Theoretical Calculation of Ca^q+ (q = 0, 1, 2) Interacting with a Krypton Atom: Electronic Structure and Vibrational Spectra Association

Abstract

The potential energy curves and spectroscopic constants of the ground and several low-lying excited states of the Ca^q+-Kr (q = 0, 1, 2) van der Waals complexes were investigated using one- and two-electron pseudopotential approaches. This treatment effectively reduces the number of active electrons in Ca^q+-Kr to a single valence electron for q = 1 and two valence electrons for q = 0, allowing the use of large and flexible basis sets for both Ca and Kr atoms. Within this work, potential energy curves (PECs) were calculated at the SCF level for the Ca⁺-Kr system, while both SCF and full configuration interaction (FCI) calculations were performed for the neutral Ca-Kr. Spin–orbit coupling effects were explicitly included in all calculations to accurately describe the fine-structure splitting of the asymptotic atomic states. The short-range core–core interaction for Ca²⁺-Kr was obtained using high-level CCSD(T) calculations. Spectroscopic constants were derived from the computed PECs and compared with available theoretical and experimental results, showing consistent trends. Furthermore, the transition dipole moments (TDM) were evaluated as a function of internuclear distances, including spin–orbit effects, to provide a comprehensive description of the electronic structure and radiative properties of these weakly bound systems.

1. Introduction

Theoretical and experimental studies of alkali and alkaline-earth metals interacting with rare-gas atoms have attracted the interest of a significant number of researchers. This scientific focus is motivated, in part, by the critical necessity of comprehending the behaviors of these metals within van der Waals complexes and, in part, by the interest in testing theoretical predictions against real experimental data on interatomic potentials [1,2,3,4,5,6,7,8,9,10,11,12].

Alkali–rare-gas collisions have been widely explored through processes such as charge transfer, energy and fine-structure exchange, and radiation redistribution. Alkaline-earth–rare-gas systems, on the other hand, offer additional richness: their stronger bonding with noble gases and the presence of multiple fine-structure components, which open distinct interaction pathways, make them especially valuable for probing collisional dynamics and benchmarking theoretical models [13,14,15,16,17]. Atomic collision processes continue to attract considerable attention, as they provide essential insight into the fundamental mechanisms of atomic interactions and enable the determination of accurate interatomic potentials [18,19,20,21]. Such potentials are particularly valuable in spectroscopy, where they form the basis for interpreting and analyzing experimental spectra.

In this context, collisional effects on the resonance line shape of the calcium atom have been investigated experimentally through absorption coefficient measurements [22] and by determining collisional redistribution of light [23]. Laser-induced excitation spectra of CaAr, SrAr, and BaAr in the vicinity of the resonance lines produced in a supersonic jet expansion were investigated by Kowalski et al. [24]. Kaup and Breckenridge have conducted pioneering investigations into metastable Ca-RG complexes (RG = Ar, Kr, Xe), utilizing two-color photoionization to precisely determine ionization thresholds and dissociation energies for both neutral and ionic species. Their findings revealed a clear trend of increasing bonding strength with the atomic mass of the rare gas, and they offered insightful comparisons to related systems such as Be⁺RG, Mg⁺RG, and Ba⁺RG [25]. Building on this foundation, Kaup and Breckenridge applied Resonance-Enhanced Two-Photon Ionization (R2PI) spectroscopy to probe a variety of excited valence states in the CaAr complex, beginning from the metastable Ca(4s4p ³P₀)Ar state. Among these, the singly excited Ca(4s4d ³D₁)Ar state is notable for its stronger bonding, primarily due to reduced electronic repulsion. Even more remarkable is the doubly excited Ca(4p4p ³S₂)Ar state, which exhibits strong binding as a result of the missing 4s electron that allows the Ar atom to approach the calcium core more closely. Despite its strong bonding, this doubly excited state is short-lived, with an ultrashort predissociation lifetime [26]. Further studies of metastable Ca-Rg complexes, both in their ground state and in the Ca(4s5s) Rydberg state via R2PI spectroscopy, reveal that Ca-Rg complexes possess weaker bonds and larger bond lengths compared to their Mg-RG counterparts. This is explained by repulsive interactions manifesting at larger internuclear separations due to the larger size of calcium. Interestingly, the Ca(4s5s) Rydberg states show binding strengths comparable to their ionic analogs, reflecting the effective penetration of the Ca(5s) orbital by the rare-gas atoms. Moreover, the higher vibrational frequencies observed in these neutral Rydberg states versus the ions are linked to narrower potential wells, a consequence of limited rare-gas penetration of the Ca(4s) orbital [27].

Cationic Ca⁺-RG (RG = Ar, Kr, Xe) complexes have also been generated using a pulsed nozzle/laser vaporization cluster source and characterized through mass-selected photodissociation spectroscopy. The molecular spectra associated with the Ca+ (²P ← ²S) atomic resonance line were observed to be red-shifted, indicating stronger binding in the excited ²P state than in the ground state. They have determined the ground state dissociation energies, vibrational constants, and the spin–orbit splittings within the ²P_1/2,3/2 state [28]. Buthelezi et al. further investigated the electronic transitions of isolated Ca⁺-Kr ions via resonant photodissociation spectroscopy, identifying an excited ²Σ⁺ state correlating with the Ca⁺3p⁶3d [²D] + Kr [¹S] atomic limit. The derived vibrational constants and diabatic binding energies were found to be consistent with analogous Ca⁺-Ar results, confirming similar ground-state bonding characteristics between Ca⁺-Kr and Ca⁺-Ar [29].

In this work, Section II outlines the computational methods employed. Section III presents and discusses the results for both ground and excited states, highlighting the effects of spin–orbit coupling. Finally, Section IV offers concluding insights and perspectives.

2. Computational Methods

The main objective of this work is to provide a detailed theoretical description of the electronic structure and dipole moment functions of the calcium atom and ion interacting with a krypton atom. The study is conducted within a model-potential framework, in which the total interaction energy is represented as the sum of three fundamental contributions: the core–core interaction between Ca²⁺ and Kr, the interaction of the valence electron(s) with the ionic core, and the spin–orbit coupling term.

$$V_{tot}=V_{C{a}^{2+}-Kr}+V_{qe-C{a}^{2+}-Kr}+V_{S.O}(q=1 or 2)$$

(1)

2.1. The Core–Core Interaction ($V_{C{a}^{2+}-Kr}\left(r\right)$)

The core–core Ca²⁺-Kr interaction is computed over a wide and dense range of internuclear distances at the CCSD(T) level of theory implemented in the Molpro package [30] using aug-cc-pv5z basis sets for the Ca and Kr atoms, respectively [31,32]. The resulting CCSD(T) interaction potentials were fitted to an analytical form using a standard nonlinear least-squares procedure, facilitating their implementation within the simulation code. Within this framework, the core–core interaction is described using a generalized Buckingham-type potential [33], which effectively combines short-range repulsion, ion-induced dipole attraction, and higher-order dispersion terms. The potential takes the following form:

$${\mathrm{V}}_{\mathrm{C}{\mathrm{a}}^{2+}-\mathrm{Kr}}\left(\mathrm{R}\right)=\mathrm{A}{\mathrm{e}}^{-\mathrm{b}\mathrm{R}}-{\displaystyle \frac{{\mathrm{D}}_4}{{\mathrm{R}}^4}}-{\displaystyle \frac{{\mathrm{D}}_6}{{\mathrm{R}}^6}}-{\displaystyle \frac{{\mathrm{D}}_8}{{\mathrm{R}}^8}}-{\displaystyle \frac{{\mathrm{D}}_{10}}{{\mathrm{R}}^{10}}}$$

(2)

Here, the parameters A and b describe the short-range Born–Mayer repulsion [34], while the D_2n (n = 2–5) terms capture long-range electrostatic and dispersion contributions. The fitted parameters of the ground state of Ca²⁺-Kr are summarized in Table 1

Table 1. Fitted parameters for Ca²⁺-Kr.

	A (Hartree)	$\mathbf{b} \left({\mathit{a}}_0^{-1}\right)$	${\mathbf{D}}_4 {(\mathit{a}}_0^4 \mathbf{H}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{r}\mathbf{e}\mathbf{e})$	${\mathbf{D}}_6 {(\mathit{a}}_0^6 \mathbf{H}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{r}\mathbf{e}\mathbf{e})$	${\mathbf{D}}_8 {(\mathit{a}}_0^8 \mathbf{H}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{r}\mathbf{e}\mathbf{e})$	${\mathbf{D}}_{10} {(\mathit{a}}_0^{10} \mathbf{H}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{r}\mathbf{e}\mathbf{e})$
Ca2+-Kr	214.188	1.52732	19.2417	1964.17	−12,723	25,333.7

Figure 1 compares the ab initio CCSD(T) potential energy with the analytical fit using Equation (2). To assess the quality of the fits, we calculated the root mean square error (RMSE), defined as follows [35]:

$$\sqrt{{\displaystyle \frac{{\sum }_{k=1}^{N_p}{\left(V_k^{fit}-V_k^{\mathrm{ab} \mathrm{initio}}\right)}^2}{N_p}}}$$

(3)

where N_p is the number of ab initio data points, and $V_k^{fit}$ $\mathrm{a}\mathrm{n}\mathrm{d} V_k^{\mathrm{a}\mathrm{b} \mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}}$ represent the fitted and computed potential energies, respectively. The resulting RMSE for the Ca²⁺ Kr is approximately 1.05 × 10⁻⁴ Hartree, confirming the excellent agreement between the analytical representation and the high-level ab initio data. This high degree of accuracy ensures that the fitted potential can reliably describe the core–core interaction, which plays a central role in modeling the ground and excited states of the Ca^q+-Kr (q = 0, 1) systems.

Figure 1. Comparison of numerical CCSD(T) and analytical potentials for the Ca²⁺-Kr molecules.

Table 2 compares the spectroscopic constants obtained in this work for the ground state of Ca²⁺-Kr with those reported by Adrian et al. [36]. As shown, the equilibrium bond distance R_e ranges between 2.87 and 2.93 Å, showing good consistency among the different methods. However, the potential well depth D_e shows more variation: the MRCI + Q method predicts the shallowest well (6264 cm⁻¹), and the CCSD(T) approach yields a slightly deeper value (6500 cm⁻¹), while Adrian et al. [36] reported the largest depth (7320 cm⁻¹) using the RCCSD(T) method. The calculated harmonic vibrational frequencies (ω_e), and rotational constants (B_e) are in close agreement across methods, with RCCSD(T) predicting the highest ω_e values.

Table 2. Spectroscopic constants for Ca²⁺-Kr. All constants are expressed in cm⁻¹ except R_e in Å.

	Re	De	ωe	ωeχe	Be	Method
Ca2+-Kr	2.93	6500	168.42	1.09	0.0724	CCSD(T)
	2.89	6264	-	-	-	MRCI + Q
	2.87	7320	179	1.14	0.0759	RCCSD(T) [36]

Overall, the CCSD(T) results consistently predict strong binding energies and short equilibrium distances for the Ca²⁺-Kr system. Compared with MRCI + Q, CCSD(T) provides a balanced compromise between computational cost and accuracy.

The basis set superposition error for related weakly bound van der Waals systems has been previously assessed [37] and found to be negligible, and it is known to decrease further when larger, augmented basis sets are employed. Accordingly, since all calculations for the Ca²⁺-Kr system were performed using the aug-cc-pV5Z basis set, no explicit BSSE correction was applied.

2.2. The Interaction Between the Electron and Ca²⁺-Kr Denoted as Vqe-Ca²⁺-Kr (q = 0 or 1)

To describe the Vqe-Ca²⁺-Kr interactions for q = 0 or 1, we used the pseudopotential of Barthelat and Durand [38] to reduce the number of active electrons. This approach provides both computational efficiency and high accuracy by restricting the electronic treatment to the valence electrons only. In the Ca⁺-Kr system, a single valence electron is active, whereas in the neutral Ca-Kr complex, two valence electrons are explicitly considered. For the one-electron Ca⁺-Kr system, calculations were performed at the SCF level of theory, while the two-electron Ca-Kr system was treated using a FCI approach. The Ca^q+-Kr complexes (considered as core–core interactions) involve three charge situations (q = 0, 1, 2); each is treated according to its electronic nature and complexity: substantial CCSD(T)/aug-cc-pV5Z for Ca²⁺-Kr, pseudopotential-based SCF + core–core interactions for Ca⁺-Kr, and core–core interactions + FCI using pseudopotentials for the neutral Ca-Kr complex. Each method is chosen to provide an accurate and balanced description of the interactions, ensuring that the resulting potential energy curves and spectroscopic constants are reliable across all charge states.

To incorporate polarization effects, we included core polarization pseudopotentials (V_CPP) following the l-dependent formulation of Foucrault et al. [39], which extends the original framework by Müller and Meyer [40]. These pseudopotentials simulate the polarization response of both the Ca²⁺ ionic core and the krypton atom. For each atom center (Ca or Kr), core polarization is represented by an effective potential that depends on the core polarizability:

$$W_{CPP}=-{\displaystyle \frac{1}{2}}\sum _{\lambda }{\alpha }_{\lambda }\overrightarrow{f_{\lambda }}{\overrightarrow{f}}_{\lambda }$$

(4)

where α_λ is the static dipole polarizability of core λ and ${\overrightarrow{f}}_{\lambda }$ denotes the electric field created at center λ produced by the valence electrons and all other atomic cores.

The core polarizabilities of Ca²⁺ and Kr, α^Ca2+ = 3.1717$a_0^3$, and α^Kr = 16.763$a_0^3$, were taken from Refs. [41,42].

The electric field $\overrightarrow{f_{\lambda }}$ acting on each center λ is given by the following:

$$\overrightarrow{f_{\lambda }} =\sum _i{\displaystyle \frac{{\overrightarrow{r}}_{i\lambda }}{r_{i\lambda }^3}}F(r_{i\lambda },{\rho }_{\lambda })-\sum _{\lambda {\lambda }^{\prime }}{\displaystyle \frac{{\overrightarrow{R}}_{\lambda {\lambda }^{\prime }}}{R_{\lambda {\lambda }^{\prime }}^3}}{Z}_{\lambda }$$

(5)

The parameters r_iλ, R_λλ′, and Z_λ represent, respectively, the distance between the valence electron and the core λ, the distance separating the two cores λ and λ^′, and the corresponding nuclear charge.

The valence electrons were described with large Gaussian basis sets. For the Ca atom, we employed the (8s, 7p, 7d)/(8s, 7p, 5d) basis set, with cut-off parameters (${\rho }_s^{{Ca}^{+}} = 1.77405, {\rho }_p^{{Ca}^{+}} = 1.81, {\rho }_d^{{Ca}^{+}} = 1.691$) [41]. For krypton, we used the uncontracted (4s, 3p) Gaussian-type basis set from Refs. [43,44] comprising s-functions with exponents 0.36894, 0.1444, 0.05, and 0.02, and p-functions with exponents 0.39486, 0.12650, and 0.02.

The inclusion of Kr basis functions is essential to properly describe the distortion of the Ca valence orbitals arising from orthogonality to the rare-gas closed shell (represented by its pseudopotential). Since krypton contributes no active electrons, its exponents were optimized to reproduce the overlap with the 3s and 3p orbitals of the rare gas and to extend toward the diffuse region. The employed Kr pseudopotential has been successfully applied in previous studies of rare-gas–metal and rare-gas–cation complexes [42,44,45], demonstrating reliable performance in describing both short-range repulsion and long-range dispersion interactions, which supports its accuracy for the weakly bound systems considered here.

Finally, for the e–Kr interaction, the analytical pseudopotential form is given by the following:

$$W_l\left(r\right)=\mathit{exp}(-\alpha r^2) {\sum }_{i=1}^n{C}_i{r}^{n_i}$$

(6)

The above formula includes the parameters α, C_i, and n_i for the different s, p, and d orbitals listed in Table 3 [45,46].

Table 3. Pseudopotential parameters of e–Kr.

l	α	Ci	ni
0	0.9251	20.0	−1
1	0.282	20.8	1
2	0.508	21.21	1

2.3. The Spin–Orbit Interaction

The spin–orbit interaction was incorporated using the semi-empirical method originally proposed by Cohen and Schneider [46], which has been successfully applied to the 2p5 and 2p5 3s configurations of Ne+ and Ne*, respectively. Within this framework, the spin–orbit coupling Hamiltonian is expressed as follows:

$$V_{S.O.}=\xi \hspace{0.17em}L\cdot S$$

(7)

where ξ denotes the spin–orbit coupling constant, L is the orbital angular momentum operator, and S is the spin angular momentum operator. For the electronic states of the Ca⁺-Kr system correlating to the Ca⁺ (4p) atomic asymptote, the spin–orbit interaction is represented by the eigenvalues of the following coupling matrix:

$$\left(\begin{array}{ccc}{E}_p{(}^2\Pi )-{\displaystyle \frac{\xi }{2}}& {\displaystyle \frac{\sqrt{2}}{2}}\xi & 0\\ {\displaystyle \frac{\sqrt{2}}{2}}\xi & E_p\left(2^2{\Sigma }^{+}\right)& 0\\ 0& 0& E_p{(}^2\Pi )+{\displaystyle \frac{\xi }{2}}\end{array}\right)$$

Here, $E_p{(}^2\Pi )$ and $E_p\left(2^2{\Sigma }^{+}\right)$ correspond to the spin-free electronic energies and $\xi$ denotes the spin–orbit coupling constant. Diagonalization of the spin–orbit Hamiltonian yields the fine-structure splitting associated with the S_1/2, P_1/2, and P_3/2 atomic limits. The explicit forms of the spin–orbit coupling matrices for the np and nd configurations have been reported elsewhere [43,46,47,48,49] and are not reproduced here. The spin–orbit coupling constants used in the present work are as follows: ξ_4p (Ca⁺) = 148.6 cm⁻¹, ξ_3d (Ca⁺) = 24.3 cm⁻¹, ξ_4p (Ca) = 105.8 cm⁻¹, ξ_3d (Ca) = 13.9 cm⁻¹ [49].

Furthermore, the intensity of electronic transitions, whether in absorption or emission, is proportional to the square of the vibrational overlap integral, which quantifies the coupling between the initial and final vibrational wavefunctions:

$$|⟨{\chi }_v {\left| \mu \left| {\chi }_{v^{\prime }}\right.⟩\right|}^2$$

(8)

where χ_v and χ_v^′ are the vibrational wavefunctions of the initial and final electronic states, respectively, and μ is the electric dipole moment operator. According to the Franck–Condon principle, the intensity of a transition reaches its maximum when the vibrational wavefunctions of the two states exhibit significant overlap.

In addition, the radiative lifetimes of vibrational levels belonging to the 2²Π electronic state were determined by considering two distinct types of electronic transitions: bound–bound and bound–free [41,43]. The lifetime associated with bound–bound transitions can be expressed as follows:

$${\tau }_{v^{\prime }}={\displaystyle \frac{1}{{\Gamma }_{v^{\prime }}}};{\Gamma }_{v^{\prime }}={\displaystyle \sum _{v=0}^{nvt}}{A}_{v{v}^{\prime }}$$

(9)

where $A_{v{v}^{\prime }}$ is the Einstein coefficient for the transition between v^′ and v vibrational states.

For bound-free transition, the spontaneous emission rate, $A_{v^{\prime }},$ was determined within the framework of the Franck–Condon approximation, following the formulation proposed by Zemke et al. [50], and is given by the following:

$$A_{v^{\prime }}={\displaystyle \frac{64{\pi }^2}{3h^4{c}^3}}{\left|\mu \left(R_{v^{\prime }+}\right)\right|}^{2}F{C}_{v^{\prime },cont}{(∆E)}_{v^{\prime },cont}^3$$

(10)

and

$$F{C}_{v^{\prime },cont}={\int \left|⟨{\chi }_{v^{\prime }}|{\chi }_E⟩\right|}^{2}dE=1-\sum _{v=0}^{nvt}{\left|⟨{\chi }_{v^{\prime }}|{\chi }_v⟩\right|}^2$$

(11)

$∆E_{v^{\prime },cont}$ represents the energy difference between the vibrational level v^′ and the corresponding asymptotic dissociation limit. The term $\mu \left(R_{v^{\prime }+}\right)$ denotes the transition dipole moment evaluated at the outer turning point of the vibrational state v^′.

3. Results and Discussion

3.1. Spectroscopic Investigation of the Ca⁺-Kr System

Using the pseudopotential technique, we computed potential energy curves for thirteen electronic states of the Ca⁺-Kr system, possessing ²Σ⁺, ²Π, and ²Δ symmetries. These states correlate asymptotically with the atomic limits Ca⁺ (4s, 3d, 4p, 5s, 4d, 5p, and 6s) + Kr. The calculations were performed over a wide and finely sampled range of internuclear distances ranging from 1.5 to 50 Å, ensuring an accurate description of both short- and long-range interactions.

The potential energy curves for both ground and excited states are presented in Figure 2. To validate the accuracy of our calculated potential energy curves, we derived their spectroscopic constants: equilibrium distance (R_e), well depth (D_e), vertical transition energy (T_e), harmonic (ω_e) and anharmonic (ω_eχ_e) frequencies, and rotational constant (B_e). These constants are presented in Table 4. The Ca⁺-Kr system features an attractive ground state, 1²Σ⁺, characterized by a potential well depth of 1141 cm⁻¹ located at an equilibrium distance of R_e = 3.34 Å. This state corresponds to a relatively stable interaction between the Ca⁺ ion and the Kr atom. In contrast, the excited state 3²Σ⁺, which dissociates into Ca⁺(²P) + Kr, exhibits a much shallower potential well D_e = 170 cm⁻¹, indicating a significantly weaker interaction. On the other hand, the 2²Σ⁺ and 4²Σ⁺ states have deeper potential wells, with depths of D_e = 1736 cm⁻¹ and 3898 cm⁻¹, respectively, suggesting greater stability.

Figure 2. Potential energy curves of the Ca⁺-Kr molecule for excited states in ²Σ⁺, ²Π, and ²Δ symmetries.

For higher excited electronic states, such as 5²Σ⁺ and 6²Σ⁺, the potential energy curves show distinct potential barriers due to the repulsive interaction between the electron and the krypton atom. Nonetheless, these states remain more attractive than the lower 1²Σ⁺ and 3²Σ⁺ states, with potential wells reaching several thousand cm⁻¹ in depth. Among all calculated states, the 7²Σ⁺ state is the deepest, with a well depth of D_e = 5232 cm⁻¹, making it the most strongly bound excited state of the Ca⁺-Kr complex. A comparative analysis with available experimental and theoretical literature reveals reasonable agreement for the ground state of the Ca⁺-Kr system.

For instance, as presented in Table 4, the equilibrium bond distance (R_e) and well depth (D_e) of the ground state 1²Σ⁺ of Ca⁺-Kr are found to be 3.34 Å and 1141 cm⁻¹, respectively. The equilibrium bond length Re is approximately 0.035–0.040 Å longer than both the experimental value (3.30 ± 0.15 Å) and the theoretical range (3.202–3.305 Å) reported in references [36,51,52]. The present R_e remains well within the experimental uncertainty and shows very good agreement with the Rydberg-state estimation and the RCCSD(T) results reported previously [36,51]. The calculated dissociation energy, D_e = 1141 cm⁻¹, also lies within the broad experimental range of 840 ± 100 to 1400 ± 150 cm⁻¹ [25,28,29,52] showing slightly better consistency with the lower experimental and theoretical estimates (1200 and 1376 cm⁻¹, respectively [36,51]). This moderate underestimation can be attributed to the approximate nature of the pseudopotential, the limitations of the employed basis sets, and the level of theory adopted. However, it remains below the mean values of both experimental and theoretical predictions.

Table 4. Spectroscopic constants for the ²Σ⁺, ²Π, and ²Δ states of the Ca⁺-Kr ionic system (neglecting spin–orbit coupling). All constants are given in cm⁻¹ except Re in Å.

Molecular State Ca+-Kr	Re	De	Te	ωe	ωeχe	Be
12Σ+ (4s 2S)	3.34	1141		65.35	0.94	0.055683
Exp. [25]		1244 ± 60 1
Exp. [28]		1400 ± 150 2
Exp. [29]		840 ± 100 2		74
Exp. [52]	3.30 ± 0.15 3	1280 ± 80 1		77 ± 2 2
Ref. [36]	3.305	1200		69.7	1.307	0.0570
Ref. [51]	3.202 4	1376 4		68 4	1.49 4
22Σ+ (3d 2D)	3.49	1736	13,048	70.54	1.40	0.050963
32Σ+ (4p 2P)	5.96	170	26,781	17.65	0.70	0.017527
42Σ+ (5s 2S)	3.09	3898	49,207	118.65	1.00	0.064802
52Σ+ (4d 2D)	3.01	4140	53,921	159.06	1.00	0.068910
62Σ+ (5p 2P)	3.08	1422	60,500	82.62	6.27	0.065527
72Σ+ (6s 2S)	3.02	5232	66,461	141.56	0.84	0.068222
12Π (3d 2D)	3.38	3214	11,576	98.34	0.63	0.054556
22Π (4p 2P)	3.70	1806	25,146	75.31	1.28	0.045388
32Π (4d 2D)	2.87	3668	54,393	196.64	1.25	0.075499
42Π (5p 2P)	3.10	4668	57,255	128.31	2.04	0.064074
12Δ (3d 2D)	2.99	6005	8785	166.34	1.46	0.069377
22Δ (4d 2D)	3.06	4262	53,800	132.27	1.38	0.066548

¹ Two-color photoionization; ² photodissociation spectroscopy; ³ estimated Rydberg state Re values of Ca-RG (ab initio calculation); ⁴ RCCSD(T) calculation.

Table 4. Spectroscopic constants for the ²Σ⁺, ²Π, and ²Δ states of the Ca⁺-Kr ionic system (neglecting spin–orbit coupling). All constants are given in cm⁻¹ except Re in Å.

Molecular State Ca+-Kr	Re	De	Te	ωe	ωeχe	Be
12Σ+ (4s 2S)	3.34	1141		65.35	0.94	0.055683
Exp. [25]		1244 ± 60 1
Exp. [28]		1400 ± 150 2
Exp. [29]		840 ± 100 2		74
Exp. [52]	3.30 ± 0.15 3	1280 ± 80 1		77 ± 2 2
Ref. [36]	3.305	1200		69.7	1.307	0.0570
Ref. [51]	3.202 4	1376 4		68 4	1.49 4
22Σ+ (3d 2D)	3.49	1736	13,048	70.54	1.40	0.050963
32Σ+ (4p 2P)	5.96	170	26,781	17.65	0.70	0.017527
42Σ+ (5s 2S)	3.09	3898	49,207	118.65	1.00	0.064802
52Σ+ (4d 2D)	3.01	4140	53,921	159.06	1.00	0.068910
62Σ+ (5p 2P)	3.08	1422	60,500	82.62	6.27	0.065527
72Σ+ (6s 2S)	3.02	5232	66,461	141.56	0.84	0.068222
12Π (3d 2D)	3.38	3214	11,576	98.34	0.63	0.054556
22Π (4p 2P)	3.70	1806	25,146	75.31	1.28	0.045388
32Π (4d 2D)	2.87	3668	54,393	196.64	1.25	0.075499
42Π (5p 2P)	3.10	4668	57,255	128.31	2.04	0.064074
12Δ (3d 2D)	2.99	6005	8785	166.34	1.46	0.069377
22Δ (4d 2D)	3.06	4262	53,800	132.27	1.38	0.066548

¹ Two-color photoionization; ² photodissociation spectroscopy; ³ estimated Rydberg state Re values of Ca-RG (ab initio calculation); ⁴ RCCSD(T) calculation.

The calculated harmonic frequency of the ground electronic state of Ca⁺–Kr, ω_e = 65.35 cm⁻¹, is slightly lower than both the experimental values (74–77 cm⁻¹ [29,52]) and the theoretical predictions (68–69.7 cm⁻¹ [36,51]). This deviation mainly reflects subtle differences in the curvature of the potential energy well near the equilibrium distance R_e, which determines the vibrational stiffness of the bond. Since pseudopotential methods simplify the treatment of core–valence correlation and short-range repulsion, small discrepancies in the potential shape may propagate into the calculated vibrational frequencies. Consequently, the lower value of ω_e can be attributed to the combined influence of potential curvature and the approximations inherent in the pseudopotential representation.

Similarly, the computed anharmonicity constant (ω_eχ_e = 0.94 cm⁻¹) is somewhat smaller than theoretical values reported in the range of 1.3–1.5 cm⁻¹ [36,52]. The rotational constant (B_e = 0.055683 cm⁻¹) agrees well with the theoretical estimate of 0.0570 cm⁻¹, confirming that the equilibrium geometry is well reproduced. Overall, the agreement between our calculated spectroscopic parameters and previously reported data is satisfactory. However, the remaining differences, particularly in ω_e, and ω_eχ_e, suggest that the pseudopotential model employed may slightly underestimate the strength of the Ca⁺-Kr interaction, likely due to its limited treatment of short-range repulsion and electronic correlation effects. Additionally, differences in computational methods and basis sets contribute to the observed deviations, further influencing the accuracy of the predicted molecular constants. As no reliable spectroscopic data could be found for the excited states, our comparison is limited to the ground state, and we present only our theoretical predictions for the excited electronic states.

For the neutral Ca-Kr molecule, Figure 3 displays the adiabatic potential energy curves for all states dissociating into the Ca(4s², 4s4p (^1,3P), 4s3d (^1,3D), and 4s4p (^1,3S)) + Kr atomic limits. These include the ^1,3Σ⁺, ^1,3Π, and ^1,3Δ symmetries, mapped over an internuclear distance range of 1.5 to 50 Å. The corresponding spectroscopic constants are summarized in Table 5.

Figure 3. Potential energy curves of the Ca-Kr molecule for excited states in ^1,3Σ⁺, ^1,3Π, and ^1,3Δ symmetries.

Table 5. Spectroscopic constants for the ^1,3Σ⁺, ^1,3Π, and ^1,3Δ states of the Ca-Kr neutral system without spin–orbit coupling. All constants are expressed in cm⁻¹ except R_e in Å.

States	Re	De	Te	ωe	ωeχe	Be
11Σ+	5.46	79		15.56	0.72	0.020865
Ref. [16]	5.05	97.5		14.79
Ref. [35]	5.11 1	102.7 1		18.98 1	0.98 1
Ref. [35]	5.08 2	111.9 2		20.10 2	0.96 2
21Σ+	6.06	54	22,317	7.66	0.07	0.016948
31Σ+	7.21	26	23,328	5.01	0.28	0.011982
Ref. [16]	6.33	31.9		8.06
41Σ+	4.44	433	23,387	29.36	0.40	0.031569
13Σ+	7.42	12	15,349
Ref. [16]	6.51	39.6		6.11
23Σ+	5.81	84	20,759	13.56	0.93	0.018413
33Σ+	4.32	495	30,927	39.83	0.99	0.033370
11Π	5.24	150	22,222	15.65	0.45	0.022602
21Π	5.44	128	23,285	16.06	0.67	0.021016
Ref. [16]	4.42	216.2		23.11
13Π	4.97	177	15,184	20.74	0.68	0.025107
Ref. [16]	3.80	267.7
23Π	4.87	235	20,607	22.56	0.48	0.026239
11Δ	3.43	718	21,646	63.25	2.91	0.052750
13Δ	3.35	1171	19,671	82.84	2.12	0.055210

¹ Dispersion parameters for the TT potential, nmax = 5; ² dispersion parameters for the TT potential, nmax = 8.

To the best of our knowledge, theoretical spectroscopic data for Ca-Kr remain limited. Therefore, we compare the present results for the ground electronic state 1¹Σ⁺ and selected excited states with the available theoretical studies reported in references [16,35]. In particular, reference [16] employed CCSD(T) calculations, while reference [35] used Tang–Toennies potentials truncated at different dispersion orders, n_max = 5 (denoted (a)) and n_max = 8 (denoted (b)). For the ground state, our calculated equilibrium distance (R_e = 5.46 Å) is slightly longer than the theoretical values, which range from 5.05 to 5.11 Å. The dissociation energy (D_e = 79 cm⁻¹) is smaller than the reported 97.5 and 111.9 cm⁻¹ [16,35]. The computed vibrational frequency (ω_e = 15.56 cm⁻¹) is also lower than those predicted for n_max = 5 (18.98 cm⁻¹) and n_max = 8 (20.10 cm⁻¹), showing better agreement with the latter. The anharmonic constant (ω_ex_e = 0.72 cm⁻¹) follows a similar trend, being slightly smaller than the values of 0.96 and 0.98 cm⁻¹ reported in [35]. Beyond the ground state, a comparison for selected low-lying excited states reveals generally consistent trends. For instance, in the 3¹Σ⁺ state, the calculated R_e = 7.31 Å is somewhat longer than the RCCSD(T) value (6.33 Å), while the dissociation energy (D_e = 26 cm⁻¹) agrees reasonably well with the reported 31.9 cm⁻¹ [16]. In the case of the 1³Σ⁺ state, our R_e = 7.42 Å again exceeds the theoretical value (6.51 Å), whereas D_e = 12 cm⁻¹ is notably smaller than the 39.6 cm⁻¹ reported in [16]. The vibrational frequency (ω_e = 5.01 cm⁻¹) for this state is also lower than the RCCSD(T) result (8.06 cm⁻¹). Such deviations likely arise from differences in the curvature of the potential energy wells, while variations in the pseudopotential and basis set choices may further contribute to these discrepancies. For the 2¹Π and 1³Π states, similar patterns are observed. In the 2¹Π, our calculated R_e = 5.44 Å and D_e = 128 cm⁻¹ contrast with 4.42 Å and 216.2 cm⁻¹ from Ref. [16]. For the 1³Π state, the deviation in R_e is more pronounced (4.97 Å vs. 3.8 Å), accompanied by a shallower potential well (D_e = 177 cm⁻¹ vs. 267.7 cm⁻¹).

Overall, despite some underestimations in binding energies and slightly longer equilibrium bond lengths, the general trends remain consistent with high-level theoretical predictions. This consistency confirms the reliability of the present approach in describing the essential features of the ground and low-lying excited electronic states of the Ca-Kr van der Waals system.

We have explored the difference potentials associated with transitions from the ground electronic states ns ²Σ⁺ [n(Ca) = 4] to the excited states ns ²Σ⁺ [n(Ca) = 5, 6], np ²Σ⁺ [n(Ca) = 4, 5], and nd (Ca) = 3, 4] in the Ca⁺-Kr complex. The resulting curves, shown in Figure 4, reveal that the ns ²Σ⁺ [n(Ca) = 4 → 6] transition is almost a flat difference potential, reflecting weak distance dependence. In contrast, the remaining transitions display pronounced maxima at shorter internuclear separations. These variations are directly related to the underlying potential energy surfaces, where humps and avoided crossings play a decisive role in shaping the difference potential profiles.

Figure 4. Excitation energies from ns ²Σ⁺ [n(Ca) = 4] to ns ²Σ⁺ [n(Ca) = 5, 6], np ²Σ⁺ [n(Ca) = 4, 5], and nd²Σ⁺ [n (Ca) = 3, 4] in the Ca⁺-Kr molecule.

3.2. Spectroscopic Investigation of the Ca^q+-Kr(q = 0,1) with Spin–Orbit Coupling

3.2.1. Excited States 4p (3²Σ⁺, 2²Π) and 3d (2²Σ⁺, 1²Π, and 1²Δ)

The spin–orbit interaction was introduced for the molecular states correlating with the Ca⁺(4p) + Kr and Ca⁺(3d) + Kr asymptotes. The atomic spin–orbit constants used in this treatment were taken from Ref [49], with ζ_4p (Ca⁺) = 148.6 cm⁻¹ and ζ_3d (Ca⁺) = 24.3 cm⁻¹. Diagonalization of the spin–orbit coupling matrix results in the splitting of the 3²Σ⁺, 2²Π states into fine-structure components. The resulting potential energy curves, corresponding to the Ca⁺-Kr system dissociating into Ca⁺ (4²P_1/2) + Kr and Ca⁺ (4²P_3/2) + Kr, are presented in Figure 5, and the associated spectroscopic parameters are summarized in Table 6. The magnitude of the spin–orbit splitting reflects the combined influence of the intrinsic atomic spin–orbit constant and the molecular interaction strength. For the states correlated with Ca⁺(4p) + Kr, the potential well depths differ by approximately 72 cm⁻¹, with D_e = 1733 cm⁻¹ for the 2²Π_1/2 state and D_e = 1805 cm⁻¹ for the 2²Π_3/2 state. While the excitation energy (T_e) is notably affected by the inclusion of spin–orbit coupling, the vibrational constants and equilibrium bond lengths of the 2²Π states remain essentially unchanged, indicating that the overall shape of the potential is preserved. In contrast, the 3²Σ⁺ state exhibits more pronounced spin–orbit effects. Upon inclusion of spin–orbit coupling, the equilibrium bond length contracts from 5.96 Å to 5.86 Å, while the dissociation energy increases from 170 cm⁻¹ to 213 cm⁻¹. The excitation energy (T_e) is also shifted by approximately 43 cm⁻¹ (see Table 5). These variations highlight the sensitivity of the Σ states to relativistic effects and demonstrate the subtle interplay between electrostatic and spin–orbit interactions in the Ca⁺-Kr system.

Figure 5. The potential energy curves of Ca⁺-Kr correlating to Ca⁺(4²P_1/2) + Kr and Ca⁺(4²P_3/2) + Kr after including the spin–orbit coupling.

Table 6. Spectroscopic constants for the ²Σ⁺, ²Π, and ²Δ states of the Ca⁺-Kr ionic system with spin–orbit coupling. All constants are expressed in cm⁻¹, except R_e in Å.

Ca+-Kr	Re	De	Te	ωe	ωeχe	Be
22Σ+ (3d 2D)	3.49	1736	13,048	70.54	1.40	0.050963
22Σ1/2+ (3d 2D1/2)	3.49	1765	13,054	70.56	1.4	0.050965
12Π (3d 2D)	3.38	3214	11,576	98.34	0.63	0.054556
12Π1/2 (3d 2P1/2)	3.38	3212	11,563	98.65	0.73	0.054555
12Π3/2 (3d 2P3/2)	3.38	3214	11,588	98.63	0.73	0.054553
12Δ (3d 2D)	2.99	6005	8785	166.34	1.46	0.069377
12Δ3/2 (3d2D3/2)	3.00	5973	8765	165.28	1.34	0.069130
12Δ5/2 (3d2D5/2)	3.00	6000	8814	165.29	1.34	0.069130
32Σ+ (4p 2P)	5.96	170	26,781	17.65	0.70	0.017527
32Σ1/2+ (4p2P1/2)	5.86	213	26,813	20.01	0.78	0.018102
22Π (4p 2P)	3.70	1806	25,146	75.31	1.28	0.045388
22Π1/2 (4p2P1/2)	3.70	1733	25,070	74.79	0.95	0.045285
22Π3/2 (4p2P3/2)	3.70	1805	25,220	74.88	0.94	0.045295

Spin-orbit coupling was also included for the molecular states correlated with the Ca⁺(3d) + Kr dissociation limit, which give rise to the ²Σ⁺, ²Π, and ²Δ symmetries. Near the equilibrium geometry, these states follow the energetic order ²Δ < ²Π < ²Σ⁺. At large internuclear separations, they correlate asymptotically with the fine-structure components of the Ca⁺(3d) atomic term: 3d ²D_1/2, 3d ²D_3/2, and 3 d²D_5/2. The corresponding spectroscopic constants for the molecular states (2²Σ⁺_1/2, 1²Π_1/2, 3d ²D_3/2, 1²Π_3/2, 1²Δ_3/2, 3d ²D_5/2, and 1²Δ_5/2) are presented in Table 6 and compared with the spin-free results.

The inclusion of spin–orbit coupling produces only minor changes in the spectroscopic parameters of these states. The equilibrium bond lengths remain essentially unchanged, while the potential well depths differ slightly by about 29, 2, 32, and 5 cm⁻¹ for the 2²Σ⁺_1/2, 1²Π_1/2, 1²Δ_3/2, and 1²Δ_5/2 states, respectively. The excitation energies T_e are shifted by 13, 12, 24, and 25 cm⁻¹ for the 2²Σ⁺_1/2, 1²Π_1/2, 1²Π_3/2, 1²Δ_3/2, and 1²Δ_5/2 states, respectively. These small energy shifts confirm that the spin–orbit interaction has only a modest influence on the potential energy surfaces associated with the Ca⁺(3d) + Kr asymptote. This weak effect is consistent with the relatively small atomic spin–orbit constant of the Ca⁺(3d) state (ξ_3d = 24.3 cm⁻¹), compared with that of Ca⁺(4p). Consequently, the splitting of the molecular potential curves correlating with Ca⁺(3d) + Kr is considerably smaller than that found for the Ca⁺(4p) + Kr limit. Notably, the energetic ordering of the ²Δ, ²Π, and ²Σ⁺ states remain unchanged upon inclusion of spin–orbit coupling, confirming the stability of the electronic structure in this region of the potential surface.

3.2.2. Excited States 4p (1³Σ⁺, 1³Π, 3¹Σ⁺, 2¹Π) and 3d (2³Σ⁺, 2³Π, 1³Δ, 2¹Σ⁺, 1¹Π, and 1¹Δ)

The potential energy curves (PECs) of the Ω states for the Ca-Kr molecule were determined by including spin–orbit coupling over a wide range of internuclear separations. The results show that the molecule dissociates into eight states for each of the Ω = 0⁻ and 0⁺ symmetries, and into seven, four, and one states for Ω = 1, 2, and 3, respectively. Figure 6 presents the PECs corresponding to the electronic states dissociating into Ca(4s4p ³P₀, ₁, ₂) + Kr and Ca(4s4p ¹P₁) + Kr, respectively. These curves are predominantly attractive and display several avoided crossings, resulting in a complex pattern characterized by multiple shallow wells. As shown in Figure 6, the 1³Π state splits into four components; 1³Π₀⁻, 1³Π₀⁺, 1³Π₁, and 1³Π₂, with an overall energy difference of approximately 55 cm⁻¹ between the lowest and highest components. In contrast, the spin–orbit interaction has a much weaker influence on the 3¹Σ⁺ and 2¹Π states, resulting in comparatively minor splittings and preserving the general shape of the potential curves.

Figure 6. The potential energy curves of the electronic states for Ca-Kr including spin–orbit coupling. The full line represents states with spin–orbit coupling, while the dashed line represents states without spin–orbit coupling.

The 1³Π state exhibits the largest spin–orbit splitting, consistent with its relatively high spin–orbit constant. However, the states dissociating into Ca(4s3d ³D_{1, 2, 3}) and Ca(4s3d ¹D₂) + Kr, (Ω = 0⁺, 0⁻, 1, 2, and 3) display much weaker splittings than those correlating with Ca(4s4p ³P_{0, 1, 2}) + Kr and Ca(4s4p ¹P₁) + Kr. This reduction arises from the smaller atomic spin–orbit constant of the 3d shell (ξ_4p(Ca) = 105.8 cm⁻¹ vs. ξ_3d(Ca) = 13.9 cm⁻¹ [49]). Table 7 summarizes the spectroscopic constants, including spin–orbit coupling effects, for the corresponding Ω excited states derived from the parent singlet and triplet states (1³Σ⁺, 1³Π, 3¹Σ⁺, 2¹Π, 2³Σ⁺, 2³Π, 1³Δ, 2¹Σ⁺, 1¹Π, and 1¹Δ). Overall, our results for the low-lying states of Ca-Kr are in reasonable agreement with those reported in Ref. [16], though some notable differences are apparent. For instance, the equilibrium bond lengths R_e for the 3¹Σ⁺₀⁺ and 2¹Π₁ states are predicted here as 7.21 Å and 5.44 Å, compared with 6.29 Å and 4.37 Å in Ref. [16], indicating slightly longer bonds. Likewise, the dissociation energies D_e are also smaller (26.4 cm⁻¹ versus 62.1 cm⁻¹ for the 3¹Σ⁺₀⁺ state and 128 cm⁻¹ versus 226.9 cm⁻¹ for the 2¹Π₁ state) suggesting weaker binding predicted by our approach. In contrast, the 1³Σ⁺₁ state shows good agreement, with R_e differing by only ~0.47 Å and D_e values of 28 and 33.9 cm⁻¹. Overall, while both studies predict similar state ordering and vibrational behavior, our calculations yield slightly longer bond lengths and smaller well depths.

Table 7. Spectroscopic constants for Ca(4s4p ^1,3P_J)Kr and Ca(4s3d ^1,3D_J)Kr with spin–orbit coupling; all constants are expressed in cm⁻¹ except R_e in Å.

Molecular State	Re	De	Te	ωe	ωeχe	Be
(3) 1Σ+0+	7.21	26.4	23,387	5.96	0.33	0.01196
Ref. [16]	6.29	62.1
(2) 1Π1	5.44	128	23,286	14.91	0.43	0.02104
Ref. [16]	4.37	22 6.9
(1) 3Σ+0−	7.04	23	15,392	6.68	0.49	0.01258
(1) 3Σ+1	6.94	28	15,386	7.37	0.48	0.01289
Ref. [16]	6.47	33.9
(1) 3Π0−	5.00	131	15,124	20.23	0.78	0.02487
(1) 3Π0+	4.95	177	15,130	20.91	0.62	0.02512
(1) 3Π1	4.99	128	15,180	20.49	0.82	0.02498
(1) 3Π2	4.99	177	15,237	20.91	0.62	0.02512
(2) 1Σ+0+	6.05	54	22,430	6.86	0.22	0.01699
(1) 1Π1	5.24	150	22,334	16.21	0.62	0.22598
(1) 1Δ2	3.80	610	21,875	94.63	3.67	0.04290
(2) 3Σ+0−	5.97	95	20,874	15.61	1.62	0.01856
(2) 3Σ+1	5.79	96	20,873	15.59	1.62	0.01855
(2) 3Π0+	4.86	235	20,713	22.22	0.55	0.02628
(2) 3Π1	4.87	228	20,720	22.24	0.56	0.02626
(2) 3Π2	4.87	248	20,727	22.39	3.07	0.00108
(1) 3Δ0−	3.35	1164	19,770	83.4	2.31	0.05520
(1) 3Δ1	3.35	1157	19,777	83.4	2.31	0.05520
(1) 3Δ2	3.35	1157	19,797	83.40	2.31	0.05520
(1) 3Δ3	3.35	1171	19,798	83.4	2.31	0.05203

3.3. Permanent and Transition Dipole Moments for Ca⁺-Kr and Ca-Kr Systems

Understanding transition and permanent electric dipole moments (TEDMs and PEDMs) is crucial for predicting lifetimes and analyzing molecular spectra and devising strategies for molecular formation. In this work, we calculate and analyze the permanent (PEDMs) and transition (TEDMs) electric dipole moments of the Ca⁺-Kr and Ca-Kr systems in their ground and low-lying electronic states. The calculations were carried out across an extensive range of internuclear distances (1.6 to 50 Å) for all considered electronic states. Specifically, the transition dipole moments were determined from the ground state (1²Σ⁺) to the excited states (3²Σ⁺ and 2²Π) with the inclusion of spin–orbit coupling. Figure 7 depicts the variation of these transition dipole moments with internuclear distance for the 2²Π and 3²Σ⁺ states. The transition dipole from the ground state to both 2²Π and 3²Σ⁺ states exhibit a peak of 2.27 a.u, occurring at 4.57 Å internuclear distance. When spin–orbit coupling is included, the 1²Σ⁺ → 2²Π transition splits into two components, 1²Σ⁺ → 2²Π_1/2 and 1²Σ⁺ → 2²Π_3/2, which show noticeable differences at intermediate distances. At long range, transitions 1²Σ⁺ → 2²Π_1/2, 1²Σ⁺ → 2²Π_3/2, and 1²Σ⁺ → 3²Σ⁺_1/2 converge toward the Ca⁺(4s) → Ca⁺(4p) atomic transition limit. In addition, the permanent and transition dipole moments of the ^1,3Σ⁺, ^1,3Π, and ^1,3Δ symmetries of the Ca-Kr neutral system were examined over the same distance range (1.6 to 50 Å). The permanent dipole moment curves for the ^1,3Σ⁺ states are presented in Figure 7a, while Figure 7b illustrates those of the ^1,3Π and ^1,3Δ symmetries. In general, the triplet ³Σ⁺ states exhibit larger permanent dipole moments than their singlet counterparts, although this trend does not consistently extend to the Π and Δ symmetries. At short internuclear separations, the permanent dipole moment curves show marked variations, before smoothly approaching zero at long range. This trend reflects the weakly bound van der Waals nature of the system and confirms the absence of charge transfer between Ca and Kr atoms.

Figure 7. Permanent dipole moments of the Ca-Kr molecular system for the ^1,3Σ⁺ (a), ^1,3Π, and ^1,3Δ (b) symmetries.

Figure 8 depicts the transition dipole moments between the ^1,3Σ⁺, ^1,3Π, and ^1,3Δ states. The curves display pronounced maxima at short and intermediate distances, then gradually decay to zero or reach a nearly constant value at large separations, reflecting the transition to pure atomic character.

Figure 8. Transition dipole moments for the Ca-Kr molecular system in the ^1,3Σ⁺, ^1,3Π, and ^1,3Δ symmetries.

3.4. Vibrational Level Spacings, Absorption Spectra, and Radiative Lifetimes

By solving the radial Schrödinger equation, we have computed the vibrational energy levels of the Ca⁺-Kr systems with spin–orbit effects. The vibrational energy level spacing (E_v+1 − E_v) as a function of v for Ca⁺-Kr in the 3²Σ⁺and 2²Π states, correlating with the Ca⁺(4p) + Kr limit, is shown in Figure 9. The common behavior is a decrease as v rises and when approaching the dissociation limit, the energy level spacing vanishes. For the 3²Σ⁺ and 2²Π states, 26 and 65 vibrational levels were identified, with well depths of D_e = 267 cm⁻¹ and 1846 cm⁻¹, respectively. The spin–orbit components 3²Σ⁺_1/2, 2²Π_1/2, and 2²Π_3/2 exhibit the same ordering.

Figure 9. Vibrational energy level spacing (E_v+1 − E_v) in terms of the vibrational number v of Ca⁺-Kr.

Figure 10 displays the absorption spectrum of the Ca⁺-Kr molecule for transitions from the ground vibrational level (v = 0) of the 1²Σ⁺ state to the vibrational levels of the 2²Π state. As can be seen, the absorption intensity increases from v = 0 to v = 3, where it reaches its maximum, and then gradually decreases and becomes very weak for higher levels. This behavior is a direct consequence of the Franck–Condon principle. Because the potential well of the excited state (2²Π) is slightly displaced relative to that of the ground state (1²Σ⁺) and their equilibrium bond lengths are relatively close (R_e = 3.34 Å and 3.70 Å for 1²Σ⁺ and 2²Π, respectively), significant vibrational overlap occurs only for the lowest levels. As a result, absorption is restricted to transitions involving the lowest v, producing the intense peaks observed at the beginning of the spectrum. These spectral features clearly highlight how strongly the absorption intensities depend on the shapes and relative positions of the potential energy curves.

Figure 10. Absorption spectrum 1²Σ⁺ (v = 0) → 2² (v = 0–65) of the Ca⁺-Kr molecule.

Figure 11 presents the vibrational radiative lifetimes calculated for the first excited 2²Π state of Ca⁺-Kr. In these calculations, both bound–bound and bound–free transitions (1²Σ⁺ → 2²Π) were taken into account to properly describe the decay of all vibrational levels supported by the potential well. As shown in Figure 12, the vibrational radiative lifetime of the lowest vibrational level (v^′ = 0) of the 2²Π state is approximately 8.3 ns. The vibrational lifetimes remain on the nanosecond scale, decreasing gradually from about 8.3 ns and approaching near 7.3 ns for higher vibrational levels (v^′ ≈ 60–65). At these high levels, the molecule behaves nearly atom-like, and the corresponding lifetimes converge toward the atomic Ca⁺(4p) → Ca⁺(4s) transition. The agreement with the experimental lifetime of the Ca⁺ 4²P state, τ = (6.72 ± 0.20) × 10⁻⁹ s [53], is good.

Figure 11. Radiative lifetimes of the vibrational levels of the first excited 2²Π state of the Ca⁺-Kr system.

Figure 12. Franck–Condon factors between vibrational levels of the ground state 1²Σ⁺ and the first excited 3²Σ⁺ (a), 2²∏ (b), 2²∏_1/2 (c), and 2²∏_3/2 (d) states of the Ca⁺-Kr system.

In addition, we computed the Franck–Condon factors (FCFs) for the Ca⁺-Kr system, including spin–orbit (SO) coupling (2²Π_1/2, 2²Π_3/2). The resulting three-dimensional intensity maps are presented in Figure 12. These FCFs quantify the overlap between the vibrational wavefunctions of the ground 1²Σ⁺ state and those of the spin–orbit split excited states. The inclusion of SO coupling leads to Franck–Condon patterns that remain very similar to those obtained in the spin-free case, indicating that SO effects only weakly perturb the vibrational structure of the Ca⁺-Kr system. The FCFs display predominantly off-diagonal transitions, with the strongest contributions occurring at intermediate and high vibrational levels of the excited states, consistent with the relative displacement of the 1²Σ⁺ and 2²Π/3²Σ⁺ potential wells.

For the neutral Ca-Kr system, the Franck–Condon factors were also evaluated. As presented in Figure 13, the FCFs are off-diagonal and the vibrational overlap is weak for the (1-3) ¹Σ⁺ states. In contrast, for the 1 ¹Σ⁺-2 ¹Π transition, the FCFs are predominantly diagonal, with values close to 1 (f₀₀ = 0.995, f₁₁ = 0.978, f₂₂ = 0.940, and f₃₃ = 0.856) for the (0-0), (1-1), (2-2), and (3-3) transitions, respectively.

Figure 13. Franck–Condon factors between vibrational levels of the ground state 1¹Σ⁺ and the first excited 2²∏ (a) and 3²Σ⁺ (b) states of the Ca-Kr system.

4. Conclusions

We have carried out a theoretical investigation of the interactions between alkaline-earth species (Ca²⁺, Ca⁺, and Ca) and a krypton atom. Semi-local pseudopotentials were employed to model the Ca²⁺ cores and electron–Kr interactions. For the Ca⁺-Kr system, electronic structure calculations were carried out using the one-electron SCF method. In the case of the neutral Ca-Kr system, the treatment was further refined with FCI calculations. From these calculations, we obtained PECs and the associated spectroscopic constants, which had shown good agreement with available experimental and theoretical results. The resulting spectroscopic constants and vibrational spacings obtained for the 2²Π_1/2, 2²Π_3/2, and 3²Σ⁺_1/2 states are in reasonably good agreement with available experimental and theoretical [25,28,29,36,50,51] results. Importantly, the inclusion of spin–orbit coupling has also induced a splitting of the transition dipole moment. Only a limited number of theoretical works have addressed excited Ca-Kr states [16,35]. The present work extends these efforts by providing transition dipole moments near avoided crossings and offering spectroscopic data that can guide future experiments. These findings can be expanded to incorporate a more systematic treatment of spin–orbit effects or to explore larger van der Waals clusters (Ca²⁺-Kr_n, Ca⁺-Kr_n, Ca-Kr_n). Moreover, the accurate PECs and dipole moments obtained for Ca⁺-Kr will be valuable for modeling Ca⁺ line broadening in collisions with Kr atoms and for evaluating transport coefficients relevant to Ca⁺ propagation in a krypton environment.

Requests for data on potential energy curves and permanent and transition electric dipole moments as a function of interatomic distance can be made to the authors.