Spectroscopic techniques can probe molecular systems non-invasively and investigate their structure, properties and dynamics in different environments and physico-chemical conditions. Different spectroscopic techniques (spanning different ranges of the electromagnetic field) and their combination can lead to a more comprehensive picture of investigated systems. However, the growing sophistication of these experimental techniques makes it increasingly complex to interpret spectroscopic results without the help of computational chemistry. Computational molecular spectroscopy, born as a branch of quantum chemistry to provide predictions of spectroscopic properties and features, emerged as an independent and highly specialized field but has progressively evolved to become a general tool also employed by experimentally oriented researchers. In this Primer, we focus on the computational characterization of medium-sized molecular systems by means of different spectroscopic techniques. We first provide essential information about the characteristics, accuracy and limitations of the available computational approaches, and select examples to illustrate common trends and outcomes of general validity that can be used for modelling spectroscopic phenomena. We emphasize the need for estimating error bars and limitations, coupling accuracy with interpretability, touch upon data deposition and reproducibility issues, and discuss the results in terms of widely recognized chemical concepts.
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In Italy, the work described in this Primer was supported by MIUR PRIN funds (grants 2015F59J3R, 2017A4XRCA), the Italian Space Agency (ASI) (‘Life in Space’ project, N. 2019-3-U.0) and SMART@SNS Laboratory for high-performance computing facilities). F.N. is extremely grateful for generous financial support by the Max Planck Society that enables us to follow curiosity-driven research. The science described in this Primer has also been supported by the German Science Foundation through the cluster of excellence programme Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) im Rahmen der Exzellenzstrategie des Bundes und der Länder — EXC 2033 — Projektnummer 390677874. A.B.M. and R.J.D. thank National Science Foundation (NSF) CHE-1856125; R.J.D. was supported by a fellowship from The Molecular Sciences Software Institute under NSF grant OAC-1547580.
The authors declare no competing interests.
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- Rotational spectroscopy
Spectroscopy using the microwave region of the electromagnetic field to study the excitation of the rotational states of molecules.
- Electron spin resonance
A spectroscopic technique equivalent to NMR but dealing with excitation of the electronic spins in open-shell systems.
- Quantum chemistry
The application of quantum mechanics to chemistry.
Isomers that can be converted into another by rotation about a formally single bond.
- Mössbauer isomer shifts
Shifts in resonance frequency of the nuclear gamma-ray transition in a Mössbauer active isotope (for example, 57Fe) caused by its interaction with the molecular environment.
- Infrared spectroscopy
Spectroscopy using the infrared region of the electromagnetic field to study the excitation of the vibrational states of molecules.
- Raman spectroscopy
Rotational or vibrational spectroscopy that exploits the Raman effect (inelastic scattering).
- Schrödinger equation
Equation associated with the Hamiltonian operator: its resolution provides the allowed energy levels (eigenvalues) and the corresponding wave functions (eigenfunctions).
- Diffusion Monte Carlo
A Monte Carlo-based approach for obtaining the exact ground-state solution to Eq. 2.
- Absolute configuration
The spatial arrangement of atoms in a chiral system and its stereochemical description.
- Energy levels
According to quantum mechanics (see above), the allowed energy for a system bound is not continuous but discretized in energy levels.
In quantum mechanics, the operator corresponding to the energy of a system.
- Position arrays
Arrays containing the coordinates of the position of a specific point in a multidimensional space.
- Wave function
A mathematical description of the quantum state of an isolated quantum system resulting from solving the corresponding Schrödinger equation.
- Born–Oppenheimer approximation
The assumption that the motion of atomic nuclei and electrons can be treated separately, based on the much larger mass of nuclei.
- Potential energy surface
A multi-dimensional, hyper-surface that describes the variations of the electronic energy of a system in terms of suitable nuclear coordinates.
- Normal coordinates
Linear combinations of mass-weighted displacement coordinates (usually Cartesian). The motion described by a normal coordinate is called a normal mode.
- Spectroscopic transitions
The passage between two energy levels, that is, from an initial to a final state, detected by a spectroscopic technique.
- Ro-vibrational spectroscopy
Spectroscopy dealing with rotational and vibrational states of molecules.
- Contact transformation
Unitary transformation with an exponential operator U = exp(iS), where S is Hermitean and antisymmetrical with respect to time reversal, thus ensuring that U is unitary and invariant to time reversal.
Deviation from the harmonic-oscillator behaviour.
- Hyperfine structure
Interactions of the molecular electric and/or magnetic fields with the nuclear or electron (for open-shell species) moments produce a splitting of the rotational energy levels, which in turn leads to a splitting of the rotational transitions. This splitting is called the hyperfine structure.
- Coriolis couplings
Interactions between the angular momentum of a vibrational mode and the rotational angular momentum.
- Rigid-rotor harmonic-oscillator model
A reference model in which a molecular system as a whole is described in terms of a rigid rotating object and in terms of decoupled harmonic oscillators for its vibrational motion.
- Fundamental bands
Vibrational transitions from the vibrational ground state to the first excited state of a given vibrational mode (a one-quanta transition).
- Vibrational circular dichroism
Vibrational version of circular dichroism.
- Vibrational perturbation theory to the second order
Exploitation of perturbation theory to the second order to treat vibrational motions.
- Normal modes
Vibrational motion of molecules where all atoms vibrate in phase with the same frequency but with different amplitudes, and the centre of mass remains fixed.
Vibrational transitions involving the excitation of two or more quanta of a given vibration mode (that is, the quantum number describing the vibrational energy levels change varies by two or more).
- Combination bands
Observed when two or more vibrations are excited simultaneously.
- Double-perturbative approach
Simultaneous perturbative treatment of the energy and one property (for example, the electric dipole moment in infrared spectroscopy) around a stationary point.
- Property surface
(Multidimensional). The variations of a property as a function of suitable nuclear coordinates.
- Large amplitude motions
Molecular vibrations whose amplitude is so large that the harmonic oscillator model is no longer a reliable zero-order approximation.
- Density functional theory
A quantum-mechanical method in which the properties of a many-electron system are determined using functionals (that is, functions of another function) of the spatially dependent electron density and, possibly, its derivatives.
- Vibrational self-consistent field
Exploitation of the self-consistent model to treat vibrational motion.
- Vibrational configuration interaction
Exploitation of the configuration interaction model to treat vibrational motions.
- Vibronic spectroscopy
Spectroscopy involving the simultaneous excitations of vibrational and electronic states of molecules.
- One-photon absorption
A spectroscopic technique in which one-photon absorption leads from the electronic ground state to an excited electronic state.
- One-photon emission
A spectroscopic technique in which one-photon emission leads from an excited electronic state to a less-excited (lower energy, usually the ground) state.
- Electronic circular dichroism
Electronic version of circular dichroism.
- Optical rotation
The rotation angle of the polarization plane of polarized light issuing from its passage through a layer or a liquid, determined by the concentration of chiral molecules and their structure in a substance.
- Raman optical activity
Vibrational spectroscopy based on the differential Raman scattering of left and right circularly polarized light due to molecular chirality.
- Optical rotatory dispersion
The variation of the optical rotation of a substance with a change in the wavelength of light.
- Line-shape function
A mathematical function (usually Gaussian, Lorentzian or a combination of both) describing phenomenologically the shape of a spectral band.
Time rotated into the imaginary plane via Wick rotation in DMC, τ = it/ℏ.
- Coupled-cluster theory
A hierarchy of electron correlation methods that, by means of an exponential Ansatz, systematically converge to the exact solution of the molecular Schrödinger equation starting from the independent particle Hartree–Fock model.
A coupled-cluster method that considers full account of single and double excitations and a perturbative treatment of triple excitations.
- Electron correlation
The effects of electron–electron interactions beyond the mean field Hartree–Fock model.
- Domain-based pair natural orbitals
Electron pair-specific localized natural orbitals expanded in a set of local atomic orbitals belonging to pair-specific domains.
- Global-hybrid or double-hybrid density functionals
Families of density functionals including a percentage of Hartree–Fock exchange (hybrid) and MP2-type correlation (only double-hybrid).
- Møller–Plesset theory to the second order
Møller–Plesset theory including many-body effects on top of the mean field Hartree–Fock reference wave function up to the second order of perturbation theory.
- Equation of motion
In a quantum chemistry context, a methodology for treatment of electronically excited or ionized states.
- Density matrix renormalization group
A very efficient numerical variational technique devised to obtain the lowest-energy wave function of a given Hamiltonian expressed in terms of a matrix product state.
A specific generalization of Møller–Plesset theory to the second order to multiconfigurational reference wave functions.
A variant of second-order multi-reference perturbation theory similar to CASPT2.
Configuration interaction (that is, mixing of ground and excited electronic states) including only single excitations from a reference Slater determinant.
- Multi-reference configuration interaction
Extension of the configuration interaction approach to multi-reference wave functions.
- Tamm–Dancoff approximation
From a practical point of view, a synonym of CIS.
Isotopic species of a molecule.
- Harmonic approximation
A model in which the vibrational motion is described in terms of masses attached to a spring, whose energy is governed by a quadratic potential.
- Small amplitude motions
Molecular vibrations whose amplitude is small enough that the harmonic oscillator is a reliable zero-order approximation.
- Innocent solvents
Solvents that do not establish specific interactions, for example, hydrogen bonds with the solute.
- Polarizable continuum model
A description of bulk solvent effects in terms of a polarizable continuum in which the solute is fully embedded.
- Cybotactic region
The region around a solute molecule including solvent molecules belonging to the first solvation shell, that is, showing close solute–solvent contacts.
- 0–0 transition
The transition between the vibrational ground states of initial and final electronic states.
- Half-width at half-maximum
Half of the width between the two points where the value of the function is its half-maximum.
- Zero-point energy
The lowest energy that a quantum system may have, which, contrary to the classical case, is non-zero due to the Heisenberg uncertainty principle.
- Ensemble of walkers
A large number of virtual copies of a single particle moving randomly over a given potential energy surface.
- Nuclear quadrupole coupling
Interaction between the quadrupole moment of a nucleus and the electric-field gradient at this nucleus. Nuclei have a quadrupole moment when the nuclear spin is greater than 1/2. This interaction produces a hyperfine structure in the rotational spectra.
- Spin–rotation interaction
The interaction between the weak magnetic field generated by the end-over-end rotation of a molecule with the nuclear magnetic moment. The nuclear magnetic moment is present when the nuclear spin is non-null. This interaction produces a hyperfine structure in the rotational spectra.
- Molecular mechanics
Classical model to predict the energy of a molecule as a function of its conformation.
- Orbital splittings
Splittings of specific orbitals due to external factors (for example, electric or magnetic field).
- Crystal field theory
The splitting of the (relativistic) many-particle multiplet states of an ion in a dn or fn configuration incurred by the electrostatic interaction with its coordinating ligands that are treated as point charges.
An ensemble of many particle states that arise from the distribution of a given number of electrons among sets of degenerate atomic or molecular orbitals under the action of the electron–electron (and perhaps the spin–orbit coupling) interaction.
- Spin–orbit coupling
The coupling between the spin and the orbital angular momenta.
- Slater determinant
Representation of a many particle ‘mean-field’ wave function in terms of the antisymmetrized products of single-electron wave functions (molecular orbitals).
- Circular dichroism
Dichroism (splitting of a beam of light into two beams with different wavelengths) involving circularly polarized light, that is, the differential absorption of left and right-handed light.
- Magnetic circular dichroism
Circular dichroism induced by a static, longitudinal external magnetic field.
A magnetometer based on superconducting loops used to measure very low magnetic fields.
- Electron paramagnetic resonance
A synonym of electron spin resonance.
- Zero-field splitting
The lifting of the degeneracy of the 2S + 1 magnetic sublevels of a spin multiplet with total spin S in the absence of a magnetic field, caused by the effects of spin–orbit coupling and electron–electron spin–spin interactions.
- Ab initio ligand field theory
A method connecting the results of ab initio calculations with the parameters entering ligand field theory.
- Ligand field theory
A semi-empirical ‘perturbed ion’ model, based on crystal field theory, that describes the electronic structure and properties of transition metal complexes.
- Template approach
A model in which the structure of a molecular system is refined with reference to suitable fragments, whose structures are accurately known.
- Force constants
Derivatives of the potential energy with respect to nuclear coordinates evaluated at the minimum structure, for example, the quadratic force constant is the second derivative.
A fundamental theory of contemporary physics that provides a description of the properties of the matter at the atomic and subatomic level. The slash is used to denote that two levels of treatments are employed and implies a partitioning of the system (QM′, a different quantum mechanics level).
- Perturbed matrix method
A perturbative model in which the environmental effects on a quantum centre are described in terms of the CIS (configuration interaction singles) method, whose elements are the energies of the isolated solute perturbed by the electric field produced by the different configurations of the solvent issuing from a molecular dynamics simulation.
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Barone, V., Alessandrini, S., Biczysko, M. et al. Computational molecular spectroscopy. Nat Rev Methods Primers 1, 38 (2021). https://doi.org/10.1038/s43586-021-00034-1