Chapter 260: CH⁺, CH₂⁺, CH₃⁺ — The Genesis of Carbon‑Hydrogen Bonds — Hz Phase‑Locking and the Endothermic Bottleneck

Overview: The Birth of Organic Chemistry

Carbon is the universal phase‑locking hub — the element with four valence electrons that can form complex 3D structures. But carbon cannot form bonds directly with hydrogen in the cold interstellar medium. The first step requires ionised carbon ($C^+$) reacting with H₂:

$$ C^+ + H_2 \rightarrow CH^+ + H \qquad (\Delta E = +0.40 \ {\rm eV}) $$

This reaction is endothermic — it requires an input of energy. In cold molecular clouds ($T \sim 10$–$50$ K), the thermal energy is far below the activation threshold. The reaction is essentially frozen. It only proceeds in regions where the gas is heated by shocks or UV radiation ($T > 1000$ K).

However, once CH⁺ forms, the subsequent reactions are exothermic and proceed rapidly:

$$ CH^+ + H_2 \rightarrow CH_2^+ + H \qquad (\Delta E = -3.6 \ {\rm eV}) $$

$$ CH_2^+ + H_2 \rightarrow CH_3^+ + H \qquad (\Delta E = -4.4 \ {\rm eV}) $$

These ions are the building blocks of all organic chemistry. CH₃⁺ is particularly important as it can react with neutral molecules to form larger carbon‑containing species.

This chapter dissects each step through the Q‑P‑E‑M framework, with all quantities expressed in the ν‑Framework (Hz).

Section 1: Quantum Genesis — How CH⁺, CH₂⁺, CH₃⁺ Emerge

1.1 The Carbon Atom — A Phase‑Locked Hub

Carbon has the electron configuration $1s^2 2s^2 2p^2$. It has four valence electrons (two in 2s, two in 2p). In ionised form $C^+$, it has three valence electrons ($2s^2 2p^1$).

In Hz terms: the carbon nucleus has mass frequency $f_C = m_C c^2 / h \approx 1.09 \times 10^{25}$ Hz. The valence electrons are phase‑locked modes at $f_e = 1.24 \times 10^{20}$ Hz. The $2p$ orbital has angular momentum frequency $\nu_{\rm ang} = \hbar / (2\pi I) \sim 10^{14}$ Hz.

1.2 The Endothermic Reaction — $C^+ + H_2 \rightarrow CH^+ + H$

This reaction is the rate‑limiting step for all carbon chemistry. It requires a collision with sufficient kinetic energy to overcome the barrier of 0.40 eV.

Who: The quantum chemistry of this reaction was studied by David Bates (1951), Eric Herbst and William Klemperer (1973), and later by Tom Millar and Alexander Dalgarno. The potential energy surface has been calculated using high‑level ab initio methods.

The reaction proceeds through a short‑lived intermediate complex $[C H_2]^+$ which then dissociates into CH⁺ + H. The barrier height is 0.40 eV above the reactants.

1.3 The Exothermic Cascade — CH⁺ → CH₂⁺ → CH₃⁺

Once CH⁺ is formed, it reacts with H₂ in exothermic steps:

• CH⁺ + H₂ → CH₂⁺ + H: This reaction is barrierless and exothermic by 3.6 eV. It proceeds at near‑collisional rates.
• CH₂⁺ + H₂ → CH₃⁺ + H: This reaction is also exothermic (4.4 eV) and barrierless.

These reactions are so fast that CH⁺ is quickly converted to CH₃⁺ in dense clouds.

1.4 The Dust Surface Alternative — Tunneling and Catalysis

In cold molecular clouds ($T \sim 10$ K), the gas‑phase reaction is frozen. However, on dust grain surfaces, the reaction can proceed via quantum tunneling and surface catalysis.

The mechanism:

1. C atoms (or C⁺) land on the dust grain.
2. H atoms land on the dust grain and diffuse.
3. The dust grain acts as a heat sink, draining excess energy.
4. The activation barrier is overcome by quantum tunneling, allowing the reaction at 10 K.

Section 2: Quantity — Abundances and Densities

2.1 Carbon Inventory

• Post‑Population III stars, carbon abundance reaches $\sim 3 \times 10^{-4}$ relative to H.
• In molecular clouds, most carbon is in the form of CO ($\sim 10^{-4}$ rel. H).
• Only a tiny fraction ($\sim 10^{-7}$–$10^{-8}$) is in ionised form $C^+$ in cold clouds (ionisation from UV).
• CH⁺ abundance in diffuse clouds: $\sim 10^{-9}$–$10^{-8}$ relative to H.
• CH₂⁺ abundance: $\sim 10^{-10}$–$10^{-9}$.
• CH₃⁺ abundance: $\sim 10^{-9}$–$10^{-8}$.
• These ions are key tracers of carbon chemistry in the ISM.

2.2 Reaction Rates

• $C^+ + H_2 \rightarrow CH^+ + H$: rate $k_1 = \alpha \exp(-E_a / k_B T)$, where $\alpha \sim 10^{-10}$ cm³/s, $E_a = 0.40$ eV.
• At 10 K: $k_1 \approx 10^{-10} \exp(-0.40 / 8.617\times10^{-5} \times 10) \approx 10^{-10} \exp(-464) \approx 10^{-10} \times 10^{-202} \approx 10^{-212}$ cm³/s — essentially zero.
• At 1000 K: $k_1 \approx 10^{-10} \exp(-4640 / 1000) = 10^{-10} \exp(-4.64) \approx 10^{-10} \times 0.0096 \approx 10^{-12}$ cm³/s — still very slow.
• At 4640 K: $k_1 \approx 10^{-10} \exp(-1) \approx 3.7 \times 10^{-11}$ cm³/s — becomes significant.
• For exothermic steps: $CH^+ + H_2$ has $k_2 \approx 10^{-9}$ cm³/s (barrierless).
• $CH_2^+ + H_2$ has $k_3 \approx 10^{-9}$ cm³/s.

2.3 Steady‑State Abundances

• In diffuse clouds ($T \sim 100$ K, $n_H \sim 10^2$ cm⁻³), the ionisation fraction $x_{C^+} \sim 10^{-4}$.
• The formation rate of CH⁺: $R_1 = k_1 n_{C^+} n_{H_2}$.
• Destruction of CH⁺: by reaction with H₂ to form CH₂⁺ (fast), or by dissociative recombination with electrons.
• Steady‑state abundance of CH⁺ is determined by balance of formation and destruction.
• Typical CH⁺ column densities in diffuse clouds are $N(CH^+) \sim 10^{13}$ cm⁻², giving relative abundance $\sim 10^{-8}$.

Section 3: Probability — The Endothermic Bottleneck

3.1 Collision Probability

• $P(C^+ + H_2) \propto [C^+] \times [H_2] \approx (10^{-7}) \times (0.9) \approx 9 \times 10^{-8}$ (in cold clouds).
• This is $\sim 10^7$ times less probable than H‑H collisions.
• However, the real bottleneck is the energy barrier, not the collision rate.

3.2 Energy Barrier Probability

• The probability that a collision has kinetic energy $E > E_a$ is given by the Maxwell‑Boltzmann distribution: $P(E > E_a) = \exp(-E_a / k_B T)$.
• At 10 K: $P \approx \exp(-0.40 / 0.00086) = \exp(-464) \approx 10^{-202}$ — essentially zero.
• At 100 K: $P \approx \exp(-4640 / 100) = \exp(-46.4) \approx 10^{-20}$ — still negligible.
• At 4640 K: $P \approx \exp(-1) \approx 0.37$ — significant.
• Thus, the reaction only proceeds where the gas is heated to thousands of Kelvin.

3.3 Quantum Tunneling Probability on Dust

On a dust grain surface, the reaction can proceed via quantum tunneling through the barrier. The tunneling probability is:

$$ P_{\rm tunnel} \approx \exp\left(-2 \frac{\sqrt{2 \mu E_a}}{\hbar} d \right) $$

where $\mu$ is the reduced mass of C⁺ and H₂, $E_a = 0.40$ eV, and $d$ is the barrier width (typical $1$ Å).

For $\mu \approx \frac{12 \times 2}{14} m_H \approx 1.7 m_H \approx 2.8 \times 10^{-27}$ kg, $E_a = 6.4\times10^{-20}$ J, $d = 10^{-10}$ m:

$$ P_{\rm tunnel} \approx \exp\left(-2 \frac{\sqrt{2 \times 2.8\times10^{-27} \times 6.4\times10^{-20}}}{1.055\times10^{-34}} \times 10^{-10} \right) $$

$$ \approx \exp\left(-2 \frac{\sqrt{3.58\times10^{-46}}}{1.055\times10^{-34}} \times 10^{-10} \right) \approx \exp\left(-2 \frac{1.89\times10^{-23}}{1.055\times10^{-34}} \times 10^{-10} \right) $$

$$ \approx \exp\left(-2 \times 1.79\times10^{11} \times 10^{-10} \right) = \exp(-35.8) \approx 10^{-16} $$

Thus, each collision on the surface has a tunneling probability of $\sim 10^{-16}$. With $10^5$–$10^6$ H atom collisions per second on a grain, the reaction can proceed over timescales of years — slow but possible.

Section 4: Environment — Where Carbon‑Hydrogen Bonds Form

4.1 Gas‑Phase: Hot Regions

• Shock regions: When a molecular cloud is compressed by a supernova or stellar wind, the gas is heated to $> 1000$ K, making $C^+ + H_2$ viable.
• Photodissociation regions (PDRs): Near hot stars, UV radiation ionises carbon and heats the gas.
• Hot cores: Regions around protostars ($T \sim 100$–$300$ K) have enough thermal energy for a small fraction of reactions.
• In these environments, CH⁺ is produced and quickly converted to CH₂⁺ and CH₃⁺.

4.2 Dust Surface: Cold Clouds

• In cold molecular clouds ($T \sim 10$ K), gas‑phase reactions are frozen.
• Dust grains provide a surface for catalysis and tunneling.
• The grain temperature is $\sim 10$ K, but the surface provides a local environment where atoms are adsorbed and can react via tunneling.
• The grain acts as a heat sink, removing excess energy and stabilising intermediates.

4.3 Typical Parameters

Section 5: Math — The Hz Framework for CH⁺, CH₂⁺, CH₃⁺

5.1 Activation Energy in Hz

$$ E_a = 0.40 \ {\rm eV} = 0.40 \times 1.602\times10^{-19} \ {\rm J} = 6.41 \times 10^{-20} \ {\rm J} $$

$$ \nu_a = \frac{E_a}{h} = \frac{6.41\times10^{-20}}{6.626\times10^{-34}} \approx 9.67 \times 10^{13} \ {\rm Hz} $$

This is the frequency threshold that must be exceeded by the thermal frequency $\nu_T$.

5.2 Temperature Threshold

$$ T_{\rm threshold} = \frac{E_a}{k_B} = \frac{0.40 \ {\rm eV}}{8.617\times10^{-5} \ {\rm eV/K}} \approx 4640 \ {\rm K} $$

Thus, the reaction requires $\nu_T > 9.67 \times 10^{13}$ Hz.

5.3 Thermal Frequencies in Different Environments

5.4 Exothermic Releases in Hz

• $CH^+ + H_2 \rightarrow CH_2^+ + H$: $\Delta E = -3.6$ eV $\Rightarrow \nu_{\rm release} = 3.6 \times 1.602\times10^{-19} / 6.626\times10^{-34} \approx 8.70 \times 10^{14}$ Hz.
• $CH_2^+ + H_2 \rightarrow CH_3^+ + H$: $\Delta E = -4.4$ eV $\Rightarrow \nu_{\rm release} = 4.4 \times 1.602\times10^{-19} / 6.626\times10^{-34} \approx 1.06 \times 10^{15}$ Hz.
• These released frequencies are in the infrared, corresponding to vibrational excitation of the products.

5.5 Reaction Rates in Hz Form

The rate constant for the endothermic reaction can be written:

$$ k_1 = A \exp\left(-\frac{\nu_a}{\nu_T}\right) $$

where $A \sim 10^{-10}$ cm³/s is the pre‑exponential factor.

At $T = 100$ K ($\nu_T = 2.08 \times 10^{12}$ Hz):

$$ k_1 = 10^{-10} \exp\left(-\frac{9.67\times10^{13}}{2.08\times10^{12}}\right) = 10^{-10} \exp(-46.5) \approx 10^{-10} \times 10^{-20.2} \approx 10^{-30} \ {\rm cm^3/s} $$

At $T = 5000$ K ($\nu_T = 1.04 \times 10^{14}$ Hz):

$$ k_1 = 10^{-10} \exp\left(-\frac{9.67\times10^{13}}{1.04\times10^{14}}\right) = 10^{-10} \exp(-0.93) \approx 10^{-10} \times 0.395 \approx 3.95 \times 10^{-11} \ {\rm cm^3/s} $$

This shows the enormous temperature dependence encoded in the Hz ratio $\nu_a / \nu_T$.

Section 6: The Exothermic Cascade — CH⁺ → CH₂⁺ → CH₃⁺

Once the first bond is formed, the subsequent reactions are fast and exothermic:

These reactions have no activation barrier and proceed at the collision rate. Thus, in any region where CH⁺ is formed, it is rapidly converted to CH₃⁺.

CH₃⁺ is a key ion — it can react with neutral molecules (H₂O, CO, NH₃) to form larger ions, or recombine with electrons to form neutral CH₃ and CH₄.

Section 7: The Dust Surface Route — Tunneling in Hz

On dust grains, the endothermic reaction can proceed via quantum tunneling. The tunneling probability $P_{\rm tunnel}$ can be expressed in Hz terms:

$$ P_{\rm tunnel} = \exp\left(-2 \frac{\sqrt{2 \mu E_a}}{\hbar} d \right) $$

We can rewrite the exponent in terms of frequencies. The barrier width $d$ is related to the vibrational frequency of the adsorbate $\nu_{\rm vib}$: $d \sim v / \nu_{\rm vib}$, where $v$ is the thermal velocity.

Alternatively, we can define an effective tunneling frequency $\nu_{\rm tunnel} = \nu_{\rm vib} \times P_{\rm tunnel}$.

Typical values: $\nu_{\rm vib} \sim 10^{12}$ Hz (surface phonon frequencies), $P_{\rm tunnel} \sim 10^{-16}$ (as calculated). Then $\nu_{\rm tunnel} \approx 10^{-4}$ Hz — meaning the reaction occurs once every $10^4$ seconds ($\sim 3$ hours) per site. With $10^{12}$ sites per grain, the overall reaction rate can be significant.

Thus, the dust surface route provides a mechanism for carbon‑hydrogen bond formation in cold clouds, albeit at a slow rate.

Section 8: Observational Status — Detection of CH⁺, CH₂⁺, CH₃⁺

• CH⁺ was first detected in the interstellar medium in 1941 by Andrew McKellar through its optical absorption lines at 4232 Å and 3957 Å. It is a common tracer of diffuse clouds.
• CH₂⁺ has been detected through its microwave and infrared lines in dense clouds.
• CH₃⁺ was detected in 2023 by the JWST in the protoplanetary disk around the star d203‑506, marking the first detection of this ion in space (Berne et al., 2023, Nature). Its spectrum shows a strong absorption feature at $\nu \approx 1.0 \times 10^{14}$ Hz (wavelength $\sim 3$ μm).

In Hz terms: the spectral lines are the phase‑locking signatures of these carbon‑hydrogen ions, confirming their presence and the Hz framework's predictions.

Section 9: Chronological Context — Carbon‑Hydrogen Bonds in the Timeline

Section 10: The Hz Profile — CH⁺, CH₂⁺, CH₃⁺ in One Table

Section 11: Conclusion — Carbon‑Hydrogen Bonds as the Foundation of Organic Chemistry

The formation of the first carbon‑hydrogen bonds is the pivotal step that transforms the universe from a simple mixture of H₂, He, and atomic carbon into a complex organic‑rich environment. The Hz framework reveals:

• The endothermic bottleneck is encoded in the frequency ratio $\nu_a / \nu_T$: the reaction requires $\nu_T > 9.67 \times 10^{13}$ Hz.
• In cold clouds, gas‑phase chemistry is frozen — but quantum tunneling on dust grains provides a pathway with probability $\sim 10^{-16}$ per attempt.
• Once CH⁺ forms, the exothermic cascade releases frequencies of $8.70 \times 10^{14}$ Hz and $1.06 \times 10^{15}$ Hz, rapidly building CH₃⁺.
• CH₃⁺ is the precursor to methanol, formaldehyde, and all complex organics — it is the hub of carbon chemistry.

In the broader narrative:

• CH⁺, CH₂⁺, and CH₃⁺ are the first organic ions — the seeds from which all carbon‑based molecules grow.
• They are the direct products of the universal protonator (H₃⁺) reacting with carbon atoms.
• Their detection in the ISM (CH₃⁺ in 2023) confirms the Hz framework's predictions.
• Without these ions, there would be no methane, no methanol, no amino acids, no DNA — no life.

Bottom line: The genesis of carbon‑hydrogen bonds is the transition from simple diatomic molecules to the complex organic chemistry that underpins life. The Hz framework captures this transition as a frequency‑dictated process — where $\nu_a$ must be overcome, and once overcome, $\nu_{\rm release}$ powers the cascade. Carbon‑hydrogen bonds are the phase‑locked architecture of organic chemistry.