1. SECOND (PLANAR) MOMENTS AND THEIR APPLICATIONS IN SPECTROSCOPY Robert K. Bohn 1, John A. Montgomery, Jr. 2, H. Harvey Michels 2, Jason Byrd 2 1. Univ. of Connecticut, Dept. of Chemistry 2. Univ. of Connecticut, Dept. of Physics. WH03 June 19, 2013

2. Second Moments and Applications 0. Joke 1. 2 nd moment definitions 2. Structure relationships from 2 nd moments vs. rotational constants 3. Number of independent parameters from isotopologs 4. 2 nd moments and scaling 5. -CH 2 -, -CH 3 groups: standard values and applications 6. Isopropyl groups: standard values and applications 7. Phenyl groups: standard values and applications 8. -CF 2 -, -CF 3 groups: standard values and applications

3. Second Moments J. Kraitchman, Am. J. of Phys. 21 (1953) 17 Moment of inertia: I a =  m i (b i 2 + c i 2 ), etc. Sum of mass times distance from the a axis squared, I a < I b < I c Rotational constant: A(s -1 ) = h / (8  2 I a ), A(MHz) = 505379.05/I a (uÅ 2 ), etc., for B and C Second moment: P aa = (I b + I c – I a )/2 =  m i a i 2, etc. for P bb and P cc. Sum of mass times distance along the a axis (or from the bc plane) squared Inertial defect:  = I c – I a – I b = -2 P cc (0 for planar molecule)

4. A 6097.201 6061.41 3876.798 3885.0 4226.363 4265.363 B 1957.506 2077.557 1557.793 1623.354 1551.864 1610.575 C 1570.705 1650.531 1237.498 1276.971 1244.250 1277.806 P aa 248.52 233.04 301.22 288.50 306.13 295.41 P bb 73.23 73.16 107.16 107.26 100.05 100.10 P cc 9.65 10.22 23.20 22.82 19.53 18.38 a b

5. Table 4.Rotational constants and second moments of Cyclopropyl Benzene and its 13 C Isotopologs. Rot.Const.Parent 13 C 1 13 C 2 13 C 6 13 C 7 13 C 3 13 C 5 13 C 4 13 C 8/9 A/MHz4228.7369(4)*4226.15(4)4190.07(2)4170.11(11)4215.10(3)4166.43(3)4193.80(2)4225.39(4)4207.57(3) B/MHz1108.1111(1)1108.0905(2) 1107.9552(1) 1106.6538(4) 1101.8516(1) 1101.6445(2) 1097.0567(1) 1092.2242(2)1090.4263(2) C/MHz 943.9661(1) 943.8275(2) 941.9146(1) 939.9710(4) 938.7546(1) 936.1569(3) 934.2059(1) 932.2520(2) 931.9663(2) P aa /uÅ 2 435.97 435.98 436.03 436.57 438.56 438.65 440.57442.60 442.82 P bb /uÅ 2 99.41 99.48100.51101.09 99.79 101.20 100.40 99.50 99.46 P cc /uÅ 2 20.10 20.10 20.10 20.10 20.11 20.10 20.10 20.10 20.66 c

6. Second moment: P aa = (I b + I c – I a )/2 =  m i a i 2, etc. Sum of mass times distance along the a axis (or from the bc plane) squared Scaling Multiply each model coordinate, a i, by [P aa (obs)/P aa (calc)] 1/2 and the new set of a i coordinates now will reproduce P aa (obs) Repeat for b i and c i. The scaled set of coordinates reproduce the observed second moments (or rotational constants).

7. 1H-Nonafluorobutane 1 2 CCCC anti/CCCH gauche CCCC anti/CCCH anti (0.0 kcal/mol, ass'd) (0.4 kcal/mol, ass'd) 3 4 5 CCCCgauche/CCCHgauche CCCCgauche/CCCHanti CCCCgauche/CCCHgauche (cis, 0.8 kcal/mol, ass’d) (0.8 kcal/mol, ass’d) (trans, 1.1 kcal/mol) a b

8. Spectroscopic Constants of 1H-nonafluorobutane Conf. 1(Anti/Gauche) Not shown: Conf's 2(AA), 3(GA), 4(GGcis) Parameter Obs'd Calc'd* A/MHz 1428.9502(2) 1435.626 B/MHz 593.32878(6) 593.583 C/MHz 546.43577(6) 547.162 P aa /uÅ 2 711.4812 711.507 P bb /uÅ 2 213.3838 212.130 P cc /uÅ 2 140.2878 139.897 *PBE0/VTZ

9. In this example, the a i, b i, and c i coordinates were produced by a PBE0/VTC density functional calculation of the AG conformation of 1H-nonafluorobutane, C 4 HF 9. Scale factors for each principal axis. S a = [P aa (obs)/P aa (model)] 1/2 = [(711.4812)/(711.507)] 1/2 = [0.99996374] 1/2 = 0.9999818 S b = [P bb (obs)/P bb (model)] 1/2 = [(213.3838)/(212.130)] 1/2 = [1.0059105] 1/2 = 1.0029510 S c = [P cc (obs)/P cc (model)] 1/2 = [(140.2878)/(139.897)] 1/2 = [1.002793)] 1/2 = 1.0013958 P aa (model) = m 1 a 1 2 + m 2 a 2 2 + … m 14 a 14 2 = 711.507 P aa (model)(S a 2 ) = m 1 (a 1 S a ) 2 + m 2 (a 2 S a ) 2 + … m 14 (a 14 S a ) 2 = = [m 1 a 1 2 + m 2 a 2 2 … + m 14 a 14 2 ](S a 2 ) = 711.4812 = P aa (obs) Similarly, repeat for the b and c coordinates of each atom. Note that scaling does not move the center of mass nor the principal axes.

10. Spectroscopic Constants of 1H-nonafluorobutane Conf. 1(Anti/Gauche) Not shown: Conf's 2(AA), 3(GA), 4(GGcis) Parameter Obs'd Calc'd* (Calc'd) * (Scale Factors) A/MHz 1428.9502(2) 1435.6261428.962 B/MHz 593.32878(6) 593.583 593.328 C/MHz 546.43577(6) 547.162 546.436 P aa /uÅ 2 711.4812 711.507 711.483 P bb /uÅ 2 213.3838 212.130 213.381 P cc /uÅ 2 140.2878 139.897 140.287 *PBE0/VTZ

11. -CH 2 -/-CH 3 groups 1.56 uÅ 2 *propane, Pcc = 4.69 (4.69/3 = 1.56) 1-fluorobutane, Pcc = 6.39 (all anti) (6.39/4 = 1.60) 1-chlorobutane, Pcc = 6.32 (all anti) (6.32/4 = 1.58) *5-hexynenitrile, Pcc = 4.692 (all anti) (4.69/3 = 1.57) butyl cyanide, Pcc = 6.286 (all anti) (6.29/4 = 1.57) 1-hexyne, Pcc = 6.2689 (all anti) (6.27/4 = 1.57) pentane, Pcc = 7.90 (all anti) (7.90/5 = 1.58) anisole, Pcc = 1.70 1.70 p-anisaldehyde, Pcc = 1.81 (anti), 1.80 (syn) 1.81, 1.80 3-hexyne, Pcc = 7.31 (7.31/4 = 1.83) *3-heptyne, Pcc = 8.86 (all anti), (8.86/5 = 1.77) propargyl benzene Pcc = 2.20 2.20 benzyl cyanide, Pcc = 2.13 2.13 a c

12. Applications of CH 2 /CH 3 2 nd moments S-methyl thiochloroformate 35 ClCOSCH 3 ABC: 6986.1 2033.35 1590.28 P aa, P bb, P cc : 247.00 70.79 1.55 P cc = 1.55 Heavy-atom-planar Ethyl Cyanoformate NCCOOCH 2 CH 3 ABC of Conf. 1: 6453.3 1500.47 1236.36 P aa, P bb, P cc : 333.63 75.13 3.18* ABC of Conf. 2: 6787.8 1549.38 1406.80 P aa, P bb, P cc : 305.48 53.76 20.70 * Conf. 1: P cc = 3.18 or 1.59 per CH 2 /CH 3, Heavy-atom-planar, C s symmetry a c a c

13. -CH(CH 3 ) 2 (Isopropyl) Second Moments, 55 uÅ 2 *(CH 3 ) 2 CHF (2-fluoropropane)P aa = 54.9 *(CH 3 ) 2 CF 2 (2,2-difluoropropane)P bb = 55.0 (CH 3 ) 2 CHOH (2-propanol)P aa = 54.7 (CH 3 ) 2 CHSH(2-propanthiol)P bb = 54.8 (CH 3 ) 2 CHCN (2-cyanopropane)P bb = 55.2 (CH 3 ) 2 CHCCH (3-methyl-1-butyne) P bb = 54.9 *(CH 3 ) 2 CHC 6 H 4 CHO (p-Isopropyl benzaldehyde) P cc = 55.5 (syn) P cc = 55.5 (anti) (Isopropyl group straddles phenyl ring) a b c b a c c a

14. Applications of isopropyl 2 nd moments: 2-methylbutane CH 3 CH 2 CH(CH 3 ) 2 P bb =57.60 [55(isopropyl)+2(1.6)=58.2] 2-methylpentane CH 3 CH 2 CH 2 CH(CH 3 ) 2 P cc =59.27[55(isopropyl)+3(1.6)=59.8] 6-methyl-3-heptyne (isobutyl ethyl acetylene) CH 3 CH 2 C≡CCH 2 CH(CH 3 ) 2 P cc = 60.00 [55(isopropyl) + 3(1.6) =59.8] Isopropyl Fluoroformate FCOOCH(CH 3 ) 2 P cc = 27.07 and not 55. C 1 symmetry a b a c a c a c

15. -C 6 H 5 (phenyl) P bb = 89.2 uÅ 2 Phenylacetylene, P bb = 89.01 Chlorobenzene, P bb = 89.12 Cyanobenzene, P bb = 89.40 Isocyanobenzene, P bb = 89.34 a a b b

16. Applications of Phenyl Second moments: Ethyl benzene; P bb = 91.46 [89.2 + 2(1.6) = 92.4] Benzyl fluoride;P bb = 90.57 [89.2 + 1.6 = 90.8] (C-F bond orthogonal to phenyl) Propargyl benzene; P bb = 114.727 C 6 H 5 CH 2 C≡CH P cc = 2.196 Benzyl Alcohol P bb = 92.57 [90.8 if C-O bond orthogonal] P cc = 12.63 [1.6 if heavy-atom-planar] (C-O bond IS NEITHER orthogonal to phenyl nor heavy-atom-planar) Isobutyl benzene P bb = 143.7 [89.2 + 55 + 1.6 = 145.8] b b a a b a b c b c

17. Perfluoromethylene/methyl (-CF 2 -/-CF 3 ), 45 uÅ 2 C 3 F 8 (C 2v ) P cc = 134.34 perfluoropropane (134.34/3 = 44.78) HCF 2 CF 2 CF 3 (C s ) P cc = 135.312 1H-perfluoropropane (135.312/3 = 45.10) C 5 F 12 (C 2 ) P cc = 229.616 perfluoropentane (229.616/5 = 45.92) helical C 6 F 14 (C 2 ) P cc = 198.3355 perfluorohexane (198.3355/6 = 46.98) helical CF 3 (CF 2 ) 6 CN (C 1 ) P cc = 341.3318 (341.3318/7 = 48.76) Perfluoroheptyl cyanide (helical) c b b c

18. Acknowledgments David Lide (NBS, NIST) Joseph Fournier (UConn, Yale) Stewart Novick (Wesleyan) Pete Pringle " Herb Pickett " Steve Cooke" and SUNY at Purchase)