Full rotation-reflection group

The simplest molecules are atoms, which belong to point group %h (often called the full rotation-reflection group). The character table (which we omit) contains irreducible representations of dimensions 1,3,5,... these representations correspond to energy levels with electronic orbital angular-momentum quantum number /=0,1,2,... we have the (2/+1)-fold degeneracy associated with different values of the quantum number... 

It can be shown that under the operations of the full rotation-reflection group in three dimensions, 0(3) ... 

The functions, here occurring in standard order, are our standard basis functions for the real irreducible representations of the full three-dimensional rotation-reflection group, Rg x I, and for its subgroups, Aoft. and Coo . All functions are normalized to 4nj(2l- -1), where / is the azimuthal quantum number. 

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... 

The three-dimensional rotation-reflection group (that is, the full group of V ) is obtained by adjoining an inversion operation to the operations of C(3). This generates an infinite number of reflection planes and the number of reps is just double that in C(3). Each character of C(3) is multiplied by +1, so instead of labeling the reps with an index I, an r " or l is used to indicate the parity under inversion. In the direct product (+) X (+) = (+) ( + ) X (-) = (-), and (-) X (-) = (+). For e.xample,... 

The 2 group consists of four symmetry operations an identity operation designated E, a rotation by one-half of a full rotation, i.e. by 180°, called a C2 operation and two planes of reflection passing through the C2 axis and called operations. Examples of molecules belonging to this point group are water, H2O ... 

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. ... 

When the 7i-electron framework of a molecule consists of n atoms, the secular determinant will be of order n x n. Frequently, the framework will possess certain symmetries, such as reflections or rotations their presence is disguised in the full secular-determinant. By systematic use of such symmetry-properties as exist, not only may the full secular-determinant be factorised into independent sub-determinants of lower order, but much more insight may be obtained into the nature of the MO s. The symmetry properties are independent of the way in which we approximate the MO s, and so the information provided by group-theoretical techniques has a significance far greater than that of any particular set of approximate MO s. 

It is evident from (3.25) and (3.26) that b is the time shift in the origin of the time axes, c is the shift in the origins of the Cartesian systems and Q (from full orthogonal group, cf. Rem. 8) expresses the rotation (detQ = 1) or reflection (detQ = —1) of the starred frame relative to the original one. We also note the inversions of change... 

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