Excitation-adapted molecular orbitals

Configuration state functions (spin-adapted Slater determinants) constructed from excitation-adapted molecular orbitals (EAMOs) possess minimal off-diagonal elements of the Hamiltonian matrix. These orbitals, which result from separate unitary transformations among the occupied and virtual MOs. offer the most concise description of electronic excited states in terms of electron jumps . For example, at the CIS/6-31 +G level of theory, a symmetric combination of Just two singly excited configurations built from EAMOs [0.7049(7 — 9) -I-0.7049(6 -> 8)] suffices to adequately describe the first triplet excited state of N2, whereas several configurations involving MOs [0.5975 (7 9) -I- 0.5975(6 8) -t- 0.3646(7 16) -l-0.3646(6 15) -I- 0.0858(7 23) -I- 0.0858(6 22)J are... 

Because CASSCF is a full Cl, it can be considered in either an atomic orbital (AO) basis (valence bond [VB] theory) or symmetry-adapted molecular orbital (MO) theory. The two pictures are equivalent, but the VB method is more powerful because (as we discuss more fully below) it can explain why geometries change in excited states and why two potential energy surfaces intersect. In this respect, the VB picture is more appropriate for the reactivity problems we discuss here, whereas MO theory is still key to spectroscopy. 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

Valuable insight, particularly with regard to the effects of electronic excitation on reaction cross sections and reaction dynamics, has also been achieved without accurate knowledge of the actual potential surfaces, through the use of molecular-orbital correlation diagrams. Adiabatic correlation rules for neutral reactions involving polyatomic intermediates were developed by Shuler 478 These were adapted and extended for ion-neutral interactions by Mahan and co-workers.192,45 479,480 Electronic-state correlation diagrams have been used to deduce the qualitative nature of the potential surfaces that control ion-neutral reaction dynamics. The dynamics of the reaction N+(H2,H)NH+ and in particular the different behavior of the N + (3P) and N + ( Z)) states,123 for example, have been rationalized from such considerations (see Fig. 62). In this case the... 

The last equation has been used to analyze occupied-unoccupied localized molecular orbital pair contributions for excitations in chiral metal complexes and metallahelicenes [260, 261], as well as in chiral organic acids derived from amino acids by substitution of the amino group with —OH and —F [170]. The analyses in terms of canonical MOs and LMOs may be considered complementary tools, with the canonical MO analysis generally leading to fewer contributions since the canonical MOs are well adapted to describe electronic excitations. The analysis in terms of LMOs allows one to focus on selected chemist s orbitals of interest, such as contributions to excitations from a given lone pair or localized n orbital, or from metal-centered orbitals, which can also be very useful. 

Fig. 20. Molecular orbital description for the electron transfer reactions within ECL processes of intramolecular donor-acceptor systems A-D to give (A) excited singlet intramolecular charge-transfer state (B) excited triplet intramolecular charge-transfer state (C) locally excited triplet and (D) ground state. Adapted from [138],...

Figure 11.2. Molecular orbital diagrams for O2 and O2. The ground state normal O2 (dioxygen) = triplet ground state) has two unpaired ir-electrons (it is a biradical) and is paramagnetic. Two excited forms of singlet oxygen O2 ( A and E ) can be generated (often by light irradiation in the presence of a sensitizer). The first product in a one-electron reduction of oxygen is the superoxide ) anion. (Adapted from Sawyer, 1991.)...

The fundamental fixed-nuclei approximation, in which the nuclei are treated as distinguishable classical particles with positions in physical space, allows for the electronic structure of ground and excited states of both atoms and molecules to be ruled by the same principles, concepts, and approximations, such as the occupation of symmetry-adapted atomic or molecular orbitals by electrons, subject to the Pauli exclusion principle. The resulting basic interpretative and computational tool is the N-electron symmetry-adapted configuration (SAC), be it atomic or molecular. It is denoted here by i. When the SAC is adopted as a conceptual and computational tool, it is possible to use the same concepts and theoretical methods in order to treat the electronic eigenfunctions of states of both atoms and small molecules for each fixed geometry. 

Fig. 3.3 Some of the aiays in which excited-state wavefunctions can be included in a configuration interaction calculation (Figtne adapted from Hehre W], L Radom, P vR Schleijer and ] A Hehre 1986 Ab initio Molecular Orbital Theory New York, Wiley)...

It is frequently considered that valence-bond theory is not easily adapted to provide qualitative descriptions of molecular excited states. No doubt this is often true. However, for some simple systems, there exists an elementary valence-bond counterpart for each molecular orbital description of the excited state. To demonstrate this point, we shall give consideration here to a few types of electronic excitations. 

Figure 7. Two-dimensional cut of the ground- and excited-state adiabatic potential energy surfaces of Li + H2 in the vicinity of the conical intersection. The Li-EL distance is fixed at 2.8 bohr, and the ground and excited states correspond to Li(2,v) + H2 and Lit2/j ) + H2, where the p orbital in the latter is aligned parallel to the H2 molecular axis, y is the angle between the H-H intemuclear distance, r, and the Li-to-H2 center-of-mass distance. Note the sloped nature of the intersection as a function of the H-H distance, r, which occurs because the intersection is located on the repulsive wall. (Figure adapted from Ref. 140.)...

A promising development in the latter direction was the implementation of a multicon-figurational DFT approach [52] with empirical parameters (in addition to those already contained in the mixed density functionals). This method can yield accurate potential energy surfaces of excited states, and has recently been adapted to perform spin-orbit Cl calculations [53], but the lack of analytic gradients has up to now prevented its use in the simulation of molecular dynamics and in geometry optimizations. A more classically... 

The present work details the derivation of a full coupled-cluster model, including single, double, and triple excitation operators. Second quantization and time-independent diagrams are used to facilitate the derivation the treatment of (diagram) degeneracy and permutational symmetry is adapted from time-dependent methods. Implicit formulas are presented in terms of products of one- and two-electron integrals, over (molecular) spin-orbitals and cluster coefficients. Final formulas are obtained that restrict random access requirements to rank 2 modified integrals and sequential access requirements to the rank 3 cluster coefficients. 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

Fig. 4 Current at films of PcZn (100 nm) vapor deposited on ITO (1 cm ) observed in contact to aqueous electrolytes during potentiostatic polarization. Illumination occurred either in the B band (3 X 10 photons cm s ) or in the Q band (7 x 10 photons cm s ). a In the presence of 0.1 M EDTA (+460 mV), b In the presence of 10 M O2 (-300 mV vs. SCE). c Schematic representation of frontier energy levels and observed photocurrents, (adapted and in part reprinted from Electrochim. Acta D. Schlettwein, E. Karmann, T. Oekermann, and H. Yanagi Wavelength- Dependent Switching of the Photocurrent Direction at the Surface of Molecular Semiconductor Electrodes Based on Orbital- Confined Excitation and Transfer of Charge Carriers from Higher Excited States , p. 4697 704, Copyright (2000), with permission from Elsevier Science)...

Approaches which consider one state at a time are often referred to as one-state or state-selective or single-root . They were first proposed in the late 1970s. A paper published by Harris [113] in 1977, entitled Coupled cluster methods for excited states, first introduced the state-selective approach. Four papers which were published in 1978 and 1979 advancing the state-selective approach parts 6 and 7 of a series of papers entitled Correlation problems in atomic and molecular systems part 6 entitled Coupled cluster approach to open-shell systems by Paldus et al. [114] and part 7 with the title Application of the open-shell coupled cluster approach to simple TT-electron model systems by Saute, Paldus and Cfzek [115], and two papers by Nakatsuji and Hirao on the Cluster expansion of wavefunction, the first paper [116] having the subtitle Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell theory and the second paper [117] having the subtitle Pseudo-orbital theory based on sac expansion and its application to spin-density of open-shell systems. 

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