Virtual excitations

Another analysis method was based on the local wave vector estimation (LFE) approach applied on a field of coupled harmonic oscillators.39 Propagating media were assumed to be homogeneous and incompressible. MRE images of an agar gel with two different stiffnesses excited at 200 Hz were successfully simulated and compared very well to the experimental data. Shear stiffnesses of 19.5 and 1.2 kPa were found for the two parts of the gel. LFE-derived wave patterns in two dimensions were also calculated on a simulated brain phantom bearing a tumour-like zone and virtually excited at 100-400 Hz. Shear-stiffnesses ranging from 5.8 to 16 kPa were assumed. The tumour was better detected from the reconstructed elasticity images for an input excitation frequency of 0.4 kHz. 

The second-order term can be interpreted as due to the interaction between two nucleons with the virtual excitation of a nucleon-antinucleon pair [17]. This interaction is a TBF with the exchange of a scalar (a) meson, as illustrated by the diagram (d) of Fig. 2. Actually this diagram represents a class of TBF with the exchange of light (7r, p) and heavy (a, u) mesons. There are, however, several other diagrams representing TBF, Fig. 2(a-c), which should be evaluated as well in a consistent treatment of TBF. 

Each sector excitation operator is, in the usual way, a sum of virtual excitations of... 

An intuitive physical interpretation of the correlation terms in the 2-RDM is that the two electrons undergo virtual excitations in such a way that when one goes from C into , the other one undergoes the opposite transition. 

The interpretation of the correlation effects in the 3-RDM is slightly more complicated than in the 2-RDM case, although the electrons also avoid each other here by imdergoing virtual excitations. In this case, two different correlation mechanisms contribute to the overall effect. In the first mechanism, one particle is in a stationary state - the ground or an excited state - while the two other particles undergo the same kind of virtual excitations as in the 2-RDM. In the second mechanism, the cycle of transitions involves three states instead of two. Finally the role played by the hole is more complex here, since the 1-HRDM elements multiply a 2-RDM element and a cycle of two transitions. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

SAP produces a set of virtual excitations from the fully occupied to the unoccupied Kohn-Sham orbitals thus producing a fictitious statistical ensemble. A thermodynamic interpretation of SAP is presented in [50] (see [51,52] as well), where two main observations are given. First, the redistribution of single particle states in the smoothing procedure leads to... 

To consider a case where the second channel is closed and the third one is opened, let us assume that E (pi/2)-E (pi/ = 0.92 MeV (Fig. 2). The energy of the nuclear transition is not sufficient for a transition of the muon to the continuum. However, it is sufficient for an excitation to the 2p state. It is important to note that this energy is not lying in the resonant range. The diagram C (Fig. 1) describes the proton transition pm-p3/2 with a virtual excitation of the muon to states of the series nd with a y quantum emission of energy ... 

In the projection operator formalism, which leads to a rigorous basis for the optical potential, the absorptive imaginary part is associated with transitions out of the elastic channel from which no return occurs. Whereas Pgl transitions are in this category, excitation transfer (ET) transitions are not, since return ( virtual excitation ) can occur during the ET collision. In the event that a localized avoided curve crossing with one other state dominates the inelastic process (expected for many endoergic transfers), the total absorption probability (opacity) can still be defined ... 

The mixing coefficients can be determined by a multiconfigurational Dirac or Hartree-Fock procedure (MCDF, MCHF). In the present case, however, numerical values are not of interest, only the fact that A0 is smaller than unity due to the presence of virtual excitations in the normalized correlated wavefunction. 

In superexchange CT reactions, no charge ever actually resides on the bridge. Those states that the molecule occupies between the time when the electron leaves the donor and arrives at the acceptor are called virtual excitations. The transmission probability of electrons to reach the acceptor in this manner is the superexchange... 

With these conventions of notation, simple formulas exist for all matrix elements of H in the basis of Slater determinants generated by virtual excitations from a reference state [72], Denoting the latter by electron operator TL = h + nj(ij u ij) for [Pg.47]

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