Unnormalized distribution

In these last examples we have chosen unnormalized distributions. If the differential number or weight distribution is normalized, the area under the curve in Figs. 2-2 and 2-4 equals unity. That is,... 

It is convenient to use the unnormalized distribution function, g(s ), since it can be related directly to the local concentration in the cell. The function g(s ) is defined in the following way... 

To further analyze the assumption that a chemical bond is formed through, and is defined by, the occurrence of a short separation between two nuclear positions, we consider now the distribution of atom-atom distances in organic crystals, derived from a survey of the Cambridge Structural Database. To obtain distance distribution functions (DDF), an appropriate subset of the CSD is selected, and all atom-atom distances between atomic species i and j in all crystal structures in that sample are calculated, up to a certain limit / max- Let Ny R) be the number of atom-atom distances within the k-th distance bin defined by a separation a radial increment dR, and a volume dV = 4n/3[(/ k + d/ ) — (i k) ]- An inspection of the unnormalized distribution function reveals the exclusion radius ) J, or the separation below which... 

In terms of Gc and Gh the distribution function (unnormalized) for the end-to-end distance, R, of a polypeptide chain consisting of N residues can be written... 

As in Section 6.1 we set the initial momenta, Pq and jo, to zero and with the help of (2.57) we eliminate Ro as initial variable. The (unnormalized) final rotational state distribution for a given energy is thus transformed into a one-dimensional integral,... 

Let us make this more quantitative using the time-independent quantum mechanical theory outlined in Section 3.1. Because the interaction potential is independent of r the potential matrix V defined in (9.2) is diagonal, i.e., different vibrational fragment states do not mutually couple. As a result, the matrix of radial wavefunctions Xri (A, r E, n), which solve the coupled equations (3.5), is diagonal as well, i.e., Xri E,n) o< firm - If we assume, in order to simplify the subsequent discussion, that the nuclear wavefunction in the ground electronic state factorizes as pr R) final state n and the unnormalized final state distribution becomes... 

In analogy to Section 9.1 and Equations (9.4) and (9.5) the (unnormalized) final rotational state distribution becomes... 

Thus f, (m) is the (unnormalized) length distribution of inactive chains formed by disproportionation, particularly in systems where disproportionation represents an exclusive or predominating termination mechanism. f2(m) corresponds to the (unnormalized) length distribution of macroradicals. 

As a general case the ratio of the first moment to the zeroth moment of any distribution defines the arithmetic mean. For an unnormalized number distribution, , is the number of moles per unit volume with molecular weight M, and the zeroth and first moments of the distribution about zero are given respectively by... 

Figure 12 Unnormalized probability distribution for the dihedral angles, measured relative to = 0° for a trans state, for internal C—C bonds at 400 K. The largest probability is arbitrarily assigned a value of one. (From Ref. 83.)...

It is important to understand the difference between the forms of Eqs. (5.25) and (6.6) for the reactive mode distribution. Equation (5.25) is the strictly one-dimensional form whose equilibrium limit is the unnormalized form, exp[ — (x, u)]. Equation (6.6) is the proper expression for the multidimensional case, obtained by integrating the overall molecular distribution over all the nonreactive coordinates and momenta. Its extension beyond the barrier region, for all (x, u), is normalized to unity because j dxdv/2nti)Q -i x,v)P x,v) = Q . The difference between Eqs. (5.25) and (6.6) accounts for the different volumes of phase space associated with the presence of the nonreactive modes. 

Let Hk- = Hk E ) and Hk+ = Hk E ) denote the values of the histogram at its left and right boundary, and Rk = Hk+/Hk characterize their ratio. After a predetermined number of Monte Carlo steps per window, the (unnormalized) probability distribution can recursively be estimated according to ... 

The arithmetic mean of an unnormalized weight distribution is likewise given by [cf. Eq. (4.10)] ... 

It is possible to use a probability density that is not normalized, but if you do this, you must modify Eq. (5.70). For an unnormalized probability distribution... 

This distribution of electron density can be accounted for by rewriting an unnormalized function in the form... 

Thus the temperature can be related to the canonical average of the kinetic energy. If we consider a Nd = Nc degree of freedom system with kinetic energy K = 2nd potential energy U qi,q2, . then we may decompose the unnormalized canonical distribution as... 

Now we begin the cluster expansion method by introducing a set of unnormalized 5-particle distribution functions... 

Let us consider a molecule placed in a cavity surrounded by a dielectric continuum (fig. 1). The relative dielectric permittivity of the continuum is assumed to be e and in the cavity it is taken as equal to the permittivity of a vacuum. In the following we shall assume that the charge distribution of the solute is represented by a single center multipole expansion. An equivalent distributed multipole 2,3] representation may be used without further difficulty. We shall use the spherical tensors formalism [4,5] for the multipoles in which the 2/4 1 components of the multipole of rank / at the origin are defined from unnormalized spherical harmonics [6] by the equation ... 

Perspective (left) and top (right) view of the unnormalized canonical probability distribution pa E, q) for S3 at r = 300K. 

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