Kinetic energy expression

Let us consider a space fixed Cartesian coordinate i stem (SFCS, see Appendix 1 on p. 971), and vector Rcm indicating the centre of mass of a molecule composed of M atoms, Fig. 6.5. Let us construct a Cartesian coordinate i stem (Body-Fixed Coordinate System, BFCS) with the origin in the centre of mass and the axes parallel to those of the SFCS (the third possibility in Appendix I). 

3 Such a vibration may mean an osdilation of the OH bond, but also a rotation of the -CH3 group or a large displacement of a molecular liagmenL 

When these velocities F are inserted into the kinetic energy T of the molecule calculated in the SFCS, we get 

The first three ( diagonal ) terms have a clear interpretation. These are the kinetic energy of the centre of mass, the kinetic energy of rotation, and the kinetic energy of vibrations. The three further terms ( non-diagonal ) denote the roto-translational, vibro-translational and vibro-rotational couplings. 

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Since the nuclear coordinates are expanded according to Eq. (5), we can write the derivatives in the kinetic energy expression as... 

The average kinetic energy expressed by Equation is kinetic energy per molecule. We find the total kinetic energy ( kinetic molar) of one mole of gas molecules by multiplying Equation by Avogadro s number ... 

Of course, all this is not new but only a recapitulation of results from Chapter 1. The important connection to density functional theory is that we now go on to exploit the above kinetic energy expression, which is valid for non-interacting fermions, in order to compute the major fraction of the kinetic energy of our interacting system at hand. 

Figure 1.9 Molecular energies follow the Maxwell-Boltzmann distribution energy distribution of nitrogen molecules (as y) as a function of the kinetic energy, expressed as a molecular velocity (as x). Note the effect of raising the temperature, with the curve becoming flatter and the maximum shifting to a higher energy...

Here KE(1) and KE(2) are classical kinetic energy expressions for isotopomer 1 and isotopomer 2 respectively, each containing terms for the kinetic energy of each atom in each of the three coordinates. For N-atomic molecules there are three Cartesian momenta for each atom, 3N Cartesian momenta for each molecule, and consequently 3NN Cartesian momenta for the N molecule system. The integrals in the numerator and denominator can thus be written as a product of 3 integrals of the type... 

We shall impose the restriction that the potential energy V depends only on the relative coordinates of the particles V= V x,y,z). Substitution of (1.214) and (1.215) into the kinetic-energy expression leads to the following expression for the classical-mechanical Hamiltonian ... 

We found it easy to deal with the nuclear kinetic-energy terms for a diatomic molecule but for a polyatomic molecule, the necessary manipulations to bring the classical-mechanical kinetic-energy expression into a form suitable for the quantum-mechanical treatment are complex. We omit these manipulations1 and simply state their result. The classical expression for the kinetic energy TN of the nuclei is the sum of several terms ... 

Let us consider now the kinetic energy expression for the transformed wave-... 

Fig. 11. Total cross section for the interaction of relativistic positronium atoms with carbon as a function of the kinetic energy expressed in the rest masses of the incident atom (T = 7—1). The solid curve is the theoretical dependence, - the measured value. The arrow marks the region (7 < 1.2) investigated in experiments on the interaction of hydrogen atoms with carbon...

The kinetic energy expression can also be written in terms of the Laplace operators ... 

At the source outlet, every ion, if we consider only the singly charged ones, has a kinetic energy expressed by the following equation ... 

The quantities (a, =x,y, z, p) are the elements of the matrix which is the inverse of the 4x4 matrix p is the determinant of the matrix [Paj ]-We have so far considered only the kinetic energy expression. We must also consider the potential energy expression V, which can be expanded for each value of p as a Taylor series in the normal coordinates Q ... 

The quantity T q,d l//8q) is the kinetic energy expressed in terms of the momentum as in the first term on the right-hand side of eqn (5.47) that is, as p /2m, and V is the potential energy. Schrodinger considered specifically the problem of the hydrogen atom for which the explicit form of the functional is... 

The results obtained for these commutators parallel those for the field-free case, eqn (8.191). Thus the first commutator yields twice the operator for the kinetic energy, expressed here in terms of n (see eqn (8.226))... 

Figure 3. Distributions of short-term kinetic energies, expressed as effective temperatures, for Ar3, at total energies corresponding to (a) 28.44 K and (b) 30.54 K. The low kinetic energies correspond to trajectory segments in the saddle region the high kinetic energy parts of the distribution are associated with motion above the deep well of the equilibrium geometry. [Reprinted with permission from T. L. Beck, D. M. Leitner, and R. S. Berry, J. Chem. Phys. 89, 1681 (1988). Copyright 1988, American Institute of Physics.]...

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

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