Second moments and rotational spectroscopy

Second moments and rotational spectroscopy

Journal of Molecular Spectroscopy 325 (2016) 42–49 Contents lists available at ScienceDirect Journal of Molecular Spectroscopy journal homepage: www...

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Journal of Molecular Spectroscopy 325 (2016) 42–49

Contents lists available at ScienceDirect

Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms

Second moments and rotational spectroscopy Robert K. Bohn a,⇑, John A. Montgomery Jr. b, H. Harvey Michels b, Joseph A. Fournier a,c a

Dept. of Chemistry, U. of Connecticut, Storrs, CT 06269-3060, USA Dept. of Physics, U. of Connecticut, Storrs, CT 06269-3046, USA c Dept. of Chemistry, James Franck Institute, U. of Chicago, Chicago, IL 60637, USA b

a r t i c l e

i n f o

Article history: Received 28 April 2016 In revised form 2 June 2016 Accepted 2 June 2016 Available online 3 June 2016 Keywords: Second moments Planar moments Microwave spectroscopy Rotational spectroscopy Zero-point molecular structures Trifluoroanisole

a b s t r a c t Although determining molecular structure using microwave spectroscopy is a mature technique, there are still simple but powerful insights to analysis of the data which are not generally appreciated. This paper summarizes three applications of second (or planar) moments which quickly and easily provide insights and conclusions about a molecule’s structure not easily obtained from the molecule’s rotational constants. If the molecule has a plane of symmetry, group second moments can verify that property and determine which groups are located on that plane. Common groups contribute predictable values to second moments. This study examines the contribution and transferability of CH2/CH3, CF2/CF3, isopropyl, and phenyl groups to molecular constants. Structures of related molecules can be critically compared using their second moments. A third application to any molecule, even those whose structures have only the identity symmetry element, determines bond lengths and angles which exactly reproduce experimentally determined 2nd moments, rotational constants, and moments of inertia. Approximate least squares methods are not needed. Ó 2016 Published by Elsevier Inc.

1. Introduction Rotational spectra are assigned and analyzed in terms of the rotational constants A, B, and C. The structural information contained in rotational constants can equally well be expressed as moments of inertia (Ia, Ib, Ic) or second moments (Paa, Pbb, Pcc, also called planar moments). The relationships among the three sets are given in Table 1. The h/8p2 conversion factor relating rotational constants (expressed in units of MHz instead of joules) and moments of inertia (units of amu-Å2 instead of kg-m2) is [A (MHz)][Ia(amu-Å2)] = 505 379.005(36) using 2006 values of the universal constants h (Planck’s constant) and NA (Avogadro’s number) [1]. The universally accepted convention for principal axis labels is that the axis with the smallest moment of inertia is the a axis and largest the c axis. It follows that A > B > C and Paa > Pbb > Pcc. We show below that second moments are often the simplest parameters to interpret and describe a molecule’s structure. The contribution of each atom in the molecule to a moment of inertia is a mass-weighted sum of the squares of two principal coordinates, i.e., the square of the radial distance of the atom from ⇑ Corresponding author. E-mail address: [email protected] (R.K. Bohn). http://dx.doi.org/10.1016/j.jms.2016.06.001 0022-2852/Ó 2016 Published by Elsevier Inc.

that axis, summed over all the atoms. Each rotational constant is defined by the reciprocal of those sums. Interpreting a molecular structure from its rotational constants is a challenge. The second moment, Paa = Rmia2i (and similarly for Pbb and Pcc), measures the extension of masses along the molecule’s a axis (or out of the bc plane). Interpreting a molecular structure in terms of second moments is simpler because each second moment is a function of only one coordinate per atom, not two, and no reciprocal is involved. Second moments are linear combinations of moments of inertia as shown in Table 1 and the information contained in the second moments or the moments of inertia or the rotational constants is equivalent. This study emphasizes the simplicity of second moments and their usefulness in understanding molecular structure. One second moment commonly considered is the inertial defect, D = Ic – Ia – Ib = 2Pcc, usually applied to examine molecular planarity in which case D is zero. If isotopolog spectra have been measured, second moments can be used to determine Kraitchman’s substitution (rs) structure [2]. Kraitchman recognized that interpreting a molecule’s structure is simpler using second moments rather than moments of inertia or rotational constants. We show below that many structural characteristics can be more easily recognized using second moments than rotational constants.

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R.K. Bohn et al. / Journal of Molecular Spectroscopy 325 (2016) 42–49 Table 1 Fundamental parameters and relationships in rotational spectroscopy. Rotational constants

Moments of inertia

Second moments

A = (h/8p Ia) B = (h/8p2Ib) C = (h/8p2Ic)

Ia = R Ib = R Ic = R

Paa = R mia2i = (Ib + Ic Pbb = R mib2i = (Ic + Ia Pcc = R mic2i = (Ia + Ib

2

mi (b2i + c2i ) mi (c2i + a2i ) mi (a2i + b2i )

= Pbb + Pcc = Pcc + Paa = Paa + Pbb

Ia)/2 Ib)/2 Ic)/2

A  Ia = 505 379.005(36)MHzamuÅ2 Inertial defect

D = (Ic

Planar molecule

Ia + Ib = Ic; Pcc and D = 0

Scaling factors

a axis; b axis; c axis;

Ia

Ib) =

2Pcc

SFa = {(Paa(exp’l)/(Paa(model)}1/2 SFb = {(Pbb(exp’l)/(Pbb(model)}1/2 SFc = {(Pcc(exp’l)/(Pcc(model)}1/2

Table 2 Second moments of CH3/CH2 groups. Pcc/amu-Å2

Refs.

CH2F2 CH2Cl2 H2FCC„CCl CH3OCFO CH3SCClO CH3CH@CH2 CH2ClCH@CH2 CH3CI@CH2 CH3CH@CF2 CH3OCH@CH2 CH3C5H4N C6H5CH2C„N C6H5CH2C„CH

1.65 1.56 1.45 1.56 1.55 1.55 1.56 1.46 1.49 1.59 1.56 2.13 2.14

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

CH3CH2CH2CH2CN HC„CCH2CH2CH2CH3 CH3CH2OCHO CH3CH2OCCNO CH3CH2OCFO CH3CH2C„CCH2CH3 CH3CH2CH2CH2CH3 CH3CH2C„CCH2CH2CH3

6.29/4 = 1.57 6.27/4 = 1.57 3.26/2 = 1.63 3.18/2 = 1.59 3.26/2 = 1.63 7.31/4 = 1.83 7.90/5 = 1.58 8.86/5 = 1.77

Compounds with single CH3/CH2 groups Methylene fluoride Methylene chloride 1-Chloro-3-fluoropropyne Methyl fluoroformate S-Methylchlorothiolformate Propene 3-Chloropropene 2-Iodopropene 1,1-Difluoropropene Methyl vinyl ether 2-Methyl pyridine Benzyl cyanide 3-Phenyl-1-propyne Compounds with multiple (n) CH3/CH2 groups Butyl cyanide (all trans form) 1-Hexyne (all trans form) Ethyl Formate (Cs form) Ethyl cyanoformate (Cs form) Ethyl fluoroformate (Cs form) 3-Hexyne (all trans form) Pentane (all trans form) 3-Heptyne (all trans form)

Pcc and Pcc/n

Table 3 Second moments of CF2/CF3 groups.

Difluoromethane Perfluoropropane 1H-heptafluoropropane 1H-nonafluorobutane Perfluoropentane Perfluorohexane

CH2F2 CF3CF2CF3 HCF2CF2CF3 HCF2CF2CF2CF3 CF3(CF2)3CF3 CF3(CF2)4CF3

2nd moment

Ref.

Pbb = 46.00/1 = 46.00 Pcc = 134.34/3 = 44.78 Pcc = 135.31/3 = 45.10 Pcc = 181.71/4 = 45.43 Pcc = 229.61/5 = 45.92 Pcc = 281.86/6 = 46.98

[4] [26] [27] [28] [26] [29]

Tables 2–6 with accompanying discussion describe the transferability and usefulness of second moments of several common chemical groups. Table 7 with discussion describes the usefulness of comparing structures of related bur different molecules. Table 8 and discussion demonstrates that experimental rotational data may include more information about molecular geometry than is usually recognized. Tables 9–11 describe scaling factors and their use to determine structures which exactly fit the experimental rotational constants. The equations in Table 1 defining moments of inertia, rotational constants, and second moments are exact for rigid molecules and nearly exact for real molecules with zero-point vibrational motions.

[17,18] [19] [20,21] [22] [23] [24] [25] [25]

There are elegant and careful studies relating zero-point effective molecular structures and equilibrium structures determined by quantum chemical computations but we are not addressing those differences [3]. Our focus is a simpler understanding of the relationship between effective rotational constants which fit observed rotational frequencies and the fundamental structural information derived from that analysis. The use of second moments to interpret and understand molecular geometry is straightforward and may reveal structural properties not easily deduced from rotational constants. An example is that a planar molecule which is not a symmetric top has only two independent rotational constants, not three, even though its three rotational constants all have different values if the structure is not a symmetric top. For molecules whose structures include at least a plane of symmetry, contributions from group second moments are transferable among molecules. We discuss transferability of second moments of CH2/CH3 groups, CF2/CF3 groups, isopropyl groups, phenyl groups, and their combinations. Second moments are useful for identifying and interpreting spectra of isotopologs and revealing which data are independent. Finally, second moments can be used to determine structures which fit rotational constants exactly using scaling factors, SFi, defined in Table 1. That application of second moments makes least squares calculations of molecular geometry unnecessary.

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R.K. Bohn et al. / Journal of Molecular Spectroscopy 325 (2016) 42–49

Table 4 Second moments of isopropyl groups.

Table 7 Structures of N-formylpyrrolidine and N-nitrosopyrrolidine.

Symmetrical isopropyl conformations 2-Fluoropropane (CH3)2CHF 2,2-Difluoropropane (CH3)2CF2 2-Cyanopropane (CH3)2CHCN 3-Methyl-1-butene (CH3)2CHCH@CH2 (shown)

p-Isopropyl benzaldehyde

(CH3)2CHC6H4CHO

Ref.

Pbb = 54.9 Paa = 55.7 Pbb = 55.2 Pbb = 54.5

[30] [31] [32] [33]

Pcc = 55.5 (syn) Pcc = 55.5 (anti)

Unsymmetrical isopropyl conformations Isopropyl fluoroformate (CH3)2CHOCFO (shown)

Isopropyl thiolfluoroformate

2nd moment

[34]

C4H8NANO Rotational constants

A/MHz B/MHz C/MHz

6097.2012(24) 1957.5059(4) 1570.7053(4)

6061.41(20) 2077.557(5) 1650.531(5)

Second moments

Second moments

Paa/amu-Å2 Pbb/amu-Å2 Pcc/amu-Å2

248.51912(3) 73.23249(5) 9.65455(8)

233.037(8) 73.156(8) 10.220(8)

[34]

Pcc = 27.1

(CH3)2CHSCFO

C4H8NACHO Rotational constants

[35]

Pcc = 26.2

[36]

2. Applications 2.1. Transferable group moments Many bond lengths and bond angles have similar values in a variety of molecules. For example, it is well known that a CAC single bond length in any molecule is 1.54 Å, decreases to 1.40 Å in an aromatic compound, to 1.35 Å in a C@C double bond, and to 1.21 Å in a C„C triple bond. Bond angles about an sp3 tetravalent

C atom are close to 110°, near 120° about an sp2 hybridized C atom, etc. The transferability of bond lengths and angles is a powerful concept since there are no restrictions, i.e., they apply to any molecule containing that parameter independent of its location in the molecule. Similarly, chemical groups in molecules, like methyl, phenyl, and isopropyl make predictable contributions to molecular second moments when located on a plane of symmetry. Consider the substituted methane H2CClF. The CClF atoms define a symmetry plane and the second moment out of that plane is solely due to the H atoms. Another example, butane, CH3CH2CH2CH3, exists in two stable conformations, trans (C2h symmetry) and gauche (C2). In the trans form, all the C atoms lie in the ab principal plane and the c coordinate of each C atom is zero. Its Pcc second moment is due solely to the out-of-plane H atoms and its information content is focused on them only. All of the three rotational constants are

Table 5 Phenyl groups (C6H5-). C6H5-groups compounds

Fluorobenzene Chlorobenzene Bromobenzene Iodobenzene Benzonitrile Phenyl acetylene 1-Chloro-4-fluoro-benzene

Rotational constants

C6H5F C6H535Cl C6H579Br C6H581Br C6H5I C6H5C„N C6H5C„CH 35 ClC6H4F 37 ClC6H4F

2nd moments

Ref.

A

B

C

Paa

Pbb

5663.72 5672.95 5667.75 5667.73 5671.89 5655.46 5680.69 5646.07 5646.22

2570.62 1576.79 994.902 984.708 750.42 1546.88 1529.746 956.738 931.979

1767.92 1233.67 846.257 838.870 662.63 1214.40 1204.959 818.118 799.947

196.61 320.54 507.00 513.26 673.50 326.75 330.41 528.23 542.26

89.25 89.11 89.20 89.20 89.14 89.40 89.01 89.51 89.50

Pcc 0.01 0.03 0.03 0.03 0.04 0.04 0.42 0.004 0.003

D 0.02 0.06 0.06 0.06 0.08 0.08 0.08 0.008 0.006

[37] [38] [39] [39] [40] [41] [42] [37] [37]

Table 6 Structures with second moment contributions from more than one group. Compound

Exp’l Pii

Predicted Pii

Ref.

4-Methyl-1-pentyne (Cs) HC„CCH2CH(CH3)2 2-Methylbutane (Cs) H3CCH2CH(CH3)2 6-Methyl-3-heptyne (Cs) H3CCH2C„CCH2CH(CH3)2 Isobutyl benzene (Cs) C6H5CH2CH(CH3)2

Pbb = 56.1

Isopropyl(55) + CH2(1.6) = 56.6

[43]

Pcc = 57.6

Isopropyl(55) + 2 CH2(3.2) = 58.2

[44]

Pcc = 60.0

Isopropyl(55) + 3(1.6) = 59.8

[45]

Pbb = 143.7

Phenyl(89.2) + isopropyl(55) + (1.6) = 145.8

[46]

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R.K. Bohn et al. / Journal of Molecular Spectroscopy 325 (2016) 42–49 Table 8 Trifluoroanisole (C6H5OCF3) Study, An example of useful 2nd moment information.

Parent C1 C2,6 13 C3,5 13 C4 13 C8 18 O 13 13

A

B

C

Paa

Pbb

Pcc

2722.215 2719.08 2699.67 2701.13 2720.96 2722.36 2697.82

702.943 702.441 701.409 696.542 692.771 700.155 700.299

632.397 632.168 630.058 626.099 624.223 630.140 631.582

666.223 666.514 667.696 672.822 676.690 669.091 667.256

132.925 132.923 134.419 134.366 132.922 132.920 132.923

52.725 52.941 52.823 52.733 52.813 52.720 54.406

Pbb(parent) – Pbb(13C) 0.002 1.494 1.441 0.003 0.005 0.002

Table 9 Spectroscopic constants of gauche 1H-heptafluoropropane (HFP).

Rotational constants A/MHz B/MHz C/MHz Second moments Paa/uÅ2 Pbb/uÅ2 Pcc/uÅ2

Exp’l

Model I (B3LYP/6-31G⁄) Raw

Model II (MP2/cc-pVTZ) Raw

Model III (BOTE) Raw

1995.4656(7) 1120.2799(4) 982.7300(5)

1969.6546 1121.4305 979.0682

1997.8935 1126.4403 987.9615

1937.0613 1142.2651 996.4879

356.0576(2) 158.2041(2) 95.0610(2)

355.0575 161.1262 95.4564

353.6163 157.9208 95.0351

Scale factors [Pii(exp’l)/Pii(model)]1/2 SFa SFb SFc Mean deviation of SFi from unity

1.001407 0.990891 0.997927 0.0042

344.34809 162.81211 98.08774

1.003446 1.000892 1.000130 0.0015

1.016860 0.985747 0.985747 0.0156

Table 10 Principal axis coordinates. Raw coordinates

Scaled coordinates

Mean deviation from unity 0.0042 a

b

Model I: B3LYP/6-31G⁄ C1 1.366527 C2 0.200998 H3 1.282415 C4 1.181100 F5 1.370918 F6 2.520758 F7 0.159659 F8 0.415652 F9 1.340390 F10 1.286611 F11 2.152094

c 0.335738 0.495059 0.457244 0.179859 1.555535 0.307206 1.664283 0.762662 0.613697 1.223560 0.695868

0.481995 0.079638 1.565406 0.056731 0.114002 0.186129 0.611724 1.387197 1.323461 0.771163 0.221970

Mean deviation from unity 0.0015 Model II: MP2/cc-pVTZ C1 1.355580 C2 0.197352 H3 1.243781 C4 1.181689 F5 1.410306 F6 2.497659 F7 0.165185 F8 0.400549 F9 1.345560 F10 1.289807 F11 2.138801

0.336250 0.482255 0.476666 0.180977 1.535106 0.331291 1.658522 0.713448 0.533315 1.262824 0.675360

0.476940 0.100930 1.549135 0.068281 0.135483 0.228888 0.557229 1.405983 1.347939 0.690461 0.264936

Mean deviation from unity 0.0156 Model III: BOTE C1 C2 H3 C4 F5 F6 F7 F8 F9 F10 F11

1.318502 0.183702 1.146009 1.181689 1.345560 1.289807 0.165185 0.400549 1.410306 2.497659 2.138801

0.337652 0.530600 0.503084 0.180977 0.533315 1.262824 1.658522 0.713448 1.535106 0.331291 0.675360

0.462347 0.112081 1.536069 0.068281 1.347939 0.690461 0.557229 1.405983 0.135483 0.228888 0.264936

SFa 1.001407

SFb 0.990891

SFc 0.997927

a

a

b

1.368451 0.201281 1.284220 1.182762 1.372848 2.524305 0.159884 0.416237 1.342276 1.288422 2.155123

0.332680 0.49055 0.453079 0.178220 1.541365 0.304408 1.649123 0.755715 0.608107 1.212415 0.689530

0.480995 0.079473 1.562161 0.056614 0.113765 0.185743 0.610456 1.384320 1.320718 0.769564 0.221510

SFa 1.003446

SFb 1.000892

SFc 1.000130

1.360251 0.198032 1.248067 1.185761 1.415166 2.506266 0.165754 0.401930 1.350197 1.294252 2.146171

0.336550 0.482685 0.477091 0.181139 1.536475 0.331586 1.660001 0.714085 0.533791 1.263950 0.675963

0.477005 0.100944 1.549346 0.068291 0.135501 0.228919 0.557305 1.406174 1.348123 0.690555 0.264972

SFa 1.016860

SFb 0.985747

SFc 0.984450

1.360251 0.186799 1.248067 1.185761 1.350197 1.294252 0.165754 0.401930 1.415166 2.506266 2.146171

0.336550 0.523037 0.477091 0.181139 0.533791 1.263950 1.660001 0.714085 1.536475 0.331586 0.675963

0.477005 0.110338 1.549346 0.068291 1.348123 0.690555 0.557305 1.406174 0.135501 0.228919 0.264972

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R.K. Bohn et al. / Journal of Molecular Spectroscopy 325 (2016) 42–49

Table 11 Selected structural parameters from the initial computed models (Raw) and the scaled parameters derived for that model of 1H-heptafluoropropane. B3LYP/6-31G⁄

C1–H3/Å C1–C2/Å C2–C4/Å C1–F5/Å C2–F7/Å C4–F9/Å \C1–C2–C4/° \H3–C1–C2/° \F5–C1–C2/° \H3–C1–C2–C4/°

MP2/cc-pVTZ

Raw

Scaled

Raw

Scaled

Raw

Scaled

1.093444 1.537567 1.544120 1.357621 1.358960 1.348404 114.2088 111.3247 109.1216 65.3724

1.091104 1.534313 1.543162 1.347101 1.349076 1.344695 114.6789 111.2503 108.8214 65.1591

1.087114 1.531462 1.539566 1.347335 1.348263 1.337356 113.9363 111.3097 109.5583 61.7066

1.087314 1.534904 1.544082 1.348337 1.349229 1.337675 114.0703 111.2364 109.5593 61.6702

1.10 1.54 1.54 1.35 1.35 1.35 109.47 109.47 109.47 60.00

1.088382 1.543979 1.554824 1.330381 1.330380 1.362995 111.0872 108.6422 111.0872 58.4869

functions of all the atoms and do not reveal structural information so directly. In the gauche form, none of the C atoms lies in a principal axis plane and no generalizations of the second moments of these groups can be made. 2.1.1. CH2/CH3 groups Consider propane, CH3CH2CH3, shown in Fig. 1. The molecule has C2v symmetry and the local symmetry about each C atom is very close to C3v for each terminal CH3 group and C2v for the central CH2 group. The principal axes are shown as dashed lines and the c axis is perpendicular to the drawing plane. The ab plane is the plane of the figure and only a pair of H atoms on each C lies outside the ab plane and contribute to Pcc. Since the moment of inertia of a 3-fold symmetric group such as a methyl group does not vary with orientation, place one of the H atoms of each methyl group in the ab plane so that it has a zero value for its c coordinate and contributes nothing to Pcc. CH3 and CH2 groups make equal contributions to Pcc. Furthermore, the Pcc second moment changes very little if a different atom is substituted for the in-plane H. The change of the CH2 second moment caused by other atoms nearby is small as shown in Table 2 which contains a few examples. For small, rigid molecules, the contribution of each CH2/CH3 group is nearly constant, about 1.6 amu-Å2. Benzyl cyanide and 3-phenyl1-propyne listed below have slightly larger second moments due to large amplitude torsional motions. The lower part of Table 2 shows compounds with multiple CH3/CH2 groups, the molecule’s Pcc, and Pcc divided by n, the number of CH3/CH2 groups, to demonstrate the reproducibility of the group moment of a single group. 2.1.2. CF2/CF3 groups CF2/CF3 groups show similar consistent contributions to the second moments if the central C atom lies in a plane of symmetry. Table 3 shows values near 45 amu-Å2. An interesting trend is revealed in the fluorocarbons having chains longer than three C atoms. All-trans hydrocarbon analogs have dihedral angles along the carbon chains of 180°. The C atoms all lie in the ab plane and

b H

H

H C1

a

BOTE

H

C3 C2

H H

H H

c Fig. 1. Propane.

have zero values for their c coordinates. In fluorocarbons, F atoms contribute to Pcc and fluorocarbons with chains of four C atoms and longer are helical and the dihedral angles of the carbon chains are about 163° instead of 180°. As the chain lengthens, the contribution to Pcc is not just from the F atoms but also the C atoms since they no longer lie in the ab plane and make contributions to Pcc increasing its value. 2.1.3. Isopropyl groups The second moment contribution from the methyls of an isopropyl group also show consistent values when the central C atom lies in a plane of symmetry. Some examples are shown in Table 4. In the cases of CH2/CH3 group moments discussed above, the group moment contributions are always to Pcc and neither Paa nor Pbb because of the small mass of the H atom. As Table 4 shows, with more massive isopropyl groups, other principal axis planes may be relevant. The bottom two entries in Table 4 are interesting because their 2nd moments rigorously demonstrate that the isopropyl orientation is not symmetrical. The second moment value shows that one of the methyl groups is in the approximate plane of the other heavy atoms in the molecule. A sketch of 3-methyl1-butene shows a symmetrical isopropyl and another of isopropyl fluoroformate shows an unsymmetrical isopropyl conformation. For the two unsymmetrical examples, note that the second moments are roughly half that of the symmetrical forms, given that only one of the methyls is out of the plane. 2.1.4. Phenyl groups Table 5 displays rotational constants and second moments of a number of benzene derivatives. The phenyl group moments, Pbb in the compounds in Table 5, have remarkably constant values. The planarity is demonstrated by the Pcc values. The conventional inertial defect is also shown. The second moments present a striking impression of the planarity, rigidity, and structural similarity of phenyl compounds. Those features do not appear if only rotational constants are reported. Consider the Pcc values of 1-chloro-4-fluorobenzene [37]. For the purpose of defining the molecular structure, a first glance at the rotational constants of the two Cl isotopologs suggests that there are 6 independent rotational constants, three from each isotopolog, which can be used to determine geometry of the compound. A closer look reveals that the A rotational constants are not independent but equal because the Cl atom lies on the a axis and contributes nothing to Ia. That leaves 5 independent rotational constants. A second look at the second moments, however, reveals that a planar molecule has only 2, not 3, independent structural parameters. So we are down to 4 independent constants for the two isotopologs. But both isotopologs have the same Pbb value (the Cl atom lies on the a axis and has zero values for its b and c coordinates.) So the rotational constants of the two isotopologs

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of 1-chloro-4-fluorobenzene reveal only three independent structural parameters, not six. 2.1.5. Combinations of group moments The examples described above each have only one predictable second moment. But there are many molecules whose second moments can include contributions from more than one group. A few examples are listed in Table 6. These compounds have multiple groups with transferable second moments. They have more than one conformer as well. Only the conformers with a plane of symmetry are included. 2.2. Comparing structures of related molecules If one wishes to compare structures of related compounds, second moments again have an advantage over rotational constants. Consider the molecular structures of the carcinogen, Nnitrosopyrrolidine, and its nontoxic isoelectronic analog, Nformylpyrrolidine [47]. The molecules are not planar but wrinkled about the ab plane. Results are given in Table 7. Comparing rotational constants of the two compounds suggests that the structures are similar but no specific conclusions can be drawn. Comparing second moments, the Paa values indicate that the nitroso compound is slightly shorter than its formyl analog. And in the directions of the b and c principal axes, the two compounds are essentially indistinguishable, a conclusion not reached if 2nd moments are ignored and only rotational constants considered. 2.2.1. Isotopologs A recent study of C6H5OCF3 [48], trifluoroanisole, was carried out to compare it to C6H5OCH3 [49], anisole, which has coplanar heavy atoms. The authors clearly show that the trifluoro compound has its CAOAC plane orthogonal to the phenyl ring unlike the hydrogen analog. See Table 8. The rotational spectra of the parent compound and all its 13C and 18O isotopologs were assigned and analyzed. Table 8 includes the rotational constants reported [48] and we added the 2nd moments to the table. The Pbb values show that the parent, 13C1, 13C4, 13C8 and 18O isotopologs have the same Pbb value, requiring that these atoms lie in the ac plane. Furthermore, the Pbb value, 132.9 amu-Å2, is expected from the sum of 89.2 (phenyl) and 44.8 for the CF3 group, 134 amu-Å2. If the CAOAC lay coplanar with the phenyl ring, Pcc would be about 45, the 2nd moment of a CF3 group, not 132.9. The fact that the 13 C2,6 and 13C3,5 atoms have equal 2nd moments is consistent with an orthogonal rather than planar orientation of the CAOAC and phenyl planes. If the rs substitution structure is computed by Kraitchman’s method, a factor in the calculation of the square of the coordinate of the substituted atom is the parent’s 2nd moment minus that of the heavier isotopolog. From Table 8, it is immediately clear that the b coordinates of the 4 coplanar atoms will be imaginary since the value of Pbb diminished with substitution by a heavier 13C mass, either due to experimental error or zeropoint vibration effects. There is further information about the structure of trifluoroanisole that can be seen in the second moment values but not from the rotational constants. The Pbb values of the 13C2,6 and 13 C3,5 isotopologs are 134.419 amu-Å2 and 134.366 amu-Å2, respectively, and differ by 0.053. The other Pbb values cover a range of 0.005 which is an estimate of the experimental accuracy of the Pbb values. Thus the 0.053 difference is about 10 times the experimental uncertainty and is a measure of the distortion of the benzene ring by the AOCF3 substituent. Reporting only the rotational constants completely misses that benzene distortion effect. From the differences in the 2nd moments, the b coordinate value for 13 C2,6 is 1.222 Å, 0.022 Å greater than the b coordinate of 13C3,5, 1.200 Å.

A recent study of complexes of perfluoropropionic acid with one and two water molecules [51] utilizes transferable second moments of CF2 groups and water molecules to cleverly interpret the structures of these rather complex species. 2.3. Scaling second moments to produce exact zero-point structures Second moments can convert a predicted geometrical model to one which reproduces the observed rotational constants. We introduced this method in our study of the molecular structure of perfluoropentane, C5F12 [26]. To fully demonstrate this method, we choose a smaller molecule, 1H-heptafluoropropane (HFP) in its gauche conformation [27] (see Fig. 2). Generate a model for the molecule and calculate the Cartesian coordinates of all the atoms in its principal axis system, i.e., a coordinate system which diagonalizes the inertial tensor. Multiply each atom’s a coordinate by the square root of the ratio of (Paa(exp’l)/Paa(model)). We call that square root value the scaling factor, SFa, for the a axis. See Table 1. Repeat for the b and c axes and coordinates. The scaling factors for each principal axis direction are usually different from each other. The scaled Cartesian coordinates remain a principal axis system and exactly reproduce the experimental rotational constants. That scaled set of Cartesian coordinates defines a structure which is not unique, however. If a different initial model is selected, a new set of scaling factors can be determined and its scaled coordinates also constitute a structure which fits the observed rotational constants. Three model structures of HFP were selected to be scaled to structures exactly fitting the experimental rotational constants. Model I was calculated using B3LYP/6-31G⁄ density functionals [50] with a modest basis set. Model II was calculated from ab initio molecular orbitals at the MP2/cc-pVTZ level [50]. Model III was generated from an extremely simple ‘‘Back-of-the-Envelope” (BOTE) model with all CAC distances set to 1.54 Å, CAF bond lengths to 1.35 Å, all bond angles 109.47°, and all dihedral angles 180° or ±60°. Since the experimental rotational constants are accurate up to 7 significant figures, it is necessary to retain the same

Fig. 2. 1H-heptafluoropropane (gauche conformation) with atom labels.

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number of digits in the scaling factors and the scaled coordinates to avoid truncation and round-off errors. All three models, after scaling, exactly reproduce the experimental rotational constants and second moments of gauche HFP. Table 9 displays the molecular constants of the three models. The columns labeled Raw are the values taken directly from the model calculations. After scaling, all three models reproduce the experimental values. Table 10 lists the principal axis coordinates calculated for each model (Raw) and then scaled to values which precisely reproduce the experimental molecular constants. Table 11 gives the values of some of the structural parameters derived from the three different models before and after scaling. Each scaled set describes a structure which accurately reproduces the experimental constants, but a unique ground state structure is not derived. All three scaled structures reproduce the experimental rotational constants and 2nd moments and are accurate zero-point structures. The bond lengths in the models here differ by a few thousandths Å. Bond angles range over tenths of a degree. The values for \H3–C1–C2–C4/° range from 58.5° to 65.4°. The scaling method described above can generate any number of molecular geometries which precisely fit the molecule’s rotational constants. How does one select the best molecular structure? We suggest two methods but neither is perfect. One is to examine the scale factors from several models and select the model for which they are closest to unity. These mean deviations from unity are listed in the bottom line of Table 9 and are reasonable, closest to unity from the MP2 Model II and farthest from unity for the crude BOTE model. In our experience, a theoretical model which predicts rotational constants very close to the experimental values is a good choice. But there may be another model not considered which predicts the experimental parameter values with scaling factors even closer to unity. A second test of a molecule’s geometry is to predict the rotational constants for an isotopolog. A good model should predict the isotopolog’s rotational constants more accurately than a poor model. But the molecular structure of an isotopolog should not be identical to that of the parent because substituting an atom with a different mass means the isotopolog is sampling a different portion of the molecule’s potential energy surface and may introduce small changes in the directions of the principal axes. Kraitchman’s rs structure method ignores these effects [1]. Recent studies on perfluoropropionic acid hydrates [51] and allyl phenyl ether [52] cleverly use second moments to interpret and understand their structures. 3. Conclusions Second moments derived from rotational spectra are useful for interpreting structures of molecules. Group moments of CH3/CH2, CF3/CF2, isopropyl, phenyl groups, and others are transferable among molecules if the groups lie on a symmetry plane. Group moments are useful alone or combined to interpret molecular structures. They can reveal structural characteristics unambiguously. Scaling principal Cartesian coordinates of a model structure with second moment scaling factors produces a model which accurately reproduces the ground state rotational constants. Not answered by this study is the troubling conclusion that there is not a unique zero-point molecular structure derivable from rotational spectra. Acknowledgments The authors thank the organizers of several conferences for opportunities to discuss second moments with many interested microwave spectroscopists. We thank the Austin Symposium on

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