Physics:Quantum molecular spectroscopy

Short description: Study of molecular energy levels using electromagnetic spectra

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Quantum molecular spectroscopy is the study of how molecules absorb, emit, or scatter radiation as they move between quantized molecular energy states. In quantum mechanics, molecular spectra arise because the allowed rotational, vibrational, and electronic states of a molecule are discrete or partly discrete. The observed spectral lines and bands therefore reveal information about bond lengths, force constants, molecular geometry, electronic structure, and interactions with the environment.

Discrete spectral lines are a visible signature of quantized energy differences. In molecules, rotational, vibrational, and electronic structure combine to produce richer spectra than in isolated atoms.

Overview

A molecule has several kinds of quantum energy. Its electrons occupy molecular electronic states, the nuclei vibrate about equilibrium geometries, and the molecule can rotate. To a useful approximation, the total molecular energy can be written as a sum of electronic, vibrational, and rotational contributions,

$E\approx E_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{c}}+E_{\mathrm{v}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}+E_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}.$

The energy spacing of these contributions is very different. Rotational transitions usually occur in the microwave or far-infrared region, vibrational transitions in the infrared, and electronic transitions in the visible or ultraviolet. Real spectra often combine these motions, producing rotational, vibrational, rovibrational, electronic, and vibronic structure.

In quantum mechanics

In quantum mechanics, the discrete spectrum of an observable corresponds to the eigenvalues of the operator used to model that observable. According to the mathematical theory of such operators, its eigenvalues may form a discrete set of isolated points, which may be either finite or countable.

Discrete spectra are usually associated with systems that are bound or confined in some sense. The position and momentum operators have continuous spectra in an infinite domain, but a discrete spectrum in a compact or confined domain. Similar spectral ideas apply to angular momentum, Hamiltonians, and other operators of quantum systems.

The quantum harmonic oscillator and the hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom the spectrum has both a continuous and a discrete part, the continuous part corresponding to ionization.

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  The discrete part of the emission spectrum of hydrogen.
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  Spectrum of sunlight above the atmosphere and at sea level, showing absorption features with discrete and continuous parts.
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  Spectrum of light emitted by a deuterium lamp, showing sharp peaks and a continuous background.

Rotational spectra

The simplest rotational model is the rigid rotor. For a diatomic molecule with reduced mass $\mu$ and bond length $r$, the moment of inertia is

$I=\mu r^2.$

The rotational energy levels are

$E_J=\frac{h^2}{8{\pi }^{2}I}J(J+1),J=0,1,2,\dots$

where $J$ is the rotational quantum number. In spectroscopic notation the same levels are often written in terms of a rotational constant $B$,

$F(J)=BJ(J+1).$

Pure rotational absorption or emission requires a changing or permanent electric dipole moment. For a simple electric-dipole rotational transition, the selection rule is

$\mathrm{\Delta }J=\pm 1.$

Rotational spectra are especially useful for determining molecular moments of inertia and therefore bond lengths and molecular geometry.

Vibrational spectra

Vibrational spectra arise from quantized nuclear motion. In the harmonic oscillator approximation, the vibrational energy levels of a diatomic molecule are

$E_v=(v+\frac{1}{2})h{\nu }_0,v=0,1,2,\dots$

where $v$ is the vibrational quantum number and ${\nu }_0$ is the natural vibrational frequency of the bond. In the simplest harmonic model, the main infrared selection rule is

$\mathrm{\Delta }v=\pm 1.$

Real molecular potentials are anharmonic, so weak overtone and combination bands may also appear. A vibration is infrared-active when it changes the molecular dipole moment. For this reason, homonuclear diatomic molecules such as ${\mathrm{N}}_{\mathrm{2}}$ and ${\mathrm{O}}_{\mathrm{2}}$ do not show ordinary pure vibrational infrared absorption, while heteronuclear molecules such as CO and HCl do.

Rovibrational spectra

In gas-phase infrared spectra, a vibrational transition is often accompanied by rotational transitions. This produces rovibrational bands rather than a single isolated line. The rotational selection rule usually gives two main branches:

• the P branch, with $\mathrm{\Delta }J=-1$;
• the R branch, with $\mathrm{\Delta }J=+1$.

In many simple diatomic spectra the direct $\mathrm{\Delta }J=0$ line is absent, leaving a gap between the P and R branches. Rovibrational spectra can be used to determine bond lengths, rotational constants, vibrational frequencies, and anharmonic corrections.

Electronic spectra

Electronic spectra involve transitions between molecular electronic states. In a molecular-orbital picture, electronic states are described by molecular orbitals extending over the molecule rather than orbitals localized on a single atom. Electronic transitions often occur together with changes in vibrational and rotational state, producing vibronic bands.

The intensities of electronic-vibrational transitions are often described using the Franck–Condon principle. Because electrons move much faster than nuclei, an electronic transition occurs so rapidly that the nuclear geometry is approximately fixed during the transition. The strongest transitions therefore occur between vibrational states whose wavefunctions overlap strongly at the same nuclear geometry.

Selection rules and transition moments

A transition is observed only if the interaction between the molecule and the radiation gives a nonzero transition probability. In electric-dipole spectroscopy this is expressed through the transition dipole moment,

$⟨{\psi }_f|\hat{\mu }|{\psi }_i⟩,$

where ${\psi }_i$ and ${\psi }_f$ are the initial and final molecular states and $\hat{\mu }$ is the dipole-moment operator. If this matrix element vanishes by symmetry, the transition is forbidden in the electric-dipole approximation.

Selection rules are therefore not arbitrary rules but consequences of molecular symmetry, angular-momentum conservation, and the form of the radiation-matter interaction. Forbidden transitions may still appear weakly when approximations break down, for example through spin-orbit coupling, vibronic coupling, collisions, or external fields.

Spectral lines and molecular information

The position, spacing, and intensity of molecular spectral lines encode physical information. Rotational line spacings can give moments of inertia and bond lengths. Vibrational frequencies reveal bond strengths and force constants. Electronic bands reveal molecular electronic structure, dissociation energies, and excited-state geometry. Isotopic substitution changes reduced masses and therefore shifts rotational and vibrational levels, providing a sensitive test of molecular models.

Spectral lines also have finite widths. Line shapes may be affected by natural lifetime broadening, Doppler broadening, pressure broadening, instrumental resolution, and interactions with surrounding matter. These effects are important in laboratory spectroscopy, atmospheric physics, astrophysics, plasma diagnostics, and chemical analysis.

Applications

Molecular spectroscopy is used to identify molecules, measure molecular constants, determine chemical composition, and probe physical conditions. Important applications include infrared absorption spectroscopy, Raman spectroscopy, microwave spectroscopy, atmospheric remote sensing, combustion diagnostics, astrochemistry, plasma diagnostics, and the study of molecular structure and reaction dynamics.

In quantum physics, molecular spectroscopy also provides direct evidence for quantized molecular motion. It connects the abstract operator language of quantum mechanics to measurable frequencies, intensities, and line shapes.

Table of contents (48 articles)

Index

Composite matter

1. Materials
2. Matter
3. Plasma and fusion physics
4. Molecules

Sub-molecular

5. Atoms
6. Particles
7. Fields
8. Vacuum and spacetime

Full contents

1. Materials (6)↑

1. Physics:Quantum materials/band structure
2. Physics:Quantum materials/phonon
3. Physics:Quantum materials/superconductor
4. Physics:Quantum materials/topological phase
5. Physics:Quantum materials/crystal lattice
6. Physics:Quantum materials/solid state

2. Matter (6)↑

7. Physics:Quantum matter/phase
8. Physics:Quantum matter/state of matter
9. Physics:Quantum matter/thermodynamic system
10. Physics:Quantum matter/plasma
11. Physics:Quantum matter/density
12. Physics:Quantum matter/temperature

3. Plasma and fusion physics (6)↑

13. Physics:Quantum Tokamak
14. Physics:Quantum Fusion
15. Physics:Quantum matter/plasma
16. Physics:Quantum magnetic confinement
17. Physics:Quantum Lawson criterion
18. Physics:Quantum Quark–gluon plasma

4. Molecules (6)↑

19. Physics:Quantum molecular structure
20. Physics:Quantum chemical bond
21. Physics:Quantum molecular orbital
22. Physics:Quantum degenerate orbitals
23. Physics:Quantum molecular spectroscopy
24. Physics:Quantum molecular vibration

5. Atoms (7)↑

25. Physics:Quantum atoms/electron
26. Physics:Quantum atoms/energy level
27. Physics:Quantum atoms/electron configuration
28. Physics:Quantum atoms/transition
29. Physics:Quantum atoms/ion
30. Physics:Quantum atoms/orbital
31. Physics:Quantum atoms/hydrogen

6. Particles (6)↑

32. Physics:Quantum particle
33. Physics:Quantum elementary particle
34. Physics:Quantum quark
35. Physics:Quantum hadron
36. Physics:Quantum boson
37. Physics:Quantum fermion

7. Fields (6)↑

38. Physics:Quantum field theory
39. Physics:Quantum electromagnetic field
40. Physics:Quantum gauge field
41. Physics:Quantum scalar field
42. Physics:Quantum vacuum field
43. Physics:Quantum wavefunction field

8. Vacuum and spacetime (6)↑

44. Physics:Quantum spacetime
45. Physics:Quantum vacuum energy
46. Physics:Quantum virtual particle
47. Physics:Quantum zero-point energy
48. Physics:Quantum curved spacetime
49. Physics:Quantum quantum gravity
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References

Author: Harold Foppele

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