Symmetry mathematical basis

Figure 9.2 shows some simple helices and their transforms. The transform of the helix in Fig. 9.2a exhibits an X pattern that is always present in transforms of helices. I will explain the mathematical basis of the X pattern later. Although each layer line looks like a row of reflections, it is actually continuous intensity. This would be apparent if the pattern were plotted at higher overall intensity. The layer lines are numbered with integers from the equator (/ = 0). Because of symmetry, the first lines above and below the equator are labeled 1=1, and so forth. 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

McWeeny has written a tribute to the valence-bond theory pioneers of 1927-1935.362 Shavitt has outlined the history and evolution of Gaussian basis sets as employed in ah initio molecular orbital calculations.363 Hargittai has interviewed Roald Hoffmann (b. 1937)364 of Cornell University and Kenichi Fukui (1918-1998)365 of Kyoto University, who were jointly awarded the Nobel Prize in Chemistry in 1981. Fukui developed the concept of frontier orbitals and recognized the importance of orbital symmetry in chemical reactions, but his work was highly mathematical and its importance was not appreciated until Robert Woodward (1917-1979) and Hoffmann produced their rules for the conservation of orbital symmetry from 1965 onwards.366... 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

The ensemble of all equivalent positions for a space group is unique and may be considered the mathematical definition of the space group. It provides the basis for manipulating objects and points related by symmetry in a digital computer. Equivalent positions are another way of stating both the space group and the Bravais lattice of a crystal. 

We therefore describe the basis of macromolecular crystallography and provide a summary of how to understand the results of a crystallographic experiment. We start with a mathematical description of what a crystal means in terms of symmetry this applies to all crystals, whether macromolecular or not. Later, we describe how protein crystals grow by using the hanging drop and sitting drop vapor diffusion methods this explains why protein crystals are so fragile and scatter X-rays very weakly. 

This type of treatment has been very useful as a basis for the interpretation of the vibrational spectra of polyatomic molecules. Symmetry considerations have been widely employed to simplify the solution of the secular equation and in that connection the branch of mathematics known as group theory has been very helpful.1... 

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