Space-fixed frame

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

Figure 5. Velocity field (space-fixed frame) inside the central object of the remnant of a neutron star coalescence. The labels at the contour fines refer to log(p), typical fluid velocities are 10s cm/s.

Hmi is the i-th cluster hamiltonian referred to its space fixed frame with axes parallel to the laboratory frame. Kcmi is the i-th cluster kinetic energy operator with total mass Mj. The existence of electronic wave functions for each system is taken for granted as their spectra can be determined in isolation. 

We consider a closed-shell dimer AB with NA electrons and nA nuclei that can be assigned to the monomer A, and NB electrons and nB nuclei that can be assigned to the monomer B. The set of electronic coordinates r , = l,... NA + NB will be denoted by r, while the nuclear coordinates, Ry, y = 1,. ..nA + nB, will be denoted in short by R. The coordinates of electrons and nuclei are defined in a space-fixed frame. The nuclear masses will be denoted by My, and atomic units me = e = h= 1 will be used throughout this chapter. The Schrodinger equation for the total wave function Tf,ot can be written as,... 

The Van der Waals constants Cn(a)A, (oB, R) depend on the Euler angles (oA and (oB specifying the orientation of the monomers in an arbitrary space-fixed frame, and on the polar angles R = (fi, a) determining the orientation of the intermolecular axis (R is assumed to join the monomer centers of mass) with respect to the same space-fixed frame. 

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. 

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

The rotational and vibrational kinetic energies of the nuclei are represented by the term —(7/2/2/x)V ( in equation (2.37) we now seek its explicit form and the relation between the momentum operators PR and Pa in equation (2.6). If we take components of Pr in a space-fixed frame, we have the straightforward relationship ... 

It should be noted that result given by equation (91) coincides with that calculated under the assumption that the orientation of atomic orbitals remains fixed during collision ( fixed-atom approximation ) [84, 87]. This is to be expected, because the strong Coriolis mixing of molecular functions in the rotating frame prevents rotation of orbitals in a space fixed frame. 

The basis of the principal frame is the eigenvectors of the inertia tensor. The matrix that transforms a vector from the space fixed frame to the principal frame is denoted by A,-, the rows of this matrix are the vectors (u,-,v,-,w,). The inverse of A,- transforms a vector from the principal frame to the space fixed frame is simply Af. 

In this contribution, within the asymptotic approach, we have elaborated the basis for the calculation either of adiabatic channel potentials (diagonalization of the full Hamiltonian in a body-fixed frame at given interfragment distances) or of axially-nonadiabatic channel potentials (diagonalization of the full Hamiltonian in a space-fixed frame at given interfragment distances). As a by-product, we have compared our asymptotic PES on different levels of approximations with available local ab initio data. In subsequent work, we envisage the calculation of low temperature rate constants for complex-formation of the title reactions. 

This matrix is nnitary, b( canse all the matrices on the right hand side of E(p(115) are unitary. It should be noted that E(i.(115) does not describe the formal transformation from the body-fixed to space-fixed frame by coordinate transformation [1], but just gives the transformation of electronic basis sets in the asymptotic region. 

D] a, P, y) depends on the Euler angles that rotate the spaced-fixed frame on to the body-fixed frame and are eigenfuetions of. The body-fixed internal angular momentum funetion in Eq. (21) is given by... 

Beginning with the dipole-dipole angular correlation projection (in the space fixed frame) = liddQ + h. it can be seen that for the bulk dipolar fluid, the long-wave length limit of this function is given by [23],... 

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). 

Each choice of the body-axes specifies a reference orientation, in which the body frame coincides with the space-fixed frame. Let us choose one of the body axes, say u 3, to be in the direction of... 

Consider two rigid molecules A and B, both of arbitrary shape. Let ft = (R, Q) = (R, 0, 4>) be the vector pointing from the center of mass of A to the center of mass of B. The coordinates of ft are measured with respect to a space-fixed frame. Let the orientation of molecule A be described by the Euler angles = (a, P, y ), which are the angles associated with an (active) rotation of the molecule from an initial position in which a reference frame fixed on A is parallel to the space-fixed frame, to its present position. Similarly, the orientation of B is determined by the Euler angles b — ( b Pb> Vb)- The interaction energy between A and B is most generally... 

In general, H(r,Q) is invariant to a translation of origin of the laboratory frame, not to be mixed up with the space fixed frame which moves with the center-of-mass commonly used [1]. Let us select a particular (arbitrary for the time being) origin with real space coordinates u. Applying the corresponding translation operator H(r,Q) changes into ... 

In this context, consider a simple molecule such as formamide (Figure 2, top). It has 6 atoms and hence 18 Cartesian coordinates. These may be combined to form 3 coordinates that specify the location of the molecular center of mass relative to a space-fixed frame of reference, 3 coordinates that specify the orientation of the whole molecule in the space-fixed frame of reference, and 12 internal coordinates that specify the structure of the molecule. Thus, only the structure of the molecule, including 12 coordinates, is required to specify the internal or strain energy. That is, from these 12 coordinates, all bond lengths. 

---