Eckart-Sayvetz equations

The Eckart-Sayvetz equations [Eqs. (2.8)] imposed on a vibrating molecule require that the condition of zero linear and angular momenta is fulfilled. The molecular motion is considered as if it is purely vibrational. Rotations and translations of the molecule as described by the six external coordinates V ignored. 3N-6... 

The Eckart Sayvetz conditions imply that, if during the vibration a small translation of the center of masses is invoked, the origin of the Cartesian reference system is displaced so that no linear momentum is produced. The second Sayvetz condition, expressed in the last diree equations of (2.8), imposes the constraint that, during vibrational displacements, no angular momentum is produced. Eq. (2.8) implies that the reference Cartesian system translates and rotates with the molecule in such a way that the displacement coordinates Ax, Ay and Az reflect pure vibrational distortions. It is evident that through Eq. (2.8) certain mass-dependency is imposed on the atomic Cartesian displacement coordinates. 

It should be noted that the zero angular momentum condition is automatically considered dirough the last two terms in Eqs. (9.33) and (9.34). The Eckart-Sayvetz conditions are implicitly introduced in the VOTR equations thus avoiding the necessity for correcting do/dSj derivatives for rotational contributions. 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

The ab initio calculations produce values of fiy, i.e., the components of the electronically averaged dipole moment along the x y z axes defined above. In order to calculate molecular line strengths, however, we must determine, as functions of the vibrational coordinates, the dipole moment components along the molecule-fixed axes xyz (see equation (23)) defined by Eckart and Sayvetz conditions [1]. 

There are 3N + 7 coordinates on the right sides of Eq. (3.4), Le., the 3N vibrational displacements the three coordinates of the center of mass, the three Euler angles 0, x and the angle p. Since there are 3N coordinates/ /a (i = 1,2,.., N ot=x,y, z) on the left sides of Eq. (3.4), the 3N vibrational displacements are subject to seven constraint equations which further specify the molecule-fixed axis system. We shall use the following set of Eckart and Sayvetz conditions for these constraint equations ... 

Equation (2.20) is called the first Sayvetz (or Eckart) condition (Wilson et al., 1955 Califano, 1976) and specifies that during a molecular vibration the center of mass of the molecule remains constant. Introducing Eqs. (2.20) and (2.21) into Eq. (2.19) causes the first two interaction terms to vanish. 

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