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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Molecular orbital energies for CH4 using group theory
I was learning how to use symmetry in molecular orbitals, I started with $\ce{CH4}$ the point group is $T_d$ I get the reducible representation for methane the reduce it to be $T_2+A_1$, the I applied ...
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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 ...
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Mulliken Labels: How do I tell apart E, doubly degenerate, symmetry apart from T, triply degenerate, symmetry for a molecular orbital?
With A and B, it's pretty simple. If you rotate the molecule about its principle axis to where the orbitals line up, and the signs change, it's Mulliken label has a B. otherwise, it's A.
I heard E ...
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When relaxing the strong field interaction, the (t2g)^2 configuration splits into 4 states. What is the symbolic configuration of the Eg state?
When relaxing the infinitely strong field, the electrons start to feel one another's presence. They will, therefore, give rise to sets of states such as T1g, T2g... for the (t2g)^2 initial ...
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Bonding and symmetry
So I have been looking at different ways of symmetry and bonding
The so called counting method is something that I don't get
For each operation in the group:
...
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Reduction of direct product representation
When we do the reduction of the reducible representation generated from the total wavefunction expressed as product of MOs of appropriate symmetry, we can find the direct sum of more than one ...
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Group theory and exact Eigenfunctions of Hamiltonian
I think I've understand why the real Eigenfunctions of Hamiltonian belong to a given irreducible representation and I've read that also MOs have to transform as irreducible representation due to the ...
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Significance of Character tables in d orbital splitting
When we have a molecule say (Co(NH2)6) hypothetically any octahedral molecule it has the point group of Oh and the character table shows the quadratic function such as ($z^2,2z^2-x^2-y^2$) ($xy, xz, ...
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How much group representation theory is used in chemistry research?
I am a mathematician working in group theory and representation theory. When I try to explain these subjects to non-mathematicians, I often fail to provide a convincing reason to study these. So I am ...
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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
...
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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 ...
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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 ...
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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{...
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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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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 ...
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How to determine the characters of px atomic orbital in C4v point group?
How to determine the characters of px atomic orbital in $\text{C}_{\text{4v}}$ point group? I understand (px,py) is a set of basis function of E, and px and py are mixed by $\text{C}_4$ and $\sigma_{\...
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Reduction formula for infinite group order?
In group theory, what do we do when we have infinite group order, like in D∞h or C∞v but we want to mathematically determine the SALCs (symmetry adapted linear combinations) using the projection ...
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IR / Raman factor group analysis (site and factor group determination)
I'm currently trying to interpret some IR and Raman spectra of solid state compounds. They show a simple molecular unit of which the expected bands are split due to the coordination in the crystal ...
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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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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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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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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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