Spectroscopic effective Hamiltonian model

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

Once the rotational spectrum of a molecule is obtained, it must be analyzed. Such an analysis insures that the transitions observed correspond to the correct energy level differences. The data is fit to a molecular model, the so-called effective Hamiltonian , which describes the quantum mechanical interactions in a given species, and spectroscopic constants are obtained. As mentioned, these constants can be used to predict rotational transitions that could not be measured. Naturally, the model must be extremely accurate for the constants to have predictive power to 1 part in 10 or 10 , A typical Hamiltonian for a radical species might be ... 

To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part. Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models. 

Equation (12) shows the key significance of the effective Hamiltonian which is directly related to the spectroscopical and dynamical observables (line-shapes and transition probabilities). The effective Hamiltonian (7) can be written as the sum of the projection into the model space of the exact Hamiltonian and of the energy-shift operator [8,9] ... 

It is known that the UV-vis absorption of the PMDA-ODA film consists of three peaks centered at 6.4 eV (194 nm), 5.9 eV (210 nm), 4.4 eV (280 nm), and a very weak absorption tail around 3.3 eV (376 nm) [37,45,60,62,63]. LaFemina et al. [60,64], on the basis of the experimentally measured absorption spectrum, computed the electronic transition energies in PMDA-ODA using the spectroscopically parameterized CNDO/S3 model and compared with the experimental results. Fig. 20 schematically depicts the orbital charge density for HOMO and LUMO in the PMDA-ODA. It should be noted that the charge at HOMO and LUMO is localized on the ODA and PMDA residues, respectively. This means that CT can take place via the one-electron HOMO LUMO transition. Bredas and Clarke [65] earlier observed similar charge segregation behavior on the valence effective Hamiltonian non-empirical method for PMDA-ODA. Matsumoto [66,67] also confirmed this phenomenon from the INDO/S MO calculation for a large model compound, PA-ODA-PMDA-ODA-PA. 

Since the open-shell term in P does not possess spherical symmetry, the effective Hamiltonian will contain a non-spherical potential and as a result, even with initial orbitals of true central-field form (i.e. with spherical-harmonic angle dependence), the first cycle of an SCF iteration will destroy the symmetry properties of the orbitals—the solutions that give an improved energy will not be of pure s and p type but will be mixtures. This is a second example of a symmetry-breaking situation, akin to the spin polarization encountered in the UHF method. The resultant many-electron wavefunction will also lose the symmetry characteristic of a true spectroscopic state there will be a spatial polarization of the Is 2s core and the predicted ground state will no longer be of pure P type, just as in the UHF calculation there will be a spin polarization and the exact spin multiplicity of the many-electron state will be lost. Of course, the many-electron Hamiltonian does possess spherical symmetry (i.e. invariance under rotations around the nucleus), and the reason for the symmetry breaking lies at the level of the one-electron (i.e. IPM-type) model—the effective field in the 1-electron Hamiltonian is a fiction rather than a reality. 

Non-rigidity has important consequences for the rotation and vibration spectra. Extensive experimental investigations exist in this domain they are based on the elaboration of model hamiltonians to describe the external motions. Recently, non-rigid molecule effects on the rovi-bronic levels of PF5 have been examined , so leading to the prediction of the spectroscopic consequences of Berry processes. 

Hamiltonian equations, 610—615 minimal models, 615-618 multi-state effects, 624 pragmatic models, 618—621 spectroscopic properties, 598-610 linear molecules ... 

A quantitative treatment of the Jahn-Teller effect is more challenging (46). A major issue is that many theoretical models explicitly or implicitly assume the Bom—Oppenheimer approximation which, for octahedral Cu(II) systems in the vibronic coupling regime, cannot be correct (46,51). Hitchman and co-workers solved the vibronic Hamiltonian in order to model the temperature dependence of the molecular structure and the attendant spectroscopic properties, notably EPR spectra (52). Others, including us, take a more simphstic approach (53,54) but, in either case, a similar Mexican hat potential energy description of the principal features of the Jahn-Teller effect in homoleptic Cu(II) complexes emerges (Fig. 13). 

The above experimental developments represent powerful tools for the exploration of molecular structure and dynamics complementary to other techniques. However, as is often the case for spectroscopic techniques, only interactions with effective and reliable computational models allow interpretation in structural and dynamical terms. The tools needed by EPR spectroscopists are from the world of quantum mechanics (QM), as far as the parameters of the spin Hamiltonian are concerned, and from the world of molecular dynamics (MD) and statistical thermodynamics for the simulation of spectral line shapes. The introduction of methods rooted into the Density Functional Theory (DFT) represents a turning point for the calculations of spin-dependent properties [7],... 

Spectroscopic parameters of a molecule are derived from experimentally determined spectra by fitting term values to a properly chosen model Hamiltonian.161 Usually, the model Hamiltonian is an effective one-state Hamiltonian that incorporates the interactions with other electronic states parametrically. In rare cases, experimentalists have used a multistate ansatz like the supermultiplet approach162 to fit the rovibronic spectra of strongly interacting near-degenerate electronic states. The safest way of comparing theoretical data to experiment is to compute the spectrum and to fit the calculated term energies to the same model Hamiltonian as the experimentalists use. 

Clearly, the MFI description does not capture all possible complicated mechanisms of ET activation in condensed phases. The general question that arises in this connection is whether we are able to formulate an extension of the mathematical MH framework that would (1) exactly derive from the system Hamiltonian, (2) comply with the fundamental linear constraint in Eq. [24], (3) give nonparabolic free energy surfaces and more flexibility to include nonlinear electronic or solvation effects, and (4) provide an unambiguous connection between the model parameters and spectroscopic observables. In the next section, we present the bilinear coupling model (Q model), which satisfies the above requirements and provides a generalization of the MH model. 

Other semiempirical Hamiltonians have also been used within the BKO model. A Complete Neglect of Differential Overlap (CNDO/2) ° study of the effect of solvation on hydrogen bonds has appeared. o The Intermediate Neglect of Differential Overlap (INDO) °2 formalism has also been employed for this purpose.2011 Finally, the INDO/S model,which is specifically parameterized to reproduce excited state spectroscopic data, has been used within the SCRF model to explain solvation effects on electronic spectra.222,310-312 jhis last approach is a bit less intuitively straightforward, insofar as the INDO/S parameters themselves include solvation by virtue of being fit to many solution ultraviolet/visible spectroscopic data.29J... 

The steepest descent method is very effective far from the minimum of , but is always much less efficient than the Gauss-Newton method near the minimum of . Marquardt (1963) has proposed a hybrid method that combines the advantages of both Gauss-Newton and steepest descent methods. Mar-quardt s method, combined with the Hellmann-Feynman pseudolinearization of the Hamiltonian energy level model, is the method of choice for most nonlinear molecular spectroscopic problems. 

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