Symmetry in chemistry

Notes Chemistry

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 2020/03/07   Share

Cambridge Part II B7 course.

Representations

The representation matrix of an operation is defined with a left contraction

$\hat{R}\phi_i=\sum_j \phi_j D_{ji}(R).$

Great Orthogonality Theorem

If $\Gamma_a$ and $\Gamma_b$ are two irreducible unitary representations, then their representation matrices satisfy

$\frac{1}{h}\sum_R D_{ip}^a(R)^* D_{jq}^b(R)=\begin{cases}0&\Gamma_a\nsim\Gamma_b,\\ \frac{1}{n_a}\delta_{ij}\delta_{pq}& \Gamma_a=\Gamma_b.\end{cases}$

Note that this formula does not apply to equivalent but not identical representations.

First orthogonality theorem for characters

Set $i=p$ , $j=q$ in the above formula.

$\begin{align} &\frac{1}{h} D_{ii}^a(R)^* D_{jj}^b(R)\\ =&\frac{1}{h}\chi^a(R)^*\chi^b(R)\\ =& \begin{cases}0&\Gamma_a\nsim\Gamma_b,\\ \frac{1}{n_a}\delta_{ip}\delta_{ip}=1& \Gamma_a=\Gamma_b\end{cases}\\ =&\delta_{\Gamma_a\Gamma_b}. \end{align}$

In terms of classes,

$\frac{1}{h}h_c\chi^a(c)^*\chi^b(c)=\delta_{\Gamma_a\Gamma_b}.$

From here, the reduction formula is simply a substitution of one representation with the reducible representation.

$\begin{align} m_a&=\frac{1}{h}\chi^a(R)^*\chi(R)\\ &=\frac{h_c}{h}\chi^a(c)^*\chi(c) \end{align}.$

First orthogonality also states that all rows are orthogonal, therefore it is deduced that there can not be more rows than columns. (As there can’t be more orthogonal bases than the dimension.)

Second orthogonality theorem for characters

$\frac{h_c}{h}\chi^\Gamma(c)^*\chi^\Gamma(c')=\delta_{cc'}.$

This states that all columns are orthogonal, and hence no more columns than rows.

Therefore, the number of classes = the number of irreducible representations.

Projection Formula

$\hat{P}_i^a=\frac{n_a}{h}\sum_R D_{ii}^a(R)^*R.$

When applied to a function, only the component $i$ of representation $\Gamma_a$ can survive.

Spherical harmonics and full rotation group

When considering whether to rotate a function or to rotate the coordinate system, one can get to the result that

$R\psi(r)=\psi(R^{-1}r).$

The angular momentum operator

Construct the operator as

$\hat{J_z}=i\hbar \lim_{\alpha \rightarrow 0}\frac{R(\alpha,z)-E}{\alpha}.$

The class $C(\alpha)$ of $SO(3)$ contains all rotations through angle $\alpha$ about any axis.
Representation $J$ has dimension $2J+1$ .
The basis functions are spherical harmonics $Y_{JM}$ .

Characters

The $\phi$ dependence of $Y_{JM}$ is $\exp{(iM\phi)}$ .

Hence use the tranformation relationship

$\begin{align} &R(\alpha,z)Y_{JM}\\ =&\Theta(\theta)\sqrt{\frac{1}{2\pi}}\exp{[iM(\phi-\alpha)]}\\ =&\exp[-iM\alpha]Y_{JM}. \end{align}$

Therefore, the character for a given $J$ is the sum over all $M$ , which is a geometric series

$\begin{align} \chi^J(\alpha)&=\sum_{M=-J}^J\exp[-iM\alpha]\\ &=\frac{\exp[(J+1)i\alpha]-\exp[-Ji\alpha]}{\exp[i\alpha]-1}\\ &=\frac{\sin(J+\frac{1}{2})\alpha}{\sin(\frac{1}{2}\alpha)}. \end{align}$

Symmetric representation

In 1st orthorgonality theorem, put $\Gamma_a=\Gamma_1$ ,

$\frac{1}{h}\sum_R \chi^\Gamma(R)=\begin{cases}0 & \Gamma\neq\Gamma_1\\ 1 & \Gamma = \Gamma_1\end{cases}.$

Since a symmetry operation can not change the value of any physical quantity,

$\mu_i=R\mu_i=\sum_j\mu_jD_{ji}^{\Gamma_\mu}(R)\\ h\mu_i=\sum_RR\mu_i=\sum_j\mu_j\sum_RD_{ji}^{\Gamma_\mu}(R).$

This is 0 unless $\Gamma_\mu$ is $\Gamma_1$ .

The number of independent non-zero components is the number of times that $\Gamma_1$ occurs in $\Gamma_\mu$ .

Direct product representation

For two sets of basis functions corresponding to two IRs, operating on a product of functions

$\begin{align} &R(\xi_i\eta_j)\\ =&\sum_k\xi_kD^{\Gamma_a}_{ki}(R)\sum_l\eta_lD^{\Gamma_b}_{lj}(R)\\ =&\sum_{kl}(\xi_k\eta_l)D^{\Gamma_a}_{ki}(R)D^{\Gamma_b}_{lj}(R). \end{align}$

Hence we get a new representation direct product representation,

$D^{\Gamma_a}_{ki}(R)D^{\Gamma_b}_{lj}(R)=D_{kl,ij}^{\Gamma_a \otimes\Gamma_b}(R).$

By setting $k,l=i,j$ , the characters are equal to the product of characters

$\chi^{\Gamma_a \otimes\Gamma_b}(R)=\chi^{\Gamma_a}(R)\chi^{\Gamma_b}(R).$

Symmetric components

If both are IRs, then direct product contains $\Gamma_1$ once iff $\Gamma_a \sim\Gamma_b$ , prove by the symmetric representation formula (or reduction formula for $\Gamma_1$ ).

For identical representations, the symmetric component is (prove by projection formula)

$(\xi_i^*\eta_j)^{\Gamma_1}=\delta_{ij}\frac{1}{n_\Gamma}\sum_k \xi_k^*\eta_k.$

e.g. $\eta_x\xi_x+\eta_y\xi_y$ .

For integrals of products

Since $\Gamma_1$ is present iff $\Gamma_a\sim\Gamma_b$ , the integral is zero unless the representations are similar.
For identical representations, from the above equation, only the product of the same component yields non-zero product.
and for all non-zero products, the values are the same.

Symmetrized and antisymmetrized square

Symmetrized is in the sense that the result is symmetric w.r.t. the exchange of basis functions, i.e. $\eta$ and $\xi$ .

The characters are

$\chi^{\Gamma\otimes\Gamma_\pm}(R)=\frac{1}{2}[(\chi^\Gamma(R))^2\pm\chi^\Gamma(R^2)].$

For spherical harmonics,

the symmetrized square are the even terms up to $2J$ , i.e. $\Gamma^{2J}\oplus\Gamma^{2J-2}...\Gamma^{0}$ ;
the antisymmetrized square are the odd terms, $\Gamma^{2J-1}\oplus\Gamma^{2J-3}...\Gamma^{1}$ .

Jahn-Teller theorem

A non-symmetric distortion of symmetry $\Gamma_Q$ will lower the energy of at least one state, provided that $(\Gamma\otimes\Gamma)_+\ni\Gamma_Q$ .

The sum of energy changes is 0 unless the distortion is totally symmetric;
The sum of the squares of the energy changes is non-zero provided that $(\Gamma\otimes\Gamma)_+\ni\Gamma_Q$ .

For a linear molecules, there is never a distortion satisfying the JT condition.

Vibrational states

For doubly excited states, only the symmetrized squares are valid, since the application of the ladder operator can be in either order.

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Author：2LalA猪

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Published on：7th March 2020, 11:27 am

Updated on：9th June 2020, 3:21 pm

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CATALOG

1. Representations
2. Great Orthogonality Theorem
3. Spherical harmonics and full rotation group
1. 3.1. The angular momentum operator
2. 3.2. Characters
4. Symmetric representation
5. Direct product representation
1. 5.0.1. Symmetric components
2. 5.0.2. For integrals of products
5.1. Symmetrized and antisymmetrized square

6. Jahn-Teller theorem

7. Vibrational states

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