1.5: Molecular Point Groups 2

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

The D point groups are distiguished from C point groups by the presence of rotation axes that are perpindicular to the principal axis of rotation.

Dn : Cn and n⊥C2 (h = 2n)

Example: Co(en)3 3+ is in the D3 point group,

In identifying molecules belonging to this point group, if a Cn is present and one ⊥C2 axis is identified, then there must necessarily be (n–1)⊥C2s generated by rotation about Cn.

Dnd : Cn, n⊥C2, nσd (dihedral mirror planes bisect the ⊥C2s)

Example: allene is in the D2d point group,

Two C2s bisect σds. The example on the bottom on pg 3 of the Lecture 4 notes was a harbinger of this point group. As indicated there, it is often easier to see these perpendicular C2s by reorienting the molecule along the principal axis of rotation.

Note: Dnd point groups will contain i, when n is odd

Dnh : Cn, n⊥C2, nσv, σh (h = 4n)

C∞v : C and ∞σv (h = ∞)

linear molecules without an inversion center

D∞h : C, ∞⊥C2, ∞σv, σh, i (h = ∞)

linear molecules with an inversion center

when working with this point group, it is often convenient to drop to D2h and then correlate up to D∞h

Td : E, 8C3, 3C2, 6S4, 6σd (h = 24)

Oh : E, 8C3, 6C2, 6C4, 3C2 (=C4 2 ), i, 6S4, 8S6, 3σh, 6σd (h = 48)

 O : E, 8C3, 6C2, 6C4, 3C2 (=C4 2 ) A pure rotational subgroup of Oh, contains only the Cn’s of Oh point group T : E, 8C3, 3C2 A pure rotational subgroup of Td, contains only the Cn’s of Td point group

Ih : generators are C3, C5, i (h = 120) $$\Longrightarrow$$ the icosahedral point group

Kh : generators are Cφ, Cφ’, i (h = ∞) $$\Longrightarrow$$ the spherical point group

Flow chart for assigning molecular point groups:

This page titled 1.5: Molecular Point Groups 2 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Daniel Nocera via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.