Zero-flux surface condition

In order for the scalar product of n, the vector normal to the surface (see Fig. 2.5), with Vp to vanish, it is necessary that the atomic surface not be crossed by any trajectories of Vp and as such it is referred to as a zero-flux surface. The state function ij/ and n, where the gradient is taken with respect to the coordinates of any one of the electrons, vanish on the boundary of a bound system at infinity. Thus, p and Vp vanish there as well and a total isolated system is also bounded by a surface satisfying eqn (2.9). Since the generalized statement of the action principle applies to any region bounded by such a surface, the zero-flux surface condition places the description of the total system and the atoms which comprise it on an equal footing. 

The coincidence of the topological and quantum definitions of an atom means that the topological atom is an open quantum subsystem, free to exchange charge and momentum with its environment across boundaries which are defined in real space and which, in general, change with time. It should be emphasized that the zero-flux surface condition is universal— it applies equally to an isolated atom or to an atom bound in a molecule. The approach of two initially free atoms causes a portion of their surfaces to be shared in the creation of an interatomic surface. Atomic surfaces undergo continuous deformations as atoms move relative to one another. They are, however, not destroyed as atoms separate. 

Recalling that the topological definition of an atom implies the zero-flux surface condition, a region of space bounded by a surface satisfying eqn (5.86) at the point of variation is henceforth called an atom. It is required that, as tends to ip, 2(( ) is continuously deformable into the region flfi/r) as.sociated with the atom. The region 2(< ) thus represents the atom in the varied total system, which is described by the trial function <, just as 1( ) represents the atom when the total system is in the state described by kp. 

Thus, the action integral for a closed quantum system described by Schrodinger s equation vanishes for any time interval At because of the zero-flux surface condition that... 

The final term on the right-hand side of this equation vanishes because of the zero-flux surface condition thereby yielding... 

Thus, the atomic Lagrangian and action integrals in the presence of an electromagnetic field, like their field-free counterparts, vanish as a consequence of the zero-flux surface condition (eqn (8.109)). These properties are common to the corresponding integrals for the total system and it is a consequence of this equivalence in properties that the action integrals for the total system and each of the atoms which comprise it have similar variational properties. 

Note that the zero-flux surface condition for this species, after discretisation with the two-point approximation, establishes that... 

Because of the dominant topological property exhibited by a molecular charge distribution—that it exhibits local maxima only at the positions of the nuclei—the imposition of the quantum boundary condition of zero flux leads directly to the topological definition of an atom. Indeed the interatomic surfaces, along with the surfaces found at infinity, are the only closed surfaces in a three dimensional space which satisfy the zero flux surface condition of equation 6. 

The so-called zero-flux surface condition stated in equation (1) is the boundary condition for the definition of a proper open system whose properties are defined by quantum mechanics, as described in Section 3. 

Bader et al. have developed a theory of molecular structure [8], based on the topological properties of the electron density p(r). In this theory, a molecule may be partitioned into atoms or fragments by using zero-flux surfaces that satisfy the condition... 

As in the full-field formulation, we assigned a zero flux boundary condition, i.e. j = 0 at the outer boundary of the domain as well as on the axis of symmetry ahead of the crack tip (Fig. 5b). Also, along the crack surface, we assumed the NILS hydrogen concentration CL to be in equilibrium... 

Since the surface is not crossed by any gradient lines, it is referred to as the surface of zero flux. As further discussed below, the virial theorem is satisfied for each of the regions of space satisfying the zero-flux boundary condition. 

Atoms A and B have to be connected by a MED path. The existence of a MED path implies a saddle point p of the electron density distribution p (r) as well as a zero-flux surface S (AB) between atoms A and B (necessary condition). 

A linked grouping of neighbouring atoms (a concept defined more fully in the following Section) as well as individual atoms are bounded by a zero-flux surface. Such a surface may be used to define a Wigner—Seitz cell in a solid, a solute in a solution, or a molecule in a molecular crystal. The zero-flux surface, eqn (2.9), is the natural boundary condition for a system defined in real space and such a surface can always be used to define the physically... 

Bader 1975) that the condition for the satisfaction of the principle of stationary action is that each subsystem lj be bounded by a surface Si satisfying a zero-flux boundary condition of the form... 

It is through a generalization of Schwinger s principle that one obtains a prediction of the properties of an atom in a molecule. The generalization is possible only if the atom is defined to be a region of space bounded by a surface which satisfies the zero flux boundary condition, a condition repeated here as equation 15,... 

For the loading step, Eq. 4 is subject to the following boundary conditions C, = Cjo at the sample reservoir and zero flux boundary conditions, n.VC, = 0, at all the other boundaries across the chip, where n is the unit normal to the surface and C,o is an initially applied concentration for the ith species. The initial conditions for Eq. 4 are C = C,o at the sample reservoir and C, = 0 at all the other places across the chip. For the dispensing step, the boundary conditions are similar to that for the loading step however, the initial conditions are the concentration distribution of the loading step when it reaches steady state, C = C/ ioading ... 

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