Quantum indistinguishability in chemical reactions

Beyond Born–Oppenheimer Approximation.

In this section we present the basic framework for our subsequent discussion of the role of symmetry and quantum indistinguishability in chemical processes involving catalytically assisted bond breaking in symmetric molecules. In molecular processes the electrons, as fast degrees of freedom, are appropriately treated as fully quantum-mechanical indistinguishable fermions. In contrast, the constraints of quantum indistinguishability on the nuclear orbital degrees of freedom when treated within the Born–Oppenheimer approximation are invariably neglected, especially in solution chemistry [although not always in molecular spectroscopy (1)]. The molecular dynamics and chemical reactions are thus assumed to be fully controlled by classical motion of the molecular collective coordinates on the Born–Oppenheimer adiabatic energy surface. While this may be adequate for some systems, we argue that it can be wholly insufficient for chemical reactions in small symmetric molecules. As we discuss, in such systems, the nuclear and electron spin degrees of freedom can induce Berry phases that constrain the molecular orbital dynamics on the adiabatic energy surface, which must then be treated quantum mechanically since these Berry phases do not enter the classical equations of motion. As we argue, the presence of Berry phases can have strong and previously unappreciated order-one effects in chemical bond-breaking processes.

Planar Symmetric Molecules.

For simplicity of presentation we focus primarily on molecules which possess only a single n-fold symmetry axis that under a 2π/n planar rigid-body rotation (implemented by the operator C^n) cyclically permutes n indistinguishable fermionic nuclei. A water molecule provides a familiar example for n=2 and ammonia for n=3, wherein the protons are cyclically permuted.

For such molecules the nuclear spin states can be conveniently chosen to be eigenstates of C^n, acquiring a phase factor (eigenvalue) ωn−τ with ωn=ei2π/n and the “pseudospin” τ taking on values τ=0,1,2,...,n−1. Due to Fermi statistics when the molecule is physically rotated by 2π/n radians around the Cn symmetry axis, the total molecular wavefunction—which consists of a product of nuclear rotations, nuclear spins, and electronic molecular states—must acquire a sign (−1)n−1 due to a cyclic permutation of fermionic nuclei. Provided the electron wavefunctions transform trivially under the Cn rotation, this constraint from Fermi statistics of the nuclei, ei(−2πτ/n+2πL/n−π(n−1))=1, implies that the collective orbital angular momentum of the molecule is constrained by τ, taking on values L=Lquasi+nZ with the quasi-angular momentum given byLquasi=τ, n odd,τ+n/2, n even.[1]

Molecular trimer of identical fermionic nuclei.

For illustrative clarity we first formulate the problem for the case of n=3, specializing to three identical fermionic nuclei with nuclear spin 1/2 and electrical charge +1. We focus on a singly ionized trimer molecule T≡A3+ undergoing a chemical reaction,A3+→A2+A+,[2]into a singly ionized atom, A⁺, and a neutral dimer molecule, D≡A₂. The process is schematically displayed in Fig. 1, where the molecular trimer is composed of the three fermionic nuclei with creation operators, F^α†, in spin state α = ↑,↓, that form a C₃ symmetric molecular configuration characterized by a collective coordinate ϕ. The nuclei part of such a molecular state is created by the three-nuclei operator,T^τ†(ϕ)=∑αβγχαβγτF^α†(ϕ)F^β†(ϕ+2π/3)F^γ†(ϕ+4π/3).[3]The three-nuclei spin wavefunction χαβγτ is chosen as a τ representation of cyclic permutations, an eigenstate of C^3,C^3χαβγτ=χγαβτ=ω3τχαβγτ,[4]with ω3=ei2π/3, required by (C^3)3=1. Thus, for this trimer molecule, τ takes one of three values, τ=0,1,2 (or equivalently, τ=0,±1). By construction T^τ(ϕ) then also forms an irreducible representation of C^3, satisfyingT^τ(ϕ+2π/3)=ω3τT^τ(ϕ).[5]We will at times refer to τ as a pseudospin that encodes both the nuclear spin and the orbital qubits, entangled through the Pauli principle of identical nuclei.

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Fig. 1.

Schematic of a bond-breaking chemical reaction. (Left) Initial state of a C3 symmetric molecule with two molecular electrons (blue) bonding the three nuclei (red) together. (Center) An intermediate state where the reaction is catalyzed by an enzyme that “grabs” two of the nuclei, weakening their bonds to the third by depletion of electronic charge. (Right) Final product state composed of a molecular dimer and an isolated atom.

Because a discrete 2π/3 rotation executes a fermionic cyclic interchange, the τ representation of the nuclear spin wavefunction imprints a nontrivial Berry phase ωτ onto the orbital degree of freedom, ϕ, which we discuss below. For simplicity we have suppressed the position coordinate describing the center of mass of the trimer molecule as well as the orientation of the normal to this planar trimer molecule.

In addition to the nuclei, a correct description of a molecule must also consist of bonding electrons that, within the Born–Oppenheimer approximation, occupy the molecular orbitals. For concreteness we consider a (singly ionized) molecule with only two electrons that form a spin singlet in the ground-state molecular orbital ψT(r;ϕ), where the subscript denotes the trimer nuclear configuration. This wavefunction transforms symmetrically under C^3, satisfying ψT(r;ϕ+2π/3)=ψT(r;ϕ). We denote the electron creation operator in this orbital asc^σ†(ϕ)=∫rψT(r;ϕ)c^σ†(r).[6]The trimer molecular state we thus consider can be written asT=∑τ∫ϕΨτ(ϕ)c^↑†(ϕ)c^↓†(ϕ)T^τ†(ϕ)vac⊗Eϕ,[7]characterized by an orbital wavefunction in the τ representation,Ψτ(ϕ+2π/3)=ω3τΨτ(ϕ).[8]Were the general orbital wavefunction Ψτ(ϕ) expanded in angular-momentum eigenstates, eiLϕ, with L∈Z, this constraint implies that L=Lquasi+3Z with a quasi-angular momentum Lquasi=τ, consistent with Eq. 1 for n=3.

In Eq. 7 the ket Eϕ denotes the initial quantum state of the environment—i.e., the solvent and enzyme—that is entangled with the molecular rotations through the angle ϕ, as generically the environment can “measure” the molecular orientation. Note that we have implicitly assumed that the initial state of the environment does not depend on τ, so that Eϕ+2π/3=Eϕ. For a molecule with thermal angular momentum LT (defined through ℏ2LT2/I=kBT, with moment of inertia I) that is much greater than one, LT≫1, the solvent is ineffective in “measuring” the molecular quasi-angular momentum, which is a small fraction of LT. Indeed, the quasi-angular momentum decoherence time should be exponentially long for large thermal angular momentum, varying as tcohτ∼t0⁡exp(cLT2) with an order one constant c and t0 a microscopic time of order 1 ps.

A Rotating Symmetric Molecule.

We begin with a precise definition of the initial quantum mechanical state of the rotating symmetric molecule, offering three representations of the accessible Hilbert space. For the case of a diatomic (n=2) molecule, these are illustrated in Fig. 2.

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Fig. 2.

Three representations of a rotating diatomic molecule composed of identical nuclei. (A) An explicit physical representation. (B) An effective model for the dynamics of the angular coordinate, φ=2ϕ∈[0,2π), for a molecule with C2 symmetry, represented as a quantum bead on a ring with Berry flux, Φ0 determined by the nuclear spin wavefunction. The bead wavefunction, ψa(φ), is discontinuous across an nth root branch cut (red squiggly line), ψ+=eiΦ0ψ−. (C) Placing the bead on an n-fold cover of the ring with φ∈[0,2πn)—a Mobius strip for the n=2 double cover—resolves any ambiguity in the placement of the branch cut.

A “Not-Rotating” Molecule.

A first step in the bond-breaking process requires stopping the molecule’s rotation as it binds to the enzyme. What does it mean for a small symmetric molecule to be not rotating and perhaps having zero angular velocity? For 1D translational motion the linear (group) velocity of a quantum particle is vg=∂Ep/∂p, suggesting that angular (group) velocity should be likewise defined, Ω=∂EL/∂L. But angular momentum is quantized in units of ℏ, so this definition is problematic.

One plausible definition of a molecule to be not rotating is for its orbital motion to be described by a real wavefunction. For odd n this is equivalent to requiring zero quasi-angular momentum, Lquasi=0, while for n even it is not since for Lquasi=n/2 ≠ 0 a real wavefunction can still be constructed. But in either case, once we allow for quantum entanglement between the molecule and the solvent/enzyme the notion of the “molecule’s wavefunction” becomes problematic.

We believe the best way to impose a not-rotating restriction of the molecule is to impose a constraint that disallows a dynamical rotation which implements an exchange of the constituent atoms. This can be achieved by inserting impenetrable potential wedges at angles centered around ϕ=0 and ϕ=π, as illustrated in Fig. 3A for the diatomic molecule. These wedges restrict the molecular rotation angle so that one of the nuclei is restricted to the upper half-plane, δ/2<ϕ≤π−δ/2, while the other resides in the lower half-plane, π+δ/2<ϕ<2π−δ/2. Rotations that exchange the nuclei are thereby strictly forbidden. And the two nuclei, while still identical, have effectively become “distinguishable”—even if their spins are aligned.

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Fig. 3.

Two representations of a nonrotating diatomic molecule composed of identical nuclei. (A) An explicit physical representation with impenetrable potential wedges inserted around ϕ=0 and π that restrict the molecular rotation angle and constrain one nucleus to the upper half-plane and the other to the lower half-plane. These wedges strictly forbid all rotational motions that dynamically exchange the two nuclei. The identical nuclei thus become effectively distinguishable. (B) An effective quantum bead model, where the ring, now restricted to the angular range, φ∈[δ,2π−δ), has been cut open—and the bead is constrained to move on a line segment.

Upon mapping to the effective coordinate, ϕ→φ/n, the wavefunction of the bead on the ring in this not-rotating configuration, which we denote as ψb(φ), is restricted to the line segment φ∈(δ,2π−δ). As illustrated in Fig. 3B, the “ring” has in effect been “cut open.” As such, there is no meaning to be ascribed to the effective flux through the ring, present in Fig. 2 for the rotating molecule. In contrast to the n-fold cover of the rotating wavefunction, ψa(φ+2πn)=ψa(φ), the nonrotating wavefunction ψb(φ) is defined on a single cover of the cut-open ring with φ∈(δ,2π−δ).

A Bond-Broken Molecule.

Once the molecule is not rotating and the identical fermions are effectively distinguishable, a breaking of the chemical bond is greatly simplified. As illustrated in Fig. 4A, with the impenetrable barriers present the molecular bond can break and the constituent atoms are free to move off into the upper and lower half-planes, respectively. Once the atoms are physically well separated, their distinguishability no longer rests on the presence of the impenetrable barriers.

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Fig. 4.

Two representations of a bond-broken diatomic molecule composed of identical but distinguishable nuclei. (A) An explicit physical representation with impenetrable potential wedges inserted at ϕ=0 and π, allowing for a bond-breaking process to the distinguishable-nuclei state. (B) An effective quantum bead on a ring model, restricted to the angular coordinate range φ∈[δ,2π−δ), with bond breaking modeled by tunneling the bead off the ring.

In the quantum-bead representation, illustrated in Fig. 4B, the bond-breaking process corresponds to the bead tunneling off the line segment. As such, the location of the bead must now be specified by both the angle φ and a radial coordinate, r, the bead wavefunction taking the form ψc(r,φ).

Since this bond-breaking process will typically be macroscopically irreversible, we assume that once the bead tunnels off the line segment it will not return. The tunneling rate for this process can be then expressed in terms of a Fermi’s golden rule, Γb→c∼|Ab→c|2, with tunneling amplitudeAb→c=∫δ2π−δdφHbc(φ)ψc*(R,φ)ψb(φ),[20]with Hbc a tunneling Hamiltonian and R the radius of the molecule.

Indistinguishable-to-Distinguishable Projective Measurement.

We now turn to the more subtle process where the “rotating” molecule transitions into the not-rotating state, which occurs when the enzyme “catches” the molecule. When the molecule is rotating, in state ψa, dynamical processes that interchange the identical fermions are allowed and will generally be present. As such, these identical fermions are truly “indistinguishable” in the rotating state. But when the enzyme catches and holds the molecule in place, in the state ψb, a projective measurement of the atomic positions has been implemented (the enzyme is the “observer”) and the identical fermions are now “distinguished.”

The transition rate for this process, Γa→b, can be expressed as a product of a microscopic attempt frequency, ωab, and a Born measurement probability, Pab, that is Γa→b=ωabPab. The Born probability can in turn be expressed as the squared overlap of the projection of the distinguishable state, ψb, onto the indistinguishable state, ψa, that is Pab=|⟨ψb|ψa⟩|2.

Since the rotating molecule wavefunction, ψa(φ), is defined on the n-fold cover of the ring (Mobius strip for n = 2) with φ∈(0,2πn), while the not-rotating wavefunction ψb(φ) lives on a single cover of the line segment, φ∈(δ,2π−δ), the two states are seemingly defined in a different Hilbert space. This ambiguity can be resolved by noting that the Born amplitude projecting from the Mobius strip onto the open line segment can occur from any of the n edges of the Mobius strip. We thus conjecture that these n processes must be summed over,⟨ψb|ψa⟩=∑m=0n−1∫δ2π−δdφ ψb*(φ)ψa(φ+2πm).[21]Upon using the 2π periodicity condition for the Mobius strip wavefunction, ψa(φ+2πm)=eimΦ0ψa(φ) with Φ0=2πLquasi/n, this can be reexpressed as⟨ψb|ψa⟩=Cn∫δ2π−δdφ ψb*(φ)ψa(φ),[22]with the amplitudeCn=∑m=0n−1eimΦ0=∑m=0n−1ei2πmLquasi/n=nδLquasi,0.[23]The Born probability Pab=|⟨ψb|ψa⟩|2 thus vanishes unless Lquasi=0, as does the rate, Γa→b, that the enzyme grabs and holds the molecule in a nonrotating distinguishable configuration.

Physically, for Lquasi ≠ 0, there is a destructive interference between the n-parallel paths, each projecting from an edge of the Mobius strip/torus onto the line segment. For the diatomic molecule this is illustrated in Fig. 5, where the two contributions with opposite signs will destructively interfere. The rate for this process which stops the molecule’s rotation, Γa→b, will vanish unless Lquasi=0—the molecule simply cannot get caught by the enzyme for nonzero Lquasi.

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Fig. 5.

The enzymatic projective measurement that induces a transition from a rotating to a not-rotating molecular state can be described as a projective overlap between the rotating state with the quantum bead on the Mobius strip, ψa, and a not-rotating state where the bead is on the cut-open outer ring, ψb. This projective measurement implements a transition from an initial state in which the identical atoms are indistinguishable to a final distinguishable state—the enzyme acting as an observer.

Since the enzymatic reaction requires both stopping the molecules rotation, with rate Γa→b, and subsequently breaking the chemical bond, with rate Γb→c ≠ 0, the full chemical reaction rate isΓa→c=Γa→bΓb→cΓa→b+Γb→c∼δLquasi,0[24]and vanishes unless Lquasi=0. Mathematically, the chemical bond-breaking process is “blocked” by the presence of the Berry phase, operative whenever the orbital molecular state is nonsymmetric.

More physically, this blocking is due to the destructive interference between the n-possible bond-breaking processes—one for each of the symmetry-related molecular orbital configurations. These considerations thus provide support for our conjectured QDS rule that states the impossibility of (directly) breaking a chemical bond of a symmetric molecule rotating nonsymmetrically.

Differential Reactivity of Para-/Orthohydrogen.

Hydrogen provides the most familiar example of molecular spin isomers, parahydrogen with singlet entangled proton spins and rotating with even angular momentum, and orthohydrogen with triplet spin entanglement and odd rotational angular momentum. Since the allowed rotational angular momentum of such homonuclear dimer molecules with S = 1/2 fermionic ions (protons) is given by L=Lquasi+2Z, the quasi-angular momentum, while zero for parahydrogen, is equal to one for orthohydrogen. The presence of the Berry phase in the rotation of orthohydrogen will suppress the bond-breaking chemical reactivity.

Many microbes in biology (7) use H₂ as a metabolite, and the enzyme hydrogenase catalyzes the bond-breaking chemical reaction, H2→2H++2e−. Based on the QDS rule we would expect a differential reactivity between para- and orthohydrogen, with the reaction rate suppressed for orthohydrogen. Indeed, if this reaction were to proceed “directly”—without an ionized intermediary or a flipping of nuclear spin—QDS would predict a complete blocking of orthohydrogen reacting. At body temperature in a thermal distribution the ortho:para ratio approaches 3:1 (set by triplet degeneracy), while it is possible to prepare purified parahydrogen where this ratio is strongly inverted (say, 1:10). One might then hope to observe different enzymatic activity for hydrogenase catalysis in these two situations, with purified parahydrogen being significantly more reactive.

Possible differential combustion of para- and orthohydrogen with, say, oxygen might also be interesting to explore, even though this reaction is not enzymatic.

Intermolecular Entanglement of Nuclear Spins.

The ability to prepare purified parahydrogen molecules in solvent and drive a bond-breaking chemical reaction enables the preparation of two protons with nuclear spins entangled in a singlet. If–when these two protons bind onto a large molecule with different chemical environments, it is sometimes possible to perform a π rotation on one of the two nuclei to create alignment of the two spins, termed hyperpolarization. These hyperpolarized proton spins can then be used to transfer spin polarization to the nuclei of atoms on the molecules to which they are bonded.

There is, of course, a long precedent for liquid-state NMR, exploiting the fact of very long decoherence times in the rapidly fluctuating liquid environment (3). Indeed, soon after Peter Shor developed his prime factoring quantum algorithm, liquid-state NMR quantum computing efforts were the first out of the gate (8). In NMR quantum computing one uses a solvent hosting a concentration of identical molecules with multiple nuclear spins (say, protons). Ideally, the chemical environments of the different nuclei are different, so that they each have a different NMR chemical shift, and can thereby be addressed independently by varying the radio frequency. In principle it is then possible to perform qubit operations on these spins. However, there are two major drawbacks to NMR quantum computing—the difficulty in scalability and the challenge of preparing sufficiently entangled initial states. As we now suggest, it is possible that both of these can be circumvented by using small symmetric molecules which are the substrate for bond-breaking enzymes.

By way of illustration, we consider the symmetric biochemical ion pyrophosphate, P₂O74− (usually abbreviated as PPi), which is important in metabolic activity. Pyrophosphate is a phosphate dimer, which consists of two phosphate ions, PO43−, that share a central oxygen. (The inorganic-phosphate ion, abbreviated as Pi, consists of a phosphorus atom tetrahedrally bonded to four oxygens.) Since the phosphorus nucleus is an S=1/2 fermion and the oxygens are S=0 bosons, the twofold symmetry of PPi, which interchanges the two ³¹P nuclei and the three end oxygens, will, like molecular hydrogen, have two isomers, para-PPi and ortho-PPi. Moreover, para- and ortho-PPi will rotate with even and odd angular momentum, respectively. Thus, ortho-PPi, with Lquasi=1, rotates with a nontrivial Berry phase.

In biochemistry there is an enzyme (called pyrophosphatase) which catalyzes the bond-breaking reaction, PPi → Pi + Pi. Due to the Berry phase term in ortho-PPi, we expect that this reaction will be strongly suppressed, if not blocked entirely. Then, provided only para-PPi reacts, the two liberated Pi ions will have nuclear spins which are entangled in a singlet. Such intermolecular entanglement of nuclear spins could, in principle, jump start liquid-state NMR quantum computing efforts, allowing for both scalability and highly entangled initial-state preparation.

QDS Mass-Independent Mechanism for Isotope Fractionation.

Isotope fractionation refers to processes that affect the relative abundance of (usually) stable isotopes, often used in isotope geochemistry and biochemistry. There are several known mechanisms. Kinetic isotope fractionation is a mass-dependent mechanism in which the diffusion constant of a molecule varies with the mass of the isotope. This process is relevant to oxygen evaporation from water, where an oxygen molecule, which has one (or two) of the heavier oxygen isotopes (¹⁷O and ¹⁸O), is less likely to evaporate. This leads to a slight depletion in the isotope ratios of ¹⁷O/¹⁶O and ¹⁸O/¹⁶O in the vapor relative to that in the liquid water.

Another mass-dependent isotope fractionation phenomenon occurs in some chemical reactions, where the isotope abundances in the products of the reaction are (very) slightly different from those in the reactants. In biochemistry this effect is usually ascribed to an isotopic mass-induced change in the frequency of the molecular quantum zero-point vibrational fluctuations when bonded in the pocket of an enzyme. This modifies slightly the energy of the activation barrier which must be crossed for the bond-breaking reaction to proceed.

However, there are known isotopic fractionation processes which are “mass independent,” a classic example being the increased abundance of the heavier oxygen isotopes in the formation of ozone from two oxygen molecules (9). In ozone isotope fractionation the relative increased abundance of ¹⁷O and ¹⁸O is largely the same. While there have been theoretical proposals to explain this ozone isotope anomaly, these are not without controversy (10).

Here, as we briefly describe, our conjectured QDS rule for chemical reactions involving small symmetric molecules leads naturally to the prediction of a mass-independent mechanism for isotope fractionation, driven by the quantum distinguishability of the two different isotopes. In the presence of isotopes that destroy the molecular rotational symmetry, the QDS rule is no longer operative and one would expect the chemical reaction to proceed more rapidly.

By way of illustration we again consider the enzymatic hydrolysis reaction, PPi → Pi + Pi. As we now detail, in this experiment one would predict a large heavy oxygen isotope fractionation effect. Indeed, if one of the six “end” oxygens in PPi is a heavy oxygen isotope, the symmetry of PPi under a rotation is broken and the reaction becomes “unblocked” (independent of the nuclear spin state).

If correct, we would then predict a very large mass-independent oxygen isotope fractionation which concentrates the heavy oxygen isotopes in the products (Pi + Pi). For the early stages of this reaction, before the isotopically modified PPi are depleted, one would, in fact, predict a factor of 4 increase in the ratios of ¹⁷O/¹⁶O and ¹⁸O/¹⁶O in the enzymatic reaction PPi → Pi + Pi.

To be more quantitative we introduce a dimensionless function, R(f), where R denotes the ratio of the heavy isotope of oxygen in the products, relative to the reactants,R(f)=[O18/16O]prod[O18/16O]react,[25]and f∈[0,1] is the “extent” of the reaction. In a conventional isotope fractionation framework, one would expect a very small effect; that is, R(f)≈1. But within our QDS conjecture, if correct, one would haveR(f)=1−(1−f)λf,[26]with λ=4. Experiments to look for this effect are presently underway.

Activity of Reactive Oxygen Species.

In biochemistry it is well known that during ATP synthesis the oxygen molecule picks up an electron and becomes a negatively charged “superoxide” ion, O2− (11). Having an odd number of electrons (with electron spin-1/2) superoxide is a “free radical.” Together with hydrogen peroxide (H2O2) and the hydroxyl radical (the electrically neutral form of the hydroxide ion) the superoxide ion is known as a reactive oxygen species (ROS). ROS ions can cause oxidative damage due in part to their reactivity. Indeed, in the free-radical theory, oxidative damage initiated by ROS is a major contributor to aging. In biology there are specific enzymes to break down the ROS to produce benign molecules (e.g., water) (12).

In contrast to the ROS, the stable state of molecular oxygen (“triplet oxygen”) is less reactive in biology. As we detail below, we propose that this difference from triplet molecular oxygen can be understood in terms of our conjectured QDS rule.

First, we note that standard analysis of electronic molecular states (that we relegate to Supporting Information) shows that under C₂ rotation the electronic states in the triplet molecular (neutral) oxygen exhibit an overall sign change. Because ¹⁶O nuclei are spinless bosons, there is no nuclear contribution to the Berry phase.

Thus, the triplet neutral oxygen molecule O₂ exhibits a purely electronic π Berry phase, despite a spinless bosonic character of the ¹⁶O nuclei. It therefore rotates with odd angular momentum, L=Lquasi+2Z, with Lquasi=1, identical to orthohydrogen. Our QDS conjecture then implies that a direct bond-breaking chemical reaction of triplet oxygen is strictly forbidden.

In contrast to triplet oxygen, the superoxide ion O2− is not blocked by the QDS rule and can thus undergo a direct chemical bond-breaking transition. Indeed, as detailed in Supporting Information, due to the electronic nonzero orbital and spin angular momenta aligned along the body axis, the two ends of the superoxide ion are distinguishable. Thus, in contrast to triplet oxygen, superoxide does not have any symmetry under a 180^○ rotation that interchanges the two oxygen nuclei. The superoxide ion can thus rotate with any integer value of the angular momentum, L=0,1,2,3,....

As a result, the QDS is not operative and thus there is no selection rule precluding a direct bond-breaking chemical reaction of the superoxide ion. We propose that it is this feature of superoxide, relative to triplet oxygen, which accounts, at least in part, for the high reactivity of superoxide and explains why it is a “ROS.”

Ortho-Water as a Quantum Disentangled Liquid.

Molecular water has a C₂ symmetry axis which exchanges the two protons. Thus, as for molecular hydrogen, water comes in two variants, para-water and ortho-water which rotate with even and odd angular momentum, respectively. QDS then predicts that the ortho-water molecule (with Lquasi=1) cannot undergo a direct chemical reaction that splinters the molecule into a proton and a hydroxide ion, OH⁻.

Since the difference between the rotational kinetic energy of a parawater and an ortho-water molecule is roughly 30 K, liquid water consists of 75% ortho-water molecules and 25% parawater molecules. In one remarkable paper (13) it was reported that gaseous water vapor can be substantially enriched in either ortho- or para-water molecules and then condensed to create ortho- and para-liquid water—although attempts to reproduce this work have been unsuccessful (14). If QDS is operative, ortho-liquid water would be quite remarkable, having zero concentration of either “free” protons or free hydroxide ions, despite these being energetically accessible at finite temperature. Theoretically, ortho-liquid water would then be an example of a “quantum disentangled liquid” in which the protons are enslaved to the oxygen ions and do not contribute independently to the entropy density (15).

Experimentally, one would predict that ortho-liquid water would have vanishingly small electrical conductivity, nonzero only due to ortho-to-para conversion. Data on “shocked” supercritical water indicate that above a critical pressure the electrical conductivity increases by nine orders of magnitude (16). Perhaps this is due to a transition from a quantum-disentangled to a thermal state, where most (if not all) of the ortho-water molecules are broken into a proton and a hydroxide ion, leading to a significant electrical conductivity?

The properties of ortho-solid ice might also be quite interesting, provided QDS is operative. While an extensive equilibrium entropy would still be expected (consistent with the ice rules), the quantum dynamics would be quite different. Rather than protons hopping between neighboring oxygen ions, in ortho-ice these processes would actually correspond to collective rotations of the water molecules. The nature of the quantum dynamical quenching of the entropy when ice is cooled to very low temperatures is worthy of future investigation.