Chapter 1
Motivation and Basic Concepts

Sandra Gómez1, Ignacio Fdez. Galván2, Roland Lindh2, and Leticia González1

1Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Vienna, Austria

2Department of Chemistry - BMC, Uppsala University, SE-751 23 Uppsala, Sweden

Abstract

This chapter describes what electronic excited states are and why they are important to study and therefore motivates the need for theoretical tools able to characterize them. Further and most importantly, in this introductory chapter, we put together in a comprehensive manner a collection of basic concepts that might be needed, depending on the background of the reader, to understand the remaining chapters of this book.

1.1 Mission and Motivation

When a photon of light strikes a molecule, the latter's electrons are promoted from the electronic ground state to higher electronic levels. Typically, the electronic ground state of a molecule is a singlet state, but depending on the number of electrons and their most favorable way of pairing, it can be a doublet, a triplet, or a state of higher multiplicity. Assuming the electronic ground state is a singlet, upon light absorption the molecule will be excited to another singlet state, as high in energy as the energy contained in the photon allows. Once excited, a number of radiative and non-radiative decay processes are possible. These are collected in the Jablonski diagram shown in Figure 1.1(a), which assumes an electronic singlet ground state.

Figure 1.1 (a) Jablonski diagram with levels. After absorption of a photon with energy , different processes can occur: radiative processes are fluorescence (F) and phosphorescence (P), non-radiative processes are internal conversion (IC) and intersystem crossing (ISC). (b) Jablonski diagram with potential energy surfaces.

Radiative processes include fluorescence or phosphorescence, depending on whether the emission of light involves a transition between two states of the same multiplicity, for example from the lowest singlet to the , or involves a change of spin, as shown in Figure 1.1, from the triplet to the . Typically, as in the example depicted, the emitted light has a longer wavelength than the absorbed radiation because luminescence occurs from lower energy levels, and thus absorption and emission spectra are easy to identify from experimental data. In this example, the molecule returns to the original ground state from where it started and thus there was no photochemical reaction, one would say that a photophysical process has taken place.

Non-radiative processes can be much more complicated to observe experimentally, as they typically involve not only the bright or absorbing state defined by the wavelength employed to irradiate, but also dark states, i.e., states that do not have a significant oscillator strength but are populated from the bright states. A transition between electronic states of the same multiplicity is known as internal conversion, e.g., from to . When two states of different multiplicities are involved, e.g., from the to , one speaks of intersystem crossing.

The electronic levels of a molecule are defined through potential energy surfaces (PES) that extend along dimensions (with the number of atoms contained in the molecule). PES are the direct consequence of invoking the Born-Oppenheimer approximation (BOA), see section 1.7. As comfortable as it might seem for a chemist to employ electronic states to envision the course of a chemical reaction from a reactant to a product, sticking to the BOA when talking about electronic excited states implies that the coupling between different PES is neglected. However, these so-called non-adiabatic couplings between PES are the "salt and pepper" of photochemistry, as they are essential to understand which states and geometrical conformations are populated after excitation. One key concept in this respect is the non-adiabatic transition around a conical intersection, see section 1.9. Named after the ideal topology two PES adopt when they intersect (see Figure 1.1(b)), a conical intersection is the molecular funnel that allows for internal conversion, and it can also be seen as the transition state in photochemistry, which connects a reactant with a product. Likewise, intersystem crossing is mediated by spin-orbit coupling, which is another form of vibronic or non-adiabatic coupling between electronic levels.

Figure 1.1(b) summarizes the radiative and non-radiative processes described before, now in terms of PES. If after the detour via the different PES, the molecule ends up at a different geometrical configuration from which it started after irradiation, one speaks of a photochemical reaction; if instead, it returns back to the electronic ground state of the reactant, the term photophysics is employed.

Be it photophysics or photochemistry, light-induced processes are all around us. As Ciamician already recognized in 19121, "reactions caused by light are so many, that it should not be difficult to find some of practical value". Indeed, just to give one representative example, the dream of using solar fuel to produce sustainable energy is keeping many scientists around the world busy. In an effort to mimic natural photosynthesis, one needs among others, to design efficient antenna complexes able to harvest the broad solar spectrum and direct the electrons towards the catalytic centers. This design requires a profound understanding of the underlying processes that take place in the molecules after light excitation. Theoretical modeling can help explain existing experiments and hopefully guide new ones. Which are the electronic states that are populated after excitation? How does the molecule evolve along the complicated PES associated to these electronic states? Often these two simple questions are not easy to answer. They imply a need to get an accurate solution of two key equations, the electronic time-independent Schrödinger equation and the time-dependent Schrödinger equation. Both equations are challenging to solve, except for very small molecules, and so approximations and numerical strategies are required. The solution of the first equation is the goal of electronic structure theory and the solution of the second, the target of chemical dynamics. Both fields have tremendously evolved in the last decades, with the emergence of many different methods that have a common objective.

The mission of this book is to keep up-to-date with the recent development in these two intertwined fields, setting the focus at solving electronic excited states and following their time evolution. Accordingly, Part I collects the most important electronic structure methods that can be used nowadays to calculate electronic excited states as well as associated PES and other electronic properties. Part II, in turn, covers the state of the art for solving molecular motion in the electronic excited states. The variety and extension of the methods collected in this book speaks for itself about how much progress has been achieved in this branch of theoretical chemistry, which undoubtedly has also massively profited in the last years from enormous advances in computational resources. It would not be fair, however, to pretend that theoretical photochemistry has reached its cusp. A deeper reading of the chapters will reveal to the reader not only how far we have come but also how much still remains to be done.

In an effort to make the contents of this book accessible to undergraduates and newcomers to the field, the rest of this chapter contains a number of basic concepts to ease the reading. All the chapters have been written in a fully consistent manner, so as to allow them to be studied independently from the others. The chapters are, nevertheless, organized such that they try to reflect a natural progression. In this respect, the chapters are grouped in two sections consisting of Part I and Part II - electronic structure theory and methods for molecular dynamics, respectively.

In the electronic structure section the selected order of the chapters tries, to some extent, to be in the order of sophistication. However, in some cases chapters are clustered together because of common grounds or methodology. In that sense, Part I starts with the chapters based on density functional theory (DFT) - the chapters on time-dependent DFT (TD-DFT) and multi-configurational DFT (MC-DFT). This is followed by chapters revolving around equation-of-motion coupled cluster theory (EOM-CC) and the algebraic-diagrammatic construction (ADC) scheme for the polarization propagator, which are grouped together due to the technical similarities of the methods. Finally, five chapters are grouped together based on the use of a configurational interaction (CI) type of wave function. Initially, the basics of the so-called complete active space SCF (CASSCF) and related methods - the foundation of multi-configurational quantum chemistry - is introduced. This is followed by two chapters on techniques describing how to solve the associated equations - the chapters on density matrix renormalization group (DMRG) and the quantum Monte-Carlo (QMC) approaches. To conclude Part I, two chapters about the inclusion of electronic dynamical correlation follow - the chapters on the multi-reference...