Ring-Puckering Potential Energy Functions for Trimethylene Sulfide

Article pubs.acs.org/JPCA

Ring-Puckering Potential Energy Functions for Trimethylene Sulfide and Its Monovalent Cation Hye Jin Chun, Esther J. Ocola, and Jaan Laane* Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, United States ABSTRACT: The spectra and ring-puckering potential energy function for trimethylene sulfide cation (TMS+) from vacuum ultraviolet mass-analyzed threshold ionization spectra have recently been reported. To provide an in-depth comparison of the potential function with that of trimethylene sulfide (TMS) itself, we have used ab initio MP2/cc-pVTZ calculations and DFT B3LYP/cc-pVTZ calculations to predict the structures of both TMS and TMS+ and then used these to calculate coordinate-dependent ring-puckering kinetic energy functions for both species. These kinetic energy functions allowed us to calculate refined potential energy functions of the puckering for both molecules based on the previously published spectra. TMS has an experimental barrier of 271 cm−1 and energy minima at ring-puckering angles of ±29°. For TMS+ the barrier is 60 cm−1 and the energy minima correspond to ring-puckering angles of ±21°. The lower barrier for the cation reflects the smaller amount of angle strain in the ring angles for TMS+.

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INTRODUCTION We have been investigating the potential energy functions for vibrations such as the ring-puckering, ring-twisting, and carbonyl wagging for five decades, and these have been discussed in several reviews.1−9 Far-infrared, mid-infrared combination band, Raman, microwave, electronic, and laserinduced fluorescence spectra have all been utilized to determine the potential energy functions, which generally do a remarkably good job of reproducing the experimental data through quantum-mechanical calculations. For ring-puckering vibrations of four-membered ring molecules the potential energy function has the form V = ax 4 + bx 2

and this is justified when all of the other vibrations are of considerably higher frequency. The far-infrared spectra10 and microwave spectra11 of the ring-puckering vibration of trimethylene sulfide (TMS), which can also be named thietane, were initially studied at the University of California, Berkeley by the Strauss and Gwinn research groups, respectively. A potential function of the form of eq 1 was utilized, but because the reduced mass for the motion was not calculated, the potential energy function was presented in reduced (dimensionless) form. In later work, our laboratory12 calculated the reduced mass to be 109.5 u based on a structure from molecular mechanics calculations and thereby determined the potential function to be

(1)

where x is the ring-puckering coordinate (Figure 1), and this has proven to be very successful in correctly predicting the

V (cm−1) = 5.61 × 105x 4 − 2.474 × 104x 2

where x is in Å. This corresponds to a barrier of planarity of 275 cm−1. Lee and coworkers13 have very recently reported the vacuum ultraviolet mass-analyzed threshold ionization (VUV-MATI) spectra of the TMS monovalent cation (TMS + ) and determined its ring-puckering energy levels. They also calculated an empirical potential energy function but apparently did not use a reduced mass in their computations. In the present study we will present calculated ab initio and density functional theory (DFT) structures for TMS and TMS+ and then used these to calculate coordinate-dependent kinetic energy functions, which can be utilized to calculate the

Figure 1. Definitions of the ring-puckering coordinate x and of the ring-puckering angle θ.

experimental frequencies. As we have shown,1−9 the constant a arises from angle-bending force constants, while b arises both from angle-bending force constants and torsional interactions. The use of this one-dimensional function assumes that there is little interaction with the other vibrations in these molecules, © XXXX American Chemical Society

(2)

Received: February 20, 2017 Revised: March 27, 2017 Published: March 28, 2017 A

DOI: 10.1021/acs.jpca.7b01659 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

Figure 2. Calculated structural parameters of TMS and TMS+. Both MP2/cc-pVTZ and B3LYP/cc-pVTZ values are given with the latter in parentheses. The bond distances are in Ångströms (Å).

potential energy surfaces in dimensioned form for those two species.

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CALCULATIONS Structure Calculations. The geometrical structures of TMS and TMS+ were obtained from ab initio MP2/cc-pVTZ and DFT/B3LYP/cc-pVTZ calculations. The Gaussian 09 program14 was used for the computations, and the Semichem AMPAC/AGUI program15 was used to visualize the structures. The calculated structures are shown in Figure 2. For both species the bond distances and angles were also calculated for the planar structures. The bond distances, as expected, are almost identical for the planar and puckered forms, but the angles are somewhat different. The change in interior ring bond angles upon puckering is required information for us to properly calculate the curvilinear model for the puckering because the values of ρ and ω are needed.16 These reflect the relative amounts of bending of the interior ring angles. For TMS ρ was calculated to be 1.345 and ω to be −0.147. For TMS+ ρ was calculated to be 1.286 and ω to be −0.125. A ω value of zero would correspond to equal bending of the angles upon puckering. The negative values we have determined here confirm that during puckering there is more bending of the CSC angle and the angle across from it than at the other two CCC angles. This is expected because the CSC bending force constant is smaller than that for a CCC angle. Kinetic Energy Calculations. Using the structures from the ab initio calculations, we have used the vector methods of the Laane RDMS4 program16 to calculate the coordinatedependent kinetic energy function for the ring-puckering of both TMS and TMS+. As discussed above, the value of ω is necessary for the calculations. The kinetic energy expressions (reciprocal reduced mass) for the two species are

Figure 3. Coordinate dependence of g44, the reciprocal reduced mass expansion for TMS and TMS+.

structure from molecular mechanics calculations.12 Here the reduced masses for TMS and TMS+ are 104.0 and 104.5 u, respectively. Potential Energy Calculations. The DA1OPTN Meinander−Laane potential energy program17 was used to fit the experimental data of Strauss10 and Gwinn11 for TMS and from Lee and coworkers13 for TMS+. The kinetic energy functions of eqs 3 and 4 were applied in the calculations, and the constants a and b in eq 1 were adjusted to give the best least-squares fits. The Hamiltonian for the problem is H (x ) =

(3)

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and g44 (x)TMS+ = 9.569 × 10−3 − 2.206 × 10−2x 2 + 7.157 × 10−2x 4 − 4.698 × 10−1x 6

(5)

where the potential function V(x) is given in eq 1. The coefficients of the harmonic oscillator basis functions were obtained from the output of the DA1OPTN Meinander−Laane program. These were then used to calculate the Franck− Condon factors and the intensities of the VUV-MATI transitions.

g44 (x)TMS = 9.614 × 10−3 − 2.256 × 10−2x 2 + 7.150 × 10−2x 4 − 4.697 × 10−1x 6

−ℏ2 ∂ ∂ g (x ) + V (x ) 2 ∂x 44 ∂x

THEORY R. P. Bell18 first proposed that the potential energy function for the ring-puckering vibration of a four-membered ring can be represented by a quartic oscillator. Laane19 later showed that it should contain both the x4 and x2 terms, as shown in eq 1. In

(4)

These are similar and are shown in Figure 3. We had previously reported the reduced mass for TMS to be 109.5 u based on a B


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Figure 4. Potential energy functions for the ring-puckering of TMS and TMS+. The experimentally observed transitions are shown for TMS, while the experimental energy level spacings are shown for TMS+.

1970 Laane20 showed that the energy levels for the puckering could be calculated using a reduced (dimensionless) coordinate using the function 4

2

V = A(Z + BZ )

Zmin = ±(B /2)1/2

and that the barrier is barrier = AB2 /4

(6)

(7)

The reduced mass μ (reciprocal of g44) is required to convert from Z to x. The constants A and B are related to a and b by A = (ℏ2 /2μ)2/3 a1/3

(8)

and B = (2μ/ℏ2)1/3 a−2/3b

In the paper by Lee and coworkers potential function as

(9) 13

they presented their

V (θ ) = Ak (θ − zk)2 (θ + zk)2

(10)

where Ak = hk /zk4

(11)

and where hk is the potential energy barrier and ± zk are the energy minima. To avoid confusion, we have subscripted the constants in the paper by Lee and coworkers13 with k. Equation 10 is equivalent to

⎡ 2(D + D ) ⎤ x θ3 θ5 1 2 ⎥x = sin θ = ⎢ =θ− + + ··· D3D4 C 3! 5! ⎣ ⎦ (17)

where the sin θ term has been expanded into a Taylor’s series and C is a constant related to the geometric structure. Therefore, eq 1 can be rewritten as

V (θ ) = Ak (θ 2 − zk2)2 = (hk /zk4)(θ 4 − 2zk2θ 2 + zk4) (12)

V = ax 4 + bx 2 ⎛ ⎞ ⎛ ⎞ 2θ 6 θ4 = aC 4⎜θ 4 − + ···⎟ + bC 2⎜θ 2 − + ···⎟ 3 3 ⎝ ⎠ ⎝ ⎠

or V (θ ) = (hk /zk4)θ 4 − 2hk θ 2/zk2 + hk

(13)

Because Lee et al. have defined their energy minima to have V = 0, and because we know that the B coefficient in eq 6 is negative, we will rewrite eq 6 in a form that can easily be compared to eq 10. This is V = AZ 4 − ABZ2 + barrier

(16)

Noting that zk = Zmin and hk = AB2/4 we readily see that the potential energy functions in eqs 13 and 14 are the same, except in the labeling of the coordinate. Equation 10 uses θ, whereas we use Z. The important point is that in either case the coordinate is in reduced (dimensionless) form even though Lee and coworkers have presented θ to be in radians. It is necessary to use a computed reciprocal reduced mass expansion (kinetic energy function) to put the puckering coordinate into a dimensioned form, and this apparently was not done. Moreover, no potential energy parameters were presented in the published work. In the next section we present our calculated results in dimensioned form both for the parent TMS and its cation based on our rigorously calculated kinetic energy function. It might be noted that the use of θ as the puckering coordinate in the paper by Lee and coworkers13 is not inherently wrong and is consistent with the quartic x4 term being the dominant term. From Figure 1, which shows x, θ, and the distances D1, D2, D3, and D4 the relationship between θ and x can be determined to be

where Z is a dimensionless coordinate that can be related to the dimensioned ring-puckering coordinate x (Figure 1) by Z = (2μ/ℏ2)1/6 a1/6x

(15)

⎛ bC 2 ⎞ 4 2 2 = ⎜aC 4 − ⎟θ + bC θ + ··· 3 ⎠ ⎝

(18)

This shows that the ring-puckering vibration also can be represented similarly in θ as x. However, all previous work in this area has used eq 1 and x as the puckering coordinate so we prefer to also do that.

(14)

As readily shown by taking the derivative dV/dx and setting it to zero to find the energy minima, we find C


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et al. reported a barrier of 48 cm−1 and the minima to be at ±18°, but it is not clear how they reached that conclusion without a reduced mass calculation. We have also calculated the Franck−Condon factors for the transitions, and these were used to calculate the relative transition intensities, which are comparted to the reported experimental values in Table 3. These agree reasonably well,

RESULTS AND DISCUSSION Figure 2 presents the structures of TMS and TMS+ from the MP2/cc-pVTZ ab initio and DFT B3LYP/cc-pVTZ calculations. These were necessary for the reduced mass calculations and are also interesting in their own right. The bond distances and angles for the two are quite similar, although the CSC ring angle in the cation can be seen to be somewhat larger. Utilizing the kinetic energy function for TMS in eq 3 and the published spectral data9,10 we have refined the TMS ringpuckering potential function in eq 2, and the improved function is VTMS(cm−1) = 5.316 × 105x 4 − 2.403 × 104x 2

Table 3. Frequencies and Relative Intensities for the Vibrational Transitions of the TMS − TMS+ Ring-Puckering Vibration transitiona

(19)

This is shown in Figure 4. As shown in Table 1, this fits the experimental far-infrared and microwave data very well. Only

a

observeda

0→1 1→2 2→3 3→4 4→5 5→6 6→7 7→8 8→9 9 → 10 10 → 11 11 → 12

0.2746 138.25 12.43 85.7 62.7 84.25 91.3 99.7 107.6 114.0 118.25 123.5

frequency (cm−1)

relative intensity

frequency (cm−1)c

relative intensityd

→ → → → → → →

0 17 82 147 227 314 409

1.18 1.00 0.67 0.35 0.15 0.05 0.01

0.0 16.6 81.7 146.5 225.8 313.2 408.8

0.70 1.00 0.54 0.35 0.14 0.04 0.01

0 1 0 1 0 1 0

calculatedb 0.30 138.48 12.74 85.96 63.06 84.57 91.66 99.74 106.50 112.62 118.20 123.50

Δ 0.0 −0.2 −0.3 −0.3 −0.4 −0.4 −0.4 0.0 +1.1 +1.4 +0.1 +0.2

except that our computations calculate the 1 → 1 transition to be stronger than the 0 → 0 transition, whereas the experimental results show the 0 → 1 to be somewhat stronger. Figure 5 shows the calculated wavefunctions for these energy states involved in the transitions. Figure 6 compares the experimental potential energy functions to those predicted by the ab initio and DFT theoretical calculations. The theoretical curves were generated by calculating the potential energy for approximately 20 values of the puckering coordinate for each species. As can be seen, the ab initio calculation predicts a barrier that is too high for each species, while the DFT barriers are too low. Table 4 compares the experimentally determined barriers and energy minima θ values for both TMS and TMS+ for those predicted theoretically.

two of the weak higher level far-infrared frequencies differ by >0.4 cm−1 from the calculated values. The energy minima of TMS are at x = ± 0.15 Å or at puckering angles of ±29°. We have also utilized the data from Lee and coworkers13 to calculate the potential function for TMS+, and in dimensioned form this is

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CONCLUSIONS Lee and coworkers presented some very nice experimental results on the ring-puckering levels of the TMS cation. They then proposed an unusual form of the potential energy function in terms of the coordinate θ, which they reported to be in radians. In principle, there is no problem in using the dihedral angle θ (ring-puckering angle) as the coordinate, and we have shown that the potential energy function would also be dominated by a quartic term if θ is used. However, the potential energy curve can only be presented in dimensioned form if the reduced mass has been calculated specifically for that coordinate. Moreover, all previous work on ring-puckering vibrations that we are aware of has used the potential function of the form of eq 1, where the out-of-plane distance x has been used as the coordinate. It is not clear to us why Lee and coworkers presented their potential function in such an unusual form that can only be understood when it has been expanded into individual polynomial terms. The work of Lee and coworkers provided valuable data, and we can now see what the effect on the structure of TMS is when an electron is stripped off. The cation becomes more

(20)

−1

This function has a barrier of 60 cm , and this is also shown in Figure 4 along with the corresponding energy levels. Table 2 presents a comparison of the calculated energy level spacings from this function to those observed by Lee and coworkers. All of these values agree within ±1 cm−1. The energy minima of TMS+ are at x = ±0.11 Å and at puckering angles of ±21°. Lee Table 2. Observed and Calculated Ring-Puckering Energy Spacings (cm−1) for TMS+

a

spacing

observeda

0−1 1−2 2−3 3−4 4−5 5−6

17 65 65 80 87 95

calculatedb 17 65 65 79 88 95

0 1 2 3 4 5 6

a Ref 13 labeled ground-state levels 0 and 1 as 0+ and 0−. bVUV-MATI. Ref 13. cFrom DA1POTN program output. dFrom Franck−Condon factors.

Ref 10. bVTMS (cm−1) = 5.316 × 105 x4 − 2.403 × 104 x2

VTMS+(cm−1) = 4.044 × 105x 4 − 0.988 × 104x 2

calculated

TMS → TMS+

Table 1. Observed and Calculated Ring-Puckering Transitions (cm−1) for TMS transition

experimentalb

Δ 0 0 0 −1 +1 0

Ref 13. bVTMS+(cm−1) = 5.044 × 105 x4 − 0.988 × 104 x2. D



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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jaan Laane: 0000-0003-4423-6122 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the Robert A. Welch Foundation (Grant A0396) for financial support. Computations were carried out on the Texas A&M University Department of Chemistry Medusa computer system funded by the National Science Foundation, Grant No. CHE-0541587. The Laboratory for Molecular Simulation provided the Semichem AMPAC/AGUI software.

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REFERENCES

(1) Laane, J. Vibrational Potential Energy Surfaces in Electronic Excited States. In Frontiers of Molecular Spectroscopy; Laane, J., Ed.; Elsevier: Amsterdam, The Netherlands, 2009; pp 63−132. (2) Laane, J. Experimental Determination of Vibrational Potential Energy Surfaces and Molecular Structures in Electronic Excited States. J. Phys. Chem. A 2000, 104, 7715−7733. (3) Laane, J. Spectroscopic Determination of Ground and Excited State Vibrational Potential Energy Surfaces. Int. Rev. Phys. Chem. 1999, 18, 301−341. (4) Laane, J. Vibrational Potential Energy Surfaces of Non-rigid Molecules in Excited Electronic States. In Structure and Dynamics of Electronic Excited States; Laane, J., Takahashi, H., Bandrauk, A., Eds.; Springer: Berlin, Germany, 1999; pp 3−35. (5) Laane, J. Vibrational Potential Energy Surfaces and Conformations of Molecules in Ground and Excited Electronic States. Annu. Rev. Phys. Chem. 1994, 45, 179−211. (6) Laane, J. Vibrational Potential Energy Surfaces of Non-Rigid Molecules in Ground and Excited Electronic States. In Structures and Conformations of Non-Rigid Molecules; Laane, J., Dakkouri, M., Eds.; Kluwer Publishing: Amsterdam, 1993; pp 65−98. (7) Cheatham, C. M.; Laane, J. The Jet-Cooled Fluorescence Excitation Spectra, Potential Energy Function, and Conformations of 2-Cyclopenten-1-one, Propanal, and Propynal in the S1(n,π*) Electronic Excited State. In Time Resolved Vibrational Spectroscopy; Takahashi, V. H., Ed.; Springer-Verlag: Berlin Heidelberg, 1992; pp 179−182. (8) Laane, J. Determination of Vibrational Potential Energy Surfaces from Raman and Infrared Spectra. Pure Appl. Chem. 1987, 59, 1307− 1326. (9) Laane, J. One-Dimensional Potential Energy Functions in Vibrational Spectroscopy. Q. Rev., Chem. Soc. 1971, 25, 533−552. (10) Borgers, T. R.; Strauss, H. L. Far-Infrared Spectra of Trimethylene Sulfide and Cyclobutanone. J. Chem. Phys. 1966, 45, 947−955. (11) Harris, D. O.; Harrington, H. W.; Luntz, A. C.; Gwinn, W. D. Microwave Spectrum, Vibration-Rotation Interaction, and Potential

Figure 5. Wavefunctions for the energy levels associated with the transitions in the VUV-MATI spectrum. Wavefunctions for even levels are shown by solid lines; those for odd levels are shown by dashed lines.

Figure 6. Experimental and theoretical potential energy functions for the ring-puckering vibration of TMS and TMS+.

nearly planar, demonstrating that the ring angle strain within the ring has been increased.

Table 4. Energy Barriers and Energy Minima θ Values for TMS and TMS+ TMS barrier (cm−1) θ (degrees) TMS+ barrier (cm−1) θ (degrees) a

experimental

MP2/cc-pVTZ

B3LYP/cc-pVTZ

literature

271 ±29

412 ±29

120 ±22

274a

60 ±21

248 ±26

3 ±9

48b 18b

Ref 10. bRef 13. E


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The Journal of Physical Chemistry A Function for the Ring-Puckering Vibration of Trimethylene Sulfide. J. Chem. Phys. 1966, 44, 3467−3480. (12) Rosas, R. L.; Cooper, C.; Laane, J. Evaluation of Molecular Mechanics Methods for the Calculation of the Barriers to Planarity and Pseudorotation of Small Ring Molecules. J. Phys. Chem. 1990, 94, 1830−1936. (13) Lee, Y. R.; Park, C. B.; Hwang, J.; Sung, B. J.; Kim, H. L.; Kwon, C. H. Observation of the Ring-Puckering Vibrational Mode in Thietane Cation. J. Phys. Chem. A 2017, 121, 1163−1167. (14) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision A.02; Gaussian, Inc: Wallingford, CT, 2009. (15) AGUI; Semichem, Inc: Shawnee, KS, online at www.semichem. com. (16) Laane, J.; Harthcock, M. A.; Killough, P. M.; Bauman, L. E.; Cooke, J. M. Vector Representation of Large-Amplitude Vibrations for the Determination of Kinetic Energy Functions. J. Mol. Spectrosc. 1982, 91, 286−299. (17) Laane, J.; Meinander, N. Computation of the Energy Levels of Large-Amplitude Low-frequency Vibrations. Comparison of the Prediagonalized Harmonic Basis and the Prediagonalized Distributed Gaussian Basis. J. Mol. Struct. 2001, 569, 1−24. (18) Bell, R. P. The occurrence and properties of molecular vibrations with V(x) = αx4. Proc. R. Soc. London, Ser. A 1945, 183, 328−337. (19) Laane, J. Origin of the Ring-Puckering Potential Energy Function for Four-Membered Rings and Spiro Compounds. A Possibility for Pseudorotation. J. Phys. Chem. 1991, 95, 9246−9249. (20) Laane, J. Eigenvalues of the Potential Function V = z4 ± Bz2 and the Effect of Sixth Power Terms. Appl. Spectrosc. 1970, 24, 73−80.

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Ring-Puckering Potential Energy Functions for Trimethylene Sulfide

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