The rotation–vibration spectrum of methyl fluoride from first principles†
Received
16th March 2018
, Accepted 15th May 2018
First published on 17th May 2018
Accurate ab initio calculations on the rotation–vibration spectrum of methyl fluoride (CH_{3}F) are reported. A new ninedimensional potential energy surface (PES) and dipole moment surface (DMS) have been generated using highlevel electronic structure methods. Notably, the PES was constructed from explicitly correlated coupled cluster calculations with extrapolation to the complete basis set limit and considered additional energy corrections to account for corevalence electron correlation, higherorder coupled cluster terms beyond perturbative triples, scalar relativistic effects, and the diagonal Born–Oppenheimer correction. The PES and DMS are evaluated through robust variational nuclear motion computations of pure rotational and vibrational energy levels, the equilibrium geometry of CH_{3}F, vibrational transition moments, absolute line intensities of the ν_{6} band, and the rotation–vibration spectrum up to J = 40. The computed results show excellent agreement with a range of experimental sources, in particular the six fundamentals are reproduced with a rootmeansquare error of 0.69 cm^{−1}. This work represents the most accurate theoretical treatment of the rovibrational spectrum of CH_{3}F to date.
1 Introduction
Methyl fluoride (CH_{3}F) was one of the first fiveatom systems to be treated by fulldimensional variational calculations^{1} after a period of pioneering studies on polyatomic molecules.^{2} The field has since gone from strength to strength and accurate rotation–vibration computations on small molecules are nowadays fairly routine.^{3} This has enabled a range of applications such as the production of comprehensive molecular line lists to model hot astronomical objects,^{4–6} to probing fundamental physics and a possible spacetime variation of the protontoelectron mass ratio.^{7–10}
The starting point of any variational calculation is the potential energy surface (PES) and its quality will largely dictate the accuracy of the predicted rovibrational energy levels, and to a lesser extent, the transition intensities. Thanks to sustained developments in electronic structure theory it is now possible to compute vibrational energy levels within ±1 cm^{−1} from a purely ab initio PES.^{11–17} To do so requires the use of a oneparticle basis set near the complete basis set (CBS) limit and the treatment of additional, higherlevel (HL) corrections to recover more of the electron correlation energy.^{18,19} Similarly, transition intensities from first principles, which requires knowledge of the molecular dipole moment surface (DMS), are now comparable to, if not more reliable in some instances, than experiment.^{20,21}
Although the rovibrational spectrum of CH_{3}F has been well documented, its theoretical description is not reflective of the current stateoftheart in variational calculations. Notable recent works include the PESs and energy level computations of Manson et al.,^{22,23} Nikitin et al.^{24} and Zhao et al.^{25,26} Theoretical CH_{3}F spectra are also available from the TheoReTS database^{27} for a temperature range of 70–300 K but details on the calculations are unpublished except for the PES.^{24} In this work, highlevel ab initio theory is used to generate a new PES and DMS for CH_{3}F. The surfaces are represented by suitable symmetrized analytic representations and then evaluated through robust variational nuclear motion calculations. Computed results are compared against a variety of experimental spectroscopic data to provide a reliable assessment of our theoretical approach.
The paper is structured as follows: in Section 2, the electronic structure calculations and analytic representation of the PES are presented. Likewise, the details of the DMS are given in Section 3. The variational calculations are described in Section 4. Our theoretical approach is then assessed in Section 5 where we compute the equilibrium geometry of CH_{3}F, pure rotational energies, vibrational J = 0 energy levels, vibrational transition moments, absolute line intensities of the ν_{6} band, and the rotation–vibration spectrum up to J = 40. Concluding remarks are offered in Section 6.
2 Potential energy surface
2.1 Electronic structure calculations
Similar to our previous work on SiH_{4}^{16} and CH_{4},^{17} the goal is to construct a PES which possesses the “correct” shape. Obtaining tightly converged HL energy corrections with respect to basis set size is less important. Using a focalpoint approach,^{28} the total electronic energy was represented as 
E_{tot} = E_{CBS} + ΔE_{CV} + ΔE_{HO} + ΔE_{SR} + ΔE_{DBOC}.  (1) 
The energy at the CBS limit E_{CBS} was computed with the explicitly correlated F12 coupled cluster method^{29} CCSD(T)F12b in conjunction with the F12optimized correlation consistent polarized valence basis sets, ccpVTZF12 and ccpVQZF12.^{30} Calculations employed the frozen core approximation with the diagonal fixed amplitude ansatz 3C(FIX)^{31} and a Slater geminal exponent value of β = 1.0a_{0}^{−1}.^{32} The OptRI,^{33} ccpV5Z/JKFIT^{34} and augccpwCV5Z/MP2FIT^{35} auxiliary basis sets (ABS) were used for the resolution of the identity and the two density fitting basis sets, respectively. Unless stated otherwise calculations were performed with MOLPRO2012.^{36}
A parameterized, twopoint formula^{32} was chosen to extrapolate to the CBS limit,

E^{C}_{CBS} = (E_{n+1} − E_{n})F^{C}_{n+1} + E_{n}.  (2) 
The coefficient
F^{C}_{n+1} is specific to the CCSDF12b or (T) component of the total CCSD(T)F12b energy and values of
F^{CCSDF12b} = 1.363388 and
F^{(T)} = 1.769474 were chosen.
^{32} The Hartree–Fock (HF) energy was not extrapolated, rather the HF + CABS (complementary auxiliary basis set) singles correction
^{29} computed in the larger basis set was used.
The contribution from corevalence (CV) electron correlation ΔE_{CV} was determined using the CCSD(T)F12b method with the F12optimized correlation consistent corevalence basis set, ccpCVTZF12.^{37} The same ansatz and ABS as in the E_{CBS} calculations were employed, however, the Slater geminal exponent was set to β = 1.4a_{0}^{−1}.
To account for higherorder (HO) correlation we employed the hierarchy of coupled cluster methods such that ΔE_{HO} = ΔE_{T} + ΔE_{(Q)}. The full triples contribution ΔE_{T} = E_{CCSDT} − E_{CCSD(T)}, and the perturbative quadruples contribution ΔE_{(Q)} = E_{CCSDT(Q)} − E_{CCSDT}. Calculations with the CCSD(T), CCSDT, and CCSDT(Q) methods were carried out in the frozen core approximation using the general coupled cluster approach^{38,39} as implemented in the MRCC code^{40} interfaced to CFOUR.^{41} The full triples and perturbative quadruples utilized the correlation consistent ccpVTZ and ccpVDZ basis sets,^{42} respectively.
Scalar relativistic (SR) effects ΔE_{SR} were obtained using the secondorder Douglas–Kroll–Hess approach^{43,44} at the CCSD(T)/ccpVQZDK^{45} level of theory in the frozen core approximation. The spin–orbit interaction was not considered for the present study as this can be ignored for light, closedshell molecules in spectroscopic calculations.^{46}
The diagonal Born–Oppenheimer correction (DBOC) ΔE_{DBOC} was computed with all electrons correlated using the CCSD method^{47} implemented in CFOUR with the augccpCVDZ basis set. Because the DBOC is mass dependent its inclusion means the PES is only applicable for ^{12}CH_{3}F.
Grid points were generated randomly using an energyweighted sampling algorithm of Monte Carlo type. The global grid was built in terms of nine internal coordinates: the C–F bond length r_{0}; three C–H bond lengths r_{1}, r_{2} and r_{3}; three ∠(H_{i}CF) interbond angles β_{1}, β_{2} and β_{3}; and two dihedral angles τ_{12} and τ_{13} between adjacent planes containing H_{i}CF and H_{j}CF. All terms in eqn (1) were calculated on a grid of 82653 points with energies up to hc·50000 cm^{−1} (h is the Planck constant and c is the speed of light) and included geometries in the range 1.005 ≤ r_{0} ≤ 2.555 Å, 0.705 ≤ r_{i} ≤ 2.695 Å, 45.5 ≤ β_{i} ≤ 169.5° for i = 1, 2, 3 and 40.5 ≤ τ_{jk} ≤ 189.5° with jk = 12, 13.
Computing the HL corrections at each grid point is computationally demanding but given the system size and chosen levels of theory is actually timeeffective. Since the HL corrections vary in a smooth manner and are relatively small in magnitude, see Fig. 1 and 2, an alternative strategy would be to compute the HL corrections on reduced grids, fit suitable analytic representations to the data and then interpolate to other points on the global grid.^{13,15} This approach can be advantageous for larger systems or more computationally intensive electronic structure calculations. However, an adequate description of each HL correction requires careful consideration and is not necessarily straightforward. These issues are avoided in the present work.

 Fig. 1 Onedimensional cuts of the corevalence (CV) and higherorder (HO) corrections with all other coordinates held at their equilibrium values.  

 Fig. 2 Onedimensional cuts of the scalar relativistic (SR) and diagonal Born–Oppenheimer (DBOC) corrections with all other coordinates held at their equilibrium values.  
2.2 Analytic representation
Methyl fluoride is a prolate symmetric top molecule of C_{3v}(M) molecular symmetry.^{48} The XY_{3}Z symmetrized analytic representation utilized in this work has previously been employed for nuclear motion calculations of CH_{3}Cl.^{15} Morse oscillator functions describe the stretching coordinates, 
ξ_{1} = 1 − exp[−a(r_{0} − r^{ref}_{0})],  (3) 

ξ_{j} = 1 − exp[−b(r_{i} − r^{ref}_{1})]; j = 2, 3, 4, i = j − 1,  (4) 
where a = 1.90 Å^{−1} for the C–F internal coordinate r_{0}, and b = 1.87 Å^{−1} for the three C–H internal coordinates r_{1}, r_{2} and r_{3}. For the angular terms, 
ξ_{k} = (β_{i} − β^{ref}); k = 5, 6, 7, i = k − 4,  (5) 

 (6) 

 (7) 
where τ_{23} = 2π − τ_{12} − τ_{13}, and r^{ref}_{0}, r^{ref}_{1} and β^{ref} are the reference equilibrium structural parameters. Values of r^{ref}_{0} = 1.3813 Å, r^{ref}_{1} = 1.0869 Å, and β^{ref} = 108.773° have been used, however, this choice is somewhat arbitrary due to the inclusion of linear expansion terms in the parameter set of the PES. Thus, the reference equilibrium structural parameters do not define the minimum of the PES and the true equilibrium values will be discussed in Section 5.1.
The potential function,

 (8) 
has maximum expansion order
i +
j +
k +
l +
m +
n +
p +
q +
r = 6 and is composed of the terms

V_{ijk…} = {ξ^{i}_{1}ξ^{j}_{2}ξ^{k}_{3}ξ^{l}_{4}ξ^{m}_{5}ξ^{n}_{6}ξ^{p}_{7}ξ^{q}_{8}ξ^{r}_{9}}^{C3v(M)},  (9) 
which are symmetrized combinations of different permutations of the coordinates
ξ_{i} that transform according to the
A_{1} representation of
C_{3v}(M). The terms in
eqn (9) are determined onthefly during the variational calculations and the PES implementation requires only a small amount of code.
^{15}
To determine the expansion parameters f_{ijk…} in eqn (8), a leastsquares fitting to the ab initio data was carried out. Weight factors of the form proposed by Schwenke^{49}

 (10) 
were used in the fit. Here,
Ẽ^{(w)}_{i} = max(
Ẽ_{i},10
000), where
Ẽ_{i} is the potential energy at the
ith geometry above equilibrium and the normalization constant
N = 0.0001 (all units assume the energy is in cm
^{−1}). Energies below 15
000 cm
^{−1} are favoured in our fitting by the weighting scheme. Watson's robust fitting scheme
^{50} was also utilized to further improve the description at lower energies and reduce the weights of outliers. The final PES was fitted with a weighted rootmeansquare (rms) error of 0.97 cm
^{−1} for energies up to
hc·50
000 cm
^{−1} and required 405 expansion parameters.
For geometries where r_{0} ≥ 1.95 Å and r_{i} ≥ 2.10 Å for i = 1, 2, 3, the respective weights were reduced by several orders of magnitude. At larger stretch distances a T_{1} diagnostic value >0.02 indicates that the coupled cluster method has become unreliable.^{51} Despite energies not being wholly accurate at these points, they are still useful and ensure that the PES maintains a reasonable shape towards dissociation. In subsequent calculations we refer to the PES as CBSF12^{HL}. The expansion parameters and a Fortran routine to construct the CBSF12^{HL} PES are provided in the ESI.†
3 Dipole moment surface
3.1 Electronic structure calculations
In a Cartesian laboratoryfixed XYZ coordinate system with origin at the C nucleus, an external electric field with components ±0.005 a.u. was applied along each coordinate axis and the respective dipole moment component μ_{A} for A = X, Y, Z determined using central finite differences. Calculations were carried out in MOLPRO2012^{36} at the CCSD(T)F12b/augccpVTZ level of theory and employed the frozen core approximation with a Slater geminal exponent value of β = 1.2a_{0}^{−1}.^{32} The same ansatz and ABS as in the explicitly correlated PES calculations were used. The DMS was computed on the same grid of nuclear geometries as the PES.
3.2 Analytic representation
The analytic representation used for the DMS of methyl fluoride was previously employed for CH_{3}Cl and the reader is referred to Owens et al.^{52} for a detailed description. To begin with, it is necessary to transform to a suitable moleculefixed xyz coordinate system before fitting an analytic expression to the ab initio data. A unit vector is defined along each bond of CH_{3}F, 
 (11) 
where r_{C} is the position vector of the C nucleus, r_{0} the F nucleus, and r_{1}, r_{2} and r_{3} the respective H atoms. The ab initio dipole moment vector μ is projected onto the molecular bonds and is described by three moleculefixed xyz dipole moment components, 
 (12) 

 (13) 
We have formed symmetryadapted combinations for μ_{x} and μ_{y} which transform according to the E representation of C_{3v}(M), while the μ_{z} component is of A_{1} symmetry. The symmetrized molecular bond representation described here is beneficial as the unit vectors e_{i} that define μ for any instantaneous configuration of the nuclei are related to the internal coordinates only, meaning the description is selfcontained.
The three dipole moment surfaces μ_{α} for α = x, y, z corresponding to eqn (12)–(14) are represented by the analytic expression

 (15) 
The expansion terms

 (16) 
have maximum expansion order
i +
j +
k +
l +
m +
n +
p +
q +
r = 6 and are best understood as a sum of symmetrized combinations of different permutations of the coordinates
ξ_{i}. Note that
Γ =
E for
μ_{x} and
μ_{y}, and
Γ =
A_{1} for
μ_{z}. For the stretching coordinates we employed linear expansion variables,

ξ_{1} = (r_{0} − r^{ref}_{0}),  (17) 

ξ_{j} = (r_{i} − r^{ref}_{1}); j = 2, 3, 4, i = j − 1,  (18) 
whilst the angular terms are the same as those defined in
eqn (5)–(7). The reference structural parameters
r^{ref}_{0},
r^{ref}_{1} and
β^{ref} had the same values as in the case of the PES.
The expansion coefficients F^{(α)}_{ijk…} for α = x, y, z were determined simultaneously through a leastsquares fitting to the ab initio data. Weight factors of the form given in eqn (10) were used along with Watson's robust fitting scheme.^{50} The three dipole moment surfaces, μ_{x}, μ_{y}, and μ_{z}, required 171, 160 and 226 parameters, respectively. A combined weighted rms error of 1 × 10^{−4} D was achieved for the fitting. Similar to the PES, the analytic representation of the DMS is generated onthefly at runtime. Its construction is slightly more complex because μ is a vector quantity and the transformation properties of the dipole moment components must also be considered.^{52} The expansion parameter set of the DMS is given in the ESI† along with a Fortran routine to construct the analytic representation.
4 Variational calculations
The general methodology of TROVE is well documented^{53–56} and calculations on another XY_{3}Z molecule, namely CH_{3}Cl, have previously been reported.^{15,52} We therefore summarize only the key aspects relevant for this work.
The rovibrational Hamiltonian was represented as a power series expansion around the equilibrium geometry in terms of the nine coordinates introduced in eqn (3)–(7). However, for the kinetic energy operator linear displacement variables (r_{i} − r_{ref}) were used for the stretching coordinates. The Hamiltonian was constructed numerically using an automatic differentiation method^{55} with both the kinetic and potential energy operators truncated at sixth order. The associated errors of such a scheme are discussed in Yurchenko et al.^{53} and Yachmenev and Yurchenko.^{55} Atomic mass values^{57} were employed throughout.
A multistep contraction scheme^{56} was used to construct the symmetrized vibrational basis set, the size of which was controlled by the polyad number,

P = n_{1} + 2(n_{2} + n_{3} + n_{4}) + n_{5} + n_{6} + n_{7} + n_{8} + n_{9} ≤ P_{max},  (19) 
and this does not exceed a predefined maximum value
P_{max}. Here, the quantum numbers
n_{k} for
k = 1,…,9 relate to primitive basis functions
ϕ_{nk}, which are obtained by solving onedimensional Schrödinger equations for each
kth vibrational mode using the Numerov–Cooley method.
^{58,59} Multiplication with symmetrized rigidrotor eigenfunctions 
J,
K,
m,
τ_{rot}〉 produced the final basis set for use in
J > 0 calculations. The quantum numbers
K and
m are the projections (in units of
ħ) of
J onto the moleculefixed
z axis and the laboratoryfixed
Z axis, respectively, whilst
τ_{rot} determines the rotational parity as (−1)
^{τrot}. As shown in
Fig. 3, the size of the Hamiltonian matrix,
i.e., the number of
J = 0 basis functions, grows exponentially with respect to
P_{max} and TROVE calculations above
P_{max} = 14 were not possible with the resources available to us. We will see in Section 5 that differently sized basis sets and basis set techniques must be utilized when computing various spectroscopic quantities due to the computational demands of variational calculations.

 Fig. 3 Size of the J = 0 Hamiltonian matrix with respect to the polyad truncation number P_{max}. Calculations were not possible above P_{max} = 14.  
5 Results
5.1 Equilibrium geometry and pure rotational energies
The equilibrium geometry derived from the CBSF12^{HL} PES is listed in Table 1. It is in excellent agreement with previous values determined in a joint experimental and ab initio analysis by Demaison et al.,^{60} which is regarded as the most reliable equilibrium structure of CH_{3}F to date. Also validating is the agreement with the refined geometry PES of Nikitin et al.,^{24} particularly as the CBSF12^{HL} PES has been generated in a purely ab initio fashion.
Table 1 The equilibrium structural parameters of CH_{3}F
r(C–F)/Å 
r(C–H)/Å 
β(HCF)/deg 
Ref. 
Approach 
1.38242 
1.08698 
108.746 
This work 
Purely ab initio PES 
1.3826(3) 
1.0872(3) 
108.69(4) 
Demaison et al.^{60} 
Experimental and ab initio analysis 
1.3827 
1.0876 
108.75 
Demaison et al.^{60} 
Ab initio calculations 
1.38240 
1.08696 
108.767 
Nikitin et al.^{24} 
Refined geometry PES 
It is more illustrative to look at the pure rotational energy levels, shown in Table 2, since these are highly dependent on the molecular geometry through the moments of inertia. Calculations with TROVE employed a polyad truncation number of P_{max} = 8, which is sufficient for converging ground state rotational energies. Despite being consistently lower than the experimental values, the rotational energies up to J ≤ 5 are reproduced with an rms error of 0.0015 cm^{−1}. The residual error ΔE(obs − calc) increases at each step up in J, however, this can be easily counteracted by refining the equilibrium geometry of the CBSF12^{HL} PES through a nonlinear leastsquares fitting to the experimental energies, for example, see Owens et al.^{17} The accuracy of the computed intraband rotational wavenumbers can be substantially improved as a result, but we refrain from doing this here as the errors for CH_{3}F are very minor and it leads to a poorer description of the vibrational energy levels.
Table 2 Comparison of computed and experimental J ≤ 5 pure rotational term values (in cm^{−1}) for CH_{3}F. The observed ground state energy levels are taken from Nikitin et al.^{24} but are attributed to Demaison et al.^{60}
J

K

Sym. 
Experiment 
Calculated 
Obs − calc 
0 
0 
A
_{1}

0.00000 
0.00000 
0.00000 
1 
0 
A
_{2}

1.70358 
1.70348 
0.00010 
1 
1 
E

6.03369 
6.03352 
0.00017 
2 
0 
A
_{1}

5.11069 
5.11040 
0.00029 
2 
1 
E

9.44074 
9.44038 
0.00036 
2 
2 
E

22.43005 
22.42946 
0.00059 
3 
0 
A
_{2}

10.22124 
10.22066 
0.00058 
3 
1 
E

14.55120 
14.55055 
0.00065 
3 
2 
E

27.54025 
27.53937 
0.00088 
3 
3 
A
_{1}

49.18585 
49.18459 
0.00126 
3 
3 
A
_{2}

49.18585 
49.18459 
0.00126 
4 
0 
A
_{1}

17.03508 
17.03411 
0.00097 
4 
1 
E

21.36493 
21.36388 
0.00105 
4 
2 
E

34.35362 
34.35235 
0.00127 
4 
3 
A
_{1}

55.99863 
55.99698 
0.00165 
4 
3 
A
_{2}

55.99863 
55.99698 
0.00165 
4 
4 
E

86.29575 
86.29356 
0.00219 
5 
0 
A
_{2}

25.55202 
25.55057 
0.00145 
5 
1 
E

29.88172 
29.88019 
0.00153 
5 
2 
E

42.86997 
42.86822 
0.00175 
5 
3 
A
_{1}

64.51425 
64.51212 
0.00213 
5 
3 
A
_{2}

64.51425 
64.51212 
0.00213 
5 
4 
E

94.81034 
94.80767 
0.00267 
5 
5 
E

133.75234 
133.74897 
0.00337 
5.2 Vibrational J = 0 energy levels
To assess the CBSF12^{HL} PES it is necessary to have converged vibrational energy levels, and one method of ensuring this is a complete vibrational basis set (CVBS) extrapolation.^{61} Similar to basis set extrapolation techniques of electronic structure theory (see e.g. Feller^{62} and references therein), the same ideas can be applied to TROVE calculations with respect to P_{max}. Vibrational energies were computed for P_{max} = {10,12,14} and fitted using the exponential decay expression, 
E_{i}(P_{max}) = E^{CVBS}_{i} + A_{i}exp(−λ_{i}P_{max}),  (20) 
where E_{i} is the energy of the ith level, E^{CVBS}_{i} is the respective energy at the CVBS limit, A_{i} is a fitting parameter, and λ_{i} is determined from 
 (21) 
The CVBS extrapolated J = 0 energies are shown in Table 3 alongside known experimental values.^{63–70} The six fundamentals are reproduced with an rms error of 0.69 cm^{−1} and a meanabsolutedeviation (mad) of 0.53 cm^{−1}. This level of accuracy extends to most of the other term values which are well within the ±1 cm^{−1} accuracy expected from PESs based on highlevel ab initio theory. Most significant perhaps is the computed ν_{4} level which shows a residual error ΔE(obs − calc) of −1.26 cm^{−1}. This is a noticeable improvement compared to the PES of Zhao et al.^{25} (ΔE_{ν4} = 3.33 cm^{−1}), which was generated at the CCSD(T)F12a/augccpVTZ level of theory, and the PES of Nikitin et al.^{24} (ΔE_{ν4} = 4.86 cm^{−1}), computed at the CCSD(T)/ccpVQZ level of theory with relativistic corrections, thus highlighting the importance of including HL corrections and a CBS extrapolation in the PES of CH_{3}F.
Table 3 Comparison of computed and experimental J = 0 vibrational term values (in cm^{−1}) for CH_{3}F. The computed zeropoint energy was 8560.2409 cm^{−1} at the CVBS limit
Mode 
Sym. 
Experiment 
Calculated 
Obs − calc 
Ref. 
ν
_{3}

A
_{1}

1048.61 
1048.88 
−0.27 
63

ν
_{6}

E

1182.67 
1182.79 
−0.12 
63

ν
_{2}

A
_{1}

1459.39 
1459.67 
−0.28 
63

ν
_{5}

E

1467.81 
1468.03 
−0.21 
63

2ν_{3} 
A
_{1}

2081.38 
2081.81 
−0.43 
64

ν
_{3} + ν_{6} 
E

2221.81 
2222.20 
−0.40 
64

2ν_{6} 
E

2365.80 
2365.96 
−0.16 
65

ν
_{2} + ν_{3} 
A
_{1}

2499.80 
2500.28 
−0.48 
65

ν
_{3} + ν_{5} 
E

2513.80 
2514.27 
−0.47 
65

2ν_{5} 
A
_{1}

2863.24 
2863.90 
−0.66 
66

ν
_{2} + ν_{5} 
E

2922.23 
2922.59 
−0.36 
66

2ν_{2} 
A
_{1}

2926.00 
2926.66 
−0.66 
66

2ν_{5} 
E

2927.39 
2927.92 
−0.53 
66

ν
_{1}

A
_{1}

2966.25 
2967.30 
−1.05 
66

ν
_{4}

E

3005.81 
3007.07 
−1.26 
66

3ν_{3} 
A
_{1}

3098.44 
3098.97 
−0.53 
67

ν
_{3} + 2ν_{5} 
A
_{1}

3905.4 
3906.39 
−0.99 
68

ν
_{1} + ν_{3} 
A
_{1}

4011 
4012.28 
−1.28 
69

ν
_{3} + ν_{4} 
E

4057.6 
4059.31 
−1.71 
70

2ν_{4} 
E

6000.78 
6003.11 
−2.33 
70

It is worth noting, at least for the values considered in Table 3, that the computed P_{max} = 14 vibrational energy levels are within 0.01 cm^{−1} of the CVBS values with the majority converged to one or two ordersofmagnitude better. A complete list of the P_{max} = 14 computed J = 0 energy levels is included in the ESI.†
5.3 Vibrational transition moments
The vibrational transition moment is defined as, 
 (22) 
where Φ^{(i)}_{vib}〉 and Φ^{(f)}_{vib}〉 are the J = 0 initial and final state eigenfunctions, respectively, and _{α} is the electronically averaged dipole moment function along the moleculefixed axis α = x, y, z. Transition moments are relatively inexpensive to compute and provide an initial assessment of the DMS. Calculations in TROVE used P_{max} = 12 and considered transitions from the ground vibrational state only (i = 0).
In Table 4, vibrational transition moments for the fundamentals are listed alongside known experimental values derived from measurements of absolute line intensities.^{71–73} The agreement is encouraging but it suggests that the DMS may overestimate the strength of line intensities. This behaviour will be confirmed in the following sections. A list of computed transition moments from the vibrational ground state up to 10000 cm^{−1} is provided in the ESI.† For the groundstate electric dipole moment of CH_{3}F we compute a value of 1.8503 D, which is close to the experimentally determined value of 1.8584 D.^{74}
Table 4 Vibrational transition moments (in Debye) for the fundamental frequencies (in cm^{−1}) of CH_{3}F
Mode 
Sym. 
ν
^{exp}_{0f}

μ
^{calc}_{0f}

μ
^{exp}_{0f}

Ref. 
ν
_{1}

A
_{1}

2966.25 
0.05205 
— 
— 
ν
_{2}

A
_{1}

1459.39 
0.01390 
0.01196 
73

ν
_{3}

A
_{1}

1048.61 
0.20020 
0.19015 
71

ν
_{4}

E

3005.81 
0.08485 
— 
— 
ν
_{5}

E

1467.81 
0.04903 
0.04976 
73

ν
_{6}

E

1182.67 
0.03085 
0.02835 
72

5.4 Absolute line intensities of the ν_{6} band
Recently, Jacquemart and Guinet^{75} generated an experimental line list of almost 1500 transitions of the ν_{6} band with absolute line intensities determined with an estimated accuracy of 5%. To compare with this study we have generated an ab initio room temperature line list for CH_{3}F. This was computed with a lower state energy threshold of hc·5000 cm^{−1} and considered transitions up to J = 40 in the 0–4600 cm^{−1} range.
Describing high rotational excitation can quickly become computationally intractable since rovibrational matrices scale linearly with J. It was therefore necessary to use a truncated J = 0 basis set. TROVE calculations were initially performed with P_{max} = 12, resulting in 49076 vibrational basis functions, which was subsequently reduced to 2153 functions by removing states with energies above hc·9600 cm^{−1}. The resulting pruned basis set was multiplied in the usual manner with symmetrized rigidrotor functions to produce the final basis set for J > 0 calculations.
Naturally, errors are introduced into our rovibrational predictions and it is hard to quantify this without more rigorous calculations. However, we have previously used truncated basis set procedures to construct a comprehensive line list for SiH_{4} without noticeable deterioration.^{76} It should be emphasised that the main advantage of truncation is the ability to retain the accuracy of the vibrational energy levels and respective wavefunctions generated with P_{max} = 12.
Absolute absorption intensities were simulated using the expression,

 (23) 
where
A_{if} is the Einstein
A coefficient of a transition with wavenumber
ν_{if} (in cm
^{−1}) between an initial state with energy
E_{i} and a final state with rotational quantum number
J_{f}. Here,
k is the Boltzmann constant,
h is the Planck constant,
c is the speed of light and the absolute temperature
T = 296 K. The nuclear spin statistical weights are
g_{ns} = {8,8,8} for states of symmetry {
A_{1},
A_{2},
E}, respectively, and for the room temperature partition function
Q(296 K) = 14587.780.
^{27} Transitions obey the symmetry selection rules
A_{1} ↔
A_{2},
E ↔
E; and the standard rotational selection rules,
J′ −
J′′ = 0, ±1,
J′ +
J′′ ≠ 0; where ′ and ′′ denote the upper and lower state, respectively. The ExoCross code
^{77} was employed for all spectral simulations.
In Fig. 4, the computed absolute line intensities of the ν_{6} band are plotted against the experimental line list of Jacquemart and Guinet^{75} alongside the percentage measure %[(obs − calc)/obs], which quantifies the error in our predicted intensities. The shape and structure of the ν_{6} band is well reproduced but the DMS overestimates the strength of line intensities. We expect that this behaviour can be corrected for by using a larger augmented basis set in the electronic structure calculations, however, the improvement in intensities may not justify the additional computational expense. As expected from the J = 0 calculations, computed line positions of the ν_{6} band had an average residual error of Δν(obs − calc) = −0.125 cm^{−1}.

 Fig. 4 Absolute line intensities of the ν_{6} band for transitions up to J = 40 at T = 296 K and the corresponding residual errors %[(obs − calc)/obs] when compared with the experimental line list from Jacquemart and Guinet.^{75}  
5.5 Overview of the rotation–vibration spectrum
A final benchmark of our rovibrational calculations and ab initio line list is a comparison with the PNNL spectral library.^{78} Absorption crosssections were simulated at a resolution of 0.06 cm^{−1} using a Gaussian profile with a halfwidth at halfmaximum of 0.135 cm^{−1}. The results are shown in Fig. 5 and 6 with the experimental PNNL spectrum, which was measured at a temperature of 25 °C with the dataset subsequently renormalized to 22.84 °C (296 K). Overall the agreement is extremely pleasing, particularly as both the strong and weak intensity features are equally well reproduced. Whilst the intensities of the ab initio spectrum are stronger, this is only slight and we have no hesitation recommending the PES and DMS for future use in spectroscopic applications.

 Fig. 5 Simulated CH_{3}F rotation–vibration spectrum up to J = 40 compared with the PNNL spectral library^{78} at T = 296 K.  

 Fig. 6 Closer inspection of the computed crosssections compared with the PNNL spectral library^{78} at T = 296 K.  
6 Conclusions
A new PES and DMS for methyl fluoride have been generated using highlevel ab initio theory and then rigorously evaluated through variational nuclear motion calculations. The computed results showed excellent agreement with a range of experimental data, which included the equilibrium geometry of CH_{3}F, pure rotational and vibrational energies, vibrational transition moments, absolute line intensities of the ν_{6} band, and the rotation–vibration spectrum up to J = 40. This work demonstrates the importance of including HL energy corrections and an extrapolation to the CBS limit in the PES to accurately describe the rovibrational spectrum of CH_{3}F from first principles.
To go beyond the accuracy achieved in this work in a purely ab initio manner will require extensive larger basis set electronic structure calculations. That said, the computational cost associated with this is unlikely to correlate with the gain in accuracy and empirical refinement of the PES is recommended instead. Although computationally intensive,^{79} refinement can lead to ordersofmagnitude improvements in the accuracy of the computed rovibrational energy levels and consequently more reliable transition intensities as a result of better wavefunctions.
Conflicts of interest
There are no conflicts of interest to declare.
Acknowledgements
Besides DESY, this work has been supported by the Deutsche Forschungsgemeinschaft (DFG) through the excellence cluster “The Hamburg Center for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” (CUI, EXC1074) and the priority program 1840 “Quantum Dynamics in Tailored Intense Fields” (QUTIF, KU1527/3), by the Helmholtz Association “Initiative and Networking Fund”, and by the COST action MOLIM (CM1405).
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Footnote 
† Electronic supplementary information (ESI) available: The expansion parameters and Fortran 90 functions to construct the PES and DMS of CH_{3}F. A list of computed vibrational energy levels and vibrational transition moments. See DOI: 10.1039/c8cp01721b 

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