden rule. A semiclassical (low-frequency) approximation applied for the nuclear degrees of freedom yields transparent, well-known formulas for the thermal transition rates. The underlying physics as well as practical consequence of the results are analyzed.