Central forces potential energy

The Finnis-Sinclair type potentials (Finnis and Sinclair 1984) are central-force potentials but have a many-body character in that the energy of a system of particles is not merely a sum of pair interactions between individual atoms. In this scheme, modified for binary alloys by Ackland and Vitek (1990), the total energy of a system of N atoms is written as... 

Similarly, in studies of lamellar interfaces the calculations using the central-force potentials predict correctly the order of energies for different interfaces but their ratios cannot be determined since the energy of the ordered twin is unphysically low, similarly as that of the SISF. Notwithstcinding, the situation is more complex in the case of interfaces. It has been demonstrated that the atomic structure of an ordered twin with APB type displacement is not predicted correctly in the framework of central-forces and that it is the formation of strong Ti-Ti covalent bonds across the interface which dominates the structure. This character of bonding in TiAl is likely to be even more important in more complex interfaces and it cannot be excluded that it affects directly dislocation cores. 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

The potential energy expressions used for force field calculations are all descendants of three basic types originating from vibrational spectroscopy (5) the generalized valence force field (GVFF), the central force field, and the Urey-Bradley force field. General formulations for the relative potential energy V in these three force fields are the following ... 

The summations in Eq. (8) and (9) usually extend over all internal parameters, independent and dependent, i.e. the potential constants in these expressions are also not all independent. For example, the nonsymmetric tetrasubstituted methane CRXR2R3R4 possesses five independent force constants for angle deformations at the central carbon atom, whereas in our calculations we sum over the potential energy contributions of the six different angles (only five are independent ) at this atom using six different potential constants for angle deformations. The calculation of the independent force constants, which is necessary for the evaluation of the vibrational frequencies, will be dealt with in Section 2.3. 

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... 

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... 

NUCLEAR POTENTIAL. The potential energy V of a nuclear particle as a function of its position in the field of a nucleus or of another nuclear particle. A central potential is one that is spherically symmetric, so that V is a function only of the distance r of the particle from the center of force. A noncentral potential, on the other hand, is one that is not spherically symmetrical, or one that depends upon the relative directions of the angular momenta associated with the particle and the center of force, as well as upon the distance r. A negative potential corresponds to an attractive force, while a positive potential corresponds to a repulsive force. 

If one adopts McLennan s [78b] interpretation, then Eq. (21) is a realization of a standard theorem of Newtonian mechanics conservation of total energy = conservation of kinetic plus potential energy (see, e.g., Chap. 4 of Kleppner and Kolenkow, [80]). The reason is simple Coulomb electric force is central, then work is path independent, and total energy is function of position only. The time derivative of total energy is of course zero, as in Eq. (21). In this interpretation Qp and Qi are manifestations of kinetic energy. 

Here F is the central force, V is the potential energy, r is the orbital radius, m is the mass, w is the angular velocity, E is the total energy, etc. 

Here we consider an optical transition between Aj and E electronic states of a center of a trigonal symmetry. To describe the vibrations of the center we use the collinear-configurational approximation [27] in which only the central forces are taken into account in the optical center (taking account of deviations from this approximation, see later). If one restricts oneself to the linear vibronic coupling in the e state, then in this approximation the potential energy operators in the Ai and E electronic states can be presented in the form ... 

We will not discuss the actual construction of potential energy surfaces. This monograph deals exclusively with the nuclear motion taking place on a PES and the relation of the various types of cross sections to particular features of the PES. The investigation of molecular dynamics is — in the context of classical mechanics — equivalent to rolling a billiard ball on a multi-dimensional surface. The way in which the forces i fc(Q) determine the route of the billiard ball is the central topic of this monograph. In the following we discuss briefly two illustrative examples which play key roles in the subsequent chapters. 

The motion in the reaction coordinate Q is, like in gas-phase transition-state theory, described as a free translational motion in a very narrow range of the reaction coordinate at the transition state, that is, for Q = 0 hence the subscript trans on the Hamiltonian. The potential may be considered to be constant and with zero slope in the direction of the reaction coordinate (that is, zero force in that direction) at the transition state. The central assumption in the theory is now that the flow about the transition state is given solely by the free motion at the transition state with no recrossings. So when we associate a free translational motion with that coordinate, it does not mean that the interaction potential energy is independent of the reaction coordinate, but rather that it has been set to its value at the transition state, Q j = 0, because we only consider the motion at that point. The Hamiltonian HXlans accordingly only depends on Px, as for a free translational motion, so... 

Most systematic studies on gas-phase SN2 reactions have been carried out with methyl halides, substrates which are free of complications due to competing elimination. Application of the Marcus rate-equilibrium formalism to the double-minimum potential energy surface led to the development of a model for intrinsic nucleophilicity in S 2 reactions233. The key quantities in this model are the central energy barriers, Eq, to degenerate reactions, like the one of equation 22, which are free of a thermodynamic driving force. 

The Electric Field Within a Metal.—We have seen in Chap. XXI, Sec. 2, that an electron in an isolated atom is acted on by a central force on the average, equal to the attraction exerted by the nucleus, diminished by a certain shielding effect on account of the other electrons. The potential energy of the electron in such an atom was illustrated in Fig. XXII-7. When such atoms are placed near each other, the potential... 

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