Energy Euler angle

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

For simplicity, let us restrict ourselves to the case when fluctuations of the anisotropic potential are connected only with variation of its direction, determined by the Euler angles X. The energy of interaction of a rotator with the directing held depends only on the difference angle... 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

Rotational Diffusion Constants (x 1010 s" ), Euler Angles, Activation Energies (kj/mol), D Values (x 1012 s" ), Correlation Coefficients (R), and Quotients of the Rotational Diffusion Constants for Compound 31... 

It is assumed that the internal coordinates of the monomers A and B used in the calculations of EA and EB are the same as within the dimer AB. Thus, the interaction energy depends on the separation R between the centers of mass of the monomers, on the Euler angles characterizing their mutual orientation, and on monomer s internal coordinates. 

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

Figure 12. Local equilibrium and saddle structures of a four-atom Morse cluster, M4. The local equilibrium structures (regular tetrahedron) and the Hrst rank saddle structures (planar rhombus) have the energy —6.00s and —5.02s, respectively. cp1 cp2, and tp3 are the hyper-angles that specify each reaction channel. When tp2 = 3, (P and cp3 cannot be determined uniquely and only the summation cp3 +

In terms of the shape coordinates, the Euler angles, and the Cartesian coordinates of the center of mass, the kinetic energy operator can be written as the sum... 

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

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... 

For linear molecules A and B, where the interaction energy does not depend on the Euler angles y and Vg, only terms with = Kg = 0 contribute and one can use (3) to obtain a simplified expression for (l) (see also table 1) ... 

The Euler angle f does not appear in the kinetic energy, and consequently it is a cyclic or ignorable variable. In the body frame xyz, the other Euler angles 9, (p are the polar coordinates of the Z axis. The determinant of fhe matrix connecting the two alternative representations in Eq. (7) is D = sin 9 jabc. 

Cartesian coordinates and momenta for a polyatomic reactant are found from the energies of its normal modes [Eq. (2.33)] and the components of its angular momentum. The procedure is given by Eqn. (210) and steps 1-3 for microcanonical normal-mode sampling in Section II.A.3.a, and is applied to both reactants. Each reactant is randomly rotated through its Euler angles, as described by Eqs. (3.24) and (3.25), and the impact parameter, center-of-mass separation, and relative velocity added as described by Eqs. (3.26) and (3.27). 

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