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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What does it mean that a state belongin to a given irrep transforms like $Rx$, $Ry$ or $Rz$

The present question is related to this other question I did few days ago. Given a point group and the list of the irreps (see for example here) the meaning of an irrep which transforms like $x$ or $x^...
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How do I show that a transition is electric dipole allowed with group theory/symmetry?

This is actually a follow up of this question The follow-up is not because of the electric instead of magnetic dipole (this is trivial). It is because I'm interested in extra info. Suppose I have a ...
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What is m.2m point group?

I'm using Bilbao Crystallographic Server to find 3D crystallographic point groups of the K points of anatase TiO2. Since the space group of anatse TiO2 is $I4_1/amd$, on this page, I found the point ...
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7 votes
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Why not just use the x,y,z as basis functions instead of linear/quadratic functions of these unit vectors?

I am going through Applications of Group Theory to the Physics of Solids by M. S. Dresselhaus. In chapter 4 he states This is with respect to the P(3) group. I understand basis functions must ...
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Why are 2D block matrices considered irreducible?

I am currently reading Applications of Group Theory by M.S. Dresselhaus On page 18 irreducible representations are defined and an example is given as follows: The author claims that these ...
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Electronic states of VO vanadium oxide in $C_{2v}$ point group

The electron configuration of $V^{2+}$: ... 3$d^{3}$ with 3 d electrons in 3 d orbitals. In $C_{\infty v}$, $d_{xz}$ and $d_{yz}$ have $\Pi$ IRREP; $d_{z}^{2}$ has $\Sigma^{+}$ IRREP; $d_{x^{2}-y^{2}}$...
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How do we find the SALCs?

I am trying to understand exactly what IRREPs are and, in order to know more about it, I started reading the chem.libretext course on molecular symmetry which up to this point seemed fairly ...
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Using symmetry and group theory arguments to explain iron(II) in a tetrahedral crystal field

I am trying to figure out how to explain $1s \rightarrow 3d$ spectroscopic transitions for $\ce{Fe^{2+}}$ in $T_\mathrm{d}$ symmetry. These transitions make up the pre-edge region in K edge X-ray ...
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Why are Methyl vibr. modes called r modes and methylene vibr. modes called d modes

I am coming from a chemical background and currently work in the field of vibrational spectroscopy. When I talk to fellow physicists or chemists I always encounter the same question. "Why are the ...
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Is there a better coordinate framework and symmetry operator for constructing molecular orbitals of molecules with higher-order geometry?

The following excerpt is taken from [1] (with a few rewording, emphasizes are mine). In an attempt to construct molecular orbital (MO) of any molecule, one needs to determine the symmetries of ...
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Tetrahedral complex and vibronic coupling or different rule for d-d transition?

I'm currently preparing a few slides for an upcoming talk in our group. I would like to mention some of the very basics regarding crystal field and ligand field theory as well, though this will not be ...
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Help determining normalisation constant in SALC of methane

We have been told that a symmetry adapted combination of four hydrogen orbitals is used to create localised hydrogen orbitals for methane. These orthonormal combinations are denoted ψa1, ψt’2, ψt’’2 ...
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How to read character table?

I am having some problems reading character table. I will describe what I understand and thought to be right. Consider the $\pi$ MOs formed from overlap of p-orbitals in benzene. These two degenerate $...
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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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MO diagram when three orbitals of the same symmetry is allowed to mix

How exactly should we combine three orbitals when we have three orbitals that are of correct symmetry to interact? Take $\ce{H2O}$ for example. $\mathrm{2s},$ $\mathrm{2p}_z$ and the $A_1$ group $(\ce{...
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What does the order of a class mean in a character table?

I am not from a chemistry background. I am trying to see how the symmetry operations affect the stiffness tensor which led me in the path of understanding character tables. I am looking at the ...
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Symmetry Operation on Product of Functions

In regards to direct product representations, I'm trying to find a proof for: $R(X_iY_j)=R(X_i)R(Y_j)$ Where $R$ is a symmetry operation of a group and $X_i$ and $Y_j$ are members of different basis ...
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Finding Equivalent representation of a molecule/crystal

Is there a website or online calculator to determine the reducible representation of a crystal/molecule using wyckoff positions without having to analyse the number of unmoved atoms/bonds for ...
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$Q_3$ is symmetric with respect to the $\sigma_v$ operation?

I am currently studying the textbook Infrared and Raman Spectroscopy: Principles and Spectral Interpretation, second edition, by Peter J. Larkin. Section 8. Symmetry: Infrared and Raman Active ...
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Is there a geometrically intuitive explanation of the equivalence of the numbers of irreducible representations and symmetry classes?

In group theory, as used in chemistry, we’re familiar with the idea that the number of irreducible representations of a symmetry group equals the number of classes of symmetry operations, e.g. three ...
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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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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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5 votes
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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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2 votes
2 answers
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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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3 votes
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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 $D_\mathrm{\infty h}$ 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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3 votes
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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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5 votes
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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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4 votes
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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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