Density functional study of hydrogen-bonded systems:
from gas-phase adducts to catalytically relevant systems (*)
P. UGLIENGO, B. CIVALLERI and E. GARRONE
Dipartimento di Chimica IFM, Universitá di Torino - Via P. Giuria 7, I-10125 Torino, Italy (ricevuto il 28 Febbraio 1997; approvato l’8 Maggio 1997)
Summary. — In this note, the performance of methods based on density functional
theory (DFT) to deal with hydrogen bonding interactions (HBI) is addressed by comparing geometries, harmonic vibrations and binding energies with those obtained by well-established ab initio techniques on the gas phase adducts (FH)2,
FH/CO and FH/NH3. Anharmonic F-H stretching mode was also studied in detail.
The best combination of functional and basis set is selected and has then been used to model the isolated hydroxyl groups at the silica surface (using H3SiOH (SIL))and
the Brønsted site present in the zeolite cavities(using H3Si(OH)AlH3 (BRO)), both
free and in interaction with NH3, H2O and CO molecules. Comparison with MP2 and
available experimental data is discussed.
PACS 31.15 – Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations).
PACS 36.40 – Atomic and molecular clusters. PACS 82.65.Jv – Heterogeneous catalysis at surfaces. PACS 01.30 – Conference proceedings.
1. – Introduction
In recent years, DF methods have received an extremely high interest from the computational chemists community. Reasons are at least threefold: i) improvement of the accuracy of the numerical procedures needed to solve the Kohn-Sham equations; ii) new definitions of accurate exchange-correlation functionals; iii) favourable speed/cost ratio of the calculations with respect to the traditional post Hartree-Fock methods. In a series of papers it has been showed that DF equilibrium geometries, frequencies as well as transition state geometries and energies are very close to those computed by high-level accuracy (and cost) CCSD(T) [1]. To reach such an accuracy, however, gradient-corrected exchange-correlation functionals associated with polarized Gaussian basis sets of at least double-zeta quality are mandatory. Interestingly enough, basis set dependence of the computed quantities is far less critical than the one associated to MP2 or CCSD(T) methods. In the realm of intermolecular interactions, few studies
(*) Paper presented at the “First International Workshop on Reactivity of Oxide Materials. Theory and Experiment”, Como, 8, 9 November 1996.
have been published so far, the majority of which deals with hydrogen bonding interactions (HBI) [2]. HBI are of extreme relevance in all fields of chemistry, playing a central role in dictating the structure of many organic liquids, molecular crystals and biological macromolecules. HBI are also relevant in the context of Brønsted catalysts, because the first steps of molecular recognition between surface sites and adsorbed molecules involve, inevitably, the formation of a hydrogen bonded complex. The same holds for the case of the interaction of molecules with siliceous materials, for instance the amorphous silica used in chromatography.
2. – Theoretical methods
A full geometry optimization of the structures shown in fig. 1 has been performed with analytical gradients of the total energy at all levels of theory adopted and the resulting harmonic frequencies have been computed by analytical second differentiation of the total energy followed by the solution of the equations of nuclear motion by standard methods. Zero point energy (ZPE) calculations and characterization of stationary points have been also carried out. Heats of formation DH0(0) are computed from the binding energies relative to the free molecular
components corrected for both BSSE and zero point energy. For DFT calculations, the normal grid integration mesh has been adopted. Anharmonic stretching frequency of both F-H (for the gas phase adducts) and O-H (for SIL and BRO adducts) modes have been calculated by a method already adopted by some of us [3]. Shifts in the (an)harmonic F-H and O-H stretching frequencies are computed as: Dn4n(X-H)adduct2
n(X-H)free( X 4O, F).
3. – Gas phase adducts
Calculations have been carried out at SCF, MP2 and DF levels. DF computations have employed the local exchange/correlation functionals S-VWN, gradient-corrected exchange-correlation functionals B-LYP, B3-LYP and B-P86 as described in the
Fig. 1. – Top: structures of the gas phase adducts. Bottom: structures of complexes SIL/CO, SIL/H2O, SIL/NH3and BRO/CO, respectively.
Gaussian-94 package. The basis sets used are either from Pople or Dunning collections and are grouped in two categories: i) polarized, hereafter indicated as POL: 6-31G(d , p), DZP, cc-pVDZ, cc-pVTZ; ii) diffused-polarized, hereafter indicated as POLD: 6-311G(d, p), 6-3111G(d, p), 6-3111G(2d, 2p), 6-3111G(3df, 2pd), aug-cc-pVDZ, aug-cc-pVTZ. For the case of CO, both FH/CO and FH/OC were considered. For the sake of brevity, the whole set of results relative to all combinations of functionals and basis sets is not reported here, being available on internet [4]. Table I shows average heats of formation together with the corresponding standard deviations computed with data resulting from the basis sets belonging to the POL and POLD sets, respectively. Standard deviations are then a measure of the convergency of the computed property with the basis set. The same procedure has been applied to compute the average values of Dn(FH) and the results are shown in table II. A few conclusions are possible: i) the local exchange/correlation S-VWN method greatly overestimates both DH0(0) and Dn(FH) with respect to both gradient corrected and MP2 methods; ii) B-LYP and B3-LYP values of aDH0( 0 )b are very close to each other,
representing a definite improvement with respect to the SCF data; iii) for the FH/FH adduct, the B3-LYP aDH0
( 0 )b value of 11.1 61 kJ/mol is in excellent agreement with both experimental measurement of 12.70 60.012 kJ/mol and values of 12.3 kJ/mol and 11.6 kJ/mol computed at CCSD(T)/TZ2P( f , d) and CCSD(T)/QZ3P, respectively [5];
TABLEI. – Average heats of formation aDH0( 0 )b corrected for BSSE and corresponding standard deviations s for all adducts obtained with data computed with POL and POLD sets, respectively. Data in kJ/mol.
Adduct Set SCF S-VWN B-LYP B-P86 B3-LYP MP2
FH/OC POL 2.261 8.261.4 0.2361.5 2 1.2361.5 1.461.3 2 0.1 61 POLD 2.260.8 1061 1.2461.0 2 0.4361 2.161 0.4261.0 FH/CO POL 2.360.8 24.262 8.362 7.862 8.162 5.961.3 POLD 2.160.3 25.560.5 7.560.4 7.960.5 7.660.4 6.560.9 FH/FH POL 9.861.0 24.864 7.265.5 6.265.5 8.565.2 7.464 POLD 8.960.8 26.660.4 1060.7 960.5 11.260.7 9.460.8 FH/NH3 POL 30.763 6865 39.965 42.264.5 41.364.4 34.763.6 POLD 29.161.4 7362 4261.7 44.761.5 42.461.7 36.862
TABLEII. – Average FH harmonic frequency shifts aDnb and corresponding standard deviations s for all adducts obtained with data computed with POL and POLD sets. Data in cm21.
Adduct Set SCF S-VWN B-LYP B-P86 B3-LYP MP2
FH/OC POL 1766 83620 18611 36611 1869 166 POLD 2764 130610 4667 6064 4567 2268 FH/CO POL 6569 329636 169625 217625 151619 96614 POLD 7967 434615 234613 286615 204612 149623 FH/FH POL 8866 362649 18768 22069 15567 11369 POLD 8767 325620 175615 210616 159616 114621 FH/NH3 POL 463632 898695 679687 758686 670672 563677 POLD 535614 1200620 936627 1020618 879627 767649
TABLEIII. – Harmonic (vh) and anharmonic (v01) Dn(FH) (cm21) for the FH/CO and FH/NH3 adducts. Basis set is 6-3111G(2d, 2p). Gas phase measurements labelled as (gp); cold matrix measurements (12 K) labelled with the corresponding atomic symbol.
Adduct 2Dn(FH) SCF B-LYP B3-LYP MP2 Experiment
FH/CO Dvh 79 232 200 157 Dv01 100 281 250 194 117(gp)/165 (Ar) (a) FH/NH3 Dvh 529 912 860 735 Dv01 674 1089 1058 914 912 (Ar) (b) (a) Reference [7]. (b) Reference [6].
iv) for FH/NH3, aDH0( 0 )b values of 42 62 kJ/mol at both B3-LYP and B-LYP are
in excellent agreement with the values of 41.5 kJ/mol and 42.1 kJ/mol at QCISD(T)/6-3111G(d, p)//MP2/6-311G(d, p) and MP4/6-3111G(d, p)//MP2/6-311 G(d , p), respectively [6]; v) for FH/CO, both B3-LYP and B-LYP values of aDH0( 0 )b
are 7.5 60.5 kJ/mol, very close to the values of 7.7 kJ/mol and 7.5 kJ/mol computed at MP2/TZ2P and QCISD(T)/6-311G(2d, 2p)//MP2/6-311G(d, p), respectively [7]: the preference for the C-down structure in comparison with the O-down one is clear-cut at both B-LYP and B3-LYP levels; vi) as to the Dn(FH), B-LYP shifts are systematically larger than the B3-LYP values, as evident from table II. Even larger are the differences between values computed with those functionals and those at MP2 level, the latter being always smaller than the former; vii) standard deviations associated to the POL set are significantly reduced with the POLD set where good convergency for both DH0( 0 ) and Dn(FH) are attained. It is worth noting that, for the case of POL set,
the standard deviations affecting the Dn(FH) at DF levels are larger than the corresponding MP2 ones: however, DF deviations decrease significantly with the POLD set showing a faster convergency of the F-H shift with basis set size.
Few computed values of the harmonic Dn(FH) are available in the literature. For the FH/FH adduct, values of 2105 and 2107 cm21 have been computed at
CCSD(T)/QZ3P and CCSD(T)/TZ2P(f , d), respectively. Experimental determination of Dn(FH) in cold matrix (Ne, 4 K) gives 2105 cm21 which decreases to 2128 cm21 in a
different matrix (Ar, 12 K) [5]. Gas phase determination gives a value of 293 cm21, the
lowest absolute shift among both computed and measured values. Our computed DF shifts are, in all cases, too large as shown by the data in table II; MP2 results are closer to both experiment and to the above reported results based on CCSD(T) theory. A similar trend is observed for the FH/CO adduct: the value of 2165 cm21is measured in
cold matrix (Ar, 12 K) whereas the value of 2117 cm21 results from gas-phase
measurements [7]. The MP2/TZ2P value of 2154 cm21 is in good agreement with our
average MP2 value of 2149623 cm21 (see also table III).
The anharmonic frequency shift Dn(FH) has also been computed following the procedure quoted in the theoretical-methods section. Table III shows results of such calculations for FH/CO and FH/NH3 adducts. Comparison with both experimental and
MP2 data shows a trend of the DF techniques to overestimate Dn(FH), probably as a consequence of the incomplete cancellation of the self-energy interaction with the present functionals, which is particularly severe for hydrogen-containing bonds. The
comparison with experiment is, however, difficult because of the large influence of either the polarization effects of the cold matrix or the roto-vibrations coupling for the gas-phase measurements on the values of Dn(FH).
4. – Models of acidic sites for silica and zeolites
Interactions of SIL with H2O, NH3and CO have been computed, whereas for BRO,
only the complex with the CO molecule has been considered. The resulting structures are shown in fig. 1. Only the B3-LYP functional has been adopted here with DZP, 6-311G(d, p) and aug-cc-pVDZ basis sets, because of its superior performance with respect to S-VWN, B-LYP and B-P86 when dealing with the properties of the gas phase adducts (see the previous section). The excellent behaviour of B3-LYP functional is mainly due to the Hartree-Fock component in the exchange contribution to the electron correlation. Table IV compares results obtained at B3-LYP and MP2 levels, with experimental measurements of the adsorption of the corresponding molecules on amorphous silica and zeolites. A few points of conclusion are: i) basis set dependence of DH0( 0 ) at both B3-LYP and MP2 is large, with a significative decrease of the DH0( 0 )
value passing from DZP to aug-cc-pVDZ basis set; ii) the Dvh(OH) values decrease
TABLE IV. – Heats of formation, DH0( 0 ) (kJ/mol), harmonic Dv
h(OH) (cm21) and fundamental
Dv01(OH) (cm21) shifts in the frequency of the OH mode for SIL and BRO in the reported adducts. Dn(CO) (cm21) is the harmonic frequency of CO for SIL/CO and BRO/CO adducts. Bare numbers are at B3-LYP level. Numbers in parentheses are at MP2 level.
Adduct Method 2DH0(0) Dvh(OH) Dv01(OH) Dn(CO)
SIL/CO DZP 5.7 (2.8) 283 (252 ) 2100 (260 ) 27 (20) 6-311G(d, p) 3.6 (3.6) 270 (248 ) 285 (259 ) 23 (16) aug-cc-pVDZ 2.8 (4.6) 265 (259 ) 283 (276 ) 19 (16) experiment (a) 11.0 278 14 SIL/H2O DZP 22.3 (19.1) 2295 (2228 ) 2358 (2279 ) 6-311G(d, p) 16.6 (15.2) 2237 (2189 ) 2290 (2236 ) aug-cc-pVDZ 14.2 (13.8) 2222 (2203 ) 2272 (2252 ) experiment (b) 25/45 2300 SIL/NH3 DZP 32.1 (26.8) 2541 (2452 ) 2677 (2577 ) 6-311G(d, p) 25.8 (23.0) 2504 (2432 ) 2639 (2562 ) aug-cc-pVDZ 22.3 (21.9) 2452 (2395 ) 2571 (2549 ) experiment (c) 30/60 2650 / 2 950 BRO/CO DZP 13.9 (14.4) 2229 (2174 ) 2285 (2224 ) 46 (33) 6-311G(d, p) 9.8 (10.9) 2190 (2141 ) 2240 (2189 ) 39 aug-cc-pVDZ 8.5 2184 2235 35 experiment (d) 13.6 / 17.2 2275 / 2 340 32 / 35 (a) Reference [3].
(b) Reference [9] for IR data; ref. [8] and references therein for heats of formation.
(c) Reference [8]. The shifts in the frequency of the OH mode have been measured at 298 K and 4 K, respectively. (d) For DH0(0) see ref. [10], for vibrational features see ref. [3].
almost monotonically with the size of the basis set: however, convergency is still not attained at the aug-cc-pVDZ and it is slower for B3-LYP than for MP2 methods; iii) anharmonic corrections to the Dvh(OH) are sizeable (see table IV, Dv01(OH)
column) and should be included for strict comparison with experimental measurements. From previous section we expect the Dv01(OH) computed with DF
methods to be too large; however, comparison with the experimental measurements carried out for both silica/molecule and zeolite/molecule complexes shows a deficincy in the clusters adopted (SIL and BRO), because too small values of the n(OH) shifts were computed; iv) the harmonic shift Dn(CO) is overestimated by B3-LYP particularly for the weaker interaction with SIL and shows a larger basis set dependence with respect to the MP2 values.
5. – Conclusions
The present results show that DF methods are extremely efficient in calculating the binding energies of H-bonded adducts at the condition that basis sets containing both diffuse and polarized functions are adopted. Harmonic and anharmonic frequency shifts computed for the F-H stretch when engaged in hydrogen-bonded adducts are overestimated when compared to both MP2 and CCSD(T) levels on the one hand, and experimental measurements on the other hand. The frequency shifts computed with any DF methods should then be taken with caution. The study has been extended to the modelling of silica and zeolites acidic sites by studying the H-bonded adducts of H2O, NH3 and CO with the cluster models SIL and BRO at B3-LYP and MP2 levels.
Energetic B3-LYP results are in good agreement with MP2 data; vibrational frequency shifts of the O-H stretch either harmonic or with anharmonic correction are over-estimated by B3-LYP method with respect to MP2 values. Comparison with experimental data shows that both minimal models as SIL and BRO are reasonable models for the isolated silica hydroxyl and for the Brønsted acidic site in zeolites even if they show a too less acidic behaviour with respect to the real materials.
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