Coordinates center-of-mass

The CM coordinates for a two-particle system are defined in a zero-momentum reference frame in which the total force on two particles that interact only with each other is zero. We can define the total force of two interacting particles as 

One consequence associated with observing elastic collisions in the CM coordinates is that the individual particle kinetic energies are unchanged by the 

For CM coordinates, Fig. 3.3, we define the system velocity, vc, such that in this coordinate system there is no net momentum change, so that 

We also define in CM coordinates a reduced mass, Mc, given by the relation 

From (3.7) we see that, for large mass difference between Adi and M2, Adc approaches the value of the lower mass. For example, if Ad2 MX. A /c = M2. 

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Transfening into the center-of-mass coordinates, where... 

Actually, it is also useful to introduce the mean-square displacement of inner monomers measured in the center of mass coordinate system of the chain... 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

The application of the Bom-Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is sufficiently simple that we can also introduce center-of-mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons. 

C. Transformation to Center-of-Mass Coordinates Permutational Symmetry... 

The number of internal degrees of freedom for any system may be reduced by a transformation to center-of-mass coordinates. For example, the system of n + 1 particles with 3(n + 1) degrees of freedom is reduced n pseudoparticles with 3n degrees of freedom, with the 3 leftover degrees of freedom describing the motion of the center of mass. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

As an initial molecular system of reference a system centered on the 01 atom, has been selected. The x axis coincides with the 01-02 bond and the three atoms 01, 02 and HI lie in the xy plane. Appendix 1 shows the equations that connect Cartesian and internal coordinates and their derivatives. From the initial Cartesian coordinates, the X, Y and Z center-of-mass coordinates and its X, Y yZ. . . derivatives are calculated. The positions of the atoms have to be referred to the center of mass ... 

Suppose we are given Cartesian coordinates a, and mass mi for each atom i. To convert the coordinates to principal axes frame of reference, first translate to center-of-mass coordinates... 

For example, a projection operator into center-of-mass translation restricts to the molecular center-of-mass coordinates x, y, and z, and a projection operator into rotation restricts it to molecular rotational coordinates. In the case of SD it is usefril to construct projection operators measuring the influence of solvent molecules according to their location relative to the solute by restrictingyto, e.g., the closest molecule, all the molecules within the first solvation shell, etc.. Once we have chosen a particular projection operator,, we can find the projected portion of the influence coefficient... 

A manner to do away with the problem is to introduce appropriate algorithms in the sense that mappings from real space to Hilbert space can be defined. The generalized electronic diabatic, GED approach fulfils this constraint while the BO scheme as given by Meyer [2] does not due to an early introduction of center-of-mass coordinates and rotating frame. The standard BO takes a typical molecule as an object description. Similarly, the wave function is taken to describe the electrons and nuclei. Thus, the adiabatic picture follows. The electrons instantaneously follow the position of the nuclei. This picture requires the system to be always in the ground state. 

Here, the pt are the permanent dipoles of molecules i = 1 and 2, and the ptj( r, i 2, Rij) are the dipoles induced by molecule i in molecule j the are the vectors pointing from the center of molecule i to the center of molecule j and the r, are the (intramolecular) vibrational coordinates. In general, these dipoles are given in the adiabatic approximation where electronic and nuclear wavefunctions appear as factors of the total wavefunction, 0(rf r) ( ). Dipole operators pop are defined as usual so that their expectation values shown above can be computed from the wavefunctions. For the induced dipole component, the dipole operator is defined with respect to the center of mass of the pair so that the induced dipole moments py do not depend on the center of mass coordinates. For bigger systems the total dipole moment may be expressed in the form of a simple generalization of Eq. 4.4. In general, the molecules will be assumed to be in a electronic ground state which is chemically inert. 

Starting with Eq. 5.22 and summing over the center of mass coordinates, Xcm(4m/>0 exp (—JtcM/kT) = 1, we get the density operator of the states of relative motion, according to... 

In this section, we apply the eigenfunctions with another normalization. First, we introduce the relative and center-of-mass coordinates... 

In order to explain the band structure for the small confinement regime the nature of the potential energy function in the Hamiltonian has been examined in the internal space. Since, for quasi-one-dimensional quantum dots, the electrons can only move along the z coordinate, their x and y dependence is neglected in the analysis. The internal space is defined by a unitary transformation from the independent electron coordinates (z, Z2, , zn) into the correlated electron coordinates (za, zp,...). The coordinate za represents the totally symmetric center-of-mass coordinate za = 7=(zi + Z2 + + zn), and the remaining correlated electron coordinates zp,..., zn represent the internal degrees of freedom of the N electrons [20,21]. In the case of two electrons the correlated coordinates are defined by... 

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