Orthogonal space

The Hamiltonian (3.4) is a function of the usual spatial coordinates x, y, z or r, 0, (j)). Electrons possess the intrinsic property of spin, however, which is to be thought of as a property in an independent, or orthogonal, space (spin space). Spin is actually a consequence of the theory of relativity but we shall merely graft on the property in an ad hoc fashion. The spin, s, of an electron (don t confuse with s orbitals ) takes the value 1/2 only. The z component of spin, m, takes (25 + 1) values of ms, ranging 5, 5-l,...-s. Thus for the single electron, = +1/2 or -1/2, also labelled a or p, or indicated by t or i. 

If u is partitioned into the primary space, a, and an orthogonal space of product operators, f, the partitioned form of the propagator matrix reduces to... 

If P is a projector, l—P is a projector onto the orthogonal space since (Pxf(I- P)x=xTPTx - xTPTPx=0... 

The P operator projects out of any function the component that lies in the model space, and the Q operator projects out the component in the orthogonal space. They also obey relation (3.6). 

Tellegen s theorem can be used in an evolving network, where the forces are allowed to change with time, and after a time interval dt, the forces become A) + dXJdt. Since, according to Tellegen s theory, the flows and forces lie in the orthogonal spaces, we have... 

Figure 13.11 Principle of PCA-based library design. The original data space X is spiked with a virtual optimum (denoted by a star in a) whereas in b) a set of reference compounds defines the optimum), thereby defining a target area for compound selection. After PCA, the relevant principal components (PC) span an orthogonal space, and distance criteria are used for picking compounds that are closest to the a) virtual optimum, or b) reference compounds.

The orthogonal space random walk simulation method... 

Zheng, L.Q., Chen, M.G., Yang, W. Random walk in orthogonal space to achieve efficient free-energy simulation of complex systems. Proc. Natl. Acad. Sci. USA 2008,105, 20227-32. 

However, mathematically speaking, the wave functions can be orthogonal, but in the physical space, this conception impeded an unfeasibility for electron transitions between the states. The decomposition of V(n) into non-combining (orthogonal) spaces was possible only if spin resonance was taken as a kind of perturbation. Its basic structure could be analyzed even at the level of the unperturbed system without spin. 

We choose a Voronoi cell V as a fundamental domain of D. When one hits a three-dimensional boundary of a Voronoi cell, by cutting Eg parallel to the observable space, one projects into the observable space E the dual (three-dimensional) boundary. The result appears as a tile in E. The dual boundary is defined as a convex hull of all lattice points of which Voronoi domains contain the hit boundary. Instead of cutting the six-dimensional Voronoi cells, one can define a procedure on a single projected Voronoi cell y(0) with a hierarchy of all its lower-dimensional boundaries into the orthogonal space E, Vx(0) = W, the window [6]. In case of Z)g, the window W has an outer shape of a triacontahedron see Figure 12-2 top). 

Some of the basis functions define an active space P and the remaining part of Hilbert space is called the orthogonal space Q = I — P. The active space is spanned by the basis functions that have a filled core and the remaining active electrons distributed over a set of active orbitals. The orthogonal complete space incorporates aU other possible basis functions that are characterized by having at least one vacancy in a core orbital. The state wavefunction in an active space is written as... 

Finally, electrodeposition is orthogonal space forproducing materials, the diversity to MBE and chemical vapor deposition of conditions under which a compound (CVD),asitinvolvesgrowthinacondensed can be formed. 

Putz, M. V. (2012b). Chemical Orthogonal Spaces, Mathematical Chemistry Monographs, Vol. 14, University of Kragujevac, Kragujevac. 

FIGURE 0.0 The featured concepts of the First Principles of Quantum Chemistry five-volume handbook as placed in the paradigmatic chemical orthogonal space of atoms and molecules. 

Note that, within the frontier view, the electronegativity and chemical hardness may be considered as two orthogonal (thus independent) chemical descriptors, see the HOMO-LUMO midlevel vs. gap of Eqs. 1.7 and 1.8, and can be therefore further used as 2D realization of the reaction coordinates to build up the chemical orthogonal space (COS) within which the chemical bond and reactivity is described. 

Acknowledgements Author thanks Prof. Mircea Diudea from Babes-Bolyai University of Cluj-Napoca for courtesy in providing the Hyper file for the Fullerene structure and to Romanian Ministry of Education and Research for supporting the present work through the CNCS-UEFISCDI (former CNCSIS-UEFISCSU) project Chemical Bond Within Orthogonal Spaces of Reactivity. Applications on Molecules of Bio-, Eco- and Pharmaco- Logical Interest>, Code PN II-RU-TE-2009-1 grant no. TE-16/2010-2011. 

A maximum overlap choice can be made for the spaces 5 " and naming the minimizing vectors Ixi) and Iq2) and their orthogonal spaces 5 " and Then one can write... 

---