Reaction Mechanism for Direct Proton Transfer from Carbonic Acid to a Strong Base in Aqueous Solution II: Solvent Coordinate-Dependent Reaction Path

Abstract

The protonation of methylamine base CH₃NH₂ by carbonic acid H₂CO₃ within a hydrogen (H)-bonded complex in aqueous solution was studied via Car–Parrinello dynamics in the preceding paper (Daschakraborty, S.; Kiefer, P. M.; Miller, Y.; Motro, Y.; Pines, D.; Pines, E.; Hynes, J. T. J. Phys. Chem. B 2016, DOI: 10.1021/acs.jpcb.5b12742). Here some important further details of the reaction path are presented, with specific emphasis on the water solvent’s role. The overall reaction is barrierless and very rapid, on an ∼100 fs time scale, with the proton transfer (PT) event itself being very sudden (<10 fs). This transfer is preceded by the acid–base H-bond’s compression, while the water solvent changes little until the actual PT occurrence; this results from the very strong driving force for the reaction, as indicated by the very favorable acid-protonated base ΔpK_a difference. Further solvent rearrangement follows immediately the sudden PT’s production of an incipient contact ion pair, stabilizing it by establishment of equilibrium solvation. The solvent water’s short time scale ∼120 fs response to the incipient ion pair formation is primarily associated with librational modes and H-bond compression of water molecules around the carboxylate anion and the protonated base. This is consistent with this stabilization involving significant increase in H-bonding of hydration shell waters to the negatively charged carboxylate group oxygens’ (especially the former H₂CO₃ donor oxygen) and the nitrogen of the positively charged protonated base’s NH₃⁺.

Graphical abstract

1. INTRODUCTION

In the preceding paper,¹ hereafter denoted as I, the protonation of the strong base methylamine CH₃NH₂ by carbonic acid H₂CO₃ in aqueous solution was studied via Car–Parrinello molecular dynamics (CP MD).^2,3

$$\text{HOCOOH}\cdots {\text{NH}}_2{\text{CH}}_3\to {\text{HOCOO}}^{-}{\cdots }^{+}{\text{HNH}}_2\text{CH}$$ (1)

This proton transfer (PT) reaction within a hydrogen (H)-bonded complex was found to be barrierless and very rapid, consistent with the strong thermodynamic driving force inferred from the difference in acid strengths of H₂CO₃ and the base’s conjugate acid CH₃NH₃⁺, a reactivity result of relevance for our proposal described in more detail elsewhere,⁴ that H₂CO₃ has the potential of being a major protonating agent of, for example, nitrogen bases in the blood.⁵

Primary attention was focused in I on the time evolution of the proton coordinate, the hydrogen bond coordinate $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ and the charge distribution’s change between the original acid–base pair and the product contact ion pair. The important influence of the aqueous solvent in the reaction was invoked at assorted points in the analysis, but this was not analyzed in detail. In the present paper, we continue our analysis of this PT reaction by presenting a detailed analysis of the reaction path, with a specific and explicit focus on the role of the water solvent.

The outline for the remainder of this paper is as follows. Section 2 details the methodology for our analysis of the reaction path in terms of the H-bond coordinate and an energy gap which has two components, one which involves the aqueous solvent contribution, which serves as a solvent coordinate, and a second contribution which serves as a coordinate related to reaction-induced geometry changes within the H-bonded complex. Section 3 analyzes the reaction path, in terms of the coordinates just mentioned, prior to the PT occurrence and for a short period of ∼50 fs thereafter, in which a necessary angular aspect of the complex’s internal H-bonding is completed, the nascent contact ion pair charge distribution evolves to its final form and solvent stabilization of the ion pair is ensured. A detailed analysis of the full aqueous solvation relaxation after the PT occurrence is then given in section 4. Here we also provide insights on the contributions to this relaxation from solvent water- ion pair atomic site H-bond changes in number and strength, for the sites where the largest charge change in the PT reaction occurs. Concluding remarks are offered in section 5.

2. TRAJECTORY ANALYSIS DETAILS

We have used the same simulated carbonic acid–methylamine PT reaction trajectories as in I, to which the reader is referred for details including construction of the simulation box containing the aqueous solution of the H-bonded acid–base pair complex, the method of preparing and collecting 10 CP MD reaction trajectories, and the obtaining of reference water solvent structures for isolated species (carbonic acid, bicarbonate, methylamine, and protonated methylamine) in aqueous solution. In the following, we describe the energy gap coordinate analysis that we will use for exploration of the reaction path with the inclusion of the water solvent.

2.1. Energy Gap Coordinate ΔE Analysis Method

In I, we focused on dynamic evolution of the proton coordinate and the H-bond separation of the acidic oxygen and basic nitrogen in our geometric description of the reaction. In order to include the water solvent in the dynamical description (and as in previous activated PT studies^6–10), we employ an energy gap coordinate ΔE. This will prove useful, despite the present PT reaction’s lack of any activation barrier. The more specialized details of this ΔE construction are relegated to the Supporting Information in section S1. For simplicity, only the salient features of the ΔE analysis are presented here. Of particular interest is that this gap is defined to include not only a solvent contribution (also familiar from solvation dynamics studies^11–28) but also a reacting pair vibrational contribution, which depends upon the acid–base complex’s geometry (not however including the proton stretch). Since ΔE, and its solvent and vibrational components defined below, only depend on the geometries of the complex and the surrounding water solvent molecules, they can be evaluated at each time step during a trajectory. The reaction path presented in section 3 will be constructed in terms of these ΔE components from an average of the 10 trajectories’ ΔE time dependence.

The gap ΔE is defined to be the interaction energy difference between the two diabatic valence bond (VB) potentials (neutral pair minus ion pair), in the presence of the water solvent; these are representative of the limiting neutral and ion pair H-bonded complexes of eq 1, and each is evaluated at its potential minimum with respect to the proton coordinate. ΔE is

$$\mathrm{\Delta }E(\mathbf{Q},\mathbf{S})=V_{\mathrm{N}}(q_{\mathrm{N},min},\mathbf{Q},\mathbf{S})-V_{\mathrm{I}}(q_{\mathrm{I},min},\mathbf{Q},\mathbf{S})$$ (2)

where q_N,min and q_I,min are the respective fixed proton location minima of the O_A–H_A and H_A–N vibrational stretches and Q represents the H-bonded complex’s intermolecular coordinates including the H-bond separation $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ and bend angles defined below.²⁹ S represents the solvent configuration, that is, the coordinates of all the solvent water molecules.

Figure 1 qualitatively depicts the gap ΔE between the neutral V_N and ionic V_I VB potentials. The VB potentials themselves, as well as the adiabatic proton potential that results from their electronic mixing (solid line), are also displayed. For PT reactions,^4–10 this ΔE gap reflects the proton potential’s evolving asymmetry; here negative ΔE values correspond to reactant configurations with the proton bound to the carbonic acid, while positive ΔE values correspond to product configurations with the proton bound to the methylamine base.

Figure 1.

Schematic energy gap ΔE definition in terms of the diabatic neutral and VB potentials for the acid–base H-bonded complex in interaction with the aqueous solvent. This figure is also intended to convey the idea that the VB states’ reference geometries have the proton coordinate positions fixed at their values at the minima of the respective potentials. The displayed situation is appropriate before, e.g, H-bond and solvent rearrangements permit the PT and ultimately an exothermic reaction situation. See Figure 2 of I.

The total energy gap ΔE coordinate can be instructively decomposed into a H-bond, vacuum gap contribution ΔE_v (which will depend on the internal coordinates of the H-bond complex) and a solvent contribution ΔE_S

$$\mathrm{\Delta }E=\mathrm{\Delta }{E}_{\mathrm{v}}+\mathrm{\Delta }{E}_{\mathrm{S}}$$ (3)

ΔE and its components are to be evaluated for each time point along each trajectory. But ΔE cannot be evaluated in the CP MD simulations using eq 2 because the VB potentials are not known. We will circumvent this difficulty and describe here an approximate ΔE evaluation protocol that, as we shall later see in section 3, provides a clear picture for the nascent reaction path in terms of coordinates that affect this asymmetry coordinate. This method utilizes electronic structure calculations of the H-bonded complex in the surrounding water solvent’s field for each time point. Two such calculations are performed for system configurations for each time point, corresponding to the two limiting VB structures, now described in more detail.

2.1.1. Diabatic Proton Positions

The two VB reference structures require assignment of the proton position in each of them. These two reference proton positions are to be determined for each configuration along a trajectory and the difference between them is just the proton position. All the other nuclei, including solvent molecules, remain at their coordinates given by that trajectory time point. The reference proton positions are defined respectively as 1 Å away from the carbonic acid O_A in the O_A-H_A direction for the HOCOOH⋯NH₂CH₃ complex in its neutral VB state and 1 Å away from the protonated base in the H_A-NH₂⁺ direction for the ionic VB state HOCOO⁻⋯⁺HNH₂CH₃.²⁹ These two positions are therefore determined from the coordinates O_A, H_A, and N. Since the H-bond complex’s internal coordinates Q, which include the H-bond length $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$, vary throughout the trajectory, so too will these reference proton positions. The result is an energy difference that is Q-dependent, consistent with the eq 2 definition.

2.1.2. Total Energy Gap ΔE

With these two reference H-bond complex VB structure configurations defined for each time point, we can now turn to details of the electronic structure information necessary for ΔE’s evaluation. We employ quantum chemistry calculations (B3LYP/6-311+G**) for the carbonic acid–nitrogen base pair, with the surrounding water molecules treated as point charges (SPC/E charges: −0.8476 for O_W and 0.4238 for H_W).³⁰ Accordingly, ΔE is the difference (neutral minus ionic) in electronic energy (B3LYP/6-311+G**) of the H-bonded complex in the field of the surrounding water molecules for the two reference proton positions indicated above.²⁹

2.1.3. Vacuum Energy Gap Contribution ΔE_v

ΔE_v in eq 3 is the ΔE energy gap in the absence of the water solvent. Hence, ΔE_v is evaluated in the manner just described for ΔE, except that the water molecules’ reaction field is ignored. ΔE_v is then the difference in electronic energy (B3LYP/6-311+G**) of the H-bonded complex, in the absence of the surrounding water molecules’ point charges, for the two reference proton positions indicated above.

2.1.4. Solvent Energy Gap Contribution ΔE_S

The solvent contribution ΔE_S can obviously be obtained as the difference between ΔE and ΔE_v via eq 3. An alternate and potentially more instructive route which we will also use is to calculate ΔE_S by an electrostatic method similar to procedures in traditional evaluations of solvent response.^12,15,23,24 It has the appeal of analytic simplicity and will allow us to conveniently calculate aspects of the dynamic hydration around desired atoms within the H-bonded acid–base complex.³¹

In the electrostatic method, ΔE_S is taken as the difference in electrostatic interaction between the different charge distributions characteristic of the neutral and ionic VB states and the surrounding water molecules’ charge distribution. For its evaluation, ESP point charges for each atom in the acid–base pair are obtained from electronic structure calculations of a series of before PT H-bonded complexes taken from the CP MD equilibration trajectory and a set of structures from the ion pair after PT. The charges are modified to remove any charge transfer between the carbonic acid proton donor and the methylamine base proton acceptor.³² The resulting charges are given in Table 1. With our definitions, the only configurational, structural difference between the neutral and ionic VB states is the proton position (just as with the ΔE (ΔE_v) calculation described above). Since the complex’s charges reside on the H-bonded complex’s individual nuclei whose positions change with time, ΔE_S is a function of the intermolecular H-bond coordinates Q.

Table 1.

ESP Atomic Charges for the Carbonic Acid/Bicarbonate-Methylamine/Protonated Methylamine Acid–Base Complex, in Its Neutral Reactant and Ionic Product Forms, Used to Evaluate the Solvent Energy Gap Coordinate ΔE_S^a

atom/moiety	neutral pair charge (e)	ion pair charge (e)
C	1.00	1.04
OA	−0.57	−0.88
OC	−0.73	−0.88
Ona	−0.68	−0.80
Hna	0.53	0.51
HCO3 moiety	−0.45	−1.01
N	−0.64	−0.14
${\overline{H}}_{\mathrm{N}}$ b	0.25	0.28
CH3	−0.01	−0.21
${\overline{H}}_{\mathrm{C}}$ c	0.05	0.12
CH3NH2 moiety	0.00	0.57
HA	0.45	0.44

^aAtom labeling is that depicted Scheme 1.

^bThe overbar indicates an overage over the two nitrogen hydrogens, both before and after PT (the transferring proton H_A’s charge is accounted for separately).

^cThe overbar indicates an overage over the three methyl hydrogens, both before and after PT.

3. REACTION MECHANISM

We now turn our attention to aspects of the reaction mechanism, extending those described in I, with a specific emphasis on how key reactive coordinates including the solvent contribute to the reaction path. Our coordinate selection focuses especially on the acid–base H-bond coordinate $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ and the solvent energy gap coordinate ΔE_S (see subsection 2.1). Our basic perspective is that, as discussed in I, the proton vibrational dynamics is faster than either of these coordinates, and the proton vibration typically adiabatically adjusts to them. However, for completeness and clarity, we also include the proton coordinate in our presentation. The reaction path is first presented in terms of these $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ and ΔE_S coordinates utilizing an average over all 10 trajectories. The time dependence of these reaction coordinates, with attention also to the proton coordinate, is then analyzed utilizing the related energy gap coordinate ΔE analysis of subsection 2.1.

3.1. Reaction Path

We begin by presenting the reaction path for the carbonic acid-methylamine PT in Figure 2b, which plots the averages of the solvent coordinate ⟨ΔE_S⟩ and H-bond coordinate $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ against each other along the trajectory, with t = 0 defined for x_PT = 0, the location of the PT’s occurrence (the same trajectory average scheme as that used to produce Figures 1c and 3 of I). The corresponding average proton’s position is displayed in Figure 2a. Plotting both coordinates’ time dependence at the same time relative to the PT occurrence yields Figure 2b. The connection of the complex’s H-bond dynamics with the variation of the H-bond length-dependent ΔE component, the vacuum potential asymmetry ΔE_v is made in Figure 2c, which displays the trajectory-averaged ⟨ΔE_v⟩ versus the average H-bond coordinate $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ (the averaging is similar to that for ⟨ΔE_S⟩).

Figure 2.

Trajectory-averaged (a) normalized proton coordinate x_PT (eq 3 of I), H-bond length $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ (dashed line), total charges of methylamine and bicarbonate moieties (solid lines) as a function of time; (b) solvent coordinate ⟨ΔE_S⟩ and (c) vacuum gap coordinate ⟨ΔE_v⟩ paths versus the averaged H-bond length $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$. The green dashed vertical line in (a) indicates t = 0 the time for PT occurrence, and the time points along each path are indicated in panels (b) and (c), with t = 0 retaining this meaning.

Figure 3.

Total, vacuum and solvent energy gap coordinates versus time for the fourth reaction trajectory. The t = 0 value here is the time at which the equilibration constraints were turned off and the dashed vertical red lines in all panels indicate the PT occurrence time when the normalized proton coordinate (eq 3 of I) x_PT = 0. (a) The total energy gap ΔE (solid line) and (b) vacuum gap ΔE_v (solid line) values versus time for the fourth reaction trajectory. (c) The solvent contribution ΔE_S versus time as calculated with the electrostatic method (dashed green line) and for the ΔE_S = ΔE − ΔE_v difference method (solid line). (Evaluation details for ΔE and its components are given in subsection 2.1.) The normalized proton coordinate x_PT (dotted blue line) is also plotted in panels (a) and (c), while the H-bond coordinate $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ is also plotted in panel (b) (dashed blue line), with values given by the right-hand ordinate axis. Each panel’s curves are rather noisy, due to thermal fluctuations for this single trajectory, especially due to the specific water solvent molecules in close proximity to the H-bonded complex. Nonetheless, we can identify specific H-bond complex vibrational motions in, e.g., panels (b) and (c); see the text.

In this subsection, we only give a first discussion of the reaction path’s main features in order to provide an overview. We then turn to further details concerning the path in the remainder of section 3 and, for later times, in section 4.

The reaction path begins with a compression of the H-bond separation within the neutral carbonic acid-methylamine base pair complex until the point of PT occurrence (t = 0). The averaged H-bond contribution ⟨ΔE_v⟩ in Figure 2c increases during the H-bond compression; in any event, this is in the direction favoring the ion pair, and as we explain in subsection 3.2, this is related to the fact that ⟨ΔE_v⟩ is anticorrelated with the average H-bond length coordinate ⟨ΔE_v⟩. During the rapid $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ compression, the water solvent has little changed (see Figure 2b), evidenced by the nearly constant ⟨ΔE_S⟩, but does change rapidly in the PT occurrence region, directly favoring the ion pair product right in the neighborhood of the proton occurrence at t = 0, a phenomenon related to establishment of a symmetric proton potential (see Figure 2b of I).

For the first 50 fs after the PT occurrence, which itself takes place after the first minimum of the H-bond compression (see Figure 1 of I) and in which the H-bond coordinate $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ is increasing, ⟨ΔE_v⟩’s behavior in Figure 2b is associated with the proton bending H_A–N–CH₃ angle adjusting to become that appropriate to the protonated methylamine base, and the solvent coordinate ⟨ΔE_S⟩ in Figure 2b increases slightly during this H-bond expansion, corresponding to the continuing initial solvent stabilization; these aspects will be further discussed in subsection 3.2. After this 50 fs period, ⟨ΔE_S⟩ increases significantly with time, as solvent stabilization of the ion pair increases (whose time scale will be discussed further in section 4), while $\langle R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}\rangle$ fluctuates about the ion pair separation, and ⟨ΔE_v⟩ primarily fluctuates around a constant value associated with the now established ion pair average structure.

3.2. Time Dependence of ΔE

In order to put the Figure 2 reaction path into a more extended perspective, we now turn to detailed analyses of the total energy gap ΔE and its vacuum and solvent components ΔE_v and ΔE_S (see eq 3). We give the analyses from the perspective of a single trajectory first and then of a trajectory average.

3.2.1. Single Trajectory Analysis of ΔE

Figure 3a presents the full energy gap coordinate ΔE evaluated for the fourth reaction trajectory. The other two Figure 3 panels display the trajectory’s vacuum and solvent gap components. Clearly all of these coordinates change considerably during the reaction, with major changes in the PT occurrence’s neighborhood, indicated by the dashed line. The corresponding results for all ten reaction trajectories are presented in Figure S1 of SI. In what follows, the general behavior for single trajectories is analyzed, using the fourth trajectory as a representative.

Total Energy Gap ΔE

It will be useful to bear in mind Figure 2 of I which displays the evolution of the proton potential in solution for key points in the reaction progress. We recall that our ΔE definition convention, proton potential in aqueous solution for the acid–base H-bond complex neutral pair minus that for ion pair, is such that with the proton localized within the acid, ΔE is negative, since the neutral potential minimum is below that of the ionic potential. The opposite situation occurs where the ion pair is more stable than the neutral pair. Thus, Figure 3a shows that ΔE is initially negative, with the proton being localized near the carbonic acid’s carbonyl group acidic oxygen; the transition from negative to positive ΔE occurs near x_PT = 0, and after the occurrence of PT the proton is bound to the nitrogen of the methylamine base and ΔE is positive.

H-Bond Contribution ΔE_v

The H-bond complex’s vacuum contribution ΔE_v’s evolution for the fourth trajectory is presented in Figure 3b. Apart from the evident higher frequency oscillations, ΔE_v within and after the PT occurrence region is strongly anticorrelated with the H-bond length $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$, also plotted there: ΔE_v is more (less) negative for larger (shorter) $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ separations. The origin of this $\mathrm{\Delta }{E}_{\mathrm{v}}-R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ connection will now be pursued, as will the high frequency oscillations in Figure 3b (and Figure 2c), which will be shown to originate from acid–base complex bending vibrations (as previewed in subsection 3.1).

We first provide some useful physical background for the vacuum gap ΔE_v. At infinite separation of the PT products, ΔE_v is just the negative of the proton affinity difference ΔPA for methylamine and bicarbonate, which is the vacuum reaction free energy ΔG_RXN for PT from carbonic acid to methylamine. The resulting value ΔE_v = −ΔPA = −125 kcal/mol^33–35 is extremely negative, so that this PT reaction is extremely difficult to effect in vacuum. This ΔE_v value increases (becomes less negative) upon formation of the H-bonded complex at the given H-bond geometry. This motivates our decomposition of ΔE_v into two contributions

$$\mathrm{\Delta }{E}_{\mathrm{v}}(\mathbf{Q})={\mathrm{\Delta }}_Q(\mathbf{Q})-\mathrm{\Delta }PA$$ (4)

where Δ_Q(Q) is the vacuum ΔE contribution for the formation of the neutral and ionic complexes at a given geometry Q, from the isolated species; it and therefore ΔE_v are Q-dependent, that is, dependent on the H-bond separation $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ and angles.²⁹ (For the aqueous solution problem of interest, those coordinates will be those relevant for that problem; see below.) From the ΔE_v values in Figure 3b and ΔPA = −125 kcal/mol, Δ_Q ∼ 100 kcal/mol, a large positive value reflecting the stronger interaction in the ion pair.³⁶ Further, as the H-bond coordinate is reduced, the electrostatic ion pair interaction becomes stronger, and Δ_Q increases. The result is that ΔE_v is less negative as the H-bond length decreases, as is seen in Figure 3b upon carboxylate-protonated methylamine pair formation.

The detailed H-bond distance and proton bending angle impacts in Figure 3b can be understood via a further, extended discussion of the H-bond geometry Q dependence of ΔE_v. We begin with influence of the O_A-N H-bond separation $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$, and for this purpose consider the on-contact carbonic acid-methylamine pair in vacuum. We obtained several of these vacuum complexes by optimizing, within a B3LYP/6-311+G** level of theory and basis set calculation, the on-contact neutral pair in vacuum by constraining $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ at different values. Proton potentials at several different H-bond lengths were generated for each of these complexes, by evaluating the electronic energy (B3LYP/6-311+G**) for several proton H_A positions along the O_A–H_A direction.

These vacuum proton potentials are displayed in Figure 4a, which shows that, as $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ decreases, there is a decrease in the relative energy for the shoulder/well, with the proton located 1 Å away from N with respect to the potential minimum. The consequence of this potential variation is the fairly linear decrease in ΔE_v with increasing $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ seen in Figure 4b for each of the optimized vacuum complexes.³⁷

Figure 4.

(a) Calculated vacuum potentials along the $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ proton stretch for the carbonic acid–methylamine complex for differing H-bond coordinate $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ values, indicating the relative destabilization of the ionic component with increasing $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$. The energy minimum for each potential is set to zero to aid in the comparison. These potentials were evaluated via B3LYP/6-311+G**electronic structure calculations. (In aqueous solution, the ionic portion of these curves is stabilized, in a fashion similar to what is seen in Figure 2c of I.) (b) The vacuum gap ΔE_v versus $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ for the vacuum H-bonded acid–base complexes in panel (a). (c) ΔE_v versus the H_A–O_A–C proton bend angle for the $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}=2.7$ Å vacuum H-bonded complex in (a). Lines in (b) and (c) are linear interpolations between data points (○). See subsection 2.1 for details of the ΔE_v calculation. (d) Trajectory-averaged H-bond coordinate $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ as a function of time. (e) Trajectory-averaged ΔE_v (red line) and two bending angles: ∠CO_AH_A (black line) and ∠H_ANCH₃ (blue line) versus time (the carbon of the protonated base’s methyl group is labeled CH₃ in the Figure for convenience).

The angular dependence of the vacuum gap ΔE_v is also relevant in Figure 3b. In particular, the time scale for the high frequency fluctuations (∼20 fs) of ΔE_v in Figure 3b suggests their association with a proton bending mode within the H-bond acid–base complex (it cannot be due to the proton stretch: the 2 VB states’ proton positions are already fixed at 1 Å from either O_A or N). To explore this issue, we select the $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}=2.7$ Å vacuum complex and evaluate ΔE_v for different proton bending H_A–O_A–C angles; this is displayed in Figure 4c. The observed nearly harmonic bend angle behavior indicates the coupling between ΔE_v and fluctuations of this H_A–O_A–C bend mode, and indeed the ΔE_v minimum at ∼103° closely corresponds to the equilibrium bend angle ∼108°. The high frequency fluctuation of ΔE_v in Figure 3b before PT occurrence is evidently due to such H_A–O_A–C proton bend oscillations.

In order to analyze the situation further for the aqueous reaction of interest, it is most convenient to examine the trajectory-averaged Figure 4d and e. The former displays the H-bond evolution previously discussed, while the latter details the evolving proton angles: ∠CO_AH_A, which is most relevant for the reactant neutral pair, and the second angle ∠H_ANCH₃, which is most relevant for the product contact ion pair. Both angles can obviously be defined during the reaction, but their relative importance clearly changes as the proton is transferred from the carbonic acid’s oxygen to the nitrogen of the methyamine base. To come back to our primary purpose of interpreting the vacuum gap coordinate’s behavior, Figure 4e also exhibits the average ⟨ΔE_v⟩ during the PT reaction. Its behavior before PT is argued in Supporting Information section S4 to be the result of the combination of the H-bond coordinate and reactant angle ∠CO_AH_A (although this is a somewhat subtle effect); it is also argued there that after PT occurrence, ⟨ΔE_v⟩’s rather pronounced oscillatory behavior in Figure 4e can be understood in terms of the combination of the anticorrelated H-bond and ion pair product angle ∠H_ANCH₃.

Water Solvent Contribution ΔE_S

We now turn to the important water solvent contribution ΔE_S to the total energy gap ΔE. Figure 3c displays the solvent coordinate ΔE_S results calculated via the electrostatic method described in subsection 2.1, plotted for the fourth trajectory as the dashed line. (ΔE_S results for all 10 trajectories are plotted in Supporting Information Figure S3.) Comparison of the three panels in Figure 3 indicates, as per our expectation given the formation of an ion pair in the PT reaction, that the aqueous solvent makes a considerable contribution to ΔE during the PT reaction trajectory. An alternate ΔE_S is also displayed in Figure 3c as the solid line. This is simply the eq 3 ΔE_S definition, the difference of ΔE – ΔE_v from Figure 3a and b. The temporal behavior for each method is similar—even extending to the common fluctuations. This similarity yields similar solvent relaxation times for the two, as discussed below in subsection 4.1. The offset visible in Figure 5c between the two methods’ is due to the errors associated with each method, as outlined in Supporting Information section S3.

Figure 5.

Average over the 10 trajectories for (a) the total energy gap, ⟨ΔE⟩ (black line), the vacuum contribution, ⟨ΔE_v⟩ (blue line), and (b) the solvent coordinate contribution, ⟨ΔE_S⟩, from the electrostatic (black line) and ΔE_S = ΔE − ΔE_v difference (blue line) methods versus time; see the text for details.

3.2.2. Trajectory-Averaged Analysis of ΔE

We now analyze the time dependence of the 10 trajectory averages for the total energy gap coordinate ΔE and its components used to generate Figure 2. The averages are taken in the same manner as that for the generation of Figure 1c of I; each ΔE is averaged over the trajectories for every time point relative to the PT occurrence time. The results are shown in Figure 5, with the ΔE and ΔE_v averages given in Figure 5a and the two ΔE_S evaluations, as in Figure 3c, are given in Figure 5b. The key features and their origin will already be familiar from our earlier discussion of the single trajectory results, so that we can be relatively brief.

The negative to positive transition for ΔE, depicting the acid–base neutral pair to contact ion pair transition, is clearly evident in Figure 5a, as is the negative ΔE_v contribution due to the intrinsic acidity measure ΔPA. The positive solvent coordinate ΔE_S contribution of Figure 5b is relatively flat before occurrence of the PT and rises just during the occurrence and continues thereafter, reflecting the aqueous solvent rearrangement stabilizing the bicarbonate-protonated amine contact ion pair; clearly the major portion of the stabilization is post the PT occurrence. (One indication of a small water-complex H-bonding changes during the proton occurrence may be found in Figure S9a and b of SI.)

Finally, we can use Figure 5 to characterize the initial energy gap response, i.e. in the first 50 fs after PT response, pointed out in subsection 3.1, to the suddenly produced contact ion pair. This response includes both proton bend adjustment and water solvent relaxation (see Figure 2). Immediately after the PT occurrence, the H-bond expands and ⟨ΔE_v⟩ becomes less negative, and exhibits an H-bond vibrational period pattern (see Figure 3b), The Figure 5b dip at ∼20 fs in ⟨ΔE⟩ and ⟨ΔE_v⟩ is consistent with this. But in Figure 5a, a higher frequency motion is also apparent after the PT occurrence. The initial rise (t ∼ 10 fs) and oscillation in ⟨ΔE_v⟩ (and ⟨ΔE⟩) after this occurrence is due to alignment of the O_A–H_A–N H-bond, becoming nearly linear, which is coupled with proton bending modes, which before the PT occurrence, was carbonic acid’s H_A–O_A–C proton bending mode and after it is the protonated amine’s H_A–N–CH₃ proton bending mode. (Details are given in Supporting Information section S4.) The steady increase of the solvent coordinate contribution ⟨ΔE_S⟩ to the energy gap after the sudden PT amounts to a sort of solvation dynamics for the newly created bicarbonate-protonated amine contact ion pair. But while the PT occurrence itself is sudden, Figure 2a shows that the produced ion pair’s charge distribution continues to develop during the initial 50 fs. The Figure 5b solvation dynamics in this period is thus occurring in an evolving charge distribution; this is an important difference from standard solvation dynamics discussions.^11–28 Thereafter, ⟨ΔE_S⟩ continues to progress and further stabilize the ion pair in the presence of its now established fixed charge distribution. This relaxation continues beyond the confines of Figure 5b and is more fully characterized in the next section.

The analysis of the reaction path in this section has covered the basic essentials for the carbonic acid-methylamine PT reaction. We can, however, push the analysis to longer times in the next Section in order to probe several further interesting reaction aspects, in particular site-specific effects in the water solvent.

4. LONGER TIME ASPECTS OF THE PT REACTION: SITE-SPECIFIC SOLVATION ASPECTS OF THE PT REACTION

As we just noted, section 3 has completed the description of the major nuclear motion aspects of the H₂CO₃–NH₂CH₃ PT reaction. We can nonetheless glean further information about the reaction process by extending our study to longer times in this section. In particular, we wish to address what is happening in the water solvent at a molecular level for the PT reaction. It is the acquisition of this level of detail that requires attention to longer times than examined in previous sections, beginning with establishment of the aqueous solvent relaxation time scale.

The electrostatic version of the solvent coordinate ΔE_S is utilized in this Section (as in the bold line in Figure 5b); the alternate difference version analyzed in Supporting Information section S3 yields a similar time scale (see Figure S4 of SI). The pairwise nature of ΔE_S’s electrostatic version allows identification of contributions from each hydration shell water molecule, so that we can then analyze the solvent structure associated with the PT reaction around key sites of the acid–base H-bonded complex.

4.1. Water Solvent Relaxation Time Scale

To begin the establishment of the water solvent relaxation time scale, we replot in Figure 6a the electrostatic solvent coordinate ⟨ΔE_S⟩ for a longer time window than in Figure 5b, beginning with t = 0 (defined for each trajectory’s PT occurrence when the proton coordinate x_PT = 0); this coincides with the ⟨ΔE_S(t)⟩’s increase due to the emergence of the ion pair.

Figure 6.

Average water solvent coordinate ⟨ΔE_S⟩ as a function of time with t = 0 set for the proton coordinate x_PT = 0, locating the PT occurrence. (b) Trajectory-averaged water relaxation function Δ(t) in the presence of the H₂CO₃–CH₃NH₂ reaction system, eq 5, as a function of time (solid line) and its linear exponential fit (blue dashed line).

In order to clarify the behavior in Figure 6a, we employ the normalized nonequilibrium response function to characterize the ⟨ΔE_S(t)⟩ time dependence^14–17,23

$$\mathrm{\Delta }(t)=\frac{\langle \mathrm{\Delta }{E}_{\mathrm{S}}(t)\rangle -\langle \mathrm{\Delta }{E}_{\mathrm{S}}(\infty )\rangle }{\langle \mathrm{\Delta }{E}_{\mathrm{S}}(0)\rangle -\langle \mathrm{\Delta }{E}_{\mathrm{S}}(\infty )\rangle }$$ (5)

shown in Figure 6b. This characterizes the water solvent relaxation from the initial value ⟨ΔE_S(0)⟩, where the solvent is equilibrated with the neutral pair HOCOOH⋯NH₂CH₃ before PT, to its equilibrium value ⟨ΔE_S(∞)⟩ for the ion pair HOCOO⁻⋯⁺HNH₂CH₃; from Figure 6a, we extract the limiting averages ⟨ΔE_S(0)⟩ and ⟨ΔE_S(∞)⟩before and after PT as 17 and 45 kcal/mol.

We recall from the end of section 3 that a period of ∼50 fs after the PT occurrence at t = 0 is required for full development of the charge distribution in the product carboxylate-protonated methylamine contact ion pair. This initial evolution period automatically makes our relaxation function Δ(t) in Figure 6b different, at least at short times, from standard water^11–22 and other^23–28 solvation dynamics theory^{12,13,15,17,20–28} and experiment.^{11,14,16,18,19} Here we make the simplest estimate of the water relaxation time for our carbonic acid PT problem by simply fitting Δ(t) in Figure 6b with a single exponential function, which yields a relaxation time τ_S ∼ 120 fs.

For perspective, experimental analysis of water dynamics in a nonreactive setting yields trimodal behavior, comprising a Gaussian with an ∼50 fs (45%) time constant and a biexponential functions, with one time scale of ∼130 fs (20%) and the other at ∼1 ps (35%).¹¹ The Gaussian originates from inertial motion of water molecules, while the other two are associated with libration and slower rotation of water molecules, respectively.^11–22 Even though there is a solute charge-evolution perturbation on the ∼50 fs time scale in our PT case, it seems reasonable to associate the Δ(t) in Figure 6b time scale ∼120 fs to a significant degree, especially in view of the large body of information on the characteristic time scale for these motions, with water librations, that is, hindered rotations. We will see that we should also include H-bond compressions in the interpretation.

Due to our limited run time of 1 ps after PT occurrence, any slower exponential component at ∼1 ps such as found in water solvation dynamics experiments^{11,14,16,18,19} is missed. A key reason for this restriction is that the ion pair attempts to dissociate at that time scale. Such attempts were observed in two of our 10 trajectories (for details, see Supporting Information section S5). Since these dissociation attempts affect ΔE_S, all data past the point of the attempts were discarded in the trajectory averaging for ⟨ΔE_S(t)⟩.

4.2. Site-Specific Solvent Rearrangement

With the time scale ∼120 fs of the water solvent relaxation after PT occurrence now established, we can now ask what the contribution to these dynamics is from various sites on the reaction product carboxylate-protonated methylamine contact ion pair. We focus on three key sites, with the numbering from Scheme 1: the O_A and O_C oxygens on the bicarbonate and the NH₃ moiety carrying the transferred proton on the protonated methylamine; Table 1 shows that these have the largest charge change in the PT reaction: the proton-donating O_A and the neighboring carboxylic O_C which is in resonance with it carry the majority of the negative charge increase in the anion, and the NH₃ moiety carries the majority of the increased positive charge in the protonated base.

Scheme 1. Neutral cis–cis Conformer on-Contact Carbonic Acid–Methylamine Pair with the Atoms Labeled^a.

^aLabels are maintained throughout the PT reaction producing the contact ion pair products.

Figure 7a displays the trajectory-averaged solvent coordinate ⟨ΔE_S⟩, again evaluated via the electrostatic method due to the interaction with six water molecules, the two nearest water solvent molecules for each of the three sites. (The choice of two molecules follows from the determination that the coordination numbers around these sites are approximately two after the PT reaction (see Supporting Information section S6).) ⟨ΔE_S⟩ increases for each site, as expected from the increased charge magnitudes. The contribution for the O_A site is dominant for the initial increase of ⟨ΔE_S⟩; the water solvent molecules around O_A respond quickly after the PT occurrence, within ∼30 fs. This initial ⟨ΔE_S⟩ increase in the ∼50 fs interval reflects the initial stabilization of the transferred proton in the incipient ion pair product, as pointed out in subsection 4.1. The O_A site contribution continues to evolve past the initial period and the other site contributions increase as well. The NH₃ site contribution in particular becomes of comparable magnitude to that of the O_A site, while the contribution O_C site, with its smaller charge change, remains less important.

Figure 7.

(a) Ten trajectory averaged solvent coordinate ⟨ΔE_S⟩ contributions (from the two nearest water molecules) for the three key sites (the proton-donating O_A, its neighboring oxygen O_C, and the NH₃ moiety for the protonated base). t = 0 is the time for the proton coordinate x_PT = 0, i.e., the PT occurrence, in each trajectory. (b) Comparison of total ⟨ΔE_S⟩ and the sum of the contributions from these three key sites (O_A, O_na, O_C, and NH₃).

These contact ion pair site-specific effects clearly should be related to H-bonds of these sites to water molecules in the pair’s hydration shell. That this is indeed the case is detailed in Supporting Information section S6 via an examination of appropriate radial distribution functions, as well as of H-bond numbers and lengths (as a measure of H-bond strength). In brief, it is shown there that, with the notable exception of the proton-donating oxygen O_A, which after PT occurrence picks up a new H-bond with a water molecule, the predominant aqueous solvent effects for the other two sites result from the strengthening of water molecule–ion pair site H-bonds already present in the carbonic acid-methylamine neutral contact pair.

Finally, Figure 7b displays the averaged total solvent coordinate ⟨ΔE_S⟩ and the sum Σ of the three Figure 7a site-specific contributions to ⟨ΔE_S⟩ and that for the solvation nearest the nonreactive hydroxyl oxygen O_na (see Scheme 1). The contributions in Σ for these four sites (O_A, O_C, O_na, and NH₃) can be used to operationally define a “first hydration shell” for the water solvent relaxation for the carboxylate–protonated methylamine ion pair complex. The ⟨ΔE_S⟩ – Σ difference then provides an estimate of the second solvation shell and beyond contribution to ⟨ΔE_S⟩. While the time scales for the ⟨ΔE_S⟩ and Σ are similar beyond the initial period, the amplitude of the first solvation shell contribution is clearly more significant, but solvation of the ion pair by outer shells is not entirely negligible.

Finally, in order to understand the solvent relaxation around the product ion pair complex in more molecular detail, we have followed the time evolution of the parameters for the H-bond rearrangement of waters next to the key sites of the complex. These key sites are O_A and the NH₂ grouping which has the maximum hydration occurring after PT occurrence, as we have already observed in Figure 7. The detailed analysis in Supporting Information subsection S6.3, which analyzes the time behavior of two angles and two H-bond distances, related to the H-bonds between the acidic O_A and a single nearest water molecule and between the base’s hydrogens N each with a single nearest water. The analysis shows that these H bond rearrangements, which are basically strong H-bond formations, occur on a time scale of 200 fs or so, with fitted exponential decay times of ∼170 fs. These observations are consistent with the 200 fs predominant solvent relaxation shown in Figure 6b and its associated average solvent energy gap coordinate in Figure 6a. The angular and H-bond distance dynamics in Supporting Information subsection S6.3 indicate that the key water solvent motions on this time scale, when the relaxation has significantly completed, are (certainly coupled) water librational motions and H-bond distance compressions associated with the formation of strong H-bonds between these water molecules and the two charge sites of the product ion pair complex.

5. CONCLUDING REMARKS

In this Article, we have detailed, beyond the geometric and charge evolution description of paper I, the reaction path for the rapid, barrierless proton transfer (PT) from carbonic acid (H₂CO₃) to the strong base methylamine (CH₃NH₂) within a hydrogen (H)-bonded complex in aqueous solution, described by Car–Parrinello molecular dynamics simulations. This path demonstrates the special importance of the H-bond coordinate distance $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ between H₂CO₃’s proton-donating oxygen O_A and the proton-accepting nitrogen of CH₃NH₂ and aqueous solvation here described by an energy gap solvent coordinate.

Compression of $R_{{\mathrm{O}}_{\mathrm{A}}\mathrm{N}}$ coordinate is a key for the reaction path prior to the PT occurrence, while water solvent rearrangement assisting this PT commences only during that occurrence. Further H-bond and particularly solvent motion helps to stabilize the nascent carboxylate-protonated methylamine contact ion pair just after this, in a transient period of ∼50 fs, during which the final charge distribution of the pair is being established. In this transient period, there is also a contribution to the reaction path from the total energy gap coordinate contribution arising from the participation of the C–O_A–H or equivalently the O_A–H–N angle change associated with the transfer of the proton. The complete further stabilization of the product contact ion pair is characterized by a time scale of ∼120 fs, implicating, for example, the water librational motions in the process. A site-specific analysis of the aqueous solvent relaxation after the PT occurrence indicates that the major solvent rearrangement occurs around the proton-donating oxygen O_A of carbonic acid and the NH₃ moiety of the protonated methylamine base, an aspect also analyzed in terms of changes in the number and strength of the H-bonds between solvent water molecules and these H-bond acceptor sites. The time scales observed account for most of the water solvent relaxation and analysis shows that not only water librations are involved, but also H-bond compressions, both motions being key in the formation of new strong H-bonds of water to the two sites.

An aspect of the reaction mechanism we have not examined here or in I is issues of intramolecular and intermolecular energy transfer for vibrations in the produced ion pair. A proper treatment would require a quantization of the proton and possibly other vibrational degrees of freedom, for example, the protonated methylamine and perhaps its anion partner. This might be attempted, perhaps via existing methodology³⁸ at some later date, but goes far beyond the scope of the present study.