Symmetry equivalent atoms

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. 

Caesium chloride is not body-centered cubic, but cubic primitive. A structure is body centered only if for every atom in the position x, y, z there is another symmetry-equivalent atom in the position x+ j,y+ j,z+ j in the unit cell. The atoms therefore must be of the same kind. It is unfortunate to call a cluster with an interstitial atom a centered cluster because this causes a confusion of the well-defined term centered with a rather blurred term. Do not say, the 04 tetrahedron of the sulfate ion is centered by the sulfur atom. 

How many sets of symmetry equivalent atoms are found in the following molecules ... 

Fig. 12.5 Twelve topological sets A,B, L made of symmetry-equivalent atoms of C3v C6QH18(a) hydride eight sets A,B,D,F,G,I,K,L have sjx no(jes each, whereas C,E,H,J include three nodes 13C-NMR theoretical spectrum in fact presents 12 lines with relative intensities 8(6) 4(3). Sets B(6) C(3) D(6) E(3) correspond to C-H bonds originating -NMR theoretical spectrum with four lines with relative intensities 2(6) 2(3). C3v C60H18(P) hydride has similar resonance patterns due to the fact that 12 hydrogen bonds switch from sets B and D to F and G (see Fig. 12.4)...

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

We might note in passing that were the Fourier equation applied to asymmetric units related by space group symmetry in a crystallographic unit cell, the expressions for symmetry equivalent atomic positions assume considerable value. Their application can reduce the number of terms in the summation by the number of symmetry equivalent positions. We need, in practice, to consider only the atoms comprising a single asymmetric unit in the actual calculations. 

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w... 

We circumvent this problem in 2fa-like methods by lumping the non-linear terms into the local V and determining its symmetry. Now we can use an expression like Eq. (20) to simplify matrix elements over atom-centered basis functions when there are many symmetry-related atoms. For a given symmetry-adapted basis function centered on atom type C we expand the basis function into terms centered on the N symmetry-equivalent atoms,... 

The bipolar (e.g. bipyramids and bicapped antiprisms) and non-polar (e.g. D2d-dodecahedron and tricapped trigonal prism) deltahedral custers are best analysed in terms of the interactions between the two sets (polar and non-polar or equatorial and non equatorial) of symmetry-equivalent atoms which make up the cluster1573. In this way it has been shown that, although by symmetry there are no degenerate L /L" pairs and therefore no symmetry-induced departures from the (n + 1) rule, the frontier orbitals of bipolar deltahedral clusters consist of two parity matched U1 and L e pairs, giving rise to possible SEP counts of (n - 1), (n + 1) or (n + 3). In the case of the non-polar deltahedra the frontier orbitals are non-degenerate 1/ and L" orbitals, giving rise to possible SEP counts of n, (n + 1) or (n + 2). 

Fig. 18.7. Distances between phosphorus atoms of adjacent guanine residues within each of the two strands (a) and between phosphorus atoms of guanine residues in different strands, defining the width of the minor groove (b), in three crystal forms of the left-handed hexamer duplex (dfCGCGCGjjj. The phosphorus atoms of one strand are numbered 2 to 6 (starting with P(2) of residue G 2), and the phosphorus atoms of the complementary strand are numbered 8 to 12 (starting with P(8) of residue G 8). Symmetry equivalent atoms are marked with asterisks. In the three crystal forms, the stacking of duplexes in a 3 -5 and 5 -3 manner leads to the formation of infinite helices with a continuous minor groove...

If we consider all the symmetry operations which are associated with a particular molecular geometry, these operations form a point group and all these operations have the property of permuting atoms in identical environments in the molecule. However, if a set of identical Cartesian basis functions is placed on each symmetry-equivalent atom then, in addition to the permutation of symmetry-equivalent basis functions, some of the symmetry operations will send these basis functions into linear combinations of themselves (it is only necessary to think of the action of a three- or five-fold rotation on a set of p basis functions to see this). 

If a basis of functions is chosen on the reasonable grounds that symmetry-equivalent atoms in the molecule have identical basis functions centred on them, then this basis will carry a representation of the molecular point group any operation of the point group Q ) will send the basis functions into linear combinations of themselves without the generation of any functions outside the basis. 

Therefore a symmetry operation can, at most, induce a transformation of the basis functions on one atom to linear combinations of the identical basis functions on another symmetry-equivalent atom (including, of course, the possibility of linear combinations of the basis functions on that atom itself). 

Thus, in deciding to incorporate the effects of molecular symmetry into an implementation of the LCAO method (for example), we must be aware of the possible pitfalls in this decision. In any case, however, the use of the same basis on symmetry-equivalent atoms seems quite innocent and we can always attempt to use this piece of information to reduce the redundant computation of molecular integrals particularly the time- and storage-consuming repulsion integrals. 

The generation of equivalents (e.g. in a toluene molecule on an inversion centre) may be prevented by assigning a negative part number. If necessary, bonds may be added to or deleted from the connectivity array using the B l ND or fre E instructions. To generate additional bonds to symmetry equivalent atoms, EQIV can be used. 

Here s is the standard uncertainty (default value is 0.02 or the first DBFS parameter). If s is negative, the absolute value is used as standard uncertainty and symmetry equivalent atoms are taken into account when deciding which atoms are connected (that can be interesting when the asymmetric unit contains fractions of a full molecule and bonds go through symmetry elements). 

The first eight residual electron density maxima as found in ti-03.res are very close to the fluorine positions (see Figure 5.13). Together with the relatively high U values of the F-atoms, this result indicates that the fluorine atoms are also disordered. Therefore we delete all current F-atoms and replace them with the new sites taken from the Q-positions. To make sure that all new F-atoms belong to the right component, one should check the Al-F distances (or Al-Q distances, respectively), which are supposed to be about 1.7 A. This is much easier after generating the symmetry equivalent atoms. 

Fig. 5.21 Highest residual electron density maxima in tol-Ol.res forming a disordered toluene molecule on a mirror. Left-hand side asymmetric unit right-hand side with symmetry equivalent atoms, revealing interpenetration toluene orientations.

Fig. 5.27 Possible hydrogen bonding pattern for one of the two independent benzoate molecuies. Atoms of symmetry equivalent atoms are labeled with an A after the original atom name.

In the approach [63] WFs are generated directly from Bloch valence states without any preliminary symmetry analysis. In this case one obtains WFs that have the centroid positions in the vicinities of some points of the direct lattice of the crystal occupied by atoms or being midpoint between the pairs of symmetry-equivalent atoms (for the oxides listed in Table 9.15, these points are near oxygen atom positions). The noncontradictory chemical interpretation of the WFs obtained is difficult due to the absence of the s3Tnmetry analysis. Indeed, it is difficult to explain the appearance in MgO crystal (see Table 9.15) of four equivalent WFs corresponding to the tetrahedral sp hybridization, while the oxygen atom is octahedrally coordinated. 

When we have two equivalent atomic orbitals, ij/i and 2 say, the former on atom 1 and the latter on a symmetry-equivalent atom, 2, symmetry demands that the electron density at a given point in is equal to the electron density at the corresponding point in 2 That is, if/l = jf at these... 

In the Dacre and Elder method, the reduction in the integral number comes from the relations between the symmetry-equivalent atoms. Consider two symmetry-equivalent atoms, A and B, each of which has two s /2 2-spinor basis functions, Xk and xx. which we label ka and kb, and A,a and Xb. The relations between the one-electron potential energy integrals (for example) that do not follow fi om Hermitian conjugation are... 

After having identified the symmetry properties of basis sets of atomic orbitals, we need to explore howto build wavefunctions with given symmetry properties from a particular basis. First we need to introduce the concept of symmetry-equivalent atoms or orbitals. Clearly, the two hydrogen Is orbitals of water 4.4 are equivalent, specifically because any symmetry operation of the C2V point group will either send Xi to itself or to xa and likewise send xa to itself or to xi- Thus, xi, Xa form a completely equivalent set The SF4 molecule of 4.7 also belongs to the point group. Here, however, no symmetry operation sends xi or xi to xs or X4. or vice... 

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