Single particle electron density

The single-particle electron density p(r, t) and the current density j(r, t), which are defined as the expectation values,... 

Here if RQM> i) represents the quantum wavepacket value at the spatial point Rqm at time t, Rc represents the classical nuclear coordinates and Pc represents the single-particle electronic density matrix. It is important to note that Eqs. (10) and... 

The spatial distribution of the electron cloud in a molecular system is investigated quantitatively through the single-particle electron density,which has served as a basic variable in the so-called density functional theory (DFT), an approach that bypasses the many-electron wavefunction, which is the usual vehicle in conventional quantum chemistry or electronic structure theory (for a review of modem developments in quantum chemistry, see reference 7). The theoretical framework of DFT is well known for the associated conceptual simplicity as well as for the computational economy it offers. Another equally important aspect of DFT is its ability to rationalize the existing concepts in chemistry as well as to give birth to newer concepts, which has led to the important field of conceptual DFT. ... 

We have described in detail the Hamiltonian formulation of a hybrid potential. This may not be the most convenient for some potentials, notably those of DFT type, for which it is easier to work with the energy directly. The central quantity in DFT is the single particle electron density, p, which is related to the square of the wavefunction as ... 

Because the many-particle Lagrangian density L° reduces to a sum of singleparticle operators, one may define an effective single-particle Lagrangian density iP°(r, t) by the usual recipe of summing over all spins and integrating over the coordinates of all the electrons but one, a step which expresses the result in terms of the charge density p(r, t) as... 

The problem of the description of the excited states within the Polarizable Continuum Model leads to two non-equivalent approaches, the approach based on the linear response (LR) approach, and the state specific (SS) approach, as already said in the Introduction. Each approach has advantages and disadvantages. The LR approach is computationally more convenient, as it gives the whole spectrum of the excited states of interest in a single calculation, but is physically biased. In fact, in the LR approach the solute-solvent interaction contains a term related to the one-particle transition densities of the solute connecting the reference state adopted in the LR calculation, which usually corresponds to the electronic groimd state, to the excited electronic state. The SS approach is computationally more expensive, as it requires a separate calculation for each of the excited states of interest, but is physically im-biased. In fact, in the SS approach the solute-solvent interaction is determined by the effective one-particle electron density of the excited state. 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

Our work demonstrates that EELS and in particular the combination of this technique with first principles electronic structure calculations are very powerful methods to study the bonding character in intermetallic alloys and study the alloying effects of ternary elements on the electronic structure. Our success in modelling spectra indicates the validity of our methodology of calculating spectra using the local density approximation and the single particle approach. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Kohn and Sham provided a further contribution to make the DFT approach useful for practical calculations, by introducing the concept of fictitious non-interacting electrons with the same density as the true interacting electrons [8]. Non-interacting electrons are described by orthonormal single-particle wavefunctions y/i (r) and their density is given by ... 

In Equations 4.1 and 4.2, the numbers before the integral signs occur due to the indistinguishability of electrons and electron pairs, respectively. The single-particle density p(x) is defined as the diagonal element of the single-particle density matrix Pi(xi xi), viz.,... 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

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