Orbitals state-specific

A very widely employed method for the measurement of spin-orbit state-specific rate constants is the time-resolved measurement of the concentrations of individual atomic levels after formation of these species from a suitable precursor, either by flash photolysis [13], or, more recently, by laser photodissociation. The concentrations of the various atomic reactant states are monitored by atomic absorption or fluorescence spectroscopy using atomic emission sources [14], or, for spin-orbit-excited states, by observation of the spontaneous infrared emission [15-18]. Recently, Leone and co-workers have utilized gain/absoiption of a colour centre and diode infrared laser to probe the relative populations of ground and spin-orbit excited halogen atoms produced in a chemical reaction [19] and also by photodissociation [20],... 

The features in Figure 15.2 are believed to be due to electron transitions from the highest occupied molecular orbital states to lowest unoccupied molecular orbital states (HOMO-LUMO). Specifically, for single molecules and dilute solutions, the absorption in the blue part of the spectrum is proposed to be caused by the aromatic structure of TNT [4], probably involving it to it transitions.1 In the solid state, where the molecules are stacked up on top of each other, interactions between the molecules occur causing the energy levels to split into higher... 

Different electronic states have in many cases veiy differently shaped orbitals and the error introduced by using a common set cannot always be fully recovered by the MR-CI treatment. A well optimized wave function is especially important for the calculation of transition properties like the transition moments and the oscillator strength. A state specific calculation of the orbitals is more important for obtaining accurate values of the transition moments than extensive inclusion of correlation. Since excited states commonly exhibit large near-degeneracy effects in the wave function an MCSCF treatment then becomes necessary. 

The Pt 6s and 6p orbitals are very diffuse, whereas the Pt 5d states are more localized orbitals with specific geometries. Due to their delocalized nature, the Pt 6s,p states will have more interaction with the support than the localized Pt 5d states. Consequently, the Pt 6s,p states are affected the most, even though the Pt 5d states are shifted to lower binding energies for basic supports. Since the Pt 6s and p states are more important for chemisorption of H than for CH3 adsorption, it can be concluded that the influence of the support is less important for the chemisorption of CH3. 

Chattopadhyay, S. Mahapatra, U. S. Mukheijee, D. Property calculations using perturbed orbitals via state-specific multireference coupled-cluster and perturbation theories, J. Chem. Phys. 1999, 111, 3820-3831. 

State-specific reactions of excited alkali metal atoms have been encountered a number of times. We have mentioned already the K -I- H2 KH - - H reaction which is turned on by excitation of potassium to the 7s level (ca. 3.7 eV above the ground state), and not by the excitation to the 5d D level, although it is only 0.01 eV below the 7s level [155, 156]. Such a change in reactivity is, of course, linked to the different potential energy surface on which the reaction is promoted, but owing to the equivalent energy of the 7s S and 5d D levels the difference in reactivity is connected with orbital symmetry. 

The second notion is concerned with aspects of the issue of the a priori identification of ND and D correlations and the choice of the state-specific set of zero-order orbitals and multiconfigurational wavefunctions in terms of which this identification is assumed and implemented. In this context, 1 use examples from published results and from new computations. 

The choice of so as to satisfy flo 1 is what the introduction of the concept of the Fermi-sea of orbitals is all about. (1 note that the methodology toward the satisfaction of this criterion is understood much better today than in the late 1960s and early 1970s.) It may involve either state-specific diabatic states, for example [35a, 35b], MCHF-type calculations, or in special cases as in the example (4.1) on the ethylene molecule and in example (4.2) on the computation of electron correlation in Be, NONCl calculations. The background and the syllogism that led to the Fermi-sea is discussed in the following sections. The implementation of the criterion is demonstrated in Section 10 with new calculations at the CASSCF (4>°) and MRCISD (flo F -h 4> ") level for the first 11 E+ states of Be2. 

On the other hand, by emphasizing the state-specific calculation of the wavefunctions of initial and final states and by taking into account orbital NON, it is possible to understand semiquantitatively multiple electron excitations in atoms even at the SCF level. Such one-photon excitations may reach doubly, triply, or even quadruply excited unstable states. Given the existing high-energy photon sources, in atoms these are measurable. Two examples of transitions whose oscillator strengths are finite and reasonable even without the inclusion of electron correlation, are as follows ... 

However, aspects of the problem of photon-induced excitation of MES can be understood from the point of view of the SPSA even at the SCF level, provided that the proper combination of state-specific orbital symmetries is present. This is because NON of the two sets of orbitals emerges naturally as a physically meaningful (it accounts for relaxation in a simple way) quantitative factor that contributes to the amplitude of the simultaneous excitation of many electrons. Of course, when good accuracy is desired, electron correlation in initial and final states must be added. 

This became possible not only by the state-specific nature of the computations but also by the realization that the natural orbitals produced from hydrogenic basis sets were the same as the MCHF orbitals that are computable for the intrashell states up to about N = 10 - 12. Therefore, for DES with very high N, instead of obtaining the multiconfigurational zero-order wavefunction from the solution of the SPSA MCHF equations (which are very hard to converge numerically if at all), we replaced the MCHF orbitals by natural orbitals obtained from the diagonalization of the appropriate density matrices with hydrogenic orbitals. 

The self-consistenfly compufed sum of < o and /mt in the formalism of [77], whose simplest implementation is the case of the Be ground state discussed earlier, can be recognized as the formal description of the widely used, in later decades, computationally powerful model of the CASSCF wavefunc-tion implemented in a very effective way by Roos and Siegbahn [1, 2]. The key question has to do with the choice of the active space of zero-order spin orbitals. In fact, these concepts follow from the general criterion of Eq. (8) that has led to the analysis of electronic structures in terms of state-specific Fermi-seas (see next section). 

Not only is the mixing different but also the extent of the 2p orbitals in the two states differs. (On the contrary, the 2s orbitals are nearly the same, with (r) = 2.5 a.u.). Specifically, for the state, r)2p = 2.87 a.u., whereas for the P" state, the 2p MCHF orbital is much more diffuse, with r) = 3.57 a.u. This fact implies that the principal state-specific correlation functions from... 

In symbols, the general compass is the form + 0 " of the wave-function of Eq. (8). In principle, the two parts are represented by different function spaces, whose elements and size depend on the problem. The zero-order wavefunction, is normally obtained self-consistently. Its orbitals belong to the state-specific Fermi-sea, see below. 

In particular, as regards its implementation, first to atomic structures and subsequently to diatomic ones, this has been done in the following way In order to secure the accuracy of the Fermi-sea orbitals, we compute via the numerical solution of the state-specific HE or, most frequently, MCHF equations. The choice of the components of 0 " that are considered relevant to the overall calculation depends on the desired level of accuracy and the property. They are expressed in terms of analytic functions, whose final optimization is done variationally to all orders. 

The introduction in the early 1970s of the concept and the methodology of the Fermi-sea as the zero-order orbital set for the construction of the state-specific multiconfigurational wavefunction played on the themes... 

F. A. Evangelista, J. Gauss, Insights into the orbital invariance problem in state-specific multireference couple cluster theory, J. Chem. Phys. 133 (2010) 044101. 

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