Collective motion of electron

For centuries, metal nanoparticles have never ceased to attract scientists and artists from many diverse cultures. In this section we briefly introduce a phenomenon of metal nanoparticles that still inspires scientists localized surface plasmon resonance (LSPR) (Hutter and Fendler, 2004). Metal nanoparticles show nonlinear electronic transport (single-electron transport of Coulomb blockade) and nonlinear/ultrafast optical response due to the SPR. Conduction electrons (—) and ionic cores (-F) in a metal form a plasma state. When external electric fields (i.e., electromagnetic waves, electron beams etc.) are applied to a metal, electrons move so as to screen perturbed charge distribution, move beyond the neutral states, return to the neutral states, and so on. This collective motion of electrons is called a plasma oscillation. SPR is a collective excitation mode of the plasma localized near the surface. Electrons confined in a nanoparticle conform the LSPR mode. The resonance frequency of the surface plasmon is different... 

Faster computers and development of better numerical algorithms have created the possibility to apply RPA in combination with semiempirical Hamiltonian models to large molecular sterns. Sekino and Bartlett - derived the TDHF expressions for frequency-dependent off-resonant optical polarizabilities using a perturbative expansion of the HF equation (eq 2.8) in powers of external field. This approacii was further applied to conjugated polymer (iialns. The equations of motion for the time-dependent density matrix of a polyenic chain were first derived and solved in refs 149 and 150. The TDHF approach based on the PPP Hamiltonian - was subsequently applied to linear and nonlinear optical response of neutral polyenes (up to 40 repeat units) - and PPV (up to 10 repeat units). " The electronic oscillators (We shall refer to eigenmodes of the linearized TDHF eq with eigenfrequencies Qv as electronic oscillators since they represent collective motions of electrons and holes (see Section II))... 

It seems that the e3q)lidtly correlated functions, in spite of serious problems at the integral level, can be generalized in future towards the collective motions of electrons, perhaps on the basis of the renormalization theory of Kenneth Wilson (introduced into chemistry for the first time by Martin Head-Gordon). ... 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

In light of tire tlieory presented above one can understand tliat tire rate of energy delivery to an acceptor site will be modified tlirough tire influence of nuclear motions on tire mutual orientations and distances between donors and acceptors. One aspect is tire fact tliat ultrafast excitation of tire donor pool can lead to collective motion in tire excited donor wavepacket on tire potential surface of tire excited electronic state. Anotlier type of collective nuclear motion, which can also contribute to such observations, relates to tire low-frequency vibrations of tire matrix stmcture in which tire chromophores are embedded, as for example a protein backbone. In tire latter case tire matrix vibration effectively causes a collective motion of tire chromophores togetlier, witliout direct involvement on tire wavepacket motions of individual cliromophores. For all such reasons, nuclear motions cannot in general be neglected. In tliis connection it is notable tliat observations in protein complexes of low-frequency modes in tlie... 

The function g describes the collective motions of the electrons kc is the cut-off vector for the plasma oscillations and is the plasma frequency (see Pines 1955, particularly p. 391) see also Section III.C. 

When the size of metals is comparable or smaller than the electron mean free path, for example in metal nanoparticles, then the motion of electrons becomes limited by the size of the nanoparticle and interactions are expected to be mostly with the surface. This gives rise to surface plasmon resonance effects, in which the optical properties are determined by the collective oscillation of conduction electrons resulting from the interaction with light. Plasmonic metal nanoparticles and nanostructures are known to absorb light strongly, but they typically are not or only weakly luminescent [22-24]. 

Just as the energy splitting within a band of levels will effect the physical properties of the cluster, so will the density of the sub-levels, the filling of the individual sub-levels, and the filling of the band of levels. For possible collective motion of the electrons within the cluster, the band must not be completely filled. In the case of AU55 this point is not completely clear. 

Note that there is no bulk absorption band in aluminum corresponding to the prominent extinction feature at about 8 eV. Indeed, the extinction maximum occurs where bulk absorption is monotonically decreasing. This feature arises from a resonance in the collective motion of free electrons constrained to oscillate within a small sphere. It is similar to the dominant infrared extinction feature in small MgO spheres (Fig. 11.2), which arises from a collective oscillation of the lattice ions. As will be shown in Chapter 12, these resonances can be quite strongly dependent on particle shape and are excited at energies where the real part of the dielectric function is negative. For a metal such as aluminum, this region extends from radio to far-ultraviolet frequencies. So the... 

The occurrence of the energy gap 2A below the Peierls transition temperature allows in principle the collective motion of the electrons under the influence of an applied electric field. This holds as long as the energy hkpvpr of the moving electrons is less than A, where upr is the velocity of the collectively moving electrons. However, this so-called Frbhlich mode [4] is very sensitive to lattice imperfections because it is a true ID movement. 

Some theoretical purists tend to view molecular mechanics calculations as merely a collection of empirical equations or as an interpolative recipe that has very little theoretical Justification. It should be understood, however, that molecular mechanics is not an ad hoc approach. As previously described, the Born-Oppenheimer approximation allows the division of the Schrodinger equation into electronic and nuclear parts, which allows one to study the motions of electrons and nuclei independently. From the molecular mechanics perspective, the positions of the nuclei are solved explicitly via Eq. (2). Whereas in quantum mechanics one solves, which describes the electronic behavior, in molecular mechanics one explicitly focuses on the various atomic interactions. The electronic system is implicitly taken into account through judicious parametrization of the carefully selected potential energy functions. 

Molecules, atoms, ions, and electrons in condensed medias are constitutive units of the material. Because of the strong interaction between those particles they lose their individual properties. They are replaced by quasiparticles, the quantum of collective motions of condensed materials, that play a fundamental role in modern solid state physics. 

Current interest in the stability of closed shells stems from several different areas of research. The subject of giant resonances [5] will be described later in the present book. It involves collective motion of all the electrons in a closed shell or subshell, and their cooperative response is therefore of interest [6]. 

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