Ionisation

A strong-field ionisation rate (ADK, PPT, molecular ADK) is evaluated from a few properties of the atom or molecule. optmat supplies those properties, curated and with their sources (TC-439, TC-444). For the noble gases it also gives the rates themselves, validated against calculations and measurements: Ionisation rates (TC-652). For the molecules it gives a rate set by all-electron calculations, and the forms that propagation codes use: Ionisation rates of molecules (TC-673, TC-677). Everything here comes from optmat.gas.ionisation.ionisation_parameters():

from optmat.gas.ionisation import ionisation_parameters

argon = ionisation_parameters("Ar")
argon.ionization_energy  # J, NIST ASD
argon.n_star, argon.l_star  # the effective quantum numbers of ADK and PPT
argon.hartree_coefficient_squared  # |C_n*l*|^2, Ammosov et al. eq. (19)
argon.structure["tong_2002"].coefficients  # C_l of the orbital, Tong et al. 2002
argon.polarizability, argon.ion_polarizability  # C m^2/V, for the Stark shift
argon.barrier_suppression_alpha  # Tong & Lin 2005
argon.barrier_suppression_zhang  # Zhang, Lan & Lu 2014: a1, a2, a3
argon.reasons  # why each value was chosen

What is provided

• The first ionisation energy. The atoms’ come from the NIST Atomic Spectra Database (TC-443). The molecules’ come from their primary determinations (TC-542):

  • H2 and D2: Liu et al. (2009, 2010), N = 0 → N⁺ = 0;

  • N2: Trickl et al. 1989;

  • O2: Tonkyn et al. 1989;

  • CH4: Berkowitz et al.’s 1987 photoion onset;

  • SF6: Frost et al.’s 1967 first threshold, whose kind the paper does not state;

  • N2O: Berkowitz & Eland 1977;

  • CO2: Wang et al. 1988.

  The vertical energies of N2O and CO2 are Turner et al.’s 1970 He I band maxima as Potts & Williams 1974 reprint them (the book itself is not in the library), and CH4’s is Heilbronner et al.’s 1977 first Jahn-Teller maximum (TC-547). The WebBook (nist_webbook_2026) is no longer the source of any recommended value. Its eV values for H2, D2 and O2 are the cm⁻¹ values converted with the 1973 CODATA factor, 8 ppm high (TC-505).

  • The recommended energy is the adiabatic one.

  • Where the vertical energy differs it is given too (O2, CH4, SF6, N2O, CO2). A tunnelling rate at fixed nuclei sees the vertical energy, and the difference, 1 eV for CH4, enters the rate’s exponent.

• The ionising orbital: its name, an atom’s valence l, and the magnetic quantum number m along a molecule’s axis. For an atom’s p shell, m = 0 is the member that ionises fastest in a linearly polarised field (Ammosov, Delone & Krainov 1986, eq. 14).

• The structure coefficients C_l of the orbital’s asymptotic tail, in the normalisation of Tong, Zhao & Lin (2002), eqs. (1)-(2): Ψ = Σ_l C_l F_l(r) Y_lm, with F_l → r^(Z/κ − 1) e^(−κr) and κ = (2 I_p)^(1/2) in atomic units.

  • Every curated tabulation is available by name, with a default. The molecules default to Zhao et al. 2010 and the atoms to Tong et al. 2002.

  • CH4 and SF6 have Zhao et al.’s 2011 coefficients, combined across degenerate orbitals; N2O has Zhao et al.’s 2017 table (TC-681, TC-684, TC-675). These are available by name, but default_structure is unset because their independent validation is limited. The molecular rate models state which set they use.

• The derived quantities of ADK and PPT: n* = Z/κ (Ammosov et al., eq. 2) and l* = n* − 1 (their eq. 17), and the Hartree coefficient |C_nl|^2 of their eq. (19). adk_coefficient_squared() is the Stirling form, their eq. (20), which is misprinted in the paper and corrected here (TC-441).

• The static polarizabilities:

  • The neutral’s: the electronic static limit of the gas’s fitted dispersion model (TC-441), with its uncertainty. The nuclei do not follow an optical field, so the infrared (vibrational) part is left out.

  • The cation’s: Wang et al. 2021’s CCSD(T) values for He+, Ne+ and Ar+. The others are requested.

  The difference α₀ − α₊ sets the Stark shift (α₀ − α₊)F²/2 of the ionisation energy in a static field F.

• The model constants. These are fitted inside one rate formula, not properties of the gas, and each is labelled as such:

  • Tong & Lin’s (2005) barrier-suppression α, fitted for He, Ne and Ar.

  • Zhang, Lan & Lu’s (2014) a1, a2, a3 for He to Xe. The rate is W_ADK exp(a1 x² + a2 x + a3), with x = F/F_b. Their eq. (8) prints the exponent’s sign as minus, but their own Table IX needs plus (TC-509, TC-542).

  • Talebpour, Yang & Chin’s (1999) effective charges for N2 (0.9) and O2 (0.53). These were fitted within PPT with l = m = 0. O2’s was fitted with I_p = 12.55 eV (O2⁺ v⁺ = 1), not with the adiabatic 12.07 eV (TC-508).

The parameters of every gas:

Gas	I_p (eV)	Source	Vertical (eV)	Orbital	n*	Structure coefficients	alpha (au)	alpha+ (au)	alpha_BSI	Zhang a1, a2, a3	Z_eff (PPT, l = 0)
He	24.5874	nist_asd_2026/ie_He_I		1s (m = 0)	0.7439	tong_2002: C0 = 3.13	1.384	0.2811	7	0.1355, -0.8621, 0.021562
Ne	21.5645	nist_asd_2026/ie_Ne_I		2p (m = 0)	0.7943	tong_2002: C1 = 2.1	2.661	1.311	9	0.10061, -1.04832, -0.07542
Ar	15.7596	nist_asd_2026/ie_Ar_I		3p (m = 0)	0.9292	tong_2002: C1 = 2.44	11.08	6.834	9	0.16178, -1.50441, 0.32127
Kr	13.9996	nist_asd_2026/ie_Kr_I		4p (m = 0)	0.9858	tong_2002: C1 = 2.49	16.76			0.1464, -1.36533, 0.02055
Xe	12.1298	nist_asd_2026/ie_Xe_I		5p (m = 0)	1.0591	tong_2002: C1 = 2.57	27.08			0.2108, -1.88482, 0.574281
H2	15.4258	liu_2009/ie_para_H2		HOMO 1 sigma_g (m = 0)	0.9392	zhao_2010: C0 = 1.78, C2 = 0.11, C4 = 0	5.433
D2	15.4665	liu_2010/ie_ortho_D2		HOMO 1 sigma_g (m = 0)	0.9379	zhao_2010: C0 = 1.78, C2 = 0.11, C4 = 0	5.364				0.72
N2	15.5807	trickl_1989/ie_N2		HOMO 3 sigma_g (m = 0)	0.9345	zhao_2010: C0 = 2.68, C2 = 1.1, C4 = 0.06	11.74				0.9
O2	12.0696	tonkyn_1989/ie_O2	12.3070	HOMO 1 pi_g (m = 1)	1.0617	zhao_2010: C2 = 0.52, C4 = 0.03	10.59				0.53
CH4	12.6100	berkowitz_1987/adiabatic_CH4	13.6000	HOMO 1 t2 (m = 0)	1.0387		17.26
SF6	15.3500	frost_1967/first_threshold_SF6	15.6900	HOMO 1 t1g (m = 0)	0.9415		30.06
N2O	12.8860	berkowitz_1977/adiabatic_N2O	12.8900	HOMO 2 pi (m = 1)	1.0275		19.7
CO2	13.7780	wang_1988/adiabatic_CO2#X	13.7900	HOMO 1 pi_g (m = 1)	0.9937	zhao_2010: C2 = 1.97, C4 = 0.4, C6 = 0.04	17.53

How well the parameters give the rates

The parameters are validated against their own primary sources (spectroscopy for the ionisation energies, quantum chemistry for the coefficients). A rate tests the parameters and the formula together, so the comparison below is reported, never fitted to. It is of the uncorrected tunnelling rate, to leading order in the field; the rates of Ionisation rates carry the correction at finite field, and are compared with the same references there.

The references are the ab initio static-field rates curated so far (tier 1 of TC-439):

• Parker et al.’s full two-electron TDSE for He;

• Majety & Scrinzi’s haCC for He, H2 and N2 (tRecX);

• Saenz’s frozen-nuclei H2;

• Zhang, Lan & Lu’s single-active-electron rates for Ar, in the GSZ model potential. This is a one-electron calculation, so it tests the formula, not the multi-electron physics.

The table gives the static rate of Tong et al.’s eq. (7) for one electron in the orbital, aligned with the field, divided by the reference, at the fields (au) each reference gives:

Gas	Reference	F = 0.03	F = 0.045	F = 0.05	F = 0.053	F = 0.06	F = 0.08	F = 0.094	F = 0.1	F = 0.12	F = 0.14
He	Parker et al. 2009, full TDSE		1.03	1.01		0.87	0.82		0.89	0.96	1.05
He	Majety & Scrinzi 2015, haCC(5)						0.83		0.90	0.98	1.07
H2	Saenz 2002, frozen nuclei, parallel				1.67	1.42	1.51		1.87	2.22
H2	Majety & Scrinzi 2015, parallel					2.22	2.06		2.21	2.49
N2	Majety & Scrinzi 2015, parallel	1.17		1.95		2.66	4.99		8.57	12.99	17.63
Ar	Zhang et al. 2014, SAE (GSZ potential)				2.18			3.75

• Helium: the one-electron rate with Tong et al.’s C_0 = 3.13 matches the exact two-electron rate to 1-3 % at the lowest fields. Between 0.06 and 0.12 au it is 4-18 % low, and it rises above the exact rate as the field approaches barrier suppression.

  • Tong & Lin’s barrier-suppression factor lowers it everywhere, to 0.63-0.89 of the exact rate.

  • So an additional occupancy factor of 2 doubles an already correct low-field rate. With the correction at finite field, which the rate needs at every field (TC-655), the exact rate is 1.3 to 1.6 times the one-electron rate between 0.06 and 0.25 au: neither 1 nor 2 (TC-658).

• H2: the one-electron rate is 1.4-2.2 times the frozen-nuclei rate and 2.1-2.5 times the fixed-nuclei one.

• N2: the one-electron rate is 1.2 times the reference at 0.03 au, and the excess grows steeply towards N2’s barrier-suppression field, κ⁴/16 = 0.08 au.

• Ar: 2.2 times the SAE rate at 0.053 au (0.64 of the barrier-suppression field) and 3.8 times at 0.094 au. With the same C_l = 2.44, Zhang et al.’s fitted correction reproduces their Table IX to within 5 % over 0.64-3.8 of that field (TC-509).

No ab initio multi-electron or calibrated measured rates are curated yet for Ne, Ar, Kr, Xe, O2, CO2, CH4, SF6 and N2O. Majety & Scrinzi’s Ne and Ar are figures only. The measured yields now in the library are figures only (TC-508, TC-510 to TC-512):

• Larochelle et al. 1998, Guo et al. 1998 and Talebpour et al. 1999 calibrate their intensities against a model rate (ADK or PPT), so they are not independent rate references.

• Walker et al. 1994 calibrate theirs on xenon photoelectron spectra, to about 30 %.

Where the literature leaves a gap, TDSE reference calculations (tRecX) will fill it, as validators ranked below published ab initio results.

Source interpretations

The parameters must be read with their definitions. Berkowitz et al.’s rotational model puts methane’s rotationless threshold at about 12.615 eV; SF6 has two photoelectron onsets. Wang et al.’s Hartree–Fock polarizabilities are 3–11 % below the correlated values. These distinctions matter when comparing parameter sets.

Talebpour, Yang and Chin fitted O2’s effective charge of 0.53 with \(I_p = 12.55\) eV. Pairing it with 12.07 eV gives a faster rate than the fitted one; their \(l = 0\) is a model choice, not O2’s \(\pi_g\), \(m = 1\) orbital. See Ionisation rates of molecules for the resulting rate models. Zhang et al.’s coefficients are read from Table XI with the plus sign in eq. (8), which reproduces their own Table IX (TC-542).

Open questions

• Which energy for tunnelling: adiabatic or vertical. The two differ by 1 eV for CH4 and 0.34 eV for SF6 (TC-444).

• Occupancy: the factor for doubly occupied or degenerate orbitals that belongs with each set of coefficients. The noble-gas defaults use an effective electron count with gas-specific evidence and limitations (TC-658, TC-848, TC-850; Noble-gas ionisation: static references). The molecular coefficients and occupancy must be considered together (Ionisation rates of molecules).

• Missing parameters: polarizabilities of Kr+, Xe+ and the molecular cations. Structure coefficients are curated for CH4, SF6 and N2O, but need independent validation. The vertical energies of N2O and CO2 rest on a reprint of Turner et al. 1970.

• Rate references: the current atomic references include static and laser-field all-electron calculations (Noble-gas ionisation: static references, Noble-gas ionisation: all-electron laser calculations). Molecular gaps and the available H2 and N2 checks are described in Molecular ionisation: validation and limitations.