The Hydrogen Bond

The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. These give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions, in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known…
— Paul Dirac¹

The claim that chemistry has been completely explained in terms of quantum theory is now received wisdom among physicists and chemists. Yet quantum physics is able neither to predict nor explain the strong association of water molecules in liquid or ice. Quantum chemistry algorithms either exclude hydrogen bonded (H-bonded) systems, or treat them by modeling a water molecule as an asymmetric tetrahedron having two positive and two negative electrical charges at its vertices. Recent calculations of the potential energy surface of the simple water dimer {H₂O}₂ yield 30,000 ab initio energies at the CCSD(T) level.² But free OH-stretches are below experimental values by 30-40cm^-1 and their dissociation energy 1.1kJ·mol^-1 below benchmark experimental values. To obtain satisfactory agreement with experiment, it is necessary to replace ab initio potentials with spectroscopically accurate measurements. This is hardly a ringing endorsement of the underlying theory.

In this essay, I defend the thesis that the hydrogen bond is an emergent property of matter resulting from a non-linear coupling between quantified energy levels of water molecules and a quantified internal electromagnetic field. Such a coupling leads to the emission of Nambu–Goldstone gauge bosons yielding Bose–Einstein condensates at relatively high temperatures, thus forming liquid water or ice. These ideas have been developed by physicists at the Universities of Milan and Naples, students in fact, or in spirit, of Hiroomi Umezawa, one of the founders of quantum field theory (QFT).³

A Short History of the Hydrogen Bond

In June 1611, a strong heat wave enveloped Tuscany; and with the heat proving stifling, Galileo Galilei decamped for the nearby villa of his friend Filippo Salviati.⁴ Since it was hot, Galileo determined to discuss things that were cold. His interlocutors were two professors from the University of Pisa, Vincenzo di Grazia and Giorgio Coresio. According to Aristotelian physics, ice, since it was obviously colder than water, should comprise water minus a certain amount of fire or heat. This led water to condense into a solid. It is in the nature of air or fire to move upwards, di Grazia and Coresio observed, no doubt persuaded that they were recounting the obvious, but in the nature of solids to move downwards. It follows that ice remained buoyant because of its large flat shape, which prevented water from penetrating its surface and dragging it down where it belonged.

Galileo demurred. The correct analysis of ice and water should begin with the assumption that ice is less dense than water. It floats because it is less dense. Galileo considered ice as water with more by way of volume; Aristotelian scholars, as water with less by way of heat.

The disagreement came to the ears of the philosopher Lodovico delle Colombe, who, having been mocked by Galileo a few years before, saw in the dispute an occasion for revenge. Delle Colombe set up a spectacular experiment in which he compared the behavior of spheres and wafers made of ebony, a substance known to be denser than water. Delle Colombe demonstrated that the spheres always sank when dropped in water; the wafers did not. This was, he supposed, proof that buoyancy was more a matter of shape than density.

Stung by these experiments, Galileo began writing a fifteen-page essay in September 1611 dedicated to his mentor, Cosimo II de’ Medici. Galileo’s treatise is the very first book devoted to the subject of hydrogen bonding in water.⁵

Ice floats in water. True enough. Modern theories affirm that in ice, water molecules are associated by linear hydrogen bonds. On melting, these bonds begin, spaghetti-like, to bend. Bending causes the liquid to become denser than the solid between 0°C and 4°C. But above 4°C, radial expansion wins over angular bending; as with most other liquids, the higher the temperature, the lower the density.

For all that, Galileo was still not able clearly to explain why ebony wafers float.

The collapse of the Aristotelian conception of nature at the end of the eighteenth century marked a new era in the study of water. A clear distinction emerged between atoms, on the one hand, and molecules, on the other. In 1796, the French chemist Joseph Louis Proust, in affirming the law of definite proportions, suggested that compounds combine by weight and in simple proportions.⁶ Proust found that 1g of hydrogen would combine with 8g of oxygen, or 2g with 16g, or 4g with 32g, but never in fractional proportions. This observation was refined in 1804 by the English chemist John Dalton. Water, Dalton argued, is HO, where H designates a hydrogen atom, and O, an atom of oxygen.⁷

This was simple, clear-cut, and wrong. It was the Italian chemist Amedeo Avogadro, who in 1811 deduced that water is, in fact, H₂O, and not HO. In order to obtain two volumes of water, Avogadro observed, it is necessary to mix two volumes of hydrogen with one volume of oxygen: 2H₂ + O₂ = 2H₂O.⁸ Avogadro’s masterful analysis gave force to the idea that, among other things, atoms can form bonds.⁹

For all the success of the atomic theory of matter, it soon became obvious that something very strange was occurring when hydrogen combined with elements such as oxygen, nitrogen, or fluorine. Studies involving the liquefaction of gases indicated that the higher their molecular weight, the higher their boiling or melting point, and the higher their latent heat of vaporization or melting. This was clearly not the case for HF with respect to HCl-HBr-HI, or H₂O with respect to H₂S-H₂Se-H₂Te, or NH₃ with respect to PH₃-AsH₃-SbH₃.¹⁰ Various anomalies were noted with respect to congelation points and the vapor density curves of several liquid mixtures.¹¹ Nor was it possible to explain on rational grounds the base constants of ammonia and its substituted amines.¹² Very soon after the discovery of X-ray diffraction, it became obvious that in ice, water molecules were strongly associated in a clear tetrahedral crystalline structure, one characterized by large hexagonal channels in which various gases were trapped.¹³

The discovery of the HF₂- ion in 1923,¹⁴ and the observation in 1925 of strong variations in the stretching frequency of the O-H bond¹⁵ (so-called stretch marks), provided clear evidence that chemical bonding was a more complex phenomenon than first thought.

It was far more complex.

During the first decades of the twentieth century, Wendell M. Latimer, Gilbert Newton Lewis, and Worth H. Rodebush, at the University of California, attempted to understand weak bonds in strong theoretical terms. They were, to a certain extent, forced to fumble. In May 1919, a graduate student at the University of California named Maurice L. Huggins conjectured that the hydrogen nucleus might be held in suspension between the octets of two other atoms. His teacher, William C. Bray, addressed him in words that were as assured as they were incorrect:

Huggins, there are several interesting ideas in this paper, but there is one you’ll never get chemists to believe: the idea that a hydrogen atom can be bonded to two other atoms at the same time.¹⁶

The British chemist Nevil Sidgwick was apparently one of the believing chemists, arguing strongly in 1924 for the existence of intramolecular hydrogen bonds.¹⁷ He seems to have had as much luck as Huggins, the English chemist Henry Edward Armstrong, rejecting, in somewhat florid terms, the very idea of what he called a bigamous hydrogen atom.¹⁸ The association between water molecules, he insisted, was a matter of oxygen-oxygen bonds. Chaste English chemists referred to the hydrogen bond with great, but understandable, reluctance during the 1920s.

It was Linus Pauling who, in the end, brought something like chemical respectability to the hydrogen bond.¹⁹ Pauling was well trained in crystallography; and fascinated by quantum mechanics. Pauling thought at first, that if the hydrogen bond really exists, it must be purely electrostatic.²⁰ In 1934, Pauling and Lawrence Brockway experimentally confirmed Latimer and Rodebush’s conjecture that carboxylic acids could indeed form hydrogen bonds;²¹ and it is in this important paper that Pauling began to doubt their electrostatic character. A resonance between ionic and covalent forms of carboxylic groups, Pauling suggested, served to establish the stability of the hydrogen bond.

Twelve months later, Pauling published a paper in which he argued that the structure and residual entropy of hexagonal ice is linked to the intrinsic asymmetry of the hydrogen bond itself.²² Pauling used only neutral molecular formulas for water.

He had discarded for good the idea of electrostatic bonding between hydrogen atoms.

It was left to William H. Zachariasen, in his study of the structure of liquid methyl alcohol, to invoke for the first time the dipolar nature of the hydrogen bond. “Every hydrogen atom,” he wrote, “is thus linked to two oxygen atoms.”²³ The crucial point now follows: it is “undoubtedly … linked more strongly to one of the oxygen atoms than to the other …”²⁴ “Naturally,” he added serenely, “if we wish to characterize the nature of these hydrogen bonds, we should employ the term dipole bonding.”²⁵

The hydrogen bond gained official recognition during a meeting of the Faraday Society held in Edinburgh in 1936. In his keynote lecture, Joel Hildebrand offered a benediction:

It is becoming evident, again, that the term “association” under which we have lumped all departures from normal behavior, must be subdivided into association arising from the interaction of dipoles, and that due to the formation of definite chemical bonds. Of these, perhaps the most interesting are the hydrogen bonds or “bridges” between oxygen, nitrogen, or fluorine atoms, a species of chemical interaction.²⁶

French, German, and Japanese scientists embraced the hydrogen bond in short order, one by one.²⁷

An Official Definition

In 2011, the International Union of Pure and Applied Chemistry (IUPAC) embraced the hydrogen bond within the folds of an official definition:

The hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X–H in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation. A typical hydrogen bond may be depicted as X–H•••Y–Z, where the three dots denote the bond. … The evidence for hydrogen bond formation may be experimental or theoretical, or ideally, a combination of both.²⁸

This definition is useless as it stands, an observation not wasted on the IUPAC. In order further to clarify their ideas, they appended to their definition a list of twelve emendations:²⁹

(E1) The forces involved in the formation of a hydrogen bond include those of an electrostatic origin, those arising from charge transfer between the donor and acceptor leading to partial covalent bond formation between H and Y, and those originating from dispersion.

(E2) The atoms X and H are covalently bonded to one another and the X–H bond is polarized, the H•••Y bond strength increasing with the increase in electronegativity of X.

(E3) The X–H•••Y angle is usually linear (180°) and the closer the angle is to 180°, the stronger is the hydrogen bond and the shorter is the H•••Y distance.

(E4) The length of the X–H bond usually increases on hydrogen bond formation leading to a red shift in the infrared X–H stretching frequency and an increase in the infrared absorption cross-section for the X–H stretching vibration. The greater the lengthening of the X–H bond in X–H•••Y, the stronger is the H•••Y bond. Simultaneously, new vibrational modes associated with the formation of the H•••Y bond are generated.

(E5) The X–H•••Y–Z hydrogen bond leads to characteristic NMR signatures that typically include pronounced proton deshielding for H in X–H, through hydrogen bond spin–spin couplings between X and Y, and nuclear Overhauser enhancements.

(E6) The Gibbs energy of formation for the hydrogen bond should be greater than the thermal energy of the system for the hydrogen bond to be detected experimentally.

(C1) The pK_a of X–H and pK_b of Y–Z in a given solvent correlate strongly with the energy of the hydrogen bond formed between them.

(C2) Hydrogen bonds are involved in proton-transfer reactions (X–H•••Y → X•••H–Y) and may be considered the partially activated precursors to such reactions.

(C3) Networks of hydrogen bonds can show the phenomenon of cooperativity, leading to deviations from pairwise additivity in hydrogen bond properties.

(C4) Hydrogen bonds show directional preferences and influence packing modes in crystal structures.

(C5) Estimates of charge transfer in hydrogen bonds show that the interaction energy correlates well with the extent of charge transfer between the donor and the acceptor.

(C6) Analysis of the electron density topology of hydrogen-bonded systems usually shows a bond path connecting H and Y and a (3,–1) bond critical point between H and Y.³⁰

It is worth noting that (E3)-(E6) and (C1)-(C6) are purely empirical, and say nothing about the physical origin of H-bonding. (E1) says only that whatever the nature of the hydrogen bond, it is not entirely covalent, a point never at issue; while (E2) refers to a putative bond strength instead of referring to its more physical and measurable stabilization energy.

According to classical physics, water is a diamagnetic substance that should display a very low sensitivity to permanent magnetic fields. At the same time, because of its high electric dipolar moment p₀ = 1.85498D in the vapor state, it should react strongly with static electric fields. But in this regard, theory and experiment are in conflict. Maxwell equations predict a static relative dielectric constant of ε_r ≈ 13, far from the experimental value of ε_r ≈ 80.

What is more, Maxwell equations suggest dipolar interaction energies of 0.05eV for direct dipole-dipole Keesom interactions; 0.03eV for Debye interactions between permanent and induced dipoles; and 0.12eV for London dispersive interactions between two induced dipoles. These calculations assume an oxygen-oxygen distance of 3.65Å, corresponding to the O-H covalent bond length (0.95Å), augmented by the sum of van der Waals radii of hydrogen (1.2Å) and oxygen (1.5Å).

Such values cannot explain the abnormally high boiling point of liquid water, or the hydrogen bond energy of about 0.22eV.

Having been invited to every party for more than a century, the hydrogen bond remains a guest without a face. We still do not understand why, with its ridiculous molecular weight, water is not a gas and why ice should float on water.

Quantum of Action

In classical physics the energy, E, of a system may become larger or smaller, but whether larger or smaller, it becomes larger or smaller continuously.

There are no jumps.

At the end of the nineteenth century, it became obvious that however plausible the principle, it was not true. In blackbody radiation, continuity lapses. The relationship between the energy E emitted by a black box, and its associated electromagnetic wave, is mediated by Planck’s constant h. The Planck–Einstein equation E = hν, draws a connection between the energy emitted by a black box and the frequency of its atomic oscillations. The connection is counterintuitive because E must be expressed as an integral multiple of hν, so that, in effect, E = nhν, where n is a non-negative integer.

In 1905, Einstein argued persuasively that light, which is an electromagnetic phenomenon, and thus wave-like in nature, makes its appearance in the world as a particle. The frequency and wavelength of a photon are governed by the equation λν = c, where c is the speed of light. The equation E = hν, now reappears as E = hc / λ, having achieved a new incarnation as the description of the energy possessed by a photon and so by a particle.

Some years later, Louis de Broglie, considering these relationships, concluded that if waves could be particles, particles could be waves. If p is a particle’s linear momentum, then the equation λ = h/p describes its associated wavelength λ.

By the third decade of the twentieth century, physicists were in possession of a far flung series of correspondences: energy and frequency, linear momentum and wavelength, angular momentum and wave angular orientation, position and momentum. These pairs are conjugate variables, each the Fourier transform of the other. In 1926, Werner Heisenberg demonstrated that conjugate variables within quantum systems are bounded by an ineliminable form of uncertainty: Δx Δp ≥ ħ, where in Dirac’s notation, ħ = h/2π ≈ 10^-34 J·s.

Twenty years after Planck and Einstein introduced physicists to the imperative of quantum action, Werner Heisenberg, Erwin Schrödinger, and Paul Dirac provided the theory that made sense of the facts. Quantum mechanical affairs are conducted within the confines of a complete infinite dimensional vector space equipped with an inner product—what is now known as a Hilbert space. It is this space that comprises all possible states of a given quantum system.

Observables of the system are represented by linear operators, and each eigenstate of an observable corresponds to an eigenvector of an operator. The associated eigenvalue is the value of the observable. These facts make for the furniture of a quantum system. What remains to be determined is the evolution of its states. Both Werner Heisenberg and Erwin Schrödinger addressed and then solved this problem, although in quite different ways; and shortly thereafter physicists came to understand that their schemes were, deep down, the same. Since particles and waves shared in quantum theory an inner, if latent, identity, it seemed perfectly natural to Schrödinger to model the evolution of a quantum system in terms of a function ψ(x,t) that determines the amplitude of a wave. The result is a partial differential equation

$$ih\frac{\partial }{\partial t}|\psi =\mathrm{H}|\psi$$

where ψ is the wave function, and H its quantum Hamiltonian.

Thereafter, quantum mechanics is governed by two assumptions.

The first specifies the probability of finding a particle in a particular place and at a particular time in terms of the squared absolute value of ψ(x,t):

$$P\left(x,t\right)=|\psi \left(x,t\right){|}^2.$$

The second establishes that quantum mechanics conforms to the principles of probability:

$$\int *\psi \left(x,t\right) \psi \left(x,t\right)=1,$$

for all times t.

What lends to quantum theory its very great strangeness is just that the wave function encodes a superposition of states, a point brought out vividly by Paul Dirac in his treatise:

The general principle of superposition of quantum mechanics applies to the states [undisturbed motions] … of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state.³¹

The classical distinction between matter and radiation (or matter and fields) now is seen to disappear. The evolution of any system on the atomic scale is limited to integral multiples of Planck’s constant.

The first incarnation of quantum mechanics was expressed in terms of Planck’s quantum of action; and it was entirely a matter of a quantized constraint placed on a particle or on its associated wave. Quantization is performed only for particles of matter and not for the fields in which they are embedded. Such fields are treated classically through Newton’s law and Maxwell’s equations for the electromagnetic field.

For all of its revolutionary implications, quantum mechanics cannot be reconciled with special relativity. Quantum mechanics is not covariant in quite the sense demanded by relativity; and far more to the point, quantum mechanics takes its particles neat, one at a time. Special relativity, as Paul Dirac realized at once, allows for the creation and annihilation of particles, something unaccounted for and so left unexplained by first quantization methods.

Molecular Orbital Approximations

Nevertheless, one may apply first quantization within the frame of molecular orbital approximations, a technique widely employed since the late 1920s. Molecular orbital theory studies molecular bonding by approximating the positions of bonded electrons via a linear combination of their atomic orbitals. In the end, there is a natural return to quantum mechanics, achieved, for example, by applying the Hartree–Fock model to Schrödinger’s equation.

Consider thus the basic C_2v-symmetry of the water molecule H₂O. Ten electrons must be distributed among five energy levels according to the following electronic configuration:

$$\left(1{\mathrm{a}}_1{)}^2\right(2{\mathrm{a}}_1{)}^2\left(1{\mathrm{b}}_2{)}^2\right(3{\mathrm{a}}_1{)}^2\left(1{\mathrm{b}}_1{)}^2\right(4{\mathrm{a}}_1{)}^0(2{\mathrm{b}}_2{)}^0$$

This does not allow the establishment of partial covalence involving the Highest Occupied Molecular Orbital (HOMO) energy level, displaying b₁-symmetry, and the Lowest Unoccupied Molecular Orbital (LUMO) energy level, displaying a₁-symmetry. Any HOMO-LUMO interaction is doomed to fail because their integral overlap is null.

One might, of course, argue that upon hydrogen bonding, symmetry is lowered, thus leading to a possible non-zero overlap. This is unsatisfying. Before hydrogen bonding, both partners display their full C_2v-symmetry with zero overlap. From experiment, we know that the final symmetry of water dimers, or higher polymers, is C_s. At what distance does the symmetry change from C_2v to C_s? The assumption that C_s-symmetry holds at every distance is of no help, or, at best, little help. HOMO would then represent one symmetry, LUMO another. The overlapping integral would again be zero. Overlap may occur through other molecular orbitals, but at both 2.75Å and 2.98Å, the overlap between the acceptor oxygen and the hydrogen-bonding proton is negative, pointing to a net anti-bonding covalent interaction.³²

X-ray emission spectroscopy is evidence for the fact that the 1b₁ HOMO-level is not affected by hydrogen bonding.³³ Instead, a strong perturbation of the 3a₁ (HOMO-1) level is observed, evidence of a rather unconventional HOMO-1/LUMO interaction. Compton scattering experiments in hexagonal ice show evidence of a neat anti-bonding, repulsive interaction between neighboring water molecules despite the quantum-mechanical, multicenter character of the wave functions.³⁴ Topological analysis of electronic density demonstrates that it is not possible to differentiate between hydrogen bonds and mere van der Waals interactions.³⁵

In fact, nothing in the standard quantum mechanical treatment of the water molecule points to the tetrahedral character of the water monomer. Obviously, owing to the linear character of quantum theory, it is easy to build localized wave functions, showing two lone pairs on the oxygen atom, from delocalized molecular orbitals. There is no theoretical reason why this view is better than the fully delocalized one.

Given the experimental photoelectron spectrum of the water molecule, its most faithful representation should display three kinds of orbitals (two σ-bonds, one 2s-type lone pair and one 2p-type lone pair), and not two (two σ-bonds and two equivalent lone pairs), as suggested by molecular orbital theory.³⁶ The only way to retrieve a physical picture involving two lone pairs and two σ-bonds approximately oriented towards the vertices of a tetrahedron, is to look at the positions of the largest eigenvalues and corresponding eigenvectors of the Hessian minima in their molecular electrostatic potential.³⁷

This means reverting to a purely electrostatic view of hydrogen bonding.

The situation is so confusing that the scientific community is today divided into two opposing camps, one camp promoting water as a random tetrahedral network with flickering hydrogen bonds,³⁸ and the other promoting water in terms of a two-state model, one tetrahedral, the other not.³⁹

One to Many

Picture liquid water as a flickering network of hydrogen bonds. Neutron scattering experiments, as well as molecular dynamic simulations, have shown that the average residence time of the hydrogen atom around a water molecule is close to 1ps at T = 300K, and increases to 20ps at T = 250K.⁴⁰ The five-site transferable intermolecular potential (TIP5P) water model has allowed computational chemists to determine a density maximum near 4°C at 1atm by fixing the electrical charge on each hydrogen atom at +0.241e.⁴¹ This model was also able to reproduce the density of water between -37.5°C and 62.5°C at 1atm with an average error of 0.006g cm^-3, and the density of liquid water at 25°C over the range 1-10,000atm with an average error of 2%. Electromagnetic laws suggest that such a charge, moving on the picosecond timescale, may be expected to generate an electromagnetic field with a frequency of about 10¹² Hz.

But electromagnetic fields are treated classically when quantization is extended only to particles.

A richer theoretical treatment is required.

It is certainly possible to quantize a field characterized by infinitely many degrees of freedom. In first quantization, separate Hilbert spaces are required for systems with differing numbers of particles. In second quantization, the familiar Hilbert space framework is expanded to a Fock space. An arbitrary Fock space is a linear combination of n-particle Hilbert spaces. An infinite number of harmonic oscillators is assigned to each point in a Fock space. These oscillators describe every possible field-excitation mode. Two operators permit the creation of new quantum waves or their annihilation. We are now in the domain of quantum field theory. For all that, it is important to stress that the principles of quantum field theory are not new. They are entirely consistent with the fundamental principles of quantum mechanics.

When a quantum field is unexcited, the total linear momentum of the field is zero. For each mode having a wave vector p, there exists a similar mode with wave vector –p. But the total energy of the field cannot be zero, if only because each harmonic oscillator is found in its ground state with zero point energy. This ground state is populated by an infinite number of virtual field excitations. From its vacuum state, a quantum field is able to produce any number of quanta. The vacuum is thus filled with unobservable virtual particles, and, yet, virtual or not, these particles are responsible for a wide range of real physical phenomena.

The idea is curious enough to merit discussion. It is hardly self-evident.

• Virtual particles prevent negatively charged electrons from collapsing onto positively charged atomic nuclei. In classical physics this is unfathomable; in quantum mechanics, the explanation is assigned to Heisenberg’s uncertainty principle. In QFT, a far richer explanation is possible.
  
  At ħ·c ≈ 200 MeV·fm, an electron falling onto a nucleus whose size is about 1fm (10^-15m) encounters an ocean of virtual photons with a maximum energy of 200MeV. This value allows for the materialization of an electron/positron pair with a rest mass of about 1MeV/c² and a maximum kinetic energy of 199MeV. Attracted by the positron’s positive charge, the falling electron disintegrates, generating 1MeV of kinetic energy that must be added to the kinetic energy of the other electron popping out of the vacuum. One falling electron disappears from the world, replaced by another electron created from the void. Since they are quanta of the same field, it is impossible to distinguish between the falling electron and the expulsed electron. Electrons appear for this reason to be stationary.
• If virtual particles explain why atoms are stable against implosion, they also explain Faraday’s line forces, which emanate from any static electrical charge or permanent magnet.⁴² A virtual photon covering a distance ∆x carries with it a certain amount of energy ∆E: ∆E·∆x ≈ ħ·c or, what amounts to the same thing, f = ∆E/∆x ≈ ħ·c/(∆x)², where f designates a force. The result is Coulomb’s law.
• The non-zero impedance of the vacuum and the finite value of the speed of light are also illustrations of the existence of virtual particles.⁴³ Virtual particles may also be invoked in order to explain the spontaneous relaxations of atoms between excited states or towards a ground state.
• Radioactive decay offers additional evidence of the existence of the vacuum’s virtual particles. Moreover, if an atom remains in an excited state, interaction with the vacuum’s virtual particles will affect the energy value at this level, a phenomenon known as the Lamb shift.⁴⁴ When an atom absorbs virtual photons emitted by another atom close by, an attraction is expected by physicists, and named the London interaction by chemists.
• The ratio ħ/e corresponds to an electric potential times a duration, or to a magnetic vector potential times a distance. Any quantum phase gradient generates a magnetic vector potential, whereas any variation in time generates an electric potential and vice versa.
  
  There thus exist effects of potentials on charged particles, even in the region where all the fields, and therefore the force on the particles, vanish. Such is the Aharonov–Bohm effect.⁴⁵
• The existence of creation/annihilation operators implies that one may also find quantum states characterized by large fluctuations in the number of quanta having well-defined quantum phases. For large fluctuations, this implies the existence of non-orthogonal coherent states. These coherent states represent a bridge of sorts to classical physics inasmuch as the dynamics of a quantum harmonic oscillator resemble closely the yin and yang of a classical harmonic oscillator. A coherence domain may also be viewed as a macroscopic condensate of Nambu–Goldstone bosons, which emerge from the spontaneous symmetry breaking of the vacuum by the dipolar field of water molecules.⁴⁶
• Virtual particles are also responsible for the static and dynamic Casimir effect.⁴⁷

Water Works

In first quantization, the physical and chemical properties of molecules are mainly dependent upon their HOMO and LUMO states; other excited states play a quite minor role, except in spectroscopy. This is not the case in QFT. Electrons of a water molecule are confined within a sphere having a diameter of about 0.3nm. The physical vacuum inside a water molecule is filled with virtual photons. Their energy is not enough to create electron/positron pairs, but it is, nevertheless, enough to self-excite the water molecule over the whole of its energy spectrum. Obviously, self-excitation is only possible for a very short time.⁴⁸ It may well happen that during relaxation towards their ground state, virtual photons initially absorbed by a water molecule fail to return to the vacuum, but are used to excite another water molecule. This process is dependent on the density, N/V, for a given volume V containing N water molecules.

Suppose that ∆E = ħω is the energy of a virtual photon emerging from the vacuum and propagating along ∆x during ∆t. If the average distance between two water molecules is d ≈ (V/N)^1/3, two cases may be encountered, and must be distinguished.

1. d = (V/N)^1/3 > λ/2π. Virtual excitations concern isolated water molecules and the virtual electromagnetic field around each water molecule fluctuates with a zero time-average.
2. d = (V/N)^1/3 < λ/2π. Virtual excitations may be shared between several molecules, thus making possible condensation towards a coherent state. The N molecules form a coherence domain.

Coherence Domains

A coherence domain may be idealized as a sphere surrounding an internal electromagnetic field in which the field’s maximum amplitude is at its center. If such coherence domains are compactly packed, they will be at a certain inter-domain distance from each other. This means that the evanescent parts of internal fields are overlapping, and a detailed analysis of this situation leads to the conclusion that the internal field should reach its minimal value for r₀ = 3π/4ω_q.⁴⁹ For an excitation at ω_q = 12.07eV, the diameter of one coherence domain is L ≈ 2r₀ ≈ 75nm, leading to a volume V_DC = πL³/6 = 220,893nm³. Assuming 100% of coherence for a liquid phase at the lowest possible temperature (T_S ≈ 228K), with a density very close to that of ice (ρ ≈ 0.92 kg·m^-3), yields N_DC = 30.8·V_CD(nm³) ≈ 6.8 million water molecules.

A coherence domain is definitively not a water cluster.

If temperatures increase, thermal fluctuations may expel a certain fraction of the water molecules from a coherence domain, thickening the intersurface zone and decreasing the radius of the coherence domain. At x = 0, boiling occurs. At room temperature, an incoherent water film separates domains.⁵⁰

It is this separation that prevents water from being a good electronic conductor. The existence of coherent and incoherent water molecules is supported by a substantial, but controversial,⁵¹ body of experimental data.⁵²

Conditions for a Coherence Domain

In order to determine the conditions under which a coherence domain may be formed, it is necessary to consider either transitions between ground states and excited discrete levels, or virtual transitions above the ionization threshold.

Experimental data suggests the existence of a coherent coupling constant between the electromagnetic field and the photons bouncing back and forth between water molecules. The principle of least action leads to three coherence equations for each discrete excitation.⁵³ The first and the second describe the absorption and emission of a virtual photon between the ground state and a given excited state, when both are mediated by the coupling constant.⁵⁴

The third describes the state of the electromagnetic field when perturbed by the presence of water molecules in their ground state. This equation may have either three real roots or one real and two complex roots. In the first case, the field amplitude will have a sinusoidal variation describing a stable perturbed state, where the internal field fluctuates around its initial null value. In the second, the amplitude may grow exponentially, as in a laser, but with a flood of virtual photons reaching a non-zero macroscopic value oscillating in phase with the water matter field. This effect involves virtual photons, which means that they remain trapped within a coherence domain, reflected back and forth between water molecules.

Hydrogen bonding is nothing less than the bonding energy welding water molecules together.

Notable Results

There are three coherence equations for each accessible discrete excited level of the water monomer. The input to each equation is the experimental excitation spectrum; the output, a set of four coherence parameters. The first parameter is the mixing angle between the ground state and a given excited level. The other two represent the amplitude reached by the trapped internal electromagnetic field, and variations in the quantum phase governing the reduced frequency oscillations of internal fields.

A coherence domain characterized by a mixing angle of sin²(β) = 0.1 indicates that electrons in water molecules spend 10% of their time on a very diffuse 5d Rydberg state of oxygen. Coherent water molecules are a little fatter than incoherent water molecules. Moreover, among the five d-states, one pair (z², x²-y²) transforms as the totally symmetric a₁-representation of the C_2v group, and could thus be mixed with the two molecular orbitals (2a₁, 3a₁). This leads to a set of four a₁-levels arranged in a more or less tetrahedral configuration to minimize electronic repulsions.⁵⁵

What is this if not an explanation of the basic tetrahedral structure of ice or liquid water?

A coherence gap protects the coherence domain from incoherent thermal fluctuations. It may be expressed as a sum of three terms: E_coh = E_em + E_mat + E_cm, where E_em is the positive energy borrowed from the vacuum and stored in the internal electromagnetic field; E_mat is the positive energy borrowed from the vacuum and used for exciting water molecules; and E_cm denotes the negative coupling energy between the electromagnetic field and the electronic currents circulating among excited water molecules.

If the first term increases linearly with matter density (N/V), the last is proportional to (N/V)^3/2. Consequently, there exists a critical density ρ* for which E_cm cancels E_em + E_mat. When this critical density is reached, condensation into a coherence domain occurs spontaneously. The increase in matter density (N/V) and in field amplitude A₀ stops as soon as the electronic clouds of the pulsating water molecules begin to overlap. At this point, repulsive energy dominates attractive energy. The coherence domain has now reached an equilibrium configuration.

This level is particularly interesting for two reasons:

1. It leads to a critical density very close to the experimental critical density of water vapor.
2. A coherence domain considered as a condensate of bosons leads to a critical temperature close to the freezing point of ice.

These are notable results.

Dissolved Gases, Ionic Species

Quantum field theory has deep consequences with respect to the status of dissolved gases or ionic species. Given the phase coherence within a coherence domain, it follows that anything that is not water, or that is unable to resonate in phase with its electromagnetic field, should be rejected outside its coherence domain. Dissolved gases accumulate at the interstices generated by coherence domain packings, and this in a highly unstable configuration favoring coalescence between bubbles. For exactly the same reason, dissolved electrolytes accumulate in competition or synergy with dissolved gases. Highly complex behavior is thus expected for the coalescence properties of nanobubbles as a function of the kind of electrolytes added to the water, a fact fully confirmed by experiments.⁵⁶

A detailed analysis shows that electrolytes do not behave according to the Debye-Hückel model derived from purely classical considerations. Instead of forming a diffuse layer characterized by a Debye–Hückel length, they form a coherent plasma oscillating in resonance with the electromagnetic field trapped inside the coherence domain.⁵⁷

The classical view of osmosis is not adequate for describing the status of electrolytes in aqueous solutions. This might explain the fact that very low electromagnetic fields oscillating with frequencies of a few Hz could induce molecules or ions of mesoscopic size to acquire coherent motion.⁵⁸

The Hofmeister series in biology might well reflect this quantum behavior.⁵⁹ The existence of coherent water stabilized by biopolymers (protein, DNA, lipidic membrane) may also explain why sodium ions that move in solution with a tightly held water shell⁶⁰ are excluded from the intracellular medium, in contrast with potassium, which binds water molecules rather weakly.

Proof of Concept

The great challenge facing quantum chemists is to accumulate experimental evidence for the existence of coherence domains in liquid water. Direct observations are apt to be very difficult. Nevertheless, indirect experimental evidence already exists. It is well known that water masers exist in intergalactic space. They justify a clear distinction between the two nuclear spin isomers of water—ortho and para water. The coherent librations of these isomers, if observed at room temperature and in a liquid state, would be strong evidence for the existence of quantum coherence. Consider four-photon Rayleigh-wing spectroscopy of coherent librations in the range 0–50 cm⁻¹ in Milli-Q liquid water. The observation that these coincide perfectly with the rotational spectrum of gaesous H₂O is thus very encouraging.⁶¹ As the ortho ↔ para conversion is rather fast in small water clusters,⁶² this shows that such clusters do not exist in liquid and behave quite differently from coherent domains embodying millions of water molecules.

From a structural viewpoint, it has recently become possible to encapsulate large water assemblies in giant polyoxomolybdate-based nanocapsules and characterize their inner cavity through single-crystal X-ray diffraction. In some nanocapsules, perfectly ordered structures based on Platonic and Archimedean polyhedrals, and displaying a full tetrahedral structure, are observed.⁶³ The {H₂O}₁₀₀ assembly with an average density ρ ≈ 0.69g·cm^-3 may help in visualizing what could be the inner core of a water coherence domain.

On the other hand, there exist other nanocapsules displaying non-ordered chain-like and non-tetrahedral water assemblies,⁶⁴ such as {H₂O}₅₉ with an average density ρ ≈ 0.36g·cm^-3, that may correspond to fragments of incoherent water. Similarly, in some molecular dynamics simulations, large vortex-like coherent patterns appear, although the orientational memory of individual molecules is quickly lost.⁶⁵

Finally, topological analysis using Voronoi polyhedra has revealed the existence of tetrahedral and non-tetrahedral patches with isosbestic points.⁶⁶

Conclusion

More than four hundred years ago, Galileo argued that ice floated on water because it had more vacuum rather than less heat. In this, he was correct. The vacuum plays an important role in hydrogen bonding. Condensates of such bosons or coherence domains are responsible for the amazing properties of liquid water.

From theory, we know that close to its maximum density temperature of 4°C, liquid water must have an overall coherence of 50%. Just for fun, let’s associate the number 1 to each ordered coherence domain, and 0 to each patch of incoherent water. Consider a 1cm³ water droplet. As the volume of one coherence domain is about 10⁵nm³, we should have approximately 10²¹/10⁵ = 10¹⁶ domains for each cm³ of liquid water. This suggests an analogy to memory based on ferromagnetic domains. Encoding all the books of all the libraries in the world would require approximately 0.2l of liquid water.

There is plenty of room at the bottom.

If the vacuum is empty, as in classical physics and in old-fashioned quantum mechanics, water will reveal only a random flickering network of hydrogen bonds. Any quantum interaction is doomed to be repulsive, leaving only weakly attractive electrostatic forces. This is why the hydrogen bond was first rejected by the scientific community, for reasons that have been forgotten or overlooked. The difference between a covalent bond, a dispersive interaction, and a hydrogen bond is now very clear.

Acknowledging the role played by the quantum vacuum in chemical bonding should have implications in biology. A living cell is 70% water by weight and 99% by mole. An elementary calculation shows that owing to the nanometric scale of cell biopolymers, this amount of water corresponds to, at most, four water layers around each component. It is much easier to keep coherence within such layers than in bulk liquid, even at a temperature close to 37°C. Coherence is a marked feature of any living system. Many years ago, Albert Szent-Györgyi wrote that:

One of my difficulties with protein chemistry was that I could not imagine how such a protein molecule can “live.” Even the most involved protein structural formula looks “stupid,” if I may say so. … It looks as if some basic fact about life were still missing, without which any real understanding is impossible.⁶⁷

The missing basic fact is just that the quantum vacuum uses water as a mediator of phase coherence between electromagnetic and matter fields. That ħc ≈ 200eV·nm indicates that within a protein cavity having a size of about 1nm, virtual photons with energy as high as 200eV are available for a very short time. Still, even if the probability of excitation by virtual photons is low, their amplification could be an explanation for the extraordinary catalytic power of various enzymes. Coherent water has a low ionization threshold, and is thus able to generate a tension close to –100mV when associated with incoherent water. Membrane potentials could then arise from water and not necessarily from ions.

Water is life. Why should anyone be surprised?