Vector operators table

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Table II lists the execution times for a DO-Loop which performs a simple addition of two vectors A and B. Times are given for both scalar and vector additions using one or two pipes as well as times for the same operation on the Cyber 174. Note that, for vector operations, increasing the array size is much less expensive than the corresponding increase on a scalar machine. For very small arrays (e.g. N 5) scalar execution on the ASC is faster because Vector overhead consumes a much higher percentage of the execution time.

In most intstances however merely being able to arrange the calculation as a series of vector operations, without worrying over the "unit address increment" requirement, makes extremely good if not maximal use of AFPP, VP or AP. As an illustration of this point Table XIII shows "normal" FORTRAN code for a pivotal condensation matrix inverter (the author is unfortunately by now anonymous) and Table XIV shows the vectorized version for the MVP-9500 at about two thirds completion. The VPLIB version is completely (as far as the author can manage at leastl) vectorized and written in assembler. Most of the vectorization is fairly obvious and only the reduction loops contain any obscurity. In order to maintain peak vector efficiency the MVP-9500 reduction loop does a little more work than is strictly necessary an alternative would ruin the vector flow. It is left as an exercise to the determined reader to unravel the full correspondence between Tables XIII and XIV. 

We call attention to several more involved vector operations listed in Table 1.3.1. These may be verified by writing out both sides of each equation in component form. 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

Here the coupling tensor between the spin vector operator I and the applied magnetic field Bq is given by the unit matrix 1. The prefactor is listed in Table 3.1.1. The energy... 

In the above formulae, spherical components of the irreducible tensor operators occur. In changing to Cartesian components one can use the transformations listed in Table 1.13 (Section 1.5.3) the following relationships hold true (a) three components of the first-rank tensor (vector) operator are... 

The problem has spherical symmetry, so the first step is to look up the spherical form of the vector operators (see Table 17.1 on page 312). The potential i(j depends only on r and not on angles 9 or spherical symmetry. Therefore, (dqj/dO) = 0and d

) = 0, and V ifj in Equation (21.30) becomes... 

In electronic and vibrational spectroscopy we can neglect both molecular dimensions relative to the wavelength and the effects of the magnetic field of light relative to those of its electric field (electric dipole approximation). Then, the interaction of a single l7-polarized photon with a molecule is described by the projection of the electric dipole moment vector operator M (Table 3) into eP (photon creation) or eP (photon annihilation). Creation... 

Table 13-1 Boond-mode flelds of weakly guiding waveguides. The form of the transverse electric field depends on the shape of the waveguide cross-section. Vector operators are defined in Table 30-1, page S92, and parameters are defined inside the back cover.

The equations satisfied by 4 and Ft are in Table 13-1, and Tp, F denote different solutions of the scalar wave equation for the same value of 1. Vector operators are defined in Table 30-1, page 592. 

In the sixth column of the main body of the character table is indicated the symmetry species of translations (7) of the molecule along and rotations (R) about the cartesian axes. In Figure 4.14 vectors attached to the nuclei of H2O represent these motions which are assigned to symmetry species by their behaviour under the operations C2 and n (xz). Figure 4.14(a) shows that... 

The H2O molecule, therefore, has three normal vibrations, which are illustrated in Figure 4.15 in which the vectors attached to the nuclei indicate the directions and relative magnitudes of the motions. Using the C2 character table the wave functions ij/ for each can easily be assigned to symmetry species. The characters of the three vibrations under the operations C2 and (t (xz) are respectively + 1 and +1 for Vj, - - 1 and + 1 for V2, and —1 and —1 for V3. Therefore... 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

Transform) the content of a given column ( vector) can be mathematically modified in various ways, the result being deposited in the (N + 1) column. The available operators are addition of and multiplication with a constant, square and square root, reciprocal, log(w), Infn), 10 , exp(M), clipping of digits, adding Gaussian noise, normalization of the column, and transposition of the table. More complicated data work-up is best done in a spreadsheet and then imported. 

Preprocessing is the operation which precedes the extraction of latent vectors from the data. It is an operation which is carried out on all the elements of an original data table X and which produces a transformed data table Z. We will discuss six common methods of preprocessing, including the trivial case in which the original data are left unchanged. The effects of each of these six types of preprocessing will be illustrated numerically by means of the small 4x3 data table from the study of trace elements in atmospheric samples which has been used in previous sections (Table 31.1). The various effects of the transformations can be observed from the two summary statistics (mean and norm). These statistics include the vector of column-means m and the vector of column-norms of the transformed data table Z ... 

The results of applying these operations to the double-closed data in Table 32.6 are shown in Table 32.7. The analysis yielded two latent vectors with associated singular values of 0.567 and 0.433. 

Note For the triplet states the functions are given only for the Ms = 1 spin component. For all degenerate levels the a components (as given by the vector coupling coefficients of Table 2) are listed first, and these transform as +1 under oxz and the b components as - 1 under the same operation. 

Table 1. Gradient and Laplacian operators for different geometries. 5 stands for a scalar and e stands for a unit vector associated to coordinate i... 

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