Vector potentials properties

If the vector potential components have the property that the derived field intensity, the YM field in Eq. (99) is nonzero, then the vector potential cannot be... 

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

The vector of properties is a kind of chromosome, a code that defines the genotype of the individual. This sequence of values specifies the "genetic makeup" that (possibly uniquely) defines the animal. In Daisy s case, not all of the animal s characteristics are defined by her biological genetic code we have included in the vector other characteristics that would help to pick her out from the kine (cattle) crowd. In a similar way, a single GA solution can be constructed as a "chromosome" or genome in vector form, which contains all the information needed to define the potential solution to a scientific problem. 

While the present discussion of the response property is limited to scalar potential only, an analogous description is also possible for the vector potential and a similar equation can be derived. 

In prineiple, this additional derivation also starts from the many-fermion Hamiltonian given in equation (75). As discussed earlier, extension of this equation to moleeules usually is done by imposing the requirement that some of these partieles are nuelei whieh are assumed to be fermions. An alternative, but less rigorous approaeh to introduce nuclei is to eonsider them as an external perturbation whieh modifies the external veetor and sealar potential in equation (75). Further approximations ean be made for non-dynamical properties by invoking the elamped nuelei approximation. For such a situation the external vector potential A can be written as... 

Another interesting property of the current density is that the total electronic energy of an arbitrary system in the presence of any external field whatever, with vector potential A, may be written [5]... 

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

J. P. Vigier, Theoretical determination of a = e2/he deduced from asymptotic group invariance properties of high energy charged dirac particles in a constant external vector potential, Lett. Nuovo Cimento 7(12) (Ser. 2) 501-506 (1973). 

Gauge invariance of magnetic properties is associated with the continuity equation (33,34), In a gauge transformation of the vector potential... 

Nakagawa, S., Massie, B. and Hawley, R.G. (2001) Tetracycline-regulatable adenovirus vectors pharmacologic properties and clinical potential. EurJ. Pharm. Sci., 13, 53-60. 

First, we note that the standard photodetection is a local measurement of the field variables (intensities). At the same time, the Aharonov-Bohm effect represents a topological measurement referred to the properties of vector potential along some loop. In the usual form, the Aharonov-Bohm effect deals with static or slowly time-varying magnetic fields [101]. The effect consists in the appearance of a persistent current in a metallic loop over which the magnetic flux passes. This current is a periodic function of magnetic flux with the period of flux quantum hc/e. Besides that, certain resistance oscillations in the loop incorporated into an external circuit with the same period can occur. 

Our efforts will be repaid if we take pause to examine the properties of the vector potential, and thereby also its derivative fields. The vector potential is self-evidently Hermitian, as befits the status of the field it represents. Its parity with respect to space-inversion is odd, since P operation reverses the sign of r, e, and k. Its character with respect to time-inversion T, which is also of interest, is less self-evident. First, this operation gives... 

In Eq. (28), A = (1/2)H X R is the value of the vector potential at the nuclear position R of atomic orbital 0 , and fv is an exponential gauge factor. It is clear that this approach allows both the LCAO coefficients and the atomic basis functions to change with the magnetic field. The important effect of using such an expansion is that calculated magnetic properties are independent of the choice of origin for any set of 0 functions.50... 

D and 3D autocorrelation vectors [70] represent intramolecular 2D topologies or 3D distances within particular molecules. An autocorrelation coefficient is a sum over all atom pair properties separated by a predefined number of bonds (2D) or distance (3D), while the entire vector represents a series of coefficients for all topological or Cartesian distances. Atomic properties involve hydrophobicity [71], partial atomic charges, hydrogen bonding potential and others. Again, a PCA is often used to reduce the number of variables. 3D autocorrelation vectors of properties based on distances calculated from 3D molecular surfaces [72] have also been applied to visually assess the diversity of different libraries [73]. 

---