Isotopologues: Highly Anharmonic Hydrogen ... - ACS Publications

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FHF− Isotopologues: Highly Anharmonic Hydrogen-Bonded Systems with Strong Coriolis Interaction Peter Sebald, Arne Bargholz, Rainer Oswald, Christopher Stein, and Peter Botschwina* Institut für Physikalische Chemie, Universität Göttingen, Tammannstraße 6, 37077 Göttingen, Germany S Supporting Information *

ABSTRACT: Explicitly correlated coupled cluster theory at the CCSD(T*)-F12b level in conjunction with the aug-ccpV5Z basis set has been used in the calculation of threedimensional potential energy and dipole moment surfaces for the bifluoride ion (FHF−). An empirically corrected analytical potential energy function (PEF) was obtained by fit to four pieces of accurate spectroscopic information. That PEF was used in variational calculations of energies and wave functions for a variety of rovibrational states of the isotopologues FHF−, FDF−, and FTF−. Excellent agreement with available data from IR laser diode spectroscopy is observed and many predictions are being made. Unusual isotope effects among the spectroscopic constants and unusual features of the calculated line spectra are discussed. most extensive treatment of the rovibrational problem in FHF− and FDF− available in the literature dates back to 1991 and was published by Špirko, Č ejchan, and Diercksen.15 These authors constructed four different analytical potential energy functions (PEFs I−IV) that were based on high-level ab initio calculations (many-body perturbation theory at fourth-order, MBPT(4), and the coupled cluster (CC) methods CCSD and CCSDT-1a). Subsequently, Watson’s isomorphic rovibrational Hamiltonian23 for linear molecules was employed in variational calculations of rovibrational energies, but the so-called Coriolis term was neglected. However, that term is of great importance for a quantitative theoretical description of the Coriolis interacting rovibrational systems occurring for isotopologues of the bifluoride ion. Using the theoretically most well founded PEF IV, Špirko et al.15 obtained very good agreement with experiment2−4 for the fundamental vibrational transitions of FHF−, the maximum deviation amounting to only 15.7 cm−1. For the combination tone ν1 + ν3, the error was even as small as 3.1 cm−1, but the wavenumber of the other observed combination tone (ν1 + ν2) was underestimated by 38.3 cm−1. The more recent theoretical work on FHF− and FDF− 16−22 did not consider rotation and provided little progress in the accuracy of the calculated vibrational energies (cf. Table 1 of ref 21 and Tables 3 and 4 of ref 22).

1. INTRODUCTION The hydrogen bifluoride ion (FHF−) constitutes one of the strongest hydrogen-bonded systems (D0(exp) = 44.4 ± 1.6 kcal mol−1)1 and is thus of general interest to various areas of chemistry and related disciplines. From the viewpoint of spectroscopy it is rather unique through the fact that highresolution infrared (IR) spectra could be observed for a variety of rovibrational transitions of both FHF− and FDF−. Already in 1986, Kawaguchi and Hirota2 used IR diode laser spectroscopy in conjunction with the magnetic field modulation technique to study a series of rovibrational transitions for each isotopologue, either in the range 1780−1853 cm−1 for FHF− or between 1366 and 1403 cm−1 for FDF−. Originally, most of the observed lines were assigned to the proton or deuteron stretching vibrations (ν3), but these assignments were revised later.3,4 For the ν3 band of FHF−, the band origin was then determined at 1331.1502(7) cm−1 and that band was found to be perturbed through Coriolis interaction with the lower-lying bending vibrational state (ν2). The band with origin at 1848.6988(18) cm−1 was reassigned to the combination tone ν1 + ν3. For FDF−, the wavenumbers of the IR active fundamentals were determined to be ν2 = 928.7303(17) cm−1 and ν3 = 934.1933(7) cm−1 and very strong Coriolis interaction was recognized between these two states. In summary, accurate spectroscopic information could be gained for as many as 6 different vibrational states of FHF− and 7 states of FDF−, inclusive of the vibrational ground state in both cases. Although FHF− has been the subject of a larger number of theoretical studies accounting for vibrational anharmonicity,5−22 accurate calculations of rovibrational states and the intensities of transitions among them are still missing. The © 2013 American Chemical Society

Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: December 15, 2012 Revised: February 19, 2013 Published: February 19, 2013 9695

dx.doi.org/10.1021/jp3123677 | J. Phys. Chem. A 2013, 117, 9695−9703

The Journal of Physical Chemistry A

Article

The present paper is devoted to a theoretical investigation of a variety of rovibrational states of FHF−, FDF−, and FTF− up to high values of the rotational quantum number J. The basis of this study is provided by an empirically corrected threedimensional near-equilibrium potential energy surface (PES). Accurate information from the diode laser IR study of FHF− 3 was employed to improve a PES calculated by explicitly correlated coupled cluster theory involving nonlinear correlation factors (see refs 24−26 for recent reviews). That theory was also employed to calculate an electric dipole moment surface for subsequent use in the calculation of rovibrational transition dipole moments and line intensities. Thereby, a detailed comparison with the available high-resolution IR data has become possible. In addition, the present paper will make a larger number of predictions for spectroscopic properties of all three isotopologues of the bifluoride ion, including the experimentally hardly accessible species FTF−. However, the latter is of quite some interest for theoretical reasons.

V − Ve =

∑ CijkS1iS2jS3k (1)

ijk

The symmetry coordinates S1−S3 are defined as follows:

S1 = S2 = S3 =

1 (R − 2re) 2

(2a)

2 x re

(2b)

2z

(2c)

R is the instantaneous distance between the fluorine nuclei, z is the elongation of the proton out of its equilibrium position along the F−F axis, and x measures the perpendicular distance of the proton from that axis. The parameters Cijk were determined by weighted least-squares fitting using weights of the form 1/(Ei + 512 cm−1)2. The standard deviation of the fit amounts to 1.56 cm−1. An empirically improved analytical PEF was obtained in the following way. Four pieces of experimental information for FHF−, namely B000, G100, G010, and G001 were used to determine four scaling factors for re, S1, S2, and S3. The values found are 0.999914, 0.998885, 0.998675, and 0.999157. Using these factors in rovibrational calculations, spectroscopic constants were obtained that agree with the experimental ones within 6 significant digits. The coefficients of the corrected PEF are supplied as Supporting Information (see Table S1). Owing to the polynomial nature of the analytical PEF, the effect of coordinate scaling can be easily incorporated into the coefficients. Adjustment of re leads to an empirically corrected FF equilibrium distance of Re = 2.27922 Å, to be compared with an experimental value of 2.27771(7) Å, with three times the standard deviation in parentheses. The systematic error of the experimental Re may be much larger, because the determination of Be from four Bv values (for the ground state and the three singly excited states) corresponds to a linear extrapolation that may be not very accurate for a highly anharmonic system such as FHF−. Contour plots of the empirically corrected PEF of FHF− are displayed in Figure 1 for four different values of the FF distance R. As is typical for symmetric strong hydrogen-bonded systems, a double minimum potential develops upon increase of R. At ΔR = 0.5 Å (see bottom of Figure 1), the barrier height between the two equivalent minima of the two-dimensional representation already amounts to 4346 cm−1. The electric dipole moment was calculated at 226 symmetryunique nuclear configurations using the finite field method (field strengths of each ±0.0003 au in x and z directions). The two components of the dipole moment, evaluated with respect to the Eckart frame of FHF−, were fitted to polynomials of the form

2. DETAILS OF CALCULATIONS AND RESULTS FOR HF Explicitly correlated coupled cluster theory at the CCSD(T*)F12b level27,28 was used in most of the electronic structure calculations of the present work. T* means that the perturbative triples contribution is scaled according to the recipe of Werner and co-workers.28,29 The use of coupled cluster variant CCSD(T*)-F12b has been recommended in conjunction with large atomic orbital (AO) basis sets.24,29 Consequently, we use the flexible Dunning-type basis set augcc-pV5Z,30,31 briefly termed AV5Z in the following. The geminal exponent β, which appears in the Slater-type correlation factors, is chosen to be β = 1.4 α0−1.32 Following the recommendations of Yousaf and Peterson,33 the present CCSD(T*)-F12b calculations make use of the additional basis sets AV5Z/OPTRI, V5Z/JKFIT,34 and AV5Z/MP2FIT.35 The 16 valence electrons are correlated in the explicitly correlated coupled cluster calculations that were carried out with the MOLPRO system of ab initio programs.36 The chosen method and basis set was first used in calculations for diatomic HF. The equilibrium distance was calculated to be re = 0.9173 Å, to be compared with an experimental value of 0.9168 Å.37 The vibrational term values Gv (v = 1, 2, ...), measured with respect to the vibrational ground state, are 3964.8, 7758.5, and 11385.4 cm−1. Compared with experiment,38 the calculated values are too large by 3.4, 7.7, and 11.6 cm−1. The corresponding values for DF are 2909.1 (2.4), 5727.1 (5.3), and 8456.0 (8.6) cm−1, deviations from experiment being given in parentheses. The relative errors for HF and DF thus range between 0.08 and 0.10% and similar accuracy may be expected for the isotopologues of the bifluoride ion. For FHF−, the present CCSD(T*)-F12b/AV5Z calculations yield a centrosymmetric equilibrium structure with a FH distance of re = 1.13971 Å and a total energy of Ve = −200.2562855 Eh. For comparison, calculations with the standard CCSD(T) method39 in conjunction with the AV6Z basis yield re = 1.13962 Å and Ve = −200.2501426 Eh. Around the corresponding equilibrium structure, 3758 symmetryunique energy points with relative energies of less than 19 000 cm−1 were calculated by CCSD(T*)-F12b and fitted to a polynomial expansion in rectilinear symmetry coordinates:

μα =

∑ Cijkα S1iS2jS3k ijk

α : parallel or perpendicular (3)

where S1, S2, and S3 are defined in eq 2. The parallel and perpendicular components of the dipole moment were fitted with a standard deviation of 3.0 × 10−4 ea0 and 3.2 × 10−5 ea0, respectively. The coefficients Cαijk are supplied as Supporting Information (Table S2). For FDF− and FTF−, an appropriate transformation is made in the rovibrational calculations. The parallel component of the dipole moment varies strongly with 9696



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Figure 1. Two-dimensional contour plots of the empirically corrected potential energy function for FHF−. Contour lines up to 9000 cm−1 are given in intervals of 500 cm−1.

Figure 2. Variation of the electric dipole moment of FHF− with the Cartesian coordinates of the proton.

the z coordinate of the proton but shows only little dependence on the FF distance (Figure 2). The graphs of μ⊥(x) at different ΔR as displayed in the inset of Figure 2 are almost linear and exhibit much smaller slopes compared to μ∥(z). Rovibrational energies and wave functions for FHF−, FDF−, and FTF− were calculated by diagonalization of Watson’s isomorphic Hamiltonian for linear molecules23 in a basis of symmetrized harmonic oscillator/rigid rotor (HO/RR) functions, involving one-dimensional HOs for the stretching and two-dimensional HOs (characterized by quantum numbers ν2 and l with restriction ν2 ≥ l) for the bending vibrations. That Hamiltonian is formulated in four normal coordinates Qk and may be written as follows: 1 Ĥ = 2

4

∑ Pk 2 + k=1

integration, respectively, whereas integration over the Euler angles was performed analytically. All rovibrational calculations of the present work were carried out with a program written by one of us.41 A carefully selected vibrational basis of 819 HO products was used in the calculations with J = 0. It guarantees convergence in the vibrational energies of the states of interest to the present work to better than 0.01 cm−1. Actually, the numerical accuracy of relative rovibrational energies within one vibrational state should be still higher. The individual rovibrational calculations were carried out with basis sets of up to 13 104 HO/RR functions, the maximum l value being set to lmax = 15 in that case. In a triatomic linear molecule of D∞h symmetry, Coriolis interaction takes place between the antisymmetric stretching vibration and the component of the degenerate bending vibration with e parity. In most molecules, both vibrations are energetically well separated and the spectroscopic formulas used to analyze spectra only deal with the issue of l-type doubling. In cases of strong Coriolis interaction as present in the bifluoride isotopologues such a treatment is not appropriate, however, and one has to consider the Coriolis interaction explicitly. Following Nielsen,42 we set up a 2 × 2 Hamiltonian matrix for each value of J:

1 1 μ(πx 2 + πy 2) + μ(Π′x2 + Π′y2) 2 2

− μ(Π′xπx + Π′yπy) + V

(4)

In eq 4, Pk are the linear momenta conjugate to the normal coordinates Qk. The first two terms of the Hamiltonian correspond to the vibrational kinetic energy part (T̂ vib), the third corresponds to the rotational contribution (Ĥ rot), the fourth is the Coriolis term (Ĥ cor), and the last term (V) is the potential energy. A more detailed description of the individual terms is given in Watson’s original paper.23 Following the pioneering work of Whitehead and Handy,40 the matrix elements of Ĥ over the two types of HO functions were computed by means of Gauss−Hermite and Gauss−Laguerre 9697



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Table 1. Calculated and Experimental Spectroscopic Constants (cm−1) for Totally Symmetric Vibrational States of FHF−, FDF−, and FTF− a FHF−

FDF−

FTF−

state

method

Gv

Bv

10 Dv

Gv

Bv

10 Dv

Gv

Bv

106Dv

(0,0°,0)

CCSD(T*)-F12b corr exp CCSD(T*)-F12b corr exp CCSD(T*)-F12b corr CCSD(T*)-F12b corr

2315.49 2312.90

0.33413 0.33418 0.33418(1) 0.33179 0.33185 0.33185(1) 0.32916 0.32921 0.32600 0.32606

0.432 0.433 0.431(4) 0.446 0.447 0.443(5) 0.470 0.472 0.515 0.516

1734.52 1732.58

0.33572 0.33577 0.33579(1) 0.33328 0.33334 0.33333(3) 0.33026 0.33032 0.32568 0.32574

0.425 0.426 0.424(6) 0.444 0.445 0.427(28) 0.490 0.491 0.634 0.636

1483.97 1482.32

0.33645 0.33651

0.422 0.423

594.13 593.50

0.33393 0.33399

0.446 0.447

1169.59 1168.38 1757.10 1755.33

0.33038 0.33044 0.32594 0.32600

0.527 0.528 0.496 0.498

(1,0°,0)

(2,0°,0) (3,0°,0)

583.66 583.05 583.05 1156.79 1155.59 1716.09 1714.33

6

591.13 590.50 588b 1168.27 1167.05 1721.07 1719.33

6

Calculated spectroscopic constants are obtained from fits to the rovibrational energies (see Supporting Information) with the expression Gv + BvJ(J + 1) − DvJ2(J + 1)2 and Jmax = 60. Gv values for the vibrational ground state correspond to zero-point energies (ZPEs). Underlined values have been adjusted to the corresponding experimental data (see text). Values in parentheses correspond to 3 times the standard deviation in terms of the least significant digit. bEstimated from the centrifugal distortion constant (ref 4).

a

Table 2. Calculated and Experimental Spectroscopic Constants (cm−1) for Ungerade Vibrational States of FHF−, FDF−, and FTF− a FHF− state (0,0°,1)

method e

CCSD(T*)-F12b corr exp (0,11,0)e CCSD(T*)-F12b corr exp η constant I, II, IIIb (0,11,0)f CCSD(T*)-F12b corr (1,0°,1)e CCSD(T*)-F12b corr exp (1,11,0)e CCSD(T*)-F12b corr exp η constant I, II, IIIb (1,11,0)f CCSD(T*)-F12b corr (2,0°,1)e CCSD(T*)-F12b corr exp (2,11,0)e CCSD(T*)-F12b corr η constant I, II, IIIb (2,11,0)f CCSD(T*)-F12b corr

Gv

106Dv

Bv

1332.27 0.31692 1331.15 0.31697 1331.15 0.31697(1) 1287.60 0.33601 1286.03 0.33607 1286.03 0.33607(1) 0.531, 0.531, 0.531 1287.60 0.33641 1286.03 0.33646 1850.16 0.31406 1848.54 0.31411 1848.70 0.31413(3) 1860.73 0.33363 1858.57 0.33369 1858.48(35) 0.33327(87) 0.346, 0.347, 0.344(5) 1860.73 0.33397 1858.57 0.33403 2354.78 0.31082 2352.68 0.31086 2421.71 2418.99 0.188, 0.188 2421.71 2418.99

FDF− Gv

0.462 0.463 0.461(4) 0.467 0.468 0.464(6) 0.457 0.458 0.473 0.474 0.476(5) 0.498 0.499 [0.464]c 0.476 0.477 0.493 0.494

0.33087 0.33093

0.541 0.542

0.33112 0.33118

0.511 0.513

Bv

935.25 0.32288 934.46 0.32293 934.19 0.32294(1) 930.01 0.33704 928.87 0.33710 928.73 0.33711(2) 0.592, 0.592, 0.592 930.01 0.33742 928.87 0.33748 1470.73 0.31984 1469.41 0.31989 1469.19(1) 0.31969(4) 1512.93 0.33446 1511.19 0.33452 1509.55(55) [0.3346]c 0.488, 0.488, 0.471(5) 1512.93 0.33488 1511.19 0.33494 1989.70 0.31622 1987.87 0.31627 1985 0.31615(3) 2079.58 0.33122 2077.27 0.33128 0.384, 0.384 2079.58 0.33158 2077.27 0.33164

FTF− 106Dv

Gv

Bv

106Dv

0.456 0.457 0.457(7) 0.449 0.450 0.445(10)

764.51 763.86

0.32563 0.32568

0.453 0.454

772.27 771.33

0.33752 0.33757

0.440 0.441

0.616, 0.616 772.27 0.33789 771.33 0.33795 1307.39 0.32244 1306.20 0.32249

0.437 0.438 0.480 0.481

1359.23 1357.67

0.33483 0.33488

0.479 0.480

0.548, 0.549 1359.23 0.33526 1357.67 0.33532 1830.41 0.31837 1828.71 0.31842

0.468 0.469 0.506 0.507

0.443 0.445 0.477 0.478 [0.454]c 0.483 0.484 [0.464]c 0.468 0.470 0.503 0.504 0.472(25) 0.560 0.561 0.530 0.532

1924.01 0.33099 1921.90 0.33105 0.462, 0.462 1924.01 0.33129 1921.90 0.33135

0.623 0.624 0.577 0.578

Calculated spectroscopic constants for e parity substates were obtained from fits to the rovibrational energies (see Supporting Information) according to the procedure described in section 2. Energies of f parity levels were fitted with the expression Gv + Bv[J(J + 1) − 1] − Dv[J(J + 1) − 1]2. Jmax = 60 was used in both types of fits. Underlined values have been adjusted to the corresponding experimental data (see the text). Values in parentheses correspond to 3 times the standard deviation in terms of the least significant digit. bI, II, and III refer to CCSD(T*)-F12b, corr, and exp, respectively. cFixed (see ref 4). a

⎛G + B X − D X 2 − η X ⎞ v v ⎜ v ⎟ ⎟ H (J ) = ⎜ − η X Gv ′ + Bv ′(X − 1) − Dv ′⎟ ⎜⎜ ⎟ (X − 1)2 ⎝ ⎠ with X = J(J + 1)

In eq 5, the two vibrational states under consideration are denoted by the collective quantum numbers v and v′, v corresponding to the “stretch-only” state. The quantities G, B, and D (with appropriate indices) stand for the “unperturbed” spectroscopic constants, η is the Coriolis interaction constant, and the eigenvalues of H(J ) are the calculated term energies

(5) 9698



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the scope of the present paper. Indication of the perturbation in the (3,0°,0) state of FTF− is also provided by the nonmonotonic sequence of Dv values for the nν1 overtone series. 3.2. Pairs of Coriolis Interacting States: (0,0°,1)/ (0,11,0), (1,0°,1)/(1,11,0), and (2,0°,1)/(2,11,0). Calculated and experimental spectroscopic constants for the three pairs of states (0,0°,1)/(0,11,0), (1,0°,1)/(1,11,0), and (2,0°,1)/(2,11,0) are listed in Table 2. The e sublevels were fitted according to the procedure described in section 2 whereas the standard formula (see footnote a) of Table 2) was employed to fit the f sublevels, Jmax = 60 being chosen in both cases. In comparison with the accurate experimental data, the CCSD(T*)-F12b results for the Gv values are too large by 1.12−2.25 cm−1 for FHF− and by 1.06−1.54 cm−1 for FDF−. This means a great improvement with respect to all previous ab initio calculations. The experimental values for G100, G110, and G201 of FDF− are only very approximately known and have been left out from the comparison. The Gv values obtained with the corrected potential are in almost perfect agreement with all precisely determined experimental data.3,4 The experimental value for G110(FHF−) is 1858.48(35) cm−1 and thus significantly less accurate than the other experimental Gv values of FHF− quoted in Table 2. The best theoretical value is 1858.57 cm−1 (corr); it should have an uncertainty of less than 0.05 cm−1. An accurate experimental value has been determined for the origin of the hot band 2ν1 + ν3 − ν1 (see the footnote to Table 2 of ref 4) and amounts to 1397.2363(13) cm−1. The corresponding theoretical values are 1398.57 cm−1 (CCSD(T*)-F12b) and 1397.37 cm−1 (corr). Combining the experimental value for the hot band origin with the theoretical value (corr) for G100(FDF−), we arrive at the prediction G201(FDF−) = 1987.74 cm−1, with an estimated uncertainty of ca. 0.05 cm−1. Throughout, the spectroscopic constants Bv, Dv, and η obtained with the corrected PEF and the procedure described in section 2 are in excellent agreement with those experimental data having low standard deviations. The accuracy of the experimental values for B110 and D110 of FHF− and FDF− is significantly lower than in the other cases and the corresponding theoretical data (“corr”) are expected to be more reliable. This also holds for the spectroscopic constants of the (1,0°,1) and (2,0°,1) states of FDF−. For the pair (0,0°,1)/ (0,11,0) of FHF− and FDF−, both theoretical η values (CCSD(T*)-F12b and corr) are in perfect agreement with the corresponding experimental values. We are therefore confident that the η values obtained with the empirically corrected PEF for the other two pairs given in Table 2 are also accurate to the three digits quoted. The energetically lowest pair of states involved in strong Coriolis interaction is (0,0°,1)/(0,11,0). According to the experiments of Kawaguchi and Hirota,2−4 the difference in term energies G001 − G010 amounts to 45.12 cm−1 for FHF− and to 5.4 cm−1 for FDF−. For FTF−, the present calculations with the empirically corrected PEF yield a negative value of −7.47 cm−1; i.e., the (0,0,1) state comes to lie below the (0,11,0) state. To see the magnitude of Coriolis interaction, it may be instructive to compare the results of calculations with and without inclusion of Ĥ cor in the rovibrational Hamiltonian (cf. eq 4). That is done graphically in Figure 3. Plotted are the relative energies ΔE with respect to the arithmetic means of the Gv values of the two states under consideration. Results obtained with the full Hamiltonian are represented by blue dots, and red dots are used for the calculations neglecting Ĥ cor. The insets show the variation of the expectation value of |l| with the

obtained with the full Hamiltonian (see eq 4). This set of nonlinear equations has been solved iteratively. The squared transition dipole moments μif 2 between two rovibrational states were calculated closely following the work of Carter et al.43 These quantities may be approximately written as a product of three factors: μif 2 ≈ μvv ′2 FHLFHW

(6)

μvv′2

In eq 6, is the squared transition dipole moment of the pure vibrational transition between vibrational states v and v′, FHL is the Hönl-London factor,44 and FHW is the Hermann− Wallis factor.45 For the P- and R-branch transitions, the latter is given by the formula46 FHW = [1 + A1m + A 2 m2]2

(7)

with m = −J for P-branch transitions and m = J + 1 for Rbranch transitions.

3. RESULTS AND DISCUSSION 3.1. Totally Symmetric Vibrational States of FHF−, FDF−, and FTF−. Rovibrational energies for ten different vibrational states of the three isotopologues of the bifluoride ion are supplied as Supporting Information (Tables S3−S5). Throughout, levels up to Jmax = 60 are considered. For the three states with l = 1, the energies of both e and f parity sublevels are quoted. Calculated and experimental spectroscopic constants for the four lowest totally symmetric vibrational states of the three isotopologues are listed in Table 1. The results obtained with the uncorrected CCSD(T*)-F12b PEF are already in excellent agreement with the few available experimental data. The term energy G100 of FHF− is overestimated by only 0.61 cm−1 or 0.1%, the rotational constants B000 and B100 are too low by 0.00005−0.00006 cm−1 and the quartic centrifugal distortion constants D000 and D100 are well within the error bars of their experimental counterparts. The corrected values involve adjustment to experimental data, are in virtually perfect agreement with experiment, and should provide very accurate predictions in all those cases where no spectroscopic data are available. Closer inspection of Table 1 exhibits some unusual isotope effects and gives hints to the presence of perturbations. Vibrational anharmonicity must be responsible for the increase in ν1 upon substitution of hydrogen by its heavier isotopes. The positive H/D shift was already predicted in the early work of Almlöf5 and is even more pronounced for FTF−, the predicted H/T shift amounting to as much as 10.45 cm−1. For the first overtone of the symmetric stretching vibration (2ν1), we calculate a H/D shift of 11.46 cm−1, in good agreement with the 10.2 cm−1 calculated by Špirko et al.15 The predicted G100 value of 590.50 cm−1 for FDF− is expected to be accurate to better than 0.1 cm−1 and thus considerably more reliable than the estimate of 588 cm−1 as calculated from the quartic centrifugal distortion constant.4 The low deuterium shift of 5.0 cm−1 calculated for 3ν1 (FDF −) is indicative of some perturbation as is the unusually large increase in D300 compared to D200. Another perturbation is predicted for the (3,0°,0) state of FTF−, with the term energy of 1755.33 cm−1 being much higher than expected. From the nν1 sequence with n = 1 and 2, a value around 1728 cm−1 would be more likely. A first analysis shows that state (3,0°,0) is mainly shifted upward through anharmonic vibrational interaction with state (0,0°,2). A quantitative treatment is more complicated because the first overtone of the bending vibration also plays a role; it is outside 9699



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3.3. Theoretical IR Spectra of FHF−, FDF−, and FTF−. Calculated squared transition dipole moments for transitions between selected rovibrational states and the corresponding rovibrational energy differences were used to compute stick spectra. Figures 4 and 5 display such spectra for the most prominent bands of FHF−, FDF−, and FTF−. Because the fluorine atoms have nuclear spin of 1/2, an intensity alternation with a ratio of 3:1 can be recognized in all the spectra. For each isotopologue, the intensities are normalized with respect to the strongest line calculated. The spectra are drawn for a temperature of 375 K, at which the intensity ratio of lines R(49) and R(15) in the ν2 band of FHF− is 0.9 and thus agrees with the corresponding experimental value.3 The intensity ratio of the almost coinciding lines ν3 P(35) and ν2 R(45) is calculated to be 2.8, in good agreement with the experimental value of 2.5. For the hot bands shown (ν1 + ν3 − ν1 and 2ν1 + ν3 − ν1), a Boltzmann population of the initial state (1,0°,0) was assumed. The hot bands ν1 + ν2 − ν1 and 2ν1 + ν2 − ν1, which also lie in the energetical range of the presented spectra, are not included because their intensities are lower by factors of about 8 and 400, respectively (cf. Table 3). Table 3. Rotationless Transition Dipole Moments μvv′ (D) for Vibrational Transitions of FHF− Isotopologuesa lower state

upper state

FHF−

FDF−

FTF−

(0,0°,0)

(0,0°,1) (1,0°,1) (2,0°,1) (3,0°,1) (0,11,0) (1,11,0) (2,11,0) (0,0°,1) (1,0°,1) (2,0°,1) (3,0°,1) (0,11,0) (1,11,0) (2,11,0)

0.854 0.383 0.144 0.050 0.222 0.015 0.001 0.593 0.621 0.465 0.230 0.017 0.223 0.021

0.801 0.250 0.072 0.019 0.171 0.008 0.001 0.471 0.698 0.330 0.125 0.011 0.172 0.011

0.765 0.191 0.048 0.010 0.141 0.006 0.001 0.423 0.712 0.259 0.090 0.008 0.141 0.007

(1,0°,0)

a

Calculated with the EDMF of Table S2 (Supporting Information) and vibrational wave functions of Ĥ − Ĥ cor. Squared rovibrational transition dipole moments were fitted according to eqs 6 and 7.

As may be seen from Figure 4a, the P-branch of the ν3 band of FHF− strongly overlaps with the R-branch of the ν2 band. As a result of the very strong Coriolis interaction with ν3 at larger J values (cf. Figure 3), the ν2 band shows a band head for line R(34) at 1300.345 cm−1. This was also observed by Kawaguchi and Hirota.3 The P-branch of the ν2 band has a lower intensity than its R-branch. As a consequence, only lines P(29), P(33), and P(39) could be observed.3 The inset of Figure 4a shows the Q-branch transitions of the ν2 band. The intensities of the strongest lines are comparable with those of the P-branch and are smaller by a factor of 4.5 compared with the strongest lines in the corresponding R-branch. As may be expected from the spectroscopic constants given in Tables 1 and 2, the hot band ν1 + ν3 − ν1 (blue lines in Figure 4a) shows a band head at line R(13) with a wavenumber of 1269.412 cm−1. In their second paper on FHF− and FDF−, Kawaguchi and Hirota3 assigned a line at 1269.5628 cm−1 to R(13) of the ν1 + ν3 − ν1 band, in very good agreement with our calculations. It is not clear, however, whether that line corresponds to the band head

Figure 3. Variation of the relative rovibrational energies of the pair of states ν3(0,0°,1) and ν2(0,11,0), with red dots describing the unperturbed situation (see the text). The insets show the variation of the expectation value of |l| with J.

rotational quantum number J, which provides a convenient diagnostic tool to study rovibrational interactions.47 Owing to the large difference in rotational constants between the interacting states, amounting to 0.01910 cm−1 for FHF− and to 0.01417 cm−1 for FDF−, the graphs describing the unperturbed situation (red dots) cross between J = 47 and J = 48 for FHF− and between J = 19 and J = 20 in the case of FDF−. In FTF−, ν3 lies below ν2 and so no crossing occurs. Inclusion of Ĥ cor in the rovibrational Hamiltonian (blue dots) leads to a mixing of the two states. A nice indicator of the amount of mixing is provided by ⟨|l|⟩, which starts at values of 0 and 1 and attains a common value of 0.5 at the crossing points occurring for FHF− and FDF−. 9700



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Figure 5 shows stick spectra of FHF− and FDF− in the region of the combination bands ν1 + ν3 and ν1 + ν2, represented by

Figure 4. Computed stick spectra of FHF− (a), FDF− (b), and FTF− (c), all at 375 K. Colors employed: ν2 (green), ν3 (red), and ν1 + ν3 − ν1 (blue). Abscissae in cm−1. Throughout, the relative intensity of the strongest line of a given isotopologue is normalized to unity.

Figure 5. Computed stick spectra of FHF− (a) and FDF− (b), both at 375 K. Colors employed: ν1 + ν2 (green), ν1 + ν3 (red), and 2ν1 + ν3 − ν1 (blue). Abscissae in cm−1. For normalization of the relative intensities see the caption of Figure 4.

because the adjacent lines R(12) and R(14) were not observable. Figure 4b shows the computed stick spectrum of FDF−. For that isotopologue, the R-branch of the ν2 band and the Pbranch of the ν3 band are overlapping completely. Actually, the most intense line of Figure 4b is R(23) of the ν2 band, with a relative intensity of unity. The intensity of the strongest line within the Q-branch is about 15 times smaller. Like for FHF−, the ν2 band of FDF− is characterized by a band head, calculated at R(11) = 931.026 cm−1. Although the existence of an experimental band head is obvious from Figure 1a of ref 4, its position could be only roughly determined by Kawaguchi and Hirota.4 The reported value of 930.8 cm−1 is clearly too low. This is not too surprising because no lines could be observed between R(7) at 930.690 cm−1 and R(15) at 930.658 cm−1. Another band head is predicted within the hot band ν1 + ν3 − ν1 of FDF− and computed to occur at R(17) = 884.576 cm−1. The hot band was searched for in the 880 cm−1 region, but no lines were observed.4 Finally, the predicted IR spectrum of FTF− is shown in Figure 4c. Again, two band heads are found in the lower region of the spectrum (680−800 cm−1), but the fundamental with band head at R(8) = 765.891 cm−1 is now ν3. Within the hot band ν1 + ν3 − ν1, the band head is predicted at R(19) = 719.033 cm−1.

red or green lines, respectively. In addition, the spectra include the hot bands 2ν1 + ν3−ν1 (blue lines). Band heads now appear in the “stretch-only” bands. For the ν1 + ν3 bands of FHF− and FDF−, they are located at R(11) and R(15), with calculated wavenumbers of 1852.093 and 1474.433 cm−1, respectively. The former may be compared with the experimental result of R(11) at 1852.2457 cm−1 (see Figure 2 of ref 2; note that the spectrum shows the ν1 + ν3 band, not the ν3 band). No band head within the ν1 + ν3 band was observed for FDF− because Rbranch lines were detectable only between R(31) and R(41).4 Note that Table 1 of ref 4 suffers from a few typographical errors; in particular, R(2)−R(4) of the ν1 + ν3 band should be replaced by P(2)−P(4). The computed positions for the band heads of the hot bands are R(14) at 1774.420 cm−1 (FHF−) and R(16) at 1402.986 cm−1 (FDF−). The results for the former stand as predictions. The assignment of the latter agrees with experiment (after correction of the early band assignment 2 ) and the calculated line position exceeds the experimental value by only 0.135 cm−1. For both FHF− and FDF−, the intensities of lines within the ν1 + ν2 combination bands are rather low because Coriolis interaction is less pronounced than for the fundamentals. 9701



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familiar double harmonic approximation (DHA), only the transitions (1,0°,0) → (1,0°,1) and (1,0°,0) → (1,11,0) would have nonzero values. It is the huge mechanical anharmonicity that gives most of the hot-band transitions μvv′ values larger than 0.1 D. Because the EDMF is almost linear in the vicinity of equilibrium (cf. Figure 2), electrical anharmonicity plays a very minor role.

In the calculation of transition dipole moments (cf. section 2), either eigenfunctions of the full Watson Hamiltonian or eigenfunctions of Ĥ − Ĥ cor (“unperturbed”) were employed. In both cases, values for (μif 2/FHL)1/2 were computed and plotted against m (cf. section 2). As an example, Figure 6 shows the

4. CONCLUSIONS On the basis of explicitly correlated coupled cluster calculations (method: CCSD(T*)-F12b) close to the basis set limit, an accurate near-equilibrium potential energy function has been constructed for the highly anharmonic bifluoride ion. The empirically corrected PEF makes use of four pieces of experimental data for the most abundant isotopologue, FHF−. Variational calculations of rovibrational energies yield spectroscopic constants that are in virtually perfect agreement with all experimental data existing for FHF− and FDF−.2−4 In several cases, the accuracy of the present calculations outperforms the precision of the available spectroscopic data and many predictions are being made. The IR spectra computed for FHF−, FDF−, and FTF− are rather unique, showing strong to extremly strong signatures of Coriolis interaction and a larger number of band heads, which are an unusual feature in rovibrational spectroscopy. As a consequence of the very strong Coriolis interaction in FDF−, rather surprisingly, it is the R(23) line of the ν2 band that has the highest intensity of any line in the IR spectrum of FDF−. The occurrence of band heads in the ν2 bands of all three isotopologues and not in the ν3 bands, as would be expected from the unperturbed B001 values (cf. Table 2), may be surprising, as well.

Figure 6. Variation of the expression (μif 2/FHL)1/2 with m for FDF−. Results for band ν2 are given in green; those for band ν3, in red. Curves of dots correspond to calculations with the full Hamiltonian (“perturbed”); curves of triangles, to calculations neglecting Ĥ cor (“unperturbed”).

results for the ν2 and ν3 bands of FDF−. Though the values calculated with the aid of the unperturbed wave functions show a very small variation with m, the change of the values calculated with the full Hamiltonian is rather dramatic. Of course, this is a consequence of the very strong Coriolis interaction discussed in section 3.2. Nevertheless, a closer inspection may be interesting. Upon a change in m from 0 to 60, the transition dipole moments for transitions within the ν3 band are reduced by a factor of 2−3, whereas the values for ν2 increase by a factor of more than 6. As a consequence, the sum of the corresponding line intensities of both transitions for a single value of m may be larger than the sum of the intensities for the unperturbed case. Such a phenomenon is well-known from Coriolis interaction in larger molecules as discussed by Mills,48 who gave an easy explanation for this observation. According to Figure 6, rotationless vibrational transition dipole moments are best obtained from the unperturbed data and we decided to fit these using eqs 6 and 7. The results are listed in Table 3. For FHF−, the transition dipole moment of the proton stretching vibration is calculated to be μ3 = 0.854 D, in quite good agreement with the previous CCSD(T)/AVQZ value of 0.825 D as obtained from 2-dimensional variational calculations.49 Deuterium or tritium substitution lowers μ3 by 6.2% and 10.9%, respectively. Comparable calculations for 35 ClH35Cl− and 35ClD35Cl− yield μ3 values of 1.418 and 1.394 D, respectively.50 As is well-known for strongly hydrogenbonded systems, the transition dipole moments for combination tones of ν3 with one quantum of intermolecular stretching vibration(ν1) are quite large, amounting to as much as 0.383 D for FHF−. This is still larger than the transition dipole moment of the proton bending vibration (μ2) by about a factor of 2 (cf. Figure 6). Transition dipole moments of hot transitions arising from vibrational state (1,0°,0) are also listed in Table 3. Within the

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ASSOCIATED CONTENT

S Supporting Information *

Two tables with parameters of the empirically corrected potential energy function (PEF) and the electric dipole moment function (EDMF) for FHF−, and three tables with rovibrational energies for the low-lying vibrational states of FHF−, FDF−, and FTF−. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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