Hydrogen Bond

N. D. Sokolov

Contents

§ 1. Brief historical survey . . . . . . . . . . . . . . . . . . . 205
§ 2. General information on the hydrogen bond . . . . . . . . . 208
§ 3. The simplest manifestations of the hydrogen bond . . . . . 213
§ 4. Spectroscopic manifestations of the hydrogen bond (vibrational spectra) . . . . . . . . . . . . . . . . . . . . . 219
§ 5. Inadequacy of the simple electrostatic concept of the hydrogen bond . . . . . . . . . . . . . . . . . . . . . . . . 231
§ 6. Quantum-mechanical treatment of the hydrogen bond . . . . . 236
§ 7. Calculation of the vibrational frequencies of the AH group in a complex with a hydrogen bond . . . . . . . . . . . . . . 243
§ 8. On the origin of the band in the vibrational spectrum of the hydrogen bond . . . . . . . . . . . . . . . . . . . . . 249
§ 9. Other physical manifestations of the hydrogen bond (electronic spectra. Magnetic spectra) . . . . . . . . . . . . . . 254
§ 10. The hydrogen bond and proton-transfer processes . . . . . . 261

Cited literature . . . . . . . . . . . . . . . . . . . . . . . . 275

§ 1. Brief Historical Survey

In nature there are quite many substances in which a hydrogen atom is capable of joining two atoms of other elements, belonging either to two different molecules or to one and the same molecule. This is indicated by a number of peculiarities possessed by these substances. These include association of molecules in the liquid phase and the related properties of liquids, elevated melting and boiling temperatures, “anomalies” in solubility and saturated-vapor pressure, peculiarities in spectra, and so on. The elements capable of being joined by means of a “hydrogen bond” are chiefly oxygen, nitrogen, and fluorine, and in some cases also certain others (for example, chlorine and sulfur).

The hypothesis that a hydrogen atom could be connected simultaneously with two other atoms was first put forward by M. A. Ilyinsky in 1887 in connection with his ideas on the divisibility of the valence of elements¹. At approximately the same time, experimental data began to accumulate on the molecular weight of various compounds,

N. D. SOKOLOV

determined by measuring the freezing or boiling temperature, as well as the saturated vapor pressure. It was observed that in such solvents as benzene, the molecular weight of substances containing hydroxyl groups noticeably exceeds the molecular weight corresponding to the chemical formula of the given compound, which indicates the presence of strong association. At the same time, in substances in which the hydroxyl OH is replaced by the alkoxy group OCH\(_3\), association is absent. In 1891 Nernst noted the differing capacity of substances for association depending on the nature of the solvent\(^2\). Considering the question of the distribution of benzoic acid between benzene and water, he concludes that in benzene the acid exists in the form of double molecules, and in water—in the form of single molecules. Beginning in 1893, Auwers\(^3\) published systematic studies of the association of various compounds and established a number of important regularities; in particular, he showed that not only substances containing hydroxyl are capable of association, but also substances containing the NH group. However, all these facts clearly did not fit within the framework of the valence theory that existed at that time and therefore were cut off from the development of the theory of chemical structure as a whole. Only at the beginning of the 20th century, with the appearance of Werner’s coordination theory, were they interpreted from the point of view of this theory, as a manifestation of the divalency of the hydrogen atom. The bond of a hydrogen atom, for example, of a hydroxyl group with a second oxygen atom was treated by Werner\(^4\) as a coordination bond and was represented by him in the following way:

\[ \begin{array}{c} \mathrm{R}\\[-0.2em] \diagdown\\[-0.2em] \mathrm{O}\;-\;-\;-\;\mathrm{H}\cdot\mathrm{X}.\\[-0.2em] \diagup\\[-0.2em] \mathrm{R} \end{array} \]

Coordination theory facilitated the systematization of the facts then available and formed the basis for the further development of ideas about the hydrogen bond.

From the standpoint of coordination theory, Pfeiffer\(^5\) investigated various intermolecular compounds, and in interpreting them assumed that the hydrogen atom possesses secondary valence. In particular, in 1914 he was the first to propose the now firmly established cyclic formula for dimers of carboxylic acids

\[ \mathrm{RC} \begin{matrix} \,\,\displaystyle \mathop{=}^{\mathrm{O}\cdots\mathrm{H}-\mathrm{O}}\\[-0.2em] \displaystyle \mathop{-}_{\mathrm{O}-\mathrm{H}\cdots\mathrm{O}} \end{matrix} \mathrm{C}\cdot\mathrm{R}. \tag{1} \]

Moore and Winmill\(^6\) in 1912 explained the incomplete ionization of aliphatic amines in water by the existence of complexes of the type

\[ \mathrm{R}_3\mathrm{N}\ldots\mathrm{H}-\mathrm{O}-\mathrm{H}. \]

In 1909 Hantzsch introduced into organic chemistry the concept of an intramolecular hydrogen bond. Later he,^7 and also Pfeiffer,^8 on this basis explained the structure and properties of a number of organic compounds, such as, for example, the enolic form of acetoacetic ester

\[ \begin{array}{c} \mathrm{O}\\[-0.3em] /\ \backslash\\[-0.3em] \mathrm{CH_3C}\quad \mathrm{H}\\ \vert\quad \vdots\\ \mathrm{HC}\quad \mathrm{O}\\ \backslash\ \mathrm{C}//\\ \vert\\ \mathrm{OC_2H_5}. \end{array} \]

Huggins,^9 and then Latimer and Rodebush,^10 gave the first electronic interpretation of the hydrogen bond on the basis of Lewis’s valence theory. Starting from Werner’s coordination theory, they proposed that the hydrogen atom in compounds with a hydrogen bond can simultaneously hold around itself two electron pairs and thus form covalent bonds with two electronegative atoms. Schematically this was represented as follows:

\[ \mathrm{H\ddot{F}:H:\ddot{F}:,} \]

where the dots denote electrons. This representation was widely used in numerous works by Sidgwick to explain various physicochemical properties of substances with hydrogen bonds (see,^11 where the relevant literature is given). In 1928 Pauling^12 criticized this point of view, asserting that the hydrogen bond is due to ionic forces. The latter point of view soon became dominant in the literature.

With the development of the newest physical methods of investigation, new manifestations of the hydrogen bond were discovered, and thereby new evidence of its existence was obtained and its properties were more fully studied. Thus, for example, the results of X-ray structural analysis of crystals not only attested beyond doubt to the presence, in certain cases, of hydrogen bonds, but also gave valuable information about its geometry. Pauling and Brockway^13 established by electron diffraction that dimers of formic acid in the gas phase do indeed have structure (I), which had been proposed for carboxylic acids earlier by Pfeiffer.

Spectroscopic investigations of complexes with a hydrogen bond made it possible to discover highly characteristic and important manifestations of it. The first systematic investigations of vibrational

spectra of complexes with a hydrogen bond, which yielded important results, were carried out by Hilbert, Wulf, Hendricks, and Liddel[^14], and by Errera and Mollet[^15] (see also[^16]). Subsequently the number of works devoted to the study of the hydrogen bond by means of infrared spectra and Raman spectra rapidly increased, and at present there is a very extensive literature on this question*). The first studies of the electronic spectra of compounds containing an intramolecular hydrogen bond belong to Bell and co-workers[^20] and to N. A. Valyashko[^21].

Spectroscopic studies showed that the chemical bonds and the electronic structure of molecules and of individual groups joined by the hydrogen bridge A—H...B undergo characteristic changes. When a hydrogen bridge is formed, the frequency, width, and intensity of the spectral band characteristic of vibrations of the AH group change strongly; in a number of cases the absorption band in the electronic spectrum is shifted appreciably toward the long-wave side.

Important information on the properties and nature of the hydrogen bond has also been obtained by measuring the dielectric constant, molecular refraction, and other physical properties of substances containing hydrogen bridges.

§ 2. GENERAL INFORMATION ON THE HYDROGEN BOND

As indicated above, hydrogen bonds A—H...B are formed predominantly in those cases where A and B are O, N, F, and sometimes Cl and S. In some molecules with an intramolecular hydrogen bond, Br and I also act as B. The C—H bond can enter into the hydrogen bridge C—H...B in very rare cases, namely when the C atom is attached to a strongly electronegative group, as, for example, in hydrogen cyanide (N≡C—H) or chloroform (Cl₃C—H).

The distances between atoms A and B joined by a hydrogen bond are appreciably shorter than in the absence of the latter. For example, the O...O distance in RO—H...OR₁ complexes is, for different substances, approximately from 2.5 to 2.8 Å. The O...O distance in the absence of a hydrogen bond, when the molecules ROH and OR₁ interact by means of ordinary van der Waals forces, may be estimated as follows. The O—H distance is equal to ~1.0 Å, and the van der Waals radii of the H and O atoms are respectively 1.2 Å and 1.4 Å; consequently, the O...O distance in this case would be ~3.6 Å, i.e., considerably greater than in reality. This fact directly testifies that

*) For a literature survey up to 1946 inclusive, see[^17]. See also[^18],[^19].

forces responsible for the formation of hydrogen bonds considerably exceed the forces of ordinary van der Waals interaction. Experience shows that if the distance A...B is significantly greater than some distance characteristic for the given pair of atoms, then no hydrogen bond A—H...B is formed between them. Consequently, the possibility of sufficient approach of atoms A and B is one of the conditions for the formation of a hydrogen bond. Table 1 gives values of the lengths A...B for some hydrogen

Table 1

Distances A...B in hydrogen bonds22

Compound	A—H...B	Distance A...B (Å)	Group affiliation A—H	Group affiliation B
Acetylene-dicarboxylic acid·2H₂O	O—H...O	2.56	Carboxyl group	Water
Oxalic acid	O—H...O	2.50	Carboxyl group	Carbonyl group
Acetylglycine	O—H...O	2.56	Carboxyl group	Carboxyl group
Salicylic acid23	O—H...O	2.63	Carboxyl group	Carboxyl group
Cytidine	O—H...O	2.59	Hydroxyl	Hydroxyl
Cytidine	O—H...O	2.74	Hydroxyl	Hydroxyl
Cytidine	O—H...O	2.83	Hydroxyl	Hydroxyl
Hydrogen peroxide	O—H...O	2.78	Hydroxyl	Hydroxyl
Urea·H₂O₂	O—H...O	2.63	Water	Water
Ice	O—H...O	2.76	Water	Water
Cytidine	O—H...N	2.87	Water	Water
Acetamide	N—H...O	2.83
Acetamide	N—H...O	2.99
Cytidine	N—H...O	2.93
Cytidine	N—H...O	3.00
Urea	N—H...O	2.99
Urea	N—H...O	3.04
Ammonium azide	N—H...N	2.94
Ammonium azide	N—H...N	2.99
Melamine	N—H...N	3.00
Melamine	N—H...N	3.02
Melamine	N—H...N	3.05
Melamine	N—H...N	3.10
Hydrazine·2HCl	N—H...Cl	3.10
Meta-toluidine·2HCl	N—H...Cl	3.10
Meta-toluidine·2HCl	N—H...Cl	3.22
Meta-toluidine·2HCl	N—H...Cl	3.26
NH₄F	N—H...F	2.63
KHF₂	F—H...F	2.26
HF23a	F—H...F	2.49

bonds, taken as examples from the summary[^22]. From Table 1, in particular, it follows that the distance A . . . B for one and the same pair of atoms A, B may vary appreciably on going from one substance to another, and for a given substance—depending on the nature of the atomic groups of which they are a part.

In principle there are two possibilities for the position of the H atom in the complex A—H . . . B.

1) The H atom is bound to A much more strongly than to B; for transfer of the proton from A to B and the formation of the complex A− . . . (HB)+, expenditure of energy is necessary. The potential energy of the system A—H . . . B as a function of the position of the proton is then represented by a curve having two minima separated by a more or less high barrier (Fig. 1).

Fig. 1. Potential curve of the system A—H . . . B with two wells.

Fig. 2. Potential curve of the system A—H . . . B with one well.

2) The H atom in the complex A—H . . . B is bound to A and B approximately to the same extent; consequently, when AH and B approach one another and the complex is formed, the A—H bond undergoes strong stretching. The dependence of the potential energy on the position of the proton will then be expressed by a curve with one minimum (Fig. 2).

According to experimental data, the hydrogen bond is usually much weaker than the covalent bond A—H (for example, for water and alcohols the energy of the H . . . O bond is approximately 20 times less than the energy of the O—H bond); moreover, as is known, the formation of a hydrogen bond upon the approach of A—H and B requires no activation energy. Taking these facts into account, one may expect in advance that the length of the A—H bond in the complex A—H . . . B differs comparatively little from the length of the isolated A—H bond and that the distance H . . . B, on the contrary, is comparatively large. Thus, for example, for the case of the linear complex O—H . . . O we find
\(R_{\mathrm{H}\ldots\mathrm{O}} \approx R_{\mathrm{O}\ldots\mathrm{O}} - R_{\mathrm{O-H}} \approx 1.7\) Å.

Experimental data confirm these expectations. As Pauling showed[^23a], the value of the residual entropy of ice, in which—

HYDROGEN BOND

of the molecule are linked by hydrogen bonds, is consistent with the notion that for each proton there are two potential wells (Fig. 1) and is incompatible with the idea of the existence of a single potential well (Fig. 2).

The new method for studying the structure of molecules by means of neutron diffraction, developed over the last 10 years, makes it possible to determine more accurately the position of the proton in hydrogen bonds. By this method, in the case of water, the validity of the first assumption was definitively proved^24. The same conclusion is reached by the application of X-ray structural analysis, whose development in recent years has made it possible, by measuring the distribution of electron density, to locate hydrogen atoms approximately*). In Fig. 7,a (see below) the internuclear distances in crystalline salicylic acid are given according to Cochran’s data^23. The interatomic distance O—H in the presence of a hydrogen bond differs little from its value in isolated molecules (~0.97 Å). An analogous result was obtained for the N—H bond by the method of nuclear magnetic resonance^24a.

In all probability, in the overwhelming majority of cases the length of the A—H bond changes only slightly upon formation of a hydrogen bridge. (On this question see, for example,^19, ^22.)

However, as an exception, there exist systems whose potential energy has one potential well. Such systems apparently include the ion \((\mathrm{FHF})^{-}\)^25, as well as the nickel dimethylglyoxime molecule^25a, ^46a.

Since, in the formation of a hydrogen bond, the H atom plays a specific role, strictly speaking, as a necessary criterion for the presence of this bond one should take not the distance A . . . B, but the distance between the atoms H and B.

In the case of an intermolecular hydrogen bond, the mutual orientation of the molecules is determined mainly by the latter and therefore the atoms A, H, and B in the complex A—H . . . B are, apparently, always located practically on one straight line, and the two criteria essentially coincide. If, however, an intramolecular hydrogen bond is in question, then the position of the H atom and the distance H . . . B are determined mainly by the structure of the corresponding part of the molecular skeleton, which probably can change only very slightly under the influence of the hydrogen bond. In the case where the H atom is situated not far from the straight line passing through A and B, for judging the possibility of bond formation it is still sufficient to know the distance A . . . B, which should have approximately the same values as in an intermolecular hydrogen

*) The error in determining the position of H atoms by this method is probably about 0.1 Å. Therefore the accuracy of the values of the O—H and H . . . O bond lengths given in Fig. 7,a is apparently overestimated.

bond. If the H atom is located to the side of the indicated straight line, it is necessary to have information about the distance H...B itself, which can often be determined from structural data.

The energy of the hydrogen bond, i.e., the energy necessary for carrying out the process \(RA — H...BR_1 \to RAH + BR_1\), is usually from 4 to 8 kcal. This energy is small in comparison with the energy of covalent bonds, amounting to 50–100 kcal, but exceeds the energy of an ordinary van der Waals interaction, amounting to 2–3 kcal.

An exact determination of the energy requires refined experimental technique and has been carried out for very few systems. A rough idea of the magnitude of the hydrogen-bond energy \(\varepsilon\) can be obtained directly from data, for example, on the heat of vaporization of liquids in which the interaction of molecules is due mainly to hydrogen bonds. Thus, in the case of water, in which each molecule is linked to others by means of two hydrogen bridges, to determine the energy of rupture of one of them the molar heat of vaporization (10.4 kcal) should be divided in half; in this way we find \(\varepsilon \simeq 5.2\) kcal. The most accurate measurements of the energies of hydrogen bonds have been made by the spectroscopic method. Table V (p. 229) gives values of \(\varepsilon\) for certain substances.

There is no single opinion in the literature on the question of the nature of the hydrogen bond. Two points of view exist. As mentioned above, according to the most widespread opinion, the hydrogen bond is due to the simple electrostatic attraction of dipoles or residual charges of the interacting groups and, for its explanation, does not require taking into account the quantum properties of electrons. According to another point of view, the indicated forces, although they do play an important role in the formation of the hydrogen bond, are in principle insufficient to explain all its manifestations. We shall return to this question below (§§ 5, 6). Here the following should be noted.

All the principal manifestations of the hydrogen bond may be divided into two groups. One of them includes those manifestations which are not connected with the nature of the hydrogen bond and for the explanation of which it is sufficient only to assume that the H atom under certain conditions can connect two other atoms, as a result of which a stable complex is formed. The study of this group of manifestations of the hydrogen bond can obviously give little information about its nature. This includes the simplest phenomena directly due to the association of molecules: an increase in the apparent molecular weight of a substance, the influence of the hydrogen bond on the temperature and heat of melting and boiling, on solubility, on the saturated-vapor pressure, on the heat of sublimation, etc. Also belonging here are the corresponding phenomena due to an intramolecular hydrogen bond.

Another group comprises those manifestations of the hydrogen bond on which its nature has a substantial effect, and for the explanation of which, in addition to the hypothesis of association of molecules (or molecular groups), a more or less detailed conception of the electronic structure and of the “mechanism” of formation of the hydrogen bond is necessary. The study of this group of phenomena can obviously provide the basic information about its nature. These include, above all, spectroscopic, as well as certain other, manifestations of the hydrogen bond.

§ 3. THE SIMPLEST MANIFESTATIONS OF THE HYDROGEN BOND

As mentioned above, already at the end of the nineteenth century, in cryoscopic and ebullioscopic measurements it was observed that the molecular weight of certain substances containing OH and NH groups is noticeably greater than the value that follows from the corresponding chemical formulas, whereas for substances not containing the indicated groups the molecular weight measured under the same conditions had a practically “normal” value.

Figure 3 shows, for a number of substances, the dependence of the average degree \((f)\) of association (the ratio of the measured molecular weight to the molecular weight corresponding to the chemical formula of the given substance) on their concentration in benzene. Obviously, the course of these curves is directly explained by the formation of hydrogen bonds between molecules containing OH or NH groups (curves 1–6) and by the absence of such bonds in the remaining substances (curves 7–8). From Fig. 3, in particular, it follows that the formation of a hydrogen bond is not determined by the dipole moment of the molecules, since, for example, nitrobenzene, which has the largest dipole moment of all the substances shown in Fig. 3, is practically not associated.

Fig. 3. Dependence of the degree of association of typical associated substances on their concentration in benzene: 1—formamide, 2—ethyl alcohol, 3—acetoxime, 4—α-oxime of camphor, 5—benzohydrol, 6—aniline, 7—nitrobenzene (for comparison), 8—naphthalene (for comparison).

Associated molecules in the liquid phase form more or less complex aggregates. Thus, according to X-ray diffraction ...

according to Zachariasen^27 and Garvey^28, methyl and ethyl alcohols in the liquid state form the complexes shown in Fig. 4. In neutral solutions at low concentrations such

Fig. 4. a) Chain association of ethyl alcohol molecules (according to Zachariasen^27 and Garvey^28). b) Arrangement of atoms in a layer of a boric acid crystal^27. Large circles are oxygen atoms, small circles are boron atoms; double lines are hydrogen bonds (hydrogen atoms are not shown).

associates are only just beginning to form. For carboxylic acids (with the exception of formic acid; see below) and oximes in the liquid phase, as also in the gas phase, cyclic dimerization is apparently typical (see example I).

Hydrogen fluoride in the gas phase forms cyclic complexes of several molecules, for example \((\mathrm{HF})_6\)^35,^37.

The degree of association of a substance depends strongly on the nature of the solvent. Thus, according to Fig. 5, the average degree of polymerization of ethyl alcohol in cyclohexane is 3–4 times greater than in benzene, while in dioxane it is practically absent. The latter circumstance is readily explained by the fact that alcohol molecules easily form hydrogen bonds with dioxane molecules, which contain oxygen atoms, and this hinders association. In this case, naturally, an equilibrium is established between the complexes “alcohol—dioxane” and “alcohol—alcohol.” A regular decrease in the degree of association and a lowering of the boiling point is observed for butyl alcohols if they are arranged in order of increasing degree of branching

Fig. 5. Average degree of association of ethyl alcohol in cyclohexane (curve 1), in benzene (curve 2), and in dioxane (curve 3) as a function of molar concentration^29.

carbon chain.

\[ \begin{array}{c@{\qquad}c} & t_{\mathrm{boil}}(^{\circ}\mathrm{C})\\[2mm] \mathrm{CH_3CH_2CH_2CH_2OH} & 116\\[2mm] \begin{array}{c} \mathrm{CH_3}\\[-1mm] \diagdown\\[-1mm] \mathrm{\ \ CHCH_2OH}\\[-1mm] \diagup\\[-1mm] \mathrm{CH_3} \end{array} & 108\\[5mm] \begin{array}{c} \mathrm{CH_3CH_2}\\[-1mm] \diagdown\\[-1mm] \mathrm{\ \ CHOH}\\[-1mm] \diagup\\[-1mm] \mathrm{CH_3} \end{array} & 99\\[5mm] \begin{array}{c} \mathrm{\ \ \ CH_3}\\[-1mm] \mathrm{\ \ \ |}\\[-1mm] \mathrm{CH_3{-}C{-}OH}\\[-1mm] \mathrm{\ \ \ |}\\[-1mm] \mathrm{\ \ \ CH_3} \end{array} & 83 \end{array} \qquad \begin{array}{c} \left.\begin{array}{c}\\[18mm]\end{array}\right\}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\\[-22mm] \text{decrease}\\ \text{in association}\\[22mm] \downarrow \end{array} \]

Such a regularity may be due to two circumstances—first, to steric hindrances that arise when two molecules approach one another; as the branching of the carbon chain increases, these hindrances, evidently, increase; second, to a decrease in the effective positive charge of hydrogen as the number of methyl groups directly bonded to the COH group increases. As we shall see below, the role of this charge in the formation of the hydrogen bond is recognized in any of the existing points of view on its nature. Let us note that the acidity of the butyl alcohols decreases in the same direction as the degree of association.

The formation of an intramolecular hydrogen bond must, evidently, sharply reduce the association of molecules with one another. Fig. 6 shows the dependence of the degree of association of ortho-, meta-, and para-oxybenzaldehydes in naphthalene on concentration (cryoscopic measurements). As might have been expected, for the ortho isomer

\[ \begin{array}{c} \text{H}\\[-1mm] \quad \mathrm{C}\\[-1mm] \diagup\quad \diagdown\\[-1mm] \text{fused benzene ring}\quad \mathrm{O}\\[-1mm] \quad\quad\quad \vdots\\[-1mm] \quad\quad\quad \mathrm{H}\\[-1mm] \quad\quad\quad /\\[-1mm] \quad\quad \mathrm{O} \end{array} \tag{II} \]

association is almost absent.

Fig. 6. Dependence of the average degree of association of ortho-, meta-, and para-oxybenzaldehydes on concentration in naphthalene\({}^{26}\).

Analysis of the experimental data shows that the most stable are those intramolecular hydrogen bridges in which the bond

H...B represents a side of a hexagon, as, for example, in ortho-oxybenzaldehyde, ortho-nitrophenol, etc. Intramolecular hydrogen bonds also exist in five-membered rings, as, for example, in ortho-methoxyphenol (guaiacol)

\[ \begin{gathered} \text{(structural formula of guaiacol with an intramolecular } \mathrm{H\cdots O} \text{ bond)} \end{gathered} \tag{III} \]

and also in seven-membered rings, for example,

\[ \begin{gathered} \text{(structural formula with the fragment } \mathrm{-CH_2-N-COCH_3} \text{ and an intramolecular } \mathrm{N-O\cdots H} \text{ bond)} \end{gathered} \]

However, here, especially in the latter case, the H...B bonds are noticeably weaker, and the distinction between the ortho isomers, on the one hand, and the meta and para isomers, on the other hand, in the sense of their degree of association is smoothed out.

The main reason for the reduced stability of five- and seven-membered rings with a hydrogen bond apparently lies in the greater distance between the atoms H and B. This distance is not easy to estimate. For example, in ortho-methoxyphenol (III) the distance H...O is approximately \(2.3\) Å) (Fig. 7, b), whereas, according to\({}^{23}\), in salicylic acid it is about \(1.7\) Å (Fig. 7, a*).

Let us note that an increase in the distance H...B and the disappearance of the intramolecular bond may occur under the influence of a new substituent. Thus, for example, if into ortho-nitrophenol a substituent \(X\), having a large molecular volume, is introduced in the position shown in scheme (IV), then under its influence the group \(\mathrm{NO_2}\) may rotate around the bond \(\mathrm{C-N}\) through a certain angle, which, obviously, may be accompanied by rupture of the bond \(\mathrm{O\ldots H}\). The same result will be obtained if into the molecule of nitroacetanilide, in the position shown in scheme (V), a substituent \(X\), which

*) In this estimate, the unknown values of bond lengths and valence angles in (III) were assumed equal to the corresponding values in other analogous molecules whose geometry has been studied. See also the footnote on p. 211.

which, because of its large volume, causes rotation of the acetanilide group about the C—N axis.

(IV)

(V)

The difference between ortho isomers and meta- and para-derivatives of phenols and amines is manifested not only in the degree of association, but also in vapor pressure, in surface tension, heat of sublimation, etc. Naturally, owing to the absence of association, for example in ortho-nitrophenol, its saturated vapor pressure is considerably higher, while the surface tension, heat of sublimation, density, and melting and boiling temperatures are lower than those of the meta- and para-isomers (Table II).

Fig. 7. Internuclear distances in compounds with a hydrogen bond: a) salicylic acid ^23, b) ortho-methoxyphenol.

The difference in the heats of sublimation of the ortho- and meta- or ortho- and para-isomers approximately represents the energy of the intermolecular hydrogen bond in nitrophenol. According to the data of Table II it amounts to 4.4–5.3 kcal.

Association of molecules in the liquid phase by means of a hydrogen bond is also reflected in the dielectric constant. If a liquid is not associated, then, as is known, between its dielectric-

Table II

Vapor pressure of the saturated vapor \((p)\), surface tension \((\sigma)\), heat of sublimation \((\lambda)\), density \((\rho)\), melting \((t_{\text{m}})\) and boiling \((t_{\text{boil}})\) temperatures of nitrophenols

	\((p\ 100^\circ\mathrm{C})\), mm Hg	\(\sigma^{31}\) \((125^\circ\mathrm{C})\)	\(\lambda^{33}\), kcal/mole	\(\rho^{33}\) \((125^\circ\mathrm{C})\)	\((t_{\text{m}},^\circ\mathrm{C})\)	\((t_{\text{boil}},^\circ\mathrm{C})\)
Ortho . . .	2.92	31.1	17.5	1.205	45	214
Meta . . .	0.196	39.5	21.9	1.259	96	—
Para . . .	0.083	42.5	22.8	1.266	113	295

between the dielectric constant \(D\) and the dipole moment \(\mu\) of its molecules there exists the following relation:

\[ \frac{D-1}{D+2}\frac{M}{\rho} = \frac{4\pi}{3}N\left(\alpha+\frac{\mu^2}{3kT}\right), \tag{3,1} \]

where \(M\) is the molecular weight, \(N\) Avogadro’s number, \(\alpha\) the polarizability. For liquids associated by means of hydrogen bonds, this relation is not fulfilled\({}^{34}\). For most such substances (alcohols, water, amides, hydrogen fluoride, hydrogen cyanide, etc.) the true value of \(D\) exceeds that calculated from formula (3,1). For example, for liquid hydrogen cyanide \(D=116\); according to Pauling\({}^{33}\), this value is approximately three times greater than would be expected for hydrogen cyanide if one starts from the value of the dipole moment of an individual molecule \((\mu \simeq 3\cdot 10^{-18}\) esu).

The experimental value of \(D\) can be explained by assuming that the HCN molecules are partially associated by means of hydrogen bonds and form chains of the type

\[ \mathrm{NC-H\ldots NC-H\ldots NC-H}, \]

with the average degree of association approximately equal to 3. For certain other substances, such as acetic, propionic, and butyric acids, the true value of \(D\) proves to be lower than that calculated by formula (3,1). This can be explained by the formation in the liquid phase of symmetric cyclic dimers of type (I), which agrees well with other experimental data. Thus, for acetic acid dissolved in benzene, the molecular-

weight corresponds to a double molecule, and the dipole moment is equal to zero. Let us note that the dielectric constant of formic acid turns out to be higher than the value calculated by formula (3.1); this apparently indicates that in the present case, in the liquid phase, along with cyclic association, chain association also occurs.

Hydrogen bonding also plays a definite role in certain adsorption phenomena. Thus, the high capacity of wool and cotton fibers to adsorb water is explained by the formation of hydrogen bonds. According to the ideas of M. M. Dubinin, A. V. Kiselev, and their co-workers \(^{35a}\), the adsorption of water by active carbons is due to hydrogen bonds formed between surface oxides, which play the role of primary adsorption centers, and water molecules, which in turn become secondary centers for further adsorption.

We see, therefore, that the phenomena considered can indeed be explained quite satisfactorily on the basis of the single assumption of the formation of a weak bond between two molecules or atomic groups through the H atom*).

We shall now turn to phenomena for the explanation of which assumptions concerning the nature of this bond are also necessary.

§ 4. SPECTROSCOPIC MANIFESTATIONS OF THE HYDROGEN BOND

(VIBRATIONAL SPECTRA)

Spectroscopic methods for studying molecules undoubtedly provide deeper information about their structure than any other physical methods of investigation. Accordingly, the study of the vibrational spectra of compounds with a hydrogen bond has proved very fruitful. It has made it possible not only to elucidate various new aspects of molecular structure, but also, to a considerable extent, to penetrate into the nature of the hydrogen bond itself.

We shall consider the principal manifestations of the hydrogen bond \(A—H \ldots B\) in vibrational spectra using as an example the case in which the group \(A—H\) is a hydroxyl group \((O—H)\).

As is well known, the hydroxyl group has a characteristic frequency of valence vibrations. Regardless of the nature of the substance into which the OH group enters (alcohols, phenols, acids, water, etc.), in monomeric molecules its fundamental tone lies near \(3600\ \mathrm{cm}^{-1}\) \((2.8\ \mu)\), the first overtone near \(7000\ \mathrm{cm}^{-1}\) \((1.4\ \mu)\), and the second overtone near \(10\,000\ \mathrm{cm}^{-1}\) \((1\ \mu)\). Thus, for example, in

*) For reviews of similar manifestations of the hydrogen bond, see \(^{26, 35, 36, 37}\).

at low concentrations of n-butyl alcohol dissolved in $\mathrm{CCl}_4$, the absorption curve in the infrared region clearly shows a peak ($\sim 3630\ \mathrm{cm}^{-1}$) corresponding to vibrations of the OH group of individual molecules (Fig. 8, a). With an increase in the concentration of the alcohol (Fig. 8, b, c), alongside this comparatively narrow band there begins to appear a broad intense band, whose maximum is shifted toward longer wavelengths by an amount $\Delta \nu \sim 250\ \mathrm{cm}^{-1}$, independent of the alcohol concentration. This band is due to vibrations of the OH group in a complex with a hydrogen bond $\mathrm{O—H\ldots O}$. The intensity of this band increases with increasing concentration, i.e., with an increase in the degree of association of the alcohol molecules, while the intensity of the initial band decreases. In pure alcohol (Fig. 8, d) the latter disappears, which means the absence of monomers.

Fig. 8. Absorption curve of n-butyl alcohol in $\mathrm{CCl}_4$ (after Hoyer$^{37}$).

A similar shift of the vibration frequency of the OH group toward longer wavelengths by several hundred $\mathrm{cm}^{-1}$ and a strong broadening of the band, observed both in infrared spectra and in spectra of combination scattering of light, are characteristic phenomena accompanying the formation of a hydrogen bond. They have been observed not only for the OH group, but also for the groups NH, HCl, HF, CH (for example in $\mathrm{CHCl}_3$), SH. (For the literature see$^{17,37}$.)

The position of the maximum, the width, and the intensity of the band corresponding to the complex $\mathrm{A—H\ldots B}$ vary with temperature. The first systematic investigations of this dependence belong to G. S. Landsberg and S. A. Ukholin$^{39}$, who studied the spectra of combination scattering of water and methyl alcohol over a wide range of temperature and density variation. As can be seen from Fig. 9$^{37}$, the intensity and width of the band decrease upon heating (see, however,$^{39a,40a}$), and its maximum shifts toward higher frequencies. At a sufficiently high temperature, corresponding to the complete breaking of the hydrogen bonds, the band disappears altogether$^{39}$.

The appearance of a shifted broad band upon the formation of a hydrogen bond is observed in the interaction not only of molecules

of one and the same substance, but also of molecules of different substances. For example, according to the data of V. M. Zezyulinskii40, in the infrared spectrum of a weak solution of phenol in CCl4 there is observed a comparatively narrow band with a frequency of 7050 cm−1, characteristic of vibrations of the OH group (Fig. 10, curve 1). When acetoacetic ester (CH3COCH2COOC2H5) is added to this solution, the intensity of the band drops sharply and at the same time a broad band appears, the maximum of which corresponds to 6000 cm−1 (curve 2, Fig. 10). A similar effect on the spectrum of phenol is produced by the addition of acetylacetone (CH3COCH2COCH3, curve 3, Fig. 10).

Fig. 9. Absorption curves of ethyl alcohol in a solution of CCl4 (layer thickness 0.5 cm, concentration 0.1 mole/l): 1 — 0° C, 2 — 20° C, 3 — 70° C (after Hoyer37).

Obviously, in both cases hydrogen bonds are formed between phenol molecules, on the one hand, and molecules of the third component, on the other.

The possibility of the formation of hydrogen bridges RAH . . . BR1 between molecules of different substances is confirmed by data on the spectra not only of the A—H group, but also of the BR1 molecule. For example, when light and heavy water are dissolved in dioxane and acetone, Gordy41 observed that the bands of pure dioxane, corresponding to 1125 cm−1 and 875 cm−1, under the influence of water shift by 10 cm−1 toward longer wavelengths, while the vibration frequency of the C=O group of acetone (1740 cm−1) shifts in the same direction by 30 cm−1, which was later confirmed by P. P. Shorygin42*). It is significant that approximately the same shift

*) The facts cited, like the entire body of experimental data on the hydrogen bond, refute the point of view of V. M. Chulanovskii43, according to which a hydrogen bond can form only between molecules of one and the same substance.

characteristic of the bands of the \(\mathrm{C{=}O}\) group in carboxylic acids upon their dimerization in the gas phase \(^{38}\)*).

Gordy and Stanford \(^{46}\) investigated the infrared spectra of heavy methyl alcohol \((\mathrm{CH_3OD})\) in various solvents \(\mathrm{BR_1}\) and established that the frequency shift of the OD group under

Fig. 10. Influence of a third component on the OH band of phenol in a \(\mathrm{CCl_4}\) solution; 1 — absorption spectrum of a \(1.0\,M\) solution of phenol in \(\mathrm{CCl_4}\); layer thickness \(d = 0.5\ \mathrm{cm}\); 2 — absorption spectrum of a \(2.25\,M\) solution of ethyl acetate in a \(1.0\,M\) solution of phenol in \(\mathrm{CCl_4}\); \(d = 0.5\ \mathrm{cm}\); 3 — absorption spectrum of a \(2.9\,M\) solution of acetylacetone in a \(1.0\,M\) solution of phenol in \(\mathrm{CCl_4}\); \(d = 0.5\ \mathrm{cm}\) (after V. M. Zezyulinsky \(^{40}\)).

the influence of the hydrogen bond \(\mathrm{CH_3O{-}D\ldots BR_1}\) with solvent molecules depends regularly on the degree of basicity of the latter. Namely, it was found that there is a linear relation between the indicated frequency shift and the logarithm of the equilibrium constant of the substance \(\mathrm{BR_1}\) with \(\mathrm{H_2O}\), characterizing the ability of \(\mathrm{BR_1}\) to attach a proton with formation of the ion \((\mathrm{HBR_1})^+\). As can be seen from Fig. 11, this regularity is fulfilled for a series of substances whose equilibrium constant varies over an enormous interval: from \(10^{-3}\) (piperidine) to \(10^{-25}\) (nitrobenzene).

* According to M. I. Batuev \(^{44}\), in the liquid phase the \(\mathrm{C{=}O}\) groups in dimers of carboxylic acids have a frequency of about \(1600\ \mathrm{cm^{-1}}\). According to Sh. Sh. Raskin and A. V. Tsekhanskaya \(^{45}\), the \(\mathrm{C{=}O}\) group of, for example, benzoic acid in the liquid phase \((250^\circ\mathrm{C})\) possesses the following frequencies: \(1656\), \(1695\) (weak band), \(1730\ \mathrm{cm^{-1}}\).

Later it was established\(^{45a,46a}\) that, for a number of crystals with hydrogen bonds RAH...BR\(_1\) (chiefly ROH...OR\(_1\)), the shift of the frequency of the AH group is the greater, the smaller the equilibrium distance A...B in the given crystal.

Table III gives some data on the influence of the medium on the frequency of the fundamental vibrations of AH and AD. As can be seen from Table III, the magnitude of the relative frequency shift \(\Delta \nu/\nu_0\) on going from the gas phase to an inert solvent (CCl\(_4\), C\(_6\)H\(_6\), etc.) is 0.01–0.02, whereas on going to the liquid phase or to a solvent forming hydrogen bonds it is 0.05–0.09 and even 0.15.

Lyuttke and Mecke\(^{53}\) carried out a systematic study of the influence of various solvents on the spectrum of the OH group of phenol. They measured the position of the maximum, the half-width, and the integrated intensity of the OH band in the region of 10,300 cm\(^{-1}\) (second overtone). Since the authors used low concentrations of phenol (0.05 mole/liter), all changes in the spectrum could be due only to the influence of the medium.

Fig. 11. Relation between the equilibrium constant of
H\(_2\)O + BR\(_1\) ⇄ OH\(^{-}\) + (HBR\(_1\))\(^{+}\)
(for various BR\(_1\)) and the frequency shift according to Gordy and Stanford\(^{46}\):
1 — nitrobenzene; 2 — benzyl benzoate; 3 — methyl benzoate; 4 — acetophenone; 5 — para-methylacetophenone; 6 — ortho-chloroaniline; 7 — meta-chloroaniline; 8 — ortho-toluidine; 9 — aniline; 10 — dimethylaniline; 11 — ethyltoluidine; 12 — methylaniline; 13 — quinoline; 14 — meta-toluidine; 15 — pyridine; 16 — quinaldine; 17 — \(\alpha\)-picoline; 18 — dimethylbenzylamine; 19 — benzylamine; 20 — tributylamine; 21 — tri-n-propylamine; 22 — piperidine.

As follows from Table IV, the magnitude of the frequency shift does not depend on the dielectric constant of the medium or on the dipole moment of the solvent molecules. The latter is especially clearly manifested in comparing chlorobenzene and benzene; when phenol is dissolved in the latter, along with the peak characteristic of inert solvents (10,330 cm\(^{-1}\)), a second peak appears, shifted to the long-wavelength side by 150 cm\(^{-1}\); this apparently indicates that, for a certain orientation, benzene and phenol molecules can combine with one another by means of

Table III

*Fundamental vibration frequencies of AH and AD in various media
(in \(cm^{-1}\)) **

Medium	H\(_2\)O\(^{19}\), i.r.s.	H\(_2\)O\(^{19}\), Raman	D\(_2\)O\(^{19,41}\), i.r.s.	D\(_2\)O\(^{19,41}\), Raman	CH\(_3\)OH\(^{39,47}\), Raman	CH\(_3\)OD\(^{46}\), i.r.s.	HCl\(^{19}\), i.r.s.	DCl\(^{19}\), i.r.s.	CH\(_3\)COOH
Vapors (%) (monomers)	3756 (3652)	3654	2785	2666	3684 (3672)	2724	2889	2091	3578\(^{51}\) (Raman)
Liquid phase	3400	3440	2507	2535	3402	2494	2701	1965	3040\(^{52}\) (i.r.s.)
Inert solvent	3705	3614	—	—	3611	2681	2865\(^{50}\)	—	—
Acetone	3620	—	2624	—	3532	2584	—	—	—
Dioxane	3570\(^{48}\)	—	2632	—	3516	2584	2469\(^{41}\)	—	—
Pyridine	3450\(^{49}\)	—	2519	—	3406	2500	—	—	—

*) It should be noted that in the original papers, for the most part, the temperature at which the measurements were made is not indicated. The abbreviation “i.r.s.” means “infrared spectra,” and “Raman” means “combination-scattering spectra.”

Table IV

Effect of the solvent on the spectrum of the phenol OH group
in the region \(10\,300\ cm^{-1}\)

Solvent	Dielectric constant of the solvent	Frequency (\(cm^{-1}\)), 20° C	Frequency (\(cm^{-1}\)), 50° C	Integral absorption, 20° C	Integral absorption, 50° C	Band half-width (\(cm^{-1}\)), 20° C	Band half-width (\(cm^{-1}\)), 50° C
1. C\(_6\)H\(_{12}\)	2.02	10 340	10 355	8.0	8.8	100	100
2. CCl\(_4\)	2.24	10 330	10 340	8.2	8.8	110	110
3. C\(_2\)HCl\(_5\)	3.82	10 285	10 300	7.7	8.0	160	130
4. C\(_6\)H\(_5\)Cl	5.69	10 250	10 270	9.3	9.9	210	200
5. C\(_6\)H\(_6\), I	—	10 175	10 200	7.8	8.2	270	290
5. C\(_6\)H\(_6\), II	2.28	10 330	10 330	7.8	8.2	—	—
6. CH\(_3\)NO\(_2\)	39.4	10 160	10 190	4.5	5.6	370	340
7. C\(_6\)H\(_5\)OCH\(_3\)	4.37	10 130	10 160	6.2	7.5	430	420
8. C\(_6\)H\(_5\)CHO	18.0	\(\sim\)9 920	\(\sim\)10 000	2.8	4.0	540	500
9. C\(_6\)H\(_5\)COCH\(_3\)	18.0	\(\sim\)9 870	\(\sim\)9 900	2.6	3.1	650	720
10. (C\(_2\)H\(_5\))\(_2\)O	4.33	\(\sim\)9 700	—	2.7	—	700	—

hydrogen bond*). Similarly to benzene, a shifted peak is also observed in the case of certain other aromatic compounds \(^{53,137}\).

It follows further from Table IV that, as the frequency is shifted toward the long-wavelength side (i.e., as the hydrogen bond becomes stronger), the integrated absorption in the band generally decreases, while the width of the band increases. With increasing temperature, as was already noted above, the band maximum is shifted somewhat toward higher frequencies, the integrated absorption increases; the width of the band upon heating in some cases increases, while in others it decreases. I. A. Yakovlev \(^{39a}\), V. I. Malyshev, and M. V. Shishkina \(^{40a}\), however, found that the width of the OH band of a series of alcohols always increases with increasing temperature. According to the data of G. S. Landsberg and F. S. Baryshanskaya \(^{54}\), in crystals of hydroxides, upon strong lowering of the temperature (down to \(100^\circ\mathrm{K}\)), the band of the OH group of water narrows sharply and is transformed into one or several narrow bands.

Similarly to intermolecular hydrogen bonding in vibrational spectra, intramolecular bonding also manifests itself. As an example, Fig. 12 gives the absorption curve of orthochlorophenol (in dilute solution \( \mathrm{CCl_4} \)), corresponding to the first overtone of the OH group.

trans  cis

\[ \text{(VI)} \]

This curve has two maxima; the position of the smaller one coincides with the position of the absorption band of the phenol molecule (\(7050\ \mathrm{cm}^{-1}\)); the second maximum is shifted toward longer wavelengths by somewhat more than \(100\ \mathrm{cm}^{-1}\). According to Pauling \(^{56}\), these bands belong to two isomeric forms of orthochlorophenol, namely to the trans- and cis-forms (see VI).

\[ \text{trans}\qquad\qquad\text{cis} \]

* It is possible that the weak hydrogen bonds formed between benzene and alcohol molecules account for the decrease in the degree of association of the latter on going from a solution in cyclohexane to a solution in benzene (see Fig. 5).

In the first case a hydrogen bond cannot be formed, owing to the large distance between the atoms H and Cl; in the second case the hydrogen bond OH...Cl is formed, under the influence of which the frequency of the O—H vibration is shifted.

An analogous picture is observed in the spectra of a number of other ortho-substituted phenols \(X \cdot C_6H_4 \cdot OH\) \(^{55}\); here the substituent \(X\) may be either a halogen atom, or an OH group, or \(OCH_3\). For the spectra of these substances the presence of two bands is characteristic, corresponding to the cis- and trans-configurations of the molecules and having approximately the same width. In all these cases the hydrogen bond closes a five-membered ring and, in accordance with this, the distance H...B, as we saw above, is rather large and the strength of the hydrogen bond is apparently comparatively small*).

There exists, however, another group of ortho-substituted phenols, in which the hydrogen bond closes a six-membered ring and

Fig. 13. Absorption curves of orthonitrophenol in \(CCl_4\); \(1\)—\(c = 0.1\); \(2\)—\(c = 0.5\); \(3\)—\(c = 1.0\) mole/l; \(4\)—pure nitrobenzene. Solid lines—\(t = 20^\circ C\), dotted lines—\(t = 50^\circ C\). The arrow indicates the position of the phenol absorption band \(^{55}\).

the spectra of which, in their character, differ strongly from the spectrum of orthochlorophenol. In these substances the substituent \(X\) is one of the groups: \(NO_2\), \(CHO\), \(COOR\), \(COR\). As a typical example, Fig. 13 shows \(^{55}\) the absorption curves of ortho-

*) According to Wulf’s data \(^{57}\), orthocyanophenol \((OH \cdot C_6H_4 \cdot CN)\) and orthophenylphenol \((OH \cdot C_6H_4 \cdot C_6H_5)\) also possess analogous spectra, which compels one to assign these compounds to the same category of substances with an intramolecular hydrogen bond \(^{55}\). It should be noted, however, that according to the old data of Auwers \(^{36,26}\), the ortho- and para-forms of cyanophenol have the same palmitization index, which would indicate the absence of an intramolecular hydrogen bond in the ortho compound. This question is subject to further experimental investigation.

nitrophenol in CCl₄. Here, instead of two comparatively narrow bands there is a broad band extending approximately from 9200 to 10 300 cm⁻¹. Such a picture indicates that in substances of this kind the intramolecular hydrogen bond differs in its nature from the hydrogen bond in substances of the orthochlorophenol type.

Apparently, the spectra of the intramolecular hydrogen bond in oxy derivatives of anthraquinone, investigated by D. N. Shigorin and N. S. Dokunikhin [58], should be assigned to an intermediate type; in these compounds the hydrogen bridge also closes a six-membered ring. For example, vapors of 1,4-dioxyanthraquinone in the region of the fundamental OH tone have a band with a maximum near 3100 cm⁻¹ and a width of about 200—300 cm⁻¹.

As was indicated above, if a substance capable of association by means of a hydrogen bond is dissolved in an inert medium, then at sufficiently low concentrations a definite equilibrium is established in the solution between monomers and polymers, which can be studied spectroscopically by measuring intensities. The most convenient is measurement of the intensity of the band of monomeric molecules, since it depends directly on the concentration of the latter (Fig. 14) and can serve as a measure of their relative concentration.

Fig. 14. Absorption curves for the OH group of phenol in CCl₄ solution at the same layer thickness and different concentrations: I — c = 0.3; II — c = 0.15; III — c = 0.075; IV — c = 0.0375 mol/l [59].

If the total concentration of the dissolved substance is known, then, knowing the intensity of the monomer band, one can determine the equilibrium constant, the degree and heat of association (the hydrogen-bond energy). Since the width of the band of monomeric molecules is usually small, instead of the integral intensity one may use the maximum coefficient ...

of absorption \(\gamma_c\), defined by the equality \(\gamma_c=(1/cd)\ln(I_0/I)\), where \(I/I_0\) is the relative intensity at the point of the absorption maximum in a solution of concentration \(c\) with layer thickness \(d\). If the absorption coefficient of an individual molecule, obtained by extrapolating \(\gamma_c\) to \(c=0\), is denoted by \(\gamma_0\), then the fraction of monomeric molecules \(a\) in the solution will be equal to \(\gamma_c/\gamma_0\). Studying the intensity of the second overtone of the OH group for phenol monomers at various concentrations, Kempter and Mecke \({}^{59}\) proposed that, along with monomers, in a carbon tetrachloride solution there also exist complex associates with all possible numbers of molecules, each of which is formed by the addition of one molecule to a complex containing one molecule fewer, the equilibrium constant \(K\) being the same for any complex. Under this assumption the quantities \(c\), \(a\), and \(K\) are connected by the following relation \({}^{59}\):

\[ ca=K(1-\sqrt{a}), \]

or

\[ c\gamma_c=a-b\sqrt{\gamma_c}, \]

where

\[ a=\gamma_0 K,\qquad b=K\sqrt{\gamma_0}. \]

If \(c\gamma_c\) and \(\sqrt{\gamma_c}\) are plotted along the coordinate axes, then, in agreement with this equation, a straight line is obtained, with \(K=0.443\ \mathrm{mol/l}\).

Fig. 15. Determination of the percentage content \((a)\) of monomeric phenol molecules in \(\mathrm{CCl}_4\) \((a=\gamma_c/\gamma_\infty)\) from the intensity of the monomeric OH band for the fundamental tone \({}^{60}\) \((1\nu_{\mathrm{OH}})\), the first overtone \((2\nu_{\mathrm{OH}})\), and the second overtone \({}^{59}\) \((3\nu_{\mathrm{OH}})\), according to Mecke \({}^{61}\).

The validity of the equation given above was confirmed in other works, in which spectra of the same system were studied in the region of the fundamental tone \({}^{60}\) and the first overtone \({}^{61}\) (see Fig. 15). Knowing \(K\), one can find the percentage content of complexes of any

of the degree of association \(f\) by the formula \(\alpha_f=f\alpha(1-\sqrt{\alpha})^{f-1}\). Thus, for example, at a concentration \(c=0.3\) mole/l the authors found that monomers amount to 47%, dimers to 29.6%, triple complexes to 14%, etc. For aliphatic alcohols and benzyl alcohol, the assumption of identity of the equilibrium constant \(K\) for a complex of any degree of association proves insufficient.^62 Making more general assumptions, it is possible here to determine the average degree of association at various concentrations. Measurements showed that molecules of methyl alcohol are associated to the greatest extent; molecules of tertiary alcohols (tert-butyl and tert-pentyl alcohols) are associated to a considerably lesser extent, as are molecules of other substances in which the OH group is shielded by neighboring methyl groups (for example, o-cresol; see Table V).

Table V

Energy \((\varepsilon)\) of the hydrogen bond A—H...B and average degree of association \((f)\) of certain substances

No.	Substance	Formula	Medium	\(\varepsilon\) (kcal/mole)	\(f\) at \(c=1\) (mole/l)	Literature
1	Methyl alcohol	\(\mathrm{CH_3OH}\)	\(\mathrm{CCl_4}\)	4.7	3.0\(_5\)	61, 63
2	Methyl alcohol	\(\mathrm{CH_3OH}\)	\(\mathrm{C_6H_6}\)	3.7	—	62
3	tert-Butyl alcohol	\((\mathrm{CH_3})_3\mathrm{COH}\)	\(\mathrm{CCl_4}\)	5.3	2.0	63
4	Benzyl alcohol	\((\mathrm{C_6H_5})\mathrm{CH_2OH}\)	\(\mathrm{CCl_4}\)	4.6	2.5	62
5	Benzyl alcohol	\((\mathrm{C_6H_5})\mathrm{CH_2OH}\)	\(\mathrm{C_6H_6}\)	4.4	—	62
6	Phenol	\(\mathrm{C_6H_5OH}\)	\(\mathrm{CCl_4}\)	4.3\(_5\)	2.3	61
7	Phenol	\(\mathrm{C_6H_5OH}\)	\(\mathrm{C_6H_6}\)	3.5\(_5\)	1.2	61
8	Phenol	\(\mathrm{C_6H_5OH}\)	\(\mathrm{C_6H_5Cl}\)	3.5	1.5	61
9	p-Chlorophenol	\(\mathrm{ClC_6H_4OH}\)	\(\mathrm{CCl_4}\)	3.7	1.9	61
10	o-Cresol	\(\mathrm{CH_3C_6H_4OH}\)	\(\mathrm{CCl_4}\)	3.8	1.7	61
11	p-Cresol	\(\mathrm{CH_3C_6H_4OH}\)	\(\mathrm{CCl_4}\)	4.4	2.3	61
12	Formic acid	\(\mathrm{HCOOH}\)	Gas phase	6.9\(_5\)	—	52
13	Acetic acid	\(\mathrm{CH_3COOH}\)	Same	8.0\(_5\)	2	52
14	Propionic acid	\(\mathrm{CH_3CH_2COOH}\)	Same	8.2\(_5\)	2	52
15	Butyric acid	\(\mathrm{C_2H_5CH_2COOH}\)	Same	8.3	2	52
16	Isobutyric acid	\(\mathrm{C_2H_5CH_2COOH}\)	Same	8.5\(_5\)	2	52
17	Isovaleric acid	\(\mathrm{C_3H_7CH_2COOH}\)	Same	8.0	2	52

Using methyl alcohol as an example, it was shown that the results of spectroscopic measurements of the degree of association at low concentrations (\(<0.2\) mole/l) agree well with the corresponding results of partial-pressure measurements.^62

Studies of the temperature dependence of the adsorption coefficient for the monomer band made it possible in a number of cases to determine the energy $\varepsilon$ of the hydrogen bond (Table V, 1–11).

The method for determining the hydrogen-bond energy from the intensity of the absorption spectrum of the dimer band was developed by V. I. Malyshev${}^{62a}$. A. A. Shubin${}^{52}$ studied the infrared absorption spectra of a series of carboxylic acids in the gas and liquid phases. He showed that the position and shape of the absorption bands characterizing the OH...O bond, as well as the molar absorption, practically do not change on going from the gas phase to the liquid phase (the exception is formic acid, for which the band in the liquid phase is somewhat broader than in the gas phase*). This gives grounds for assuming that the acids studied in the liquid phase are associated according to the very same type as in the gas phase, for which, as was noted above, by analogy with formic and acetic acids${}^{13}$ dimeric association should be expected. The temperature dependence of the absorption at the maximum of the dimer band made it possible to determine the heat of dissociation of a series of carboxylic acids, which proved, for all the cases studied (with the exception of formic acid), to be approximately the same and equal to $\sim 8.3$ kcal (Table V).

Let us note that the presence of a broad band in the spectra of dimers of carboxylic acids, studied by many authors, indicates that its appearance upon formation of a hydrogen bridge is not connected with the presence of a liquid phase, but is inherent in this complex as such.

The manifestation of the hydrogen bond in vibrational spectra consists not only in changes in the spectra of the interacting molecules, but also in the appearance of new frequencies due to vibrations of the molecules RAH and $\mathrm{BR}_1$ in the complex $\mathrm{RA—H...BR}_1$ relative to one another.

Such vibrations have been most fully studied for water molecules. The corresponding band in the region 150–225 cm$^{-1}$ in the combination scattering spectrum of water was first observed by Boll${}^{64}$. His data were confirmed by Magat${}^{65}$, who, using a calculation based on the Bernal and Fowler model of water${}^{66}$, showed that this band should be assigned to translational intermolecular vibrations. The band in the region 550 cm$^{-1}$, first observed by Kabani and de Rool${}^{67}$, was assigned to hindered rotation (librations) of water molecules. Cartwright${}^{68}$ showed that, with this assignment, the change in these frequencies on going from light water to heavy water is in agreement with theoretical calculation,

*) This circumstance is possibly connected with the fact that, as noted above, formic acid in the liquid phase has partially chain association.

performed by Bernal and I. E. Tamm.^69 Thus it was finally established that, in the case of water, the frequencies of molecular vibrations along the hydrogen-bond line lie in the region of \(200\ \mathrm{cm}^{-1}\). Studies of the spectra of combination scattering of light and heavy ice, and of crystallization water in crystalline hydrates, carried out by E. F. Gross and V. I. Valkov,^70 established the presence, in the region \(100\text{–}200\ \mathrm{cm}^{-1}\), of a number of frequencies which were interpreted as fundamental frequencies or overtones of translational vibrations of molecules joined by a hydrogen bond. Thus, for example, in the spectrum of gypsum \((\mathrm{CaSO_4}\cdot 2\mathrm{H_2O})\) the authors observed the following frequencies: \(92,\ 110,\ 122,\ 134,\ 148,\ 164,\ 182,\ 210\ \mathrm{cm}^{-1}\). In the case of other substances, frequencies lying in this region cannot always be assigned to intermolecular vibrations. Thus, for example, according to the data of Sh. P. Raskin and A. V. Sechkarev,^45 the frequencies in the region of \(180\ \mathrm{cm}^{-1}\), observed in the spectrum of benzoic acid, must be assigned to intramolecular vibrations. One of the arguments in favor of such a conclusion is the circumstance that these same frequencies are also observed if one passes from benzoic acid to substances in which, instead of the OH group, there are \(\mathrm{CH_3}\), \(\mathrm{Cl}\), \(\mathrm{H}\), or \(\mathrm{OCH_3}\).

V. M. Chulanovskii and P. D. Simova^71 investigated the infrared spectra and the spectra of combination scattering of certain carboxylic acids and methyl alcohol in the liquid state. The presence of several bands in the region of \(200\ \mathrm{cm}^{-1}\) they interpreted as a combination of translational and deformation vibrations of the hydrogen bond. The distance between the separate bands, amounting to approximately \(30\ \mathrm{cm}^{-1}\), the authors consider equal to the frequency of deformation vibrations of the H atom perpendicular to the bond line \(\mathrm{O}\ldots\mathrm{O}\).

§ 5. INADEQUACY OF THE SIMPLE ELECTROSTATIC CONCEPTION OF THE HYDROGEN BOND

As noted above, the explanation of the spectroscopic manifestations of the hydrogen bond requires certain assumptions concerning its nature.

Pauling, who was the first to express the idea that the hydrogen bond is due to simple electrostatic attraction, wrote in his monograph:^35 “As a result of the development of the quantum-mechanical theory of valence it became known that a hydrogen atom, having only one stable orbital, cannot form more than one purely covalent bond and that the attraction of two atoms observed in the formation of a hydrogen bond must be due to ionic forces.”

In support of his opinion Pauling did not give any special calculations or sufficiently convincing experimental evidence, and argued only by reference to

Pauli principle in its approximate formulation*), applicable, strictly speaking, only to the case when the interaction of electrons may be neglected. Nevertheless, Pauling’s views became widely disseminated (see, for example,\(^{72}\)).

Within the framework of the electrostatic model, repeated calculations were made both of the energy \((\varepsilon)\) of the hydrogen bond\(^{73-80}\) and of the shift \((\Delta \nu)\) of the vibrational frequency of the AH group upon formation of the complex\(^{79-83}\). These calculations, as a rule, led to values in satisfactory agreement with experimental data, which created the impression that the model itself was correct.

However, it is known that calculations of the interaction of atoms and molecules on the basis of classical electrostatics reflect the true relations very incompletely. Thus, within electrostatics it is impossible consistently to calculate the energy of ions at small internuclear distances, at which the ions begin to repel one another. In some calculations of hydrogen-bond energy (in the works of Melvin-Hughes, Davies, Harms, and others\(^{73}\), and also in paper\(^{84}\)) this repulsion is not taken into account at all; in others it is included formally, by analogy with calculations of ionic lattices. Namely, in addition to the energy of attraction of the residual charges of the atoms (or dipoles of the molecules), the polarization energy and the dispersion interaction, an additional positive term of the form \(b/r^n\) is introduced into the general expression for the energy of the system RAH...BR\(_1\), where \(b\) is determined from the requirement of an energy minimum. The exponent \(n\), in essence, is chosen so that the calculated value of the energy should agree as closely as possible with the experimental value. In the papers of Davies\(^{76}\) and Brgleb\(^{79}\) it is assumed that \(n = 7\); in another work by Brgleb\(^{74}\), \(n = 9\) is adopted; in his third work\(^{77}\) and in the papers of Verwey and Roulinson\(^{75}\), \(n = 12\); in the papers of Malade\(^{78-80}\), \(n = 16\); in Stockmayer’s paper\(^{85}\), \(n = 24\), and so on. Owing to the impossibility, within classical electrostatics, of consistently taking account of the repulsion of atoms, and also because of the general crudeness of such calculations, it must be acknowledged that, strictly speaking, by this method one can correctly calculate only the order of magnitude of \(\varepsilon\). For this reason electrostatic calculations of the energy of the hydrogen bond are not indicative, and leave open the question of its nature. As for electrostatic calculations of the vibrational frequency of AH in complexes with a hydrogen bond, then, as will be shown below, they are altogether erroneous.

Nevertheless, there are examples of the successful application of electrostatic concepts for the qualitative explanation of certain—

*) The formulation in question is that according to which no more than two electrons may be present in each quantum cell, and with antiparallel spins.

…aspects of the hydrogen bond. This applies, for example, to the assertion that an electrical asymmetry of the interacting molecules connected by a hydrogen bond is necessary*), to the prediction, for some systems, of a definite relation between the saturated-vapor pressure over a solution and the dielectric constant of the solvent\(^{86}\), etc. Since, however, such examples are few in number and are qualitative in character, they cannot serve as convincing proof that, for understanding the nature of the hydrogen bond as a whole, classical electrostatics is sufficient and that the quantum properties of the electrons may be disregarded.

The chief of the indicated spectroscopic manifestations of the hydrogen bond is the shift of the characteristic frequency \(\Delta\nu\) toward longer wavelengths. Can this shift be explained from the standpoint of a simple electrostatic picture, or is it necessary, in order to explain it, to proceed from a quantum-mechanical treatment of the hydrogen bond?

Within the framework of a simple electrostatic concept, in order to calculate \(\Delta\nu\) it is first necessary to write down an expression for the energy of the system \(A — H \ldots B\). If, for the interatomic distances, we introduce the notation shown in Fig. 16, then this energy may be expressed, for example, in the following form:

Fig. 16. Arrangement of atoms in a complex with a hydrogen bond.

\[ U = W(r) + \frac{Z_A Z_B e^2}{R} - \frac{Z_H Z_B e^2}{r_{HB}} - \frac{Z_H^2 e^2 \alpha}{2 r_{HB}^4} + \frac{b}{r_{HB}^n}, \tag{5.1} \]

where \(W\) is the energy of the isolated \(A — H\) bond; \(Z_A\), \(Z_H\), and \(Z_B\) are the effective (residual) charges of atoms \(A\), \(H\), and \(B\), respectively; \(\alpha\) is the polarizability of atom \(B\); the last term in equation (5.1) is the repulsion energy (see above). To the expression for \(U\) there is sometimes added the energy of the dispersion interaction and other higher-order terms. All variants of the expressions for \(U\) used in electrostatic calculations can be written in the following form**):

\[ U = W(r) + w_1(r_{HB}) + w_2(R), \tag{5.2} \]

where \(w_1\) and \(w_2\) depend, respectively, only on \(r_{HB}\) and on \(R\).

* As we have seen, however, some spectroscopic data (Table IV, example 5) argue against this assertion (see p. 225).

** When the other atoms of the complex \(RA — H \ldots BR_1\) are taken into account, the corresponding terms of the molecular interaction energy will enter either into \(w_1(r_{HB})\) or into \(w_2(R)\).

It is not difficult to show that, in a sequential calculation on the basis of an expression of the form (5.2), the frequency shift either is absent or is obtained in the direction opposite to reality. We give here the course of the reasoning for the special case when \(w_2=0\) (for the general treatment, see the paper \(^{87}\)).

Making the substitution \(r_{\mathrm{HB}}=R-r\) and putting \(w_2=0\), equation (5.2) can be rewritten as follows\(^*\):

\[ U=W(r)+w_1(R-r). \tag{5.3} \]

The function \(W(r)\) has a minimum at \(r\) equal to the equilibrium length \(r_0\) of the isolated bond A—H.

Since the complex AH … B is in a state of minimum energy, two equalities must be satisfied:

\[ \left(\frac{\partial U}{\partial r}\right)_{r'_0,R_0}=0, \tag{5.4} \]

and

\[ \left(\frac{\partial U}{\partial R}\right)_{r'_0,R_0}=0, \tag{5.5} \]

where \(r'_0\) and \(R_0\) are the corresponding equilibrium values in the complex A—H … B. Since \(W(r)\) does not depend on \(R\), equality (5.5) can be rewritten in the form

\[ \left(\frac{\partial w_1}{\partial R}\right)_{r'_0,R_0}=0. \tag{5.6} \]

The vibration frequency of A—H in the complex is determined by the equality

\[ \nu=\frac{1}{2\pi} \left[ \frac{ \left(\dfrac{\partial^2 U}{\partial r^2}\right)_{r'_0,R_0} }{m^*} \right]^{1/2}, \tag{5.7} \]

where \(m^*\) is the reduced mass. According to (5.3),

\[ \left(\frac{\partial^2 U}{\partial r^2}\right)_{r'_0,R_0} = \left(\frac{d^2 W}{d r^2}\right)_{r'_0} + \left(\frac{\partial^2 w_1}{\partial r^2}\right)_{r'_0,R_0}. \tag{5.8} \]

Since the vibration frequency \(\nu_0\) of the unperturbed AH group is determined by the quantity \(\left(\dfrac{d^2 W}{d r^2}\right)_{r_0}\), it follows from equality (5.8) that the change in frequency under the influence of the hydrogen bond may be due to two factors: first, to a change in the equilibrium

\(^*\) Let us note that the coordinates \(r\) and \(R\) differ negligibly little from the normal coordinates \(x_1\) and \(x_2\) of the A—H … B system, which have the form \(x_1=r\), \(x_2=a r+r_{\mathrm{HB}}\). Here \(a=m_A/(m_A+m_H)\), where \(m_A\) and \(m_H\) are the masses of atoms A and H. Since \(m_H\ll m_A\), \(a\approx 1\) and \(x_2\approx R\).

distances (if \(r'_0 \ne r_0\)) and, secondly, by the influence of the term \(\left(\dfrac{\partial^2 w_1}{\partial r^2}\right)_{r'_0,R_0}\).

It is not difficult, however, to show that, by virtue of equalities (5.3)—(5.5), \(r'_0=r_0\), and, consequently, the first term in equation (5.8) plays no role in the change of frequency. Indeed, equality (5.4) may be rewritten as

\[ \left(\frac{dW}{dr}\right)_{r'_0}+ \left(\frac{\partial w_1}{\partial r}\right)_{r'_0,R_0}=0. \]

Since, by assumption, \(w_1\) depends only on the difference \((R-r)\), we have

\[ \left(\frac{\partial w_1}{\partial r}\right)_{r'_0,R_0} = -\left(\frac{\partial w_1}{\partial R}\right)_{r'_0,R_0}. \]

By virtue of equality (5.6), the latter expression is equal to zero; consequently, \(\left(\dfrac{dW}{dr}\right)_{r'_0}=0\). This means that the equilibrium length \(r'_0\) of the A—H bond in the complex A—H ... B coincides with its equilibrium length \(r_0\) in the absence of a hydrogen bond, as was required to be proved. Further, since the function \(w_1\) at \(R=R_0\) and \(r=r'_0\) has a minimum, the quantity \(\left(\dfrac{\partial^2 w_1}{\partial r^2}\right)_{r'_0,R_0}\) is greater than zero. Hence, according to (5.8), it follows that, contrary to the facts, the frequency of the AH group, under the influence of the atom \(B\), is shifted, under our assumptions, toward shorter waves.

Thus, any expression of the form (5.2) for the energy of the complex A—H ... B, in particular expression (5.1), leads to a contradiction with the experimental data.

What, then, is the reason for the aforementioned satisfactory agreement with experimental data of the results of simple electrostatic calculations of the frequency shift? This agreement is explained by the fact that in all published works proceeding from one or another expression for the energy of the form (5.2), strangely enough, the condition of an energy minimum (5.5) was not observed. For this reason all these works are erroneous.

In the article\(^{81}\) devoted to the spectroscopic manifestations of the hydrogen bond, Bauer and Magat, on the basis of electrostatic concepts, calculated the shift in the vibrational frequency of the OH group of a water molecule upon transition from vapor to the condensed phase. Assuming the structure of water according to Bernal and Fowler\(^{66}\), they took into account the following factors: 1) the interaction of residual atomic charges, assumed to be unchanged, 2) the mutual polarization of molecules, 3) stretching of the OH bond, and 4) the influence of the medium. In doing so, the authors obtained a result in good agreement with experiment. Since all interatomic distances, except \(r\) and \(R\), were naturally regarded as fixed in the calculation, their expression for the energy can be represented in the form (5.2). However, in their calculation Bauer and Magat completely ignored the energy of repulsion of the molecules (with the excep—

by terms expressing the Coulomb repulsion from one another of the hydrogen atoms), even omitting terms of the form \(Z_A Z_B e^2/R\) (see \(^{81}\), note 1). It follows from this that their expression for the energy does not satisfy the equilibrium condition (5.5). If, however, one forces it to satisfy this condition, then, as we have seen, all agreement with experiment disappears.

In an analogous way, Burr-Maale calculated the vibrational frequencies of AH for carbonic acid \(^{8}\) and for acetamide \(^{82}\). Although in calculating the bond energy she took into account the repulsion energy and the equilibrium condition, nevertheless, as follows from the text of her articles, this was not done when calculating the vibrational frequency. Thus, these works suffer from the same defect as the paper by Bozer and Maga.

In Coggeshall’s paper \(^{83}\), devoted to a quantum-mechanical solution of the problem of vibrations of the perturbed O—H bond in methyl alcohol, the perturbation energy \(\omega_1\) is taken to be equal to \(-qE_p z\), where \(z = r - r_0\), \(q\) is the fixed residual charge of the H atom, and \(E_p\) is the electric field acting on it from the neighboring molecule, assumed to be homogeneous. In essence, such an expression for \(\omega\) is a linear term in the expansion, in a series in \(r-r_0\), of the energy of the Coulomb interaction of the residual charges of two molecules. Since the repulsive forces are not taken into account here, the equilibrium condition (5.5) is not fulfilled, and therefore Coggeshall’s entire calculation is erroneous.

A similar error is also contained in some other works analyzing the question of the causes of the frequency shift under the influence of the hydrogen bond (see, for example, \(^{88}\)).

Thus, the simple electrostatic model proves insufficient for explaining the shift of the AH frequency under the influence of the hydrogen bond. Consequently, this model in general incompletely reflects the nature of the hydrogen bond. To solve the problem, it is evidently necessary to take account of the quantum properties of the electron, i.e. a quantum-mechanical treatment of the question is necessary.

§ 6. QUANTUM-MECHANICAL TREATMENT OF THE HYDROGEN BOND \(^{89,90}\)

Until recently there was no quantum-mechanical solution in the literature of the problem of the interaction of a polar molecule A—H (or, in the more general case, a molecule A—X) with an atom B having a filled electron shell. Precisely in this circumstance one may see the chief reason for the spread of the view of the hydrogen bond as a simple electrostatic interaction. The question of the nature of the hydrogen bond can be resolved only by relying on a quantum-mechanical treatment of the indicated problem.

Moreover, it is evident that only in this way can one determine exactly what factor causes the dynamic interaction of the bonds A—H and H . . . B in the complex A—H . . . B, which

necessary for shifting the frequency of the \(A—H\) vibrations toward longer wavelengths.

As was indicated above (§2), in the presence of a hydrogen bond the distance between the atoms \(H\) and \(B\) in the system \(A—H \ldots B\) is considerably smaller than the distance between the same atoms bound only by ordinary van der Waals forces. This testifies to an essentially different character of the interaction of the atoms \(H\) and \(B\) in the presence of a hydrogen bond.

A simple qualitative argument makes it possible to clarify wherein the peculiarity of this interaction consists \(^{91}\).

We shall assume that in the molecule \(AH\), in place of hydrogen there stands an atom \(X\) having one \(s\)-electron in its outer shell. Let there be a set of monovalent atoms \(A\) of different electronegativity, which form with atom \(X\) a series of bonds \(AX\), arranged in order of increasing polarity, beginning with the purely homopolar \(A—X\) and ending with the limiting ionic \(A^-X^+\). In principle any of the intermediate degrees of polarity may exist, and we shall assume that in our set the polarity varies continuously, although, of course, in reality only some of its members exist.

Let an atom \(B\), having in its outer shell two electrons with saturated spins, approach the bond \(AX\) from the side of \(X\). In the limiting case of a homopolar bond, after a slight attraction at large distances due to dispersion forces, repulsion will, as is known, arise (Fig. 17, a). In the limiting case of an ionic bond, on the contrary, a donor–acceptor bond*) of atom \(B\) with the cation \(X^+\) will arise, associated with a change in the distribution of the electron density of atom \(B\) (Fig. 17, c). In this case the interaction energy of \((A^-X^+)\) and \(B\) will be approximately equal to the interaction energy of \((X:B)^+\) (with a correction for repulsion from \(A^-\)) and, as is known, may reach 30–50 kcal.

Fig. 17. Curves of the potential energy of interaction of molecule \(A—X\) with atom \(B\):
a) bond \(A—X\) homopolar, b) bond \(A—X\) polar, c) bond \(A—X\) ionic \((A^-X^+)\).

*) The simplest example of a typical donor–acceptor bond is the bond in the molecular ion \((\mathrm{He:H})^+\), which dissociates into a helium atom, He (“donor”), and a proton \(H^+\) (“acceptor”).

If one makes the natural assumption that the interaction energy \(AX + B\) changes continuously and monotonically with the change in the polarity of \(AX\), then one may assert that:

1) for a sufficiently small deviation from a purely ionic bond, the decrease in the absolute value of the interaction energy of \(AX\) with atom \(B\) will also be small, and the energy curve will still have a sufficiently deep minimum (Fig. 17, \(b\));

2) as the polarity of \(AX\) decreases, the depth of the minimum will decrease and, beginning with some state, there will remain only an insignificant minimum due solely to dispersion forces (Fig. 17, \(a\));

3) with a further decrease in polarity, repulsion will increase, making itself felt at ever greater distances.

Thus, one may conclude that, with sufficiently strong polarity of the group \(AX\), the molecule \(RAX\) can combine with another molecule \(BR_1\) by means of a bond which, by its nature, is very close to a donor–acceptor bond and is associated with a redistribution of the density of the electron shell, mainly of atom \(B\) (atom \(B\) acts as an “electron donor,” atom \(X\) as an “electron acceptor”). Naturally, along with the redistribution of electron density, the Coulomb interaction of the residual charges of the atoms must also play an important role in the stability of the complex.

Modern computational methods of quantum mechanics do not make it possible to determine with any reliability numerical values of the interaction energy of atoms and molecules. However, with the aid of these methods one may hope to establish an approximate dependence of the bond energy on the properties of the interacting atoms and on interatomic distances, which is perhaps more valuable than obtaining isolated numerical data.

Let us consider an ideal linear system of three atoms with four electrons \(A : X + : B\), where \(A : X\) is a polar molecule with a \(\sigma\)-bond, \(A\) is an electronegative atom, \(X\) is a monovalent electropositive atom (in particular, it may be an H atom), and \(B\) is an atom having two unshared electrons. We shall assume that atom \(B\) approaches the molecule \(A - X\) from the side of \(X\). The problem is to establish the nature of the dependence of the energy of their interaction on the properties of atoms \(A\), \(X\), and \(B\), and on the interatomic distances.

To obtain qualitative conclusions, one may use the usual approximate method of quantum-chemical calculations, according to which the wave function of a system of approaching atoms is composed in the form of a linear combination of the wave functions of the system of isolated atoms.

For our purposes it is necessary that the wave function reflect the three possible modes of decomposition of the complex \(A - X \ldots B\) into

individual atoms or ions:

\[ \begin{array}{ccc} \mathrm{A}^{-}: & \mathrm{X}^{+} & :\mathrm{B}\quad(\Psi_1),\\ \mathrm{A}\cdot & \cdot\mathrm{X} & :\mathrm{B}\quad(\Psi_2),\\ \mathrm{A}: & \cdot\mathrm{X} & \cdot\mathrm{B}^{+}\quad(\Psi_3). \end{array} \]

In accordance with this, the complete wave function of the system \(\mathrm{A}-\mathrm{X}\ldots \mathrm{B}\) must be a linear combination of three wave functions \(\Psi_1\), \(\Psi_2\), and \(\Psi_3\), each of which is a proper function of the system of three separated particles with the corresponding distribution of electrons*). Thus,

\[ \Psi=\Psi_1+\lambda\Psi_2+\mu\Psi_3. \]

Here the parameters \(\lambda\) and \(\mu\) are determined from the condition of an energy minimum:

\[ \frac{\partial E}{\partial \lambda}=\frac{\partial E}{\partial \mu}=0. \]

Since the distance \(\mathrm{X}\ldots \mathrm{B}\) in complexes with a hydrogen bond is large in comparison with the distance \(\mathrm{A}-\mathrm{X}\), one may assume that \(\mu\) is small.

Representing the functions \(\Psi_1\), \(\Psi_2\), and \(\Psi_3\) in the form of antisymmetrized sums of products of the atomic wave functions of each electron, one can find the general expression for the energy of the system, containing various integrals of interaction of the atoms \(\mathrm{A}\), \(\mathrm{X}\), \(\mathrm{B}\). This expression is considerably simplified if it is expanded in a series in the small exchange integral between \(\mathrm{X}\) and \(\mathrm{B}\), retaining only terms of second order of smallness. As a result, for the interaction energy of the molecule \(\mathrm{AX}\) with the atom \(\mathrm{B}\) one obtains the following expression:

\[ w=Q'_1(1-\omega)+Q'_2\omega+P_1\omega-P_2, \tag{6,1} \]

where \(Q'_1\) is the energy of the Coulomb interaction of the cation \(\mathrm{X}^{+}\) with atom \(\mathrm{B}\); \(Q'_2\) is the energy of the Coulomb interaction of atom \(\mathrm{X}\) with atom \(\mathrm{B}\); \(P_1\) is the repulsion energy that would arise between \(\mathrm{AX}\) and \(\mathrm{B}\) if the bond \(\mathrm{A}-\mathrm{X}\) were homeopolar; \(-P_2\) \((P_2>0)\) is the attraction energy of \(\mathrm{AX}\) and \(\mathrm{B}\), expressed through exchange integrals between \(\mathrm{X}\) and \(\mathrm{B}\) and dependent on \(\omega\). The latter quantity \((\omega)\) approximately represents the fraction of an electron falling on the atom \(\mathrm{X}\); consequently, \(1-\omega=Z\) may be interpreted as the positive effective charge of the atom \(\mathrm{X}\). If the bond \(\mathrm{AX}\) is almost ionic, then \(\omega\ll 1\). In this case the quantity \(P_2\) may be represented in the following form:

\[ P_2=\bar{P}_2-\omega f, \tag{6,2} \]

\[ \text{*) In the first approximation we do not take into account excited states of the atoms.} \]

where \(\widetilde P_2>0\) and \(f>0\) do not depend on \(\omega\) or on the distance \(r\) between A and X. From (6.1) and (6.2) it is seen that in the limiting case of the ionic bond A\(^{-}\)X\(^{+}\), when \(\omega=0\),

\[ w=Q'_1-\widetilde P_2, \]

which corresponds to the donor–acceptor interaction of the cation X\(^{+}\) of the molecule A\(^{-}\)X\(^{+}\) with atom B. In the limiting case of a homeopolar bond A—X, when \(\omega=1\), it can be shown\(^{9}\) that

\[ w=Q'_2+P_1. \]

This means the presence of strong repulsion between AX and B.

In the general case of a polar molecule A—X, when \(0<\omega<1\), expression (6.1) shows that the energy of the system A—X ... B will be less than in the case of a homeopolar molecule, owing to the presence at \(P_1\) of the factor \(\omega<1\), and also because an additional attraction \((P_2)\) arises between AX and B, associated with a change in the density distribution of the cloud of atom B under the influence of the positive charge of atom X. This latter circumstance gives grounds for saying that in the stability of the complex A—X ... B (in the special case—the complex with a hydrogen bond AH ... B) an important role is played by the donor–acceptor interaction of X and B (H and B).

Equation (6.1), in principle, makes it possible to find the character of the dependence of the energy \(w\) on internuclear distances. The most essential point here is the fact that \(\omega\), and consequently also the effective charge \(Z\) of atom X (H), turn out to depend on the distance \(r\). Owing to this, the interaction energy \(w\) of AH and B proves to depend not only on \(r_{\mathrm{HB}}\) and \(R\) (Fig. 16), but also directly on \(r\). This result is fundamentally new in comparison with a simple electrostatic calculation (equations (5.1) and (5.2)).

Thus, we arrive at the conclusion that, along with factors that are approximately reflected in electrostatic calculations (the interaction of residual charges), the quantum-mechanical treatment also takes account of others that lie beyond classical electrostatics. Namely, the quantum-mechanical calculation naturally takes into account the repulsion energy of atoms H and B and leads to the conclusion that this energy depends on the degree of polarity of the AH bond, more precisely on the residual charge \(Z\) of atom H. The calculation also shows that, under the influence of this charge, the distribution of the electron density of atom B changes considerably, which cannot be reduced to simple electrostatic polarization. This redistribution substantially lowers the energy of the system and signifies the formation of a weak donor–acceptor bond.

From the calculation it follows that the exceptional position of the H atom among atoms with one outer electron, in terms of their ability to form complexes \(RA - X \ldots BR_1\), is due, first, to the absence of inner electrons in the hydrogen atom and, second, to the comparatively high value of its ionization potential.

Since the donor–acceptor bond is determined by the lone electrons of the atom \(B\), the stability of the hydrogen bond \(RA - H \ldots BR_1\) must depend on their configuration, which in turn depends on the structure of the molecule \(BR_1\). If the cloud of the lone pair has an axis of symmetry (\(p\)-orbital or hybrid orbitals), then the energetically most favorable configuration should be that \(RA - H \ldots BR_1\) in which the bond line \(A - H\) approximately coincides with the direction of this axis*), since with such an orientation the overlap of the orbitals of atoms \(B\) and \(H\) will evidently be greatest. Thus, if \(BR_1\) is a water molecule, then the axes of the clouds of the two lone pairs of electrons are directed approximately toward the vertices of a tetrahedron (\(sp^3\)-hybridization of orbitals, angle

Fig. 18. Orbitals of the lone electrons of an oxygen atom in its various valence states: a) tetrahedral hybridization (two covalent bonds OH or \(O - R\), located outside the plane of the drawing, are not shown); b) trigonal hybridization (for example, in the CO group); in compounds of the type \(R - O - R\), one of the depicted \(sp^2\)-orbitals forms the bond \(O - R\), while the second lone pair is placed in a \(p\)-orbital whose axis is perpendicular to the drawing; c) linear hybridization.

between their axes \(109^\circ\), Fig. 18, a; in a free water molecule the valence angle \(H - O - H\) is \(105^\circ\)). In accordance with this, an ice crystal, in which the molecules are joined to one another by hydrogen bonds, has a tetrahedral structure. Apparently, the lone electrons of the oxygen atom have the same configuration in molecules of the form \(R - O - R_1\), in which the atom \(O\) is directly bonded to two atoms, for example, in alcohols, phenols, ethers, etc. There are, however, facts indicating that in such compounds the oxygen atom may have not

*) In the case of a hybrid orbital, this conclusion is valid not only for donor–acceptor interaction, but also for simple electrostatic interaction of residual charges, since the center of gravity of the negative charge in a hybrid orbital does not coincide with the center of gravity of the positive one (see Fig. 18)\(^{84}\).

not only a tetrahedral, but also a trigonal configuration (\(sp^2\)-hybridization, the angle between the axes of the orbits is \(120^\circ\), Fig. 18, b). This is indicated, for example, by the structure of the crystal of boric acid (see Fig. 4, b), in which each of the oxygen atoms forms three bonds (one of them by means of an unshared pair of electrons), arranged at an angle of \(120^\circ\) to one another. The same trigonal configuration of unshared electrons is also possible in the oxygen atom entering into the group \(C = O\), which, apparently, is realized in carboxylic acids. For the group \(C = O\), however, other possibilities are not excluded, the limiting one of which corresponds to \(sp\)-hybridization; in this case the axis of the cloud of one of the unshared electron pairs of the O atom is directed along the line of the \(C = O\) bond on the side opposite to atom C, while the axis of the other pair is perpendicular to the line of the \(C = O\) bond (Fig. 18, c).

In the formation of an intramolecular hydrogen bond, the presence of a definite configuration of unshared electrons may affect the strength of the bond. Thus, for example, in the formation of six-membered rings (see Fig. 7, a) the angle \(C = O \ldots H\) is about \(109^\circ\) and, consequently, lies in the middle of the possible (for the given class of compounds) interval of variation of the hybridization angle (90 and \(120^\circ\)); in the case of five-membered rings (see Fig. 7, b), however, the angle \(C — O \ldots H\) is about \(75^\circ\) and, consequently, deviates strongly from the possible (for the given class of compounds) values of the hybridization angle (from 109 to \(120^\circ\)). This circumstance naturally lowers the stability of five-membered rings with a hydrogen bond.

Above, in the interpretation of the hydrogen bond, only one pair of electrons of atom B was taken into account, namely the unshared electrons whose cloud has an axis directed, in the case of an intermolecular bond, approximately along the line \(B \ldots H — A\). With a tetrahedral configuration of atom B such a restriction is quite natural. In fact, for example, in the case of an oxygen atom its second unshared pair, just like the electrons forming the \(O — R\) bonds, has a cloud whose axis is directed at an angle of \(109^\circ\) to the line of the hydrogen bond and therefore overlaps only slightly with the cloud of atom H. We shall call this interaction the \(\sigma\)-donor-acceptor interaction. If, however, one is dealing with an \(sp^2\)- or \(sp\)-configuration of atom B, as, for example, in the case of the oxygen atom in the group \(C = O\), then the situation changes in some sense. Namely, in addition to the \(\sigma\)-donor-acceptor interaction, a comparatively weak donor-acceptor interaction may arise with the aid of the \(\pi\)-electrons of the \(C = O\) bond, using the \(p\)-orbital of atom H. We shall call such an interaction the \(\pi\)-donor-acceptor interaction. Obviously, it will bring about an additional lowering of the energy of the complex \(A — H \ldots B\) and, consequently, promote stabilization of the hydrogen bond. Apparently, it is precisely this circumstance that accounts for the increased

value of the heat of the hydrogen bond in dimers of carboxylic acids (see Table V). Since the atom A usually has lone-pair electrons, naturally the π-electrons participating in the formation of the hydrogen bond cannot, in the general case, be considered in isolation from the latter. All the π-electrons of the two molecules can form a single system, which apparently has its own specific manifestations (see § 9).

The same may be said of compounds with an intramolecular hydrogen bond. The ortho-substituted phenols with six-membered rings considered above (o-nitrophenol, etc.), as well as some hydroxy derivatives of anthraquinone, are compounds in which π-donor–acceptor interaction can take place. It is possible that the strong broadening of the band of the AH group in the infrared spectra mentioned above is connected precisely with this circumstance (see § 8).

In compounds with five-membered rings (o-chlorophenol, etc.) π-interaction is in all probability practically absent, owing to the excessively large H . . . B distance noted above, which prevents any appreciable overlap of the π-orbitals (the σ-orbitals of B and H nevertheless overlap because of their greater extension in the direction of the bond*).

§ 7. CALCULATION OF THE VIBRATIONAL FREQUENCIES OF THE AH GROUP IN A COMPLEX WITH A HYDROGEN BOND

For the theory of the spectroscopic manifestations of the hydrogen bond, the most important result of the quantum-mechanical calculation of the system A—H . . . B set out above is the clarification of which factor is responsible for the dynamic interaction of the A—H and H . . . B bonds in the complex and for the shift of the A—H frequency toward longer wavelengths. As we shall see below, this factor proves to be the dependence of \(Z\) on the distance \(r\).

In practice, sufficiently simple expressions for an explicit dependence of \(w\) on the internuclear distances can be obtained only for the case of an almost ionic A—H bond. Under this condition, the quantity \(\omega\) may be represented approximately in the form\(^ {87}\)

\[ \omega = \omega_0 e^{-b(r-r_0)}, \tag{7,1} \]

where \(b\) is a theoretical constant equal to \(3.78\ \text{\AA}^{-1}\). The remaining terms entering into the expression for \(w\) can be approximated by simple exponential functions of \(r_{\mathrm{HB}}\). The potential ener-

* Lüttke and Mecke\(^ {55}\) attempted to explain the difference between these two types of compounds by invoking the concept of inductive and mesomeric effects. It should be thought, however, that in comparison with the influence of the difference in the distances H . . . B, the influence of these effects is small.

The energy of the system, now having the form (compare (5.2) and (5.3))

\[ U = W(r) + w_1(R-r,r) + w_2(R), \tag{7.2} \]

must evidently satisfy, in addition to the conditions for an energy minimum (5.4) and (5.5), also the condition

\[ (U)_{r_0',\,R_0}' = -D-\varepsilon, \tag{7.3} \]

where \(D\) is the dissociation energy of the isolated bond AH, and \(\varepsilon\) is the energy of the hydrogen bond, equal to the heat of the reaction \(RA-H + BR_1 \to RA-H\ldots BR_1\). In the first approximation we shall not explicitly take into account the Coulomb interaction of the residual charge of atom B with the residual charges of atoms A and H (in particular, we shall assume \(w_2=0\)). In implicit form this interaction will nevertheless be taken into account to some extent, since condition (7.3) will be satisfied. In this case the following expression is obtained for \(U\) \(^{87}\):

\[ U = W(r)+\varepsilon p\left(e^{-2c(R-R_0)-(b-2c)(r-r_0')} -2e^{-c(R-R_0)+c(r-r_0')}\right), \tag{7.4} \]

where

\[ p = 1-\frac{D}{\varepsilon}\left(2e^{-a(r_0'-r_0)}-e^{-2a(r_0'-r_0)}-1\right), \]

and the energy \(W(r)\), as usual, is expressed by the Morse function

\[ W(r)=D\left(e^{-2a(r-r_0)}-2e^{-a(r-r_0)}\right). \tag{7.5} \]

The parameter \(c\) is determined from the frequency of the intermolecular vibrations \(A\ldots B\) (or \((RAH)\ldots(BR_1)\)). Calculating the vibration frequency of \(A-H\) by formula (5.7) with the aid of function (7.4), we find

\[ \frac{\Delta \nu}{\nu_0}=-\frac{\varepsilon}{D}\rho_1, \tag{7.6} \]

where

\[ \rho_1=\frac{1}{2a^2}(3ab-b^2+4bc-2c^2). \tag{7.6a} \]

For a numerical estimate of the coefficient \(\rho_1/D\) we shall use the following parameter values, referring to the system \(OH\ldots O\):

\[ a=2.30\ \text{\AA}^{-1},\quad c=1.6\ \text{\AA}^{-1*}),\quad D=110\ \text{kcal}, \]

then \(\rho_1/D \simeq 1.3\cdot10^{-2}\ \text{kcal}^{-1}\). We thus see that the frequency shift has the correct sign.

*) The adopted value of \(c\) is an average for several systems \(O-H\ldots O\). Variation of \(c\) has little effect on the value of \(\rho_1/D\).

HYDROGEN BONDING

In making a quantitative comparison with experimental data, it should be borne in mind that, owing to the approximate nature of the initial model, and in particular because the influence of the medium has not been taken into account, one can require of the theoretical calculation only that it give the correct order of magnitude. From formula (7.6), using the data of Table V, for methyl alcohol we find \((\varepsilon = 4.6\ \text{kcal})\), \(\Delta\nu = -220\ \text{cm}^{-1}\), whereas according to experimental data (Table III) \(\Delta\nu = 280\ \text{cm}^{-1}\) and \(\rho_1/D = 0.016\). For carboxylic acids the experimental value of \(\Delta\nu\) varies in the interval \(440\text{--}540\ \text{cm}^{-1}\) \(^{51,92}\), while \(\varepsilon\), as we have seen (Table V), has approximately the same value \((8\ \text{kcal})\); hence we find that \(\rho_1/D\) is \(0.016 \div 0.019\ \text{kcal}^{-1}\).

When the above-neglected Coulomb interaction of the residual charges of the atoms is explicitly taken into account, the calculated value of the coefficient \(\rho_1/D\) increases. An estimate is difficult, since the charge distribution even in the relatively simple water molecule is unknown*). However, under any estimate the order of magnitude of \(\rho_1/D\) remains correct.

As an analysis of the corresponding formulas \(^{87}\) shows, for a series of complexes \(\mathrm{RAH}\ldots\mathrm{BR}_1\), in which only the radical \(R_1\) varies, while the molecule \(\mathrm{RAH}\) and the atom \(B\) remain unchanged, an approximate proportionality should be observed between the frequency shift \(\Delta\nu\) and the hydrogen-bond energy \(\varepsilon\). In the absence of corresponding experimental data for \(\nu\) and \(\varepsilon\), a direct test of this conclusion is difficult. We note, however, that Badger and Bauer \(^{93}\), proceeding, it is true, from not very reliable data, found an approximate proportionality between \(\Delta\nu\) and \(\varepsilon\) for a series in which both molecules varied simultaneously. According to the data of these authors, the proportionality coefficient is approximately \(0.012\ \text{kcal}^{-1}\), i.e. close to the theoretical value given above. Indirect confirmation of the indicated dependence may be seen in the fact that, according to \(^{138}\) (see also \(^{46}\)), the frequency shift of the OD group of heavy methanol upon its interaction with various amines depends linearly on the heat of mixing of these amines with chloroform.

Let us note that if, in explicitly taking account of the interaction of residual charges, the parameter \(b\) were several times smaller (this would mean a less strong dependence of \(Z\) on \(r\), which is closer to the truth), then \(\Delta\nu/\nu_0\) would be close to the experimental value. If \(b = 0\), i.e. if the residual charge \(Z\) did not depend at all on \(r\), then, in agreement with § 5, the frequency shift would practically turn out to be zero (according to formulas (7.6) and (7.6a), it would have the sign opposite to that observed experimentally).

*) If the dipole moment of the \(\mathrm{H_2O}\) molecule is assigned to the \(\mathrm{O—H}\) bonds, then for \((\omega)_r\) one obtains a value \(\sim 0.3e\); in this case \(\rho_1/D \sim 4 \cdot 10^{-2}\ \text{kcal}^{-1}\). If one assumes that a considerable part of the dipole moment is due to unshared electrons (see, for example, the article by Pople \(^{84}\)), then \(\rho_1/D\) decreases.

The condition (5.4) makes it possible to calculate the amount of elongation \(r_0-r'_0\) of the A—H bond upon formation of a hydrogen bridge.

The calculation shows that

\[ r'_0-r_0=\frac{b\varepsilon}{2a^2D}\simeq 0.02\ \text{Å}. \]

When the Coulomb interaction of the charges is explicitly taken into account, the elongation obtained is somewhat larger. The increase in the equilibrium length of the A—H bond plays an important role in the frequency shift. As we saw above (§ 5), if \(r'_0=r_0\), then the frequency shift is obtained in the direction opposite to the actual one, or equal to zero. A necessary prerequisite for the increase in the equilibrium length of the A—H bond upon complex formation is the anharmonicity of its vibrations. As the calculation shows\(^{87}\), for a harmonic oscillator, for which

\[ \frac{d^3W}{dr^3}=0, \]

\(r'_0\) coincides with \(r_0\). The important role of anharmonicity in the change of the frequency under the influence of the hydrogen bond was first noted by B. I. Stepanov*) \(^{94,95}\). However, in a qualitative consideration of the question he did not succeed in establishing that, for a shift of the frequency, it is necessary that, upon formation of the complex, elongation of the AH bond should occur, the equilibrium length of which he assumed to be unchanged.

Calculation by the classical formula (5.7), as is known, makes it possible to determine only the fundamental frequency of the vibrations. To calculate all the vibrational levels of the system A—H...B and the transition probabilities, it is necessary to solve the corresponding wave equation. For the case of a linear complex A—H...B it has the form:

\[ \left[-\frac{\hbar^2}{2\mu_1}\nabla_r^2-\frac{\hbar^2}{2\mu_2}\nabla_R^2+U(r,R)\right]\Psi=E\Psi, \tag{7,7} \]

where \(\mu_1\) and \(\mu_2\) are the corresponding reduced masses.

If the function (7.4) is used as \(U(r,R)\), then it is possible to find both the eigenvalues and the eigenfunctions of equation (7.7)\(^{96}\). The eigenvalue depends on two quantum numbers, one of which \((v)\) corresponds to the valence vibrations of the A—H bond, and the other \((m)\) to the longitudinal intermolecular vibrations of the hydrogen bond \((\mathrm{RAH})\ldots(\mathrm{BR}_1)\). For the shift of the fundamental vibration frequency of the AH group \((v=0\to v=1,\ m\to m)\), the following expression is obtained:

\[ \frac{\Delta\nu}{\nu_0} = -\frac{\varepsilon}{D}\rho_1 -\frac{\varepsilon}{D}\frac{h\nu_0}{D}\rho_2 -\left(\frac{\varepsilon}{D}\right)^2\rho_3 +\frac{h\widetilde{\omega}}{4D}\rho_1(2m+1), \tag{7,8} \]

where the coefficients \(\rho_j\ (j=1,2,3)\) depend only on \(a,b\) and \(c\);

*) B. I. Stepanov was the first to point out that, in the interpretation of the vibrational spectroscopy of the hydrogen bond, it is necessary to consider simultaneously the vibrations of both the A—H bond and the A...B bond.

HYDROGEN BOND

\(\tilde{\omega}\) represents the natural frequency of oscillations of \((\mathrm{RAH})\ldots(\mathrm{BR}_1)\).

Since \(\hbar \tilde{\omega}<\varepsilon\), it follows from (7, 8) that the shift \(\Delta \nu\) is negative. For the system \(\mathrm{O}-\mathrm{H}\ldots\mathrm{O}\), calculation gives \(\rho_1=1.46\), \(\rho_2=2.06\), \(\rho_3=2.40\). Since \(\varepsilon/D\) is of the order of \(5\cdot 10^{-2}\), the term with \((\varepsilon/D)^2\) in equation (7,8) plays practically an insignificant role. Taking into account that for small \(m\) the last term in (7,8) may be neglected, we find:

\[ \frac{\Delta \nu}{\nu_0} \approx -\frac{\varepsilon}{D} \left( \rho_1+\frac{\hbar \nu_0}{D}\rho_2 \right) \approx -0.016\varepsilon . \tag{7,9} \]

Thus, the ratio \(\dfrac{\Delta \nu}{\nu_0\varepsilon}\) proves to be close to the experimental value. Since, as was noted above, agreement with experiment can be required of the calculation performed only as to order of magnitude, the agreement obtained with the experimental data should not be overestimated.

In formula (7,9), as above, the proportionality coefficient between \(\Delta \nu\) and \(\varepsilon\) may be regarded as constant only for a series of complexes \(\mathrm{RAH}\ldots\mathrm{BR}_1\), in which only \(\mathrm{R}_1\) changes.

Solving equation (7,7) makes it possible to find the frequency not only of the principal transition, but also of other allowed, although less probable, transitions caused by a combination of the valence vibrations \(\mathrm{AH}\) and the intermolecular vibrations of the hydrogen bond (Fig. 19). The calculation shows that, for the transition \(v=0\to v=1\), \(m\to m+\Delta m\) \((\Delta m=\pm 1,\pm 2,\ldots)\), these frequencies are equal to

\[ \nu_{m\to m+\Delta m} \approx \nu_0-(\Delta \nu)_{m\to m} +\Delta m\cdot \tilde{\omega} \]

\[ (\Delta m=\pm 1,\pm 2,\ldots), \tag{7,10} \]

Fig. 19. Scheme of combined vibrational transitions in the system \(\mathrm{A}-\mathrm{H}\ldots\mathrm{B}\) \((v\) is the quantum number of the \(\mathrm{A}-\mathrm{H}\) bond vibrations, \(m\) and \(m'\) are the quantum numbers of intermolecular vibrations \((\mathrm{RAH})\ldots(\mathrm{BR}_1))\).

where \((\Delta \nu)_{m\to m}\) is the frequency shift according to formula (7,8). It follows from expression (7,10) that, in the spectrum of the hydrogen bond, alongside the principal intense band of frequency \(\nu\), there should exist several weak lines (or bands) shifted with respect to this band by approximately an integer multiple of the quantum \(\hbar\tilde{\omega}\), as in the sto-

toward both shorter and longer wavelengths*). This conclusion also follows from the qualitative scheme of B. I. Stepanov^94,97. An estimate of the intensity of the indicated additional bands for the case of water and alcohol shows that the one nearest to the main band should have an intensity approximately 10 times, and the next one \(10^3\) times, smaller than the intensity of the latter^96.

Investigations of the combination-scattering spectrum of gypsum \((\mathrm{CaSO}_4\cdot 2\mathrm{H}_2\mathrm{O})\), carried out by A. I. Stekhanov^98, confirmed the conclusion that additional frequencies exist. He found that the spectrum of the OH group in a gypsum crystal at \(-200^\circ\mathrm{C}\) has a fine structure. In addition to two principal bands \(\nu_1 = 3408\) and \(\nu_2 = 3486\ \mathrm{cm}^{-1}\), corresponding to the symmetric and antisymmetric vibrations of the perturbed water molecule, the spectrum contains six more low-intensity bands, arranged symmetrically with respect to the two principal bands. If one assumes that, of several frequencies^70 observed in the long-wave region of the gypsum spectrum, the frequencies \(\bar{\omega}_1 = 146\) and \(\bar{\omega}_2 = 132\ \mathrm{cm}^{-1}\) correspond to the transitions \(m=0 \to m=1\) and \(m=1 \to m=2\), which characterize intermolecular vibrations of the hydrogen bond, then the indicated six bands can be satisfactorily explained by a combination of the valence vibrations OH and these intermolecular vibrations^98 (see Table VI and Fig. 19).

Table VI

Combination of frequencies	Observed \((\mathrm{cm}^{-1})\)	Calculated \((\mathrm{cm}^{-1})\)
\(\nu_2 - (\bar{\omega}_1 + \bar{\omega}_2)\)	3230	3208
\(\nu_1 - \bar{\omega}_1\)	3273	3262
\(\nu_2 - \bar{\omega}_1\)	3349	3340
\(\nu_1\)	3408	—
\(\nu_2\)	3486	—
\(\nu_1 + \bar{\omega}_1\)	3564	3554
\(\nu_2 + \bar{\omega}_1\)	3626	3632
\(\nu_1 + (\bar{\omega}_1 + \bar{\omega}_2)\)	3673	3686

Weak bands that can be explained in an analogous way were observed earlier in the spectra of carboxylic acids by M. I. Batuev^99.

From what has been said above it follows that the quantum-mechanical treatment of the hydrogen bond correctly conveys its principal spectroscopic manifestation—the shift of the characteristic frequency of the OH group toward longer wavelengths—and also predicts the presence of additional weak bands in this region. The agreement of theory with experiment indicates that the theory, in the main, correctly reflects the essence of the hydrogen bond. Thus the latter is not reducible to a simple electrostatic inter—

* According to calculations^96, for such molecules as water and alcohols, on the long-wave side there exist only two or three bands.

effect, but is a weak donor–acceptor bond, in which, of course, the Coulomb interaction of the residual charges of the atoms also plays a definite role.

§ 8. ON THE ORIGIN OF THE BAND IN THE VIBRATIONAL SPECTRUM OF THE HYDROGEN BOND

As was noted above, upon formation of a hydrogen bond the narrow band characteristic of the AH group in an isolated molecule is transformed, at ordinary and high temperatures, into a diffuse band whose half-width amounts to several hundred $\text{cm}^{-1}$. The position of the maximum of the band shifts, with increasing temperature, toward shorter waves. At sufficiently low temperatures the broad band degenerates into several narrow bands or even into one relatively narrow line (crystals).

The main difficulty in explaining the origin of the broad band consists in the fact that, as noted above, it is observed not only in the liquid phase, where polymolecular association is possible, but also in the gas phase (dimers of carboxylic acids). For this reason the suggestion encountered in the literature that the broad band is due to the presence of polymers (chains)18 is, in any case, insufficient and apparently does not convey the principal cause of the broadening of the AH band upon formation of the hydrogen bond.

On the question of the origin of the band in the spectrum of the hydrogen bond there exist two main points of view: the “fluctuation” one (Boer and Maga81, G. S. Landsberg54, M. I. Batuev100) and the “predissociation” one (B. I. Stepanov94, 95).

A characteristic feature of the “fluctuation” concept is the assumption that the appearance of the diffuse band and the temperature dependence of the effect are caused by “fluctuations” of the mutual positions of the perturbed and perturbing molecules. A characteristic feature of the “predissociation” concept consists in the assumption that the diffuse band appears as a result of the transition of the excitation energy of the AH bond to the H...B bond, which causes dissociation of the latter and blurring of the energy levels of the excited state of the A—H bond (predissociation); in other words, upon excitation of the AH vibrations there is a finite probability that the system AH...B will find itself not in a state with discrete energy, but in a state with continuous energy corresponding to the dissociation of A...B (Fig. 20). Obviously, a necessary condition for predissociation is the requirement that the magnitude of the vibrational quantum of A—H be greater than the heat of dissociation $(\varepsilon)$ of the hydrogen bond.

In connection with the question of the origin of the structure in the band and of the reason for the degeneration of the broad band at low temperatures into several narrow bands, M. I. Batuev put forward the “chas-”

frequency-modulation theory.”^99 Its essence reduces to the assumption that the above-mentioned additional weak bands observed in the hydrogen-bond spectrum at low temperatures are caused by modulation of the vibration frequency of the A—H bond by the frequencies of slow vibrations of the intermolecular vibrations \((\mathrm{RAH})\ldots(\mathrm{BR}_1)\), while the blurring of the structure into a band at higher temperatures is explained by “fluctuation” effects. It is not difficult to see, however, that the frequency-modulation point of view is, in essence, an attempt to interpret, on the basis of classical concepts, effects which by their nature are quantum-mechanical and therefore, naturally, can find only a very approximate reflection in the classical theory.

Fig. 20. Scheme of an optical transition in the system A—H…B, accompanied by predissociation (the region of the continuous energy spectrum is indicated by dots).

There can be no doubt, for example, that the “frequency-modulation theory” is completely untenable in the question of the distribution of intensities in the spectrum of complexes with a hydrogen bond. At the same time, the quantum-mechanical treatment set forth above not only explains the same effects that are envisaged by the frequency-modulation concept (the appearance of additional bands), but also predicts (semiquantitatively) the distribution of their intensities.

As was noted above, the principal question on which the “fluctuation” and “predissociation” points of view diverge is the question of the origin of the band.

It can hardly be doubted that the “predissociation” effect, which, according to Stepanov, causes the appearance of a broad band in the spectra of complexes with a hydrogen bond, really exists. The only question that remains open is whether this effect is always practically realized and what its magnitude and role are

in different cases. It cannot be regarded as excluded in advance that in reality it may be comparatively small. Nevertheless, the possibility of this effect must be taken into account in any other, more perfect theory, regardless of those particular assumptions that will underlie it for explaining the whole aggregate of the properties of the hydrogen bond.

What assumptions, then, necessary for explaining the origin of the band and its temperature dependence, underlie the ideas of B. I. Stepanov? There are four such assumptions (apart from the idea of predissociation):

1) The true complex with a hydrogen bond \(RA—H\ldots BR_1\), which has many degrees of freedom, may be replaced by an ideal system of three linearly arranged atoms \(A—H\ldots B\)*), and it is sufficient to take into account only two degrees of freedom of this system—its longitudinal vibrations.

2) In solving the Schrödinger equation for the linear system \(A—H\ldots B\), the interaction of the \(AH\) group with the atom \(B\) of a neighboring molecule may be regarded as a small perturbation, and the methods of perturbation theory may be applied.

3) To the vibrations \(A\ldots B\)**) of the system \(AH\ldots B\) we shall apply the Franck–Condon principle in its usual formulation, with the sole difference that the transition considered is not between electronic states, but between different vibrational states of the \(A—H\) bond.

4) The energy of interaction of the \(A—H\) and \(H\ldots B\) bonds is so large that the matrix element of the “interaction” of levels of the discrete and continuous spectra of the \(A\ldots B\) vibration is greater than the distance between the discrete levels of the \(A\ldots B\) vibration.

Not all these assumptions are equally plausible. Moreover, owing to the qualitative character of all the arguments, it remains unproved whether these assumptions are necessary and sufficient for a quantitative explanation of the regularities observed in the spectrum of the hydrogen bond.

It is not difficult to see that assumption 1) excessively simplifies the true relations and does not create the necessary prerequisites for explaining the cause of the broadening of the characteristic line into a broad band and the temperature dependence of the latter.

In the original version of B. I. Stepanov’s concept \(^{94}\), the origin of the broad band corresponding to the transition of the \(AH\) group from the ground vibrational state to an excited one was explained entirely by predissociation, i.e. by the broadening of the excited level because of its “interaction” with the continuous energy levels of the \(A\ldots B\) bond. However, such an explanation did not make it possible—

* In the monograph \(^{95}\), instead of atoms \(A\) and \(B\), groups \(RA\) and \(BR_1\) are considered, each of which is treated as a material point.
** More precisely \((RAH)\ldots(BR_1)\).

to explain the appearance of several narrow bands at low temperatures and required an implausibly strong “interaction” of the indicated levels, necessary to provide a band width of several hundred \(\mathrm{cm}^{-1}\).

Therefore, subsequently it was assumed that the broad band is the result of the superposition of a number of narrower bands, each of which is due to transitions from one of the vibrational levels \(A\ldots B\) of the ground state \(AH\) to the corresponding levels of the excited state[^95][^97]. If, however, in such an interpretation one takes into account that the intensity of transitions from different initial levels is proportional to the corresponding Boltzmann factor—which the authors[^95][^97] did not take into account—then it turns out that the half-width of the band at ordinary temperatures must be determined only by the transition from the ground level \((m = 0)\). This means, in fact, a return of the theory to its original version.

The difficulty encountered by the theory is connected with the neglect of all degrees of freedom except two vibrations along the bonds \(A—H\) and \(A\ldots B\), which oversimplifies the true relations. For a correct interpretation of the spectra of hydrogen-bonded complexes, it is necessary to take into account other degrees of freedom of the complex, which will significantly change the picture.

First of all, it is necessary to take into account the deviation of the structure of the complex \(A—H\ldots B\) from a straight line, i.e., its deformation (angular) vibrations. In the literature there are separate general statements noting the need to take such vibrations into account[^101].

It is very probable that the smallest distance between the energy levels of deformation vibrations is considerably less than \(200\ \mathrm{cm}^{-1}\). As was noted above, according to V. M. Chulanovskii and P. D. Simova[^71], it is about \(30\ \mathrm{cm}^{-1}\). Taking into account such a small magnitude of the quanta of these vibrations, it is easy to see that, when the latter are included, not only is the indicated difficulty connected with the Boltzmann factor removed, but assumption 4) also becomes more probable, since it is no longer necessary to postulate that the magnitude of the matrix element of the “interaction” of the levels of the discrete and continuous spectra is of the order of \(200\ \mathrm{cm}^{-1}\). When deformation (and other) degrees of freedom are taken into account, it is sufficient to assume that this matrix element has an order of magnitude of \(30\ \mathrm{cm}^{-1}\), and may even be smaller.

Let us now clarify the question of how the concept of B. I. Stepanov, modified in this way, relates to the “fluctuation” ideas on the origin of the band.

According to the latter, in a substance containing hydrogen bonds there are all possible complexes, differing in the mutual arrangement of the molecules forming them, and consequently also in the values of the equilibrium distances and the natural frequencies of vibrations

disturbed group A—H. The higher the temperature, the broader the range of its possible vibrational frequencies.

It is not difficult to see that taking into account the various orientations of the two molecules forming a complex that arise under the influence of “fluctuations” is, in essence, equivalent to taking into account the deformation vibrations of this complex, as well as the rotation of one of its parts (molecules) relative to the other. These degrees of freedom, generally speaking, are quantized. Each vibrational and rotational state has its own equilibrium distance between the molecules of the complex and, naturally, its own absorption frequency of the AH group.

Thus, taking account of deformation vibrations, which, as we have seen, is necessary in order to explain the broad band and the temperature dependence within the framework of the predissociation concept, is in essence equivalent to taking account of the main part of the effects to which the “fluctuation” point of view appeals. In other words, to explain the regularities in the spectrum of the hydrogen bond, a certain combination of “predissociation” and “fluctuation” representations is necessary.

A measure of the proper predissociation effect in complexes with a hydrogen bond could be the half-width of the band caused by the optical transition from one definite level of the ground state AH. It would seem that this width could be judged from spectra recorded at sufficiently low temperatures, when there is no thermal excitation even of the states nearest to the ground state. However, in all probability, under these conditions, when the substance is in the crystalline state, the proper predissociation effect practically disappears altogether, since in this case the continuous part of the energy spectrum A…B, corresponding to rupture of the hydrogen bond, begins at energy values greater than the energy of a vibrational quantum of the group A—H.

These considerations agree with the results of investigations by G. S. Landsberg and F. S. Baryshanskaya⁵⁴ (and also⁹⁸) mentioned in § 4, from which it follows that in crystals at low temperatures (100° K) the spectrum of the OH group forming a hydrogen bond consists of one or several rather narrow lines.

In the liquid phase, under the condition that long chains or other complex aggregates of associated molecules are formed, the theoretical interpretation of the origin and temperature dependence of the band becomes considerably more complicated. In such aggregates, their own normal vibrations are possible, which substantially affect the band width¹⁸. With increasing temperature, some of these complexes break up, which is also reflected in the spectra. A theory of these complex phenomena does not yet exist.

The interpretation of spectra of an intramolecular hydrogen bond is made difficult by the circumstance that the vibration \(A\ldots B\) is not characteristic, since in the general case it depends on many normal vibrations of the molecule. Therefore, without special consideration of each case, the preceding interpretation of the origin of the band and, in particular, the idea of predissociation cannot be extended to the intramolecular hydrogen bond. Nevertheless, one assumption can be made about the cause of the difference described above in the spectra of the OH group in ortho-substituted phenols with five- and six-membered rings.

It is not excluded that the difference in the spectra of, for example, orthochlorophenol and orthonitrophenol is connected not only with the stronger interaction, noted above, of OH with the \(\mathrm{NO_2}\) group than with the Cl atom, but also with the possibility of hindered rotation of the \(\mathrm{NO_2}\) group about the \(C—N\) axis, which is absent in the case of Cl. It may be supposed that, with a sufficiently strong rotation of the \(\mathrm{NO_2}\) group, the system finds itself in a quasi-continuous region of the energy spectrum (rupture of the hydrogen bond), which makes possible, in principle, the effect of predissociation upon excitation of the AH group and the corresponding broadening of its spectral band.

§ 9. OTHER PHYSICAL MANIFESTATIONS OF THE HYDROGEN BOND

Electronic spectra

The hydrogen bond manifests itself not only in vibrational spectra, but in certain cases also in the electronic absorption spectra of molecules. These spectra have been studied much less than vibrational spectra, and their interpretation is not always unambiguous. The influence of an intramolecular hydrogen bond on electronic spectra has been investigated to the greatest extent.

The influence of substituents on the electronic structure of benzene derivatives depends, as is known, on their mutual arrangement. Therefore, unlike vibrational spectra, comparison of the electronic spectra, for example, of para- and ortho-derivatives for the purpose of investigating the influence of a hydrogen bond is not always indicative. A considerably more characteristic feature is the change in the electronic spectrum when an alkoxyl group OR in the ortho position is replaced by a hydroxyl group OH capable of forming an intramolecular hydrogen bond. It is known that such a replacement in monosubstituted benzene derivatives is reflected only slightly in the spectrum. For example, the positions of the long-wavelength absorption bands of phenol \((\mathrm{C_6H_5OH})\) and methoxybenzene \((\mathrm{C_6H_5OCH_3})\) in hexane practically coincide (2730 and 2720 Å). As is seen from Table VII, the same coincidence is observed for meta-hydroxybenzaldehyde and meta-methoxybenzaldehyde.

At the same time, in the case of ortho derivatives, replacement of the methyl of the methoxy group by a hydrogen atom, both in hexane and in alcohol,

Table VII^103

Electronic spectra of hydroxy and methoxy derivatives of benzaldehyde and acetophenone

Substance	\(\lambda\) (Å), in hexane	\(\lg \gamma\), in hexane	\(\Delta\lambda\), in hexane	\(\lambda\) (Å), in alcohol	\(\lg \gamma\), in alcohol	\(\Delta\lambda\), in alcohol
1. Ortho-methoxybenzaldehyde; benzene ring with \(OCH_3\) and \(CHO\) in ortho positions	3100	3.75	185	3195	3.63	55
2. Ortho-hydroxybenzaldehyde; benzene ring with \(OH\) and \(CHO\) in ortho positions	3285	3.51	185	3250	3.48	55
3. Meta-methoxybenzaldehyde; benzene ring with \(OCH_3\) and \(CHO\) in meta positions	3090	3.46	-10	3145	3.45	15
4. Meta-hydroxybenzaldehyde; benzene ring with \(OH\) and \(CHO\) in meta positions	3080 \((3.3 \times 10^{-4} M)\)	3.69	-10	3160	3.46	15
5. Ortho-methoxyacetophenone; benzene ring with \(OCH_3\) and \(COCH_3\) in ortho positions	3000	3.58	290	3050	3.58	220
6. Ortho-hydroxyacetophenone; benzene ring with \(OH\) and \(COCH_3\) in ortho positions	3290	3.63	290	3270	3.50	220
7.^104 2-Hydroxy-6-methoxyacetophenone; benzene ring with \(OH\), \(OCH_3\), and \(COCH_3\) substituents	3345	3.85	75	3355	3.78	70
8.^104 2,6-Dihydroxyacetophenone; benzene ring with two \(OH\) groups and \(COCH_3\) substituent	3420	3.85	75	3425	3.70	70

causes a considerable shift of the band maximum toward longer wavelengths (Table VII).

As is known, an analogous shift of the long-wave band in the electronic spectrum is also observed in some benzene derivatives, if the chromophoric group (for example, HCO or COCH$_3$) is taken out of the plane of the benzene ring under the influence of an ortho substituent having a large molecular volume (see, for example, $1^{v2}$). Since the methoxyl group OCH$_3$ has a significantly larger van der Waals radius than the hydroxyl OH, the question naturally arises whether the shift to the long-wave side in examples 1, 2, and 5, 6 (Table VII) is due to the removal of the CHO and COCH$_3$ groups from coplanarity. From examples 7 and 8 (Table VII), however, it is evident that even if this effect does occur, in any case it is appreciably smaller than the influence of hydrogen bonding. In fact, the comparatively small shift of the band by $\Delta\lambda = 75$ Å upon replacing OCH$_3$ by OH can be explained only on the assumption that in 2-hydroxy-6-methoxyacetophenone a hydrogen bond already exists and that, consequently, the COCH$_3$ group is coplanar with the benzene ring. The indicated value of $\Delta\lambda$ could be regarded as the result of the direct influence of replacing the OCH$_3$ group by OH in the ortho position.

The formation of intermolecular hydrogen bonds manifests itself in the electronic spectra in an analogous way. Namely, it also leads to a shift of the band toward longer wavelengths. As is seen from Table VIII, the shift observed upon going from hexane

Table VIII

Change in $\lambda_{\max}$ upon going from hexane to alcohol

Substances without intramolecular hydrogen bonds	$\Delta\lambda$ (Å)	Substances with an intramolecular bond	$\Delta\lambda$ (Å)
Ortho-methoxybenzaldehyde . .	95	Ortho-hydroxybenzaldehyde	—35
Meta-hydroxybenzaldehyde . . .	80	Ortho-hydroxyacetophenone	—20
Meta-methoxybenzaldehyde . .	55
Ortho-methoxyacetophenone . .	50

to alcohol is somewhat smaller than in the case of the formation of intramolecular bridges, but is nevertheless of the same order of magnitude. At the same time, the position of the corresponding bands of substances with an intramolecular bond changes comparatively little, and the shift occurs in the opposite direction.

If, upon dissolving in hexane, for example, the concentration of meta-hydroxybenzaldehyde is increased sufficiently for complexes with a hydrogen bond to be able to form, then the absorption band

HYDROGEN BOND

also shifts toward the long-wavelength side (when the concentration is changed from \(3.3\cdot 10^{-4}\ M\) to \(10^{-3}\ M\), the shift of the band maximum is equal to \(100\ \text{\AA}\) \(^{103}\)). Figure 21 gives the absorption curves of ortho- and para-nitrophenol, illustrating the indicated changes in the spectrum on going from hexane to water.

The shift of the absorption-band maximum toward the long-wavelength side upon formation of a hydrogen bond can be explained qualitatively in the following way. The system of \(\pi\)-electrons in a molecule with conjugated bonds (butadiene, benzene, nitrobenzene, etc.), responsible for the absorption of light in the visible and ultraviolet regions, can approximately be considered separately from the rest of its electrons, and it may be assumed that they are in a one-dimensional potential box whose dimensions are determined by the length of the chain of conjugated bonds in the molecule \(^{105}\). If, in accordance with the Pauli principle, two electrons with antiparallel spins are placed on each quantum energy level of such a potential box, then the change in energy of the system when one electron passes from the highest occupied level to the nearest free level is equal to \(^{105*}\)

Fig. 21. Curves of the ultraviolet spectrum of nitrophenols: 1 — para-nitrophenol in hexane, 2 — ortho-nitrophenol in hexane, 3 — para-nitrophenol in water, 4 — ortho-nitrophenol in water \(^{103a}\).

\[ \Delta E=\frac{h^{2}}{8ml^{2}}(N+1), \tag{9,1} \]

where \(l\) is the length of the (open) conjugated chain of bonds and \(N\) is the number of \(\pi\)-electrons.

According to this formula, the quantity \(\Delta E\) decreases as the length of the conjugated chain increases (Fig. 22, \(a,b\)). In other words, the absorption maximum, whose position is determined by the formula \(\lambda=\dfrac{hc}{\Delta E}\), shifts, with increasing \(l\), toward longer wavelengths.

As we saw above, the formation of an intramolecular hydrogen bond, closing a six-membered ring, is due in part to the \(\pi\)-donor–acceptor interaction of the hydrogen atom of the AH group

*) Formula (9,1) assumes that there is one electron for each atom of the open conjugated chain.

with the group \(BR_1\) at the expense of the \(\pi\)-electrons of the latter. Such an interaction is, to some extent, equivalent to a lengthening of the conjugated chain*). Modeling the latter by a potential box, we obtain a qualitative explanation of the principal change in the electronic spectrum that occurs in this case. In an analogous way one can explain the displacement of the absorption band toward longer wavelengths upon the formation of an intermolecular hydrogen bond.

This picture can be refined by assuming that, in reality, upon the formation of a hydrogen bond the potential box for the \(\pi\)-electrons not only lengthens, but also changes its profile somewhat: between the atoms H and B a potential barrier arises (Fig. 22, c), and the bottom of the right-hand part of the box (the region of the AH bond) lies higher than that of the left-hand part (the region of the initial conjugated system). In passing from potential box a (Fig. 22) to a box with such a profile, there occurs, as is known, a splitting of the initial quantum levels, as a result of which the frequency of the principal transition decreases \(^{106}\) (Fig. 22, c).

Fig. 22. Scheme of quantum energy levels in potential boxes of different length and shape.

Similar ideas concerning the convergence of energy levels upon conjugation in complexes with a hydrogen bond are in accord with data on the molecular refraction of certain substances. Thus, for example, the molecular refraction of p-dimethylaminobenzaldehyde \(((\mathrm{CH}_3)_2\mathrm{N}\cdot \mathrm{C}_6\mathrm{H}_4\cdot \mathrm{CHO})\) in benzene is 51.4, and in alcohol is 54.3 \(^{107}\). However, the refraction of diethyl ketone \(((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{CO})\) upon going from benzene to alcohol practically does not change.

Magnetic spectra

In the study of the proton magnetic resonance of alcohols, the influence of the hydrogen bond on the magnetic spectra of these substances was discovered.

It is known that a magnetic particle situated in a constant magnetic field \(H\) absorbs quanta of an alternating electromagnetic field of magnitude

\[ h\nu=\gamma \mu_{\mathrm{n}} H, \tag{9,2} \]

*) In this case there is no need to assume that a delocalization of all the \(\pi\)-electrons of the six-membered ring occurs with the hydrogen bond, although it is possible that in reality it does take place.

where \(\gamma\) is the Landé nuclear factor (nuclear gyromagnetic ratio), \(\mu_{\mathrm{яд}}\) is the nuclear magneton of Bohr, equal to \(eh/4\pi m_p c\) (\(m_p\) is the mass of the proton, \(c\) is the speed of light). It was established that the value of \(\gamma\), calculated from this relation, depends on the chemical nature of the substance of which the given nucleus is a part \({}^{108}\). This dependence is due to the fact that an external magnetic field induces in the atom under consideration a diamagnetic moment directed oppositely \({}^{109}\). As a result, the effective field in which the atomic nucleus is situated is always smaller than \(H\).

This phenomenon, which has received the name of magnetic shielding, has found application in investigations of the structure of the electron shells of molecules. If the magnetic resonance of, for example, a proton is studied, then, depending on the character of the bond of the hydrogen atom in the molecule under investigation, the resonance condition (9.2) will be satisfied at different values of the magnetic-field strength \(H\). As a result, the magnetic spectrum of the given substance is obtained on the oscilloscope screen; in the general case it consists of several peaks (signals), each of which corresponds to a definite group of chemically equivalent hydrogen atoms (Fig. 23). The smaller the value of the field strength \(H\) at which the absorption maximum is observed, the weaker is the magnetic shielding of the given group of nuclei. Empirically it has been established that often the degree of shielding changes in parallel with the electron density at the nucleus. Thus it becomes possible to judge the distribution of electron density in the molecule.

Fig. 23. Oscillogram of the proton magnetic resonance of ethyl alcohol \({}^{108a}\). The peaks refer (from left to right) to OH, CH\(_2\), CH\(_3\).

Investigation of the proton magnetic resonance of methyl and ethyl alcohols in the liquid phase showed, first, that the proton in the OH group is shielded more weakly than in the CH\(_2\) and CH\(_3\) groups,

(Fig. 23), which corresponds to the generally accepted view of the greater degree of polarity of the OH bond in comparison with C—H. Secondly, this investigation made it possible to establish that the position of the signal from the proton belonging to the OH group depends on temperature \(^{109a}\); namely, when the latter is increased, the OH signal is gradually shifted toward larger values of \(H\), approaching the signal of the \(\mathrm{CH_2}\) group, whose position does not depend on temperature

Fig. 24. Shift of the OH signal of ethyl alcohol relative to the \(\mathrm{CH_2}\) signal as a function of temperature (symbols of different kinds refer to different series of experiments) \(^{109a}\).

(Fig. 24). It was suggested that this phenomenon is caused by hydrogen bonds between alcohol molecules; when the temperature is raised these bonds are broken, which is accompanied by a change in shielding \(^{110}\). If the mean lifetime of the two states—\(\mathrm{ROH}\ldots\mathrm{OHR}\) and \(\mathrm{ROH}+\mathrm{OHR}\)—is sufficiently small, then the position of the signal on the oscillogram will correspond to a certain average shielding between these states. The magnitude of this shielding will be determined by the relative concentration of dimers and monomers. This assumption was confirmed by measurements of the dependence of the position of the OH signal of methyl and ethyl alcohols on concentration upon dissolution in \(\mathrm{CCl_4}\) \(^{109a}\). It turned out that with decreasing concentration, i.e., as the relative number of hydrogen bridges decreases, the OH signal gradually approaches the \(\mathrm{CH_2}\) signal, i.e., dilution is equivalent to an increase in temperature. (See also \(^{110a}\).)

If it is assumed that the magnitude of the magnetic shielding of the proton is directly related to the magnitude of the electron density near the nucleus, then from the facts indicated it follows that the formation of hydrogen bonds is accompanied by a decrease in the electron density near the H nucleus. If by \(\omega\) we denote the fraction of the electron density of the AH (OH) bond falling on the proton, and by \(\omega'\) the fraction of the density of the unshared electrons of atom B (O) falling on it, then this circumstance means that the decrease in \(\omega\) in absolute magnitude is greater than the increase in \(\omega'\), caused by the donor–acceptor interaction H...B. The conclusion that the effective positive charge of the H atom increases upon formation of a hydrogen bond confirms the assumption made above (§ 7) concerning the decrease of \(\omega\) with increasing \(r\) in the complex A—H...B (formula (7.1)).

§ 10. HYDROGEN BONDING AND PROTON-TRANSFER PROCESSES

As was indicated in § 5, under the influence of Pauling, the opinion became established in the chemical literature according to which the hydrogen bond is due to the simple electrostatic interaction of the molecules forming the complex RA—H...BR\(_1\). Thus, the hydrogen bond, by its very nature, was regarded as fundamentally different from the covalent (donor–acceptor) bond in the molecular ion \((HBR_1)^+\), arising, for example, as a result of proton transfer from the RAH molecule to the BR\(_1\) molecule. From this point of view, the possibility of the existence of an internal interrelation between the formation of the complex RAH...BR\(_1\) and the process of proton transfer from A to B was ruled out in advance. It is no accident that a number of chemists (see, for example, \(^{111}\)) pointed to the difficulty of applying the generalized concept of acid (according to Lewis) to the interpretation of the properties of protonic acids. In the light of the electrostatic concept of the hydrogen bond, certain facts seemed completely inexplicable. Thus, for example, the experimentally found linear dependence between the logarithm of the equilibrium constant of the process

\[ \mathrm{RAH} + \mathrm{BR}_1 \rightleftharpoons \mathrm{RA}^- + (HBR_1)^+ \tag{10,1} \]

and the shift of the frequency of the AH group upon formation of a complex with a hydrogen bond \(^{46}\) (Fig. 11) was of a purely empirical character. And although, under the influence of such facts, separate general statements appeared in the literature concerning the role of the hydrogen bond in the process indicated (see, for example, \(^{112}\)), these statements nevertheless remained theoretically unfounded and did not acquire concrete content.

In the theory of the question of the mechanism of intermolecular proton transfer, for a long time the ideas put forward in 1935 by Goriuti and Polanyi \(^{113}\) prevailed. According to these authors,

change in the energy of the system \(AH + B\) in the process of proton transfer, assumed to be adiabatic, is depicted by potential curves, as shown in Fig. 25*). Curve \(a\) represents the potential energy of the isolated \(A-H\) bond in the gas phase; curve \(b\), the potential energy of the \(+H-B\) bond in the absence of \(A^-\), and both energies are approximated by Morse functions (see (7,5)). Near the point of intersection of these curves, owing to degeneracy, splitting of the energy level occurs and, thus, an adiabatic transition across the energy barrier from one equilibrium position to another is ensured. True, in practice the authors always neglect this splitting. The difference in energy between the two equilibrium positions is taken by them to be equal to the heat effect of the reaction \(Q\). The influence of the solvent on the form of the potential curves manifests itself only in the fact that for curve \(b\) the factor \(D\) in the Morse function is taken to be equal to the sum of the molecule’s affinity for the proton and the hydration energy of the ion \(+HB\).

Fig. 25. Potential curves for proton transfer according to Gurney and Polanyi \(^{113}\). When the nature of one of the molecules changes, the corresponding curve is displaced vertically parallel to itself.

The authors believe that if the molecules \(AH\) and \(B\) are capable of forming a complex with the hydrogen bond \(AH \ldots B\), then the distance \(A \ldots B\) is thereby so small (in the case of water \(\sim 2.5 \text{ Å}\)) that the potential barrier disappears and the proton has only one equilibrium position. A potential barrier, in their opinion, arises under the condition that the distance \(A \ldots B\) is about \(3 \text{ Å}\), when formation of a hydrogen bond cannot be expected. However, such a point of view, as we saw above, is erroneous.

Using the indicated method of constructing the curves, the authors \(^{113}\) assume that a small change in the properties of one of the interacting molecules (for example, \(A\)) causes a vertical displacement of the corresponding potential curve (in our example, curve \(a\)),

*) Recently it has been shown quite convincingly that intermolecular proton transfer under ordinary conditions occurs according to the laws of classical mechanics over the potential barrier, and not by quantum-mechanical tunneling. If the tunnel effect played an important role, then in the corresponding reactions there would have to be observed, first, definite deviations from the Arrhenius equation and, second, a significant increase in the pre-exponential factor upon replacement of the proton by a deuteron, which is not observed \(^{114}\).

but it does not change the shape or position of the other curve, owing to which it follows directly from the graphical construction that

\[ \Delta E=-\alpha \Delta Q, \tag{10,2} \]

where \(\Delta E\) and \(\Delta Q\) are the corresponding changes in the activation energy and heat of reaction (Fig. 25), and, obviously, \(0<\alpha<1\).

In Bell’s book\(^{115}\) essentially the same picture of potential curves is adopted as in the work\(^{113}\).

The picture described above of the potential curves for the proton-transfer process is very approximate, because it ignores the deformation of these curves under the influence of the interactions \(AH\) with \(B\) (hydrogen bond) and \(A^-\) with \(+HB\) (mainly an ionic bond), when the proton is in each of the two equilibrium positions\(^{116}\). It is natural to expect that a change in the nature of, for example, \(B\) must change not only the position of curve \(b\), but also its shape simultaneously with the change in the shape and position of curve \(a\). In addition, in the indicated treatment the influence of the medium on the proton-transfer process is not sufficiently reflected.

The proton-transfer process (10,1) ends with the attachment of the proton to \(BR_1\) by means of a donor-acceptor bond effected by the lone pair of electrons of \(B\). On the other hand, as was indicated in § 6, the hydrogen bond \(RA — H\ldots BR_1\) is to a considerable degree due to the same donor-acceptor interaction of \(H\) and \(B\) with the aid of the same electrons, i.e., it represents, as it were, an incipient form of the bond that is formed as a result of the transfer. This comparison naturally leads to the hypothesis that the proton-transfer process (10,1) always has as its first stage the formation of the hydrogen bridge \(RA — H\ldots BR_1\)*).

According to this point of view, the donor-acceptor interaction between \(H\) and \(B\) is realized at all “stages” of the motion of the proton from \(A\) to \(B\), and as the proton passes from the electron shell of the \(A — H\) bond into the shell, deformed by it, of the lone pair of electrons of atom \(B\), the \(H\ldots B\) bond is strengthened. As a result atom \(A\) becomes negatively charged, while the \(HB\) group

*) In 1948 this hypothesis was put forward by N. D. Sokolov, proceeding from ideas about the nature of the hydrogen bond (see § 6)\(^{117}\). Independently of him, and at almost the same time, an analogous suggestion was made by Wirtz\(^{118}\) (in connection with studies of energy transfer in proteins and of the anomalous mobility of the hydrogen ion in water) and by Mecke and Cahnwerp\(^{119}\) (in connection with the study of anomalies in the electrical conductivity of certain associated substances). These authors\(^{118,119}\), however, did not consider the mechanism of proton transfer from the standpoint of the nature of the hydrogen bond and thus assigned to the latter only the role of a factor facilitating the favorable mutual orientation of the molecules between which the proton passes. Ideas close to those indicated have been expressed beginning in 1947 (Davis\(^{132}\)).

is charged positively; in other words, the process

\[ \mathrm{AH}\ldots \mathrm{B}\to \mathrm{A}^{-}\ldots(\mathrm{HB}^{+})^{*} \]

takes place.

Thus, the role of the hydrogen bond in the process of proton transfer consists not only in ensuring a favorable mutual orientation of the molecules. More important is the circumstance that the donor–acceptor bond \(\mathrm{H}\ldots \mathrm{B}\) is strengthened in the course of the motion of the proton from A to B, which leads to a lowering of the energy barrier and to the transfer precisely of a positive charge, and not of a neutral atom.

It is not difficult to see, however, that in the gas phase and in nonpolar solvents the process of proton transfer (10,1), leading to the formation of two ions \(\mathrm{RA}^{-}\) and \((\mathrm{HBR}_{1})^{+}\), is strongly endothermic and practically does not occur. This endothermicity is due to the fact that the heat of removal of the proton from the neutral molecule \((\mathrm{RAH})\) is, in absolute value, much greater than the heat of its attachment to the neutral molecule \((\mathrm{BR}_{1})^{**}\). For this reason, intermolecular proton transfer (10,1) can in practice take place only in a polar solvent, which ensures strong solvation of the ions formed and thereby considerably lowers the energy of the final state.

The polarization of the solvent, which ensures solvation of the ions, also strongly affects the very process of proton transfer. As is known, the polarization of a polar solvent is mainly determined by the orientation of its dipolar molecules, whereas the electronic polarization is comparatively small. The motion of the proton from A to B, accompanied by the formation of two oppositely charged ions \(\mathrm{RA}^{-}\) and \((\mathrm{HBR}_{1})^{+}\), can occur only in the presence of a definite orientation of the surrounding molecules of the medium. Since the time of the actual motion of the proton from A to B, of the order of \(10^{-12}\)—\(10^{-13}\) sec, is apparently less than or approximately equal to the time of reorientation of molecules (the time of dipole relaxation), of the order of \(10^{-11}\)—\(10^{-12}\) sec, one should expect that the polarization of the medium cannot “follow” the motion of the proton, but arises beforehand, owing to a random fluctuation. Consequently, proton transfer from A to B takes place only in those complexes \(\mathrm{RA}-\mathrm{H}\ldots \mathrm{BR}_{1}\),

*) Since the mass of the hydrogen atom is considerably smaller than the masses of the group \(\mathrm{RA}\) and the molecule \(\mathrm{BR}_{1}\), to a first approximation one may assume that in the process of this transition the distance \(\mathrm{A}\ldots \mathrm{B}\) remains unchanged.

**) For example, in the case where \(\mathrm{RAH}\) is water and \(\mathrm{BR}_{1}\) is ammonia, the first quantity in the gas phase is approximately 390 kcal, and the second 209 kcal\({}^{120}\). In this case the change in energy in the process
\(\mathrm{RAH}\ldots \mathrm{BR}_{1}\to \mathrm{RA}^{-}\ldots{}^{+}\mathrm{HBR}_{1}\)
is about 70 kcal\({}^{121}\). It is possible that in the gas phase proton transfer is impossible not only thermodynamically, but also kinetically, because of the necessity of overcoming a considerable potential barrier.

in the vicinity of which this sort of random reorientation of molecules has occurred*). At the same time, while the effective charges of the groups \(RA\) and \(H\ldots BR_1\) are small (the distance \(r\) differs little from the equilibrium one), their interaction energy with the medium is small and the motion of the proton is associated with an expenditure of energy. At a certain \(r\), the interaction with the medium becomes so strong that, upon further motion of \(H^+\), the energy of the system begins to decrease, i.e., the proton finds itself on the other side of the top of the potential barrier.

In Fig. 26, curve \(I\) depicts the potential energy of the isolated bond \(A—H\) in the gas during its dissociation into atoms.

Fig. 26. Potential curves for proton transfer. \(I\) — energy of the isolated bond \(A—H\) in the gas; \(II\) — energy of the bond \(A—H\) in the complex \(RA—H\ldots BR_1\) in the gas; \(III\) — the same in a solvent in the absence of a change in the polarization of the medium; \(III'\) — the same, taking into account the change in the polarization of the medium.

Curve \(II\) depicts the dependence of the potential energy of the complex \(RAH\ldots BR_1\) on the distance \(r\) in the gas phase during the transfer of a proton from \(A\) to \(B\); the minimum of curve \(II\) is shifted downward relative to curve \(I\) by the value of the hydrogen-bond energy; the energy value in the state \(A^{-}\ldots{}^{+}HB\) is conventionally taken as equal to \(\sim 70\) kcal (see the footnote on p. 264). Curves \(III\) and \(III'\) depict the potential energy of the complex \(RA\ldots H\ldots BR_1\) in solution. Curve \(III\) corresponds to the imaginary case in which, in the course of the proton’s motion, the polarization of the medium is the same as in the equilibrium state. Curve \(III'\) corresponds to the case in which polarization of the medium has occurred**).

*) It is not difficult to see that such an orientation of the dipoles of the medium in itself usually requires a certain expenditure of energy, which is included as a term in the measured experimental activation energy. In the case, for example, of such solvents as water or alcohols, this kind of orientation of molecules requires a partial rupture of intermolecular hydrogen bonds.

**) The energy required for the reorientation of the molecules of the medium is not taken into account in Fig. 26.

Near the equilibrium position both curves coincide and are shifted downward with respect to curve II by the solvation energy of the complex \(\mathrm{RAH\ldots BR_1}\). As \(r\) increases, the interaction of the forming charges with the polarized medium increases, curve III′ deviates more and more from curve III and, finally, at \(r \simeq 1.8\ \text{Å}\) has a minimum. A further lowering of the energy of the system is connected with separation of the ions \(\mathrm{RA^-}\) and \((\mathrm{HBR_1})^+\) and with an increase in solvation.

It follows from the foregoing that the representation of the potential curve of the system \(\mathrm{A\ldots H\ldots B}\) as consisting of two curves “interacting” only near their point of intersection is inadmissible even as a first approximation. In fact, as is seen from Fig. 26, the form of the potential curve of the A—H bond (curve I) under the influence of the \(\mathrm{BR_1}\) molecule and of the medium changes not only near the activated state, but already near the equilibrium position (curve III′). As a result, a change in the nature of one of the molecules affects not only its “own” branch of the potential curve (see Fig. 25), but also the entire curve as a whole.

Fig. 27. Change in the potential energy of the system \(\mathrm{RA\ldots H\ldots BR_1}\) upon changing the substituent \(\mathrm{R_1}\) \((\Delta E_a\)—change in activation energy, \(\Delta Q\)—change in the heat effect of the reaction, \(\Delta \varepsilon\)—change in the energy of the hydrogen bond).

In particular, when the nature of \(\mathrm{BR_1}\) changes, this influence on the “foreign” branch is expressed in a change in the energy of the hydrogen bond \(\mathrm{RA\ldots H\ldots BR_1}\) (Fig. 27).

The picture drawn makes it possible to establish a relation between the heat \(Q\) of process (10,1) and the energy \((\varepsilon)\) of the hydrogen bridge in the complex \(\mathrm{RAH\ldots BR_1}\). Indeed, since the latter is partly due to the same donor–acceptor interaction of the H and B atoms which wholly determines the bond in the ion \((\mathrm{HBR_1})^+\), it is natural to assume that, for a series of reactions (10,1) differing only in the nature of \(\mathrm{R_1}\), the following relation is fulfilled:

\[ Q = a - b\varepsilon, \tag{10,3} \]

where \(a\) and \(b\) are certain constants (for the endothermic direction \(b > 0\)). If we now assume that the change in entropy \((\Delta S)\) in process (10.1) is the same for the indicated series, and neglect the difference in the solvation energy of the ions \((\mathrm{HBR_1})^+\) with a change in \(\mathrm{R_1}\), then it is not difficult to show\(^{117,122,123}\) the validity of the following relation:

\[ RT \ln K = - (Q - T\Delta S) \simeq A' + B'\varepsilon \qquad (B' > 0). \]

Taking into account that, according to equality (7.6) (or (7.9)), for the series under consideration the hydrogen-bond energy \(\varepsilon\) is approximately proportional to the frequency shift, we find that at fixed temperature

\[ \lg K = A + B\frac{\Delta \nu}{\nu_0}, \tag{10.4} \]

where \(\Delta \nu = \nu - \nu_0\) is the shift of the vibrational frequency of the AH group in the \(\mathrm{R_1AH}\) molecule under the influence of the hydrogen bond, and \(A\) and \(B\) are constant quantities, with \(B > 0\).

This equation expresses nothing other than the above-mentioned dependence, found empirically by Gordy and Stanford\(^{46*}\) (Fig. 11). The experimental points fit the straight line rather well

\[ \lg K = -27.2 + 272 \frac{\Delta \nu}{\nu_0}. \tag{10.5} \]

The most probable reason for the deviations that occur lies in the change of the entropic term.

The existence of a linear dependence between \(\lg K\) and the frequency shift \(\Delta \nu\), fulfilled over a wide interval of variation of \(\lg K\), may serve as indirect confirmation of the proportionality between the hydrogen-bond energy and this shift\(^{**}\).

\(*\) As indicated above, the quantity \(\Delta \nu / \nu_0\) in equation (10.4) is the relative frequency shift of the AH group of the \(\mathrm{R_1AH}\) molecule that participates in reaction (10.1). On the other hand, Gordy and Stanford\(^{46}\) measured the shift of the OD frequency of the \(\mathrm{CH_3OD}\) molecule. However, as the same authors\(^{46}\) showed, the relative frequency shifts of different AH (or AD) groups under the influence of the same substances \(\mathrm{BR_1}\) are linearly related to one another. For example, between the frequency shifts of heavy water \((\Delta \nu'/\nu_0')\) and \(\mathrm{CH_3OD}\) \((\Delta \nu' / \nu_0')\) there is the following relation:

\[ \Delta \nu / \nu_0 = 0.023 + 1.1\,\Delta \nu' / \nu_0'. \]

\(**\) It should be noted that relation (10.5) is obeyed by a larger number of compounds than might have been expected in advance. Indeed, as can be seen from Fig. 11, bases \(\mathrm{BR_1}\) lie on one and the same straight line, differing not only in the nature of substituent \(\mathrm{R_1}\), but also in the nature of atom B. This indicates, in particular, that in reality the proportionality between \(\Delta \nu\) and \(\varepsilon\) occurs under less stringent conditions than was assumed in § 6. (See, however,\(^{133}\).)

It is not difficult to see that, analogously to equation (10,4), one can obtain the corresponding relation between the frequency shift and the logarithm of the rate constant for proton transfer. If it is assumed that, for the same series of reactions (10,1), in which only \(R_1\) changes, the pre-exponential factor in the Arrhenius equation

\[ k = k_0 e^{-\frac{E}{RT}} \]

remains approximately constant, then the change in \(k\) will be determined entirely by the activation energy. Taking into account that, as the proton is removed from A, its bond with atom B becomes stronger, and that the nature of this bond is basically the same as in the initial state, it is natural to conclude that the top of the potential barrier is located closer to the energy of the initial state the greater the interaction energy \(\varepsilon\) of H and B in the latter. In a first approximation one may put

\[ E = a_1 - b_1 \varepsilon . \]

Hence, taking (7,6) into account, we find\(^{117,122,123}\)

\[ \lg k = A_1 + B_1 \frac{\Delta \nu}{\nu_0} \quad (B_1 > 0). \tag{10,6} \]

The theoretical estimate of the magnitude \(B_1\) for the case of proton transfer from one oxygen atom to another \((\mathrm{RO}\ldots\mathrm{H}\ldots\mathrm{OR}_1)\) gives \(B_1 = 76 \div 116^{122}\).

Fig. 28. Relation between \(\Delta \nu/\nu_0\) and the rate constant of semicarbazone formation (straight line \(I\), according to\(^{46}\)): 1—furfural, 2—benzaldehyde, 3—acetone, 4—cyclohexanone, 5—cinnamaldehyde, 6—heptaldehyde, 7—butyraldehyde.

Experimentally, a linear dependence between \(\lg k\) and \(\Delta \nu/\nu_0\) was found by Gordy and Stanford\(^{46}\) for the reaction of formation of semicarbazones \((\mathrm{RR'C}=\mathrm{N}-\mathrm{NH}-\mathrm{CO}-\mathrm{NH}_2)\) from semicarbazide \((\mathrm{NH}_2-\mathrm{NH}-\mathrm{CO}-\mathrm{NH}_2)\) and certain aldehydes and ketones \((\mathrm{RR'C}=\mathrm{O})\) in a buffered aqueous solution of phosphoric acid (Fig. 28, straight line \(I\))*). However, these authors did not propose any explanation of this regularity. The straight line they gave is based on kinetic data for only two aldehydes and two keto-

*) In Figs. 28 and 29, along the abscissa axis are plotted the values of the same quantity as in Fig. 11, namely the relative frequency shifts \(\Delta \nu/\nu_0\), which cause molecules of various bases at the OH bond in the \(\mathrm{CH_3OD}\) molecule when a hydrogen bond is formed with it.

gens[^124]. More reliable data for the same reaction, occurring in a mixture of water with methyl glycolate[^125], show that a linear relationship is observed only in the case of aldehydes (Fig. 28, line II), whereas the data for ketones lead to a considerable scatter of points.

For a complex reaction proceeding through several stages, a regularity of the form (10.6) can be observed only provided that proton transfer is the slowest stage. Thus, from the presence of regularity (10.6) one may draw definite conclusions about the mechanism of the reaction under consideration. For example, consideration of the above-mentioned reaction of semicarbazide with aldehydes in the presence of an acid, when there is a linear relation between \(\lg K\) and \(\Delta \nu/\nu_0\), leads to the conclusion that, contrary to the prevailing opinion[^112], the slowest stage is the following process (\(\mathrm{H_2NB}\) is semicarbazide, \(\mathrm{HA}\) the catalyzing acid):

\[ \mathrm{H_2NB + R'RCO\ldots HA \rightarrow \left[ \begin{array}{c} \mathrm{R'}\\ |\\ \mathrm{R-C-OH}\\ |\\ \mathrm{H_2NB} \end{array} \right]^+ + A^- .} \]

This stage is not as simple as ordinary elementary chemical acts, since it includes simultaneously proton transfer and addition of the \(\mathrm{H_2NB}\) molecule to the C atom, which is accompanied by a considerable redistribution of electron density. It is possible that this stage has a somewhat different mechanism, in which the trimolecular activated complex has a cyclic structure[^134]. In this case as well, a synchronous displacement of several nuclei occurs. The existence of such synchronicity in elementary acts had already been assumed earlier in a number of other cases (see, for example,[^126,^127]).

Line II in Fig. 28 can be expressed by the equation

\[ \lg K = -2.36 + 97\frac{\Delta \nu}{\nu_0}. \]

Consequently, for the reaction of semicarbazone formation the coefficient \(B_1\) in equation (10.6) is equal to 97. The theoretical estimate of this quantity given above agrees with this result.

There are very few experimental data in the literature needed to verify equality (10.6). In addition to the example considered, some data on the rate and on the corresponding frequency shifts are available for the decomposition reaction of nitramide catalyzed by various amines. As was established long ago1, in this reaction the slowest stage is the transfer of a proton from the nitramide molecule to the catalyst molecule—the amine.

Therefore, one may expect in advance the presence of a linear dependence between $\lg k$ and $\Delta \nu/\nu_0$, which does in fact exist (Fig. 29).

Let us note that, by eliminating the quantity $\Delta \nu/\nu_0$ from equations (10.4) and (10.6), a linear relation is obtained between the logarithms of the rate constant and the equilibrium constant,

\[ \lg k = a + \alpha \lg K. \]

Fig. 29. Relation between $\Delta \nu/\nu_0$ and the rate constant for decomposition of nitramide (catalysis by amines: 1—ortho-chloroaniline, 2—meta-chloroaniline, 3—ortho-toluidine, 4—aniline, 5—meta-toluidine).

This relation, established empirically for a number of reactions catalyzed by acids and bases, is called the Brønsted relation1. Fulfillment of this relation may serve as a weighty argument in favor of the proposition that the slowest stage of the reaction under consideration is proton transfer. In those cases where the measurement of the principal constant $K$ encounters great difficulties (as, for example, in the case of aldehydes), this relation may be replaced by the dependence (10.6), which, as we have seen, can in the indicated sense play the same role in investigations as the Brønsted relation.

The transfer of a proton from one neutral molecule to another (10.1), evidently, represents one of the possible cases of the proton-transfer process. It is encountered mainly in the electrolytic dissociation of acids, and also in acid-base catalysis, in tautomeric transformations, etc. In addition to process (10.1), the following proton transfers are also of great importance:

\[ (\mathrm{RAH})^{+} + \mathrm{BR}_{1}^{-} \to \mathrm{RA} + \mathrm{HBR}_{1}, \tag{10.7} \]

\[ (\mathrm{RAH})^{+} + \mathrm{BR}_{1} \to \mathrm{RA} + (\mathrm{HBR}_{1})^{+}, \tag{10.8} \]

\[ \mathrm{RAH} + \mathrm{BR}_{1}^{-} \to \mathrm{RA}^{-} + \mathrm{HBR}_{1}. \tag{10.9} \]

Process (10.7), evidently, is the process inverse to (10.1). Transition (10.8) plays a role, for example, in catalysis by $\mathrm{H_3O^+}$ ions and determines the anomalously high mobility of protons in water and in certain alcohols, etc. Process (10.9) occurs, for example, in catalysis by anions, determines the anomalous mobility of hydroxyl anions in water, etc. Since ions participate in all these reactions, they are practically realized only

HYDROGEN BOND

in polar solvents. At the initial stage of these processes (at the moment when the particles approach one another), a donor–acceptor interaction arises between H and B, which is preserved to one degree or another also in the course of the proton transfer. However, this interaction is not a hydrogen bond in the usual sense.

For many reactions catalyzed by acids and bases, the proton-transfer stage is the slowest, as is shown, in particular, by the fulfillment of the Brønsted relation \(^{115}\). In this case the activation energy of the reaction may be regarded as the activation energy of proton transfer. Experimental data show \(^{115}\) that, for processes of type (10,1), activation energies of \(15\text{–}20\ \text{kcal/mole}\) are characteristic, whereas for processes (10,8) they are \(4\text{–}5\ \text{kcal/mole}\). Such a difference is apparently due primarily to the fact that detachment of a proton from a neutral molecule (process (10,1)) requires considerably more energy than detachment from a molecular cation (10,8). In addition, in the first case an expenditure of energy is needed for polarization of the medium, while in the second polarization already exists in the initial state.

In addition to processes in which the transfer of a single proton occurs in complexes with a hydrogen bond, it is possible that there exist processes of simultaneous, synchronous transfer of many protons in a chain of molecules connected by hydrogen bridges \(^{126}\). Such a process, as a result of which charge is transferred from one end of the polymer chain to the other, must be accompanied by a corresponding rearrangement of the electron shells of all molecules in the chain, for example,

\[ \begin{gathered} \mathrm{\ \ \ \ \ \ \ \ \ \ R}\\[-0.2em] \mathrm{\ \ \ \ \ \ \ \ \ /}\\[-0.2em] \cdots \mathrm{H-O-C=O\cdots H-O-C=O\cdots H} \longrightarrow \\[0.6em] \mathrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R \qquad\qquad\qquad R}\\[-0.2em] \mathrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ / \qquad\qquad\qquad /}\\[-0.2em] \longrightarrow \mathrm{-H\cdots O=C-O-H\cdots O=C-O-H\cdots} \end{gathered} \]

In the case of water and alcohols this rearrangement is simplified because the same oxygen atom represents both the donor and the acceptor of the proton:

\[ \begin{gathered} \mathrm{\ \ \ R \qquad R \qquad R}\\[-0.2em] \mathrm{\ \ \ / \qquad / \qquad /}\\[-0.2em] \mathrm{H-O\cdots H-O\cdots H-O\cdots} \longrightarrow \\[0.6em] \mathrm{\ \ \ \ \ \ \ \ \ \ \ R \qquad R \qquad R}\\[-0.2em] \mathrm{\ \ \ \ \ \ \ \ \ \ / \qquad / \qquad /}\\[-0.2em] \longrightarrow \mathrm{-H\cdots O-H\cdots O-H\cdots O-H\cdots} \end{gathered} \]

In this case the whole chain, with the exception of the terminal atoms, remains

electrically neutral. For such a cooperative action it is evidently necessary that the normal vibration of the initial system, corresponding to a simultaneous symmetrical stretching and contraction of all bonds A—H, should acquire a certain excess energy equal to the activation energy of the given process.

In the presence of an external electric field, transitions along the field will predominate over transitions in the opposite direction. Such a synchronous shift of protons is probably realized, for example, in ferroelectrics at temperatures in the region between two Curie points \(^{129,12}\). In some associated liquids, for example in water, alcohols, etc., a similar process, together with the process in dimers (10,8), may account for the anomalously high mobility of the proton \(^{118,126}\). In order that, in this case, the shift of protons should not be limited to a single act, as in the case of polarization of a ferroelectric, but should be repeated many times in one and the same direction and thereby ensure the existence of a continuous electric current, it is evidently necessary that the molecules of the chain, after each act of acceptance and transfer of a proton, rotate through a definite angle. In the case of water this angle is \(109^\circ\).

Fig. 30. System of hydrogen bonds in polypeptides. The arrows indicate the direction of the polypeptide chain. The hydrogen bridges are perpendicular to it. \(A\)—ketone form, \(B\)—enol form \(^{118}\).

An analogous transfer of a proton is possible, for example, in proteins, which are polypeptide chains oriented in a definite manner, the links of which are connected with one another by hydrogen bridges (Fig. 30). If a proton or some other cation is attached from outside to the corresponding terminal atom of the hydrogen-bond chain, a displacement of charge occurs in this chain and the ketone form is converted into the enol form, or conversely.

The question of the activation energy of such processes of simultaneous proton transitions in chains is not clear. If the height of the potential barrier that each proton must overcome is equal to the height of the barrier corresponding to the transition of a single ...

proton (for example, (10,1) or (10,8)), then the resultant activation energy for \(n\) protons will be \(n\) times greater. In this case such processes, if they exist at all, are probably only for \(n = 2—4\). It is not excluded, however, that, owing to the specificity of the cooperative transition, the resultant activation energy of the process is considerably less than the sum of the activation energies of \(n\) single transitions. In this case the process is also possible at large \(n\).

The cooperative process of proton transfer along an open chain of molecules should, in the general case, take place in a medium capable of strong polarization, since ions are then always present. If, however, such a process takes place in a closed cycle of several molecules linked by hydrogen bridges, then the presence of such a medium is not obligatory. Such cooperative proton transfer can in principle occur both in the gas phase and in nonpolar liquids, since it is not associated with the formation of ions.

A. I. Brodskii \(^{130}\) suggested that the very rapid isotopic exchange of hydrogen observed, for example, in compounds containing hydroxyl is due to the simultaneous transition of two protons in a bimolecular complex formed by two hydrogen bonds

\[ \begin{array}{c} R_1 \backslash \\ \quad O - H \\ \quad \vdots \quad \vdots \\ \quad D - O \\ \qquad \backslash R_2 \end{array} \]

(the direction of the arrows coincides with the direction of the hydrogen bonds \((\mathrm{H}\ldots \mathrm{O}\ \text{and}\ \mathrm{D}\ldots \mathrm{O}))\).

It is not difficult to see, however, that the formation of such a double complex is not very probable. In fact, first, the hydrogen bonds \(\mathrm{H}\ldots \mathrm{O}\) \((\mathrm{D}\ldots \mathrm{O})\), arranged at right angles to the \(\mathrm{H—O}\) \((\mathrm{D—O})\) bonds, either cannot form at all or are only slightly stable because the axes of the lone-pair clouds form in water and alcohols an angle of \(109—120^\circ\), not \(90^\circ\). Secondly, in such a configuration there is strong repulsion of the oxygen atoms from one another; if, for example, one assumes that the distances \(\mathrm{O}\ldots \mathrm{H}\) and \(\mathrm{O}\ldots \mathrm{D}\) are equal to the normal distances in complexes with a hydrogen bond (1.7 Å), then, with the length of the \(\mathrm{O—H}\) \((\mathrm{O—D})\) bond \(\sim 1\) Å, the distance \(\mathrm{O}\ldots \mathrm{O}\) will be about 2.0 Å, whereas the sum of the van der Waals radii of oxygen atoms is 2.8 Å.

A more plausible assumption would be that isotopic exchange occurs in a cyclic complex of three or more

number of molecules. In this case the indicated difficulties in the formation of the complex would hardly arise. In the case, for example, of three molecules, the process of simultaneous transfer of three protons would be represented as follows:

\[ \begin{array}{c} \text{cyclic complex of three } \mathrm{ROH} \text{ molecules with hydrogen bridges} \\ (R_1O\!-\!H\cdots O R_2,\; R_2O\!-\!H\cdots O R_3,\; R_3O\!-\!H\cdots O R_1) \;\longrightarrow\; \text{intermediate cyclic arrangement} \;\longrightarrow\; (R_1O\cdots H\!-\!O R_2,\; R_2O\cdots H\!-\!O R_3,\; R_3O\cdots H\!-\!O R_1). \end{array} \]

It has recently been found that the rates of hydrogen–deuterium exchange in hydroxyl compounds are, within the experimental error, the same both in the liquid and in the gaseous phase \(^{131}\). To explain this fact the authors assume that the exchange takes place in a cyclic complex consisting of several molecules linked by hydrogen bridges. At the same time, the authors erroneously believe that neutral atoms, and not protons or deuterons, are exchanged (transferred).

Taking into account the nature of the hydrogen bond, one may assert that the formation of the latter cannot facilitate the transfer of a neutral atom

\[ \mathrm{A-H\cdots B \to A\cdots HB}, \]

since the H atom will always be strongly repelled by the unshared electron pair of atom B*).

It is not excluded that simultaneous proton transfers in a cycle of several molecules occur not only in isotope exchange, but also in other chemical reactions.

In a number of cases such a mechanism should be preferred to an ionic mechanism. Thus, for example, it appears more probable than the mechanism proposed by Svezhom \(^{133}\) for a number of reactions which proceed in benzene and, according to the author’s supposition, go through the formation of ions. The supposition that many reactions proceed through a cyclic complex of several molecules was first made by E. A. Shilov \(^{134}\).

*) I. P. Gragerov and A. I. Brodskii \(^{132}\) measured the activation energy \((E)\) for the transfer of a hydrogen atom by the example of equilibrium between hydroquinone and quinone. In calculating per one H atom they found \(E = 32\ \text{kcal/mol}\), i.e. a value considerably higher than the activation energy for proton transfer. It is not excluded, however, that this process proceeds through several stages and that the cited value represents an effective activation energy. (On the role of the hydrogen bond in chemical reactions, see the papers of N. M. Emanuel’ and D. G. Knorre \(^{135}\) and A. I. Shatenshtein \(^{136}\).)

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