# Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries

## Abstract

**:**

## Abstract

## Keywords

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## 1. Introduction

## 2. Irreducible Representations and Quantum Numbers

**Figure 1.**Eigenstates labeled with the energy and the irreps. A degeneracy still remains due to the degeneracy of the irreps.

## 3. The Eigenfunction Approach

**Table 1.**Character table of the group ${\mathcal{D}}_{3}$. The notation for the classes is the following: ${K}_{1}=\left\{E\right\},{K}_{2}=\{{C}_{3},{C}_{3}^{2}\}$ and ${K}_{3}=\{{C}_{2}^{a},{C}_{2}^{b},{C}_{2}^{c}\}$.

${\mathcal{D}}_{3}$ | ${K}_{1}$ | ${K}_{2}$ | ${K}_{3}$ |
---|---|---|---|

${A}_{1}$ | 1 | 1 | 1 |

${A}_{2}$ | 1 | 1 | $-1$ |

E | 2 | $-1$ | 0 |

**Table 2.**λ’s table for the group ${\mathcal{D}}_{3}$ obtained from the character table and the relation (29).

${\mathcal{D}}_{3}$ | ${\lambda}_{1}^{\nu}$ | ${\lambda}_{2}^{\nu}$ | ${\lambda}_{3}^{\nu}$ |

${A}_{1}$ | 1 | 2 | 3 |

${A}_{2}$ | 1 | 2 | $-3$ |

E | 1 | $-1$ | 0 |

Irrep | Eigenvalue | Eigenvector |

${A}_{1}$ | 3 | $(1,1,1)$ |

E | 0 | $(1,0,-1)$ |

E | 0 | $(1,-1,0)$ |

**Table 4.**Irreps of ${\mathcal{C}}_{2}^{a}$ contained in the irreps of the group ${\mathcal{D}}_{3}$.

${\mathcal{C}}_{2}^{a}$ | E | ${C}_{2}^{a}$ | |

A | 1 | 1 | |

B | 1 | $-1$ | |

${A}_{1}$ | 1 | 1 | A |

${A}_{2}$ | 1 | $-1$ | B |

E | 2 | 0 | $A\oplus B$ |

**Table 5.**Eigenvalues associated with the operators ${\widehat{K}}_{3}$ and ${\widehat{k}}_{2}={\widehat{C}}_{2}^{a}$ corresponding to the chain of groups ${\mathcal{D}}_{3}\supset {\mathcal{C}}_{2}^{a}$.

${\mathcal{D}}_{3}$ | ${\lambda}_{3}^{\nu}$ | ${\mathcal{C}}_{2}^{a}$ | ${\lambda}_{2}^{\mu}$ | $\phantom{\rule{4pt}{0ex}}{\lambda}_{3}^{\nu}+{\lambda}_{2}^{\mu}$ |

${A}_{1}$ | $+3$ | A | $+1$ | $+4$ |

${A}_{2}$ | $-3$ | B | $-1$ | $-4$ |

E | 0 | A | $+1$ | $+1$ |

E | 0 | B | $-1$ | $-1$ |

**Table 6.**Eigensystem associated with the matrix representation of the class operator ${\widehat{C}}_{II}$.

Irreps | Eigenvalue | Eigenvector |

${A}_{1}$ | $+4$ | $(1,1,1)$ |

$(E,A)$ | $+1$ | $(2,-1,-1)$ |

$(E,B)$ | $-1$ | $(0,1,-1)$ |

**Step 1**. From the character table of the symmetry group G, the λ’s table is generated using (29).

**Step 2**. We proceed to identify the column that distinguish the irreps of the group G. In general more than one column is needed to achieve this goal. In such case a linear combinations of columns are selected in such a way that the eigenvalues are all different. This process defines the linear combination of classes, which we shall identify with the operator $\widehat{C}$.

**Step 3**. A subgroup H is proposed in such a way that the irreps of G are not contained more than once in H.

**Step 4**. From the character table of H, we construct the λ’s table and identify the columns that distinguish the irreps. The columns involved define a linear combination of classes of the subgroup that define the operator $\widehat{C}\left(s\right)$. This operators is the equivalent of $\widehat{C}$ in the group G.

**Step 5**. A table of eigenvalues associated with G and H is constructed (Table 5 in our example). A linear combination of λ’s is identified to define a new operator ${\widehat{C}}_{II}$.

**Step 6**. The representation of the operator ${\widehat{C}}_{II}$ is generated using the space to be projected. The diagonalization of the matrix representation $\Delta \left({C}_{II}\right)$ provides the symmetry adapted functions.

## 4. An Example: Stretching Degrees of Freedom of Methane

**Step 1**. From the character table of the group ${\mathcal{T}}_{d}$ given in Table 7, we obtain the λ’s table using the relation (29), which is displayed in Table 8.

${\mathcal{T}}_{d}$ | ${K}_{1}$ | ${K}_{2}$ | ${K}_{3}$ | ${K}_{4}$ | ${K}_{5}$ |

E | $4{C}_{3},4{C}_{3}^{2}$ | $3{C}_{2}$ | $3{S}_{4},3{S}_{2}^{3}$ | $6{\sigma}_{d}$ | |

${A}_{1}$ | 1 | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |

${A}_{2}$ | 1 | $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$ |

E | 2 | $-1$ | $\phantom{-}2$ | $\phantom{-}0$ | $\phantom{-}0$ |

${F}_{1}$ | 3 | $\phantom{-}0$ | $-1$ | $\phantom{-}1$ | $-1$ |

${F}_{2}$ | 3 | $\phantom{-}0$ | $-1$ | $-1$ | $\phantom{-}1$ |

${\mathcal{T}}_{d}$ | ${\lambda}_{1}^{\nu}$ | ${\lambda}_{2}^{\nu}$ | ${\lambda}_{3}^{\nu}$ | ${\lambda}_{4}^{\nu}$ | ${\lambda}_{5}^{\nu}$ |

${A}_{1}$ | 1 | 8 | 3 | 6 | 6 |

${A}_{2}$ | 1 | 8 | 3 | $-6$ | $-6$ |

E | 1 | $-4$ | 3 | 0 | 0 |

${F}_{1}$ | 1 | 0 | $-1$ | 2 | $-2$ |

${F}_{2}$ | 1 | 0 | $-1$ | $-2$ | 2 |

**Step 2**. From the λ’s table we note that the either the class ${K}_{4}$ or ${K}_{5}$ are suitable to be chosen as $\widehat{C}$ since in both cases all the eigenvalues are different (both contain generators of the group). We propose

**Step 3**. We propose the subgroup ${\mathcal{C}}_{2v}$ to distinguish the different components of the irreps E, ${F}_{1}$ and ${F}_{2}$. To check if the group ${\mathcal{C}}_{2v}$ is appropriate we proceed to find the irreps of this subgroup contained in the irreps of the group using (42). To this end we write down the character table of the subgroup ${\mathcal{C}}_{2v}$ including the irreps of the group ${\mathcal{T}}_{d}$, together with the reductions obtained through (42), as indicated in Table 9. Since there is no repetition of the irreps of ${\mathcal{C}}_{2v}$ contained in the irreps of ${\mathcal{T}}_{d}$, the proposed subgroup ${\mathcal{C}}_{2v}$ is suitable to label the states.

**Step 4**. From the character table of ${\mathcal{C}}_{2v}$ we construct the corresponding λ’s table as given in Table 10. It is clear that there is no column that is able to distinguish the irreps by itself, as expected. So we have to propose a linear combination of them. In fact we propose ${\lambda}_{2}^{\nu}+3{\lambda}_{3}^{\nu}$, whose values are included in the table. This means that the operator $C\left(s\right)$ is defined as

${\mathcal{C}}_{2v}$ | E | ${C}_{2}^{z}$ | ${\sigma}_{d}^{I}$ | ${\sigma}_{d}^{II}$ | |

${A}_{1}$ | 1 | 1 | 1 | 1 | |

${A}_{2}$ | 1 | 1 | $-1$ | $-1$ | |

${B}_{1}$ | 1 | $-1$ | 1 | $-1$ | |

${B}_{2}$ | 1 | $-1$ | $-1$ | 1 | |

${A}_{1}$ | 1 | 1 | 1 | 1 | ${A}_{1}$ |

${A}_{2}$ | 1 | 1 | $-1$ | $-1$ | ${A}_{2}$ |

E | 2 | 2 | 0 | 0 | ${A}_{1}\oplus {A}_{2}$ |

${F}_{1}$ | 3 | $-1$ | $-1$ | $-1$ | ${A}_{2}\oplus {B}_{1}\oplus {B}_{2}$ |

${F}_{2}$ | 3 | $-1$ | 1 | 1 | ${A}_{1}\oplus {B}_{1}\oplus {B}_{2}$ |

${\mathcal{C}}_{2v}$ | ${\lambda}_{1}^{\nu}$ | ${\lambda}_{2}^{\nu}$ | ${\lambda}_{3}^{\nu}$ | ${\lambda}_{4}^{\nu}$ | ${\lambda}_{2}^{\nu}+3{\lambda}_{3}^{\nu}$ |
---|---|---|---|---|---|

${A}_{1}$ | 1 | 1 | 1 | 1 | $+4$ |

${A}_{2}$ | 1 | 1 | $-1$ | $-1$ | $-2$ |

${B}_{1}$ | 1 | $-1$ | 1 | $-1$ | $+2$ |

${B}_{2}$ | 1 | $-1$ | $-1$ | 1 | $-4$ |

**Step 5**. We now proceed to display the Table 11 of eigenvalues associated with ${\mathcal{T}}_{d}$ and ${\mathcal{C}}_{2v}$.

**Table 11.**λ’s table of the groups ${\mathcal{T}}_{d}$ and ${\mathcal{C}}_{2v}$ to determine the CSCO-II.

ν | m | $\nu +3m$ |

6 | 4 | 18 |

$-6$ | $-2$ | $-12$ |

0 | $-2$ | $-6$ |

$-2$ | $-4$ | $-14$ |

2 | $-4$ | $-10$ |

**Step 6**. We now consider the space of stretching coordinates of methane with the labeling displayed in Figure 3. The matrix representation of the of the operator ${C}_{II}$ in this basis is given by

Irreps | Eigenvalue | Eigenvector |

${A}_{1}$ | $+18$ | $(1,1,1,1)$ |

$({F}_{2},{A}_{1})$ | $+14$ | $(1,1,-1,-1)$ |

$({F}_{2},{B}_{1})$ | 8 | $(-1,1,0,0)$ |

$({F}_{2},{B}_{2})$ | $-10$ | $(0,0,-1,1)$ |

## 5. Summary and Conclusions

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Lemus, R.
Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries. *Symmetry* **2012**, *4*, 667-685.
https://doi.org/10.3390/sym4040667

**AMA Style**

Lemus R.
Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries. *Symmetry*. 2012; 4(4):667-685.
https://doi.org/10.3390/sym4040667

**Chicago/Turabian Style**

Lemus, Renato.
2012. "Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries" *Symmetry* 4, no. 4: 667-685.
https://doi.org/10.3390/sym4040667