Internuclear, definition

As has been mentioned above, the inclusion of basis functions (49) with high power values, nik, is very essential for the calculations of molecular systems. It is especially important for highly vibrationally excited states where there are many highly localized peaks in the nuclear correlation function. To illustrate this point, we calculated this correlation function (it corresponds to the internuclear distance, r -p = r ), which is the same as the probability density of pseudoparticle 1. The definition of this quantity is as follows ... 

The answer is yes, in a very general way, as has been discussed before [62,63]. Consider any parameter in the external potential, called y. For definiteness, we choose the internuclear separation in a diatomic molecule. Then the exchange-correlation energy depends parametrically on this quantity. Now imagine making an infinitesimal change in y. The differential change in is... 

For the strong interactions dealt with in the last section, interatomic potentials are determined by overlap-dependent terms which have an inverse exponential variation with internuclear distance (R). At large separations the interactions are independent of overlap and may be represented by inverse powers ofR. There will be between these two an intermediate region, which encompasses the van der Waals minimum for non-bonded interactions, and we will take this to be contained in our definition of a weak-interaction. 

In contrast to outer-sphere reactions, the simple observation that a reaction occurs by an inner-sphere mechanism necessarily introduces an element of structural definition. The relative dispositions of the oxidizing and reducing agents are immediately established and, except for structurally flexible bridging ligands such as NC5H4(CH2) C5H4N, the internuclear separation between redox sites can be inferred from known bond distances. Even so, bimolecular inner-sphere reactions necessarily occur by a sequence of elementary steps (Scheme 2) and the observed rate constant may include contributions from any of the series of steps. 

Py(r) dr is the probability of finding the atom pair i and j at an internuclear distance interval between r and r+dr. The integral over r, although for convenience written without specified limits of integration, should be considered as a definite integral. Py(r) is always limited to a rather narrow r-interval beyond which it is negligible in value. 

Note the appearance of the AB equilibrium geometry (Rm, which can be obtained by a self-consistent procedure136) and Le Roy s parameter142 [R0, which represents the smallest value of the internuclear distance for which the asymptotic series of the dispersion energy is still a good representation of the damped series (49)] in the definition of the reduced coordinate x represents the expectation value of the square of the radial coordinate for the outermost valence electrons, which is tabulated in the literature143 for atoms with 1 120. Other important parameters in the dispersion damping... 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

Figure 3. Evolution of the spin-orbit couplings involving the S2 (4Ag) state along the internuclear distance. See Table 3 for the definition of these terms.

The concept of softness is associated with polarizability. The larger the chemical system is, the softer it will be. This correlation of softness with polarizability can be found directly from a bond charge model [65 68] where softness is found to be proportional to the internuclear distance of a molecule [69-72]. To extend this definition (Eq. (28)) to open systems, the system is considered as a member of a grand canonical ensemble with bath parameters /t, v( T ), and temperature 0. This definition of S in such an ensemble can be written in terms of a number fluctuation formula [64] ... 

Born-Oppenheimer approximation The Born Oppenheimer approximation consists of separating the motion of nuclei from the electronic motion. An often used physical picture is that the nuclei being so much heavier than electrons may be treated as stationary as the electrons move around them. The Schrodinger equation can then be solved for the electrons alone at a definite internuclear separation. The Born-Oppenheimer approximation is quite good... 

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