Symmetry reduction

The structural and magnetochemistry of RE3Ni7B2 phases confirms an earlier structure for Mr = Gd, Tb, Dy, Ho, Er, Tm, Lu (U3CO7B2 type, P63/mmc). For Mre = Sm, Eu, Yb, however, a symmetry reduction is claimed (P6/mmm) corresponding to a different site occupation, which actually represents an occupation variant of the CeCo4B type insofar as Eu + Ni statistically occupy the 2c sites . 

Potential Energy Surfaces All potential energy surfaces generated with this theory should be smooth. No discontinuities due to symmetry reduction should occur. 

The anomalously reduced stabilities of certain nonalternant hydrocarbons and higher members of [4n+2] annulenes arise from their seemingly peculiar geometrical structures in which a strong bond distortion often accompanied by a molecular-symmetry reduction occurs. 

We now move to the predictions based on the symmetry rule. Comparing the lowest excitation energies (the sixth column) with the results concerning the symmetry reduction obtained on the basis of the dynamic theory (the last column), we can draw a very clear-cut criterion for the symmetry reduction ... 

The symmetries of the lowest excited states listed in Table 1 are nothing but the symmetries to which the most soft second-order bond distortions belong. It is seen that the types of symmetry reduction predicted using the symmetry rule are in complete agreement with those obtained on the basis of the dynamic theory. 

Molecule (symmetry) Hiickel SCF Lowest excited state Symmetry reduction ... 

It should be noted, on the other hand, that a symmetry reduction is predicted even in molecules II, VIII and XIII whose peripheral skeletons correspond to 4n-l-2 cyclic polyenes. The transannular bonds in these molecules are different in nature from those mentioned above. For example, the introduction of the transannular bonds between atoms 2 and 8 and between 3 and 7 of cyclododecapentaene to form bowtiene (II) (Fig. 3) brings about the splitting of the top filled degenerate orbitals of the unperturbed system into two levels, one with its energy raised and... 

On the other hand, in the anion radical of fulvalene and the cation radical of heptafulvalene, the energy gaps between the ground and lowest excited state (which is in both cases doubly degenerate in the Hiickel approximation (Fig. 4)) are predicted to be reasonably large (1.4 and 1.7 eV, respectively), so that these radicals would not suffer a symmetry reduction. 

Inspection of Table 2 reveals that all those molecules that suffer a molecular-symmetry reduction in the ground state possess (E2 — E1) values considerably larger than the critical value, so that they should have a fully-symmetrical nuclear configuration in their first excited states. On the other hand, there are cases where a molecule has an ( , — Eg) value significantly higher than the critical value, but has a relatively smaller (Ej— i) value. The ( 2 i) value of the pentalene dianion (I ) is of the same order of magnitude as the critical value and those for the peri-condensed nonalternant hydrocarbon, XVII, the fulvalenes, XXI, XXII and XXIII, and the dianions, IVand VII are significantly smaller than the critical value ( 0.6eV). 

The symmetry of the most soft distortion in the lowest excited state is given by the direct product of the symmetry of the first excited state (shown in Table 1) and that of the second excited state (shown in Table 2). These symmetries are b3g(R ) for 1 and VII 2(1 ) for XVII and IV- hi (z) for XXI and XXIII, and fli(z) for XXII. The symmetries of the lowest excited states are then predicted to be Cj, Q, and C2 , respectively. It should be noted that despite the strong vibronic coupling with the second excited state, the first excited state of sesqui-fulvalene (XXII) does not undergo a symmetry reduction. 

Using the same method as described in II.B, Binsch and Heil-bronner have examined the second-order bond distortion in the lowest excited states of nonalternant hydrocarbons (I, IV—VII, X, XI, XIII — XV and XVII), and have shown that, of the molecules examined, only VI and XVII suffer a molecular-symmetry reduction in the lowest... 

The cation radical heptafulvalene, on the other hand, undergoes no symmetry reduction. Both the rings show a moderate double-bond fixation (Fig. 7), and the unpaired spin is delocalized throughout the molecule (Table 3). 

In the anion and cation radicals of fulvalene (XXI) the situation turns out to be quite reversed. Removal of an electron from the neutral molecule to produce the cation radical results in a symmetry reduction (Dj -> C2 ), the stabilization energy being calculated to be 17.8 kcal mole . On the other hand, addition of an electron to form the anion radical leaves the molecular symmetry unaffected. 

Fulvalene and heptafulvalene are predicted, in agreement with the result obtained using the symmetry rule, to suffer a symmetry reduction 2v in their lowest excited states. The longest wave-length electronic absorption bands of these molecules are expected to be relatively broad. This seems to be what is observed . On the other hand, the lowest excited state of sesquifulvalene is predicted not to undergo symmetry reduction, which again supports the prediction based on the symmetry rule. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

A suitable way to represent group-subgroup relations is by means of family trees which show the relations from space groups to their maximal subgroups by arrows pointing downwards. In the middle of each arrow the kind of the relation and the index of the symmetry reduction are labeled, for example ... 

In all cases we start from a simple structure which has high symmetry. Every arrow (= -) in the preceding examples marks a reduction of symmetry, i.e. a group-subgroup relation. Since these are well-defined mathematically, they are an ideal tool for revealing structural relationships in a systematic way. Changes that may be the reason for symmetry reductions include ... 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

The symmetry reduction can be continued. A (non-maximal) subgroup of F43m is I42d with doubled lattice parameter c. On the way F43m —174 2d the Wyckoff position of the zinc atoms splits once more and can be occupied by atoms of two different elements. 

The symmetry reduction to the mentioned hettotypes of diamond is necessary to allow the substitution of the C atoms by atoms of different elements. No splitting of Wyckoff positions, but a reduction of site symmetries in necessary to account for distortions of a structure. Let us consider once more MnP as a distorted variant of the nickel arsenide type (Fig. 17.5, p. 197). Fig. 18.4 shows the relations together with images of the structures. 

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