Property-specific wavefunctions

The SSEA, which is the subject of this review, was introduced in 1993-1994 with the purpose of exploring the potential and the efficiency of using state-specific wavefunctions in solving from first principles TDMEPs related to laser pulse-induced properties and phenomena. In a large number of applications since then, it has been demonstrated that this fundamental, yet conceptually simple, approach to such TDMEPs can indeed be realized computationally. 

The wavefunctions of Eq. (36) are in a form suitable for compufation. However, they do not constitute an orthonormal set. This is rectified by choosing a particular linear combination, the so-called eigenchannel wavefunctions, which has the desired property. Specifically, the orthonormal set has the form... 

It turns out that the CSP approximation dominates the full wavefunction, and is therefore almost exact till t 80 fs. This timescale is already very useful The first Rs 20 fs are sufficient to determine the photoadsorption lineshape and, as turns out, the first 80 fs are sufficient to determine the Resonance Raman spectrum of the system. Simple CSP is almost exact for these properties. As Fig. 3 shows, for later times the accuracy of the CSP decays quickly for t 500 fs in this system, the contribution of the CSP approximation to the full Cl wavefunction is almost negligible. In addition, this wavefunction is dominated not by a few specific terms of the Cl expansion, but by a whole host of configurations. The decay of the CSP approximation was found to be due to hard collisions between the iodine atoms and the surrounding wall of argons. Already the first hard collision brings a major deterioration of the CSP approximation, but also the role of the second collision can be clearly identified. As was mentioned, for t < 80 fs, the CSP... 

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

The symmetry classification of wavefunctions is based on the symmetry properties of molecules. Most small molecules possess certain symmetry elements such as a plane (a), or an w-fold axis (OJ, or a centre of symmetry (i), or perhaps a variety of these elements in combination. In order to be as definite as possible we shall develop the argument in terms of a specific example. The ground state of... 

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. 

The last fundamental aspect characterizing PCM methods, i.e. their quantum mechanical formulation, is presented by Cammi for molecular systems in their ground electronic states and by Mennucci for electronically excited states. In both contributions, particular attention is devoted to the specific aspect characterizing PCM (and similar) approaches, namely the necessity to introduce an effective nonlinear Hamiltonian which describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. 

Both the g-value distribution and the hyperfine splittings of the g = 2.0055 defect are consistent with the expected properties of dangling bonds. Consistency, however, does not constitute proof of the structure, and other possibilities have been proposed, which are discussed below. The ESR parameters do provide quantitative constraints that must be met by alternative models and, at present, are the only specific experimental information that we have about the defect wavefunctions. 

In (II) the reaction field of the dipole is included in the molecular Hamiltonian, so that the QM calculation, at whatever level, is modified to give a new molecular wavefunction for one molecule at the centre of a cavity. This calculation can be carried out in the absence of an applied macroscopic field and would give the unperturbed properties (dipole moment, energy states etc) of a solvated molecule. The macroscopic field has then usually been applied in a finite field calculation of the hyperpolarizability. One source of uncertainty in this procedure arises from the fact that when the reaction field is introduced into the hamiltonian it appears in a specific form,... 

The quantum mechanical indices described above are obtained from calculated molecular wavefunctions. The quality of the wave-function and, consequently, of the indices depends entirely on the formalism and the level of approximation one uses. Because the molecular wavefunction is often described as a linear combination of atomic orbitals (LCAO), one can easily obtain some of the indices mentioned above. These include the atomic net charges, sigma and i charges, frontier electron densities, Euqmq and LEMO t ie superdelocalizability parameters. Within the LCAO approximation one can assign the reactivity indices to specific atoms or bonds in the molecule. These indices reflect the stationary reactivity properties of the atoms and bonds in the molecule as described by charges or orbital energies and can therefore serve only as indicators of the reactivity. 

While semiempirical models which can be applied to molecules the size of 1 and 2 are necessarily only approximate, we were searching for trends rather than absolute values. In concept, the design of semiempirical quantum mechanical models of molecular electronic structure requires the definition of the electronic wavefunction space by a basis set of atomic orbitals representing the valence shells of the atoms which constitute the molecule. A specification of quantum mechanical operators in this function space is provided by means of parameterized matrices. Specification of the number of electrons in the system completes the information necessary for a calculation of electronic energies and wavefunctions if the molecular geometry is known. The selection of the appropriate functional forms for the parameterization of matrices is based on physical intuition and analogy to exact quantum mechanics. The numerical values of the parameters are obtained by fitting to selected experimental results, typically atomic properties. 

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