Several new vibrational bands of the [12.5] Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} and the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transitions have been observed in high resolution absorption measurements recorded using Intracavity Laser Spectroscopy (ILS). These new bands have been rotationally analyzed and incorporated into a comprehensive PtS dataset that was fit to a mass-independent Dunham expression using PGOPHER. The comprehensive dataset included all reported field-free, gas phase spectroscopic data for PtS, including 32 Fourier transform microwave transitions (estimated accuracy: 1 kHz), 9 microwave/optical double resonance transitions (25 kHz), 51 millimeter and submillimeter transitions (25–50 kHz), 469 molecular beam-laser induced fluorescence transitions (0.003 cm^{−1}), and 4870 ILS transitions (0.005 cm^{−1}). The determined equilibrium constants have been used with the Rydberg-Klein-Rees method to produce potential energy curves for the four known electronic states of PtS. Isotopic shifts in electronic transition energy beyond expectations from the Born-Oppenheimer approximation were observed and treated as electronic field-shift effects due to the difference in the nuclear charge radius between Pt isotopes. The magnitude and sign of the determined field-shift parameters are rationalized through the analysis of the previously reported *ab initio* calculations.

## I. INTRODUCTION

In 1932, Dunham^{1} outlined a method to describe the rovibrational energy levels of the electronic state for a diatomic molecule using a single analytical expression. While the derived expression adequately characterizes the rovibrational structure of isolated electronic states in molecules containing heavy atoms, the simple expression and associated mass relationships fail to describe systems containing light atoms or strongly interacting electronic states. The mass-dependent deviations result from the underlying assumption that electronic and nuclear motions can be decoupled, i.e., the Born-Oppenheimer approximation is valid. In 1974, Ross *et al.*^{2} modified Dunham’s expression to include terms that account for these mass-dependent deviations from the Born-Oppenheimer approximation and used the modified expression to characterize the emission from rare isotope CO lasers, fitting the data from ^{12}C^{16}O, ^{13}C^{16}O, ^{12}C^{18}O, and ^{13}C^{18}O to a single mass-independent equation. These mass-dependent correction terms that account for deviations from the Born-Oppenheimer approximation were derived from theory and justified by Watson^{3} in 1980. In several papers published in the 1980s, Tiemann and others^{4–9} observed that Watson’s Born-Oppenheimer breakdown (BOB) parameters were anomalously large for Tl-halides and Pb-chalcogenides when compared to the analogous parameters for compounds containing lighter Group 13 and Group 14 atoms. They surmised that these anomalies could be explained analogously to the observed field shift effects that have been well reported in atomic spectroscopy,^{10} leading Schlembach and Tiemann^{6} to derive correctional terms from theory to account for variations in spectroscopic parameters that result from the difference in the nuclear size between isotopes. These terms scale with the effective electron density at the nucleus, *ρ*_{el}, and its first (*ρ*_{el}′|_{re}) and second (*ρ*_{el}″|_{re}) derivatives with respect to the internuclear distance for the electronic, rotational, and vibrational corrections, respectively. Interestingly, Tiemann^{8} later revised the original derivation^{6} of the molecular field shift for rotation to include an additional factor of π^{2}.

There have been several examples in the literature of these molecular field shifts affecting experimentally determined rotational constants, including the mass-independent Dunham analyses of WO,^{11} PtO,^{12} PtSi,^{13} PtS,^{14} PtCl,^{15} TlF,^{5} TlCl,^{5} TlBr,^{5} TlI,^{5} PbO,^{16} PbS,^{5} PbSe,^{5,17} and PbTe.^{5,17} In some cases, these analyses have been supported by computational investigations that have characterized the electronic distribution in the molecular systems. Cooke *et al.*^{18} outlined an approach to calculate *ρ*_{el} and its derivatives using density functional theory (DFT), and the calculated values were used with Tiemann’s revised expression^{8} to estimate the magnitude of the rotational field shift effect for the molecules listed above. In general, their predictions agreed well with the experimentally determined values.

More recently, Saue and others^{19,20} derived expressions for electronic, rotational, and vibrational field shift effects and suggested that Tiemann’s revision^{8} was not justified. They used coupled-cluster singles-and-doubles (CCSD) methods, as well as DFT with a wide variety of functionals, to calculate *ρ*_{el} and its derivatives, estimating rotational field shift effects with their derived expressions, and these agreed with Schlembach and Tiemann’s original derivation. In their manuscript on PbS,^{20} theoretical values for ρ_{el} were not provided, and thus, the level of agreement between *V*_{00} and *ρ*_{el} cannot be evaluated. Their CCSD results also agree reasonably well with the experimentally determined values, and they suggest that Cooke *et al.*^{18} inadvertently employed a nonrelativistic Hamiltonian and the relativistic corrections were coincidentally of the same magnitude as the π^{2} scaling factor incorporated in Tiemann’s revised expression.^{8}

In contrast to the molecular field shift effects of rotational energy, relatively few examples of experimentally determined vibrational or electronic field shifts have been reported. The analysis of the [18.9] *A* Ω = 0^{+}–*X*^{1}Σ^{+} transition of PbS observed with microwave/optical double resonance (MODR) by Knöckel *et al.*^{8} resulted in the only determination of a vibrational field shift effect present in the literature to the best knowledge of the authors. Electronic field shifts have only been reported for PbS,^{8,9} YbCl,^{21–23} YbBr,^{23} PtF,^{24} and PtCl.^{15} The recent observations of electronic field shifts in PtF and PtCl by our group, and the wealth of spectroscopic data in the literature for PtS provide a strong incentive to investigate the known electronic states of PtS using a mass-independent Dunham-type Hamiltonian to determine if its electronic structure also exhibits molecular field shift effects.

The first reported spectroscopic transitions of gas-phase PtS were recorded using Molecular Beam-Laser Induced Fluorescence (MB-LIF) and pump/probe MODR in 1995 by Li *et al.*^{25} Medium-resolution (∼1 cm^{−1}) LIF scans spanned the 13 000–18 600 cm^{−1} region, and high-resolution (∼0.001 cm^{−1}) scans were collected for the 15 850–15 950 cm^{−1}, 17 950–18 050 cm^{−1}, and 18 360–18 460 cm^{−1} regions. Four bands were observed in the high resolution scans and were assigned as the (0, 0) band of the [15.9] *A* Ω ≠ 0–*X*^{3}Σ^{−}_{Ω=0+} transition, the (0, 0) band of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition, and the (1, 0) and (0, 0) bands of the [18.0] *C* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition. Two additional bands were identified in the medium-resolution scans at 16 366 and 16 805 cm^{−1} and were assigned as the (1, 0) and (2, 0) bands of the *B*-*X* transition, but were not measured at high-resolution for rotational analysis. The (0, 0) band of the *B*-*X* transition and the (1, 0) and (0, 0) bands of the *C*-*X* transition were rotationally analyzed, but a secure rotational assignment was not reported for the *A*-*X* transition. The R(5), R(6), and R(8) lines of the (0, 0) band of the *B*-*X* transition were used to measure the pure rotational R(4), R(5), and R(7) transitions of the 3 most abundant PtS isotopologues in their ground states using MODR in the 44–71 GHz range. Molecular constants for the *X* state, *B* state, and *C* state were determined in their analysis and reported for ^{194}PtS, ^{195}PtS, and ^{196}PtS. In the same year, the same group also measured optical Stark spectra of PtO, PtS, and PtC and reported a dipole moment of 1.78(2) D for PtS in its *X*^{3}Σ^{−}_{Ω=0+} state.^{26}

In 2004, Cooke and Gerry used Fourier Transform Microwave (FTMW) spectroscopy to extend the known pure rotational spectrum of PtS to cover the region below 20 GHz. Transitions for 8 PtS isotopologues (^{192}Pt^{32}S: 0.74% natural abundance; ^{194}Pt^{32}S: 31.30%; ^{195}Pt^{32}S: 32.12%; ^{196}Pt^{32}S: 23.96%; ^{198}Pt^{32}S: 6.80%; ^{194}Pt^{34}S: 1.41%; ^{195}Pt^{34}S: 1.45%; and ^{196}Pt^{34}S: 1.08%) and from 3 vibrational levels (v = 0–2) were observed with an estimated measurement accuracy of ±1 kHz. The 32 FTMW transitions and 9 MODR transitions were incorporated into a Dunham type fit and mass-independent molecular constants were obtained. A significant BOB in the rotational constant was observed due to both Pt and S isotopic substitutions. The breakdown due to S substitution was treated as mass-dependent BOB, but the primary contribution to breakdown upon Pt substitution was assumed to arise from field shift effects.

In 2009, Liang *et al.*^{27} estimated a PtS vibrational frequency of 500 cm^{−1} based on infrared spectra collected for a cryogenic Ar matrix containing products from the reaction of S_{2} with laser ablated Pt. In 2010, Handler *et al.*^{28} reported a rotational analysis of the [12.5] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition, observed in the near-infrared using Intracavity Laser Spectroscopy (ILS). The lack of observed isotopologue separation led to the assignment of the transition as a (0, 0) vibrational band. The most recent spectroscopic investigation of PtS was a millimeter- and submillimeter-wave absorption study reported by Okabayashi *et al.*^{29} in 2012, expanding the known pure rotational spectrum of PtS into the 150–406 GHz range. Transitions of ^{194}Pt^{32}S, ^{195}Pt^{32}S, ^{196}Pt^{32}S, and ^{198}Pt^{32}S were observed, and the four isotopologues were fit independently, with transitions from the FTMW and MODR spectra for each isotopologue included in the fit.

In this study, additional vibrational bands of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} and [12.5] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transitions of PtS have been recorded in absorption using ILS. These vibrational bands have been rotationally analyzed and fit to a mass-independent Dunham-type Hamiltonian using PGOPHER.^{30} All of the field-free, gas phase spectroscopic data for PtS^{14,25,28,29} were included in the fit, encompassing 8 isotopologues and 4 electronic states. Electronic field-shift effects resulting from the difference in the nuclear charge radius between Pt isotopes were observed for the *B*-*X*, *C*-*X*, and *Y*-*X* transitions. The field-shift parameters are used to determine the change in effective electron density upon electronic excitation at the Pt nucleus, Δ*ρ*_{el}, and the results are compared with those for PbS,^{8,9} PtF,^{24} and PtCl.^{15} Finally, the determined Dunham parameters are used to construct potential energy curves for the *X*, *Y*, *B*, and *C* states of ^{195}Pt^{32}S using the Rydberg-Klein-Rees (RKR) method.

## II. EXPERIMENTAL METHODS

Intracavity laser absorption spectra were collected using the dye^{31} (visible) and Ti:Sapphire^{32,33} (near-infrared) ILS systems at the University of Missouri-St. Louis that have been described in detail elsewhere, and only a brief description is provided here. The PtS molecules were produced in the plasma discharge of a Pt-lined Cu hollow cathode. The cathode was located within the resonator chamber of a Coherent Verdi^{TM} V-10 pumped tunable laser system: a dye laser using DCM (14 500 cm^{−1}–16 400 cm^{−1}) or R6G (16 000–17 200 cm^{−1}), or a Ti:Sapphire laser (11 850–11 950 cm^{−1}; 12 400–12 525 cm^{−1}; and 12 850–13 000 cm^{−1}). In this configuration, molecular absorption occurring during initial laser evolution is amplified by the laser action and very long effective path lengths can be obtained.

The vibrational bands of the [12.5] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition were observed in 2011 when collecting spectra of PtF,^{32} using trace SF_{6} in Ar. The (1, 0) and (0, 1) bands of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition of PtS were initially observed when collecting spectra of PtF,^{24} using a trace amount of SF_{6} as the reagent gas in an Ar/He sputter gas mixture. Spectra for the (0, 0), (0, 2), (1, 3), (2, 0), (3, 1), and (4, 1) bands of the *B*-*X* transition were collected using a trace amount of CS_{2} as the reagent gas to limit spectral contamination due to the presence of PtF. Conditions were optimized for each observed band within the following ranges: ILS effective path length of 0.5–2 km, RF discharge current of 0.30–0.80 A, reagent gas composition of 0.25%–5.0%, and total pressure of 0.30–1.5 Torr.

Plasma spectra were calibrated by comparing the spectra from an extracavity I_{2} cell (collected in conjunction with each plasma spectrum) to the reference data from Salami and Ross^{34} for the visible spectra and to the atlas of Gerstenkorn *et al.*^{35} for near-infrared spectra. Measurement accuracy for the ILS data is estimated to be ±0.005 cm^{−1}. Low signal-to-noise and a limited number of calibration lines limited the measurement accuracy for the (1, 0) and (2, 1) bands of the [12.6] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition and the (0, 2) and (1, 3) bands of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition to ±0.0075 cm^{−1}. After calibration, individual spectra (∼7 cm^{−1} wide) were concatenated to produce a single spectrum (∼100 cm^{−1} wide) using an in-house^{31} Microsoft Excel workbook programmed with Visual Basic for applications. Portions of the concatenated ILS spectra for the (1, 0) band of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition and (0, 1) band of the [12.5] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition of PtS are provided in Figs. 1(a) and 1(b), respectively.

## III. RESULTS

### A. The comprehensive PtS data set

Several techniques have been used to study PtS, including FTMW,^{14} MODR,^{25} millimeter- and submillimeter-wave absorption,^{29} MB-LIF,^{25} and ILS^{28} spectroscopies. As such, these data must be appropriately weighted in the comprehensive mass-independent Dunham analysis to ensure fidelity to the estimated measurement accuracy of the original analyses when performing the simultaneous fit. The 32 FTMW transitions were assigned an uncertainty of ±1 kHz, the MODR and millimeter-/submillimeter-wavetransitions for ^{194}Pt^{32}S and ^{196}Pt^{32}S were assigned an uncertainty of ±10 kHz, and the MB-LIF transitions were assigned an uncertainty of ±0.003 cm^{−1}. The MODR and millimeter-/submillimeter-wave transitions were deweighted to ±50 kHz because hyperfine splitting due to the ^{195}Pt nucleus (I = 1/2) was not resolved for these measurements. The data recorded with ILS included the (4, 1), (3, 1), (2, 0), (1, 0), (0, 0), (0, 1), (1, 2), (0, 2), and (1, 3) vibrational bands of the [15.9] *B* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition and the (2, 1), (1, 0), (0, 0), (0, 1), and (1, 2) vibrational bands of the [12.6] *Y* Ω = 0^{+}–*X*^{3}Σ^{−}_{Ω=0+} transition. Isotopologue shifts were only partially resolved for the Δv = 0 transitions, and line positions for these bands are assigned an uncertainty of ±0.01 cm^{−1} in the fit. The (1, 0) and (2, 1) bands of the *Y*-*X* transition and the (0, 2) and (1, 3) bands of the *B*-*X* transition are assigned an uncertainty of ±0.0075 cm^{−1} in the fit due to low signal-to-noise and a limited number of reference line positions for the I_{2} spectra collected in the spectral region where these bands occur. The remaining ILS transitions were assigned an uncertainty of ±0.005 cm^{−1}. With the inclusion of the original line positions reported by Handler *et al.*,^{28} 4870 ILS observations were added to the comprehensive dataset, including transitions from ^{194}Pt^{32}S, ^{195}Pt^{32}S, ^{195}Pt^{32}S, and ^{198}Pt^{32}S. The resulting spectroscopic dataset contained 5432 observations involving 8 PtS isotopologues, 4 electronic states, and 14 vibrational levels of these states.

### B. The mass-independent Dunham Hamiltonian

The comprehensive PtS dataset was fit to a mass-independent Dunham-type Hamiltonian in PGOPHER^{30} using the constrained-variable method outlined by Breier and co-authors.^{36,37} For this method, the individual rovibrational parameters for each electronic state, vibrational band, and isotopologue (which are entirely uncorrelated using PGOPHER’s^{30} default settings) are constrained to the relationships initially outlined by Dunham^{1} and expanded upon by Ross *et al.*,^{2} Watson,^{3} and Tiemann and others.^{4–9} The form of the mass-independent Dunham expression used in this study followed the linear equation outlined by Le Roy,^{38} which has several advantages over the expressions of Watson^{3} and Tiemann^{6} (as is explained in Le Roy’s publication). The known electronic states of PtS are well described as Hund’s case (c) Ω = 0^{+} states. Le Roy’s equation^{38} can be separated into several components that effectively characterize these simple electronic states,

where *H*_{Ω=0} is the mass-independent Dunham-type Hamiltonian for a given electronic state, *H*_{elec} is the electronic component of the Hamiltonian, *H*_{rovib} is the rovibrational energy expression outlined by Dunham,^{1}^{,}*H*_{BOB} contains the mass-dependent deviations from the Born-Oppenheimer approximation first derived by Watson,^{3}^{,}*H*_{fs} contains the nuclear-radius dependent field-shift deviations from the Dunham^{1} model derived by Tiemann and co-authors,^{4–9} and *H*_{HF} addresses the hyperfine interactions with the ^{195}Pt nucleus.

where A and B refer to the two atoms of the diatomic molecule; 1 and α refer to the reference and relative isotopologues, respectively; $Y001$ is the independent variable for the electronic energy of the reference isotopologue for a given electronic state; *l* and *m* refer to the *l*th-vibrational and *m*th-rotational components in the expansion; $Ylm1$ are the independent rovibrational Dunham parameters for the reference isotopologue 1; *μ* is the reduced mass for the reference (1) and relative (α) isotopologues; v is the vibrational quantum number; *J* is the rotational quantum number; δ⟨r^{2}⟩_{A}^{α1} is the change in mean squared nuclear charge radii^{10,40} between reference (1) and relative (α) isotopes of atom A; *M*_{A}^{α} is the mass of isotopologue α; *ΔM*_{A}^{α1} is the difference in mass between the relative (α) and reference (1) isotopologues; *δ*_{lm}^{A} are the independent variables that treat mass-dependent Born-Oppenheimer breakdown (BOB), which are closely related to the Δ_{lm} parameters derived by Watson;^{3}^{,}*f*_{lm}^{A} are the independent variables that treat deviations from the Dunham^{1} model due to the variation in nuclear size between isotopes, closely related to the field-shift parameters (*V*_{lm}^{A}) introduced by Tiemann and co-authors;^{4–9}^{,}*C*_{I}^{1} is the nuclear spin-rotation coupling constant^{39} for the reference isotopologue. The linear BOB and field-shift coefficients *δ*_{l,m}^{A,B} and *f*_{l,m}^{A,B} of Le Roy^{38} are related to the more commonly used nonlinear coefficients Δ_{l,m}^{A,B} of Watson^{3} and *V*_{l,m}^{A,B} of Schlembach and Tiemann^{6} by the expressions

where *m*_{e} is the mass of an electron, and all other terms are defined above. It should be noted that Schelmbach and Tiemann’s^{6} *V*_{00}^{A,B} parameter is an exception to Eq. (8) and has the same form in the Hamiltonian as the *f*_{00}^{A,B} parameter of Le Roy^{38} defined by Eq. (5).

In the constrained-variable method,^{36,37} the band-by-band parameters for each isotopologue are constrained to the relationships outlined in Eqs. (2)–(6). Each vibrational level for each state of each isotopologue was defined in PGOPHER^{30} as a ^{1}Σ^{+} state [as is appropriate for Hund’s case (c) Ω = 0^{+} state], and the band-by-band parameters for the states were the *Origin*, *B*, *D*, and *H* parameters. The isotopologues of ^{195}Pt had an additional C_{I} parameter to treat the hyperfine splitting observed in the FTMW^{14} measurements. As an example of the constraints used, the *B* value for v = 0 in the *X* state of ^{194}Pt^{34}S (A = Pt, B = S, 1 = ^{195}Pt^{32}S, and α = ^{194}Pt^{34}S) was constrained to the expression

### C. Fitting the comprehensive PtS data set

The *X*^{3}Σ^{−}_{Ω=0+}, [12.5] *Y* Ω = 0^{+}, [15.9] *B* Ω = 0^{+}, and [18.0] *C* Ω = 0^{+} states of PtS were each fit to a unique set of parameters in the mass-independent Dunham fit that reflect the spectroscopic observations that characterize these states. The *X*^{3}Σ^{−}_{Ω=0+} ground state is the best characterized; it is involved in all 5432 observed transitions, which encompasses v = 0–3, J values as low as 0 and as high as 121, 8 isotopologues, resolution of hyperfine splitting due to the ^{195}Pt nucleus, and 32 measurements^{14} reliable to 1 kHz. The high precision of the FTMW^{14} pure rotational transitions required the incorporation of both mass-dependent and field-shift deviations from the first order rotational Dunham^{1} constant Y_{01} in order to accurately fit the experimental data. When both the *f*_{01}^{Pt} and *δ*_{01}^{Pt} parameters were allowed to float in the fit, they were highly correlated and, as a result, were very poorly determined. To combat this issue, it was assumed that the magnitude of the mass-dependent BOB would be very similar for Pt and S, and, correspondingly, the Δ_{l,m} parameters for both atoms should be nearly equal (as was postulated by Watson^{3} and reiterated by Cooke and Gerry^{14}). To implement this assumption, Δ_{01}^{Pt} was set to be equal to Δ_{01}^{S}, and solving this equality in terms of Eq. (7) yielded the following constraint:

In this way, mass-dependent BOB for Pt can be treated approximately, and the field-shift effect on molecular rotation can be estimated to a high degree of precision. This approach is very similar to that applied by Cooke, Gerry, and others^{11–14,18} in their series of mass-independent Dunham analyses, and identical to the approach used in our mass-independent analysis of PtCl.^{15}

Transitions involving the excited electronic states of PtS were observed with a measurement accuracy of ±0.003 cm^{−1} for the MB-LIF data^{25} and ±0.005 cm^{−1} for the ILS data. At this level of precision, deviations from the Dunham^{1} model for rotation and vibration are expected to be insignificant (absent strong interactions with nearby electronic states). However, the uncertainty in the determined electronic transition energy is less than 1 part in 10^{7}, and deviations from the Dunham^{1} model were observed in the ILS data. The isotopic shift in electronic energy from ^{194}Pt^{32}S to ^{196}Pt^{32}S was 0.0283 cm-^{1} for the *Y*-*X* transition and −0.0048 cm^{−1} for the *B*-*X* transition. Considering the case of PtS where only Pt-isotopic substitution is observed, the difference in term energy, *T*_{e}, between two electronic states is

where ′ and ″ refer to the excited and lower electronic states, respectively. The electronic energy of the ground state, $Y001$″, is zero by definition for the *X*^{3}Σ^{−}_{Ω=0+} state, but the ground and excited state BOB and field-shift corrections are indistinguishable. The effective difference in *T*_{e} is therefore given by

where $\Delta \delta 00Pt$ and $\Delta f00Pt$ are defined by

Similar to the case with the rotational corrections for the ground state, when both $\Delta \delta 00Pt$ and $\Delta f00Pt$ were allowed to float in the fitting process, the parameters were highly correlated and poorly determined. Without observations of isotopic substitution of sulfur to estimate the mass-dependent contribution to the isotope shift, the parameters were floated independently, and the fits were evaluated. These fits reveal that the data are somewhat better described when the isotope shift is treated as a field-shift effect (weighted fit error = 1.0278) rather than a mass-dependent BOB (weighted fit error = 1.0310). This is consistent with the observed atomic electronic isotope shifts,^{10,40} where field-shift effects dominate mass-shifts for atoms with Z ≥ 58 (Z_{Pt} = 78).

While the observation of several vibrational bands of the *Y*-*X* and *B*-*X* transitions allow the separation of the electronic isotopologue shift from the mass-dependent changes in vibrational energy, the observation only of the (1, 0) and (0, 0) bands of the *C*-*X* transition in the MB-LIF^{25} precludes the separation of this energetic contributions to the [18.0] *C* Ω = 0^{+} state. The first order mass-dependence of the vibrational frequency for the *C* state can be reasonably well determined from the experimental ΔG_{1/2}. The second order mass-dependence can be estimated using the Pekeris relationship,^{41}

where the conventional terms^{42} (*B*_{e}, *α*_{e}, *ω*_{e}, and *ω*_{e}*x*_{e}) have been replaced by their corresponding Dunham^{1} parameters (Y_{01}, −Y_{11}, Y_{10}, and −Y_{20}). The rotational parameters should be well determined from the MB-LIF data,^{25} and Y_{10} can be determined from the experimental ΔG_{1/2} with the relation

While the Pekeris relationship^{41} is rigidly correct only for a Morse potential and the Dunham^{1} parameters are only approximately equivalent to the conventional^{42} terms, the estimated value should adequately compensate for the second order mass-dependence of the vibrational energy levels. In the fit, Y_{20} of the [18.0] *C* Ω = 0^{+} state was constrained to Eq. (15) so that the isotope shift in electronic energy could be approximated.

In total, 38 floated and 2 constrained parameters were included in the effective Hamiltonian to fit the 5432 observations. The floated parameters for the *X*^{3}Σ^{−} state were Y_{00}, Y_{10}, Y_{20}, Y_{01}, Y_{11}, Y_{21}, Y_{02}, Y_{12}, Y_{03}, *f*_{01}^{Pt}, *δ*_{01}^{S}, and C_{I}^{1}. The floated parameters for the [12.5] *Y* Ω = 0^{+} state were Y_{00}, Δ*f*_{00}^{Pt}, Y_{10}, Y_{20}, Y_{01}, Y_{11}, Y_{02}, Y_{12}, and Y_{03}. The floated parameters for the [15.9] *B* Ω = 0^{+} state were Y_{00}, Δ*f*_{00}^{Pt}, Y_{10}, Y_{20}, Y_{30}, Y_{01}, Y_{11}, Y_{21}, Y_{02}, Y_{12}, Y_{22}, and Y_{03}. The floated parameters for the [18.0] *C* Ω = 0^{+} state were Y_{00}, Δ*f*_{00}^{Pt}, Y_{10}, Y_{01}, Y_{11}, and Y_{03}. The two constrained parameters were δ_{01}^{Pt} of the *X* state and Y_{20} of the *C* state, constrained to Eqs. (10) and (15), respectively. The determined parameters are reported in Table I with their associated uncertainties. A comparison between the band constants derived from this fit and the molecular constants for PtS previously determined in the literature is provided in Table II. The root mean squared (RMS) values for the individual datasets were 1.001 kHz for the FTMW data,^{14} 6.8 kHz for the resolved MODR data,^{25} 10.9 kHz for the resolved millimeter- and submillimeter-wave data,^{29} 0.0026 cm^{−1} for the MB-LIF data,^{25} and 0.0060 cm^{−1} for the ILS data, with an average weighted error from the fit of 1.0278. The rms values are provided by the isotopologue, band, and dataset in Table S1 of the supplementary material. The .pgo file and input file (which is a text file containing the constraints necessary for the Dunham fit and the experimental line list) used to perform this analysis are also provided in the supplementary material.

Parameter . | Ground state X^{3}Σ^{−}_{Ω=0}
. | Y state [12.5] Ω = 0^{+}
. | B state [15.9] Ω = 0^{+}
. | C state [18.0] Ω = 0^{+}
. |
---|---|---|---|---|

Y _{00}(T_{e}) | 0^{b} | 12 475.893 42(94) | 15 961.615 31(67) | 18 047.785 98(60) |

Δf_{00}(Pt)^{a} | −0.377 8(36) | 0.064 1(24) | 0.155 6(56) | |

Y _{10}(ω_{e}) | 517.932 05(52) | 481.179 3(12) | 443.270 9(11) | 441.145 73(47) |

Y _{20}(−ω_{e}x_{e}) | −1.810 51(15) | −1.955 90(41) | −1.843 78(48) | −2.132 2(39)^{c} |

Y_{30}(ω_{e}y_{e}) × 10^{3} | −7.064(64) | |||

Y _{01}(B_{e}) | 0.147 498 507 0(71) | 0.140 737 32(64) | 0.138 358 88(39) | 0.136 479 3(16) |

f(_{01}Pt) × 10^{3}^{a} | −0.001 098(18) | |||

δ(_{01}Pt) × 10^{3} | 0.025 987(20)^{d} | |||

δ(_{01}S) × 10^{3} | 0.158 47(12) | |||

Y _{11}(-α_{e})× 10^{3} | −0.635 047(13) | −0.653 13(41) | −0.667 64(40) | −0.749 88(92) |

Y _{21}(γ_{e})× 10^{6} | −0.596 6(48) | −1.310(93) | ||

Y _{02}(−D_{e})× 10^{6} | −0.048 035 9(42) | −0.047 80(16) | −0.059 604(66) | −0.0544(12) |

Y _{12}(β_{e})× 10^{6} | −0.000 171 9(62) | −0.000 529(56) | 0.001 663(49) | |

Y _{22}× 10^{9} | −0.316(12) | |||

Y _{03}(H_{e})× 10^{12} | −0.006 95(81) | −0.041(12) | 0.147 3(28) | |

Hyperfine Parameters^{e} | ||||

C_{I}(^{195}Pt^{32}S) × 10^{6} | −2.247(20) | |||

C_{I}(^{195}Pt^{34}S) × 10^{6} | −2.133(19)^{e} | |||

Vibrational levels | 0, 1, 2, 3 | 0, 1, 2 | 0, 1, 2, 3, 4 | 0, 1 |

Techniques | FTMW, MODR, mm | ILS | MB-LIF, ILS | MB-LIF |

and sub-mm, MB-LIF, ILS |

Parameter . | Ground state X^{3}Σ^{−}_{Ω=0}
. | Y state [12.5] Ω = 0^{+}
. | B state [15.9] Ω = 0^{+}
. | C state [18.0] Ω = 0^{+}
. |
---|---|---|---|---|

Y _{00}(T_{e}) | 0^{b} | 12 475.893 42(94) | 15 961.615 31(67) | 18 047.785 98(60) |

Δf_{00}(Pt)^{a} | −0.377 8(36) | 0.064 1(24) | 0.155 6(56) | |

Y _{10}(ω_{e}) | 517.932 05(52) | 481.179 3(12) | 443.270 9(11) | 441.145 73(47) |

Y _{20}(−ω_{e}x_{e}) | −1.810 51(15) | −1.955 90(41) | −1.843 78(48) | −2.132 2(39)^{c} |

Y_{30}(ω_{e}y_{e}) × 10^{3} | −7.064(64) | |||

Y _{01}(B_{e}) | 0.147 498 507 0(71) | 0.140 737 32(64) | 0.138 358 88(39) | 0.136 479 3(16) |

f(_{01}Pt) × 10^{3}^{a} | −0.001 098(18) | |||

δ(_{01}Pt) × 10^{3} | 0.025 987(20)^{d} | |||

δ(_{01}S) × 10^{3} | 0.158 47(12) | |||

Y _{11}(-α_{e})× 10^{3} | −0.635 047(13) | −0.653 13(41) | −0.667 64(40) | −0.749 88(92) |

Y _{21}(γ_{e})× 10^{6} | −0.596 6(48) | −1.310(93) | ||

Y _{02}(−D_{e})× 10^{6} | −0.048 035 9(42) | −0.047 80(16) | −0.059 604(66) | −0.0544(12) |

Y _{12}(β_{e})× 10^{6} | −0.000 171 9(62) | −0.000 529(56) | 0.001 663(49) | |

Y _{22}× 10^{9} | −0.316(12) | |||

Y _{03}(H_{e})× 10^{12} | −0.006 95(81) | −0.041(12) | 0.147 3(28) | |

Hyperfine Parameters^{e} | ||||

C_{I}(^{195}Pt^{32}S) × 10^{6} | −2.247(20) | |||

C_{I}(^{195}Pt^{34}S) × 10^{6} | −2.133(19)^{e} | |||

Vibrational levels | 0, 1, 2, 3 | 0, 1, 2 | 0, 1, 2, 3, 4 | 0, 1 |

Techniques | FTMW, MODR, mm | ILS | MB-LIF, ILS | MB-LIF |

and sub-mm, MB-LIF, ILS |

^{a}

Platinum-isotope based Born-Oppenheimer Breakdown parameter that scales with a mean nuclear charge radius, δ⟨r^{2}⟩.

^{b}

Held to zero in the fit, ∴ T_{0} = ΔG_{1/2}.

^{c}

Constrained to the Pekeris relationship in the fit [Eq. (15)]. The value in parenthesis corresponds to 1σ (in the units of the last digit) propagated from the uncertainty in the values determined by the fit for *Y*_{01}, *Y*_{11}, and *Y*_{10}.

^{d}

Constrained in the fit to Eq. (10). The uncertainty reported in parentheses reflects the propagated error from the floated δ_{01}(S) parameter.

^{e}

Hyperfine coupling to the ^{195}Pt nucleus (I = 1/2) was observed in the FTMW spectrum of Cooke and Gerry.^{14} C_{I} mass scaling follows (μ_{1}/μ_{α}), where 1 is the reference isotopologue $Pt32195S$ and α is the relative isotopologue $Pt34195S$. The error reported for C_{I} $Pt34195S$ is propagated from the uncertainty of the fitted parameter C_{I} $Pt32195S$.

. | X^{3}Σ^{−}_{Ω=0+} state
. | [12.5] Y Ω = 0^{+} state
. | [15.9] B Ω = 0^{+} state
. | [18.0] C Ω = 0^{+} state
. | ||||
---|---|---|---|---|---|---|---|---|

. | This work . | Literature . | This work . | Literature . | This work . | Literature . | This work . | Literature . |

^{194}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.487 6(12) | N/A | 15 924.259 66(91) | 15 924.253 1(5)^{b} | 18 009.292 8(12) | 18 009.290 4(5)^{b} |

B_{0} | 0.147 287 754 4(98) | 0.147 287 754 7(40)^{c} | 0.140 512 84(68) | N/A | 0.138 125 08(44) | 0.138 138(4)^{b} | 0.136 203 3(17) | 0.136 186(3)^{b} |

D_{0} × 10^{6} | 0.048 191 9(53) | 0.048 191 0(43)^{c} | 0.048 13(17) | N/A | 0.058 937(71) | 0.078(7)^{b} | 0.054 5(12) | 0.054(2)^{b} |

H_{0} × 10^{12} | −0.006 97(81) | −0.00 75(12)^{c} | −0.041(12) | N/A | 0.147 6(28) | ^{d} | ||

^{195}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.480 7(12) | 12 457.480 4(13)^{e} | 15 924.275 56(91) | 15 924.269 6(5)^{b} | 18 009.312 4(12) | 18 009.310 0(5)^{b} |

B_{0} | 0.147 180 834 1(98) | 0.147 180 833 7(63)^{c} | 0.140 410 76(68) | 0.140 411 639(88)^{e} | 0.138 024 74(44) | 0.138 060(4)^{b} | 0.136 104 4(17) | 0.136 104(3)^{b} |

D_{0} × 10^{6} | 0.048 121 8(52) | 0.048 117 0(60)^{c} | 0.048 06(16) | 0.048 32(12)^{e} | 0.058 852(71) | 0.088(7)^{b} | 0.054 4(12) | 0.055(2)^{b} |

H_{0} × 10^{12} | −0.006 95(81) | −0.008 4(16)^{c} | −0.041(12) | ^{d} | 0.1473(28) | ^{d} | ||

C_{I} × 10^{6} | −2.247(20) | −2.228(52)^{f} | ||||||

^{196}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.472 6(12) | N/A | 15 924.291 49(91) | 15 924.285 6(5)^{b} | 18 009.332 3(12) | 18 009.331 1(5)^{b} |

B_{0} | 0.147 075 208 2(98) | 0.147 075 193 3(53)^{c} | 0.140 309 91(68) | N/A | 0.137 925 61(44) | 0.137 962(4)^{b} | 0.136 006 6(17) | 0.136 004(3)^{b} |

D_{0} × 10^{6} | 0.048 052 6(52) | 0.048 036 9(57)^{c} | 0.047 99(16) | N/A | 0.058 768(71) | 0.093(7)^{e} | 0.0543(12) | 0.052(2)^{b} |

H_{0} × 10^{12} | −0.006 94(80) | −0.011 2(16)^{c} | −0.041(12) | N/A | 0.146 9(28) | ^{d} |

. | X^{3}Σ^{−}_{Ω=0+} state
. | [12.5] Y Ω = 0^{+} state
. | [15.9] B Ω = 0^{+} state
. | [18.0] C Ω = 0^{+} state
. | ||||
---|---|---|---|---|---|---|---|---|

. | This work . | Literature . | This work . | Literature . | This work . | Literature . | This work . | Literature . |

^{194}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.487 6(12) | N/A | 15 924.259 66(91) | 15 924.253 1(5)^{b} | 18 009.292 8(12) | 18 009.290 4(5)^{b} |

B_{0} | 0.147 287 754 4(98) | 0.147 287 754 7(40)^{c} | 0.140 512 84(68) | N/A | 0.138 125 08(44) | 0.138 138(4)^{b} | 0.136 203 3(17) | 0.136 186(3)^{b} |

D_{0} × 10^{6} | 0.048 191 9(53) | 0.048 191 0(43)^{c} | 0.048 13(17) | N/A | 0.058 937(71) | 0.078(7)^{b} | 0.054 5(12) | 0.054(2)^{b} |

H_{0} × 10^{12} | −0.006 97(81) | −0.00 75(12)^{c} | −0.041(12) | N/A | 0.147 6(28) | ^{d} | ||

^{195}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.480 7(12) | 12 457.480 4(13)^{e} | 15 924.275 56(91) | 15 924.269 6(5)^{b} | 18 009.312 4(12) | 18 009.310 0(5)^{b} |

B_{0} | 0.147 180 834 1(98) | 0.147 180 833 7(63)^{c} | 0.140 410 76(68) | 0.140 411 639(88)^{e} | 0.138 024 74(44) | 0.138 060(4)^{b} | 0.136 104 4(17) | 0.136 104(3)^{b} |

D_{0} × 10^{6} | 0.048 121 8(52) | 0.048 117 0(60)^{c} | 0.048 06(16) | 0.048 32(12)^{e} | 0.058 852(71) | 0.088(7)^{b} | 0.054 4(12) | 0.055(2)^{b} |

H_{0} × 10^{12} | −0.006 95(81) | −0.008 4(16)^{c} | −0.041(12) | ^{d} | 0.1473(28) | ^{d} | ||

C_{I} × 10^{6} | −2.247(20) | −2.228(52)^{f} | ||||||

^{196}Pt^{32}S | ||||||||

T_{00} | 0^{a} | 0^{a} | 12 457.472 6(12) | N/A | 15 924.291 49(91) | 15 924.285 6(5)^{b} | 18 009.332 3(12) | 18 009.331 1(5)^{b} |

B_{0} | 0.147 075 208 2(98) | 0.147 075 193 3(53)^{c} | 0.140 309 91(68) | N/A | 0.137 925 61(44) | 0.137 962(4)^{b} | 0.136 006 6(17) | 0.136 004(3)^{b} |

D_{0} × 10^{6} | 0.048 052 6(52) | 0.048 036 9(57)^{c} | 0.047 99(16) | N/A | 0.058 768(71) | 0.093(7)^{e} | 0.0543(12) | 0.052(2)^{b} |

H_{0} × 10^{12} | −0.006 94(80) | −0.011 2(16)^{c} | −0.041(12) | N/A | 0.146 9(28) | ^{d} |

## IV. DISCUSSION

### A. Evaluation of the mass-independent Dunham fit

In principle, the analytical expression derived by Dunham^{1} can be used to fit any set of spectroscopic observations if an appropriate number of parameters are included in the model. The spectroscopic data for PtS include 13 vibrational levels, for which at least 3 isotopologues are identified. At minimum, *T*_{v}, *B*, and *D* are necessary to characterize each vibrational level for each isotopologue, so a band-by-band fit would require more than 114 parameters (with one *T*_{v} fixed for each isotopologue). From a sense of parameter reduction, the applied Dunham Hamiltonian is quite effective in describing the data with only 40 parameters. The ability of the Dunham^{1} Hamiltonian to represent physical characteristics of the electronic states of PtS, however, must be evaluated by examining the relationships between the determined parameters.

One such example is the determination of the centrifugal distortion corrections from the characterization of the rovibrational structure. Watson^{3} outlined three theoretical relationships that should hold if the Dunham model is appropriately applied,

and Eq. (17) is the well-known Kratzer relation.^{42} These equations have been evaluated along with Eq. (15) to examine the validity of the Dunham^{1} fit. The calculated and experimentally determined parameters are reported in Table III.

. | X^{3}Σ^{−}_{Ω=0+}
. | [12.5] Y Ω = 0^{+}
. | [15.9] B Ω = 0^{+}
. | [18.0] C Ω = 0^{+}
. |
---|---|---|---|---|

U_{20}(exp.) | −49.730 6(43) | −53.724(11) | −50.644(13) | −58.57(11) |

U_{20}(calc.)^{a} | −50.074 3(15) | −51.344(47) | −48.614(42) | −58.79(11) |

U_{02}(exp.) × 10^{6} | −1.3194 3(12) | −1.312 9(45) | −1.637 2(18) | −1.495(34) |

U_{02}(calc.) × 10^{6}^{b} | −1.319 251 2(42) | −1.322 805(19) | −1.481 032(14) | −1.435 219(51) |

U_{12}(exp.) × 10^{9} | −4.72(17) | −14.5(15) | 45.7(13) | ^{c} |

U_{12}(calc.) × 10^{9}^{d} | −4.964 20(74) | −8.263(24) | −8.946(27) | −10.230(63) |

U_{03}(exp.) × 10^{12} | −0.191(22) | −1.12(33) | 4.045(78) | ^{c} |

U_{03}(calc.) × 10^{12}^{e} | −0.971 113(23) | −1.083 84(75) | −1.342 82(89) | −1.48 91(20) |

. | X^{3}Σ^{−}_{Ω=0+}
. | [12.5] Y Ω = 0^{+}
. | [15.9] B Ω = 0^{+}
. | [18.0] C Ω = 0^{+}
. |
---|---|---|---|---|

U_{20}(exp.) | −49.730 6(43) | −53.724(11) | −50.644(13) | −58.57(11) |

U_{20}(calc.)^{a} | −50.074 3(15) | −51.344(47) | −48.614(42) | −58.79(11) |

U_{02}(exp.) × 10^{6} | −1.3194 3(12) | −1.312 9(45) | −1.637 2(18) | −1.495(34) |

U_{02}(calc.) × 10^{6}^{b} | −1.319 251 2(42) | −1.322 805(19) | −1.481 032(14) | −1.435 219(51) |

U_{12}(exp.) × 10^{9} | −4.72(17) | −14.5(15) | 45.7(13) | ^{c} |

U_{12}(calc.) × 10^{9}^{d} | −4.964 20(74) | −8.263(24) | −8.946(27) | −10.230(63) |

U_{03}(exp.) × 10^{12} | −0.191(22) | −1.12(33) | 4.045(78) | ^{c} |

U_{03}(calc.) × 10^{12}^{e} | −0.971 113(23) | −1.083 84(75) | −1.342 82(89) | −1.48 91(20) |

^{c}

Not determined in the fit.

In general, the calculated expansion coefficients agree quite well with their experimentally determined counterparts. This agreement is exceptional for the *X*^{3}Σ^{−}_{Ω=0+} state. However, the experimentally determined *U*_{12} and *U*_{03} parameters for the [15.9] *B* Ω = 0^{+} state differ significantly from their theoretical counterparts, and the corresponding parameters are not determined for the [18.0] *C* Ω = 0^{+} state. These deviations could indicate that the *B* state is interacting with a nearby electronic state, and is, therefore, not perfectly described by the Dunham potential associated with the determined constants. This is discussed further in Sec. IV B.

### B. Potential energy curves for the known electronic states of PtS

The *Y*_{l,m} parameters determined in the mass-independent Dunham fit of the known electronic states of PtS can be used to generate potential energy surfaces for the analyzed transitions. These determined parameters were used with the Rydberg-Klein-Rees (RKR) method implemented by a program provided by Tellinghuisen^{43} to produce potential energy curves for the *X*^{3}Σ^{−}_{Ω=0+}, [12.5] *Y* Ω = 0^{+}, [15.9] *B* Ω = 0^{+}, and [18.0] *C* Ω = 0^{+} states of ^{195}Pt^{32}S. These potential energy curves are provided in Fig. 2. The deviations in the higher order centrifugal distortion coefficients determined for the *B* state from their theoretically expected values can be rationalized by examining the curves for the *B* and the *C* states. The v = 5 and 6 levels of the *B* state are predicted to lie very close in energy to the v = 0 and 1 levels of the *C* state, and extrapolation to v = 7 and v = 2 for the *B* and *C* states leads to a pseudodegeneracy of these vibrational levels. It is likely that these two states interact to some degree and that the magnitude of that interaction would be vibrationally dependent. As such, it is reasonable to speculate that the observed deviations result from this additional vibrational dependence, which is not included in the simple Dunham model.

### C. Electronic field-shift effects

As mentioned in Introduction, the electronic field-shift has not been well reported for molecular systems. This work clearly demonstrates isotope dependent shifts in electronic energy for 3 electronic transitions of PtS can be well described as field-shift effects. The magnitude of the electronic field-shift, *V*_{00}^{A}, has the form^{6,20}

where *Z*_{A} is the nuclear charge of atom A, *e* is the elementary charge, ε_{0} is the permittivity of free space, $\rho \xafeA$ is the effective electron density at nucleus A (in Å^{−3}), and *V*_{00}^{A} is in units of J fm^{−2}. As mentioned earlier, only the difference in electronic field-shifts can be determined from an observed transition, and, consequently, the determined Δ*f*_{00}^{A} values (in cm^{−1} fm^{−2}) are proportional to the change in effective electron density, $\Delta \rho \xafeA$,

where *h* is Planck’s constant, *c* is the speed of light (in cm s^{−1}), and

In the case where $\rho \xafe$ values have not been explicitly evaluated with *ab initio* methods, the change in electron density at the nucleus can be estimated from the degree of metal *s*-orbital character in the electronic states associated with the experimentally determined Δ*f*_{00} values.

With the inclusion of this work, electronic field-shifts have been reported for PtF,^{24} PtCl,^{15} PtS, PbS,^{8,9} YbCl,^{21} and YbBr.^{21} A comparison of the determined Δ*f*_{00} values and their corresponding $\Delta \rho \xafe$ values is provided in Table IV. The observed field-shifts for the Pt-halides are both positive, indicating a greater electron density at the nucleus in the excited states of these transitions, while the field shifts observed for PtS are negative for the *Y*-*X* transition and positive for the *B*-*X* and *C*-*X* transitions. The PbS transitions both have positive Δ*f*_{00}^{Pb} values and the Yb-halides have negative Δ*f*_{00}^{Yb} values.

MX . | Transition . | $\Delta f00M$ (cm^{−1} fm^{−2})
. | $\Delta \rho \xafeM$ (Å^{−3})^{a}
. |
---|---|---|---|

PtF | [15.8 + x] Ω = 5/2–B^{2}Δ_{5/2} | 0.2910(36)^{b} | 153.4(19) |

PtCl | [13.8] Ω = 3/2–X^{2}Π_{3/2} | 0.5577(59)^{c} | 293.9(31) |

PtS | [12.5] Y Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | −0.3778(36)^{d} | −199.1(19) |

[15.9] B Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | 0.0641(24)^{d} | 33.8(13) | |

[18.0] C Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | 0.1556(56)^{d} | 82.0(30) | |

PbS | [18.9] A0^{+}–X^{1}Σ^{+} | 0.1004(15)^{e} | 50.34(75) |

[21.8] B1–X^{1}Σ^{+} | 0.0848(16)^{f} | 42.51(80) | |

YbCl | [18.6] A^{2}Π–X^{2}Σ^{+} | −0.2154(83)^{g} | −126.5(49) |

[19.9] B^{2}Σ^{+}–X^{2}Σ^{+} | −0.2486(83)^{g} | −146.0(49) | |

YbBr | [19.7] B^{2}Σ^{+}–X^{2}Σ^{+} | −0.2403(83)^{g} | −141.1(49) |

MX . | Transition . | $\Delta f00M$ (cm^{−1} fm^{−2})
. | $\Delta \rho \xafeM$ (Å^{−3})^{a}
. |
---|---|---|---|

PtF | [15.8 + x] Ω = 5/2–B^{2}Δ_{5/2} | 0.2910(36)^{b} | 153.4(19) |

PtCl | [13.8] Ω = 3/2–X^{2}Π_{3/2} | 0.5577(59)^{c} | 293.9(31) |

PtS | [12.5] Y Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | −0.3778(36)^{d} | −199.1(19) |

[15.9] B Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | 0.0641(24)^{d} | 33.8(13) | |

[18.0] C Ω = 0^{+}–X^{3}Σ^{−}_{Ω=0+} | 0.1556(56)^{d} | 82.0(30) | |

PbS | [18.9] A0^{+}–X^{1}Σ^{+} | 0.1004(15)^{e} | 50.34(75) |

[21.8] B1–X^{1}Σ^{+} | 0.0848(16)^{f} | 42.51(80) | |

YbCl | [18.6] A^{2}Π–X^{2}Σ^{+} | −0.2154(83)^{g} | −126.5(49) |

[19.9] B^{2}Σ^{+}–X^{2}Σ^{+} | −0.2486(83)^{g} | −146.0(49) | |

YbBr | [19.7] B^{2}Σ^{+}–X^{2}Σ^{+} | −0.2403(83)^{g} | −141.1(49) |

^{a}

Calculated with Eq. (22).

^{b}

Reference 24.

^{c}

Reference 15.

^{d}

This work.

^{e}

Reference Reference. 8.

^{f}

Reference Reference. 9.

^{g}

Reference 23.

The electronic states of PtX (X = F, Cl, and S) molecules for which electronic field shifts have been observed are well described as Hund’s case (c) Ω-states. The electronic structure of these molecules has been investigated with *ab initio* methods by Zou and Suo^{44} (PtF and PtCl) and Zou^{45} (PtS). These studies used high-level relativistic multireference configuration interactions with single and double excitation (MRCISD+Q) calculations with perturbative spin-orbit coupling corrections, and were able to accurately predict the spectroscopic constants, term energies, and transition dipole moments for the observed electronic transitions of the PtX molecules. The predicted Ω-states are often derived from multiple Λ-S states, which are often themselves multiconfigurational. The degree of Pt 6*s* character in the resultant Ω-states was estimated by tracing the lineage of the observed electronic states through their predicted Λ-S and configurational origins. The predicted Ω-states are reported with the percent composition from their dominant Λ-S states, and the Λ-S states are reported with the percent composition from their dominant electron configurations. The primary molecular orbitals that determine the character of the low-lying states of the PtX molecules have been termed the 1σ, 2σ, 3σ, 1π, 2π, and 1δ orbitals in the MRCISD+Q studies,^{44,45} where the 2σ and 3σ orbitals are hybrids of the Pt 5*d*_{σ}, Pt 6*s*, and X n*p*_{σ} atomic orbitals with the 2σ predominantly Pt 5*d*_{σ} and the 3σ predominantly Pt 6*s* in character. As such, the degree of Pt 6*s* character (and, by extension, the relative $\rho \xafePt$ values) can be estimated from the electronic population in the 2σ and 3σ molecular orbitals. Approximate molecular orbital populations for the predicted Ω-states were calculated as weighted averages from Λ-S and configurational compositions. These values are presented inTable V (the Λ-S and configurational compositions are provided in Table S2 in the supplementary material for reference).

. | 1σ^{a}
. | 2σ^{b}
. | 3σ^{c}
. | 1π^{d}
. | 2π^{e}
. | 1δ^{f}
. |
---|---|---|---|---|---|---|

PtF | ||||||

B^{2}Δ_{5/2}–(1) 5/2 | 2.00 | 2.00 | 0.00 | 4.00 | 4.00 | 3.00 |

[15.8 + x] Ω = 5/2–(4) 5/2 | 2.00 | 1.89 | 1.00 | 4.00 | 3.00 | 3.11 |

PtCl | ||||||

X^{2}Π_{3/2}–(1) 3/2 | 2.00 | 2.00 | 0.00 | 3.87 | 3.33 | 3.80 |

[13.8] Ω = 3/2–(4) 3/2 | 2.00 | 1.46 | 1.00 | 3.82 | 2.97 | 3.49 |

PtS | ||||||

X^{3}Σ^{−}_{Ω=0+}–(1) 0^{+} | 2.00 | 2.00 | 0.00 | 3.82 | 2.18 | 4.00 |

[12.5] Y Ω = 0^{+}–(3) 0^{+} | 2.00 | 1.59 | 0.00 | 3.96 | 2.83 | 3.61 |

[15.9] B Ω = 0^{+}–(4) 0^{+} | 2.00 | 1.74 | 0.16 | 3.88 | 2.48 | 3.74 |

[18.0] C Ω = 0^{+}–(5) 0^{+} | 2.00 | 1.89 | 0.78 | 3.72 | 1.73 | 3.89 |

. | 1σ^{a}
. | 2σ^{b}
. | 3σ^{c}
. | 1π^{d}
. | 2π^{e}
. | 1δ^{f}
. |
---|---|---|---|---|---|---|

PtF | ||||||

B^{2}Δ_{5/2}–(1) 5/2 | 2.00 | 2.00 | 0.00 | 4.00 | 4.00 | 3.00 |

[15.8 + x] Ω = 5/2–(4) 5/2 | 2.00 | 1.89 | 1.00 | 4.00 | 3.00 | 3.11 |

PtCl | ||||||

X^{2}Π_{3/2}–(1) 3/2 | 2.00 | 2.00 | 0.00 | 3.87 | 3.33 | 3.80 |

[13.8] Ω = 3/2–(4) 3/2 | 2.00 | 1.46 | 1.00 | 3.82 | 2.97 | 3.49 |

PtS | ||||||

X^{3}Σ^{−}_{Ω=0+}–(1) 0^{+} | 2.00 | 2.00 | 0.00 | 3.82 | 2.18 | 4.00 |

[12.5] Y Ω = 0^{+}–(3) 0^{+} | 2.00 | 1.59 | 0.00 | 3.96 | 2.83 | 3.61 |

[15.9] B Ω = 0^{+}–(4) 0^{+} | 2.00 | 1.74 | 0.16 | 3.88 | 2.48 | 3.74 |

[18.0] C Ω = 0^{+}–(5) 0^{+} | 2.00 | 1.89 | 0.78 | 3.72 | 1.73 | 3.89 |

^{a}

X n*p*_{σ} + Pt 5*d*_{σ}.

^{b}

Pt 5*d*_{σ} + Pt 6*s* + Xn*p*_{σ}.

^{c}

Pt 6*s* + Pt 5*d*_{σ} + X n*p*_{σ}.

^{d}

X n*p*_{π} +Pt 5*d*_{π}.

^{e}

Pt 5*d*_{π} +X n*p*_{π}.

^{f}

Pt 5*d*_{δ}.

The electron configurations of the PtX molecules indicate that the 2σ orbital is fully occupied and the 3σ orbital is empty for the lower states of the observed electronic transitions, i.e., the *B*^{2}Δ_{5/2} state of PtF, the *X*^{2}Π_{3/2} state of PtCl, and the *X*^{3}Σ^{−}_{Ω=0+} state of PtS. In the case of the platinum halides, electronic excitation yields a singly occupied 3σ orbital and a partially depleted 2σ orbital. Because the 3σ is primarily derived from the Pt 6*s* atomic orbital and the 2σ orbital has only minor contribution from Pt 6*s*, this electronicrearrangement would yield a greater electron density at the Pt nucleus upon excitation and a positive Δ*f*_{00}^{Pt} value. For the *Y*-*X* transition of PtS, the 2σ orbital is partially depleted and the 3σ orbital remains unoccupied, leading to a decrease in Pt 6*s* character and the expectation of a negative Δ*f*_{00}^{Pt} value. For the *B*-*X* and *C*-*X* transitions of PtS, the 3σ orbital becomes partially occupied and the 2σ population is slightly decreased. Again, because the 3σ orbital has significantly more Pt 6*s* character than the 2σ orbital, a positive Δ*f*_{00}^{Pt} is expected, with a larger field-shift for the *B*-*X* transition than the *C*-*X* transition due to the relative population of the molecular orbitals.

The interpretation of the determined Δ*f*_{00}^{Yb} values is much more straightforward. The low-lying states of lanthanide monohalide molecules have been investigated using a ligand-field approach with the atomic Ln^{+} orbitals.^{46} These calculations predict that the *X*^{2}Σ^{+} ground states for YbCl and YbBr are derived from the 4*f*^{14}6*s*^{1} Yb^{+} configuration and the *A*^{2}Π and *B*^{2}Σ^{+} states are derived from the 4*f*^{14}6*p*^{1} Yb^{+} configuration. This assignment of the *A* and *B* states is supported by spectroscopic evidence that indicates that these states form a unique perturber pair that is predominantly Yb^{+} 6*p* in origin.^{21–23,47} Consequently, promotion from the *X* state in these molecules to the *A*/*B* states would decrease the electron density at the Yb nucleus, resulting in a negative Δ*f*_{00}^{Yb} parameter.

While the 1st and 2nd derivatives of $\rho \xafePb$ with respect to the equilibrium bond length, *r*_{e}, have been calculated and reported for the ground state of PbS,^{19,20} these predictions do not include the reported values for the electron density itself or any values for the excited electronic states of PbS. As such, the reported values^{8,9} cannot be meaningfully evaluated at this time and are provided merely as another example of the electronic field-shift effect.

## V. CONCLUSIONS

The high resolution spectroscopic data for platinum sulfide, PtS, have been fit to a mass-independent Dunham^{1} type Hamiltonian using PGOPHER.^{30} The resulting molecular parameters closely follow the theoretical relationships outlined by Watson,^{3} and these parameters have been used with the Rydberg-Klein-Rees method to produce potential energy curves for the known electronic states of ^{195}Pt^{32}S. These potential energy curves indicate that the excited vibrational levels of the [15.9] *B* Ω = 0^{+} and [18.0] *C* Ω = 0^{+} states likely overlap, possibly accounting for large centrifugal distortion terms in the Dunham^{1} Hamiltonian of the *B* state. Isotopic shifts in electronic transition energy deviating from the Born-Oppenheimer approximation were identified for the three known transitions of PtS, and these shifts were treated as field-shift effects. The electronic field-shift has only been reported for PtF,^{24} PtCl,^{15} PbS,^{8,9} YbCl,^{23} YbBr,^{23} and, with this work, PtS. These observed shifts are rationalized with the assistance of *ab initio* calculations.^{44–46} Since the rotational and vibrational field shifts depend on the 1st and 2nd derivatives of the electron density as a function of bond length, additional computational efforts will be necessary to provide a full evaluation of the observed field-shift effects.

## SUPPLEMENTARY MATERIALS

See supplementary material for the root mean squared (rms) values from the mass-independent Dunham fit, partitioned by the isotopologue and vibrational bands (Table S1); the Λ-S and configurational compositions of the *ab initio* Ω-states for the PtX molecules (Table S2); and the .pgo (PtS_SM.pgo), .ovr (PtS_ILS_Data.ovr), and input text file (PtS_Fit.txt) used to perform the mass-independent Dunham analysis. The constraints used to define the Dunham-type Hamiltonian and the experimental line positions and assignments can be found in the input text file.

## ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation, Grant Nos. CHE-1566454 (JOB) and CHE-1566442 (LOB). J. Harms would like to gratefully acknowledge the UMSL Graduate School for supporting this work through a Dissertation Fellowship.

## REFERENCES

^{−1}