Complicated potential energy terms

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

The adiabatic approximation is made in a curvilinear coordinate system, and although the potential energy term is simple, the kinetic energy term is complicated by factors dependent upon the curvature of the reaction path [21,25,27]. As shown by Skodje et al. 

The situation is somewhat more complicated when the perturbation is a magnetic field. An electric field interacts directly with the charged particles (electrons and nuclei), and adds a potential energy term to the Hamiltonian operator. A magnetic field. 

Absorption and emission spectroscopies provide experimental values for the quantized energies of atomic electrons. The theory of quantum mechanics provides a mathematical explanation that links quantized energies to the wave characteristics of electrons. These wave properties of atomic electrons are described by the Schrddinger equation, a complicated mathematical equation with numerous terms describing the kinetic and potential energies of the atom. 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

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... 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

---