Nonorthogonal

Menon M and Subbaswamy K R 1994 Transferable nonorthogonal tight-binding scheme for silicon Phys. Rev. B 50 11 577... 

Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

The hexahedral tessellation presents more of a problem because of its nonorthogonality. Bayes [bayes87b] suggests two update schemes, both based on situating the sites on a cubic lattice (1) update according to... 

Pauling, L. Correlation of Nonorthogonality of Best Hybrid Bond Orbitals with Bond Strength of Orthogonal Orbitals Proc. Natl. Acad. Sci. (USA) 1976, 73, 274-275. 

Extended Hiickel calculations are performed with a nonorthogonalized AO basis set therefore, the spin densities are to be evaluated by gross atomic populations and not simply by squares of expansion coefficients. 

Figure 5. Schematic of nonorthogonal transformations used in finite-element/Newton algorithms for calculating cellular interfaces, (a) Monge cartesian representation for almost planar interfaces, (b) Mixed cylindrical/cartesian mapping for representing deep cells.

Feauveau, J. C., Nonorthogonal multiresolution analysis using wavelets. In Wavelets—A Tutorial in Theory and Applications (C. K. Chui, ed.). Academic Press, San Diego, CA, p. 153. 1992. 

FIGURE 2.5 Effective nonorthogonal 2D retention space from Liu (1995). Reprinted with permission from the American Chemical Society. 

The question of describing the target states and whether or not the nonorthogonality of the target states in our present analytical solution, really makes a significant difference requires further research. Yet the qualitative agreement between the CDW-EIS model and experiment leads to the conclusion that it does represent the physics of the process correctly. 

For compactness and clarity, Eq. (2.11) is written in matrix notation. It is similar to the more familiar case of a time-independent basis set expansion but with two important differences The AIMS basis is time-dependent and nonorthogonal. As a consequence, the proper propagation of the coefficients requires the inverse of the (time-dependent) nuclear overlap matrix... 

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonorthogonal basis with electron translation factors. 

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonorthogonal spin... 

The determinantal wave function in Eq. (21) is built [23] from complex dynamical spin orbitals Even when the basis orbitals ut in Eq. (22) are orthogonal these dynamical orbitals are nonorthogonal, and for a basis of nonorthogonal atomic orbitals based on Gaussians as those in Eq. (24) the metric of the basis becomes involved in all formulas and the END theory as implemented in the ENDyne code works directly in the atomic basis without invoking transformations to system orbitals. 

Stuchebrukhov AA (1996) Tunneling currents in electron transfer reaction in proteins. II. Calculation of electronic superexchange matrix element and tunneling currents using nonorthogonal basis sets. J Chem Phys 105(24) 10819-10829... 

Stuchebrukhov AA (1997) Tunneling currents in proteins nonorthogonal atomic basis sets and Mulliken population analysis. J Chem Phys 107(16) 6495-6498... 

The ESQC method allows a fast calculation of the electronic transmission from an exact solution of the TISE. The basis set here is supposed orthogonal, but the method can be generalized to nonorthogonal ones. Following [31], solving this equation is equivalent to the propagation of an initial wave packet located on the... 

Hoerl, A. E., Kennard, R. W. Ridge regression Biased estimation for nonorthogonal problems. Technometrics 1970, 12, 55-67. 

The possibility even exists of including dynamical effects with time-dependent friction terms (plus random forces at finite temperatures).77-80 Flowever, it may not be advisable to take advantage of this possibility, as the simulation would become increasingly slow with increasing number of time steps. Moreover, the simulation will slow down considerably in higher dimensions because of the nonorthogonality of the dynamical coupling in reciprocal space. 

By virtue of the orthogonality of the transformation matrix T, the orbitals vk will then also form an orthonormal set. Nonorthogonal molecular orbitals... 

Figure 16.8 Three different kinds of bolaamphiphiles containing different apo-lar spacers and/or polar head groups. The cartoons representthe orthogonal and nonorthogonal arrangements proposed to explain the fiber structures observed in solution...

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