Potential energy central force problem

For a one-particle central-force problem, the potential energy is a function only of the particle s distance from the origin V=V r) from (1.156), the Hamiltonian is... 

Since the potential energy in (4.4) depends only on R, we have a central-force problem, and the results of Section 1.11 show that... 

For a one-particle system with potential energy a function of r only [V=V(r), a central-force problem], the stationary-state wave functions have the form tf/ = R(r)Y (0, ), where R r) satisfies the radial equation (6.17) and Y" are the spherical harmonics. 

The problem of the preferred conformation of cyclodecane has been extensively studied by Dunitz et al. (46). In the crystals of seven simple cyclodecane derivatives (mono- or 1,6-disubstituted cyclodecanes) the same conformation was found for the ten-membered ring (BCB-conformation, Fig. 9). It follows from this that the BCB-conformation is an energetically favourable conformation, possibly the most favourable one. Numerous force field calculations support this interpretation Of all calculated conformations BOB corresponded to the lowest potential energy minimum. Lately this picture has become more complicated, however. A recent force field calculation of Schleyer etal. (21) yielded for a conformation termed TCCC a potential energy lower by 0.6 kcal mole-1 than for BCB. (Fig. 9 T stands for twisted TCCC is a C2h-symmetric crown-conformation which can be derived from rrans-decalin by breaking the central CC-bond and keeping the symmetry.) A force field of... 

The problem of obtaining a relationship between W r) and the intermolecular potential u(r) is central to the statistical mechanics of non-ideal gases and liquids. Various methods have been described for making this connection as discussed in the statistical mechanical literature [G1-G4]. As with any potential energy, one may obtain the force acting on the central particle in the direction of any other particle by differentiation with respect to the distance vector rj. Thus, the mean force acting on the central particle due to a molecule at rj is... 

The hydrogen atom is a typical case of the central-field problem. As was shown in Fig. 19.5, the proton is at the center with a charge + e while the electron is at a distance r with a charge —e. The coulombic force acts along the line of centers and corresponds to a potential energy, V(r) = —e /4ncQr. 

Throughout this textbook we will be studying the Coulomb force interactions of various particles atomic nuclei and electrons, atoms and other atoms, molecules and other molecules. From the potential energy function and the total energy, we can in principle determine all the possible results of these interactions. Because the total energy is conserved, the potential energy becomes the key to many central problems throughout physical chemistry. Keep an eye on it. 

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