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{{Short description|Study of molecular energy levels using electromagnetic spectra}}
{{Quantum matter backlink|Molecules}}

'''Quantum molecular spectroscopy''' is the study of how [[Physics:Quantum molecular structure|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.<ref name="Bernath">{{cite book |last=Bernath |first=Peter F. |title=Spectra of Atoms and Molecules |edition=3rd |publisher=Oxford University Press |year=2016 |isbn=978-0-19-938257-6}}</ref><ref name="Herzberg">{{cite book |last=Herzberg |first=Gerhard |title=Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules |publisher=Van Nostrand Reinhold |year=1950 |isbn=978-0-442-03385-8}}</ref>

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">
[[File:Hydrogen spectrum.svg|400px]]
<div style="font-size:90%;">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.</div>
</div>

=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,<ref name="Bernath" /><ref name="Herzberg" />

:<math>E \approx E_\mathrm{electronic}+E_\mathrm{vibrational}+E_\mathrm{rotational}.</math>

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.<ref name="BritannicaMolecularSpectra">{{cite web |title=Spectroscopy - Theory of molecular spectra |website=Encyclopaedia Britannica |url=https://www.britannica.com/science/spectroscopy/Theory-of-molecular-spectra |access-date=6 May 2026}}</ref>

=In quantum mechanics=
In [[Physics:Quantum mechanics|quantum mechanics]], the discrete spectrum of an observable corresponds to the eigenvalues of the [[Physics:Quantum methods/operator|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.<ref name="LL">{{cite book |last1=Landau |first1=L. D. |last2=Lifshitz |first2=E. M. |title=Quantum Mechanics: Non-Relativistic Theory |series=Course of Theoretical Physics |volume=3 |publisher=Pergamon Press |year=1965 |url=https://archive.org/details/QuantumMechanics_104}}</ref> Similar spectral ideas apply to angular momentum, Hamiltonians, and other operators of quantum systems.<ref name="LL" />

The [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]] and the [[Physics:Quantum atoms/hydrogen|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.

<gallery mode=packed>
Hydrogen spectrum.svg|The discrete part of the emission spectrum of hydrogen.
Solar Spectrum.png|Spectrum of sunlight above the atmosphere and at sea level, showing absorption features with discrete and continuous parts.
Deuterium lamp 1.png|Spectrum of light emitted by a deuterium lamp, showing sharp peaks and a continuous background.
</gallery>

=Rotational spectra=
The simplest rotational model is the rigid rotor. For a diatomic molecule with reduced mass <math>\mu</math> and bond length <math>r</math>, the moment of inertia is

:<math>I=\mu r^2.</math>

The rotational energy levels are

:<math>E_J=\frac{h^2}{8\pi^2 I}J(J+1), \qquad J=0,1,2,\ldots</math>

where <math>J</math> is the rotational quantum number.<ref name="BritannicaMolecularSpectra" /> In spectroscopic notation the same levels are often written in terms of a rotational constant <math>B</math>,

:<math>F(J)=BJ(J+1).</math>

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

:<math>\Delta J=\pm 1.</math>

Rotational spectra are especially useful for determining molecular moments of inertia and therefore bond lengths and molecular geometry.<ref name="Bernath" />

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

:<math>E_v=\left(v+\frac{1}{2}\right)h\nu_0,\qquad v=0,1,2,\ldots</math>

where <math>v</math> is the vibrational quantum number and <math>\nu_0</math> is the natural vibrational frequency of the bond.<ref name="BritannicaMolecularSpectra" /> In the simplest harmonic model, the main infrared selection rule is

:<math>\Delta v=\pm 1.</math>

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 <math>\mathrm{N_2}</math> and <math>\mathrm{O_2}</math> do not show ordinary pure vibrational infrared absorption, while heteronuclear molecules such as CO and HCl do.<ref name="LibreSelection">{{cite web |title=Selection rules and transition moment integral |website=Chemistry LibreTexts |url=https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Spectroscopy/Fundamentals/Selection_rules_and_transition_moment_integral |access-date=6 May 2026}}</ref>

=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 <math>\Delta J=-1</math>;
* the R branch, with <math>\Delta J=+1</math>.

In many simple diatomic spectra the direct <math>\Delta J=0</math> 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.<ref name="LibreRovib">{{cite web |title=Rovibrational Spectroscopy |website=Chemistry LibreTexts |url=https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Spectroscopy/Rotational_Spectroscopy/Rovibrational_Spectroscopy |access-date=6 May 2026}}</ref>

=Electronic spectra=
Electronic spectra involve transitions between molecular electronic states. In a molecular-orbital picture, electronic states are described by [[Physics:Quantum atoms/orbital|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.<ref name="BritannicaMolecularSpectra" />

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.<ref name="Bernath" /><ref name="LibreSelection" />

=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,

:<math>\langle \psi_f|\hat{\mu}|\psi_i\rangle,</math>

where <math>\psi_i</math> and <math>\psi_f</math> are the initial and final molecular states and <math>\hat{\mu}</math> is the dipole-moment operator. If this matrix element vanishes by symmetry, the transition is forbidden in the electric-dipole approximation.<ref name="LibreSelection" />

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.<ref name="Bernath" />

=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.<ref name="Herzberg" /><ref name="Bernath" />

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.<ref name="Bernath" />

=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.<ref name="Bernath" /><ref name="BritannicaMolecularSpectra" />

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.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Matter}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum molecular spectroscopy|1}}
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Physics:Quantum molecular spectroscopy - Revision history

imported>WikiHarold: Replace raw Quantum Collection backlink with BM backlink template

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