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 ν3 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 experimentally7,8 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 0920 and GAMESS21 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 ΔE0 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
where ΔEelec is the difference in total energies for the reaction (1), and ΔZPE is the zero-point energy correction. For comparison, the W1BD28 and G429 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 (Re) 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 ν3 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.
aChange in bond distance relative to that of HX molecule (X = F, Cl, and Br). bRef. 8. cF–F internuclear distance calculated from the rotational constant. dRef. 15. eCl–Cl internuclear distance calculated from the rotational constant. fRef. 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 ν1 and ν2 fundamental bands and the ν1 + ν2 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 ν3 and ν1 + ν3 frequencies. The levels of theory and basis sets affect the PT2-VSCF frequencies differently for each band, although the effects are small. The ν2 and ν1 + ν2 frequencies are most significantly affected by the basis set employed. For the VPT2 method, the agreement of the ν1 and ν3 fundamental frequencies with experimental data are less satisfactory than the PT2-VSCF frequencies, although the agreement for the ν1 + ν3 combination band appears to be much better. The basis set effect is the most significant for the ν3 and ν1 + ν3 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 ν3 and ν1 + ν3 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 ν3 and ν1 + ν3 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.
aCalculated by the VSCF method. bCalculated by the PT2-VSCF method. cThe values in parentheses are the infrared intensities in km/mol. dRef. 8. eRef. 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 ν3 and ν1 + ν3 bands. As in the case of FHF−, the ν3 and ν1 + ν3 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 ν3 frequency by the VPT2 method is even larger. Also, it is found that the ν3 and ν1 + ν3 frequencies calculated depend very much on the level of theory or the basis set. For example, the PT2-VSCF frequencies for ν3 and ν1 + ν3 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 ν3 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 ν3 and ν1 + ν3 frequencies with experimental data. It is also seen in the Tables 3 and 4 that the VPT2 method predicts the ν1 and ν1 + ν2 frequencies considerably smaller, compared to the PT2-VSCF method.
aCalculated by the VSCF method. bCalculated by the PT2-VSCF method. cThe values in parentheses are the infrared intensities in km/mol. dRef. 15. eCalculated from the observed centrifugal distortion constant. fEstimated from the ν3/ν2 perturbation analysis. gRef. 14b. h(ν1 + ν3) - ν3. iRef. 14a.
aCalculated by the VSCF method. bCalculated by the PT2-VSCF method. cThe values in parentheses are the infrared intensities in km/mol. dRef. 18b. e(ν1 + ν3) − ν3. fRef. 17a. gRef. 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 ν3 and its combination bands of XHX− anions. In many theoretical studies for XHX− anions, the n3 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 ν3 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 ν3 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 ν3 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, ΔH298 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−.
aUncorrected values without counterpoise calculations. bRef. 5. cRef. 31. dRef. 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 D0 = 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 ν3 and the combination ν1 + ν3 frequencies. The VPT2 method performs less satisfactory in predicting the anharmonic frequencies than the PT2-VSCF method. For ClHCl− and BrHBr−, the ν3 and ν1 + ν3 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 ν3 and ν1 + ν3 bands, suggesting that the potential energy function along the ν3 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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