Normalized integral charge density

The second property in Equation 4.5 normalizes p(r) to the total number of electrons in the system by integrating over the whole 3D space. Note that in atomic units (used throughout this chapter unless otherwise mentioned), the number density p(r) becomes the electronic charge density, thereby paving the way to various useful, interpretative approaches as described below. 

The normalization expression (3.26) is appropriate for wave functions. For a charge density function, a different normalization must be used, because the charge is given by the integral over the first power of the function. The density functions in general use are labeled d,mp, and are defined by the normalization... 

For large distances, M T r / >oo, the XC terms vanish, and the integral describes the coulomb interaction of two spherical, normalized charge densities, which reduce basically to I /Rv-,v that is, we get... 

The charge density distributions of a single neutron, p (r), and of all neutrons (the second integral in Elq. (3)) both integrate to zero. Usually, the contribution of the neutrons is omitted in Eq. (3). In any case, normalization of this nuclear charge density distribution correctly yields the total nuclear charge Q = J d r p r) = Ze. 

The normalization condition for the charge density distribution now reduces to a radial integral,... 

Other quantities are also usefully expressed in terms of the eigenvectors. For example, the contribution of the ith orbital to the charge density is just the integrand of the normalization integral 0i(r) =... 

Fig. 21 Density profiles p z) and running integrals Njo (z) of the ion densities for cations (full lines) and anions (dashed lines) at three different surface charge densities in units of pCcm as indicated. Left NaCi solutions right CsF solutions. The top graphs show, for reference, the corresponding oxygen density profile near the uncharged surface. Density profiles are normalized to the bulk densities corresponding to 2.2 molal solutions in each case.

The behavior of the component of D normal to the boundary may be treated similarly, except that the integrals are not necessarily zero. In this case the charge density p must be taken into account. In the transition from the volume element to the surface element, the volume density becomes a surface density, Psurf, given by... 

The Zemach correction is essentially a nontrivial weighted integral of the product of electric and magnetic densities, normalized to unity. It cannot be measured directly, like the rms proton charge radius which determines the main proton size correction to the Lamb shift (compare the case of the proton size correction to the Lamb shift of order Za) in (6.13) which depends on the third Zemach moment). This means that the correction in (11.4) may only conditionally be called the proton size contribution. 

Two electrochemical techniques are directly based on the expression for the faradaic current density jF, namely chronoamperometry and normal pulse polarography. A third technique, named chronocoulometry, deals with the integral of jF, giving the charge transferred per unit area via the faradaic process as a function of time. The general expression obtained... 

Dose normally ev/gm. Usually calculated from quoted dose to air in same source assuming catalyst absorbs y as does H2O (55.2 x IQi ev/gm H2O, roentgen). Reactor doses frequently given only as integrated flux of neutrons, nvt (density x velocity x time) or cm-2. Charged particle doses sometimes given as flux and time or current and time. 

Since the density is normalized within the unit cell, and the integral in Eq.(4.48) is over the spherical cell, the monopole moments have to be renormalized, and this is done by adding/subtracting the constant SSCA in accordance with charge neutrality of the spherical cell. 

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