Gas-phase high-resolution molecular spectroscopy for LAV molecules

1 Introduction

Spectroscopy in general provides a wide range of information on the structure, potential energy surface and internal dynamics of molecules. The most accurate data is obtained from high-resolution rovibrational spectra. The main goal of rotational and vibration-rotational spectroscopy is spectrum interpretation, i. e., pointing out the states between which a transition occurs. In order to extract information from rovibrational spectra, a detailed and scrupulous assignment and analysis is needed. The difference between rotational and rovibrational spectroscopy is as follows. In rotational spectroscopy, one can observe transitions between rotational energy levels within the same vibrational level, whereas in vibration-rotation spectroscopy observed transitions are between piles of rotational energy levels belonging to two different vibrational levels. As vibrational transitions can be observed in the liquid or solid state of the sample, the rotational ones only in the gas phase. The assignment and analysis of such spectra are carried on using theoretical models describing molecular motions in the context of quantum mechanics. After the spectral assignment has been done, one can determine the shape of multidimensional potential energy surface and molecule geometry. It is not straightforward though, even for rigid molecules whose potential energy surface consists of one deep minimum only. An equilibrium configuration is a configuration that corresponds to such a minimum. For molecules with several minima of similar in energy, it is a challenge to determine the molecular geometry. It is a case for non-rigid molecules performing large amplitude motions which take a molecule from one configuration to another.

Methylamine, which has been of spectroscopic interest for many years, is considered as a nonrigid (floppy) molecule with two large amplitude motions. Thus, in order to describe spectroscopy and dynamics of methylamine, its equilibrium structure is not enough, since the molecule undergoes two tunneling motions, the inversion (umbrella motion) of the amino group and internal rotation of the methyl group with respect to the rest of the molecule. The high-resolution spectra of such nonrigid molecules like methylamine are often very complicated because of transition tunneling splittings but at the same time very rich in information about their barrier heights or arrangement of their functional groups.

To be able to understand spectra of floppy molecules, it is essential to use group theory. The group theory is used among others to assign symmetry to energy levels, to develop selection rules for transitions or to calculate the overall shape of tunneling splitting. The size of tunneling splittings is dependent on the barrier height of a given large amplitude motion, which may be calculated from ab initio methods. Having the barrier heights calculated, one can predict the inversion–torsion–rotation levels and finally the spectrum of a molecule. There are two group theories used in high-resolution molecular spectroscopy, i. e. the point group theory and the permutation-inversion group theory. The point group symmetry is used for so-called rigid molecules like water or carbon dioxide, i. e., the molecules with one equilibrium configuration only. The point group theory is widely described in the spectroscopic text books [1, 2, 3, 4]. As for the permutation-inversion (PI) group, it is used to describe a symmetry of nonrigid molecules like methanol, methylamine, hydrazine, 2-methylmalonaldehyde since such molecules have more than one equilibrium configuration and the point group symmetry cannot be applied any longer. In the PI groups the symmetry elements consist of the permutations of identical nuclei with or without inversion [4]. Once internal rotation or inversion is feasible in a molecule, the point group fails and the PI theory has to be applied. For instance, methylamine molecule, CH₃NH₂, belongs to the point group symmetry C_s (in its rigid equilibrium geometry) and to the G₁₂ permutation-inversion symmetry group.

In this review, the studies of different high-resolution FT spectral regions of a methylamine molecule will be presented. The molecule performs two large amplitude motions which reflect strongly in its rovibrational spectra. The published results will be summarized to describe the current state of knowledge on the high-resolution IR spectra of methylamine.

2 Methylamine molecule

The methylamine molecule (CH₃NH₂) is a seven-atom organic molecule and the simplest primary alkylamine in chemistry. Its CH₃ group is connected to NH₂ group by C-N bond. Methylamine performs two large amplitude motions, the CH₃-torsional motion (internal rotation) and the NH₂-wagging motion (inversion), which makes the CH₃NH₂ a prototype for complex non-rigid molecules. A schematic view of the methylamine molecule is presented in Figure 1. The coupling between internal rotation of methyl group and inversion of two hydrogen atoms in the amine group gives rise to the tunneling splitting. Since the barriers in the potential functions are of intermediate height, the observed splittings are relatively large even in the ground state and the assignment and fitting of the spectrum is a very tedious task.

Figure 1:

A schematic view of the methylamine molecule with inversion ρ and torsion τ coordinates.

Both large amplitude motions are hindered by potential barriers of 23.2 kJ/mol for the wagging, and 8.6 kJ/mol for the torsion [5]. Figure 2 presents six equivalent minima in the potential energy surface of methylamine. The minima are separated by inversion and torsional barriers. The barriers to inversion and to torsion are low enough to allow a tunneling between minima, which leads to splittings of energy levels. In each such minimum, a rotation-inversion-torsion function is localized.

Figure 2:

Potential energy surface of methylamine where ρ and τ are inversion and torsional angles, respectively, given in [deg] (taken from paper [6]).

Methylamine as a non-rigid molecule has a very high symmetry and is described by permutation-inversion symmetry group G₁₂ which is isomorphic to the C_6v point group [7]. Each energy level is labeled by the value of the usual quantum numbers J and K = K_a and by a torsion-wagging-rotation symmetry species Г corresponding to an irreducible representation of the G₁₂ group. The group G₁₂ consists of six components: A₁, A₂, B₁, B₂, E₁, E₂. Each K = 0 rovibrational transition splits into four (A, B, E₁, E₂) lines. For 0 < K ≤ 4 the lines of A and B symmetry are further split due to asymmetry (A₁, A₂, B₁, B₂). The symmetry labels of doubly degenerate levels E₁ and E₂ have additional ±1 labels to distinguish between two levels with the same J and K_a values, so finally for K ≠ 0 there are A₁, A₂, B₁, B₂, E₁₊₁, E_1-1, E₂₊₁ and E_2-1 symmetry species with respective intensities, 1:1:3:3:3:3:1:1. For the K = 0 levels, the +1 and −1 labels have no meaning since there is only one E₁ and E₂ level. Therefore, all K = 0 E levels are designated as +1 levels. When asymmetry splitting disappears for K > 5, the (A₁, A₂) and (B₁, B₂) components overlap and their intensities are doubled. The torsion barrier tunneling causes a splitting between states of A or B symmetry and E₁ or E₂ symmetry. The tunneling through the inversion barrier splits states of A and B symmetry or E₁ and E₂. Thus, the following transitions are only allowed: A₁ ↔A₂, B₁ ↔B₂, E₁ ↔E₁ and E₂ ↔E₂ [7], and they are called the symmetry selection rules, which determine between which energy levels’ transitions can occur.

Asymmetric rotor spectra are usually classified as near-prolate or near-oblate. It depends on which limit of the symmetric rotors they are at. The methylamine molecule is considered as a near prolate asymmetric rotor. The spectral bands are further categorized as parallel or perpendicular type. This distinction is made on the basis of the transition dipole moment which may be parallel or perpendicular to the near-top axis. Since the methylamine molecule is a near prolate asymmetric rotor, a type A band is a parallel band and type B and C bands are considered as two perpendicular bands.

The methylamine molecule possesses 15 normal vibrational modes. It comes out from the 3N-6 formula which calculates the degrees of freedom of the non-linear molecule (for linear molecules the formula is 3N-5). The experimental harmonic vibration frequencies (in cm⁻¹) are presented in Table 1.

In this review in the order of appearance four vibrations in methylamine will be described, i. e. the wagging, CN stretching, torsional and CH₃ stretching vibrations. The wagging vibration represented by symbol ν₉, with its experimental frequency of 780 cm⁻¹ has the A′ symmetry, which means that this vibration is symmetric with respect to reflection in the symmetry plane, which passes through the C-N axis and bisects the NH₂ angle (Figure 1).The C-N stretch vibration, ν₈, with the frequency of 1044 cm⁻¹ has also the A′ symmetry. The lowest vibration in frequency, the torsional vibration, ν₁₅, being antisymmetric has A′′ symmetry and appears at 264 cm⁻¹. And finally, the CH₃ stretch vibration is also antisymmetric with respect to reflection in the symmetry plane (A′′ symmetry) with the experimental frequency of 2985 cm⁻¹.

3 Measurements of spectra

For chemists, the most desirable aspects of research are molecular structures, vibrational energies and excited electronic states of the molecules of interest. To obtain these information, both rotational and rovibrational spectra are needed. High-resolution gas phase IR spectra provide information on vibrational levels and rotational structures as well. Moreover, using this technique, it is also possible to study molecules with zero permanent dipole moment, whereas rotational spectra (microwave and millimeterwave) require a permanent dipole moment. The IR spectral region extends from 100 to 4000 cm⁻¹.

The molecular spectra of methylamine which are going to be described in the next chapters are the high-resolution spectra recorded at the University of Oulu in Finland using Bruker IFS-120HR Fourier transform spectrometer. The resolution due to MOPD (Maximum Optical Path Difference) was 0.00125 cm⁻¹ and the relative wavenumber precision was almost one order of magnitude better than the respective resolution, due to careful calibration of the spectra. The pressure was 0.036 Torr and the path length of 3.2 m in the optimized White cell.

4 Assignments and analysis of rovibrational spectra with Loomis-Wood for Windows program

The high-resolution rovibrational spectra of polyatomic molecules are often very complex. A clue to the spectra assignment is to put labels with proper quantum numbers on a given transition, and this is the initial step of any analysis. To perform that, some special techniques need to be applied. One such technique is checking the assignment by ground or excited state combination differences.

Figure 3 illustrates the rovibrational transitions, where a transition between two vibrational levels v′′ (lower) and v′ (upper) is ruled by the formula Δv = ± 1. Between these two vibrational levels rotational transitions occur with the selection rule ΔJ = ± 1, where J is the angular momentum quantum number (J′′ and J′ denote lower and upper states, respectively). The group of lines with ΔJ = +1 correspond to the R branch with increasing wavenumber and with ΔJ = –1 to P branch with decreasing wavenumber. Each transition is labeled R(J) or P(J), where the value J corresponds to J′′. To derive information about a series of lower and upper states, between which transitions occur, differences of spectral line frequencies between transitions with a common upper state, like in Figure 3 series R(0) and P(2), depend on properties of the lower state. Similarly, for transitions with the same lower state, like in Figure 3 series R(2) and P(2), differences between these transitions are dependent on properties of the upper states only. Thus, the rotational energy gap in the vibrational lower state can be obtained by lower state combination differences (LSCDs), ν˜[R(0)]-ν˜[P(2)], or similarly the energy interval in the excited state by upper state combination differences (USCDs), for instance, ν˜[R(2)]-ν˜[P(2)]. It is worth mentioning that by lower state one can understand any lower state, but in most cases it is a ground state (usually well determined) and then the name of LSCDs becomes GSCDs (ground state combination differences). If there is an assignment of a hot band, then obviously the lower state is not the ground state but the appropriate lower state (it will be discussed in more details in one of the chapters below, where the assignment and analysis of the first excited state of methylamine are described).

Figure 3:

Schematic view of P and R branches along with combination differences.

Among many computer programs for assigning the spectra, the Loomis-Wood for Windows (LLW) package was built just on the basis of combination differences [11]. This is the graphical software, which is extensively used in the investigations and is very helpful, especially in troublesome assignments. The program is dedicated to assigning Fourier Transform Infrared rotation-vibration spectra. The LWW package can be also used for molecules with large amplitude motions. The LWW is based on the idea of the Loomis-Wood (LW) algorithm which was presented for the first time in 1928 [12]. The algorithm of the LW uses the periodicity of a pattern of lines from the spectrum. The lines are arranged in such a way that the periodicity is converted into recognizable spectral pattern in the LW diagram. In other words this symbolic representation of sequences of transitions in LW diagrams translates the numerical information from the wavenumber listings of branches into a graphical representation. Figure 4 shows such a LW diagram from LWW package, where lines (peaks) of a given series (for instance ^RR₅ of A₂ symmetry of the first torsional band of methylamine) shown in a fragment of the spectrum above are presented in the LW diagram by small triangles arranged one by one. The spacing between adjacent R branch lines and P branch lines is close to 2B, where B is the rotational constant [11].

Figure 4:

LW diagram from LWW software presenting a series of lines and the corresponding spectral lines above.

To start the assignment with LWW, it is necessary to have an energy file of rotational sublevels. Usually, the starting rovibrational energies are of poor quality, but it can be improved after the first assignments are correctly done. The assignments in the LWW are confirmed by Lower State Combination Differences (LSCDs), which are the built-in part of the program. This is a big advantage while assigning the spectral lines since after a line has been indicated, it is possible to check at once the correctness of the assignment by LSCDs. The program allows simultaneous display of several LW diagrams that are mutually connected by LSCD, i. e. a couple of series of branches that belong to the same “family of branches”, which means the series sharing the common upper energy level. These series appear as visually recognizable patterns of the same shape in the LW diagram. In Figure 5 three LW diagrams of the series belonging to such a “family” are presented, i. e. series ^RQ₅, ^RR₅ and ^PP₇ of A₂ symmetry species with the same upper state of K = 6. If it happens that one of the series has a different shape, the assignment is wrong and should be corrected. In other words, the label attached is not the right one. The lower state of the band under assignment (not necessarily the vibrational ground state) should be determined with accuracy allowing for the LSCD checking. Along with the LW diagrams, one has the access to the plot of the IR spectrum with labels of predicted or assigned lines in the so-called spectrum window. This plot is directly linked with its peaklist (table of wavenumbers and intensities). The LWW program allows a simple calculation of a correction function to upper state energies once the assignments have been verified by LSCD checking and Figure 6 presents the same series as in Figure 5 after they have been fitted (all lie in the middle of the LW diagrams along the central line). Once the spectral lines have been assigned, they can be exported to a file, which can be used in external fitting procedures with complex Hamiltonian models. The software gives a possibility of choosing the series for exporting or any other purpose. The rovibrational energies refined by least-squares fitting external program can be imported back into the LWW program where they replace previous energies. The assignment can be carried on this way until the entire range of the spectrum is satisfactory assigned.

Figure 5:

LW diagrams showing the “family of branches” assigned and confirmed by the LSCDs.

Figure 6:

LW diagrams showing the “family of branches” after the fit.

The LWW software has a lot of other advantages and useful tools like a possibility to merge two spectra in different spectral ranges, to generate the whole set of branches to be assigned or to plot energy files which help find the line crossing. There are also many functions which enable a LWW user to perform very useful actions when assigning spectra like moving the assignment of a present series to another one (with different K value) or assigning a given line to a different J value. To sum up, the LWW software facilitates a lot a hard work of assignment and there is a special website dedicated to the program where a user can find an utter description of all the possibilities the LWW software offers [13].

5 Theoretical model

In order to calculate the positions and intensities of molecular absorption or emission of infrared lines, an effective Hamiltonian approach is commonly used. An effective Hamiltonian is derived from the complete Hamiltonian through perturbation theory and almost in all cases effective Hamiltonians are constructed by considering Born-Oppenheimer approximation, which separates electronic, vibrational and rotational motion. Among all these motions, the electronic one is believed to be the fastest, then in turn the vibration of the nuclei and finally the slowest is rotation of the molecule. Thus, the complete Hamiltonian can be expressed by the following formula:

The purely rotational Hamiltonian for a rigid rotor without taking into account any interactions can be presented by rotational angular momentum operators [2]. For linear or symmetric molecules, one can solve the Schrödinger equation and the energy eigenvalues are obtained directly, whereas for asymmetric molecules it is not that straightforward, as the asymmetric rotor wavefunctions are expanded in a basis set of symmetric wavefunctions and the Hamiltonian matrix is then diagonalized.

There is no single theory which can be applied to all types of molecules. Different programs are used for linear, spherical, symmetric or asymmetric top molecules. The problems become even more difficult for floppy molecules, in which case a separate theory should be used in almost each individual case. For symmetric top molecules, for instance, the program SIMFIT can be used for simultaneous fitting of vibration-rotation and rotational spectra [14].

For the analysis of asymmetric rotor spectra, the powerful package SPFIT/SPCAT written by H. M. Pickett [15] is commonly applied. These programs were originally aimed at cataloguing rotational spectra of many different molecules. This was the reason for a very general way of setting up the molecular Hamiltonian and efficient factorization of its energy matrix. Thus, the input of the SPFIT program consists of a set of assigned experimental transitions or energy levels and a file with molecular parameters. A least square procedure is applied to the assigned data. From the obtained set of parameters the SPCAT program calculates back the transition frequencies and intensities.

It is not possible to use the standard fitting programs for molecules performing large amplitude vibrations. Thus, it is necessary to create individual approaches dedicated to the specific group of LAV molecules or sometimes even to a single molecule. Among others there are some programs written particularly for LAV molecules: XIAM (internal rotation program for up to three symmetric internal rotors and up to one quadrupole nucleus [16]), ERHAM (Effective Rotational HAMiltonian program for molecules with up to two periodic large-amplitude motions [17]), BELGI (The BELGian Internal Rotor Program [18]), RAM36 (the Rho Axis Method for 3-and 6-fold barriers [19]). More detailed description of each of the program is available at the link cited in [20].

As the object of this article is a methylamine molecule and its spectra, mainly the quantum theory concerning this molecule will be under consideration. Thus, in most studies concerning methylamine spectra, the phenomenological Hamiltonian based on the group-theoretical high-barrier tunneling formalism was used [7]. This formalism is based on the assumption that a complete set of vibration-rotation states, which are localized in one minimum of the potential surface, interacts with the equivalent complete set of vibration-rotation states localized in neighboring minima. The minima in the potential surface in Figure 2. correspond to the six configurations (frameworks) shown in Figure 7. The coordinate ρ stands for the inversion angle, whereas the τ coordinate is the internal-rotation angle. These six minima are tied together by two types of tunneling path. Both types entail some torsion (internal rotation), whereas only one involves inversion. For example, when the torsion of the CH₃ group occurs only (angle τ changes by 120°) and the inversion does not take place (coordinate ρ does not change), this case is illustrated by tunneling motion from position 1 to 3 in Figure 7. If inversion takes place (change from positive value of ρ to negative), then the internal rotation is involved as well, since coordinate τ changes by 60° and this situation is shown by tunneling motion from framework 1 to 2 in Figure 7. The system of 6 minima is repeated m times to minimize the torsion-rotation coupling in the molecule-fixed axis system. This is achieved by counterrotation of the molecular axis system by -ρτ when the methyl rotor is rotated by +τ [7].

Figure 7:

Schematic view of the six equivalent configurations in methylamine (frameworks) and the path between subsequent configurations under the symmetry operation a = (123)(45).

The formalism proposed for methylamine by Ohashi and Hougen [7] treats each analyzed state as an isolated one, which is split into several sublevels of A₁, A₂, B₁, B₂, E₁ and E₂ symmetry and the Hamiltonian operator is defined as follows:

where higher order term represents expansion of terms in J(J + 1) and/or K² as it is in a standard expression for the asymmetric top molecule (ordinary centrifugal distortion terms). For instance, the term h₁ has the following form:

The Hamiltonian contains effective terms which include ΔK = 0, ±1, ±2, ±4 operators. The coefficients h, f, s, r, f⁽²⁾ depend on vibrational coordinates and are multiplied by rotational operators.

Moreover, each term in eq. (2) is expanded in a Fourier series describing different interactions between multiple minima in the inversion-torsion potential surface (Figure 2).

The c = h, f, r and s coefficients are expended as a cosine series presented below

and the q coefficient as a sine series

where n is the ordinal number of a molecular configuration interacting with the configuration number 1, and τ the angle depending on the type of interaction and symmetry species.

In eq. (4) and eq. (5), the n-th term represents a tunneling from framework 1 to framework n. The subscript n = 1 corresponds to nontunneling motion (τ angle is 0 in Figure 7). The even subscripts n = 2,4,6 … describe tunneling involving the inversion of the NH₂ group whereas the odd ones n = 3,5,7 … refer to tunneling involving mainly the internal rotation of the CH₃ group. The respective parameters used in the fitting model (molecular constants describing the observed spectrum of methylamine) are defined as h_n, f_n, q_n etc.

The experimental data are the frequencies of the transitions between the energy levels of a studied molecule and the role of the theoretical model is to reproduce the observed frequencies. As a result, the spectroscopic parameters with information on the energy structure of the system are set. The Ohashi–Hougen Hamiltonian operator was applied successfully in fitting the ground vibrational state of methylamine [21] and its first excited torsional state [6, 22, 23]. For other states of methylamine, i. e. the wagging or C-N stretching states, the results were not as good. In these cases it was not possible to fit the line within experimental accuracy. The main reason for the failure is the coupling between the wagging, ν₉, and the third excited torsional, 3ν₁₅, states or between C-N stretching and the fourth excited torsional (4ν₁₅) states, which will be in more detailed described in the subsequent chapters. Thus, in order to obtain a better fit result, it is necessary to take into account at least two states, which interact with each other and a global fit should be conducted then. One can think also of another reason for which the tunneling model gives worse fitting results. For higher rovibrational states the inversion-torsion splittings get higher than for the ground state or the first excited torsional state, i. e. the higher are the rovibrational states, the smaller are the barriers between the equilibrium minima and the inversion torsion functions are no longer well localized in each minimum, but get dispersed. The inversion torsion functions can be presented by the probability density, and Figure 8 shows the situation described above.

Figure 8:

The probability density of the inversion torsion functions for J = K = 0 rotational states of A₁ symmetry in (a) the ground vibrational state, and (b) first and (c) second excited torsional states of methylamine.

6 Assignment and analysis of the wagging state, ν₉

The rotational structure of the ground state of methylamine has been extensively studied since 1957 [24]. An analysis of tunneling-rotational levels of the first excited inversion (ν₉ wagging) state of methylamine was needed for understanding of a complex dynamics of the molecule. The first high-resolution study of the ν₉ wagging band was performed in 1992 [25]. Since the transition moment of the wagging vibration is neither perpendicular nor parallel to axis of the top, it consists of both components. The wagging band has a hybrid structure which is further modified through splitting due to inversion and internal rotation of the methylamine molecule.

The spectrum of the ν₉ wagging band spreads from 640 to 960 cm⁻¹ (Figure 9). The wagging band consists of perpendicular and parallel components. The first identified series were the ^RR series belonging to perpendicular component of the band on the high frequency side of the spectrum along with their weaker ^PP counterparts on the low frequency side [25]. At the band centre, ^QQ branches spread over some region of the spectrum, which is the effect of the torsion and inversion. As for the parallel component of the wagging band, the ^QR and ^QP branches are generated and depending on the K values, some of them are even stronger in intensity than the perpendicular components but to assign these series was not that straightforward since they have their beginnings at the very center of the band, where the lines are overlapped and really dense.

Figure 9:

A part of the wagging band, ν₉, of methylamine in the FTIR spectrum.

In the first attempt to assign and analyze the wagging band of methylamine, the best fit that could be achieved was of standard deviation of 0.0095 cm⁻¹ for almost 5000 lines and K ≤ 10 and J ≤ 15 [25]. Many line series have been found, identified and fit, even those of A and E₂ symmetry (weaker components). Still some unanswered questions remained, even for stronger components in intensities of B and E₁ symmetry. Some series were missing. In an independent approach [26, 27] about 8800 transitions were assigned but still some transitions could not be identified.

It was believed that some levels of the wagging state were in a strong resonance with some “dark” states. By the “dark” state one can understand the state which resonates with so called “bright” state (in this case it is the ν₉ state). Perturbations are the main complication in the analysis of asymmetric rotor bands and the mostly encountered perturbations are caused by Coriolis interaction and/or Fermi-type resonance. Each perturbation expresses the mixing of a “bright” rotational level of the vibrationally excited state with rotational levels of nearly located “dark” vibrational states. When a perturbation mixing takes place, the “dark” rotational levels borrow intensities from the bright state, hence they become observable transitions in the spectrum. Perturbations cause frequency shifts of the observed transitions. The candidates for “dark” states in the wagging region seemed to be the third excited torsional state (especially the upper part of the torsional splitting) and the fourth torsional state (the lower part of the torsional splitting), which might interact with the wagging state (Figure 10).

Figure 10:

Energies of the inversion-torsional states of CH₃NH₂ (presented in paper [6]).

Having this insight into possible interactions, the wagging band of methylamine was put through the analysis in a new high-resolution spectrum [28]. In general, the assignment of the band was extended to about 13,000 transitions up to J = 40 in most series and many of the missing lines in previous studies were identified. The only two series that still were not found were of the A (K = 0) and E₂₊₁ (K = 3). The reason for which these two could not be assigned is the perturbation. The resulting standard deviation was 0.018 cm⁻¹, which is not within the experimental accuracy and again the perturbation is blamed for not perfect result of the fit. Figure 11 shows one of the series that was luckily found and assigned to K = 0 of B symmetry, the ^PQ₁(B) branch. One can notice that the branch changes its way twice, which is also reflected in its counterparts’ shapes, the braches P and R. That made it very difficult to find these series. The series of K = 0 of B symmetry are shifted of 4 cm⁻¹ because of the strongest perturbation taking place in the wagging band. Going back to the indicated perturbers (“dark” states), the 3ν₁₅ can be involved only in local Coriolis interactions since it has a different symmetry with respect to the reflection in the symmetry plane of the NH₂ group. As for the 4ν₁₅ state, it can cause both the Coriolis and the Fermi types of interactions with the wagging state, because it has the same symmetry as the wagging state [29].

Figure 11:

Part of the methylamine spectrum showing the structure of the ^PQ₁(B) branch. The labels show the values of J′(taken from paper [28]).

To overcome the difficulties with the fitting of all the assigned and perturbed data, the theoretical model should consist of two or even three interacting states. The effective Hamiltonian for inversion-torsion coupling in methylamine has been already created and will be implemented soon. Such a Hamiltonian treats explicitly the coupling between two or three states. The approach is based on the idea of the group-theoretical Hamiltonian applied simultaneously to two mutually perturbing states.

Concluding, the inversion band has been successfully reanalyzed and completed [28].

7 Assignment and analysis of the C-N stretching band of methylamine, ν₈

The analysis of the C-N stretching band was the next step of a systematic study on the high-resolution IR spectrum of methylamine, CH₃NH₂. The C-N stretching band spreads in the spectrum from 960 to 1200 cm⁻¹.

The C-N stretching vibration is a parallel A-type band with the change of dipole moment along the axis of least moment of inertia. Thus, the K-selection rule is ΔK = 0 and the J-selection rules are ΔJ = −1, 0 or +1 and they correspond to IR transitions labeled as P, Q or R, respectively. In the spectrum, the C-N stretching band turns up as a strongly overlapped structure with lines of relatively weak intensities, which makes the assignment rather hard (Figure 12). The C-N stretch band is centered at about 1044 cm⁻¹ with dense J-multiplets in the P and R branches and a tight Q branch. It was not straightforward to assign K values to transitions properly, and before the unambiguous results were obtained with help of the GSCDs, a lot of attempts of assignment had to be made.

Figure 12:

A part of the C-N stretch band, ν₈, of methylamine in the FTIR spectrum.

Despite many difficulties with correct labeling of the transitions, over 3500 lines with a resolution of 0.001 cm⁻¹ for K from 0 to 12 have been assigned [30, 31]. All allowed transitions for B symmetry species were well assigned up to K′ = 12 (K′ denotes K value of the higher state), since the B lines are of the highest intensities. The assignment of E₁ transitions was more troublesome comparing to the B symmetry species, although the lines of both symmetries are more or less of similar intensities. Just to remind, the relative intensities of individual lines presented by statistical weights are the following: A₁:A₂:B₁:B₂:E₁₊₁:E_1-1:E₂₊₁:E_2-1 = 1:1:3:3:3:3:1:1. Finally the transitions up to K′ = 11 and 9 for E₁₊₁ and E_1-1 symmetries were assigned, respectively. These difficulties with identification could be caused by many irregularities appearing in the spectrum and shown in Figure 13 and Figure 14 (taken from paper [31]) and by some resonances in the C-N stretch region.

Figure 13:

Part of the C-N stretching band in the – range showing the J = 4←5 manifold of the ^QP branch consisting of lines for K′ = 1 values and different symmetry species.

Figure 14:

Part of the ν₈ band in the – range showing the structure of the ^QP branch for J = 8 and B symmetry for different K values.

As for irregularities mentioned, in Figure 13 one can see a part of the C-N stretching band with the J = 4←5 manifold of the ^QP branch consisting of lines for K′ = 1 values and different symmetry species. It can be noticed that due to accidental overlapping some lines have higher intensities than predicted from the statistical weights. Moreover, in Figure 14 a structure of the ^QP branch for J = 8 and B symmetry for different K values is presented where there is no simple correlation between the position of the line and the value of K.

Apart from identification of the lines of B and E₁ symmetries, also the low-intensity transitions to which A and E₂ symmetry species transitions belong were observed and successfully assigned. For A symmetry, all series up to K′ = 9 have been identified whereas for E₂₊₁ and E_2-1 symmetry up to K′ = 8. There is only lack of the K′ = 5 series for the E_2-1 symmetry. All assignments have been confirmed by the GSCD and are consistent with the experimental laser lines [32, 33, 34], and are available in the paper on the analysis of the C-N stretching band [31].

Not only the transitions belonging to the ν₈ band were identified but also the transitions, which were assigned to K′ = 4 series of E_2-1 symmetry species of a “dark” state. To check possible resonances in the excited states of methylamine the inversion-torsion-rotation energy levels were calculated for selected J = K values from the two-dimensional potential surface [5].

In Figure 15 the energies of the inversion-torsional states are presented: the first excited torsional state-ν₁₅, the second excited torsional state-2ν₁₅, the third excited torsional state-3ν₁₅, the fourth excited torsional state-4ν₁₅, the excited inversion state-ν₉, the combination state-ν₉+ν₁₅ and the C-N stretching state-ν₈. The energies were calculated for values up to J = K = 5. For the C-N stretching state, its energies were calculated using the effective Hamiltonian and all experimental data. Figure 15 shows a diagram with energy ladders for all indicated states for K = 4 where the nearest neighboring states to the C-N stretching state are the combination state ν₉+ν₁₅ and the 4ν₁₅ state, thus these states were taken into consideration as “dark” states.

Figure 15:

Calculated energy levels schemes of CH₃NH₂ for J = K = 4 for different symmetry species (GS – ground state; ν₁₅, 2ν₁₅, 3ν₁₅, 4ν₁₅ – excited torsional states; ν₉ – excited wagging; ν₈ – C-N stretching state; ν₉+ν₁₅, ν₉ + 2ν₁₅ – combination states).

The perturbation observed in the C-N stretching for K′ = 4 for the E_2-1 symmetry seemed to be a Fermi-type resonance since all lines of the series are shifted of about 0.4 cm⁻¹. It was proved that a Fermi-type resonance is allowed only between states of the same type, for A′-A′ or A′′-A′′ states [29]. The C-N stretching state, ν₈, is an A′ type state, the inversion-torsion combination state, ν₉+ν₁₅, is an A′′ type state, and the fourth excited torsional state, 4ν₁₅, is an A′ type state. Concluding, the state responsible for the perturbation in K′ = 4 series of E_2-1 symmetry, is the 4ν₁₅. Between the ν₈ and the ν₉+ν₁₅ states only Coriolis coupling is possible. Many other perturbations are also observed, some Fermi-type resonances as well as local Coriolis interactions.

Taking into account all the experimental data and the resonances that occur in the C-N stretch spectrum, the fit was performed using the Hougen–Ohashi single state model with the final result of 0.04 cm⁻¹ [31]. The band centre was determined at 1044.8134 cm⁻¹. The group theoretical Hamiltonian is dedicated to fit the data of one state only. As the observed vibrational state is in resonance with other vibrational states, the Hougen–Ohashi Hamiltonian gives the results for the C-N stretching state, ν₈, which differ from the experimental data.

In any case, it was possible to assign the transitions in the C-N stretching region of the spectrum for the first time and fit all the identified data in spite of the difficulties caused by weak line intensities, perturbations and strong lines overlapping.

At the same time, the C–N stretching infrared fundamental of CH₃NH₂ has been investigated by R. M. Lees et al. [35] using high-resolution laser sideband and Fourier transform synchrotron spectroscopy. In that work, many sub-bands have been assigned for K values ranging also up to 12 for the stronger B and E₁ inversion species for the v_t = 0 torsional state, along with many of the weaker sub-bands of the A and E₂ species. Both ground and C–N stretch origins were fitted to a phenomenological Fourier series model. It was noticed that the amplitude of the torsional energy oscillation increased substantially for the C–N stretching, while the amplitude of the inversion energy oscillation did not change. The C–N stretching vibrational energy was determined at 1044.817 cm⁻¹, whereas the effective upper state B-value at 0.7318 cm⁻¹. Several anharmonic resonances with v_t = 4 ground-state levels (the fourth excited torsional state) have been observed and partially identified. The interaction coupling constants were determined from J-localized level-crossing resonances.

8 Assignment and analysis of the first excited torsional state, ν₁₅

The torsional vibration, ν₁₅, is the lowest frequency vibration in the methylamine molecule. In comparison to the assignments of the wagging band and C-N stretching, the assignment of the first torsional band is different since it is not much perturbed. As it is shown in Figure 10, the first excited torsional state is well isolated from other vibrational states, i. e. the second, third and fourth excited torsional state (2ν₁₅, 3ν₁₅, 4ν₁₅), the inversion state, ν₁₅ or the C-N stretch, ν₈.

The ν₁₅ spectrum of CH₃NH₂ shown in Figure 16 ranges from 100 to 360 cm⁻¹. The center of the band is determined at 264 cm⁻¹. The band is a B-type band, since the changing dipole moment lies in the axis of intermediate moment of inertia (b axis) and thus the symmetry rules are with ΔK = ±1 and ΔJ = 0, ±1. The respective IR transitions are labeled as P, Q or R. The type B bands do not have a strong central Q branch. The Q lines spread over and overlap the P and the R branches. The high-frequency side (the right part) of the spectrum is dominated by strong, well resolved ^rQ branches and ^rR branch series. The low-frequency side (the left part) of the band consists of a group of weak lines, mainly of the ^pP an ^pQ series. Near the band center, the Q branches spread over a spectrum region and many weak lines (of A and E₂ symmetry) are covered by stronger lines, which makes the identification process hard.

Figure 16:

A part of the first torsional band of methylamine, ν₁₅, in the FTIR spectrum.

The analysis of the first torsional band started in 1988 [22], then a year later was reinvestigated [23], and much improved in 2016 [6]. Since some of the series of the first torsional band were already identified, the reassignment began with the previously assigned lines, which most of them were the intense ones of B and E₁ symmetry. After the lines have been reassigned, the energy file needed for the LWW was recalculated and the next step of identification started, to find completely new lines. Although the lines, especially the weak ones, were hidden in a dense spectrum, it was possible to assign almost 12,000 transitions and finally fit them with the standard deviation σ = 0.00079 cm⁻¹ using the single state effective Hamiltonian with the set of 88 molecular parameters [6]. It should be mentioned at this point, that apart from the IR data assigned in the first torsional band, also the data from previously measured MW and IR pure rotational transitions were taken into account in the fitting [22, 23]. All the assignments were confirmed by GSCDs. The single state model worked properly, mostly because of the fact that the ν₁₅ state lies in the isolated region and is not expected to be in the interaction with other states.

In the region of the first torsional band, several hot bands are observed, ν₁₅→2ν₁₅, ν₁₅→3ν₁₅, ν₁₅→4ν₁₅ [36, 37]. In order to analyze these hot bands, it is essential to have the precise knowledge on rovibrational energies of the first torsional state. The effective parameters obtained from the final fit of ν₁₅ will be used for calculations of rovibrational energies to facilitate the analysis of the hot bands. Without this information, the Lower State Combination Differences could not confirm the assignment of the hot bands originating from the first torsional state.

9 Assignment and analysis of the asymmetric CH-stretching state, ν₁₁

As far as assignment of higher frequency vibrational bands of LAV molecules is concerned, it is still a challenge. In that region (near 3000 cm⁻¹) the density of vibrational states is so high that there are many interactions between them, resonances, which hinder identification and assignment. For methylamine, only one of the high frequency bands was studied, i. e. the asymmetric CH stretching band, ν₁₁ (in Table 1 it is denoted as CH₃ d-stretch with its experimental vibrational frequency of 2985 cm⁻¹) [38].

The high-resolution infrared spectrum of methylamine in the CH stretching fundamental band region was recorded using slit-jet direct absorption spectroscopy with the resolution of 0.0025 cm⁻¹ [38]. The region covered by the ν₁₁ band ranges from 2965 to 3005 cm⁻¹. The transition dipole moment of the ν₁₁ band moves along the b-rotational axis, thus the rules for the b-type transitions are as follows: ΔK = ±1 and ΔJ = 0, ±1. The spectrum in this region was difficult to assign since apart from the complexity caused by torsion and inversion tunneling, like in lower frequency vibrations (NH₂ wagging, CN stretching and the torsion), it is strongly perturbed by “dark” states. In the spectrum, there are multiplets of 2 or 3 mixed bright and dark states [38]. Since many perturbations occur in that region, one does not observe the complete series of lines, only fragments of the branches are visible. With help of the deperturbation theory and LSCDs [38], it was possible to assign over 600 lines in the ν₁₁ band. It was found out that in the CH stretching band the torsion-inversion levels appeared in a different order than in the ground state (in the ground state the order of the energy levels tunneling multiplet follows the order: A₁<B₁<E₁<E₂ [39]), which is also the case for other vibrational states, but in the ν₁₁ state, the levels order does not correspond to the observed one. It is so, because in so perturbed region probably not all the dark states were identified and taken into account.

10 Summary

High-resolution IR rovibrational spectra of floppy molecules like methylamine are still, despite the advances in the experimental techniques, very complex to analyze. Each rovibrational analysis requires individual approach, the assignments and studies of LAV molecules spectra cannot be unified. It was seen in preceding chapters describing analyses of the methylamine rovibrational spectra that each spectral region was different and involved diverse problems. It is very important for complex studies to do the assignments in a systematic way. For instance, with a well-determined ground state of methylamine [21, 39], it was possible to perform further analyses (for instance for NH₂ wagging [25, 28], CN stretching [30, 31, 35]). After the successful studies on the first excited torsional state [6, 22, 23], one can gradually move toward the next spectral regions, for instance, the studies of the torsional overtones like the second excited torsional state, 2ν₁₅ [36, 37, 40] then in turn the third excited torsional state, 3ν₁₅, etc. As the next step the analyses of hot bands appearing in these regions should be performed and finally it would be worthwhile going back to resonances in the region of about 1000 cm⁻¹ where the previous fit results were not perfect in spite of some resonances [28, 31].