Infrared laser spectroscopy of the n-propyl and i-propyl radicals: Stretch-bend Fermi coupling in the alkyl CH stretch region

Propyl is archetypical of the alkyl radicals prevalent in combustion systems. The barrierless association of propyl and O₂ produces an oxygen-centered C₃H₇OO^∙ peroxy radical, which can decompose to bimolecular products or undergo unimolecular rearrangements to produce carbon-centered radicals, such as the β- or γ-hydroperoxy propyl (i.e., QOOH) species.^1–5 Propyl peroxy radicals are the smallest alkyl species that can isomerize to QOOH via intramolecular H-atom transfer pathways having barriers that lie below the alkyl + O₂ reaction asymptote.^6,7 The propyl + O₂ reaction is therefore a prototype of low-temperature hydrocarbon oxidation chemistry, as it represents the prototypical alkyl system undergoing transformations characteristic of larger species.⁸ Indeed, many experimental and computational studies have been reported with the aim of characterizing the potential energy surface and detailed kinetics of the propyl +O₂ reaction.^9–14 On the other hand, detailed spectroscopic analyses of isolated propyl and propyl peroxy radicals remain incomplete.¹⁵ Here, we present a mid-infrared (IR) spectroscopic characterization of the n- and i-propyl radicals in the CH stretching region, which is complicated by the presence of multiple anharmonic resonance polyads.

Fessenden reported in the 1960s a series of seminal electron spin resonance (ESR) studies of alkyl radicals produced via photolysis of precursors dissolved in liquid cyclopropane solution at −140 °C.^16,17 Comparing the relative magnitudes of β-proton hyperfine coupling constants extracted from pentyl, butyl, and propyl ESR spectra, they arrived at the n-propyl equilibrium structure shown in Scheme 1, in which the singly-occupied p-orbital makes a $π/6$ angle with the β-CH bonds. Moreover, from the temperature dependence of the β-coupling constants, a barrier to hindered rotation between equivalent C_s structures was estimated to be ∼140 cm⁻¹. Using the stationary photolytic technique,^18,19 Krusic and co-workers assigned line broadening in their ESR spectra to the exchange of magnetically non-equivalent α hydrogen atoms,²⁰ an interpretation which is consistent with the n-propyl structure proposed by Fessenden. The equilibrium structure of n-propyl and the barrier to internal rotation were the focus of several early electronic structure computations, which gave qualitatively different results depending on the level of theory, often in conflict with conclusions arrived at via experimental ESR spectra.^21–24 Modern coupled-cluster theory was recently applied to the n-propyl system,²⁵ and a zero-Kelvin enthalpic barrier of 137 cm⁻¹ was found for hindered rotation. Moreover, the most probable structure at zero Kelvin was predicted to have C_s symmetry. Both findings are in remarkable agreement with the experimental results inferred from the 1960s ESR spectra of alkyl radicals produced in liquid cyclopropane solutions.

Other than the aforementioned ESR studies, Pacansky’s argon matrix isolation experiments represent the only other spectroscopic work carried out on propyl radicals.^26,27 In separate experiments, dibutyryl peroxide and diisobutyryl peroxide precursors were co-deposited with argon on a cold window and photolyzed to form the n- and i-propyl radicals (and CO₂), respectively. Infrared spectra were measured with a dispersive instrument, and a few bands could be definitively assigned to propyl radicals. For n-propyl, the α-$CH2$ wagging mode was observed at 530 cm⁻¹. Two other strong bands at 3100 and 3118 cm⁻¹ were assigned to symmetric and antisymmetric stretching vibrations of the α-$CH2$ group. These three features were later described as being the characteristic signatures of primary alkyl radicals.²⁸ The remaining spectrum was congested and more difficult to assign, especially in the CH stretching region. However, Pacansky did note an “anomalous band” located near 2812 cm⁻¹. This rather strong band in the CH stretch region was viewed as “anomalous” simply due to its characteristically large redshift from typical alkyl CH stretch vibrations. Pacansky assigned this band to the symmetric β-$CH2$ stretch and rationalized the red shift as being due to a hyperconjugative stabilization and softening of the β-CH bonds, i.e., an electron transfer from β-CH σ-bonding orbitals into the half-filled p-orbital localized on the α carbon atom.

Ultraviolet absorption spectra that probe the $B∼$-$X∼$ transition have been reported for gas-phase propyl peroxy radicals, and these are broad and featureless due to the dissociative nature of the excited state.^29,30 More recently, a near-IR study was reported by Miller and co-workers,¹⁵ in which cavity ringdown spectroscopy was used to probe the non-dissociative $A∼$-$X∼$ transition. This study was followed by a detailed theoretical analysis of the interconversion pathways connecting the various rotamers of the n- and i-propyl peroxy systems.³¹ 

In contrast to the peroxy species, spectroscopic studies of gas-phase propyl radicals are completely lacking. Here we report IR spectra in the CH stretching region for both the n- and i-propyl radicals, which have been produced in the gas-phase by flash-vacuum pyrolysis and isolated in helium nanodroplets. The spectra are sufficiently resolved so as to reveal a vibrational complexity that can be assigned to strong anharmonic coupling between the CH stretches and the overtones and combinations of CH_n bends (referred to here as scissor modes). Resonance polyads are characterized with effective Hamiltonian approaches that are constructed in either a normal or local mode representation. The origin of the extensive anharmonic coupling is revealed via comparisons of experimental spectra to local mode Hamiltonian predictions.^32–38 The weakly interacting nature of superfluid helium allows for a direct comparison between experimental band origins and computed spectra. Moreover, the helium droplet spectra reported here provide a robust starting point for future high resolution gas-phase spectroscopic studies.

The details of the helium droplet methodology have been discussed elsewhere.^39–42 Liquid He droplets consisting of 4500 He atoms on average are formed in a cryogenic nozzle expansion (35 bars, 17 K, $5μm$ orifice diameter).³⁹ The droplet expansion is collimated into a beam by a 0.4 mm conical skimmer prior to entering a differentially pumped chamber containing propyl radicals at a density of about 10¹⁰ cm⁻³ over an ∼1 cm path. Propyl radicals are generated in a high-temperature, effusive, flash-vacuum pyrolysis source consisting of a quartz tube wrapped with a Ta filament.^41,43 Precursor molecules colliding with the hot tube decompose, and the products of this decomposition (referred to hereafter as the pyrolysate) effuse through the droplet beam path. Collisions between droplets and gas-phase pyrolysate molecules result in solvation and cooling to ∼0.4 K via He atom evaporation.^44,45

The precursor molecule used for the production of n-propyl radicals is n-butyl nitrite (n-BN), which was synthesized by NaNO₂ addition to an aqueous solution of n-butanol and sulfuric acid.⁴⁶ The pyrolysis of R− CH₂ONO nitrite molecules has been shown to lead to the formation of NO, H₂CO, and R radicals (R = C₂H₅, C₃H₃, C₃H₅).^43,47,48 Thermal decomposition of n-BN produces the n-propyl radical, as shown in Eq. (1). Gas-phase i-propyl radicals were produced similarly via the thermal decomposition of i-butyl nitrite (i-BN),

Droplets are detected by electron ionization in a quadrupole mass spectrometer equipped with a crossed-beam ionizer. The mechanism for dopant ionization via He⁺ charge transfer has been described elsewhere.⁴⁵ Ionization and fragmentation of dopant molecules yields desolvated, gas-phase ions that provide mass spectrometer signatures of the upstream droplet doping process. Moreover, the cross-section for droplet ionization provides the action necessary for measuring IR spectra of He-solvated molecules and clusters.

The mid-IR idler output from a continuous-wave optical parametric oscillator (cw-OPO) is spatially overlapped with the droplet beam in a counter-propagating configuration. The tuning and calibration of the cw-OPO are discussed elsewhere.⁴⁹ Vibrational excitation of dopant molecules effects a geometric cross-section reduction of the droplet via evaporative cooling. Because this cooling is fast in comparison to the time between excitation and mass spectrometer detection, resonant vibrational excitation is measured as a reduction in the average ionization cross-section of the helium droplets, which can be detected in judiciously chosen mass channels to provide quasi-species-selective IR spectra. The laser beam is mechanically chopped at 80 Hz, and the mass spectrometer ion current is processed with a lock-in amplifier, providing background-free laser-induced depletion signals as the IR radiation is tuned with approximately 20 MHz resolution. Infrared spectra are normalized to the mid-IR idler power.

Geometric parameters at the electronic global minima were optimized using CFOUR software⁵⁰ at the CCSD(T)/ANO0 (n-propyl) and CCSD(T)/ANO1 (i-propyl) levels of theory. The electronic global minima for n- and i-propyl radicals have C₁ and C_s symmetry, respectively; however, we note that the minimum energy structures on the zero-Kelvin enthalpic surfaces have C_s and C_2v symmetry, respectively. The zero-Kelvin enthalpic surfaces are defined as the vibrationally adiabatic surfaces for large-amplitude motion about the high-symmetry structures; these motions correspond to either CH₃ hindered rotation for n-propyl²⁵ or out-of-plane $CHα$ bending for i-propyl (Fig. S1 of the supplementary material). For the n-propyl radical, the C₁ electronic global minimum structure differs from the zero-Kelvin most probable structure by an ∼8° pyramidal distortion and an ∼25° internal rotation of the methylene group α-$CH2)$⁠.²⁵ The C_s symmetry i-propyl electronic minimum differs from the most probable C_2v structure by an ∼17° distortion of the $CHα$ bond out of the plane containing the three carbon atoms, along with an ∼11° rotation of the methyl groups in the opposite sense of the $CHα$ distortion. On the n-propyl zero-Kelvin enthalpic surface, the energy of the C₁ structure described above lies only 10 cm⁻¹ above the C_s structure, indicative of the rather floppy nature for motion about the high-symmetry geometry.²⁵ Similarly for i-propyl, at zero-Kelvin, the C_s electronic minimum lies ∼35 cm⁻¹ above the C_2v structure. For each electronic global minimum structure, harmonic vibrational frequencies and a semi-diagonal quartic force field were computed at the same level of theory. Additionally, to obtain infrared active transition intensities, a geometry optimization followed by a harmonic frequency computation was performed using the next largest size of ANO basis set: ANO1 for n-propyl²⁵ and ANO2 for i-propyl.

Second order vibrational perturbation theory with resonances (VPT2+K)⁵¹ was implemented using an in-house script written in Mathematica 9.⁵² Historically, anharmonic resonances have been identified by an inspection of the force constant magnitudes and zeroth-order energy differences between states. Often an arbitrary threshold is set that weighs one or both of these factors. There are, however, less arbitrary ways to decide when the interaction between vibrational states is strong.^53,54 The Martin test is one of these.⁵⁴ It provides an estimate of how the frequencies of two interacting states would be affected if their interaction were to be treated explicitly. Another more sophisticated scheme for resonance identification would be to use the harmonic derivatives method developed by Matthews and Stanton.⁵³ Simulations were performed, identifying resonances on the basis of their Martin test values. The results from these simulations and further discussion of the Martin test are available in the supplementary material. Tables I and II contain harmonic and VPT2 computational results for the CH stretch vibrations of i- and n-propyl radicals, respectively, along with the eigenvalues of resonance polyads identified via the Martin test.

To facilitate a fair comparison to the local mode Hamiltonian procedure described in Section III B, the anharmonic couplings in n-propyl and i-propyl were modeled via effective Hamiltonians of dimension 22 and 28, respectively. The associated basis states are the CH stretching fundamentals and all overtones/combinations of normal modes that have primarily CH_n scissoring character. The energies of these states were first computed via the full VPT2 treatment. Anharmonicity constants were then “deperturbed,” which was accomplished by removing all terms in the summation that couple states explicitly included within each effective Hamiltonian. Deperturbed VPT2 frequencies were subsequently computed using the modified anharmonicity constants and placed along the diagonals of the Hamiltonian matrices.

Fermi-type coupling was accounted for by cubic force constants multiplied by the appropriate numerical factors. Higher-order coupling was included via the Darling-Dennison resonance constants proposed by Lehmann.⁵⁵ The function of these is to recover some of the correlation from the fourth order of vibrational perturbation theory. Expressions for the resonance constants that allow for high-order coupling between fundamentals and between 2 quanta states were (carefully) obtained from the literature.^56–59 It is important to realize that the resonance constants have denominators containing differences of harmonic frequencies, just as the anharmonicity constants do; thus, they also require deperturbation, or else artefactual values will be obtained. This scheme for treating resonances is known as the VPT2+K method.⁵⁸ Hamiltonian matrices and additional discussion of computational techniques are provided in the supplementary material.

A reduced-dimensional variational treatment of the anharmonicity was performed by simply diagonalizing each effective Hamiltonian. The eigenvalues are the corrected anharmonic frequencies. We assume that transitions to dark states derive their intensity entirely via intensity-borrowing from “bright” fundamental CH stretch basis states. In this simple scheme, transition intensities were obtained by summing the squared eigenvector components, which are multiplied by their associated harmonic intensities. Moreover, we did not account for intensity arising from higher-order terms in the dipole moment expansion, i.e., we assumed electrical harmonicity.

The local mode approach employed here has been described previously,^36,37 and we provide only a brief overview. As a starting point, harmonic frequencies were computed for the high-symmetry structures of n-propyl (C_s) and i-propyl (C_2v), which were obtained via symmetry-constrained geometry optimizations. Computations were carried out at the B3LYP/6-311++G(d,p) level of theory, using the Gaussian 09 suite of programs.⁶⁰ The procedure described below for constructing the local mode effective Hamiltonian is analogous to the “simplified model” described in Ref. 37.

We performed an orthogonal transformation of the normal modes to obtain a set of local CH stretch and HCH scissor modes. A central step in constructing this transformation is a series of intermediate computations that involve calculating normal modes that have been localized.³⁸ This simply involves increasing by a factor of two the masses of atoms not involved in the local motions. The quadratic force constant matrix was then re-mass-weighted and diagonalized, and this procedure was repeated to localize all CH stretch and HCH scissor modes. The resulting set of localized modes is used to transform the quadratic normal mode Hamiltonian and dipole derivatives.³⁸ 

The effective Hamiltonian in the local mode basis was constructed by including only those states necessary for a satisfactory simulation of the CH stretching region. Based on experience developed over several investigations,^35–37 the basis states required are the CH stretch fundamentals and the HCH scissor overtones/combinations. The combination tones are composed of one quantum in each of the two scissor modes. The resulting effective Hamiltonians for n-propyl and i-propyl are 22- and 28-dimensional, respectively.

In the localized representation, basis states are coupled by both quadratic and higher-order terms. Quadratic couplings were determined directly from the normal mode calculations. However, anharmonic terms, which include couplings between CH stretches and scissor overtones/combinations, were not evaluated for either n- or i-propyl. Instead, these couplings have been calculated previously using 4th order perturbation theory for a set of model systems.³² The couplings have been found to be similar for all model systems studied. Due to this similarity, average values of these coupling terms were inserted into the local mode Hamiltonians as non-adjustable parameters. For example, the cubic coupling term that mixes a CH stretch with its contiguous HCH scissor overtone is set equal to 22 cm⁻¹. The magnitude of the various stretch-scissor coupling terms is tabulated elsewhere.³⁷ 

To incorporate the effects of missing basis functions, CH stretch anharmonicities, and deficiencies in the electronic structure calculations used to determine the normal modes, we apply select scaling factors to the CH stretch and HCH scissor frequencies. All of the scalings were refined in a previous study of alkylbenzene chains,³⁷ and are not altered for the present study. The effective Hamiltonians were diagonalized to determine the transition energies and intensities. The localized CH stretch fundamentals are the only basis states that contribute oscillator strength, and the dipole derivatives are not scaled.

Droplet beam mass spectra were measured for both the n-BN precursor flowing through the room-temperature pyrolysis source (Fig. 1(a)) and with the pyrolysis source operated at approximately 1000 K (Fig. 1(b)). In addition to (He_n)⁺ ions originating from the ionization of neat droplets,⁴⁵ various peaks between 27 and 72 u are observed due to the ionization and fragmentation of He-solvated n-BN. As the pyrolysis source is heated, notable changes in the mass spectrum include reduced intensity at 41 and 43 u and enhanced intensity at 15, 27, 29, 30, and 39 u, all of which is consistent with n-BN thermal decomposition (Eq. (1)) and He droplet solvation of the pyrolysate. This pyrolysis signature is similar to that previously observed for other nitrite precursors used to generate ethyl, propargyl, and allyl radicals.^43,47,48 We find via difference mass spectra (vide infra) that the measurement of laser-induced depletion signal in channel 39 u proves effective at discriminating against droplets that contain pyrolytic decomposition products other than propyl radicals. It is important to emphasize that the density of molecules in the pyrolysis tube is optimized to dope droplets with single molecules. Under these source conditions, we expect less than five percent of the droplet ensemble to capture more than one molecule. Therefore, cluster formation is not expected to appreciably contribute to the infrared spectra reported here.

The laser-induced depletion spectrum of the precursor molecule shown at the bottom of Fig. 2 was measured with the quadrupole set to pass only ions having m/z = 43 u, which is one of the most intense peaks in the MS recorded with cold pyrolysis conditions. The precursor absorptions lie between 2850 and 3000 cm⁻¹ and exhibit relatively weak and broad features. Because the m/z = 39 u signal is significantly enhanced upon n-BN pyrolysis, this mass channel was chosen to record the depletion spectrum of the n-BN decomposition products (top spectrum of Fig. 2). Finally, the middle spectrum of Fig. 2 is that of the C₃H₆ molecule, which was recorded by metering a pure propene sample into a differentially pumped pickup cell.

The top spectrum in Fig. 2 shows weak signals (marked by *) that correspond to the positions of the most intense propene vibrations; however, the vast majority of the “pyrolysis-on” spectrum cannot be explained by either n-BN precursor or propene vibrations. There are two intense vibrations above 3000 cm⁻¹ that are in the vicinity of symmetric and antisymmetric α-$CH2$ stretch vibrations previously assigned to the n-propyl radical in Pacansky’s solid-Ar matrix isolation study.²⁶ A third distinct vibration falls near 2823 cm⁻¹, which is near the alkyl radical “anomalous band.”^26,28,61 The congestion between these two regions, however, is difficult to interpret. We considered the possibility of a droplet sequentially capturing both an n-propyl radical and another component of the pyrolysate. However, when the flux through the pyrolysis source is reduced by ∼40%, all bands between 2810 and 2960 cm⁻¹ decrease in intensity together (but do not disappear entirely), indicating that the spectrum arises predominantly from the pickup of a single molecule. We therefore conclude that the features in the congested “pyrolysis-on” spectrum in Fig. 2 are due mostly to the pickup and detection of a single n-propyl radical.

Figure 3 shows difference mass spectra (DMS) obtained by fixing the laser frequency to the peak of an isolated vibrational band and scanning the quadrupole from 10 to 82 u in 0.01 u increments. The ion signal is processed with a lock-in amplifier, giving the difference in the overall mass spectrum with and without laser excitation. This modulation scheme provides an essentially background-free mass spectrum of helium droplets containing the species being resonantly excited by the laser. The bottom DMS corresponds to the most intense precursor vibration at 2979 cm⁻¹ (cold pyrolysis source condition), revealing significant depletion signal on masses greater than 43 u (the mass of the n-propyl radical). The top DMS was measured with the laser frequency fixed to the symmetric α-$CH2$ stretch of n-propyl near 3026.9 cm⁻¹ (hot pyrolysis source condition), and this DMS shows no significant depletion signal beyond 43 u, with the exception of (He_n)⁺ (n ≥ 11) peaks. DMS scans were obtained for each band in the n-propyl radical spectrum (top of Fig. 2), and each of these bands shows a similar intensity pattern between m/z = 39 and 43 u, with no significant contribution from higher mass channels. This confirms that the spectrum obtained in mass channel 39 u with hot pyrolysis source conditions is due almost entirely to the n-propyl radical.

The successful detection of the n-propyl radical led to the expectation that pyrolysis of isobutyl nitrite [i-BN, (CH₃)₂CHCH₂ONO] under the same experimental conditions would yield a clean IR spectrum of the i-propyl radical. The IR spectra shown in Fig. 4 correspond to the precursor, propene, and pyrolytic decomposition products (bottom, middle, and top, respectively). The “pyrolysis-on” spectrum (recorded on m/z = 39 u) contains many sharp bands that can be assigned to propene (marked by asterisks) and many others that cannot. The IR spectrum of i-BN decomposition products contains an intense absorption at 3062 cm⁻¹ consistent with the lone $CHα$ stretch of the i-propyl radical, as assigned in Pacansky’s solid-Ar matrix isolation study.²⁷ The propene spectrum was scaled and subtracted from the “pyrolysis-on” spectrum, and a series of DMS and pressure dependence studies (as discussed above) indicates that the remaining absorptions arise predominantly from droplets containing single i-propyl radicals. Figure 5 displays both the n-propyl (top) and subtracted i-propyl (bottom) spectra with asterisks denoting residual propene features.

It is perhaps not unreasonable to expect propene to be produced more efficiently via thermal decomposition of i-BN (i.e., compared to n-BN), as the mechanism is likely due to the unimolecular dissociation of the propyl radicals themselves. Miller and Klippenstein have reported the C₃H₇ potential surface and theoretical kinetics of propyl radical dissociation.⁶² Although the barrier height for dissociation via C–H beta scission to produce propene + H is approximately the same for both n- and i-propyl radicals, n-propyl can dissociate via a low barrier C–C beta scission pathway to produce ethylene and methyl radical; in contrast, i-propyl cannot. For n-propyl dissociation, theory finds the C–C beta scission pathway to dominate at the temperature and pressure conditions relevant to the pyrolysis source used here. Indeed, the C–C beta scission barrier lies 5 kcal/mol below that for C–H beta scission. The appearance of moderately intense MS features at 15 and 28 u following the thermal decomposition of n-BN provides some experimental evidence for this (see top of Fig. 1). We note, however, that the vibrational bands of the n-propyl dissociation products (C₂H₄ and CH₃) do not appear in the spectrum because they are discriminated against by the choice of mass channel (39 u) to record the laser-induced depletion. In contrast, vibrational excitation of helium-solvated propene results in a large depletion signal in channel 39 u.

Figures 6–9 compare experimental spectra of the i- and n-propyl radicals to computations employing VPT2+K effective Hamiltonians. The theoretical spectra presented here were produced with a VPT2 treatment that includes all vibrational modes. We note, however, that it is sometimes useful to ignore contributions from torsional modes in the VPT2 treatment, because torsional modes are often poorly approximated as harmonic oscillators at zeroth-order and can have unphysically large force constants. The difference between anharmonic frequencies obtained with and without the treatment of torsions can be taken as a measure of how strongly coupled a vibration is to the torsional degrees of freedom.⁴³ By extension, the frequency difference is a metric of the inaccuracy of VPT2 for those transitions, brought about by the poor theoretical treatment of torsional modes.²⁵ When we exclude low frequency torsional modes, the simulated CH stretch spectra do not change qualitatively, although the magnitudes of a few frequencies change by as much as 10-25 cm⁻¹ (Tables I and II).^25,43 A detailed discussion of simulations that employ both approaches can be found in the supplementary material (Figs. S2-S5).

Figure 6 compares the VPT2 (red) and VPT2+K (blue) simulations to the experimental (black) i-propyl spectrum. The higher frequency region of the simulated spectrum (above 2950 cm⁻¹) has a similar appearance with and without resonance treatments. The VPT2 analysis alone is capable of reproducing the band origin for the lone $CHα$ stretch. With an explicit resonance treatment, the near degeneracy of the two $CHβ$ transitions is lifted, and the transitions are blue shifted into rather good agreement with the experimental spectral features. The lower frequency $CHγ$ region changes dramatically after resonant interactions are taken into account. This illustrates that the VPT2 analysis is insufficient for this spectral region; mechanical anharmonicity and intensity borrowing must be explicitly accounted for when modeling the low frequency CH stretching regions of hydrocarbons. Indeed, it is well-known that this spectral region is prone to resonance interactions involving CH stretches and overtones/combinations of CH_n bending modes that have fundamentals in the 1350–1500 cm⁻¹ range.^33–35,37

Dipole decomposition³⁷ offers a useful way to interpret the results of the VPT2+K simulations (Figs. 7 and 9). This entails separating the total intensity of each peak into contributions from different bright state CH oscillators. In the i-propyl spectrum (Fig. 7), the a′ CH_α oscillator located at the radical site gives rise to the single peak at highest frequency. The next highest frequency features are assigned to fundamental transitions of the methyl $CHβ$ oscillators, the higher frequency transition having a″ symmetry and the lower frequency transition having a′ symmetry. These fundamentals do not exhibit strong coupling to any two-quanta states; and thus, they are qualitatively reproduced by all of our simulations. The lower frequency $CHγ$ region proves to be a challenge for the VPT2+K approach. A significant portion of the total transition intensity is distributed between 2900 and 2930 cm⁻¹. The experimental spectrum, however, contains no spectral features in this region. The spectral pattern below 2950 cm⁻¹ is rather sensitive to the resonance criteria and to the values of the Darling-Dennison resonance constants. Many examples of this sensitivity can be found in the supplementary material.

Figure 8 compares the VPT2 (red) and VPT2+K (blue) simulations to the experimental n-propyl spectrum (black). The two highest frequency peaks at 3027 and 3110 cm⁻¹ are well-predicted, and as with i-propyl, these features have a similar appearance with and without the explicit resonance treatment. These bands are definitively assigned to the symmetric and antisymmetric α-$CH2$ stretching fundamentals localized on the radical center. The antisymmetric γ-$CH3$ stretch fundamentals around 2975 cm⁻¹ agree well with predictions from full VPT2; however, the predicted frequencies are blue-shifted when resonances are considered, and an additional low intensity peak is predicted near 3010 cm⁻¹, which is not observed experimentally. In contrast to predictions for i-propyl, VPT2+K does a reasonable job of simulating the lower frequency region.

As a result of the more local nature of the n-propyl CH stretches, we can assign labels to the bright state normal modes that correspond to either α-$CH2$⁠,β-$CH2,$orγ-$CH3$⁠. The dipole decomposition shown in Fig. 9 is obtained by analyzing the various $α,β,γ$-bright state contributions to the eigenvalues obtained from the VPT2+K analysis. From this we find that the n-propyl spectral complexity between 2800 and 2960 cm⁻¹ derives from resonant interactions involving either the β-$CH2$ or γ-$CH3$ oscillators. Contributions from oscillators localized on the β and γ carbons are spread out, but in general, β-$CH2$ contributions are observed in the lower frequency region, with the γ-$CH3$ intensity contributions falling largely between 2900 cm⁻¹ and 3000 cm⁻¹. For a few of the simulated peaks that closely match experiment, 2827 cm⁻¹, 2951 cm⁻¹, and 2873 cm⁻¹, dipole decomposition indicates that they derive their transition intensity from the β-$CH2$ oscillators, the γ-$CH3$ oscillators, and a mixture of both, respectively. The remaining peaks are predicted less confidently, and definitive assignments of these are not possible with the normal mode VPT2+K treatment.

For the case of VPT2+K, increasing the number of states in the effective Hamiltonian can improve the description of the anharmonic coupling that dominates in the low frequency CH stretching region (for both n- and i-propyl; results shown in Figs. 6 and 8). However, this can come at the cost of accurate high frequency stretch predictions. This point has been made by Stanton and co-workers.⁵⁷ It is appealing to think that, as the effective Hamiltonian matrix grows in size, and the eigenvalues approach the variational solution to the semi-diagonal quartic potential, the accuracy would improve, but this is often not the case. The reduced accuracy of frequencies computed with large effective Hamiltonians, for relatively uncoupled CH stretching modes (e.g., α-$CH2$ modes in n-propyl), can be rationalized as owing to an upset of the fortuitous error-cancellation, from which the full VPT2 treatment benefits. Also, using our effective Hamiltonians, some interactions with states of lower frequency than the CH stretch fundamentals are treated variationally, yet all interactions with higher frequency states (excluding the typically negligible 1:1 resonances with other single-quantum CH stretching states) are still treated perturbatively. This has a biasing effect upon the anharmonic corrections because interactions that raise and lower the energy of the vibrational states are treated in different ways. Perhaps the best example of this effect, in our study, is seen for the γ-$CH3$ asymmetric stretching modes of n-propyl, which are observed experimentally at 2973 and 2978 cm⁻¹. When relying upon the Martin test to identify resonances,⁵⁴ requiring smaller effective Hamiltonians, these modes are predicted rather accurately (Fig. S3 in the supplementary material). The Martin test with a threshold value of 0.5 cm⁻¹ does not predict any resonances involving these modes, so they receive the full VPT2 treatment, not requiring deperturbation. Expansion of the effective Hamiltonian to include all CH stretch fundamentals and scissor overtones/combinations significantly degrades the quality by which these transitions are predicted (Fig. 9).

Figures 10 and 11 compare the local mode model Hamiltonian predictions to the experimental n- and i-propyl spectra, respectively. As described in Section III B, the basis states consist of the entire set of localized CH stretch fundamentals and the overtones/combinations of localized HCH scissor modes. The sizes of the effective Hamiltonians are the same as for the VPT2+K treatment, namely 22- and 28-dimensional for n- and i-propyl, respectively. We emphasize again that both the scale factors and the Fermi coupling parameters used to generate the effective Hamiltonians were refined in previous studies of alkyl CH stretch spectra,³⁷ and they are used here without modification.

For the n-propyl spectra shown in Fig. 10, the format is the same as in Fig. 9. The dipole decompositions for each CH_n group are shown as red traces, and the blue trace corresponds to the results from the “Full Model” local mode predictions. Overall, there is good agreement between experiment and theory, with the largest differences being in the higher frequency region (above 3000 cm⁻¹). The quality of the agreement in the high frequency region is a measure of the transferability of the scale factors that adjust the harmonic frequencies.³⁷ We find the local mode frequencies of the α-$CH2$ group to be underestimated using these scale factors. In addition, the splitting between the symmetric and antisymmetric α-$CH2$ stretches is overestimated, indicating that the model overestimates the quadratic coupling between stretches localized on the radical site. The agreement in the low frequency region is sensitive to both the transferability of the scale factors and the accuracy of the Fermi coupling parameters. The local mode model successfully predicts the number of lower frequency features in the spectrum. Other than the lowest frequency band at 2820 cm⁻¹, which is underestimated by the model, the agreement is very good below 3000 cm⁻¹, despite there being zero adjustable parameters.

Figure 11 compares the local mode model to the experimental i-propyl spectrum. Overall, the local mode Full Model spectrum (blue) satisfactorily reproduces most of the spectral features, although the $CHα$ stretch is underestimated, and the agreement in the lower frequency region is less quantitative than observed for n-propyl. In general, the relative intensities of bands comprising the lower frequency region are predicted by the model, although a few discrepancies are apparent in the ordering of the higher and lower intensity transitions. In comparison to the VPT2+K approach, the local mode model is far better at predicting the lower frequency region of the i-propyl spectrum. Again, this lower frequency region is especially sensitive to stretch-scissor anharmonic coupling interactions, as illustrated by the VPT2+K simulations shown in Fig. 6.

The red traces in Fig. 11 correspond to dipole decompositions for the $CHα$ bright state and one of the equivalent CH₃ groups (i.e., $CHβ,γ)$⁠. To highlight the spectral features arising from the coupling between CH₃ groups, we constructed a local mode effective Hamiltonian for the CH₃CHCD₃ isotopologue. For the latter isotopologue, the Hamiltonian matrix is of reduced dimension; it includes only those basis states associated with the $CHα$ group and the CH₃ group. Comparing the resulting spectrum (purple trace) to the CH₃CHCH₃ Full Model (blue trace), we see that many of the finer features arising in the 2800-2900 cm⁻¹ region are the result of mixing between the two CH₃ groups. As discussed below, these states mix extensively despite the relatively small coupling terms that directly connect them.

The local mode model Hamiltonian allows for a detailed analysis of characteristic alkyl radical signatures present in the CH stretch region, some of which were described qualitatively by Pacansky in his matrix isolation study.²⁶ The open shell nature of the system can be appreciated in a qualitative sense via inspection of the n-propyl dipole decompositions (Fig. 10), where the radical site leads to distinct $α,β,γ$-CH stretch contributions to the spectrum. In previous work on α-alkyl benzyl radicals,⁶³ the radical site has been shown to dramatically influence the CH oscillators localized on adjacent carbon atoms. For CH oscillators localized on non-adjacent carbon atoms, the radical effect was observed to be far smaller. The CH stretch Hamiltonian for n-propyl is

Descending along the diagonal, the first three states correspond to the local mode γ-CH stretches (the 2952 cm⁻¹ mode corresponds to the in-plane stretch), the fourth and fifth states are the β-CH stretches, and the last two states correspond to the α-CH stretches. Full local mode model Hamiltonian matrices for both propyl radicals are provided in the supplementary material, whereas we focus here on the stretch Hamiltonians. The local mode α-CH stretches are >100 cm⁻¹ higher in frequency than is “typical” for alkyl CH stretches (i.e., ∼2950 cm⁻¹). Moreover, the quadratic coupling between the α-CH stretches is more than twice the magnitude observed for alkyl moieties within closed shell systems.³⁷ In contrast, β-CH frequencies are >100 cm⁻¹ below what is typically observed for alkyl CH stretches, and the coupling between them is reduced to almost zero. The former effect was recently noted for α-alkyl benzyl radicals and was shown to depend sensitively on the orientation of the CH stretch relative to the radical site.⁶³ Finally, we find the 3 × 3 γ-CH stretch block to be typical of the CH₃ stretch Hamiltonians constructed for closed shell systems.³⁷ As observed in the α-alkyl benzyl study,⁶³ the n-propyl radical site significantly affects only those CH oscillators localized on or adjacent to it.

Turning to the i-propyl CH stretch Hamiltonian (Eq. (3)), we find many of the same effects due to the radical site,

The first state corresponds to the lone $CHα$ stretch, whereas the remaining states are due to the methyl groups. The CH₃ blocks have two different types of stretches, an in-plane $(CHβ)$ stretch and two out-of-plane $(CHγ)$ stretches. The $CHγ$ stretches are shifted down in frequency to 2855 cm⁻¹. This shift is close in magnitude to the shift observed for the β-CH stretches in the n-propyl Hamiltonian (Eq. (2)). On the other hand, the $CHβ$ stretches at 2963 cm⁻¹ fall in the range typical of CH₃ stretches in closed shell systems. These differences in the $CHβ,γ$ stretch frequencies are a direct manifestation of the orientation dependence described above. The in-plane $CHβ$ oscillators are oriented perpendicular to the singly occupied p-orbital localized on the α carbon, whereas each $CHγ$ oscillator makes an ∼$π/3$ angle with the symmetry plane. These observations are consistent with the hyperconjugative stabilization hypothesis, in which electron density in $CHγ$σ-bonds is partially transferred to the half-filled p-orbital, leading to a concomitant reduction in the $CHγ$ oscillator’s quadratic force constant.

The 3 × 3 CH₃ stretch block reveals an interesting result associated with the intramonomer quadratic couplings. The down-shifted $CHγ$ states are coupled to each other by only 6 cm⁻¹, which is similar in magnitude to the coupling observed between the n-propyl β-CH oscillators. However, the $CHγ$ stretches are still strongly coupled to the “unshifted” $CHβ$ stretch (21 cm⁻¹). The larger coupling constant is identical to the matrix elements that couple the three γ-CH oscillators in the n-propyl radical, showing that the presence of one unshifted CH stretch is sufficient to produce unmodified intramonomer couplings. Finally, we note that the direct coupling between the two CH₃ group stretches is relatively small (with a maximum of 2 cm⁻¹), yet the final spectra are still clearly sensitive to the combined presence of both CH₃ groups, as shown in Figure 11. Rather than being due to direct stretch-stretch couplings, this mixing appears to arise from the coupling between scissor combination modes that involve both CH₃ groups. The coupling between the harmonic local mode scissors is much larger (near 6 cm⁻¹). The effects of these scissor combination couplings appear to be the primary mechanism by which the two CH₃ groups are mixed.

High-quality infrared spectra of n- and i-propyl radicals in the CH stretching region were obtained via the helium droplet isolation method. Quasi-species-selective spectroscopy was achieved through the detection of ionization cross-section modulations in judiciously chosen mass channels. Assignments of spectral features to propyl radicals were confirmed by recording the difference in the mass spectrum with and without laser excitation. In the limit of 3N−6 uncoupled oscillators, one predicts seven CH stretch vibrations for both n- and i-propyl. However, the resolution achieved in the experiment reveals a vibrational complexity that demands a treatment beyond the harmonic approximation.

Second-order vibrational perturbation theory, VPT2, accurately predicts the high-frequency stretching vibrations localized on the radical site α-$CH2$ for n-propyl; $CHα$ for i-propyl). The CH stretch vibrations localized on carbon atoms adjacent to the radical center are red shifted, due to a hyperconjugative stabilization of the system and concomitant softening of the CH oscillators (β-$CH2$ for n-propyl; $CHγ$ for i-propyl). The associated red shifts drive these CH stretch modes into resonance with the overtones and combination tones of CH_n bending modes. This effect contributes substantially to the spectral complexity observed between 2800 and 3000 cm⁻¹. Clearly, VPT2 alone cannot account for the complexity that emerges in this lower frequency region.

The pervasive anharmonic coupling and intensity borrowing evident in the CH stretch region are modeled with two separate effective Hamiltonian approaches. (1) The VPT2+K approach treats Fermi and Darling-Dennison resonances explicitly via the diagonalization of an effective Hamiltonian matrix. The matrix contains deperturbed diagonal elements and off-diagonal coupling terms derived from quartic force fields computed at the CCSD(T)/ANO0 (n-propyl) or CCSD(T)/ANO1 (i-propyl) levels of theory. The effective Hamiltonian is represented in a normal mode basis consisting of CH_n stretching fundamentals and CH_n bending overtones/combinations. (2) The local mode effective Hamiltonian approach employs a localization scheme that takes as input a harmonic frequency computation at the B3LYP/6-311++G(d,p) level of theory. The localized basis states correspond to CH stretching fundamentals and overtones/combinations of HCH scissor modes. Refined harmonic scale factors and anharmonic coupling terms are taken from previous studies of closed shell hydrocarbon CH stretch spectra and are transferred to the local mode model without modification.³⁷ Both approaches generate Hamiltonian matrices that are 22-dimensional for n-propyl and 28-dimensional for i-propyl. The computational cost of the local mode approach is far lower than the VPT2+K method, because it does not require a quartic force field as input.

Local mode predictions are generally in very good agreement with experiment, despite there being zero adjustable parameters in the model. The success of the local mode model indicates a rather robust transferability of the anharmonic coupling terms.³⁷ The presence of a radical center apparently does not significantly affect the cubic coupling between localized CH stretch and HCH scissor modes. On the other hand, the quadratic force field is strongly affected by the radical site. For example, the two α-$CH2$ stretches are shifted to higher energy and coupled more strongly by quadratic terms in the Hamiltonian. In contrast, the β-$CH2$ stretches are largely decoupled from each other and shifted to lower energy. Both observations are completely consistent with the hybridization $(α-CH2)$ and hyperconjugation $(β-CH2)$ arguments put forth by Pacansky to rationalize the matrix IR spectra of n-propyl.^23,25

With regard to predicting the number and relative intensities of transitions within the spectrally congested 2800 to 3000 cm⁻¹ region, the local mode approach is generally superior to VPT2+K. This is especially evident in the i-propyl spectrum, where VPT2+K distributes the transition intensity in a qualitatively incorrect manner, a fairly large portion of it being between 2910 and 2925 cm⁻¹, where nothing is observed experimentally. The discrepancy may partly be due to the fact that the local model Hamiltonians are evaluated at the C_s (n-propyl) and C_2v (i-propyl) geometries, whereas the VPT2+K computations were carried out at the C₁ (n-propyl) and C_s (i-propyl) electronic minima. We chose the high-symmetry structures for the local mode approach because these are the most probable structures and the minima on the zero-Kelvin enthalpic surfaces, and the model only requires as input a harmonic frequency calculation of localized CH stretch and HCH scissor vibrations. Moreover, the impact of methyl torsions is expected to be somewhat mitigated by computing frequencies at the most probable structure. The geometries for the VPT2+K procedure were fixed to the electronic minima out of necessity because performing VPT2 with anharmonic force fields computed at non-minimum energy structures is not a well-defined practice. The frequencies of CH stretches depend on the dihedral angle of the CH bond to the radical site and thus the model spectrum is sensitive to the geometry. For example, the computed CH stretch harmonic frequencies at the CCSD(T) level of theory are shown in Tables I and II for both geometries considered here (see also Tables S4 and S5). The average deviation in going from the electronic minimum to the high-symmetry structure is 15.3 cm⁻¹ for i-propyl and 6.5 cm⁻¹ for n-propyl. It is less obvious how the force fields should depend on this geometry change; however, we note again that in a previous study of closed shell alkyl benzene systems, it was found that the anharmonic contributions to the spectra were very similar for all molecules and conformations.³⁷ Therefore, the choice of geometry is likely not the only factor leading to the observed discrepancy between the two effective Hamiltonian approaches; for example, the choice of representation, local versus normal mode, should result in different convergence behavior. The coupling between basis states in the local mode model more accurately reflects the salient interactions responsible for the experimental spectral complexity (stretch-scissor coupling), and is therefore more easily converged. Collectively, these observations suggest that we should expect the local mode model to accurately predict the CH stretch spectra of larger alkyl radical systems, and because of the low cost of such computations, this approach provides an excellent alternative to the more expensive VPT2+K method.

See supplementary material for the detailed discussion of the VPT2+K method, vibrationally adiabatic potential curves for the i-propyl radical (Fig. S1), the VPT2+K results and simulated spectra using Martin test resonance criteria (Tables S1-S3; Figs. S2 and S3), the VPT2+K results and simulated spectra using 22- and 28-dimensional effective Hamiltonians (Figs. S4 and S5), and the results of harmonic frequency calculations at the high- and low-symmetry structures described in the text (Tables S4 and S5). All Hamiltonian matrices described here are included in csv format.

G.E.D. acknowledges support from the Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division of the US Department of Energy (DOE) under Contract No. DE-FG02-12ER16298. D.P.T. and E.L.S. acknowledge support from NSF Grant No. CHE-1566108. H.F.S. acknowledges support from US Department of Energy, Office of Basic Energy Science (Grant No. DE-FG02-97ER14748) and computing resources at the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231. We dedicate this work to Jacob Pacansky.