All Questions
Tagged with symmetry group-theory
70
questions
2
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1
answer
80
views
Identifying Type of Symmetry Operations
From the given set of symmetry operations for a space group, say, P-3m1 or Cmma, how can one identify the type of all symmetry operations. For example, how to distinguish the axis of rotation symmetry ...
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0
answers
62
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How do we determine the local symmetry of anatase TiO2?
In this physical review B paper, the authors mentioned that the local symmetry of anatase TiO2 is D2d point group. Also, in this applied physics letters paper, the authors mentioned that the symmetry ...
- 273
1
vote
1
answer
91
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Electronic configuration and symmetry labels of 3d$^7$ ions
In 3d$^7$ ions, e.g. Co$^{2+}$, the ground state is $^4$F, and it contains different levels due to the crystal field, such as $^4T_{1g}$, $^4T_{2g}$ and $^4A_{2g}$. Can anyone tell me what exactly ...
- 11
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0
answers
49
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How do I assign functions for cartesian coordinates and rotations to irreducible representations of a character table?
I do not really understand the intuition behind the functions assigned to irreducible representations.
For example, in H₂O, the A₂ representation is given by
...
- 128
2
votes
1
answer
211
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Is there a difference between $A_{1g}$ modes and $A_{g}$ modes?
I know that both are totally symmetric and that the '$1$', according to the Mulliken table, refers to symmetry around the $C_2$ axis but I was reading some papers and found that modes were sometimes ...
- 65
3
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1
answer
50
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Are A1g modes reduced at 180 degrees polarisation in Raman spectra?
I have read that A1g modes are identifiable in Raman spectra by looking at cross-polarised Raman spectra and seeing peaks that strongly reduce in intensity. So essentially, the intensity of A1g modes ...
- 65
7
votes
2
answers
101
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Point group of harmonic oscillator
The book Molecular Quantum Mechanics by Atkins and Friedman [1] says the point group of a harmonic oscillator is $C_\mathrm{s},$ composed by the identity operator $E$ and a reflection $\sigma_\mathrm{...
- 461
2
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0
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77
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Factor Group Analysis of Na2SO4 - Site Group Symmetry
I'm still trying to understand how to construct a factor group analysis for a given molecular point group and the resulting solid state group symmetry. For this I searched for some examples and found ...
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3
votes
1
answer
305
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Relation between symmetry of a reciprocal vector and lattice planes perpendicular to it
What is the relation the symmetry of a high-symmetry point in the first Brillouin zone and lattice planes perpendicular to it? Are the two symmetries equivalent?
I have this question because I want to ...
- 273
1
vote
1
answer
81
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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^...
- 267
7
votes
1
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412
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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 ...
- 267
2
votes
1
answer
411
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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 ...
- 273
3
votes
1
answer
1k
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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 ...
- 131
10
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0
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141
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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 ...
- 101
2
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0
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59
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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 ...
- 639
1
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0
answers
202
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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 ...
- 4,322
3
votes
1
answer
2k
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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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0
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89
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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 ...
- 41
1
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0
answers
102
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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 ...
- 1,083
5
votes
1
answer
560
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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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1
answer
91
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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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4
votes
1
answer
636
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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, ...
- 223
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0
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62
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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 ...
- 606
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0
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107
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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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2
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508
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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 ...
- 21
2
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0
answers
54
views
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 ...
- 479
3
votes
1
answer
157
views
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 $C_\infty$, $\...
- 1,548
2
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1
answer
1k
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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 ...
- 5,685
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1
answer
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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 ...
- 63
2
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0
answers
56
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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 ...
- 453
3
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1
answer
587
views
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 $...
- 453
2
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1
answer
185
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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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1
answer
614
views
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 ...
- 59
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3
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5k
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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 ...
- 327
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1
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2k
views
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?
- 9
6
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2
answers
344
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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 ...
- 4,322
1
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1
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738
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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 ...
- 884
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1
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2k
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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?
- 1,098
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1
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504
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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 - ...
- 1,267
1
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0
answers
2k
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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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3
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1
answer
1k
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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 ...
- 1,267
2
votes
1
answer
18k
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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 ...
- 105
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1
answer
1k
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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 ...
7
votes
1
answer
423
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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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5
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1
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1k
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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 $\sigma_\mathrm d$? ...
- 531
2
votes
1
answer
637
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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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1
answer
2k
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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 (...
- 1,120
10
votes
1
answer
2k
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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}...
- 1,120
2
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0
answers
265
views
How is group theory (and matrices) used in symmetry operations? [closed]
I am a high school IB student. For my 2-year math course, there is an internal assessment called "exploration" which should be max. 12 pages long. The goal of this assignment is for students to ...
25
votes
1
answer
4k
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What is the mathematical basis behind the Jahn-Teller effect?
Both first-order and second-order Jahn-Teller distortions play a very important role in chemistry.
It is often said that the Jahn-Teller effect is based on symmetry arguments, and hence nothing can ...
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