Internuclear separation

Finally we have to remember to add on the nuclear repulsion and, if we repeat the calculation for a range of values of the internuclear separation, we arrive at the potential energy curves shown in Figure 4.3 for the ground-state (singlet)... 

The problem of the interaction between two hydrogen atoms a and b in their ground state at large internuclear separation R has also been treated (Hirschfelder and Lowdin, 1958), and the inter-... 

Fig. 5.—Plot of the Relative Relaxation Efficiency between Two Protons versus the Inverse Sixth Power of Their Internuclear Separation. (Derived from measurements of Dreiding stereomodels reproduced, with permission, from Ref. 43.)...

The polarlsablllty of a molecule will vary during vibrations which change the Internuclear separations. Thus the vibrations of a molecule sitting In an electrical field will be coupled to the field via the polarlsablllty. This should be particularly noticeable for a molecule adsorbed on an electrode surface where the field strength Is typically In the range 10 -10 V cm, The dipole, perpendicular to the surface, Induced In the molecule by the static electric field will fluctuate In step with the normal mode vibrations of the molecule. 

Thus, the current semi-empirical methods (MNDO, AMI and PM3) differ in the way in which core-repulsions are treated. Within the MNDO formalism the corerepulsion ( asmndo) is expressed in terms of two-centre, two-electron integrals (Eq. 5-4), where Za and Zb correspond to the core charges, Rab is the internuclear separation, and a a and aB are adjustable parameters in the exponential term [19]. 

The relationships between bond length, stretching force constant, and bond dissociation energy are made clear by the potential energy curve for a diatomic molecule, the plot of the change in the internal energy AU of the molecule A2 as the internuclear separation is increased until the molecule dissociates into two A atoms ... 

Schematic representation of the potential energy of a system comprised of three atoms in a linear configuration as a function of the internuclear separation distances. The dashed line represents the reaction X + YZ — XY+ Z.

Another remarkable aspect of LF, seen in the simulations, is that it can actually vibrationally cool the molecules. In particular, this means that the average internuclear separation Ravg should decrease as a function of increasing intensity (Fig. 1.6). This is exactly what we observe experimentally in Fig. 1.7, even as the amplitude of the vibrations is increasing [29]. 

Fig. 3. The Hydrogen Molecule optimised orbital exponents for small internuclear separation...

In everything that follows in this section we tacitly use a realistic model for the molecular electronic structure at all internuclear separations (e.g. UHF or electron-pair). Deferring questions of numerical techniques for the moment to a later section, we can investigate these functions as basis functions for the calculation of molecular electronic structure and investigate the spatial symmetry constraint in a wider context than the hydrogen molecule. 

A second test involves the role of internuclear separation between the electron donor and acceptor sites. Rearrangement of eq 8 leads to the prediction of a -1/d dependence for E. ... 

He also investigated the dependence of the properties on the internuclear distance in the vicinity of the equilibrium internuclear separation R. However, our goal was not to look for the most accurate values or to compare with them, but to study the performance of various linear response methods. In particular we wanted to find out which level of correlation is necessary in order to reproduce the correct dependence on the internuclear distance for a wider range of R. In Fig. 6, it can be seen that the CCSD dipole polarizability curve is sufficiently close to the results of the calculations. The SOPPA... 

The reason why one chose to follow the main liquid-crystalline to gel phase transition in DPPC by monitoring the linewidth of the various or natural abundance resonance is evident when we consider the expressions for the spin-lattice relaxation time (Ti) and the spin-spin relaxation time T2). The first one is given by 1/Ti oc [/i(ft>o) + 72(2ft>o)] where Ji coq) is the Fourier transform of the correlation function at the resonance frequency o>o and is a constant related to internuclear separation. The relaxation rate l/Ti thus reflects motions at coq and 2coq. In contrast, the expression for T2 shows that 1/T2 monitors slow motions IjTi oc. B[/o(0) -I- /i(ft>o) + /2(2u>o)], where /o(0) is the Fourier component of the correlation function at zero frequency. Since the linewidth vi/2 (full-width at half-maximum intensity) is proportional to 1 / T2, the changes of linewidth will reflect changes in the mobility of various carbon atoms in the DPPC bilayer. 

Table I. Ground-state energies of the N2 molecule, as described by the DZ basis set (113), for a few internuclear separations...

Figure 1. Diagrams of potential energy, V, versus Internuclear separation, q, for a molecule undergoing vibrational excitation by (a) the Raman effect or (b) a resonance Raman effect (hVfj-hvg) or a pre-resonance effect h >

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