Article pubs.acs.org/JPCA
First Predictions of Rotationally Resolved Infrared Spectra of Dideuteromethane (12CH2D2) From Potential Energy and Dipole Moment Surfaces Michael̈ Rey,*,† Andrei V. Nikitin,‡ and Vladimir G. Tyuterev† †
Groupe de Spectrométrie Moléculaire et Atmosphérique, UMR CNRS 7331, BP 1039, F-51687, Reims Cedex 2, France Laboratory of Theoretical Spectroscopy, Institute of Atmospheric Optics, SB RAS, 634055 TOMSK, Russia
‡
ABSTRACT: We report the variationally computed infrared spectrum of 12 CH2D2 using our recent potential energy and dipole moment methane surfaces, which have been initially derived in the irreducible tensor representation adapted to the tetrahedral symmetry of the major isotopologue 12CH4. The nuclear motion calculations are accomplished by combining the normal-mode Eckart−Watson Hamiltonian with isotopic and symmetry transformations. Our direct vibrational calculations are compared to the 93 observed band centers up to 6300 cm−1. Except for two outliers the root-mean-square deviation is 0.22 cm−1 and the maximum error is 0.7 cm−1 without empirical adjustment of parameters. The work aims at filling the gap concerning missing line strength information for this molecule. Theoretical spectra predictions are given up to J = 25 and, for the very first time, ab initio intensity predictions for rovibrational line transitions are in good qualitative agreement with available experimental spectra.
I. INTRODUCTION Up to a few years ago, accurate quantum mechanical spectra predictions using full dimensional ab initio potential energy surfaces (PES) and dipole moment surfaces (DMS) were only feasible for molecules with N ≤ 4 atoms.1−13 This limitation was essentially due to a large number of degrees of freedom with increasing N that have to be treated simultaneously, making computations both for the electronic and nuclear motion calculations very demanding. The main challenge concerns (i) electronic structure calculations on an extensive grid of nuclear configurations and (ii) the efficiency of theoretical methods for global vibration−rotation spectra calculations using sufficiently large basis sets. For N ≥ 5, the very high dimensionality of the full nuclear motion problem requires a development of efficient optimization methods for converging vibrational levels even at relatively low energy ranges.14−27 This implies using appropriately defined sets of coordinates for the PES and for the kinetic energy terms to perform variational or perturbative calculations. It has been recognized for many years that the systems of coordinates may be described either using curvilinear (bondlength, bond-angles, Jacobi, Radau, or hyperspherical)28−38 or rectilinear, normal coordinates.39 The full account of all molecular symmetry properties is also one of the key issues for efficient calculations. To this end, a method based on the extensive use of irreducible tensor operators25,40−42 for computing rotationally resolved spectra using the Eckart− Watson normal-mode Hamiltonian43 has been recently developed. With the increasing power of modern computers, variationally computed infrared spectra at room temperature of 5 and 6 atomic systems as ketene,16 methane and its isotopologues,23−25 or ethylene18,44,45 are now feasible. © XXXX American Chemical Society
Accurate knowledge of spectra for methane isotopic species is important in various domains of science and applications, including chemistry and dynamics,46−49 laser excitations, atmospheric optics, planetology,50,51 and astrophysics.52,53 Experimental spectra analyses commonly employ effective Hamiltonian models using empirically fitted parameters to observed data.54,55 Such models are usually defined on a small, finite-dimensional subspace spanned by a set of strongly interacting vibrational levels-the so-called polyads-making thus possible numerically exact calculations. A significant progress has been achieved in spectra assignments and data reduction both for line positions and intensities,56−59 particularly for 12CH4 and 12 CH3D. However, these empirical models covering various spectral ranges as well as isotopomers generally have a relatively large number of parameters. Many of them may be poorly defined, particularly for states coupled by resonance interactions: a determination of coupling parameters from experimental levels is known to be mathematically ill-defined problem.60,61 Extrapolation capabilities of effective spectroscopic models critically depend on the range of available data included in the fit, limited information for inactive bands, dark state perturbations and various ambiguity issues. Better understanding of the methane dynamics would benefit of detailed studies of exited ro-vibrational states of low-symmetry isotopologue CH2D2 as much additional information can be extracted from spectra compared to its symmetric-top and Received: January 20, 2015 Revised: April 21, 2015
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DOI: 10.1021/acs.jpca.5b00587 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A spherical-top isotopic counterparts.62,63 However, high resolution spectra for the “mother molecule” are generally much more investigated and ro-vibration parameters are better determined on larger wavenumber range than those of isotopic species.55 For low-lying vibrational bands of CH2D2 there exists many experimental studies and spectra analyses,63−73 but this is in contrast with a limited knowledge of line intensities, except for measurements and calculations of vibrationally induced permanent dipole moment.62,74 Information on line parameters of CH2D2 is totally missing in HITRAN55,75 and in other specialized spectroscopic databases76,77 as well. The present work aims at studying the isotopic effects under the double H → D substitution and at filling the gap for CH2D2 line strength information in the infrared by global variational predictions. This could be also helpful for a better understanding of intramodes coupling, for the characterization of dark states and for modeling experimental rotationally resolved spectra at various temperature conditions.
12
A
B
CH4 (Td) → 12CH4 (C2v) → 12CH 2D2 (C2v)
will be explicitly considered hereafter and useful relations corresponding to steps A and B will be also derived. A comparison of energy levels with experimental data is given in section IV and the discussion of results in section V.
III. TREATMENT OF METHANE 12CH4 IN C2V SUBGROUP: STEP A Frames and Axis Switching. Degenerate vibrational modes of the “mother” CH4 methane isotopologue are decomposed into nondegenerate ones under the double H → D substitution. CH2D2 has nine vibrational normal modes qk transforming according to one-dimensional irreducible representations (irreps) of C2v point group as A1 (q1, q2, q3, q4), A2 (q5), B1 (q6, q7) and B2 (q8, q9). As it will be described below the correlations with the Td symmetry types of CH4 will be readily established by means of appropriate molecular fixed-frame rotations. Isotopic relations for molecular parameters have been discussed in many works, particularly in the frame of the local-mode86,87 and Expanded Local Mode approximations.88,89 Our approach is different. For the variational normal mode computation of isotopic spectra, a systematic algorithm for all transformations involving internal and normal coordinates which are necessary to transpose PES and DMS expressions for isotopic species in a numerically exact way at any orders was necessary. Let us first consider body-fixed Eckart frames90 associated with standard axis conventions for Td and C2v molecules. The orthogonal matrix C, specifying the Eckart frame is a pure rotation matrix and, in cases of nonlinear and nonplanar molecules, reads91,92
II. VARIATIONAL COMPUTATION OF METHANE ISOTOPIC SPECTRA Many ab initio studies have been devoted to the determination of methane PES62,78−81 and DMS.81,82 Large amplitude nuclear motion in methane in relation with isotopic effects and dynamics has been considered by Marquart, Quack, and co-workers.62 In our previous works we have reported accurate variational rotationally resolved spectra calculations for 12CH4,24 13CH4,23 CH3D,25 and CD424 both for line positions and intensities. Theoretical line lists for hot methane have been generated for astrophysical applications: ExoMol linelist in UCL London83 and Reims−Tomsk RNT linelist,84 the latter one being calculated up to 2000 K. Recently Wang and Carrington22 have empirically refined ab initio methane PES of Schwenke80 by fitting it to 40 experimental band centers and computed vibrational levels of Td and C3v isotopic species using contracted basis functions with Lanczos-type eigensolvers. Variational calculations of CH2D2 spectra are not yet available. Because of the lower C2v symmetry, the sizes of the basis sets required for the convergence of rovibrational states are larger and variational calculations are more demanding than for Td and C3v species. In this work we report first ab initio calculations of CH2D2 intensities in the infrared spectral range using our recent NRT PES81 and DMS.85 Though the calculations were carried out in the framework of the Born− Oppenheimer (BO) approximation using the same PES and DMS for all methane isotopic species, a preliminary study of axes and coordinate transformations was necessary (section II). This is because our PES and DMS have been initially derived in the irreducible tensor representation adapted to the tetrahedral symmetry of the major isotopologue 12CH4. Another reason is that we use normal mode mass-dependent coordinates which have to be precisely transformed to their counterparts under isotopic substitutions. In this work we employ the approach similar to that previously developed for C3v species25 and applied to CH3D but adapted here to C2v symmetry. This consists in the decomposition of the problem in two steps (A + B). First, the Td → C2v symmetry transformations is artificially considered for the mother molecule. With this step (A) we were able to produce line position and intensity calculations of 12CH4 in C2v subgroup and to check the validity of all analytical transformations by comparison with our previous Td calculations.24 Then we will focus on the isotopic transformation in the normal coordinate framework (section III). Accordingly, the following transformation
̃ )−1/2 C = F(FF
(1)
where ∼ is the transpose of the matrix. F is a matrix whose columns are the Eckart vectors, and F̃F is the symmetric Gram matrix positively defined. For tetrahedral molecules belonging to the Td point group we follow the standard conventions which are used in the spectroscopy literature to define the frame (x, y, z) attached to the molecule (MFF1). The quantization axis (Oz) is taken as one of the S4 axes such that the coordinate of the corresponding hydrogen atom (labeled as 1) is positively defined. The C2v molecular fixed frame (MFF2) (x′,y′,z′) is attached to the molecule using the Ir representation.93 We thus choose the direction (Ox′) along the C2 symmetry axis which intersects the two planes 7 1 and 7 2 containing atoms 1,2 and 3,4 respectively. The (Oy′) axis is in the plane 7 1 and (Oz′) in 7 2. Both MFF1 and MFF2 are linked together through a π/4 rotation around the z = x′ axis. We thus write (x′, y′, z′)t = 9 x −1 × (x , y , z)t
(2)
with ⎛ ⎜0 ⎜ 9x = ⎜ ⎜0 ⎜⎜ ⎝1
1 1 ⎞ − ⎟ 2 2⎟ 1 1 ⎟ ⎟ 2 2 ⎟ ⎟ 0 0 ⎠
(3)
In terms of the group theory, the (x,y,z) system spans the F2 irrep of the Td point group while x′, y′ and z′ transform as the A1, B1 and B2 irreps of C2v, respectively. The total angular momentum J is a axial vector and transforms as the F1 irrep of Td which B
DOI: 10.1021/acs.jpca.5b00587 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A subduces as Td ↓ C2v = A2+B1+B2. Its Cartesian components are expressed in the MFF2 as (Jx(A′ 2) , J y(B′ 2) , Jz(′B1))t = 9 x −1 × (Jx(F1) , J y(F1) , Jz(F1))t
Watson Hamiltonian adapted to the Td point group simply transforms as 9 x,
