Projected tensors geometric projection

Exactly as in all affine cases we can now interpret the different projective tensors geometrically. Say, for instance, that Aa is a projective covariant vector. Then... 

The structural picture that was envisaged to represent the temperature-dependent fluctuations of the EFG tensor [15] is based on the X-ray structure of MbOa that exhibits a geometric disorder of Fe02 with two different positions of the terminal O-atom [28]. Within this stmcture, the projection of the 0-0 bond on the heme plane is rotated by about 40° in position 2 compared to 1 (Fig. 9.10). Conventional Mossbauer studies of single crystals of Mb02 have shown that the principal component of the FFG tensor lies in the heme plane and is oriented along the projection of the 0-0 bond onto this plane [29]. If the terminal O-atom is located in position 2, the EFG should be of the same magnitude as in position 1, but its orientation is different. The EFG fluctuates between positions 1 and 2 with a rate that depends on temperature. 

Note that the soft reciprocal vectors b are expanded in a basis of tangent vectors, and so are manifestly parallel to the constraint surface (as indicated by the use of a tilde), while the hard reciprocal vectors ihi are expanded in normal vectors, and so lie entirely normal to the constraint surface (as indicated by the use of a caret). These basis vectors may be used to construct a geometric projection tensor... 

The product of any 3N vector with this geometric projection tensor isolates the soft component of that vector. The geometrical projection tensor is a symmetric tensor, like the Euclidean identity and unlike the dynamical projection tensor. To reflect this fact, its bead indices are written directly above and below one another, with no offset to indicate whether the implicit Cartesian index associated with each bead index acts to the right or left. 

The inertial and geometrical projection tensors, and associated reciprocal vectors, are identical for models with equal masses for all beads, in which the mass tensor is proportional to the identity. 

Because appears contracted with in the equation of motion, the hard components of have no dynamical effect, and are arbitrary. The values of the soft components of F depend on the form chosen for the generalized projection tensor, and reduce to the metric pseudoforce found by Fixman and Hinch in the case of geometric projection. 

Geometrically projected random forces, which were introduced by Hinch [10], have a variance given by the geometrically projected friction tensor... 

Cartesian kinetic SDEs with unprojected, geometrically projected, and inertially projected random forces require the same correction forces in certain special cases. Inertial and geometric projections are completely equivalent for models with an equal bead mass m for all beads, for which the mass tensor m v = is proportional to the identity. Unprojected and geometrically projected random forces require identical correction forces in the case of local, isotropic friction with an equal friction coefficient for all beads, as in the Rouse or Kramers model, for which the friction tensor ... 

We then compare Eq. (2.418) to the second line on the RHS of Eq. (2.390), for the case of a generalized projection tensor = P , which is the same as the inertial or geometric projection tensor in this simple class of models. After some straightforward algebra, we find that, for the class of models to which Liu s algorithm applies, the last term in Eq. (2.390) may be written more explicitly as... 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

Because the vibrations that underlie IR absorption spectra must affect the electric dipole of a molecule, we would expect the frequencies of these modes to be sensitive to local electric fields, and this is indeed the case. Shifts in vibration frequencies caused by external electric fields can be measured in essentially the same manner as electronic Stark shifts, by recording oscillations of the IR transmission in the presence of oscillating fields. The Stark tuning rate is defined as S = dv/dE , where v is the wavenumber of the mode and is the projection of the field (E) on the normal coordinate [88, 89]. To a first approximation, is given by —u Aft + E Aa)/hc, where ti is a unit vector parallel to the normal coordinate, Afi is the difference between the molecule s dipole moments in the excited and groxmd states, and Aa is the difference between the polarizability tensors in the two states (Sect. 4.13, Box 4.15 and Box 12.1). However, anharmonicity and geometrical distortions caused by the field also can contribute to vibrational Stark effects [90, 91]. 

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