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. 2023 May 16;127(20):4473–4482. doi: 10.1021/acs.jpca.3c02056

Excited Electronic States of Sr2: Ab Initio Predictions and Experimental Observation of the 21Σu+ State

Jacek Szczepkowski †,*, Marcin Gronowski ‡,*, Anna Grochola †, Włodzimierz Jastrzebski †, Michał Tomza ‡, Paweł Kowalczyk §
PMCID: PMC10226130  PMID: 37192534

Abstract

graphic file with name jp3c02056_0006.jpg

Despite its apparently simple nature with four valence electrons, the strontium dimer constitutes a challenge for modern electronic structure theory. Here we focus on excited electronic states of Sr2, which we investigate theoretically up to 25000 cm–1 above the ground state, to guide and explain new spectroscopic measurements. In particular, we focus on potential energy curves for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu states computed using several variants of ab initio coupled-cluster and configuration-interaction methods to benchmark them. In addition, a new experimental study of the excited 21Σu+ state using polarization labeling spectroscopy is presented, which extends knowledge of this state to high vibrational levels, where perturbation by higher electronic states is observed. The available experimental observations are compared with the theoretical predictions and help to assess the accuracy and limitations of employed theoretical models. The present results pave the way for future more accurate theoretical and experimental spectroscopic studies.

Introduction

Diatomic molecules at ultralow temperatures are a perfect platform for research touching upon the very fundamentals of quantum physics and chemistry.1 Ultracold polar molecules have been proposed and employed for a plethora of ground-breaking experiments ranging from quantum-controlled collisions and chemical reactions2 to quantum simulations3 and precision measurements of fundamental constants and their spatiotemporal variation.4 After spectacular successes with alkali-metal molecules, which can be efficiently formed from ultracold atoms using magnetoassociation5 followed by optical stabilization,6 the production of ultracold molecules containing alkaline-earth-metal atoms has emerged as another important research goal.

Recently, an ultracold gas of Sr2 dimers in their absolute ground state was obtained using all-optical methods, where weakly bound singlet-state molecules were formed in an optical lattice by narrow-line photoassociation and transferred to the ground rovibrational level by the stimulated Raman adiabatic passage (STIRAP).7 Fast chemical reactions between such dimers were observed close to the universal limit. Nevertheless, ultracold Sr2 molecules have already been employed in a series of exciting experiments ranging from studying asymptotic physics in subradiant states8 to photodissociation with quantum state control.9 Very recently, a new type of molecular lattice clock based on ultracold Sr2 dimers with long vibrational coherence has also been established.10,11 This paves the way for upcoming applications of these molecules in quantum simulation,12 quantum metrology,13 and precision measurements probing the fundamental laws of nature.14,15

Exciting developments and applications of ultracold molecules described above would not have been feasible without thorough experimental spectroscopic analysis and substantial theoretical ab initio electronic structure evaluations of the underlying molecular structure. The required level of accuracy varies for each application. Generally, precise measurements can provide more accurate outcomes than theoretical calculations. However, ab initio quantum-chemical calculations of potential energy curves, permanent and transition electric dipole moments, and other molecular couplings are frequently necessary to propose, guide, and explain experimental endeavors.

Alkaline-earth-metal diatomic molecules, despite their apparently simple nature with four valence electrons and closed-shell ground electronic state, have constituted a challenge for modern electronic structure theory. Already the simplest Be2 dimer presents unusually strong bonding and unique shape of the ground-state potential energy curve,16 which accurate theoretical description required highly correlated methods.17 Confirming the existence of elusive vibrational states of the ground-state Mg2 dimer also needed state-of-the-art quantum-chemical calculations.18 Thus, it is not surprising that the accurate theoretical description of the Sr2 dimer in the ground and excited electronic states may require careful treatment, similar to lighter neutral dimers or charged Sr2+ molecular ion.19

The ground X1Σg+ and excited 21Σu and 31Πu electronic states of Sr2 were initially investigated experimentally with absorption and laser-induced fluorescence spectroscopy,20−22 followed by high-resolution Fourier-transform laser-induced fluorescence spectroscopy of the X1Σg+ state23,24 and the minimum region of the excited 11Σu, 11Πu, and 21Σu+ states.25 Recently, highly accurate measurements with ultracold Sr2 allowed for improving the accuracy of rovibrational spectra of the X1Σg and 11Σu+ states.7 The ground and excited electronic states of Sr2 were also investigated theoretically using different computational approaches, including large-core semiempirical pseudopotentials,26,27 small-core relativistic pseudopotentials,28 and all-electron relativistic Hamiltonian.29 The challenging character of calculations for excited molecular electronic states could be seen in contradictory dissociation energies for the lowest-excited 11Σu and 11Πu states reported without detailed estimates of computational uncertainties. Several other studies focused solely on the ground X1Σg+ electronic state,30−35 which is already well understood.

In this work, we investigate the excited electronic states of the Sr2 molecule. We start with the computational evaluation of the complete molecular electronic spectrum up to the excitation energy of around 25000 cm–1. Next, we compute potential energy curves for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu states using several variants of advanced ab initio methods. New experimental measurements of the excited 21Σu+ state using polarization labeling spectroscopy are presented, extending the range of observed vibrational levels to higher energies. The corresponding Dunham coefficients and experimental potential energy curve are reported. The observed perturbations in the recorded spectrum give preliminary information on higher-excited electronic states. The comparison of the experimental observations with several theoretical predictions helps to assess and benchmark the accuracy and limitations of employed theoretical models.

Electronic Structure Calculations

Computational Methods

Several computational approaches were used in the electronic structure calculations to assess and benchmark their accuracy. The all-electron computations employed the eXact-2-Component Hamiltonian36 and the equation-of-motion coupled cluster method37 with single and double excitations (EOM-CCSD)38 with the relativistic correlation-consistent core–valence quadruple-ζ basis sets (aug-cc-pwCVQ-X2C)39 in the version implemented in the Molpro 2022.1 program.40 We explored the impact of the core–electron correlation on the results by correlating only valence electrons (denoted as x2cCCSDv), valence and 4s4p electrons (denoted as x2cCCSDbc), as well as valence, 4s4p, and 3s3p3d electrons (denoted as x2cCCSDsc).

Other equation-of-motion coupled cluster computations used the small-core relativistic energy-consistent ECP28MDF pseudopotential41,42 with the quadruple- and quintuple-zeta pseudopotential-based correlation-consistent polarized core–valence basis sets (aug-cc-pCVQZ-PP and aug-cc-pCV5Z-PP, donated as QZ and 5Z, respectively)39 in the Cfour 2.1 software.43 We obtained the complete basis set (CBS) limit with two-point 1/X3 extrapolation.44 To estimate the role of the higher excitations, we compared EOM-CCSD (denoted as ecpCCSD) and EOM-CCSDT-345,46 (denoted as ecpCCSDT3).

We also performed multireference computations with the standard Davidson correction.47 We described the valence-electron correlation by the multiconfiguration reference internally contracted configuration interaction method48−50 with the active space composed of 20 (sMRCI+Q) or 24 (MRCI+Q) orbitals. Such a sizable active space is necessary to correctly describe the 21Πu state of Sr2. The orbitals were optimized at the complete active space self-consistent field (CASSCF) level.51 Additionally, we employed the hybrid CIPT2+Q method,52 which adds the core–electron correlation to MRCI+Q by the multireference Rayleigh–Schrödinger second-order perturbation theory. All multireference computations used the aug-cc-pwCV5Z-PP basis set39 and were performed with the Molpro 2022.1 program. Since we had a problem converging calculations for monomers in the dimer basis set, we assumed that the basis set superposition error for multireference methods is the same as for ecpCCSD. This assumption is well justified as the total basis set superposition error is relatively small for Sr2.

We shifted the computed interaction energies by the sum of the appropriate experimentally measured atomic excitation energies from the NIST database53 and 1081.64 cm–1, corresponding to the molecular ground state’s depth.24 This procedure guarantees that the reported energies are relative to the ground state’s minimum and tend to the corresponding atomic values in their asymptotes.

Theoretical Results

Figure 1 shows an overview of potential energy curves (PECs) computed with the sMRCI+Q/5Z approach, which is often the method of choice to study the excited electronic states of diatomic molecules. However, in the case of Sr2, this approach has some shortcomings, as we shall discuss later. The main conclusion from the overview of excited states is that the 21Σu+ state is fairly well separated from other states, and, thus, a relatively small number of perturbations from other states should be expected.

Figure 1.

Figure 1

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Potential energy curves for the ground and excited electronic states of Sr2 obtained in the nonrelativistic spin-free sMRCI+Q/5Z computations with the scalar-relativistic small-core pseudopotential. The states are labeled by symmetry and asymptote or only by symmetry in the case of states involving asymptotes for which numerical difficulties prevent obtaining whole PECs.

The strontium dimer contains 76 electrons. Therefore, a precise quantum-mechanical description is challenging. Additionally, the large charge of its nuclei limits the applicability of nonrelativistic quantum mechanics. Thus, an accurate description of Sr2 has to include: (i) extensive orbital basis set, (ii) valence-electron correlation, (iii) core–electron correlation, (iv) scalar relativistic contribution, (v) spin-related relativistic effects, like fine and hyperfine couplings, and (vi) leading quantum electrodynamic corrections. However, it is currently only feasible to simultaneously account for some of these effects for many-electron molecules. Here, we neglect spin-related and quantum electrodynamic contributions (v and vi) and explore the sensitivity of the potential energy curves to the remaining contributions (i–iv), which are usually the most crucial for reaching quantitative description of any molecule. The spin–orbit coupling, the largest neglected contribution, can be perturbatively added in the next steps.28,54

Figure 2 presents the PECs for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu electronic states obtained at several different levels of theory. Corresponding spectroscopic parameters are collected in Table 1. We selected these singlet ungerade states for detailed computational tests because they are the most relevant for parallel spectroscopic measurements.

Figure 2.

Figure 2

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Potential energy curves for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu electronic states of Sr2 obtained with different computational methods. See the text for details. Thre potentials for 11Σu+, 21Σu, and 11Πu are compared with the experimental curves from the present work and ref (25).

Table 1. Origin of the State Te with Respect to the Minimum of the Ground Electronic State, Dissociation Energy Ee, Equilibrium Bond Length Re, Harmonic Frequency ωe, Equilibrium Rotational Constant Be, and Distortion Constant De for Selected States of Sr2, as Predicted by Various Quantum Chemical Computationsa.

state method Te (cm–1) Ee (cm–1) Re (Å) ωe (cm–1) Be (cm–1) De (10–9 cm–1)
11Σu+ ecpCCSDT3/QZ 13013 8219 4.035 74.94 0.02356 9.32
11Σu+ ecpCCSDT3/5Z 13010 8221 3.982 78.17 0.02419 9.27
11Σu+ ecpCCSDT3/CBS 12984 8247 3.928 81.54 0.02486 9.24
11Σu+ ecpCCSD/5Z 13693 7538 4.017 76.41 0.02376 9.20
11Σu+ sMRCI+Q/5Z 12649 8582 4.209 71.25 0.02164 7.99
11Σu+ MRCI+Q/5Z 12910 8321 4.181 71.12 0.02194 8.36
11Σu+ CIPT2+Q/5Z 12502 8729 3.93 77.82 0.02484 10.0
11Σu+ x2cCCSDv 13373 7858 4.272 69.86 0.02102 7.61
11Σu+ x2cCCSDbc 13731 7500 4.037 75.28 0.02354 9.20
11Σu+ x2cCCSDsc 13673 7559 4.044 74.81 0.02345 9.22
11Σu+ theory26 12363   3.850 79 0.0259  
11Σu+ theory27   5490 4.01 80.21    
11Σu+ theory29 17269 5475 4.02 88    
11Σu+ theory28   8433 3.99      
11Σu+ exp.25 12796(2)   3.95(1) 80.71(3) 0.024794(2)  
11Δu ecpCCSDT3/QZ 16605 4627 3.939 85.34 0.02472 8.30
11Δu ecpCCSDT3/5Z 16580 4652 3.921 85.74 0.02494 8.44
11Δu ecpCCSDT3/CBS 16550 4681 3.903 86.34 0.02518 8.56
11Δu ecpCCSD/5Z 17626 3605 3.974 80.18 0.02429 8.92
11Δu sMRCI+Q/5Z 15864 5367 4.041 85.22 0.02349 7.14
11Δu MRCI+Q/5Z 15646 5585 4.031 84.99 0.02361 7.28
11Δu CIPT2+Q/5Z 15786 5446 3.902 113.1 0.02519 5.00
11Δu x2cCCSDv 17393 3839 4.106 78.01 0.02275 7.74
11Δu x2cCCSDbc 17635 3596 3.982 79.82 0.02418 8.88
11Δu x2cCCSDsc 17595 3636 3.98 80.09 0.02421 8.84
11Δu theory26 16158   3.868 82 0.0257  
11Πu ecpCCSDT3/QZ 17331 3901 4.123 84.6 0.02256 6.42
11Πu ecpCCSDT3/5Z 17510 3722 4.107 84.67 0.02274 6.56
11Πu ecpCCSDT3/CBS 17695 3536 4.089 84.79 0.02293 6.71
11Πu ecpCCSD/5Z 19324 1907 4.217 75.01 0.02157 7.13
11Πu sMRCI+Q/5Z 14741 6490 4.173 85.77 0.02203 5.81
11Πu MRCI+Q/5Z 14530 6701 4.159 85.96 0.02217 5.90
11Πu CIPT2+Q/5Z 15933 5298 3.992 89.6 0.02407 6.94
11Πu x2cCCSDv 17530 3701 4.278 80.66 0.02096 5.66
11Πu x2cCCSDbc 19299 1932 4.226 75.57 0.02148 6.94
11Πu x2cCCSDsc 19219 2012 4.223 75.88 0.0215 6.91
11Πu theory26 16243   3.952 96 0.0246  
11Πu theory29 18658 4081 3.93 72    
11Πu exp.25 16617.86(2)   4.0473(2) 86.300(3) 0.023415(2) 6.943
21Σu+ ecpCCSDT3/QZ 18193 4587 4.192 82.78 0.02183 6.07
21Σu+ ecpCCSDT3/5Z 18000 4780 4.19 82.17 0.02185 6.18
21Σu+ ecpCCSDT3/CBS 17797 4983 4.187 81.52 0.02187 6.30
21Σu+ ecpCCSD/5Z 18858 3922 4.276 71.96 0.02097 7.13
21Σu+ sMRCI+Q/5Z 17674 5106 4.273 82.66 0.021 5.42
21Σu+ MRCI+Q/5Z 17182 5599 4.277 84.23 0.02096 5.19
21Σu+ CIPT2+Q/5Z 17145 5635 4.145 88.54 0.02233 5.68
21Σu+ x2cCCSDv 19297 3484 4.324 74.9 0.02051 6.16
21Σu+ x2cCCSDbc 18914 3866 4.283 72.26 0.02091 7.01
21Σu+ x2cCCSDsc 18959 3821 4.276 72.81 0.02097 6.96
21Σu+ present exp. 17358.70(10)   4.176(1) 84.2169(29) 0.021990(14) 7.12(6)
21Σu+ exp.25 17358.75(1)   4.1783(1) 84.215(1) 0.021969(1) 5.98
21Πu CCSDT-3/CBS 23846 –1066 5.715 7.442 0.01174 117
21Πu sMRCI+Q/5Z 20805 1975 3.739 104 0.02743 7.64
21Πu MRCI+Q/5Z 19921 2859 3.677 137.1 0.02836 4.86
21Πu CIPT2+Q/5Z 19241 3539 3.533 142.9 0.03074 5.69
21Πu present exp. >19100          
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a

These values are compared with experimental results whenever possible. The values in parentheses are experimental uncertainties in units of last digits.

The first observation is that the PECs exhibit relatively low sensitivity to the orbital basis set size, meaning that calculations in the quadruple- and quintuple-zeta basis sets are already close enough to the complete basis set limit. This can be demonstrated by comparing the results in the quintuple-zeta basis set with the estimated CBS (ecpCCSDT3/5Z vs ecpCCSDT3/CBS). The difference in the dissociation energy is of the order of 30 cm–1 for the 11Σu+ and 11Δu states and of the order of 200 cm–1 for the 11Πu and 21Σu states.

Next, we observe that correlating only valence electrons is insufficient. We systematically investigate the effect of the core–electron correlation by changing the size of the frozen core in the all-electron equation-of-motion coupled-cluster computations with the eXact-2-Component Hamiltonian (x2cCCSDv vs x2cCCSDbc vs x2cCCSDsc). The lack of the core correlation not only alters the depth of PEC by hundreds of cm–1, but mainly elongates the equilibrium distance. Surprisingly, the core correlation also affects the asymptotic region of the 11Σu+ and 21Σu states. On the other hand, the correlation of the valence and 4s4p electrons is sufficient, and the correlation of the electrons occupying the lower orbitals (replaced by the small-core pseudopotential) is not necessary.

Finally, we address the relativistic effects. Our calculations include the scalar relativistic effects only. We do not observe substantial differences between the all-electron x2cCCSDsc and small-core-pseudopotential ecpCCSD results. Therefore, we conclude that using the small-core ECP28MDF pseudopotential to account for the scalar relativistic effects is justified and sufficient, which is in agreement with other studies.28,55 Our calculations do not account for the spin-related part of the Dirac-Coulomb-Breit Hamiltonian. It is necessary to go beyond this approximation to correctly describe the crossing between states of different spin multiplicity coupled by spin–orbit coupling, which can be added to our curves perturbatively. For example, this coupling is important for the 11Σu+ state at distances larger than 4.5 Å. Kotochigova29 included the spin–orbit part in her computations directly, but her results for 11Σu significantly deviate from modern experimental results25 and our present calculations due to her approximate treatment of the electron correlation by the configuration interaction valence bond self-consistent-field approach. In contrast, Skomorowski et al.28 included the electron correlation directly and spin–orbit interaction perturbatively for 11Σu+ with the coupled cluster method and small-core pseudopotential, and obtained a much better agreement with the experiment.

Overall, the primary factor determining the accuracy of the calculations for Sr2 is the inclusion of high excitations in the description of the valence-electron correlation. For the analyzed electronic states, the difference between the ecpCCSDT3 and ecpCCSD results, that is, the inclusion of triple excitations, is more significant than the effect of the core–electron correlation. Among the methods used, the ecpCCSDT3 and CIPT2+Q approaches include the core–electron correlation and a substantial part of excitations higher than doubles. The CIPT2+Q method is a multireference approach that includes all possible excitations within the active space. Thus, CIPT2+Q accounts well for the static correlation but gives only an approximate description of the core–electron dynamic correlation. This approximation was necessary since an alternative approach, based on the multireference configuration interaction method with a large active space that correlates core electrons, goes beyond the technical capabilities of modern quantum-chemical programs. On the other hand, the single-reference ecpCCSDT3 approach accounts for the dynamic correlation of core and valence electrons but only includes a fraction of triple and higher excitations.

The importance of higher excitations and multireference nature can be seen by analyzing the wave functions. We inspected the squares of reference coefficients obtained with the sMRCI+Q method at R = 4.13 Å, close to respective equilibrium distances. We found that the electronic wave function of the 11Σu+ state consists mostly of single-excited determinants (72%), and the role of excitations higher than double is negligible (3%). Therefore, it is not surprising that for this state, we observe the smallest difference between energies obtained with the ecpCCSDT3 and CIPT2+Q approaches, which also agree well with another single-reference calculation reported by Skomorowski et al.28 The slightly smaller role of single-excited determinants is visible for the 21Σu and 11Δu states (about 66%), where the difference between PECs calculated with the ecpCCSDT3 and CIPT2+Q methods is larger. Still, the role of triple excitations for these states is below 5%, and higher excitations are an order of magnitude less important. For the 11Πu state, we observe similar contributions from single- and double-excited determinants (49% and 39%), that explains the significant difference between the ecpCCSD and ecpCCSDT3 results. The role of triple and quadruple excitations for this state is relatively low, accounting for about 5% and 0.65%, respectively. Therefore, we can conclude that the application of the full configuration interaction method may be unnecessary to obtain accurate results for the 11Σu+, 21Σu, 11Δu, and 11Πu states. We can also assume that the ecpCCSDT3 and CIPT2+Q approaches properly set the boundaries for the PEC shapes. Indeed, near the minima, the experimental PECs for the 11Σu+, 21Σu, and 11Πu states lie between the curves obtained with the ecpCCSDT3 and CIPT2+Q methods.

The 21Πu state deserves special attention and particular comment because the variation of PECs obtained for this state with different methods is the largest, and this is the only state analyzed in which double excitations are dominant (75%). Additionally, it exhibits the highest contribution from the quadrupole excitations (2%). It corresponds to the 1P + 1S asymptote, and for large internuclear distances, it is repulsive. We can observe its two avoided crossings with other states of the same symmetry. The one at a larger interatomic separation involves a state from the 3P + 3P asymptote. The assignment of the second crossing is far more complex, as numerical difficulties prevent obtaining whole PECs for all states from the 1D(4d5p) + 1S(5s2), 1P(5s5p) + 1S(5s2), 1D(5s5d) + 1S(5s2) and 3D(5s4d) + 3P(5s5p) asymptotes. We suppose that doubly excited states, 1D(4d5p) + 1S(5s2) and 3D(5s4d) + 3P(5s5p), play a crucial role here. The equation-of-motion coupled cluster method with single and double excitations does not describe them accurately, so the PEC predicted at that level is mostly repulsive. The inclusion of some higher excitations by the ecpCCSDT3 method allows for a poor description of 1D(4d5p) state of Sr, where the excitation energy is overestimated by nearly 2000 cm–1 (see Table 2). The MRCI+Q/5Z method predicts the energy of 1D(4d5p) in a reasonably good agreement with the experiment. On the other hand, MRCI+Q/5Z tends to overestimate the depth of the potential for other states of Sr2. Additionally, our active space is too small to fully account for the 1P(5s5p) + 1S(5s2) and 1D(5s5d) + 1S(5s2) asymptotes. Overall, the expected position of the minimum in the 21Πu potential is in the wide range between 19000 cm–1 given by the CIPT2+Q/5Z method and 24000 cm–1 from the ecpCCSDT3/CBS computation above the minimum of the ground electronic state (see Table 1). This range covers the value Te > 19100 cm–1, estimated based on our present experimental observations (vide infra). Our computations do not confirm the existence of the avoided crossing between 11Πu and 21Πu predicted by Boutassetta et al.26 We indeed observe that 11Πu and 21Πu approach each other in the repulsive part of the PECs, but a possible crossing may occur only in the region experimentally insignificant. We suppose that a small basis set, with an insufficient number of high angular momentum components, could have significantly decreased the precision of Boutassetta et al. results for highly excited states. However, their approach accounts well for static correlation and thus reproduces the general shape of the PECs. Czuchaj et al.27 reported PEC, which differs from our and Boutassetta et al. results. However, their active space in multireference computations was smaller than ours and did not allow for an accurate description of static correlation. We believe that the accurate description of 21Πu state is a major computational challenge. Most likely, the only way to obtain its reliable and accurate description is to use the full configuration interaction with a large basis set and proper account for the core and core–valence correlation method, which is out of the scope of this work. Therefore, for now, we must assume that the exact shape of the curve is unknown and falls somewhere between the curves predicted by the ecpCCSDT3/CBS and CIPT2+Q/5Z approaches.

Table 2. Excitation Energies (in cm–1) of Singlet Electronic States of the Sr Atom Obtained with Different Quantum Chemistry Methods.

method 1D 5s4d 1P 5s5p 1S 5s6s 1D 4d5p 1P 5s6p 1D 5s5d
exp.53 20149.685 21698.452 30591.825 33826.899 34098.404 34727.447
x2cCCSDv 20144 20817 29076 33569 32494 33623
x2cCCSDbc 20659 22612 30982   34811 35355
x2cCCSDsc 20855 22652 30993   34837 35405
ecpCCSDT3/5Z 20489 21845 30517 35776 34150 34820
MRCI+Q/5Z 20162 20849 29095 33609    
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We provide the potential energy curves for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu states obtained with the ecpCCSDT3/CBS and CIPT2+Q/5Z methods, which are the most accurate among the investigated apporaches, and for the states presented in Figure 1 obtained with the sMRCI+Q/5Z method in the Supporting Information accompanying this paper.

Experiment

Experimental Setup

The Sr2 molecules were produced in a three section heat-pipe oven56 of 1 m length filled in the central part with 15 g of strontium. The central part with a length of 20 cm was heated to 1020 °C, while external parts were maintained at 720 °C. In the case of strontium a proper circulation of the metal inside the heat-pipe is a challenge. To solve this problem, 1.5 g of metallic magnesium was added to its central section (strontium and magnesium form an alloy with substantially lower melting point than its constituents57). A steel mesh was placed separately in each section and the heat-pipe was filled with 15 Torr of argon buffer gas.

A polarization labeling spectroscopy (PLS) technique was employed to obtain spectra of Sr2 molecules. The PLS is a pump–probe experimental technique, which takes advantage of an optical anisotropy created in a chosen group of molecules in the sample to limit the number of observed spectral lines.58 In the present experiment a NarrowScan dye laser of a spectral line width of 0.07 cm–1 pumped with a XeCl excimer laser (Light Machinery) was used as a pump laser and its wavelength was scanned between 18400 and 20300 cm–1, covering transitions from the ground state of Sr2 to the upper part of the 21Σu+ state. As a probe laser a ring dye laser (Coherent 899, pumped with Sprout laser) was employed, and its wavelength was controlled with HighFinesse WS-7 wavemeter. The laser was working on Rhodamine 6G, what enabled tuning its light within a spectral range 16800–17600 cm–1. The probe laser wavelength was fixed on selected transitions from the ground X1Σg state of Sr2 molecule to low levels of the 21Σu+ state known from the Supporting Information of the publication describing the bottom part of this state.25

Reference signals for wavenumber calibration of the molecular spectra were needed, therefore two auxiliary signals were recorded, namely transmission fringes from a Fabry-Pérot interferometer with FSR = 1 cm–1, and optogalvanic spectrum from argon and neon hollow-cathode lamps. This ensured that the uncertainty of wavenumbers determined in this way is below ±0.1 cm–1.

Analysis of the Spectra

As the bottom part of the 21Σu+ state has been characterized in the Fourier-transform spectroscopy experiment by Stein et al.,25 we have concentrated on higher vibrational levels of this state. A typical example of the recorded spectrum of Sr2 is presented in Figure 3. Our experiment provided information about rovibrational levels with quantum numbers ranging from v′ = 13 to 52 and J′ from 43 to 149, solely in the most abundant 88Sr2 isotopologue. We supplemented the database with levels of 88Sr2 measured in,25 however to avoid too strong influence of these levels on subsequent fits of molecular constants we limited the borrowed levels to J′<150 and assigned them with the same accuracy of 0.1 cm–1 as our own data. This resulted in 714 levels taken from25 and 1760 levels from our own measurements. The term values of all levels were calculated by adding the measured transition energies to the energies of the initial X1Σg (v″, J″) levels obtained from the highly accurate molecular constants reported in ref (23).

Figure 3.

Figure 3

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A portion of the experimental spectrum of Sr2 coresponding to transitions from rovibrational level v″ = 4, J″ = 90 in the ground X1Σg+ state to consecutive rovibrational levels (v′ = 19–28) of the excited 21Σu state.

Originally we tried to fit the term energies to the standard Dunham expansion

graphic file with name jp3c02056_m001.jpg 1

but the rms error of the fit amounted to 0.5 cm–1, i.e., five times more than our experimental accuracy. This result suggested strong perturbations in the 21Σu+ state, particularly that the misbehaving levels, all of them corresponding to v′ ≳ 19, were centered around isolated (v′, J′) values and the deviations fell into patterns characteristic for perturbations. Figure 4 displays term values of several levels of the 21Σu state plotted against J(J + 1) and open squares localize regions of the observed perturbations. Figure 5 visualizes typical pattern of deviations between the observed and predicted line positions versus rotational quantum number J.

Figure 4.

Figure 4

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Reduced term values Ered = E – 0.019 × J(J + 1) cm–1 of part of the observed rovibrational levels in the 21Σu+ state (dots) plotted against J(J + 1) . The value of 0.019 cm–1 is an approximate rotational constant Bv for vibrational levels v′ = 19–29 displayed in the figure. Open squares indicate approximate positions of centers of perturbations. A gap in the data near J(J + 1) = 13000 is due to the lack of data on possible labeling transitions for J = 110–114 listed by Stein et al.25

Figure 5.

Figure 5

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Observed shifts of the rotational energy levels in the 21Σu+ state from their predicted positions for vibrational levels v′ = 24 and 25.

In the preliminary analysis presented here which aims primarily to test accuracy of theoretical predictions based on various computational methods, we decided to remove the apparently perturbed levels from the database and to fit Dunham coefficients to the remaining levels. For this purpose we originally fitted all the observed levels with various sets of Dunham coefficients and then gradually discarded these levels which differed from the expected positions by more than 0.3 cm–1, irrespectively to the number of coefficients used in a fit. When fitting energies of somewhat arbitrarily chosen 1636 levels (out of the total 2474) we obtained rms error 0.08 cm–1, this time consistent with the experimental accuracy. The Dunham coefficients have been rounded to minimize the number of digits by a procedure described by Le Roy.59 They are listed in Table 3 together with the equilibrium bond length Re calculated from the rotational constant and reduced mass of strontium nuclei. A rotationless potential energy curve for the 21Σu+ state was constructed by a standard Rydberg-Klein-Rees (RKR) method. The vibrational term energies Gv and turning points R– and R+ are given in Table 4 and the potential curve is displayed in Figure 2 along with the theoretical predictions. Our work extends the range of experimentally determined potential to 3.5 Å < R < 6.1 Å and more than doubles the range of covered energies.

Table 3. Dunham Coefficients That Describe the 21Σu+ State of 88Sr2 in the Range 0 ≤ v′ ≤ 52, J′ ≤ 149a.

constant value (cm–1)
Te 17358.70(10)
Y10 84.2169(27)
Y20 –0.26729(21)
Y30 –0.10850(6) × 10–2
Y40 –0.10760(6) × 10–4
Y01 0.021990(14)
Y11 –0.6808(18) × 10–4
Y21 –0.4860(10) × 10–6
Y31 –0.1030(13) × 10–7
Y02 –0.712(6) × 10–8
Re (Å) 4.176(1)
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a

The numbers in parentheses give uncertainties in the last quoted digits (one standard deviation).

Table 4. Rotationless RKR Potential Energy Curve for the 21Σu+ State of Sr2a.

v Gv (cm–1) R– (Å) R+ (Å) v Gv (cm–1) R– (Å) R+ (Å)
  0 4.176          
0 42.024 4.084 4.275 26 2018.540 3.630 5.153
1 125.703 4.020 4.352 28 2150.854 3.616 5.210
2 208.838 3.978 4.408 30 2279.868 3.603 5.268
4 373.445 3.917 4.496 32 2405.468 3.590 5.327
6 535.786 3.870 4.571 34 2527.538 3.579 5.388
8 695.797 3.832 4.638 36 2645.955 3.568 5.451
10 853.411 3.799 4.701 38 2760.596 3.558 5.516
12 1008.555 3.770 4.760 40 2871.329 3.549 5.583
14 1161.155 3.744 4.818 42 2978.022 3.540 5.653
16 1311.129 3.721 4.875 44 3080.537 3.532 5.726
18 1458.395 3.700 4.930 46 3178.731 3.524 5.803
20 1602.863 3.680 4.986 48 3272.459 3.517 5.884
22 1744.442 3.662 5.041 50 3361.570 3.510 5.970
24 1883.034 3.646 5.097 52 3445.910 3.504 6.061
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a

The first line refers to the bottom of the potential curve, and the corresponding R value is the equilibrium distance. The full list of turning points of the potential is given in the Supporting Information accompanying this work.

It must be noted that in the range v′ = 0–18 the 21Σu+ state is free of (strong) perturbations which become visible only from v′ = 19. The most likely perturber is the 11Πu state, as the outer limb of its potential curve gradually approaches potential of the 21Σu state. However, at v′ = 25 apparently an additional perturbing state emerges, since perturbations become more frequent (v′ = 25 is perturbed at least around three J′ values, see Figure 5). From analysis of the theoretical potential energy curves it follows that another perturber appearing here must be the 21Πu state, the only other singlet ungerade state which is expected to be present in the vicinity. Therefore our observation indicates that the bottom of its potential cannot be located higher than approximately 19100 cm–1 above the minimum of the ground state potential, what can serve as another test for validity of theoretical predictions.

Conclusion

In this study, we investigated the excited electronic states of the strontium dimer. We theoretically obtained the complete molecular electronic spectrum up to the excitation energy of around 25000 cm–1 using the multireference configuration interaction method. Next, we studied in detail potential energy curves for the 11Σu+, 21Σu, 11Πu, 21Πu, and 11Δu states using several advanced electronic structure methods. We evaluated the importance of the orbital basis set size, core- and valence-electron correlation, and scalar relativistic effects. Theoretical results had been motivated by our ongoing spectroscopic studies. We presented new experimental measurements of the excited 21Σu+ state using polarization labeling spectroscopy, extending the range of observed vibrational levels to higher energies. We reported the corresponding Dunham coefficients and experimental potential energy curve. We observed perturbations in the recorded spectrum that give preliminary information on higher-excited electronic states. We compared the available experimental observations with the theoretical predictions to assess the accuracy and limitations of employed theoretical models. Our findings provide valuable insights into the complex electronic structure of Sr2, paving the way for future, more accurate theoretical and experimental spectroscopic studies.

The challenging nature of excited electronic states of the Sr2 dimer makes them a perfect testbed and playground for near-future developments of the electronic structure theory and computation. In the following work, we plan to present calculations at the valence full configuration interaction level with large basis sets. Such converged calculations in versions with both small-core and large-core pseudopotentials and approximately included core and core–valence correlation may resolve the nature of most problematic states such as 21Πu.

A more rigorous deperturbation procedure is also needed to clarify the experimental observations. It would require a coupled channels treatment involving (at least) the 21Σu+, 11Πu and 21Πu interacting states and such an analysis is in future plans of our group. However, more precise theoretical predictions of the relevant potential energy curves are needed to serve as a starting point for deperturbation procedure. At the present stage we show the approximate results, accuracy of which is more than sufficient for comparison with theoretical models. All the experimental term energies are listed in the Supporting Information accompanying this paper.

Acknowledgments

Financial support from the National Science Centre Poland (Grant No. 2021/43/B/ST4/03326) is gratefully acknowledged. The computational part of this research has been partially supported by the PL-Grid Infrastructure (Grant No. PLG/2021/015237).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.3c02056 and at http://dimer.ifpan.edu.pl.

  • Potential energy curves of electronic states of Sr2 molecule calculated with the CIPT2Q5Z method (ZIP)

  • Potential energy curves of electronic states of Sr2 molecule calculated with the ecpCCSDT3/CBS method (ZIP)

  • Potential energy curves of electronic states of Sr2 molecule calculated with the sMRCI/5Z method (ZIP)

  • Rotationless RKR potential energy curve for the 21Σu+ state of Sr2 molecule (see Table 4) (TXT)

  • Experimental energies of the observed rovibrational levels in the 21Σu+ state (TXT)

The authors declare no competing financial interest.

Special Issue

Published as part of The Journal of Physical Chemistry virtual special issue “Cold Chemistry”.

Supplementary Material

jp3c02056_si_001.zip (5.1KB, zip)
jp3c02056_si_002.zip (5.4KB, zip)
jp3c02056_si_003.zip (38.2KB, zip)
jp3c02056_si_004.txt (49.2KB, txt)
jp3c02056_si_005.txt (3.1KB, txt)

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Supplementary Materials

jp3c02056_si_001.zip (5.1KB, zip)
jp3c02056_si_002.zip (5.4KB, zip)
jp3c02056_si_003.zip (38.2KB, zip)
jp3c02056_si_004.txt (49.2KB, txt)
jp3c02056_si_005.txt (3.1KB, txt)

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