Hybridization tensors

The original rs-method is based on equations presented by Kraitchman [4], He had noticed that the numerically dominant part of the inertial moment tensor for an isotopomer in which (exactly) one atom has been substituted, can be replaced by the known inertial moments of the parent molecule, and that the remainder of the tensor then depends only on the Cartesian coordinates of the substituted atom. Equating the roots of the secular equation for this hybrid tensor to the experimental inertial moments of the isotopomer, Kraitchman obtained his famous equations which, quite in contrast to any r0-type method, always aim at the determination of the Cartesian coordinates of just this one substituted atom. The rest of the molecule is irrelevant, it does not influence the result and need not be known. Costain has checked, for several small molecules, the invariance of -determined bond lengths when employing different pairs of parent molecule and isotopomer [7]. He found it superior to the comparable r0-data and recommended to prefer -structures for... 

Neese, F. 2007. Calculation of the zero-field splitting tensor on the basis of hybrid density functional and Hartree-Fock theory. The Journal of Chemical Physics 127 164112/1-9. 

The QM/MM methodology [1-7] has seen increasing application [8-16] and has been recently reviewed [17-19], The classical solvent molecules may also be assigned classical polarizability tensors, although this enhancement appears to have been used to date only for simulations in which the solute is also represented classically [20-30], The treatment of the electronic problem, whether quantal, classical, or hybrid, eventually leads to a potential energy surface governing the nuclear coordinates. 

The spin densities deduced from the formula aftft, = ( A1 + 2 A )/3 are presumed to be positive because of direct delocalization of the unpaired electron from the (a3da + b4s) hybrid orbital. It should be noted, however, that Aft1 and A , measured for rotating rings at T = 150 K, are not principal values of the proton hfs tensor, i.e. a , determined in this way does not exactly correspond to the isotropic coupling obtained by the common formula aftft, = (A" + A" + A )/3. 

Using Neese s coupled-perturbed Kohn-Sham hybrid density functional (UPBEO, UBILYP) techniques, the gyromagnetic (g) and hyperfine tensor components of the thiophene-1,3,2-dithiazoly radical (TDTA) were computed and found to be in very good agreement with values from experimental determination <2006CPL(418)30>. 

The UBILYP hybrid density functional (FIDF) method has been found suitable for accurate reproduction of gyromag-netic (g) tensors of TTTA to within 140-750 ppm of their experimental value. The report is significant since it represent the first accurate calculation of g tensors of this class of radicals without using post-FIF techniques <2005CPL(405)382>. 

Step 7 is likely to be the most computationally demanding step of an MCD calculation based on transition moments. Step 5 will take very little computer time as will step 4 if a functional that does not include exact exchange is used and step 8 will be quick if the influences of the -tensor and ZFS are neglected or not needed. Steps 2 and 5 will take more time and steps 1, 4 (with a hybrid functional the magnetic CPKS equations must be solved), and 8 (if the g-tensor or ZFS is calculated) may take a significant amount of computer time. Step 3 will consume a large proportion of the computational time of an MCD calculation and may be the most expensive step under some circumstances. 

The authors also reconsidered radical 20, which, in the current work, was formed by x-irradiation and studied at 77 K. The experimentally determined hyperfine coupling tensors for the two methylene protons have the same principal values and agreed with the tensors found in the earlier work.16 These tensors are presumed to be the average of the two individual proton tensors. By assuming that C5 is sp2 hybridized the authors were able to calculate, from the observed average tensors, that Ami = 1.06 mT and Amai = 3.45 mT for the methylene protons. Using the individual tensors for the methylene protons and a 3.6 mT isotropic coupling for the C4(H) proton, a satisfactory simulation of the... 

Within the framework of the bond polarization the shift components of the unpolarized bond and the parameter Aai giving the polarization influence on the chemical shift are determined empirically from solving a set of linear equations 1 for a number of substances where both the chemical shift tensor and the molecular structure are known. The bond polarization energies Vai are calculated as effect of surrounding net atomic charges qx on atomic hybrids %. With the bond polarity parameter the polarization energy can be calculated. 

Table n. Principal values of the chemical shift tensor calculated with uncorrected and corrected hybrids ... 

The details of electron distribution in atoms characterized by higher multipoles (dipoles and quadrupoles) are defined by the deviations S P . As it has been already mentioned the vector parts of the HOs centered at each given atom vA transform as 3-vectors under the molecule/space rotations, and the hybrid densities sAvA transform as 3-vectors as well. On the other hand, the diadic products v,A vA under 3-dimensional rotations transform as a sum of a scalar and of the rank two tensor of the 3-dimensional space. The values of these momenta are obtained by averaging their standard definitions ... 

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