Eckart condition

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

Eckart conditions, Renner-Teller effect, triatomic molecules, 610-615 Ehrenfest dynamics, direct molecular dynamics error sources, 403—404 Gaussian wavepacket propagation, 378-383 molecular mechanics valence bond (MMVB), 409-411... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

For harmonic force field calculations it is usual to express the force field in terms of internal co-ordinates Ri defined through a linear transformation from cartesian displacement co-ordinates 8xm (a = x, y, or z) in the molecule-fixed axis system.24 The molecule-fixed axes are located using the Eckart conditions.24 The equations are written in the form... 

The superscript of Polo s p°-vectors indicates that the Eckart conditions are satisfied [64], which is required for the PAS coordinates. For the calculation of the vectors... 

Let the n 1 three-dimensional vectors zSi (i = 1,1) be the mass-weighted Jacobi vectors for a reference molecular configuration. The reference configuration is usually set to be a local equilibrium structure of the molecule oriented to a certain orientation. The Eckart subspace is defined as a (3n — 6)-dimensional subspace in the (3n — 3)-dimensional translation-reduced configuration space, which is parameterized by Jacobi vectors pf (/ = 1,..., m - 1) with three additional constraint conditions called the Eckart conditions,... 

This factorization of the ju-tensor has also been observed in the standard theory of small amplitude motion57, S8, where I = 1°, the inertial tensor of the equilibrium configuration. Below it is shown how this particular result is following from the Eckart conditions. 

The assertion that the PAS is convenient for separating rotations and vibrations can be rejected, therefore. We shall see below (Sect. 4) that the small amplitude vibrations are always treated most simply using Eckart conditions, whereas large amplitude motions must be specially taken care of. Principal inertial axes may only be relevant in relation to the reference structure of the Eckart conditions. 

Eckart System (ES). It is well-known how the axis convention proposed by Eckart39 enters the standard vibration-rotation theory2 49 52) in the alternative method of deriving the kinetic energy the Eckart conditions are used in formulating rotational s-vectors. For this purpose we may proceed exactly as in the PAS example above. 

In this section we shall see how the principles outlined above are applied to evaluate the Wilson-Howard Hamiltonian1,2 However, most of the derivation may be worked out without explicitly assuming that rectilinear internal coordinates are used. We shall take advantage of this in that we will also examine the general consequences of the Eckart conditions as opposed to the special properties connected with the introduction of linearized coordinates. As an intermediate result we will therefore obtain a Hamiltonian which is exactly equivalent to the one which Quade derived for the case of geometrically defined curvilinear coordinates7 ... 

The Eckart conditions play an important role in this connection. We shall discuss this in more detail below, since the arguments presented apply equally well to the treatment of nonrigid molecules. Hence, to study the basis of introducing Eckart conditions, let us for a moment go back to an earlier stage where axis conventions were not yet formulated. We recapitulate that we are looking for the conditions required in order that the atomic position coordinates, rag, can be given as unique functions of 3 N-6 internal coordinates, or equivalently stated, in order that the expansion [Eq. (3.6)] can be determined as a unique inverse of Eq. (3.5). 

The two sets of rotational s-vectors are easily expressed using the corresponding constraint vectors of the Eckart conditions as described in Sect. 2.2.3.2.2,... 

As rotational constraints we retain the three Eckart conditions and the corresponding constraint vectors [Eq. (2.68) or (3.35)], here depending on p. These constraints have been discussed by Hougen17) as well. 

Field-dependent geometry optimization and Eckart conditions... 

The Cartesian coordinates of the resulting structure were then formed at the center-of-mass and were rotated to obey the Eckart conditions with the previous structure (the path was defined for torsional angles in half-degree intervals). Note that in general this approach does not generate the true reaction path as defined by Miller et al. [60] in that the path does not conform to that of steepest descent it is expected to be very close to it, however, and their Hamiltonian is stiU valid. The choice of H2O2 was used for the initial test of this RPH version of MULTIMODE, where it was shown to produce results in excellent agreement with previous exact calculations [61,62]. 

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... 

Quasi-Rigid Model-Simplifying by Eckart Conditions Approximation Decoupling of Rotations and Vibration Spherical, Sjfminetric, and Asymmetric Tops Separation of Translational, Rotational, and Vibrational Motions... 

Quasi-Rigid Model—Simplifying by Eckart Conditions... 

We completely get rid of the redundancy if we agree the second Eckart condition is introduced (equivalent to three conditions for the coordinates) ... 

The nuclear relaxation (NR) contributions were computed using a finite field approach [73,74]. In this approach one first optimizes the geometry in the presence of a static electric field, maintaining the Eckart conditions. The difference in the static electric properties induced by the field can then be expanded as a power series in the field. Each coefficient in this series is the sum of a static electronic (hyper) polarizability at the equilibrium geometry and a nuclear relaxation term. The terms evaluated in Ref. [61] were the change of the dipole moment up to the third power of the field, and that of the linear polarizability up to the first power ... 

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