Symmetry operators spatial rotation

As a particular example of materials with high spatial symmetry, we consider first an isotropic chiral bulk medium. Such a medium is, for example, an isotropic solution of enantiomerically pure molecules. Such material has arbitrary rotations in three dimensions as symmetry operations. Under rotations, the electric and magnetic quantities transform similarly. As a consequence, the nonvanishing components of y(2),eee, y 2)-een and y,2)jnee are the same. Due to the isotropy of the medium, each tensor has only one independent component of the xyz type ... 

In physics and chemistry there are two different forms of spatial symmetry operators the direct and the indirect. In the direct transformation, a rotation by jr/3 radians, e.g., causes all vectors to be rotated around the rotation axis by this angle with respect to the coordinate axes. The indirect transformation, on the other hand, involves rotating the coordinate axes to arrive at new components for the same vector in a new coordinate system. The latter procedure is not appropriate in dealing with the electronic factors of Born-Oppenheimer wave functions, since we do not want to have to express the nuclear positions in a new coordinate system for each operation. 

The symmetry elements (Section 1.18) of a molecule aid in locating its principal axes. Clearly, a molecular symmetry operation must carry the momental ellipsoid to an orientation indistinguishable from its original position. Consider a Cn rotation (n= M) such a rotation about any axis other than one of the three principal axes of the momental ellipsoid will send the ellipsoid to a nonequivalent spatial orientation. Hence a Cn symmetry axis of the molecule must coincide with one of its principal axes. (The converse is, of course, not true every molecule has three principal... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

The point symmetry group of the molecule is denoted by 9 (Dnu or Cnv in the present case), and it is necessary to produce from the functions (35) wavefunctions which form bases for irreducible representations A of rd. We note first of all that since all the orbitals are localized on one or other of the atoms forming the molecule, the application of a spatial symmetry operation 52 of rS is equivalent to a permutation of the orbitals on the equivalent atoms amongst themselves, possibly multiplied by a rotation of the orbitals on the central atom. Hence with every operation 52 we may associate a certain permutation of the orbitals, Pr, in which the bar emphasizes that one permutes the orbitals themselves and not the electron co-ordinates. Thus,... 

Spatial symmetry operations are linear transformations of a coordinate function space. When choosing the space in orthonormal form, symmetry operations will conserve orthonormality, and hence all transformations will be carried out by unitary matrices. This will be the case for all spatial representation matrices in this book. When all elements of a unitary matrix are real, it is called an orthogonal matrix. As unitary matrices, orthogonal matrices have the same properties except that complex conjugation leaves them unchanged. The determinant of an orthogonal matrix wiU thus be equal to 1. The rotation matrices in Chap. 1 are all orthogonal and have determinant -I-1. 

The symmetry operations that we have encountered are either proper or improper. Proper symmetry elements are rotations, also including the unit element. The improper rotations comprise planes of symmetry, rotation-reflection axes, and spatial inversion. All improper elements can be written as the product of spatial inversion and a proper rotation (see, e.g.. Fig. 1.1). The difference between the two kinds of symmetry elements is that proper rotations can be carried out in real space, while improper elements require the inversion of space and thus a mapping of every point onto its antipode. This can only be done in a virtual way by looking at the structure via a mirror. From a mathematical point of view, this difference is manifested... 

The operations used for describing molecular symmetry are E, the identity operation Cn, rotation of angle 2it/n about some axis /, inversion a, reflection and S , rotation-reflection. The a and S operations may be expressed as products of rotations and the inversion, and they need not be treated separately in detail. Inversion turns out to require special consideration in connection with the Dirac equation, so for the next few sections we will only consider spatial rotations, and we will return to inversion as a separate theme later. 

For instance, through the combinations (are noted with x ) between the rotations and the reflections are obtained new symmetry operations, the roto-reflections, which generate spatial changes, so that the final results overlapped to the initial structure. 

Spatial symmetry may be utilized in a similar way. According to the theorems in Appendix 3 (p. 541), the expansion of any wavefunction W of given symmetry species contains only symmetry functions of the same species. The situation is precisely analogous to that which arises in the case of spin for the eigenvalues (S, M) are in fact the labels that define the different basis functions (Af = S, S - 1,... -S) of a (2S + 1)-dimensional representation Dj of the group of rotations in spin space, and therefore correspond to the species labels (or, i) used in Appendix 3. Functions of pure symmetry species, with respect to spatial symmetry operations, may again be built up by linear combination of the basic determinants for molecules, this is easily accomplished by the methods of Appendix 3, and adequate illustrations appear in later sections. 

To incorporate spatial symmetry requirements, we note that all functions are invariant under spatial rotations about the bond axis, so that we need consider only how they behave under the inversion i, which is the only non-trivial operation of the appropriate symmetry subgroup C, = E, i. The effect of i on the orbitals is simply to interchange a and b, and hence (remembering that interchange of the columns changes the sign of a determinant) we obtain... 

The effect of symmetry operations on the spatial wavefunctions can be obtained without any difficulty. To manipulate the effect of the symmetry operations on the spin wavefunctions, on the other hand, requires rather tedious procedures. The spin functions a and are transformed by rotation through an angle

direction cosines are A, p and v in the following manner [21] ... 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

If the atom or moleeule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear moleeules and point group symmetry for nonlinear polyatomies), the trial wavefunetions should also eonform to these spatial symmetries. This Chapter addresses those operators that eommute with H, Pij, S2, and Sz and among one another for atoms, linear, and non-linear moleeules. 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

What happens with the interaction between the rotational and spin symmetries once the system is characterized as being defined by at least different spinors Wigner and von Neumann [10] combined both types of symmetries with the permutation aspect [11]. They intuitively reached the idea using atomic spectroscopy that the H operator has to be constructed by two terms H, resulting from the spatial motion of the single electron only (and the electromagnetic interaction with the field of the atomic core), and (//2), which has to visualize the electron spin. For simplicity, we can consider the eigenvalue problem of the spinless wave function i r without the second term as... 

Factorization of the Cl space is difficult in the relativistic case. The first problem is the increase in number of possible interactions due to the spin-orbit coupling. The second problem is the rather arbitrary distinction in barred and unbarred spinors that should be used to mimic alpha and beta-spinorbitals. Unlike the non-relativistic case the spinors can not be made eigenfunctions of a generally applicable hermitian operator that commutes with the Hamiltonian. If the system under consideration possesses spatial symmetry the functions may be constrained to transform according to the representations of the appropriate double group but even in this case the precise distinction may depend on arbitrary criteria like the choice of the main rotation axis. 

The total parity of a given class of levels (F fine structure component for E-states, upper versus lower A-doublet component for II-states) is found to alternate with 7. The second type of label, often loosely called the e// symmetry, factors out this (—l) 7 or (—l)-7-1/2 7-dependence (Brown et al., 1975) and becomes a rotation-independent label. (Note that e/f is not the parity of the symmetrized nonrotating molecule ASE) basis function. In fact, for half-integer S, it is not possible to construct eigenfunctions of crv in the form [ A, S, E) —A, S, — E)], because, for half-integer S, vice versa.) The third type of parity label arises when crv is allowed to operate only on the spatial coordinates of all electrons, resulting in a classification of A = 0 states according to their intrinsic E+ or E- symmetry. Only A = 0) basis functions have an intrinsic parity of this last type because, unlike A > 0) functions, they cannot be put into [ A) — A)] symmetrized form. The peculiarity of this E symmetry is underlined by the fact that the selection rule for spin-orbit perturbations (see Section 3.4.1) is E+ <-> E, whereas for all types of electronic states and all... 

There is a close connection between symmetry and the constants of the motion (these are properties whose operators commute with the Hamiltonian H). For a system whose Hamiltonian is invariant (that is, doesn t change) under any translation of spatial coordinates, the linear-momentum operator p will commute with H, and p can be assigned a definite value in a stationary state. An example is the free particle. For a system with H invariant under any rotation of coordinates, the operators for the angular-momentum components commute with H, and the toted angular momentum and one of its components are specifiable. An example is an atom. A linear molecule has axial symmetry, rather than the spherical synunetry of an atom here only the axial component of angular momentum can be specified (Chapter 13). 

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