Space partitioned

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

Gutdeutsch U, Birkenheuer U, Kruger S and Rdsch N 1997 On cluster embedding schemes based on orbital space partitioning J. Chem. Phys. 106 6020... 

What is the relationship between rule space partitions and differences in behavior ... 

The time evolution of the function f is thus replaced by a sequence of discrete symbols labeling the bins visited by each point of the orbit. Because of the coarse-graining of the phase space, however, detailed knowledge of the actual orbits is generally lost i.e. many different orbits may yield the same symbolic sequence. Different state-space partitionings also generally give rise to different symbolic representations. 

Applications of electron propagator methods with a single-determinant reference state seldom have been attempted for biradicals such as ozone, for operator space partitionings and perturbative corrections therein assume the dominance of a lone configuration in the reference state. Assignments of the three lowest cationic states were inferred from asymmetry parameters measured with Ne I, He I and He II radiation sources [43]. 

Some minor discrepancies between theory and experiment on tetrafluoroterephthalonitrile remain to be resolved. The peak densities in the bonds are slightly but systematically lower in the theoretical than in the experimental maps. Analysis of the second moments of the pseudoatoms from the Hirshfeld space partitioning (chapter 6) indicate a greater contraction into the molecular plane in the theoretical than in the experimental study. Whether such discrepancies are artifacts of the refinement model, the result of inter-molecular interactions, or have another origin, is a question of considerable interest. 

Space Partitioning and Topological Analysis of the Total Charge Density... 

Space Partitioning Based on the Atom-Centered Multipole Expansion... 

FIG. 6.3 Definition of vectors used in discrete boundary space partitioning. 

Though rare, there are cases in which the total density shows minor maxima at non-nuclear positions. As all (3, — 3) critical points are attractors of the gradient field, basins occur which do not contain an atomic nucleus. These non-nuclear basins (which have been found in Si—Si bonds1 in Li metal, and some other cases, distinguish the zero-flux partitioning from other space partitioning methods. 

We will use an example as illustration. The dipole moment vector for formamide has been determined both by diffraction and microwave spectroscopy. As the diffraction experiment measures a continuous charge distribution, the moments derived are defined in terms of the method used for space partitioning, and are not necessarily equal. Nevertheless, the results from different techniques (Fig. 7.1) agree quite well. 

Inspection of these integrals indicates that they are amenable to a space partitioning— like that involved in the atomic real-space core-valence separation described in Chapter 3—simply by selecting the appropriate limits of integration. Briefly, we can approach the study of core and valence regions with the help of Cl wavefunctions. 

I (T + 2V ), of the valence energy in the present real-space partitioning scheme, where T and V are, respectively, the relevant kinetic and potential energies. 

Hybrid correlation models based on active-space partitioning Correcting MP2 theory for bond-breaking reactions ... 

Fig. 2.2. Space partitioning in EPE embedded cluster calculations. I - internal region treated at a QM level II - shell model enviromnent of the QM cluster subdivided into regions of explicit optimization (Ila), of the effective (Mott-Littleton) polarization (lib) and of the external area (lie). The sphere indicates an auxiliary surface charge distribution which represents the Madelung field acting on the QM cluster (dashed line).

It should be noted that Eq. (26) does not contain an -dependent matrix element on its left-hand side as do the MPn EOM equations (Eq. (14)) or the equivalent GE equations. This is because the EOM-CC equations are usually not subjected to the operator-space partitioning that the GF and MPn-based EOM theories commonly employ. It should also be noted that the operators appearing on the left-hand side of Eq. (26) produce non-Hermitian matrices. As a result, there arise non-orthogonal left- and right-eigenvectors in solving Eq. (26). As discussed in Refs. [34,35], it is important to compute both sets of eigenvectors if one wishes to compute, e.g. photo-electron intensities. 

Partitioning or cell-based methods are based on the definition of a relatively small finite number of bins on the axes of a multidimensional property space. The bins are then combined to define a set of cells that cover the whole property space. Partitioning involves the identification of a set of characteristics that are of importance in combinatorial libraries, for example, molecular properties that are... 

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