Coordinate system body-fixed

Where XYZ stands for a specified Cartesian coordinate frame. Thus once a Cartesian coordinate frame is chosen, the nine spherical components can be determined using Eq. (7.4.1) from the nine Cartesian components. Clearly the spherical components and the Cartesian components change if the coordinate axes are rotated. Suppose we know the values of the Cartesian components of the polarizability tensor in a coordinate frame rigidly fixed within the molecule6 (the body-fixed frame OXY Z). Then the problem confronting us is to determine the Cartesian components of the polarizability tensor in a coordinate system rigidly fixed in the laboratory (the laboratory frame OX Y Z ). The relative orientations of the molecular and laboratory-fixed... 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

Space. Some fixed reference system in which the position of a body can be uniquely defined. The concept of space is generally handled by imposition of a coordinate system, such as the Cartesian system, in which the position of a body can be stated mathematically. 

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

Fig. 1. Definition of body-fixed and space-fixed coordinate systems. Refer to Table 1 and to the text for a detailed explanation...

Fig. 11.1. Center-of-mass or Jacobi coordinates R and r used to describe the fragmentation of a triatomic molecule ABC into A and BC. S and s are the centers-of-mass of ABC and BC, respectively, v is the relative velocity of the recoiling fragments in the center-of-mass system. The space-fixed z-axis is parallel to the vector Eo of the electric field, while the body-fixed z -axis is parallel to the scattering vector R at all times. The azimuthal angle ip, which is not indicated in the figure, describes rotation in the plane perpendicular to R.

In order to facilitate the evaluation of the overlap between the bound and the continuum wavefunctions it is advisable to represent the continuum wavefunction, like the wavefunction of the parent molecule, in the body-fixed (bf) coordinate system although the boundary conditions are most simply expressed in the space-fixed system. The transformation from the space- to the body-fixed system is rather lengthy and has been worked out in detail by Balint-Kurti and Shapiro (1981). The result is... 

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... 

Equations (1-124) and (1-133) are valid in an arbitrary space-fixed coordinate system. However, since the angular functions A A (a)A, coB, R) are invariant with respect to any frame rotation162, a specific choice of the coordinate system may considerably simplify Eq. (1-125). In particular, in the body-fixed coordinate system with the z axis along the vector R the polar angles R = (/ , a) are zero. Using the fact that (r = (0,0)) = 8Mfi 14S, one gets,... 

Here, the operator J is the total angular momentum operator in the space-fixed frame, and Tx, X=A and B, is defined by Eq. (1-262). Note, that the present coordinate system corresponds to the so-called two-thirds body-fixed system of Refs. (7-334). Therefore, the internal angular momentum operators jA and jB, and the pseudo angular momentum operator J do not commute, so the second term in Eq. (1-265) cannot be factorized. 

It ensues from the property (11) that it is sufficient to define (r R) and n(r)> only within the domain of internal nuclear coordinates R. The replacement of R by R = Rj>, where Rj = Xj,Yj,Zj>, which results in the removal of three degrees of freedom (two for linear molecules), corresponds to adopting a rotating ("body-fixed") coordinate system in place of the fixed ("space-fixed") one. Various definitions of the former coordinate system are possible, the most natural involving the requirement that the... 

Choosing a body-fixed (BF) coordinate system with the Z axis parallel to r, and a fixed value of the intramolecular distance r, the ground-state of the bound cluster of N He atoms is obtained by solving the Schrodinger equation... 

Let us apply the above general formalism to two simple examples that are central to this book chapter, namely that of a bulk fluid and a fluid confined to a slit-pore (see Sections 1.3.2 and 1.3.3). In both cases, we take as the reference system a rectangular prism of volume Vo = SxoSyoSzo, where a body-fixed coordinate system is employed such that the faces of the prism coincide with the planes x = ,Sxo/2, y = Syo/2, and = .Szo/2. If the rmstrained system is exposed to an infinitesimally small compressional or shear strain, Vo —> 1/ = SxSy z- This implies that a mass element originally at a point To in the unstrained. system changes position to a point r in the strained system. 

Figure 1. Center-of-mass coordinate systems for atom-diatom van der Waals molecules. The primed axes are used in the space-fixed (SF) formulation, the unprimed axes in the body-fixed (BF) formulation.

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