Vanishing Dipole Moment Integrals and Selection Rules

The integrals of the form

$$I = \int f_1 f_2 f_3 \,d\tau$$ (29)

are very important in quantum mechanics as they include transition matrix elements.

For dipole transitions in molecules under influence of electromagnetic radiation, $f_1$ and $f_3$ are the molecular wavefunctions of the initial and the final quantum states and $f_2$ is a component of the molecular dipole moment, $\mu_x$, $\mu_y$, or $\mu_z$. In case of electronic transitions, the components of the dipole moment are just the coordinates of the optical electron, $x$, $y$, and $z$.

The conditions when the transition matrix elements (29) are necessary zero lead to the transition selection rules. As shown in the previous section, the integral (29) can be nonzero only if the product $f_1 f_2 f_3$ spans totally symmetric representation $A_1$, or its equivalent. In order to test whether this condition is fulfilled, the characters of all three functions should be multiplied together and the resulting characters should be analyzed.

As an example, let us investigate whether an electron in an $a_1$ orbital in $H_2O$ can make an electric dipole transition to a $b_1$ orbital. Having in mind that $H_2O$ molecule belong to the $C_{2v}$ group, we should examine all three $x$, $y$, and $z$ components of the transition dipole moment. Reference to the $C_{2v}$ character table in Table 4 shows that these three components transform as $B_1$, $B_2$, and $A_1$, respectively. The calculation runs as shown in Table 6.

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Table 6: Optical Transition in Water

	$x$-component				$y$-component				$z$-component
	$E$	$C_2$	$\sigma_v$	$\sigma_v'$	$E$	$C_2$	$\sigma_v$	$\sigma_v'$	$E$	$C_2$	$\sigma_v$	$\sigma_v'$
$f_1$($B_1$)	1	$-1$	1	$-1$	1	$-1$	1	$-1$	1	$-1$	1	$-1$
$f_2$	1	$-1$	1	$-1$	1	$-1$	$-1$	1	1	1	1	1
$f_3$($A_1$)	1	$1$	1	$1$	1	$1$	$1$	1	1	1	1	1
$f_1 f_2 f_3$	1	1	1	1	1	1	$-1$	$-1$	1	$-1$	1	$-1$

It is seen that the product with $f_2=x$ spans $A_1$, the product with $f_2=y$ spans $A_2$, and the product with $f_2=z$ spans $B_1$. Thus, only the $x$-component of the transition dipole moment may be nonzero. Therefore, we conclude that the electric dipole transition between $a_1$ and $b_1$ is allowed and that $x$-polarization of the radiation can be absorbed, or emitted in this transition. Note that the electric vector of this radiation is perpendicular to the molecular plane.

Continuing this analysis we can build similar table for any of the $a_1$, $a_2$, $b_1$, and $b_2$ orbitals of the $C_{2v}$ symmetry molecule and for all $x$, $y$, and $z$ directions of the transition dipole moment. The result is that the $B_1 \leftrightarrow B_2$ and $A_1 \leftrightarrow A_2$ transitions are forbidden, while the transition between all other states are allowed for certain component of the dipole moment each. Particularly, the transitions between the states of the same symmetry $A_1\leftrightarrow A_1$, $B_1 \leftrightarrow B_1$, ets. are possible for $z$ component of the dipole moment which is parallel to the $C_2$ axis, while the transitions between different symmetry states are possible either for $x$, or $y$ components of the dipole moment. Other selection rules can be obtained using a similar procedure for all other molecular symmetry groups.

The obtained selection rules for a $C_{2v}$ molecule are analogues of the $\Sigma \leftrightarrow \Sigma$, $\Pi\leftrightarrow \Pi$, ($\Delta M = 0$) and $\Sigma \leftrightarrow \Pi$ ( $\Delta M = \pm 1$) selection rules for the electronic transitions in diatomic molecules we studied before.

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