Rotation-reflection invariance

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

The first term on the right-hand side represents the total sum of squares of Y, that obviously does not depend on R. Likewise, the last term represents the total sum of squares of the transformed X-configuration, viz. XR. Since the rotation/reflection given by R does not affect the distance of an object from the origin, the total sum of squares is invariant under the orthogonal transformation R. (This also follows from tr(R" X XR) = tr(X rXRR T) = tr(X XI) = tr(X" X).) The only term then in eq. (35.2) that depends on R is tr(Y XR), which we must seek to maximize. 

The many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. This means that the wave function, with a wave vector k, is left invariant under the symmetry elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k. 

The molecular Hamiltonian is invariant under all orthogonal transformations (rotation-reflections) of the particle variables in the frame fixed in the laboratory. The usual potential energy surface is similarly invariant so it is sensible to separate as far as possible the orientational motions of the system from its purely internal motions because it is in terms of the internal motions that the potential energy surface is expressed. The internal motions comprise dilations, contractions and deformations of a specified configuration of particle variables so that the potential energy surface is a function of the molecular geometry only. 

The Hamiltonian /lclcc(f f) has the same invariance under the rotation-reflection group 0(3) as does the full translationally invariant Hamiltonian (6), and it has a somewhat extended invariance under nuclear permutations, since it contains the nuclear masses only in symmetrical sums. Since it contains the translationally invariant nuclear coordinates as multiplicative operators, its domain is of... 

Point group - A group of symmetry operations (rotations, reflections, etc.) that leave a molecule invariant. Every molecular conformation can be assigned to a specific point group, which plays a major role in determining the spectrum of the molecule. 

Although the Cartesian coordinate system provides a simple and unequivocal description of atomic systems, comparisons of structures based on it are difficult the list of coordinates can be ordered arbitrarily, or two structures might be mapped to each other by a rotation, reflection or translation. Hence, two different lists of atomic coordinates can in fact represent the same or very similar structures. In a good representation, permutational, rotational and translational symmetries are built in explicitly, i.e. the representation is invariant with respect to these symmetries, while retaining the faithfulness of the Cartesian coordinates. If a representation is complete, a one-to-one mapping is obtained between the genuinely different atomic environments and the set of invariants comprising the representation. 

In diatomic molecules of the second and third periods, both s and p orbitals (2s, 2p and 3s, 3p, respectively) come into play, for example, in Nj, O2, Na2, F2, and CI2. MOs have to satisfy certain symmetry conditions, since the Hamiltonian is invariant under rotations, reflections, and other possible symmetry operations that leave the molecule invariant. 

It is easily established that the Coulomb Hamiltonian is invariant imder the coordinate transformations that correspond to imiform translations, rotation-reflections, and permutations of particles with identical masses and charges. Because of the symmetry of the Coulomb Hamiltonian its eigenfunctions will be basis functions for irreducible representations (irreps) of the translation group in three dimensions, the orthogonal group in three dimensions, and for the various symmetric groups corresponding to the sets of identical particles. 

The Hamiltonian (O Eq. 2.23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spin-orbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling of orbitals that form bases for representations of the rotation group SO(3). Spin-eigenfunctions too are achieved by vector coupling. ... 

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