Translational and Rotational Invariance

In this section we shall explain how this problem can be handled. Additionally we introduce the notation Pj with 

3-vectors t. We rewrite the invariance property in the form of a one-dimensional function 

Because xeE is arbitrary, g and H fulfill the same condition of translational invariance as E, i.e. 

The scalar product in Eg.(65) is an orthogonality condition. For all peE the gradient vector g(p) lies in the subspace T clR orthogonal 

Each teT is a vector in the kernel of the Hessian matrix H(p). Because dim(T) 3, with three coordinates of the displacement tclR, the translational invariance implies the existence of three zero eigenvalues of the Hessian matrix. Equation (67) also means that the rows (or the colimns by symmetry) of H(p) belong to T . 

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One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

The equality of Eq. (3a) arises because of the translational and rotational invariance of the energy. At the weak minimum we also have... 

According to Noether s theorem (Arnold (1989)) symmetries of a mechanical system are always accompanied by constants of the motion. According to Section 3.1, system symmetries can be obvious (e.g. geometric) or hidden . Examples for obvious symmetries that lead to constants of the motion are invariance with respect to time translations, spatial translations and rotations. Invariance with respect to time leads to the conservation of energy, spatial and rotational symmetries lead to the conservation of linear and angular momentum, respectively (see, e.g., Landau and Lifschitz (1970)). Hidden symmetries cannot be associated with... 

A property of the autocorrelation function is that it does not change when the origin of the x variable is shifted. In effect, autocorrelation descriptors are considered TRI descriptors, meaning that they have translational and rotational invariance. 

Baumann, K. (2002b) Distance profiles (DiP) a translationally and rotationally invariant 3D structure descriptor capturing steric properties of molecules. Quant. Struct. -Act. Relat., 21, 507-519. 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

Our final remark in this area concerns the application of translational and rotational invariance in gradient calculations. This is incorporated in Q-Chem [20] and, like symmetry, is easy to handle outside the compute-intensive region. 

Code should be independent of translation or rotation of a structure (translational and rotational invariance). 

The elements of this matrix, transcribed in terms of g only [i.e. in the form ° [x(a)]= (g), which is unrestrictrdly possible because of the translation and rotation invariance of the expressions °g(x)l, are to be used in the exact expression of the quantum kinetic energy operator, obtained in following the so-called gOO-approach, in the general form of Eq.(3), Ref.[32]. They play a role in the part of the Hamiltonian operator that accounts for pure internal deformations, which can be entirely depicted with the help of the 3N-6 internal coordinates. Indeed, the overall exact quantum-mechanical kinetic energy operator can be written as [32] ... 

Thermodynamical ensembles are generally defined without the constraint of Hamiltonian translational and rotational invariance, in which case the previous statement is not entirely correct. In the present article, however, the terminology of Table 1 will be (loosely) retained to encompass ensembles where this invariance is enforced. The statistical mechanics of these latter ensembles must be adapted accordingly [76, 77, 78, 79, 80, 81]. This requires in particular the introduction of a modified definition for the instantaneous temperature, relying solely on internal degrees of freedom and kinetic energy (Sect. 3). 

There are other conditions on these forces, related to the translational and rotational invariance of the electronic energy (which are always exactly fulfilled also for approximate wave functions). In particular, in a diatomic molecule, there are three translational conditions - two rotational conditions and one condition provided by the electronic virial theorem. Taken together, these conditions determine the diatomic nuclear force field completely. The only nonvanishing force acts along the molecular axis and may be obtained directly from (he kinetic and potential enei ies if the wave function is fully variational with respect to a scaling of both the electronic and the nuclear coordinates. 

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