Jet-Cooled Laser-Induced Fluorescence ... - ACS Publications

Article pubs.acs.org/JPCA

Jet-Cooled Laser-Induced Fluorescence Spectroscopy of Cyclohexoxy: Rotational and Fine Structure of Molecules in Nearly Degenerate Electronic States Jinjun Liu† and Terry A. Miller*,‡ †

Department of Chemistry and Conn Center for Renewable Energy Research, University of Louisville, 2320 South Brook Street, Louisville, Kentucky 40292, United States, and ‡ Laser Spectroscopy Facility, Department of Chemistry and Biochemistry, The Ohio State University, 120 West 18th Avenue, Columbus Ohio 43210, United States S Supporting Information *

ABSTRACT: The rotational structure of the previously observed B̃ 2A′ ← X̃ 2A″ and B̃ 2A′ ← Ã 2A′ laser-induced fluorescence spectra of jet-cooled cyclohexoxy radical (c-C6H11O) [Zu, L.; Liu, J.; Tarczay, G.; Dupré, P; Miller, T. A. Jet-cooled laser spectroscopy of the cyclohexoxy radical. J. Chem. Phys. 2004, 120, 10579] has been analyzed and simulated using a spectroscopic model that includes the coupling between the nearly degenerate X̃ and Ã states separated by ΔE. The rotational and fine structure of these two states is reproduced by a 2-fold model using one set of molecular constants including rotational constants, spin-rotation constants (ε’s), the Coriolis constant (Aζt), the quenched spin− orbit constant (aζed), and the vibronic energy separation between the two states (ΔE0). The energy level structure of both states can also be reproduced using an isolated-state asymmetric top model with rotational constants and ef fective spin-rotation constants (ε’s) and without involving Coriolis and spin−orbit constants. However, the spin−orbit interaction introduces transitions that have no intensity using the isolated-state model but appear in the observed spectra. The line intensities are well simulated using the 2-fold model with an out-of-plane (b-) transition dipole moment for the B̃ ← X̃ transitions and in-plane (a and c) transition dipole moment for the B̃ ← Ã transitions, requiring the symmetry for the X̃ (Ã ) state to be A″ (A′), which is consistent with a previous determination and opposite to that of isopropoxy, the smallest secondary alkoxy radical. The experimentally determined Ã −X̃ separation and the energy level ordering of these two states with different (A′ and A″) symmetries are consistent with quantum chemical calculations. The 2-fold model also enables the independent determination of the two contributions to the Ã −X̃ separation: the relativistic spin−orbit interaction (magnetic effect) and the nonrelativistic vibronic separation between the lowest vibrational energy levels of these two states due to both electrostatic interaction (Coulombic effect) and difference in zero-point energies (kinetic effect).

1. INTRODUCTION

The near degeneracy of these two states introduces a significant pseudo-Jahn−Teller (pJT) interaction,7 which just as the Jahn−Teller (JT) interaction in polyatomic molecules with high symmetry, e.g., methoxy and t-butoxy, distorts the potential energy surfaces (PESs) of both states and changes their vibronic structure. The energy level splitting due to this Coulombic effect is coupled with other interactions that change the relative energies of these two nearly degenerate states, namely, the relativistic spin−orbit (SO) interaction (magnetic effect) and the zero point energy (ZPE) of vibrational motion (kinetic effect). The total separation of the vibrationless levels of the Ã and X̃ states due to the nonrelativistic effects (Coulombic and kinetic) is denoted ΔE0. In addition to ΔE0, the relativistic SO effect contributes to the experimentally

Alkoxy radicals are important reaction intermediates in atmospheric chemistry.1−4 The lowest two electronic (X̃ and Ã ) states of all alkoxy radicals, except those with C3v symmetry and doubly degenerate X̃ 2E states, e.g., methoxy (CH3O) and tbutoxy (t-C4H9O), have small energy separations5 of ≲400 cm−1. The Ã and X̃ states separation (ΔE) is especially small in secondary alkoxies, for example, 61.1(1) cm−1 for the isopropoxy radical (i-C3H7O) as reported in the preceding paper.6 As all alkoxy radicals are alkyl derivatives of the methoxy radical, the near degeneracy of the X̃ and Ã states can be viewed as originating from the degeneracy of the X̃ 2 E state of the methoxy radical, for which the degeneracy of the px and py orbitals of the oxygen atom remains. Any asymmetric alkyl substitution alters the energies of these two orbitals, and if Cs symmetry is maintained it splits the X̃ 2 E state into an A′ state with a half-filled px orbital in-plane and an A″ state with an outof-plane, half-filled py orbital. © XXXX American Chemical Society

Special Issue: David R. Yarkony Festschrift Received: May 2, 2014 Revised: July 22, 2014

A

dx.doi.org/10.1021/jp504326p | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

Article

Figure 1. Moderate-resolution LIF spectra of cyclohexoxy under (a) hard jet expansion and (b) moderate jet expansion conditions and (c) highresolution LIF spectra of selected vibronic bands taken under hard jet expansion conditions. Spectra are reprinted with permission from Figures 2 and 7 of ref 27. Copyright 2004 AIP.

pure parallel type transition, implying an A′ symmetry for the X̃ state as the B̃ state is well determined to have A′ symmetry. The spectrum can be well simulated using a standard Hamiltonian for an asymmetric top consisting of rotational kinetic energy and spin-rotation interaction. However, the experimentally derived spin-rotation constants, especially the nonzero value for the spin-rotation tensor component εaa, with the a axis perpendicular to the Cs plane, cannot be explained by either ab initio calculations or semiempirical calculations based on second-order perturbation theory and the spin-rotation constants of a reference molecule.26 This anomaly was attributed6 to the interaction between the nearly degenerate Ã 2A″ and X̃ 2A′ states. A 2-fold model that includes the Coriolis interaction, the SO interaction, and the vibronic separation was built to simulate the spectrum.26 It reproduces the rotational and fine structure of the B̃ ← X̃ origin band as well as the observed rotational contour of the B̃ ← Ã origin band.6 Such a spectroscopic model can be applied to any molecule with Cs symmetry and nearly degenerate electronic states. In the present paper, the 2-fold model is implemented to simulate the previously reported LIF spectra of the cyclohexoxy radical (c-C6H11O).27 The cyclohexoxy radical was studied by LIF and dispersed fluorescence (DF) spectroscopy under jetcooled conditions.27 The DF spectra reveal an Ã −X̃ separation of ∼60 cm−1, very close to that of isopropoxy and suitable for strong pJT interaction. In the LIF experiments, vibronic bands of both the B̃ −X̃ and B̃ −Ã transitions were recorded with rotational and fine structure resolved and two types of rotational structures were observed. An initial attempt to simulate the rotational and fine structure of the LIF spectra was unsuccessful for several reasons. First, the effect on the rotational structure of the vibronic interaction between the X̃ and Ã states was not well understood. Neither the effective

observed separation, ΔE, of the zero-point levels of the two states. To separate relativistic and nonrelativistic effects the rotational and fine structure of the two states must be resolved and analyzed. So far the majority of experimental studies of JT and pJT effects have been performed on the vibronic structure of observed spectra. Fewer studies have been performed on the rotational structure and mostly limited to the E ⊗ e problem in C3v molecules.8 However, Hougen did investigate theoretically the rotational energy levels of nearly degenerate A′ and A″ pairs of vibronic states of C3v molecules.9 He showed that in the limiting case of large A′ and A″ separation compared to the Coriolis interaction, the two vibronic states are characterized by their respective ef fective rotational A constants, the vibronic state lying higher (lower) in energy always having the larger (smaller) values of Aeff. In the limiting case of small separation, the two vibronic states can be regarded as accidentally degenerate and are characterized by the same effective Coriolis constant. A more comprehensive theoretical investigation of the rotational structure in vibronically coupled systems was carried out by Mayer and Cederbaum10 and applied to the analysis of the rotational structure of Li3.11 Ernst and co-workers have theoretically analyzed the rotational structure and Coriolis interaction of alkali trimers.12−14 Rotational structure has also been analyzed for CH4+ and other molecular ions by Merkt and co-workers.15−17 At present, experimentally resolved rotational and finestructure electronic spectra have been reported for methoxy (CH3O)18−22 and its isotopic and alkyl derivatives: CD3O,23 CH2DO and CHD2O,24 ethoxy(C2H5O),25 and isopropoxy (iC3H7O).26 The B̃ ← X̃ rovibronic spectra of isopropoxy obtained by laser-induced fluorescence (LIF) spectroscopy26 is closely related to the present work. The origin band shows a B



Article

expansion spectrum. The vibronic bands that have their “satellite” bands observed in the moderate-expansion spectrum are assigned to transitions from the lowest vibrational level of the X̃ state to the lowest vibrational level of the B̃ state (band A) and its vibrationally excited levels of the symmetric (a′) modes (bands C, E, F, and G). Their satellite bands (bands A′, C′, E′, F′, and G′) are therefore transitions from the lowest vibrational level of the Ã state to the same vibrational levels of the B̃ state. The vibronic bands without “satellite” bands (bands B and D) are assigned to the transitions from the lowest vibrational level of the X̃ state to vibrational levels of the B̃ state with antisymmetric (a″) vibrations. The transition scheme is illustrated in Figure 2 and can be explained physically by invoking the pJT effect. Because of the

Hamiltonian nor transition intensity formulas were available then. Second, it was not possible to calculate the spin-rotation constants ab initio or semiempirically with sufficient accuracy to aid a spectral analysis. Moreover, for molecules in nearly degenerate electronic states, the ef fective spin-rotation constants can be large, comparable to the rotational constants, and give rise to complicated rotational and fine structure patterns. With the recent progress in understanding the asymmetrically deuterated methoxy24 (CH2DO and CHD2O) and isopropoxy spectra,26 we revisit the cyclohexoxy radical in the present work and report here the analysis of its rotational and fine structure.27 Modern electronic structure calculations can provide meaningful estimates of the ΔE0 splitting of its X̃ and Ã states due to chemical substitution. Effective spin-rotation constants of cyclohexoxy can be now predicted using isopropoxy as a more suitable reference molecule. Alternatively, the spin-rotation as well as SO parameters can roughly be estimated by electronic structure calculations. Using the estimated values, the rotationally resolved B̃ −X̃ and B̃ −Ã LIF spectra can be assigned. Further refinement of the parameters leads to spectral fits and simulations using both a conventional isolated-state Hamiltonian for an asymmetric top with spinrotation interaction and the 2-fold effective Hamiltonian that was first proposed for isopropoxy.26 In the following we will first briefly summarize the previous experimental results of the LIF spectra of the cyclohexoxy radical (section 2). Prediction of the molecular constants, including the rotational constants, the spin-rotation constants, and the Ã − X̃ separation is presented in section 3. The two spectroscopic models used to simulate the spectra are described in section 4. The predicted molecular constants in section 3, either ab initio or semiempirical, were used as initial estimations in the spectral simulation and fits (section 5) with both models. The properties of the spectroscopic models and the physical significance of the molecular constants determined in fitting the spectra are discussed in section 6. Conclusions (section 7) complete this article.

Figure 2. Schematic diagram of potential energy curves of the electronic states of cyclohexoxy along the pJT-active vibrational coordinate with the low-lying vibronic states and observed LIF transitions between them indicated. The dashed curves represent the undistorted potentials without the pJT effect.

pJT effect, the ground X̃ 2A″ state is distorted and its symmetry is lowered from Cs to C1 at the minimum of the potential energy curve (see section 3.2). Transitions from its lowest vibrational level to both the symmetric (bands A, C, E, F, and G) and antiasymmetric (bands B and D) levels are therefore allowed both electronically and vibrationally (Franck−Condon allowed). For the Ã 2A′ state, although the curvature of the potential energy curve is increased due to the pJT effect, the molecule retains Cs symmetry at its minimum. Although the pJT effect mixes the A′ and A″ electronic wave functions (see section 6.3), the vibrational wave function of the vibrationless level of the Ã state is still of a′ symmetry. Transitions from the lowest vibrational level of the Ã states to the totally symmetric (a′) modes of the B̃ state are allowed (bands A′, C′, E′, F′, and G′). The B̃ ← Ã transitions to the a″ vibrational levels of the B̃ state, however, remain vibrationally forbidden, which explains the experimental observation that bands B and D do not have satellite bands. Rotational and fine structure of the vibronic bands have been resolved in LIF spectra taken under hard jet-expansion conditions with a high-resolution (Δν̃ ∼ 200 MHz) excitation laser (Figure 1c). Two types of rotational structure were found: The bands in the hard-expansion spectrum with and without hot satellite bands are of different rotational structure and are called type I and type II bands, respectively. For instance, bands A, C, E, F, and G in Figure 1a have a type I rotational structure, while bands B and D have a type II structure. Despite the lower signal-to-noise ratio compared to the other bands, we also obtained a rotationally resolved spectrum of one of the

2. LASER-INDUCED FLUORESCENCE SPECTRA OF CYCLOHEXOXY AND VIBRONIC ASSIGNMENT In an LIF experiment,27 the cyclohexoxy radicals were generated by photolysis of cyclohexyl nitrite28 in “moderate” and “hard” free-jet expansions using a previously described apparatus and procedures.29−31 Experimentally obtained spectra under these conditions with a moderate-resolution (Δν̃ ∼ 0.1 cm−1) excitation laser are illustrated in Figure 1a,b. The rotational temperature of the hard expansion is estimated to be ≈1 K, and that of the moderate expansion ≈100 K by simulating the rotational contours of the B̃ −Ã and B̃ −X̃ transitions (see below). Only transitions from the lowest vibronic state (X̃ ) have significant intensities under hard expansion conditions. The B̃ ← X̃ vibronic bands are labeled A, B··· G in Figure 1a. Under moderate expansion, transitions from both the X̃ and the Ã states to the B̃ state are possible. New vibronic bands from the Ã state were observed and are labeled with primed letters: A′, C′, E′, F′, and G′ in Figure 1b, corresponding to the labeling of the corresponding, equally blue-shifted bands under the hard-expansion conditions, which remain in the moderate-expansion spectrum. The separations between the vibronic bands from the Ã and the X̃ states, for example, bands A and A′, are ∼62 cm−1, corresponding to the Ã −X̃ energy separation. Not all B̃ ← X̃ vibronic bands are accompanied by B̃ ← Ã “satellite” bands in the moderateC



Article

Figure 3. Ab initio calculated PESs (top) and PES cuts along the pJT coordinates (bottom) of cyclohexoxy (left) and isopropoxy (right). In panels a and c, the dashed lines delineate cuts through the restrained PES scans. The minima (min1, min2, min3), the transition barriers between them (TS1, TS2, TS3), and the conical intersection (CI) are marked on the two-dimensional projection below the three-dimensional PES. In panels c and d, solid lines are for fully relaxed scans and dashed lines for partially restrained scans. Blue thick lines are for A′ states and red thin lines for A″ states. Horizontal lines are the calculated vibrational levels. Note, however, that the position of the vibrationless level depends not only on the ZPE of the pJT mode but all the vibrational modes. Molecular figure inserts illustrate the motion of the pJT-active modes. Panels c and d are reprinted with permission from Figures 1b,d of ref 26. Copyright 2013 AIP.

study on cyclohexoxy strongly relies on the previous work on isopropoxy.26 It is therefore interesting and instructive to compare them in terms of their calculated PESs and geometries before presenting the detailed computational results for cyclohexoxy. Figure 3a,c illustrates the CASSCF (Complete Active Space Self-Consistent Field) calculated PESs of the X̃ and Ã states of cyclohexoxy and isopropoxy, respectively. The PESs of these two molecules are very similar to each other: The upper and lower PES sheets are close to each other in energy. Along the coordinates of the pJT-active vibrational modes, they both have a singular point or “conical intersection” where the two PESs meet (CI in Figure 3a and 3c). For both molecules, the energy difference between the three PES minima (min1, min2, min3), the height of the CI, and the energy barriers (TS1, TS2, TS3) within the “moat” around the CI are all very similar. A strong correspondence between their vibronic and rotational structures is therefore expected. There are, however, also significant differences between these two molecules: The global minima of the PESs of the cyclohexoxy radical (the equivalent min2 and min3 in Figure 3a) are two C1 geometries on each side of the saddle point with A″ symmetry (TS1), while the global minimum of isopropoxy (min1 in Figure 3c) is a Cs geometry with A′ symmetry. These two molecules

electronically hot bands, band F′, under hard-expansion conditions. It has a type II structure as do bands B and D. The experimentally observed, rotationally resolved spectra of bands A, B, F, and F′ are compared in Figure 1c.

3. PREDICTION OF MOLECULAR CONSTANTS In order to gain some perspective on the nature of the pJT effect in cyclohexoxy, it is extremely useful to perform highlevel electronic structure calculations for it. We have carried out various quantum chemical calculations on cyclohexoxy, including (i) geometry optimization of the ground (X̃ 2A″) and the first excited (Ã 2A′) electronic states, calculation of their rotational and spin-rotation constants, as well as the Ã −X̃ nonrelativistic energy separation (ΔE0), and ordering of the A′ and A″ states; (ii) determination of the ZPEs of these two states by numerically solving the Schrödinger equation using the ab initio potential energy curves to determine the ZPE of the pJT-active mode and using calculated harmonic frequencies for the other modes; (iii) geometry optimization and frequency calculation of the second excited state (B̃ 2A′) and the B̃ −X̃ transition energy. All ab initio calculations have been performed by the Gaussian 09 program package.32 3.1. Potential Energy Surface Overview and Comparison to Isopropoxy. Because of their similarities, the current D



Article

the ground-state chair-equatorial conformer was optimized at the B3LYP/6-31+G(d) level of theory. The calculation predicts that the ground state of cyclohexoxy is a Cs molecule and has A″ symmetry, in contrast to that of isopropoxy, which has A′ symmetry. This means a reversal of the energy ordering of the electronic configurations of the half-filled pπ-orbitals of the oxygen atom lying within the Cs plane (px) and perpendicular to the Cs plane (py): in the case of cyclohexoxy, py has higher energy, while for isopropoxy, px has higher energy. Such a difference is attributed to a different degree of overlap of the pπ orbitals with the orbitals of the ring carbons in the case of cyclohexoxy. (Note that the ethoxy and all-trans conformers of primary alkoxy radicals with Cs symmetry have ground electronic states with A″ symmetry as does cyclohexoxy.) The rotational constants of ground-state cyclohexoxy are listed in Table 2. Calculations also show that Cs geometry of ground state cyclohexoxy is a saddle point, indicated by an imaginary vibrational frequency. This imaginary-frequency mode corresponds to −CHO wagging of the oxygen and the hydrogen atoms that are connected to the common carbon atom moving out of the Cs plane in-phase (see the inset of Figure 3d). This is the pJT-active mode that connects the Ã and X̃ states (see below). By moving the oxygen atom slightly out of the Cs plane to either direction and optimizing the geometry, now with C1 symmetry, two stable geometries, mirror images of each other, can be found, with a stabilization energy of ∼90 cm−1, which is close to the calculated pJT stabilization energy of the Ã 2A″ state of isopropoxy (∼98 cm−1).26 The calculated rotation constants for the C1 geometry are close to those of the Cs geometry. 3.3. Ã State and Ã −X̃ Separation. The Ã ← X̃ excitation of the alkoxy radicals with symmetry lower than C3v is known to be a promotion of one electron from the completely filled pπorbital to the half-filled pπ-orbital of the oxygen atom. (These two orbitals are degenerate in methoxy and t-butoxy if the JT effect is not considered.) This can be implemented in the calculations by swapping the HOMO and LUMO of the β electrons of the ground electronic state. Geometries of the Ã state of cyclohexoxy were hence optimized at the B3LYP/631+G(d) level of theory. As expected, the Ã state of cyclohexoxy has A′ symmetry. Moreover, the Cs geometry of Ã state cyclohexoxy is stable. The rotational constants of the Ã 2A′ state of cyclohexoxy are given in Table 2. The Ã −X̃ separation of cyclohexoxy between the minimum of the Ã 2A′ state and maximum of the X̃ 2A″ state JT barrier,

therefore are complementary cases in the analysis of molecules with strong pJT effects. Another difference between isopropoxy and cyclohexoxy is that the former is a near oblate top while the latter is a near prolate top. This induces some difference in notation related to the principal coordinates (a, b, c) and the “internal coordinates” (x, y, z) that are defined according to symmetry. For cyclohexoxy as in our previous work on methoxy and its derivatives22−24 including isopropoxy,26 the z axis is defined as along the CO bond. The y axis is perpendicular to the Cs plane. The x axis is within the Cs plane and perpendicular to the other two axes. Because of the difference in moments of inertia, the principal axes of isopropoxy and cyclohexoxy are related to the internal axes in different ways. In isopropoxy, the CO bond is tilted from the b-principal axis by ∼17.6°, while the CO bond of cyclohexoxy is tilted from the a principal axis by ∼18.7° (see section 3.2). The y axis corresponds to a and b axis in isopropoxy and cyclohexoxy, respectively. Table 1 compares the correspondence between the principal axes and the internal axes of these two molecules as well as that of the methoxy radical. Table 1. Correspondence between Principal Axes (a,b,c) and Internal Symmetry Axes (x,y,z) for Methoxy, Isopropoxy and Cyclohexoxya x y z

methoxy

isopropoxyb

cyclohexoxyc

b/c c/b a

∼c a ∼b

∼c b ∼a

a

For methoxy, the b and c axes are equivalent since CH3O is a symmetric top. bCalculated geometric tilt angle between the z axis and b axis is 17.6° for the ground electronic state. cCalculated geometric tilt angle between the z axis and a axis is 18.7°.

3.2. Geometry Optimization and Rotational Constants of the X̃ State. The conformational landscape of cyclohexoxy was discussed in ref 27. It has two low-energy stable conformers that might be populated under jet-cooled conditions using photolysis of cyclohexyl nitrite: chair-equatorial and chair-axial. Reference 27 attributes all observed vibronic bands to the lowest-energy conformer chair-equatorial based on quantum chemical calculations, vibronic analysis of the moderateresolution LIF and DF spectra, and the rotational structures of the high-resolution LIF spectra. Therefore, the geometry of

Table 2. Calculated Molecular Constants of Cyclohexoxy (in GHz unless Otherwise Indicated)a X̃ state A B C εaa εbb εcc |εac + εca|/2 ΔE (cm−1) θ (deg)

Ã state

b

X̃ /Ã state

b

4.267 2.330b 1.659b ab initio −0.428b 0.001b −0.040b 0.099b

s.e. −2.401e 1.087e 0.575e 0.907e 18.7

c

4.254 2.299b 1.633b ab initio −0.451b −0.027b −0.007b 0.063b

4.311 2.316c 1.653c

B̃ state 4.317d 2.268d 1.633d

s.e. −2.645e 1.087e 0.670e 0.629e 119f 13.1b

b

a s.e. = semi-empirical. bAt the B3LYP/6-31+G(d) level of theory. cAt the CASSCF(7,5)/6-31+G(d) level of theory. dAt the CIS/6-31+G(d) level of theory. ePredicted based on the transferability of the mass-independent spin-rotation tensor using isopropoxy as the reference molecule in the internal axis system. fAt the B3LYP/6-31+G(d) level of theory with ZPE correction.

E



Article

both of which are of Cs symmetry, is ∼125 cm−1. When the molecule is allowed to distort from Cs symmetry in geometry optimization, the pJT stabilization energy (90 cm−1) increases the Ã −X̃ separation to ∼215 cm−1, which is the energy separation between the minima of the two states. Similar to isopropoxy, the Ã −X̃ separation of cyclohexoxy is small, comparable to its vibrational structure, which is described below. 3.4. Zero-Point Energy Corrections. The Ã −X̃ separation calculated above does not include the ZPE, which represents a kinetic contribution to ΔE0. Since the Ã and X̃ states are coupled by the pJT effect and the PESs of both states are highly anharmonic along the pJT-active mode, an accurate calculation of the ZPEs would require detailed vibronic analysis and can be tedious given the relatively large size of the molecule and its 48 vibrational modes. We therefore limit our objective to obtaining a rough estimate of the relative ZPE contributions of the two states and to check if the ZPE corrections reverse the energy ordering of the A′ and A″ states, i.e., change the sign of ΔE0, which is defined as positive (negative) if the ground vibronic level has a″ (a′) symmetry. To calculate the ZPE using the Gaussian package, the harmonic frequencies at the Cs geometry minima of each state are calculated excluding the pJT active mode, i.e., the −CHO wagging, of the X̃ 2A″ state. To account for the pJT-active mode, one-dimensional PES scans along the normal coordinates of this mode for both states were carried out at the B3LYP/6-31+G(d) level of theory. The results are shown in Figure 3. Two kinds of scans were performed: fully relaxed ones and ones that have the oxygen atom restrained to move in the plane that is perpendicular to the original Cs plane. For the Ã 2A′ state, the geometry optimization in the fully relaxed scan always led to the X̃ 2A″ state when the oxygen atom is ∼±2° out of the original Cs plane. The fully relaxed scan outside of this range were therefore discarded. Nevertheless, it has been shown in Figure 3b that the restrained scans do not change the calculated energies significantly from those of the fully relaxed ones. Scans of the PES for isopropoxy are also shown in Figure 3d for comparison. The ab initio calculated potential energy curve along the pJT-active mode in the Ã 2A′ state is taken as the reference, and the corresponding vibrational frequency is used to convert the scanning coordinate to the dimensionless normal coordinate. The vibrational energy levels of the X̃ 2A″ state are calculated by solving the Schrödinger equation for nuclear motion numerically in the same coordinates, following the approach outlined by Chan et al.33 The numerically calculated ZPE contribution of the pJT mode was then included into the calculation of the overall ZPE calculations and the ZPE-corrected Ã −X̃ separation. Note that modes other than the pJT-active one have similar values for the two states, and computational errors are expected to largely conceal out. The resulting value of ΔE0 = 119 cm−1 again suggests that the magnitude of ΔE0 (and hence ΔE as well) does not change greatly when the ZPE correction is included and the energy ordering is that the A″ state is below the A′ state. 3.5. Geometry Optimization at the Conical Intersection. In the present work, global fittings were performed in which the rotational constants of the Ã and X̃ states are assumed to be the same (see section 5). As a comparison, the minimum energy point of the conical intersection seam between these two states was determined in a CASSCF calculation using the basis set of 6-31+G(d). The active space contains 7 electrons in 5 orbitals: the four energetically highest

occupied orbitals (the O 2sσ orbital, the C−O σ bond, the O 2px and 2py orbitals) and the lowest virtual orbital (the C−O σ* antibonding orbital). Following refs 34 and 35, the two states are averaged with equal weights in the calculation of the conical intersection. The calculated rotational constants are given in Table 2. Details of the CASSCF calculations and related results will be presented in future publications. 3.6. Spin-Rotation Constants. It has been shown in the case of isopropoxy26 and 2-butoxy36 that the experimentally derived spin-rotation constants of secondary alkoxy radicals can have values that are comparable to their rotational constants. It is, however, important to point out that the experimentally derived values for the spin-rotation constants are effective values and there are significant contributions to these constants from third or even higher order perturbations with the upper vibrational levels of both Ã and X̃ states.26 It has been shown that ab initio calculated spin-rotation constants of isopropoxy are significantly smaller than the experimentally derived ones.26 As cyclohexoxy has an electronic structure similar to isopropoxy, despite the reversal of the A′ and A″ ground states, it is instructive to calculate its spin-rotation constants both from electronic structure theory and semiempirically using isopropoxy as the reference molecule. In addition, in simulating the experimentally observed spectra, it is useful to determine the magnitudes of the second and higher-order perturbation contributions to the spin-rotation constants and the transferability of the spin-rotation constants between homologues (section 6.2). The spin-rotation constants of both the X̃ 2A″ and Ã 2A′ states were calculated at the B3LYP/6-31+G(d) level of theory, and the results are given in Table 2. It is also possible to predict spin-rotation constants from a chosen reference molecule with known values based on the transferability of the spin-rotation tensor.37 For instance, in the previous work of 1-propoxy38 and other larger primary and secondary alkoxy radicals,36,39,40 the spin-rotation constants were predicted using ethoxy as the reference molecule.41 The method is based on the fact that contributions of the electronic and vibronic perturbations to the spin-rotation constants are proportional to the rotational constants and the assumption that the SO interaction is dominated by components along the CO bond and is little affected by alkyl substitution. A massindependent spin-rotation tensor can therefore be defined as ε′ = Iε where I is the moment of inertia tensor. The massindependent spin-rotation tensor ε′ is assumed to be conserved upon chemical and isotopic substitution in the internal coordinate system defined in section 3.1. The spin-rotation constants of the substituted molecule can therefore be calculated as εS = IS−1US−1UR IR εR UR −1US

(1)

where εR (εS) are the spin-rotation constants of the reference and (substituted) molecules, similarly IR and IS are moment of inertia tensors, UR and US are the matrices of the unitary transformations that convert the principal axis systems, in which the IR and IS are diagonal, respectively, into the common internal axis system in which the mass-independent spinrotation tensor is conserved upon substitution. Because of the similarity between the electronic configuration and PESs of isopropoxy and cyclohexoxy, the spinrotation constants of cyclohexoxy are determined using isopropoxy as the reference molecule. The experimentally determined ground-state effective spin-rotation constants of isopropoxy26 are used, and those of the Ã -state constants are F



Article

simulated in the principal-axis (a,b,c) system using an asymmetric-top Hamiltonian

assumed to have the same values. The unitary transformation matrices are determined using the ab initio calculated geometries of isopropoxy and cyclohexoxy. The results are given in Table 2. The semiempirical results are significantly larger than the ab initio ones. Moreover, the ab initio calculated spin-rotation constants are all negative, whereas the semiempirically calculation yields both positive and negative values. 3.7. B̃ State. The B̃ state geometry was optimized at the CIS/6-31+G(d) level of theory. The rotational constants at the equilibrium are given in Table 2. The calculation also gives the excitation energy between the vibrationless levels of the B̃ and X̃ states. Vibrational frequencies of the B̃ state were calculated as well.

/ROT = ANa 2 + BNb 2 + CNc 2

in which A, B and C are rotational constants and N is the “spinless” rotational angular momentum, N = J − S, where J is the total angular momentum of the molecule and S is the electron spin (S = 1/2 for the present work). Since the rotational temperature in the (hard) jet expansion is known to be low (∼1 K) (see section 5), and only the transitions from the lowest J and K levels were observed (K is the projection of N in the molecule-fixed coordinates), centrifugal distortion terms were not included in the effective Hamiltonian for the present analysis. The fine structure due to the coupling of the electron spin with the rotation of the molecule is modeled by the spinrotation term /SR :43−46

4. THEORY Theory of the rotational structure of pJT molecules is usually built upon that of JT molecules. An important case is the reduction by isotopic or chemical substitution of the E ⊗ e JT effect in C3v molecules to the pJT effect in Cs molecules. The appropriate JT molecule in this case is methoxy, CH3O or CD3O, for which the rotational structure has been well studied.22,23 The effect of symmetry lowering by isotopic substitution is well illustrated by the rotationally resolved spectra of CH2DO and CHD2O.24,42 These spectra show how the effect of unequal masses in the nuclear kinetic energy part of the Hamiltonian can remove the degeneracy of the e vibronic levels of methoxy by ZPE effects. In the case of chemical substitution like methyl groups for H’s in isopropoxy, asymmetry also occurs for the PESs. The energy splitting in the PESs combines with the asymmetry in the kinetic energy to give rise to the vibronic splitting ΔE0, which is increased further by SO interaction to produce ΔE, the experimentally observable separation between the lowest vibrational levels of the X̃ and Ã states. To treat the case of isopropoxy, an effective Hamiltonian was proposed to describe the energy levels in two close lying states of Cs molecules coupled by the pJT effect and it was applied to simulate the LIF spectra of isopropoxy.6,26 It was constructed by adding SO and Coriolis coupling between the pJT eigenfunctions, to complement the conventional rotational Hamiltonian for an asymmetric top. In the present work on cyclohexoxy, these terms are derived slightly differently from the previous work on isopropoxy (ref 26) to reflect the relation and similarity between the asymmetrically deuterated methoxy radical and the symmetryreduced cyclohexoxy as well as isopropoxy radicals. We will first briefly summarize the effective Hamiltonian for asymmetric tops, i.e., the isolated-state Hamiltonian, and then introduce the extra terms due to the coupling between the two nearby states. The Hamiltonian matrix elements are derived in both Hund’s case (a) and (b) spin-rotational basis sets. 4.1. Isolated-State Hamiltonian. In the isolated-state model, the two states (A′ and A″) separated by ΔE are treated independently. Previously, the spin and rotational structure of the high-resolution LIF spectra of either the A′ or A″ state of the alkoxy radicals (other than methoxy) were simulated using an effective Hamiltonian that consists of two parts:36,38 / = /ROT + /SR

(3)

/SR =

1 2

∑ εαβ(NαSβ + NβSα) α ,β

(4)

where εαβ (α, β = a,b,c) are the elements of the spin-rotation tensor in the principal coordinate system. It is worth noting that the spin-rotation constants derived from simulating experimentally observed spectra of molecules are reduced molecular constants and for a Cs molecule like cyclohexoxy, only four reduced spin-rotation constants are determinable: ε̃xx, ε̃yy, ε̃zz, and |(ε̃zx + ε̃xz)/2|, with the y-axis being the principal axis that is perpendicular to the symmetry plane.47 In the case of cyclohexoxy, the out-of-plane axis is the principal b-axis, and the four determinable reduced spin-rotation constants are therefore ε̃aa, ε̃bb, ε̃cc, and |(ε̃ac + ε̃ca)/2|. In the present work, the small difference in ε and ε̃ is neglected, and the symbol ɛ is used for simplicity. 4.2. 2-Fold Hamiltonian. All larger alkoxy radicals can be regarded as alkyl group substitutions of the methoxy radical. These substitutions split the doubly degenerate X̃ 2 E ground state of methoxy into two states with A′ and A″ symmetries. Because of the lowering in symmetry, the orbital and vibrational angular momenta are quenched. The quenching, however, is not complete, especially in the case of nearly degenerate states since they are coupled by SO and Coriolis interactions. Such coupling, as well as the lifting of the vibronic degeneracy, is specifically included into the 2-fold spectroscopic model. Experimentally, the two states (A′ and A″) of larger alkoxy radicals are separated by an energy gap of ΔE that combines several contributions. However, now the different contributions are considered explicitly in the calculation of the spin-rotational eigenpairs. The SO interaction also exists in E ⊗ e JT molecules such as CH3O and splits the ground electronic state of methoxy into E1/2 and E3/2 components separated by an effective spin− orbit constant aζed. The SO interaction is completely diagonal in a symmetrized |Λ = ±1⟩ basis set (see section 4.3). The remaining Coulombic and kinetic (ZPE) effects are introduced by chemical and/or mass substitution and their physical consequences: different potential energy curves along the x and y axes and correspondingly a unique force field for each of the states. These two effects produce the vibronic quenching of methoxy’s degenerate 2E state and are diagonal in the vibronic basis set of |Γ⟩ with Γ = A′ or A″. Both Coulombic and kinetic effects shift all rotational energy levels in each state by the same amount ΔE0 and hence are not separable.

(2)

where /ROT represents the rotational part of the Hamiltonian and /SR the spin-rotation part. The rotational structure can be G



Article

Sz = Sa cos θ + Sc sin θ

Following our previous work on the asymmetrically deuterated methoxy radicals, the center of the energy gap between the A′ and A″ states is chosen as the zero point of vibronic energy. The difference between the lowest vibronic levels of a′ and a″ symmetry is defined as ΔE0 = Ea ′ − Ea ″

where θ is the tilt angle between the in-plane z-axis (CO bond) and a axis, which is calculated to be 15.9° for the geometry of cyclohexoxy at its equilibrium. The SO Hamiltonian in the principal-axis system is therefore of the form:

(5)

HSO = aζed(Sa cos θ + Sc sin θ )3 z

With this definition, the vibronic quenching Hamiltonian can be expressed as /Q =

ΔE0 (|A′⟩⟨A′| − |A″⟩⟨A″|) 2 ΔE0 (3+2 + 3−2) 2

(6)

HCor = −2Bzz Nz(Lz + Gz)

1 (|Λ = +1⟩ + |Λ = −1⟩) 2

|A″⟩ =

1 (|Λ = +1⟩ − |Λ = −1⟩) i 2

(7)

HCor = −2ζt(A cos θNa + C sin θNc)3 z

(8)

| J , P , S , Σ , W⟩ =

1 [|Λ = +1, J , W , S , Σ⟩ 2

+ W( −1) J − P + S −∑ |Λ = −1, J , −P , S , −Σ⟩]

(15)

where W is the symmetry of the total wave function with respect to the reflection plane, or its “parity”. Both the SO and Coriolis terms of the effective Hamiltonian are diagonal in this basis set while the vibronic quenching term (HQ) is purely off-diagonal. Their elements in the J , P , S , Σ , W basis set were derived before42 and are summarized in the Supporting Information (section S.1.1). In the case of cyclohexoxy, or pJT molecules with Cs symmetry in general, it is more convenient to use Hund’s case (b) basis set since the rotational and the spin-rotation Hamiltonian elements have been derived previously in this basis set36,38 and can be directly used. We therefore adopt the Hund’s case (b) basis set |Γ, J, N, K, S⟩. The Hamiltonian elements in this basis set can therefore be determined and are given in the Supporting Information (section S.1.2). 4.4. Transition Types. Both the B̃ 2A′ ← Ã 2A′ and B̃ 2A′ ← 2 X̃ A″ spectra with rotational resolution were simulated simultaneously in the present work of cyclohexoxy. It is therefore important to examine the relation between the intensities of these two transitions. The weight of the a, b, and c transition dipole moment (TDM) components can be determined by symmetry. For A′ ← A″ electronic transitions, the TDM must be perpendicular to the Cs plane, i.e., with A″ symmetry so that ⟨B̃ |dZ|Ã /X̃ ⟩ is nonzero, where dZ is the spacefixed Z component of the TDM. In cyclohexoxy, this corresponds to a TDM component perpendicular to the plane along the b axis. For A′ ← A′ electronic transitions, the TDM must be of A′ symmetry and within the Cs plane, which for cyclohexoxy, implies TDM components along the a and c axes.

(9)

where L is the orbital angular momentum and 3z ≡ Λ = + 1 Λ = + 1 − Λ = − 1 Λ = − 1 i s t h e Hougen pseudo-operator,42,48 which can be written equivalently as (see eq 8): 3 z = i(|A″⟩⟨A′| − |A′⟩⟨A″|)

(14)

where ζt is the expectation value of the vibronic angular momentum (L + G)z. Ergo, the Coriolis interaction also couples the A′ and A″ electronic states. The corresponding Hamiltonian terms are off-diagonal and purely imaginary. 4.3. Basis Sets and Hamiltonian Elements. Previously, for both the JT-active methoxy radical (CH3O) and its pJTactive asymmetrically deuterated isotopomers, a symmetrized Hund’s case (a) basis set was used:18,48

following the conventional definition of |Λ = ± 1⟩ = 1/√2(|A′⟩ ± i|A″⟩). The SO interaction also splits the zero-point levels of the two states. The SO interaction can be treated easily in its “internal axis system”, in which the orbital angular momentum as well as the vibrational angular momentum (see below) is quantized. In the case of alkoxy radicals, the SO interaction mainly originates from an unpaired electron in a p-type orbital of the oxygen atom, whose axis is perpendicular to the CO bond. The “natural” z-axis of the SO interaction is therefore the CO bond. The x- and y-axis components of the SO interaction do not make a first-order contribution to the spin-vibronic Hamiltonian based on time-reversal consideration.48,49 The SO interaction therefore has the same form as the methoxy radical: HSO = aLzSz = aζedSz 3 z

(13)

where Bzz is the rotational constant along the z axis and G is the vibrational angular momentum. Using the same technique of the projection above, the Coriolis Hamiltonian can be constructed in the principal axis system as

where 3±2 ≡ Λ = ± 1 Λ = ∓ 1 are Hougen’s artificial ladder operators that connect the two states (3±2 Λ = ∓ 1 = Λ = ± 1 ).48 It is easy to prove that the two definitions of HQ (eqs 6 and 7) are equivalent using the relation: |A′⟩ =

(12)

If it is assumed that the vibrational angular momentum is also quantized along the internal z axes like the orbital angular momentum, the only Coriolis term is48

Note that this term was previously defined as42 /Q =

(11)

(10)

Hence 3 z connects the A′ and A″ states and the Hamiltonian term HSO has only imaginary off-diagonal elements in the basis set of |Γ⟩. In methoxy, the “natural” (x,y,z) axes defined above coincide with the principal inertial axes (a,b,c) with z = a. There is no difference between the internal or principal axis system. Unlike the E ⊗ e JT-active molecules, however, the “natural” axis of the SO interaction in a Cs molecule is not coincident with a principal axis. Since it is convenient to construct the rotational and spin-rotation Hamiltonian in the principal-axis system (section 4.1), we project the vibronic and spin angular momenta onto the principal axes:42 H



Article

In the isolated-state model, the X̃ and Ã states are treated separately. There is therefore no relation between the TDMs between the B̃ 2A′ ← X̃ 2A″ and B̃ 2 A′ ← Ã 2A′ transitions as Ã and X̃ are independent electronic states. On the contrary, the 2fold model relates the X̃ 2A″ and Ã 2A′ states of cyclohexoxy to the |ev±⟩ states of methoxy. This connection can be used to estimate the magnitudes of the TDMs for transitions from X̃ and Ã states. For methoxy, the states |ev±⟩ are related to the states |evx,y⟩ as follows: 1 (|evx⟩ ± i|evy⟩) 2

|ev±⟩ ≡

parameters were held at zero since they are highly correlated with the SO and Coriolis constants. Similarly the Ã state values were held at zero except for εbb which was estimated by simulation of the rotational contour of B̃ 2 A′ ← Ã 2A″ transition6 to be ≈ −3.3 GHz. Note however that since the εbb values in the two states are highly correlated, this is only an effective value. This ambiguity in determining the spin-rotation constants arises from the strong correlation between the molecular constants, which was previously observed in the analysis of the doubly degenerate X̃ 2 E ground state of methoxy as well. In the case of cyclohexoxy, the correlation may be broken partially (but not always completely) by fitting both the B̃ 2A′ ← X̃ 2A″ and B̃ 2A′ ← Ã 2A′ transitions simultaneously, which is possible due to the improved quality of the B̃ 2A′ ← Ã 2A′ spectrum. In this section, we present the spectral simulation and fitting of the molecular constants using the aforementioned two models. The isolated-state model will first be implemented due to its relative simplicity, followed by the 2-fold model, which yields physically more meaningful values for the molecular constants. All simulations are done using the SpecView software.51 The Hamiltonian matrix elements in both the Hund’s case (a) and (b) basis sets were used in simulating the spectra and gave the same results. The simulations presented in the paper were generated using the case (b) basis set. Four of the observed vibronic bands: A, B, F′, and F, are simulated and fit using both spectroscopic models. Although not reported here, other observed bands can be reproduced in the same fashion. A rotational temperature (Trot) of 1 K is used in all simulations to reproduce the intensities of the rotationally resolved spectra. An electronic temperature (Tel) in the hard expansion was determined to be ∼30 K based on the intensity ratio between band F and band F′. 5.1. Isolated-State Model. 5.1.1. Simulation of Individual Vibronic Bands. In simulations using the isolated-state model, the ab initio calculated values of the rotational constants and the values of spin-rotation constants predicted using the semiempirical method listed in Table 2 are used as initial estimates. Simulation of band A (the origin band, type I structure) and band B (type II structure) are performed using the relative values for a, b, and c transition types determined in section 4.4. These results are illustrated in Figure 4 and are compared with experimental spectra. The experimental spectrum of band A does not resemble any of the three simulated spectra. Close inspection, however, reveals that the b-type transition well simulates both wings of the spectrum, but the strong center portion of the spectrum is not reproduced in the b-type simulation. Because of the symmetry of the two electronic states (X̃ 2A″ and B̃ 2A′), a TDM perpendicular to the Cs plane, i.e., a b-type transition, is expected for band A. As will be shown in section 6.3, the abnormally strong transitions in the center portion can be explained by the coupling between the X̃ 2A″ and Ã 2A′ states by SO interaction. In the fitting using the isolatedstate model, the question of interstate coupling is ignored, and the center portion of band A is not included in the simulation and fits. As illustrated in Figure 4, the a type simulation reproduces the experimental spectrum of band B the best. In cyclohexoxy, the a principal axis lies in the Cs plane and approximately along the CO bond (z-axis in the internal axis system). Since the upper level of band B is of a″ symmetry, the observed transition type is consistent with the assigned symmetries.

(16)

where |evx⟩ and |evy⟩ transform, respectively, as a′ and a″. The spherical components of the dipole operator are defined as50 T01(μ) = μz T±1 1(μ) = ∓

1 (μ ± iμy ) 2 x

(17)

Using the symmetry requirement, Γg ⊗ Γμ ⊗ Γe ⊇ A′, where g and e stand for ground and excited electronic states, respectively, and given that the excited (B̃ ) state is of A′ symmetry, the nonvanishing TDM elements are (see Supporting Information section S.1.3 for detailed derivations.) M =

1 ⟨evx|μz |A′⟩ 2

M⊥+ − M⊥− = −⟨evx|μx |A′⟩ M⊥+ + M⊥− = −⟨evy|μy |A′⟩

̃2

(18)

̃2

For the B A′ ← X A′′ transition of cyclohexoxy, the nonvanishing TDM is μy, i.e., perpendicular to the Cs plane, while for the B̃ 2A′ ← Ã 2A′ transition, the TDM is within the Cs plane, i.e., with μx and μz elements. For methoxy radical, due to symmetry, M⊥− = 0, i.e., ⟨evx|μx|A′⟩ = ⟨evy|μy|A′⟩. For cyclohexoxy, because the tilt angle of the CO bond (z-axis) from the a principal axis is small (z ∼ a and x ∼ c) and the yaxis is coincident with the b-axis, the relation above should hold approximately. It is therefore expected that ⟨Ã 2A′|μc|B̃ 2A′⟩ ≈ ⟨X̃ 2A′′|μb|B̃ 2A′⟩.

5. SIMULATION OF HIGH-RESOLUTION LIF SPECTRA In the earlier work on isopropoxy,26 the rotational and finestructure resolved origin band of the B̃ 2A′ ← X̃ 2A′ transition was simulated and fit using two spectroscopic models: a traditional isolated-state model for asymmetric tops with spinrotation interactions and a newly derived 2-fold model that includes SO and Coriolis interactions. These two models yield nearly identical values for the rotational constants of the X̃ state and identical values for the B̃ state. The values for the spinrotation constants derived in the first method have to be regarded as effective but are useful in initial simulation of the spectrum and assignment of the transitions. In our initial analysis26 of the isopropoxy spectrum using the 2-fold model, both the SO and the Coriolis constants are fit whereas all spinrotation constants were set to zero. In our subsequent work, the upper limit of the values of the ground state spin-rotation constants were estimated based on perturbation theory as εzz ∼ εbb ∼ −1.6 GHz, εxx ∼ εcc ∼ −0.2 GHz, εyy = εaa ∼ −1.6 GHz, and εzx ∼ εbc ∼ −0.5 GHz. As these values are small, the I



Article

Given the simulation using the predicted molecular constants, most of the strongest rotational lines in the experimental spectra can be assigned (except for the center portion of band A). Both the rotational and spin-rotation constants can be optimized to achieve a final fit to the spectra. It was found that the off-diagonal spin-rotation constant |εca + εac|/2 cannot be determined with any significant precision as it makes a small contribution to the observed frequencies. It is therefore fixed to zero in the fits. By adjusting the values of the other molecular constants the simulation quality improves and more lines can be assigned and included. The spectra were hence fit in an iterative way until nearly all observed lines (except for the center portion of band A) were assigned and simulated. The simulated spectra of bands A and B are shown in Figure 5 and compared with the respective experimental spectra. The fit values and the statistical one standard deviation, number of transitions assigned, and the one-standard deviation, σ, of the fit are given in Table 3. The standard deviatons are well within the expected experimental uncertainties of the transition frequencies. One other thing to note is that the quoted statistical error values should be reasonable estimates for all except the T00 values and values derived from them like ΔE, Gv, etc. The absolute frequencies are limited by calibration to the I2 spectrum with quoted52 uncertainty is ±0.002 cm−1, much greater than the statistical error bars which reflect the large number of measurements contributing to each T00 value. Such considerations obviously apply universally to Tables 3−5. Bands F and F′ were simulated and fit in a similar fashion. The simulated spectra are shown in Figure 5 and compared with experimental spectra. Note that the simulations are generally good except that the center portion of band F is not reproduced. The fit values, error bars of the molecular constants, number of transitions assigned, and σ for the fit

Figure 4. Simulation of bands A and B of cyclohexoxy using calculated rotational and spin-rotation constants with different transition types (a−c) compared with the experimental spectra (top and bottom). Trot = 1 K.

Figure 5. Simulation of bands A, B, F′, and F using the isolated-state model. Trot = 1 K. J



Article

Table 3. Experimentally Determined Molecular Constants of Cyclohexoxy (in GHz unless Otherwise Indicated) in Fitting Individual Vibronic Bands Using the Isolated-State Modela band A A″ B″ C″ εaa″ εbb″ εcc″ |εac″ + εca″|/2 A′ B′ C′ ΔE (cm−1) T00 (cm−1) Gv (cm−1) no. of lines σ (MHz) μa/μb/μc a

band B

4.334 2.365 1.654 −2.219 0.768 0.471 0 4.275 2.280 1.612

(5) (4) (2) (17) (20) (11) fixed (3) (4) (2)

26754.0620

(5)

81 68 0:1:0

band F

band F′

4.381 2.368 1.667 −2.169 0.762 0.450 0 4.407 2.273 1.632

(40) (3) (3) (9) (10) (7) fixed (40) (2) (2)

4.329 2.385 1.668 −2.137 0.760 0.435 0 4.356 2.287 1.648

(5) (8) (3) (23) (34) (17) fixed (4) (7) (3)

301.5135 75 41 1:0:0

(3)

687.7988 67 92 0:1:0

(7)

4.302 2.391 1.693 −2.216 0.772 0.452 0 4.323 2.292 1.658 61.3312

(96) (6) (5) (12) (17) (12) fixed (95) (4) (4) (4)

52 44 1:0:0

Numbers in parentheses are uncertainties in the last digit.

Table 4. Molecular Constants of Cyclohexoxy Determined in Combined Fitting of Bands A, B, F′, and F Using the 2-Fold Model (in GHz unless Otherwise Indicated)a X̃ /Ã state

B̃ state

000 exptl

000 calcd

A B C

4.322 2.361 1.678

(3) (2) (1)

εaa(X̃ ) εaa(Ã ) εbb εcc |εac + εca|/2 ζt ΔE (cm−1) aζed (cm−1) ΔE0 (cm−1) T00 (cm−1)h Gv (cm−1)

0 −4.640 0 0 0 0.218 61.3307 −33.7 51.2

fixed (20) fixed fixed fixed (1) (4) (1) g

4.311b 2.316b 1.653b ab initio −0.428d −0.451d 0.001/−0.027d −0.040/−0.007d 0.099/0.063d

exptl

4710 calcd

4.272 2.275 1.617

(3) (3) (2)

26723.4108

(5)

4.317c 2.268c 1.633c

2210

exptl

exptl

4.340 2.269 1.636

(3) (3) (2)

4.346 2.272 1.656

(2) (2) (1)

301.5131 band B 135 55 1:0:0.15

(6)

687.7962 (5) bands F′/F 58/125 97/84 1:0:0.15/0:0.15:0

s.e. −2.401e −2.645e 1.087/1.087e 0.575/0.670e 0.907/0.629e

119f

band A 111 91 0:0.15:0

no. of lines σ (MHz) μa/μb/μc

Global Fit 3 with εaa″ (X̃ ) = 0. Numbers in parentheses are uncertainties in the last digit. bAt the CASSCF(7,5)/6-31+G(d) level of theory. cAt the CIS/6-31+G(d) level of theory. dAt the B3LYP/6-31+G(d) level of theory. ePredicted based on the transferability of the mass-independent spinrotation tensor using isopropoxy as the reference molecule in the internal axis system. fAt the B3LYP/6-31+G(d) level of theory with ZPE correction. gFixed to ΔE0 = {(ΔE)2 − (aζed)2}1/2. hDefined as the energy between the middle point of the vibrationless levels of the X̃ and Ã states and the vibrationless level of the B̃ state. a

and B̃ 2A′) and two vibrational levels of the B̃ 2A′ state. Bands A, B, and F share the same lower level (vibrationless level of the X̃ 2A″ state), while bands F and F′ share the same upper level (CO stretch level of the B̃ 2A′ state). It is therefore possible to carry out a global fitting that combines all four bands. The simulations are of similar quality to the simulations of individual bands. The simulated spectra are therefore not shown here. The fit values and error bars of the molecular constants, number of transitions assigned, and σ for the fit are given in Table S.2 of the Supporting Information. Note that the

are given in Table 3. Lower-state rotational and spin-rotation constants derived from fitting of all bands are close to each other. Note that the A rotational constants determined in fitting bands B and F′ have larger error bars due to the ΔK = 0 selection rule of type-a transitions. The other two excited-state rotational constants of bands F and F′ are close to each other because they share the same upper vibronic level. 5.1.2. Global Fitting. With transitions of all four vibronic bands (A, B, F, F′), five vibronic levels are involved: the vibrationless level of the three electronic states (X̃ 2A″, Ã 2A′, K



Article

Table 5. Molecular Constants of Cyclohexoxy Determined in Combined Fitting of Bands A, B, F′, and F Using the 2-Fold Model (in GHz unless Otherwise Indicated)a X̃ /Ã state

B̃ state

000 exptl

000 calcd

A B C

4.323 2.361 1.679

(3) (2) (1)

εaa(X̃ ) εaa(Ã ) εbb εcc |εac + εca|/2 ζt ΔE (cm−1) aζed (cm−1) ΔE0 (cm−1) T00 (cm−1)h Gv (cm−1)

−2.307 −2.307 0 0 0 0 61.3305 −33.0 51.7

fixed (8) fixed fixed fixed fixed (4) (1) g

exptl

b

4.311 2.316b 1.653b ab initio −0.428d −0.451d 0.001/−0.027d −0.040/−0.007d 0.099/0.063d

calcd

4.272 2.276 1.618

(3) (3) (2)

26723.4106

(5)

c

4.317 2.268c 1.633c

4710

2210

exptl

exptl

4.341 2.268 1.637

(3) (2) (1)

4.347 2.271 1.659

(2) (2) (1)

301.5133 band B 135 63 1:0:0.15

(6)

687.7962 (5) bands F′/F 58/125 102/87 1:0:0.15/0:0.15:0

s.e. −2.401e −2.645e 1.087/1.087e 0.575/0.670e 0.907/0.629e

119f

band A 111 88 0:0.15:0

no. of lines σ (MHz) μa/μb/μc

Global Fit 4 with εaa″ (X̃ ) = εaa″ (Ã ). Numbers in parentheses are uncertainties in the last digit. bAt the CASSCF(7,5)/6-31+G(d) level of theory. At the CIS/6-31+G(d) level of theory. dAt the B3LYP/6-31+G(d) level of theory. ePredicted based on the transferability of the mass-independent spin-rotation tensor using isopropoxy as the reference molecule in the internal axis system. fAt the B3LYP/6-31+G(d) level of theory with ZPE correction. gFixed to ΔE0 = {(ΔE)2 − (aζed)2)}1/2. hDefined as the energy between the middle point of the vibrationless levels of the X̃ and Ã states and the vibrationless level of the B̃ state. a c

global fit narrows the error bar of the rotational constant A as both parallel and perpendicular transitions are included. It also directly determines the Ã −X̃ separation as 61.3301(9) cm−1. This fit is called Global Fit 1. Both theoretical calculations (section 3) and spectral simulations show that the X̃ and Ã states have very similar rotational constants and spin-rotation constants. The number of fit parameters can be further reduced by assuming that they are the same. This fit is called Global Fit 2 and results of it are summarized in Table S.3 of the Supporting Information. The determined constants in Global Fits 1 and 2 are consistent within the expected experimental error, confirming the aforementioned assumption. 5.2. 2-Fold Model. 5.2.1. Combined Fitting and Simulation of Bands F and F′. In the 2-fold model, it is assumed that two nearly degenerate vibronic states are connected by SO and Coriolis interactions. We first implement this model to simulate the rotational and fine structure of bands F and F′ because their lower levels, the vibrationless levels of the X̃ 2A″ and Ã 2A′ states are separated by only ≈60 cm−1 and hence strongly coupled and they share the same upper level. As an initial estimate, values of the molecular constants were set to those determined in Global Fit 2 (see Table S.3 of the Supporting Information), while SO and Coriolis constants were fixed to zero. This allows direct assignment of the transitions based on the simulation using the isolated-state model. The spin-rotation constants were then decreased gradually while the SO and Coriolis constants were allowed to float in the fitting. This effectively converts the isolated-state model fitting to the 2-fold model fitting. The tilt angle (θ) is fixed to the mean of the calculated values for the Ã and X̃ state (15.9°). The TDM of the B̃ 2A′ ← X̃ 2A′′ and B̃ 2A″ ← Ã 2A′ transitions were

adjusted to reproduce the experimentally observed spectra while keeping μb of the former and μc of the latter at the same value as suggested in section 4.4. It was found that the Coriolis constant is strongly correlated to the spin-rotation constant εaa ∼ εzz while the SO constant is strongly coupled to the other two spin-rotation constants εbb = εyy and εcc ∼ εxx. On the basis of the estimation in ref 26, εbb and εcc were fixed to zero while εaa, the spin-rotation constant that is expected to have the largest value, was allowed to float in the fitting. Perturbation theory requires εaa of both the X̃ 2A″ and the Ã 2A′ states to be negative. The higher-state spinrotation constant εaa(Ã ) is expected to have a larger absolute value than that of the lower state εaa(X̃ ) because it is closer to the lowest vibrationally excited levels, which results in stronger perturbation and hence larger spin-rotation constants. We therefore varied the ratio εaa(X̃ )/εaa(Ã ) from 0 to 1 and carried out combined fitting of bands F and F′ for each ratio. All fittings have very similar standard deviation (

Jet-Cooled Laser-Induced Fluorescence ... - ACS Publications

Recommend Documents