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Coordinate systems Euler angles

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. [Pg.186]

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). 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 angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... [Pg.345]

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]. [Pg.253]

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS ... [Pg.124]

Figure 1 Definition of the coordinate systems Oxyz and OXYZ and Euler angles 6, cp, and ij/. Figure 1 Definition of the coordinate systems Oxyz and OXYZ and Euler angles 6, cp, and ij/.
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... [Pg.302]

The equations of internal motion defined within the local (molecular) fixed coordinate system have to be transformed to the laboratory fixed coordinate system in which all experiments are performed. Thus, introducing Euler s angles, (b, 0, and X, the coordinates of the electrons (e) and nuclei (n) are transformed in the following... [Pg.150]

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. [Pg.123]

Here, the subscript (c) is short for the set of expansion parameters (c) = (2i, 22, A, L, oi, u2) r, is the vibrational coordinate of the molecule i R is the separation between the centers of mass of the molecules the Q, are the orientations (Euler angles a, jS y,) of molecule i Q specifies the direction of the separation / the C(2i22A M[M2Ma), etc., are Clebsch-Gordan coefficients the DxMt) are Wigner rotation matrices. The expansion coefficients A(C) = A2i22Al u1u2(ri,r2, R) are independent of the coordinate system these will be referred to as multipole-induced or overlap-induced dipole components - whichever the case may be. [Pg.147]

If one performs a turn of the coordinate system x y z by Euler angles a, 0,7, thus transferring it to the position xyz, then the components Ay and A of the vector A in the new coordinate system may be expressed through the components Ay>, A11 in the initial coordinate system by means of Wigner D-functions... [Pg.243]

The operators in (2.36) are easily re-expressed in the molecule-fixed coordinate system since the V" operator merely becomes the V, operator in the new coordinate system and, as mentioned earlier, Fei,nuci becomes independent of the Euler angles. We must also consider the transformation of the partial differential operators 3/3[Pg.52]

In molecular quantum mechanics, we often find ourselves manipulating expressions so that one of a pair of interacting operators is expressed in laboratory-fixed coordinates while the other is expressed in molecule-fixed. A typical example is the Stark effect, where the molecular electric dipole moment is naturally described in the molecular framework, but the direction of an applied electric field is specified in space-fixed coordinates. We have seen already that if the molecule-fixed axes are obtained by rotation of the space-fixed axes through the Euler angles (, 6, /) = >, the spherical tensor operator in the laboratory-fixed system Tkp(A) can be expressed in terms of the molecule-fixed components by the standard transformation... [Pg.167]

Here, as usual, the suffices p and q refer to space- and molecule-fixed components respectively and > stands for the three Euler angles (. 0. /) which relate the two coordinate systems. From equations (5.155) and (5.156) we have... [Pg.169]

In chapter 2 we introduced the molecule-fixed axis system (x, y, z) which rotates in space with the molecule. This axis system is related to the non-rotating (but translating) axis system (X, Y,Z) by the three Euler angles (0,0,/). The coordinates of a point i in the molecule-fixed axis system are related to its coordinates expressed in the space-fixed axis system by the transformation... [Pg.245]

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values <f>, 9, x)by <ji = n + <ji,G = n — 0, and x = n X-...
In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. [Pg.249]

To obtain the isomorphic Hamiltonian for a diatomic molecule, x is introduced as an independent variable and the coordinates of the particles which make up the molecule are measured in an axis system (x, y, z) whose orientation is described by the Euler angles (<-/>, 0, x) in the (X, Y, Z) axis system. We recall that we chose x to be zero in constructing the true Hamiltonian. The (x, y, z) axes are therefore obtained by rotation of the (x7, y, z ) axis system about the z (= z) axis through the angle x As a result, we have... [Pg.322]

The analysis is not yet complete because we have to consider the left-hand term in equation (8.495) which relates the space-fixed and molecule-fixed coordinate systems. Although we have already selected the q = 0 component, we will retain q as a variable in order to facilitate later discussion. First we note that and can be expressed in terms of the Euler angles 0 and x as follows ... [Pg.571]


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See also in sourсe #XX -- [ Pg.365 , Pg.565 ]




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