Mass, centers of

In a crossed-beam experiment the angular and velocity distributions are measured in the laboratory coordinate system, while scattering events are most conveniently described in a reference frame moving with the velocity of the centre-of-mass of the system. It is thus necessary to transfonn the measured velocity flux contour maps into the center-of-mass coordmate (CM) system [13]. Figure B2.3.2 illustrates the reagent and product velocities in the laboratory and CM coordinate systems. The CM coordinate system is travelling at the velocity c of the centre of mass... 

Consider a polyatomic system consisting of N nuclei (where > 3) and elecbons. In the absence of any external fields, we can rigorously separate the motion of the center of mass G of the whole system as its potential energy function V is independent of the position vector of G (rg) in a laboratory-fixed frame with origin O. This separation introduces, besides rg, the Jacobi vectors R = (R , , R , .. , Rxk -1) = (fi I "21 I fvji) fot nuclei and electrons,... 

We will omit the kinetic energy operator Tg of the center of mass <5, since no external fields act on the system and consider only its internal kinetic energy operator 7 " given by [26]... 

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... 

H at m energy of 1.2 eV in the center-of-mass frame. By using an atomic orbital basis and a representation of the electronic state of the system in terms of a Thouless determinant and the protons as classical particles, the leading term of the electronic state of the reactants is... 

The mass weighted position of a single nucleus v in the center-of-mass frame of a molecule with N atomic nuclei at time points t is obtained from an END trajectory and can be expressed as... 

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

In special cases (as in colloidal solutions) some particles can be considered as essential and other particles as irrelevant , but in most cases the essential space will itself consist of collective degrees of freedom. A reaction coordinate for a chemical reaction is an example where not a particle, but some function of the distance between atoms is considered. In a simulation of the permeability of a lipid bilayer membrane for water [132] the reaction coordinate was taken as the distance, in the direction perpendicular to the bilayer, between the center of mass of a water molecule and the center of mass of the rest of the system. In proteins (see below) a few collective degrees of freedom involving all atoms of the molecule, describe almost all the... 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

The qualitative solution behavior becomes more apparent when going to local coordinates, i.e., we rewrite the equations of motion in terms of the center of mass... 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

Translation of the center of mass and rotation of the internal coordinate system as in Step2... 

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... 

As an illustration, we consider a simple example of a top with a fixed point at the center of mass moving in an applied field not dissimilar from those encountered in molecular simulations. Specifically, we used... 

The plane is specified as a plane parallel to the screen and offset from the center of mass of the current selection towards the user by a specified n u m her of An gsLroin s. Tor exam pie, if you m ahe n o selection and specify an offset of 0.0, th e plan e is th rough thecen-... 

If we think about two masses connected by a spring, each vibrating with respect to a stationary center of mass Xc of the system, we should expect the situation to be vei similar in form to one mass oscillating from a fixed point. Indeed it is, with only the substitution of the reduced mass p for the mass m... 

Figure 4-2 Two Masses Vibrating Harmonically with Respect to their Center of Mass. The center of mass may be stationary or moving with respect to an external coordinate system.

To an observer at the center of mass, the overall motion of the system (translation) is irrelevant. The only important motions are those motions relative to the center of mass. Distances from the center of mass to each particle are internal coordinates of the system, usually denoted r and r2 to emphasize that they are internal coordinates of a molecular system. 

In the case of a polyatomic molecule, rotation can occur in three dimensions about the molecular center of mass. Any possible mode of rotation can be expressed as projections on the three mutually perpendicular axes, x, y, and z hence, three moments of inertia are necessar y to give the resistance to angular acceleration by any torque (twisting force) in a , y, and z space. In the MM3 output file, they are denoted IX, lY, and IZ and are given in the nonstandard units of grams square centimeters. 

COORDINATES TRANSLATED TO NEW ORIGIN WHICH IS CENTER OF MASS... 

For homonuclear molecules (e.g., O2, N2, etc.) the inversion operator i (where inversion of all electrons now takes place through the center of mass of the nuclei rather than through an individual nucleus as in the atomic case) is also a valid symmetry, so wavefunctions F may also be labeled as even or odd. The former functions are referred to as gerade (g) and the latter as ungerade (u) (derived from the German words for even and odd). The g or u character of a term symbol is straightforward to determine. Again one... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

The electronic spatial extent is a single number that attempts to describe the size of a molecule. This number is computed as the expectation value of electron density times the distance from the center of mass of a molecule. Because the information is condensed down to a single number, it does not distinguish between long chains and more globular molecules. 

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