Combined symmetry operation

It was easy to show that we can formulate the method also in the case of a combined symmetry operation (for instance helix operation = translation + rotation) instead of simple translation ( ). In this case k is defined on the combined symmetry operation and from going from one cell to the next one, one has (1) to put the nuclei in the positions required by the symmetry operation and (2) one has to rotate accordingly also the basis set. 

Each operation must have an inverse that, when combined with the operation, yields the identity operation (sometimes a symmetry operation may be its own inverse). Note By convention, we perform combined symmetry operations from right to left as written. 

We can apply the formalism developed in the preceding section also in the case of a combined symmetry operation. To show this let us consider a helix in which we pass from one unit to the next by a translation t and simultaneous rotation a. We can then introduce the helix operator... 

Results (1.65) and (1.66) enable the ab initio SCF LCAO CO program for linear chains to be modified and hence applicable also in the case of a combined symmetry operation. This modified program has been applied for the nucleotide base stacks (see Section 2.3 in the next chapter). 

In the calculations of the electronic structure of these polypeptides, we applied the all-valence electron MINDO crystal-orbital method in its MINDO/3 parametrization. The calculations involved atomic coordinates corresponding to the antiparallel-) -pleated-sheet structure of Pauling and Corey S hence we had a combined symmetry operation in the case of poly( y) each gly unit was obtained in this polymer from its neighboring one by a translation along the helical axis through 3.25 A and by a simultaneous 180° rotation around it. In order to preserve the cyclic property of the Fock hypermatrix of the whole polymer (which is... 

HF forms a zigzag chain in the crystal and, in the chain direction, the unit cell contains 2HF molecules (see Figure 9.3). The internal coordinates at successive HF molecules obey a screw axis symmetry. Applying this symmetry, several authors have suggested that a single HF molecule forms the unit cell and only four internal coordinates (ri, r, a, and P) need be optimized. Application of the combined symmetry operation leads to a great decrease in the amount of computational work necessary to determine the equilibrium geometry. 

Finally one should mention that with the help of simple group theoretical arguments one can show that the described formalism can be applied (in the ID case) also for combined symmetry operations (for instance to the helix operation which contains a translation and a rotation going from one unit to the next one) instead of a simple translation. As the detailed derivation shows in the case of a helix operation one has to put the nuclei in the next cell in the right position and rotate the basis functions which do not point in the direction of the polymer axis (say z axis) or are not spherically symmetric. Thus one has to rotate the and py functions and d, etc., functions. In this way in the case of a DNA helix (in DNA B) one can take a single nucleotide and not 10 of them as unit cell. 

Finally, it should be mentioned that the formalism described here is valid also for the case of a repeated combined symmetry operation (for instance helix operation). As group theoretical considerations show it in this case 1.) one has to put the nuclei into the right positions by moving from one cell to the next and 2.) one has to rotate correspondingly also the basis functions /14/. 

The formalism of the a initio Hartree-Fock CO method first proposed about 15 years ago (2,3) has been discussed in details elswhere (1,4) (see also the contribution of J. Ladik in this Volume), It has to be noted that in the case of a helix one steps from unit to unit, not with a simple translation but with a combined symmetry operation (translation along the helix axis and rotation around it). Therefore, one has to rotate both the atoms in the elementary cell and also the atomic orbitals centered on these atoms. This means that, having chosen the z axis along the main axis of the helix, the atomic orbitals with components in the xy plane have to be rotated (18). Second neighbors interactions have been included with a correct electrostatically balanced cutoff (19) of the different types of integrals. 

For a function to transform according to a specific irreducible representation means that the function, when operated upon by a point-group symmetry operator, yields a linear combination of the functions that transform according to that irreducible representation. For example, a 2pz orbital (z is the C3 axis of NH3) on the nitrogen atom... 

A geometric object can have several symmetry elements simultaneously. However, symmetry elements cannot be combined arbitrarily. For example, if there is only one reflection plane, it cannot be inclined to a symmetry axis (the axis has to be in the plane or perpendicular to it). Possible combinations of symmetry operations excluding translations are called point groups. This term expresses the fact that any allowed combination has one unique... 

When two symmetry operations are combined, a third symmetry operation can result automatically. For example, the combination of a twofold rotation with a reflection at a plane perpendicular to the rotation axis automatically results in an inversion center at the site where the axis crosses the plane. It makes no difference which two of the three symmetry operations are combined (2, m or T), the third one always results (Fig. 3.6). 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

However, to construct the appropriate linear combinations of the n orbitals, it is sufficient to choose a subgroup of > whose symmetry operations permute all... 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

This process could be continued so that all the combinations of symmetry operations would be worked out. Table 5.3 shows the multiplication table for the C3 point group, which is the point group to which a pyramidal molecule such as NH3 belongs. 

Multiplication tables can be constructed for the combination of symmetry operations for other point groups. However, it is not the multiplication table as such which is of interest. The multiplication table for the C2v point group is shown in Table 5.2. If we replace E, C2, and cryz by +1, we find that the numbers still obey the multiplication table. For example,... 

For a given molecule belonging to a particular point group, it is possible to consider the various symmetry species as indicating the behavior of the molecule under symmetry operations. As will be shown later, these species also determine the ways in which the atomic orbitals can combine to produce molecular orbitals because the combinations of atomic orbitals must satisfy the character table of the group. We need to give some meaning that is related to molecular structure for the species A1( B, and so on. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Taking into account these symmetry operations together with those corresponding to the translations characteristic of the different lattice types (see Fig. 3.4), it is possible to obtain 230 different combinations corresponding to the 230 space groups which describe the spatial symmetry of the structure on a microscopic... 

Linear combination of the two lone-pair orbitals n and n.2) with reference to the relevant symmetry operation leads to a symmetric (n+) and an antisymmetric combination (n ). 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

The 230 three-dimensional space groups are combinations of rotational and translational symmetry elements. A symmetry operation S transforms a vector r into r ... 

A group consists of a set (of symmetry operations, numbers, etc.) together with a rule by which any two elements of the set may be combined - which will be called, generically, multiplication - with the following four properties ... 

Problem 4-3. In Figure 4.1c, what symmetry operation or combination of symmetry operations takes the x just above the horizontal line (just above the 3 o clock line) into (a) the o closest to the top of the circle (b) the x closest to the top of the circle, (c) the x next to the x in part (b) of this problem. 

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