Wavefunction similarity

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

It is clear, that there are two types of energy behavior with respect to ft. For the states with wavefunctions similar to the eigenstates of hp = - A+fiw r), one may suppose dpEk(P) —1. But for the states, similar to the eigenstates of H, that is localized mainly within 2W, one may suppose dpEk ((J>) 0. 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

Tliis leads to two sets of wavefunctions, one for each spin, similar to UHF theory. 

As an alternative, we might want to plot the square of the wavefunction rather than the wavefunction itself, on the physical grounds that such quantities have a direct interpretation. The form of the graph is very similar to Figure 3.4, and I haven t shown it. 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

Regarding the emission properties, AM I/Cl calculations, performed on a cluster containing three stilbene molecules separated by 4 A, show that the main lattice deformations take place on the central unit in the lowest excited state. It is therefore reasonable to assume that the wavefunction of the relaxed electron-hole pair extends at most over three interacting chains. The results further demonstrate that the weak coupling calculated between the ground state and the lowest excited state evolves in a way veiy similar to that reported for cofacial dimers. 

The perturbed energies and wavefunctions for the i-th system state can be expressed in a similar way as in scalar perturbation theory ... 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

The chemistry of superheavy elements has made some considerable progress in the last decade [457]. As the recently synthesized elements with nuclear charge 112 (eka-Hg), 114 (eka-Pb) and 118 (eka-Rn) are predicted to be chemically quite inert [458], experiments on these elements focus on adsorption studies on metal surfaces like gold [459]. DFT calculations predict that the equilibrium adsorption temperature for element 112 is predicted 100 °C below that of Hg, and the reactivity of element 112 is expected to be somewhere between those of Hg and Rn [460, 461]. This is somewhat in contradiction to recent experiments [459], and DFT may not be able to simulate accurately the physisorption of element 112 on gold. More accurate wavefunction based methods are needed to clarify this situation. Similar experiments are planned for element 114. 

It is possible that the complexes benzene- -HX can be described in a similar way, but in the absence of any observed non-rigid-rotor behaviour or a vibrational satellite spectrum, it is not possible to distinguish between a strictly C6v equilibrium geometry and one of the type observed for benzene- ClF. In either case, the vibrational wavefunctions will have C6v symmetry, however. 

Similar to Eq. (2), the time-dependent wavefunction is expanded in terms of the BF parity-adapted rotational basis functions ... 

Similar to the diatom-diatom reaction, the initial wavefunction is chosen as the direct product of a localized translational wavepacket for R and a specific (JMe) state for the atom-triatom system with a specific rovibrational eigenstate (z/o, Lo,Bo) f°r the triatom ABC ... 

H. Kuhn developed a model which shows how it is possible to proceed in small, clear, calculable steps from one development phase to the next. Starting from certain situations or states of the system, possible conditions for moving to the next steps are estimated. In the development of his model, Kuhn proceeds in a manner similar to that involved in quantum mechanics here, suitable test functions were generated which provided approximate solutions for wavefunctions in order to be able to explain chemical bonding phenomena better. 

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