Introduction

Spectroscopies based on molecular vibrations of studied samples can generally be divided to Raman spectroscopy and Infrared spectroscopy (IR). In IR spectroscopy, the studied system is excited with IR radiation and the absorbed radiation is measured. In Raman, the system is excited with laser and the shift of the emitted photons is measured. In both vibrational techniques the excited states are similar and in the same range of wavenumbers. Yet based on the selection rules and intensities of the vibrational modes, the two spectra can be very different. Vibrational modes are IR active due to a change of the dipole moment, and Raman active due to a change in polarizability. Interpreting solid-state IR and Raman spectra can often be a demanding task, especially for novel yet structurally complex compounds, and thus quantum chemical methods are often used to aid the interpretation of the spectra of especially new compounds and materials. ^[1]

The scope of this article is to briefly describe the main quantum chemical approaches that enable interpretation of solid-state vibrational spectra. Practical examples and useful interpretation tools are also included. In addition to vibrational spectra, for example nuclear magnetic resonance (NMR) spectroscopy can also be interpreted with the aid of quantum chemical methods, but will not discussed here.

Calculating vibrational spectra

Harmonic approximation

Based on the quantum mechanical harmonic oscillator approximation it is possible to study molecular vibrations and vibrational modes of IR spectra by treating them as the second derivative of the potential energy with respect to bond stretching at equilibrium geometry. However, in order to model the change in polarizability with respect to the vibrations a mixed third derivative has to be studied. This makes the intensity of Raman spectra even more sensitive to technical aspects in the calculations.

The use of the harmonic approximation gives rise to an error that has to be addressed. The most critical difference caused by this approximation is that real bonds dissociate as they are stretched, yielding a limited number of vibrational levels. The differences between the harmonic oscillator approximation and the true molecular system are the largest for the higher vibrational levels, even at shorter distances the potential enerfgy will be smaller than that predicted by the harmonic approximation. Thus, the harmonic frequencies will always be larger than true frequencies. In practice, this can be taken into account when comparing quantum chemically modeled spectra with experimental spectra by scaling the wavenumbers. ^[2]

ab initio methods

To understand how quantum chemical calculations can be used to model vibrational spectra it is good to also know the basics of the main methods in use, namely the Hartree-Fock (HF) theory and density functional theories (DFT). But first, let us define another term, ab initio. Both of these approaches are ab initio or first principles methods, meaning that the models do not take into account experimental parameters and neither can the results be obtained with them being fitted to experimental models or values. HF theory is principally the basis of all modern quantum chemical methods. Using the HF method the wave function of a system can be approximated by a single Slater determinant. However, HF only models electron exchange energy and doesn't take into account electron correlation, leading to inaccurate approximations. The computation of correlation is quite expensive with traditional methods and another approach, the density functional theory is needed.^[2] 

The theory behind DFT is based on the fact that the ground state of an electronic system depends only on its electron density. The resulting wave function is much easier to solve than the true wave function, albeit it has to be solved with an approximation. The simplest DFT approximation, local density approximation (LDA) depends only on the local density, but makes the approximation of an even electron spread in the molecule. The next DFT level, generalized gradient approximation (GGA) a gradient of the density is introduced to correct the changes in the electron densities.^[2]  

In many examples of modern solid-state quantum chemistry, a combination of these two, a hybrid functional is often used. The inclusion of the HF exact exchange improves the accuracy of many properties, including vibrational frequencies. Examples of a widely used ones are B3LYP^[3] and PBE0 ^[4].

Other practical considerations

When comparing the HF and DFT methods in the calculation of vibrational spectra it is good to discuss the errors. The HF errors are usually large as HF method overemphasizes bonding, resulting in larger force constants and thus frequencies. This can, however, be corrected by using a scaling factor. DFT rarely requires a scaling factor, but the errors might be random around the experimental values. Hybrid functionals require at least some scaling to take into account the effect of the HF character. 

Even though it would be possible to calculate the line broadening for a given function, this is rarely done as there are several complication especially related to ordering and surfaces. Structural disorder in bond lengths, angles etc. will all modify the widths of the frequencies more significantly than the theoretical broadening. Thus, often in computational studies the frequencies and intensities are simply listed for comparison to observed spectra, and the peaks are broadened with the use of a Gaussian or Lorentzian function maintaining the overall integrated intensity. A term often appearing related to the broadening is full width at half maximum (FWHM), which is an expression for the width of the peak between points on the curve that are half of the maximum amplitude.^[2]  [5]

Furthermore, it is good to keep in mind that in all computational chemistry studies, a certain balance needs to be found between the accuracy of the method, the size of the simulated model and the computational cost of the calculations. Higher-accuracy calculations are naturally more computationally demanding, and become increasingly so if the model is also large. The choice of the right model and right level of theory thus largely depends on the scientific question. One additional interesting aspect of computational study of vibrational modes is the fact that the computational study might result in an imaginary frequency corresponding to a decrease in energy. This indicates a structural instability in the computed model. 

Practical examples of calculated spectra

In this section examples of spectral assignments from peer-reviewed publications on novel inorganic compounds are introduced. These examples are based on hybrid density functional methods and the CRYSTAL^[6] software package. 

PbF[Br₂F₇]

To confirm the structure of the first synthesized fluoridobromate(III) of a p-block element, PbF[Br₂F₇], quantum chemical calculations with PBE0 hybrid density functional were carried out to model its vibrational spectra^[7]. The computed and experimental IR and Raman spectra are presented in Figures 1 and 2, respectively. 

Figure 1. Experimentally observed (black) and quantum chemically calculated (red) IR spectra of PbF[Br₂F₇]. (Figure from^[7], license: CC-BY 4.0)

Figure 2. Experimentally observed (black) and quantum chemically calculated (red) Raman spectra of PbF[Br₂F₇]. (Figure from^[7], license: CC-BY 4.0)

In this example, the wavenumbers have been scaled by a factor of 0.97 in order to take into account the harmonic overestimation. The IR spectrum was widened with a Lorentzian line shape with FWHM of 8 cm^-1 and the Raman spectrum with a pseudo-Voigt line shape with FWHM of 8 cm^-1. The peak assignment in this example was carried out with visual inspection of the normal modes with the Jmol program. 

BaOsF₆

In another study, barium hexafluoridoosmate(IV) was obtained. It was characterized with the help of quantum chemical calculations likewise using the PBE0 hybrid density functional^[8]. In this example the IR spectrum was broadened using a Lorenzian peak profle with FWHM of 8 cm^-1. A comparison of the observed and calculated spectra is presented in Figure 3.

Figure 3. Experimentally observed (black) and quantum chemically (red) calculated IR spectra of BaOsF₆ (Figure from^[8], license CC-BY 4.0)

Ba(BrF₄)₂

As structural distortions often lead to changes in the observed and theoretical spectra, it is sometimes useful to carry out quantum chemical calculations also for the gas-phase equivalents of building blocks of the larger solid-state crystal structures. In an example study on barium tetrafluoridobromate (III) ^[9]the the full assignment of the vibrational modes was enabled by calculations of the molecular vibration modes of [BrF₄]^- with the help of TURBOMOLE program package^[10]. In the structure the ideal symmetry of the anion is reduced from D_4h to S₄, causing splitting of the vibrational modes. and shifts of the ideal frequencies. An example of this assignment of the split modes is presented in Figure 4.

Figure 4. Assigning split vibrational modes of the [BrF₄]^- anion (Figure: Antti Karttunen).

Tools for carrying out the visual interpretation

In this section, a practical introduction to interpretation is presented using the calculated Raman spectra of strontium borate hydride Sr₅(BO₃)₃H as an example (Wylezich et al., Chem. Eur. J. 2020, 26, 11742-11750, https://doi.org/10.1002/chem.202002273). 

With Jmol software

Jmol^[11] is an open-source program that can be used for the viewing of a large variety of three-dimensional chemical structures, including solid-state ones. It also includes tools for the visualization and thus interpretation of vibrational spectral using quantum chemical calculations. In order to visualize the vibrational modes, first a file containing the quantum chemically calculated vibrational data has to be opened (for example, CRYSTAL, Orca, or Gaussian output file). 

The visualization tools for vibrational modes can be found under Tools / AtomSetChooser (Figure 5):

Figure 5. (Figure: Kim Eklund).

Individual modes can be found from the Frequencies tab. Then a mode can simply be chosen and played (Figure 6):

Figure 6. (Figure: Kim Eklund).

It can be useful to use the Amplitude and Period sliders to change the amplitude and the speed of the vibrations to make them easier to observe (Figure 7):

Figure 7. (Figure: Kim Eklund).

A twofold increase in the vibration animation can also be done easily with a right click on the main screen when the vibration mode is on (Figure 8):

Figure 8. (Figure: Kim Eklund).

Another helpful trick is to increase the size of the model by opening the console from FIle → Console and typing, for example,

load "" {2 2 2}

which could help in interpreting the vibrations 

With CRYSPLOT

There is also an online tool CRYSPLOT^[12] for the interpretation of the spectra, provided by the developers of the CRYSTAL program package that is often used for solid-state quantum chemical calculations. It can open CRYSTAL output directly in an interactive browser-based viewer. Note that the file containing the vibrational data has to be saved with .out (or .xyz) file extension. In addition to the visualization of the vibrations, this tool also includes the plotted spectra, a listing of the modes and the possibility to filter them (Figure 9). 

Figure 9. A screenshot from the CRYSPLOT vibrational assignment tool (Figure: Kim Eklund). 

References