Orthogonalized operators

Product operator A set of orthogonal operators that can be connected together by a product. The product operator approach can be used to analyze pulse sequences by applying simple mathematical rules to weakly coupled spin systems. 

The mean angular moment (in units of %) is fit = dqjdv and the mean dipole-dipole energy is ED — — dq/dp. In the limit of high temperatures we may expand the exponential and retain only the linear terms, v and /9 may then be considered as the coefficients of the expansion of p in orthogonal operators, since Tr(VmMa) — 0. In this limiting case it then becomes evident that v and / are independent. It is easily shown that the thermodynamical temperature, defined by... 

The last term Tt is the rest of the development, involving all the orthogonal operators, with the exclusion of M. Caspers assumes that the contribution of the terms which are not principal diagonal may be neglected in the case of large fields. He then finds that for large H and t, Q fulfils the integral equation... 

Unitary operators are also known as complex orthogonal operators. If we use the standard basis and the standard complex scalar product on C , then the columns of the matrix of a unitary operator are all mutually perpendicular and have length one. In other words, a transformation T C" C" is unitary if and only if T T = I. 

One can also define orthogonal operators, if a complex-conjugation operator has been defined on the vector space. The last three classifications are basis-set specific and cannot be meaningfully applied to general operators. 

If we are interested in only real orbitals, k can be restricted to i k1 yielding the special orthogonal operators,... 

The so-called method of orthogonal operators, mentioned in the Introduction, looks fairly promising in semi-empirical calculations [20,34,134]. Its main advantage is that the addition of new parameters practically does not change the former ones. As a rule this approach allows one to reduce the root-mean-deviation of calculated values from the measured ones by an order of magnitude or so in comparison with the conventional semi-empirical method [135]. Unfortunately it requires the calculation of complex matrix elements of many-electron operators. 

The Cluj-distance index CJD is obtained from the symmetric Cluj-distance matrix CJD as the sum of the matrix entries corresponding to all the P order paths (i.e. bonds) above the main diagonal, i.e. applying the Wiener operator "iPto the P order sparse symmetric Cluj-distance matrix CJD or from the sparse unsymmetric Cluj-distance matrix CJDjj applying the Wiener orthogonal operator "W ... 

The Cluj-detour index CJA is analogously obtained from the P order sparse symmetric Cluj-detour matrix CJA applying the Wiener operator IV or from the unsymmetric Cluj-detour matrix CJAu applying the Wiener orthogonal operator VF ... 

Wiener-type indices, called Szeged fragmental indices, are derived from these matrices applying the orthogonal operator "VV" to both dense and the sparse matrices ... 

For acyclic graphs, the hyper-Wiener index is also obtained by applying the Wiener operator to the symmetric Cluj-distance matrix CJD and the -+ Wiener orthogonal operator to the Cluj-distance matrix CJDu-... 

Molecular descriptors calculated by applying the - Wiener operator to symmetric matrices derived from molecular graphs or the - Wiener orthogonal operator to unsymmetric matrices. 

Including effects of SECs using orthogonalized operator basis 15.57 17.96 18.86... 

The terms are present because an orthogonalized operator basis set is not employed in these calculations. The last row of Table III presents results for calculations that include both the third-order contributions to the partitioned EOM equation due to single excited configurations and use an operator basis in which the 3-block operators are Schmidt-orthogonalized to the 1-block (therefore, D - =0) through second order. These results differ from the results using the nonorthogonal operator basis by only 0.0 to 0.01 eV for each symmetry. 

Central to the development is an ensemble operator T akin to the one considered in the previous chapter and constructed from the appropriate system hamiltonian H of the reacting system and a bias operator B. A self-adjoint orthogonal operator B is constructed as the difference between a projector onto the product states and a projector Qr onto the reactant states, such that... 

Stretching, Fig. 2 Orthogonal operating directions of tensile force and forming machine (DIN 8585-2). 1 disk carrier, 2 work piece, 3 disk... 

By drawing a correspondence between nuclear states classified by isospin and atomic operators classified by quasi-spin, Leavitt (1987) has provided a complete group-theoretical listing for the f shell of orthogonal operators scalar with respect to S and L. 

This equation refers to the strong orthogonality operator Eq. (75), where the projection space is spanned by the full computational space of MOs. In the case of An3 the analogous space involves the full occupied space (active + inactive MOs), therefore the summation index q in terms on the r.h.s. of Eq. (96) is replaced with J, where J belongs to the full occupied space. The An3 formula can be written as... 

If X is a column positional vector, the orthogonalization operation is written as xiorthogonal) = P xifractional). One possible form of the P matrix, the orthogonalization matrix, is ... 

Since the introduction of new parameters may alter the values of the parameters already fitted, Judd and Crosswhite (1984) have introduced orthogonalized operators. These yield parameters which are more precisely defined and more stable than the conventional ones. 

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