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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34
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4answers
2k views

Why is the letter J omitted in the spdf… sequence?

Under spherical symmetry, the irreducible representations corresponding to $L = 0, 1, 2, \cdots$ are assigned the letters $\mathrm{S}, \mathrm{P}, \mathrm{D}, \mathrm{F}, \cdots$ after which the ...
21
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1answer
2k views

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 ...
20
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3answers
998 views

Group theoretical condition for an integral to be zero

In many aspects of chemistry it is of interest to find out whether a given integral $\int \psi \,\mathrm{d}\tau$ vanishes, i.e. becomes zero. Group theory provides us with a handy condition: If $\...
18
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1answer
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What are t2g and eg in CFT?

In the Crystal Field Theory, when the splitting of the d-orbital occurs, it gets divided into two parts... The upper part with higher Energy is the $t_{2g}$ and the lower part with lower Energy is ...
16
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5answers
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Why do we need the identity operator, E

In group theory, in order to assign a point group, one must first identify the symmetry elements present in the molecule. To give an example, water has: a rotational axis, labelled C2 since it is a ...
14
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4answers
847 views

How do I determine the molecular vibrations of linear molecules?

I know how to do it for just about every other point group, but the $D_{\infty \mathrm h}$ and $C_{\infty \mathrm v}$ character tables aren't as straightforward. In particular, I'm interested in the ...
14
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1answer
5k views

Using group theory to make molecular orbital diagrams

Because of Orthocresols' meta post I ask this question about application of group theory to molecular orbital theory. In my undergraduate lecturers, we've seen many examples of making qualitative MO ...
13
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1answer
15k views

How can I find the symmetry labels of atomic orbitals in a molecule?

What are the symmetry labels for the p and d orbitals of platinum in $\ce{[PtCl4]^2-}$? I understand how to find the point group of a molecule, but am not sure how to use the character table to find ...
13
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2answers
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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 ...
13
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1answer
1k views

What are point groups and the benefits of using them?

I have heard vague information about point groups and symmetry before but I am unsure how they actually help me predicting physical properties of molecules. How do I assign point groups and what good ...
12
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1answer
519 views

What is it that is “symmetry-adapted” about symmetry-adapted perturbation theory?

I think this is a fairly straightforward question which I should know the answer to, but surprisingly cannot find a discussion of online. Symmetry-adapted perturbation theory (SAPT) is a fairly ...
11
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1answer
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How are point group character tables typeset correctly?

I want to know how to typeset character tables of point groups correctly. Consider the following two extremes using the $C_{\text{2v}}$ (or is it $\text{C}_{\text{2v}}$?) point group as an example. ...
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How does one recognized Td/Oh symmetry in molecules?

The identification of point groups of a molecule is usually done following a strict scheme, either manually or algorithmically. In all textbooks I could find, however, the first step of the scheme is ...
10
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2answers
696 views

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 ...
9
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1answer
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Symmetry representations

I'm Having a real struggle trying to understand symmetry representations matrices and character tables. Is there anyone who would be kind enough to help me out here? I understand it is probably a ...
9
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1answer
901 views

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 & & \...
8
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1answer
714 views

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}...
8
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1answer
6k views

How to get the reducible representations of the B-B and the B-Cl bond of B2Cl4

I just can't seem to get the correct answer to this symmetry question in my book. "Introduction to molecular symmetry J.S.Ogden." I am trying to find the reducible representations of both the $\ce{B-...
8
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1answer
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Determination of +/- reflections in sigma molecular term symbols

This page, which depicts the molecular term symbols for the $\ce{O2}$ and $\ce{O2-}$ (Edit: Mistaken on $\ce{O2-}$) molecules, perhaps best summarizes the full scope of my questions. In general, I don'...
8
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1answer
284 views

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 ...
7
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3answers
279 views

Why does (2z²-x²-y², x²-y²) have -1 trace in the character table of Td for C3 rotation?

I'm trying to understand the irreducible rep for $(2z^2-x^2-y^2, x^2-y^2)$ in the $T_\mathrm{d}$ point group. Specifically, for $C_3$ rotations, a trace of $-1$ is listed in the character table for ...
7
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1answer
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How do I determine the crystal field splitting for an arbitrary point group?

How can I figure out the degeneracy of the d orbitals for a site that has a given point group? Specifically I'm interested in $D_{3d}$ and $D_{3h}$, but it would be good to know how to do it in the ...
7
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1answer
154 views

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 ...
7
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1answer
1k views

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 (...
7
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1answer
768 views

Are these two space groups in a group–subgroup relationship?

Given two crystal space groups, how can one determine if they are in a group-subgroup relationship? The specific case at hand is P 21/m 2/m 2/a (aka P m m a, #51) and P m m 2 (#25), but knowing how to ...
7
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2answers
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Examples of molecules with D4 and C9 point groups

Trying to come up with examples of (hypothetical) molecules/shapes that have $D_4$ and $C_{9h}$ point groups. For $D_4$, I thought a cyclobutane structure, but with 4 substituents coming off of the ...
7
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0answers
178 views

Vanishing integral rule for two-electron integrals

I understand the vanishing integral rule for a one-electron integral to be: $$\langle i | \hat{O} | j \rangle = 0 \hspace{1cm} \mathrm{if} \hspace{0.2cm} \Gamma_O \neq \Gamma_i \otimes \Gamma_j$$ ...
7
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1answer
197 views

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 ...
6
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2answers
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Identifying the C3, C4, S4, and S6 symmetry operations in the Oh point group

The Oh character table has ten symmetry classes. I am trying to identify them, but my stumbling block is in identifying the order of each class. I will use this diagram as a reference, from Sydney ...
6
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2answers
116 views

Accessible statement on the intuition and usefulness of group theory to chemistry

I am writing a lesson plan for high school students who are studying group theory, and I am having difficulty in relating the subject matter to the real world. I have quite a few examples using the ...
6
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1answer
180 views

Symmetry and SALC: choosing coordinate basis

When we construct symmetry adapted linear combinations of orbitals (aka SOs) to build an MO diagram, we choose the z axis as the axis of highest symmetry and all other axis to make symmetry operations ...
6
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0answers
109 views

Equivalent Wyckoff position in different groups

I am not a chemist, I am a physicist, so please bare with me if what I am asking is rather obvious. If I have two symmetry groups ($P4/mmm$ and $P\!-\!1$), is there a way to find their respective ...
5
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1answer
3k views

Understanding group theory easily and quickly [closed]

Please can you explain me the group theory especially the reducible and irreducible representation and class and subclass and how to tell they are in the same group or not. I know it is a tedious task ...
5
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1answer
86 views

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 $D_{3d}$ ...
5
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1answer
4k views

Molecular term symbols for excited state oxygen

What are the possible term symbols for excited state oxygen with configuration $(1\pi_u)^3$,$(1\pi^*_g)^3$ (all other occupied orbitals are closed shell)? Based on the possible values of S and $\...
5
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1answer
3k views

MO-Scheme of SCN- and its bonding properties when used as a ligand

I tried to figure out the MO-scheme of the tetragonal-bipyramidal complex trans-$\ce{[Co(en)2(NCS)2]SCN}$ in which the isothiocyanate ligands are bound to the $\ce{Co^3+}$-Ion in $\eta^{1}$-mode (en = ...
5
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1answer
353 views

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?
5
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1answer
286 views

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 $\sigma_\mathrm d$? ...
5
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1answer
755 views

Non-bonded orbitals in water

General Chemistry perspective: Looking at the molecular orbitals of water, we can see that the oxygen is $sp^3$ hybridized. Oxygen forms two sigma bonds with hydrogens, and there are two lone pair ...
5
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1answer
85 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 ...
5
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0answers
137 views

Finding the irreducible representation which an orbital transforms under [duplicate]

I have some major confusion on irreducible representations. An example of a problem I have is Based on the character table for the point group of each of the follwing molecules, which irreducible ...
5
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0answers
390 views

Determining the lowest rank non-vanishing multipole moment [closed]

I have been told that the multipole moments of rank 0,1,2,3,...n have respectively 1,3,5,7,...2n+1 components which transform like the spherical harmonics of rank 0,1,2,3,...n. How do I use the ...
4
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2answers
132 views

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
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1answer
77 views

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 ...
4
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1answer
650 views

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 $C_{2v}$ ...
4
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0answers
51 views

Imposing particle number subspace on second quantized Hamiltonians

Here is my question: Second-quantized Hamiltonians act on any state, regardless of how many electrons the state has. They are also block diagonal on different particle-number subspaces, as they ...
4
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0answers
169 views

Decomposition of axial vector and vector representions of C4v group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way $$ \vec{V} ...
3
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2answers
416 views

Deriving the irreps of the d-orbitals under C3 and S6 operations in octahedral symmetry

Contextual background: I am trying to reproduce the symmetries in the graph on this page (MO diagram for $\ce{ML6}$ complex). I have reproduced the metal s and p orbital representations as well as the ...
3
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1answer
494 views

Theoretical Calculation of Lattice Parameter from Space Group

If I am given a compound which belongs to, say the cubic $\text{Ia}\bar 3\text{d}$ space group, is there a known way to calculate the lattice parameter of the compound? In general, can lattice ...
3
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1answer
123 views

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 ...