Theoretical Study of Water Cluster Catalyzed Decomposition of Formic

Article pubs.acs.org/JPCA

Theoretical Study of Water Cluster Catalyzed Decomposition of Formic Acid Satoshi Inaba*,†,‡ †

School of International Liberal Studies, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan Department Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road NW, Washington, D.C. 20015-1305, United States

‡

S Supporting Information *

ABSTRACT: We have performed a number of quantum chemical simulations to examine water cluster catalyzed decomposition of formic acid. The decomposition of formic acid consists of two competing pathways, dehydration, and decarboxylation. We use the Gaussian 4 method of the Gaussian09 software to locate and optimize a transition state of the decomposition reaction and obtain the activation energy. The decomposition starts by transferring a proton of a formic acid to a water molecule. The de Broglie wavelength of a proton is similar to the width of the potential barrier of the decomposition reaction at low temperature. The tunneling, in which a proton penetrates the potential barrier, enhances the decomposition rate. Water molecules serve as the catalyst in the decomposition and reduce the activation energy. The relay of a proton from a water molecule to a neighboring water molecule is accomplished with little change of the geometry of a molecule, resulting in the reduction of the activation energy. Two water molecules are actively involved in the decomposition reaction to reduce the activation energy. We have also examined the effect of water clusters with three, four, and five water molecules on the decomposition reaction. The noncovalent distance between a hydrogen atom of a water molecule and an oxygen atom of a neighboring water molecule decreases in a water cluster due to the cooperative many-body interactions. A water molecule in a water cluster becomes a better proton donor as well as a better proton acceptor. The activation energy of the decomposition is further decreased by the catalytic effect of a water cluster. We calculate the reaction rate using the transition state theory corrected by the tunneling effect of a proton. The calculated reaction rate of the decarboxylation is smaller than that of the dehydration when less than three water molecules are included in the simulation. However, the major product of the decomposition of a formic acid becomes carbon dioxide and hydrogen molecule formed by the decarboxylation when a water cluster with more than four water molecules serves as catalyst in the decomposition of formic acid.

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INTRODUCTION A water gas shift reaction is one of the important reactions in industry and contains formic acid as an intermediate.1 A carbon monoxide reacts with a water molecule to form a carbon dioxide and a hydrogen molecule. A water gas shift reaction in supercritical water is a method to produce hydrogen molecules effectively for fuels.2 It was proposed to use formic acid to store hydrogen molecules safely for fuel.1 A water gas shift reaction is a part of a reaction network involving single carbon compounds in subseafloor hydrothermal systems.3 The reaction network generates the energy to support life in the hydrothermal environments. Reactions involving formic acid are very important and were studied extensively by experiment and theory. The decomposition of formic acid in a gas phase diluted in Ar is described by the molecular elimination processes.4 The decomposition of formic acid to radicals was found to be negligible.4,5 Formic acid has two isomers: trans-HCOOH and cis-HCOOH. Two hydrogen atoms are on the same side with respect to the CO bond in a cis-HCOOH, whereas they are on the opposite side in a trans-HCOOH. The molecular elimination processes starting from different isomers consist © 2014 American Chemical Society

of two parallel competitive pathways, decarboxylation and dehydration: cis‐HCOOH → CO2 + H 2 trans‐HCOOH → CO + H 2O

decarboxylation

(1)

dehydration

(2)

Many experimental results showed that the main pathway of the decomposition of formic acid in a gas phase is the dehydration, leading to the major product of CO with an order of magnitude smaller CO2 yield.4−6 A number of theoretical calculations were carried out to find the activation energies for the decarboxylation and the dehydration in a gas phase.5,7−10 Similar activation energies for the decarboxylation and the dehydration were obtained; however, the activation energy for the dehydration is a little smaller than that for the decarboxylation. Hu et al.11 examined the effect of H2 formed by the decarboxylation on the decomposition of formic acid. They found that the hydrogen molecule opens a new channel Received: March 1, 2014 Revised: April 3, 2014 Published: April 15, 2014 3026

dx.doi.org/10.1021/jp5021406 | J. Phys. Chem. A 2014, 118, 3026−3038

The Journal of Physical Chemistry A

Article

mobility in liquid water, due to the hydrogen bond network. Water dimer, trimer, tetramer, and pentamer were confirmed using the far-infrared vibrational−rotational-tunneling spectroscopy experiments.24−27 The energy of a water cluster is smaller than the sum of the energies of the individual water molecules by the hydrogen bonds in a water cluster. The covalent bond distance between a hydrogen atom and an oxygen atom in a water cluster increases while the noncovalent bond distance in a water cluster decreases by the cooperative many-body interactions. It is expected to reduce the activation energy of the decomposition by the catalytic effect of a water cluster because the less energy is required to transfer a proton in a water cluster. We examine the decomposition of a formic acid catalyzed by water clusters with three, four, and five water molecules.

of the dehydration, resulting in the enhancement of the decomposition of formic acid to CO. In the new channel of the dehydration, formic acid reacts with H2 to form formaldehyde and then it decomposes into CO. Experimental studies showed that the decomposition of formic acid in aqueous solution was also described by the molecular elimination processes.12 The decarboxylation becomes the main decomposition process in an aqueous solution, producing CO2 as the major product of the decomposition, whereas the dehydration yields an order of magnitude lower CO.12 Ruelle et al.13,14 proposed a new mechanism of the decomposition of formic acid in which water molecules produced as a product of the dehydration serve as catalyst in the decomposition for formic acid and reduce the activation energy of the decomposition. Melius et al.15 performed quantum chemical simulations with water solvation effect and found that transition states are stabilized by the water, reducing the activation energy of the decomposition reaction. Tokmakov et al.16 also studied the water catalyzed decomposition as well as the self-catalyzed decomposition process involving the dimer of formic acid. They found the activation energy for the decomposition of the dimer of formic acid are larger than that for the water catalyzed decomposition of formic acid. On the other hand, Wang et al.17 found the smaller activation energy by the dimer of formic acids and implied the catalytic effect of the dimer of formic acids. The catalytic effect of water molecules is very important and is included in theoretical studies of other reactions as well (e.g., the hydration of carbon dioxide18 and the oxidation reaction of ethanol19). Akiya and Savage20 examined the decomposition of formic acid with two water molecules, taking into account of the isomers of formic acid. A trans-HCOOH is more stable than a cis-HCOOH due to the long distance between hydrogen atoms. The isomerization of formic acid proceeds through the internal rotation of the OH bond around the CO bond. They found that the activation energy for the isomerization of formic acid is determined independently of water molecules. On the other hand, they obtained the reduced activation energy for the decomposition by water molecules. Chen et al.10 increases the number of water molecules included in quantum chemical simulations to three water molecules. They used the Gaussian 2 method in the Gaussian software because the Gaussian 2 method is the most reliable method to obtain the energy of a molecule at that time.21 It is necessary to adopt a sophisticated method to obtain the accurate activation energy for the decomposition of a formic acid even though it takes long to perform numerical simulations. Inexpensive methods, on the other hand, have the difficulty to describe the transfer of a proton accurately, leading to the incorrect conclusion on the decomposition of formic acid. They showed that the activation energy is further reduced by the additional third water molecule. In the present study we apply the Gaussian 4 method of the Gaussian09 software22 to locate and optimize the geometry of transition states, reactant compounds, and product compounds of the decomposition reaction. The Gaussian 4 method was shown to give the enthalpy of formation of molecules the most accurately among available methods.23 We increase the number of water molecules included in a simulation to five and find if more water molecules can play an important role to reduce the activation energy. A water molecule in a liquid phase tends to have hydrogen bonds with neighboring water molecules. A proton has the high

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COMPUTATIONAL METHODS We perform a number of quantum chemical simulations with the Gaussian 09 software.22 We use the Gaussian 4 method to optimize the geometry of a molecule and to calculate the vibration frequencies of the molecule. We chose the Gaussian 4 method because it was shown that it gives the small error in the enthalpy of formation by comparing with the 454 experimental energies (0.83 kcal/mol in average).23 We perform quantum chemical simulations of a supermolecule that consists of a formic acid and water molecules. We make quantum chemical simulations to obtain the electronic energy of a molecule. The thermal correction is added to the electronic energy to obtain the total energy of the molecule. The thermal correction consists of the corrections due to the vibration motion of the atomic nuclei, the rotational motion of the molecule, and the translation motion of the molecule. The atomic nuclei vibrate in the electric field formed by the electrons and the atomic nuclei. The vibration frequencies of the atomic nuclei in the molecule give the thermal correction by the vibration. The electronic energy difference of a reactant compound and a transition state corresponds to the energy barrier of a reaction. A particle is required to overcome the electric repulsion for a reaction to proceed. The reaction rate is dependent on the difference of the electronic energies corrected by the zero point vibration energy, ΔE0. The transition state theory expresses the reaction rate as kclassical =

q‡ k T B exp( −ΔE0 /kBT ) qrc h

(3)

where q‡ and qrc are the partition functions of a transition state and a reactant complex, respectively, and kB, T, and h are the Boltzmann constant, the temperature, and the Planck constant. The partition function factors into a product of the individual partition functions due to translation, electric motion, rotational motion, and vibrational motion. The reaction rate depends on the partition functions of the rotational and vibrational motions because a reactant complex and a transition state have the same partition functions of translation and electric motion. The rotational partition function is given by qr = 3027

πT 3 ΘxΘyΘz

(4)



Article

where Θx, Θy, and Θz are the rotational temperatures. Assuming vibrational modes are independent of each other, we obtain the vibrational partition function as qv =

∏ i

1 1 − exp( −hνi /kBT )

where

(5)

where νi is the frequency of the ith vibrational mode. We find the rotational temperatures and the vibrational frequencies in the output file of the Gaussian 4 and calculate the partition functions of a reactant compound and a transition state with eqs 4 and 5. A reaction proceeds only when the energy of a particle is larger than the potential energy barrier in the classical mechanics. However, the quantum mechanics suggests that even a particle with the energy less than the potential energy barrier is able to penetrate the barrier to ignite a reaction. The effect referred to as the tunneling effect is crucial to evaluate the reaction rate.28−30 The potential barrier is well approximated by the unsymmetrical Eckart potential with three parameters, A, B, and L, and is given by V=−

Ay By − 1−y (1 − y)2

⎞−1 ⎟⎟ ⎠

ν=

−F * m

1 2π

kquantum = Γ*kclassical

α2 =

2πV2 hν

(14)

The transmission probability of a particle with the energy, E, is given by κ (E ) =

cosh(2π (a + b)) − cosh(2π (a − b)) cosh(2π(a + b)) + cosh(2πd)

(21)

NUMERICAL RESULTS We use the Gaussian 4 method to obtain the optimized geometry of molecules. Figure 1 shows the optimized geometry of the molecules we consider in this study. The reactants for the decomposition of a formic acid are the isomers of a formic acid, cis-HCOOH, and trans-HCOOH, and the products are four inorganic molecules, CO2, H2, CO, and H2O. The computer program, MOLDEN, is used to display the geometry of the molecules.31 The optimized geometry of the transition states for the decomposition of a cis-HCOOH and a trans-HCOOH is shown in Figure 1 as well. It is not difficult to identify the pathway for the decomposition of a formic acid from the geometry of a transition state. We could confirm eqs 1 and 2 that the decomposition pathways from a cis-HCOOH and a transHCOOH correspond to the decarboxylation and the dehydration, respectively. A transition state for the decomposition of a formic acid is found using the synchronous transit guided quasi Newton method. In the Gaussian 09 we choose the option of qst2 and give tentative structures of reactant and product compounds to locate a transition state. A transition state is located at a saddle point of the potential energy surface and has only one imaginary frequency of vibration. Once a transition state is found, we use the option of IRC to follow the intrinsic reaction coordinate. By finding reactant and product compounds at both ends of the intrinsic reaction coordinate, we confirm the transition state lies on the intrinsic reaction coordinate that actually connects the two energy minimum points on the potential energy surface. The obtained reactant and product compounds are optimized further to find the vibration frequencies of the compounds. This last step is required because the energy difference of a transition state and a reactant compound corresponds approximately to the activation energy of the decomposition.20

(12)

(13)

exp(−E /kBT )κ(E)d(E /kBT )

■

(11)

2πV1 hν

∞

We compare the classical decomposition rate of a formic acid with the quantum decomposition rate and show the significant tunneling effect at the low temperature.

(10)

α1 =

∫0

We evaluate the above integral numerically to obtain Γ*. The reaction rate including the tunneling effect is finally given by

where V1 and V2 are the differences of the potential energies between a transition state and a reactant complex and that between a transition state and a product complex, respectively, and F* is the second derivative of the potential energy evaluated at the transition state. Rather than the above three parameters, V1, V2, and F*, the transmission probability of a particle with mass, m, is expressed in a simple formula with the following three parameters, u, α1, and α2:

u = hν /kBT

(19)

(20)

(9)

1 V2

(18)

E V1

Γ* = exp(V1/kBT )

and 2 ⎛ 1 ⎜⎜ + L = 2π − F * ⎝ V1

4α1α2 − π 2

The function cosh (2πd) in eq 15 is replaced by cos(2π|d|) if d is imaginary. Assuming that particles follow the Boltzmann distribution, the tunneling correction factor, Γ*, is given by

(8)

V2 )2

(17)

ξ=

where L is the half width of the potential barrier. The three parameters are determined by the shape of the potential energy distribution:

B = ( V1 +

2πb = 2 ( −1 + ξ)α1 + α2 (α1−1/2 + α2−1/2)−1

and

(7)

A = V1 − V2

(16)

2πd =

(6)

y = −exp(2πx /L)

2πa = 2 α1ξ (α1−1/2 + α2−1/2)−1

(15) 3028



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Figure 1. Optimized geometry of cis-HCOOH, trans-HCOOH, CO2, H2, CO, H2O, and transition states for the decomposition of a cisHCOOH and a trans-HCOOH. TScn and TStn stand for transition states for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by n water molecules, respectively. The Gaussian 4 method is used to obtain the optimized geometry. Gray, brown, and red solid circles correspond to a hydrogen atom, a carbon atom, and an oxygen atom, respectively. Atoms with a bond in a molecule are chemically connected with each other. The number near a bond is the bond length in the unit of angstrom. We add thin solid lines to show the distances between atoms in angstrom.

Figure 2. Rate constant for the decomposition of (a) a cis-HCOOH (decarboxylation) and (b) a trans-HCOOH (dehydration). The tunneling effect of a proton enhances the rate constant at low temperature.

dynamical simulation is not useful to study the formation and breaking of molecular bonds. A formic acid and about 2700 water molecules interact with each other in a box, in which the temperature and the pressure are regulated to be constant values, T = 300 K and 1 atm. A three point model of TIP3P is used to describe a water molecule. All the molecules are located in a box with the side of 48 Å. Periodic boundary conditions and the particle mesh Ewald method are adopted to include the electric forces of molecules outside of the box. We make molecular dynamical simulations for 100 ps and output the coordinates of all the molecules every 0.01 ps. The trajectory analysis is carried out with the use of cpptraj in AmberTools 13.34 By averaging 10 000 coordinate data of water molecules, we obtain the radial distribution of water molecules around a formic acid. Figure 3a shows the radial distribution of the number density of water molecules around the carbon atom of a formic acid. The number density is normalized by the background number density of water molecules. Any water molecule cannot approach a formic acid within about 3 Å due to the electric repulsion force by the formic acid. The normalized number density of water molecules increases with the distance from the formic acid and becomes the maximum at 3.5 Å from the formic acid, forming the first hydration shell. The second hydration shell seems to appear at 6.3 Å from the trans-HCOOH in spite of the small number density. The effect of a formic acid is reduced with an increase in the distance from the formic acid and the number density of water molecules approaches the background number density. The cumulative number of water molecules is shown in Figure 3b. The cumulative number of water molecules is the total number of water molecules that exist within the radius from the carbon atom of a formic acid. The cumulative number increases rapidly with an increase in the distance from the formic acid. The cumulative number of water molecules around a cis-HCOOH is

The classical rate constants, kclassical, for the decomposition of a cis-HCOOH and a trans-HCOOH are calculated using eq 3 as shown in Figure 2a,b. The rate constants increase with an increase in the temperature. The energy barrier for the decomposition of a cis-HCOOH is 66 kcal/mol whereas the thermal energy of a proton is 0.9 kcal/mol at 25 °C. The huge difference of the energies makes the decomposition rate to be a very small value at the low temperature. The de Broglie wavelength of a proton transferred during the decomposition of a formic acid is 1.5 Å at 25 °C, whereas the width of a potential barrier for the decomposition of a cis-HCOOH is 1.8 Å. A proton is able to penetrate the potential barrier without having the larger energy than the potential barrier by the tunneling effect because the de Broglie wavelength is similar to the width of the potential barrier. We calculate the rate constants, kquantum, for the decomposition of a formic acid by taking into account the tunneling effect of a proton, as shown in Figure 2a,b. The rate constant is enhanced by the tunneling effect at the low temperature, on the other hand, the tunneling effect does not play an important role at the high temperature because the de Broglie wavelength of a proton becomes much shorter than the width of the potential barrier. The width of the potential barrier for the decomposition of a trans-HCOOH is wider than that of a cis-HCOOH, resulting in the weaker tunneling effect. It is essential to include interactions of a formic acid with water molecules to examine the decomposition of a formic acid in aqueous solution.32,33 We use Amber 1234 to perform molecular dynamical simulations to find the distribution of water molecules around a formic acid even though a molecular 3029



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at 0 K is smaller than the sum of the total energies of the individual reactant molecules due to the hydrogen bonds in the reactant compounds. The decomposition process starts by transferring a proton from a formic acid to a water molecule. In the decomposition of a cis-HCOOH, a proton of the OH bond in a cis-HCOOH moves to a neighboring water molecule. The excess proton of the water molecule is transferred to another neighboring water molecule when two or three water molecules are included in the simulation. The hydrogen atom with negative charge in the CH bond of the HCOO and the excess proton in the water molecule combine to form a hydrogen molecule, leaving CO2, H2, and water molecules used as the catalyst. All the water molecules in a ring structure seem to get involved in the decomposition of a cis-HCOOH; however, IRC calculation shows that only one water molecule serves as the catalyst in the decomposition of a cis-HCOOH with four and five water molecules in a ring structure. Only the water molecule adjacent to a hydrogen atom of the OH bond in the cis-HCOOH is involved directly in the decomposition. The water molecule receives a proton from the OH bond of the cis-HCOOH. The water molecule with the excess proton rotates with respect to the hydrogen bond with the next water molecule and transfers the excess proton to the proximity of a hydrogen atom of the HCOO to form a hydrogen molecule. The maximum number of water molecules involved in the decomposition of a cisHCOOH in a ring structure is three. On the other hand, the decomposition of a trans-HCOOH starts with the transfer of a proton of the CH bond in a transHCOOH to a water molecule. The excess proton of the water molecule moves to a neighboring water molecule and finally attached to the oxygen atom of the OH bond of the COOH. The decomposition is completed by detaching the newly formed water molecule from CO. We made several simulations with a trans-HCOOH and four water molecules in a ring structure using different initial structures of a supermolecule to examine if more water molecules can be involved in the decomposition reaction. We could not find a transition state of a trans-HCOOH with four water molecules in a ring structure and speculate that the maximum number of water molecules involved in the decomposition of a trans-HCOOH in a ring structure is three. Figure 6 shows the optimized geometry of the product compounds of a cis-HCOOH and a trans-HCOOH with water molecules in ring structures. A hydrogen molecule, a carbon dioxide, and a carbon monoxide have nearly the same bond length with that shown in Figure 1, indicating that the stable structure of the molecules are recovered. Clusters of water molecules are seen in the product compounds of the cisHCOOH with four and five water molecules and that of the trans-HCOOH with three water molecules. The total energy of the product compounds is reduced due to the hydrogen bonds when water molecules form a water cluster. Square and pentagon structures with nearly the same length of sides are formed by four and five water molecules. A cluster of water molecules weakly connect with a hydrogen molecule, a carbon dioxide, and a carbon monoxide. We consider the unimolecular decomposition reaction of a formic acid from a reactant compound as done by Akiya and Savage.20 The transition state theory is applied to calculate the classical decomposition rate of a formic acid with eq 3. The classical decomposition rate is corrected by taking into account of the tunneling effect of a proton with eq 21. The electrical

Figure 3. Radial distribution of (a) the number density of water molecules normalized by the background number density and (b) the cumulative number of water molecules around a cis-HCOOH (red) and a trans-HCOOH (green) as a function of the distance from the carbon atom of a formic acid in angstrom.

nearly the same as that around a trans-HCOOH. About 15 and 8 water molecules are located within 5 and 4 Å from the carbon atom of a formic acid. The water molecules around a formic acid modifies the electric field of the formic acid. It is important to include the interactions of a formic acid and water molecules to study the decomposition of a formic acid in aqueous solution. We include water molecules in the simulations as a part of a supermolecule that consists of a formic acid and water molecules because water molecules are known to serve as catalyst for the decomposition of a formic acid.10 A proton is transferred from a water molecule to another water molecule within a supermolecule.20 We consider a ring structure formed by a formic acid and water molecules connected with each other by hydrogen bonds. The optimized geometry of the transition states for the decomposition of a cis-HCOOH and a transHCOOH are shown in Figure 4. Up to five and three water molecules are included in the decomposition of a cis-HCOOH and a trans-HCOOH, respectively, because we failed to find a transition state in a ring structure of a formic acid with more water molecules. We follow the intrinsic reaction coordinate from a transition state to obtain the geometry of the reactant and product compounds. The obtained geometry of the reactant and product compounds is optimized to find the vibration frequencies of the compounds using the Gaussian 4 method. Figure 5 shows the optimized geometry of the reactant compounds of a cis-HCOOH and a trans-HCOOH with water molecules in a ring structure. An oxygen atom in a water molecule faces a neighboring hydrogen atom, forming a hydrogen bond. The sum of the electrical energy and the zero point energy of vibration gives the total energy of a supermolecule at 0 K. The total energy of a reactant compound 3030



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Figure 4. Optimized geometry of transition states for the decomposition of a cis-HCOOH and a trans-HCOOH. Water molecules and a formic acid form a ring structure. The Gaussian 4 method is used to optimize transition states. Water molecules serve as catalysts of the decomposition and up to five water molecules are included in the simulations. TScn and TStn stand for transition states for the decomposition of a cis-HCOOH and a transHCOOH catalyzed by n water molecules. We add thin solid lines to show the distances between atoms in angstroms.

the major decomposition pathway even though the very small difference of the activation energies is found (1 kcal/mol when two water molecules are included in the simulations). The simulations expect the major products of the decomposition of a formic acid in aqueous solution come from the dehydration and they are CO and H2O when a formic acid and water molecules form a ring structure. This seems to disagree with the experimental results. However, the major product of the decomposition cannot be determined by the activation energy alone. Instead, it is necessary to compare the Gibbs energy of the product compounds of the decarboxylation with that of the dehydration. The Gibbs energy of the product compounds of the decarboxylation is always smaller than that of the dehydration regardless of the number of water molecules involved in the decomposition, leading the decarboxylation to the main decomposition path in experiments for long time. Water clusters are found in the product complexes in Figure 6. A cluster of water molecules plays a crucial role in the decomposition of a formic acid. Small water clusters with two, three, four, and five water molecules were confirmed by the farinfrared vibrational−rotational-tunneling spectroscopy experiments.24−27 Figure 8 shows the optimized geometry of small cyclic water clusters with three, four, and five water molecules35 obtained with the Gaussian 4 method. One of hydrogen atoms in a water molecule faces an oxygen atom of a neighboring water molecule, forming a cyclic network of hydrogen bonds. We obtain the larger energy if we consider isomers of the water

energy with the zero point energy shown in Table 1 is one of the most important quantities to determine the decomposition rate. The difference of the energies between a reactant compound and a transition state corresponds to the approximate activation energy because the ratio of the partition function of a reactant compound to that of a transition state weakly depends on the temperature. The approximate activation energies for the decarboxylation from a cisHCOOH and the dehydration from a trans-HCOOH are shown in Figure 7. Both activation energies decrease by about 20 kcal/mol if a water molecule is included in the simulations, indicating the active role of a water molecule as the catalyst. The activation energy for the decomposition of a cis-HCOOH becomes smallest, 44 kcal/mol, when two water molecules are included in the simulation and increases by less than 2 kcal/mol if more water molecules are included in the simulation. The decreasing activation energy for the decomposition of a transHCOOH approaches a constant value of 43 kcal/mol when two water molecules are included in the simulations. The number of water molecules involved in the decomposition of a cis-HCOOH and a trans-HCOOH as the effective catalyst is two when a formic acid and water molecules form a ring structure. The activation energy for the decomposition of a cisHCOOH is a little larger than that for the decomposition of a trans-HCOOH. The smaller activation energy for the decomposition of a trans-HCOOH leads the dehydration to 3031



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Figure 5. Optimized geometry of reactant compounds for the decomposition of a cis-HCOOH and a trans-HCOOH. The Gaussian 4 method is used to optimize reactant compounds. Up to five water molecules are included in the simulations as catalysts. RCcn and RCtn stand for reactant compounds for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by n water molecules in a ring structure.

clusters, in which two hydrogens of a water molecule are used for hydrogen bonds while another water molecule donates no hydrogen atom for the hydrogen bond in a water cluster. The covalent bond distance between the oxygen and the hydrogen of a water molecule is increased by the hydrogen bond. On the other hand, the noncovalent distance between the oxygen of a water molecule and the hydrogen of a neighboring water molecule bonded by the hydrogen bond decreases. This distance further decreases if more water molecules are included in a water cluster by the cooperative many-body interactions, as shown in Figure 8. We calculate the bond energy by subtracting the energy of a water cluster from the sum of the energies of the individual water molecules: 8.0, 17.3, and 23.9 kcal/mol for the water clusters with three, four, and five water molecules. The energy of each bond for the water clusters with three, four, and five water molecules are 2.7, 4.3, and 4.8 kcal/mol, respectively. A water molecule becomes a better proton donor as well as a better proton acceptor due to the shorter distance of the water molecules in the water cluster. Two water molecules on a side of a water cluster act as the catalyst of the decomposition, while the rest of water molecules in a water cluster work together to keep the shape of the water cluster. The optimized geometry of the transition states for the decomposition of a cis-HCOOH and a trans-HCOOH with water clusters are shown in Figure 9. A water cluster with two

water molecules (e.g., a water dimer) also serves as the catalyst for the decomposition reaction. However, we have already showed them in Figure 4 and do not repeat them here. We examine the decomposition of a formic acid catalyzed by water clusters with three, four, and five water molecules. It is seen that a formic acid interacts with two water molecules of the water cluster in the transition state while the rest of the water molecules are not directly involved in the decomposition process. This can also be interpreted as the following situation. A formic acid is decomposed by the catalyst of the two water molecules in a ring structure that interact with other background water molecules and have hydrogen bonds. We follow the intrinsic reaction coordinate from a transition state and find the geometry of the reactant and product compounds. We use the Gaussian 4 method to optimize the geometry of the reactant and product compounds. Figure 10 shows the optimized geometry of the reactant compounds and the product compounds. A formic acid attracts a water cluster with two hydrogen bonds in a reactant compound. We could not find a reactant compound of a cis-HCOOH and a water cluster with three water molecules; instead they form a ring structure. As in the decomposition of a formic acid with water molecules in a ring structure, a proton of the OH bond in a cisHCOOH moves to a neighboring water molecule of a water cluster, while a proton of the CH bond in a trans-HCOOH is 3032



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Figure 6. Optimized geometry of product compounds for the decomposition of a cis-HCOOH and a trans-HCOOH. The Gaussian 4 method is used to optimize product compounds. Up to five water molecules are included in the simulations as catalyst. PCcn and PCtn stand for product compounds for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by n water molecules in a ring structure.

Table 1. Relative Electrical Energy Corrected by the Zero Point Energy (kcal/mol) of the Reactant Compounds, the Transition States, and the Product Compounds with Respect to That of a trans-HCOOH and nH2O for the Decomposition of a Formic Acid with Water Moleculesa structure

n=0

n=1

n=2

n=3

n=4

n=5

RCcn TScn PCcn WRCcn WTScn WPCcn RCtn TStn PCtn WRCtn WTStn WPCtn

3.921 70.002 −6.455

−2.297 47.996 −5.427

−6.480 37.351 −10.464

−25.075 20.749 −25.166 −21.352 19.955 −24.937

−30.914 13.966 −32.451 −29.234 13.145 −32.405

0 67.485 3.964

−1.323 48.807 1.060

−6.413 36.258 −4.646

−17.043 29.304 −16.438 −16.970 28.095 −15.713 −12.527 30.629 −14.064 −11.402 31.352 −13.354

−20.223 20.250 −20.934

−27.344 12.134 −25.211

a

The relative electrical energy of a cis-HCOOH, CO2 + H2, and CO + H2O with respect to a trans-HCOOH can be found in the reactant compound, RCc0, the product compound, PCc0, and the product compound, PCt0, respectively. The reactant compounds, the transition states, and the product compounds of a formic acid with water molecules in a ring structure are denoted by RC, TS, and PC, whereas that with a water cluster are denoted by WRC, WTS, and WPC. A trans-HCOOH with four and five water molecules in a ring structure is not found because water molecules form a water cluster.

HCOOH. In the decomposition of a trans-HCOOH, the excess proton is transferred to the oxygen of the OH bond of the COOH, forming a water molecule that is moving away from CO. During the decomposition process, other water molecules

transferred to a water cluster. The excess proton of the water molecule in a water cluster is transferred to the next water molecule. The excess proton of the water molecule combines with the hydrogen atom from the CH bond of the HCOO to form a hydrogen molecule in the decomposition of a cis3033



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Figure 7. Approximate activation energies for the decomposition of a cis-HCOOH and a trans-HCOOH when a formic acid and water molecules form a ring structure. The approximate activation energy is calculated from the difference of the electrical energies corrected by the zero point energies between the reactant compound and the transition state. Up to five and three water molecules are included in the simulations as catalyst of the decomposition of a cis-HCOOH and a trans-HCOOH, respectively.

Figure 9. Optimized geometry of transition states for the water cluster catalyzed decomposition of a cis-HCOOH and a trans-HCOOH. The Gaussian 4 method is used to locate and optimize transition states. Water clusters with three, four, and five water molecules are included in the simulations. WTScn and WTStn stand for transition states for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by a water cluster with n water molecules, respectively. Figure 8. Optimized geometry of water clusters with three, four, and five water molecules. The Gaussian 4 method is used to obtain the optimized geometry.

molecules in a ring structure is nearly the same as that for the decomposition of the formic acid with a water cluster with three water molecules. With an increase in the number of water molecules, the activation energy is reduced when a water cluster is introduced as the catalyst of the decomposition. The lowest activation energy becomes 41 kcal/mol when a water cluster with four water molecules is included in the decomposition of a cis-HCOOH. This is about 3 kcal/mol smaller than the smallest activation energy obtained when a cis-HCOOH and water molecules form a ring structure. On the other hand, the activation energy for the decomposition of a trans-HCOOH continues to decrease and becomes the smallest when the decomposition of a trans-HCOOH is catalyzed by a water cluster with five water molecules. The activation energy is reduced by about 3 kcal/mol by considering the catalytic effect of the water cluster. In the simulations of the decomposition of a cis-HCOOH catalyzed by water clusters with four and five water molecules, all of the four and five water molecules are located within 4 Å from the carbon atom of a cis-HCOOH. Numerical results of molecular dynamical simulations show the average number of water molecules within 4 Å from a formic acid is eight as shown in Figure 3b. It might be necessary to include more water

keep the hydrogen bonds to maintain the shape of the water cluster. A product compound of the decomposition of a transHCOOH consists of a carbon monoxide and a water cluster including a water molecule formed by the decomposition. For example, the decomposition of a trans-HCOOH catalyzed by a water cluster with five water molecules leaves six water molecules in a book structure. It was shown that a water cluster with six water molecules, the water hexamer, have five isomers with nearly the same energies.36 It might be possible to find another water cluster form if we adopt a different transition state. Water clusters are recovered in a product compound and act as the catalyst of another decomposition reaction. The approximate activation energies for the decarboxylation and the dehydration are shown in Figure 11 when water molecules form a water cluster. For comparison the approximate activation energy is also shown when a formic acid and water molecules form a ring structure. The activation energy for the decomposition of a formic acid with three water 3034



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Figure 10. Optimized geometry of reactant and product compounds for the water cluster catalyzed decomposition of a cis-HCOOH and a transHCOOH. The Gaussian 4 method is used to optimize reactant and product compounds. WRCcn and WRCtn stand for reactant compounds for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by a water cluster with n water molecules, whereas WPCcn and WPCtn stand for product compounds for the decomposition of a cis-HCOOH and a trans-HCOOH catalyzed by a water cluster with n water molecules.

decrease further if a water cluster with more than six water molecules is included in the decomposition of a trans-HCOOH. However, it is beyond the present study to consider the decomposition of a trans-HCOOH catalyzed by a water cluster with six water molecules. It was shown that a water cluster with six water molecules has five isomers with nearly the same energies.36 It requires large amount of time to examine the decomposition of a formic acid catalyzed by the five isomers of the water clusters. Parts a and b of Figure 12 show the rate constants for the decomposition of (a) a cis-HCOOH and (b) a trans-HCOOH catalyzed by up to five water molecules as a function of the temperature. The rate constant of the decomposition of a formic acid catalyzed by a water cluster is shown when more than three water molecules are included in the simulations because a water cluster decomposes a formic acid more effectively than water molecules in a ring structure. Both rate constants for the decomposition of a cis-HCOOH and a transHCOOH increase with an increase in the temperature regardless of the number of water molecules included in the simulations. The rate constant is enhanced by the catalytic effect of water molecules when a water molecule is introduced in the simulations. Figure 12a shows that the rate constant increases with an increase in the number of water molecules

Figure 11. Same as Figure 7 but the activation energy of the decomposition of a formic acid catalyzed by water clusters. Water clusters with three, four, and five water molecules are considered.

molecules in the simulations to find the maximum decomposition rate. However, the activation energy for the decomposition of a cis-HCOOH catalyzed by a water cluster with four water molecules is smaller than that with five water molecules by 1 kcal/mol. On the other hand, the activation energy for the decomposition of a trans-HCOOH catalyzed by a water cluster with five water molecules becomes smaller than that with four water molecules. The activation energy might 3035



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interactions between a formic acid and water molecules to examine the decomposition of a formic acid in aqueous solution. We consider a supermolecule that consists of a formic acid and water molecules. We use quantum chemical simulations to describe the breaking and formation of bonds between atoms in the supermolecule. We use the synchronous transit guided quisi Newton method in the Gaussian09 to locate a transition state of the decomposition. The approximate activation energy of the decomposition is obtained from the difference of the electrical energies corrected by the zero point energies between the reactant compound and the transition state. The activation energy is used to calculate the classical decomposition rate of a formic acid with the transition state theory. It is required to include the tunneling effect of a proton to obtain the quantum decomposition rate. The transfer of a proton is the most important process in the decomposition of a formic acid and needs to be treated very accurately. Even a proton with the energy less than the activation energy for the decomposition can penetrate the potential barrier to ignite the decomposition process by the tunneling. The de Broglie wavelength of a proton becomes long at the low temperature, increasing the tunneling correction factor for the decomposition. However, the tunneling effect on the decomposition becomes weak at the high temperature due to the short de Broglie wavelength compared with the width of the potential barrier. We also found that the activation energy for the decomposition of a formic acid is significantly reduced by water molecules included in the simulations. The decomposition process starts by the transfer of a proton in a formic acid to a water molecule. The excess proton of the water molecule is transferred to a neighboring water molecule bonded by the hydrogen bond. The excess proton from the water molecule combines with the hydrogen atom of the CH bond in a cis-HCOOH to form a hydrogen molecule in the decarboxylation, while the excess proton is captured by the oxygen atom of the OH bond in a trans-HCOOH to form a carbon monoxide and an extra water molecule in the dehydration. Two water molecules serve as the catalyst most effectively in the decomposition when a formic acid and water molecules form a ring structure. The rate constant is further increased if we consider the decomposition of a formic acid catalyzed by a water cluster. The noncovalent distance between the oxygen of a water molecule and the hydrogen of a neighboring water molecule bonded by the hydrogen bond in a water cluster becomes short by the cooperative many-body interactions. Water molecules in a water cluster becomes a better proton donor as well as a better proton acceptor. Two water molecules of a water cluster act as the catalyst in the decomposition while the other water molecules keep the shape of the water cluster by the hydrogen bonds. A water cluster plays a significant role in the decomposition of a formic acid. The rate constant for the decarboxylation is larger than that for the dehydration when a water cluster serves as catalyst of the decomposition, leading the decarboxylation to the main decomposition pathway. We also found that the Gibbs energy of the product compound from the decarboxylation, CO2 and H2 with water molecules, is always smaller than that from the dehydration, CO and H2O with water molecules. The main decomposition products are expected to be CO2 and H2, from the decarboxylation and agree with experiments.

Figure 12. Rate constant for the decomposition of (a) a cis-HCOOH (decarboxylation) and (b) a trans-HCOOH (dehydration) catalyzed by water clusters with three, four, and five water molecules and water molecules in a ring structure. The decomposition is catalyzed by one and two water molecules that form a ring structure with a formic acid. A formic acid is decomposed more effectively by a water cluster when more than three water molecules are included in the simulation.

and approaches the maximum value when water clusters with four and five water molecules are available for the decomposition of a cis-HCOOH. On the other hand, the little increase in the rate constant is found when more than two water molecules are included in the decomposition of a transHCOOH as shown in Figure 12b. Figure 11 shows that the activation energy for the decomposition of a trans-HCOOH decreases with the increasing number of water molecules. However, the ratio of the partition function of a transition state to that of a reactant compound decreases and offsets the decreased activation energy.

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SUMMARY We studied the decomposition of a formic acid catalyzed by water clusters with three, four, and five water molecules as well as water molecules in a ring structure. We made a number of quantum chemical simulations using the Gaussian 4 method in the Gaussian09 software to locate and optimize the geometry of transition states, reactant compounds, and product compounds. The Gaussin 4 method is known to have the least errors when we calculate the enthalpy of formation for molecules. A formic acid decomposes in two competitive pathways: decarboxylation and dehydration. A formic acid in aqueous solution decomposes mainly into CO2 and H2 by the decarboxylation with a small amount of CO and H2O by the dehydration. We made molecular dynamical simulations using the Amber 12 software to find the distribution of water molecules around a formic acid. The radial distribution of water molecules is modified by a formic acid, showing the first hydration shell at 3.5 Å from the carbon atom of the formic acid. The average number of water molecules within 4 Å from the formic acid is about eight. It is crucial to include the 3036



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Some experiments showed that the decomposition of a formic acid in aqueous solution is dependent on the pH value and the pressure of the solution.37,38 At low temperature the reaction rate of the dehydration is enhanced in an acidic condition whereas that of the decarboxylation is nearly independent of the solution pH value.37 Further theoretical studies are required to examine the decomposition of formic acid in an acidic condition.

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ASSOCIATED CONTENT

S Supporting Information *

Table of the rate constants (1/s) for the decomposition of a cisHCOOH and a trans-HCOOH catalyzed by n water molecules is given as a function of the temperature (Table S1). This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 81-3-5286-1730. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We performed most of the computations in Research Center for Computational Science, Okazaki, Japan. We acknowledge Shohei Ohara for helpful discussions. We also express our appreciation to Alan Boss for providing facilities at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington during the 2013−2014 sabbatical year in the U.S.A.

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REFERENCES

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Theoretical Study of Water Cluster Catalyzed Decomposition of Formic

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