Tunneling Isomerization of Small Carboxylic Acids and Their

Article pubs.acs.org/JPCA

Tunneling Isomerization of Small Carboxylic Acids and Their Complexes in Solid Matrixes: A Computational Insight Masashi Tsuge*,†,‡ and Leonid Khriachtchev*,† †

Department of Chemistry, University of Helsinki, P.O. Box 55, Helsinki FI-00014, Finland Department of Applied Chemistry, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan

‡

S Supporting Information *

ABSTRACT: We have studied hydrogen-atom tunneling in the cis-to-trans conformational change of some carboxylic acid monomers and formic acid (FA) complexes and dimers at the MP2(full) and CCSD(T) levels of theory within the Wentzel−Kramers−Brillouin approximation. The barrier for the minimum energy path, where the OH bond length and the COH bending angle are optimized, is found to be a good approximation for providing the highest barrier transparency. The matrix effect on the transmission coefficients of cis-FA monomer, trans−cis FA dimer (tc1), and cis-acetic acid monomer are modeled by the polarizable continuum model (PCM) at the MP2(full) level of theory in different environments. For the cis-FA monomer and trans−cis FA dimer (tc1), the calculated transmission coefficients agree with the experimental lifetimes observed in noble-gas solids. However, this method cannot reproduce the experimental results obtained for cis-acetic acid. Moreover, the long lifetime of cis-FA and cis-acetic acid in the N2 environment cannot be reproduced either, which is most probably due to specific interactions that are not included in the PCM. The calculation for cis-HCOOD shows a strong decrease of the barrier transparency compared to that for cisHCOOH, which is consistent with the experiments. In general, good agreement is observed between the calculated barrier transparency (including PCM) and experimental tunneling rate. However, some exceptions are found, which shows that additional factors influence the tunneling rate.

1. INTRODUCTION Conformational isomerism is one of the most important phenomena in chemistry and biology.1−4 Many conformational changes that involve a movement of a hydrogen atom or proton are known to proceed via quantum tunneling.5,6 Even movements of heavier atoms (carbon and oxygen) can be enhanced by quantum tunneling.7−9 The transmission of a particle through a potential barrier depends on the barrier width and height. For example, it has been demonstrated in a tunneling reaction of matrix-isolated methylhydroxycarbene that it preferably forms acetaldehyde through the barrier (28.0 kcal mol−1), which is higher but thinner than the barrier to form vinyl alcohol (22.6 kcal mol−1).10 Formic acid (HCOOH, FA) is the smallest organic acid, and it has two (trans and cis) conformers, the cis conformer being higher in energy than the trans conformer by ca. 1365 cm−1.11 Because of the energy difference, the amount of the higherenergy conformer is very small at normal conditions. In 1997, Pettersson et al. showed that the cis conformer could be prepared in large amounts by vibrational excitation of the trans conformer in low-temperature matrixes.12 The vibrational excitation method has been applied to various carboxylic acids and their complexes to obtain spectroscopic information on the higher-energy conformers as well as to study conformational changes.13 The cis-to-trans conversion of carboxylic acids was observed in the dark in low-temperature matrixes.13 This reaction occurs © XXXX American Chemical Society

via quantum tunneling of the hydrogen atom, which is evidenced by the low-temperature limit of this process and by the huge H/D isotope effect. The cis-to-trans tunneling rate is found to be strongly dependent on the environment (matrix material, matrix temperature, and interaction with other atoms or molecules).13−23 In almost all cases, the intermolecular interaction decreases the rate of the cis-to-trans conversion of cis-FA. Only in the FA···Xe (non-H-bonded) complex, the tunneling rate increases.22 Tunneling reactions have been observed for a number of other species in solid matrixes.24−29 Different lifetimes of cis-FA in various systems have been explained in terms of the tunneling barrier heights.18,19,21,22 Unfortunately, the computational methods in those works have been different, which makes the systematic comparison of these calculations with each other and with the experimental results difficult. Moreover, only the barrier heights have been considered in those works whereas the barrier widths have been assumed unchanged. The second simplification in the previous works concerns the calculation of the tunneling barriers that were obtained by changing torsional angle without changes of the OH bond length and the COH bending angle.18,19 Special Issue: Markku Räsänen Festschrift Received: September 24, 2014 Revised: November 12, 2014

A

dx.doi.org/10.1021/jp509692b | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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computed in the gas phase using the geometry optimized in a medium.36 The geometry optimization and harmonic vibrational analysis at the CCSD(T) level were performed using the Molpro program.37 Other calculations including the single-point energy calculations and the BSSE correction at the CCSD(T) level were done using the Gaussian 09 program.38 The potential energy curves for the cis-to-trans conversion were calculated by scanning the torsional angle (τ) from 180° (the cis form) to 0° (the trans form) typically with steps of 10°. In this procedure, the coordinates of all atoms excluding the tunneling H atom were frozen at the minima of the cis conformers, which corresponds to the adiabatic approximation where the heavy atoms do not move during the fast H atom/ proton tunneling.39 The time of tunneling (t) is roughly estimated by the tunneling velocity equation v = (2U/m)1/2 and relation t = l/v, where l, U, and m are the barrier width, barrier height, and mass of the tunneling particle, respectively.40 By use of typical parameters for the cis-to-trans isomerization of FA (U = 2800 cm−1, width l = 2 Å, and m = 1.673 × 10−27 kg), the time of tunneling is estimated to be 25 fs. This time is comparable with the period of the harmonic oscillations in the molecule, and probably, the change of the molecular structure cannot occur during this time. However, this model requires further theoretical justification and it is accepted as a working hypothesis. The transmission coefficients of the tunneling H atom through the barrier were estimated using the WKB approximation.30 In the WKB approximation, the transmission coefficient is given as

Intuitively, it is possible that some other tunneling paths can lead to a higher tunneling probability. Finally, the treatment of solvation and zero-point vibrational energy was insufficient in the previous studies of H-atom tunneling in carboxylic acids. In the present work, these questions are studied more accurately. We consider different tunneling paths and solvation effects and estimate the transmission coefficients of the tunneling barriers using the Wentzel−Kramers−Brillouin (WKB) approximation30 to compare these calculations with the available experimental results. We consider the following systems (Figure 1a−j): the

Figure 1. Structures of the systems studied in the present work: (a) cis-FA, (b) cis-propionic acid (PnA), (c) cis-propiolic acid (PlA), (d) cis-acetic acid (AA), (e) trans−cis FA dimer tc1, (f) trans−cis FA dimer tc4, (g) cis-FA···Xe (non H-bonded) complex, (h) cis-FA···Xe (H-bonded) complex, (i) cis-FA···N2 complex, and (j) cis-FA···CO2 complex.

T=

⎛ ⎜1 + ⎝

1 4

x2

x1

(

2m (V (x ) ℏ2

dx

exp −2 ∫

x2

x1

dx

− E)

2m (V (x ) ℏ2

)

− E)

⎞2 ⎟ ⎠

)

(2)

where V(x) is the potential energy, E is the energy of the tunneling particle, and x1 and x2 are the boundaries of the potential barrier (i.e., V(x1) = V(x2) = E). According to Vigneron,41 this approximation is very often used in the literature although it has some limitations but, recently, a good agreement between experimental and computational tunnelling rates for carbenes has been reported.10,42 Five kinds of the cis-to-trans tunneling barriers were examined for the cis-FA monomer: (i) The barrier for “simple torsional path” where the OH bond length ROH and the COH bending angle θ are unchanged (i.e., these values are the same as in the cis-FA monomer). (ii) The barrier for “shorter torsional path I” where ROH is varied as a function of the torsional angle τ and the bending angle θ is fixed at the value of the cis-FA monomer,

cis-FA (HCOOH, DCOOH, and HCOOD), cis-propionic acid, cis-propiolic acid, and cis-acetic acid monomers, the cis-FA···Xe (non H-bonded), cis-FA···Xe (H-bonded), cis-FA···N2, and cisFA···CO2 complexes, and two trans−cis FA dimers (tc1 and tc4 in notations of ref 21), for which the experimental data are available.14−19,21,22,31−33

2. COMPUTATIONAL DETAILS The calculations were performed at the MP2(full) and CCSD(T) levels of theory with the standard split valence basis set 6-311++G(2d,2p) for H, C, and O atoms and the def2-TZVPPD basis set for a Xe atom. After the geometry optimization, the harmonic vibrational analysis was performed to confirm that the obtained structures are true minima with no imaginary frequencies or first-order transition states (TS) with one imaginary frequency. The matrix environments were modeled by using the polarizable continuum model (PCM)34 at the MP2(full)/6-311++G(2d,2p) level of theory. Because PCM is not available for the CCSD(T) level of theory, the matrix effect was taken into account for the CCSD(T) calculations using the MP2 correction: ECCSD(T) + PCM = ECCSD(T) + (EMP2 + PCM − EMP2)

(

exp −2 ∫

R OH(τ ) = R OH(τ =180°) × (1 − A × sin(τ ))

(3)

where A is a constant. (iii) The barrier for “shorter torsional path II” where θ is varied as a function of τ while ROH is fixed at the value of the cis-FA monomer, θ(τ ) = θ(τ =180°) + B × sin(τ )

(1)

(4)

where B is a constant. (iv) The barrier for “bending path” in the molecular plane with ROH unchanged (i.e., only θ is scanned in this case). (v) The barrier for “minimum energy path” where ROH and θ are optimized for the H atom.

The basis set superposition error (BSSE) of the complexes was corrected by the counterpoise method.35 Because the procedure for the BSSE correction is not well-established with PCMs, the BSSE correction for the species in a medium is B



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Table 1. Transmission Coefficients for Five Tunneling Paths of cis-FA Monomer Calculated at the MP2(full)/6-311++G(2d,2p) Level of Theory in Vacuum (i) simple torsional path

(ii) shorter torsional path I (A = 0.01)

(iii) shorter torsional path II (B = 3)

(iv) bending path

(v) minimum energy path

3242 2.858 1.940 6.6 × 10−17

3245 2.840 1.919 1.1 × 10−16

3207 2.826 1.915 1.5 × 10−16

9848 2.468 1.996 6.3 × 10−30

3204 2.855 1.920 1.4 × 10−16

barrier height (cm−1) distance between minima (Å) barrier thickness (Å)a transmission coefficientb a

The tunneling distance corresponding to x2 − x1 in eq 2. bCalculated according to eq 2 using E = ZPVE(cis) − ZPVE(TS) = 380 cm−1.

results as seen later. For the cis-FA···N2, cis-FA···CO2, and cisFA···Xe (H-bonded) complexes where the tunneling H atom is involved in the intermolecular hydrogen bond, the fully optimized TS was not obtained because the structure of the fully optimized TS originating from the cis-FA···X complex should involve large intermolecular motions. The intermolecular relaxation cannot probably occur at the time scale of the tunneling event. That is why we estimate the energy of the H atom as a difference of the ZPVE in the cis-FA···X complex and the sum of the ZPVEs of the FA monomer in the fully optimized TS and of the X monomer. The list of frequencies calculated for all the examined systems are given in Tables S1−S10 (Supporting Information).

In case i, the tunneling H atom travels along the arc with a radius of R′ = ROH × sin θ0 (θ0 is COH bending angle for the cis-FA monomer, which is equal to 108.9° at the MP2(full)/ 6-311++G(2d,2p) level of theory). Thus, the travel distance is shorter for smaller ROH and larger θ. In cases ii and iii, eq 3 and eq 4 are written in a way that R′ becomes smaller during tunneling for positive constants A and B. Constants A and B are used in intervals from 0.005 to 0.05 and from 1 to 4, respectively (with increments of 0.005 and 1). The use of constants A and B outside these intervals do not lead to an increase of the barrier transparency. In all these cases, the potential energy curve is considered as a function of the scanned angles. To obtain the potential energy curve V(x) as a function of the tunneling coordinate x, the scanned angles should be connected with x. In cases i−iii and v, the hydrogen atom tunnels via an arc with radius of R′(τ). The tunneling coordinate x is connected with R′(τ) via an arc length formula written in the polar coordinates as x(τ ) =

∫τ

τ

dτ1

0

⎧ dR′(τ1) ⎫2 ⎬ {R′(τ1)} + ⎨ ⎩ dτ1 ⎭

3. RESULTS 3.1. Tunneling Path Effect. Table 1 presents the tunneling barrier parameters of the cis-FA monomer for the various tunneling paths. The barriers are relatively low for the shorter torsional path II (3207 cm−1 for B = 3) and the minimum energy path (3204 cm−1) without taking the ZPVE into account. For the simple torsional path and the shorter torsional path I (A = 0.01), the barriers are somewhat higher (3242 and 3245 cm−1, respectively). For the bending path, the barrier is very high, as can be expected (9848 cm−1). The path length is the shortest for the shorter torsional path II (1.915 Å) whereas the path lengths are similar for the shorter torsional path I and the minimum energy path (1.919 and 1.920 Å, respectively). For the simple torsional path, the path length is longer (1.940 Å). It is interesting that the path length is even longer for the bending path (1.996 Å), which is not intuitively obvious. The transmission coefficient obtained in vacuum for the simple torsional path of the cis-FA monomer is 6.6 × 10−17 (Table 1) using an H atom energy of 380 cm−1. For the shorter torsional path I, the transmission coefficients are 1.11 × 10−16, 1.13 × 10−16, and 1.07 × 10−16 for A = 0.005, 0.01, and 0.015, respectively, suggesting that A = 0.01 is quite close to the maximum transparency. In the case of the shorter torsional path II, the use of B = 2, 3, and 4 gives slightly larger transmission coefficients of 1.4 × 10−16, 1.5 × 10−16, and 1.4 × 10−16, respectively. The bending path has an extremely small transmission coefficient (6.3 × 10−30), and it should not contribute to the isomerization. We also tried to find the nonadiabatic TS of the bending path but it was a second-order TS (vibrational analysis gave two imaginary frequencies). The minimum energy path leads to a transmission coefficient of 1.4 × 10−16, which is close to the highest value obtained for the shorter torsional path II (B = 3). Therefore, the minimum energy path can be considered as a good approximation for the cis-to-trans tunneling path with maximum transparency and we use this path for more complicated systems. In fact, the obtained transmission coefficient of 1.4 × 10−16 is reasonable for the cis-FA monomer. The tunneling rate is a

2

(5)

where τ0 = 180° and R′(τ) is described by the low of cosines as {R OH(τ )}2 + {R 0}2 − 2·R OH(τ ) ·R 0·cos θ(τ )

R′(τ ) =

(6)

where R0 = ROH(τ0) × cos θ0. The integral in eq 5 is evaluated numerically. The derivative at a given point is calculated by averaging the slopes between this and two neighboring points. In case iv, the hydrogen atom tunnels via an arc with a fixed radius of ROH, and the tunneling coordinate is connected with the COH bending angle as x(θ ) = 2πR OH ×

θ − θ0 360°

(7)

where θ0 corresponds to cis-FA. Then, the discrete data for V(x) were fitted by the function V (x ) =

1 2

4

∑ Vn(1 − cos(nπx/L)) n=1

(8)

where L is the travel distance of the hydrogen from cis-FA to trans-FA. This fitted curve was used to calculate the transmission coefficient by eq 2. The energy of the tunneling H atom was estimated as the zero-point vibrational energy (ZPVE) difference between the cis and TS configurations, which were fully optimized. Indeed, the TS for tunneling (adiabatic TS) is different from the fully optimized TS because all atoms excluding the tunneling particle are assumed to be at the same positions as in the cis form. This means that the used energy of the tunneling H atom is an approximation; however, this approach leads to reasonable C



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mental results (Figure 3). The calculated transmission coefficient in an N2 environment is between Ne and Ar environments as expected from the dielectric constants, which strongly disagrees with the experimental data (Table 2). For the trans−cis FA dimer (tc1), the transmission coefficients in Ne (1.01 × 10−18), Ar (5.69 × 10−19), and Kr (4.48 × 10−19) are in agreement with the experimental decay time. For the cis-acetic acid monomer, the transmission coefficients are 3.60 × 10−16 (Ar), 2.28 × 10−16 (Kr), 1.24 × 10−16 (Xe), and 7.94 × 10−16 (N2), which cannot reproduce the experimental results even for the noble-gas matrixes (Table 2). Thus, a qualitative agreement between the calculations and experiments is seen only for cis-FA and the tc1 dimer in noble-gas matrixes. We will discuss these discrepancies later. 3.3. Different Systems. The transmission coefficients for the minimum energy paths of various systems were investigated at the MP2(full) and CCSD(T) levels of theory with the PCM for an Ar matrix. The studied systems include cis-propionic acid, cis-acetic acid, cis-propiolic acid, cis-FA (HCCOH, DCOOH, and HCOOD) monomers, trans−cis FA dimers (tc1 and tc4), and the 1:1 cis-FA···Xe (non-H-bonded and H-bonded), 1:1 cis-FA···N2, and 1:1 cis-FA···CO2 complexes (Figure 1). The computational results at the CCSD(T) and MP2(full) levels of theory are presented in Table 3 and Table S11 (Supporting Information), respectively. In general, the transmission coefficient correlates with the experimentally observed decay time (Figures 4 and S1, Supporting Information). The calculation for cis-HCOOD shows a strong decrease of the transparency of the barrier compared to that for cis-HCOOH, in reasonable agreement with the experiment.15 In contrast, the calculations for cis-DCOOH give the same transparency as for cis-HCOOH whereas a 3-fold decrease of the lifetime for cis-DCOOH was experimentally observed.14

product of the transmission coefficient and the frequency of attempts with accuracy of a term depending on the vibrational energy relaxation. If we take the torsional frequency as the frequency of attempts (2 × 1013 Hz), we obtain a tunneling rate of 3 × 10−3 s−1, i.e., a lifetime of 5 min, which is just slightly shorter than the experimental value of 7 min (measured at the level of 1/e). Similar estimations have been done for H atom tunneling reactions of phenylcarbene and methylhydroxycarbine, in a good agreement with the experimental lifetime in Ar matrixes.10,42 Of course, we do not overestimate the importance of this oversimplified evaluation. The tunneling barrier for the minimum energy paths of the cis-FA monomer and the cis-FA···CO2 complex are shown in Figure 2 for the case of vacuum. The kinetic energies of the

4. DISCUSSION In the previous studies of tunneling in cis-FA, the tunneling barriers were calculated for fixed ROH and θ (simple torsional path).18,19,22 In the present work, ROH and θ are varied, and it appears that the transmission coefficient for the simple torsional path of the FA monomer (6.6 × 10−17) is twice smaller than that for the minimum energy path (T = 1.4 × 10−16) (Table 1). A slightly higher transmission coefficient is obtained for the shorter torsional path II of FA (1.5 × 10−16) where θ is varied as a function of τ (eq 4). Thus, the minimum energy path seems to be a good approximation to estimate transmission coefficients for tunneling in carboxylic acids, and this convenient approach was used for the more complicated systems. Even a higher transmission coefficient can be probably found; however, it is beyond the scope of the current work because our main motivation is to systematically compare the transmission coefficients calculated within the same model and the experimentally determined cis-to-trans tunneling lifetimes for the various systems. The strict theoretical justification of the used model is outside the scope of the present work. Moreover, other unsolved problems exist (see later). Decay of the cis-FA monomer (HCOOH), trans−cis FA dimer (tc1), and cis-acetic acid monomer has been studied experimentally in different matrixes.14−17,32,43 The lifetime of the cis-FA monomer in Ne, Ar, Kr, Xe, and N2 environments are 5, 378, 670, 3700, and 21000 s, respectively. If we exclude the N2 environment, the experimental lifetime correlates with the dielectric constant of matrix media; i.e., 1.24 (Ne), 1.66 (Ar), 1.88 (Kr), and 2.22 (Xe). This effect was explained

Figure 2. Tunneling barrier for the minimum energy path of (a) cis-FA monomer and (b) cis-FA···CO2 complex in vacuum calculated at the MP2(full)/6-311++G(2d,2p) level of theory. The dotted lines show the kinetic energy of the tunnelling H atom estimated as ZPVE(cis) − ZPVE(TS).

tunneling H atom in cis-FA monomer and the cis-FA···CO2 complex are calculated to be 380 and 444 cm−1, respectively, and they are shown in Figure 2 by dotted lines. The increase of the barrier height in the FA···CO2 complex (4106 cm−1) compared to that for the FA monomer (3204 cm−1) mainly originates from the interaction energy of the cis-FA···CO2 complex (−830 cm−1).18 3.2. Matrix Effect. We have studied the matrix effect on the minimum energy path of cis-FA monomer, trans−cis FA dimer (tc1), and cis-acetic acid monomer using the PCM at the MP2(full) level of theory (Table 2). For the cis-FA monomer, the barrier (without taking ZPVE into account) increases from 3204 cm−1 (vacuum) to 3262 cm−1 (Ne), 3340 cm−1 (Ar), 3374 cm−1 (Kr), and 3425 cm−1 (Xe) whereas the changes in the kinetic energies of the tunneling H atom (ΔZPVE) are not significant, from 379 cm−1 (Ne) to 385 cm−1 (Xe). The tunneling paths also become longer for the larger dielectric constants (Table 2). The transmission coefficients for the minimum energy path of cis-FA in these noble-gas environments are 6.50 × 10−17 (Ne), 2.37 × 10−17 (Ar), 1.55 × 10−17 (Kr), and 8.80 × 10−18 (Xe) in qualitative agreement with the experiD



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Table 2. Transmission Coefficients (T) for Minimum Energy Path of cis-FA Monomer, trans−cis FA Dimer (tc1), and cis-Acetic Acid Monomer Calculated in Different Environments with the PCM at the MP2(full)/6-311++G(2d,2p) Level of Theory environment (dielectric constant)

vacuum (1.00)

barrier height (cm−1)a barrier thickness (Å)b ΔZPVE (cm−1) transmission coefficient, T experimental decay time (s)c

3204 1.920 380 1.40 × 10−16

barrier height (cm−1)a barrier thickness (Å)b ΔZPVE (cm−1) transmission coefficient experimental decay time (s)c

3802 1.989 438 1.51 × 10−18

barrier height (cm−1)a barrier thickness (Å)b ΔZPVE (cm−1) transmission coefficient experimental decay time (s)c

2860 1.859 331 3.02 × 10−15

Ne (1.24)

Ar (1.66)

cis-FA Monomer 3262 3340 1.945 1.974 379 379 6.50 × 10−17 2.37 × 10−17 5 378 trans−cis FA Dimer (tc1) 3844 3901 2.006 2.019 446 459 1.01 × 10−18 5.69 × 10−19 3 1.6 × 10 1.7 × 103 cis-Acetic Acid 2991 1.930 310 3.60 × 10−16 45

Kr (1.88)

Xe (2.22)

N2 (1.37)

3374 1.988 380 1.55 × 10−17 670

3425 2.005 385 8.80 × 10−18 3.7 × 103d

3290 1.954 379 4.58 × 10−17 2.1 × 104

3063 1.967 306 1.24 × 10−16 10

2938 1.904 314 7.94 × 10−16 2.9 × 104

3925 2.027 463 4.48 × 10−19 2.2 × 103 3023 1.944 308 2.28 × 10−16 53

Energy difference between τ = 180° (cis conformer) and τ = 90° (adiabatic TS), where ZPVE is not taken into account. bThe tunneling distance corresponding to x2 − x1 in eq 2. cExperimental decay times are from following references: FA(Ne), ref 17; FA(Ar, Kr, Xe), ref 14; FA(N2), ref 16; tc1(Ne, Ar), ref 17; tc1(Kr), previously not reported; acetic acid (Ar, Kr, Xe), ref 32; and acetic acid (N2), ref 16. dAveraged value for decay time for two sites. a

The experimental decay time of the cis-FA and cis-acetic acid monomers in an N2 matrix is much longer than in noble-gas matrixes,16 in contradiction with the PCM calculations. This disagreement is contributed by the dipole−quadrupole interactions between the embedded molecule and surrounding N2 molecules, which are not included in the PCM. Because the dipole moment of the cis forms is bigger than that in the TS, an additional stabilization of the cis forms is expected for the N2 environment. As the first approximation, this effect can be discussed in terms of specific 1:1 interactions with the OH group.16 The strong interaction of an N2 molecule with the OH group in the 1:1 cis-FA···N2 complex studied in an Ar matrix leads to a longer lifetime than for the cis-FA monomer (by a factor of 6.5).19 The importance of specific interaction is also supported by the longer decay time of the cis-FA···N2 complex compared to that of the H-bonded cis-FA···Xe complex studied in an Ar matrix (by a factor of 2). Indeed, for the H-bonded structures, the interaction energy of the cis-FA···N2 complex (−4.8 kJ mol−1) is bigger than that of the cis-FA···Xe complex (−2.2 kJ mol−1).19,22 Simulation of the tunneling processes in solid nitrogen is a challenge for theory. As shown in Figure 4 and Table 3, the calculated transmission coefficients (CCSD(T)) correlate with the experimental decay time for different systems studied in an Ar matrix. For example, both theoretically and experimentally, the tc4 dimer and cis-FA monomer have similar decay times whereas the tc1 dimer is significantly more stable. The inclusion of the PCM(Ar) improves the agreement of the calculated and experimental results, especially between the cis-propionic acid and cis-acetic acid monomers. The ratio of the transmission coefficients T(propionic acid)/T(acetic acid) are 3.3 and 1.1 with and without PCM(Ar), respectively, whereas the ratio of the experimental decay rates is ca. 3.32,33 This improvement is due to the change in ΔZPVE: the ΔZPVE of propionic acid increases with the PCM(Ar) whereas that of acetic acid

Figure 3. Experimental tunneling decay time versus the calculated transmission coefficient for the cis-FA monomer (square), trans−cis FA dimer tc1 (circle), and cis-acetic acid monomer (triangle) in noble-gas (Ne, Ar, Kr, and Ne) and N2 matrixes. The matrixes are modeled by the PCM at the MP2(full) level of theory. The dotted lines are for guiding the eye.

by stronger solvation in more polarizable environments.14 The same qualitative trend is obtained in the present calculations; however, the quantitative agreement is lacking. From Ne to Xe environments, the calculated transmission coefficient changes by a factor of ∼10 whereas the experiment shows a significantly larger difference. In contrast, for FA dimer tc1, the change in the transmission coefficient from Ne to Kr (by a factor of 2.3) is larger than the experimental difference (by a factor of 1.4). The transmission coefficients calculated for cis-acetic acid monomer disagree with the experimental decay time in Ar, Kr, and Xe matrixes (Table 2 and Figure 3). Indeed, a shorter lifetime is observed in Xe (10 s) than in Kr (53 s) and Ar (45 s) matrixes, in contrast to the larger dielectric constant of solid Xe. E



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Table 3. Barrier Height, ΔZPVE, Effective Barrier Height, and Transmission Coefficients (T) for the Minimum Energy Path Calculated at the CCSD(T) Level of Theorya CCSD(T)

cis-propionic acid cis-acetic acid cis-DCOOH cis-FA···Xe (non-H-bonded) cis-FA cis-propiolic acid trans−cis FA dimer (tc4) cis-FA···Xe (H-bonded) trans−cis FA dimer (tc1) cis-FA···N2 cis-FA···CO2 cis-HCOOD

barrier height

ΔZPVEd

effective barrier heighte

2745 2729 3040 2935 3040 3068 3227 3143 3637 3757 3947 3040

350 320 363 355 367 341 457 367g 439 409g 432g 284

2395 2409 2677 2580 2677 2727 2770 2776 3198 3328 3514 2756

CCSD(T) + PCM(Ar) T 8.84 7.70 4.05 8.30 4.35 8.79 3.38 1.14 5.64 3.52 4.38 2.57

× × × × × × × × × × × ×

10−15 10−15 10−16 10−16 10−16 10−17 10−16 10−16 10−18 10−19 10−20 10−23

barrier height

ΔZPVEd

effective battier heighte

2862 2859 3177 3116 3177 3086 3330 3236 3736 3788 4055 3177

381 300 362 376 366 344 409 380g 460 409g 449g 284

2481 2559 2815 2740 2811 2742 2921 2856 3276 3359 3606 2893

exp decay time (s) in Ar matrixesc

b

T 3.00 9.21 6.60 1.39 7.02 5.23 3.67 3.82 2.04 2.00 1.09 1.96

× × × × × × × × × × × ×

ref 10−15 10−16 10−17 10−16 10−17 10−17 10−17 10−17 10−18 10−19 10−20 10−24

17 45 105f 144 378 400 480 1290 1440 2880 4.8 × 104 1.3 × 106

33 32 14 22 14 31 21 22 21 19 18 15

Barrier height, ΔZPVE, and effective barrier height are in cm−1. bThe effect of an Ar matrix calculated at the MP2/6-311++G(2d,2p) level with PCM(Ar) was taken into account (eq 1). cAlthough the experimental temperature ranges from 4.3 to 10 K, all data correspond to the lowtemperature limit. dDifference between zero-point vibrational energies of the cis conformer and that of the TS. eEffective barrier height is the barrier height reduced by ΔZPVE. fAveraged value for decay time for two sites. gZPVE of the TS was calculated differently (see Computational Details). a

τCOH/2 value but the correlation between the calculations and experiments is not affected much. The transmission coefficients calculated at the MP2(full) level show a trend similar to those calculated at the CCSD(T) level (Figures 4 and S1, Supporting Information). The deviations of the cis-FA···N2 and cis-FA···CO2 complexes in an Ar matrix at the MP2(full) level are bigger than those at the CCSD(T). Nonetheless, for the calculation of transmission coefficients, the MP2(full) level of theory seems to be a good compromise between accuracy and computational cost. The general correlation found between the transmission coefficients and experimental decay time confirms that the barrier height and width are essentially important in determining the tunneling rate. However, as mentioned above, this is not the 1:1 correlation. For species in an Ar matrix (Figure 4), the experimental rate changes much more slowly compared with the calculated transmission coefficient. Moreover, as seen in Figures 3 and 4, there are other deviations between theory and experiment, implying that there are other factors that should be considered. The most remarkable disagreement between the experimental and computational results is a relatively fast decay of cis-acetic acid in a Xe matrix. These results show that there are other factors that influence the tunneling rate. According to Pettersson et al.14 and Domanskaya et al.,15 these factors include different accepting modes with different couplings to the initial state and the environment, different number of phonons needed for the energy dissipation, and different reorganization energies of the tunneling process. In addition, the observed distinctions can be at least partially connected with the used model because the application of the PCM to solid matrixes is somewhat questionable; e.g., the dispersion interaction is not modeled. These complicated questions are outside the scope of the present work.

Figure 4. Experimental tunneling decay time in Ar matrixes versus the transmission coefficient calculated at the CCSD(T)/6-311++G(2d,2p) level of theory including PCM(Ar). PnA, PlA, AA, and FA denote propionic acid, propiolic acid, acetic acid, and FA (HCOOH) monomers, respectively. A dotted line is for guiding eye.

decreases, resulting in the relative lowering of the effective barrier of propionic acid (Table 3). The calculated transmission coefficient depends on the way to estimate the energy of the tunneling H atom. In the present work, the energy is obtained as the ZPVE difference between the cis molecule and the nonadiabatic TS, excluding the FA···N2, FA···CO2, and FA···Xe (H-bonded) complexes for which a somewhat different method is used (see Computational Details). The use of ΔZPVE as the kinetic energy leads to a better agreement between the calculated and experimental decay times. If the τCOH/2 value is used for the H atom energy as in ref 18, the transmission coefficient for propionic acid (8.9 × 10−17) is the same as for acetic acid (9.0 × 10−17) at the MP2(full) level with the PCM(Ar), which disagrees with the experimental difference by a factor of 3. In the other systems, the transmission coefficients calculated with the ΔZPVE are about 1 order of magnitude larger than those calculated with the

5. CONCLUSIONS We present computational studies of hydrogen-atom tunneling in the cis-to-trans isomerization reactions of carboxylic acids (formic acid, acetic acid, propionic acid, propiolic acid) and formic acid dimers and complexes. The calculations of the F



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tunneling transmission coefficients have been performed at the MP2(full) and CCSD(T) levels of theory with the 6-311++G(2d,2p) basis set under the Wentzel−Kramers− Brillouin (WKB) approximation and using the PCM approach. The main results follow: (1) The barrier for the minimum energy path, where the OH bond length and the COH bending angle are optimized, is found to be a good approximation for the tunneling process. The transmission coefficient for the minimum energy path is twice as high as that for the simple torsional path, which has been considered in the previous studies. (2) The PCM calculations reproduce the experimental trend observed for the cis-FA monomer and trans−cis FA dimer (tc1) in noble-gas solids. However, this methodology cannot explain the experimental data obtained for cisacetic acid in noble-gas solids. Moreover, the very long lifetime of the cis forms of FA and acetic acid in N2 matrixes cannot be reproduced either, which is attributed to the lack of theoretical description within the PCM. In particular, specific interactions with the surrounding N2 molecules should be accurately simulated. (3) The use of the PCM improves the agreement between theory and experiment for cis-acetic and cis-propionic acid monomers in an Ar matrix. (4) The calculation for cis-HCOOD shows a strong decrease of the barrier transparency as compared to that for cisHCOOH, in a reasonable agreement with the experiments in an Ar matrix. In contrast, the calculations for cis-DCOOH do not explain the experimental results. (5) The calculated transmission coefficient depends on the method to estimate the energy of the tunneling H atom. The use of ZPVE of the cis and TS configurations leads in some cases to a better agreement with experiment. (6) Despite a good general agreement, there are some discrepancies between theory and experiment. It follows that other more complicated factors influence tunneling rate. Some of these factors have been discussed in the literature.14,15

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Calculated frequencies of all the system considered (Tables S1−S10), barrier height, ΔZPVE, effective barrier height, and transmission coefficients calculated at the MP2(full) level with and without PCM(Ar) (Table S11 and Figure S1). This material is available free of charge via the Internet at http://pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Authors

*Masashi Tsuge: E-mail: [email protected]. Tel: +886-357121212#56579. *Leonid Khriachtchev: E-mail: leonid.khriachtchev@helsinki.fi. Tel: +358-294150310. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.T. thanks the Academy of Finland for a postdoctoral grant (No. 1139105). This work is a part of the Project KUMURA of the Academy of Finland (No. 1277993). Kseniya Marushkevich is thanked for studies of the tc1 dimer in a Kr matrix. The CSCIT Center for Science is thanked for computational resources. G



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