Space overlap

So how is it possible to ensure this Examining the phase space overlap relationship of the reference and the target systems will give insight towards answering this question. 

There are two types of bone (a) compact or cortical bone and (b) trabecular or cancellous bone. Cortical bone is found principally in the shafts (diaphyses) of long bones. It consists of a number of irregularly spaced overlapping cylindrical units termed Haversian systems. Each consists of a central Haversian canal surrounded by concentric lamellae of bony tissue. Trabecular bone is found principally at the ends of long bones and flat bones. It is composed of a meshwork of trabeculae within which... 

Consider again non-relativistic fermions. Their BCS spectrum (for homogeneous systems) is isotropic when the polarizing field drives apart the Fermi surfaces of spin-up and down fermions the phase space overlap is lost, the pair correlations are suppressed, and eventually disappear at the Chandrasekhar-Clogston limit. The LOFF phase allows for a finite center-of-mass momentum of Cooper pairs Q and the quasiparticle spectrum is of the form... 

Then the classification is made for each object, taking into account that an object can fit the model of just one category or more than one category (when two or more category spaces overlap, at least partially) or it can be outside every category space, an outlier. 

Similar to the fluoiesence discussed in Appendix D, we can define doorway and window wavepackets and write the signal as their phase-space overlap. Assuming that the delay time between the probe and the pump pulses is to, we obtain Eqs. (4.5)-(4.8). 

The most general expression for the recombination rate, Rr, involves a double integration over the phase space overlap of the clouds, thus incorporating terms arising from their spatial overlap and the velocity... 

The primitive VB model is defined in terms of overlap and Hamiltonian matrix elements over the basis states of eqn. (2.1.3). For fixed there are 2N possible spin-product functions so that this gives the dimension of the model s space. Indeed (though not originally formulated in this manner) the model may be mathematically represented entirely in spin space, despite the fundamental spin-free nature of the interactions. One may introduce a spin-space overlap operator by integrating out the spin-free coordinates... 

A general interpretation of this phenomenon was given, taking into account the space overlap 2p , 2p ) of carbon atomic orbitals which induces in these molecules a three-dimensional conjugation. 

Overlapping a-boundaries. For closely spaced overlapping boundaries on parallel planes, the total phase shift a will be the sum of the individual phase shifts. Two overlapping boundaries with a = 2x/3 have a - 4ir/ 5, and the fringe pattern is essentially the same as for a single boundary. However, for three overlapping boundaries a = 2x, and the boundary is out-of-contrast. 

Figure 4.15 (a) Enhanced through-bond overlap in DBA via the frontier orbitals of the bridging units (two in this example, shown in blue) (b) weaker through-space overlap between the tails of the D and A wavefunctions (frj-) and at the same D-A edge-to-edge distance... 

The UV absorption spectra of 97 and 99 show multiple bands with tails extending out almost to 300 nm [19]. Though it is difficult to say how much electronic interaction among the acetylenic units occurs via orbitals on the phosphorus atoms and how much occurs via through-space overlap of the in-plane p-orbitals, there can be little doubt that these JV-phos-pha[7V]pericyclynes enjoy a substantial degree of cyclic electron delocalization. 

However, EXP becomes inefficient when phase space overlap is low, that is, when the configurations sampled with one Hamiltonian are very improbable in the other, and visa versa. Because the statistical uncertainty of EXP grows exponentially as the overlap in phase space between the states decreases, this method requires an impractical amount of uncorrelafed dafa fo produce reasonable estimates when overlap is poor [24-26]. Introducing a number of alchemical inferme-diate states can allow much more efficient computation of free energy differences befween Hamiltonians with little phase space overlap ... 

To compute free energy differences between states with little overlap, it will usually be more efficient to compute the free energy along a pathway of intermediate states. Free energy calculations can be made significantly more efficient by optimizing the choice of intermediate states for increased phase space overlap [36, 59-61],... 

Wu, D., Kofke, D.A. Phase-space overlap measures. 1. Fail-safe bias detection in free energies calculated by molecular simulation, J. Chem. Phys. 2005,123, 054103. 

The collection Aloc is endorsed with relations of time precedence -<, time overlapping o, space overlapping o, and inclusion Cj, Cg, C, between locations. In some versions of Situation Theory, the space-tile locations can be given by complex objects. E.g., a simple option (equivalent to the above) is that space-time locations are pairs of two components, one for space locations, and one for time points or periods. 

In bacteriochlorin dimers 1 and 2, a through space overlap of the unreduced pyrroles LUMOs explains the enhanced mixing of monomer-like Qx transitions. This overlap is in addition to r-7r interactions through the phenylene bridge that are also observed for 3 and 4. This is important regarding the use of porphyrin chromophores to model the properties of natural systems, because it implies that BC dimers in the photosynthetic apparatus are able to perform excitation energy transfer via the LUMO. This is not possible in model porphyrin dimers. 

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