Describing excited states

The Hartree-Fock method adequately describes the ground state of most molecules. However, the exact wave function itself should take into account the fact that electrons repel each other and need breathing space. The electrons should be allowed to make use of energy levels which are normally empty in the ground state to maintain this breathing space. In other words, to add terms describing excited states in the ground state wave function. 

With regard to the former, one would like to include as many important configurations as possible. Unfortunately, the definition of an important configuration is often debatable. One popular remedy is the full-valence complete active space SCF (CASSCF) approach in which configurations arising from all excitations from valence-occupied to valence-virtual orbitals are chosen. [29] Since this is equivalent to performing a full Cl within the valence space, the full-valence CASSCF method is limited to small systems. Nevertheless, the CASSCF approach using a well-chosen (often chemically motivated) subspace of the valence orbitals has been shown to yield a much improved depiction of the wave function at all points on a potential surface. Furthermore, the choice of an active space can be adjusted to describe excited state wave functions. 

Spectroscopists usually use notations based upon the symmetry properties of wave functions to describe excited states. Since many molecules have no special symmetry properties, such devices are not really strictly applicable in general. However, symmetry notation applicable to related molecules of high symmetry is often extended to unsymmetrical systems. The procedure has been treated formally by Platt (7) who introduced the concept of local symmetry. ... 

Standard wavefunction methods (i.e., other than DFT), which have been extensively applied both to the computation of vertical (i.e., at ground state equilibrium geometry) excitation energies and excited state reaction paths are the current preferred method for applications in this field. Wavefunction methods that are used in studying photochemical mechanisms are limited to those that can describe excited states correctly. Unfortunately, standard methods for the evaluation of the ground state PES such as SCF and DFT cannot describe excited states because they are restricted to the aufbau principle. 

In the frame of the hybrid methods it must be computed by a QM method. The Schrodinger equation with the effective Hamiltonian Hjf has multiple solutions, which describe excited states of the R-system provided the M-system is frozen in its ground state. Electronic energy of the system in the state expressed by the wave function eq. (1.231), has the form [29,30] ... 

For years, the capabilities of TDDFT to describe excited states have been limited to isolated molecules, despite the fact that a large part of the spectroscopic experiments probe molecules in liquids. However, in the last few years there have been several extensions of the TDDFT to describe excited state of molecules in solution. These extensions are of particular interest as they will allow to expand the areas of application of the TDDFT to several photophysical and photochemical processes in condensed phase. 

The few attempts at describing excited states in transition metal complexes within the restricted Hartree Fock (RHF) formalism were rapidly abandoned because of computational difficulties (convergence of the low-lying states in the open-shell formalism) and theoretical deficiencies (inherent lack... 

Let T be the wavefunction of the above described excited state of the crystal, satisfying the stationary Schrodinger equation... 

The few attempts at describing excited states in transition metal complexes within the Restricted Hartree Fock (RHF) formalism were rapidly abandoned due to the computational difficulties (convergence of the low-lying states in the open-shell formalism) and theoretical deficiencies (inherent lack of electronic correlation, inconsistent treatment of states of different multiplicities and d shell occupations). The simplest and most straightforward method to deal with correlation energy errors is the Configuration Interaction (Cl) approach where the single determinant HF wave function is extended to a wave function composed of a linear combination of many de-... 

An analysis of the time-resolved fluorescence in excimer-forming polymer systems can also prove informative in elucidating the conformations of macromo-lecular chains in solution these data are particularly relevant in terms of the structures adopted by polyelectrolytes in aqueous media. In this latter context, it is useful at this stage to briefly overview the development of the kinetic schemes to describe excited state interactions in fluorescent polymer solutions. 

As computational methods for describing excited states have been refined, additional understanding of the structures has developed. Relatively early computational studies provided some indication of the geometries associated with the butadiene excited states.The ground state has a maximum at a twist of 90° about the C(l)-C(2) bond. This structure, which can be approximately described as a singlet methylene-allyl diradical, is found at about 2.3 eV and is more stable than a structure with 90° twist at both terminal groups (3.1 eV). There is no major pyramidalization of the methylene groups in this second structure. The spectroscopic (Franck-Condon) state is about... 

As long as a satisfactory multireference coupled-cluster theory is missing, there are various options for states that need a zeroth-order multiconfiguration wave function. One possibility is to start from an MC-SCF calculation and to improve this by selected Cl. Since the MC-SCF part is basically extensive, while the Cl part is not, and since one can hardly go beyond external double excitations, one tends to include as many configurations in the MC-SCF part as possible. However, MC-SCF is usually of CAS (complete active space) [154] type, e.g. like full Cl, which restricts the possible size of the active space. Such multireference Cl scheme have been very popular for describing excited states, reaction barriers, dissociation processes etc. 

The other orbitals, which are empty in the Hartree product, can be used to describe excited states. 

Of the three approximations outlined, the one-electron approximation, i.e., the local density approximation, is the weakest. This approximation fails to replicate features such as image charges at a surface. Also, it fails to describe excited states properly, but this is not a concern for structural energies. Despite such shortcomings, the local density remains one of the most successful approaches to calculating many-body energies in solids. 

Valence bond (VB) theory. In VB theory one starts with the occupied atomic orbitals of the atoms and constructs a many-electron wave function to describe bonding directly in terms of these atomic orbitals. Although similar to MO, the differences will become transparent (below). VB theory is most useful for describing reactions and bond dissociations because the many-electron states of the atoms are built into VB. However, VB is computationally much more complicated than MO, and it is much less obvious how to describe excited states in terms of VB. Important chemical concepts such as resonance are based on VB concepts. 

Configurations involving the orbital describe excited states of the molecule. As an example of an excited state, you can write the orbital diagram... 

In order to calculate an absorption spectrum, knowledge of the ground state, however, is not enough. We need also a proper description of the excited state, since excitations are processes that involve both. Here, the above-mentioned methods break down. In some cases it is possible to use external constraints, like an externally fixed (non-ground-state) multiplicity, to describe excited states with a ground-state method. But besides these exceptions we need to use more elaborate approaches to get a good picture of an excitation. 

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