1.13: Незведені уявлення та види симетрії

Last updated
Save as PDF

Page ID: 17547

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

Дві одновимірні незвідні уявлення, охоплені\(s_N\) and \(s_1'\) are seen to be identical. This means that \(s_N\) and \(s_1'\) have the ‘same symmetry’, transforming in the same way under all of the symmetry operations of the point group and forming bases for the same matrix representation. As such, they are said to belong to the same symmetry species. There are a limited number of ways in which an arbitrary function can transform under the symmetry operations of a group, giving rise to a limited number of symmetry species. Any function that forms a basis for a matrix representation of a group must transform as one of the symmetry species of the group. The irreducible representations of a point group are labeled according to their symmetry species as follows:

1D зображення маркуються\(A\) or \(B\), depending on whether they are symmetric (character \(+1\)) or antisymmetric (character \(-1\)) under rotation about the principal axis.
Двовимірні зображення позначені\(E\), 3D representations are labeled \(T\).
У групах, що містять центр інверсії,\(g\) and \(u\) labels (from the German gerade and ungerade, meaning symmetric and antisymmetric) denote the character of the irreducible representation under inversion (\(+1\) for \(g\), \(-1\) for \(u\))
У групах з горизонтальною дзеркальною площиною, але без центру інверсії, нескорочувані уявлення задаються простими та подвійними простими мітками, щоб позначити, чи є вони симетричними (символьними).\(+1\) or antisymmetric (character \(-1\)) under reflection in the plane.
Якщо потрібна подальша різниця між нескорочуваними уявленнями, індекси\(1\) and \(2\) are used to denote the character with respect to a \(C_2\) rotation perpendicular to the principal axis, or with respect to a vertical reflection if there are no \(C_2\) rotations.

Незведене подання 1D в\(C_{3v}\) point group is symmetric (has character \(+1\)) under all the symmetry operations of the group. It therefore belongs to the irreducible representation \(A_1\). The 2D irreducible representation has character \(2\) under the identity operation, \(-1\) under rotation, and \(0\) under reflection, and belongs to the irreducible representation \(E\).

Іноді виникає плутанина з приводу зв'язку між функцією\(f\) and its irreducible representation, but it is quite important that you understand the connection. There are several different ways of stating the relationship. For example, the following statements all mean the same thing:

"\(f\) has \(A_2\) symmetry"
"\(f\) transforms as \(A_2\)"
"\(f\) has the same symmetry