Anharmonic Vibrational Frequency Shifts upon Interaction of Phenol

Article pubs.acs.org/JPCA

Anharmonic Vibrational Frequency Shifts upon Interaction of Phenol(+) with the Open Shell Ligand O2. The Performance of DFT Methods versus MP2 Vancho Kocevski and Ljupčo Pejov* Institute of Chemistry, Faculty of Science, “Sts. Cyril and Methodius University”, P.O. Box 162, 1001 Skopje, Republic of Macedonia ABSTRACT: Anharmonic vibrational frequency shifts of the phenol(+) O−H stretching mode upon complex formation with the open-shell ligand O2 were computed at several DFT and MP2 levels of theory, with various basis sets, up to 6-311++G(2df,2pd). It was found that all DFT levels of theory significantly outperform the MP2 method with this respect. The best agreement with the experimental frequency shift for the hydrogen-bonded minimum on the potential energy surfaces was obtained with the HCTH/407 functional (−93.7 cm−1 theoretical vs −86 cm−1 experimental), which is a significant improvement over other, more standard DFT functionals (such as, e.g., B3LYP, PBE1PBE), which predict too large downshifts (−139.9 and −147.7 cm−1, respectively). Good agreement with the experiment was also obtained with the mPW1B95 functional proposed by Truhlar et al. (−109.2 cm−1). We have attributed this trend due to the corrected long-range behavior of the HCTH/407 and mPW1B95 functionals, despite the fact that they have been designed primarily for other purposes. MP2 method, even with the largest basis set used, manages to reproduce only less than 50% of the experimentally detected frequency downshift for the hydrogen-bonded dimer. This was attributed to the much more significant spin contamination of the reference HF wave function (compared to DFT Kohn−Sham wave functions), which was found to be strongly dependent on the O−H stretching vibrational coordinate. All DFT levels of theory outperform MP2 in the case of computed anharmonic OH stretching frequency shifts upon ionization of the neutral phenol molecule as well. Besides the hydrogen-bonded minimum, DFT levels of theory also predict existence of two other minima, corresponding to stacked arrangement of the phenol(+) and O2 subunits. mPW1B95 and PBE1PBE functionals predict a very slight blue shift of the phenol(+) O−H stretching mode in the case of stacked dimer with the nearly perpendicular orientation of oxygen molecule with respect to the phenolic ring, which is entirely of electrostatic origin, in agreement with the experimental observations of an additional band in the IR photodissociation spectra of phenol(+)−O2 dimer [Patzer, A.; Knorke, H.; Langer, J.; Dopfer, O. Chem. Phys. Lett. 2008, 457, 298]. The structural features of the minima on the studied PESs were discussed in details as well, on the basis of NBO and AIM analyses.

1. INTRODUCTION Rationalization and in-depth understanding of a number of phenomena in contemporary physical sciences has become almost impossible without a solid theoretical background in certain cases. One of the most exploited issues in physical chemistry nowadays is the problem of noncovalent bonding between molecular species.1 The reason for this lies in the fact that clusters built up by noncovalently bonded molecular species are typical model systems that bridge the gap between isolated molecules and the corresponding condensed phases formed by them.1 Studies of molecular clusters offer the exceptional possibility to study the evolution of various properties going from isolated molecules, through their increasingly larger associates, and finally − the condensed phases. Among the vast variety of noncovalent intermolecular interaction types, those involving aromatic molecules are of exceptional importance, especially in relation to chemical and biological recognition phenomena.2 For example, clusters of phenol with various molecules (ligands) © 2012 American Chemical Society

are typical noncovalently bonded systems which offer an exceptional possibility to study two competing types of noncovalent interactions − hydrogen bonding and stacking (π-bonding).3−19 This possibility is due to the fact that the ligand molecules can interact either with the aromatic ring or with the acidic hydrogen atom from the OH group. In the former case, the noncovalent interaction is of π-bonding type (stacking), whereas in the latter it could be either a hydrogen bond (conventional or unconventional, depending on the ligand type and the X−H oscillator) or even electrostatic interaction (dipole−quadrupole, quadrupole−quadrupole, etc.). Whereas the π-bonding interaction is favored in the case of nonpolar ligands (such as, e.g., rare gas atoms, methane, etc.), due to the predominance of dispersion forces between the ligand and the highly polarizable Received: October 12, 2011 Revised: January 25, 2012 Published: January 25, 2012 1939

dx.doi.org/10.1021/jp209801s | J. Phys. Chem. A 2012, 116, 1939−1949

The Journal of Physical Chemistry A

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π-electronic system, hydrogen-bonding interaction is typical in the case of polar ligands. It has been demonstrated that ionization of phenol makes the H-bonding interaction preferable in the case of both types of previously mentioned ligands, as a result of the advent of induction forces and their dominance over the dispersion ones.3−19 In a recent study, Dopfer et al. have investigated a particularly interesting case of phenol(+) radical cation interacting with an open-shell ligand O2.19 This study was the first one in which a diradical ligand interacting with aromatic system has been considered. Using infrared photodissociation (IRPD) technique, Dopfer et al. have recorded the IR spectral patterns of phenol(+)-(O2)n clusters, n ranging from 1 to 4. In the IR spectra recorded in the case of the smallest cluster (phenol(+)−O2 dimer), these authors have observed two bands which could be assigned to the OH stretching vibrations of the phenol moiety. One of these bands, which was red-shifted by 86 cm−1 with respect to the OH stretching mode of the free phenol radical cation and also characterized by large half-width, was attributed to the structural arrangement of subunits corresponding to a typical conventional hydrogen-bonding interaction, in which an openshell ligand is involved as a proton-acceptor. Another band, blue-shifted by 8 cm−1 with respect to free phenol(+) and characterized by much smaller half-width, was attributed to a π-bonded arrangement between molecular subunits within the dimer. To support the empirical band assignments, as well as to get a further insight into certain geometrical features of the studied phenol(+)-(O2)n clusters, the authors have also undertaken a B3LYP/6-311G(2df,2pd) study of the simplest cluster, characterized with n = 1.19 Though their computed harmonic vibrational frequency shift of the OH stretching mode upon hydrogen bonding of phenol(+) with O2 was in good qualitative agreement with the experiment (−127 vs −86 cm−1) in the sense of support of the basic band assignment, the overestimation of the apparent red shift is obvious. The authors have, however, demonstrated that the employed theoretical level is sufficient to obtain good agreement between theoretical and experimental vibrational frequencies (upon scaling of theoretical harmonic values). In experimental studies of such noncovalently bonded clusters, most often the vibrational frequency shifts with respect to the free molecular species have been used as indicators for the existence and type of noncovalent bonding within the cluster.3−19 Therefore, establishing a computational methodology that would enable accurate calculations of vibrational frequency shifts for certain intramolecular modes is a particularly important issue in studies of noncovalently bonded molecular clusters. It has been actually recognized that in numerous cases the straightforwardly calculated harmonic vibrational frequency shifts can lead to fortuitous very good agreement with the experimental spectroscopic data.20−31 The fundamental reason behind this lies in the fact that errors of various origin tend to mutually cancel. The most important errors contributing to the disagreement between theoretical and experimental vibrational frequencies (and consequently, frequency shifts) are (i) the deficiency in the longrange behavior of the “standard” functional combinations that are mostly used in computations, which, on the other hand, leads to inappropriate description of the low-density highgradient region of the electronic density within the molecular clusters,20 (ii) the inadequacy of harmonic approximation for the oscillators involving motion of low-mass particles, such as hydrogen, for example.24,27 The second error becomes especially significant in cases of hydrogen-bonded systems,

where the very existence of the hydrogen bond is often judged on the basis of experimentally measured X−H frequency shift occurring due to cluster formation with respect to the free X−H oscillator. Previous literature studies have shown that the anharmonic corrections to the total X−H vibrational frequency shifts can be as large as 30−40% even for weakly bonded systems,20 which further emphasizes the need to correct for the anharmonicity of these vibrational modes in the computational approaches. On the other hand, the previous studies within our and other research groups have demonstrated that, if one considers the anharmonic vibrational frequency shif ts computed by DFT methodologies, they are too large as compared to experimental data.20−31 Contrary to DFT, we have demonstrated that usually the MP2 anharmonic frequency shifts are in excellent agreement with the experimental data.21,23,26,29−31 The last statement holds in cases when the reference HF wave function is not spin-contaminated to a larger extent.21,23,26,29−31 The computational cost of the MP2 methodology is, however, rather high and it would be therefore highly desirable to establish a DFT-based method for computation of accurate anharmonic vibrational frequency shifts. As the DFT methods suffer much less from problems arising due to spin contamination than MP2 (see refs 32−36 and references therein), this would be especially important in cases of radical clusters. As a continuation of our previous systematic computational studies related to the above-mentioned issues,21−31,37 in the present paper we report the results of a thorough theoretical study of the phenol(+)−O2 dimer in its ground electronic state. We consider various aspects of the noncovalent interactions that build up this dimer. Various combinations of exchange and correlation functionals are examined (along with the MP2 methodology) with respect to accurate prediction of exact, i.e., anharmonic O−H stretching vibrational frequency shifts of the phenol moiety, arising as a result of both its ionization and interaction with the open-shell ligand O2. The electrostatic + polarization contribution to the overall anharmonic vibrational frequency shifts is analyzed by employing the charge field perturbation (CFP) methodology. Energetics of this interaction is examined by the approach suggested by Xantheas and the electron density of the noncovalently bonded dimer is characterized by NBO (including second-order perturbation theory analysis of the DFT analog of the Fock matrix) and Bader (AIM) analysis. We also address the issue concerning the influence of the spin contamination on the OH stretching vibrational potential computed by DFT and MP2 approaches. The main aim of undertaking such a comparative DFT vs MP2 study is 2-fold. First, from a practical aspect, in a sense of supporting the experimental observations, in this way we were able to analyze the performances of various functional combinations (and the MP2 method) with respect to the correct prediction of various experimentally measured properties of the noncovalently bonded molecular cluster (i.e., to their practical usage for predicting the energetics and vibrational spectroscopic properties relevant to the experimental detection of the studied system). Second, from a more fundamental aspect, the DFT methods are tested with respect to their fundamental correctness (especially with respect to the asymptotic behavior of the used functionals); this is possible to do with extensive numerical testing of the employed methodologies against a good reference set of experimental data (which are available for the presently studied system).19 Moreover, in addition to the previously mentioned aspects, we also provide certain further in-depth physical insight into the studied noncovalent interaction, related to both the 1940



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structure of dimer and its vibrational spectroscopic and energetic characteristics. We employ molecular electrostatic reasoning in combination with the NBO analysis to achieve this aim. We also extend and implement our charge field perturbation methodology in a very specific manner for the present case to get an insight into the factors governing the OH stretching frequency shift in various mutual orientations of the subunits within the dimer.

functional is of the same general form as the other GGA HCTH functionals,50,51 i.e., for the molecule m: N

E xc, m =

q

(2)

where

2. COMPUTATIONAL DETAILS

xm σ 2 =

2.1. DFT Approach, XC Functional Combinations, and General Methodology Aspects. In the DFT approach, in the present study we have used the following combinations of exchange and correlation functionals: B3LYP, PBE1PBE, mPW1B95, and HCTH/407. The first methodology is based on a combination of Becke’s three-parameter adiabatic connection exchange functional (B3)38 with the Lee−Yang−Parr (LYP,39 B3-LYP) correlation one, and it has exhibited a remarkable performance for variety of purposes.40−42 The second DFT methodology was based on PBE1 exchange functional combined with the PBE correlation one43 and is characterized by certain further improvements over the other DFT functionals constructed with the main aim of improving the well-known deficiency in their long-range behavior, as it provides an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling as well as a smoother potential.43 The mPW1B95 functional has been semiempirically constructed by the group of Truhlar.44,45 The parametrization of this nonstandard functional has been done as follows. The exchange−correlation energy in DFT may be written in the following form:41,45 ⎛ X ⎞⎟ S X HF E xc = ⎜1 − (E x + ΔE xGCE) + Ex ⎝ ⎠ 100 100 Y + EcLSDA + ΔEcGCC 100

∑ cq ∫ fq (ρmα ,ρmβ ,xmα2 , xmβ2) dr

(∇ρmσ )2 ρm8/3 σ

(3)

The total number of terms in 2, N is 15, 5 for exchange, 5 for like spin correlation and 5 for αβ correlation. Exact forms of functions f in 2 have been initially given by Becke;52 f1, f 2, and f 3 have the LDA functional form. The 15 parameters cq have been determined by minimization procedure as described in details in ref 49, against a training set of 407 systems. All of the HCTH functionals have been shown to perform remarkably well particularly in the case of hydrogen-bonded systems, the HCTH/407 one being even more flexible.49−51 The standard Pople-type 6-31++G(d,p) basis set was employed for orbital expansion in all DFT calculations, solving the Kohn−Sham (KS) SCF equations iteratively for each particular purpose of this study, with an “ultrafine” (99, 590) grid for numerical integration (99 radial and 590 angular integration points). Though the used basis sets is rather small on an absolute scale, it has been shown to perform remarkably well in combination with DFT methods,21−31,37,53,54 giving results nearly converged with respect to the basis set size. In the case of MP2 method, aside from the 6-31++G(d,p) basis set, calculations were also performed with two larges basis sets,6-311++G(d,p) and 6-311++G(2df,2pd). This was done because with many-body perturbation theory-based methods, larger basis sets are usually required to achieve convergence of the molecular properties. The potential energy surfaces (PESs) of the phenol(+)−O2 cationic dimer were explored using the previously mentioned DFT and MP2 levels of theory. Schlegel’s gradient optimization algorithm was used for geometry optimizations (employing analytical computation of the energy derivatives with respect to nuclear coordinates).55 The modified GDIIS algorithm was actually implemented in productive searches for stationary points on the studied PESs,56 as it has been found to be much more effective in the cases of weak noncovalent intermolecular interactions. The character of located stationary points on the studied PESs of the dimers was tested by harmonic vibrational analysis. The absence of imaginary frequencies (negative eigenvalues of the Hessian matrices) served as an indication that a true minimum on the particular PES is in question. Harmonic vibrational analyses could be used for further calculation of harmonic vibrational frequency shifts of the relevant intramolecular modes, but because these have been often shown to be in fortuitous agreement with the experimental data,21−31,37,53,54 in the present paper we focus our discussion on the more reliable anharmonic vibrational frequency shifts. The harmonic frequency shifts are discussed here only for comparison purposes. Complexation of phenol(+) cation (with total spin S = 1/2) with O2 (total spin S = 1) can in principle give rise to either doublet (S = 1/2) or quartet (S = 3/2) molecular dimer state. Our calculations have shown that it is always the quartet state that is characterized with larger (in absolute value) interaction energy, and we have therefore focused on the PESs corres-

(1)

where ExS is the local spin density approximation (LSDA) to the exchange energy (the Slater term),46,47 ExGCE is the exchange energy gradient correction, ExHF is the Hartree−Fock exchange energy, EcLSDA is the LSDA approximation to the correlation energy, ΔEcGCC is the gradient correction to the correlation energy, and X and Y are parameters. These two factors actually determine the percentage of HF exchange and the percentage of gradient correction to the correlation energy that are included in a particular functional. For nonhybrid (i.e., “pure DFT”) methods, X = 0. As described in ref 44, aiming to develop meta hybrid DFT methods for thermochemistry, thermochemical kinetics, and noncovalent interactions, Truhlar et al. have optimized the value of X in 1, whereas the ExGCE was kept the same as in the mPW exchange functional,48 and the sum EcLSDA + ΔEcGCC was the same as in Becke95 functional.41 Optimization of X against the AE6 representative atomization energy database has led to the mPW1B95 functional (X = 31), whereas adjusting X to minimize the RMSE for the Kinetics9 database has led to the mPWB1K functional (X = 44).45 We have chosen to use, however, only the mPW1B95 functional in our present study, because it has been recommended as a more general purpose functional, whereas the mPWB1K was found to be more suitable for kinetics. Finally, the HCTH/40749 1941



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level of theory at series of points selected applying the CHelpG point-selection algorithm.65 The charge distribution of the infield molecular oxygen was also computed by fitting to its MEP sampled at series of points chosen by the CHelpG algorithm.65 Subsequently, the O−H stretching vibrational potential of the phenol(+) unit was calculated within the inhomogeneous electrostatic field generated by the molecular oxygene charge distribution. The CFP approach for modeling the electrostatic + polarization influence on the X−H vibrational frequencies accounts completely for all terms in the perturbation-theory expansion of the energy as a function of the field:

ponding to this electronic state. Our conclusions with this respect are therefore analogous as those presented in ref 19. 2.2. Calculation of Anharmonic O−H Stretching Frequencies and CFP Analysis. In the present study, we put a special emphasis on the OH stretching frequency shifts induced upon noncovalent interaction (hydrogen bonding or stacking) of phenol(+) cation with oxygen. X−H vibrations are known to be inherently considerably anharmonic, and because anharmonic contributions to the overall observed vibrational frequency shifts have often been found to be as high as 30− 40%,20−31,37 to compute the corresponding frequencies with a sufficient accuracy, it is necessary to go beyond the widely used harmonic approximation in quantum chemistry. It has been shown in the literature that theoretically calculated harmonic X−H vibrational frequency shifts may in some cases even be in fortuitous excellent agreement with the experimental data, due to cancellation of errors.20−31,37,53,54 Aside from using various variants of the scaled quantum mechanical force fields methods57−61 and the vibrational self-consistent field (VSCF) approaches,62,63 which have been developed in the literature, in our approach, we accounted explicitly for the anharmonicity of the O−H stretching motion, using a computationally feasible approach, described in details in our previous papers.20−31,37,53,54 Briefly, to obtain the vibrational potential energy function (V = f (rOH)) for an intramolecular OH oscillator within a free phenol(+) monomer or noncovalently bonded dimer, a series of 15 pointwise DFT energy calculations were performed, varying the O−H distances from 0.850 to 1.550 Å with a step of 0.05 Å. The nuclear displacements corresponding to the O−H stretching vibration were generated by keeping the center-of-mass of the vibrating OH group fixed (to mimic as closely as possible the corresponding vibrational mode). The obtained energies were leastsquares fitted to a fifth-order polynomial in ΔrOH (Δr = r − re): V = V0 + k2Δr 2 + k3Δr 3 + k4Δr 4 + k5Δr 5

1 1 1 −ΔE(F ⃗) = μ ⃗ 0 ·F ⃗ + Θ·∇F ⃗ + Ω·∇2 F ⃗ + Φ·∇3F ⃗ 2! 3! 4! + 1 + ··· + α·F ⃗ + ··· (6) 2!

(where Θ, Ω, Φ, ... are the quadrupole, octupole, and hexadecapole tensor functions of second, third, fourth, etc. order, and μ⃗ 0 and α are the dipole moment vector of the free molecule and the dipole polarizability function, respectively, the last quantity being a second order tensor). All quantum chemical calculations were performed with the Gaussian03 series of programs66 (and the NBO code,67 included as Link 607 in G03).

3. RESULTS AND DISCUSSION 3.1. Structures and Interaction Energies. Complexes of phenol (and similar organic systems) with various ligands enable the possibility to study two competing noncovalent intermolecular interactions that are possible in this case. The first possibility is that ligand can interact with the acidic proton of the aromatic OH group, leading to hydrogen-bonding arrangement. The second possibility involves interaction with the π-electronic cloud of the aromatic ring, leading to π-bonding or stacking arrangement. Which of these two possible interactions will be preferred strongly depends on the nature of ligand (i.e., its polarity), as well as on the electronic excitation or ionization of the aromatic subunit. As has been pointed out in the Introduction, the specificity of the O2 ligand lies in its open-shell (3Σg−) character, which enables investigation of the influence of such diradical character of the ligand on the phenol-ligand interaction. All levels of theory implemented in the present study predict that the global minimum on the phenol(+)−O2 PES corresponds to a nearly linear hydrogen-bonded structure depicted in Figure 1a), in which both interacting subunits are coplanar, with trans configuration. Similarly as reported in ref 19, we could not detect stable cis isomer on any of the PESs considered (though careful searches have been carried out, involving even analytical computations of the Hessian matrices in each optimization step). The MP2 level with the smallest basis set, however, predicts a global minimum corresponding to a structure shown in Figure 1b). In this structure, the H···O−O angle is close to 180°, and the oxygen molecule does not lie within the phenol(+) plane. Enlargement of the basis set size (to 6-311+ +G(d,p)), however, leads first to a structure shown in Figure 1c), in which the O2 ligand is more significantly tilted and though being still out of the phenol(+) plane, it is actually positioned closer to it. Finally, the predicted global minimum structure at MP2/6-311++G(2df,2pd) level of theory (Figure 1d)) resembles the one predicted at DFT levels, shown in Figure 1a). Though quantum chemical calculations basically

(4)

The resulting potential energy functions were subsequently cut after fourth order and transformed into Simons−Parr−Finlan (SPF) type coordinates:64 ρ = 1 − rOH,e/rOH

(5)

(where rOH,e is the equilibrium, i.e., the lowest-energy, value). The one-dimensional vibrational Schrödinger equation was solved variationally. The fundamental anharmonic O−H stretching frequency (corresponding to the |0⟩ → |1⟩ transition) was computed from the energy difference between the ground (|0⟩) and first excited (|1⟩) vibrational states. The CFP (charge-field perturbation) approach is applied in the present study (analogously as in our previous studies of various hydrogen-bonded systems)20−31,37,53,54 to get an insight into the electrostatic (+ polarization) contribution to the total shift of each anharmonic O−H vibrational frequency upon the studied noncovalent interaction. Implementation of the CFP approach in the present study was performed in the following way. The molecular oxygen unit was represented by a set of two point charges placed at the corresponding nuclear positions within the dimeric geometry. Because the O2 molecule is significantly polarized by the phenol(+) cation within the dimer, especially in the hydrogen-bonded arrangement, its charge distribution was computed when placed in the field generated by the phenol(+) subunit represented by set of point charges chosen such to reproduce the molecular electrostatic potential (MEP) calculated from the corresponding KS densities for each 1942



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In eq 7, Q is the ion charge, Θ the quadrupole moment of O2, R is the distance between the center of mass of O2 molecule and the charge, and α⊥ and α|| are perpendicular and parallel volume polarizabilities of the ligand molecule. The induction contribution to V is always attractive, whereas the chargequadrupole term (first one in eq 7) depends on the orientation of the molecular subunits within the dimer. In the case of negative quadrupole moment (as in the case of O2), the linear arrangement would be favored by both the first and second term in eq 7, as can be straightforwardly verified by plotting the θ dependence. As can be seen from Figure 1, however, we actually get a bent orientation, θ being close to 130°, similarly as described in ref 19 (e.g., 132° at HCTH level, 128° at mPW1B95, etc.). As implied in ref 19, this is due to the interplay of charge-transfer interactions between monomers. This aspect is discussed further in the paper, in the context of NBO analysis. As there are no experimental data for the structure of this dimer, theoretical structural data are not presented in details (though these are available from the authors upon request). We put the emphasis in the present paper on the comparison of our theoretically computed anharmonic vibrational frequencies with the existing experimental ones, which have been obtained in the recent study by Dopfer et al.19 Besides the hydrogen-bonded arrangement of the molecular subunits within the dimer, which, as mentioned before, corresponds to the global minimum on all of the studied PESs, several levels of theory have also predicted existence of other minima. Thus, the PBE1PBE, mPW1B95 and MP2/6-31++G(d,p) levels of theory also predict an existence of a π-bonded minimum, similar to the one reported in the paper by Dopfer et al.19 This minimum is shown in Figure 1 e), and will hereby be referred to as the 90° π-bonded minimum (as the angle between the O−O axis and the phenol(+) plane is close to 90°); i.e., the O2 molecule is placed nearly perpendicularly to the phenol(+) plane. mPW1B95/6-31++G(d,p) level of theory, however, also predicts a third π-bonded minimum on the phenol(+)−O2 PES, depicted in Figure 1 f). We will refer to this structure as the 0° π-bonded minimum (as the O−O axis is nearly parallel to the phenol(+) plane). As there are no explicit experimental structural data for the phenol(+)−O2 dimer in the gas phase, we will judge on the reliability of the predicted structures on the basis of the computed anharmonic vibrational frequency shifts (which have been obtained experimentally in the paper by Dopfer et al.).19 Note, however, that our preliminary computational studies have shown that upon basis set enlargement to 6-311++G(2df,2pd), all four DFT levels of theory also predict the existence of a stable 90° π-bonded minimum on the PESs. Interaction and dissociation energies were computed in the following way. Using the notation of Xantheas,70 the interaction energy of a dimer AB (not accounting for the basis set extension effects, i.e., basis set superposition error, BSSE) is given by

Figure 1. Structures.

incorporate all of the relevant effects which determine the structure of the minima on the phenol(+)−O2 dimer PES, it is illustrative to compare the predictions of simple electrostatic modeling with the exact QM results. The long-range forces between phenol(+) cation and O2 ligand are dominated by two terms: charge-quadrupole and induction. The contribution of these two terms to the potential energy of the intermolecular interaction may be represented by68,69 V (R ,θ ) =

QΘ 3

8πε0R −

α∪β ΔE = EAB (AB) − EAα (A) − EBβ(B)

(3 cos2 θ − 1)

2 2 2 1 Q (α cos θ + α⊥ sin θ ) 2 4πε0R4

(8)

(the superscripts denoting the basis set and the subscripts referring to the geometry used for energy calculation). Usually, however, ΔE is defined and calculated in the following manner: α∪β β α ΔE′ = EAB (AB) − EAB (A) − EAB (B)

(7) 1943

(9)



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Table 1. Interaction and Dissociation Energies for the Global Minimum on the Phenol-O2(+) Cationic Dimer Potential Energy Hypersurfaces Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) Basis Set (1) and at the MP2 Level with the 631++G(d,p) (1), 6-311++G(d,p) (2), and 6-311++G(2df,2pd) (3) Basis Sets (All Values in kJ mol−1)a MP2

−ΔE′ −ΔE(fCP) −ΔE(BSSE) D0 Edef(C6H5OH) Edef(CO) Edef BSSE a

B3LYP (1)

PBE1PBE (1)

mPW1B95 (1)

HCTH/407 (1)

(1)

(2)

(3)

−11.58 −9.66 −9.57 7.43 0.09 0.01 0.10 1.92

−13.38 −11.02 −10.93 8.71 0.09 0.01 0.10 2.35

−12.24 −9.98 −9.92 7.47 0.06 0.00 0.06 2.26

−12.71 −11.10 −11.06 9.30 0.04 0.01 0.05 1.61

−11.58 −5.58 −5.31 −4.54 0.02 0.26 0.28 6.00

−13.03 −7.38 −7.14 −1.45 0.02 0.21 0.23 5.65

−12.68 −8.54 −8.18 3.53 0.06 0.31 0.37 4.14

See text for details and definitions.

i.e., as a “vertical” (in a sense) interaction energy. The (full) function counterpoise corrected energy (by the Boys−Bernardi method)71 for the BSSE is given by α∪β α∪β α∪β ΔE(fCP) = EAB (AB) − EAB (A) − EAB (B)

Table 2. Interaction and Dissociation Energies for the π-Bonded Minima on the Phenol-O2(+) Cationic Dimer Potential Energy Hypersurfaces Calculated at mPW1B95 and MP2 Levels of Theory with the 6-31++G(d,p) Basis Set (All Values in kJ mol−1)a

(10)

MP2

and BSSE is, thus, often calculated by BSSE = ΔE(fCP) − ΔE′

(11) −ΔE′ −ΔE(fCP) −ΔE(BSSE) D0 Edef(C6H5OH) Edef(CO) Edef BSSE

Because eqs 9 and 10 do not converge to the same result upon approaching the complete basis set limit due to the fact that the reference energies of monomeric units are calculated at different geometries, following Xantheas,70 it is suitable to include the deformation (i.e., relaxation) energy in the following manner: α (A) + E β (B) ΔE(BSSE) = ΔE(fCP) + Edef def

(12)

In the last equation, the deformation energies are given by

a

α (A) = E α (A) − E α (A) Edef AB A

(13)

β β β Edef (B) = EAB (B) − EA (B)

(14)

mPW1B95 90°

90°

0°

−5.18 −3.38 −3.37 2.65 0.00 0.00 0.00 1.80

−12.01 −3.47 −3.05 −9.52 0.01 0.40 0.41 8.55

−11.58 −3.41 −3.37 −2.51 0.00 0.03 0.03 8.18


charge-transfer (CT) between monomers, we have carried out a natural bond orbital (NBO) analysis72−78 of the minima found on the studied mPW1B95/6-31++G(d,p) phenol(+)−O2 PESs. For these purposes, a second-order perturbation theory (SOPT) analysis of the Kohn−Sham’s one-electron analog of the Fock matrix within the NBO basis was carried out.72−78 The results from this analysis are summarized in Table 3. As can be seen, similarly as in the case of other hydrogen-bonded systems, the CT from the proton-accepting to the protondonating unit is much larger than that occurring in the opposite direction, in the sense of both energetic effects and the actual quantity of charge transferred. Moreover, most of the CT occurs from the proton-acceptor nonbonding n(O) orbital to the proton-donor σ*(O−H) orbital, whereas much less from the proton-acceptor bonding σ(O−O) orbital to the protondonor hydroxyl group H atom Rydberg (extra-valence) states. This is in line with the qualitative supposition implied in ref 19 that the charge-transfer interaction plays a significant role in the sense of determination of the minimum on the studied PES of phenol(+)−O2 dimer. The energetic effects due to the CT interactions were estimated by the second-order perturbation theoretical expressions of the form:72−78

The results obtained at all levels of theory implemented in the present study are summarized in Table 1. As can be seen, all DFT levels of theory predict similar interaction and dissociation energies. The MP2 level, on the other hand, with the 6-31+ +G(d,p) and even with the larger 6-311++G(d,p) basis sets predicts negative dissociation energies, which implies that the zero-point vibrational energy would be large enough to disrupt the dimer. Only with the largest (6-311++G(2df,2pd)) basis does the MP2 level predict the existence of a stable phenol(+)− O2 dimer, with positive D0 value, albeit much smaller than the value predicted by the DFT methods. Table 2, on the other hand, summarizes the interaction and dissociation energies for the π-bonded minima on the studied PESs. As can be seen, the only structure that can be considered stable is the 90° π-bonded minimum at the mPW1B95 level of theory. As accurate experimental value of the interaction/dissociation energy of the title complex is not available, we will actually judge the accuracy of the implemented methodologies on the basis of anharmonic vibrational analysis of the O−H stretching mode, which is discussed in the next section. To characterize the charge-transfer contribution to the overall intermolecular interaction, its influence on the adopted structure of the dimer (implied before), and moreover, to derive conclusions about the direction and magnitude of the actual

ΔE ψ(2) donor → ψ acceptor ≈ − 2 · 1944

* |F |̂ ψ ⟩2 ⟨ψdon acc εacc − εdon

(15)



Article

Table 3. Results from the Second-Order Perturbation Theory Analysis of the Kohn−Sham Analog of the Fock Matrix within the NBO Basis for the Three Minima on the Phenol(+)−O2 mPW1B95/6-31++G(d,p) PESa donor orbital

a

acceptor orbital

ΔE(2)/(kJ mol−1)

(Eacc − Edon)/au

Hydrogen-Bonded Minimum 17.07 1.01

⟨ψ*don|F̂|ψacc⟩/au

qdonor→acceptor/e

N(O)O2

σ*(O−H)

0.081

0.0129

N(O)O2

σ*(O−H)

1.72

1.26

0.029

0.0011

σ(O−O)O2

Ry*(H)

0.71

1.88

0.022

0.0003

N(O)O2

σ*(C−H)

0.42

1.31

0.015

0.0003

N(O)O2

Ry*(H)

0.38

1.71

0.016

0.0002

σ(O−O)O2

Ry*(H)

0.33

2.27

0.017

0.0001

N(O)O2

σ*(O−H)

0.29

0.88

0.010

0.0003

σ(O−O)O2

Ry*(H)

0.21

1.32

0.010

0.0001

σ(O−H)

σ*(O−O)O2

0.59

0.88

0.014

0.0005

σ(O−H)

Ry*(O)O2

0.21

1.67

0.011

0.0001

σ(C−O)

Ry*(O)O2

0.21

0.74

0.008

0.0002

N(O)O2

Ry*(C)

π-Bonded Minimum (90°) 0.13 1.66

0.008

0, indicating a local depletion of electronic charge, the kinetic energy G(r) 1945



Article

the basis set size. When the absolute OH stretching frequency of free phenol(0) is in question, on the other hand, the B3LYP methodology performs best, along with the MP2 methods. Other DFT levels are somewhat inferior in the case of neutral phenol monomer. B3LYP results are also in excellent agreement with the experimental data when the O−H stretching frequency of the free phenol(+) unit is in question. In this case, however, MP2 methodology performs significantly worse. This can be attributed to the spin-contamination problems inherent to the MP2 method, which are of minor importance in density functional theory, as discussed in a more quantitative manner below. In Table 6, the computed anharmonic O−H stretching frequencies of free phenol(+) and phenol(+)−O2 dimeric

being in local excess over the value determined by the average virial ratio. According to the actual ρ(rc) and ∇2ρ(rc) values at the intermolecular (3,−1) BCP of the O−H···O contact in the case of the hydrogen-bonded minimum on the mPW1B95/ 6-31++G(d,p) PES, when the proposed AIM-based criteria for hydrogen bonding are applied,80 the present hydrogen bond may be classified as a medium-strength one. The hydrogen bond strength is also revealed by a comparison of the ρ(rc) and ∇2ρ(rc) values corresponding to the intramolecular (3,−1) BCPs in the case of free phenol(+) (the BCP corresponding to the “shared” O−H interaction), indicating a ρ(rc) (and |∇2ρ(rc)|) value decreases at the O−H BCP upon hydrogen bonding with the open-shell ligand O2. 3.2. Anharmonic Vibrational Frequencies and Frequency Shifts. The main emphasis in the present paper is put on the vibrational frequency shift of the O−H stretching mode of the phenol(+) moiety within the dimer, induced upon interaction with the open-shell ligand O2. However, at the beginning we will also consider the absolute anharmonic O−H stretching f requency of the free phenol(+) monomeric unit and its nonionized counterpart, phenol(0), as well as the frequency shift induced by ionization of phenol. The issue concerning vibrational frequency shifts upon ionization of organic molecules has been only scarcely treated in the literature, whereas it can serve as a good indicator of the appropriateness of a given theoretical method for a description of structural and force field changes upon molecular ionization. The previous comment is particularly valid when anharmonic vibrational frequency shifts are in question. According to the literature data, upon ionization of bare phenol unit, a significant downshift of the O−H stretching frequency occurs, of −123 cm−1 (from 3657 to 3534 cm−1).17,18 In Table 5, absolute anharmonic O−H stretching frequencies of

Table 6. Anharmonic OH Stretching Vibrational Frequencies and Frequency Shifts for the Three Minima Located on the Phenol-O2(+) Cationic Dimer PES Calculated with the Full Wavefunction Approach at Four DFT as Well as at MP2 Levels of Theory with the 6-31++G(d,p) (1) Basis Set and at the MP2 Level with the 6-311++G(d,p) (2) or 6-311++G(2df,2pd) (3) Basis Set, Together with the Available Experimental Dataa A

Table 5. Anharmonic OH Stretching Vibrational Frequencies for Free Phenol(0) and Phenol(+) and the Frequency Shifts upon Ionization of the Free Phenol to Phenol(+) Calculated at Four DFT Levels of Theory and at MP2 with the 6-31++G(d,p) (1) Basis Set and Also at the MP2 Level with the 6-311++G(d,p) (2) or 6-311+ +G(2df,2pd) (3) Basis Set, Together with the Available Experimental Dataa phenol(0)

B3LYP (1) PBE1PBE (1) mPW1B95 (1) HCTH/407 (1) MP2 (1) MP2 (2) MP2 (3) experimental

r0/Å

ν/cm−1

r0/Å

3646.2 3708.0 3720.8 3612.0 3676.0 3686.5 3675.9 3657

0.9671 0.9637 0.9612 0.9651 0.9680 0.9635 0.9626

3536.9 3593.0 3607.8 3502.5 3587.2 3588.5 3577.0 3534

0.9761 0.9727 0.9704 0.9743 0.9749 0.9704 0.9694

ν/cm−1

ν/cm−1


3397.0 3445.3 3498.6 3408.8 3582.0 3574.8 3537.2 3448

−139.9 −147.7 −109.2 −93.7 −5.2 −19.5 −39.8 −86

a

C (0°)

ν/cm−1

ν/cm−1

ν/cm−1

ν/cm−1

3594.4 3608.5

+1.4 +0.7

3607.4

−0.4

3586.9

−0.3

3542

+8

3542

+8


structures corresponding to various minima on the corresponding PESs are presented. Table 7, on the other hand, contains Table 7. Anharmonic OH Stretching Vibrational Frequencies and Frequency Shifts for the Three Minima Located on the Phenol-O2(+) Cationic Dimer PES Calculated with the Charge Field Perturbational (CFP) Approach at Four DFT as Well as at MP2 Levels of Theory with the 6-31++G(d,p) (1) Basis Set and at the MP2 level with the 6-311++G(d,p) (2) or 6-311++G(2df,2pd) (3) Basis Set, Together with the Available Experimental Dataa

phenol(+)

ν/cm−1

B (90°)

minimum

(0) − (+) Δν/cm−1 −113.0 −109.3 −115.0 −109.5 −88.8 −98.0 −98.9 −123

A minimum B3LYP (1) PBE1PBE (1) mPW1B95 (1) HCTH/407 (1) MP2 (1) MP2 (2) MP2 (3) experimental

a

The r0 values, computed from the 1D DFT or MP2 OH stretching potentials are given as well. See text for details and definitions.

free phenol(+) and phenol(0) are presented, together with the corresponding frequency shifts upon ionization. Considering the frequency shift upon ionization ((0)-(+)), the mPW1B95 and B3LYP levels of theory perform best, though the other DFT levels give also similar results. The MP2 level, on the other hand, is significantly outperformed by all of the DFT methods that have been used in the present study, regardless of

a

ν/cm

−1

3500.8 3551.5 3572.1 3477.1 3568.5 3498.9 3546.1 3448

B (90°) ν/cm

−1

−36.1 −41.5 −35.7 −25.4 −18.7 −25.3 −30.9 −86

C (0°)

−1

ν/cm

3594.1 3608.9

+1.1 +1.1

3588.1

+0.9

3542

+8

ν/cm

−1

ν/cm−1

ν/cm−1

3607.0

−0.8

3542

+8


the analogous values computed by the CFP approach. In their experimental paper,19 Dopfer et al. have detected two fundamental vibrational transitions corresponding to the in-dimer 1946



Article

is almost completely electrostatically induced, as can be seen from the results of CFP analysis given in Table 7. This is exactly what one would certainly expect for such an arrangement of the currently considered subunits in the π-bonded dimeric structure. The MP2 level of theory fails to predict even the qualitative trends in the case of this minimum. The second π-bonded minimum (the one in which the O−O axis is nearly parallel to the phenol(+)), which has been located only on the mPW1B95 PES, should be characterized with a very small, almost negligible frequency downshift of less than 0.5 cm−1 (Table 4), which is of purely electrostatic origin (Table 5). Because, experimentally, a blue shift of the O−H stretching frequency has been detected, and assumed to be due to the π-bonded subunit arrangement within the dimer, our theoretical data seem to further support the interpretation given in the original work of Dopfer et al.,19 according to which the observed π-bonded structure corresponds the nearly orthogonal arrangement of phenol(+) and O2 subunits. One can, of course, not disregard the possibility for an existence of the alternative “parallel” (0°) minimum. To further confirm the possibility for existence of such (presumably rather shallow) minimum on the PES of the studied open-shell dimer, higher level computations would be required, and even consideration of the free-energy hypersurfaces, instead of the standard PESs. As implied before, one of the possible reasons behind the apparent superiority of the DFT methods over the MP2 level of theory is the problem of spin contamination. It has been recognized that whereas the MP2 level of theory suffers from spin contamination problems, and due to this problem can lead to various erroneous predictions, density functional theory is not sensitive to this issue, at least not that significantly as MP2.33−36 To get a clearer insight into the reasons behind various performances of the DFT and MP2 methods in the presently studied case of a radical dimer, we hereafter analyze the spin contaminations to a somewhat greater extent. In the present study, all calculations have been carried out with the unrestricted Kohn−Sham and MP2 formalisms. The degree of spin contamination was therefore checked for in each case by comparing the obtained expectation values of the total spin squared operator ⟨Ŝ2⟩ with the theoretical (exact) one, s(s + 1). Note in this context, however, that such a test has been shown as rather useful even in cases when DFT methods are applied, despite the fact that the interpretation of ⟨S2̂ ⟩ is not quite straightforward for unrestricted DFT as in the case of UHF (UMP2).81−85 We have found in the present case of phenol(+)−O2 dimer that the HF wave function used for perturbation theory treatment for the minimum on MP2/6-31++G(d,p) PES is characterized with ⟨Ŝ2⟩ = 3.9973 before annihilation and 3.7752 after annihilation, whereas the exact one is 3.7500, which gives the relative error of about 6.7%, which is actually not very large (in the case of organic systems, often a spin contamination expressed as a relative error of less than 10% is considered as negligible).30 Increasing the basis set size (to 6-311++G(d,p) and 6-311++G(2df,2pd)) does not change this value significantly. The relative errors in the discussed quantity in the case of DFT methods are negligible, often less than 0.7%. In comparison to the analogous values found in our previous study, the present ones are much smaller, and also the errors due to this are expected to be much smaller than those reported in our previous paper. Still, as mentioned before, even with the largest basis set, the MP2 level of theory manages to reproduce only less than 50% of the experimentally detected frequency downshift in the case of the hydrogen bonded phenol(+)−O2 dimer. As we have discussed

O−H stretching motion of the phenol(+) subunit. The first one, which is downshifted by 86 cm−1 with respect to the free phenol(+) value, they have attributed to the hydrogen-bonded structure, which is in line with intuitive expectations. The other one, however, which is upshifted by 8 cm−1 compared to that of free phenol(+), has been attributed to a less-stable π-bonded minimum. As can be seen from Table 6, the (so to say) most standard B3LYP and PBE1PBE functionals predict too large frequency shifts upon dimerization for the hydrogen-bonded structure. This behavior of most widely used DFT functionals has already been recognized in the literature. As we have mentioned before, it is due to the inappropriate asymptotic behavior of the functionals in question, which is mostly pronounced in the regions of low density and high gradient. It is exactly this region that is of substantial importance for noncovalent intermolecular interactions. On the other hand, if one considers the harmonic frequency shifts (given in Table 8), in the two Table 8. Anharmonic vs Harmonic OH Stretching Vibrational Frequency Shifts for the Three Minima Located on the Phenol-O2(+) Cationic Dimer PES Calculated with the Full Wavefunction Approach at Four DFT Levels of Theory with the 6-31++G(d,p) (1) Basis Set and at the MP2 Level with the 6-311++G(d,p) (2) or 6-311++G(2df,2pd) (3) Basis Set, Together with the Available Experimental Dataa A Δν/cm−1 minimum

anharm.

harm.


−139.9 −147.7 −109.2 −93.7 −5.2 −25.3 −39.8 −86

−110.3 −116.7 −86.7 −68.9 −7.0 −17.0 −42.1

a

B (90°) Δν/cm−1

C (0°) Δν/cm−1

anharm.

harm.

anharm.

harm.

+1.4 +0.7

+2.0 +0.5

−0.4

−0.5

−0.3

−2.0

+8

+8


mentioned cases they are in significantly better agreement with the experimental data, due to the fortuitous cancellation of errors inherent to harmonic approximation and inappropriateness of the DFT functionals. The newer mPW1B95 and HCTH/407 functionals, which have been constructed having in mind the previously mentioned points relevant to noncovalent interactions, however, give significantly better agreement with the experiment, particularly HCTH/407. MP2 methodology performs rather poorly, managing to cover only less than 50% of the experimentally detected frequency downshift. As can be seen from Table 7, the CFP analysis indicates that the electrostatic + polarization contribution to the overall (anharmonic) O−H stretching frequency shift is typically less than 25% in the case of all DFT methods. At MP2 level, the situation is quite opposite, in two cases the CFP shift being even larger that the one computed from the “full-wavefunction” approach. In the case of the first π-bonded minimum located in the present study (the one in which the O−O axis intersects the phenol(+) plane at nearly 90°), both PBE1PBE and mPW1B95 levels of theory predict small frequency upshifts of about 1− 2 cm−1 (Table 6). Though the agreement with experimental data19 is not quantitative, the trend is qualitatively correctly predicted. Such a small blue shift of the O−H stretching mode 1947



Article

in our previous study,37 the spin contamination artifacts may in principle affect each point of the computed 1D OH stretching potential in a rather different manner, which perhaps hampers a bit the relatively simple explanation for the performances of the applied methods outlined before. To get further insight into this problem, we have explored the dependence of spin contamination on the vibrational coordinate rOH. Figure 2 shows the

respectively, the last value in particular being in excellent agreement with the experimental data. We have attributed the observed trend in performances of the DFT functionals to the corrected long-range behavior of mPW1B95 and HCTH/407, which is essential for a correct description of the low-density− high-gradient intermolecular region and therefore for accurate scanning of the corresponding vibrational potentials. Therefore, despite the fact that the last two functionals have been constructed for quite different purposes, they are recommended for exact computations of anharmonic vibrational frequency shifts. The MP2 method, on the other hand, regardless of the flexibility of the basis set used, was found to perform significantly poorer than essentially all DFT methods with this respect, managing to reproduce only less than 50% of the experimentally detected frequency downshift for the hydrogenbonded dimer. We have attributed this observation to the much more significant spin contamination of the reference HF wave function (compared to DFT Kohn−Sham wave functions), which was found to be strongly dependent on the O−H stretching vibrational coordinate. We have also addressed the issue concerning O−H stretching frequency shifts upon ionization of phenol molecule and found that all DFT levels of theory outperform MP2 in this case as well. DFT levels of theory also predict existence of two other minima (besides the most stable hydrogen-bonded one), corresponding to stacked arrangement of the phenol(+) and O2 subunits. mPW1B95 and PBE1PBE functionals predict a very slight blue shift of the phenol(+) O−H stretching mode in the case of stacked dimer with the nearly perpendicular orientation of oxygen molecule with respect to the phenolic ring, which is entirely of electrostatic origin, in agreement with the experimental observations of an additional band in the IR spectra of phenol(+)−O2 dimers.19 We have also discussed the structural features of the minima on the studied PESs in detail, on the basis of NBO (including second-order perturbation theory analysis of the Kohn−Sham analog of the Fock matrix) and AIM analyses.

Figure 2. Dependence of the relative error of ⟨S2⟩ on the OH stretching coordinate for the reference HF wavefunction and for the mPW1B95 combination of functionals.

dependence of the relative error of ⟨Ŝ2⟩ on the OH stretching coordinate for the reference HF wave function, which has been used for MP2 calculations and for the mPW1B95 combination of functionals (at both levels, the 6-31++G(d,p) basis set has been used). As can be seen, whereas at the DFT level the spin contamination remains essentially constant with variation of rOH, the situation is opposite at MP2 level. This could certainly be a reason for the previously discussed erroneous behavior of MP2 with respect to computation of anharmonic OH stretching frequencies on the basis of 1D potentials.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

4. CONCLUSIONS We have undertaken a thorough theoretical analysis of the phenol(+)−O2 dimer, with a main emphasis on methodological aspects concerning exact computations of the anharmonic vibrational frequency shifts of the O−H stretching mode upon dimerization, a parameter of crucial importance for experimental detection of the type and strength of this noncovalent intermolecular interaction. Several DFT levels of theory were applied, along with the MP2 method, with various basis sets, up to 6-311++G(2df,2pd). We have found that all DFT levels of theory significantly outperform the MP2 method when the prediction of the vibrational frequency shift of the phenol cation O−H stretching mode upon hydrogen-bond formation with the open-shell ligand O2 is in question. Although widely used “standard” functionals like B3LYP and PBE1PBE significantly overestimate the experimentally measured frequency red shift of −86 cm−1 (the corresponding values being −139.9 and −147.7 cm−1, respectively), the mPW1B95 and HCTH/ 407 functionals perform significantly better, leading to theoretical anharmonic frequency shifts of −109.2 and −93.7 cm−1,

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