Rotational transition

In quantum mechanics, a rotational transition is an abrupt change in angular momentum. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

Rotational transitions are important in physics due to the unique spectral lines that result. Because there is a net gain or loss of energy during a transition, electromagnetic radiation of a particular frequency must be absorbed or emitted. This forms spectral lines at that frequency which can be detected with a spectrometer, as in rotational spectroscopy or Raman spectroscopy.

Diatomic molecules

Molecules have rotational energy owing to rotational motion of the nuclei about their center of mass. Due to quantization, these energies can take only certain discrete values. Rotational transition thus corresponds to transition of the molecule from one rotational energy level to the other through gain or loss of a photon. Analysis is simple in the case of diatomic molecules.

Nuclear wave function

Quantum theoretical analysis of a molecule is simplified by use of Born–Oppenheimer approximation. Typically, rotational energies of molecules are smaller than electronic transition energies by a factor of Template:Math ≈ 10⁻³–10⁻⁵, where Template:Math is electronic mass and Template:Math is typical nuclear mass. From uncertainty principle, period of motion is of the order of the Planck constant Template:Math divided by its energy. Hence nuclear rotational periods are much longer than the electronic periods. So electronic and nuclear motions can be treated separately. In the simple case of a diatomic molecule, the radial part of the Schrödinger Equation for a nuclear wave function Template:Math, in an electronic state Template:Math, is written as (neglecting spin interactions) $$[-\frac{{\hslash }^2}{2\mu R^2}\frac{\partial }{\partial R}\left(R^2\frac{\partial }{\partial R}\right)+\frac{⟨{\mathrm{\Phi }}_s|N^2|{\mathrm{\Phi }}_s⟩}{2\mu R^2}+E_s(R)-E]F_s(𝐑)=0$$ where Template:Math is reduced mass of two nuclei, Template:Math is vector joining the two nuclei, Template:Math is energy eigenvalue of electronic wave function Template:Math representing electronic state Template:Math and Template:Math is orbital momentum operator for the relative motion of the two nuclei given by $$N^2=-{\hslash }^2[\frac{1}{\sin \mathrm{\Theta }}\frac{\partial }{\partial \mathrm{\Theta }}(\sin \mathrm{\Theta }\frac{\partial }{\partial \mathrm{\Theta }})+\frac{1}{{\sin }^2\mathrm{\Theta }}\frac{{\partial }^2}{\partial {\mathrm{\Phi }}^2}]$$ The total wave function for the molecule is $${\mathrm{\Psi }}_s=F_s(𝐑){\mathrm{\Phi }}_s(𝐑,{𝐫}_1,{𝐫}_2,\dots ,{𝐫}_N)$$ where Template:Math are position vectors from center of mass of molecule to Template:Mathth electron. As a consequence of the Born-Oppenheimer approximation, the electronic wave functions Template:Math is considered to vary very slowly with Template:Math. Thus the Schrödinger equation for an electronic wave function is first solved to obtain Template:Math for different values of Template:Math. Template:Math then plays role of a potential well in analysis of nuclear wave functions Template:Math.

Vector addition triangle for orbital angular momentum of a diatomic molecule with components of orbital angular momentum of nuclei and orbital angular momentum of electrons, neglecting coupling between electron and nuclear orbital motion and spin-dependent coupling.Since angular momentum Template:Math of nuclei is perpendicular to internuclear vector Template:Math, components of electronic angular momentum Template:Math and total angular momentum Template:Math along Template:Math are equal.

Rotational energy levels

The first term in the above nuclear wave function equation corresponds to kinetic energy of nuclei due to their radial motion. Term Template:Math represents rotational kinetic energy of the two nuclei, about their center of mass, in a given electronic state Template:Math. Possible values of the same are different rotational energy levels for the molecule.

Orbital angular momentum for the rotational motion of nuclei can be written as $$𝐍=𝐉-𝐋$$ where Template:Math is the total orbital angular momentum of the whole molecule and Template:Math is the orbital angular momentum of the electrons. If internuclear vector Template:Math is taken along z-axis, component of Template:Math along z-axis – Template:Math – becomes zero as $$𝐍=𝐑\times 𝐏$$ Hence $$J_z=L_z$$ Since molecular wave function Ψ_s is a simultaneous eigenfunction of Template:Math and Template:Math, $$J^2{\mathrm{\Psi }}_s=J(J+1){\hslash }^2{\mathrm{\Psi }}_s$$ where J is called rotational quantum number and Template:Math can be a positive integer or zero. $$J_z{\mathrm{\Psi }}_s=M_j\hslash {\mathrm{\Psi }}_s$$ where Template:Math.

Also since electronic wave function Template:Math is an eigenfunction of Template:Math, $$L_z{\mathrm{\Phi }}_s=\pm \mathrm{\Lambda }\hslash {\mathrm{\Phi }}_s$$ Hence molecular wave function Template:Math is also an eigenfunction of Template:Math with eigenvalue Template:Math. Since Template:Math and Template:Math are equal, Template:Math is an eigenfunction of Template:Math with same eigenvalue Template:Math. As Template:Math, we have Template:Math. So possible values of rotational quantum number are $$J=\mathrm{\Lambda },\mathrm{\Lambda }+1,\mathrm{\Lambda }+2,\dots$$ Thus molecular wave function Template:Math is simultaneous eigenfunction of Template:Math, Template:Math and Template:Math. Since molecule is in eigenstate of Template:Math, expectation value of components perpendicular to the direction of z-axis (internuclear line) is zero. Hence $$⟨{\mathrm{\Psi }}_s|L_x|{\mathrm{\Psi }}_s⟩=⟨L_x⟩=0$$ and $$⟨{\mathrm{\Psi }}_s|L_y|{\mathrm{\Psi }}_s⟩=⟨L_y⟩=0$$ Thus $$⟨𝐉.𝐋⟩=⟨J_z{L}_z⟩=⟨{L_z}^2⟩$$

Putting all these results together, $$\begin{array}{cc}⟨{\mathrm{\Phi }}_s|N^2|{\mathrm{\Phi }}_s⟩F_s(𝐑)& =⟨{\mathrm{\Phi }}_s|(J^2+L^2-2𝐉\cdot 𝐋)|{\mathrm{\Phi }}_s⟩F_s(𝐑)\\ & ={\hslash }^2[J(J+1)-{\mathrm{\Lambda }}^2]F_s(𝐑)+⟨{\mathrm{\Phi }}_s|({L_x}^2+{L_y}^2)|{\mathrm{\Phi }}_s⟩F_s(𝐑)\end{array}$$

The Schrödinger equation for the nuclear wave function can now be rewritten as $$-\frac{{\hslash }^2}{2\mu R^2}[\frac{\partial }{\partial R}\left(R^2\frac{\partial }{\partial R}\right)-J(J+1)]F_s(𝐑)+[{E^{\prime }}_s(R)-E]F_s(𝐑)=0$$ where $${E^{\prime }}_s(R)=E_s(R)-\frac{{\mathrm{\Lambda }}^2{\hslash }^2}{2\mu R^2}+\frac{1}{2\mu R^2}⟨{\mathrm{\Phi }}_s|({L_x}^2+{L_y}^2)|{\mathrm{\Phi }}_s⟩$$ E′_s now serves as effective potential in radial nuclear wave function equation.

Sigma states

Molecular states in which the total orbital momentum of electrons is zero are called sigma states. In sigma states Template:Math. Thus Template:Math. As nuclear motion for a stable molecule is generally confined to a small interval around Template:Math where Template:Math corresponds to internuclear distance for minimum value of potential Template:Math, rotational energies are given by, $$E_r=\frac{{\hslash }^2}{2\mu {R_0}^2}J(J+1)=\frac{{\hslash }^2}{2I_0}J(J+1)=BJ(J+1)$$ with $$J=\mathrm{\Lambda },\mathrm{\Lambda }+1,\mathrm{\Lambda }+2,\dots$$ Template:Math is moment of inertia of the molecule corresponding to equilibrium distance Template:Math and Template:Math is called rotational constant for a given electronic state Template:Math. Since reduced mass Template:Math is much greater than electronic mass, last two terms in the expression of Template:Math are small compared to Template:Math. Hence even for states other than sigma states, rotational energy is approximately given by above expression.

Rotational spectrum

Template:Main When a rotational transition occurs, there is a change in the value of rotational quantum number Template:Math. Selection rules for rotational transition are, when Template:Math, Template:Math and when Template:Math, Template:Math as absorbed or emitted photon can make equal and opposite change in total nuclear angular momentum and total electronic angular momentum without changing value of Template:Math.

The pure rotational spectrum of a diatomic molecule consists of lines in the far infrared or microwave region. The frequency of these lines is given by $$\hslash \omega =E_r(J+1)-E_r(J)=2B(J+1)$$ Thus values of Template:Math, Template:Math and Template:Math of a substance can be determined from observed rotational spectrum.

See also

• Vibrational transition

References

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