Atom-centered means approximation

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

The crystal structures of Rb. sphaeroides reaction centers from the carotenoidless strain R-26 and from the wild-type strain 2.4.1. were initially determined at 2.8 A and 3.0 A resolution, respectively. The structure of the Rb. sphaeroides R-26 reaction center is shown by the model in Fig. 8. Major structural features, such as a hydrophobic core containing 11 transmembrane helices from the L-, M- and H-subunits, the approximate rotational symmetry of the L- and M-subunits, the position and orientation of cofactors, etc., are all apparently conserved. All cofactors are associated with the LM-complex. Note that the helices and their connecting segments are designated with upper- and lower-case letters, respectively, and the meaning of the underlining is the same as that for the Rp. viridis model in fig. 7. The dot in the circle slightly below the center represents the nonheme iron atom, which is approximately equidistant from the two quinone molecules. 

The molecular structure consists of three discrete AICI4 tetrahedra and a benzene ring bound to the uranium atom. The coordination around the metal center can be viewed roughly as a pentagonal bip o amid with five equatorial chlorines, an apical chlorine and an apical benzene ring. This coordination is only approximate since the uranium atom lies roughly 0.5 A out of the plane of the equatorial chlorines. The mean AI—Cl distance for the chlorines bonded to the uranium is 2.18 A and 2.09 A for the others. The mean U—C distance is 2.91 A. 

In listing the ways in which metal ions may promote organic reactions, the requirement that the metal ion be suitably positioned within the substrate molecule was emphasized. Specific complexation or chelation of the metal ion with the substrate appears to be an absolute requirement of metal ion catalysis. In many cases chelation appears to be the rule, which usually means that the substrate must contain a donor atom in addition to the reactive center of the molecule with which the metal ion also complexes, or must contain two donor atoms in addition to the reactive center. Many attempts have been made to correlate the effectiveness of catalysis by a series of metal ions with the relative formation constants of the complexes. Such correlations have been successful in a number of reactions, but unsuccessful in others. In the successful correlations the complex chosen for the correlation closely approximates the transition state of the reaction. This indicates that the metal ion complex must stabilize the transition state of the reaction in order to assist the reaction effectively, and that metal ion complex formation in the ground state can have an effect exactly opposite to that of catalysis, since in such a case the ground state becomes stabilized. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

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