Density-Functional Methods for Excited States

We review the theoretical foundation of constricted variational density functional theory and illustrate its scope through applications.

Ensemble density functional theory (DFT) is a novel time-independent formalism for obtaining excitation energies of many-body fermionic systems. A considerable advantage of ensemble DFT over the more common Kohn–Sham (KS) DFT and time-dependent DFT formalisms is that it enables one to account for strong non-dynamic electron correlation in the ground and excited states of molecular systems in a transparent and accurate fashion. Despite its positive aspects, ensemble DFT has not so far found its way into the repertoire of methods of modern computational chemistry, probably because of the perceived lack of practically affordable implementations of the theory. The spin-restricted ensemble-referenced KS (REKS) method is perhaps the first computationally feasible implementation of the ideas behind ensemble DFT which enables one to describe accurately electronic transitions in a wide class of molecular systems, including strongly correlated molecules (biradicals, molecules undergoing bond breaking/formation), extended π-conjugated systems, donor–acceptor charge transfer adducts, etc.

Recent advances in reduced density matrix functional theory (RDMFT) and linear response time-dependent reduced density matrix functional theory (TD-RDMFT) are reviewed. In particular, we present various approaches to develop approximate density matrix functionals which have been employed in RDMFT. We discuss the properties and performance of most available density matrix functionals. Progress in the development of functionals has been paralleled by formulation of novel RDMFT-based methods for predicting properties of molecular systems and solids. We give an overview of these methods. The time-dependent extension, TD-RDMFT, is a relatively new theory still awaiting practical and generally useful functionals which would work within the adiabatic approximation. In this chapter we concentrate on the formulation of TD-RDMFT response equations and various adiabatic approximations. None of the adiabatic approximations is fully satisfactory, so we also discuss a phase-dependent extension to TD-RDMFT employing the concept of phase-including-natural-spinorbitals (PINOs). We focus on applications of the linear response formulations to two-electron systems, for which the (almost) exact functional is known.

This chapter gives an overview of the description of the optical and dielectric properties of bulk insulators and semiconductors in time-dependent density-functional theory (TDDFT), with an emphasis on excitons. We review the linear-response formalism for periodic solids, discuss excitonic exchange-correlation kernels, calculate exciton binding energies for various materials, and compare the treatment of excitons with TDDFT and with the Bethe–Salpeter equation.

We present a review of different computational methods to describe time-dependent phenomena in open quantum systems and their extension to a density-functional framework. We focus the discussion on electron emission processes in atoms and molecules addressing excited-state lifetimes and dissipative processes. Initially we analyze the concept of an electronic resonance, a central concept in spectroscopy associated with a metastable state from which an electron eventually escapes (electronic lifetime). Resonances play a fundamental role in many time-dependent molecular phenomena but can be rationalized from a time-independent context in terms of scattering states. We introduce the method of complex scaling, which is used to capture resonant states as localized states in the spirit of usual bound-state methods, and work on its extension to static and time-dependent density-functional theory. In a time-dependent setting, complex scaling can be used to describe excitations in the continuum as well as wave packet dynamics leading to electron emission. This process can also be treated by using open boundary conditions which allow time-dependent simulations of emission processes without artificial reflections at the boundaries (i.e., borders of the simulation box). We compare in detail different schemes to implement open boundaries, namely transparent boundaries using Green functions, and absorbing boundaries in the form of complex absorbing potentials and mask functions. The last two are regularly used together with time-dependent density-functional theory to describe the electron emission dynamics of atoms and molecules. Finally, we discuss approaches to the calculation of energy and angle-resolved time-dependent pump–probe photoelectron spectroscopy of molecular systems.

Measuring the nonlinear response of electrons and nuclei to attosecond broadband X-ray radiation has become possible by newly developed free electron lasers and high harmonic generation light sources. The design and interpretation of these novel experiments poses considerable computational challenges. In this chapter we survey the basic description of nonlinear X-ray spectroscopy signals and the electronic structure protocols which may be used for their simulation.

In this chapter we present applications of TD-DFT aiming at reproducing and rationalizing the optical signatures of molecules, and, more precisely, the absorption and fluorescence spectra of conjugated compounds belonging to both organic and inorganic families. We particularly focus on the computations going beyond the vertical approximation, i.e., on the calculation of 0–0 energies and vibronic spectra with TD-DFT, and on large applications performed for “real-life” structures (organic and inorganic dyes, optimization of charge-transfer structures, rationalization of excited-state proton transfer, etc.). We present a series of recent applications of TD-DFT methodology for these different aspects. The main conclusions of TD-DFT benchmarks aiming at pinpointing the most suited exchange-correlation functionals are also discussed.

Absorption spectroscopy, emissive properties, and ultrafast intersystem crossing processes in transition metal complexes are discussed in the light of recent developments in time-dependent density functional theory (TD-DFT) , spin-orbit coupling (SOC) effects, and non-adiabatic excited states dynamics. Methodological highlights focus on spin-orbit and vibronic couplings and on the recent strategies available for simulating ultra-fast intersystem crossings (ISC).

The role of SOC in the absorption spectroscopy of third-row transition metal complexes is illustrated by two cases studies, namely Ir(III) phenyl pyridine and Re(I) carbonyl bipyridine complexes.

The problem of luminescence decay in third-row transition metal complexes handled by TD-DFT linear and quadratic response theories including SOC is exemplified by three studies: (1) the phosphorescence of Ir(III) complexes from the lowest triplet state; (2) the emissive properties of square planar Pt(II) complexes with bidentate and terdentate ligands characterized by low-lying metal-to-ligand-charge-transfer (MLCT) and metal-centered (MC) states; and (3) the ultra-fast luminescence decay of Re(I) carbonyl bipyridine halides via low-lying singlet and triplet charge transfer states delocalized over the bipyridine and the halide ligands.

Ultrafast ISC occurring in spin crossover [Fe (bpy)₃]²⁺, in [Ru (bpy)₃]²⁺, and [Re (Br)(CO)₃(bpy] complexes are deciphered thanks to recent developments based on various approaches, namely non-radiative rate theory within the Condon approximation, non-adiabatic surface hopping molecular dynamics, and quantum wave packet dynamics propagation.

Nonadiabatic dynamics simulation of electronically-excited states has been a research area of fundamental importance, providing support for spectroscopy, explaining photoinduced processes, and predicting new phenomena in a variety of specialties, from basic physical-chemistry, through molecular biology, to materials engineering. The demands in the field, however, are quickly growing, and the development of surface hopping based on density functional theory (SH/DFT) has been a major advance in the field. In this contribution, the surface hopping approach, the methods for computation of excited states based on DFT, the connection between these methodologies, and their diverse implementations are reviewed. The shortcomings of the methods are critically addressed and a number of case studies from diverse fields are surveyed.

Conical intersections are perhaps the most significant mechanistic features of chemical reactions occurring through excited states. By providing funnels for efficient non-adiabatic population transfer, conical intersections govern the branching ratio of products of such reactions, similar to what the transition states do for ground-state reactivity. In this regard, intersections between the ground and the lowest excited states play a special role, and the correct description of the potential energy surfaces in their vicinity is crucial for understanding the mechanism and dynamics of excited-state reactions. The methods of density functional theory, such as time-dependent density functional theory, are widely used to describe the excited states of large molecules. However, are these methods suitable for describing the conical intersections or do they lead to artifacts and, consequently, to erroneous description of reaction dynamics? Here we address the first part of this question and analyze the ability of several density functional approaches, including the linear-response time-dependent approach as well as the spin-flip and ensemble formalisms, to provide the correct description of conical intersections and the potential energy surfaces in their vicinity. It is demonstrated that the commonly used linear-response time-dependent theory does not yield a proper description of these features and that one should instead use alternative computational approaches.