Reaction force constant and projected force

Chemical Physics Letters 456 (2008) 135–140

Contents lists available at ScienceDirect

Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Reaction force constant and projected force constants of vibrational modes along the path of an intramolecular proton transfer reaction Pablo Jaque a,c,*, Alejandro Toro-Labbé a, Peter Politzer b, Paul Geerlings c a

Laboratorio de Química Teórica Computacional (QTC), Facultad de Química, Pontifica Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile Department of Chemistry, University of New Orleans, New Orleans, LA 70148, USA c Eenheid Algemene Chemie (ALGC), Vrije Universiteit Brussel (VUB), Faculteit Wetenschappen, Pleinlaan 2, 1050 Brussels, Belgium b

a r t i c l e

i n f o

Article history: Received 3 January 2008 In final form 13 March 2008 Available online 29 March 2008

a b s t r a c t We have explored the relationships between the reaction force F(n), the reaction force constant j(n) and the projected force constants of the intramolecular proton transfer HON@S ? O@NSH along the intrinsic reaction coordinate n. The structural changes and energetics associated with the reaction are analyzed in terms of the three regions defined by F(n): reactant, transition and product. The significance of the similarity between j(n) and the variation of the force constant associated to the reaction coordinate mode, kn(n), is discussed in detail. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction One of the main goals of quantum chemical modeling is to provide information about the energy barrier, thermodynamic driving force and mechanism of a chemical reaction. This can lead to an improved capability for controlling the process, e.g. its rate and its selectivity. A starting point for such modeling is the Born–Oppenheimer approximation [1], in terms of which a potential energy surface (PES) for any system of interest can be obtained, be it that the computation of accurate PESs for polyatomic systems is still a continuous challenge. The PES describes changes in the total energy E as a function of the nuclear arrangement {Qi}. The number of internal degrees of freedom is equal to (3N 6) for a nonlinear system of N atoms. Key points on the PES are those where the gradient of E is zero with respect to each of the (3N 6) degrees of freedom. If such points correspond to global or local energy minima, then they represent stable states of the system. If the point is such that the energy decreases along one of the degrees of freedom, then it is a first-order saddle point and is a transition state between two of the energy minima. An effective way in which to depict the path followed by a system in going from one minimum on the PES to another (e.g. from reactants to products in a chemical reaction) is by means of the intrinsic reaction coordinate (IRC), n [2,3]. This is the path of lowest energy (steepest-descent), expressed in mass-weighted Cartesian coordinates, from the transition state to the reactants and to the

* Corresponding author. Address: Eenheid Algemene Chemie (ALGC), Vrije Universiteit Brussel (VUB), Faculteit Wetenschappen, Pleinlaan 2, 1050 Brussels, Belgium. Fax: +32 2 629 3317. E-mail address: [email protected] (P. Jaque). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.03.054

products. The IRC allows the variation in the potential energy V(n) of a (3N 6)-dimensional space to be condensed into a twodimensional profile such as is displayed in Fig. 1a, which shows the computed V(n) for the proton transfer: HO—N@S ! O@N—SH:

ð1Þ

The ground states of HON@S and O@NSH correspond to the minima at the left and right ends of V(n), respectively. From Fig. 1a, O@NSH is predicted to be approximately 7 kcal/mol more stable than HON@S, and the activation barriers in the forward and reverse processes are about 27 and 34 kcal/mol. It has been shown that information about isotope effects can be obtained from the widths of potential energy barriers [4]. In order to gain further insight from a V(n) profile concerning the mechanism of a process, we have introduced the concept of the reaction force [5–15]. This is defined by FðnÞ ¼

oVðnÞ : on

ð2Þ

For a V(n) profile such as that in Fig. 1a, taking the reactants to be at the left, F(n) has the form shown in Fig. 1b. For the reverse process, n would be increasing in the opposite direction and F(n) would be the negative of that in Fig. 1b. F(n) is always zero at the positions of the reactants and products, nR and nP, where V(n) is a minimum, and at the transition state, nTS, where it is a maximum. F(n) has a minimum between the reactants and the transition state, at n1, and a maximum between the transition state and the products, at n2. These are the points of inflection of V(n). The points n1 and n2 divide a reaction into three regions along the IRC. Our experience has been that each of these regions tends, in general, to emphasize a certain phase of the process [5–15]. In the first, ‘reactant’ region, from reactants R to the force minimum,

136

P. Jaque et al. / Chemical Physics Letters 456 (2008) 135–140

a

the focus is on structural distortions in the reactants (bond stretching, angle bending, rotations, etc.) which prepare them for subsequent steps. These changes are resisted by the reactants, and so the reaction force F(n) is negative and therefore retarding. To overcome this resistance requires an amount of work given by Z n1 WðnR ! n1 Þ ¼ FðnÞ dn: ð3Þ

ξ1 ξ2

30

25

20

Potential Energy, ΔV

nR

15

10

5

0

-5

-10 -3

b

-2

-1

0

1

2

3

4

5

30

Reaction Force, F(ξ)

20

10

nR

0

-10

-30

-3

-2

-1

0

1

2

3

4

5

jðnÞ ¼

60

oFðnÞ : on

ð5Þ

The profile of j(n) is in Fig. 1c. It is positive in the reactant and product regions (nR ? n1 and n2 ? nP) and negative in the transition region (n1 ? nTS ? n2). It is zero at the F(n) minimum and maximum, n1 and n2, and goes through maxima and a minimum at the inflection points of F(n). In the present work, we have chosen, as our model system, the intramolecular proton transfer shown in Eq. (1) and displayed in Scheme 1. In this process the CS symmetry is maintained along the reaction path. Our objective is to gain insight into the mechanism, thermodynamics and kinetics of the process, from analyzing the potential energy, reaction force, reaction force constant and projected force constants along the IRC. For an earlier study of this proton transfer, also in terms of the reaction force, see GutiérrezOliva et al. [10]. The present communication is organized as follows: In Section 2, we give the details of the computations; in Section 3, the results are presented and in Section 4 they are discussed.

40

Reaction Force Constant, κ(ξ)

n1

The first term, as mentioned earlier, is the work needed for the structural preparation of the reactants; the second is the amount of work involved in the first zone of the transition region. We have shown, for specific reactions, that decomposing activation energies into these two contributions can help to elucidate the effects of solvents and catalysts upon reaction rates, i.e. whether these effects are primarily structural or electronic [12,14,15]. Another interesting aspect of the reaction force concept, which we introduce here, is the reaction force constant j(n):

-20

c

At the force minimum, n1, the reactants can be described as being in an activated state R*. The second, ‘transition to products’ region is composed of two zones, n1 ? nTS and nTS ? n2. It is in these zones that the most significant electronic reordering normally occurs (e.g. changes in electrostatic potentials, bond populations, dipole moments, Fukui functions, etc.). This produces a positive driving force, which gradually overcomes the retarding one; they are equal in magnitude at the transition state TS, at nTS, after which the driving force dominates. At the force maximum, n2, the system has reached what may be viewed as an activated or distorted state of the products, P*. In the final, ‘products’ region, n2 ? nP, these relax structurally to their equilibrium forms in the products P. The preceding summary was intended to describe what we have found to be, in general, the key features of each phase of an elementary process. However there is certainly overlapping; structural changes do occur in the transition region, and there are some electronic effects in the reactant and product regions. In terms of the reaction force, the activation energy Eact is the sum of two terms: Z n1 Z nTS FðnÞ dn FðnÞ dn: ð4Þ Eact ¼

20

0

-20

-40

-60

-80

-100 -3

-2

-1

0

1

2

3

4

5

IRC Fig. 1. (a) Potential energy (kcal/mol), (b) reaction force and (c) reaction force constant profiles of the intramolecular proton transfer HON@S ? O@NSH. The points n1 and n2 correspond to the minimum and maximum of the force, the vertical lines divide the process into the reactant, transition and product regions.

2. Computational procedure Geometry optimizations for each stationary state were performed using the hybrid exchange-correlation functional B3LYP [16–18] with a split-valence triple-zeta plus polarization basis


137

Scheme 1.

set, 6-311G(d,p) [19]. The performance of this computational scheme has been recently tested in other intramolecular hydrogen shift reactions, where both the reaction energy and activation barrier were shown to be comparable with those predicted with high level post-Hartree–Fock methodologies [20]. The stationary states were confirmed through harmonic vibrational analyses. The IRC for the process was obtained with a gradient reaction step size of 0.10 amu1/2 bohr. We also extended the vibrational analyses to nonstationary states along the IRC. As discussed by Miller et al. [21], this involves choosing the (3N 6) internal coordinates to be the reaction coordinate (the IRC) plus those corresponding to (3N 7) normal modes of vibration orthogonal to the IRC. Motion along the reaction coordinate, as well as infinitesimal rotations and translations, must be projected out of the force constant matrix. It is important to stress some computational aspects related to the harmonic frequency calculation at nonstationary structures along the reaction coordinate. This will depend upon the choice of coordinate system, Natanson et al. [22,23] have discussed in detail this point. In this work, the vibration modes perpendicular to the reaction coordinate was done in mass-weighted Cartesian coordinates, the overall conclusions can be expected not to be dependent on this choice. One of our objectives is to gain insight into the dynamical coupling between the reaction coordinate and the vibrational modes as the reaction advances from reactants to products. All calculations were carried out with GAUSSIAN 03 [24]. 3. Results 3.1. Structural and energetic data In Fig. 1 are the variations of V(n), F(n) and j(n) along the IRC of the reaction in Eq. (1). Table 1 shows how the structure of the system changes between the five key points: nR (reactants, R), n1 (force minimum, R*), nTS (transition state, TS), n2 (force maximum, P*) and nP (products, P). The work associated with each of these steps – W(nR ? n1), W(n1 ? nTS), W(nTS ? n2) and W(n2 ? nP) – is in Table 2, which also includes the overall energy change DE and the activation energy Eact. For the reverse process, O@NSH ? HON@S, the energies would have the same magnitudes but opposite signs. In the reactant region, nR ? n1, a major event is the decrease of the ONS angle from 117.1° to 106.7° [10]. This brings closer together the atoms that will donate and accept the hydrogen atom.

Table 1 Geometrical parameters of the structures at key points on the reaction pathway of HON@S ? O@NSH Structure

rNO

rNS

rOH

rSH

hONS

R R* TS P* P

1.337 1.275 1.257 1.237 1.165

1.595 1.650 1.661 1.677 1.923

0.985 1.141 1.372 1.653 2.586

2.426 1.848 1.664 1.463 1.353

117.1 106.7 107.0 107.4 116.2

Bond distances (r) and bond angles (h) are given in Å and in degrees, respectively.

Table 2 Energetic quantities associated with the process HON@S ? O@NSH, computed at the B3LYP/6-311G(d,p) level DE Eact W(nR ? n1) W(n1 ? nTS) W(nTS ? n2) W(n2 ? nP)

7.0 26.5 17.8 8.7 10.1 23.4

All values are in kcal/mol.

Concomitantly with this, the OH bond stretches as the hydrogen begins to move away from the oxygen and toward the sulfur; in response, the NO bond shortens, as it begins to acquire double bond character, while the NS lengthens, moving away from a double bond. All of this produces the state R*, and requires a work input, W(nR ? n1), of 17.8 kcal/mol. The transition region (n1 ? nTS ? n2) is characterized structurally largely by the movement of the hydrogen from the oxygen to the sulfur. This is where the OH bond breaks and the SH is formed, giving the state P*. Finally, in the product region, n2 ? nP, the ONS bond angle again opens, the newly-formed but stretched SH bond shortens, and the N@O and NS bonds complete their transitions to double and single-bond character, respectively. To some extent, these observations together with bond dissociation enthalpies (BDE) may be explaining the exothermicity of the reaction under study. The average BDEs values for N@O and N@S bonds are 150.61 ± 0.03 and 110.82 ± 5.02 kcal/mol [25], respectively. These values are suggesting that the double bond in the product region is stronger than it is in the reactant region. On the other hand, the BDEs data for OH and SH are, respectively, 101.76 ± 0.07 [26] and 84.1 ± 0.2 kcal/mol [27]. These data are indicating that the OH bond confers more stability to the reactant R (HO–N@S) that the SH bond to the product P (O@N–SH). We can note from this analysis that the thermodynamic driving force is mainly given by the formation of the NO double-bond. For the reverse process, the opposite of each of these steps occurs in each region. The activation energy is W(nP ? n2) + W(n2 ? nTS) = W(n2 ? nP) W(nTS ? n2) = 33.5 kcal/mol. For both the forward and the reverse processes, the largest part of the activation energy is the work needed for the initial structural changes. 3.2. Vibrational analysis Fig. 2 displays graphical representations of the displacement vectors for the normal coordinates at the transition state, while Table 3 gives the frequencies and their assignments for both the reaction coordinate and (3N 7) vibrational modes orthogonal to it for stationary and nonstationary structures. The (3N 7) vibrational modes orthogonal to the IRC do not remain the same in proceeding from reactant to product. For example, OH stretching is observed only in the reactant, SH stretching just in the product. However, a vibrational correlation diagram might be established along n, it correlates the vibrational modes of the reference structures

138

P. Jaque et al. / Chemical Physics Letters 456 (2008) 135–140 Table 3 Harmonic frequencies (cm1), calculated at the B3LYP/6-311G(d,p) level, for vibrational modes of both stationary (R, TS, and P) and nonstationary (R* and P*) states of the HON@S ? O@NSH reaction

Fig. 2. Schematic representation of the displacement vectors of the normal modes at the transition state; for each mode the irreducible representation of the CS group is indicated.

through the modes of the transition state. This is very significant in the identification of the reactive modes, i.e. the generalized normal modes that contribute to the motion along the IRC [22,23,28]. Contrarily to the frequencies, the Hessian matrix (matrix of force constants) contains the local information about the interaction and chemical bonding between every atom–atom pair. Therefore, the force constants correlation diagram or profile provides information on how the chemical bonding evolves along the reaction path. Only the mode corresponding to that identified as the reaction coordinate in Fig. 2a, which is the motion of the hydrogen between the oxygen and the sulfur, can be more or less followed nearly along the entire IRC, from just after the reactant position to just before the product. However, it closely resembles the IRC throughout the entire region between n1 and n2, i.e. in the TS region; this mode has been identified as n by us which is characterized through its force constant, kn(n). In Fig. 3a is shown the variation of the force constant kn(n) for the reaction coordinate along the IRC. Note the striking similarity to the j(n) plot in Fig. 1c: kn(n) is positive with a maximum in the reactant region, zero very near to n1, then negative and passing through a minimum at the TS structure, zero again by n2 and then positive with a maximum in the product region. This will be discussed in the next section. In Fig. 3b and c, respectively, can be seen how the force constants of the (3N 7) orthogonal normal modes of vibrational and the average of the force constants of the (3N 6) vibrational

Mode

Frequency

R A0 A0 0 A0 A0 A0 A0

Coupled bending (SNO and NOH angles) Torsion (out-of-plane H motion) In-plane deformation (SNO bending plus H motion) In-plane deformation (mainly NO stretch) Mainly in-plane H rocking OH stretching

506 737 925 1057 1405 3445

R* A0 A0 A0 A0 0 A0 A0

Reaction coordinate (H motion between O and S) ONS in-plane bending In-plane deformation (NOH backbone) Out-of-plane H motion NO stretching plus H motion in molecular plane H motion perpendicular to the reaction coordinate

774i 721 904 1142 1335 1987

TS A0 (a) A0 (b) A0 (c) A0 0 (d) A0 (e) A0 (f)

Reaction coordinate (H motion between O and S) In-plane deformation (SNO bending plus H motion) In-plane deformation (ONS backbone) Out-of-plane H motion NO stretching plus H motion parallel to NO bond H motion perpendicular to the reaction coordinate

1736i 704 805 1017 1330 1729

P* A0 A0 A0 0 A0 A0 A0

Reaction coordinate (H motion between O and S) In-plane deformation (SNO bending plus H motion) Out-of-plane H motion In-plane deformation (ONS backbone) NO stretching plus H motion in molecular plane H motion perpendicular to the reaction coordinate

406i 757 786 826 1431 1902

P A0 A0 0 A0 A0 A0 A0

Primarily NS stretching Torsion (out-of-plane H motion) In-plane deformation (ONS backbone), in-plane H rock NSH bending NO stretching SH stretching

300 411 513 852 1741 2647

P3N6 k1 i¼1 modes, hki ¼ 3N6 , changes along the IRC. This shows very clearly the significance of the points n1 and n2, at which hki has sharp and deep minima very near to those points. Fig. 3b shows the force constants for (3N 7) vibrational modes perpendicular to the IRC path. Notice that the retarding force related to the reactant region (nR ! nR ) is considerably coupled with the third, fourth, and sixth vibrational modes due to their large force constants; the fourth mode is particularly important because its force constant is increasing until the activated reactant R* is reached. These normal modes are essentially a combination of deformations in the molecular plane and OH stretching (see Table 3 for the assignment). In agreement with the coupling between the reaction path and the normal modes of vibration orthogonal to it, we can consider that these normal modes are also involved in the kinetic control of the proton transfer indicated in Eq (1), whereas the forward force associated to the product region (nP ! nP ) is mainly coupled with the N@O and S–H stretchings (see Table 3 for the assignment). These results could be interpreted in the sense that the thermodynamic control is mainly exerted by the NO stretching as it was discussed before on the bases of BDEs data. 4. Discussion and conclusions The profiles of kn(n) in Fig. 3a and j(n) in Fig. 1c are very similar, but they do differ, particularly in the relative heights of the two maxima. What is significant is that kn(n) changes sign at the same points on the IRC as does j(n), namely n1 and n2, the positions of the


a

minimum and maximum of F(n), which divide a reaction into separate regions, as discussed earlier. A transition state is characterized by the energy gradient being zero with respect to all degrees of freedom but the force constant for one of these being negative. Thus the fact that both j(n) and kn(n) are negative not only at the TS but throughout the entire region between n1 and n2 provides support for identifying this as the transition region, the focus being on the whole region rather than just the TS. In this region, j(n) and kn(n) have very much the same profiles, and both have minima at the TS, which is due to the fact that in the TS region the normal mode identified as n is a reactive one whereas the differences between j(n) and kn(n) in the reactant and product regions can be interpreted as evidence for the mixing of the (3N 7)- and n normal modes. It is further interesting to recognize that the potential energy V(n) is composed of nuclear and electronic contributions, V(n) = P VN(n) + Ve(n); where VN is the nuclear potential ðV N ¼ A>B ZRAABZB Þ,

ξ1 ξ2

3

2

kζ

1

0

-1

-2

-3

b

-2

-1

0

1

2

3

4

5

30

25

20

k2 (A") k3 (A') k4 (A') k5 (A') k6 (A')

k (3N-7)

15

10

5

0

-3

-2

-1

0

139

1

2

3

4

5

c 5.5

5.0

and the electronic term is given by: Ve = Te + Vee + VNe; Te being the kinetic energy of electrons, Vee the electron–electron interaction, and VNe the electron–nuclei attraction energy. Thus the reaction force constant j(n), being the second derivative of V(n), can similarly be written as j(n) = jN(n) + je(n). At n1 and n2, j(n) = 0, meaning that jN = je. This is consistent with the dominant term being a positive nuclear jN in the reactant and product regions, which emphasizes structural changes, and a negative electronic je in the transition region, in which electronic factors play a major role. Finally, we stress the information that can be extracted from the mean force constant profile displayed in Fig. 3c. It is well-known that k is a property that carries information about the chemical bonding, rigidity, and electronic population of the chemical bonds as well as the specific interactions stabilizing the molecular system. Notice from Fig. 3c that hki is higher in the product region than in the reactant region which helps to explain the thermodynamic driving force of the intramolecular proton transfer reaction. Otherwise stated, the product is mainly stabilized by stronger through-bond interactions than the reactant. The identification of these kind of specific interactions allows to rationalize the amounts of work listed in Table 2, especially those involved in the formation of the activated reactant and product, R* and P*, respectively. The information contained in hki suggests us that it is a complementary property to the electronic properties profiles [6,8,10] which have already shown to be very useful in the identification of specific interactions controlling the activation and relaxation processes in a chemical reaction.

4.5

Acknowledgements 4.0

We greatly appreciate discussions with Dr. Jane S. Murray, Dr. Soledad Gutiérrez-Oliva, Dr. Bárbara Herrera, and Dr. Jorge M. Seminario. The authors also appreciate the constructive comment of one referee. P.J. and A.T.-L are grateful for financial support from FONDECYT project #1060590 and by FONDAP project # 11980002 (CIMAT). P.G. thanks the Fund for Scientific Research Flanders (FWO) and the VUB for continuous support of his research group. P.J. acknowledges also to the Chile-Flanders Bilateral Agreement for a postdoctoral stay at the ALGC (VUB) group.

3.5

3.0

2.5

2.0 -3

-2

-1

0

1

2

3

4

5

IRC

References Fig. 3. (a) Force constant for reaction step (mdyne/Å) and (b) force constants for the (3N 7) normal modes of vibration orthogonal to the reaction path, and (c) average of projected force constants of the intramolecular proton transfer HON@S ? ONSH along the IRC. The points n1 and n2 correspond to the minimum and maximum of the force, the vertical lines divide the process into the reactant, transition and product regions.

[1] [2] [3] [4] [5]

M. Born, J.R. Oppenheimer, Ann. Physik. 84 (1927) 457. K. Fukui, Acc. Chem. Res. 14 (1981) 363. C. González, H.B. Schlegel, J. Phys. Chem. 94 (1990) 5523. S. Kato, K. Fukui, J. Am. Chem. Soc. 98 (1976) 6395. A. Toro-Labbé, J. Phys. Chem. A 103 (1999) 4398.

140


[6] P. Jaque, A. Toro-Labbé, J. Phys. Chem. A 104 (2000) 995. [7] J. Martínez, A. Toro-Labbé, Chem. Phys. Lett. 392 (2004) 132. [8] A. Toro-Labbé, S. Gutiérrez-Oliva, M.C. Concha, J.S. Murray, P. Politzer, J. Chem. Phys. 121 (2004) 4570. [9] P. Politzer, A. Toro-Labbé, S. Gutiérrez-Oliva, B. Herrera, P. Jaque, M.C. Concha, J.S. Murray, J. Chem. Sci. 117 (2005) 467. [10] S. Gutiérrez-Oliva, B. Herrera, A. Toro-Labbé, H. Chermette, J. Phys. Chem. A 109 (2005) 1748. [11] P. Politzer, J.V. Burda, M.C. Concha, P. Lane, J.S. Murray, J. Phys. Chem. A 110 (2006) 756. [12] E. Rincón, P. Jaque, A. Toro-Labbé, J. Phys. Chem. A 110 (2006) 9478. [13] B. Herrera, A. Toro-Labbé, J. Phys. Chem. A 111 (2007) 5921. [14] J.V. Burda, A. Toro-Labbé, S. Gutiérrez-Oliva, J.S. Murray, P. Politzer, J. Phys. Chem. A 111 (2007) 2455. [15] A. Toro-Labbé, S. Gutiérrez-Oliva, J.S. Murray, P. Politzer, Mol. Phys. 105 (2007) 2619. [16] A.D. Becke, Phys. Rev. A 38 (1988) 3098.

[17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27] [28]

A.D. Becke, J. Chem. Phys. 98 (1993) 5648. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. R. Krishman, J.S. Binkley, R. Seeger, J.A. Pople, J. Chem. Phys. 72 (1980) 650. N. González-Rivas, A. Cedillo, J. Chem. Sci. 117 (2005) 555. W.H. Miller, N.C. Handy, J.E. Adams, J. Chem. Phys. 72 (1980) 99. G. Natanson, B.C. Garrett, T.N. Truong, T. Joseph, D.G. Truhlar, J. Chem. Phys. 94 (1991) 7875. C.F. Jackels, Z. Gu, D.G. Truhlar, J. Chem. Phys. 102 (1995) 3188. M. Frisch et al., GAUSSIAN 03, Revision C.02, Gaussian, Inc., Wallingford, CT, 2004. J.A. Kerr, in: D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics 1999– 2000: A Ready-Reference Book of Chemical and Physical Data (CRC Handbook of Chemistry and Physics), 81st edn., CRC Press, Boca Raton, FL, USA, 2000. B. Ruscic, D. Feller, D.A. Dixon, K.A. Peterson, L.B. Harding, R.L. Asher, A.F. Wagner, J. Phys. Chem. A 105 (2001) 1. K.M. Ervin, V.F. DeTuri, J. Phys. Chem. A 106 (2002) 9947. J.-Y. Wu, J.-Y. Liu, Z.-S. Li, X.-R. Huang, C.-C. Sun, Phys. Chem. Chem. Phys. 4 (2002) 2927.