# Questions tagged [group-theory]

The systematic discussion of molecular symmetry is known as group theory. Group theory is a systematic way of describing the symmetry of molecules using concepts/rules borrowed from mathematics. This tag should be applied to any question which involves the theory or application of point groups, symmetry elements, or character tables.

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### Example of an achiral molecule without a plane of symmetry or inversion center?

Some websites, such as this textbook, mention how generally a plane of symmetry or inversion center in a molecule is enough to consider it achiral. Nonetheless, this source clarifies but if these ...
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### Why do most character tables include an $(x^2 + y^2)$ term?

Why do most character tables (e.g. $C_{3h}$ but not $C_{2h}$) include an $(x^2 + y^2)$ term? Is it an abbreviated form for $d_{z^2}$, applying only where $3d$ (and higher) orbitals might be involved, ...
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### Reducing power of ammoniated electrons

I came across a question asking which of the following conversion/reductions can be accomplished using ammoniated electrons. $\ce{O2}$ to $\ce{O2^{2-}}$ $\ce{K2[Ni(CN)4]}$ to $\ce{K4[Ni(CN)4]}$ ...
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### How to define axes if all rotation axes pass through the same number of atoms?

I was wondering how one can define the axes for a molecule with several equal axes of rotation, all of which pass through the same atom. I know that the $z$-axis is usually defined as the principal ...
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### Connection between irreducible representations and electronic states in diatomic molecules

I am trying to understand the connection between irreducible representations and molecular electronic states. To explain the problem: I found some potential energy curves (PECs) for $\ce{N2+}$ in the ...
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### Symmetries and degeneracies of the s,p, and d orbitals

Consider the molecular AB_8 (complex with central atom A and 8 B ligands (coordination number 8), for which I determined the point group D_4d) and determine the symmetries and degeneracies of the s, p,...
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### Grouping of symmetry classes in chemistry character tables

In the C3v point group, it lists three sigma Vs together in one column (class), and it doesn't list the cVs separately. Consequently there are three irreducible representations. In the C2v point ...
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### Screw axis in P21c space group

I am having some trouble with understanding glide axis. The following is a question asking to reconstruct the full unit cell: And the following is the answer to the question: I have highlighted an ...
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### Why does dipole must lie parallel to a rotation symmetry axis?

While studying group theory, I've got confused by the following statement: For a molecule to have a permanent dipole moment, it must have an asymmetric charge distribution. The point group of the ...
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### How to deal with non-symmorphic space groups for orbital/vibrational analysis?

I am learning about how to apply group theory analysis of MO diagrams and vibrational in the solid state context (not isolated molecules). I think the strategy for calculating representations and ...
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### C5v E1 SALC production using Full Projection

I have another SALC question. @Andrew and @user5713492 did a phenomenal job clarifying my previous confusion with Methane T2 SALC production, so I attempted to apply this to a new point group, and ...
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### How do I interpret characters that are not 1 or -1 in a point group table?

When the character in a point group table is listed as 1 or -1, I understand how an object/orbital etc. transforms to give either itself or the negative of itself. When the character in a point ...
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### Methane T2 SALCs

So I have a question on the form of the $T_2$ SALCs of methane. Below I show the $T_d$ character table and the accompanying reducible representation for the sigma framework (using, in the case of ...
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### What happens when a symmetry element is added or removed from a point group? [closed]

What new point group is formed if symmetry element i is added to C3 point group?
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### How to get $\Gamma_{vib}$ for hexafluoroethane?

An exercise asks me to find $\Gamma_{vib}$ for that molecule. I know that: $$\Gamma_{cart}=\Gamma_{vib}+\Gamma_{translations}+\Gamma_{rot}$$ Since the molecule I'm considering belongs to the $D_{3d}$ ...
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### Vibronic coupling: How do I determine the new point group after vibration?

So I was recently discussing the transitions in Egyptian Blue ($\ce{CaCu[Si4O10]}$) with some of my students, who had to prepare this compound. What I like in particular in this case is how, at least ...
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### Does "higher symmetry" mean any symmetry group of higher order?

What is the precise definition of a higher symmetry in group theory? I see this term scattered around, such as in Cotton's Chemical Applications of Group Theory: ...In the cyclic groups each of the ...
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### Cs Point Group's "Horizontal" Reflection Plane

Why is it that $C_s$ character tables always list a $\sigma_h$ operation despite the definition of $C_s$ not necessitating the reflection plane be horizontal? In fact, how can the reflection plane ...
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### Vanishing integral rule for two-electron integrals

I understand the vanishing integral rule for a one-electron integral to be: $$\langle i | \hat{O} | j \rangle = 0 \hspace{1cm} \mathrm{if} \hspace{0.2cm} \Gamma_O \neq \Gamma_i \otimes \Gamma_j$$ ...
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### Why does benzene have 3 π–π* transitions in the UV range, but ethene, butadiene, and hexatriene each have only 1?

Are the π and π* orbitals of benzene much closer together in energy which allows for multiple transitions? Why is this not the case for hexatriene if they both contain the same number of pi bonds?
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### Are there any descent symmetry tables for character tables of linear structures?

I know about tables containing information about descending in symmetry, i.e. from larger point groups to their sub-groups. We can see such tables in this answer, or in this PDF. My question is - ...
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I was looking at this tutorial about reductions of representations. At first, the basis consists of four orbitals $(s_N, s_1, s_2, s_3)$ and the representations look like this and we want to reduce ...