AppendixE.pdf

Point Group Symmetry EIt is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry elements are (e.g., planes, axes of rotation, centers of inversion, etc.). For the reader who feels, after reading this appendix, that additional background is needed, the texts by Cotton and EWK, as well as most physical chemistry texts can be consulted. We review and teach here only that material that is of direct application to symmetry analysis of molecular orbitals and vibrations and rotations of molecules. We use a specific example, the ammonia molecule, to introduce and illustrate the important aspects of point group symmetry.I. The C3v Symmetry Group of Ammonia - An ExampleThe ammonia molecule NH3 belongs, in its ground-state equilibrium geometry, to the C3v point group. Its symmetry operations consist of two C3 rotations, C3, C32 (rotations by 120° and 240°, respectively about an axis passing through the nitrogen atom and lying perpendicular to the plane formed by the three hydrogen atoms), three vertical reflections, σv, σv', σv“, and the identity operation. Corresponding to these six operations are symmetry elements : the three-fold rotation axis, C3 and the three symmetry planes σv, σv' and σv“ that contain the three NH bonds and the z-axis (see figure below).Nσv'σvσxz is σv''H3H2H1C3 axis (z)x- axisy- axisThese six symmetry operations form a mathematical group . A group is defined as a set of objects satisfying four properties.1.A combination rule is defined through which two group elements are combined to give a result which we call the product. The product of two elements in the group must also be a member of the group (i.e., the group is closed under the combination rule).2.One special member of the group, when combined with any other member of the group, must leave the group member unchanged (i.e., the group contains an identity element).3.Every group member must have a reciprocal in the group. When any group member is combined with its reciprocal, the product is the identity element.4.The associative law must hold when combining three group members (i.e., (AB)C must equal A(BC)).The members of symmetry groups are symmetry operations; the combination rule is successive operation. The identity element is the operation of doing nothing at all. The group properties can be demonstrated by forming a multiplication table. Let us label the rows of the table by the first operation and the columns by the second operation. Note that this order is important because most groups are not commutative . The C3v group multiplication table is as follows:EC3C32σvσv'σv“second operation EEC3C32σvσv'σv“ C3C3C32Eσv'σv“σv C32C32EC3σv“σvσv' σvσvσv“σv'EC32C3σv'σv'σvσv“C3EC32 σv“σv“σv'σvC32C3E First operationNote the reflection plane labels do not move. That is, although we start with H1 in the σv plane, H2 in σv'', and H3 in σv“, if H1 moves due to the first symmetry operation, σv remains fixed and a different H atom lies in the σv plane.II. Matrices as Group RepresentationsIn using symmetry to help simplify molecular orbital or vibration/rotation energy level calculations, the following strategy is followed: 1. A set of M objects belonging to the constituent atoms (or molecular fragments, in a more general case) is introduced. These objects are the orbitals of the individual atoms (or of the fragments) in the m.o. case; they are unit vectors along the x, y, and z directions located on each of the atoms, and representing displacements along each of these directions, in the vibration/rotation case. 2. Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be “block diagonal“. That is, objects of different symmetry will not interact; only interactions among those of the same symmetry need be considered.To illustrate such symmetry adaptation, consider symmetry adapting the 2s orbital of N and the three 1s orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3v point group. The act of each of the six symmetry operations on the four atomic orbitals can be denoted as follows:(SN,S1,S2,S3)→E(SN,S1,S2,S3)→C3 (SN,S3,S1,S2)→C32 (SN,S2,S3,S1)→σv (SN,S1,S3,S2)→σv“ (SN,S3,S2,S1)→σv'(SN,S2,S1,S3)Here we are using the active view that a C3 rotation rotates the molecule by 120°. The equivalent passive view is that the 1s basis functions are rotated -120°. In the C3 rotation, S3 ends up where S1 began, S1, ends up where S2 began and S2 ends up where S3 began.These transformations can be thought of in terms of a matrix mul。