Kinetic energy description

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

Therefore, it is yet to be clarified whether the description of the particle destruction process with Eqs. (2-4) or the simpfification in the estimate of energy dissipation from the measured turbulent kinetic energy produces these differ-... 

In fact, it turns out that the compensation between the kinetic energy and the nuclear attraction does lead to a qualitative description of the optimum orbitals in molecular systems, but only in the frame of the following restrictive conditions. 

This description results from the fact that the optimum orbitals are essentially determined in the region surrounding each atom by the compensation between the kinetic energy T of the electron and the Coulomb attraction of the electron by the nucleus of that atom. This compensation implies that the orbital is very weakly dependent of the environment of the atom in the molecular system so that it is essentially determined by atomic conditions (Valley theorem). 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

So far, the description has been limited to the case of an isolated molecule. In practice, however, the organic chemist typically deals with molecules in solution, or in gas phase at relatively high pressures. The medium then acts as a heat sink and efficiently removes excess vibrational energy. These are the conditions to which we shall limit our attention in the following. The simplest description would be that the overall motion of the wavepacket is slowed down by friction so that the nuclei never acquire very much kinetic energy in spite of the acceleration they receive from the hypersurface corresponding to the excited state. Attempts at calculations, even crude, become even more complicated. More realistic pictures of the effect of the heat bath presently appear to be hopelessly complex for detailed calculations. 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Much of the theoretical work in turbulent flows has been concentrated on the description of statistically homogeneous turbulence. In a statistically homogeneous turbulent flow, measurable statistical quantities such as the mean velocity2 or the turbulent kinetic energy are the same at every point in the flow. Among other things, this implies that the turbulence... 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

With the purpose of evaluate not only the energy but also the electron density itself, Ashby and Holzman [15] performed calculations in which the relativistic TF density was replaced at short distancies from the nucleus from the one obtained for the 1 s Dirac orbital for an hydrogenic atom, matched continuously to the semiclassical density at a switching radius rg where the kinetic energy density of both descriptions also match. 

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