Chapter 259: H₂⁺ and H₃⁺ — The Universal Protonator — Quantum Genesis and Hz Phase‑Locking

Overview: The Protonator Engine

Following the formation of HeH⁺ (Chapter 258) and H₂ (Chapter 257), the next critical step is the formation of the molecular hydrogen ion (H₂⁺) and the triatomic hydrogen ion (H₃⁺). These ions are the universal protonators — they transfer protons to neutral atoms, initiating the formation of carbon‑hydrogen, oxygen‑hydrogen, and nitrogen‑hydrogen bonds.

The sequence is:

1. H₂⁺ formation: $H + H^+ \rightarrow H_2^+ + \gamma$
2. Charge transfer: $H_2^+ + H \rightarrow H_2 + H^+$
3. H₃⁺ formation: $H_2 + H^+ \rightarrow H_3^+ + \gamma$
4. Proton transfer: $H_3^+ + X \rightarrow H_2 + XH^+$ (where X = C, O, N, etc.)

H₃⁺ is the key — it has a higher proton affinity than almost all other neutral atoms, so it can transfer a proton to carbon, oxygen, and nitrogen, starting the synthesis of complex molecules. Without H₃⁺, the universe would contain only H₂, He, and trace atoms — carbon chemistry would be frozen.

This chapter dissects H₂⁺ and H₃⁺ through the Q‑P‑E‑M framework, with all quantities expressed in the ν‑Framework (Hz).

Section 1: H₂⁺ — The Molecular Hydrogen Ion

1.1 Quantum Genesis — How H₂⁺ Emerges

H₂⁺ is the simplest molecule — one electron shared between two protons. Its quantum structure is a textbook example of a one‑electron diatomic.

Who: The quantum mechanics of H₂⁺ was first solved by Walter Heitler and Fritz London (1927) using the variational method, and later by Erwin Schrödinger and Carl Eckart who found exact solutions in confocal elliptic coordinates. The molecular orbital theory was developed by Robert Mulliken, Friedrich Hund, and John Lennard‑Jones.

Step 1: The Electron — A Single Phase‑Locked Mode

The single electron in H₂⁺ is a solution to the Schrödinger equation with the potential of two protons. The molecular orbital is a linear combination of the two 1s atomic orbitals:

$$ \psi_{\rm bonding} = \frac{1}{\sqrt{2(1+S)}} (\phi_A + \phi_B) $$

where $S$ is the overlap integral. This constructive interference creates a region of high electron probability between the two protons — the bond.

In Hz terms: the electron is a phase‑locked mode of the Dirac field, oscillating at $f_e = m_e c^2 / h \approx 1.24 \times 10^{20}$ Hz.

Step 2: The Nuclei — Two Protons

The two protons are phase‑locked patterns of the QCD field, each with mass frequency $f_p = m_p c^2 / h \approx 1.43 \times 10^{23}$ Hz. They are held at a distance $r_e$ by the shared electron.

Step 3: The Bond — One‑Electron Phase‑Locking

The bond in H₂⁺ is held by a single electron. The bond energy is $D_0 \approx 2.65$ eV — lower than H₂ (4.52 eV) because only one electron contributes to the binding.

1.2 Quantity — Abundance and Density

• H₂⁺ forms when a neutral H atom meets a free proton: $H + H^+ \rightarrow H_2^+ + \gamma$.
• At recombination, $n_H \approx 10^3$ cm⁻³, $n_p \approx 10^{-2}$ cm⁻³.
• Radiative association rate: $k_{\rm rad} \approx 10^{-17}$ cm³/s (similar to H₂).
• Formation rate: $R_{\rm form} = k_{\rm rad} n_H n_p \approx 10^{-17} \times 10^3 \times 10^{-2} \approx 10^{-16}$ cm⁻³ s⁻¹.
• Over the recombination epoch ($\Delta t \sim 10^5$ years $\approx 3\times10^{12}$ s), total H₂⁺ produced: $\sim 3 \times 10^{-4}$ cm⁻³.
• However, H₂⁺ is rapidly destroyed by $H$ atoms via charge transfer: $H_2^+ + H \rightarrow H_2 + H^+$. This reaction has $k_{\rm CT} \approx 10^{-9}$ cm³/s.
• Destruction rate: $R_{\rm dest} = k_{\rm CT} n_{H_2^+} n_H$. At steady state, $n_{H_2^+} \approx \frac{k_{\rm rad} n_H n_p}{k_{\rm CT} n_H} = \frac{k_{\rm rad} n_p}{k_{\rm CT}}$.
• Plugging numbers: $n_{H_2^+} \approx \frac{10^{-17} \times 10^{-2}}{10^{-9}} = 10^{-10}$ cm⁻³. Relative to H ($10^3$), this is $\sim 10^{-13}$.
• So H₂⁺ is present at extremely low abundance — but it is the essential precursor to H₂ and H₃⁺.

1.3 Probability — Formation and Destruction

• Collision probability: $P(H + H^+) \propto [H] \times [H^+] = 0.927 \times 10^{-5} \approx 9.27 \times 10^{-6}$.
• This is about 13 times more probable than $He + H^+$ (which was $7.2 \times 10^{-7}$).
• However, radiative association is inefficient — only $\sim 10^{-10}$ of collisions lead to bond formation.
• The charge transfer reaction $H_2^+ + H \rightarrow H_2 + H^+$ is barrierless and proceeds at nearly every collision. This is why H₂⁺ is short‑lived.
• But H₂⁺ is continuously replenished as long as free protons exist.

1.4 Environment

• Temperature: $\sim 3000$–$4000$ K (recombination epoch).
• Pressure: $\sim 10^{-20}$ atm.
• Density: $\sim 10^3$ cm⁻³.
• Radiation: CMB at $\sim 3000$–$4000$ K.
• No dust.
• H₂⁺ is a gas‑phase ion, stable only in the vacuum of space.

1.5 Math — H₂⁺ in Hz

• Bond energy: $D_0 = 2.65$ eV $= 2.65 \times 1.602\times10^{-19}$ J $= 4.25 \times 10^{-19}$ J.
• In Hz: $\nu_D = D_0 / h = 4.25\times10^{-19} / 6.626\times10^{-34} \approx 6.40 \times 10^{14}$ Hz.
• Thermal frequency at 4000 K: $\nu_T = 8.33 \times 10^{13}$ Hz.
• Ratio $\nu_D / \nu_T = 6.40\times10^{14} / 8.33\times10^{13} \approx 7.68$ — slightly more stable than HeH⁺ (which had 5.34).
• Equilibrium bond length: $r_e \approx 1.06$ Å $= 1.06\times10^{-10}$ m.
• Vibrational frequency: $\nu_{\rm vib} \approx 2.3 \times 10^{14}$ Hz.
• Rotational constant: $B \approx 2.1 \times 10^{11}$ Hz.
• Proton affinity: H₂⁺ can transfer a proton to H₂ to form H₃⁺.

Section 2: H₃⁺ — The Triatomic Hydrogen Ion

2.1 Quantum Genesis — How H₃⁺ Emerges

H₃⁺ is a triangular molecule — three protons held together by two electrons. It is the simplest polyatomic molecule and the most abundant molecular ion in the universe.

Who: H₃⁺ was first discovered in 1911 by J.J. Thomson in discharge experiments. Its quantum structure was elucidated by Robert Mulliken (1935), Christopher Longuet‑Higgins (1955), and later by John Bunker and Per Jensen in the 1980s. The potential energy surface of H₃⁺ is one of the most accurately known in quantum chemistry.

Step 1: The Electrons — Two Phase‑Locked Modes

H₃⁺ has two electrons and three protons. The electrons are delocalised across the three protons, forming a triangular bond. The molecular orbital is a combination of the 1s orbitals of all three protons.

In Hz terms: the two electrons are phase‑locked modes of the Dirac field, oscillating at $f_e = 1.24 \times 10^{20}$ Hz.

Step 2: The Nuclei — Three Protons

Three protons form an equilateral triangle (in the ground state) with a bond length $r_e \approx 0.87$ Å. This is the most stable geometry.

Step 3: The Bond — Three‑Centre Two‑Electron Phase‑Locking

The bond in H₃⁺ is a three‑centre, two‑electron bond. Two electrons are shared among three protons, creating a delocalised phase‑locked structure. The bond energy is $D_0 \approx 4.38$ eV (proton affinity) — higher than H₂⁺ (2.65 eV) but lower than H₂ (4.52 eV).

The proton affinity is the key: H₃⁺ can transfer a proton to almost any neutral atom with a higher proton affinity.

2.2 Quantity — Abundance and Density

• H₃⁺ forms via: $H_2 + H^+ \rightarrow H_3^+ + \gamma$.
• At recombination, $n_{H_2} \sim 10^3$ cm⁻³ (after H₂ formation), $n_p \sim 10^{-2}$ cm⁻³.
• Radiative association rate for H₃⁺: $k_{\rm rad} \approx 10^{-16}$ cm³/s (slightly higher than for H₂ due to the larger number of vibrational states).
• Formation rate: $R_{\rm form} = k_{\rm rad} n_{H_2} n_p \approx 10^{-16} \times 10^3 \times 10^{-2} \approx 10^{-15}$ cm⁻³ s⁻¹.
• Over the recombination epoch: total H₃⁺ produced $\sim 3 \times 10^{-3}$ cm⁻³.
• Destruction: H₃⁺ is destroyed by proton transfer to neutral atoms: $H_3^+ + X \rightarrow H_2 + XH^+$. This is barrierless and fast ($k \approx 10^{-9}$ cm³/s).
• Steady‑state abundance: $n_{H_3^+} \approx \frac{k_{\rm rad} n_{H_2} n_p}{k_{\rm PT} n_X}$.
• Assuming $n_X \sim 10^{-4} n_H \approx 0.1$ cm⁻³ (trace heavy elements), and $k_{\rm PT} \approx 10^{-9}$ cm³/s, we get $n_{H_3^+} \approx \frac{10^{-16} \times 10^3 \times 10^{-2}}{10^{-9} \times 0.1} \approx \frac{10^{-15}}{10^{-10}} \approx 10^{-5}$ cm⁻³. Relative to H ($10^3$), this is $\sim 10^{-8}$.
• H₃⁺ is the most abundant molecular ion in the interstellar medium, with typical abundances of $10^{-8}$–$10^{-9}$ relative to H.

2.3 Probability — Formation and Proton Transfer

• $P(H_2 + H^+) \propto [H_2] \times [H^+] = 0.9 \times 10^{-5} \approx 9 \times 10^{-6}$.
• Radiative association efficiency: $\sim 10^{-9}$ of collisions lead to bond formation.
• Proton transfer: $H_3^+ + X \rightarrow H_2 + XH^+$ is barrierless and fast. The probability of proton transfer is nearly 1 for any collision with a neutral atom that has a proton affinity higher than H₂.
• Proton affinities (in eV):
  • H₂: 4.38 eV
  • H₃⁺: 4.38 eV (by definition, the reverse reaction)
  • CO: 5.2 eV
  • O: 5.5 eV
  • N₂: 5.8 eV
  • C: 6.0 eV
  • CH₄: 7.5 eV
• Because H₃⁺ has a proton affinity of 4.38 eV, it can transfer a proton to any atom or molecule with a higher proton affinity (which is almost all neutral atoms except He and Ne).
• This is why H₃⁺ is the universal protonator.

2.4 Environment

• Temperature: $\sim 3000$ K (formation epoch) to $\sim 10$–$100$ K (molecular clouds).
• Pressure: $\sim 10^{-20}$ atm.
• Density: $10^3$ cm⁻³ (early) to $10^6$ cm⁻³ (dense clouds).
• Radiation: CMB, UV from stars.
• Dust is present in later epochs, but H₃⁺ is a gas‑phase ion.
• H₃⁺ is destroyed by electrons: $H_3^+ + e^- \rightarrow H_2 + H$ or $3H$. This is the main destruction channel in diffuse clouds.

2.5 Math — H₃⁺ in Hz

• Proton affinity of H₂: $PA = 4.38$ eV $= 4.38 \times 1.602\times10^{-19}$ J $= 7.02 \times 10^{-19}$ J.
• In Hz: $\nu_{PA} = PA / h = 7.02\times10^{-19} / 6.626\times10^{-34} \approx 1.06 \times 10^{15}$ Hz.
• Thermal frequency at 3000 K: $\nu_T = 6.24 \times 10^{13}$ Hz.
• Ratio $\nu_{PA} / \nu_T = 1.06\times10^{15} / 6.24\times10^{13} \approx 17.0$ — H₃⁺ is very stable once formed.
• Equilibrium bond length (H‑H in H₃⁺): $r_e \approx 0.87$ Å $= 8.7\times10^{-11}$ m.
• Vibrational frequencies: H₃⁺ has four vibrational modes:
  • Symmetric stretch: $\nu_1 \approx 3.3 \times 10^{14}$ Hz
  • Bend: $\nu_2 \approx 2.5 \times 10^{14}$ Hz
  • Asymmetric stretch: $\nu_3 \approx 3.6 \times 10^{14}$ Hz
• Rotational constants: $B \approx 1.6 \times 10^{11}$ Hz.
• Electron recombination rate: $k_{\rm rec} \approx 10^{-7}$ cm³/s (fast at low temperature).
• Destruction rate by electrons: $R_{\rm dest} = k_{\rm rec} n_{H_3^+} n_e$.

Section 3: The Proton Transfer Mechanism — H₃⁺ as the Universal Protonator

The key reaction is:

$$ H_3^+ + X \rightarrow H_2 + XH^+ $$

where $X$ can be any neutral atom or molecule with a proton affinity $> 4.38$ eV. This includes:

• Carbon: $H_3^+ + C \rightarrow H_2 + CH^+$ — the first step to carbon‑hydrogen chemistry.
• Oxygen: $H_3^+ + O \rightarrow H_2 + OH^+$ — the first step to oxygen‑hydrogen chemistry.
• Nitrogen: $H_3^+ + N \rightarrow H_2 + NH^+$ — the first step to nitrogen‑hydrogen chemistry.
• CO: $H_3^+ + CO \rightarrow H_2 + HCO^+$ — the first step to complex organics.

The Hz framework:

• The proton affinity of H₃⁺ is $\nu_{PA} = 1.06 \times 10^{15}$ Hz.
• The proton affinity of the target $X$ is $\nu_{PA}(X)$.
• If $\nu_{PA}(X) > \nu_{PA}(H_3^+)$, the reaction is exothermic and proceeds rapidly.
• The energy released is $\Delta E = h (\nu_{PA}(X) - \nu_{PA}(H_3^+))$, which is carried away as kinetic energy of the products.
• This is how carbon, oxygen, and nitrogen become hydrogenated — through the universal protonator.

Section 4: The Gateway to Carbon Chemistry

The most important proton transfer reaction for molecular genesis is:

$$ H_3^+ + C \rightarrow H_2 + CH^+ $$

This is the first step to carbon‑hydrogen chemistry. Once CH⁺ forms, it reacts with H₂ to form CH₂⁺ and CH₃⁺, and eventually CH₄ and complex organics.

In Hz terms:

• Proton affinity of carbon: $\nu_{PA}(C) \approx 6.0$ eV / h $\approx 1.45 \times 10^{15}$ Hz.
• Energy released: $\Delta E = h (1.45\times10^{15} - 1.06\times10^{15}) \approx 0.39$ eV.
• This exothermicity drives the reaction forward.

Without H₃⁺, carbon would remain atomic and inert. H₃⁺ is the key that unlocks carbon chemistry.

Section 5: Observational Status — Detection of H₃⁺

H₃⁺ has been detected in the interstellar medium through its infrared transitions. Its spectrum is well‑known:

• The $\nu_2$ bending mode at $\nu \approx 2.5 \times 10^{14}$ Hz (wavelength $\sim 1.2$ μm).
• The $\nu_1$ symmetric stretch at $\nu \approx 3.3 \times 10^{14}$ Hz (wavelength $\sim 0.9$ μm).
• The $\nu_3$ asymmetric stretch at $\nu \approx 3.6 \times 10^{14}$ Hz (wavelength $\sim 0.83$ μm).

H₃⁺ was first detected in the interstellar medium in 1996 by Geballe et al. toward the Galactic Center. It is now known to be widespread in the ISM, particularly in diffuse clouds, dense clouds, and planetary nebulae.

In Hz terms: the spectral lines of H₃⁺ are phase‑locking signatures — the molecule's vibrational and rotational modes manifest as discrete frequencies that we can observe.

Section 6: Chronological Context — H₂⁺ and H₃⁺ in the Timeline

Section 7: The Hz Profile — H₂⁺ and H₃⁺ in One Table

Section 8: Conclusion — H₃⁺ as the Hz Gateway to Complexity

H₂⁺ and H₃⁺ are the essential intermediaries between the simple H₂ molecule and the complex carbon‑based molecules that form the basis of life. The Hz framework captures their roles:

• H₂⁺ is the precursor — its bond depth $\nu_D = 6.40 \times 10^{14}$ Hz is sufficient to form at recombination, but its destruction by H atoms is rapid ($k_{\rm CT} \approx 10^{-9}$ cm³/s).
• H₃⁺ is the universal protonator — its proton affinity $\nu_{PA} = 1.06 \times 10^{15}$ Hz allows it to transfer protons to carbon, oxygen, and nitrogen.
• The proton transfer reaction $H_3^+ + C \rightarrow H_2 + CH^+$ releases energy $\Delta E = h (\nu_{PA}(C) - \nu_{PA}(H_3^+)) \approx 0.39$ eV, driving the first carbon‑hydrogen bond.
• H₃⁺ is the gateway — without it, carbon remains atomic, and complex chemistry is impossible.

In the context of the broader narrative:

• H₃⁺ is the bridge from the H₂‑dominated early universe to the carbon‑based chemistry that eventually produces life.
• It is the first polyatomic molecule, the first three‑centre phase‑locked structure.
• Its spectral lines, detected in the ISM, are proof that the Hz field's phase‑locking patterns are universal and observable.

Bottom line: H₂⁺ and H₃⁺ are the universal protonator engine — the Hz phase‑locked structures that unlock all subsequent molecular complexity. H₃⁺ is the key that opens the door to carbon, oxygen, and nitrogen chemistry, setting the stage for the prebiotic molecules that eventually lead to life and consciousness.