Three-particle configuration interaction

For 4f" configurations with three or more f electrons, the Hamiltonian is expanded with the term tiP (i = 2, 3, 4, 6, 7, 8) to take the three-particle configuration interaction into account, ti are the three-particle operators and T are the parameters (Judd 1966). Variation of the T parameters in a fitting procedure has to be done carefully, since these parameters are only sensitive to particular levels. If the level for which a... 

Model Hartree-Fock calculations which include only the electrostatic interaction in terms of the Slater integrals F0, F2, F and F6, and the spin-orbit interaction , result in differences between calculated and experimentally observed levels596 which can be more than 500 cm-1 even for the f2 ion Pr3. However, inclusion of configuration interaction terms, either two-particle or three-particle, considerably improves the correlations.597,598 In this way, an ion such as Nd3+ can be described in terms of 18 parameters (including crystal field... 

When talking about the NM dose, one has to make a distinction between three dose metrics (1) the administered dose (particle mass, number, or surface area administered per volume media at the onset of an experiment), (2) the delivered dose (particle mass, number, or surface area to reach the cell monolayer via diffusion and sedimentation over the duration of an experiment), and (3) the cellular dose (particle mass, number, or surface area internalized by the cells during the experiment). The determination of the cellular and delivered dose of NMs is essential for accurate interpretation of results derived from in vitro particle-cell interaction studies (e.g., particle uptake, cytotoxicity, biokinetic studies) [36], Using two different cell culture configurations, upright and inverted, Cho et al. [37] have recendy shown that the uptake of NPs is gready sensitive to the position in which cells are cultured and strongly... 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

What Lowdin published in 1955 (the manuscripts were received in July that year) was a series of three articles under a common heading Quantum theory of many-particle systems. The three parts had the following subtitles I. Physical interpretation by means of density matrices, natural spin-orbitals, and convergence problems in the method of configuration interaction II. Study of the ordinary Hartree-Fock approximation and III. Extension of the Hartree-Fock scheme to include... 

In recent analyses of lanthanide spectra, the term Fci in eq. (24.1) has included the effects of configuration interaction as expressed in the Trees correction aL(L+l), and the parametrized Casimir operators PGiGz) and 70(67) (Trees, 1964 Rajnak and Wybourne, 1963). The additional terms represent those effects of configuration interaction that can be accounted for by two-body effective operators that do.not transform as the / in eq. (24.1). For configurations of three or more equivalent f-electrons, the three-particle operators of Judd (1966a), T t<... 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

Nuclear magnetic resonance (NMR) spectroscopy is a nonin-vasive and nondestructive spectroscopic technique that allows determination of the constitution and relative configuration of molecules, the characterization of the dynamic three-dimensional (3D) conformation of molecules, and their interaction with other molecules. NMR spectroscopy detects the characteristics of nuclear spins the most commonly studied nuclei are the spin-i/z-particles H, N, and NMR observables... 

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