Simulating Infrared Spectra of Matrix Isolation Experiments: What Works Best?

Matthew H V Graneri; Dino Spagnoli; Duncan A Wild; Allan J McKinley

doi:10.1002/jcc.70518

. 2026 Oct 3;47(26):e70518. doi: 10.1002/jcc.70518

Simulating Infrared Spectra of Matrix Isolation Experiments: What Works Best?

Matthew H V Graneri ^1,^2,^✉, Dino Spagnoli ¹, Duncan A Wild ^1,³, Allan J McKinley ¹

PMCID: PMC13633506 PMID: 42827410

ABSTRACT

Matrix isolation is an experimental technique which allows spectroscopists to observe highly reactive or weakly bound systems by trapping them in frozen noble gas matrices. In this work, we studied the viability of using ab initio molecular dynamics (AIMD) to simulate several H₂S·amine complexes isolated in solid argon crystals, comparing the results with conventional density functional theory (DFT) harmonic and anharmonic calculations to assess the most practical and accurate method to calculate the vibrational spectrum of such systems. While AIMD accurately predicted the intensity ratios of certain peaks, the frequencies themselves were no more accurate than conventional harmonic DFT calculations. Modeling an explicit argon matrix did not improve the results. Based on the computational cost of each method, a gas phase anharmonic or scaled harmonic frequency calculation at an appropriate level of theory was determined to be the most accurate and efficient way to simulate matrix isolation spectra.

Keywords: ab initio molecular dynamics, anharmonicity, matrix isolation, vibrational spectra

This study investigated how to best reproduce infrared spectra of matrix isolated species accurately and efficiently. Anharmonicity was found to be far more influential than the argon matrix itself. We also found that AIMD frequencies were no more accurate than conventional harmonic values at an equivalent level of theory.

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1. Introduction

Matrix isolation (MI) is a technique where analyte species (guests) are trapped in a solid, inert matrix (host) for observation [1]. Gas mixtures of guest and host species, with a significant excess of the host (typically with a ratio of more than 1:100), are sprayed onto a cold surface where they rapidly freeze. The guest species then become trapped in an inert cage. Several layers of inert host molecules between each guest molecule ensure that no interactions between the guest molecules can take place. This technique allows molecules such as highly reactive chemical intermediates or weakly bound molecular complexes to be observed spectroscopically for periods far beyond their natural lifetimes [2]. It is common practice to use a host species which provides as little interaction with the guest as possible; neon, argon, krypton, nitrogen (N₂), and para‐hydrogen (p‐H₂) are often chosen, due to their rigidity, transparency, and low polarizability.

Trapping molecules in inert matrices introduces several “matrix effects” which are reflected in their spectra [3]. One of these matrix effects is a slight shift of the transition frequencies compared to the gas phase where the magnitude depends on properties of the host and guest species [4]. While these shifts are often small (some being on the order of a few wavenumbers) [5, 6], the shift is not always consistent, and different vibrations of a given molecule may be shifted by different amounts. This can make assignment of some vibrations quite challenging.

Many different studies have been conducted attempting to characterize the effects of matrices on guest molecules computationally. Molecular dynamics (MD) simulations have been used to study the dynamics of HCl trapped in argon matrices, showing significant translation and rotation within the matrix cavity [7]. MD was also used to simulate matrix depositions to investigate the effects of window temperature and cooling on site structure [8, 9]. Several ab initio studies were reported by Jordan, Del Bene, and colleagues investigating the influence of the matrix environment on the proton transfer reaction between ammonia and hydrogen halide molecules [10, 11, 12, 13]. In more recent years, density functional theory (DFT) calculations have been used to explicitly model molecules trapped in matrices to investigate site structures, vibrational frequencies, and the effect of a matrix on guest molecules themselves [14, 15, 16]. Ito performed calculations to explicitly model the formic acid dimer embedded in up to 26 argon atoms using B971/6‐31++G(3df, 3dp) level of theory with both conventional DFT and QM/MM simulations [17, 18]. Perhaps most notably, Müller et al. were able to show that heavy atom quantum tunneling occurred in a neon matrix by simulating the [F₅]⁻ anion complex surrounded by neon atoms [19].

Both MD and DFT calculations have limitations for simulating matrix effects. MD simulations are based on classical mechanics and force fields and do not generally simulate effects such as instantaneous polarizability [20]. DFT calculations, on the other hand, are computed on a static system at 0 K. In true matrix experiments, guest species can cause the host molecules in their proximity to become slightly polarized and are performed at temperatures where both the guest and host molecules undergo limited movement. A combination of DFT and MD might therefore be needed to simulate matrix effects accurately.

Ab initio molecular dynamics (AIMD) is a technique whereby DFT and MD are combined to model large‐scale dynamic systems over time. The method simulates finite temperature systems where the forces between molecules are calculated “on the fly” by electronic structure calculations [20]. This ensures that the movement of particles is determined by the instantaneous electronic environment, while also allowing molecules the freedom to move over the course of a simulation.

One additional potential advantage of AIMD over conventional DFT is the possibility that some anharmonic effects might be captured over time [21]. To run explicit anharmonic DFT calculations on a system isolated in an inert gas matrix is highly impractical. The size of the system involved prohibits the use of techniques such as second order vibrational perturbation theory (VPT2) [22] or vibrational self‐consistent field (VSCF) theory [23], both of which are very computationally demanding even for medium‐sized molecules. While it is possible to explicitly calculate a potential energy surface of the vibrational modes of interest and fit it to an anharmonic potential function [12, 19], this is not an automated procedure.

The purpose of this study was to compare several different methods of simulating vibrational spectra of matrix isolated species, and to determine which method is the most accurate, practical, and efficient at replicating experimental values. The systems chosen for this study—complexes of H₂S with methylamine (MA), dimethylamine (DMA), and trimethylamine (TMA)—have been investigated in a previous experimental study [24], where evidence of anharmonicity was observed in the form of strong Fermi resonances in the H₂S subunit. The H₂S·amine systems, therefore, provide an opportunity to test both the utility of AIMD as a method of simulating argon matrix experiments as well as anharmonicity.

2. Methods

For each complex, three types of calculations were compared: (1) AIMD simulations of the H₂S·amine complexes embedded in an argon crystal; (2) static DFT calculations of the complexes in the same argon crystal and in the gas phase; and (3) anharmonic DFT calculations of the complexes on their own (i.e., no argon atoms present). All methods were calculated with the B3LYP‐D3 functional [25, 26] to ensure a fair comparison between each simulation. The initial argon crystal was generated in VESTA [27], based on the crystal data from Dobbs et al. [28] The corners of the initial cubic structure were pruned, and two approximately‐central argon atoms were removed from the structure, leaving 114 argon atoms in the final crystal. The H₂S·amine complexes were then placed within the central cavity. The coordinates used to graph the computed spectra presented in this work were generated in ChimeraX using the SEQCROW package [29, 30].

2.1. AIMD

AIMD calculations were conducted in CP2K [31] using the Gaussian and plane wave Quickstep method [32]. The initial equilibration was done by running an AIMD simulation at BLYP‐D3 level of theory [26, 33], with aug‐DZVP‐GTH basis sets [32] and GTH‐BLYP pseudopotentials [34], over 20,000 frames, with a time step of 0.5 fs in a NVT (constant number of atoms, temperature and volume) ensemble with a massive Nosé–Hoover chain thermostat. Periodic boundary conditions were not enabled (i.e., the crystal was treated as discrete, not continuous). The temperature was set to 10 K to match the experimental conditions, and a time constant of 10 fs was used. This was followed by a further 20,000 steps at the same level of theory, but with a global thermostat, and the time constant set to 100 fs. Finally, a further 10,000 frames were calculated at B3LYP‐D3 level of theory [25] to relax the system for the production run. B3LYP‐D3 calculations were conducted using the auxiliary density matrix method (ADMM) approximation, with ADMM‐DZP auxiliary fitting basis sets [35]. The production run consisted of 40,000 frames with timesteps of 0.5 fs (i.e., a total of 20 ps was simulated). From this trajectory, the electron density was calculated at 2.5 fs intervals. This was then exported in BQB format [36] and processed in TRAVIS [37], where IR spectra were simulated [38, 39]. As TRAVIS does not produce a list of vibrational frequencies, the resulting spectra were analyzed by hand (i.e., peak positions and intensities were determined “manually”). Power spectra, normal vibrations and coordinates, and diffusion coefficients were also calculated using TRAVIS. Analyses of the matrix cavities throughout the simulations were conducted with the CageCavityCalc (C3) software [40]. The energy profiles of the final equilibration and the production run for each AIMD simulation can be found in Section S2.

2.2. Static Harmonic

A starting structure for each complex isolated in argon was selected from a random step in the AIMD production run. Following on, geometry optimizations and analytical frequency calculations were run in ORCA 5.0 [41] using the B3LYP‐D3 functional with the RIJCOSX approximation (henceforth RI‐B3LYP‐D3) [42, 43]. Due to the size of the systems under examination, the only basis sets considered during this work were the def2‐SVPD sets (and the corresponding def2/J auxiliary basis sets) [44, 45]. The DEFGRID3 integration grid was used. The “verytightscf” and “verytightopt” keywords were used, and their convergence criteria were enforced. Frequency calculations of the entire systems revealed no imaginary frequencies. Where indicated, these frequencies were scaled by a factor of 0.9671, as prescribed by Kesharwani et al. [46].

2.3. Static Anharmonic

Anharmonic frequency calculations were conducted in Gaussian 16 [47] at B3LYP‐D3 level of theory using def2‐QZVPPD basis sets [44, 48]. Only the most stable conformers for each complex reported in Graneri et al. [24] were considered in this work. Geometries were optimized using the “vtight” keyword. Anharmonic frequencies were calculated using generalized second order vibrational perturbation theory (GVPT2) [49, 50], as implemented in Gaussian 16. The “superfine” integration grid was used for all DFT calculations. ¹ For completeness, VPT2 [22] frequencies were calculated at RI‐B3LYP‐D3/def2‐SVPD level of theory in ORCA 5.0 for each complex without inclusion of the argon atoms. The difference between GVPT2 and VPT2 is that GVPT2 identifies and treats resonances between vibrations explicitly by removing the offending force constants and reintroducing them variationally [49], whereas in VPT2, resonances are not treated at all, which can cause the resonant frequencies to become very inaccurate [52].

3. Results and Discussion

3.1. Structure

The structures of each complex, with and without argon, calculated at RI‐B3LYP‐D3/def2‐SVPD level of theory are presented in Figure 1. A thorough discussion of the structures of the H₂S·amine complexes can be found in Graneri et al. [24]. Briefly, calculations at multiple high levels of theory found that all three complexes form a S—H⋯N hydrogen bond, matching experiment. These results have been reproduced at RI‐B3LYP‐D3/def2‐SVPD level of theory, as shown in Figure 1a–c. Following an examination of Figure 1d–f, it is also clear that the S—H⋯N binding motif is preserved when the complex is embedded in the argon matrix. Indeed, embedding the complexes in argon resulted in very little structural change; as outlined in Table 1, the largest change in any significant bond length or angle amounts to only 3.6%. This suggests that there is very little interaction between the guest species and the host, which is to be expected when using a noble gas matrix. It is notable, however, that the largest changes are in the intermolecular parameters, indicating that the matrix has a greater effect on the overall complex than it does on its substituents, possibly demonstrating a stabilizing effect of the matrix on the complexation of the molecules.

Structures of H₂S complexes with MA, DMA, and TMA optimized at RI‐B3LYP‐D3/def2‐SVPD level of theory in the gas phase (a–c) and embedded in argon matrices (d–f).

TABLE 1.

Structural properties of H₂S·amine complexes calculated at RI‐B3LYP‐D3/def2‐SVPD level of theory. ^a

	Gas phase			Ar matrix			Effect of matrix (%)
	MA	DMA	TMA	MA	DMA	TMA	MA	DMA	TMA
S—H_b	1.377	1.383	1.389	1.381	1.384	1.387	+0.3	+0.1	−0.1
H_t—S	1.352	1.352	1.352	1.348	1.347	1.348	−0.3	−0.3	−0.3
∠H_b—S—H_t	93.4	93.4	93.4	93.5	92.8	93.0	+0.1	−0.6	−0.4
H_b⋯N	2.030	1.960	1.909	1.956	1.911	1.923	−3.6	−2.5	+0.7
∠S—H_b⋯N	168.0	167.8	175.9	173.7	163.3	171.4	+3.4	−2.7	−2.5

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Note: Bond lengths are in units of Å, and angles are in degrees.

^{^a}

H_b refers to the hydrogen atom in H₂S which interacts with the amine, while H_t refers to the noninteracting hydrogen atom.

H₂S·amine complexes in the argon matrix also retained their S—H⋯N structures throughout AIMD simulations. Overall, the complexes move very little in the matrix over the simulation period. In the H₂S·MA and H₂S·TMA complexes, the H₂S molecule appears to be undergoing a hindered rotation. A comparison between plots of the mean square displacement (MSD) for each molecule throughout each simulation (Figure 2) shows that the H₂S molecule in those complexes is far more mobile than in the H₂S·DMA complex (Figure 2a). This may be due to the matrix cavities being large enough to allow a greater degree of hindered motion of the H₂S. Figure 2b shows that the TMA subunit is moving within the matrix cage—even more so than the H₂S molecule. This could be due to the symmetry of the TMA molecule allowing it to rotate and librate in the cavity without moving a significant distance from its reference position.

Mean square displacement (MSD) plots of each molecule in the AIMD calculations, where (a) shows the MSD for the H₂S subunit and (b) shows the MSD of the amine subunit in each of the simulations.

To characterize the matrix cavity throughout the simulation, we performed an analysis of the cavity volume using the C3 program, the results of which are presented in Figure 3. The average cavity volumes and their standard deviations throughout the simulations are 23.46 ± 2.77, 43.88 ± 3.21, and 56.47 ± 5.02 Å³ for the MA, DMA, and TMA systems, respectively. These values match those of the static argon matrices calculated at the RI‐B3LYP‐D3 level of theory, with volumes of 24.38, 45.38, and 56.25 Å³, respectively. For each complex, the cavity sizes appear to be stable throughout the simulation; however, the size of the cavity in the H₂S·TMA simulation seems to be steadily shrinking over time. It is possible that the system had not reached equilibrium before the production run had begun. Regardless, the cavity size of the matrix in the H₂S·MA simulation is significantly smaller than those in the H₂S·DMA and H₂S·TMA simulations, the latter two being almost the same size toward the end of the simulation. This may indicate that the nature of each site is somewhat different. If the substitutional site of an argon matrix (i.e., a site where an argon atom has been removed from the crystal lattice to create a cavity) has a diameter of 3.755 Å [53], the volume of the cavity should be approximately 27.72 Å³. This being the case, it is likely that the H₂S·MA complex occupies a single substitutional site in the argon matrix and the H₂S·TMA complex occupies a double substitutional site. The H₂S·DMA complex may be settling into a more restricted space, where the argon atoms are constricting its movements. This would explain why MSD of H₂S is lower in the H₂S·DMA simulations than in the H₂S·TMA simulation.

Analysis of the matrix cavity sizes over the course of the AIMD simulations.

3.2. Spectra

Figure 4 presents the vibrational spectra of the H₂S·MA complex calculated by each theoretical method, as well as the experimental difference spectrum reported in Graneri et al. [24], each normalized with respect to the most intense peak. At first glance, the RI‐B3LYP‐D3/def2‐SVPD static harmonic calculation of the gas phase H₂S·MA complex shows remarkable accuracy. On closer inspection, however, the agreement is less compelling. A comparison between each of the different methods with experiment (Table 2) shows that the root mean square deviation (RMSD) of the static harmonic calculation from the experimental values is 81.7 cm⁻¹. VPT2 anharmonic corrections do little to improve the situation, giving an RMSD of 84.6 cm⁻¹, where a large negative frequency was calculated for ν₂ of MA (−4607.8 cm⁻¹, see Table S1). Several tests were run to identify the cause of the negative frequency. The results suggest that the size of the grid used for the RIJCOSX approximation in ORCA 5.0 is insufficient for the calculation of third and fourth derivatives of the energy (see the Supporting Information for details) [54]. As this deficiency only affects two of the intramolecular vibrations (see below), they are treated as outliers while the remaining frequencies are used in our assessments. Static calculations of the H₂S·MA complex within an argon crystal show worse agreement with experiment, with an RMSD of 109.1 cm⁻¹. In contrast, the GVPT2 calculations were very accurate, with an RMSD of only 33.7 cm⁻¹ relative to experiment. Scaling the harmonic RI‐B3LYP‐D3 calculations by a factor of 0.9671 reduced the RMSDs for the calculations with and without argon atoms to 70.6 and 39.6 cm⁻¹, respectively. While the former value is still quite high, the scaled gas phase RI‐B3LYP‐D3 values are surprisingly accurate, only marginally outperformed by the far more computationally expensive GVPT2 calculation.

Spectra of the H₂S·MA complex calculated using DFT and AIMD compared with experiment. Trace (a) was calculated at RI‐B3LYP‐D3/def2‐SVPD level of theory without argon atoms; (b) was calculated at RI‐B3LYP‐D3/def2‐SVPD level of theory with argon atoms; (c) is the spectrum calculated from the AIMD simulation; (d) was calculated at GVPT2/B3LYP‐D3/def2‐QZVPPD level of theory; and (e) is the experimental difference spectrum of a mixture of 1:1:500 H₂S:MA:Ar, reported in Reference [24], where positive peaks indicate species created upon co‐condensation of H₂S and MA, and negative peaks indicate species that have been depleted. All spectra have been normalized with respect to the most intense peak.

TABLE 2.

Root mean square deviations (RMSDs) of the calculated frequencies for each H₂S·amine complex relative to the experimental values [24].

	B3LYP‐D3 ^a	RI‐B3LYP‐D3/def2‐SVPD			AIMD
	GVPT2	Harmonic (scaled)	VPT2	Ar matrix (scaled)	TRAVIS	Power
H₂S·MA	33.7	81.7 (39.6)	84.6	109.1 (70.6)	97.8	98.0
H₂S·DMA	36.2	87.9 (45.9)	59.9 ^b	109.4 (67.6)	94.3	112.9
H₂S·TMA	41.1	85.8 (57.2)	63.7	108.6 (65.3)	97.5	93.1
Combined RMSD	38.2	85.6 (51.2)	67.7 ^c	108.9 (67.0)	96.7	100.4

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Note: All values are in wavenumber units. Values in parentheses were calculated from frequencies which were scaled by a factor of 0.9671 [46].

^{^a}

Calculated using the def2‐QZVPPD basis set.

^{^b}

Including ν₁₆ of DMA from the calculation increases the RMSD to 503.6 cm⁻¹.

^{^c}

Including ν₁₆ of DMA from the calculation increases the combined RMSD to 266.9 cm⁻¹.

Results from AIMD calculations on the H₂S·MA complex are also presented in Table 2. Due to the nature of the TRAVIS output, the vibrational transitions of the complex could not all be identified due to their low intensity or possible overlap with other peaks. Normal modes for each vibration are, however, printed following a normal coordinate analysis; therefore, they are complete. Unfortunately, the normal modes identified by TRAVIS did not resemble those calculated in either Gaussian 16 or ORCA, and therefore assignments were based on similarity to values calculated from static frequency calculations.

The AIMD calculations produced frequencies with RMSDs of 97.8 and 98.0 cm⁻¹ for the TRAVIS and power spectra, respectively. These are better than the static argon matrix results, but not as accurate as the gas phase calculations. It is interesting to note, however, that only the AIMD spectrum predicted the most intense peak to be that of ν₉ of the MA subunit (around 820 cm⁻¹), as opposed to the stretching vibration of H₂S (around 2350 cm⁻¹). Finally, it is worth noting that only the GVPT2 calculation produced a signal for the 2ν₂ bending vibration of H₂S, which undergoes a Fermi resonance with the symmetric stretching vibration (ν₁), increasing its intensity. While overtones are not produced from harmonic frequency calculations, we were surprised that the AIMD simulation did not recover the effect. We had expected to observe the overtone due to the resonance effect having a classical mechanics equivalent to Fermi resonance [55]. As such, it is possible that the overtone was not simulated by the AIMD simulation due to an exclusively quantum effect, such as facilitation of the resonance by tunneling or zero point vibration. Tunneling can be ruled out, as the experimental spectra showed no evidence of any isotope effect when exchanging H₂S for D₂S [24]. Zero point vibration could be a factor in the appearance of the overtone in the GVPT2 spectra (and not in the AIMD spectra), but there is a more compelling explanation for this which is outlined below.

The results for the H₂S·DMA complex are less clear. The spectra calculated for this system are presented in Figure 5. The RMSDs of all methods are high, apart from the GVPT2 calculation, which achieved an RMSD of 36.2 cm⁻¹ (Table 2). The RMSD of the VPT2 calculation is particularly high, with a value of 503.6 cm⁻¹. An examination of the data shows that the VPT2 method contained an outlier, where the value calculated for ν₁₆ of DMA is 1386.7 cm⁻¹, whereas the experimental value is 2802.3 cm⁻¹. This also appears to be a result of the RIJCOSX grid being too small (see Supporting Information for details). Removing this outlier reduces the RMSD to only 59.9 cm⁻¹.

Spectra of the H₂S·DMA complex calculated using DFT and AIMD compared with experiment. Traces (a–e) are the DMA equivalents of those presented in Figure 4. Asterisks (*) indicate signals from TMA impurities in the gas mixture.

All three calculations which included explicit argon atoms (RI‐B3LYP‐D3, TRAVIS, and power spectrum) performed poorly, with RMSDs of around 100 cm⁻¹. The TRAVIS frequencies are slightly more accurate than the RI‐B3LYP‐D3 and power spectrum results, by 15.1 and 18.6 cm⁻¹, respectively. Scaling the static harmonic frequencies again greatly improved the accuracy of the RI‐B3LYP‐D3 calculations, reducing the RMSDs from 109.4 and 87.9 cm⁻¹ to 67.6 and 45.9 cm⁻¹ with and without argon atoms, respectively. The AIMD simulations did not reproduce the experimental intensity distributions, and instead predicted intensities more in line with the other theoretical models. As with the H₂S·MA calculations, only GVPT2 produced a frequency for the 2ν₂ vibration of H₂S.

Simulations of H₂S·TMA spectra reproduce the experimental results slightly more accurately. The GVPT2 calculations show a surprising level of accuracy, with an RMSD of only 41.1 cm⁻¹, closely followed by the VPT2 calculations with an RMSD of 63.7 cm⁻¹ (Table 2). As for the previous two complexes, the static argon matrix frequencies have RMSDs just below 110 cm⁻¹. The AIMD simulations performed poorly, with RMSDs of 97.5 and 93.1 cm⁻¹ for the TRAVIS and power spectra, respectively. Unfortunately, as can be seen in the spectra (Figure 6), it is difficult to determine the precise location of the ν₁ stretching vibration of the H₂S subunit in the TRAVIS spectrum, so no exact position could be determined. The intensities are, however, reasonably accurate.

Spectra of the H₂S·TMA complex calculated using DFT and AIMD compared with experiment. Traces (a–e) are the TMA equivalents of those presented in Figure 4.

Overall, our results indicate that the inclusion of an explicit argon matrix does not improve the calculated frequencies. Cumulative RMSDs for each method across all three complexes (Table 2) suggest that the inclusion of an argon crystal decreases the accuracy of the calculated frequencies; the RMSDs of the frequencies from the RI‐B3LYP‐D3/def2‐SVPD calculations with and without argon atoms are 108.5 and 85.6 cm⁻¹, respectively. It was surprising that the frequencies from the AIMD simulations did not seem to be any more accurate than the static frequency calculations, with cumulative RMSDs of 96.7 and 100.4 cm⁻¹ for the TRAVIS and power spectra, respectively, despite there being no restrictions on the mixing of vibrations and the ability for all atoms to move freely within the simulation space.

What is clear from our results is that the strongest influence on the performance of the calculation is the inclusion of anharmonicity. The cumulative RMSDs for the GVPT2 and VPT2 calculations are 38.2 and 67.7 cm⁻¹, respectively. This is a significant increase in accuracy compared to the AIMD and harmonic frequency values, despite not considering the influence of the argon matrix. However, reasonable accuracy can also be achieved by simply scaling the harmonic frequencies: the cumulative RMSDs of the scaled RI‐B3LYP‐D3 frequencies for the systems with and without argon atoms are 67.0 and 51.2 cm⁻¹, respectively. Overall, our results suggest that whether one approximates it with perturbation theory or a frequency scaling approach, anharmonicity has a far greater impact on predicting frequencies accurately than the simulation of an argon matrix.

Our results also show that AIMD simulations of the argon matrix systems do not inherently recover anharmonic effects. Table 3 presents the RMSDs for the static frequency calculations relative to the AIMD (TRAVIS) frequencies. The values in Table 3 show that all three raw harmonic frequency calculations deviate from the AIMD results by between 18 and 37 cm⁻¹. In contrast, the lowest RMSD for a calculation that takes anharmonicity into account is the scaled RI‐B3LYP‐D3 calculation for H₂S·TMA, with a value of 50.7 cm⁻¹. The combined RMSDs for each method paint an even clearer picture, with the RMSDs for the harmonic calculations all falling between 28.3 and 30.3 cm⁻¹, whereas the most rigorous explicit treatment of anharmonicity (GVPT2) deviates from the AIMD values by 96.2 cm⁻¹, clearly showing that the AIMD results converge on the harmonic frequencies, not the anharmonic frequencies. This may be due to the temperature at which the AIMD calculation was run. The utility of AIMD to simulate anharmonic effects depends on its ability to sample the potential energy surface (PES) from the atomic displacements [56]. This, in turn, depends on the temperature: at low temperatures, atomic displacements are small and only a small portion of the PES is sampled; at high temperatures, atomic displacements are larger and a greater portion of the PES is sampled. This could explain why the overtone peak is missing. It is only at temperatures of over 1000 K where the displacements begin to sample the PES to an extent where anharmonic effects may be recovered [56]. Unfortunately, such high simulation temperatures rule out AIMD as a useful method for simulating MI experiments.

TABLE 3.

RMSDs of the frequencies calculated from static calculations relative to AIMD for the H₂S·amine complexes.

	B3LYP‐D3/def2‐QZVPPD		RI‐B3LYP‐D3/def2‐SVPD (gas phase)			RI‐B3LYP‐D3/def2‐SVPD (Ar Matrix)
	Harmonic	GVPT2	Harmonic	Scaled	VPT2	Harmonic	Scaled
H₂S·MA	34.3	94.1	24.0	78.1	92.6	19.2	59.7
H₂S·DMA	23.7	98.4	18.7	71.2	84.7	21.1	55.7
H₂S·TMA	32.6	95.5	36.9	72.1	92.0	36.9	50.7
Combined RMSD	30.3	96.2	28.6	73.3	89.7	28.3	54.8

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Note: All values are in wavenumber units.

There is one other aspect of the calculations which deserves some attention. As indicated earlier, only the AIMD simulations were able to replicate the experimental intensity ratios for the H₂S·MA and H₂S·TMA spectra, showing that the amine bending vibration is more intense than the H₂S stretching vibration. We are not entirely certain about why the static DFT calculations do not reproduce this trend. It is unlikely to be the level of theory, as a CCSD(T)/aug‐cc‐pV(T+d)Z calculation conducted on the H₂S·MA complex produced a similar trend in the intensity ratio of approximately 3:1 in favor of the H₂S stretching vibration (see Table S5 and Reference [24] for details). It is also not a result of the double harmonic approximation, as the anharmonic spectra exhibit the same deficiency. It may be a matrix effect which the static DFT calculations are unable to reproduce, or an unexpected side‐effect of the argon atoms being allowed to move freely within the AIMD simulation space. As none of the other calculations captured this effect, it is worth investigating if there are other systems where AIMD succeeds where conventional DFT fails to replicate experimental intensities, but that is beyond the scope of this work.

3.3. Computational Cost

While the accuracy of a calculation is important, the practicality of running it determines if it should be attempted. As such, it is worth discussing the computational cost of each method to assess its value for predicting frequencies for matrix isolated molecules. The GVPT2 calculations cannot be assessed in this context as they were conducted in Gaussian 16. Each calculation was run with the amount of memory and processors deemed appropriate for the system under investigation, given the resources available. As such, it is important to stress that this assessment is not based on a minute‐to‐minute race, nor is it a comment on the performance of either ORCA or CP2K. We are merely analyzing the practicality of using each method based on the time scales involved.

The AIMD calculations were all assigned 25 cores and 120 GB of RAM. The RI‐B3LYP‐D3 argon matrix calculations were given between 16 and 20 processors and between 3 and 10 GB of RAM per processor. The RI‐B3LYP‐D3 calculations without argon were given between 8 and 12 processors and 1 GB RAM per processor (for VPT2 calculations, the number of processors was determined by the number of hessians which needed to be calculated).

The durations of each calculation are presented in Table 4. As expected, the calculations without argon atoms took significantly less time to complete than those with argon atoms. Even the VPT2 calculations were two orders of magnitude faster than the harmonic argon matrix calculations. It is, however, the AIMD calculations that took the longest to complete, such that the production runs took over a month to complete on their own (not including the equilibration runs or the generation of the electron density files). Given the fact that the AIMD frequencies were no more accurate than the harmonic, nonargon matrix frequency calculations, the time taken to complete such large calculations is excessive and, apparently, unnecessary.

TABLE 4.

Total durations of each calculation.

	RI‐B3LYP‐D3/def2‐SVPD			AIMD ^c
	Harmonic ^a	VPT2 ^b	Ar matrix ^a	AIMD ^c
H₂S·MA	0.15	0.63	238.50	960.97
H₂S·DMA	0.23	1.63	305.85	915.61
H₂S·TMA	0.35	3.65	519.97	886.83

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Note: All values in hours.

^{^a}

Timings for the optimizations and frequency calculations.

^{^b}

Timings refer to the VPT2 calculations only.

^{^c}

Timings refer to the production runs only.

4. Conclusion

This study set out to computationally replicate a MI experiment to determine which methods provide the most accurate frequency predictions in the most efficient way possible. We compared high level GVPT2, low level gas phase harmonic and VPT2 calculations with calculations in which our sample systems—H₂S·amine complexes—were isolated in an explicitly modeled argon matrix, both with conventional static DFT and the more dynamic AIMD methods.

Our results suggest that, at least computationally, the presence of the argon matrix has very little influence on the accuracy of the calculated frequencies; instead, appropriate consideration of anharmonicity appears to be far more important. In our simulations, AIMD did not seem to recover any anharmonic effects, to the extent that the AIMD frequencies were almost identical to the static harmonic frequencies. This is probably because the simulation temperature was too low for the atoms to have sampled a large enough region of the PES to recover any anharmonic effects. Unfortunately, the temperatures required to sample larger portions of the PES are far too high to simulate a MI experiment.

Disregarding the GVPT2 calculations, the most economical and accurate method to simulate a system isolated in a solid argon matrix appears to be a gas phase, scaled harmonic frequency calculation. These calculations, while not theoretically rigorous, will give reasonably accurate results, given an appropriate scaling factor. They will not, however, be able to recover true anharmonic effects, such as combination and overtone bands, or nonlinear changes in frequencies and intensities arising due to resonances. For that, it is necessary to use GVPT2 or another explicit anharmonic treatment.

Funding

This work was supported by the Department of Education, Australian Government.

Supporting information

Data S1: jcc70518‐sup‐0001‐TableS1‐S11‐FigureS1‐S6.docx.

Section S1: VPT2 analysis in ORCA 5.0.

Table S1: Results from VPT2 calculations on the H₂S·MA complex, all calculated at B3LYP‐D3/def2‐SVPD level of theory in ORCA 5.0. All values in units of wavenumbers. The ν₂ frequencies of MA are indicated in bold.

Section S2: Energy profiles of the AIMD simulations.

Figure S1: Kinetic energy of the H₂S·MA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Kinetic energy is presented in Kelvin.

Figure S2: Potential energy of the H₂S·MA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Potential energy is presented in a.u.

Figure S3: Kinetic energy of the H₂S·DMA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Kinetic energy is presented in Kelvin.

Figure S4: Potential energy of the H₂S·DMA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Potential energy is presented in a.u.

Figure S5: Kinetic energy of the H₂S·TMA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Kinetic energy is presented in Kelvin.

Figure S6: Potential energy of the H₂S·TMA argon matrix system over the course of the AIMD simulation. The blue segment of the trace corresponds to the final equilibration, while the orange portion corresponds to the production run. Potential energy is presented in a.u.

Section S3: Tables of vibrational frequencies calculated for each complex.

Table S2: Vibrational transitions of the H₂S·MA complex calculated at various levels of theory. All values are in wavenumber units.

Table S3: Vibrational transitions of the H₂S·DMA complex calculated at various levels of theory. All values are in wavenumber units.

Table S4: Vibrational transitions of the H₂S·TMA complex calculated at various levels of theory. All values are in wavenumber units.

Table S5: Frequencies calculated for the H₂S·MA complex at CCSD(T)/aug‐cc‐pV(T+d)Z level of theory, as reported in Graneri et al. [24]. Frequencies are listed in cm⁻¹ and intensities are listed in km mol⁻¹.

Section S4: Tables of coordinates for optimized structures.

Table S6: Structure of the H₂S·MA complex optimized at B3LYP‐D3/def2‐SVPD (RIJCOSX, DefGrid3) level of theory in Cartesian coordinates. All values in units of Å.

Table S7: Structure of the H₂S·MA complex optimized at B3LYP‐D3/def2‐QZVPPD level of theory in Cartesian coordinates. All values in Å.

Table S8: Structure of the H₂S·DMA complex optimized at B3LYP‐D3/def2‐SVPD (RIJCOSX, DefGrid3) level of theory in Cartesian coordinates. All values in Å.

Table S9: Structure of the H₂S·DMA complex optimized at B3LYP‐D3/def2‐QZVPPD level of theory in Cartesian coordinates. All values in Å.

Table S10: Structure of the H₂S·TMA complex optimized at B3LYP‐D3/def2‐SVPD (RIJCOSX, DefGrid3) level of theory in cartesian coordinates. All values in Å.

Table S11: Structure of the H₂S·TMA complex optimized at B3LYP‐D3/def2‐QZVPPD level of theory in Cartesian coordinates. All values in Å.

JCC-47-0-s001.docx^{(1,000.1KB, docx)}

Acknowledgments

The authors acknowledge the use of Kaya, the High‐Performance Computing facility at The University of Western Australia, and the technical assistance provided by the UWA IT–HPC team (https://doi.org/10.26182/dh5s‐v482). The authors would like to thank the HPC Service of FUB‐IT, Freie Universität Berlin, for computing time and user support (https://doi.org/10.17169/refubium‐26754). We would like to thank Prof Frank Nesse and Prof Juerg Hutter for their consent to use calculation‐timing data from ORCA and CP2K in this work. M.H.V.G. would like to acknowledge the Australian Government for the provision of a research training program (RTP) scholarship. Open access publishing facilitated by The University of Western Australia, as part of the Wiley ‐ The University of Western Australia agreement via the Council of Australasian University Librarians.

Endnotes

^¹

These calculations have been rerun in this work and differ from those reported in Graneri et al. [24] as the authors have since been made aware of a bug in Gaussian 16 which affects the calculation of anharmonic frequencies if run as part of the same job as the optimization. For details, see appendix C in Nejad [51].

Data Availability Statement

Input files required to reproduce this work, together with output files, and the full trajectories for the AIMD production runs reported in this work can be found in the Zenodo repository at DOI: https://doi.org/10.5281/zenodo.20624236.

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