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Microwave Spectra ofgaucheCH2DCH2OH Including Excited States of the –OH Torsion
JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO.
188, 1–8 (1998)
MS977478
Microwave Spectra of gauche CH2DCH2OH Including Excited States of the –OH Torsion Chun Fu Su and C. Richard Quade* Department of Physics, Mississippi State University, Mississippi State, Mississippi 39762; and *Department of Physics, Texas Tech University, Lubbock, Texas 79409 Received January 3, 1997; in revised form October 27, 1997
Previous studies of the microwave rotational spectra of gauche CH2DCH2OH have been extended for the torsional ground state and new studies are reported for the first excited states of the – OH torsion. The extended studies for the ground state hydroxyl gauche, methyl symmetric conformation made it possible to determine the product of inertia coefficients D and E as well as the rigid rotor A , B , and C and the centrifugal distortion coefficients DJ and DJK . Likewise for the hydroxyl gauche, methyl asymmetric conformations I and II in the ground state rotational coefficients and selected centrifugal distortion coefficients have been determined. Rotational coefficients B and C and centrifugal distortion coefficients DJ and DJK have been determined for all of the excited states. A significant result of the spectroscopic search was that c -dipole transitions were not observed within the range of our spectrometer for either the methyl symmetric or methyl asymmetric conformations in the excited states. For the methyl asymmetric conformation, this means that the tunnelling energy for the excited state has overcome the torsional potential energy term that suppressed the pure hydroxyl tunnelling, that localizes the molecule into conformations I and II for the ground state. The very small tunnelling energy within conformations I and II has been predicted. The role of internal rotation – overall rotation Coriolis coupling including denominator corrections from the rotational energy for hydroxyl gauche, methyl symmetric CH2DCH2OH is shown in the Appendix to contribute to the effective rotational coefficients. q 1998 Academic Press
assign additional ground state lines and these also are reported in this work along with the appropriate analyses. These additional assignments made it possible to determine the product of inertia coefficients D and E that mix the hydroxyl gauche / and 0 substates for the methyl symmetric conformation and the high J lines made it possible to determine the centrifugal distortion coefficients DJ and DJK in the ground state. Centrifugal distortion coefficients have also been determined for the ground state hydroxyl gauche, methyl asymmetric conformations I and II and for all substates of the first excited state of the hydroxyl torsion for the gauche conformation. A significant amount of time was spent on the search for the c -dipole transitions of the excited states. It proved to be a negative search. This is consistent with our expectation that the tunnelling energy for the excited state of the hydroxyl gauche torsion should be of the order 1.5 THz ( 1a, Table VIII ) . The cancellation of rotational and torsional energies is not enough for the term values to fall within the range of our spectrometer. This negative result also confirms that for the hydroxyl gauche, methyl asymmetric conformations in the first excited state of the hydroxyl torsion the pure hydroxyl tunnelling is not suppressed by the Vs 1s 2 sin a1 sin a2 potential energy term ( 1a, p. 4306 ) , which indicates that Vs 1s 2 is not significantly larger than 8 cm01 or 240 GHz. Although the mechanism that suppresses the hydroxyl tunnelling in the hydroxyl gauche, methyl asymmetric con-
INTRODUCTION
Many years ago, Kakar, Seibt, and Quade reported the microwave torsional–rotational spectra of normal ethyl alcohol and of a few isotopic species in the hydroxyl gauche conformation and the torsional ground state (1). From these studies it was possible to determine the components of the dipole moment, hydroxyl gauche tunnelling energy, trans– gauche energy difference, and the potential energy coefficients hindering the methyl (1a) and hydroxyl (2) internal rotations. Further, the product of inertia terms, D(Px Py / Py Px ) and E(Px Pz / Pz Px ), that mix hydroxyl gauche / and 0 substates for the normal species were determined (1a). Finally, analysis of the rotational coefficients AI and AII for conformations I and II of the hydroxyl gauche, methyl asymmetric forms of CH2DCH2OH and –OD gave the potential energy term that suppresses the pure –OH or –OD tunnelling as 8 and 4 cm01 , respectively (1a, pp. 4305–07). It was not possible to resolve the tunnelling between the symmetric and antisymmetric substates within conformations I and II (see Fig. 1). In the present work, the microwave studies are extended to the first excited states of the hydroxyl gauche torsion of CH2DCH2OH. These states are the hydroxyl gauche / and 0 states for the methyl symmetric conformation and also the hydroxyl gauche / and 0 substates for the methyl asymmetric conformation. Only a-dipole lines were found in all cases. In the course of the study it was necessary to 1
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TABLE 1 Microwave Torsional–Rotational Spectra of Sym–CH2DCH2OH in the Ground State
formations of CH2DCH2OH has been discussed in detail by Kakar and Quade ( 1a, pp. 4305 – 4307 ) under the approximation that the methyl asymmetric tunnelling is too small to be resolved, in this work we redevelop the formalism including only torsional energies, rotational energies neglected, to include the methyl asymmetric tunnelling. This leads to a prediction of the tunnelling energy between the symmetric and antisymmetric substates of conformations I and II. EXPERIMENTAL
All measurements were made on the Mississippi State University Hewlett–Parkard spectrometer. Frequencies above 40 GHz were obtained using SpaceKom (Honeywell)
*Denotes the lines reported in (1a) and remeasured by Suenram at NIST. / Denotes the lines in (1a) and remeasured at Mississippi State. Units are in MHz. In the least squares fit the data to determine the B’s, C’s, DJ’s, and DJK’s all a-dipole lines were used except 303 R 202 and 321 R 220 for the 0 state and the K01 Å 3 lines for both states and no c-dipole lines were used. The A’s and D were determined from the c-dipole lines but not by a least squares method. The calculated values in the table do include contributions from DJ and DJK as well as the contributions from D and E for the c-dipole transitions.
FIG. 1. Hydroxyl gauche, methyl symmetric and asymmetric isomers of CH2DCH2OH.
doublers. The measurements were made at room temperature. The Stark effect was used to facilitate the assignments. The computer program for the analysis of the data in Tables 1–4 in terms of rotational coefficients A, B, and C
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MICROWAVE SPECTRA OF gauche CH2DCH2OH
and centrifugal distortion coefficients DJ , DJK , DK , dJ , and dK is a nonlinear least squares fit. The plus and minus uncertainty to each coefficient in the Tables is one standard deviation except for D A/ , D A0 , and D in Table 1.
TABLE 3 Microwave Rotational Spectra of Methyl Asym–CH2DCH2OH in the Torsional Ground State for Conformations I and II
SPECTRA AND ANALYSIS
Hydroxyl gauche, Methyl Symmetric Ground State As a precursor to assigning the spectral lines for the first excited state of the hydroxyl gauche torsion, it was necessary to assign additional ground state lines for both the methyl symmetric and the methyl asymmetric conformations. These lines for the methyl symmetric conformation are reported in Table 1 along with the lines previously measured by Kakar and Quade (1a, Table IV). In the course of the work at NIST on the following paper, it was found that some of the frequencies reported by Kakar and Quade were systematically high by 0.15–0.30 MHz especially in P and K bands. All of these lines have been remeasured either at National Institute of Standards and Technology (NIST) or Mississippi State and the revised frequencies are given in Table 1. The Hamiltonian that was used to analyze the data for hydroxyl gauche, methyl symmetric CH2DCH2OH has the form TABLE 2 Microwave Rotational Spectra of Sym–CH2DCH2OH in the First Excited State of the –OH Torsion
*Denotes the lines either assigned and measured or remeasured from (1a) by Suenram at NIST. / Denotes the lines reported in (1a) remeasured at Mississippi State. Units are in MHz. The fit is not sensitive to D IK , D JII , and d JII .
( sÉHTRÉs ) Å AsP 2z / BsP 2y / CsP 2x 0 DJsP 4 0 DJKsP 2 P 2z 0 DKsP 4z 0 2dJsP 2 (P 2x 0 P 2y ) 0 dKs( P 2z (P 2x 0 P 2y ) 0 (P 2x 0 P 2y )P 2z ) / Es ,
[1] a
with s Å / or 0 for the hydroxyl gauche substates. The
Units are in MHz.
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TABLE 4 Microwave Rotational Spectra of Asym–CH2DCH2OH in the First Excited State of the –OH Torsion
Note. In this case, the –OH tunnelling energy is so large that the molecule is no longer localized in the conformations I and II. A / and 0 symmetry notation is used for the torsional states. Units are in MHz.
Hamiltonian also has terms mixing the / and 0 substates of the form ( /ÉHTRÉ0 ) Å D(Px Py / Py Px ) / E(Px Pz / Pz Px ),
[2]
where D and E are odd in the hydroxyl gauche internal rotation angle and pure imaginary. It should be kept in mind that there are Coriolis terms linear in Py and Pz that couple the hydroxyl gauche / and 0 substates but it is shown in the Appendix how these terms contribute to the effective As , Bs , and Cs , including cross-mixing with the product of inertia terms and the third-order rotational denominator corrections. This approach to the analysis, taking As , Bs , Cs , D, E, and D as spectroscopic parameters, is based upon the framework fixed axis method (FFAM) for studying internal rotation– overall rotation interactions in the asymmetric–asymmetric molecules (3). Although there are two internal rotors, the hydroxyl and methyl groups, we use a single rotor approach,
fixing the deuterated methyl group conformation to make the framework symmetric. A Van Vleck transformation is used to isolate the torsional 3 1 3 energy matrix for the hydroxyl trans and gauche substates. Since the hydroxyl trans substate is well below the two hydroxyl gauche substates because of the V1 and V2 terms in the effective potential energy, we neglect any residual trans–gauche interactions (1a, Table VIII). However, the / and 0 hydroxyl gauche substate remain strongly interacting by an HTR term that involves Pz , Py , Px Py / Py Px and Px Pz / Pz Px . As shown in the Appendix, most, but not all, of these terms are incorporated into the effective rigid rotor coefficients. There are some things that we wish to note from the analysis. First, the analysis is sensitive to the centrifugal distortion terms DJ and DJK only. Second, the K01 Å 2 to 3 c-dipole transitions do not fit well in the analysis. We thought this occurred because the FFAM, like the principal axis method (PAM) for symmetric internal rotor (4), is a perturbation approach and the pPz coupling is large enough to cause the approach to break down for K1 Å 3. However, the internal axis method (IAM) (5) was tried in the analysis and it did not improve the situation. Right now we do not know the source of difficulty with these lines. It should be noted that the a-dipole K01 Å 3 lines for J Å 3 to 4 also show a poor fit. The problem energy levels appear to be those with K01 Å 3 for the hydroxyl gauche / state. Third, the J Å 3 a-dipole lines, 202 r 303 and 220 r 321 , for the 0 substate that interact through the rotational asymmetry show deviations from their expected positions. Since there are no problems with the lines from / substate, we do not think the deviations comes from ( / ÉHTRÉ 0 ). Also, since all internal rotation–overall rotation interaction are diagonal in J, we do not expect interaction from the hydroxyl trans, methyl symmetric conformation for the low values of J. Further, since the lines for the hydroxyl gauche, methyl asymmetric conformations do not appear to be perturbed, we do not feel the shift is methyl symmetric–methyl asymmetric interaction in origin. Therefore we are left with the idea that there is some unaccounted for interaction within the ground state hydroxyl gauche 0 manifold. These levels are interacting pairs in the Wang representation. These two lines as well as the c-dipole K01 Å 2 to 3 lines and a-dipole K01 Å 3 lines were not included in the least squares fit. Finally, both D and E are of the order 25% smaller for hydroxyl gauche, methyl symmetric CH2DCH2OH than CH3CH2OH. In either case, the empirical D and E are not represented well and were not expected to be represented well, from calculations based upon molecular structure because of the lumping of all internal rotation effects into the empirical spectroscopic constants. Hydroxyl gauche, Methyl Symmetric –OH Excited State For the hydroxyl gauche, methyl symmetric first excited state of the –OH torsion, complete sets of the a-dipole lines
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were identified and assigned. The intense search for c-dipole lines that would be tunnelling transitions as for the ground state was not fruitful, which is consistent with our expectation that the hydroxyl gauche tunnelling energy for this state is large, predicted to be of the order of 1.5 THz. The large tunnelling completely washes out effects from the D and E product of inertia terms. Also, without c-dipole lines, it is not possible to determine the As of Eq. [1]. The a-dipole lines were analyzed on the basis of
aspects of tunnelling (7). For hydroxyl gauche, methyl asymmetric CH2DCH2OH the lowest symmetry potential energy term has the form Vs 1s 2 sin a1 sin a2 , where a1 is the internal rotation coordinate for the hydroxyl group and a2 is that for the methyl group. The theoretical development in Ref. (1a, pp. 4305–07) shows how the effective rigid rotor Hamiltonian evolves in the form ( sÉHTRÉs ) Å AsP 2z / BsP 2y / CsP 2x 0 DJsP 4
( sÉHTRÉs ) Å BsP 2y / CsP 2x 0 DJsP 4 0 DJKsP 2 P 2z ,
[3]
with a data fit as good as that for the ground state lines. The assigned spectral lines and spectroscopic constants are given in Table 2. It should be noted that the difference in the Bs and Cs between the torsional substates is smaller for the excited state than for the ground state. We interpret this to mean that the hydroxyl gauche torsional wavefunctions are less localized in the excited state than in the ground state. Therefore the internal rotation angle dependence of the moment of inertia dominates over the second Coriolis interaction from pPx and pPy terms. Hydroxyl gauche, Methyl Asymmetric Ground State For the hydroxyl gauche, methyl asymmetric conformations there are four possible states: two with the methyl D and hydroxyl H on the same side of the molecular plane, conformation I, and two with the methyl D and hydroxyl H on opposite sides of the molecular plane, conformation II (see Fig. 1). The spectra clearly indicate, as was determined previously (1a, Table V), that the hydroxyl gauche ground state tunnelling is suppressed and the molecule so to speak localizes into conformations I and II with twofold degeneracy for each conformation. Tunnelling exists between the symmetric and antisymmetric substates of conformations I and II but this splitting is unresolvable with our spectrometer. In this tunnelling the hydroxyl gauche hydrogen and the methyl asymmetric deuterium move in tandem across the molecular plane. This suppression of the pure hydroxyl gauche tunnelling can be caused by two things in the torsional Hamiltonian. The first is a kinetic energy term odd in both a1 and a2 with low symmetry of the form sin a1 sin a2 p1 p2 (6). For hydroxyl gauche, methyl asymmetric CH2DCH2OH, this term is not large enough (1a, p. 4306) to suppress the pure hydroxyl tunnelling even though it couples the two internal rotations in the proper way. It is not unusual for the internal rotation coordinate dependence in the kinetic energy not to be large enough to suppress tunnelling for most molecules. On the other hand, it has been found for molecules with partially deuterated methyl groups that the internal rotation coordinate dependence in the zero point energy of the other vibrations of the molecules contributes to the effective internal rotation potential energy in a manner to suppress certain
0 DJKsP 2 P 2z 0 DKsP 4z 0 2dJsP 2 (P 2x 0 P 2y ) 0 dKs( P 2z (P 2x 0 P 2y ) 0 (P 2x 0 P 2y )P 2z ),
[4] with s Å I and II and centrifugal distortion added. The isolation of the effective Hamiltonian in Ref. (1a) is correct as is the effective energy matrix. However, the transformed torsional wavefunctions are labeled incorrectly. Also, Ref. ( 1a ) does not include the methyl asymmetric tunnelling energy. Both of these things are developed later in the present work. The previously and presently assigned spectral lines and the results of the analysis are given in Table 3. The lines follow rigid rotor behavior, including centrifugal distortion, in the sense that the hydroxyl gauche tunnelling has been suppressed and does not appear in the c-dipole transitions. The assignment of the additional lines made possible a centrifugal distortion analysis. In the course of the work on the following paper, where frequency measurements were made at NIST, it was found that many of the lines, especially those in P and K bands, reported by Kakar and Quade (1a, Table V) were systematically off in frequency by 0.15–0.30 MHz in a high direction. The previously assigned lines were remeasured, some at NIST and the rest at Mississippi State. The revised frequencies are given in Table 3. The centrifugal distortion analyses have been carried out using all five coefficients in Eq. [4]. However, the analyses were not sensitive to all the coefficients. For example, the analysis for conformation I is not sensitive to DK and that for conformation II is not sensitive to DJ and dJ . The uncertainties in all coefficients, rotational and centrifugal distortion, are based upon the number of analyzed lines and which centrifugal distortion coefficients are used. All coefficients have been found to change more than the uncertainties when new lines were assigned and added to the fit. The coefficients are fair for prediction. The J Å 5, 6, and 7 Q-type lines were predicted with preliminary coefficients and then found at NIST without searching. Hydroxyl gauche, Methyl Asymmetric –OH Excited State It was expected that the hydroxyl gauche tunnelling energy of the first excited state, of the order 1.5 THz, would be enough to override the 8 cm01 Å 240 GHz energy that
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localizes the molecule into conformations I and II in the ground state. Such has been found to be the case. This means that the excited hydroxyl gauche states may be labeled by / and 0 and there are two methyl asymmetric states for each of these. The search for the c -dipole lines, which would be combined rotational and torsional transitions, was futile. The assignments of the a-dipole lines are given in Table 4. One might expect four sets of lines, rather than the observed two, to occur for the excited states. We interpret this to mean that the difference in Bs and Cs for the two methyl asymmetric substates is too small to be resolved. At the same time, as for the hydroxyl gauche, methyl symmetric CH2DCH2OH, the excited state Bs and Cs are closer together than for the conformations I and II of the ground state. Again, it is presumed that this arises from the averaging of the moments of inertia over the torsional angle that dominates over the smaller Coriolis interaction pPx and pPy . The Hamiltonian of Eq. [3] was used to analyze the data. DISCUSSION
The spectra and analyses in this work provide some interesting features for the six conformations of hydroxyl gauche of CH2DCH2OH. The relative large disagreement between calculated and observed frequencies is consistent with those for other molecules with asymmetric internal rotors—much larger than for those molecules with symmetric internal rotors. Also, it should be kept in mind that the spectroscopic coefficients determined in the data analysis contain a lot of hidden effects due to internal rotation. An important aspect of the experimental and theoretical portions of this work concerns the suppression of the hydroxyl gauche tunnelling energy so that the molecule localizes so to speak into conformations I and II for the hydroxyl gauche, methyl asymmetric forms. At this point we work through the transformation of the torsional energy matrix as was done in Ref. (1a) (pp. 4305–07), but here neglecting the rotational Hamiltonian and including the methyl asymmetric tunnelling energy. We take as the zeroth-order torsional basis Cij Å wi ( a1 ) cj ( a2 ),
IIS
IIS 1 e / (D / d) 2
IS
[5]
where w is the wavefunction for the hydroxyl gauche torsion and c the wavefunction for the methyl asymmetric torsion; i , j r /, 0 for the symmetry of the respective torsional states. In this basis, the hydroxyl gauche substates are separated by the tunnelling energy D and the methyl asymmetric substates by the tunnelling energy d. These substates are mixed by the interaction energy e å Vs 1s 2 ( //Ésin a1 sin a2É00 ) . The resulting 4 1 4 torsional energy matrix is // 00 /0 0/
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1 CIS Å q [( // ) 0 ( 00 )] 2
1 CIIA Å q [( /0 ) 0 ( 0/ )] 2
1 CIA Å q [( /0 ) / ( 0/ )], 2 [7]
where conformation I has hydroxyl H and methyl D on the same side and conformation II has hydroxyl H and methyl D on opposite sides of the molecular plane (see Fig. 1). In checking these properties, it must be noted that w0 and c0 are exponential sums and to get a sine function to show the oddness of the 0 substate must be multiplied by i. The transformed energy matrix is
IA
IIA
0
0
0
0
1 0e / (D / d) 2
1 0 (D 0 d) 2 1 e / (D / d) 2
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1 CIIS Å q [( // ) / ( 00 )] 2
IIA
AID
00 /0 0/ e 0 0 D/d 0 0 d 0e D
The matrix elements of ( /Ésin ai É0 ) are pure imaginary, which controls the sign of the interaction terms in Eq. [ 6 ] . Since the spectral data show that the hydroxyl gauche, methyl asymmetric CH2DCH2OH so to speak localizes into conformations I and II, the off-diagonal term e must be much larger than D and d to suppress the pure hydroxyl tunnelling energy. Therefore the energy matrix of Eq. [6] is best transformed to the new basis
IS 1 0 (D / d) 2 1 0e / (D / d) 2
IA
// 0
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MICROWAVE SPECTRA OF gauche CH2DCH2OH
Second-order perturbation theory gives the energy of separation of conformations I and II as 2e /
D2 . 4e
[9]
We have determined (1a, pp. 4305–07) that the energy difference of Eq. [9] is of the order 12 cm01 for CH2DCH2OH. Further, the off-diagonal matrix elements in Eq. [8] show that the two substates of conformation I and the two substates of conformation II are split by DÅ
dD . 2e
these terms are incorporated into the effective rotational coefficients of Eq. [1]. The complete matrix element mixing the hydroxyl gauche / and 0 substates for CH2DCH2OH has the form ( /ÉHTRÉ0 ) Å aPz / bPy / ic(Px Py / Py Px ) / id(Px Pz / Pz Px ),
where all of the coefficients in Eq. [A1] are internal rotation dependent. Second-order perturbation theory,
[10]
From the barrier to methyl internal rotation for hydroxyl gauche CH3CH2OH, d is estimated to be less than 0.15 MHz while D is estimated to be the order 97 000 MHz from Table 1 of hydroxyl gauche, methyl symmetric CH2DCH2OH. With e of the order 12 cm01 , D is estimated to be of the order 0.04 MHz. Even though this splitting is beyond the resolution of our spectrometer, we did carry the gas pressure to extremely low levels in case e might be considerably smaller than what was determined in Ref. 1a. These experiments gave a null result. Finally, on the basis of Eq. [7] the only c-dipole transitions allowed in first order are IS r IA and IIS r IIA. Transitions from I r II are forbidden in first order. In the following paper, the tunnelling transitions within conformation I and conformation II are reported using the high resolution microwave spectrometer at NIST. This makes it possible to determine an improved and more accurate value for e and therefore Vs 1s 2 . And finally, we wish to speak to our assignment of the excited state hydroxyl gauche, methyl asymmetric lines to the excitation of the hydroxyl group only. For example, a possibility is that the methyl asymmetric group is excited while the hydroxyl group remains in the ground state. In this case, d* would be 73 times larger than d but still small enough to speak to conformations I and II. Then D *, Eq. [10], would be 73 times larger than D and be clearly resolvable for cdipole transitions since D * would be of the order of a few MHz. However, in this case it would be expected that the increased tunnelling for the excited methyl asymmetric group would also lead to a resolution of the Bs and Cs within conformations I and II. Therefore we would expect to see four lines for the a-dipole transitions rather than the two observed. This fact and the fact that our search for c-dipole lines, while difficult, was fruitless lead us to the conclusion that our assignment, which is hydroxyl gauche excited state, methyl asymmetric ground state, is correct. Further, we are convinced that we have never seen a-dipole lines of the excited state of methyl torsion, even for CH3CH2OH (2).
( /ÉHTRÉ0 )( 0ÉHTRÉ/ )/ D,
( /É(HTRÉ0 )( 0ÉHTRÉ0 ) Å a 2 P 2z / b 2 P 2y / (ab / bc 0 ad)(Py Pz / Pz Py ) / 2ac(P 2x 0 P 2y ) / 2bd(P 2z 0 P 2x ) / c 2 (Px Py / Py Px ) 2 2
1 (Px Pz / Pz Px ) / (Px Pz / Pz Px )(Px Py / Py Px )].
Inspection shows that the first five terms of Eqs. [A2] and [A3] are rigid rotor in nature and are incorporated into the effective As , Bs , and Cs . The third term amounts to a small rotation of axes and such terms also occurred after the original Van Vleck transformation. The last three terms are nonrigid rotor in nature and arise from the product of inertia terms that are odd in a and mix the / and 0 substates; these arise from the D and E terms from Eq. [2]. The last term in Eqs. [A2] and [A3] has a comparable size coefficient to the c 2 and d 2 terms, but while the latter make contributions to the diagonal energy matrix elements, (Px Py / Py Px )(Px Pz / Pz Px ) / rrr has only off-diagonal matrix elements in K01 and therefore makes a much smaller contribution when again treated by second-order perturbation theory. Therefore we have not included this term in our analysis. In the perturbation treatment of the D and E terms in Eq. [2], we do use the rotational energy denominator corrections in the calculation of the perturbation contributions. However, Eqs. [A2] and [A3] give the second-order corrections that include only the hydroxyl gauche / and 0 energy difference. We wish to point out that inclusion of these so-called denominator corrections is equivalent to going to third order in the Van Vleck transformation in the FFAM. Specifically, these third-order corrections are of the form [( sÉHTRÉs * )( s *ÉHRÉs * )( s *ÉHTRÉs ) 0
1 ( sÉHTRÉs * )( s *ÉHTRÉs )( sÉHRÉs ) 2
0
1 ( sÉHRÉs )( sÉHTRÉs * )( s *ÉHTRÉs )]/ D 2 , 2
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[A3]
2
/ d (Px Pz / Pz Px ) / cd[(Px Py / Py Px )
In this Appendix, results are presented that show why we neglect pPy and pPz Coriolis coupling in Eq. [2] and how
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where it can be taken that ( /ÉHRÉ/ ) Å ( 0ÉHRÉ0 ) for purposes of the calculation. What should be noted is that if only Coriolis terms from Py and Pz in Eq. [A1] are included, then these third-order terms reduce to rigid rotor contributions. For example, if the AP 2z term from HR and the bPy term from HTR are considered, it is straightforward to show from the commutation relations that
after application of the commutation relations. The rotation to the principal axis system will make the coefficient of Py Pz / Pz Py and therefore the coefficient of the whole term vanish. Hence Eq. [A6] is contained in the empirical rigid rotor coefficients. More importantly and more simply, none of the terms in Eq. [A6] has contributions diagonal in K01 . ACKNOWLEDGMENTS
Py P 2z Py 0
1 2 2 1 2 2 P y P z 0 P z P y Å P 2x 0 P 2z . 2 2
[A5]
Other terms can be reduced in a similar manner. What this means is that the third-order terms from the Coriolis interaction—the rotational denominator corrections—are rigid rotor in nature and are included in the empirical A, B, and C. The cross-terms between the Coriolis coupling and the products of inertia can also be reduced. For example, the cross-term between Py and (Px Py / Py Px ) from HTR and the P 2y term from HR reduces to i[ 0 (Py Pz / Pz Py ) / /
1 2 P y (Py Pz / Pz Py ) 2
1 (Py Pz / Pz Py )P 2y / Py (Py Pz / Pz Py )Py ] 2
[A6]
The authors wish to thank Dr. Richard Suenram of NIST for making the measurements of the tunnelling frequencies within conformations I and II of hydroxyl gauche, methyl asymmetric CH2DCH2OH that are reported in the following paper. In the course of his work on this project, it was found that some of the lines reported in Ref. (1a) Tables IV and V needed remeasurement and many of the remeasurements were made by him.
REFERENCES 1. (a) R. K. Kakar and C. R. Quade, J. Chem. Phys. 72, 4300–4307 (1980). (b) R. K. Kakar and P. J. Seibt, J. Chem. Phys. 57, 4060– 4061 (1972). 2. C. F. Su and C. R. Quade, J. Mol. Spectrosc. 175, 390–394 (1996). 3. C. R. Quade and C. C. Lin, J. Chem. Phys. 38, 540–550 (1963). 4. C. C. Lin and J. D. Swalen, Rev. Mod. Physics 31, 841–892 (1959). 5. M. Liu and C. R. Quade, J. Mol. Spectrosc. 146, 228–251 (1991). 6. J. V. Knopp and C. R. Quade, J. Chem. Phys. 53, 1–10 (1970). 7. (a) T. L. Chang and C. R. Quade, J. Mol. Spectrosc. 111, 398–402 (1985). (b) G. L. Walker and C. R. Quade, J. Chem. Phys. 52, 6427– 6428 (1970).
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MS977478
Microwave Spectra of gauche CH2DCH2OH Including Excited States of the –OH Torsion Chun Fu Su and C. Richard Quade* Department of Physics, Mississippi State University, Mississippi State, Mississippi 39762; and *Department of Physics, Texas Tech University, Lubbock, Texas 79409 Received January 3, 1997; in revised form October 27, 1997
Previous studies of the microwave rotational spectra of gauche CH2DCH2OH have been extended for the torsional ground state and new studies are reported for the first excited states of the – OH torsion. The extended studies for the ground state hydroxyl gauche, methyl symmetric conformation made it possible to determine the product of inertia coefficients D and E as well as the rigid rotor A , B , and C and the centrifugal distortion coefficients DJ and DJK . Likewise for the hydroxyl gauche, methyl asymmetric conformations I and II in the ground state rotational coefficients and selected centrifugal distortion coefficients have been determined. Rotational coefficients B and C and centrifugal distortion coefficients DJ and DJK have been determined for all of the excited states. A significant result of the spectroscopic search was that c -dipole transitions were not observed within the range of our spectrometer for either the methyl symmetric or methyl asymmetric conformations in the excited states. For the methyl asymmetric conformation, this means that the tunnelling energy for the excited state has overcome the torsional potential energy term that suppressed the pure hydroxyl tunnelling, that localizes the molecule into conformations I and II for the ground state. The very small tunnelling energy within conformations I and II has been predicted. The role of internal rotation – overall rotation Coriolis coupling including denominator corrections from the rotational energy for hydroxyl gauche, methyl symmetric CH2DCH2OH is shown in the Appendix to contribute to the effective rotational coefficients. q 1998 Academic Press
assign additional ground state lines and these also are reported in this work along with the appropriate analyses. These additional assignments made it possible to determine the product of inertia coefficients D and E that mix the hydroxyl gauche / and 0 substates for the methyl symmetric conformation and the high J lines made it possible to determine the centrifugal distortion coefficients DJ and DJK in the ground state. Centrifugal distortion coefficients have also been determined for the ground state hydroxyl gauche, methyl asymmetric conformations I and II and for all substates of the first excited state of the hydroxyl torsion for the gauche conformation. A significant amount of time was spent on the search for the c -dipole transitions of the excited states. It proved to be a negative search. This is consistent with our expectation that the tunnelling energy for the excited state of the hydroxyl gauche torsion should be of the order 1.5 THz ( 1a, Table VIII ) . The cancellation of rotational and torsional energies is not enough for the term values to fall within the range of our spectrometer. This negative result also confirms that for the hydroxyl gauche, methyl asymmetric conformations in the first excited state of the hydroxyl torsion the pure hydroxyl tunnelling is not suppressed by the Vs 1s 2 sin a1 sin a2 potential energy term ( 1a, p. 4306 ) , which indicates that Vs 1s 2 is not significantly larger than 8 cm01 or 240 GHz. Although the mechanism that suppresses the hydroxyl tunnelling in the hydroxyl gauche, methyl asymmetric con-
INTRODUCTION
Many years ago, Kakar, Seibt, and Quade reported the microwave torsional–rotational spectra of normal ethyl alcohol and of a few isotopic species in the hydroxyl gauche conformation and the torsional ground state (1). From these studies it was possible to determine the components of the dipole moment, hydroxyl gauche tunnelling energy, trans– gauche energy difference, and the potential energy coefficients hindering the methyl (1a) and hydroxyl (2) internal rotations. Further, the product of inertia terms, D(Px Py / Py Px ) and E(Px Pz / Pz Px ), that mix hydroxyl gauche / and 0 substates for the normal species were determined (1a). Finally, analysis of the rotational coefficients AI and AII for conformations I and II of the hydroxyl gauche, methyl asymmetric forms of CH2DCH2OH and –OD gave the potential energy term that suppresses the pure –OH or –OD tunnelling as 8 and 4 cm01 , respectively (1a, pp. 4305–07). It was not possible to resolve the tunnelling between the symmetric and antisymmetric substates within conformations I and II (see Fig. 1). In the present work, the microwave studies are extended to the first excited states of the hydroxyl gauche torsion of CH2DCH2OH. These states are the hydroxyl gauche / and 0 states for the methyl symmetric conformation and also the hydroxyl gauche / and 0 substates for the methyl asymmetric conformation. Only a-dipole lines were found in all cases. In the course of the study it was necessary to 1
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TABLE 1 Microwave Torsional–Rotational Spectra of Sym–CH2DCH2OH in the Ground State
formations of CH2DCH2OH has been discussed in detail by Kakar and Quade ( 1a, pp. 4305 – 4307 ) under the approximation that the methyl asymmetric tunnelling is too small to be resolved, in this work we redevelop the formalism including only torsional energies, rotational energies neglected, to include the methyl asymmetric tunnelling. This leads to a prediction of the tunnelling energy between the symmetric and antisymmetric substates of conformations I and II. EXPERIMENTAL
All measurements were made on the Mississippi State University Hewlett–Parkard spectrometer. Frequencies above 40 GHz were obtained using SpaceKom (Honeywell)
*Denotes the lines reported in (1a) and remeasured by Suenram at NIST. / Denotes the lines in (1a) and remeasured at Mississippi State. Units are in MHz. In the least squares fit the data to determine the B’s, C’s, DJ’s, and DJK’s all a-dipole lines were used except 303 R 202 and 321 R 220 for the 0 state and the K01 Å 3 lines for both states and no c-dipole lines were used. The A’s and D were determined from the c-dipole lines but not by a least squares method. The calculated values in the table do include contributions from DJ and DJK as well as the contributions from D and E for the c-dipole transitions.
FIG. 1. Hydroxyl gauche, methyl symmetric and asymmetric isomers of CH2DCH2OH.
doublers. The measurements were made at room temperature. The Stark effect was used to facilitate the assignments. The computer program for the analysis of the data in Tables 1–4 in terms of rotational coefficients A, B, and C
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and centrifugal distortion coefficients DJ , DJK , DK , dJ , and dK is a nonlinear least squares fit. The plus and minus uncertainty to each coefficient in the Tables is one standard deviation except for D A/ , D A0 , and D in Table 1.
TABLE 3 Microwave Rotational Spectra of Methyl Asym–CH2DCH2OH in the Torsional Ground State for Conformations I and II
SPECTRA AND ANALYSIS
Hydroxyl gauche, Methyl Symmetric Ground State As a precursor to assigning the spectral lines for the first excited state of the hydroxyl gauche torsion, it was necessary to assign additional ground state lines for both the methyl symmetric and the methyl asymmetric conformations. These lines for the methyl symmetric conformation are reported in Table 1 along with the lines previously measured by Kakar and Quade (1a, Table IV). In the course of the work at NIST on the following paper, it was found that some of the frequencies reported by Kakar and Quade were systematically high by 0.15–0.30 MHz especially in P and K bands. All of these lines have been remeasured either at National Institute of Standards and Technology (NIST) or Mississippi State and the revised frequencies are given in Table 1. The Hamiltonian that was used to analyze the data for hydroxyl gauche, methyl symmetric CH2DCH2OH has the form TABLE 2 Microwave Rotational Spectra of Sym–CH2DCH2OH in the First Excited State of the –OH Torsion
*Denotes the lines either assigned and measured or remeasured from (1a) by Suenram at NIST. / Denotes the lines reported in (1a) remeasured at Mississippi State. Units are in MHz. The fit is not sensitive to D IK , D JII , and d JII .
( sÉHTRÉs ) Å AsP 2z / BsP 2y / CsP 2x 0 DJsP 4 0 DJKsP 2 P 2z 0 DKsP 4z 0 2dJsP 2 (P 2x 0 P 2y ) 0 dKs( P 2z (P 2x 0 P 2y ) 0 (P 2x 0 P 2y )P 2z ) / Es ,
[1] a
with s Å / or 0 for the hydroxyl gauche substates. The
Units are in MHz.
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TABLE 4 Microwave Rotational Spectra of Asym–CH2DCH2OH in the First Excited State of the –OH Torsion
Note. In this case, the –OH tunnelling energy is so large that the molecule is no longer localized in the conformations I and II. A / and 0 symmetry notation is used for the torsional states. Units are in MHz.
Hamiltonian also has terms mixing the / and 0 substates of the form ( /ÉHTRÉ0 ) Å D(Px Py / Py Px ) / E(Px Pz / Pz Px ),
[2]
where D and E are odd in the hydroxyl gauche internal rotation angle and pure imaginary. It should be kept in mind that there are Coriolis terms linear in Py and Pz that couple the hydroxyl gauche / and 0 substates but it is shown in the Appendix how these terms contribute to the effective As , Bs , and Cs , including cross-mixing with the product of inertia terms and the third-order rotational denominator corrections. This approach to the analysis, taking As , Bs , Cs , D, E, and D as spectroscopic parameters, is based upon the framework fixed axis method (FFAM) for studying internal rotation– overall rotation interactions in the asymmetric–asymmetric molecules (3). Although there are two internal rotors, the hydroxyl and methyl groups, we use a single rotor approach,
fixing the deuterated methyl group conformation to make the framework symmetric. A Van Vleck transformation is used to isolate the torsional 3 1 3 energy matrix for the hydroxyl trans and gauche substates. Since the hydroxyl trans substate is well below the two hydroxyl gauche substates because of the V1 and V2 terms in the effective potential energy, we neglect any residual trans–gauche interactions (1a, Table VIII). However, the / and 0 hydroxyl gauche substate remain strongly interacting by an HTR term that involves Pz , Py , Px Py / Py Px and Px Pz / Pz Px . As shown in the Appendix, most, but not all, of these terms are incorporated into the effective rigid rotor coefficients. There are some things that we wish to note from the analysis. First, the analysis is sensitive to the centrifugal distortion terms DJ and DJK only. Second, the K01 Å 2 to 3 c-dipole transitions do not fit well in the analysis. We thought this occurred because the FFAM, like the principal axis method (PAM) for symmetric internal rotor (4), is a perturbation approach and the pPz coupling is large enough to cause the approach to break down for K1 Å 3. However, the internal axis method (IAM) (5) was tried in the analysis and it did not improve the situation. Right now we do not know the source of difficulty with these lines. It should be noted that the a-dipole K01 Å 3 lines for J Å 3 to 4 also show a poor fit. The problem energy levels appear to be those with K01 Å 3 for the hydroxyl gauche / state. Third, the J Å 3 a-dipole lines, 202 r 303 and 220 r 321 , for the 0 substate that interact through the rotational asymmetry show deviations from their expected positions. Since there are no problems with the lines from / substate, we do not think the deviations comes from ( / ÉHTRÉ 0 ). Also, since all internal rotation–overall rotation interaction are diagonal in J, we do not expect interaction from the hydroxyl trans, methyl symmetric conformation for the low values of J. Further, since the lines for the hydroxyl gauche, methyl asymmetric conformations do not appear to be perturbed, we do not feel the shift is methyl symmetric–methyl asymmetric interaction in origin. Therefore we are left with the idea that there is some unaccounted for interaction within the ground state hydroxyl gauche 0 manifold. These levels are interacting pairs in the Wang representation. These two lines as well as the c-dipole K01 Å 2 to 3 lines and a-dipole K01 Å 3 lines were not included in the least squares fit. Finally, both D and E are of the order 25% smaller for hydroxyl gauche, methyl symmetric CH2DCH2OH than CH3CH2OH. In either case, the empirical D and E are not represented well and were not expected to be represented well, from calculations based upon molecular structure because of the lumping of all internal rotation effects into the empirical spectroscopic constants. Hydroxyl gauche, Methyl Symmetric –OH Excited State For the hydroxyl gauche, methyl symmetric first excited state of the –OH torsion, complete sets of the a-dipole lines
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were identified and assigned. The intense search for c-dipole lines that would be tunnelling transitions as for the ground state was not fruitful, which is consistent with our expectation that the hydroxyl gauche tunnelling energy for this state is large, predicted to be of the order of 1.5 THz. The large tunnelling completely washes out effects from the D and E product of inertia terms. Also, without c-dipole lines, it is not possible to determine the As of Eq. [1]. The a-dipole lines were analyzed on the basis of
aspects of tunnelling (7). For hydroxyl gauche, methyl asymmetric CH2DCH2OH the lowest symmetry potential energy term has the form Vs 1s 2 sin a1 sin a2 , where a1 is the internal rotation coordinate for the hydroxyl group and a2 is that for the methyl group. The theoretical development in Ref. (1a, pp. 4305–07) shows how the effective rigid rotor Hamiltonian evolves in the form ( sÉHTRÉs ) Å AsP 2z / BsP 2y / CsP 2x 0 DJsP 4
( sÉHTRÉs ) Å BsP 2y / CsP 2x 0 DJsP 4 0 DJKsP 2 P 2z ,
[3]
with a data fit as good as that for the ground state lines. The assigned spectral lines and spectroscopic constants are given in Table 2. It should be noted that the difference in the Bs and Cs between the torsional substates is smaller for the excited state than for the ground state. We interpret this to mean that the hydroxyl gauche torsional wavefunctions are less localized in the excited state than in the ground state. Therefore the internal rotation angle dependence of the moment of inertia dominates over the second Coriolis interaction from pPx and pPy terms. Hydroxyl gauche, Methyl Asymmetric Ground State For the hydroxyl gauche, methyl asymmetric conformations there are four possible states: two with the methyl D and hydroxyl H on the same side of the molecular plane, conformation I, and two with the methyl D and hydroxyl H on opposite sides of the molecular plane, conformation II (see Fig. 1). The spectra clearly indicate, as was determined previously (1a, Table V), that the hydroxyl gauche ground state tunnelling is suppressed and the molecule so to speak localizes into conformations I and II with twofold degeneracy for each conformation. Tunnelling exists between the symmetric and antisymmetric substates of conformations I and II but this splitting is unresolvable with our spectrometer. In this tunnelling the hydroxyl gauche hydrogen and the methyl asymmetric deuterium move in tandem across the molecular plane. This suppression of the pure hydroxyl gauche tunnelling can be caused by two things in the torsional Hamiltonian. The first is a kinetic energy term odd in both a1 and a2 with low symmetry of the form sin a1 sin a2 p1 p2 (6). For hydroxyl gauche, methyl asymmetric CH2DCH2OH, this term is not large enough (1a, p. 4306) to suppress the pure hydroxyl tunnelling even though it couples the two internal rotations in the proper way. It is not unusual for the internal rotation coordinate dependence in the kinetic energy not to be large enough to suppress tunnelling for most molecules. On the other hand, it has been found for molecules with partially deuterated methyl groups that the internal rotation coordinate dependence in the zero point energy of the other vibrations of the molecules contributes to the effective internal rotation potential energy in a manner to suppress certain
0 DJKsP 2 P 2z 0 DKsP 4z 0 2dJsP 2 (P 2x 0 P 2y ) 0 dKs( P 2z (P 2x 0 P 2y ) 0 (P 2x 0 P 2y )P 2z ),
[4] with s Å I and II and centrifugal distortion added. The isolation of the effective Hamiltonian in Ref. (1a) is correct as is the effective energy matrix. However, the transformed torsional wavefunctions are labeled incorrectly. Also, Ref. ( 1a ) does not include the methyl asymmetric tunnelling energy. Both of these things are developed later in the present work. The previously and presently assigned spectral lines and the results of the analysis are given in Table 3. The lines follow rigid rotor behavior, including centrifugal distortion, in the sense that the hydroxyl gauche tunnelling has been suppressed and does not appear in the c-dipole transitions. The assignment of the additional lines made possible a centrifugal distortion analysis. In the course of the work on the following paper, where frequency measurements were made at NIST, it was found that many of the lines, especially those in P and K bands, reported by Kakar and Quade (1a, Table V) were systematically off in frequency by 0.15–0.30 MHz in a high direction. The previously assigned lines were remeasured, some at NIST and the rest at Mississippi State. The revised frequencies are given in Table 3. The centrifugal distortion analyses have been carried out using all five coefficients in Eq. [4]. However, the analyses were not sensitive to all the coefficients. For example, the analysis for conformation I is not sensitive to DK and that for conformation II is not sensitive to DJ and dJ . The uncertainties in all coefficients, rotational and centrifugal distortion, are based upon the number of analyzed lines and which centrifugal distortion coefficients are used. All coefficients have been found to change more than the uncertainties when new lines were assigned and added to the fit. The coefficients are fair for prediction. The J Å 5, 6, and 7 Q-type lines were predicted with preliminary coefficients and then found at NIST without searching. Hydroxyl gauche, Methyl Asymmetric –OH Excited State It was expected that the hydroxyl gauche tunnelling energy of the first excited state, of the order 1.5 THz, would be enough to override the 8 cm01 Å 240 GHz energy that
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localizes the molecule into conformations I and II in the ground state. Such has been found to be the case. This means that the excited hydroxyl gauche states may be labeled by / and 0 and there are two methyl asymmetric states for each of these. The search for the c -dipole lines, which would be combined rotational and torsional transitions, was futile. The assignments of the a-dipole lines are given in Table 4. One might expect four sets of lines, rather than the observed two, to occur for the excited states. We interpret this to mean that the difference in Bs and Cs for the two methyl asymmetric substates is too small to be resolved. At the same time, as for the hydroxyl gauche, methyl symmetric CH2DCH2OH, the excited state Bs and Cs are closer together than for the conformations I and II of the ground state. Again, it is presumed that this arises from the averaging of the moments of inertia over the torsional angle that dominates over the smaller Coriolis interaction pPx and pPy . The Hamiltonian of Eq. [3] was used to analyze the data. DISCUSSION
The spectra and analyses in this work provide some interesting features for the six conformations of hydroxyl gauche of CH2DCH2OH. The relative large disagreement between calculated and observed frequencies is consistent with those for other molecules with asymmetric internal rotors—much larger than for those molecules with symmetric internal rotors. Also, it should be kept in mind that the spectroscopic coefficients determined in the data analysis contain a lot of hidden effects due to internal rotation. An important aspect of the experimental and theoretical portions of this work concerns the suppression of the hydroxyl gauche tunnelling energy so that the molecule localizes so to speak into conformations I and II for the hydroxyl gauche, methyl asymmetric forms. At this point we work through the transformation of the torsional energy matrix as was done in Ref. (1a) (pp. 4305–07), but here neglecting the rotational Hamiltonian and including the methyl asymmetric tunnelling energy. We take as the zeroth-order torsional basis Cij Å wi ( a1 ) cj ( a2 ),
IIS
IIS 1 e / (D / d) 2
IS
[5]
where w is the wavefunction for the hydroxyl gauche torsion and c the wavefunction for the methyl asymmetric torsion; i , j r /, 0 for the symmetry of the respective torsional states. In this basis, the hydroxyl gauche substates are separated by the tunnelling energy D and the methyl asymmetric substates by the tunnelling energy d. These substates are mixed by the interaction energy e å Vs 1s 2 ( //Ésin a1 sin a2É00 ) . The resulting 4 1 4 torsional energy matrix is // 00 /0 0/
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1 CIS Å q [( // ) 0 ( 00 )] 2
1 CIIA Å q [( /0 ) 0 ( 0/ )] 2
1 CIA Å q [( /0 ) / ( 0/ )], 2 [7]
where conformation I has hydroxyl H and methyl D on the same side and conformation II has hydroxyl H and methyl D on opposite sides of the molecular plane (see Fig. 1). In checking these properties, it must be noted that w0 and c0 are exponential sums and to get a sine function to show the oddness of the 0 substate must be multiplied by i. The transformed energy matrix is
IA
IIA
0
0
0
0
1 0e / (D / d) 2
1 0 (D 0 d) 2 1 e / (D / d) 2
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1 CIIS Å q [( // ) / ( 00 )] 2
IIA
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00 /0 0/ e 0 0 D/d 0 0 d 0e D
The matrix elements of ( /Ésin ai É0 ) are pure imaginary, which controls the sign of the interaction terms in Eq. [ 6 ] . Since the spectral data show that the hydroxyl gauche, methyl asymmetric CH2DCH2OH so to speak localizes into conformations I and II, the off-diagonal term e must be much larger than D and d to suppress the pure hydroxyl tunnelling energy. Therefore the energy matrix of Eq. [6] is best transformed to the new basis
IS 1 0 (D / d) 2 1 0e / (D / d) 2
IA
// 0
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Second-order perturbation theory gives the energy of separation of conformations I and II as 2e /
D2 . 4e
[9]
We have determined (1a, pp. 4305–07) that the energy difference of Eq. [9] is of the order 12 cm01 for CH2DCH2OH. Further, the off-diagonal matrix elements in Eq. [8] show that the two substates of conformation I and the two substates of conformation II are split by DÅ
dD . 2e
these terms are incorporated into the effective rotational coefficients of Eq. [1]. The complete matrix element mixing the hydroxyl gauche / and 0 substates for CH2DCH2OH has the form ( /ÉHTRÉ0 ) Å aPz / bPy / ic(Px Py / Py Px ) / id(Px Pz / Pz Px ),
where all of the coefficients in Eq. [A1] are internal rotation dependent. Second-order perturbation theory,
[10]
From the barrier to methyl internal rotation for hydroxyl gauche CH3CH2OH, d is estimated to be less than 0.15 MHz while D is estimated to be the order 97 000 MHz from Table 1 of hydroxyl gauche, methyl symmetric CH2DCH2OH. With e of the order 12 cm01 , D is estimated to be of the order 0.04 MHz. Even though this splitting is beyond the resolution of our spectrometer, we did carry the gas pressure to extremely low levels in case e might be considerably smaller than what was determined in Ref. 1a. These experiments gave a null result. Finally, on the basis of Eq. [7] the only c-dipole transitions allowed in first order are IS r IA and IIS r IIA. Transitions from I r II are forbidden in first order. In the following paper, the tunnelling transitions within conformation I and conformation II are reported using the high resolution microwave spectrometer at NIST. This makes it possible to determine an improved and more accurate value for e and therefore Vs 1s 2 . And finally, we wish to speak to our assignment of the excited state hydroxyl gauche, methyl asymmetric lines to the excitation of the hydroxyl group only. For example, a possibility is that the methyl asymmetric group is excited while the hydroxyl group remains in the ground state. In this case, d* would be 73 times larger than d but still small enough to speak to conformations I and II. Then D *, Eq. [10], would be 73 times larger than D and be clearly resolvable for cdipole transitions since D * would be of the order of a few MHz. However, in this case it would be expected that the increased tunnelling for the excited methyl asymmetric group would also lead to a resolution of the Bs and Cs within conformations I and II. Therefore we would expect to see four lines for the a-dipole transitions rather than the two observed. This fact and the fact that our search for c-dipole lines, while difficult, was fruitless lead us to the conclusion that our assignment, which is hydroxyl gauche excited state, methyl asymmetric ground state, is correct. Further, we are convinced that we have never seen a-dipole lines of the excited state of methyl torsion, even for CH3CH2OH (2).
( /ÉHTRÉ0 )( 0ÉHTRÉ/ )/ D,
( /É(HTRÉ0 )( 0ÉHTRÉ0 ) Å a 2 P 2z / b 2 P 2y / (ab / bc 0 ad)(Py Pz / Pz Py ) / 2ac(P 2x 0 P 2y ) / 2bd(P 2z 0 P 2x ) / c 2 (Px Py / Py Px ) 2 2
1 (Px Pz / Pz Px ) / (Px Pz / Pz Px )(Px Py / Py Px )].
Inspection shows that the first five terms of Eqs. [A2] and [A3] are rigid rotor in nature and are incorporated into the effective As , Bs , and Cs . The third term amounts to a small rotation of axes and such terms also occurred after the original Van Vleck transformation. The last three terms are nonrigid rotor in nature and arise from the product of inertia terms that are odd in a and mix the / and 0 substates; these arise from the D and E terms from Eq. [2]. The last term in Eqs. [A2] and [A3] has a comparable size coefficient to the c 2 and d 2 terms, but while the latter make contributions to the diagonal energy matrix elements, (Px Py / Py Px )(Px Pz / Pz Px ) / rrr has only off-diagonal matrix elements in K01 and therefore makes a much smaller contribution when again treated by second-order perturbation theory. Therefore we have not included this term in our analysis. In the perturbation treatment of the D and E terms in Eq. [2], we do use the rotational energy denominator corrections in the calculation of the perturbation contributions. However, Eqs. [A2] and [A3] give the second-order corrections that include only the hydroxyl gauche / and 0 energy difference. We wish to point out that inclusion of these so-called denominator corrections is equivalent to going to third order in the Van Vleck transformation in the FFAM. Specifically, these third-order corrections are of the form [( sÉHTRÉs * )( s *ÉHRÉs * )( s *ÉHTRÉs ) 0
1 ( sÉHTRÉs * )( s *ÉHTRÉs )( sÉHRÉs ) 2
0
1 ( sÉHRÉs )( sÉHTRÉs * )( s *ÉHTRÉs )]/ D 2 , 2
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[A3]
2
/ d (Px Pz / Pz Px ) / cd[(Px Py / Py Px )
In this Appendix, results are presented that show why we neglect pPy and pPz Coriolis coupling in Eq. [2] and how
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where it can be taken that ( /ÉHRÉ/ ) Å ( 0ÉHRÉ0 ) for purposes of the calculation. What should be noted is that if only Coriolis terms from Py and Pz in Eq. [A1] are included, then these third-order terms reduce to rigid rotor contributions. For example, if the AP 2z term from HR and the bPy term from HTR are considered, it is straightforward to show from the commutation relations that
after application of the commutation relations. The rotation to the principal axis system will make the coefficient of Py Pz / Pz Py and therefore the coefficient of the whole term vanish. Hence Eq. [A6] is contained in the empirical rigid rotor coefficients. More importantly and more simply, none of the terms in Eq. [A6] has contributions diagonal in K01 . ACKNOWLEDGMENTS
Py P 2z Py 0
1 2 2 1 2 2 P y P z 0 P z P y Å P 2x 0 P 2z . 2 2
[A5]
Other terms can be reduced in a similar manner. What this means is that the third-order terms from the Coriolis interaction—the rotational denominator corrections—are rigid rotor in nature and are included in the empirical A, B, and C. The cross-terms between the Coriolis coupling and the products of inertia can also be reduced. For example, the cross-term between Py and (Px Py / Py Px ) from HTR and the P 2y term from HR reduces to i[ 0 (Py Pz / Pz Py ) / /
1 2 P y (Py Pz / Pz Py ) 2
1 (Py Pz / Pz Py )P 2y / Py (Py Pz / Pz Py )Py ] 2
[A6]
The authors wish to thank Dr. Richard Suenram of NIST for making the measurements of the tunnelling frequencies within conformations I and II of hydroxyl gauche, methyl asymmetric CH2DCH2OH that are reported in the following paper. In the course of his work on this project, it was found that some of the lines reported in Ref. (1a) Tables IV and V needed remeasurement and many of the remeasurements were made by him.
REFERENCES 1. (a) R. K. Kakar and C. R. Quade, J. Chem. Phys. 72, 4300–4307 (1980). (b) R. K. Kakar and P. J. Seibt, J. Chem. Phys. 57, 4060– 4061 (1972). 2. C. F. Su and C. R. Quade, J. Mol. Spectrosc. 175, 390–394 (1996). 3. C. R. Quade and C. C. Lin, J. Chem. Phys. 38, 540–550 (1963). 4. C. C. Lin and J. D. Swalen, Rev. Mod. Physics 31, 841–892 (1959). 5. M. Liu and C. R. Quade, J. Mol. Spectrosc. 146, 228–251 (1991). 6. J. V. Knopp and C. R. Quade, J. Chem. Phys. 53, 1–10 (1970). 7. (a) T. L. Chang and C. R. Quade, J. Mol. Spectrosc. 111, 398–402 (1985). (b) G. L. Walker and C. R. Quade, J. Chem. Phys. 52, 6427– 6428 (1970).
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