The pure rotational spectrum of pyrimidine (*m*-C_{4}H_{4}N_{2}), the *meta*-substituted dinitrogen analog of benzene, has been studied in the millimeter-wave region from 235 GHz to 360 GHz. The rotational spectrum of the ground vibrational state has been assigned and fit to yield accurate rotational and distortion constants. Over 1700 distinct transitions were identified for the normal isotopologue in its ground vibrational state and least-squares fit to a partial sextic S-reduced Hamiltonian. Transitions for all four singly substituted ^{13}C and ^{15}N isotopologues were observed at natural abundance and were likewise fit. Deuterium-enriched samples of pyrimidine were synthesized, giving access to all eleven possible deuterium-substituted isotopologues, ten of which were previously unreported. Experimental values of rotational constants and computed values of vibration–rotation interaction constants and electron-mass corrections were used to determine semi-experimental equilibrium structures (*r*_{e}^{SE}) of pyrimidine. The *r*_{e}^{SE} structure obtained using coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] corrections shows exceptional agreement with the *r*_{e} structure computed at the CCSD(T)/cc-pCV5Z level (≤0.0002 Å in bond distance and ≤0.03° in bond angle). Of the various computational methods examined, CCSD(T)/cc-pCV5Z is the only method for which the computed value of each geometric parameter lies within the statistical experimental uncertainty (2σ) of the corresponding semi-experimental coordinate. The exceptionally high accuracy and precision of the structure determination is a consequence of the large number of isotopologues measured, the precision and extent of the experimental frequency measurements, and the sophisticated theoretical treatment of the effects of vibration–rotation coupling and electron mass. Taken together, these demanding experimental and computational studies establish the capabilities of modern structural analysis for a prototypical monocyclic aromatic compound.

## INTRODUCTION

Our interest in rotational spectroscopy of polar aromatic compounds derives from this subject’s relevance to astrochemistry and radioastronomy,^{1–3} its wide-ranging usefulness in exploring intramolecular potential functions and dynamics,^{4–6} and its ability, especially in conjunction with modern computational methods, to determine the molecular structure with a high precision.^{7} In the current investigation, we measure rotational constants and centrifugal distortion terms for sixteen isotopologues of pyrimidine (Fig. 1) and report a semi-experimental equilibrium structure (*r*_{e}^{SE}) for this molecule using state-of-the-art methods.

Astronomical detection of new molecules relies heavily on laboratory rotational spectra of potential targets.^{8} Pyrimidine is a molecule of particular interest because it may serve as an additional polar tracer for aromatic species. The recent detection of benzonitrile in Taurus Molecular Cloud 1 (TMC-1) represents the first identification of a simple benzene derivative in interstellar space by radioastronomy.^{9} Using IR spectroscopy, benzene has been detected in protoplanetary nebula CRL 618,^{10} and C_{60} and C_{70} have been identified by Cami *et al.* in protoplanetary nebula Tc 1.^{11} The assignment of two diffuse interstellar bands (DIBs) has been confirmed to arise from absorption bands of C_{60}^{+} by Campbell *et al.*^{12} Polycyclic aromatic hydrocarbons (PAHs) and nitrogen-containing analogs (PANHs) are thought to be responsible for unidentified infrared bands (UIBs) and/or diffuse interstellar bands (DIBs).^{13–15} Observing rotational transitions from a polar aromatic heterocycle, such as pyrimidine, would be an important step toward understanding the potentially rich chemistry of aromatic molecules in space.

Pyrimidine has an appreciable dipole moment (*μ*_{b} = 2.334 D)^{16} and is perhaps the most thermodynamically stable of three dinitrogen-containing analogs of benzene (Fig. 2).^{17} It is a viable candidate for radioastronomical detection and arguably the most likely member of the *c*-C_{4}H_{4}N_{2} family to be detectable in the interstellar medium (ISM). Photochemical conversion of pyrazine to pyrimidine has been observed, but not the reverse transformation.^{18} Although susceptible to photodestruction by UV radiation, Peeters *et al.*^{19} predicted pyrimidine to have a lifetime on the order of 10–100 years in the interstellar medium and upward of 10^{6} years in dense molecular clouds. To date, six pyrimidine rotational transitions between 329 GHz and 363 GHz have been searched for, unsuccessfully, in the star-forming regions Sgr B2(N), Orion KL, and W51 e1/e2.^{20,21} In light of these considerations, new astronomical searches in a dense molecular cloud would seem to be warranted. In addition to serving as a potential tracer molecule for aromatic species in space, pyrimidine may also hold relevance for prebiotic chemistry. Pyrimidine is the core structural unit in the nucleobases cytosine, uracil, and thymine, and it is possible that some fraction of these prebiotics were first formed extra-terrestrially before their appearance on the Earth.^{22} Experiments by Nuevo *et al.*^{23,24} demonstrated that UV irradiation of pyrimidine, frozen in a variety of astronomically relevant ices, readily forms nucleobases, including uracil and cytosine.

Pyrimidine was first studied using rotational spectroscopy in 1959 by Schneider,^{25} who determined the rotational constants of the normal isotopologue and a trideuterio-substituted isotopologue, leading to the estimation of three structural parameters. Later investigation by Blackman *et al.*^{16} in 1970 improved the determination of the rotational constants of the normal isotopologue and included an analysis of the hyperfine splitting. In 1991, Caminati and Damiani^{26} fit the spectra for the [4-^{13}C] and [1-^{15}N] isotopologues to a quartic centrifugally distorted-rotor Hamiltonian. In 1999, Kisiel *et al.*^{27} determined the rotational constants of the normal isotopologue and all four singly substituted ^{13}C and ^{15}N isotopologues, resulting in a substitution structure (*r*_{s}) that determined the positions of the six heavy atoms in the ring using Kraitchman’s equations^{28} for single isotopic substitution.

In the present work, the rotational spectra of pyrimidine and fifteen of its ^{13}C, ^{15}N, and ^{2}H isotopologues were studied from 235 GHz to 360 GHz. The normal isotopologue and three ^{13}C and one ^{15}N singly substituted isotopologues were all observable at natural abundance. A simple base-catalyzed deuterium enrichment of pyrimidine gave access to all eleven of the possible ^{2}H-substituted isotopologues. In total, the transitions of sixteen isotopologues were observed and least-squares fit to a Hamiltonian, including quartic or sextic centrifugal distortion constants. The rotational constants of all isotopologues were corrected for the effects of vibration–rotation coupling and contributions of electron mass and then used to determine the equilibrium structure of pyrimidine (*r*_{e}^{SE}). This structure is discussed in the context of recent computational and semi-experimental equilibrium structures (*r*_{e}^{SE}) of pyrimidine and other heterocyclic aromatic compounds.^{29–31}

## EXPERIMENTAL METHODS

A millimeter-wave spectrometer described previously was used to collect broadband spectra from 235 GHz to 360 GHz.^{32,33} All lines reported are estimated to have an uncertainty of ±50 kHz. A sample of pyrimidine, purchased from Sigma-Aldrich, was used without further purification (≥98.5% purity), and the spectra were collected at pressures of 15–30 mTorr. Detailed scans of smaller regions were obtained at 18 mTorr. The data were analyzed using the Assignment and Analysis of Broadband Spectra (AABS) program suite.^{34–36} Deuterium-enriched samples of pyrimidine were synthesized using the method previously employed to generate mixtures of [^{2}H]-pyridazines using *t*-BuOD/*t*-BuOLi.^{33} *Caution: this method involves heating a volatile liquid in a sealed container, which creates high pressure.* Pyrimidine samples with varying degrees of deuterium incorporation were obtained using reaction times of 1 h, 2.5 days, and 25 days. In comparison to pyridazine,^{33} deuteration of pyrimidine proved to be more difficult, requiring a greater quantity of *t*-BuOD and longer reaction times. This method is advantageous because it generates a sample containing varying amounts of mono-, di-, tri-, and tetradeuterio species, allowing for the study of all eleven of the possible deuterium-containing isotopologues. The rotational spectra were used to identify and characterize the deuterio-isotopologues, of which only [2,4,6-^{2}H]-pyrimidine had been studied previously.^{25}

## COMPUTATIONAL METHODS

Electronic structure calculations using the coupled-cluster theory were carried out with a development version of the CFOUR program.^{37} Optimized geometries were determined using analytic gradients.^{38,39} The frozen-core approximation was implemented in coupled-cluster single double triple [CCSD(T)]^{40} calculations with the ANO1, ANO2,^{41} and cc-pVTZ^{42} basis sets, while calculations with the cc-pCVXZ (X = T, Q, 5) basis sets^{42,43} included correlation of all electrons. Anharmonic vibrational frequencies and vibration–rotation interaction constants (*α*_{i}) were determined at CCSD(T)/ANO1 by calculating cubic force constants using second derivatives at displaced points.^{44–46} The electron-mass contributions to the observable rotational constants^{47} were determined at CCSD(T)/ANO1 via a rotational *g*-tensor calculation for each isotopologue. Density functional theory calculations were performed with Gaussian 16^{48} using the WebMO^{49} interface. Optimized geometries, anharmonic vibrational frequencies, vibration–rotation interaction constants, and electron-mass contributions were determined at B3LYP/6-311+G(2d,p)^{50–52} for each isotopologue. Summaries of electronic structure calculations and analyses can be found in the supplementary material.

## STRUCTURE DETERMINATION

The effective ground state (*r*_{0}) and semi-experimental equilibrium (*r*_{e}^{SE}) structures were determined by least-squares fitting in the xrefit module of CFOUR.^{37} In this work, all three moments of inertia for all sixteen isotopologues were used with equal weighting. Although all three moments are not independent for a planar molecule (inertial defect, *Δ*_{i}, precisely equal to zero), the best way to analyze the data, statistically, is include all three moments in the structure determination. To obtain the *r*_{0} structure, the ground-state rotational constants (S-reduced Hamiltonian, III^{r} representation) of all isotopologues were used without further corrections. Semi-experimental equilibrium structure determinations for the ground-state structure of pyrimidine are drawn from a combination of experimental and computational data. Structures were obtained from a fit to rotational constants and corrected to equilibrium values by removing from the experimental constants the effects of vibration–rotation interaction and electron-mass contributions. These corrections were calculated for all isotopologues using the procedure discussed previously.^{45} The computed vibrational corrections, which are equal in magnitude to one-half the sum of the vibration–rotation interaction constants (*α*_{i}),

are completely free of effects of the Coriolis resonance. This is important as this would not be true even in the improbable event that all of these corrections could be measured experimentally. The inclusion of the vibrational corrections has the effect of reducing the ground vibrational state inertial defects of the isotopologues to values that are close to that for a rigid planar molecule (*Δ*_{i} = 0) (see Table S2 in the supplementary material), which would be satisfied by the true equilibrium constants.

An additional contribution to the rotational constants is associated with the breakdown of the assumption that electron masses can be subsumed into the nuclear position. This oft-neglected contribution is particularly important for molecules with conjugated π-electron systems due to their significant out-of-plane electron distribution.^{29,30,33,53} The electron-mass correction may be derived from the diagonal elements of the electronic contribution to the rotational *g*-tensor, *g*^{bb}, as shown in Eq. (2), where *η* is the electron–proton mass ratio,

Inclusion of this correction further reduces the magnitude of the associated inertial defects, which become closer to zero (Table S2).

## ANALYSIS OF ROTATIONAL SPECTRA OF PYRIMIDINE AND ITS ISOTOPOLOGUES IN THEIR GROUND VIBRATIONAL STATES

Pyrimidine exhibits a rotational spectrum characteristic of a near-oblate, asymmetric top (*κ* = 0.869). As its dipole moment is coincident with the *b*-inertial axis connecting C(2) and C(5), the normal isotopologue of pyrimidine displays solely *b*-type transitions. The spectrum, a segment of which is shown in Fig. 3, contains easily recognizable bandhead structures, consisting of intense R-branch (^{b}R_{1,1} and ^{b}R_{−1,1}) transitions with associated, but significantly less intense, Q-branch (^{b}Q_{1,−1}) transitions, which repeat at intervals of approximately 2*C* (∼6168 MHz). The transitions in our frequency range lack observable hyperfine splitting due to coupling with the nitrogen nuclei, although the quadrupole coupling constants are well known from previous microwave work.^{16,27} The R-branch transitions initially appear as pairs of ^{b}R_{1,1} and ^{b}R_{−1,1} transitions (degenerate in *K*_{c}) at a low frequency and lose degeneracy as *K*_{a} increases. Similarly, the Q-branch series appears as degenerate pairs of ^{b}Q_{1,−1} transitions at lower frequency than the corresponding R-branch series. The R-branch transitions observed in this work range from *J′* = 16 to *J′* = 54, and the Q-branch transitions range from *J′* = 40 to *J′* = 100. A summary of the types of transitions included in the least-squares fits for each isotopologue can be found in Table I. For the normal isotopologue in the ground vibrational state, a data distribution plot of all measured transitions included in the least-squares fit is presented in Fig. 4.

Type . | C_{4}H_{4}N_{2}
. | [2-^{13}C]
. | [4-^{13}C]
. | [5-^{13}C]
. | [1-^{15}N]
. | [2-^{2}H]
. | [4-^{2}H]
. | [5-^{2}H]
. | [2,4-^{2}H]
. | [2,5-^{2}H]
. | [4,5-^{2}H]
. | [4,6-^{2}H]
. | [2,4,5-^{2}H]
. | [2,4,6-^{2}H]
. | [4,5,6-^{2}H]
. | [2,4,5,6-^{2}H]
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{a}Q_{0,−1} | 0 | 0 | 222 | 0 | 25 | 0 | 289 | 171 | 0 | 0 | 202 | 0 | 0 | 0 | 0 | 0 |

^{b}Q_{1,−1} | 1980 | 77 | 446 | 28 | 71 | 0 | 578 | 0 | 0 | 0 | 404 | 0 | 0 | 0 | 105 | 0 |

^{a}Q_{2,−1} | 0 | 0 | 222 | 0 | 25 | 0 | 289 | 171 | 0 | 0 | 202 | 0 | 0 | 0 | 0 | 0 |

^{a}R_{0,1} | 0 | 0 | 508 | 0 | 396 | 260 | 684 | 575 | 426 | 463 | 656 | 0 | 582 | 0 | 0 | 488 |

^{b}R_{1,1} | 416 | 193 | 328 | 171 | 255 | 0 | 407 | 0 | 198 | 0 | 336 | 357 | 255 | 182 | 280 | 0 |

^{b}R_{1,−1} | 131 | 35 | 64 | 1 | 28 | 0 | 79 | 0 | 0 | 0 | 51 | 51 | 0 | 17 | 34 | 0 |

^{b}R_{−1,1} | 319 | 180 | 292 | 155 | 229 | 0 | 355 | 0 | 198 | 0 | 295 | 291 | 257 | 158 | 245 | 0 |

^{a}R_{2,−1} | 0 | 0 | 22 | 0 | 0 | 0 | 20 | 65 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |

^{b}R_{3,−1} | 67 | 9 | 30 | 7 | 0 | 0 | 23 | 0 | 0 | 0 | 10 | 0 | 0 | 0 | 3 | 0 |

^{b}R_{−1,3} | 20 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Total independent transitions | 1741 | 310 | 706 | 249 | 364 | 141 | 904 | 590 | 228 | 247 | 807 | 423 | 349 | 210 | 404 | 289 |

Type . | C_{4}H_{4}N_{2}
. | [2-^{13}C]
. | [4-^{13}C]
. | [5-^{13}C]
. | [1-^{15}N]
. | [2-^{2}H]
. | [4-^{2}H]
. | [5-^{2}H]
. | [2,4-^{2}H]
. | [2,5-^{2}H]
. | [4,5-^{2}H]
. | [4,6-^{2}H]
. | [2,4,5-^{2}H]
. | [2,4,6-^{2}H]
. | [4,5,6-^{2}H]
. | [2,4,5,6-^{2}H]
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{a}Q_{0,−1} | 0 | 0 | 222 | 0 | 25 | 0 | 289 | 171 | 0 | 0 | 202 | 0 | 0 | 0 | 0 | 0 |

^{b}Q_{1,−1} | 1980 | 77 | 446 | 28 | 71 | 0 | 578 | 0 | 0 | 0 | 404 | 0 | 0 | 0 | 105 | 0 |

^{a}Q_{2,−1} | 0 | 0 | 222 | 0 | 25 | 0 | 289 | 171 | 0 | 0 | 202 | 0 | 0 | 0 | 0 | 0 |

^{a}R_{0,1} | 0 | 0 | 508 | 0 | 396 | 260 | 684 | 575 | 426 | 463 | 656 | 0 | 582 | 0 | 0 | 488 |

^{b}R_{1,1} | 416 | 193 | 328 | 171 | 255 | 0 | 407 | 0 | 198 | 0 | 336 | 357 | 255 | 182 | 280 | 0 |

^{b}R_{1,−1} | 131 | 35 | 64 | 1 | 28 | 0 | 79 | 0 | 0 | 0 | 51 | 51 | 0 | 17 | 34 | 0 |

^{b}R_{−1,1} | 319 | 180 | 292 | 155 | 229 | 0 | 355 | 0 | 198 | 0 | 295 | 291 | 257 | 158 | 245 | 0 |

^{a}R_{2,−1} | 0 | 0 | 22 | 0 | 0 | 0 | 20 | 65 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |

^{b}R_{3,−1} | 67 | 9 | 30 | 7 | 0 | 0 | 23 | 0 | 0 | 0 | 10 | 0 | 0 | 0 | 3 | 0 |

^{b}R_{−1,3} | 20 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Total independent transitions | 1741 | 310 | 706 | 249 | 364 | 141 | 904 | 590 | 228 | 247 | 807 | 423 | 349 | 210 | 404 | 289 |

^{a}

Each transition in a set of degenerate transitions is counted individually for each transition type.

Assignment of the rotational spectrum of pyrimidine was initially based on published rotational as well as quartic and sextic distortion constants.^{27} The range of our spectrometer does not allow for measurement of transitions with low-*J* values for pyrimidine, so it was crucial to include many non-degenerate high-*J* transitions to achieve an independent determination of *A*_{0} and *B*_{0}. Using the five types of R-branch transitions (^{b}R_{1,1}, ^{b}R_{−1,1}, ^{b}R_{1,−1}, ^{b}R_{−1,3}, and ^{b}R_{3,−1}) and the Q-branch transitions, it was possible to fit the spectral data of the normal isotopologue to a partial-sextic Hamiltonian with all included terms well determined. The final least-squares fit contained over 1700 individual transitions, including transitions from previous works;^{26,27} the resulting constants (S-reduction, III^{r} representation) with their uncertainties are provided in Table II. The output file of the fit, as well as all other output files, can be found in the supplementary material. Additional least-squares fits, including only transitions from this work, are also provided. Rotational transitions from nine vibrationally excited states were also observed and will be described, in detail, elsewhere.^{54} (Features from excited states ν_{16} and ν_{11} are visible in Fig. 3.)

. | m-C_{4}H_{4}N_{2}
. | [2-^{13}C]
. | [4-^{13}C]
. | [5-^{13}C]
. | [1-^{15}N]
. |
---|---|---|---|---|---|

A_{0} (MHz) | 6276.827 750 (79) | 6152.678 55 (26) | 6256.097 20 (19) | 6132.817 27 (29) | 6253.958 22 (38) |

B_{0} (MHz) | 6067.165 761 (78) | 6067.539 26 (21) | 5957.218 66 (19) | 6067.359 63 (25) | 5954.152 35 (38) |

C_{0} (MHz) | 3084.449 22 (10) | 3054.251 59 (18) | 3050.845 69 (23) | 3049.30 507 (19) | 3049.530 46 (51) |

D_{J} (kHz) | 1.472 131 (62) | 1.443 354 (79) | 1.442 40 (11) | 1.442 503 (94) | 1.443 11 (31) |

D_{JK} (kHz) | −2.452 567 (30) | −2.403 99 (16) | −2.401 16 (17) | −2.401 83 (26) | −2.406 89 (44) |

D_{K} (kHz) | 1.104 972 (39) | 1.082 47 (10) | 1.081 28 (13) | 1.081 54 (18) | 1.085 02 (29) |

d_{1} (kHz) | 0.011 730 (16) | 0.020 83 (13) | 0.001 600 (80) | 0.022 13 (15) | 0.001 94 (11) |

d_{2} (kHz) | 0.021 904 2 (89) | 0.018 247 8 (39) | 0.013 929 (33) | 0.018 260 (40) | 0.013 830 (33) |

H_{J} (Hz) | 0.000 661 (14) | [0] | 0.000 775 (21) | [0] | 0.000 486 (65) |

H_{JK} (Hz) | −0.002 710 8 (27) | [0] | −0.002 788 (27) | [0] | −0.002 788 (99) |

H_{KJ} (Hz) | 0.003 333 4 (60) | [0] | 0.003 698 (37) | [0] | 0.002 15 (10) |

H_{K} (Hz) | −0.001 319 3 (81) | [0] | −0.001 629 (28) | [0] | [0] |

Δ_{i} (u Å^{2})^{a} | 0.034 992 7 (56) | 0.035 454 (11) | 0.035 547 (13) | 0.035 395 (12) | 0.035 671 (29) |

κ^{b} | 0.868 648 | 0.945 04 | 0.813 51 | 0.957 54 | 0.812 88 |

N_{lines} | 1741 | 310 | 706 | 249 | 364 |

σ_{fit} | 0.030 | 0.039 | 0.035 | 0.041 | 0.039 |

μ_{a} (D)^{c} | 0 | 0 | 0.424 | 0 | 0.451 |

μ_{b} (D)^{c} | 2.334 | 2.334 | 2.295 | 2.334 | 2.29 |

. | m-C_{4}H_{4}N_{2}
. | [2-^{13}C]
. | [4-^{13}C]
. | [5-^{13}C]
. | [1-^{15}N]
. |
---|---|---|---|---|---|

A_{0} (MHz) | 6276.827 750 (79) | 6152.678 55 (26) | 6256.097 20 (19) | 6132.817 27 (29) | 6253.958 22 (38) |

B_{0} (MHz) | 6067.165 761 (78) | 6067.539 26 (21) | 5957.218 66 (19) | 6067.359 63 (25) | 5954.152 35 (38) |

C_{0} (MHz) | 3084.449 22 (10) | 3054.251 59 (18) | 3050.845 69 (23) | 3049.30 507 (19) | 3049.530 46 (51) |

D_{J} (kHz) | 1.472 131 (62) | 1.443 354 (79) | 1.442 40 (11) | 1.442 503 (94) | 1.443 11 (31) |

D_{JK} (kHz) | −2.452 567 (30) | −2.403 99 (16) | −2.401 16 (17) | −2.401 83 (26) | −2.406 89 (44) |

D_{K} (kHz) | 1.104 972 (39) | 1.082 47 (10) | 1.081 28 (13) | 1.081 54 (18) | 1.085 02 (29) |

d_{1} (kHz) | 0.011 730 (16) | 0.020 83 (13) | 0.001 600 (80) | 0.022 13 (15) | 0.001 94 (11) |

d_{2} (kHz) | 0.021 904 2 (89) | 0.018 247 8 (39) | 0.013 929 (33) | 0.018 260 (40) | 0.013 830 (33) |

H_{J} (Hz) | 0.000 661 (14) | [0] | 0.000 775 (21) | [0] | 0.000 486 (65) |

H_{JK} (Hz) | −0.002 710 8 (27) | [0] | −0.002 788 (27) | [0] | −0.002 788 (99) |

H_{KJ} (Hz) | 0.003 333 4 (60) | [0] | 0.003 698 (37) | [0] | 0.002 15 (10) |

H_{K} (Hz) | −0.001 319 3 (81) | [0] | −0.001 629 (28) | [0] | [0] |

Δ_{i} (u Å^{2})^{a} | 0.034 992 7 (56) | 0.035 454 (11) | 0.035 547 (13) | 0.035 395 (12) | 0.035 671 (29) |

κ^{b} | 0.868 648 | 0.945 04 | 0.813 51 | 0.957 54 | 0.812 88 |

N_{lines} | 1741 | 310 | 706 | 249 | 364 |

σ_{fit} | 0.030 | 0.039 | 0.035 | 0.041 | 0.039 |

μ_{a} (D)^{c} | 0 | 0 | 0.424 | 0 | 0.451 |

μ_{b} (D)^{c} | 2.334 | 2.334 | 2.295 | 2.334 | 2.29 |

^{a}

*Δ*_{i} = *I*_{c} − *I*_{b} − *I*_{a}.

^{b}

*κ* = (2*B* − *A* − *C*)/(*A* − *C*).

^{c}

Estimated using the experimental dipole moment^{16} and inertial axes obtained from the CCSD(T)/ANO1 geometry for each isotopologue.

All four singly substituted ^{13}C and ^{15}N isotopologues were detectable at natural abundance. The assignment of these species was facilitated by previously published constants.^{26,27} All R-branch band origins were found within 1–20 MHz of the frequency predicted from those constants, with the exception of the [5-^{13}C] isotopologue, which was further away. Although the band-origin transitions for isotopologues [5-^{13}C] and [1-^{15}N] are in close proximity, the assignments are nevertheless unambiguous based on the relative intensities, theoretical constants, previous experimental constants, and types of transitions observed. Isotopic substitution in the [4-^{13}C] and [1-^{15}N] isotopologues lies off the principal axis coordinates of the normal isotopologue of pyrimidine. As a result, the dipole moment vector is shifted off the *b*-inertial axis, which results in observable *a*-type and *b*-type transitions, producing quartets at the point where each series loses degeneracy, rather than the doublets observed for the normal isotopologue. The least-squares fit rotational and distortion constants for each of these isotopologues, based on transitions observed in this work and those from previous works,^{26,27} are also shown in Table II, and the complete output files can be found in the supplementary material.

As a consequence of the synthetic method utilized, samples of the deuterium-enriched materials exhibit highly congested spectra with transitions from many different isotopologues and their excited vibrational states. This method was beneficial for the convenient production and detection of many deuterated isotopologues and was significantly less challenging than synthetic routes used previously.^{25} The spectral congestion, however, limited the number of measurable transitions to only the most intense R-type and Q-type transitions. Many of these deuterated isotopologues exhibit a dual turnaround in the R-branch progressions near the band origin, which significantly reduced the number of R-type transitions that could be resolved. The isotopologue abundance, spectral congestion, and turnarounds in each band effectively reduced the quantity of measurable lines for all but the most abundant deuterated isotopologues to approximately 200–400 transitions, most of which were degenerate sets of R-branch transitions. Due to the large effects of centrifugal distortion on these high-*J* transitions, it was necessary to include approximately 200 transitions in the least-squares fit to accurately determine the rotational constants using a quartic centrifugally distorted Hamiltonian. Several hundred more transitions were required to obtain a satisfactory sextic fit. Due to the low number of transitions, most deuterated isotopologues were only fit to a full quartic Hamiltonian. The least-squares fit rotational constants for the 11 deuterated isotopologues studied are shown in Table III, and output files can be found in the supplementary material.

. | [2-^{2}H]
. | [4-^{2}H]
. | [5-^{2}H]
. | [2,4-^{2}H]
. | [2,5-^{2}H]
. |
---|---|---|---|---|---|

A_{0} (MHz) | 6066.853 7 (27) | 6242.397 29 (15) | 6066.961 71 (23) | 5934.805 9 (20) | 6066.664 5 (89) |

B_{0} (MHz) | 5871.752 2 (36) | 5694.506 25 (14) | 5840.148 71 (21) | 5604.131 6 (21) | 5479.254 2 (85) |

C_{0} (MHz) | 2983.300 09 (51) | 2977.377 53 (20) | 2975.120 51 (30) | 2881.893 48 (36) | 2878.529 75 (27) |

D_{J} (kHz) | 1.338 38 (47) | 1.333 658 (93) | 1.337 87 (17) | 1.203 05 (32) | 1.227 5 (15) |

D_{JK} (kHz) | −2.223 6 (11) | −2.208 42 (13) | −2.220 42 (27) | −1.996 20 (84) | −2.020 4 (21) |

D_{K} (kHz) | 0.998 84 (64) | 0.988 982 (96) | 0.996 65 (17) | 0.897 21 (49) | 0.896 98 (68) |

d_{1} (kHz) | [−0.059 34]^{a} | −0.044 526 (44) | −0.064 526 (80) | [0.026 27]^{a} | −0.119 0 (56) |

d_{2} (kHz) | [0.007 54]^{a} | −0.007 317 (19) | 0.008 069 (18) | [0.016 75]^{a} | [−0.005 997]^{a} |

H_{J} (Hz) | [0] | 0.000 570 (17) | 0.000 603 (36) | [0] | [0] |

H_{JK} (Hz) | [0] | −0.002 229 (23) | −0.002 253 (61) | [0] | [0] |

H_{KJ} (Hz) | [0] | 0.002 828 (36) | 0.002 768 (73) | [0] | [0] |

H_{K} (Hz) | [0] | −0.001 158 (24) | −0.001 074 (42) | [0] | [0] |

Δ_{i} (u Å^{2})^{b} | 0.031 469 (72) | 0.032 002 (12) | 0.032 937 (18) | 0.028 707 (51) | 0.029 19 (19) |

κ^{c} | 0.873 5 | 0.664 39 | 0.853 28 | 0.783 4 | 0.631 5 |

N_{lines} | 141 | 904 | 590 | 228 | 247 |

σ_{fit} | 0.045 | 0.037 | 0.037 | 0.038 | 0.042 |

μ_{a} (D)^{d} | 2.334 | 0.756 | 2.334 | 1.533 | 2.334 |

μ_{b} (D)^{d} | 0 | 2.208 | 0 | 1.76 | 0 |

. | [2-^{2}H]
. | [4-^{2}H]
. | [5-^{2}H]
. | [2,4-^{2}H]
. | [2,5-^{2}H]
. |
---|---|---|---|---|---|

A_{0} (MHz) | 6066.853 7 (27) | 6242.397 29 (15) | 6066.961 71 (23) | 5934.805 9 (20) | 6066.664 5 (89) |

B_{0} (MHz) | 5871.752 2 (36) | 5694.506 25 (14) | 5840.148 71 (21) | 5604.131 6 (21) | 5479.254 2 (85) |

C_{0} (MHz) | 2983.300 09 (51) | 2977.377 53 (20) | 2975.120 51 (30) | 2881.893 48 (36) | 2878.529 75 (27) |

D_{J} (kHz) | 1.338 38 (47) | 1.333 658 (93) | 1.337 87 (17) | 1.203 05 (32) | 1.227 5 (15) |

D_{JK} (kHz) | −2.223 6 (11) | −2.208 42 (13) | −2.220 42 (27) | −1.996 20 (84) | −2.020 4 (21) |

D_{K} (kHz) | 0.998 84 (64) | 0.988 982 (96) | 0.996 65 (17) | 0.897 21 (49) | 0.896 98 (68) |

d_{1} (kHz) | [−0.059 34]^{a} | −0.044 526 (44) | −0.064 526 (80) | [0.026 27]^{a} | −0.119 0 (56) |

d_{2} (kHz) | [0.007 54]^{a} | −0.007 317 (19) | 0.008 069 (18) | [0.016 75]^{a} | [−0.005 997]^{a} |

H_{J} (Hz) | [0] | 0.000 570 (17) | 0.000 603 (36) | [0] | [0] |

H_{JK} (Hz) | [0] | −0.002 229 (23) | −0.002 253 (61) | [0] | [0] |

H_{KJ} (Hz) | [0] | 0.002 828 (36) | 0.002 768 (73) | [0] | [0] |

H_{K} (Hz) | [0] | −0.001 158 (24) | −0.001 074 (42) | [0] | [0] |

Δ_{i} (u Å^{2})^{b} | 0.031 469 (72) | 0.032 002 (12) | 0.032 937 (18) | 0.028 707 (51) | 0.029 19 (19) |

κ^{c} | 0.873 5 | 0.664 39 | 0.853 28 | 0.783 4 | 0.631 5 |

N_{lines} | 141 | 904 | 590 | 228 | 247 |

σ_{fit} | 0.045 | 0.037 | 0.037 | 0.038 | 0.042 |

μ_{a} (D)^{d} | 2.334 | 0.756 | 2.334 | 1.533 | 2.334 |

μ_{b} (D)^{d} | 0 | 2.208 | 0 | 1.76 | 0 |

. | [4,5-^{2}H]
. | [4,6-^{2}H]
. | [2,4,5-^{2}H]
. | [2,4,6-^{2}H]
. | [4,5,6-^{2}H]
. | [2,4,5,6-^{2}H]
. |
---|---|---|---|---|---|---|

A_{0} (MHz) | 5913.951 77 (19) | 6082.417 37 (33) | 5809.659 9 (11) | 5692.647 99 (96) | 5680.013 44 (22) | 5457.392 48 (51) |

B_{0} (MHz) | 5602.393 35 (19) | 5457.759 88 (35) | 5351.085 3 (15) | 5457.513 65 (94) | 5457.638 49 (21) | 5330.696 99 (55) |

C_{0} (MHz) | 2876.488 87 (27) | 2876.116 03 (14) | 2785.074 18 (25) | 2785.908 67 (52) | 2782.890 77 (10) | 2696.315 71 (31) |

D_{J} (kHz) | 1.213 50 (13) | 1.208 29 (16) | 1.108 53 (25) | 1.098 11 (26) | 1.100 339 (52) | 1.006 22 (14) |

D_{JK} (kHz) | −2.009 25 (24) | −1.994 28 (38) | −1.826 39 (45) | −1.817 72 (76) | −1.820 68 (10) | −1.661 93 (38) |

D_{K} (kHz) | 0.900 31 (15) | 0.89 066 (22) | 0.813 63 (23) | 0.815 65 (42) | 0.816 410 (65) | 0.743 65 (24) |

d_{1} (kHz) | −0.038 973 (61) | −0.041 43 (11) | −0.074 0 (12) | 0.004 13 (27) | 0.003 510 (86) | −0.041 08 (28) |

d_{2} (kHz) | −0.012 243 (22) | 0.020 393 (39) | 0.006 23 (44) | 0.011 62 (10) | 0.012 150 (32) | 0.002 762 (68) |

H_{J} (Hz) | 0.000 464 (29) | [0] | [0] | [0] | [0] | [0] |

H_{JK} (Hz) | −0.001 771 (53) | [0] | [0] | [0] | [0] | [0] |

H_{KJ} (Hz) | 0.002 186 (68) | [0] | [0] | [0] | [0] | [0] |

H_{K} (Hz) | −0.000 887 (38) | [0] | [0] | [0] | [0] | [0] |

Δ_{i} (u Å^{2})^{b} | 0.029 937 (17) | 0.029 032 (12) | 0.026 164 (36) | 0.025 529 (41) | 0.026 896 (85) | 0.023 262 (25) |

κ^{c} | 0.794 86 | 0.610 36 | 0.696 8 | 0.838 21 | 0.846 49 | 0.908 23 |

N_{lines} | 807 | 423 | 349 | 210 | 404 | 289 |

σ_{fit} | 0.037 | 0.035 | 0.033 | 0.039 | 0.035 | 0.037 |

μ_{a} (D)^{d} | 1.55 | 2.178 | 2.178 | 0 | 0 | 2.334 |

μ_{b} (D)^{d} | 1.67 | 0.838 | 0.838 | 2.334 | 2.334 | 0 |

. | [4,5-^{2}H]
. | [4,6-^{2}H]
. | [2,4,5-^{2}H]
. | [2,4,6-^{2}H]
. | [4,5,6-^{2}H]
. | [2,4,5,6-^{2}H]
. |
---|---|---|---|---|---|---|

A_{0} (MHz) | 5913.951 77 (19) | 6082.417 37 (33) | 5809.659 9 (11) | 5692.647 99 (96) | 5680.013 44 (22) | 5457.392 48 (51) |

B_{0} (MHz) | 5602.393 35 (19) | 5457.759 88 (35) | 5351.085 3 (15) | 5457.513 65 (94) | 5457.638 49 (21) | 5330.696 99 (55) |

C_{0} (MHz) | 2876.488 87 (27) | 2876.116 03 (14) | 2785.074 18 (25) | 2785.908 67 (52) | 2782.890 77 (10) | 2696.315 71 (31) |

D_{J} (kHz) | 1.213 50 (13) | 1.208 29 (16) | 1.108 53 (25) | 1.098 11 (26) | 1.100 339 (52) | 1.006 22 (14) |

D_{JK} (kHz) | −2.009 25 (24) | −1.994 28 (38) | −1.826 39 (45) | −1.817 72 (76) | −1.820 68 (10) | −1.661 93 (38) |

D_{K} (kHz) | 0.900 31 (15) | 0.89 066 (22) | 0.813 63 (23) | 0.815 65 (42) | 0.816 410 (65) | 0.743 65 (24) |

d_{1} (kHz) | −0.038 973 (61) | −0.041 43 (11) | −0.074 0 (12) | 0.004 13 (27) | 0.003 510 (86) | −0.041 08 (28) |

d_{2} (kHz) | −0.012 243 (22) | 0.020 393 (39) | 0.006 23 (44) | 0.011 62 (10) | 0.012 150 (32) | 0.002 762 (68) |

H_{J} (Hz) | 0.000 464 (29) | [0] | [0] | [0] | [0] | [0] |

H_{JK} (Hz) | −0.001 771 (53) | [0] | [0] | [0] | [0] | [0] |

H_{KJ} (Hz) | 0.002 186 (68) | [0] | [0] | [0] | [0] | [0] |

H_{K} (Hz) | −0.000 887 (38) | [0] | [0] | [0] | [0] | [0] |

Δ_{i} (u Å^{2})^{b} | 0.029 937 (17) | 0.029 032 (12) | 0.026 164 (36) | 0.025 529 (41) | 0.026 896 (85) | 0.023 262 (25) |

κ^{c} | 0.794 86 | 0.610 36 | 0.696 8 | 0.838 21 | 0.846 49 | 0.908 23 |

N_{lines} | 807 | 423 | 349 | 210 | 404 | 289 |

σ_{fit} | 0.037 | 0.035 | 0.033 | 0.039 | 0.035 | 0.037 |

μ_{a} (D)^{d} | 1.55 | 2.178 | 2.178 | 0 | 0 | 2.334 |

μ_{b} (D)^{d} | 1.67 | 0.838 | 0.838 | 2.334 | 2.334 | 0 |

^{a}

Fixed to values calculated at the CCSD(T)/ANO1 level.

^{b}

*Δ*_{i} = *I*_{c} − *I*_{b} − *I*_{a}.

^{c}

*κ* = (2B − A − C)/(A − C).

^{d}

Estimated using the experimental dipole moment^{16} and inertial axes obtained from the CCSD(T)/ANO1 geometry for each isotopologue.

## STRUCTURE DETERMINATIONS OF PYRIMIDINE

The large number of isotopologues included in our study allowed the determination of complete *r*_{s}, *r*_{0}, and *r*_{e}^{SE} structures. The *r*_{e}^{SE} structure derived using CCSD(T) corrections and the *r*_{0} structure were evaluated using the ground-state rotational constants reported in Tables II and III (S-reduced Hamiltonian, III^{r} representation). The *r*_{e}^{SE} structure derived from B3LYP corrections was evaluated using “determinable constants” ($A0\u2032\u2032$, $B0\u2032\u2032$, and $C0\u2032\u2032$),^{55} as described in the supplementary material. In total, the sixteen isotopologues yield 32 independent moments of inertia, which produce a highly redundant determination of the nine independent structural parameters of pyrimidine (*C*_{2v}) by least-squares fitting in xrefit. In this work, all three moments of inertia for all sixteen isotopologues were used with equal weighting. A comparison of each of the experimentally determined structural parameters using the best fully corrected constants, including their uncertainties, to values determined from a variety of computational methods is shown in Table IV and illustrated graphically in Fig. 5. In a striking result, *all* of the equilibrium structural parameters predicted at the CCSD(T)/cc-pCV5Z level fall within the statistical error bars of the semi-experimental equilibrium structure [*r*_{e}^{SE} CCSD(T)]. From Fig. 5, it can be seen that the structural parameters predicted by other high levels of theory are generally good but occasionally fall outside the statistical error bars of the best semi-experimental structure.

. | Expt. . | Computational . | |||||||
---|---|---|---|---|---|---|---|---|---|

. | r_{e}^{SE}
. | r_{e}^{SE}
. | r_{e}^{SE}
. | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | B3LYP/ . |

Parameter . | Ref. 31 ^{a}
. | DFT^{b}
. | CCSD(T)^{c}
. | cc-pCV5Z . | cc-pwCVQZ^{29}
. | cc-pCVQZ . | CBS+CV^{57}
. | ANO1 . | 6-311+G(2d,p) . |

R_{C(2)–H} (Å) | 1.0820^{d} | 1.081 05 (61) | 1.081 65 (39) | 1.081 74 | 1.0818 | 1.0819 | 1.0820 | 1.0840 | 1.0854 |

R_{C(2)–C(5)} (Å) | 2.6534 (3)^{e} | 2.654 57 (81) | 2.653 95 (52) | 2.654 14 | 2.6546^{f} | 2.6548 | 2.6540^{f} | 2.6623 | 2.6589 |

R_{C(5)–H} (Å) | 1.0794^{d} | 1.078 75 (57) | 1.079 29 (36) | 1.079 19 | 1.0793 | 1.0794 | 1.0795 | 1.0814 | 1.0820 |

R_{C(2)–N(1)} (Å) | 1.3333 (3) | 1.333 73 (59) | 1.333 33 (38) | 1.333 26 | 1.3338 | 1.3340 | 1.3329 | 1.3395 | 1.3336 |

R_{C(4)–C(5)} (Å) | 1.3868 (2) | 1.387 16 (70) | 1.386 75 (45) | 1.386 96 | 1.3873 | 1.3876 | 1.3868 | 1.3924 | 1.3876 |

R_{C(4)–H} (Å) | 1.0825^{d} | 1.081 64 (48) | 1.082 24 (30) | 1.08232 | 1.0825 | 1.0825 | 1.0826 | 1.0846 | 1.0858 |

θ_{N(1)–C(2)–H} (deg) | 116.32 (1) | 116.331 (38) | 116.332 (24) | 116.334 | 116.30 | 116.30 | 116.36 | 116.15 | 116.51 |

θ_{C(4)–C(5)–H} (deg) | 121.65 (1) | 121.661 (35) | 121.665 (22) | 121.677 | 121.68 | 121.68 | 121.67^{f} | 121.70 | 121.69 |

θ_{C(5)–C(4)–H} (deg) | 121.21 (1)^{e} | 121.259 (59) | 121.261 (38) | 121.228 | 121.21 | 121.21 | 121.26 | 121.17 | 121.22 |

. | Expt. . | Computational . | |||||||
---|---|---|---|---|---|---|---|---|---|

. | r_{e}^{SE}
. | r_{e}^{SE}
. | r_{e}^{SE}
. | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | CCSD(T)/ . | B3LYP/ . |

Parameter . | Ref. 31 ^{a}
. | DFT^{b}
. | CCSD(T)^{c}
. | cc-pCV5Z . | cc-pwCVQZ^{29}
. | cc-pCVQZ . | CBS+CV^{57}
. | ANO1 . | 6-311+G(2d,p) . |

R_{C(2)–H} (Å) | 1.0820^{d} | 1.081 05 (61) | 1.081 65 (39) | 1.081 74 | 1.0818 | 1.0819 | 1.0820 | 1.0840 | 1.0854 |

R_{C(2)–C(5)} (Å) | 2.6534 (3)^{e} | 2.654 57 (81) | 2.653 95 (52) | 2.654 14 | 2.6546^{f} | 2.6548 | 2.6540^{f} | 2.6623 | 2.6589 |

R_{C(5)–H} (Å) | 1.0794^{d} | 1.078 75 (57) | 1.079 29 (36) | 1.079 19 | 1.0793 | 1.0794 | 1.0795 | 1.0814 | 1.0820 |

R_{C(2)–N(1)} (Å) | 1.3333 (3) | 1.333 73 (59) | 1.333 33 (38) | 1.333 26 | 1.3338 | 1.3340 | 1.3329 | 1.3395 | 1.3336 |

R_{C(4)–C(5)} (Å) | 1.3868 (2) | 1.387 16 (70) | 1.386 75 (45) | 1.386 96 | 1.3873 | 1.3876 | 1.3868 | 1.3924 | 1.3876 |

R_{C(4)–H} (Å) | 1.0825^{d} | 1.081 64 (48) | 1.082 24 (30) | 1.08232 | 1.0825 | 1.0825 | 1.0826 | 1.0846 | 1.0858 |

θ_{N(1)–C(2)–H} (deg) | 116.32 (1) | 116.331 (38) | 116.332 (24) | 116.334 | 116.30 | 116.30 | 116.36 | 116.15 | 116.51 |

θ_{C(4)–C(5)–H} (deg) | 121.65 (1) | 121.661 (35) | 121.665 (22) | 121.677 | 121.68 | 121.68 | 121.67^{f} | 121.70 | 121.69 |

θ_{C(5)–C(4)–H} (deg) | 121.21 (1)^{e} | 121.259 (59) | 121.261 (38) | 121.228 | 121.21 | 121.21 | 121.26 | 121.17 | 121.22 |

^{a}

Semi-experimental structure with corrections for vibration–rotation interaction and electron mass (B2PLYP/cc-pVTZ) and with all C–H bond lengths fixed. Uncertainty, in parentheses, as reported in the study.

^{b}

Semi-experimental structure with corrections for vibration–rotation interaction and electron mass [B3LYP/6-311+G(2d,p)]. Uncertainty (2σ) from xrefit structure determination.

^{c}

Semi-experimental structure with corrections for vibration–rotation interaction [CCSD(T)/ANO1] and electron mass [CCSD(T)/ANO1]. Uncertainty (2σ) from xrefit structure determination.

^{d}

Structural parameter fixed at this value.

^{e}

Value and uncertainty calculated from reported geometrical parameters.

^{f}

Value calculated from reported geometrical parameters.

The current study of pyrimidine, along with our other recent studies of pyridazine^{33} and hydrazoic acid (HN_{3}),^{56} underscores the benefits of incorporating multiply substituted isotopologues in the structure determination. Historically, Kraitchman’s analysis for structure determination relies on a single isotopic substitution,^{28} and the approach of utilizing singly substituted isotopologues has generally been carried forward in the current era of semi-experimental structure determination. We find that the inclusion of multiply substituted isotopologues contributes, in significant measure, to the high accuracy and precision of the structure determinations of pyrimidine, pyridazine, and hydrazoic acid. For the purpose of comparison, we determined the substitution (*r*_{s}) structure (Kraitchman’s analysis) and the effective ground-state (*r*_{0}) structure of pyrimidine, which are presented in the supplementary material. The *r*_{0} structural parameters are depicted in Fig. 5.

## THE “BEST” THEORETICAL PREDICTION THE FOR EQUILIBRIUM STRUCTURE OF PYRIMIDINE

While the all-electron CCSD(T) geometry optimizations performed for pyrimidine are extensive—that using the cc-pCV5Z basis set involved 1080 basis functions—they are certainly not complete in the sense that the corresponding results are not “exact.” This is a somewhat important point in the present context, as agreement with the semi-experimental equilibrium structure obtained from the experimental ground-state rotational constants and the CCSD(T) vibrational and electronic corrections is excellent. It is fair to say that the use of the word excellent is an understatement in this context; the bond lengths obtained by the two approaches agree to roughly 0.0001 Å, which is comparable to a nuclear diameter. As in the present work, studies of this sort that have been done in the past have generally featured theoretical corrections and ground-state data to obtain an *r*_{e}^{SE} structure, and then, high-level quantum chemical calculations are usually carried out to “confirm” the structure.^{7} This has been a successful approach for obtaining molecular structures, but the bond lengths typically obtained in such studies are generally about *an order of magnitude less precise* than the level that appears to have been achieved here. Indeed, it is natural for the skeptical reader to wonder if CCSD(T)/cc-pCV5Z is fortuitously almost coincident with the results and to what degree additional theoretical improvements would alter this situation. If agreement to roughly 0.001 Å was in question, the answer would be clear; CCSD(T)/cc-pCV5Z equilibrium structures are likely correct to that level of accuracy. However, the smaller differences like those seen here should be assumed to be fortuitous, until the question of “inexactness” is satisfactorily addressed.

To this end, the leading sources of residual error in the calculated equilibrium structure have been addressed. In an approximate order of importance, these are as follows:

Residual electron correlation effects beyond the CCSD(T) treatment.

Residual basis set effects beyond cc-pCV5Z.

Effects of scalar (mass–velocity and Darwin) relativistic effects.

The so-called diagonal Born–Oppenheimer correction (DBOC).

The first two effects are certain to cancel to some degree as additional electron correlation is expected to lengthen bonds, while further basis set expansion will shorten them.^{58} The relativistic contribution to bond lengths is generally negative (they contract). Again, the opposite is true with the DBOC, although the magnitude of these two latter contributions is decidedly smaller than the correlation and basis set effects, at least for molecules involving first-row atoms.

To address the issues raised above, the following procedure was adopted:

Residual correlation effects are assessed by performing geometry optimizations at the CCSDT(Q) level

^{59}and then estimating the correlation correction as

As calculations at the CCSDT(Q) level of theory are quite expensive, the two parameters above are obtained with the cc-pVDZ basis in the frozen-core approximation.

To estimate the basis set limit, equilibrium structural parameters obtained with the cc-pCVTZ, cc-pCVQZ, and cc-pCV5Z basis sets were extrapolated using the empirical exponential formula that is common in quantum chemistry and has also previously been advocated for similar purposes by Puzzarini,

^{60}

where *R*(X) are the values of the parameters obtained using the various basis sets (X = 3 for cc-pCVTZ, X = 4 for cc-pCVQZ, etc.), and *R*(∞) is the desired basis set limit estimate. Along with A and B, it can be directly determined if three different values of X are used. The correction to the structure due to residual correlation is then estimated as follows:

The minor contributions from relativistic effects and the DBOC are obtained according to the following recipe:

For the relativistic correction to the parameters, we use

where the former structure is computed with the X2C-1e variant of the coupled-cluster theory,^{61–63} and the latter is the corresponding standard non-relativistic calculation.

For the diagonal Born–Oppenheimer correction (DBOC),

^{64}we use

Here, the former value is obtained by minimizing the DBOC-corrected SCF energy with respect to nuclear positions, and the latter is again the traditional calculation.^{65}

The final estimates of the equilibrium structural parameters are as follows:

. | Correlation . | Basis set . | Relativity . | DBOC . | Sum of . | CCSD(T)/ . | Best theoretical . |
---|---|---|---|---|---|---|---|

. | correction [Eq. (3)] . | correction [Eq. (5)] . | correction [Eq. (6)] . | [Eq. (7)] . | corrections . | cc-pCV5Z . | estimate [Eq. (8)] . |

R_{C(2)–H} (Å) | 0.000 034 | −0.000 040 | −0.000 120 | 0.000 140 | 0.000 014 | 1.081 74 | 1.081 75 |

R_{C(2)–C(5)} (Å) | 0.000 857 | −0.000 178 | −0.000 560 | 0.000 071 | 0.000 190 | 2.654 14 | 2.654 33 |

R_{C(5)–H} (Å) | 0.000 152 | −0.000 060 | −0.000 120 | 0.000 130 | 0.000 103 | 1.079 19 | 1.079 29 |

R_{C(2)–N(1)} (Å) | 0.000 708 | −0.000 211 | −0.000 090 | 0.000 020 | 0.000 427 | 1.333 26 | 1.333 69 |

R_{C(4)–C(5)} (Å) | 0.000 505 | −0.000 177 | −0.000 250 | 0.000 012 | 0.000 090 | 1.386 96 | 1.387 05 |

R_{C(4)–H} (Å) | 0.000 054 | −0.000 055 | −0.000 120 | 0.000 141 | 0.000 020 | 1.082 32 | 1.082 34 |

θ_{N(1)-C(2)-H} (deg) | −0.010 954 | 0.011 223 | −0.014 000 | 0.001 660 | −0.012 071 | 116.334 | 116.322 |

θ_{C(4)–C(5)–H} (deg) | −0.010 614 | −0.007 200 | −0.001 000 | −0.000 371 | −0.019 186 | 121.677 | 121.658 |

θ_{C(5)–C(4)–H} (deg) | 0.010 986 | 0.016 690 | 0.005 000 | 0.002 483 | 0.035 159 | 121.228 | 121.263 |

. | Correlation . | Basis set . | Relativity . | DBOC . | Sum of . | CCSD(T)/ . | Best theoretical . |
---|---|---|---|---|---|---|---|

. | correction [Eq. (3)] . | correction [Eq. (5)] . | correction [Eq. (6)] . | [Eq. (7)] . | corrections . | cc-pCV5Z . | estimate [Eq. (8)] . |

R_{C(2)–H} (Å) | 0.000 034 | −0.000 040 | −0.000 120 | 0.000 140 | 0.000 014 | 1.081 74 | 1.081 75 |

R_{C(2)–C(5)} (Å) | 0.000 857 | −0.000 178 | −0.000 560 | 0.000 071 | 0.000 190 | 2.654 14 | 2.654 33 |

R_{C(5)–H} (Å) | 0.000 152 | −0.000 060 | −0.000 120 | 0.000 130 | 0.000 103 | 1.079 19 | 1.079 29 |

R_{C(2)–N(1)} (Å) | 0.000 708 | −0.000 211 | −0.000 090 | 0.000 020 | 0.000 427 | 1.333 26 | 1.333 69 |

R_{C(4)–C(5)} (Å) | 0.000 505 | −0.000 177 | −0.000 250 | 0.000 012 | 0.000 090 | 1.386 96 | 1.387 05 |

R_{C(4)–H} (Å) | 0.000 054 | −0.000 055 | −0.000 120 | 0.000 141 | 0.000 020 | 1.082 32 | 1.082 34 |

θ_{N(1)-C(2)-H} (deg) | −0.010 954 | 0.011 223 | −0.014 000 | 0.001 660 | −0.012 071 | 116.334 | 116.322 |

θ_{C(4)–C(5)–H} (deg) | −0.010 614 | −0.007 200 | −0.001 000 | −0.000 371 | −0.019 186 | 121.677 | 121.658 |

θ_{C(5)–C(4)–H} (deg) | 0.010 986 | 0.016 690 | 0.005 000 | 0.002 483 | 0.035 159 | 121.228 | 121.263 |

## COMPUTATIONAL CORRECTIONS FOR VIBRATION–ROTATION INTERACTION AND ELECTRON MASS

Recent work by Kisiel,^{66} Piccardo *et al.*,^{30} Penocchio *et al.*,^{31} and Császár *et al.*^{29} revealed that corrections calculated using DFT methods can be a useful alternative to coupled-cluster calculated values for providing *r*_{e}^{SE} structures.^{67} To explore this, we also calculated vibration–rotation interaction and electron-mass corrections using DFT methods similar to those employed by Piccardo *et al.*^{30} and used these to determine a semi-experimental equilibrium structure. The vibration–rotation correction to the rotational constants leads to a rather similar inertial defect at both the CCSD(T)/ANO1 and B3LYP/6-311+G(2d,p) levels (see Table S2 in the supplementary material). Both computational methods provide harmonic force fields with sufficient accuracy that the quartic centrifugal distortion constants are useful for spectroscopic analysis and interpretation (Table VI). Both computational methods exhibit excellent agreement with experimental data and are of great utility for predicting spectra of vibrational satellites and making definitive assignments to particular states.^{54} Nonetheless, it is very clear from the data depicted in Fig. 5 that the coupled-cluster vibrational corrections afford a semi-experimental structure with smaller uncertainty and better agreement with theoretical predictions than the DFT corrections.

Quartic distortion term . | Expt. value . | CCSD(T)/ANO1 . | B3LYP/6-311+G(2d,p) . |
---|---|---|---|

D_{J} (kHz) | 1.472 131 (62) | 1.44 | 1.43 |

D_{JK} (kHz) | −2.452 567 (30) | −2.40 | −2.38 |

D_{K} (kHz) | 1.104 972 (39) | 1.08 | 1.07 |

d_{1} (kHz) | 0.011 730 (16) | 0.0070 | 0.0133 |

d_{2} (kHz) | 0.0219 042 (89) | 0.021 | 0.0198 |

Quartic distortion term . | Expt. value . | CCSD(T)/ANO1 . | B3LYP/6-311+G(2d,p) . |
---|---|---|---|

D_{J} (kHz) | 1.472 131 (62) | 1.44 | 1.43 |

D_{JK} (kHz) | −2.452 567 (30) | −2.40 | −2.38 |

D_{K} (kHz) | 1.104 972 (39) | 1.08 | 1.07 |

d_{1} (kHz) | 0.011 730 (16) | 0.0070 | 0.0133 |

d_{2} (kHz) | 0.0219 042 (89) | 0.021 | 0.0198 |

## IMPLICATIONS FOR ROTATIONAL SPECTROSCOPY

The current work provides important experimental validation for the accuracy of the computational methods used in this study. As a result of this high accuracy, an *a priori* spectral prediction generated solely from the high-quality computational data reported herein—rotational constants for the equilibrium structure at CCSD(T)/cc-pCV5Z, along with vibration–rotation interaction constants, quartic and sextic centrifugal distortion constants, and electron-mass corrections at CCSD(T)/ANO1—reproduces the rotational band origins of pyrimidine from 235 to 375 GHz with a remarkable accuracy of between 6 MHz and 12 MHz. Most of the residual discrepancy is due to the error in the predicted value of *C*_{0}, which is approximately +70 kHz. This level of accuracy would have been more than sufficient for an initial search and assignment of the transitions, had the previous microwave work been unavailable. Although this ability to predict the spectrum is not of great consequence for a stable molecule like pyrimidine, it becomes critical for the study of unstable species, such as high-energy isomers, molecular ions, and radicals, with much lower-intensity spectra.

## CONCLUSION

The current study of pyrimidine enables us to establish the accuracy by which the molecular structure of a prototypical organic molecule of moderate size can be determined using the current best practice for experimental rotational spectroscopy and computational quantum chemistry. We invested a great deal of experimental effort to prepare sixteen isotopologues of pyrimidine and measure highly accurate ground-state rotational constants for them. Using either *ab initio* or density functional values to correct the experimental constants for vibration–rotation interactions and electron-mass, we determined semi-experimental equilibrium molecular structures for pyrimidine. The *r*_{e}^{SE} structure obtained using CCSD(T) corrections shows exceptional agreement with the *r*_{e} structure computed at the CCSD(T)/cc-pCV5Z level. Of the various computational methods examined, CCSD(T)/cc-pCV5Z is the only method for which the computed value for each geometric parameter lies within the experimental uncertainty (2σ) of the semi-experimental measurement. The quite close agreement in this case simultaneously confirms that the *r*_{e}^{SE} structural parameters are as accurate as they are precise and provides a very useful experimental benchmark for the accuracy of the CCSD(T)/cc-pCV5Z method. The exceptionally high accuracy and precision of the *r*_{e}^{SE} structure determination is a consequence of the large number of isotopologues measured, the precision and extent of the experimental frequency measurements, and the sophisticated theoretical treatment of the effects of vibration–rotation coupling and electron mass. Taken together, these demanding experimental and computational studies establish the powerful capabilities of the currently available structural analysis methods for species in the size range of monocyclic aromatic compounds.

## SUPPLEMENTARY MATERIAL

See the supplementary material for least-squares fitting files of pyrimidine, output files from computations, and details concerning structure determinations.

## ACKNOWLEDGMENTS

We gratefully acknowledge funding from the National Science Foundation for this project (Grant No. CHE-1664912) and for sharing departmental computing resources (Grant No. NSF CHE-0840494). J.F.S. was supported by the Department of Energy Basic Energy Sciences (Grant No. DEFG02-07ER15885).

## REFERENCES

_{6}H

_{5}CN) in the interstellar medium

_{4}H

_{2}, C

_{6}H

_{2}, and benzene in CRL 618

_{60}and C

_{70}in a Young Planetary nebula

_{60}

^{+}as the carrier of two diffuse interstellar bands

_{3}and H

_{2}O+NH

_{3}ices

^{13}C and

^{15}N isotopic species of pyrimidine

_{3})

_{2}

_{2}

_{3}isomers

_{4}H

_{4}N

_{2}) by millimeter-wave spectroscopy

_{3}) by millimeter-wave spectroscopy

The astute reader might realize that there is a potential ambiguity associated with using DBOCs in this context (at this level of theory, the equilibrium geometries of distinct isotopologues are no longer the same), which is, indeed, an issue given the magnitude of the effect under consideration. The DBOC uniformly expands all bond distances, with the contribution for C–H and N–H distances being about 0.000 14 Å, while those for heavy atom distances being at least a factor of two smaller. The issue here is that the entire procedure to fit equilibrium structures to rotational constants from which vibrational and electron-mass corrections have been removed assumes that the underlying molecular structure is isotope independent. This is not the case if the potential energy surface that defines the equilibrium structure includes the DBOC. However, of course, the assumption of isotope-independent structures is made in any fit to rotational constants, even when the ground-state constants (thereby yielding *r*_{0} effective structures) are used. An interesting topic for the future might be to make some allowance for the isotopic dependence of the equilibrium structure in the refinement procedure, but such an effort is clearly outside the scope of this work.