On the Assessment of Some New Meta-Hybrid and Generalized

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On the Assessment of Some New Meta-Hybrid and Generalized Gradient Approximation Functionals for Calculations of Anharmonic Vibrational Frequency Shifts in Hydrogen-Bonded Dimers Vancˇo Kocevski and Ljupcˇo Pejov* Institute of Chemistry, Faculty of Science, Sts. Cyril and Methodius UniVersity, P.O. Box 162, 1001 Skopje, Republic of Macedonia ReceiVed: NoVember 6, 2009; ReVised Manuscript ReceiVed: February 11, 2010

The performance of some recently proposed DFT functionals by Truhlar’s group (mPW1B95, mPWLYP1W, PBELYP1W, and PBE1W [Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 317. Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 6908.]) was tested primarily with respect to computation of anharmonic vibrational frequency shifts upon hydrogen bond formation in small molecular/ionic dimers. Five hydrogenbonded systems with varying hydrogen bond strengths were considered: methanol-fluorobenzene, phenol-carbon monoxide in ground neutral (S0) and cationic (D0) electronic states, phenol-acetylene, and phenol-benzene(+). Anharmonic OH stretching frequency shifts were calculated from the computed vibrational potentials for free and hydrogen-bonded proton-donor molecules. To test the basis set convergence properties, all calculations were performed with 6-31++G(d,p) and 6-311++G(2df,2pd) basis sets. The mPW1B95 functional was found to perform remarkably better in comparison to more standard functionals (such as B3LYP, mPW1PW91, PBE1PBE) in the case of neutral dimers. In the case of cationic dimers, however, this is not always the case. With respect to prediction of anharmonic OH stretching frequency shifts upon ionization of free phenol, all DFT levels of theory outperform MP2. Some other aspects of the functional performances with respect to computation of interaction and dissociation energies were considered as well. 1. Introduction In the ongoing quest for an efficient computational methodology for modeling of condensed phases and molecular clusters, the density functional theory (DFT)-based approaches are of essential importance. This is due to the significantly lower cost of the DFT computations as compared to the wave functionbased methods (such as Mpn, CC, and QCI). However, although the density functional theory accounts for the dynamical electron correlation effects, these are included in a more or less semiempirical manner. Ever since the advent of the DFT approach, various combinations of exchange (X) and correlation (C) functionals have been tested for a variety of purposes.1 Although some of the XC functionals combinations have been proposed for a wide variety of purposes, for rather specific computational tasks, even these may appear quite inefficient. Actually, an a priori approach, which would predict the performance of any combination of XC functionals for a particular computational aim, does not seem to exist. Usually, very extensive careful numerical testing is required, with an emphasis on various aspects in specific systems. For the reasons mentioned before, the refinement of functionals for DFT-based computational methodologies is a very active area of research in theoretical physics and chemistry. One of the two alternative approaches in this area, the nonempirical one, seems to be favored in physics, due to its exactness.2-4 The other one, which is essentially based on choosing a flexible mathematical functional, which depends on several parameters, and subsequent determination of the values of these parameters, usually by a fitting procedure, has been very successful for a * To whom correspondence should be addressed. E-mail: ljupcop@ iunona.pmf.ukim.edu.mk.

wide range of “chemical” problems.5-7 Of course, apart from the term “empirical” for the latest approach, it should be kept in mind that it is actually only partly empirical. This is so since the general form of the functional is usually guided by theory. A particularly important issue in studies of noncovalently bonded molecular clusters is to establish a computational methodology that would enable accurate calculations of vibrational frequency shifts for certain intramolecular modes. This is due to the fact that in experimental studies of such 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.8-10 It has been recognized that the straightforwardly calculated harmonic vibrational frequency shifts can lead to fortuitous very good agreement with the experimental spectroscopic data.11-22 This happens due to cancellations of various types of error. The most important of these are: (i) the known deficiency in the long-range behavior of the more often used “standard” functional combinations, which leads to inappropriate description of the low-density high-gradient region of the electronic density within the molecular clusters,18 (ii) the inadequacy of harmonic approximation for the oscillators involving motion of low-mass particles, such as hydrogen for example.15,18 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. It has been demonstrated in the literature that the anharmonic corrections to the total X-H vibrational frequency shifts can be as large as 30-40% even for weakly bonded systems.11 On the other hand, the previous studies within our and other research groups have demonstrated that if one

10.1021/jp910587y  2010 American Chemical Society Published on Web 03/10/2010

Anharmonic Vibrations in Hydrogen-Bonded Dimers considers the anharmonic Vibrational frequency shifts computed by DFT methodologies, they are too large as compared to experimental data.11-22 Contrary to DFT, we have demonstrated that usually the MP2 anharmonic frequency shifts are in excellent agreement with the experimental data, provided that the starting HF wave function is not spin-contaminated to a larger extent.12-22 However, due to the rather significant computational cost of the MP2 methodology, it would be highly desirable to establish a DFT-based method for computation of accurate anharmonic vibrational frequency shifts. This would be especially important in cases of radical clusters, as the DFT methods suffer much less from problems arising due to spin contamination than MP2. Recently, in the group of professor Truhlar, certain nonstandard combinations of XC DFT functionals have been developed that were shown to perform significantly better than the more standard ones, for several particular purposes.23,24 In the present study, we aim to further test the applicability and performance of some of these functionals for calculation of the anharmonic vibrational frequency shifts in certain hydrogen-bonded dimers (both neutral and cations) with varying hydrogen bond strengths, and to make comparisons with the performances of more standard XC functional combinations and also with the MP2 method. 2. Computational Details 2.1. The DFT Methodology and XC Functional Combinations. As mentioned in the Introduction, in the present study we employ the semiempirically constructed functionals by the group of Truhlar.23,24 The parametrization of nonstandard functionals has been done as follows. The exchange-correlation energy in DFT may be written in the following form:6,25

(

Exc ) 1 -

X X HF (ES + ∆EGCE )+ + E + ELSDA x c 100 x 100 x Y ∆EGCC (1) 100 c

)

where ExS is the local spin density approximation (LSDA) to is the exchange the exchange energy (the Slater term),26,27 EGCE x energy gradient correction, ExHF is the Hartree-Fock exchange energy, EcLSDA is the LSDA approximation to the correlais the gradient correction to the correlation tion energy, ∆EGCC c 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 which are included in a particular functional. For nonhybrid (i.e., “pure DFT”) methods, X ) 0. In ref 23, aiming to establish an efficient pure DFT methodology for a variety of purposes, the authors have optimized the value of Y, while setting X ) 0. Optimization of Y was carried out against a set of data for 28 water dimers and 8 trimers, for which highly accurate energetic data were available. The resultant nonstandard functionals were named as mPWLYP1W (Y ) 88), PBE1W (Y ) 74), and PBELYP1W (Y ) 54), where 1W denotes a one-parameter method optimized for water.23 In another study,24 on the other hand, aiming to develop meta hybrid DFT methods for thermochemistry, thermochemical kinetics, and noncovalent interactions, the same authors have optimized the value of X in eq 1, while the ExGCE was kept the same as in the mPW exchange functional,28 while + ∆EGCC was the same as in Becke95 functhe sum ELSDA c c 29 tional. Optimization of X against the AE6 representative atomization energy database has led to the mPW1B95 functional

J. Phys. Chem. A, Vol. 114, No. 12, 2010 4355 (X ) 31), while adjusting X in order to minimize the RMSE for the Kinetics9 database has led to the mPWB1K functional (X ) 44).24 We have chosen to use, however, only the mPW1B95 functional in our present study, since it has been recommended as a more general purpose functional, while the mPWB1K was found to be more suitable for kinetics. It seems relevant to mention in this context that, although it has been found that the density functionals are much more sensitive to the chosen value of X than to Y, it has also been appreciated that the sensitivity to Y is sometimes much greater than it is widely thought.23 Therefore, it is certainly worthwhile to test the functional dependence on this parameter, especially in cases when one aims to construct the best nonhybrid functional with X ) 0. 2.2. Chosen Systems and General Methodology Aspects. To test the performance of mentioned DFT functionals with respect to calculation of anharmonic vibrational frequency shifts, we have chosen a test set of several hydrogen bonded dimers with varying hydrogen bond strengths. For the chosen systems, both reliable experimental data8-10 and computational results with more standard DFT functionals are available for comparison purposes (most of the computational results were actually obtained within our group).12,16,21,30,31 The experimental data for the hydrogen-bonded systems that we refer to in the present study have been obtained by the infrared-ultraviolet (IR-UV) double-resonance technique for the IR spectroscopy in the S0 states, and using infrared photodissociation (IRPD) spectroscopy for the D0 states.8-10 Another useful experimental method that has been efficiently applied for studies of intermolecular hydrogen bonding is a combination of near-infrared spectroscopy (NIR) technique with dielectric measurements.32,33 The test set included the following dimers: (a) methanol-fluorobenzene; (b) phenol-carbon monoxide; (c) phenol-acetylene; (d) phenolcarbon monoxide radical cation, and (e) phenol-benzene radical cation. The potential energy surfaces (PESs) of the mentioned dimers (in their ground electronic states) were explored using the nonstandard combinations of exchange and correlation DFT functionals described in the previous Section. Schlegel’s gradient optimization algorithm was used for geometry optimizations (employing analytical computation of the energy derivatives with respect to nuclear coordinates).34 The modified GDIIS algorithm was actually implemented in productive searches for stationary points on the studied PESs,35 as it was found to be much more effective in the cases of weak noncovalent intermolecular interactions. Standard Pople-type 6-31++G(d,p) and 6-311++ G(2df,2pd) basis sets were employed for orbital expansion, solving the Kohn-Sham (KS) SCF equations iteratively for each particular purpose of this study, using the “ultrafine” (99, 590) grid for numerical integration (99 radial and 590 angular integration points) in all DFT calculations. Although the first of the used basis sets is rather small on an absolute scale, it has been shown to perform remarkably well in combination with DFT methods,12-22,30,31 giving results which, as we demonstrate in the present study, are nearly converged with respect to the basis set size. 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. In the present study, we have focused our attention on the hydrogen-bonded minima. For a more thorough insight into the nature of intermolecular interactions in the mentioned dimers, and the existence of other minima on various PES, the reader

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is referred to our earlier publications.12,16,21,30,31 Although harmonic vibrational analyses could be used for further calculation of harmonic vibrational frequency shifts of the relevant intramolecular modes, in the present paper we focus our discussion on the more reliable anharmonic vibrational frequency shifts, and we discuss the harmonic frequency shifts only for comparison purposes. Except for the phenol-benzene(+) cationic radical dimer, for all other systems the anharmonic vibrational frequency shifts have already been calculated the MP2/6-31++G(d,p) level of theory and were available for comparison purposes. These results have already been presented in our previous publications.12,16,21,30,31 Since such data were missing in the case of phenol-benzene(+), for the purpose of the present study we have also located the global minimum on its PES at MP2/6-31++G(d,p) level, and subsequently carried out harmonic vibrational analysis and computed the anharmonic OH stretching vibrational frequency. 2.3. Calculation of Anharmonic O-H Vibrational Frequencies. In the present study we are primarily interested in the OH frequency shifts upon hydrogen bonding by methanol and phenol in various dimers. Since all X-H vibrations are known to be large-amplitude motions, and therefore, inherently considerably anharmonic, to compute the corresponding frequencies with a sufficient accuracy it is necessary to go beyond the widely used harmonic approximation in quantum chemistry. Anharmonic contributions to the overall observed vibrational frequency shifts, on the other hand, may be as high as 30-40%.11-22,30,31 To account for the anharmonicity effects (and other systematic deficiencies of computational methods), various variants of the scaled quantum mechanical force fields methods have been developed in the literature.36-40 This approach, is, however, based on system-specific “magic” scaling factors, which are by any means artificial. In the last years, also the vibrational self-consistent field methodology (VSCF) has become widely available and implemented in computer codes.41,42 The second-order perturbative approach has also been efficiently automated by Barone for building the anharmonic force constants and evaluation of vibrorotational parameters and implemented in the Gaussian series of codes.43 An efficient local mode approximation-based method for calculation of NIR and NIR-VCD spectra up to the second X-H stretching overtone region has recently been proposed by Abbate et al.44,45 All of these methodologies are, however, much more computationally demanding. In our approach, instead of using an arbitrary scaling procedure, we accounted explicitly for the anharmonicity of the O-H stretching motion, using a computationally feasible approach. To obtain the vibrational potential energy function (V ) f(rOH)) for an intramolecular OH oscillator within a free monomer or hydrogen-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 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 least-squares fitted to a fifth-order polynomial in ∆rOH (∆r ) r - re):

V ) V0 + k2∆r2 + k3∆r3 + k4∆r4 + k5∆r5

(2)

The resulting potential energy functions were subsequently cut after fourth order and transformed into Simons-Parr-Finlan (SPF) type coordinates:46

F ) 1 - rOH,e /rOH

(3)

where rOH,e is the equilibrium, that is, the lowest-energy, value. The one-dimensional vibrational Schro¨dinger equation was solved variationally. Usage of only 15 harmonic oscillator eigenfunctions as a basis was shown to lead to excellent convergence of the computed vibrational frequencies. Superiority of the SPF-type coordinates over the ordinary bond stretch ones when a variational solution of the vibrational Schro¨dinger equation is sought has been well established, as they allow for a faster convergence (with the number of basis functions used) and a greatly extended region of convergence.46 The fundamental anharmonic O-H stretching frequency (corresponding to the |0〉f|1〉 transition) was computed from the energy difference between the ground (|0〉) and first excited (|1〉) vibrational states. 2.4. Charge Field Perturbation Calculations. The CFP (charge-field perturbation) approach is applied in the present study (analogously as in our previous studies of various hydrogen-bonded systems)12-22,30,31 to get a deeper insight into the electrostatic (+polarization) contribution to the total shift of each anharmonic O-H vibrational frequency upon the studied hydrogen bonding interaction. Implementation of the CFP approach in the present study was performed in the following way: the proton-accepting unit was represented by a set of point charges placed at the corresponding nuclear positions within the dimeric geometry, and the O-H stretching vibrational potential of the proton-donor methanol or phenol unit was calculated within the inhomogeneous electrostatic field generated by the proton-acceptor charge distribution. The sets of point charges were chosen such as to reproduce the molecular electrostatic potential of the proton-acceptor unit calculated from the corresponding KS densities for each level of theory at series of points selected applying the CHelpG point-selection algorithm.47 It is worth reminding at this point that 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:

1 b) ) f b + 1 Ω · ∇2b F+ -∆E(F µ 0·b F + Θ · ∇F 2! 3! 1 1 Φ · ∇3b F + · · ·+ Θ·b F+ + · · · 4! 2!

(4)

jj jj jj (where Θ , Ω, Φ, ... are the quadrupole, octupole, hexadecapole tensor functions of second, third, fourth, etc. order, while b µ0 j and R j are the dipole moment vector of the free molecule and the dipole polarizability function, respectively, the last quantity being a second-order tensor). All calculations were performed with the Gaussian03 series of programs.48 3. Results and Discussion 3.1. Structures and Interaction Energies. The minima located on mPW1B95/6-31++G(d,p) PESs of the studied dimers are shown in Figure 1. It can be seen that in the case of methanol-fluorobenzene and phenol-benzene(+) dimers there is a direct O-H · · · π contact with the aromatic nucleus, indicative of a π-type hydrogen bond. Phenol-acetylene dimer is also characterized by a π-type hydrogen bonding interaction, though in this particular case it is the triple-bond π-electronic cloud that serves as a proton-acceptor. The neutral and cationic phenol-CO dimers are, on the other hand, characterized by O-H · · · C contact, indicating a conventional hydrogen bonding

Anharmonic Vibrations in Hydrogen-Bonded Dimers

J. Phys. Chem. A, Vol. 114, No. 12, 2010 4357 interaction. As implied before, in the present paper we will focus solely on the global minima on the studied PESs of the considered dimers. More detailed analyses of the PESs are available in our previous papers.12,16,21,30,31 The interaction energies for the studied hydrogen bonded dimers were computed in the following way. Using the notation of Xantheas,49 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: R∪β ∆E ) EAB (AB) - EAR (A) - EBβ (B)

(5)

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

(6)

that is, as a “vertical” (in a sense) interaction energy. The (full) function counterpoise corrected energy (by the Boys-Bernardi method)50 for the BSSE is given by: R∪β R∪β R∪β ∆E(f CP) ) EAB (AB) - EAB (A) - EAB (B)

(7)

and BSSE is, thus, often calculated by:

BSSE ) ∆E(f CP) - ∆E′

(8)

Since eqs 5 and 7 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,49 it is suitable to include the deformation (i.e., relaxation) energy in the following manner:

∆E(BSSE) ) ∆E(f CP) + ERdef(A) + Eβdef(B)

(9)

In the last equation, the deformation energies are given by:

Figure 1. The minima located on mPW1B95/6-31++G(d,p) PESs of the studied dimers: (a) methanol-fluorobenzene, (b) phenol-carbon monoxide, (c) phenol-acetylene, (d) phenol(+)-carbon monoxide radical cation, and (e) phenol(+)-benzene radical cation.

R ERdef(A) ) EAB (A) - EAR (A)

(10)

β Eβdef(B) ) EAB (B) - EAβ (B)

(11)

The results obtained at all levels of theory implemented in the present study, for all of the considered hydrogen-bonded dimers are given in Tables 1-5. We will start the discussion with the methanol-fluorobenzene dimer. Unfortunately, experimental data for the interaction energies are not available for this dimer and we can compare the performances of the presently implemented methodologies only with other theoretical data. In our previous studies,16 we obtained values of -5.58, -5.54, and 3.62 kJ mol-1 for ∆E(fCP), ∆E(BSSE) and D0 at B3LYP level of theory with the smaller (6-31++G(d,p)) basis set. The corresponding values at mPW1PW91 level are -8.65, -8.58, and 6.67 kJ mol-1 as well as -9.86, -9.80, and 3.74 kJ mol-1 at MP2. As can be seen from Table 1, therefore, the presently reported data are similar to those obtained with the mPW1PW91 combination of functionals. It may be also concluded that the interaction and dissociation energy values are well-converged already with the

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TABLE 1: Interaction and Dissociation Energies for Methanol-Fluorobenzene Dimer Calculated at Three DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set (All Values in kJ mol-1)a mPWLYP1W

PBELYP1W

PBE1W

MP2

methanol-fluorobenzene

small

big

small

big

small

big

small

-∆E′ –∆E(fCP) –∆E(BSSE) D0 Edef(C6H5OH) Edef(CO) Edef BSSE

–9.31 –7.15 –7.10 5.09 0.03 0.02 0.05 2.16

–8.12 –6.56 –6.52 4.18 0.02 0.02 0.04 1.55

–10.44 –8.16 –8.09 6.10 0.03 0.03 0.07 2.28

–9.24 –7.74 –7.69 5.54 0.02 0.02 0.04 1.50

–10.69 –8.38 –8.33 6.38 0.03 0.02 0.05 2.31

–9.21 –7.48 –7.43 5.24 0.03 0.03 0.06 1.73

–21.88 –9.86 –9.80 3.74 0.03 0.03 0.06 12.01

a See text for details and definitions. The values with mPW1B95 combination of functionals are not given due to severe convergency problems when using non-atom-centered basis functions.

TABLE 2: Interaction and Dissociation Energies for Phenol-CO Dimer (The C-Bonded Minimum on the PES) Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set (All Values in kJ mol-1)a mPW1B95

mPWLYP1W

PBELYP1W

PBE1W

MP2

phenol-CO

small

big

small

big

small

big

small

big

small

–∆E′ –∆E(fCP) –∆E(BSSE) D0 Edef(C6H5OH) Edef(CO) Edef BSSE

–9.25 –7.51 –7.30 4.18 0.18 0.04 0.22 1.74

–8.33 –7.27 –7.09 4.10 0.14 0.03 0.17 1.06

–10.83 –8.70 –8.57 5.52 0.10 0.03 0.13 2.13

–9.59 –8.56 –8.43 5.42 0.11 0.03 0.13 1.03

–11.88 –9.75 –9.63 6.59 0.09 0.03 0.12 2.14

–10.42 –9.49 –9.34 6.10 0.12 0.03 0.15 0.93

–11.22 –9.25 –9.10 6.06 0.11 0.04 0.15 1.96

–10.14 –9.04 –8.89 5.87 0.13 0.03 0.15 1.10

–12.72 –8.52 –8.45 4.95 0.05 0.02 0.07 4.21

a

See text for details and definitions.

TABLE 3: Interaction and Dissociation Energies for Phenol-Acetylene Dimer Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set (All Values in kJ mol-1)a mPW1B95

mPWLYP1W

PBELYP1W

PBE1W

MP2

phenol-acetylene

small

big

small

big

small

big

small

big

small

-∆E′ –∆E(fCP) –∆E(BSSE) D0 Edef(C6H5OH) Edef(CO) Edef BSSE

–13.46 –12.00 –11.87 3.85 0.11 0.02 0.13 1.46

–12.31 –11.47 –11.33 3.73 0.12 0.02 0.14 0.83

–13.43 –11.47 –11.30 3.65 0.14 0.02 0.16 1.97

–11.61 –10.78 –10.61 3.54 0.15 0.03 0.17 0.82

–14.72 –12.66 –12.52 4.84 0.12 0.02 0.15 2.06

–12.54 –11.85 –11.70 4.39 0.13 0.02 0.15 0.69

–14.47 –12.73 –12.56 4.97 0.15 0.02 0.17 1.75

–12.84 –12.02 –11.82 4.76 0.17 0.02 0.19 0.82

–20.97 –11.44 –11.35 3.32 0.08 0.01 0.09 9.53

a


smaller basis set. The ∆E(BSSE) values obtained by the presently applied functional combinations are closer to the MP2 values using the same basis set. On the other hand, when one compares the ZPVE-corrected D0 value, it appears that the B3LYP level is in closest agreement with MP2. We next consider the weakly bonded phenol-carbon monoxide dimer (in particular, the C-bonded minimum on the PES). The experimental dissociation energy is nearly 8 kJ mol-1.8 In our previous study devoted in much more details to this system, we have found that B3LYP level of theory underestimates the D0 value, giving 3.71 and 3.85 kJ mol-1 with the 6-31++G(d,p) and 6-311++G(3df,3pd) basis sets correspondingly.30 This, however, implies that the B3LYP values have converged with the basis set size even with the smaller basis used in our previous study. MP2 results, on the other hand, were 4.95 and 6.78 kJ mol-1 with the 6-31++G(d,p) and 6-311++G(3df,3pd) basis sets, respectively.30 Testing the performance of the HCTH/407 functional has lead us to the values of 6.67 and 6.44 kJ mol-1 with the 6-31++G(d,p) and 6-311++G(3df,3pd) basis sets, which was quite comparable to the quality of MP2 results even with the smaller basis.30 Our results in the present study (see

Table 2) show that all of the presently used functionals lead to an improved performance as compared to B3LYP, with the PBELYP1W being closest to the MP2 and experimental results. Although the experimentally estimated dissociation energy for the phenol-acetylene dimer is as high as 16.7 kJ mol-1,9 considering the experimentally detected OH stretching frequency shifts upon intermolecular interaction in this and in the case of previously discussed dimer, one would expect similar interaction energies as well. As can be seen from Table 3, while the ∆E(BSSE) values are indeed higher in the case of phenol-acetylene as compared to phenol-carbon monoxide, upon inclusion of the ZPVE corrections (i.e., comparing the actual dissociation energies) the D0 values are very similar for the two systems. Our previously reported values for the last quantity calculated at B3LYP, mPW1PW91, PBE1PBE, and MP2/6-31++G(d,p) levels of theory were 1.96, 3.34, 5.29, and 3.32 kJ mol-1, respectively.12 The presently presented results are thus similar to those obtained with the mPW1PW91 combination of functionals and are close to the MP2 ones reported in our previous study.


J. Phys. Chem. A, Vol. 114, No. 12, 2010 4359

TABLE 4: Interaction and Dissociation Energies for Phenol-CO(+) Dimer (The C-Bonded Minimum on the PES) Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set (All Values in kJ mol-1)a mPW1B95

mPWLYP1W

PBELYP1W

PBE1W

MP2

phenol-CO(+)

small

small

big

small

big

small

big

small


–31.12 –28.95 –27.71 23.65 0.89 0.34 1.24 2.17

–36.08 –33.46 –31.71 27.83 1.43 0.32 1.75 2.62

–35.15 –33.73 –31.99 28.04 1.45 0.28 1.74 1.43

–36.40 –33.81 –32.25 28.36 1.26 0.29 1.56 2.59

–35.28 –33.99 –32.41 28.52 1.31 0.27 1.59 1.29

–37.79 –35.32 –33.30 29.55 1.68 0.34 2.02 2.47

–37.13 –35.56 –33.51 29.65 1.74 0.30 2.05 1.57

–32.73 –27.57 –26.82 18.54 0.62 0.13 0.75 5.16

a See text for details and definitions. The values with mPW1B95 combination of functionals with the big basis set are not given due to severe convergency problems when using non-atom-centered basis functions.

TABLE 5: Interaction and Dissociation Energies for Phenol-Benzene(+) Dimer Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set (All Values in kJ mol-1)a mPW1B95

mPWLYP1W

PBELYP1W

PBE1W

MP2

phenol-CO(+)

small

small

big

small

big

small

big

small


–57.04 –53.96 –52.60 51.02 0.70 0.66 1.36 3.07

–65.71 –62.85 –60.64 60.55 1.13 1.08 2.21 2.86

–62.99 –61.10 –58.97 58.42 1.12 1.02 2.13 1.89

–66.82 –63.98 –61.77 61.73 1.12 1.09 2.22 2.84

–63.88 –62.21 –60.06 59.60 1.11 1.04 2.15 1.67

–67.01 –64.11 –62.03 61.89 1.09 0.99 2.09 2.90

–64.60 –62.45 –60.43 59.82 1.09 0.93 2.02 2.15

–66.42 –47.31 –45.72 33.42 1.07 0.52 1.59 19.11

a See text for details and definitions. The values with mPW1B95 combination of functionals with the big basis set are not given due to severe convergency problems when using non-atom-centered basis functions.

Upon ionization of the phenol-carbon monoxide dimer, which practically leads to charge localization solely in the phenyl ring, the interaction and dissociation energies increase drastically8 due to the ion-multipole and ion-induced-multipole forces. As can be seen from Table 4, especially the mPWLYP1W, PBELYP1W, and PBE1W functionals lead to dissociation energies that are in excellent agreement with the available experimental data (∼29.0 kJ mol-1).8 All DFT values are superior to MP2 ones with the presently used basis set. Similar discussion is valid for the phenol-benzene radical cation dimer. The dissociation energy values presented in Table 5 are all consistent with the experimental estimations that this parameter should be higher than 37 kJ mol-1.9 mPWLYP1W, PBELYP1W, and PBE1W functionals all predict dissociation energy of this dimer of about 60 kJ mol-1, which is higher than the value predicted by more standard functionals such as B3LYP and mPW1PW91.21 A more accurate experimental value for this quantity would be highly desirable to be able to further test the performance of the presently implemented DFT functionals in this respect. 3.2. Anharmonic Vibrational Frequencies and Frequency Shifts. Although the main focus of this chapter will be put on the anharmonic vibrational frequency shifts, we will also consider the absolute anharmonic frequencies of the free monomeric units (phenol and methanol) computed in the present study. All values relevant to the present discussion are tabulated in Tables 6-11. As can be seen, concerning the absolute anharmonic OH stretching frequencies of free phenol and methanol monomers, the mPW1B95 values are in closest agreement with both the experimental and MP2 data. We have demonstrated in our previous study that the absolute anharmonic OH stretching frequencies of phenol computed at both DFT and MP2 levels of theory are sufficiently converged with the basis set size even when using the 6-31++G(d,p) basis.16,20,30,31 For

TABLE 6: Anharmonic OH Stretching Vibrational Frequencies of the Free Methanol, Phenol, and Phenol(+) Units Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT, and Experimental Dataa methanol (ν/cm-1)

mPW1B95 mPWLYP1W PBELYP1W PBE1W B3LYPb mPW1PW91b PBE1PBEb HCTHb OLYPb MP2b experimental

phenol (ν/cm-1)

phenol(+) (ν/cm-1)

small

big

small

big

small

big

3735.7 3496.7 3500.3 3535.7 3652.2

3734.7 3512.2 3516.6 3545.4

3720.8 3485.3 3490.1 3519.6 3646.2 3713.0 3708.0 3612.0 3585.4 3676.0

3716.4 3495.4 3500.6 3523.5

3607.8 3386.4 3390.9 3416.5 3536.9 3599.6 3593.0 3502.5 3477.7 3587.2

3597.2 3388.7 3394.3 3414.6

3721.0

3710.3 3681

3657

3534

a See text for details and definitions. b Values taken from our previous studies (refs 16, 20, 30, 31). Experimental values taken from refs 8-10.

example, passing from 6-31++G(d,p) to the 6-311++G(2df,2pd) or even 6-311++G(3df,3pd) basis set affects the free phenol monomer OH stretching frequencies by only a few cm-1. Upon ionization of the bare phenol unit, a significant OH frequency downshift of -123 cm-1 occurs (from 3657 to 3534 cm-1).8,9 Concerning the absolute anharmonic OH stretching frequency of the phenol radical cation monomer, of all the presently implemented DFT functionals, the mPW1B95 one performs best, even quite comparable to MP2 (Table 6). It is particularly notable that the OH stretching frequency shift upon ionization of neutral phenol is remarkably well reproduced at the mPW1B95/ 6-31++G(d,p) level of theory, significantly better than at the MP2 level (Table 7). Actually, all of the currently used

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TABLE 7: Anharmonic OH Stretching Vibrational Frequency Shifts upon Ionization of the Free Phenol to Phenol(+) Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT, and Experimental Dataa

TABLE 9: Anharmonic OH Stretching Vibrational Frequency Shifts for Cationic Dimers Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT and Experimental Dataa

∆ν/cm-1 mPW1B95 mPWLYP1W PBELYP1W PBE1W B3LYPb mPW1PW91b PBE1PBEb HCTHb OLYPb MP2b experimental

phenol-CO(+) (∆ν/cm-1)

small

big

–113.0 –98.9 –99.2 –103.1 –109.3 –113.4 –115.0 –109.5 –107.7 –88.8

–121.4 –106.7 –106.3 –108.9

–123

a

b

See text for details and definitions. Values taken from our previous studies (refs 20, 30, 31). Experimental values taken from refs 8, 9.

TABLE 8: Anharmonic OH Stretching Vibrational Frequency Shifts for Neutral Dimers Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT, and Experimental Dataa methanolfluorobenzene (∆ν/cm-1)

mPW1B95 mPWLYP1W PBELYP1W PBE1W B3LYPb mPW1PW91b PBE1PBEb HCTHb OLYPb MP2b experimental

phenolCO (∆ν/cm-1)

phenolC2H2 (∆ν/cm-1)

small

big

small

big

small

big

-25.6 -47.8 -39.2 -58.8 -41.5

-31.1 -49.4 -40.4 -59.9

-54.9 -62.7 -56.6 -80.6 -61.2 -76.1 -85.6 -28.8 -5.3 -42.0

-60.5 -68.6 -61.6 -86.3

-97.6 -109.1 -98.9 -127.3 -101.2 -122.9 -127.8

-101.9 -107.3 -97.1 -127.9

-49.5 -20.5 -20

-70.2 -33

-68

a See text for details and definitions. b Values taken from our previous studies (refs 12, 16, 30). Experimental values taken from refs 8-10.

nonstandard DFT functionals outperform the MP2 level in this respect. This is a particularly important result, as the problem of accurate prediction of anharmonic vibrational frequency shifts upon ionization of organic molecules has not been solved yet. According to the results presented in this study, as well as according to the results of our other ongoing studies, it seems that DFT performs better in this respect in comparison to MP2. Concerning the OH stretching frequency shifts upon dimer formation, in the case of weakly-bonded methanol-fluorobenzene dimer the closest agreement with the experimental10 (and also with the MP2)16 data was obtained with the mPW1B95 functional (Table 8). This particular functional leads to very significant improvement in comparison to the B3LYP and PBE1PBE values obtained with the 6-31++G(d,p) basis when anharmonic frequency shift is in question. Also, the PBELYP1W functional leads to significantly improved performance as compared to B3LYP and PBE1PBE. The mPWLYP1W performs essentially identical to PBE1PBE, at the same time being inferior to B3LYP. The PBE1W functional, on the other hand, is inferior to both B3LYP and PBE1PBE.

mPW1B95 mPWLYP1W PBELYP1W PBE1W B3LYPb mPW1PW91b PBE1PBEb HCTHb OLYPb MP2 experimental

phenol-benzene(+) (∆ν/cm-1)

small

big

small

big

–446.7 –542.0 –512.4 –601.1 –475.9 –526.8 –538.7 –465.9 –423.6 –328.8

–467.3 –545.7 –517.3 –613.1

–400.1 –336.6 –293.1 –415.7 –513.9 –554.2

–420.6 –319.0 –407.4

–321.0 –211

–475

a See text for details and definitions. b Values taken from our previous studies (refs 21, 31). Experimental values taken from refs 8, 9.

When the C-bonded arrangement of the monomeric units in the case of phenol-CO dimer is in question, again the mPW1B95 functional performs best (Table 8). Both mPW1B95 and PBELYP1W outperform the more standard functionals such as B3LYP, mPW1PW91, and PBE1PBE. mPWLYP1W performs very similarly to B3LYP, while PBE1W outperforms the PBE1PBE, at the same time being inferior to B3LYP and mPW1PW91. All DFT levels perform inferior to MP2/631++G(d,p) for this system. However, considering the basis set size convergence properties of MP2 and DFT methodologies, it actually appears that the values that may be regarded as practically converged with the basis set size at DFT levels are superior to the corresponding MP2 ones. Similar arguments are valid for the neutral phenol-acetylene dimer as well. mPW1B95 and PBELYP1W perform best (though this time with a quite similar performance), both slightly outperformingtheB3LYPfunctional(Table8).WhilemPWLYP1W outperforms both mPW1PW91 and PBE1PBE, at the same time being inferior to B3LYP, PBE1W performs very similarly to PBE1PBE in this particular case. The situation is drastically different when one considers the cationic dimers, which are characterized by an unpaired electron, that is, an overall spin multiplicity of 2 (Table 9). For such cases, if the HF wave function is spin-contaminated to a significant degree, the perturbation theory expansion does not converge quickly, so the MP2 level of theory is often insufficient to obtain accurate results.51-53 On the other hand, the DFT methodology often does not suffer from significant spincontamination. This makes the DFT methods even more attractive for such cases, aside from their much lower computational cost. In the case of the C-bonded minimum on the PES of phenol-CO+ dimer, as can be seen, MP2 methodology outperforms all DFT levels implemented in the present study. However, the mPW1B95 combination of functionals performs much better than all other functionals tested in this paper and is also superior to B3LYP, mPW1PW91, and PBE1PBE.31 In comparison to the performances of the MP2 methodology for spin-restricted cases (neutral dimers), however, here it is much less accurate. This means that although spin-contamination problem affects significantly the MP2 results, as discussed below, it is still the method of choice in comparison to the DFT


J. Phys. Chem. A, Vol. 114, No. 12, 2010 4361

TABLE 10: Anharmonic OH Stretching Vibrational Frequency Shifts for Neutral Dimers Calculated with the CFP Method at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p) Dataa

mPW1B95 mPWLYP1W PBELYP1W PBE1W MP2 a

methanolfluorobenzene (∆ν/cm-1)

phenol-CO (∆ν/cm-1)

phenol-C2H2 (∆ν/cm-1)

–19.3 –15.5 –13.3 –16.0 –20.5

1.2 1.8 0.5 –0.1 –9.1

–42.5 –34.2 –34.0 –37.8 –51.1


approach. When one considers the phenol-benzene radical cationic dimer, the MP2 results with the small basis, which have been obtained for the particular purpose of the present study, are significantly inferior to all DFT data, except for PBELYP1W. The new functionals, however, all show inferior performance in comparison to the B3LYP and even mPW1PW91.21 To get a clearer insight into the reasons behind various performances of the DFT and MP2 methods in case of the two studied radical dimers, we hereafter analyze the spin contaminations in these cases. Since the unrestricted Kohn-Sham and MP2 formalisms were applied, the degree of spin contamination was checked for in each case by comparing the obtained expectation values of the total spin squared operator 〈Sˆ2〉 with the theoretical (exact) one: s(s + 1). Note in this context, however, that such test has been shown as a rather useful even in cases when DFT methods are applied, despite the fact that the interpretation of 〈Sˆ2〉 is not quite straightforward for unrestricted DFT as in the case of UHF (UMP2). In the case of phenol-CO+ dimer, the HF wave function used for perturbation theory treatment for the minimum on MP2/6-31++G(d,p) PES is characterized with 〈Sˆ2〉 ) 0.9651 before annihilation, 0.7796 after annihilation, while the exact one is 0.7500, which gives the relative error of about 28.68%, which is quite large (in case of organic systems, often a spin contamination expressed as a relative error of less than 10% is considered as negligible).21 The relative errors in the discussed quantity in the case of DFT methods are less than 1% (2% for mPW1B95). In the case of phenol-benzene cation dimer, the relative spin contamination at the HF level is 31.79% (the value computed for the minimum on the corresponding MP2/6-31++G(d,p) PES), which is quite larger than in the case of phenol-CO+. The DFT level values for this system are all below 1% (except in the case of mPW1B95 method, where it is about 2% for both systems). Of course, 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. However, since the computed potential energy functions corresponding to the OH stretching vibrations were all rather smooth, it seems that for a particular OH stretching curve, the introduced errors are fairly systematic and one-directional (this aside the fact that such conclusion is not true in general). As can be seen from Tables 10 and 11, the CFP analysis of the anharmonic OH stretching potentials leads to conclusions that are quite analogous to those reported in our previous studies.11-22,30,31 In the case of methanol-fluorobenzene, essentially the whole frequency shift may be attributed to classical electrostatic interaction with the proton acceptor. The situation, however, seems to be quite the opposite in the case of neutral phenol-CO dimer. Electrostatic contribution to the overall shift

TABLE 11: Anharmonic OH Stretching Vibrational Frequency Shifts for Cationic Dimers Calculated with the CFP Method at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p) Dataa phenol-CO(+) (∆ν/cm-1)

phenol-benzene(+) (∆ν/cm-1)

–45.6 –48.1 –46.2 –55.6 –52.5

–128.5 –91.8 –81.9 –105.2 –122.9

mPW1B95 mPWLYP1W PBELYP1W PBE1W MP2 a


TABLE 12: Anharmonic vs Harmonic OH Stretching Vibrational Frequency Shifts for Neutral Dimers Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT, and Experimental Dataa methanolfluorobenzene (∆ν/cm-1)

mPW1B95 mPWLYP1W PBELYP1W PBE1W experimental a

phenol-CO (∆ν/cm-1)

phenolC2H2 (∆ν/cm-1)

anh.

harm.

anh.

harm.

anh.

harm.

–25.6 –47.8 –39.2 –58.8

–27.9 –36.5 –29.2 –52.7 –20

–54.9 –62.7 –56.6 –80.6

–49.4 –48.6 –43.9 –64.0 –33

–97.6 –109.1 –98.9 –127.3

–84.0 –86.1 –77.9 –101.2 –68


TABLE 13: Anharmonic vs Harmonic OH Stretching Vibrational Frequency Shifts for Cationic Dimers Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) Basis Set, Together with the Reference MP2/6-31++G(d,p), Several Other DFT, and Experimental Dataa phenolCO(+) (∆ν/cm-1)

mPW1B95 mPWLYP1W PBELYP1W PBE1W experimental a

phenolbenzene(+) (∆ν/cm-1)

anh.

harm.

anh.

harm.

–446.7 –542.0 –512.4 –601.1

–358.2 –407.9 –402.0 –427.2 –211

–400.1 –336.6 –293.1 –415.7

–313.2 –224.8 –193.0 –284.4 –475


in neutral phenol-acetylene dimer, on the other hand, seems to lie somewhere in between the previous two cases, accounting for some 30-40% of the overall full wave function data. Upon ionization of the phenol-CO dimer (i.e., in the case of the corresponding cationic radical dimer), the electrostatic contribution to overall ∆ν rises to some 10% (Table 11), while this contribution in the case of phenol-benzene(+) is about 20-30%. Finally, in Tables 12 and 13, the computed harmonic frequency shifts (by diagonalization of Hessian matrices) are compared to the anharmonic ones obtained from the 1D vibrational potentials. As can be seen, in the case of neutral dimers, at essentially all levels of theory the harmonic shifts are in better agreement with the experimental data. To get further insight into the computational appropriateness of our approach, as well as to check the reliability of the computed 1D vibrational potentials for the O-H oscillator of the free phenol molecule, we have subsequently computed the anharmonicity constant characterizing the mentioned vibrational mode (X). This was done on the basis of the following equation:

ν˜ OH ) ω0,OH + 2X

(12)

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TABLE 14: Anharmonicity Constant for the OH Stretching Vibration of the Free Phenol Calculated at Four DFT Levels of Theory with the 6-31++G(d,p) (small) or 6-311++G(2df,2pd) (big) Basis Set, Together with the Reference MP2/6-31++G(d,p), MP2/6-311++G(2df,2pd), and Several Other DFT and Experimental Dataa -X/cm–1 mPW1B95 mPWLYP1W PBELYP1W PBE1W B3LYPb mPW1PW91b PBE1PBEb HCTHb OLYPb MP2b exp. NIDb exp. conventionalb fit to exp. NID datab

Small

big

92.7 99.3 99.6 98.6 93.9 92.1 92.2 99.4 98.5 94.5

91.4 97.8 98.2 97.4 93.0 97.8 94.1 84.5 85.0 84.4

a See text for details and definitions. b Values taken from our previous studies (refs 20, 30, 31). Experimental NID and conventional data, as well as the data obtained from the fit to NID data taken from ref 54.

ω0,OH being the harmonic eigenvalue (obtained from the harmonic force constant k2). Our calculated values for this quantity were compared to the experimental and theoretical data published in the works by Kjaergaard et al.54,55 The experimental data in these studies were obtained using the supersonic jet experiments, implementing the technique of nonresonant ionization detected infrared and near-infrared spectroscopy (NID-IR/ NIR). At the same time, these authors have carried out a thorough theoretical analysis of the obtained results within the local mode approach.54,55 Table 14 summarizes the theoretical data obtained in the present study and compares them with those from ref 54. As can be seen, the values computed by our approach are in excellent agreement with the experimental NIDIR/NIR spectroscopic data, which further confirms the reliability of the approach and the quality of the computed 1D vibrational potentials for the free phenol monomer. As can be seen, the results at both DFT and MP2 levels of theory are converged with the basis set size already with the small (6-31++G(d,p)) basis set. The mPW1B95 functional is again superior to all of the other nonstandard functionals studied in this paper, and also even superior to MP2 with this respect. 4. Conclusions We have analyzed the performance of some recently developed DFT functionals (mPW1B95, mPWLYP1W, PBELYP1W, and PBE1W) primarily with respect to computation of anharmonic vibrational frequency shifts, but also with respect to prediction of interaction and dissociation energies in a series of small hydrogen-bonded dimers with various types and strengths of the noncovalent bond. The most important conclusions from our study may be summarized as follows: (1) In the case of neutral weakly bonded dimers, the mPW1B95 functional generally performs better than the other functionals of this group, and it also outperforms most of the other standard combination of functionals (such as, e.g. B3LYP, mPW1PW91, and PBE1PBE) when the prediction of anharmonic OH stretching frequency shift upon hydrogen bonding is in question;

(2) The absolute anharmonic OH frequencies, as well as the frequency shifts upon hydrogen bond formation computed with the novel functionals, are well-converged with the basis set size even with the modest-size 6-31++G(d,p) basis set; (3) All DFT levels of theory significantly outperform the MP2 level when the prediction of the anharmonic OH stretching frequency shift upon ionization of neutral phenol molecule is in question; (4) In the case of the considered cationic dimers, the MP2 level of theory performs significantly poorer with respect to accurate prediction of the anharmonic OH stretching frequency shifts upon hydrogen bond formation in comparison to the neutral ones, which is due to significant spin contamination of the reference HF wave function. Although some DFT levels were found to outperform MP2 in this respect for certain cases, the present results imply that this does not seem to be the case in general. (5) Wherever experimental data for the dissociation energies are available, the DFT levels of theory outperform MP2 level, especially in the case of ca`tionic dimers. With this respect, the PBELYP1W and PBE1W functionals seem to perform best. References and Notes (1) Sousa, S. F.; Fernandes, P. A.; Ramos, M. J. J. Phys. Chem. A 2007, 111, 10439, and references therein. (2) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Phys. ReV. Lett. 2003, 91, 146401. (3) Perdew, J. P.; Schmidt, K. In Density Functional Theory and its Applications to Materials; Doren, V., Alsenoy, C. V., Geerlings, P., Eds.; American Institute of Physics: New York, 2001. (4) Mattsson, A. E. Science 2002, 298, 759. (5) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (6) Becke, A. D. J. Chem. Phys. 1996, 104, 1040. (7) Wilson, P. J.; Bradley, T. J.; Tozer, D. J. J. Chem. Phys. 2001, 115, 9233. (8) Fujii, A.; Ebata, T.; Mikami, N. J. Phys. Chem. A 2002, 106, 10124. (9) Fujii, A.; Ebata, T.; Mikami, N. J. Phys. Chem. A 2002, 106, 8554. (10) Fujii, A.; Okuyama, S.; Iwasaki, A.; Maeyama, T.; Ebata, T.; Mikami, N. Chem. Phys. Lett. 1996, 256, 1. (11) Silvi, B.; Wieczorek, R.; Latajka, Z.; Alikhani, M. E.; Dkhissi, A.; Bouteiller, Y. J. Chem. Phys. 1999, 111, 6671. (12) Pejov, L.; Solimannejad, M.; Stefov, V. Chem. Phys. 2006, 323, 259. (13) Solimannejad, M. Pejov, L. J. Phys. Chem. A 2005, 109, 825. (14) Pejov, L.; Ivanovski, Gj. Chem. Phys. Lett. 2004, 399, 247. (15) Pejov, L.; Spångberg, D.; Hermansson, K. J. Phys. Chem. A 2005, 109, 5144. (16) Solimannejad, M.; Pejov, L. Chem. Phys. Lett. 2004, 385, 394. (17) Pejov, L. Chem. Phys. Lett. 2003, 376, 11. (18) Pejov, L.; Hermansson, K. J. Chem. Phys. 2003, 119, 313. (19) Pejov, L. Int. J. Quantum Chem. 2003, 92, 516. (20) Pejov, L. Chem. Phys. 2002, 285, 177. (21) Pejov, L. Chem. Phys. Lett. 2002, 358, 368. (22) Pejov, L. Chem. Phys. Lett. 2001, 339, 269. (23) Dahlke, E. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 317. (24) Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 6908. (25) Lynch, B. J.; Fast, P. L.; Harris, M.; Truhlar, D. G. J. Phys. Chem. A 2000, 104, 4811. (26) Dirac, P. A. M. Proc. Cambridge Philos. Soc. 1930, 26, 376. (27) Slater, J. C. Quantum Theory of Matter, 2nd ed.; McGraw-Hill: New York, 1968. (28) Adamo, C.; Barone, V. J. Chem. Phys. 1998, 108, 664. (29) Becke, A. D. J. Chem. Phys. 1996, 104, 1040. (30) Kocevski, V.; Pejov, L.; Hermansson, K., work in progress. (31) Kocevski, V. ; Pejov, L. , work in progress. (32) Czarnecki, M. A. J. Phys. Chem. 2003, 107, 1941. (33) Czarnecki, M. A.; Orzechowski, K. J. Phys. Chem. 2003, 107, 1119. (34) Schlegel, H. B. J. Comput. Chem. 1982, 3, 214. ¨ .; Schlegel, H. B. J. Chem. Phys. 1999, 111, 10806. (35) Farkas, O (36) Blom, C. E.; Altona, C. Mol. Phys. 1976, 31, 1377. (37) Pulay, P.; Fogarasi, G.; Pongor, G.; Boggs, J. E.; Vargha, A. J. Am. Chem. Soc. 1983, 105, 7037. (38) Fogarasi, G. Spectrochim. Acta A 1997, 53, 1211. (39) Szasz, G.; Kovacs, A. Mol. Phys. 1996, 2, 161.

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