Rovibrational levels

Radial Hamiltonian

The X and B states both have Ω = 0, so J is the rotational quantum number of the nuclear motion. For each state the level energies E_vJ and wavefunctions ψ_vJ(R) are solutions of

\[ \left[-\frac{\hbar^2}{2\mu}\frac{d^2}{dR^2} + V(R) + V_\text{ad}(R) + \frac{\hbar^2\,[1+\alpha(R)]}{2\mu R^2}\,J(J+1)\right]\psi_{vJ}(R) = E_{vJ}\,\psi_{vJ}(R), \tag{1} \]

where R is the internuclear distance, μ the reduced nuclear mass and V(R) the Born–Oppenheimer potential. The functions V_ad(R) and α(R) describe the adiabatic and nonadiabatic corrections to the Born–Oppenheimer approximation (Salumbides et al. 2008):

\[ V_\text{ad}(R) = \left(1-\frac{\mu_\text{ref}}{\mu}\right)\left(\frac{2R_m}{R+R_m}\right)^{5}\sum_i v_i\,\xi^i, \qquad \alpha(R) = \frac{\mu_\text{ref}}{\mu}\,\frac{2R_m}{R+R_m}\sum_i \alpha_i\,\xi^i , \tag{2} \]

with μ_ref the reduced mass of ¹²⁷I₂ and ξ defined in Eq. (3). V_ad vanishes for ¹²⁷I₂ and shifts the levels of the other isotopologues; α(R) modifies the effective rotational constant. Both corrections are used for the B state only. The constant term v₀ of V_ad was refitted in this work to the BIPM tables of ¹²⁹I₂ and ¹²⁷I¹²⁹I, which reduced their rms deviation from 7.1 MHz to 2.4 MHz.

Potential-energy curves

Two analytical forms are used. Each is used only for the levels where it has been tested against measurements.

Power-series potentials (Salumbides et al. 2008)

Within the range R_I ≤ R ≤ R_O the potential is a power series in the reduced coordinate ξ:

\[ V(R) = \sum_{i} a_i\,\xi^i,\qquad \xi = \frac{R-R_m}{R+b\,R_m}. \tag{3} \]

For R < R_I it is continued as A_I exp[−B_I(R − R_I)], and for R > R_O as D_e − Σ_n C_n/Rⁿ − A_O exp[−B_O(R − R_O)]. The coefficients are those of Salumbides et al. (2008): 14 coefficients a_i for X and 32 for B. They were fitted to data for X v″ ≤ 17 and B v′ ≤ 43, and within this range they reproduce the precision measurements to a few MHz. They are referred to below as the 2008 potentials.

Origin of the functional form

Expansions of a diatomic potential in a reduced coordinate that stays finite as R → ∞ predate their use for iodine. Simons, Parr & Finlan (1973) expanded in (R − R_e)/R. Ogilvie (1981) introduced 2(R − R_e)/(R + R_e) and, more generally, the family W_mn = (m + n)(R − R_e)/(mR + nR_e) with integer m and n, which is ξ of Eq. (3) up to a constant factor, with b = n/m. Šurkus, Rakauskas & Bolotin (1984) generalized it to (R^p − R_e^p)/(R^p + bR_e^p); p = 1 is ξ with b a free real parameter. The model used here combines this series with an exponential inner wall and a dispersion-plus-exchange outer branch, joined smoothly at R_I and R_O. That construction is due to Tiemann and co-workers (for example Samuelis et al. 2000, for Na₂, where b is chosen to minimize the number of coefficients). For I₂ it was fitted by Knöckel, Bodermann & Tiemann (2004) and refitted, with the Born–Oppenheimer correction functions of Eq. (2), by Salumbides et al. (2008).

Morse/long-range potentials

Outside the range of the fit the 2008 potentials deviate strongly from measurements: by +7.8 cm⁻¹ at X v″ = 48, −21 cm⁻¹ at X v″ = 54 and about 2 cm⁻¹ at B v′ = 58. These levels have their outer turning points where the potential is determined by the exponential term that joins the power series to the long-range expansion, and this term is not constrained by the data. For these levels i2spec uses Morse/long-range (MLR) potentials (Le Roy & Henderson 2007) fitted in this work:

\[ V(R) = D_e\left[1-\frac{u(R)}{u(R_e)}\,e^{-\beta(R)\,y_p^\text{eq}(R)}\right]^2,\qquad u(R)=\sum_n\frac{C_n}{R^n},\qquad y_p^{r}(R)=\frac{R^p-r^p}{R^p+r^p}, \tag{4} \]

\[ \beta(R) = y_p^\text{ref}\,\beta_\infty + \left(1-y_p^\text{ref}\right)\sum_i \beta_i\,\left(y_q^\text{ref}\right)^i,\qquad \beta_\infty=\ln\frac{2D_e}{u(R_e)} . \tag{5} \]

For any values of the fitted coefficients β_i, Eq. (4) satisfies V(R_e) = 0 and V(R) → D_e − u(R) at large R. The dissociation energy D_e and the dispersion coefficients C_n therefore keep their measured values, and the long-range behaviour does not depend on the fit.

The X potential is fitted to the levels v″ ≤ 17 of the 2008 potential, to the levels v″ = 48, 53 and 54 derived from emission measurements, and to the levels v″ = 18–25 derived from the Orsay atlas. The B potential is fitted to all precision measurements and to 2 009 lines of the Orsay atlas Partie IV (Gerstenkorn & Luc 1983), which reaches v′ = 79, 4 cm⁻¹ below the dissociation limit. Its centrifugal term carries one fitted constant beyond 5 Å in addition to the published Born–Oppenheimer function, which was fitted only inside the well. The potential alone reproduces the atlas to 30–35 MHz (median) at v′ = 51–66 and 80–100 MHz at v′ = 67–79; level corrections from the atlas bring every measured level to its data.

Figure 1. Difference between the 2008 potentials (Salumbides et al.) and the MLR potentials for X (left) and B (right). The shaded interval is the range of turning points of the levels included in the 2008 fit. Within it the two potentials agree; outside it they differ by up to several hundred cm⁻¹ for X and several tens of cm⁻¹ for B.

Table 1 lists the origin of the levels used by the model.

levels	potential	tested against
X v″ = 0–17	Salumbides et al. (2008)	frequency-comb measurements, 0.01–1 MHz
X v″ = 18–25	MLR, with level corrections from the Orsay atlas	3 700 atlas lines, 15–50 MHz per level
X v″ = 26–47	MLR	levels of Martin et al. (1986), 0.1–0.3 GHz
X v″ = 48–54	MLR	emission lines, 7–19 MHz
B v′ = 0–43	Salumbides et al. (2008)	frequency-comb measurements
B v′ = 44–50	MLR, with level corrections	frequency-comb measurements, 1–15 MHz
B v′ = 51–79	MLR, with level corrections from the Orsay atlas Partie IV	2 009 atlas lines and 3 comb lines, 20–250 MHz per level
B v′ = 80–86	MLR	nothing measured

Table 1. Source of the level energies.

Numerical solution

Equation (1) is solved variationally in a basis of B-splines of order 10 with a knot spacing of about 0.01 Å. The 2008 potentials have a discontinuous second derivative at R_I and R_O; with a uniform grid this limits the accuracy near these points to about 0.3 MHz. The B-spline basis therefore has a knot of reduced continuity (C³) at each join, and the matrix elements are integrated separately on each interval by Gauss–Legendre quadrature. The generalised eigenvalue problem Hc = ESc is then converged to about 0.1 kHz.

The size of the radial box is chosen for each vibrational level as the smallest box that reproduces the energies of a larger box to better than 1 kHz. It ranges from 3.6 Å for X v″ ≤ 17 to 12 Å for the highest B levels, which lie within a few cm⁻¹ of the dissociation limit. The wavefunctions needed for the transition moments are computed with a sinc discrete-variable representation (Colbert & Miller 1992) on a common grid; their energies are not used for line positions.

Empirical level corrections

After the potentials are applied, the frequency-comb measurements still differ from the model by 1–6 MHz. The differences vary smoothly with J within a vibrational level and change from one level to the next. They are therefore described as a correction to each level,

\[ \delta E(\text{state}, v, J) = \sum_{k=0}^{K} c_k\,y^k,\qquad y = J(J+1)/10^4, \tag{6} \]

fitted to all precision measurements by robust regression with a ridge penalty. A correction changes all lines that share the level. It is applied to ¹²⁷I₂ only, and only within 15 units of J of the measured lines; outside that range the terms of first order in y are extrapolated and the higher-order terms are held at their values at the boundary.

With the corrections the frequency-comb data sets are reproduced to 0.02–0.3 MHz, and lines left out of the fit are predicted to 0.1–0.8 MHz.