Article pubs.acs.org/JPCA
Molecular-Beam Optical Stark and Zeeman Study of the [17.8]0+− X1Σ+ (0,0) Band System of AuF Timothy C. Steimle,* Ruohan Zhang, and Chengbing Qin† Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604, United States
Thomas D. Varberg Department of Chemistry, Macalester College, 1600 Grand Avenue, St. Paul, Minnesota 55105, United States ABSTRACT: The [17.8]0+−X1Σ+ (0,0) band of gold monofluoride, AuF, has been recorded at a resolution of 40 MHz both field free and in the presence of a static electric and magnetic field. The observed Stark shifts were analyzed to determine the permanent electric dipole moment, μel, of 2.03 ± 0.05 D and 4.13 ± 0.02 D, for the [17.8]0+(v = 0) and X1Σ+(v = 0) states, respectively. The small magnetic tuning observed for the [17.8]0+(v = 0) state is attributed to rotational and magnetic field mixing with the [17.7]1 state and has been successfully modeled using an effective Hamiltonian for the 3Π state. A comparison with the numerous published theoretical predictions is made.
I. INTRODUCTION Unlike chemically inactive bulk gold, small gold clusters are highly reactive and play an increasingly important technological role. Since the seminal work by Haruta et al.,1 which revealed that small supported gold clusters catalyze the oxidative elimination of carbon monoxide, such media have become the catalyst of choice for many chemical processes.2 The increased chemical applications of small gold clusters have stimulated an immense increase in theoretical efforts to correctly predict the properties of gold containing molecules.3 A major facet of those theoretical efforts is to correctly account for the large relativistic and electron correlation effects. Relativistic effects cause the 6s orbital to strongly contract and the 5d orbital to slightly expand. In addition, the 6s orbital is significantly stabilized and the 5d orbital slightly destabilized. The relativistic effects for Au are considerably larger than either of the adjacent elements Pt or Hg.4 Simple gas-phase gold containing diatomic molecules serve as ideal venues for testing the various computational methodologies being developed to describe the large relativistic and electron correlation effects because the properties of these molecules can be precisely derived from high-resolution spectroscopic measurements. A comparison of predicted and experimentally determined molecular electric dipole moments, μel, and magnetic dipole moments, μm, are particularly insightful for assessing predicted electronic wave functions. The electric and magnetic dipole moments are readily extracted from the analysis of the Stark and Zeeman effects, respectively. Both properties are primarily dependent on the nature of the chemically relevant valence electrons. In the most general terms, bond polarity is garnered from μel and the coupling of the orbital and spin angular momenta garnered from μm. As demonstrated by Schwerdtfeger et al.,5,6 μel is particularly insightful for assessing relativistic © 2013 American Chemical Society
methodologies for electronic structure predictions because molecular properties (bond lengths, dissociation energies, force constants, etc.) strongly correlate with bond polarity. Specifically, in an ideal Au+X− molecule, where the 6s orbital is not occupied, a small relativistic bond length expansion is expected, whereas in an ideal Au−X+ molecule, which has 6s orbital occupation, a strong bond relativistic length contraction is predicted. Hence, there should be a correlation between relativistic contribution to the bond lengths and reduced dipole moment (≡ μel/Re, Re = bond length) for diatomic Au−X molecules. Similarly, a strong correlation between the relativistic contribution to the bond polarity and dissociation energy, De, is predicted.5,6 The polarity of the Au−X bond is described as the difference in the electronegativity (≡ using Mulliken’s definition as the arithmetic mean of the ionization potential and electron affinity7) of the ligand, ENX, and the electronegativity of Au, ENAu. The relativistic effect increases ENAu to 2.4 from a nonrelativistic value, which is similar to that for isovalent Cu and Ag (∼2.0). Accordingly, relativistic effects are predicted to decrease the ionic contribution to bonding for electronegative ligands (i.e., for ENX > ENAu) whereas increase the bonding contribution for electropositive ligands (ENX < ENAu). The large electronegativity of the F ligand (ENF = 4.0) suggests that the relativistic contributions will only slightly decrease the predicted dissociation energy for AuF because the relativistic contribution to the difference between ENF and ENAu is comparatively small (i.e., ENF−ENAu nonrelativistic value of 4.0−2.0 vs a relativistic value of 4.0−2.4). Alternatively stated, the relativistic stabilizaSpecial Issue: Curt Wittig Festschrift Received: February 27, 2013 Revised: April 15, 2013 Published: April 15, 2013 11737
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factor, gL, can differ significantly from unity because of nonadiabatic contributions. The value of the anisotropic gfactors, gl and gl′, are difficult to predict a priori and need to be considered as variables. The parity dependent term gl′ is only required for states with nonzero electronic orbital angular momenta. Here, we report on the analysis of the Stark and Zeeman effect in the [17.8]0+−X1Σ+ (0,0) band of AuF. Experimental studies of AuF are rather limited. The first identification of gas-phase AuF was reported in 1992 by Saenger and Sun using visible emission spectroscopy and a discharge plasma source.9 In 2000, the emission spectrum from a gold hallow cathode discharge source was recorded, assigned to progressions in A1Π−X1Σ+and B1Σ+− X1Σ+bands, and analyzed to produce the first vibrational parameters for the A1Π, B1Σ+, and X1Σ+ states.10 This same study performed a relativistic density functional theory (DFT) calculation to predict Re, μel, Te, and ωe for the ground and five low-lying excited states. Soon thereafter the pure rotational spectrum was recorded using Fourier transform microwave (FTMW) techniques and analyzed to produce precisely determined rotational and hyperfine parameters for the X1Σ+(ν = 0) state.11 A more extensive study of the pure rotational spectrum, recorded using millimeter- and submillimeter-wave spectroscopy was performed by Okabayashi et al.12 Ninety-four transitions, involving v = 0 through v = 13, in the 189−402 GHz range were recorded and fit to a Dunham expression to determine Re, ωe, and ωexe. High-resolution, laser-based studies of the visible spectrum were recently performed by the Varberg group.13,14 The A1Π−X1Σ+and B1Σ+−X1Σ+ bands were reassigned as [17.7]1−1Σ+ and [17.8]0+−X1Σ+ (vide infra). The (0,0), (1,1), (0,1), and (1,2) bands of the [17.7]1−X1Σ+ and [17.8]0+−X1Σ+ electronic transitions and the (0,0) band of the newly detected [14.0]1−X1Σ+ electronic transition were recorded at Doppler limited resolution to produce fine structure parameters.14 The [17.7]1−X1Σ+(0,0) band was also recorded at sub-Doppler resolution13 from which the 197Au(I = 3/2) and 19 F(I = 1/2) hyperfine parameters for the [17.7]1(v = 0) and X1Σ+(v = 0) states were determined. The large negative value for the 197Au magnetic hyperfine parameter determined for the [17.7]1(v = 0) state suggested that this state has a major contribution from a Hund’s case (a) 3Δ1 state. In contrast to the limited number of experimental studies, there have been numerous reported electronic structure calculations,6,10,15−25 many of which6,10,17−23 have predicted μel for the X1Σ+ state using a variety of methods. Two of these calculations10,18 have also predicted excited state properties. The most extensive investigation of μel for the X1Σ+ state was performed first by Goll et al.19 and more recently by Schwerdtfeger et al.22 In the earlier study,19 a variety of pure DFT approaches, pure wave function based approaches, and methods that mix wave function/DFT based approaches were investigated. The wave function based methods were used to describe the short-range (SR) interaction and DFT for the longrange (LR) interactions. In total, 22 approaches were used to predict μel (X1Σ+). Because of the lack of an experimental μel value, the performance of these various methodologies was assessed by making a comparison with the μel value (= 4.37 D) predicted using a complete basis set limit coupled cluster singles and doubles excitation with additional perturbative triples (CBS/ CCSD(T)) method. The internuclear separation was fixed to the experimental value (= 1.9184 Å). It was determined that the mixed LR/SR methods, which gave values of 4.44 and 4.43 D for the local-density approximation (LDA) and Perdew−Burke−
tion of the 6s orbital is for the most part irrelevant in describing the ground state properties (De, Re, and μel) of the highly polar AuF molecule because, to a first approximation, the metal center configuration is Au+(5d10). In contrast, the relativistic effects in the excited states, where the metal center configuration is Au+(5d96s1), will be much greater. Simultaneously correctly predicting the De, Re, and μel for the ground and excited states of AuF is a critical test of electronic structure predictions. Although less often measured, the Zeeman effect also provides insight into the nature of the electronic wave function. At the simplest level of approximation, the Zeeman effect provides a method of assigning the 2S+1Λ term symbol for a diatomic molecule. For example, if it is assumed that the molecule is in the Hund’s case (a) limit, where the eigenfunction is a product of an electron orbital, electronic spin, and a rotational function, Ψ = |nΛ⟩|SΣ⟩|JΩMJ⟩, then the Zeeman effect is often approximated as the expectation value:8 Zee ⟨J ΩMJ|⟨S Σ|⟨n Λ|Ĥ |n Λ⟩|S Σ⟩|J ΩMJ⟩
= ⟨J ΩMJ|⟨S Σ|⟨n Λ| − μm⃗ ·B⃗ |n Λ⟩|S Σ⟩|J ΩMJ⟩ = μB BMJ Ω[gL Λ + gSΣ]/[J(J + 1)]
(1)
In eq 1, Λ and Σ are the projections of the electronic orbital and spin angular momenta on the internuclear axis, Ω = Λ + Σ, J is the total angular momentum, and gL and gS are the magnetic gfactors taken as 1.000 and 2.002, respectively. The use of eq 1 for modeling the observed Zeeman effect for a series of spin−orbit substates allows for the determination of Λ and Σ. At the next level of approximation for modeling the Zeeman effect, Λ and Σ are considered to be rigorous quantum numbers, as implied by the assignment of a 2S+1Λ term symbol, and the energies are obtained from the diagonalization of the sum of matrix representations of the Zeeman, Ĥ Zee, spin−orbit, Ĥ s.o. rotational, Ĥ rot, and other fine and hyperfine structure operators. At this level of approximation, gL and gS are still assumed to be 1.000 and 2.002, respectively. The above approach is often inadequate for precise determination of the energy levels of late transition metal containing radicals because the high density of electronic states and the large spin−orbit and/or rotational mixing among these states diminishes the goodness of Λ and Σ . A common procedure for modeling the energy levels of such mixed states is to use an effective Hamiltonian operator where both gL and gS are allowed to deviate from 1.000 and 2.002 and adding additional small, nonadiabatic terms to account for the magnetic field induced mixing of electronic states. Ignoring the small magnetic moments associated with the nuclear spin and rotational contributions, the effective Zeeman Hamiltonian is8 Zee Ĥ (eff ) = gSμB BZ T01(S) + gL μB BZ T01(L)
+ gl μB BZ
∑
D0,1 q(ω)Tq1(S) + gl′μB
q =±1
BZ
∑ q =±1
e−2iqφD0,1 q(ω)Tq1(S) (2)
where and refer to the molecule fixed components of the electronic spin and orbital angular momentum operators expressed in spherical tensor form, respectively, BZ is the magnetic field in the laboratory Z-direction, and ϕ is the azimuthal angle of the electronic coordinates. The electronic spin g-factor, gS, of the effective Hamiltonian is typically found to be very close to the free electron value of 2.002, but the orbital gT1q(L)
T1q(S)
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Ernzerhof (PBE) functionals, respectively, were indeed better than either the pure DFT or wave function based approaches. The Coulomb-attenuated Becke three-parameter Lee−Yang− Parr hybrid functional, CAM-B3LYP, which predicted a value of 4.24 D, outperformed the other pure DFT methods. The more systematic study of μel (X1Σ+) by Schwerdtfeger et 22 al. was part of an investigation of the breakdown of the pseudopotential approximation to describe the response of the electronic wave function to an external electric or magnetic field. Three small core pseudopotentials (SCPP), one large core pseudopotential (LCPP), and five different functionals were used in DFT calculations and compared with a second-order Douglas−Kroll all electron (DK-AE) DFT calculation. The internuclear separation was fixed to the experimental value (=1.9184 Å). The resulting (DK-AE) DFT calculations with the LDA, PW91, PBE, M06, and B3LYP functionals predicted μel (X1Σ+) values of 3.585, 3.578, 3.576, 4.317, and 4.029 D, respectively. The 20 relativistic pseudopotential values ranged from a low of 3.456 D to a high of 4.992 D. Guichemerre et al.18 has performed the most extensive theoretical prediction for the excited states of AuF. In that study, the potential energy curves and various spectroscopic properties (De, Re, μel, Te, and ωe) for the X1Σ+ and nine lowlying electronic states were computed. These states included the 1,3 + Σ and 1,3Π states that correlate at infinite separation to the lowest energy Au(2S: 6s15d10) plus F(2P: 2p5) asymptote and 1,3 + 1,3 Σ , Π, and 1,3Δ states that at infinite separation correlate to the lowest excited Au(2D: 6s25d9) plus F(2P: 2p5) asymptote. There are an additional 12 states (2 × 1,3Σ−, 2 × 1,3Π, 1,3Φ, and 1,3 Δ) that correlate to the Au(2D: 6s25d9) plus F(2P: 2p5) asymptote. These states should be significantly higher in energy and not relevant to the visible spectrum. Near the equilibrium bond distance (∼1.92 Å), seven of the 10 electronic states considered correlate to either the ionically bound Au+(5d10) plus F−(2p6) X1Σ+ state or the ionically bound 1,3Σ+, 1,3Π, and 1,3Δ low-lying Au+(5d96s1) plus F−(2p6) states. The remaining predicted three states, which are labeled as 21Π, 23Π, and 23Σ+ in ref 18, are weakly covalently bound and have a large equilibrium bond distance (∼2.40 Å). These states should not be relevant to the description of the visible spectrum. Guichemerre et al.18 also predicted the diagonal and off-diagonal spin−orbit interactions between the 10 electronic states. The energies of the predicted low-lying excited states with and without spin−orbit interaction are compared with the observed energies in Figure 1. It is noteworthy that the Λ-doubling in the 13Π0 state (i.e., the splitting of the 13Π0+ and 13Π0− substates) is predicted to be more than 7000 cm−1 and that the predicted energies for the 3Π0+ and 3 Π1 substates are very close to those for the observed [17.7]1 and [17.8] 0+ states.
Figure 1. Predicted term energies from ref 18 without (left) and with (center) spin−orbit interaction and the observed term energies (right) for the low-lying excited states of AuF. The strong interaction between the 21Σ+ and 13Π states results in large splitting between the 13Π0+ and 13Π0− (i.e., Λ-doubling) substates. Rotational and magnetic field mixing of the 13Π0+ and 13Π1 states is responsible for the observed Zeeman tuning of the [17.8] 0+ state.
560 ± 10 nm bandpass filter and detected with a cooled GaAs photomultiplier tube. Photon counting techniques were used to process the signal. Absolute wavelength calibration was obtained by simultaneously recording the sub-Doppler absorption spectrum of I2. The small spectral shifts and splitting were precisely determined by simultaneously monitoring the transmission of an actively stabilized, 750.956 MHz free spectral range (f.s.r.), confocal etalon and an unstabilized, 74.9 MHz f.s.r., confocal etalon. Static electric field strengths of up to 3010 V/cm were generated by application of a voltage across a pair of conducting plates straddling the region of molecular fluorescence. One plate was a solid polished stainless steel disk 10 cm in diameter and 0.5 cm thick. The second plate was a 5 × 5 cm square neutral density filter that transmitted approximately 90% of the light. The spacing between the plates was precisely measured to be 1.960 ± 0.003 cm in the central region. The voltage was measured to an accuracy of approximately 0.2% using a commercial voltmeter. A polarization rotator and polarizing filter were used to orient the electric field vector of the linearly polarized laser radiation either parallel, ∥, or perpendicular, ⊥, to that of the applied field. The systematic errors due to current leakage and edge effects were tested by reversing the polarity of the voltages applied to the two plates and reducing their spacing. The resulting Stark shifts were identical to within the measurement uncertainty (±20 MHz). The systematic error introduced by spectral calibration procedures and electric-field strength determined is estimated to be approximately 2%. The homogeneous magnetic fields used in the optical Zeeman measurements were generated by a pair of 25 mm diameter rareearth permanent magnets with 5 mm holes attached to a soft iron yoke.28 The field was calibrated using a commercial Gauss meter. A polarization rotator was used to align the electric field vector of the linearly polarized laser radiation either perpendicular (ΔMJ = ± 1) or parallel (ΔMJ = 0) to the static magnetic field vector. The systematic errors arising from the magnetic field calibration and the measurement uncertainties in the Zeeman shifts are estimated to be less than 2%.
II. EXPERIMENT The molecular beam optical Zeeman and optical Stark spectrometer used here have been previously described.26−28 A rotating 6 mm diameter gold tube was ablated at 20 Hz with approximately 5 mJ of 532 nm output from a Q-switched, second harmonic, pulsed Nd:YAG laser. The ablation products reacted with a free-jet expansion of 5% sulfur hexafluoride (SF6) and 95% argon carrier gas from a backing pressure of approximately 3000 kPa. Approximately 10 mW of focused (f l = 750 mm) power derived from a single longitudinal mode cw-dye laser was used to excite the [17.8] 0+−X1Σ+ (0,0) band. The resulting laserinduced fluorescence (LIF) was viewed on resonance through a 11739
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Figure 2. P(1) line of the [17.8]0+−X1Σ+ (0,0) band of AuF recorded field free and in the presence of a 2366 V/cm static electric field with parallel (ΔMJ =0) and perpendicular (ΔMJ = ± 1) orientation. The Stark tuning of the associated energy levels and assigned transitions are also presented. Stark Ĥ = −μel̂ ·E ⃗
III. OBSERVATION The R(0) (ν = 17756.156 cm−1), R(1) (ν = 17756. 642 cm−1), and P(1) (ν = 17755.125 cm−1) lines were selected for optical Stark measurements. Spectra of the P(1) line recorded field free and in the presence of a 2366 V/cm static electric field with parallel (ΔMJ =0) and perpendicular (ΔMJ = ± 1) orientation are presented in Figure 2. The Stark tuning of the associated energy levels and assigned transitions are also presented. A precise measurement of the Stark induced shift was obtained by scanning over the field-free or Stark-shifted component and then turning on or off the field and continuing the scan. A total of 29 Stark shifts were precisely measured and are presented in Table 1 along with the assignment and the difference between the observed and calculated shifts. The R(0) (ν = 17756.156 cm−1), R(1) (ν = 17756. 642 cm−1), R(2) (ν = 17757.106 cm−1), P(1) (ν = 17755.125 cm−1), P(2) (ν = 17754.577 cm−1), and P(3) (ν = 177554.009 cm−1) lines were selected for optical Zeeman measurements. These six branch features were all measured in a magnetic field of 4650 G with both parallel (ΔMJ = 0) or perpendicular (ΔMJ = ± 1) orientation. The R(1) and P(3) lines, both of which have the [17.8]0+(v = 0) J = 2 rotational level as the upper energy terminus, are presented in Figure 3. The Zeeman tuning of the associated energy levels and assigned transitions is also presented. A comparison of the various Zeeman spectra revealed that there was no detectable tuning of the rotational levels of the X1Σ+(v = 0) state, as expected. The measured shifts and energy level assignment and difference between the observed and calculated (vide infra) shifts are presented in Table 2. It is evident that the shifts of a given rotational level are very asymmetric (e.g., Δν (J = 1, MJ = −1) = −89 MHz, whereas Δν (J = 1, MJ = +1) = 137 MHz) and that this asymmetry becomes less pronounced with increasing J .
(3)
where μ̂el is the dipole moment operator and E⃗ is the applied static electric field vector. The matrix representation of Ĥ Stark is block diagonal in MJ, the projection of the total angular momentum along the electric field axis, and of infinite dimension. For the purposes of the Stark analysis, the [17.8]0+ (v = 0) state was treated as a 1Σ+ state. The matrices were truncated to include the J = 0−7 rotational levels for both the [17.8]0+(v = 0) and X1Σ+(v = 0) states to ensure that the error introduced by truncation is significantly less than the measurement error of the spectral shifts (±20 MHz). The resulting 8 × 8 matrices were numerically diagonalized to produce eigenvalues and eigenvectors. A nonlinear least-squares fitting procedure, using the observed shifts given in Table 1 as input, produced μel values of 2.03 ± 0.05 and 4.13 ± 0.02 D for the [17.8]0+(v = 0) and X1Σ+(v = 0) states, respectively. The uncertainty estimates represent one standard deviation of the random error, which is slightly less than the estimated upper limit of 2% for the systematic error. The correlation coefficient was −0.37. Modeling the behavior of the energy levels of the [17.8]0+(v = 0) state in the presence of a static magnetic field, which would be negligible for an isolated Ω = 0 state, was more complicated. To model this Zeeman effect, a 3Π state description has been used. Guichemerre et al.18 predicts that the [17.8]0+(v = 0) state is primarily the 13Π0+ substate (Figure 1), which is spin−orbit mixed with the 21Σ+state, and that the [17.7]1(v = 0) state is primarily the 13Π1 substate, which is spin−orbit mixed with the 11Π and 13Σ+1 states. In the effective Hamiltonian model for a 3Π state, these mixings are accounted for by the Λ-doubling and spin−spin interaction terms: 2 2 2 eff 2 Ĥ (3 Π) = Tv ′ v ″ + AL̂Z SẐ + BR̂ + λ(3SẐ − S ̂ ) 3 2 2 1 + (o + p + q)(S+̂ + S−̂ ) 2
IV. ANALYSIS The behavior of the energy levels in the presence of the applied electric field was modeled by including the operator: 11740
(4)
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Table 1. Observed Stark Shifts for the [17.8]0+−X1Σ+ (0,0) Band branch, pol R (0), ∥
R (0), ⊥
R (1), ∥
R (1), ⊥
P (1), ∥ P (1), ⊥
field (V/cm)
assigna
2751 2751 2745 2745 2472 2931 2565 2565 2360 1991 1991 2777 2366 1991 1991 1991 2366 2777 2739 2485 2490 2739 2485 2490 2366 2745 2360 2360 2497 SD = 12 MHz
A A A A A A B B B A A A A C C C C C B B B D D D A A E E E
shift (MHz)
obs − calcdb
779 775 769 771 619 870 522 550 460 −220 −224 −366 −272 135 126 127 160 216 −352 −310 −291 184 159 163 −432 −569 30 30 34
6 2 0 2 −8 −4 −17 11 2 −22 −26 9 4 20 11 12 −2 −7 25 4 24 4 11 14 −10 −6 5 5 6
adjustable parameters were attempted. In the end, gL and gS were constrained to 1.000 and 2.002, respectively, and gl was allowed to vary because the correlation coefficient between gS and gl was near unity, and the data were relatively insensitive to gL and gl′. The optimized value for gl is 0.82(20). The Zeeman shifts predicted with gL = 1.000, gS= 2.002, gl′ = 0, and gl = 0.82 resulted in the residuals given in the fourth column of Table 2. The standard deviation of 32 MHz is still slightly larger than the estimated uncertainty (20 MHz), and a slight systematic trend in the residuals remains.
V. DISCUSSION The determined μel values for the [17.8]0+(v = 0) and X1Σ+(v = 0) states are compared with the selected predicted values of refs 13, 18, and 22 in Table 3. A perusal of Table 3 reveals that all of the high level predictions for the X1Σ+ state (i.e., the DFTCCSD, CAM-B3LYP, and CBS-CCSD(T) values of ref 19 and the DK-AE and SCPP values of ref 22) are within ±0.4 D of the experimental value. This generally good agreement may be a reflection of the small relativistic contributions expected for this highly polar molecule as discussed earlier. Note that the DFT/ B3LYP, nonrelativistic, all electron (NR-AE) prediction for μel (X1Σ+) (= 5.305 D) is only 1.3 D larger than the equivalent relativistic DK-AE predicted value of 4.029 D. Table 2 also reveals that the more recent high level small core pseudopotential (SCPP) predictions22 perform better than the mixed LR/SR methods.19 Furthermore, in contrast to the previously reached conclusion,19 the pure DFT method using CAM-B3LYP hybrid functional performs better than the mixed LR/SR methods. A comparison of the experimental μel (X1Σ+) value (= 4.13 ± 0.02 D) with that of ref 18 (= 4.89 D) is more difficult because the μel value was predicted at the optimized Re value (= 1.95 Å) and not the experimental value of 1.9184 Å. Using the plot of μel as a function of internuclear separation given in ref 18, μel decreases by approximately 0.16 D in going from 1.95 to 1.9184 Å. The estimated μel (X1Σ+) value at the experimental Re of 4.73 D is still in relatively poor agreement with the observed value of 4.13 D. The μel values for the [17.8]0+(v = 0) state (= 2.03 ± 0.05 D) is significantly less than that for the X1Σ+(v = 0) state (= 4.13 ± 0.02 D) even though the bond distance for the [17.8]0+ state (= 1.955 Å) is longer than that of the X1Σ+ state. The dominant ionic contribution for the [17.8]0+(v = 0) state is Au+(5d96s1), whereas that for the X1Σ+(v = 0) state is Au+(5d10). The 6s orbital is more readily back-polarized in comparison to the 5d orbital, and thus, there is a large reduction in the μel upon excitation. The poor agreement between the observed μel and prediction18 (= 2.68 D) is a reflection of the inability of the employed CC-SRPP method to account for both increases in relativistic and polarization effects. The prediction was obtained at an optimized Re of 1.96 Å, which is close to the experimental value. The magnetic tuning of the [17.8]0+(v = 0) state, which would be negligible for a state in the pure Hund’s case (a) limit, is a result of rotational and magnetic field mixing with the [17.7]1(v = 0) state, which lies only approximately 100 cm−1 to lower energy. As is evident from column 3 of Table 2, the tuning of the levels is faster than that expected for an isolated 3Π state having gL = 1.000 and gS = 2.002. This is consistent with the Guichemerre et al.18 prediction that the [17.7]1(v = 0) state is primarily the 13Π1 substate but with a large spin−orbit mixed contribution from the 11Π and 3Σ+1 states. Specifically, the mixing with the 3Σ+1 state increases the expected magnetic tuning because gJ (≡ Ω[gLΛ + gSΣ]/[J(J + 1)]) (see eq 1) for a 3Σ+1 state is 2.002 times larger than that for a 3Π1 or a 1Π1 state. The faster tuning is
A, MJ″=0 → MJ′=0; B, MJ″=0 → MJ′=1; C, MJ″=1 → MJ′=1; D, MJ″=1 → MJ′=2; E, MJ″=1 → MJ′=0. bCalculated shifts obtained using optimized μel values of 2.03 and 4.13 D for the [17.8]0+ and X1Σ+ states, respectively. a
The predicted separations18 between the 13Π2, 13Π1, 13Π0+, and 13Π0− substates are 1936, 484, and 7033 cm−1, respectively. The observed splitting between the [17.8]0+(v = 0) (J = 1, “-parity”) and [17.7]1(v = 0)(J = 1, “-parity”) levels is 98.8 cm−1 as determined from the difference between the R(0) (17756.120 cm−1) line of the [17.8]0+−X1Σ+ (0,0) band and Re(0) (17657.327 cm−1) line of the [17.7]1−X1Σ+ (0,0) band. The predicted separation of 1936 and 7033 cm−1 and the observed separation of 98.8 cm−1 can be reproduced using the 3Π effective Hamiltonian model (eq 3) with A = −2780 cm−1, (o+p+q) = 3520.0 cm−1, and λ= 419.3 cm−1 and constraining the rotational constant, B, and Ω-doubling constant, q, to the experimental values14 of 0.254 and 0.00324 cm−1, respectively. The field-free energies were modeled by numerically diagonalizing 6 × 6 matrices constructed in a Hund’s case (a) basis set. The Zeeman effect was modeled by numerically diagonalizing a 42 × 42 fieldfree matrix representation for the J = 0−6 rotational levels augmented by the matrix elements for Ĥ Zee. The expression for the matrix elements was taken from ref 8. The Zeeman shifts predicted assuming that gL = 1.000, gS= 2.002, and gl = gl′ = 0 resulted in the residuals given in the third column of Table 2. The standard deviation of 62 MHz is significantly larger than the estimated uncertainty (20 MHz), and a large systematic trend in the residuals is observed. Various fits using gL, gS, gl′, and gl as 11741
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Figure 3. R(1) and P(3) branch features of the [17.8]0+−X1Σ+ (0,0) band of AuF recorded in the presence of a 4500-Gauss static magnetic field with parallel (ΔMJ = 0) and perpendicular (ΔMJ = ± 1) orientation. The Zeeman tuning of the associated energy levels and assigned transitions are also presented.
⎡⎛ 3 gl = −gL ⎢⎜ Σ ⟨13 Π(v = 0)|Σaî T±1 q(li)̂ |1 Σ+(v = 0)⟩ ⎣⎝q =±1 i ⎤ ⎞ 3 ⟨1 Σ+(v = 0)|T∓1 (li)̂ |13 Π(v = 0)⟩⎟ /ΔE ⎥ ⎦ ⎠
Table 2. Observed Zeeman Shifts for the [17.8]0+(v = 0) Levels J, MJ
obs shifta
obs − calcdb
obs − calcdc
0, 0 1, −1 1, +1 2, −2 2, −1 2, 0 2, +1 2, +2 3, −3 3, −2 3, −1 3, 0 3, +1 3, +2 3, +2
10 −89 137 −173 −86 7 119 233 −298 −195 −108 0 102 214 324
−9 −45 46 −63 −35 −7 35 74 −122 −80 −56 −14 19 60 97 SD = 63
−28 −40 −4 −32 −23 −20 −9 −6 −64 −43 −44 −28 −24 −14 −12 SD = 31
where ΔE ≡ E(13Π(ν = 0)) − E(13Σ+(ν = 0)) and T1± q(l)̂ i is the spherical tensor operator for the orbital angular momentum of the individual electrons. The dominant configurations for the 13Π and 13Σ+ states are ...π +π ̅ +π −σ → 3 Πi and 1Π ...σ ′σ →
1,3 +
Σ
(6) (7)
where the π, σ′, and σ orbitals are essentially 5d±1 (Au+), 6s (Au+), and 5d0 (Au+) orbitals, respectively. Under these assumptions: gl =
Observed shifts in MHz for a magnetic field of 4650 G. bCalculated using gS = 2.0023 and gL = 1.0. cCalculated using optimized gS = 2.84 and gL = 1.0. a
gL ⟨π −|al̂ −̂ |σ ⟩⟨σ |l+̂ |π −⟩ 2 ΔE
(8)
Using the pure precession approximation29 for the evaluation of the matrix elements
inconsistent with the previous conclusion from the interpretation of the hyperfine parameters14 that the [17.7]1(v = 0) state has a major contribution from a Hund’s case (a) 13Δ1 state because gJ ≈ 0 for such a state. The determined large value for gl (= 0.82(20)) is a reflection of the 13Σ+1 /13Π mixing. The gl term in the effective Hamiltonian is a result of the cross-product of the orbital Zeeman term Zee Ĥ (orbital) = gL μB L̂ ·B̂
(5)
gl =
3gL aπ ΔE
(9)
Given that the 13Σ+1 state is below the 13Π state, ΔE is greater than zero and gl is predicted to be positive, consistent with observation. The energy denominator is calculated to be approximately 4700 cm−1 (ref 18) and aπ ≈ −2A ≈ 5560 cm−1. The unrealistically large predicted gl (≈ 4) value using these assumptions is a reflection, among other things, of the inappropriateness of second order perturbation theory due to the large spin−orbital interaction and small state separation. The assumption that there is a unique perturber is also questionable given the density of excited electronic states.
(4)
and the spin−orbit operator. Assuming that the [17.7]1(v = 0) state is the 13Π1(ν = 0) state and that this state interacts solely with the 13Σ+(ν = 0)state, then the expression for gl, derived from second order perturbation theory,8 is 11742
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Table 3. Observed and Predicted AuF μel Values method
μel (Debye)
ref
method
μel (Debye)
ref
DK-AE/LDA DK-AE/PW91 DK-AE/PBE DK-AE/M06 DK-AE/B3LYP SC-SRRP-Sb SC-SRRP-LANLb SC-SRRP-CEPb LC-SRRP-LANLb
3.585 3.578 3.576 4.317 4.029 4.046 3.939 4.071 4.872
22a 22 22 22 22 22 22 22 22
X1Σ+ exptl CBS/CCSD(T) DFT/LDA DFT/PBE DFT-CCSD(T)/LDA DFT-CCSD(T)/PBE CAM-B3LYP
4.13 ± 0.02 4.37 3.58 3.62 4.44 4.43 4.24
19a 19 19 19 19 19
CC-SRPP
4.89c
18c [17.8]0+
exptl CC-SRPP
2.03 ± 0.05 2.68
18c
a
Predicted at the experimental Re value (= 1.9184 Å). bDFT calculation using the B3LYP functional. cPredicted at the theoretical Re value (= 1.95 Å).
VI. CONCLUSIONS The electric and magnetic tuning of the [17.8]0+(v = 0) and X1Σ+(v = 0) states has been experimentally determined from which μel values and magnetic g-factors have been determined. This is the first experimental determination of these properties for any Au-containing molecule. The recent high-level relativistic predictions for μel (X1Σ+) are in generally good agreement with the experimental value. The sole prediction for μel ([17.8]0+) is in relatively poor agreement with the experimental value. This poorer agreement is most likely due to expected greater relativistic contributions in the [17.8]0+ state relative to the X1Σ+ due to occupation of the 6s orbital in the former state. The unexpectedly large magnetic tuning observed for the [17.8]0+(v = 0) state can be qualitatively understood as the rotational and magnetic fields mixing with the observed nearby [17.7]1(v = 0) state and a predicted 3Σ+1 state. The Zeeman tuning does not support the suggestion14 of a large 3Δ1 state contribution to the [17.7]1(v = 0) state. Further insight into the nature of the [17.7]1(v = 0) state can be realized by optical Zeeman studies of the [17.7]1−X1Σ+ transition, which are currently in progress. Furthermore, the application of a magnetic field should enhance the 13Π2−X1Σ+ electronic transitions. Analysis of the magnetic hyperfine structure in the 13Π2−X1Σ+ band will provide insight into why the hyperfine structure of the [17.7]1 state is inconsistent with the observed magnetic tuning. It is expected that there will be larger relativistic contributions to the properties of less polar Au−X diatomic molecules, and as such, predicting properties for such species will be more challenging. Attempts are being made in our laboratory to measure the μel values and magnetic g-factors for Au−X (X = C, S, and O).
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ACKNOWLEDGMENTS
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REFERENCES
This research has been supported by a grant from the Fundamental Interactions Branch, Division of Chemical Sciences, Office of Basic Energy Sciences, Department of Energy (DE-FG02-01ER15153-A003). C.Q. thanks the State Scholarship Fund of China Scholarship Council for financial support. T.D.V. thanks the National Science Foundation for support under grant no. CHE-0844405. The authors acknowledge the helpful discussions with Dr. Lan Cheng (Department of Chemistry, U of Texas-Austin)
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AUTHOR INFORMATION
Corresponding Author
*(T.C.S.) E-mail:
[email protected]. Phone: (480) 965-3265. Present Address
† Visiting from Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
Notes
The authors declare no competing financial interest. 11743
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