Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

4
votes
2answers
49 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 ...
2
votes
1answer
38 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 ...
1
vote
1answer
29 views

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 ...
2
votes
0answers
35 views

Vanishing integral rule for two-electron integrals

I understand the vanishing integral rule for a one-electron integral to be: $$\langle i | \hat{O} | j \rangle = 0 \hspace{1cm} \mathrm{if} \hspace{0.2cm} \Gamma_O \neq \Gamma_i \otimes \Gamma_j$$ ...
5
votes
1answer
85 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?
3
votes
1answer
43 views

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 - ...
3
votes
0answers
38 views

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 ...
1
vote
0answers
59 views

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 $D_{2h}$ ...
3
votes
1answer
69 views

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
vote
2answers
278 views

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 ...
1
vote
1answer
93 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 ...
0
votes
0answers
46 views

Why does the bending vibration of H2O span the A1 (rather than B1) irreducible representation?

Defining the $H_20$ molecule as being in the xz plane with the z axis being the principal axis. Why does water have a bending vibration from the $A_1$ irrep and not $B_1$? Since $B_1$ contains the $...
1
vote
1answer
304 views

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 ...
5
votes
1answer
88 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 ...
1
vote
0answers
108 views

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
1answer
124 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 ...
12
votes
2answers
813 views

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 ...
5
votes
1answer
129 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 $\sigma_\mathrm d$? ...
2
votes
1answer
117 views

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 ...
8
votes
1answer
281 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 ...
10
votes
2answers
418 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 ...
0
votes
0answers
326 views

Mulliken Symbols in group theory

How does one assign subscripts 1 and 2 in the B representations for the D4 symmetry group. There are two classes of C2 axis perpendicular to the principle C4 axis. I referred to F. Albert Cotton's '...
9
votes
1answer
603 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 & & \...
5
votes
1answer
378 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 $C_{2v}$ ...
7
votes
1answer
715 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 (...
8
votes
1answer
510 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}...
2
votes
0answers
163 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 ...
12
votes
1answer
476 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 ...
19
votes
3answers
552 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 $\...
14
votes
1answer
3k 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 ...
16
votes
5answers
3k views

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 ...
21
votes
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 ...
1
vote
0answers
222 views

Does Ni(H2O)6 have Oh symmetry or Th?

I optimized a geometry of $\ce{Ni(H2O)6}$ using quantum chemistry and got a molecule which is symmetrical for the $\ce{Ni-O}$ distances as shown here: In the literature it says that this complex has ...
15
votes
4answers
625 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 ...
2
votes
0answers
639 views

What is the symmetry of the cuboctahedron (FCC metal)?

The background to this is that I've recently given a tutorial wherein we had to go through the determination of point groups for atoms in various lattices (BCC, FCC/CCP and HCP). BCC and HCP, I have ...
33
votes
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 ...
6
votes
1answer
154 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 ...
3
votes
2answers
3k views

How can the ground state electron configuration for a d3 complex correspond to an F term symbol?

The ground-state term symbol from a $\mathrm{d^3}$ Tanabe-Sugano diagram is $\mathrm{^4F}$. My question is how the total orbital quantum number $\Lambda=3$, or $\mathrm{F}$ term, arises. For a $\...
8
votes
1answer
1k views

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'...
6
votes
0answers
69 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 ...
4
votes
2answers
243 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 ...
7
votes
2answers
6k views

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 ...
4
votes
0answers
50 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 ...
7
votes
2answers
971 views

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 ...
6
votes
1answer
2k 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
votes
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 $\...
2
votes
1answer
165 views

What does a molecules color have to do with its bond/orbital energies?

For example, elemental iodine is deep violet. Its sigma bond or perhaps the lone pairs are capable of absorbing all visible light frequencies except violet which is why we see it as that color. ...
13
votes
1answer
974 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 ...
7
votes
3answers
229 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 ...
2
votes
1answer
372 views

Point Groups and MO Theory

I am a little confused about how point groups give information about MO of polyatomic molecules. Is it true that all molecules of a given point group such as, $\mathrm{C_{2v}}$, will have the same MO ...