Functions, potential Terms Links

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally... 

---