Vectors, three body system

Let us consider a three-body system with Hill type stability. The corresponding R, r point is of course in the zone of possible motion of a suitable figure as Figure 7 (conditions (32) and (34)) and the corresponding radius vectors R and r and velocity vectors V and v satisfy the equations (27) and (28) of the two integrals of motion. However notice that, in order to satisfy the condition (34) it is sufficient to find two velocity vectors V and v such that ... 

Within the hyperspherical method, new quantum numbers K, T and A are introduced to describe two-electron correlations. Both K and T are angular correlation numbers (omitted here for simplicity, see [333]), while A = 0, 1 is a radial quantum number, often written as 0,+,— because it is related to the + and — classification of Cooper, Fano and Pratts [323] described in section 7.10. Another quantum number which is often used is v = n — 1 — K — T, where n is the principal quantum number. The number v turns out to be the vibrational quantum number of the three-body system, or the number of nodes contained between the position vectors ri and r2 of the two electrons [334]. 

For a three-body Coulomb explosion event, the total number of momentum components determined is nine (three for each fragment ion) in the laboratory frame. However, the number of independent momentum parameters required to describe the Coulomb explosion event in the molecular frame is reduced to three under conditions of conserved momentum. This is because three degrees of freedom in the momentum vector space are reserved to describe the translational momentum vector of the center of mass, and another three are used for the overall rotation of the system that describes the conversion from the laboratory frame to the molecular frame. In other words, the nuclear dynamics of a single Coulomb explosion event of CS, CS —> S+ + C+ + S+ in the molecular frame can be fully described in the three-dimensional momentum space specified by a set of three independent momentum parameters. There... 

For a prototypical three body problem, the Helium-like atom, the procedure is well known since the early days of quantum mechanics. More recently, Fano, Macek and Klar [18-23] identified a near separable variable p = (rf -I- rD, where rj and T2 are the two Jacobi vectors of the system, named hyperradius , corresponding to the radius of a six-dimensional hypersphere pareuneterized by five hyperangles . However, note that the hyperradius is independent of the numbering of particles and is therefore very useful for rearrangement problems. 

The procedure can clearly be extended to treat more than three particles, and this is done, e.g. in ref. [24]. It has also to be pointed out the fact that the hyperradius is a measure of the total inertia of the n-body system, and this can be a physical motivation for its candidacy as a proper nearly separable variable, invariant with respect to the choice of the set of Jacobi vectors. 

As noted by Bom and Green, a superposition approximation for the three-body distribution function, similar to Kirkwood s form in equilibrium systems, Eq. (4.57), is needed for further analysis of Eq. (6.48). Using the solution form Eq. (6.47), we can, however, immediately write down expressions for the so-called auxiliary conditions and the property flux vectors. First, we have to O(e )... 

The method has been applied to vinyl bromide and silicon clusters. For Sis clusters a truncation after the three-body term has been found to be sufficient. The employed NN is shown schematically in Fig. 6. Like the HDMR method this approach is not constrained to a fixed system size and Si clusters with 3 to 7 atoms have also been fitted. For each A-body term all interatomic distances are used as input vector without an explicit incorporation of the symmetry. For the vinyl bromide molecule the energy has been expressed using five two-body terms, six three-body terms, and five four-body terms. Still, in applications to larger systems the efficiency of the method is low because of the large number of interactions, which have to be evaluated by NNs. 

The dipole operator d is a vector defined in the body-fixed frame of the molecule. Consequently, the transition dipole moment /a defined in (2.35) is a vector field with three components each depending — like the potential — on R, r, and 7. For a parallel transition the transition dipole lies in the plane defined by the three atoms and for a perpendicular transition it is perpendicular to this plane. Following Balint-Kurti and Shapiro, the projection of /z, which is normally calculated in the body-fixed coordinate system, on the space-fixed z-axis, which is assumed to be parallel to the polarization of the electric field, can be written as... 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

When we evaluate the Green-Kubo relations for the transport coefficients we solve the equations of motion for the molecules. They are often modelled as rigid bodies. Therefore we review some of basic definition of rigid body dynamics [10]. The centres of mass of the molecules evolve according to the ordinary Newtonian equations of motion. The motion in angular space is more complicated. Three independent coordinates a, =(a,a,-2, ,3), i = 1, 2,. . N where N is the number of molecules, are needed to describe the orientation of a rigid body. (Note that a, is not a vector because it does not transform like a vector when the coordinate system is rotated.) The rate of change of a, is... 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as... 

The orientation of a nonlinear molecule can be described by three Euler angles <, 0, j, because it takes two angles to describe the orientation of any body-fixed vector and takes one angle to describe the orientation of the body about that vector. The Euler angles relate the orientation of an orthonormal molecule-fixed axis system u), u 2, u to some standard orthonormal space-fixed frame u 1, u2,113 (see Fig. 1 and Eq. (A73) in Appendix A, Section 3.c). 

Let x = x, X2, , xn) give the configuration, in phase space at any instant, of the system under consideration (e.g. the vector having as components the co-ordinates and components of momentum in the relative motion of the three, or more, bodies) and suppose then that the equations of motion of the full system are written... 

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