Electronic Rydberg States of Cd2 from ab Initio ... - ACS Publications

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1 3 3 Potentials of the D10þ u (6 S0) and F 1u(6 P2) Electronic Rydberg States of Cd2 from ab Initio Calculations and Laser-Induced Fluorescence Excitation Spectra

J. Koperski,*,† M. Strojecki,‡ M. Krosnicki,§ and T. Urbanczyk† †

Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiellonski, ul. Reymonta 4, 30-059 Krakow, Poland Instytut Katalizy i Fizykochemii Powierzchni, Polska Akademia Nauk, ul. Niezapominajek 8, 30-239 Krakow, Poland § Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdanski, ul. Wita Stwosza 57, 80-952 Gdansk, Poland ‡

ABSTRACT: The method of supersonic free-jet expansion beam combined with techniques of laser spectroscopy was used in an investigation of vibronic and isotopic structures in the D10uþ(61S0) and F31u(63P2) electronic energy Rydberg states of Cd2. Laser-induced fluorescence excitation spectra recorded using the D10uþ r X10gþ(51S0) and F31u r X10gþ transitions in the region of 206-218 nm provided spectroscopic characteristics of the excited states and allowed constructing of their intratomic potentials. Isotopic structures recorded in the (υ0 ,υ00 ) bands of the D10uþ r X10gþ transition were used in determination of the D10uþ state vibrational characteristics (ωe0xe0 , ωe0xe0 ) and υ0 assignment. The ν0,0 recorded directly in the F31u r X10gþ transition enabled determination of the bottom of the F31u state potential well. Valence ab initio calculations of Cd2 interatomic potentials were performed with relativistic and spin-orbit effects taken into account. The experimental results were compared with results of the ab initio calculations. A free-jet expansion of Cd2 as a source of entangled atoms for a test of Bell’s inequality was analyzed.

’ INTRODUCTION Laser spectroscopy of 12-group van der Waals (vdW) molecules produced and ro-vibrationally cooled in a free-jet expansion beam is one of methods for investigation of molecular energy structure.1 It was applied, among others, in experimental studies 3 3-5 3 1 8,9 of the a31u(53P1)2, b30þ c 1u(53P2),6,7 A10þ u (5 P1), u (5 P1), 1 1 10,11 1 þ 1 and B 1 u (5 P 1 ) excited as well as the X 0g (5 S 0 ) ground 6,8-10 electronic-energy states of Cd 2 (see Figure 1 for reference). On the basis of the experimental results, i.e., laserinduced fluorescence (LIF) excitation and dispersed emission spectra, for all of the studied potentials analytical representations were proposed except for that of the a31u state, for which a weak 3 dipole transition moment between the X10þ g and a 1u states 2 prevented excitation. According to the experimental results,4-6 3 b30þ u and c 1u state potentials were represented with respective Morse functions near the bottom of the potential well, i.e., for υ0 vibrational levels from 0 to 14 and from 0 to 4, respectively (a Lennard-Jones (6-12) function was also used for the c31u state7). Similarly, a Morse function was used to represent the A10þ u state potential8,9 (however, only for υ0 from 19 to 54). In the case of the more complex B11u state potential,10,11 the υ0 r υ00 = 0 excitation was possible only in a narrow range of υ0 from 33 to 40. Beyond the υ0 = 40, the excitation reaches a potential barrier and allows determination of its position and height.10 In order to extend the characterization resulting in construction of interatomic potentials for the Cd2 excited states over a large region of internuclear separations R, an inverse perturbation approach (IPA) was used providing numerical representations of the A10þ8,9 and B11u10 states. u r 2011 American Chemical Society

Characterization of the X10þ g state was reported by Czajkowski and Koperski.5 The study was based on the detection of so1 þ called “hot” bands recorded using the b30þ u r X 0g transition. 8,9 Studies of yukomski et al. concluded with Morse-vdW and Born-Mayer hybrid representations of the X10þ g -state bound well up to the υ00 = 3 level and repulsive wall (from the A10þ u (υ0 = 39) f X10þ g LIF dispersed emission), respectively. Similar conclusions were derived by Ruszczak et al.;11 however, they used the B11u(υ0 = 38) f X10þ g transitions. Characterization of the ground-state bound well was extended up to the υ00 = 8 due to the observation of the υ0 f υ00 = 0-8 bound f bound transitions in the LIF dispersed emission spectra.6 The rotational energy structures of Cd2 were reported from our laboratory by yukomski et al.12,13 and Strojecki et al.14 In the former references,12,13 rotational profiles in a single (υ0 ,υ00 ) = 1 þ (45,0) vibrational band of the A10þ u r X 0g transition in the 228 114 Cd2 (consisting of unresolved ( Cd)2 and 112Cd116Cd) isotopologue were studied. From the analysis of the R-branch, 114 0 (A10þ Cd)2 were the B00υ=0 and Bυ=0 u ) rotational constants of ( 00 determined. This allowed estimation of the Re and Re0 (A10þ u) bond lengths. In the latter reference,14 a complex multivibrational-band rotational analysis (i.e., for (υ0 ,υ00 ) = (26,0), (27,0), (42,0), (45,0), (46,0), (48,0)) and multirotational-branch Special Issue: J. Peter Toennies Festschrift Received: November 28, 2010 Revised: February 10, 2011 Published: March 16, 2011 6851

dx.doi.org/10.1021/jp1112922 | J. Phys. Chem. A 2011, 115, 6851–6860

The Journal of Physical Chemistry A

Figure 1. Ungerade interatomic potentials of Cd2 obtained as a result of ab initio calculations of this work (see Theoretical Calculations). The 1 3 3 interatomic potentials of the D10þ u (6 S0) and F 1u(6 P2) states analyzed in this work are plotted with thick solid lines while the also 3 discussed C31u(63S1), E31u(63P1), and G30þ u (6 P1) potentials are drawn with thick dashed lines. The upper part shows M 2 dipole transition moments squared calculated for transitions between the ground 3 3 3 þ 1 þ and C31u, D10þ u , E 1u, and F 1u excited states (for the G 0u -X 0g 2 transition, M function is not plotted as it is on the order of magnitude of that for the C31u-X10þ g transition). Experimentally determined poten3 tials of the D10þ u (blue thick line) and F 1u (red thick line) states are plotted for comparison. The energy range probed in this experiment is represented with ranges of the υ0 vibrational energy levels (horizontal lines) recorded in the excitation spectra. 1 þ analysis (i.e., for the P- and R-branches) of the A10þ u r X 0g 228 114 transition in the Cd2 (consisting of unresolved ( Cd)2 and 112 Cd116Cd) isotopologue was presented. Also, the B11u state was rotationally characterized.10 The rotational profiles in a single (υ0 ,υ00 ) = (38,0) vibrational band of the B11u r X10þ g transition in the 228Cd2 isotopologue were studied. From the analysis of the P-, Q-, and R-branches, the Bυ0 =38(B11u) rotational constant of (114Cd)2 was determined. It is important to acknowledge investigations of Cd2 energy structure done by other researchers. A comprehensive review was published in 200315 and updated in our most recent article.6 What is more, we draw one’s attention to the studies of Eden and co-workers16,17 related to LIF bound f free dispersed emission spectra as well as LIF excitation spectra recorded using a pumpand-probe technique. Their experimental method allowed characterizing the A10þ u state. Work done by Grycuk and coworkers18-20 deserves attention as well. Observing absorption profiles and emission spectra, they investigated the long-range 18 as well as the behavior of the a31u and b30þ u state potentials 3 1 þ tentative position of the C 1u and D 0u Rydberg-state potentials correlating with the 63S1 and 61S0 atomic asymptotes,

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respectively.20 Roughly estimated locations of the Rydberg states on an absolute energy scale were obtained by an investigation of the molecular emission of dense Cd vapor following laser excitation at λ = 326.1 nm.19 However, the conclusion of those studies clearly indicates that in order to obtain accurate and full characterization of the lowest Rydberg states, it is desirable to excite their energy structures directly from the ground X10þ g state. In this article we report on experiments in which we extend our studies of Cd2 electronic energy states using excitation wavelengths shorter than 221 nm, i.e., the wavelength at which evidence of the B11u state potential barrier was found.10,11 LIF excitation spectra recorded in the range from 206 to 218 nm 1 1 þ 3 3 1 þ using the D10þ u (6 S0) r X 0g and F 1u(6 P2) r X 0g transitions are presented. The molecules were produced in a continuous free-jet expansion beam crossed with a pulsed dye-laser beam. In the excitation to the D10þ u state, well-resolved isotopically structured vibrational components of the υ0 = 3554 r υ00 = 0 progression were recorded. In the excitation to the F31u state, transitions to the lowest υ0 = 0-16 levels were recorded allowing determination of the minimum of the excited state potential well. Analyses of the spectra provided characterization of the excited states involved in the transition and determination of ωe0 (vibrational frequencies), ωe0 xe0 (anharmonicities), De0 (well depths), D00 (dissociation energies), and Re0 (bond 0 lengths), as well as information on the D10þ u state υ assignment. Valence ab initio calculations of the interatomic potentials for the ground and excited states were performed with relativistic and spin-orbit effects taken into account. Also, M(R) transition dipole moments for the investigated transitions were calculated using the ab initio calculated potentials, allowing a proper interpretation of the observed spectra. Finally, energies of υ0 levels in the F31u state were obtained using an inverse perturbation approach (IPA). The experimental results were compared with results of ab initio calculations and the result of the IPA analysis.

’ THEORETICAL CALCULATIONS Ab initio calculations of Cd2 interatomic potentials were carried out by Czuchaj and co-workers21 without spin-orbit coupling. The spin-orbit coupling was taken into consideration by Czuchaj,22 and Czuchaj and Krosnicki23 in two latter calculations. In the very weakly bound ground state of the group-12 homoatomic molecules (Zn2, Cd2, Hg2), the long-range interaction is dominated by pure dispersion forces as expected from a simple consideration of the closed-shell atomic configurations.1 Ab initio calculations of the interaction-energy components in the ground state of Hg224 showed that short-range covalent effects play a significant role in the stabilization of the molecule. Therefore, Hg2 may be regarded as an intermediate case between a weakly bound vdW molecule and a chemically bound species. The same behavior has been inferred from ab initio calculations of Zn2 and Cd2.25,26 Studies of Dolg and co-workers25-28 resulted in a clear conclusion that the group-12 homoatomic molecules, besides a vdW-type interaction, exhibit the presence of significant covalent contributions to the bonding. Generally, in our studies1 the motivation for the calculations is usually 2-fold. The calculated potentials as well as M transition dipole moments are used in planning experiments. In this study, as the experiment involved unexplored transitions, the calculated potentials of the electronic states under consideration as well as M were employed in order to find the spectral regions where the molecular transitions may occur. Moreover, the M helped us to 6852

dx.doi.org/10.1021/jp1112922 |J. Phys. Chem. A 2011, 115, 6851–6860

The Journal of Physical Chemistry A choose those transitions that might be easily excited with laser radiation from the ground state. The other aspect of our systematic studies is the chemical bonding in dimers of 12-group elements which represents a special case of a subtle mixture of van der Waals and covalent contributions. Spectroscopic measurements for the ground and excited states are of significance for quantum chemistry in order to judge the performance of computational methods and basis sets. What is more, recently the (111Cd)2 cadmium diatom has been considered as a candidate in an experimental verification of Bell’s inequality for entangled atoms created in 111Cd-111Cd pairs after a controlled process of two-step laser dissociation of the molecule.29 The scheme, originally proposed for (199Hg)2,30,31 requires knowledge about interatomic potentials of the states involved in the process. In ab initio calculations of this work which were performed using a MOLPRO package,32 20 electrons of the Cd atom were treated explicitly while the rest of the core electrons were replaced by the effective core pseudopotential.33 In the calculations we used an augmented correlation-consistent polarized valence quadruple-ζ (aug-cc-pVQZ) atomic basis set34 augmented by three s, two p, and one d even tempered, diffuse basis set functions. Exponents can be found in ref 6. The molecular orbitals used in the calculations of the excited triplet and singlet states were separately optimized for gerade and ungerade symmetry states in the state averaged complete-active-space multiconfiguration self-consisted field (CASSCF) method35,36 for all triplet states correlating with the (5p)3P, (6s)3S, (6p)3P and all singlet states correlating with the (5s)1S, (5p)1P, (6s)1S, and (5d)1D atomic asymptotes, respectively. Additionally, in the case of ungerade states we took into account the 3Δu state correlating with the (5d)3D atomic asymptote. This state was likely to perturb the 3Σuþ(63P) state. The calculations where performed after reduction of the D¥h point group to the D2h point group which is implemented in MOLPRO. In the D2h point group each irreducible representation is one-dimensional; therefore, after reduction of symmetry the two-dimensional representation of 3 Δu state splits into one-dimensional Au and B1u irreducible representations of the D2h. The 3Δu(53D) component belonging to the Au irreducible representation was optimized with state average weight equal to 1 because it was the single one state having triplet multiplicity in the Au irreducible representation. The second component of the 3Δu state belonging to the B1u irreducible representation of the D2h was optimized with state average weight equal to zero in order to avoid accumulation of computation error, when the B1u component of 3Δu(53D) interacts with 3Σuþ(53D) state belonging also to the B1u irreducible representation. The resulting wave functions were used as references in the following complete-active-space multireference second-order perturbation method (CASPT2).37,38 In the case of CASPT2 calculations, the active space was formed in the same way as it was for the CASSCF method by distributing the 20 outermost electrons of the Cd atom into 4s4p4d outer core, 5s5p valence, and 6s6p5d orbitals. The outer core 4d orbitals were kept doubly occupied and were optimized during CASPT2 calculations; i.e., these orbitals were correlated through single and double excitations. On other hand, because of technical reasons, the 4s, 4p core orbitals where kept as a core and were not correlated. Overall, this resulted in a total of 18 closed-shell and 26 active molecular orbitals, with the 40 electrons distributed among the molecular orbitals. The resulting wave functions were used as references in the following CASPT2. The CASPT2 eigenenergies were employed in the subsequent spin-orbit (SO)

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calculations39 as diagonal elements of the SO matrix. For both components of the 3Δu(53D) state, the CASPT2 energy calculated in the Au irreducible representation was used. The off-diagonal elements of the SO operator were calculated using reference wave functions taken from the state averaged CASSCF calculation. In order to obtain dipole transition moments between the ground and ungerade excited states under consideration, we used CCSD(T) energy of the ground state as an additional diagonal element of the SO matrix. The ground-state wave function was formed from molecular orbitals optimized within the CASSCF method for ungerade states. Figure 1a presents ab initio calculated ungerade adiabatic interatomic potentials of Cd2 correlating with the (5p)3P1, (5p)3P2, (5p)1P1, (6s)3S1, (6s)1S0, (6p)3P1, and (6p)3P2 atomic asymptotes. The potentials discussed here, i.e., C31u(63S1), 1 3 3 3 3 3 þ 3 D 10 þ u (6 S0), E 1u(6 P1), F 1u(6 P2), and G 0u (6 P1) are plotted with thick (solid and dashed) lines. In order to determine transitions that are most likely to be excited from the ground state using excitation in the range from approximately 45800 to 48500 cm-1 (i.e., for 206-218 nm region), M functions were 1 þ 1 þ 3 1 þ 3 calculated for the C31u r X10þ g , D 0u r X 0g , E 1u r X 0g , F 1u 1 þ 3 þ 1 þ r X 0g , and G 0u r X 0g transitions showing that only D10þ u r and F31u r X10þ transitions have to be conX10þ g g sidered as plausible candidates (see upper part of Figure 1).

’ EXPERIMENTAL SECTION Details of the experimental procedure were reported elsewhere,8,9,11 so only the most relevant modifications applied in this study are presented. A molecular beam source (consisting of an oven body and nozzle cartridge) was fabricated from stainless steel. The source was filled with Cd (Aldrich, purity 99.999%, natural abundance) whereas Ar (Linde, purity 99.999%), Kr (Linde, purity 99.99%), and 5% Xe/He mixture (Linde, purity 99.99%) were used as carrier gases at pressures from 6 to 13 bar. The molecular beam source was heated up to 950 K. The Cd atoms seeded in carrier gas were injected through the nozzle orifice (diameters D, 160, 180, 200 μm) into an evacuated expansion chamber (note: in this study, Kr and 5% Xe/He were used to verify that the observed spectra do originate from Cd2 and not from CdRg complexes, Rg = rare gas). The Tvib vibrational temperature in the beam was estimated to be approximately 5-50 K.31 The molecules in the beam were irradiated with a doubled dye laser beam in short (206-212 nm) and long (209-218 nm) wavelength regions of the spectra (a LCR1 dye laser of Sopra was employed using Exalite 416 in pDioxane and Stilbene 3 in methanol, respectively) at a distance X varying from 5 to 9 mm from the nozzle. The dye laser was pumped with the third harmonic output of a Ndþ:YAG laser (Powerlite of Continuum, Series 7010). The frequency of the dye laser was doubled using an Autotracker system (Radiant Dyes Laser & Accessories) with a BBO-I crystal. The frequency calibration of the dye-laser fundamental output was verified to within a 0.005 nm uncertainty against a pulsed wavemeter (WA4500, Burleigh) and, in a part of the long-range region (212-216 nm), with an optogalvanic cell filled with Ar or Kr (Sirah). The spectral line width of the dyelaser fundamental output was monitored using a Fabry-Perot etalon and estimated to be between 0.5 and 1.0 cm-1. The excitation spectra were recorded by focusing the total LIF from the interaction region on the cathode of a photomultiplier (PM) tube (type 9893QB/350 of Electron Tubes, with nearly flat sensitivity in the 190-500 nm range) while tuning the laser wavelength. Due to the sensitivity of the PM 6853



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1 þ 1 1 þ Figure 2. LIF excitation spectra recorded using the F31u(63P2) r X10þ g and D 0u (6 S0) r X 0g transitions for (a) D = 200 μm, pAr = 11 bar, X = 9 mm, and (b) D = 200 μm, pAr = 13 bar, X = 6 mm (the part of the spectrum between 207 and 208 nm was recorded in a separate experiment). (c) Simulation43,44 of the excitation spectrum recorded using the F31u r X10þ g transition; for the simulation, parameters from Table 2 as well as a rotational temperature Trot = 5 K, a maximum rotational quantum number Jmax = 25 and combined Lorentzian (Δlas = 0.5 cm-1) and Gaussian (ΔDopp = 0.7 cm-1) profiles were used. The Δlas and ΔDopp represent full widths at half-maximum corresponding to the experimental values responsible for the bandwidth of the excitation laser and residual Doppler broadening associated with a transverse divergence of the molecular beam, respectively. (d) Simulation43,44 of 1 þ the excitation spectrum recorded using the D10þ u r X 0g transition; for the simulation, parameters from Table 2 as well as Trot = 5 K, Jmax = 25, and combined Δlas = 0.5 cm-1 and ΔDopp = 0.7 cm-1 were used. (e) Two simulations of (c) and (d) combined together to illustrate the complexity of the total excitation spectrum. Simulations did not include the wavelength dependence of the PM sensitivity. (f) Laser power curves plotted to show an influence of nonuniform dye-laser intensity. Components marked with asterisks are enlarged.

tube, the total LIF recorded in the experiment was dominated by the emission terminating at the X10þ g state. The dye-laser second harmonic intensity measured with a photodiode was used to normalize the recorded LIF. The procedure decreased the influence of the laser shot-to-shot instability on the measured signal.

’ RESULTS Figure 2 presents excitation spectra recorded in the region from 206 to 218 nm. According to our ab initio calculated interatomic

potentials reported here and previous experimental studies,1,2,5-14 the spectra have to be assigned as a result of an excitation from the lowest υ00 levels to the υ0 levels belonging to the electronic states lying above the B11u state energy barrier situated at the Eat(51P1) þ Eb(B11u), where Eat(51P1) = 43692.5 cm-1 40 is the B11u state atomic asymptote and Eb(B11u) = 1190 ( 20 cm-1 11 is a height of the B11u state energy barrier. The F31u(63P2) r X10þ g Transition. The LIF excitation spectrum recorded using the F31u(63P2) r X10þ g transition with Ar 6854



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1 a Table 1. Experimental Frequencies (vexpt) of the υ0 r υ00 = 0 Progression Recorded for the F31u(63P2) r X10þ g (5 S0) Transition

transition υ0 r υ00 = 0

vexpt

vexpt - vsim

vsim

vIPA

vexpt - vIPA

0

47105.60 ( 0.10

47105.60

0.00

47105.60

0.00

1

47196.76 ( 0.09

47195.20

þ1.56

47196.76

0.00

2

47285.69 ( 0.41

47283.83

þ1.86

47285.67

þ0.02

3

47373.73 ( 0.31

47371.49

þ2.24

47373.87

-0.14

4

47461.51 ( 0.26

47458.19

þ3.32

47461.30

þ0.21

5

47547.26 ( 0.25

47543.91

þ3.35

47547.43

-0.17

6

47631.99 ( 0.14

47628.67

þ3.32

47631.98

þ0.01

7 8

47715.21 ( 0.48 47797.27 ( 0.17

47712.46 47795.28

þ2.75 þ1.99

47715.09 47797.23

þ0.12 þ0.04

9

47878.66 ( 0.22

47877.14

þ1.52

47878.71

-0.05

10

47959.30 ( 0.26

47958.02

þ1.28

47959.38

-0.08

11

48039.06 ( 0.56

48037.94

þ1.12

48038.97

þ0.09

12

48118.13 ( 0.73

48116.89

þ1.24

48117.28

þ0.85

13

48194.24 ( 0.51

48194.87

-0.63

48194.42

-0.18

14

48270.28 ( 1.03

48271.89

-1.61

48270.89

-0.61

15 16

48347.52 ( 0.76 48423.60 ( 0.96

48347.93 48423.01

-0.41 þ0.59

48347.24 48423.69

þ0.28 -0.09

Experimental errors were obtained from a fit of a Gaussian function to the experimental points. For larger υ0 , the influence of the isotopic effect additionally increases the errors. vsim and vIPA are frequencies obtained in simulation43 and from inverse perturbation approach (IPA), respectively. All frequencies in (cm-1). a

Figure 3. Birge-Sponer plot of the υ0 r υ00 = 0 progression recorded using the F31u(63P2) r X10þ g transition in Cd2 (ΔGυþ1 is the separation between frequencies of successive vibrational transitions obtained from experimental data). The insert shows details of the plot. A linear fit (red line) allowed determination of the ωe0 (an intercept with vertical axis) and ωe0 xe0 (a slope) of the F31u state. Frequencies vexpt of the υ0 r υ00 = 0 transitions from Table 1 were used.

employed as a carrier gas is shown in the short-wavelength part of Figure 2. The recorded υ0 r υ00 = 0 progression extends from 206 to 212.2 nm (i.e., from 47110 to 48528 cm-1 with respect to the energy of υ00 = 0 level) and consists of over 17 vibrational components (see Table 1) with increasingly resolved internal (isotopic) structure. The progression starts from the most prominent component centered at 212.221 nm (47105.60 cm-1) which was assumed to be its origin, i.e., υ0 = 0 r υ00 = 0 transition with frequency ν0,0 (see for example our earlier study of the c31u(53P2) state potential6,7). Having this, it was immediately concluded that the bottom of the F31u state potential has to be located at the E(υ0 = 0) energy of the υ0 = 0 level lowered by the value of (ωe0 /2 - ωe0 xe0 /4),

where ω0 e and ωe0 xe0 are the F31u state vibrational frequency and anharmonicity, respectively. In order to determine the ωe0 and ωe0 xe0 in the energy region corresponding to the recorded spectrum, a Birge-Sponer (B-S) plot41 was constructed (see Table 1 and Figure 3) revealing its linearity, which suggests the adequacy of a Morse-function representation for the F31u state potential. However, the representation is justified only from the bottom of the potential well up to the E(υ0 = 16), energy of the υ0 = 16 level which corresponds to the energy of approximately 1363 cm-1 (ν16,0 - ν0,0 þ ωe0 /2 - ωe0 xe0 /4) above the bottom of the potential well. For the F31u state potential, which possesses a large well depth, the determined ωe0 = 90.57 cm-1 and ωe0 xe0 = 0.48 cm-1 cannot be used to approximate the potential well depth using relationship De0 ≈ ωe0 2/4ωe0 xe0 as it gives a value of 4272 cm-1, which is smaller than that deduced from the ν0,0, D00 0 = 317.4 cm-1 6 ground-state dissociation energy and Eat(63P2) = 58635.7 cm-1 40 energy of the 63P2 atomic asymptote De 0 ¼ D0 0 þ ðωe 0 =2 - ωe 0 xe 0 =4Þ ¼ 11892:7 cm-1

ð1Þ

where D0 0 ¼ Eat ð63 P2 Þ þ D000 - v0, 0 ¼ 11847:5 cm-1

ð2Þ

is the F31u state dissociation energy. All the above illustrates limitations of the B-S plot procedure in determination of the De0 . Similar conclusions were drawn in our previous studies, e.g., of the A10þ u(51P1) state in Cd28,9 or the E31u(63P2) state in Hg2.42 Table 2 collects all the F31u state potential parameters determined in this study. Using the determined ν0,0, ωe0 , ωe0 xe0 , and De0 , a simulation of the recorded υ0 r υ00 = 0 progression in the excitation spectrum was performed43 including the isotopic structure of each vibrational component44 (see Figure 2c). With an assumption of R00e = 3.78 Å,6 the simulation provided R0e = 3.54 Å for the F31u excited state (note: the obtained R0e as well as R00e6 allowed also including the rotational structure (despite the fact that it was not resolved in the experiment, being at least 1 or 2 orders of magnitude smaller than the isotopic 6855



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1 Table 2. Parameters of the F31u(63P2) and D10þ u (6 S0) Electronic Energy Rydberg States Determined in This Study

designation

1 D10þ u (6 S0)

ωe0 (cm-1)

117.31 ( 0.64a

ωe0 xe0 (cm-1)

87.9 ( 0.4 0.22 ( 0.02a

F31u (63P2) 90.57 ( 0.62b

l

-1

-1

D00 (cm )

0.480 ( 0.025b

11288.72 ( 226.10

g

11847.5 ( 3.20d

De0 (cm )

11347.32 ( 226.42

f

11892.7 ( 4.40c

Re0 (Å)

2.82 ( 0.03

i

3.54 ( 0.01i

2.86 ( 0.03

j

ν0,0 (cm-1) -1 k

Te (cm )

42338.84 ( 223.00e

47105.60 ( 0.10h

42290.85 ( 223.00

47071.05 ( 0.54

For υ0 = 35 to υ0 = 54, linear B-S plot of Figure 5a, Morse approximation. b For υ0 = 0 to υ0 = 16, linear B-S plot of Figure 3, Morse approximation. c Equation 1. d Equation 2. e Assuming linearity of B-S plot down to υ0 = 0. f As in footnote e, analogue of eq 1. g As in footnote e, analogue of eq 2. h Experiment. i Simulation of the excitation spectrum with M(R). j Simulation of the excitation spectrum without M(R). k ν0,0þ(ωe00 /2 - ωe00 xe00 /4) - (ωe0 /2 - ωe0 xe0 /4). l Reference 20. a

structure). The determined F31u state interatomic potential is plotted in Figure 1 (red thick line) for comparison with result of the ab initio calculations. As one may conclude, the experimentally determined potential is deeper by 1016 cm-1. Moreover, its position in R implies that it represents the outer (deeper) well of the ab initio calculated potential. Another conclusion one may draw from the comparison is that if the ab initio potential is slightly deeper and the laser excitation reaches the υ0 lying close to the potential’s inflection point, the first-order anharmonicity can be influenced and one observes a departure from the linearity of the B-S plot (see Figure 3, and also the discussion in Conclusions). Figure 2b shows the shortwavelength part of the excitation spectrum recorded under different experimental conditions (larger pAr, smaller X) preferring detection of transitions terminating at the υ0 being in the vicinity of the inflection point. In order to confirm that the υ0 r υ00 = 0 progression recorded using the F31u r X10þ g transition has its origin in an excitation of Cd2 molecule, the spectrum was also recorded using two different carrier gases: Kr and 5% Xe/He mixture (see Experimental Section) revealing similar υ0 r υ00 = 0 progressions as that recorded using Ar and, what is interesting, additional υ0 r υ00 = 1, υ0 r υ00 = 2, etc., “hot” progressions while using 5% Xe/He (see Figure 4). Analysis of 5,6 the latter confirmed the characterization of the X10þ g ground state proving a uniformity and consistency of the analysis presented here. The B-S plot for the υ0 r υ00 = 0, 1, 2, 3, ..., progressions confirmed previous characterization of the ground state ωe00 = 21.4 cm-1 and ωe00 xe00 = 0.35 cm-1 (compare with ωe00 = 21.4 ( 0.2 cm-1 and ωe00 xe00 = 0.35 ( 0.02 cm-15,6). 1 1 þ The D10þ u (6 S0) r X 0g Transition. Figure 2a shows also a 1 1 þ LIF excitation spectrum recorded using the D10þ u (6 S0) r X 0g transition. The recorded υ0 r υ00 = 0 progression extends from 208.5 to 218 nm (i.e., from 45858 to 47947 cm-1 with respect to the energy of υ00 = 0 level) and consists of 20 vibrational components with well-resolved isotopic structure (Table 3). In its short-wavelength part, the spectrum overlaps with the υ0 r υ00 = 0 progression of the F31u r X10þ g transition (see discussion in previous section). A large isotopic shift (which increases from 7 to 12 cm-1 while υ0 increases) indicates relatively high υ0 levels reached in the excitation from the υ00 = 0. The analysis shown in 0 0 0 Figure 5 enabled determination of the D10þ u state ωe and ωe xe

Figure 4. (a) The υ0 r υ00 = 0, 1, 2, 3, ..., progressions (“hot bands”) recorded for the F31u(63P2) r X10þ g transition and D = 180 μm, pXe/He = 8 bar, X = 5 mm. (b) Simulation43,44 of the excitation spectra recorded using 1 þ 1 þ both the F31u r X10þ g and D 0u r X 0g transition; for the simulation, parameters from Table 2 as well as Tvib = 50 K, Trot = 5 K, Jmax = 25, and combined Δlas = 0.5 cm-1 and ΔDopp = 0.7 cm-1 were used. The influence of the υ0 = 42-44 components belonging to the υ0 r υ00 = 0 progression of 1 þ the D10þ u r X 0g transition is clearly visible.

Table 3. Experimental Frequencies (v expt) of the υ 0 rυ 00 =0 1 1 þ 1 Progression Recorded for the D 1 0þ u (6 S 0 ) r X 0 g (5 S 0 ) Transition a transition υ0 r υ00 = 0

vexpt

35 36

vsim

vexpt - vsim

46176.02 ( 0.32

46176.02

þ0.00

46278.26 ( 0.30

46277.49

þ0.77

37

46380.37 ( 0.21

46378.52

þ1.85

38

46480.28 ( 0.21

46479.11

þ1.17

39

46581.04 ( 0.26

46579.26

þ1.78

40 41

46680.02 ( 0.21 46779.81 ( 0.20

46678.97 46778.24

þ1.05 þ1.57

42

46878.32 ( 0.23

46877.07

þ1.25

43

46976.73 ( 0.19

46975.46

þ1.27

44

47073.77 ( 0.35

47073.41

þ0.36

45

47171.26 ( 0.56

47170.92

þ0.34

46

47267.68 ( 0.55

47267.99

-0.31

47

47362.13 ( 0.45

47364.62

-2.49

48 49

47458.95 ( 0.45 47553.16 ( 0.51

47460.81 47556.56

-1.86 -3.40

50

47650.16 ( 0.46

47651.87

-1.71

51

47744.82 ( 0.44

47746.74

-1.92

52

47839.85 ( 0.47

47841.17

-1.32

53

47933.43 ( 0.43

47935.16

-1.73

54

48028.30 ( 0.50

48028.71

-0.41

a

vexpt are those for one of the most abundant 226Cd2 isotopologue. Experimental errors were obtained from a fit of a Gaussian function to the experimental points. For υ0 from the region where the progression overlaps with that of the F31u(63P2) r X10þ g transition, the errors increase. vsim are frequencies obtained in simulation.43 All frequencies in (cm-1).

(assuming that the ωe00 and ωe00 xe00 are known;5,6 see caption of Figure 5). First, the B-S plot was drawn for the υ0 r υ00 = 0 progression of the 226Cd2 (consisting of unresolved 112Cd114Cd, 110 Cd116Cd, and (113Cd)2) isotopologue (Figure 5a), giving the 6856



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Figure 5. (a) Birge-Sponer plot of the υ0 r υ00 = 0 progression 1 þ recorded using the D10þ u r X 0g transition (frequencies vexpt used to construct the plot were those of 226Cd2, see Table 3, ΔGυþ1 as in Figure 3). Insert shows details of the plot. A linear fit (red solid line) allowed determination of the ωe0 (an intercept with the vertical axis) and ωe0 xe0 (a slope) of the D10þ u state. (b) The measured (full squares with error bars) and calculated for υ0 (our assignment, red solid line) and for υ0 ( 3 (black dashed lines) isotopic shifts Δνij (eq 3) are between two isotopologues, i.e., 226Cd2 and 227Cd2 isotopic components recorded in the υ0 r υ00 = 0 progression shown in Figure 2. See text for details.

ωe0 and ωe0 xe0 as the representative values in the energy region corresponding to the excitation. Next, an analysis of the Δνij isotopic shift between the vibrational components of the 226Cd2 and 227Cd2 isotopologue, where A1 þ A2 = 226 and 227, respectively, and A is the molecular mass number, was performed according to the formula41 Δvij ðυ0 , υ00 Þ ¼ ð1 - Fij Þ½ωe 0 ðυ0 þ 1=2Þ - ωe 00 ðυ00 þ 1=2Þ - ð1 - Fij 2 Þ½ωe 0 xe 0 ðυ0 þ 1=2Þ2 - ωe 00 xe 00 ðυ00 þ 1=2Þ2 ð3Þ where Fij = (μj/μi)1/2 is an “isotopic ratio” in which μi and μj (μ = m1m2/(m1 þ m2), m1 and m2 are atomic masses) are reduced masses of ith and jth isotopologue, respectively. It is assumed that for different (A1 þ A2)i, both (ωe)i and (ωexe)i scale with Fij like (ωe)i = Fij (ωe)j and (ωexe)i = Fij2 (ωexe)j.41 The plot of Δνij vs (υ0 þ 1/2), in which the ωe0 = 117.31 cm-1 and ωe0 xe0 = 0.22 cm-1 from Figure 5a as well as ωe00 = 21.4 cm-1 and ωe00 xe00 = 0.35 cm-15,6 were used, is shown in Figure 5b. The plot reveals a possible uncertainty in the υ0 assignment (υ0 ( 3) and it had to be taken into account in estimating uncertainties of determined D10þ u state

characteristics. It is necessary to emphasize that both analyses in Figure 5 are very consistent proving a uniformity of the entire procedure. It has to be mentioned, however, that the ωe0 of the D10þ u state was determined assuming a linearity of the B-S plot from υ0 = 35 down to the bottom of the potential well, i.e., toward υ0 = 0 (see Figure 5a). This allowed calculation of the ν0,0 frequency (see Table 2) and representing the D10þ u state potential with a Morse function from the bottom of the well up to the energy of υ0 = 54 (see Figure 1, blue thick line). We have no explanation for the large difference between the ωe0 value obtained in this study and that of work of Kutner et al.20 (ωe0 = 87.9 cm-1). In this case, as it was for the F31u state, one deals with a potential having a large well depth. Therefore, the determined ωe0 and ωe0 xe0 cannot be used to approximate De0 (see section ), and it is necessary to use eqs 1 and 2 having in mind, however, that the ν0,0 was not recorded in the experiment. Using Eat(61S0) = 53310.16 cm-1,40 D000 6 and ν0,0, from eq 2 we ob-1 0 tained the D10þ u state dissociation energy D0 = 11288.7 cm , 1 þ 0 and then the D 0u state potential well depth De = 11347.3 cm-1. Table 2 collects all the D10þ u state potential parameters determined in this study. Using the determined ν0,0, ωe0 , ωe0 xe0 , and De0 , a simulation of the recorded υ0 r υ00 = 0 progression in the excitation spectrum was performed43 including the isotopic structure in each vibrational component44 (see Figure 2d). With the assumption of the R00e = 3.78 Å,6 the simulation provided R0e = 2.86 Å which, together with the De0 , is in a very good agreement with result of ab initio calculations of this work (refer to Figure 1). Note: similarly as for the F31u state, the obtained R0e allowed also including a rotational structure in the simulation despite the fact that the rotational structure is at least 1 order of magnitude smaller that the isotopic structure. The rotational structure was not resolved in the experiment, but it had, however, an influence on the shape and amplitude of the isotopic and vibrational components in spectra. Inverse Perturbation Approach Applied to the F31u(63P2) Electronic Rydberg State. Energies of the F31u state υ0 levels were also obtained quantum mechanically from an inverse perturbation approach (IPA)45,46 which was previously applied 1 8,9 and in our laboratory for determination of the A10þ u (5 P1) 1 1 10,11 3 3 state potentials of Cd2 and B 1(5 P1) state B 1u(5 P1) potential of CdKr.47 A computer program written by Pashov et al.48 was employed for the calculations and all details of the method can be found there. The basic idea of the IPA is to start from a certain initially approximate potential U0(R) and to find a correction δU(R) providing that the set of eigenvalues obtained by solving the Schr€odinger equation with the U0(R) þ δU(R) would agree well with the set of experimental energies in the least-squares approximation sense. The potential U0(R) þ δU(R) is treated then as a better approximation of the real potential and the whole procedure is repeated until certain convergence criterion is met. In our case, the U0(R) is that represented by ab initio calculated, 1000 cm-1 down shifted F31u state potential along with a set of experimental energies obtained directly from the excitation spectrum. The obtained energies of the υ0 levels -1 are listed in Table 1. An agreement (within þ0.85 -0.61 cm ) between experimental energies and those obtained in the calculations was obtained. One may conclude that the IPA increased accuracy of determination of υ0 energies as compared with the LEVEL 0 program.41 Energies of the D10þ u state υ levels were not obtained using the IPA. The method works better for irregular interatomic potentials for which experimental data cover energy regions close to a bottom of the potential well. 6857



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’ CONCLUSIONS 3 The D10þ u and F 1u State Potentials. In this article we

reported on measurements with which we extend studies of Cd2 electronic Rydberg states using a free-jet expansion beam and laser excitation at wavelengths shorter than 221 nm—the limit attained in our previous studies.10,11 The LIF excitation spectra recorded in the 206-218 nm range using the D10þ u3 3 1 þ (61S0) r X10þ g and F 1u(6 P2) r X 0g transitions were analyzed and simulated. In the excitation to the D10þ u state, isotopically resolved vibrational components of the υ0 = 35-54 r υ00 = 0 progression were recorded allowing determination of the υ0 assignment. In the excitation to the F31u state, transitions to the lowest υ0 = 0-16 levels were measured allowing determination of the minimum of the excited state potential well. Analyses of the spectra provided characterization of the D10þ u and F31u state potentials and determination of the ωe0 and ωe0 xe0 (vibrational constants), De0 (well depths), D00 (dissociation energies), and Re0 (bond lengths). Valence ab initio calculations of the interatomic potentials of the excited states were performed with relativistic and spin-orbit effects taken into account. In order to determine transitions responsible for the recorded spectra, M transition dipole moments were calculated using the ab initio potentials allowing also a proper interpretation of the experimental data. The obtained results show an encouraging agreement with results of ab initio calculations for the D10þ u state interatomic potential. The minimum of the D10þ state potential u well (i.e., De0 ) and Re0 differ only slightly, illustrating an agreement between theory and experiment. For the second investigated Rydberg state, the experimentally determined F31u state potential is deeper by approximately 8% of its De0 and its position in R implies that it represents the outer deeper well of the ab initio calculated potential (refer to Figure 1). An interesting observation was made after simulating the excitation spectra with the calculated M(R) transition dipole 1 þ 1 þ moments for the F31u r X10þ g and D 0u r X 0g transitions taken into account (see upper part of Figure 1). For the former transition, this increased the calculated intensities by a factor of 10 as well as lowered the intensities in the short-wavelength part of the spectrum with respect to the long-wavelength part which is consistent with difficulties in recording the υ0 r υ00 = 0 transitions for larger υ0 . For the latter, the calculated M caused a slight change in the R0e (D10þ u ), as judged from an envelope of the intensities for the υ0 r υ00 = 0 progression, lowering it by 0.04 Å, i.e., down to R0e = 2.82 Å. Free-Jet Expansion of Cd2 - A Source of Entangled Cd Atom Pairs. A sophisticated application of a supersonic free-jet expansion of dimers which is currently realized in our laboratory is associated with fundamental tests of quantum mechanics, i.e., test of Bell’s inequality for neutral atoms.49 In 1995, Fry et al.30,50 proposed a loophole-free experimental realization of Bohm’s spin-1/2 particle version51 of the Einstein-Podolsky-Rosen (EPR) experiment for (199Hg)2. In the proposal, photodissociation of a diatomic molecule is a critical initial step to create two atoms in an entangled state. For a detailed discussion the reader is referred to the original article.30 Here, we follow the idea of Fry et al. and extend the proposal for another isotope of 12-group atoms, namely, 111Cd. Two 111Cd atoms in the (5s)1S0 state, each with nuclear spin (I = 1/2), are produced in an entangled state with total nuclear spin equal zero (Itot = 0). (A 111Cd atom in its (5s)1S0 ground state has properties essential for the present experiment: the nuclear spin I = 1/2, and all other angular momenta

Figure 6. (a) Diagram of the electronic energy states of Cd2 and the relevant stimulated Raman adiabatic passage (STIRAP) scheme be1 1 þ 1 tween the A10þ u (5 P1) and X 0g (5 S0) states. It consists of the excitation and dissociation of the molecule at 257.1 and 305.0 nm, respectively, followed by creation of the pair of entangled Cd atoms. Diagram of the rotating (υ00 = 0, J00 = 6), ro-vibrating (υ0 = 40, J0 = 5) and dissociating (with a 90 separation angle and 0.78 eV CM kinetic energy) molecules is included. (b) Spin state selective two-photon excitation-ionization (TPEI) scheme allowing detection of the mF = -1/2 or mF = þ1/2 components of F total atomic angular momentum using two 326.2 and 230.7 nm wavelengths - see text for details. The dashed arrow shows an excitation that is impossible due to the 0.2 cm-1 HFS splitting and sufficiently narrow laser spectral bandwidth.

(S, electronic spin, and L, orbital) are zero. Therefore, the total electronic angular momentum Jtot = S þ L is zero and the Ftot = I þ J angular momentum is 1/2. Thus, the ground state 111Cd atom is a spin 1/2 fermion particle with two possible mF = (1/2 components, and the (111Cd)2 in the ground state is a boson.) Such a state is obtained by dissociation of the (111Cd)2 isotopologue produced in its X10þ g ground state in a pulsed free-jet expansion beam. The dissociation is achieved using stimulated Raman adiabatic passage (STIRAP).52 In the process (111Cd)2 are selectively excited at 6858



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)

frequencies close to a chosen (υ0 ,J0 ) r (υ00 ,J00 ) ro-vibrational 1 transition using laser radiation at 257.1 nm (the A10þ u (5 P1) r 1 þ X 0g electronic transition, see Figure 6a) and then stimulated to the photodissociation continuum of two ground state atoms using a 1 þ second laser beam at 305.0 nm (A10þ u f X 0g ). The former 0 00 excitation utilizes the P branch J = 5 r J = 6 of the rotational transition in the υ0 = 40 r υ00 = 0 vibrational band.8,9 The latter process constitutes a stimulated transition from the (υ0 = 40, J0 = 5) level directly to the repulsive (dissociative) part of the X10þ g state potential. This results in two 111Cd atoms with a center-of-mass (CM) kinetic energy 0.78 eV. Due to momentum conservation, the atoms fly apart in exactly opposite directions in the CM frame. In the laboratory frame (LAB), however, as a consequence of a defined translational velocity of molecules in the free-jet expansion (vll = 111 Cd atoms separate at 90 ( 5 angle, with 789þ115 -217 m/s), the two velocities vLAB = 1116þ86 -111 m/s, and arrive at appropriately located separate analyzers/detectors in two so-called detection planes (the separation distance between the planes is set between 0.6 and 0.7 m). Finally, the measurement of nuclear spin correlations between the two atoms in the entangled state is achieved by detection of the atoms using a spin state selective two-photon excitation-ionization (TPEI) scheme. For example, in order to detect the 111Cd atom arriving into the detection plane with mF = -1/2(or þ1/2), it is subjected to a σþ(or σ-)-polarized laser pulse with a resonance frequency corresponding to the (5s2)1S0, F = 1/2 f (5s5p)3P1, F = 1/2 transition (at 326.2 nm) followed by an ionizing π-polarized laser pulse with a frequency corresponding to the (5s5p)3P1,F = 1/2 f (5p2)3P0, F = 1/2 transition (at 230.7 nm) (see Figure 6b). As a result of the TPEI process, the (e-)-(111Cdþ) pairs are created and their components are detected in coincidence using two separate detectors. A similar process occurs in the second detection plane. The selective character of the TPEI process is guaranteed by a magnitude of the HFS splitting (approximately 0.2 cm-1) in the intermediate (5s5p)3P1 atomic state, which is easily resolved with the excitation and ionization lasers. The crucial aspect of the proposed experiment is that the 111Cd atoms in a given pair are “born together” from one (111Cd)2 molecule which was earlier produced in a free-jet expansion beam. This automatically creates an entanglement between the two atoms in a pair. The free-jet beam is used as a source of the (111Cd)2 molecules. Because of their vdW character, i.e., shallow electronic ground state with small dissociation energy D000 , the process of creation requires using a cold environment of the beam. Consequently, due to internal cooling, most of the molecules occupy the lowest υ00 = 0 level as well as lowest J00 levels. Therefore, the (111Cd)2 molecules in the (υ00 = 0, J00 = 6) level, from which the STIRAP process starts, have a relatively high population in the beam. Also, in the free-jet expansion the (111Cd)2 molecules have well-defined v which is determined by the expansion parameters such as the carrier gas pressure, the diameter of the nozzle orifice, the pressure in the expansion chamber, and the temperature of the source. The experimental setup is being constructed in our lab presently.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The research was financed from 2007-2010 funds for science of Polish Ministry of Science and Higher Education (research

project N N202 2137 33). The research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (Contracts POIG.02.01.00-12-023/08 and POIG.02.02.00-00-003/08). The help of Professor W. Jastrze-bski in applying a pointwise IPA procedure is highly appreciated.

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’ NOTE ADDED AFTER ASAP PUBLICATION This article posted ASAP on March 16, 2011. Part of the text was missing in the abstract in the pdf version only. The correct version posted on March 22, 2011.

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Electronic Rydberg States of Cd2 from ab Initio ... - ACS Publications

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