Computational Quantum Chemistry - Universidad AutÃ³noma

Research: Science and Education

Computational Quantum Chemistry: A Reliable Tool in the Understanding of Gas-Phase Reactions Annia Galano* and J. Raúl Alvarez-Idaboy Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 007730, México D.F., México; *[email protected] Annik Vivier-Bunge Universidad Autónoma Metropolitana, Iztapalapa, 09340, México D.F., México

The rapid development of computers in the last few decades has remarkably increased the potential and reliability of computational quantum chemistry as a useful and accessible tool in the understanding of chemical reactions. Over time, the application of quantum chemical methods has moved from academic questions of theoretical interest towards real-world applications used to investigate molecular structure and properties, kinetics, and reactivity. Unfortunately, for some students computational quantum chemistry remains an abstract matter that might dissuade them from becoming researchers in this fascinating field. The dilemma between accuracy and computing time remains a significant issue in computational quantum chemistry, as most of the chemical problems of interest involve a large number of atoms. Consequently, it is necessary to compromise and to establish a methodology for performing reliable calculations in a reasonable period of time. Two important choices must be made to use computational quantum chemistry for the calculation of rate constants. The first choice involves the selection of an appropriate level of theory to determine the electronic structure of reactants, transition states, and products. It is usually made based on previous experience with similar systems or by testing several methods and comparing the calculated results with available experimental data. The second choice concerns the kinetic method for the determination of rate constants, which depends on the characteristics of the particular potential energy surface. A large quantity of information is available in the literature describing computational kinetics methods (1–4), and will not be discussed here in detail. A free access virtual laboratory (5), housed on a computer server at the University of Utah, allows researchers to study, among other things, the kinetics and the thermodynamic properties of gas-phase reactions. In this article we propose to set aside the intricate mathematics and physics involved in computational quantum chemistry and to focus on its reliability as a tool for studying the kinetics of gas-phase reactions that are relevant to tropospheric chemistry. Technical explanations and formulations will be avoided, when possible, to make this work accessible to everyone regardless of his or her field of expertise. However, this article is mainly aimed at senior-level undergraduate and graduate students. We use some specific examples studied in our laboratory, to illustrate the reliability and the predictive character of this field of chemistry. Theoretical Basis The rate of a reaction, υ, may be defined either in terms of the rate of decrease in the concentration of one of the reactants or of the rate of increase in the concentration of one www.JCE.DivCHED.org

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of the products. Therefore, for the following hypothetical chemical reaction,

aA + b B + …

… + yY + zZ

and assuming that the volume does not change during the course of the reaction, a general definition of υ can be given as

υ = −

1 d[ B] 1 d[ A ] = − = ... b dt a dt

= ... =

1 d[ Y ] 1 d [Z ] = y dt z dt

(1)

In many cases υ can also be written as υ = k [A ]

α

[B]β ...

(2)

where k, α, and β are independent of concentration and time. The parameters α and β, which need not be whole numbers, are the partial orders of reaction with respect to compounds A and B, respectively. The rate constant k is unique for each chemical reaction. Several expressions relating rate constants and temperature have been proposed. For a vast number of reactions it is empirically found that −B T

(3)

k = Ae

where A and B are constants. This relationship was originally proposed by van’t Hoff (6) and Arrhenius (7) in the form k = Ae

−

Ea RT

(4)

where R is the gas constant and A and Ea are known as the pre-exponential factor and the activation energy, respectively. It is known as the Arrhenius equation. The range of applicability of the Arrhenius equation is amazingly wide. It may be used not only for elementary chemical reactions but also in completely different processes, such as the creeping of ants, the brain wave rhythm, and the rate of aging (8). A theoretical approach to reaction rates was published almost simultaneously by Eyring (9) and by Evans and Polanyi (10). It is known as the conventional transition-state theory (TST) and is the basis on which more complicated and accurate theories are built. The TST rate equation for a bimolecular reaction can be written as

Vol. 83 No. 3 March 2006

‡

k =

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q‡ k BT − E e RT q q h R1 R2

Journal of Chemical Education

(5)

481


where kB and h are the Boltzmann and Planck constants, respectively . The quantities q‡, qR1, and qR2 are the molecular partition functions, per unit volume, of the transition state and the reactants 1 and 2, respectively, and E‡ is the difference between the energy of the transition state and the sum of the energies of the reactants. Both the partition functions and the energy E‡ can be obtained from quantum mechanical calculations. It should be noticed that Ea and E‡ represent different quantities: E‡ is the energy difference between reactants and transition state at 0 K, while Ea accounts for the temperature dependence of k and can be obtained from the slope of the Arrhenius plots. The electronic structure calculations may be performed using a commercial system of programs such as Gaussian 98 (11). For a given molecular system, the program yields the optimum structure, its total electronic energy, its vibrational frequencies, and its partition functions, as well as many other properties that are not relevant to computational kinetics. It is recommended that some corrections be included in eq 5 in order to improve the k values. These will be described next. According to the wave–particle duality of matter, as first postulated by de Broglie (12), a particle of mass m and velocity v is characterized by a wavelength λ = h兾mv. As long as λ is very small compared with the dimensions of interest, the motion of a particle can be describe with enough accuracy by classical mechanics. However, on a molecular scale, m and the relevant dimensions are so small that λ may become of the same order of magnitude as them, and considerable deviations from the classical behavior are to be expected. For example, according to the classical treatment all reacting molecules must overcome the energy barrier to reach the products state, while in quantum mechanics a finite probability exists that molecules whose energy is smaller than the energy barrier will succeed in reaching the final state. This quantum penetration is known as the tunneling effect and can be critical to predict an accurate rate constant. Since tunneling is a quantum effect, its probability of occurrence is greater the less classical the behavior of the particle is, and its magnitude depends on the features of each particular reaction. The lighter the particle and the thinner the barrier, the larger the extent of the quantum tunneling effect. On the other hand, the higher the barrier (E‡), the larger the number of particles with energy less than E‡, that is, the larger the number of particles susceptible to pass trough the barrier. Within the frame of TST, the tunnel effect is introduced in the rate constant as a multiplicative factor, κ,

where x − x0 β

(8)

∑ E react

(9)

Y = e V0 =

A and B are independent parameters, Ereact is the energy of the reactants, x0 determines the location of the maximum of V(x) along the x axis and b is a range parameter. Ereact includes the zero-point energy correction (ZPE), which accounts for the vibrational energy at 0 K. In addition, a symmetry number, σ, must be introduced to account for different, but equivalent, reaction paths. It is obtained by imaging all identical atoms and by counting the number of equivalent arrangements that can be made by rotating, but not reflecting, the molecule. Including the symmetry number in eq 7, the final expression for the rate constant is ‡

k = κσ

q‡ kBT − E e RT h qR1 qR2

To conclude this section we would like to discuss the topic of internal rotations. This term refers to the torsional motion involving a group of atoms that rotates around a bond. Such a rotor can be treated in three ways, depending upon its barrier to rotation. If the barrier to rotation is much less than kBT, then the rotor may be considered freely rotating. For a symmetric rotor such as a methyl group, the partition function, Qfree-rot, is given by Q free-rot =

( 8 π 3 I ⬘ kB T ) σint h

q‡ kBT − E e RT h qR1 qR2

V (x ) =

482

AY BY + + V0 1+ Y (1 + Y )2


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−

(6)

The parameter κ depends on the temperature, the shape of the potential surface (in particular the height and width of the barrier), and the mass of the particle being transferred during the reaction: the lighter the particle and the thinner the barrier, the greater the probability of tunneling (see ref 13 for more details). The most popular way of calculating the transmission coefficient is by using the asymmetric Eckart potential barrier (14) (7)

1

2

(11)

where σint is the internal symmetry number, which is equal to the number of minima (or maxima) in the torsional potential energy curve, and I´ is the reduced moment of inertia for the internal rotation. The second case is the most common case, when the torsional barrier (V ) is comparable to kBT. The computing of the partition function is much more complex in this case and it can be done in different ways (15–18). In the third approach, for barriers to internal rotations much greater than kBT, the torsion can be considered as a non-rotating, harmonic oscillator and the partition functions can be calculated as for any vibration,

‡

k = κ

(10)

Q vib = 1 − e

kν k BT

(12)

where ν represent the vibrational frequency of the corresponding normal mode. The correct calculation of internal rotations is critical to obtain reliable values of the rate constant. This is especially important for internal rotations in the transition state, for which one (or more) rotation axis is a new bond that does not exist in the reactants. Let us use the methanol and OH reaction to illustrate this point. In Figure 1, the rotation labeled as 1 appears in both the reactant and the transition state. Consequently, it does not matter whether this motion is treated as an internal rotation or as a harmonic vibration


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Figure 1. Internal rotations in methanol, and in methanol–OH transition state.

since the corresponding partition functions cancel each other in eq 10. On the contrary, proper modeling of rotation 2 is essential for the calculation of the rate constant. Examples Computational quantum chemistry may be used to find an explanation to a variety of questions associated with chemical reactions. The excellent agreement that can be achieved between experimental and theoretical rate constants (k) is shown in Figure 2. However, it must be pointed out that the accuracy of a theoretical rate constant is strongly dependent on the level of theory used to model the reaction. The accuracy of the electronic energies is especially relevant, since the energy differences used in eq 10 affect the value of k exponentially. Other aspects that influence the precision of calculated rate constants are the following: (i) the method used to calculate the vibrational frequencies, (ii) whether the tunneling correction is included or not, (iii) the treatment of internal rotations and (iv) the level of the kinetic theory used in the calculation of the rate constant. In the above correlation, the following reactions were included: α-dicarbonyls– OH (19), alkanes–NO 3 (20), alkanes–OH (21), alcohols–OH (22), aldehydes–OH (23), aldehydes–NO3 (24), ketones–OH (25), dimethylether–OH (26), formic acid–OH (27), haloalkanes–OH (26), sulfur compounds– OH (26). All of them are gas-phase reactions relevant to tropospheric chemistry. We shall discuss four of them, to briefly illustrate how computational kinetics can help in the understanding of chemical phenomena.

Figure 2. (top) Correlation of calculated vs experimental rate constants for several OH gas-phase reactions. (bottom) Correlation of calculated vs experimental rate constants for several NO3 gas-phase reactions.

Formic Acid–OH Reaction Formic acid (HCOOH) is the most abundant carboxylic acid in the troposphere (28). It is well established that it reacts with OH radicals by means of a hydrogen abstraction mechanism that leads to the formation of water and the corresponding radical. Two different paths are possible (Figure 3): abstraction of the H atom bonded to the O atom:

HCOOH + •OH

HCOO• + H2O

(I)

or abstraction of the H atom bonded to the C atom: HCOOH + •OH

•COOH

+ H 2O

(II)

Despite the fact that the C⫺H bond strength is weaker than the O⫺H bond strength by as much as 14 kcal兾mol, www.JCE.DivCHED.org

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Figure 3. Reaction paths of the formic acid–OH reaction.


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483

Research: Science and Education Table 1. Tunneling Factor (κ), Rate Coefficients (k), and Formyl Branching Ratio (ΓCH) at 298.15 K Parameter

Calc

Exp

κOH

14251.8

---

kCH

52.7

---

kOH /(108 L mol᎑1 s᎑1)

1.98

---

kCH/(108 L mol᎑1 s᎑1)

0.191

kTot = (kOH + kCH)/ (108 L mol᎑1 s᎑1)

2.17

ΓCH = kCH/kTot a

Ref 29.

b

Ref 30.

--(2.78 ± (2.97 ± (2.69 ± (2.23 ± (2.71 ±

0.08–0.15c

0.08 c

Ref 31.

d

Ref 33.

0.47)a 0.17)b 0.17)c 0.24)d 1.12)e

e

Ref 34.

experimental results (29–31) have definitely proved that abstraction of the hydrogen atom from the acidic group (Path I) is the main channel. Singleton et al. (31) have shown that substitution of the acidic H by a D atom has a dramatic effect on the reactivity, while a similar substitution at the formyl group does not influence the reaction rate. Thus, we performed electronic calculations for the reactants, the transition states, and the products along both channels, using the PMP2/6-311++G(2d,2p)//MP2/6311++G(2d,2p) level of theory. TST was used for the kinetic study. Results are shown in Table 1. In this table, subscripts OH and CH refer to paths I and II, respectively, indicating the abstraction site. As explained above, the extent of tunneling depends directly on the height of the barrier and inversely on its width. A very illustrative software is available in this Journal (32) that helps visualize the dependence of tunneling on the dimensions of the barrier. In our study of the formic acid–OH reaction, we found that the energy barrier for the acidic channel (O⫺H) is higher and narrower than the one for the formyl channel (C⫺H). These features of the potential energy surface lead to a huge tunneling effect (κOH = 14251.8) and to an acidic rate constant that is larger than the formyl rate constant. Thus, although the calculated energy barrier of the acidic path is larger than the one for the formyl path, abstraction from OH is favored. The overall calculated rate coefficient is in excellent agreement with experimental results.

Aldehydes–NO3 Reaction The reaction of an aldehyde with an NO3 radical occurs mainly by H abstraction from the aldehydic site and corresponds to the formulation, CnH2nO + [NO3]•

HNO3 + [CnH2n − 1O]•

The purpose of the investigation was to explain the experimentally observed fact that the rate constants increase “abnormally” fast as the size of the aldehydes increases (35). Based on the changes in the dissociation energies, a smoother increase of the rate constant is expected as the aldehydes become larger. As a consequence of this behavior, no correlation is observed between the rate coefficients of the aldehydes hydrogen abstraction by OH and by NO3. 484


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In this case, the explanation requires careful consideration of the internal rotations partition functions. The crucial internal rotation is the one around the ⫺C⫺H⫺O⫺ axis (Figure 4). Let us name the two rotating fragments as top A, for the aldehyde moiety, and top B, for the NO3 group. The internal rotation partition function depends upon the square root of reduced moment of inertia (I´, eq 11),

I⬘ =

IA IB IA + I B

(13)

which is always smaller than the intrinsic moment of inertia of any of the tops. For two tops with similar masses I´ ≅ IA兾2 ≅ IB兾2, while as the difference among their masses increases the I´ value approaches the smallest value between IA and IB. In the formaldehyde–NO3 transition state the NO3 is the heaviest top, but as the aldehyde size increases, this group becomes the heaviest top (Figure 4). Thus, the partition function of the internal rotation varies significantly from one aldehyde to another (Table 2) As the size of the aldehydes increases, IA becomes much larger than IB and, in the limit, I´ tends to be equal to the moment of inertia of the lighter top (NO3) (Table 2). Accordingly, if the increase of the rate constant in the aldehydic series arises from the increase of the reduced moment of inertia, a plot of the k values versus the size of the aldehyde should present an asymptotic tendency. Indeed this behavior is observed in Figure 5, where the experimental rate coefficients (34, 36, 37) for C1 to C6 aldehydes have been plotted versus the number of carbon atoms. To avoid mixing different factors in the discussion, only straight chain aldehydes

Figure 4. Main internal rotation in some aldehyde–NO3 transition states.

Table 2. Variation of the Moments of Inertia and Internal Rotation Partition Functions with the Size of the Aldehydes Number of C Atoms

IA

IB

I´

Qfree-rot

1

016.7

115.3

14.6

23.8

2

047.9

111.8

33.5

36.0

3

074.2

111.4

44.5

41.5

4

170.5

111.1

67.3

51.0

5

266.1

111.1

78.4

55.0


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have been included in this figure. Our prediction was recently confirmed experimentally by Noda et al. (38), by measuring the rate coefficients for C6–C10 aldehydes. Thus the atypical behavior of the hydrogen abstraction reactions by the NO3 radical in aldehydes was found to be a consequence of the increase of entropy and hence of the partition functions in the transition states due to the internal rotations. The aldehyde–OH reaction does not have the same behavior because the OH moiety is always the lighter top, and therefore the

Figure 5. Plot of logarithm of experimental rate constant vs number of carbon atoms in the n-aldehyde–NO3 reactions: C1 (34), C2 (36), and C3–6 (37).

reduced moment of inertia of an internal rotation involving the OH group and any of the aldehydes is very similar to the OH intrinsic moment of inertia.

Alcohols–OH Reaction Previous to the work discussed in this section, there were open questions concerning the reactivity of alcohols with OH radicals. Oh and Andino (39) had reported that the rate constants (k) of the OH reaction with methanol, ethanol, and 1-propanol increased about 23–32% under atmospheric conditions in the presence of polar aerosols, while k of C4–C6 aliphatic alcohols remained almost unchanged. This finding suggested a difference in the mechanism depending on the size of the alcohols. In our theoretical work, C1 to C4 alcohols were studied and all possible reaction paths were modeled over the temperature range 290 to 350 K. According to our results, hydrogen abstraction mainly occurs from the alpha site of C1–C3 alcohols, and from the gamma site of 1-butanol. The larger kγ in the C4 alcohol can be explained by the geometry of the corresponding transition structure (Figure 6). It can be seen that in the alpha (TS α), beta (TS β) and gamma (TS γ) transition states the hydrogen bond-like stabilizations become increasingly strong, with the corresponding interaction distances being equal to 2.98, 2.17, and 1.93 Å, respectively. The distances have been explicitly included in Figures 6 and 7 since the perspective in the drawing might be misleading. The much shorter distance in TS γ causes a larger stabilization, implying a lower energy barrier and a larger rate coefficient for this path. As 1-butanol is the smallest alcohol presenting a secondary gamma carbon, a similar behavior may be expected for larger members of the series. These results explain the behavior described in ref 39. In addition, the results led us to propose new factors for the structure–reactivity relationship (SAR) of Kwok and Atkinson (40). Previously, Wallington and Kurylo (41) had shown that, in ketones, a significant enhancement of group reactivity towards an OH radical is observed when moving from the α to the β positions, and this had been taken into account in the SAR method. As can be seen in Figure 7, β sites in ketones and γ sites in alcohols are very similar, and similar arguments may be applied. α-Dicarbonyls–OH Reaction

Figure 6. Transition state structures for the OH hydrogen abstraction reaction from n-C4H9OH.

Figure 7. BHandHLYP/6-311G(d,p) fully optimized geometries of 1-butanol and 2-pentanone .

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Despite the important role of simple α-dicarbonyl compounds as key intermediate products in the atmospheric oxidation of several hydrocarbons, few studies concerning their subsequent fate have been performed. According to the chemical kinetics database supported by the National Institute of Standards and Technology (NIST) (42), only one experimental value of the rate constant for the OH–glyoxal reaction at 298 K has been reported (43), and no data are available in the literature for its temperature dependence. Consequently, no activation energy has been reported for this reaction. Arrhenius parameters are available, however, for the reaction of methylglyoxal with OH radicals. Thus, in our work, both reactions were studied, and methylglyoxal was used to validate the methodology. Arrhenius parameters for the glyoxal–OH reaction are reported for the first time.


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485


Figure 8. Computed energy profiles of the OH reactions with glyoxal and methylglyoxal.

Figure 9. Schematic trajectories, showing alternative ways of crossing the dividing surface at the col separating the reactants valley from the products valley.

Table 3. Arrhenius Parameters: Pre-exponential Factor (A) and Activation Energy (Ea) A/(10᎑13 cm molecule᎑1 s᎑1)a

Ea/ (kcal/mol)

Glyoxal–OH (CVT/SCT)

9.31

᎑1.04

Methylglyoxal–OH (CVT/SCT)

3.70

᎑2.13

8.4 ± 1.2

᎑1.65 ± 0.6

System

3

Methylglyoxal–OH (Exp)b a

Temperature range is 260–330 K .

486

b

Ref 58.


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According to the reaction profiles of the modeled paths for the OH reaction with glyoxal and methylglyoxal (Figure 8), barriers are very low. Thus, conventional TST is not appropriate for the kinetics calculations in this case, and the canonical variational theory (CVT) (3, 44–49) should be used (2, 9). Let us discuss briefly the differences between TST and CVT theories. The TST theory assumes that all molecular systems that have surmounted the col (barrier) in the direction of the products can not turn back and form reactant molecules again. It also assumes that the concentration of the transition-state structures can be calculated using the equilibrium theory. However, as shown in Figure 9, there are several ways in which a system may cross a potential energy surface from reactants to products. Trajectories 1 and 4 involve a direct crossing of the barrier, with no turning back, while all the other trajectories cross the dividing surface more than once. If all the possible trajectories were similar to 1 and 4, there would be no error in the quasi-equilibrium hypothesis. In Figure 9, points a to f indicate six crossings from left to right. TST theory counts them all as contributing to the reaction, although only trajectories 1 and 3 represent a successful conversion of reactants into products. Thus, as a result of the recrossing effect, TST leads to rate constants that are too high. This overestimation gets worse as the barrier lowers and the temperature increases. CVT (3, 46, 48–52) is an extension of TST that minimizes the errors owing to recrossing trajectories (1, 3, 53) by effectively moving the dividing surface along the reaction path so as to minimize the rate. It has been shown by Garret and Truhlar (54–56) that CVT is equivalent to locating the transition state at the position of maximum Gibbs energy rather than at the maximum potential energy. Comparison of our results for the methylglyoxal–OH reaction with the experiment show that the level of theory that best describes the kinetics of the system is CVT with small curvature tunneling (SCT) corrections and CCSD(T)// BHandHLYP electronic calculations. The results obtained at this level show an excellent agreement with the available experimental data (Table 3). Thus one can expect that the values obtained for the glyoxal–OH reaction (Table 3) using the same method of calculation should be equally reliable. Conclusions Computational quantum chemistry provides adequate tools to perform kinetic studies of gas-phase reactions with an accuracy similar to the one in experimental determinations, provided that the proper methodology is chosen. The predictive character of computational quantum chemistry is demonstrated in four examples of atmospheric chemical reactions in which apparently contradictory phenomena have been explained. Hence, this field of chemistry is a very useful tool in the understanding of chemical reactions. An emphasis in real-world applications of computational quantum chemistry could help in demystifying this field of science to spread its applicability. Acknowledgments The authors gratefully acknowledge the financial support from the Instituto Mexicano del Petróleo (IMP) through program D00179. We also thank the IMP Computing Cen-


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