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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How to find a symmetry group of a system if all the symmetry transformations do not obey closure and don't form a group?

For instance, consider a system with $p_x$ and $p_z$ orbitals at the vertices of a square (on xy-plane). A square by itself would have $D_4$ symmetry. However, because of the $p_x$ orbital; the $90^\...
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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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Symmetry of glide planes

I am trying to understand the symmetry elements of space group number 194 (P$6_3$/mmc), which is hexagonal and has 24 symmetry operations. In the table of symmetry operations it says that it has ...
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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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What does “orthorhombic” actually mean? [duplicate]

I'm trying to understand what it means for a molecule to have orthorhombic symmetry. According to Di Lauro's "Rotational Structure in Molecular Infrared Spectra," a molecule is orthorhombic if the ...
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How can I determine the type of IR active mode on a complex molecule from molecular geometry arguments?

I am trying to synthesize 2,6,9,10-tetracyanoanthracene and I need to discriminate it from other regioisomers. I am hoping to take advantage of the C2h symmetry under an IR spectrometer. I like to ...
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Character table and symmetry operations

It's been a while since I've dealt with these, but do character tables include all possible symmetry operations? I am looking at the $D_{\infty h}$ point group table and it lists $C_\infty$, $\...
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Pncn space group diagram

I'm trying to make a diagram of the Pncn space group. Firstly, I know that this space group belongs to the orthorhombic system and derives from the point group (2/m 2/m 2/m). Therefore, although the ...
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Molecular Orbitals (only pi system) of Allyl system using group theory

I have just started reading group theory and their applications in the MO theory. I tried to get the equations of the pi molecular orbitals of allyl system using group theory, but I am stuck at the ...
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MOs of Benzene using group theory

I am trying to analyze the $\sigma$ orbitals of benzene molecule using group theory, doing the same thing that is done to the $\pi$ orbitals. The symmetry group of benzene is the $D_{6h}$; considering ...
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Point group from mono to bulk MoS2

Many articles say even numbers of layers or bulk crystal MoS2 have D6h point group and odd numbers of layers (including monolayer) MoS2 have D3h point group But whenever looking at the figure of ...
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How do I get irreducible representations?

I'm currently trying to do a problem where it asks me to "show that the irreducible representations spanned by the atomic x-, y-, and z- coordinates is independent of the choice of local coordinate ...
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Misunderstanding about the spin part of the Jahn Teller effect

In the paper Stability if Polyatomic Molecules in Degenerate Electronic States II - Spin Degeneracy the author wrote: Thus a twofold degeneracy cannot produce instability for a molecule containing ...
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What is a “symmetrical product” of an irreducible representation with itself?

In their classic paper outlining the Jahn–Teller theorem (Proc. R. Soc. Lond. A 1937, 161 (905), 220–235), the authors wrote "where $[\phi^2]$ denotes the representation of the symmetrical product of $...
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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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Reducible representations - choice of new basis

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 ...
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N2 symmetry representation in higher symmetry groups (D3h…)

I'd like to describe the symmetry of the ground state of $N_2$ in the higher symmetry groups like $D_{3h}$ and $D_{\infty h}$. In this answer the topic is covered very good, but only for $D_{2h}$ ...
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Reducible representation of planar molecule N2H2 with bond lengths as a basis

I'm studying about molecular symmetry and its representations. Today I got a little confused about the planar $\ce{N2H2}$ molecule. It looks like this The basis in the example was chosen as $\Delta ...
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Understanding Equivalent Wyckoff positions

I'm somewhat confused by the concept of Wyckoff positions in crystal structures. From Quantum Chemistry of Solids by Robert Evarestov, the definition of a Wyckoff position is "all the crystallographic ...
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265 views

Would an (NH3)2+ molecule be trigonal planar like BH3 rather than trigonal pyramidal?

I've been learning about using MO theory to explain why $\ce{BH3}$ and $\ce{NH3}$ have different geometries and by following the line of reasoning used to rationalise the differences in geometry I ...
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How to systematically find the reducible representation of a molecule? (eg: XeF4)

I am attempting to use the reduction formula to find the irreducible representation of $\ce{XeF4}$ to determine its IR stretching vibrations. I know the point group of $\ce{XeF4}$ is $D_{4h}$ and have ...
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Notation for excited states

I am reading some papers about quantum chemistry and in one I found a notation for excited states of a molecule that I don't understand at all. In the picture below you can see the potential curves ...
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Whats the symmetric, anti-symmetric and non-symmetric part of the direct product of irreducible representations?

I am trying to understand how to see if a vibrational mode is Jahn-Teller active or not. According to the group theoretical description of the Jahn-Teller effect one needs to check if the symmetric ...
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Determining overtones bands of degenerate modes

I have a question of a more mathematical nature on the mathSE (Symmetric Direct Product Distributive?) that received a good answer, but I think an answer more oriented to chemists would be a useful ...
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Molecular orbital diagram and irreducible representations for dinitrogen

I'm trying to understand how to draw molecular orbital diagrams, since I'd like to use the molecular orbitals to determine the total state of the molecule (i.e. $A_1$, $B_1$ etc.), so I prefer the ...
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Why are there no dihedral mirror planes in the D3h point group?

Consider the molecule $\ce{BH3}$, which belongs to the $D_\mathrm{3h}$ point group. Why are the three mirror planes in this point group labelled as $\sigma_\mathrm v$ instead of $\sigma_\mathrm d$? ...
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D3d point group matrices

I am trying to construct all the matrices for the point group D3d operations. I find that the matrices of E, i, S6, inverse of S6, C3, and inverse of C3 are easy, but I do not know how to construct ...
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How to dock aggregated structures comprised of “elementary” protein units like LEGO pieces?

There are proteins with intrinsic symmetries. For example: I was wondering how to use transformations such as: rotations, translation, replications to construct possible structures using 2 types of ...
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Irreducible representations and system states connection

Let's say I have the molecule $\ce{N2+}$. Its symmetry point group is clearly $D_{\infty\mathrm h}$. But I'm confused by its irreducible representations. I know there are 8 of them, but as I ...
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Why are there only two irreducible representations in the C3 group?

There are three elements in a $C_3$ group and there are three classes in this group cause it is an Abelian group. \begin{array}{c|ccc|cc} \hline C_3 & E & C_3 & C_3^2 & & \...
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How do we calculate a character table from its basis functions

This question is a actually a homework question, I'm given the basis functions of all the irreducible representations of $C_{2v}$ group, then I'm asked to complete the character table of $C_{2v}$ ...
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Is there a possible distortion of XeF6 from Oh point group to reduced symmetry?

Below the molecular orbital diagram for $\ce{XeF6}$ (from What is the molecular structure of xenon hexafluoride?) I am trying to figure out if a distortion from $O_h$ can take place and to which (...
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How do I show that a transition is magnetic dipole allowed with group theory/symmetry?

I am taking an introductory course in symmetry and bonding and have so far learnt about SALCs and various point groups etc... one of my problems asks me to show a $\mathrm{^{2}T_{2g}\rightarrow\ ^{2}...