Eckart-Sayvetz conditions

In a normal mode analysis, the Eckart-Sayvetz conditions are observed for the whole of a system and they are not, therefore, in general satisfied for computed nuclear motions on a fragment only. The dyads in the above expressions will thus contain translational and rotational components. If and LqALqA are the dyads for the translational and... 

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. 

The Eckart-Sayvetz conditions can easily be expressed in tmms of die coordinates q g (g = X, y, z). Summarizing, the vibrational motion of an N-atomic molecule widi 3N-6 vibrational degree of freedom can be described by 3N nuclear Cartesian displacement coordinates forming a column matrix X. Six degrees of fireedmn are related widi translational and rotational motions of the molecule. These motions can be described by the external corndinates p (diree translations and duee rotations). In a transposed form die different types of vibrational coordinates may be presented as follows... 

The Eckart-Sayvetz conditions were explicitly presented vrith regard to the set of Cartesian displacement coordinates [Eqs. (2.8)]. From the relations R = BX and S = UR it is clear that the conditions of zero linear and angular momenta are also imposed on the coordinates Ri and Sj. Thus, certain mass-dependency is implicit in the definition of... 

It should be emphasized drat die condition of zero angular momentum is implicit in Eqs. (3.23), (3.27) and (3.28). In particular it is contained in die seamd term of Eq. (3.28). Thus, the usually troublesome problem associated with die cmiqiensatoiy molecular rotations as required by the Eckart-Sayvetz conditions is treated in an elegant way. 

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]. 

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]. 

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 ... 

The Eckart- and Sayvetz-conditions constitute a set of conventions for the reference structures which are particularly useful, since they allow us to use rectilinear coordinates for the small amplitude motions (Sect. 3.3). However, the introduction of reference structures, depending on the large amplitude coordinates only, leaves us with the question of how the molecular axes should be oriented within an arbitrary set of atomic reference positions. This question was only briefly commented on in Sect. 4.6, since it is special to the molecule under consideration. Some examples may illustrate types of solutions. 

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... 

C. Eckart, American physicist (1904-)- These conditions are sometimes attributed to Aaron Sayvetz, American physicist <1917—>. 

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. 

The second Sayvetz (or Eckart) condition is chosen so that whenever the atomic displacements in a molecular vibration tend to produce a rotation of the molecule, the rotating system reorients in order to eliminate this component of the motion (Califano, 1976). This condition, which is given by... 

Let us construct another space in order to determine the vibrational displacements of the molecule without referring the external coordinates, that is, without the data of the center-of-mass (COM) and the rotations. In the general case, the dimension of the latter space is 3N - 6, according to the well-known Sayvetz (or, Eckart) conditions [5-7] (the special case of the linear molecules is exceptional with its 3N — 5 dimension). Hereafter, this Euclidean space will be denoted by and let us... 

---