The Role of Electrostatic Induction in Secondary Isotope Effects on Acidity: Theory and Computational Confirmation

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Introduction
The inductive effect is defined in the IUPAC Gold Book as "an experimentally observable effect … of the transmission of charge through a chain of atoms by electrostatic induction". The initial observation that substitution of hydrogen by deuterium in the alkyl group of a carboxylic acid reduces its acidity and that a similar substitution in an alkylamine increases its basicity was attributed to electrostatic induction, in analogy to the very much larger inductive effects of chemically distinct substituent atoms and groups. 1.2.3 Accordingly, it was concluded that deuterium bound to a carbon atom is effectively more electropositive than protium. The reduced acidity of deuterioformic acid relative to normal formic acid could be attributed to differential polarization of deuterium relative to protium by the negatively charged carboxylate ion, and the reduced acidity of methyl -deuterated acetic acid to transmission of the induced negative charge to the methyl H/D atoms via the C-atom as the one-link "chain".
Analogous secondary deuterium isotope effects (SIEs) had already been observed on the rates of solvolysis and ascribed to less effective hyperconjugative electron release from deuterium than from protium 4 , 5 These SIEs, as well as others that showed up soon thereafter in a variety of reactions, were discussed in similar terms: inductive, hyperconjugative, steric and hybridization-dependent, and were reviewed by this author half a century ago 6 While it is universally recognized that isotope effects are vibrational in origin, the practice of discussing secondary deuterium isotope effects in the terms that characterize genuine electronic, stereoelectronic and steric substituent effects persists in the current physical organic and biochemical literature. 7 Charles Perrin and his co-workers have carried out an extensive re-investigation of secondary deuterium isotope effects on the acidity of carboxylic acids 8 , and the basicity of amines. 9,10 While the authors` experimental data are unexceptionable, Perrin and Flach's conclusion that "[there is] no contribution of an inductive effect to secondary deuterium isotope effects on acidity", 11 which was emphasized in Perrin's review of the field 12 and is already being cited in textbooks as established fact, 13 bears scrutiny.
The theoretical justification for assigning a dominant role to electrostatic induction in SIEs on acidity is outlined in the following section; the mathematical proof is relegated to an Appendix (Section 5). In the subsequent sections the results of computations on secondary deuterium isotope effects on three prototypical acid-base equilibria: are presented and compared with published experimental data. The computations were carried out on the protio-and deuterio isotopologs of each acid and its conjugate base, and the isotopic free energy differences (G 0 298 ) between them were correlated with their structural and polar parameters. In each case, the computations were carried out first with the bare acid and its conjugate base and then on variously hydrated reactants, in order to gain insight into the factors that modify the IE in aqueous solution. As a compromise between the high accuracy required and the large number of computations that had to be carried out routinely, they were performed with the Gaussian G03W program 14 at the MP2 level with the 6-311G** basis set, which allows for polarization of the hydrogen atoms.
Although it is possible to compute isotope effects without separating the energy into its nuclear and electronic components, as for example by Hammes-Schiffer's "nuclear-electronic approach", 16 the conventional procedure is to separate them. 17 It is assumed that the reaction occurs on a single adiabatic potential energy surface based on the Born-Oppenheimer approximation, and that relativistic effects can be neglected. If the vibrations are harmonic and the rotations are regarded as classical, all IEs can be expressed as ratios of the relevant vibrational partition functions, whichfor the isotopes of hydrogen at moderate temperaturesare dominated by differences in their zero-point vibrational energy (ZPVE). However, the fact that this approximation constrains isotopologs to have the same electronic energy does not mean that electronic phenomena cannot give rise to isotope effects. It merely requires that these effects, including those that are due to the anharmonicity of vibrational modes, are incorporated in isotope-dependent shifts of their frequencies. This is true of all so called isotopic substituent effects: inductive, hyperconjugative, steric, hybridization-dependent, etc.
The mechanism by which this occurs in the case of electrostatic polarization was demonstrated with a simple model reaction, developed in discussions with Professors Ruben Pauncz and Max Wolfsberg at the Uppsala University Summer School in Lidingö, Sweden in the summer of 1959, and summarized in Reference 6. An amplified version of the formal proof Is given below in the Appendix (Section 5) First order perturbation theory is applied to an idealized "reaction", in which an isolated vibrating H-X bond is perturbed by a positive charge placed along its axis towards its X end (Fig. 1). It is postulated that electrical anharmonicity can be neglected, i.e. that the perturbing potential (ax) is effectively linear over the vibrational amplitude. If the one-dimensional potential is purely harmonic, the potential energy curve is shifted upwards or downwards, so that the equilibrium bond length is increased or decreased (Fig 2),but its curvature remains the same. Thus, the quadratic force constant

Deuterium isotope effect on the acidity of formic acid
3.1.1 The gas phase reaction. We consider protonation of formate ion in terms of the model illustrated in Fig 1. The perturbation is the positive charge of the proton attached to its carboxylate group. In our model it is entirely electrostatic, due to interaction of the perturbing charge with the dipole of the C-H bond , vibrating in its anharmonic potential. The CH-bond dipole, , is: i.e., half the difference in atomic charge between the H and C atoms multiplied by the distance between them. In a diatomic molecule, in which the charges are equal and opposite,  would be the molecular dipole moment. We assume it to be an intrinsic property of the bond that varies monotonically between the anion and the acid, and postulate that d/dr is the same in both the acid and anion 21 This allows us to regardr e , the difference between the values of the C-H bond dipole in the acid the anion as a linear approximation to d/dr. Table 1, the long CH-bond of formate ion is shortened on protonation and its stretching frequency is increased accordingly. Correlation of the frequency change, and hence of the IE on acidity, with r e should go a long way to confirm its electrostatic origin. The Gaussian program provides two estimates of atomic charge distribution, the familiar Mulliken scale and the APT (Atomic Polar Tensor) scale. 22 According to both, the bond is polarized in the sense C δ+ H δ-, which will henceforth be referred to as positive. The Mulliken and APT charge distributions differ widely in both the acid and the anion, 23 but both schemes yield a larger C-H bond dipole in the anion than in the acid (= -0.583 and -1.537 Debye/Å respectively) Evidently, the bond-moment derivative, d/dr, is positive, i.e. the dipole moment increases with bond length. 24 Protonation of the anion interacts repulsively with the C-H dipole, reducing it, shortening the CH-bond and raising the force constant for the CH-stretching mode of formic acid (6.272 mdyn/Å) above that of formate ion (3.930 mdyn/Å). Consequently, the frequency of the CH/CD-stretching mode is increased for both isotopologs: HCOOH: 3120.7 cm -1 and DCOOH: 2319.9 cm -1 vs. HCOO : 2511.1 cm -1 and DCOO -:1828.5 cm -1 . Thus, the corresponding increase in ZPVE is less for deuterated formate ion (=246 cm -1 ) than for its protio isotopolog =305 cm -1 ), so it is the stronger base; accordingly, deuterated formic acid is a weaker acid than normal formic acid. i.e., they apply to the gas-phase reaction at 0K, at which the present computation yields pK a = -0.202 (K H K D =1.59). All of the computations predict an IE in the gas phase that is much larger than that measured in aqueous solution 25  H /K D =1.082; pK a =0.0342), where solvation evidently has a strong moderating effect.

The effect of hydration.
Hydrogen bonding of water molecules to the carboxylate O-atoms would be expected to act in the same direction as protonation, decreasing the bond's dipole moment, shortening it and raising the frequency of its stretching mode. As a consequence, the isotopic zero point energy differences should decrease as hydration of the anion becomes more efficient. Hydration of the acid molecule is expected to be less significant but cannot be ignored.
Seven hydrates of formate ion were optimized: a monohydrate (1), two dihydrates (2,3), two trihydrates (4,5) ,) and two tetrahydrates (6,7). Seven hydrates of formic acid were similarly optimized: a monohydrate (8), two dihydrates (9,10), three trihydrates (11,12,13) and a tetrahydrate (14). Seven hydrates of formic acid were similarly optimized: a monohydrate (8), two dihydrates (9,10), three trihydrates (11,12,13) and a tetrahydrate (14 pronounced than that of the anion, so it was taken as a working hypothesis that the anion hydrates (1-7) and acid hydrates (8)(9)(10)(11)(12)(13)(14) would suffice to indicate a trend. Their relevant properties are summarized in Tables 3 and 4. Table 3. Dependence on hydration of the thermochemical properties of formate ion and formate-d ion (MP2/6-311G**)    Tables 3 and 4. The slope of the plot, r e = 12.0 Debye/Å, is quite close to the value derived above for the bare anion and acid, r e = 11.2 (thin blue line). This supports the assumption that r e is an adequate approximation to the bond dipole moment derivative, d/dr, whichwithin our rather wide limits of precisionis constant over the entire range. The relative values for hydrates with the same n depend on the detailed structure of the hydrate, e.g., how large is the angle between the CH-bond and the molecular dipole moment, whether the water Formic acid hydrates G 0 hydration a ZPVE a H 0 a TS 0 a G 0 a r e (Å) molecules are H-bonded directly to the carboxylate group or to other hydrating water molecules, whether or not they are close enough to the C-H bond to interact with them directly. For example, the three points above the line (2, 4 and 7), refer to hydrates in which the C-bonded H-atom is surrounded by hydrogen atoms of the hydrating water molecules, and can be presumed to be in steric interaction with them, raising the C-H stretching frequencyand no-doubt the bending frequencies as wellabove the increase caused by electrostatic induction.   Tables 3 and 4, of the isotopic free energy difference , G 0 , against the CH-bond dipole moment, Mulliken) 30 . Although G 0 and are plotted for convenience along the X and Y axis respectively,is clearly the independent and G 0 the dependent variable. The plot is reasonably linear; the rather low coefficient of determination (R 2 = 0.88) is due to the three anomalous formate points: 2, 4 and 7. The formic acid points are not clumped together as closely as in Fig. 3. The points for monohydrate (8) , dihydrate (10) and trihydrate (13) are very close to one another, but those for dihydrate (9), trihydrates (11) and (12), and tetrahydrate (14) extend to higher values of G 0 than would be expected on the basis of alone. Note that in all of the latter four hydrates H-atoms of the bound water molecules are contiguous to the H-atom of the C-H bond. The consequent steric repulsion is evidently responsible for raising the frequencies of the C-H vibrations. Steric repulsion also appears to be the reason why, unlike 10 and 13, none of the sterically hindered polyhydrates has a significantly larger free energy of hydration than the monohydrate (8).
The linear correlation between  and G 0 illustrated in Fig. 4 constitutes strong evidence that hydration reduces the magnitude of the SIE by moderating the electron demand of the carboxylate group in the formate ion, thus reducing its C-H stretching frequency. Departures from linearity occur for those hydrates of formate ion (2, 4 and 7).and formic acid (9, 11, 12 and 14) in which the ZPVE is increased by superposition of a steric isotope effect.
In order to compare the results of gas phase computations on the various hydrates of formic acid and formate ion with the experimental results, it must be decided which of the specific hydrates of the acid and anion are most likely to be present in dilute aqueous solution. We recognize that hydration of the anion is exergonic, favoring the higher hydrates. However, the standard free energy of hydration for the equilibrium: refers to the gas phase reaction at 25 0 C and standard pressure, and does not take into account clustering of the n water molecules, as would occur in the liquid phase. Therefore, computations at the same computational level were carried out for a representative water dimer, trimer and tetramer. The respective free energies of dimerization, trimerization and tetramerization of water are: +0.856, -1.323 and -5.091 kcal/mol. They are much smaller in magnitude than the corresponding free energies of hydration of the anion (Table 3), so they do not upset the sequence favoring the tetrahydrates in aqueous solution. It follows that the dominant anion in solution would be a tetrahydrate. We note that 6 is, in effect, dihydrate 3 externally hydrated by two water molecules, whereas the four water molecules in 7 are H-bonded directly to the carboxylate group, filling its inner hydration shell and leaving room for further hydration. It can thus be assumed that 7 would be a better choice than 6 as the main form of the anion in solution. External hydration of 7 to form a loose hexahydrate, would be expected to raise G 0 slightly above the computed value, 2.089 kcal/mol.
It would not be correct to conclude similarly that the main form of the acid in solution would also be the tetrahydrate.
In dilute aqueous solution the equilibrium in Equation 1 would indeed be shifted to the right, favouring the higher hydrates, However, from the values in Table 4, the standard free energy difference for the equilibrium: (6) is: G 0 298 = 6.75 kcal/mol. According to the model computations on water clusters mentioned above, the standard free energy difference for the equilibrium: is ca. -3.75 kcal/mol. Thus, to the extent that extrapolation to the liquid phase is justified, the tetrahydrate (14) would be disfavoured relative to the most stable trihydrate (13) by over 10 kcal/mol, enough to guarantee that the latter would be the predominant hydrate of formic acid even in dilute solution.
Adopting the values G 0 298 = 2.164 kcal/mol for the acid (13 in Table 4) and 2.089 kcal/mol for the anion (7 in Table 3), we obtain G 0 298 = 0.075 kcal/mol (pK a = -0.055; K H K D =1.135), only slightly larger than the experimental result for the dissociation of formic acid in water (pK a = -0.034; K H K D = 1.082). As noted above, external hydration of 7 would be expected to bring it even closer.

Digression.
As noted above, the average potential energy at the zeroth level of an anharmonic oscillator is close to half of the ZPVE. The Gaussian program provides for computation of the molecular geometry averaged over the ground vibrational level and of the potential energy of the molecule frozen in that geometry. Table 5 lists these values for the isotopologs of formic acid and formate ion. The corresponding bond dipole moments are estimated with the equation:

Deuterium isotope effect on the acidity of acetic acid
3.2.1. The gas phase reaction. The alkyl CH bond is normally polarized C δ--H δ+ (negative) and its dipole derivative is negative, i.e., its magnitude increases on compression. 31 It is reversed to C δ+ -H δwhen attached to a sufficiently electronegative substituent, in which case the dipole derivative changes sign: the bond dipole increases with expansion and decreases with compression, as in formic acid. Therefore, whether the C-H dipole is positive or negative, bond extension shifts negative charge from carbon to hydrogen and contraction shifts it from hydrogen to carbon. Table 6 lists the values of the projection on the C-C axis of the CH 3 -group dipole moment,  g  computed with Mulliken and APT charges as the vector sum of the three CH-bond moments. Despite the disparity between the dipole moments computed for each species separately, both scales yield the same difference between the acid and its conjugate base: a reduction of 0.3 Debye. The mean equilibrium bond length of the three methyl bonds, r m , is some 0.004Å shorter in the acid than in the anion, so  g r m the quantity analogous to the bond dipole moment ,r e , would be an enormous 75 Debye/Å.  Nevertheless g r m cannot be regarded as the methyl group's dipole moment derivative, because the variation in length of the C-H bonds within each methyl group is no smaller than the difference between the two mean values (e.g., in the acid: r e = 1.0881, 1.0931, 1.0931 Å). Nor does it take their varying angles relative to the C-C axis into account, and moreover, ignores the contribution of the CH-bending modes to  g ; which is difficult to isolate because the bending modes mix with CH-stretching modes of the same symmetry speciesparticularly in the deuterated isotopologsas well as with skeletal modes of appropriate symmetry and energy. Early LCAO-MO computations suggest that varying hybridization due to change of the CH-bending angle does not shift electronic charge markedly, 32 so the CH-bending modes need not be considered separately in the present context. It should be noted that they achieve crucial importance when there is an empty p-orbital on the adjacent carbon atom, as in CH 3 CH 2 + , in which case they are a major contributor to the hyperconjugative SIE. 33 Protonation of the acetate ion interacts favourably with each the C-H bonds shortens it and lowers the potential well in which it is vibrating butif the potential were harmonicits vibrational force constant would not be affected. Since the well is anharmonic and the dipole derivative is negative, the extension of the bond is disfavoured relative to contraction, the curvature of the potential well increases in order to reduce its amplitude and increase its force constant. The frequency of the methyl stretching modes and their isotopic differences are increased, as illustrated in Table 6 for the "symmetric" stretch 34 .
The computed thermochemical data for acetic acid and acetate ion are listed in Table 7. As in formic acid, the IE is dominated by the ZPVE differences, but the direct contribution of the CH 3 /CD 3 stretching modes is only about 33%. The entropy of both the isolated acid and its conjugate base rises substantially on thermal excitation from 0 to 298 K. It is higher for the deuterated isotopologs of both, due to the lower frequencies of their methyl modes, and is only partially canceled by the concomitant increase in enthalpy. Nevertheless, the influence of entropy on the isotope effect is slight. The contribution of the ZPVE to the isotope effect (0.179 kcal/mol) is somewhat larger than that computed by Perrin and Dong 8 with B3LYP/6-31G* (0.152 kcal/mol), from which they obtain pK a = -0.111. The IE derived fromthe G 0 value in Table 7 (pK a = -0.103) is fortuitously close to theirs, but both are much larger than the experimental value, pK a = -0.0139. 25 Evidently, here too solvation reduces the IE on acidity drastically.

The effect of hydration.
One monohydrate (15) and two dihydrates (16,17) of acetate ion were optimized, as were one monohydrate (18) and two dihydrates (19,20) of acetic acid. The computation of trihydrateslet alone tetrahydrateswould have been excessively time consuming. Nevertheless, the results list ed in Tables 8 and 9  Hydration of both acid and base is exergonic, more strongly so in the latter. Here too, the main factor is the electrophilic interaction of successive water molecules with the anion, a partial protonation that raises the CH-stretching frequencies and increases PVE and G 0 .  Table 9. Dependence on hydration of the thermochemical properties of acetic acid and acetic-d 3 acid (MP2/6-311G**) Acetic acid hydrates G 0 hydration a, b  The striking similarity of the hydrates of acetate to those of formate, makes it probable that the predominant species in solution would be a tetrahydrate homologous to 7, in which the inner solvation shell is complete. In order to make an educated guess as to its thermochemistry, we draw a semi-quantitative analogy with formate, as follows: G 0 of formate increases by 0.103 kcal/mol on going from the bare ion to the more stable dihydrate 3. This is 56% of the total increase to tetrahydrate 7 (0.185 kcal/mol). When the same ratio is adopted for acetate, the increase to dihydrate 17 (0.064 kcal/mol) suggests that G 0 of acetate would increase by 0.115 kcal/mol to 6.39 kcal/mol or thereabouts for the tetrahydrate, perhaps a bit higher due to external hydration.
In 18, as in formic acid monohydrate (8), hydration is ambiphilic. Here too electron attraction is more effective than electron donation, so the C-H stretching frequencies are raised slightly andwith them -the isotopic differences in ZPVE and free energy. The second water molecule in dihydrate 19 acts only as an electrophile, whereas both water molecules in the more stable dihydrate 20 are ambiphilic and are loosely bonded to one another. G 0 of 18 and 20 is a virtually identical 6.43 kcal/mol. Turning back to Table 4, we note that the corresponding monohydrate (8) and dihydrate (10) of formic acid also have same G 0 value (2.16 kcal/mol). So does the trihydrate (13), which was recognized above as probably being the prevalent hydrate of formic acid in solution. It can reasonably be assumed that G 0 of the homologous trihydrate of acetic acid will also be close to that of 18 and 20: 6.43 kcal/mol. On the assumption that the anion in solution is predominantly a tetrahydrate analogous to 7 and the acid a trihydrate analogous to 13, the estimated value ofG 0 298 for the IE on the ionization of acetic acid in aqueous solution becomes -0.04 kcal/mol (pK a = 0.03; K H /K D =1.07), and would be reduced a bit more by external hydration of the tetrahydrated anion. This is as close to the experimental value (pK a =-0.0139; K H /K D =1.032) as can be anticipated.
A feature that has not been taken into account is the effect of hydrophobic interaction between the methyl hydrogen atoms and water molecules that are not H-bonded to the carboxylate or carboxyl group. RISM computations referred to above 29 indicate that a large number of water molecules are involved in it-on the average 8.9 in the acid and 10 in the ion. While the internal motions of these molecules may well contribute to the enthalpy and entropy of solvation, they evidently do not make a significant contribution to the isotope effect on the acidity of acetic acid.. a Projection the C-C axis; b Mean equilibrium C-H bond length c Frequency of "symmetric" C-H-stretching mode.

The gas phase reaction
As shown in Table 10, the bond dipole moments computed with the Mulliken and APT charge distributions differ not only quantitavely but in their direction as well. For both the base and the cation the dipole is negative (C δ-H δ+ ) according to the former and positive (C δ+ -H -) according to the latter. We consider the protonation of the amine separately from each of the two points of view.: 1. Mulliken: The charge-dipole interaction at the methyl carbon on protonation is attractive. The moderately increased magnitude (0.225 Debye) of the (negative) group dipole moment indicates that electron release from the methyl H-atoms is greater than from the C-atom, even though they were less negatively charged to begin with. Since g is positive and  g /r m is negative, the CH-bonds become shorter and the methyl stretching frequencies are increased.

APT:
The charge-dipole interaction is repulsive. The greatly reduced magnitude (0.657 Debye) of the (positive) group dipole moment indicates substantial electron release from the negatively charged H-atoms to the positively charged C-atom.
Since both  and d/dr are positive, the decreased dipole moment is accompanied by shorter CH-bonds and higher methyl stretching frequencies.
Thus, whichever of the two schemes is correct, both produce the same qualitative result: a net shift of negative charge within the methyl group from hydrogen to carbon, a consequent increase in the CH-stretching frequencies, and an inverse isotope effect. Therefore, in this context, the expression "deuterium is effectively more electron-releasing than protium" is not a solecism. The computed thermodynamic properties are summarized in Table 11.  (23) by successive H-bonding to a water molecule of each of three N-bonded H-atoms. The trihydrate is threefold symmetric, like the parent anion, but the H-bonded water molecules are tilted away from strict C 3v symmetry into either of two C 3 enantiomers.
Methylamine is hydrated to a stable monohydrate (24) and a one stable dihydrate (25). A second dihydrate that optimized with one imaginary frequency and a trihydrate with three will be disregarded.

22 23 24 25
The relevant thermochemical properties of the methylammonium ion and its conjugate base are summarized in Tables 12 and  13 respectively.    30% larger than the experimental value of the IE on the basicity of methylamine in aqueous solution (G 0 298 = 0.070 kcal/mol, pK b = 0.051). 9 The contribution of entropy of solvation to the isotope effect is negligible. Thus, gas phase computation of the hydrated species provides a good approximation to the secondary isotope effect on amine basicity in aqueous solution.
These results are consistent with RISM computations 35 indicating that, in solution, methylamonium ion forms three hydrogen bonds to water, and methylamineon averageforms 2.5. For both the base and the cation, these computations also indicate the presence of many water molecules in hydrophobic interaction with the methyl group. Here too, as in in the case of acetic acid, its contribution to the isotope effect is evidently negligible.

Conclusions
The mechanism by which electrostatic induction alters the frequency of C-H/C-D stretching modes and produces secondary isotope effects, is described in terms of a one-dimensional model. Within the constraints of first-order perturbation theory, the interaction between a perturbing charge and the polar bond has two separate effects: 1. A lengthening or shortening of the equilibrium bond length (r e ) and an upward or downward shift of the potential energy minimum depending on whether the interaction is attractive or repulsive.

2.
In an anharmonic potential, an increase or decrease of the harmonic force constant, according to whether the sign of the bond dipole derivative, ddr, is positive or negative.
This model is applied to the protonation of formate ion. The computations show that C-H bond is shorter in formic acid than in the formate ion and its dipole moment is smaller; evidently d/dr is positive.r e , the difference between the CH-bond dipole moments of the acid and its conjugate base divided by the difference between their equilibrium bond lengths, is thus a linear approximation to d/dr. Protonation increases the vibrational frequency of the bond andin consequence -the isotopic differences in its zero-point energy (ZPVE) and Gibbs free energy(G 0 ) increase as well, producing a normal isotope effect on acidity: HCOOH is a stronger acid than DCOOH. The inductive origin of the isotope effect is confirmed by computations on various hydrates of formic acid and formate ion. The slope of a linear plot of  vs. r e is in good agreement with r e , and a plot of G 0 vs.is also adequately linear. In both plots, the linear correlation is improved when steric interaction between the C-H bond and the H-atoms of hydrating water molecules in several of the hydrates is taken into account.
Similar computations were carried out on the acid-conjugate base pairs; CH 3 CO 2 H-CH 3 CO -2 and CH 3 NH 3 In both cases the methyl C-H bonds are shorter in the acid than in than in its conjugate base and their frequencies higher, leading to a normal SIE on the acidity of acetic acid and an inverse SIE on the basicity of methylamine. Bond shortening is accompanied by a reduction of  g , the projection of the methyl-group dipole moment on the C-C axis, representing a shift of electron density from H to C. The plot of  g against G 0 of hydration is adequately linear when steric interactions are taken into account.
The SIEs computed with the anhydrous acids and their conjugate bases are larger than the published experimental values in solution; much larger for the carboxylic acids, slightly larger for methylamine. Binding an increasing number of water molecules, especially to the ionic component, reduces the isotope effect. In all three cases: the SIE computed with the polyhydrates in the gas phase at 298K is reduced to within a factor of two of the experimental value in aqueous solution. The contribution of entropy of solvation to the SIE is negligible. Hydration reduces the SIE primarily by modifying electron release 18 | J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 2012 from the carboxylate group in the first case and electron demand from the ammonium group in the second, confirming its inductive origin.

Appendix
The essential features of the proof in Reference 6 are as follows: If a harmonic oscillator (½kx 2 , where x = r-r e ) is acted upon by a linear perturbation (V e + ax), the potential energy function becomes: V' = V e + ax + ½kx 2 (A-1) We note immediately that the force constant, i.e. the second derivative of the potential energy, is unchanged: Since 12ga is much smaller than k 2 , the square root can be expanded in good approximation to (1-6ga/k 2 ), yielding, after substitution and collection of terms, the same displacement to the new minimum derived above for the harmonic potential: x -a/k. Introducing the new value into Equation (A-6), we obtain: The harmonic force constant, vibrational frequency and isotopic zero point energy difference have thus been changed by the linear perturbation, but only because the original potential energy function includes a cubic anharmonic term.
The nature of the perturbation, implied in Reference 6 but not stated there explicitly, is charge-dipole interaction. For the interaction of a charge q and dipole  aligned as in Figure 1 The minute change in 1/ r 2 between the equilibrium lengths of the perturbed and unperturbed C-H bond is negligible, so C/r 2 can be seen as a constant coefficient.
The second term in brackets is of the order of 1 Debye/Å, whereas d/dr is somewhere between 10 and 30 Debye/Å, depending on the charge distribution scheme used to estimate it. It follows that the coefficient a, whichalong with gis responsible for the isotope effect, can be regarded, for the semi-quantitative purposes of the present paper, as directly proportional to the dipole moment derivative.: a C' d/dr (A-11) It is an implicit assumption of the model that the bond-dipole moment derivative is an intrinsic function of bond length that is not affected by the perturbation, i.e. that charge-induced dipole interaction can be neglected, so that the same value of d/dr applies to both the perturbed and unperturbed bond.