#### Journal Information

Journal ID (publisher-id): chemical

Title: Journal of the Korean Chemical Society

Translated Title (ko): 대한화학회지

ISSN (print): 1017-2548

ISSN (electronic): 2234-8530

Publisher: Korean Chemical Society대한화학회

Hydrogen bonds are important for the structure, function, solvation, and dynamics of a large number of chemical systems in
many fields of chemistry and biological science.^{1,2} Hydrogen-bond systems often display a large anharmonicity in their vibrational structures and possess irregular vibrational
energy patterns. A normal mode harmonic treatment of the vibrations are not appropriate in describing the vibrational structures
and frequencies of such systems, thereby stimulating a number of theoretical studies.^{3,4}

Hydrogen bihalide anions, XHY^{−} (X, Y = halogen), one of the simplest hydrogen-bonded complexes, provide an interesting example of hydrogen bond due to their
strong hydrogen bond stability. For example, FHF^{−} anion has the strongest known hydrogen bond with the hydrogen-bond enthalpy of 45.8 kcal/mol,^{5} much larger than ordinary hydrogen-bond energy of 3–10 kcal/mol. Hydrogen bihalide anions were also utilized in the negative
ion photo-detachment experiments to prove directly the transition state region of the reactive potential energy surface.^{6}

The spectroscopic characterization of bihalide anions has proven challenging due to a strong anharmonicity in their vibrational
structures, particularly in the ν_{3} antisymmetric stretching mode, and this interesting feature leads to a number of experimental and theoretical studies of
the vibrational structures of these anions. In particular, the vibrational spectra and frequencies of FHF− anion have been
extensively studied experimentally^{7,8} and theoretically.^{9–12} There are less information available for ClHCl^{−9b,10,11a,13–16} and BrHBr^{−} anions.^{10,11a,13,17–19}

In the present work, we have studied the symmetric hydrogen bihalide anions, FHF^{−}, ClHCl^{−}, and BrHBr^{−} by high level *ab initio* quantum chemical methods. *Ab initio* calculations at several different levels are performed using large basis sets to determine the molecular structure, vibrational
frequencies, and hydrogen-bond energies of bihalide anions. We aim to provide the consistent sets of these data at sufficiently
high levels for all three anions. In particular, the anharmonic frequencies are calculated at high levels by two different
approaches, the vibrational self-consistent field (VSCF) and the second-order vibrational perturbation theory (VPT2) methods,
and the results are compared with the experimental frequencies. The hydrogen-bond energies and enthalpies are also calculated
at several different levels including the W1BD and G4 composite methods of yielding highly accurate thermochemical data.

In the present study, several levels of *ab initio* calculations were performed using the Gaussian 09^{20} and GAMESS^{21} electronic structure programs. *Ab initio* calculations were performed at the levels of Hartree-Fock (HF), second-order Møller–Plesset (MP2), and coupled cluster with
single, double, and noniterative triple substitutions [CCSD(T)] theories. The two different basis sets of valence triple-zeta
quality, 6-311++G(2df,2p) and aug-cc-pVTZ, were used in the calculations. The equilibrium geometries of hydrogen bihalide
anions were fully optimized with no constraint on the geometry with the tight convergence criteria. Each optimized structure
was characterized by harmonic vibrational frequency calculations.

In order to obtain anharmonic corrections to the frequencies of hydrogen bihalide anions, two different approaches were used:
firstly, the VSCF method,^{22} and its correlation-corrected extension via second-order perturbation theory (referred as cc-VSCF or as PT2-VSCF in the literature),^{23 } and secondly, the VPT2 method.^{24} In the VSCF method, the full vibrational wavefunction is factorized into single mode wavefunctions corresponding to the different
normal modes, and a pairwise coupling approximation is used for the potential energy function in the normal mode representation,
where the potential energy of the system is represented by the sum of separable terms and pair coupling terms, neglecting
triple couplings of normal modes and higher-order interactions. The VSCF and PT2-VSCF methods, as implemented in GAMESS, utilize
direct calculation on the *ab initio* potential energy surfaces and a grid representation for the potential energy terms.^{25} The VPT2 approach uses quadratic, all relevant cubic and quartic force constants to create a quartic force field. The derivatives
are calculated with an *ab initio* potential energy. The VPT2 frequency calculations were performed by the routines implemented in Gaussian 09. The VSCF and
VPT2 methods also allow to determine the frequencies of combination bands as well as fundamental bands.

The hydrogen-bond energy of hydrogen bihalide anions, XHX can be defined as the energy change in dissociation reaction of

Accordingly the hydrogen-bond energy ΔE_{0} was calculated as the difference in total energies of the reactant XHX^{−} and products X^{−} + HX of the above reaction with the zero-point energy corrected. The basis set superposition error (BSSE) was corrected by counterpoise
calculations.^{26} For a complex AB formed from the free monomers A and B, the counterpoise correction is given by

where
^{27} In the present calculations for XHX^{−}, A refers to X^{−} and B refers to HX. Thus, the hydrogen-bond energy ΔE_{0} was calculated as

where ΔE_{elec} is the difference in total energies for the reaction (1), and ΔZPE is the zero-point energy correction. For comparison, the
W1BD^{28} and G4^{29} composite methods, as embedded within Gaussian 09, were also employed to determine the hydrogen-bond energies of the anions.
These methods use extrapolation schemes to achieve highly accurate thermochemical data by a series of calculations with different
levels of accuracy and basis sets.

*Table* 1 present the optimized equilibrium geometries and harmonic vibrational frequencies of hydrogen bihalide anions, XHX^{−} (X = F, Cl, and Br). Although structures of these anions have not been determined experimentally, gas-phase infrared spectroscopy
of FHF^{−8} and ClHCl^{−15} strongly supports that these anions have a linear symmetric (D_{∞h}) structure. For FHF^{−}, the calculations at all levels including the HF level predict that the anion has a linear symmetric structure. However,
the results obtained at the HF level for ClHCl^{−} and BrHBr^{−} suggest that these anions have an asymmetric structure with two X−H bonds of different length, while the MP2 and CCSD(T)
correlated methods suggest that both anions have a symmetric structure, the same as in previous *ab initio* studies of these anions.^{11,16,19} The chemical bonding in XHX^{−} may be regarded as a four-electron-three-center bond, but this type of bonding cannot be described properly at the level
of HF theory,^{19} and results from the correlated methods should provide better estimate of true structures of these anions.

We can estimate the accuracy of the bond lengths computed at various different levels by comparing with those from the analysis
of infrared spectra of FHF^{−8} and ClHCl^{−15}. In those studies, the F−F and Cl−Cl internuclear distances (R_{e}) were calculated to be 2.27771 and 3.1122 Å, respectively. Therefore assuming the linear symmetric structure for these anions,
the H−F and H−Cl bond lengths should be 1.139 and 1.556 Å, respectively. It can be seen in *Table* 1 that the H−F bond lengths obtained at HF levels deviate considerably from the value of 1.139 Å, while the results obtained
at the MP2 or CCSD(T) level are very close to this value. Similarly, the H−Cl bond length of 1.556 Å agrees very well with
those from the MP2 or CCSD(T) calculation. Therefore, the H−Br bond length of ~1.70 Å obtained from the MP2 and CCSD(T) calculations
should be a good estimate for BrHBr^{−}. It is also noted that the bond lengths of ClHCl^{−} and BrHBr^{−} calculated at the CCSD(T) level are slightly longer than those at the MP2 level.

Also present in *Table* 1 are the harmonic frequencies calculated at various levels along with the experimental frequencies. As can be seen in *Table* 1, the harmonic frequencies calculated at all levels exhibit large deviations from the experimental frequencies. The harmonic
frequencies, especially for the ν_{3} antisymmetric stretching mode, or asymmetric vibration of H atom, are also very sensitive to the levels of theory and the
basis set used, suggesting that the vibrations of XHX^{−} anions be highly anharmonic.

^{a}Change in bond distance relative to that of HX molecule (X = F, Cl, and Br). ^{b}Ref. 8. ^{c}F–F internuclear distance calculated from the rotational constant. ^{d}Ref. 15. ^{e}Cl–Cl internuclear distance calculated from the rotational constant. ^{f}Ref. 18b.

*Table* 2 presents the anharmonic vibrational frequencies of FHF^{−} calculated by the VSCF/PT2-VSCF and VPT2 methods. In addition to fundamental bands, the anharmonic frequencies of a few combination
bands for which the experimental frequencies have been reported are also presented. For the VSCF and its correlation-corrected
PT2-VSCF methods, the frequency calculations were conducted at the MP2 and CCSD(T) levels, but for the VPT2 method, the frequency
calculations were performed only at the MP2 level, since the VPT2 calculation at the CCSD(T) level was not available in the
Gaussian package. It is seen in *Table* 2 that the VSCF method generally overestimates the frequencies by 20–30 cm^{−1} compared to the correlation- corrected PT2-VSCF method, due to the lack of dynamical correlation among the vibrational modes.
The PT2-VSCF frequencies show better agreements with the experimental frequencies available.^{7,8} In particular, the PT2-VSCF frequencies for the ν_{1} and ν_{2} fundamental bands and the ν_{1} + ν_{2} combination band obtained using larger aug-cc-pVTZ basis set are quite comparable to the experimental frequencies. However,
there are still significant deviations in the ν_{3} and ν_{1} + ν_{3} frequencies. The levels of theory and basis sets affect the PT2-VSCF frequencies differently for each band, although the
effects are small. The ν_{2} and ν_{1} + ν_{2} frequencies are most significantly affected by the basis set employed. For the VPT2 method, the agreement of the ν_{1} and ν_{3} fundamental frequencies with experimental data are less satisfactory than the PT2-VSCF frequencies, although the agreement
for the ν_{1} + ν_{3} combination band appears to be much better. The basis set effect is the most significant for the ν_{3} and ν_{1} + ν_{3} frequencies in the VPT2 method.

The anharmonic frequencies calculated, particularly by the PT2-VSCF method, are found to be much improved compared to the
harmonic frequencies. Although inclusion of anharmonic corrections improves the agreement with experiments, the discrepancy
from the experimental frequencies, especially for the ν_{3} and ν_{1} + ν_{3} bands, is still quite large. In order to test whether this discrepancy is due to the fact that the potential energy function
of the system is represented by the sum of separable terms and pair coupling terms in the VSCF method, the VSCF and PT2-VSCF
calculations were performed with inclusion of third-order coupling terms in the representation of the potential energy function.
Although inclusion of third-order coupling (denoted as Coupling = 3 in *Table* 2) yields small improvement for the ν_{3} and ν_{1} + ν_{3} frequencies, especially at the CCSD(T) level, it appears that the neglect of the third-order terms is not the main cause
of the discrepancy observed.

^{a}Calculated by the VSCF method. ^{b}Calculated by the PT2-VSCF method. ^{c}The values in parentheses are the infrared intensities in km/mol. ^{d}Ref. 8. ^{e}Ref. 7c.

*Tables* 3 and 4 present the anharmonic frequencies calculated by the VSCF and VPT2 methods for ClHCl^{−} and BrHBr^{−}, respectively. For these anions, the experimental frequencies are observed only for the ν_{3} and ν_{1} + ν_{3} bands. As in the case of FHF^{−}, the ν_{3} and ν_{1} + ν_{3} band frequencies calculated deviate significantly from the experimental frequencies. The PT2-VSCF frequencies for these bands
show deviations in the range of 100–200 cm^{−1}, and the deviation of the ν_{3} frequency by the VPT2 method is even larger. Also, it is found that the ν_{3} and ν_{1} + ν_{3} frequencies calculated depend very much on the level of theory or the basis set. For example, the PT2-VSCF frequencies for
ν_{3} and ν_{1} + ν_{3} calculated at the CCSD(T) level are considerably smaller than those at the MP2 level, and the VPT2 frequencies for these
bands vary considerably depending on the basis set used. This suggest that the potential energy function for the ν_{3} mode and the coupling of this mode into other modes are quite difficult to describe accurately. As in the case of FHF^{−}, it is found that inclusion of the third-order terms in the VSCF calculation does not improve the agreement of the ν_{3} and ν_{1} + ν_{3} frequencies with experimental data. It is also seen in the *Tables* 3 and 4 that the VPT2 method predicts the ν_{1} and ν_{1} + ν_{2} frequencies considerably smaller, compared to the PT2-VSCF method.

^{a}Calculated by the VSCF method. ^{b}Calculated by the PT2-VSCF method. ^{c}The values in parentheses are the infrared intensities in km/mol. ^{d}Ref. 15. ^{e}Calculated from the observed centrifugal distortion constant. ^{f}Estimated from the ν_{3}/ν_{2} perturbation analysis. ^{g}Ref. 14b. ^{h}(ν_{1} + ν_{3}) - ν_{3}. ^{i}Ref. 14a.

^{a}Calculated by the VSCF method. ^{b}Calculated by the PT2-VSCF method. ^{c}The values in parentheses are the infrared intensities in km/mol. ^{d}Ref. 18b. ^{e}(ν_{1} + ν_{3}) − ν_{3}. ^{f}Ref. 17a. ^{g}Ref. 17b.)

The cc-VSCF method including the PT2-VSCF method usually claims an accuracy of 30–50 cm^{−1},^{30} however, the present results exhibit much larger discrepancies in predicting anharmonic frequencies, particularly for the
ν_{3} and its combination bands of XHX^{−} anions. In many theoretical studies for XHX^{−} anions, the n_{3} frequencies calculated were shown to deviate significantly from the experimental frequencies, and this large deviation was
often attributed to the neglect of the bending-stretching coupling in two-dimensional potential energy surface.^{9b,10a,11,12b,19b} However, the present VSCF and PT2-VSCF calculations show that large deviation of the ν_{3} frequencies still exists, even though couplings among all vibrational modes are included in the calculation. The similar
VSCF calculation at the MP2 level for these anions also shows large discrepancy for the ν_{3} frequency.^{12a} Therefore, it is suspected that the perturbative corrections in the PT2-VSCF and VPT2 methods are not quite effective in
the treatment of anharmonic mode-mode couplings of XHX^{−} anions. Also for ClHCl^{−} and BrHBr^{−}, variation of the ν_{3} and its combination frequencies with the basis set employed appears to become larger, and this may be an indication that
larger basis sets than in the present study are needed in accurate description of the potential energy surfaces of these anions.

*Table* 5 presents the hydrogen-bond energies and enthalpies of XHX^{−} calculated at several different levels of theory with the zero-point energy and BSSE corrections. The enthalpy, ΔH_{298} is the value calculated at 298 K. For these anions, the experimental enthalpies at 298 K have been reported,^{5,31,32} and thus the calculated hydrogen-bond enthalpies can be directly compared with the experimental values. Also it turns out
that the counterpoise corrections are relatively large for these anions, implying that the basis set superposition error (BSSE)
is significant. For FHF^{−} and ClHCl^{−}, the BSSE are larger for the 6-311++G(2df,2pd) basis set than for the aug-cc-pVTZ basis set, while the reverse is true for
BrHBr^{−}.

^{a}Uncorrected values without counterpoise calculations. ^{b}Ref. 5. ^{c}Ref. 31. ^{d}Ref. 32.

For FHF^{−}, there are two different experimental hydrogen-bond enthalpies reported, that is, 45.8 and 38.6 kcal/mol.^{5,31} The present calculations at all levels result in the hydrogen-bond enthalpies of over 40 kcal/mol, and thus the experimental
value of 45.8 kcal/mol is more likely. The recent theoretical determination of the bond dissociation energy of D_{0} = 43.3 kcal/mol for FHF^{−} also supports this value.^{33} Among all levels of calculation for FHF^{−}, the W1BD method yields the closest value to the experimental value, however, the results obtained at the CCSD(T) and MP2
levels with the aug-cc-pVTZ basis set as well as the G4 method are within 2 kcal/mol from the experimental value.

For ClHCl^{−} and BrHBr^{−}, the hydrogen-bond energies and enthalpies calculated at various levels are not much different from one another as shown
in *Table* 5. It is also seen that for these anions, the hydrogen-bond energies obtained at the CCSD(T) level are consistently smaller
than those obtained at the MP2 level. For ClHCl^{−}, there are two experimental enthalpies available, which are rather close to each other.^{31,32} The enthalpy values obtained at the CCSD(T)/aug-cc-pVTZ level and the G4 method agree very well with the experimental values.
Similarly, for BrHBr^{−}, the enthalpy value by the G4 method is in a very good agreement with the experimental value,^{32} and the CCSD(T)/aug-cc-pVTZ enthalpy is also close to the experimental one. Therefore, it appears that the hydrogen-bond
energies of bihalide anions can be accurately determined at the CCSD(T) level, if sufficiently large basis set is employed,
and the G4 composite method is also very effective in determination of the hydrogen-bond energies.

In the present study, high level *ab initio* calculations have been performed to characterize symmetric hydrogen bihalide anions, FHF^{–}, ClHCl^{–}, and BrHBr^{–}. The geometries and vibrational frequencies of these anions are computed at several different levels of theory using large
basis sets. Although the HF calculations yield the unequal H−X bond lengths for ClHCl^{–} and BrHBr^{–}, the correlated MP2 and CCSD(T) calculations suggest that all these anions are of linear and symmetric structures, and the
computed bond lengths are consistent with experimental data.

The harmonic frequencies calculated at all levels exhibit large discrepancy from the experimental frequencies reported, suggesting
that the vibrations of these anions are very anharmonic. The anharmonic frequencies are calculated by two different approaches,
the PT2-VSCF and VPT2 methods. For FHF^{−}, the anharmonic frequencies obtained by the PT2-VSCF method agree reasonably well with the experimental frequencies, although
the discrepancy is rather large for the ν_{3} and the combination ν_{1} + ν_{3} frequencies. The VPT2 method performs less satisfactory in predicting the anharmonic frequencies than the PT2-VSCF method.
For ClHCl^{−} and BrHBr^{−}, the ν_{3} and ν_{1} + ν_{3} frequencies by both PT2-VSCF and VPT2 methods show quite large discrepancies with the experimental values. The PT2-VSCF calculations
at the MP2 and CCSD(T) levels yield considerably different frequencies for the ν_{3} and ν_{1} + ν_{3} bands, suggesting that the potential energy function along the ν_{3} mode and the coupling of this mode into other modes are quite sensitive to the levels of theory. It is also possible that
the perturbative correction alone is not sufficient for accurate description of the potential functions of bihalide anions.

The hydrogen-bond energies and enthalpies of bihalide anions are estimated with zero-point energy and BSSE corrections. The
W1BD and G4 composite methods are also employed for comparison. The hydrogen-bond enthalpies calculated at the CCSD(T) level
using large basis set agree quite well with the experimental values available, within a discrepancy of 1–2 kcal/mol. For FHF^{−}, there are two experimental hydrogen-bond enthalpies reported, and the present calculation supports the experimental value
of 45.8 kcal/mol over the other value of 38.6 kcal/mol. Also, the W1BD and G4 composite methods are found to be very effective
in predicting the hydrogen-bond enthalpies of bihalide anions.

This work was supported by Research Grant of Incheon National University in 2014 to B.-S. Cheong.

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