External-field dependence

Thus, whether or not a closed mechanical expression for Jl exists for a confined fluid (as well as for any other system exposed to an external field) depends critically on the natme of the external potenial. It must be such that the confined fluid is homogeneous in at least one spatial direction. In other words, the conclusion that is a homogeneous function of degree 1 in one or more of its extensive variables is ineluctably coupled to considerations of the symmetry of the external potential representing the confining substrate. 

The Fermi contact, or isotropic contact or isotropic hyperfine contact mechanism applies when there is finite unpaired electron density at the nucleus. This either adds to, or subtracts from the external field, depending upon the sign and magnitude of (the hyperfine coupling constant). 

Under certain conditions, cellulose derivatives possessing the characteristics of cholesteric liquid crystals present cholesteric helical structures dissolution and transition from the cholesteric to the nematic phase [98]. When shear is over, the system is relaxed over a determined time and intense, shifting to a transition state, where the energy of deformation is minimal and the orientation ordering is maintained, causing the appearance of band structures. When the external field is removed, the shear-induced anisotropy is affected by the inevitable relaxation of the macromolecular chains. Structural relaxation after removal of the external field depends on the shear history and relaxation mechanism [99,100]. Moreover, literature suggests a possible competition between the order induced by shear and thermodynamically, and also a correlation between the viscosity peak and the appearance of the anisotropic phase at low shear rates [ 101,102]. 

EFUT External field dependent unitary transformation... 

Equation (7) implies that the molecular response to an external field depends, in addition to the field itself, on the derivatives with respect to the nuclear coordinates of the molecular multipole moments. These derivatives are the intrinsic molecular observables that should translate into the empirical force field parameters. 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

The locations of the maxima of the -field and the E-field are different depending on the mode chosen for the EPR experuuent. It is desirable to design the cavity in such a way that the B field is perpendicular to the external field B, as required by the nature of the resonance condition. Ideally, the sample is located at a position of maxuuum B, because below saturation the signal-to-noise ratio is proportional to Simultaneously, the sample should be placed at a position where the E-field is a minimum in order to minimize dielectric power losses which have a detrimental effect on the signal-to-noise ratio. 

Even for a single radical tire spectral resolution can be enlianced for disordered solid samples if the inliomogeneous linewidth is dominated by iimesolved hyperfme interactions. Whereas the hyperfme line broadening is not field dependent, tire anisotropic g-matrix contribution scales linearly with the external field. Thus, if the magnetic field is large enough, i.e. when the condition... 

If, as is eommon, the atomie orbital bases used to earry out the MCSCF energy optimization are not explieitly dependent on the external field, the third term also vanishes beeause (9xv/3)i)o = 0. Thus for the MCSCF ease, the first-order response is given as the average value of the perturbation over the wavefunetion with X=0 ... 

Conservation of linear and angular momentum. If the potential function U depends only on particle separation (as is usual) and there is no external field applied, then Newton s equation of motion conserves the total linear momentum of the system, P,... 

This hamiltonian is time-dependent if the external field A%(x) is time-dependent. 

Now many physical properties depend mainly on the behaviour of the electron in the outer part of its orbit. As an example we may mention the mole refraction or polarizability of an atom, which arises from deformation of the orbit in an external field. This deformation is greatest where the ratio of external field strength to atomic field strength is greatest that is, in the outer part of the orbit. Let us consider such a property which for hydrogen-like atoms is found to vary with nrZ t. Then a screening constant for this property would be such that... 

Once the Fock operators have been constructed from a set of MSOs, this matrix equation is linear in its unknowns. Its coefficients are dependent on time in a way determined by the forces driving the electrons. These forces are the nuclear Coulomb potentials in molecular collisions or dynamics, but they could also be weak external fields. 

---