Example: comparing two fairly symmetrical molecules and asking which is more symmetric or packing-efficient.
Why symmetry matters in real properties:
Melting points can differ due to packing efficiency driven by symmetry.
n-pentane vs neopentane: n-pentane melts at −130∘C, neopentane at −16∘C.
Boiling points differ as well: n-pentane T<em>b=10∘C, neopentane T</em>b=36∘C.
The larger difference in melting points (ΔTm ≈ 114 °C) compared with the boiling point difference (ΔTb ≈ 26 °C) is largely due to packing differences stemming from symmetry.
Spectroscopy connection:
More symmetrical isomers tend to have simpler spectra (e.g., 1H NMR for ortho- vs para-xylenes).
Group theory enables spectral prediction (not covered in full here; example discussed later).
Overall framework: symmetry of a molecule is described by a set of symmetry operations; there are 5 physical operations used to describe total symmetry (E, C_n, σ, i, Sn).
Basic Symmetry Elements and Operations
Identity, E:
This operation does not change anything; all atoms stay in place.
All molecules possess this symmetry element.
Rotational axes, C_n:
An imaginary line through the molecule about which the molecule can be rotated.
If a rotation yields a configuration indistinguishable from the original, the molecule has a rotation axis.
Rotation angle for a C_n axis is θ=n360∘.
Example: water with a C_2 axis: rotation by 180∘ regenerates the original orientation (Hydrogen atoms labeled for visualization are identical in reality).
A molecule may have more than one C_n axis; the axis with the largest n is called the principal rotation axis.
PtCl4^{2-} provides a case with coincident C4 and C2 axes perpendicular to the molecular plane through Pt; there are four other C2 axes in the molecular plane. Coincident axes are real.
Mirror planes, σ:
An imaginary plane cutting through the molecule; reflection across the plane exchanges positions of atoms.
If the reflected arrangement is indistinguishable from the original, the molecule has a mirror plane.
PtCl_4^{2-} has 5 mirror planes:
(i) through Pt and all four Cl atoms;
(ii) through Pt, Cl1, Cl3 (⊥ to the molecular plane) and likewise through Pt, Cl2, Cl4;
(iii) through Pt and bisecting the angles Cl1–Pt–Cl2 and Cl3–Pt–Cl4; (and the corresponding planes perpendicular to the molecular plane)
Designations: a mirror plane including the principal axis is σv (vertical); planes perpendicular to it are σh (horizontal).
Inversion center, i:
A point at the center of the molecule where all points (x, y, z) invert to (−x, −y, −z).
If the molecule retains its structure after this inversion, it has an inversion center.
PtCl4^{2-} has an inversion center; H2O does not.
Improper rotation axis, S_n:
A compound operation consisting of a rotation about an axis by some angle followed by reflection through a plane σ_h perpendicular to that axis.
The book’s illustration includes an S4 axis; S1 and S_2 axes do not exist as distinct entities (they coincide with σ and i, respectively).
S_n describes a combined rotation-reflection symmetry element.
Point symmetry and crystallography:
If a molecule has a point that remains unchanged under all symmetry operations, it possesses point symmetry (centered on i if present, or at the intersection of all symmetry elements otherwise).
Individual molecules have point symmetry; collections in a crystal lattice may not.
Point Groups and Molecular Symmetry
Notation and scope:
Some very high-symmetry groups: Ih (icosahedral), Oh (octahedral), Td (tetrahedral).
Very low symmetry groups: C1 (E only), Ci (E and i), Cs (E and σ).
Groups based on an n-fold rotation axis:
Cn: only E and Cn;
Cn h: E, Cn, σ_h;
Cn v: E, Cn, σ_v.
Dihedral groups, Dn: E, Cn, C_2(⊥) (perpendicular to the principal axis);
Dnd: E, Cn, C2(⊥), σv;
Dnh: E, Cn, C2(⊥), σh;
D_∞h: linear molecule with an inversion center.
Among these, Dnh is particularly important; Cnv is quite common; many molecules fall into C1, Cnv, or D_nh.
Typical examples:
Water, H2O, is C2v (elements: E, C_2, and two σ planes).
CH2Cl2 and cis-Cr(CO)4(PH3)2 share the same symmetry elements (E, Cn, and two σ planes).
A flow-chart approach to assigning point groups by inspection is recommended (flow chart referenced on page 58).
Special note:
While several point groups exist, the vast majority of chemically relevant molecules fall into C1, Cnv, or D_nh.
Linear molecules with no inversion center fall under C∞v; linear molecules with inversion center fall under D∞h.
Irreducible and Reducible Representations
Interdependent symmetry operations:
In a given point group, applying two different operations often yields a third operation within the same group (closure property).
Example: For water (C2v), applying C2 to swap H atoms and then a σ plane operation can yield the same result as applying another symmetry operation in the group.
Character Tables and irreducible representations:
Every point group has an associated Character Table (Appendix D in the book).
The top row lists symmetry elements; the rest of the table provides characters for each irreducible representation (rows) under those symmetry operations (columns).
Each row corresponds to an irreducible representation (irrep).
The irreps describe how molecular properties (orbitals, vibrations, rotations) transform under symmetry operations.
Some irreps are linked to specific molecular motions (e.g., rotation about the z-axis) as discussed in the initial pages of this section.
Practical use:
Reducible representations can be decomposed into a direct sum of irreducible representations to determine how sets of functions (e.g., vibrational modes) transform.
The notes indicate Experiment 2 will cover irreducible representations in more detail.
Uses of Point Group Symmetry
Optical activity and chirality:
A molecule is chiral if it is not superimposable on its mirror image.
While many organic compounds use a chiral carbon, chiral atoms are not required for chirality.
Inorganic example: [Co(en)_3]^{2+} (en = ethylene diamine): no single atom is chiral, yet the complex as a whole is chiral.
In this context, chirality is defined as the absence of an Sn axis. This includes the absence of a σ plane corresponding to S1 or an inversion center corresponding to S_2.
Resource note: Marion Cass maintains visualization resources for such complexes (e.g., a Carlton College site).
Dipole moments:
Dipole moment requires nonsymmetric electron density; a molecule’s symmetry can forbid a dipole moment entirely.
A quick rule: the two operations that exclude a dipole moment are inversion (i) and a horizontal mirror plane (σ_h).
The book lists the point groups that permit dipole moments; practical hint: i and σ_h are the primary exclusions to memorize.
Infrared and Raman spectroscopy:
This area is the most important practical application of symmetry in chemistry, but it is complex enough to be treated as its own course.
The book suggests skipping this section in places and revisiting it in the lab Experiment 2 for spectral predictions and analysis.
chirality in inorganic complexes as a theme for symmetry analysis:
The absence of an Sn axis (and related symmetry elements) is a key criterion for identifying chirality in complexes such as [Co(en)3]^{2+}.
Experimental Context and Visualization Aids
Visualization tools mentioned:
Two-dimensional images (as shown in Experiment 2) to help visualize symmetry operations.
An excellent website linked on the course page provides 3-D pictures to aid visualization of symmetry elements.
Practical pointer:
The flow chart on page 58 of the text helps in assigning point groups by inspection; using these visual tools can simplify identifying symmetry operations in molecules.
Linear cases: C∞v (no inversion center) or D∞h (with inversion for linear molecules).
Quick Notes and Exam Preparation Tips
Be able to assign point groups by inspection using the flow chart approach; focus on C1, Cnv, and D_nh for many real molecules.
Remember key distinctions:
σv vs σh designation relative to the principal axis.
The principal axis is the C_n axis with the largest n.
Sn axes are combinations of rotation and reflection; S1 and S_2 do not appear as distinct entities beyond σ and i.
Common illustrative molecules to review:
H2O: E, C2, σv, σv' → C_2v.
PtCl4^{2-}: multiple C2 and C_4 axes; multiple σ planes; i center; highly symmetric example for studying multiple symmetry elements.
Co(en)3^{2+}: chiral despite lacking chiral atoms; absence of an Sn axis.
CH2Cl2 and cis-Cr(CO)4(PH3)_2: share the same basic symmetry elements (useful for applying the same point group analysis).
Experimental resources:
Experiment 2 in the lab manual will discuss irreducible representations in more detail.
Appendix D (Character Tables) provide a practical reference for assigning irreps to molecular motions.
Overall takeaway:
Group theory provides a structured, predictive framework for understanding molecular symmetry, spectroscopy, and physical properties through a finite set of symmetry elements and their associated representations.
Notation cheat sheet (brief)
E: Identity
C_n: n-fold rotation axis, rotation by θ=n360∘
σ: Mirror plane; σv (through principal axis), σh (perpendicular to principal axis)
i: Inversion center; (x,y,z) → (−x, −y, −z)
Sn: Improper rotation (rotation by θ=n360∘ followed by reflection in σh)
Point groups mentioned: C1, Ci, Cs, Cn, Cnh, Cnv, Dn, Dnd, Dnh, D∞h, C_∞v, Ih, Oh, Td