Understanding the character table of C2v is fundamental for anyone studying molecular symmetry in quantum chemistry or group theory. This specific point group describes molecules with a unique two-fold rotation axis and two distinct vertical mirror planes, creating a mathematical framework that dictates how atomic orbitals combine.
Defining the C2v Point Group
The C2v point group is one of the simplest non-abelian point groups, characterized by containing a C2 rotation axis and two mirror planes that intersect along this axis. Water (H2O), hydrogen sulfide (H2S), and formaldehyde (CH2O) are classic examples of molecules belonging to this symmetry class. The presence of these symmetry elements means that certain physical properties, like molecular orbitals or vibrational motions, remain unchanged under specific symmetry operations.
Symmetry Operations and Elements
The C2v group consists of exactly four symmetry operations: the identity operation (E), a 180-degree rotation around the principal axis (C2), and two reflections across vertical planes (σv and σv'). The identity operation does nothing, leaving the molecule identical to its original state. The C2 rotation flips the molecule 180 degrees around the z-axis, while the two mirror planes slice the molecule into symmetrical halves, one typically defined as the xz-plane and the other as the yz-plane.
Matrix Representation of Operations
Each symmetry operation within the C2v point group can be represented by a matrix that acts on the coordinates of the molecule. The 3x3 matrices for E, C2, σv(xz), and σv'(yz) operate on the x, y, and z components of vectors. For instance, the C2 matrix around the z-axis is [[-1, 0, 0], [0, -1, 0], [0, 0, 1]], effectively inverting the x and y coordinates while leaving the z-coordinate unchanged.
The Character Table Structure
The character table organizes the traces (characters) of the matrices representing each operation for every irreducible representation (irrep) of the group. For C2v, there are four irreps, labeled A1, A2, B1, and B2, corresponding to the totally symmetric representation and others that transform differently under the symmetry operations. The top row lists the symmetry operations, and the first column lists these irreps, with the characters filling the intersection of rows and columns.
