Reduced density matrix treatment

Reduced density matrix treatment of spin-orbit interaction terms in niany-electron systems 63... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

Reduced Density Matrix Equations for Combined Instantaneous and Delayed Dissipation in Many-Atom Systems, and their Numerical Treatment... 

Many apparent anomalies in the spectra produced by double resonance experiments can be resolved by a density matrix treatment. Thus in A- X experiments with a reduced amplitude of the irradiated field the normal effect of the NOE upon the intensities of the transitions may be severely modified and both emission and absorption may be observed. (58) It turns out that the overall behaviour depends upon... 

From a quantum mechanical perspective, the treatment of the laser-adsorbate interaction can be straightforwardly included in the equations of motion for the reduced density matrix. Since the molecules are only of modest size compared to the wavelength of the incoming electric field, F t), the interaction can be treated within the semi-classical dipole approximation, i.e. the system Hamiltonian is replaced by... 

Note that, since the coherences of the reduced density matrix vanish on long timescales, the coupling to the STM electric field disappears from the equations of motion and only the incoherently driven evolution of the system mediated by the non-adiabatic couplings remains. This is consistent with the picture of an ensemble of localized states in quasi-thermal equilibrium that diffuses towards other local minima. Since the zeroth-order vibrational states are non-local by constmction, both tunneling and above-threshold excitations are seamlessly included in this treatment. 

Heat treatment of related glasses melted under reducing conditions can yield a unique microfoamed material, or "gas-ceramic" (29). These materials consist of a matrix of BPO glass-ceramic filled with uniformly dispersed 1—10 p.m hydrogen-filled bubbles. The hydrogen evolves on ceranarning, most likely due to a redox reaction involving phosphite and hydroxyl ions. These materials can have densities as low as 0.5 g/cm and dielectric constants as low as 2. 

This approach, based on a complex-valued realization of the PCM algorithm, reduces to a pair of coupled integral equations for real and imaginary parts of apparent charge density for tr(f,to) [13]. An alternative technique avoiding explicit treatment of the complex permittivity has been also derived [14,15]. The kernel K(f,f, t) of operator K does not appear explicitly. However, its matrix elements can be computed for any pair of basis charge densities p1(r) and p2(r) px k p = Jp2(j) (r, f)d3r, where tp(r, t), given by Equation (1.137), corresponds to p(r) = p2(r). 

The density g(f) in Eqs. (7.1) and (7.2) came from a semi-classical treatment employing scattering centers. Quantum mecha,nically one frequently obtains an identical expression. In the quantum mechanical treatment one starts off with the matrix element between identical initial and final states of all the nucleons involved in the process. In those cases where the matrix element can be reduced to a sum of individual nucleon matrix elements and where plane waves can be employed for the scattered particle (Born approximation) one obtains expression (7.1) and (7.2). If the calculation are carried through considering all... 

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