Center-of-mass motion

3 Center-of-Mass Motion From Eqs. 3.119 and 3.142, the center-of-mass displacement in time t is calculated as 

It is proportional to t. The center of mass of the bead-spring chain makes a diffu-sional motion on all time scales. From this equation, we obtain the center-of-mass diffusion coefficient Dq. 

The centroid motion of the bead-spring chain is identical to the motion of a particle that receives a friction of o- The latter is also evident in Eq. 3.131 with i = 0. 

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Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

The problem of separating the center-of-mass motion in a molecular system is an intricate one that has no implications in the present work the interested reader is referred to [40] for details. 

This way we take the quasi-particle energies which are described by an effective mass and a self-energy shift and solve the Schrodinger equation for the separable Yamaguchi potential. Separating the center of mass motion, with energy p2/2 from the relative motion, with reduced mass M M /(M +... 

From this expression one has to remove the center-of-mass motion. The corresponding Hamiltonian can then be rewritten in terms of the intrinsic variables and the Euler angles. The general form of the Hamiltonian operator is, in the Eckart frame,1... 

Of the 3n coordinates needed to describe an n-atom molecule, three are used for center of mass motion, three describe angular displacement (rotation, hindered rotation, or libration) (two if the molecule is linear, 0 if monatomic), the remaining 3n—6 (3n—5, if linear, 3n—3 = 0, if monatomic) describe atom-atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from rotation and vibration, but when the separation can be made it is a convenience. Elementary treatments assume... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

Another approach has been proposed and employed to a number of molecules by Handy et al. (Handy et al., 1986 loannou et al., 1996 Handy and Lee, 1996). In this method the adiabatic correction is computed without separation of center of mass motion and ... 

The thermal properties of AU55 are treated in Sect. 3, using especially the results of MES measurements [24,25,42]. These are discussed in connection with the concept of bulk versus surface modes in small particles. An explanation of the temperature dependence of the MES [42] absorption intensities and the Cv results [25] on the basis of a model using the site coordination and the center-of-mass motion are briefly reviewed. The consequences of the Mossbauer results for surface Debye temperatures and for the melting temperature of small gold particles are also discussed. 

This in itself is proof of the importance of the center-of-mass motion in treating the thermal properties of small particles at low temperature. 

The experimental observation that one has different Debye temperatures for the three distinct surface sites of the AU55 cluster makes the use of a continuum-model picture for discussing the thermal behavior questionable. Indeed, for such small particle sizes, where the surface structure is so manifest, the use of the concept of surface modes becomes dubious, and is certainly inadequate to explain the observed temperature dependence of the f-factors. None the less, it has proven possible to describe the low temperature specific heat of AU55 quite well using such a continuum-model, when the center-of-mass motion is taken into account [99],... 

The T phonon term [25], corresponding to the center-of-mass motion of the clusters, can be safely extrapolated for temperatures below 2 K, and the linear term can be estimated from the known value of 7 for bulk gold [142] in the same temperature range. These two terms would be approximately equal for a temperature of about 40 mK. 

From the Mossbauer results on Aujj, the thermodynamic behavior of AU55, as well as AugLi " and AuiiL7(SCN)j, at temperatures between 2 K and 30 K can be understood on the basis of the center-of-mass motion of the whole gold core of the material, within the matrix formed by the surrounding ligands. Furthermore, none of these clusters are in the liquid (fluxional) state on the MES time scale, up to at least 30 K. 

The coordination numbers based on this structure work extremely well for describing the microscopic physical properties of this material, including the Mossbauer I.S.s of the surface sites and of the specific heat of the clusters below about 65 K. No linear electronic term in the specific heat is seen down to 60 mK, due to the still significant T contribution from the center-of-mass motion still present at this temperature. The Schottky tail which develops below 300 mK in magnetic fields above 0.4 T has been quantitatively explained by nuclear quadrupole contributions. 

This result is also correct for the center of mass motion of the pair if Ai is replaced by Acm- We will also use this expression for the sum over states Zr = Z22XqM/v of relative motion, where s substituted for... 

The Schrodinger equation for the interacting pair of linear molecules is again separated into the equations of center-of-mass motion and relative motion, exactly as this was done in Chapter 5. The equation of the center-of-mass motion is of little interest and will be ignored. The Schrodinger equation of relative motion is given by [354]... 

The convection flow corresponding to a center of mass motion a>. nmnuniing to ,tu... 

The other important case is a center-of-mass motion of molecules between the leads (Fig. 10). Here not the internal overlap integrals, but the coupling to the leads Vik<7,a(x) is fluctuating. This model is easily reduced to the general model (137), if we consider additionaly two not flexible states in the left and right leads (two atoms most close to a system), to which the central system is coupled (shown by the dotted circles). 

The translational motion of the particles as a whole (i.e., the center-of-mass motion) can be separated out. This is done by a change of variables from riab, R ab to i CM and r, R, where Rcm gives the position of the center of mass and r, R are internal coordinates that describe the relative position of the electrons with respect to the nuclei and the relative position of the nuclei, respectively. This coordinate transformation implies... 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

That is, the Maxwell-Boltzmann distribution for the two molecules can be written as a product of two terms, where the terms are related to the relative motion and the center-of-mass motion, respectively. After substitution into Eq. (2.18) we can perform the integration over the center-of-mass velocity Vx. This gives the factor y/2iVksTjM (IZo eXP( —ax2)dx = sjnja) and, from the equation above, we obtain the probability distribution for the relative velocity, irrespective of the center-of-mass motion. 

Since the outcome of the collision only depends on the relative motion of the reactant molecules, we begin with an elimination of the center-of-mass motion of the system. From classical mechanics it is known that the relative translational motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass fj, = rri nif)/(m + mB), relative to a fixed center of force. This result can be generalized to molecules by introducing proper relative coordinates, to be described in detail in Section 4.1.4. 

The dynamics of the two-particle problem can be separated into center-of-mass motion and relative motion with the reduced mass /i = morn s/(rnp + me), of the two particles. The kinetic energy of the relative motion is a conserved quantity. The outcome of the elastic collision is described by the deflection angle of the trajectory, and this is the main quantity to be determined in the following. The deflection angle, X, gives the deviation from the incident straight line trajectory due to attractive and repulsive forces. Thus, x is the angle between the final and initial directions of the relative velocity vector for the two particles. 

In a collision process, it is the relative position of the atoms that matters, not the absolute positions, when external fields are excluded, and the potential energy E will depend on the distances between atoms rather than on the absolute positions. It will therefore be natural to change from absolute Cartesian position coordinates to a set that describes the overall motion of the system (e.g., the center-of-mass motion for the entire system) and the relative motions of the atoms in a laboratory fixed coordinate system. This can be done in many ways as described in Appendix D, but often the so-called Jacobi coordinates are chosen in reactive scattering calculations because they are convenient to use. The details about their definition are described in Appendix D. The salient feature of these coordinates is that the kinetic energy remains diagonal in the momenta conjugated to the Jacobi coordinates, as it is when absolute position coordinates are used. 

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