Reordering

This ionic potential is periodic. A translation of r to r + R can be acconnnodated by simply reordering the sunnnation. Since the valence charge density is also periodic, the total potential is periodic as the Hartree and exchange-correlation potentials are fiinctions of the charge density. In this situation, it can be shown that the wavefiinctions for crystalline matter can be written as... 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

The stmcture of activated carbon is best described as a twisted network of defective carbon layer planes, cross-linked by aHphatic bridging groups (6). X-ray diffraction patterns of activated carbon reveal that it is nongraphitic, remaining amorphous because the randomly cross-linked network inhibits reordering of the stmcture even when heated to 3000°C (7). This property of activated carbon contributes to its most unique feature, namely, the highly developed and accessible internal pore stmcture. The surface area, dimensions, and distribution of the pores depend on the precursor and on the conditions of carbonization and activation. Pore sizes are classified (8) by the International Union of Pure and AppHed Chemistry (lUPAC) as micropores (pore width <2 nm), mesopores (pore width 2—50 nm), and macropores (pore width >50 nm) (see Adsorption). 

A modem optimizing compiler would recognize that X and Z are independent and would probably reorder the code to be... 

Compute and examine the orbitals at the RHF/3-21G level in order to select the active space. We will be performing a 4-electron CAS, using 4 and 6 active orbitals. The orbitals we want are those corresponding to the rt system (where the excited electrons go) therefore, the orbitals we want will be pairs of symmetry A2 and Bp Reorder the orbitals so that six appropriate orbitals make up the active space. 

To interpret the magnitude of the PIA signal, the rale equation for the photogeneration of states will be discussed. It describes the change of the number of states, n, with respect to time, /, depending on the generation rate kp (A=const., excitation density) and the decay rate In1 /=inverse lifetime, (/reorder of kinetics) and can be written as follows ... 

First of all, we perform an important reordering, introducing the new discrete variables... 

Some invaluable isolated actions were taken from time to time in the last twenty to twenty five years that prevented air quality from declining at an even faster rate. For example, energy demand was met (electricity and refined oil products) by expanding capacity of plants located outside of the MCMA a metro was built the vehicular traffic system was reordered and new hydrotreating, reforming and catalytic cracking units reduced lead and sulfur in refined products. 

Fig. 10 a UV-Vis DRS spectra of TS-1 (curve 1, full line), immediately after contact with H2O2/H2O solution (curve , dotted line), after time elapse of 24h (curve 3, dashed line) and after subsequent H2O dosage (curve 4, scattered squares), b as for a for the XANES spectra, c as for a for the -weighted, phase imcorrected, FT of the EXAFS spectra. Spectra 2-4 of b and c have been reordered at liquid nitrogen temperatime. Adapted from [49] with permission. Copyright (2004) by ACS... 

Now the procedure to get the energy of the singlet is outlined after reordering equation (5-35) and changing to energies rather than determinants we have... 

The role of a boundary in a manifold with boundary can be interpreted with reference to a hyperplane within a Euclidean space E using the concept of halfspace, where the hyperplane is in fact the boundary of the half-space. By appropriate reordering of the coordinates, a half-space Hn becomes the subset of a Euclidean space En containing all points of En with non-negative value for the last coordinate. 

For medium and large networks, the occurrence matrix that is of the same structure (isomorphic) as the coefficient matrix of the governing equations is usually quite sparse. For example, Stoner (S5) showed a 155-vertex network with a density of 3.2% for the occurrence matrix (i.e., 775 nonzeros out of a total of 1552 entries) using formulation C. Still lower densities have been observed on larger networks. In these applications it is of paramount importance that the data structure and data manipulations take full advantage of the sparsity of the governing equations. Sparse computation techniques are also needed in order to capture the full benefit of cycle selection and row and column reordering. 

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