Comparison of one-particle basis set extrapolation to explicitly

THE JOURNAL OF CHEMICAL PHYSICS 133, 244108 (2010)

Comparison of one-particle basis set extrapolation to explicitly correlated methods for the calculation of accurate quartic force fields, vibrational frequencies, and spectroscopic constants: Application to H2 O, N2 H+ , NO2 + , and C2 H2 Xinchuan Huang,1 Edward F. Valeev,2 and Timothy J. Lee3,a) 1

SETI Institute, 189 Bernardo Ave., Suite 100, Mountain View, California 94043, USA Department of Chemistry, 107 Davidson Hall, Virginia Tech, Blacksburg, Virginia 24061, USA 3 NASA Ames Research Center, Moffett Field, California 94035-1000, USA 2

(Received 25 February 2010; accepted 6 October 2010; published online 28 December 2010) One-particle basis set extrapolation is compared with one of the new R12 methods for computing highly accurate quartic force fields (QFFs) and spectroscopic data, including molecular structures, rotational constants, and vibrational frequencies for the H2 O, N2 H+ , NO2 + , and C2 H2 molecules. In general, agreement between the spectroscopic data computed from the best R12 and basis set extrapolation methods is very good with the exception of a few parameters for N2 H+ where it is concluded that basis set extrapolation is still preferred. The differences for H2 O and NO2 + are small and it is concluded that the QFFs from both approaches are more or less equivalent in accuracy. For C2 H2 , however, a known one-particle basis set deficiency for C–C multiple bonds significantly degrades the quality of results obtained from basis set extrapolation and in this case the R12 approach is clearly preferred over one-particle basis set extrapolation. The R12 approach used in the present study was modified in order to obtain high precision electronic energies, which are needed when computing a QFF. We also investigated including core-correlation explicitly in the R12 calculations, but conclude that current approaches are lacking. Hence core-correlation is computed as a correction using conventional methods. Considering the results for all four molecules, it is concluded that R12 methods will soon replace basis set extrapolation approaches for high accuracy electronic structure applications such as computing QFFs and spectroscopic data for comparison to high-resolution laboratory or astronomical observations, provided one uses a robust R12 method as we have done here. The specific R12 method used in the present study, CCSD(T)R12 , incorporated a reformulation of one intermediate matrix in order to attain machine precision in the electronic energies. Final QFFs for N2 H+ and NO2 + were computed, including basis set extrapolation, core-correlation, scalar relativity, and higher-order correlation and then used to compute highly accurate spectroscopic data for all isotopologues. Agreement with high-resolution experiment for 14 N2 H+ and 14 N2 D+ was excellent, but for 14 16 N O2 + agreement for the two stretching fundamentals is outside the expected residual uncertainty in the theoretical values, and it is concluded that there is an error in the experimental quantities. It is hoped that the highly accurate spectroscopic data presented for the minor isotopologues of N2 H+ and NO2 + will be useful in the interpretation of future laboratory or astronomical observations. © 2010 American Institute of Physics. [doi:10.1063/1.3506341] I. INTRODUCTION

The calculation of accurate quartic force fields (QFFs) along with rovibrational spectroscopic constants and transition energies has been of significant interest over the last 20 years (for example, see Refs. 1–7 and references therein). Singles and doubles coupled-cluster theory with a perturbational estimate of connected triple excitations [CCSD(T)] (Ref. 8) has become the method of choice in most of these studies. Although the residual higher-order electron correlation effects can be significant, CCSD(T) is a good compromise between the cost and accuracy for most practical cases. Corrections for so-called small effects, such a) Author to whom correspondence should be addressed. Electronic mail:

[email protected]. 0021-9606/2010/133(24)/244108/15/$30.00

as core-correlation, scalar relativity, post-Born–Oppenheimer, and even quantum electrodynamics,9 have been included in the calculation of accurate spectroscopic data in one form or another for more than a decade. Recently, approaches have been developed for the calculation of accurate QFFs in which some small effects were either included explicitly or as corrections in the QFFs, such as core-correlation, extrapolation to the basis set limit, inclusion of the effect of diffuse functions in the one-particle basis set, and scalar relativity, while other small effects were generally included only as corrections, such as higher-order valence electron correlation and the diagonal Born–Oppenheimer correction.10–13 The purpose of these approaches was to explicitly include all such corrections directly into the QFFs, thus making it possible to include all effects in accurate variational rovibrational calculations of term state energies and transition energies or in

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perturbation theory approaches which lead to spectroscopic constants often used in the fitting of experimental spectra. Further, having all small effects in the QFF itself means that accurate spectroscopic data may be computed for all isotopologues of interest. The QFFs that have been computed using this approach have proved to be very accurate, but in many cases there still remains a measurable error compared to the best experiments (for example, 1–3 cm−1 for fundamental vibrational frequencies).10–13 In addition, the approach becomes relatively expensive computationally for larger molecules due to the need to use triple-zeta (TZ), quadruple-zeta (QZ), and quintuple-zeta (5Z) quality basis sets. This is especially true if diffuse functions are needed, such as for molecular anions, since the series of basis sets required in order to be able to extrapolate the energies to the one-particle basis set limit includes a diffuse function in each atomic shell. We have proposed a method to include the effect of diffuse functions approximately,12 but the need for TZ, QZ, and 5Z basis set calculations remains a potential bottleneck for larger molecules. It has been known for many years that the standard ab initio methods (based on Slater determinants) need large one-particle basis sets because of their poor modeling of the wave function at short interelectronic distances. This behavior is related to the inability of standard wave functions to describe the electron–electron cusps of the exact wave functions. Thus a nonempirical alternative to oneparticle basis set extrapolation is to make the wave function depend on the interelectronic coordinates, rij , so that the cusp is described properly. The R12 (or F12) methods, pioneered by Kutzlenigg,14, 15 are currently the most practical among such approaches. Several groups, including one of the current authors, have recently introduced efficient coupledcluster R12 methods: the CCSD(F12) method of Klopper and co-workers,16 the CCSD-F12x methods of Werner and coworkers,17 and the CCSD(2)R12 method of Valeev.18 The performance of these methods is comparable and all can be extended straightforwardly to include the (T) correction. The latter two methods are currently included in publicly available software packages. The consensus is that the R12 methods improve the energies by at least two cardinal numbers of the correlation-consistent basis sets if the exponential (Slatertype) correlation factor is used. Thus the R12 methods offer a sound nonempirical alternative to extrapolation methods, provided the approximations that go into the various R12 approaches have been systematically improved enough. The purpose of the present study, then, is to compare a state-of-the-art CCSD(T)-R12 method with one-particle basis set extrapolation for the calculation of accurate QFFs, spectroscopic constants, and rovibrational transition energies. In several QFF studies over the last two years,11–13 we have examined a few extrapolation approaches and generally we have found that use of a three-point formula19 with one-particle basis sets of TZ, QZ, and 5Z quality performs best by comparison to experimental fundamental vibrational frequencies, and that is the approach adopted here. As a brief summary, a twopoint formula with the QZ and 5Z basis sets is usually only slightly worse than the three-point formula, but the two-point

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formula with the TZ and QZ basis sets is significantly worse, so we have concluded that use of the 5Z basis set is needed for basis set extrapolation. (This is consistent with previous studies of basis set extrapolation, though not specifically aimed at fundamental vibrational frequencies—see Ref. 20, for example). Since including the TZ basis set with the QZ and 5Z basis sets increases the cost only marginally, we prefer the threepoint formula with the TZ, QZ, and 5Z basis sets. However, we note that the basis set extrapolation formula is never exact, and in fact one measure of the uncertainty in the extrapolation may be given by the difference found when extrapolating two series of basis sets where one includes diffuse functions and one does not. It has been shown that basis set extrapolation for these two cases leads to different asymptotes, often with the series that does not include diffuse functions leading to a lower energy.12, 21, 22 In effect, the question to be answered is whether use of the CCSD(T)-R12 with modest basis sets can improve on the uncertainty of the basis set extrapolation. We have chosen to apply these methods to the H2 O and C2 H2 molecules, for which very accurate experimental data are available, and to the N2 H+ and NO2 + molecular cations. N2 H+ has been identified in many sources in the interstellar medium (ISM) as well as in the ionosphere of Titan.23–26 However, since the main isotopologue of NO2 + does not possess a permanent dipole moment, it has not been identified in any astronomical observations to date. This situation may change when several telescopes with increased sensitivity and/or greater precision in the infrared (IR) will begin operation soon, such as the Herschel Space Observatory (launched last year), the Stratospheric Observatory for Infrared Astronomy (SOFIA), the James Webb Space Telescope (JWST), and the Atacama Large Millimeter Array (ALMA). These telescopes should allow for the identification of symmetric molecules in the IR or the identification of less abundant isotopologues at longer wavelengths. A CCSD(T) QFF was reported for NO2 + in 1992, and this was the first accurate prediction of its equilibrium molecular structure, fundamental vibrational transition energies, and other spectroscopic constants.27 Not long thereafter, its fundamental vibrational frequencies were determined experimentally using zero-kinetic-energy photoelectron spectroscopy.28–31 In contrast, N2 H+ has been widely studied both theoretically and experimentally (see Refs. 32–42 and references therein), and its fundamentals, overtones, and combination bands up to 10 600 cm−1 are known to high precision.35 There are also considerable theoretical and experimental spectroscopic data available for the deuterated species, N2 D+ (Refs. 36 and 42–44). Both isotopologues have been studied theoretically very recently using high levels of theory including Werner’s CCSD(T)-F12x methods.17, 45 Brites and Hochlaf32 computed QFFs at various levels of theory and compared these to experiment. While they obtained good agreement with experiment for some transition frequencies and concluded that the CCSD(T)-F12x approach was the method of choice, their findings can be hardly definitive since important effects such as core-correlation, scalar relativity, and higher-order electron correlation were not included. More importantly, they did not compare with one-particle basis set extrapolation, so there is no basis on

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which to judge these two approaches for attaining basis set completeness. We note that Rauhut et al. have recently computed harmonic and fundamental vibrational frequencies for several small molecules using Werner’s CCSD(T)-F12x methods, though they also did not compare to one-particle basis set extrapolation so there was no attempt to compare the two approaches for basis set completeness.46 The theoretical approach is described in Sec. II, followed by results and discussion. Our conclusions are presented in Sec. IV.

II. THEORETICAL APPROACH A. Details of the conventional ab initio methods

We first describe details of the conventional CCSD(T) calculations plus details of the corrections that have been included. In all calculations we have used Dunning’s correlation consistent basis sets.47–49 We will denote the cc-pVXZ (X = T, Q, or 5) basis sets as TZ, QZ, or 5Z. For H2 O and N2 H+ , core-correlation was explicitly included in the calculation of the QFFs, and thus the basis sets include tight corecorrelating functions49 —these basis sets are denoted CXZ (X = T, Q, or 5). For N2 H+ and NO2 + , diffuse functions were included in the basis sets, while for H2 O diffuse functions were not included—see Ref. 11 for H2 O QFFs where diffuse functions were included in the one-particle basis set. For C2 H2 , two series of basis sets were used—one without diffuse functions and one with diffuse functions. When diffuse functions are included in the basis set, these47 are denoted AXZ or ACXZ (X = T, Q, or 5) depending on whether or not corecorrelating functions are also included. QFFs have been determined according to the prescription described previously.11 For C2 H2 and N2 H+ , a reference geometry was determined at the CCSD(T)/C5Z level of theory. For NO2 + , a similar procedure was used, but a correction for scalar relativity was included, while for H2 O, we used the reference structure determined previously.11 A grid of geometries centered on this reference structure was then used for all calculations. The number of unique geometries was 101, 75, 55, and 32 for C2 H2 , H2 O, N2 H+ , and NO2 + , respectively. CCSD(T) energies are extrapolated to the oneparticle basis set limit using a three-point formula that experience has shown to be reliable.11–13 A correction for scalar relativity is evaluated at the CCSD(T)/TZ level of theory using the Douglas–Kroll approximation.50 Higher-order valence electron correlation was evaluated using either the averaged coupled-pair functional (ACPF),51 the multireference configuration interaction (MRCI), or the MRCI+Q (MRCI plus the multireference analog of the Davidson correction) method. For all of these multireference methods, a complete active space self-consistent field reference function was used in which the active space consisted of the full valence space. The internally contracted version of these methods was utilized. Scalar relativity and higher-order correlation corrections were included here only for N2 H+ and NO2 + —see our earlier study in which these corrections were included in a QFF for H2 O.11 We note that the higher-order correlation correction is meant to approximate a full configuration interac-

J. Chem. Phys. 133, 244108 (2010)

tion (FCI) result and not CCSDT, i.e., exact coupled-cluster triples. It is well established that for some molecular properties, such as vibrational frequencies, the difference between CCSDT and CCSD(T) is opposite to the difference between CCSDT and FCI, showing that CCSD(T) often benefits from a cancellation of errors. All of these calculations were performed with the MOLPRO 2006.1 program.52 Fundamental vibrational frequencies were computed using either an exact variational method (VAR) or second-order perturbation theory (PT).53 The MULTIMODE program54 was used for the VAR calculations, while the SPECTRO program55 was used for the PT calculations and for computing other spectroscopic constants. The QFF was transformed into a Morse-cosine coordinate system according to Ref. 56 for the exact variational calculations. We note that transformation of the QFF into a Morse-cosine coordinate system is important, otherwise some fundamental vibrational frequencies, in particular stretching frequencies, can be too high by tens of wavenumbers (cm−1 ). As discussed in Ref. 56, the Morsecosine coordinate system serves to build in the correct limiting behavior for the potential function. B. Details of the R12 methods

The CCSD(T)R12 method is defined as CCSD corrected perturbatively for the effect of triples and R12 geminals.57 Unlike the other coupled-cluster (CC) R12 methods in the literature, in CCSD(T)R12 the standard CC energy and wave function are corrected a posteriori without the need to modify the CC equations. The CCSD(T)R12 energy is evaluated as E CCSD(T)R12 = E CCSD + E (T) + E (2)R12 ,

(1)

where the first two terms add up to the standard CCSD(T) energy and the latter is a second-order R12 correction for the two-particle basis set incompleteness of the CCSD energy xy

xy

ij

ij 1 E (2)R12 = 18 Ti j (B (i j) )zw x y Tzw + 2 Ti j V x y . xy

(2)

In Eq. (2) Ti j are the first-order amplitudes of the geminal (rij -dependent) terms in the wave function and matrices B and V represent the geminal matrix elements of the zerothorder and first-order Hamiltonians, respectively. These matrices formally involve three- and four-electron integrals, which are avoided in the R12 method via judicious use of the resolution of the identity using the complementary auxiliary basis set (CABS) approach.58 The definition of matrix B is exactly the same as in the Møller–Plesset second-order perturbation theory (MP2) R12 method (see Ref. 15 for a detailed discussion of the MP2-R12 method and explicit expressions for B), whereas V is the V matrix of MP2-R12 “dressed” by the CCSD amplitudes in a complicated manner. Expressions for these matrices are too complicated to be given here. To keep the focus of the manuscript on the application of the R12 method to QFF construction we refer interested readers to the supplementary materials59 for the precise expressions used in this work. We only note that the definition of matrix V slightly differs from that employed originally in Refs. 57 and 60 and the effects of these differences are examined in detail in the supplementary materials.

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Just as with the standard CCSD(T) computations, to obtain the CCSD(T)R12 energy with precision sufficient for constructing a meaningful QFF requires extra care. All one- and two-electron integrals were computed as precisely as possible in standard 64-bit machine precision. Our experiments have convinced us that it is possible to compute the CCSD(T)R12 energy with the same precision as that of the underlying CCSD(T) energy. The effect of various “parameters” of R12 methods on the precision of the computed energies was considered in detail; the results are presented in the supplementary materials. In one set of R12 calculations, the specialized cc-pVXZF12 basis sets of Peterson and co-workers61 were utilized for the orbital expansions, the recommended Slater geminal parameters, and the associated cc-pVXZ-F12-CABS (Ref. 62) to construct the CABS basis using the CABS+ approach.58 Each cc-pVXZ-F12 computation utilized the cc-pV(X+1)ZRI basis sets for the density fitting. Slater-type geminal correlation factors were expanded in six Gaussian geminals using the standard fitting procedure. In a second set of R12 calculations, the standard aug-cc-pVXZ basis sets were used for the orbital expansion; density fitting utilized the aug-ccpV(X+1)Z-RI basis sets. In all R12 calculations reported here, core-correlation was not included. The supplementary material59 contains some R12 calculations with corecorrelation explicitly included; our conclusion is that with the current formulation of the R12 methods it is not safe to use all-electron R12 calculations, even with the aug-cc-pCVXZ basis sets. The computational cost of evaluating Eq. (2) is comparable to the cost of the CCSD(T) calculation itself, demonstrating that R12 methods are more cost effective relative to basis set extrapolation since smaller one-particle basis sets can be used. As an example, a recent study on the cyclopropenylidene molecule63 showed that on going from a TZ to a QZ basis set, the central processing unit (cpu) time for one CCSD iteration increased by about a factor of 11 and the (T) step increased by a factor of almost 16 (using the Titan coupledcluster programs, Refs. 64 and 65). Further, these times do not include system time and input/output wait times which were shown to increase at a higher rate than the cpu time on going to larger one-particle basis sets. Thus, at only a factor of two increase in the computational cost, the CCSD(T)R12 method is highly cost effective relative to one-particle basis set extrapolation. All R12 computations used the open-source MPQC package utilizing the CCSD(T) computations of the PSI package. All technical details can be investigated by checking-out the MPQC source code from http://www.mpqc.org/.

III. RESULTS AND DISCUSSION A. Comparison of R12 and basis set extrapolation: H2 O

Contained in Table I are spectroscopic data for H2 O computed at various levels of theory. See Refs. 66–70 for previous experimental and theoretical studies. Included are the molecular structure (both equilibrium and vibrationally averaged),

J. Chem. Phys. 133, 244108 (2010)

harmonic frequencies, fundamental vibrational frequencies, and vibrationally averaged rotational constants. We have also included the fitting error for the QFF (in atomic units) given as the sum of the squared residuals for the QFF grid. Making sure that the fitting error is consistent over all methods and that it is comparable to the machine precision is one test we routinely use to confirm the accuracy and reliability of each QFF. For example, performing this test led us to determine that the one-electron scalar relativity integrals lose precision beyond the TZ basis set (see Refs. 11 and 71). In the present study, this test was essential to assure that the CCSD(T)R12 energies were as precise as the standard CCSD(T) methods, as discussed in the methods section. The first set of R12 results, denoted AXZ-R12, utilize the standard Dunning aug-cc-pVXZ basis sets (see the methods section for details). The second set of R12 results, denoted XZ-F12, differ in that they use the specialized cc-pVXZ-F12 basis sets of Peterson et al. (see methods section for details). We note that the reformulation of the V matrix in the R12 method mentioned in the methods section leads to better overall convergence behavior on going to larger one-particle basis sets—this is discussed in the supplementary material. Examination of the XZ-F12 results shows that there is generally a monotonic convergence on going from X = D through X = Q for most molecular parameters, and in most cases the (X = T – X = D) difference is larger than the (X = Q – X = T) difference. There are minor exceptions, such as for ω1 , ν 1 , and ν 3 , but in all cases the (X = T – X = D) and (X = Q – X = T) differences are small, and in fact the (X = Q – X = D) difference is small. In contrast, examination of the AXZ-R12 results shows that the convergence pattern on going from X = D to X = Q is almost always oscillatory and while the (X = Q – X = T) difference is relatively small (for example, less than 4 cm−1 for harmonic and fundamental frequencies), the (X = T – X = D) difference is usually larger (greater than 8 cm−1 in some cases). Perhaps the main conclusion to be drawn from Table I is that the difference between the QZ-F12 and AQZ-R12 results is generally small (for example, always less than 2 cm−1 for harmonic and fundamental frequencies), although in each case fundamental vibrational frequencies from the QZ-F12 calculations where corecorrelation is included, denoted QZ-F12+coreA, are slightly closer to the experimental fundamentals relative to the AQZR12+coreA values. We note that the core-correlation contribution included in both sets of R12 results is computed as a correction from the non-R12 CCSD(T) method (see footnote to Table I for more details), since as of now R12 calculations in which core-correlation is included explicitly exhibit erratic convergence behavior with respect to one-particle basis set improvements. This is discussed in more detail in the supplementary material. Turning to a comparison of the R12 methods with basis set extrapolation, the two rows to compare are those labeled QZ-F12+coreA and CTQ5. The QZ-F12+coreA equilibrium bond distance is only 0.000 094 Å longer than the CTQ5 value, and the difference for the equilibrium bond angle is only –0.197o . The harmonic frequencies exhibit differences ranging from 2.0 to 2.7 cm−1 , while the range for the fundamental vibrational frequencies is similar: 2.2–

0.52 0.52 0.50 0.50

0.52 0.53 0.51 0.51 0.50 0.50 0.49

DZ-F12 TZ-F12 QZ-F12 QZ-F12+coreAa

TZ CTZ QZ CQZ 5Z C5Z CTQ5 104.52b 104.48c

0.9572b 0.95777c

104.337 104.405 104.452 104.574

104.523 104.422 104.465 104.587

(HOH)

103.582 103.684 104.116 104.225 104.372 104.492 104.771



0.9594258 0.9584302 0.9578893 0.9571187 0.9580396 0.9571217 0.9573941

0.9588495 0.9586201 0.9583248 0.9574035

0.9579676 0.9583799 0.9581368 0.9572162

R(O–H)

3840.90 3841.66 3844.45 3850.74 3840.05 3846.93 3839.60

3835.13 3834.23 3835.37 3842.27

3842.55 3834.03 3837.51 3844.40

ω1 (ss)

1668.84 1667.54 1659.28 1658.02 1653.36 1652.58 1646.92

1652.54 1650.68 1650.02 1649.36

1649.37 1649.85 1650.37 1649.68

ω2 (bend)

3945.51 3944.68 3951.41 3958.40 3949.32 3956.65 3951.32

3943.80 3944.28 3945.96 3953.28

3953.51 3944.17 3948.11 3955.43

ω3 (as)

27.8878b 27.8806d

27.0420 27.1680 27.5106 27.6308 27.6706 27.8080 27.9713

27.5921 27.6575 27.7092 27.8485

27.7763 27.6855 27.7337 27.8732

A

14.5092b 14.5218d

14.6495 14.6595 14.5766 14.5782 14.5218 14.5255 14.4654

14.5057 14.4979 14.4970 14.5003

14.4947 14.5012 14.4987 14.5021

B

9.2869b 9.2777d

9.2594 9.2781 9.2834 9.2977 9.2785 9.2955 9.2882

9.2629 9.2669 9.2723 9.2893

9.2789 9.2713 9.2759 9.2928

C

0.9724e

0.97419 0.97317 0.97264 0.97183 0.97278 0.97182 0.97207

0.97361 0.97336 0.97305 0.97210

0.97267 0.97312 0.97286 0.97190

R0

104.50e

103.40 103.51 104.00 104.11 104.25 104.38 104.64

104.205 104.283 104.334 104.460

104.402 104.303 104.354 104.481

A0

3657.05f

3666.99 3667.23 3666.82 3672.07 3662.67 3668.10 3661.36

3659.44 3657.95 3658.73 3664.27

3665.44 3658.23 3660.16 3665.72

ν1 (ss)

1594.75f

1618.02 1617.33 1606.37 1605.16 1600.33 1599.35 1593.68

1599.95 1598.75 1597.41 1596.56

1597.92 1597.85 1597.41 1596.54

ν2 (bend)

3755.93f

3758.22 3757.65 3760.84 3766.80 3758.71 3764.73 3759.83

3754.97 3754.81 3756.14 3762.06

3763.31 3755.19 3757.53 3763.47

ν3 (as)

a The core-correlation corrections were based on three-point extrapolated CCSD(T) (with and without core-correlation) energies computed with T/Q/5Z quality basis sets that include diffuse functions and core-correlating functions—see Ref. 11 for basis details. b Reference 66. c Reference 67. d Reference 68. e Reference 70. f Reference 69.

Exp

0.51 0.51 0.50 0.50

Fitting rms ×10−16

ADZ-R12 ATZ-R12 AQZ-R12 AQZ-R12+coreAa

QFF

TABLE I. Least-squares fitting error (a.u.2 ), equilibrium geometry(Å/◦ ), harmonic frequency (cm−1 ), vibrationally averaged rotational constants (cm−1 )/structure (Å/◦ ), and vibrational fundamentals (cm−1 ) computed on various H2 O QFFs. See text for details of these QFFs.

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2.9 cm−1 . The (VQZ-F12+coreA – CTQ5) differences for the rotational constants are relatively small, ranging from −0.1228 (A) to 0.0011 (C) cm−1 . This level of agreement is significantly better than we found when trying to include explicitly core-correlation into the R12 calculations, especially for the rotational constants (see the supplementary material). We are evaluating recently published core-correlation extensions of the cc-pVXZ-F12 basis sets72 as well as investigating improved methods for describing core-correlation effects in R12 methods (the results will be reported separately). Comparison to the high-resolution experimental values given in Table I suggests that it is not possible to say which is better, QZ-F12+coreA or our CTQ5 extrapolated basis set results, especially since higher-order correlation effects were not included. The main point, however, is that given that the QZF12+coreA calculation requires significantly fewer computational resources, this is a very encouraging development. B. Comparison of R12 and basis set extrapolation: N2 H+

Table II contains molecular and spectroscopic data computed for N2 H+ at the various levels of theory. Fundamental vibrational frequencies have been computed using both second-order perturbation theory and variationally. Examination of the AXZ-R12 results shows that there is not really a monotonic improvement on going from X = D to T to Q, but that the differences are generally small. By comparison, the XZ-F12 results usually show a monotonic improvement, but again the differences are generally small. The core-correlation correction was computed in two ways, one in which diffuse functions were included and one without diffuse functions (see the footnote to Table II for more details). Except for the bend fundamental, ν 2 , the difference between the two approaches is small. Comparing the AQZR12+coreA and the QZ-F12+coreA sets of results shows very good agreement. Specifically, the range of differences for the fundamental vibrational frequencies (variational results) is 0.7–2.8 cm−1 , while the differences for the two bond distances are only 0.000 017 and 0.000 166 Å for the N–N and N–H bonds, respectively. For the basis set extrapolation approach, we have included results from both a 2pt extrapolation73 and the 3pt approach19 described earlier. Consistent with our earlier findings,11–13 we see that the 2pt ACQ5 results are similar to the 3pt ACTQ5 quantities, while the 2pt ACTQ values exhibit larger differences—as much as 8.8 cm−1 for ν 1 . It has been noted previously that at least the 5Z basis set is needed in the extrapolation procedure in order to obtain accurate results, and since the additional cost of the TZ calculations is small, the 3pt extrapolation formula is recommended. Moving to an evaluation of R12 methods versus oneparticle basis set extrapolation, we compare the AQZR12+coreA results with the 3pt ACTQ5 values. The differences in the equilibrium structure bond distances are similar to those found with H2 O, specifically the (AQZ-R12+coreA – ACTQ5) difference is 0.000 21 and –0.000 53 Å for R(N– N) and R(N–H), respectively. The differences for the fundamental vibrational frequencies are somewhat larger, be-

J. Chem. Phys. 133, 244108 (2010)

ing 7.0, 5.1, and 0.0 cm−1 for ν 1 , ν 2 , and ν 3 , respectively, using the variational results (the differences are essentially the same for the perturbation theory quantities). Interestingly, however, the differences for the harmonic frequencies are somewhat different, being 11.2, 3.1, and 0.4 cm−1 , for ω1 through ω3 , respectively, suggesting some cancellation of errors. The (AQZ-R12+coreA – ACTQ5) difference for the rotational constant, B0 , is smaller than that found with H2 O, being only –0.000 42 cm−1 . On the whole, the agreement between AQZ-R12+coreA and ACTQ5 for the experimental observables is slightly worse than found for H2 O, but it is still very good. For N2 H+ , we have included corrections for scalar relativity and higher-order valence electron correlation (corecorrelation was already included explicitly), and these results are also given in Table II. The higher-order correlation correction was computed in three ways: the internally contracted, multireference averaged coupled-pair functional method (denoted +ACPF in Table II); the internally contracted multireference configuration interaction method (denoted +MRCI); and the MRCI plus the multireference analog of the Davidson correction (denoted +MRCIQ). All of these were computed with the AQZ one-particle basis set. Comparison of the spectroscopic data for the three approaches to including the higher-order correlation correction shows that for most of the data the +ACPF and +MRCIQ methods yield very similar results, with the one exception being the fundamental frequency for the bend ν 2 . The two methods give nearly identical values for the harmonic frequency, ω2 , so this is due entirely to differences in the cubic and quartic parts of the QFF. The +MRCI quantities exhibit the usual problem in that the higher-order correlation correction is underestimated. Comparison of the +ACPF and +MRCIQ results with experiment shows that the +ACPF method yields a much more accurate prediction for the fundamental frequency for the bend and that they are similar otherwise. Based on these observations, the best computed spectroscopic data for N2 H+ given in Table II are for the +ACPF row (which is in full ACTQ5+rel+ACPF). A more detailed comparison with experiment is given later. Comparison of our +ACPF results with earlier theoretical studies shows that the best previous theoretical values are given by Brites and Hochlaf.32 The agreement between our +ACPF quantities and Brites and Hochlaf’s values is reasonable except for the equilibrium structure and the rotational constant where the deviations are somewhat larger. However, this agreement is somewhat fortuitous since they did not include corrections for scalar relativity, core-correlation, nor higher-order correlation. C. Comparison of R12 and basis set extrapolation: NO2 +

The next example is given by computing QFFs at the various levels of theory for NO2 + . The molecular constants and spectroscopic data computed for NO2 + from the various QFFs are presented in Table III. For NO2 + , core-correlation was added as a correction for the R12 methods and the basis set extrapolation methods. It was computed using a 2pt

0.97 0.95 0.95 0.94 0.94

0.99 0.95 0.95 0.92 0.94 0.95 0.95 0.96

0.96

0.95

DZ-F12 TZ-F12 QZ-F12 QZ-F12+coreCa QZ-F12+coreAa

ACTZ ACQZ AC5Z ACTQ ACQ5 ACTQ5 ACTQ5+rel +ACPF

+MRCI

+MRCIQ 1.031b 1.034c 1.0342d 1.0336f 1.034 60g

1.094b 1.096c 1.0942d 1.0928f 1.092 698g 701.5d 693.23e

3400.23e

694.70

699.74

688.20 699.83 700.46 708.18 701.10 699.02 698.37 695.47

704.14 702.96 699.54 703.48 703.65

686.00 696.97 698.00 701.92 702.09

ω2 (bend)

3400.1d

3397.97

3402.76

3401.33 3413.31 3408.92 3422.13 3404.31 3400.12 3398.07 3398.59

3401.25 3402.20 3402.89 3407.94 3407.97

3405.97 3402.91 3406.23 3411.26 3411.30

ω1 (N–H)

2269.15e

2299.9d

2296.68

2303.60

2288.59 2301.76 2306.05 2311.29 2310.54 2309.61 2308.89 2297.74

2292.34 2297.44 2299.35 2308.86 2308.90

2298.90 2296.56 2300.48 2309.98 2310.02

ω3 (N–N)

1.562 16g

1.562 320

1.56440

1.55375 1.56060 1.56283 1.56560 1.56518 1.56468 1.56527 1.56256

1.55516 1.55768 1.55861 1.56428 1.56428

1.55616 1.55759 1.55873 1.56439 1.56440

Be

1.553 92e 1.553 97g,h 1.554 05k

1.550 93b 1.5587d

1.554 19

1.55631

1.54562 1.55239 1.55469 1.55735 1.55711 1.55665 1.55724 1.55447

1.54691 1.54943 1.55040 1.55605 1.55605

1.54816 1.54942 1.55058 1.55623 1.55623

B0

1.097 22

1.09641

1.10063 1.09795 1.09699 1.09600 1.09598 1.09614 1.09589 1.09712

1.09994 1.09894 1.09858 1.09640 1.09640

1.09952 1.09903 1.09856 1.09638 1.09638

R0 (N–N)

1.030 73

1.03040

1.03146 1.03083 1.03089 1.03035 1.03094 1.03110 1.03112 1.03069

1.03246 1.03219 1.03190 1.03107 1.03107

1.03169 1.03166 1.03159 1.03076 1.03076

R0 (N–H)

b

CoreC and coreA corrections are computed as E(3pt CCSD(T) with core) – E(3pt CCSD(T) no core) using CT/Q/5Z and ACT/Q/5Z basis sets, respectively. Reference 33. c Reference 34. d Reference 32 (CCSD(T)-F12b/aug-cc-pV5Z results are quoted). e References. 35–36. f Reference 39. g Reference 40. h Reference 37. i Reference 41. j Reference 42. k Reference 38.

1.0333352

1.0328798

1.0342001 1.0331086 1.0331939 1.0323228 1.0332846 1.0335407 1.0335852 1.0332789

1.0345792 1.0342889 1.0341154 1.0331871 1.0331788

1.0345408 1.0340252 1.0339474 1.0330208 1.0330125

Re (N–H)

1.0928736

1.0920948

1.0962345 1.0936134 1.0926806 1.0917063 1.0917033 1.0918606 1.0916105 1.0927763

1.0955842 1.0945982 1.0942458 1.0920899 1.0920891

1.0951767 1.0946812 1.0942272 1.0920730 1.0920722

Re (N–N)

688.37e 686.80j

3233.95i

687.52b 685.6d

679.81

686.33

689.59 688.97 689.53 688.88 690.09 690.42 690.38 692.94

687.82 687.25 687.52 690.63 694.88

684.27 686.73 688.71 691.75 696.02

ν 2 (bend) PT

3233.96e

3233.60b 3232.9d

3230.25

3235.22

3236.79 3239.51 3237.26 3241.85 3234.89 3233.41 3231.87 3232.10

3231.57 3231.28 3232.02 3236.28 3237.36

3236.75 3233.75 3234.77 3239.03 3240.12

ν 1 (N–H) PT

2257.87e

2247.07b 2262.7d

2258.60

2265.68

2252.41 2264.03 2268.39 2272.51 2272.97 2272.36 2271.61 2260.62

2254.97 2260.10 2261.81 2270.88 2271.15

2261.47 2259.40 2263.05 2272.12 2272.38

ν 3 (N–N) PT

3233.95i

3233.96e

3233.60b 3232.9d

3227.41

3232.56

3233.94 3236.88 3234.62 3239.25 3232.38 3230.55 3228.99 3229.55

3228.81 3228.63 3229.38 3233.54 3234.69

3233.98 3231.03 3232.16 3236.45 3237.52

ν 1 (N–H) VAR

686.80j

688.37e

687.52b 685.6d

676.80

683.12

683.85 685.27 685.88 686.24 686.48 686.56 686.45 688.20

685.01 684.31 684.02 687.25 690.93

679.09 683.01 684.79 687.94 691.63

ν 2 (bend) VAR J = 1

2257.87e

2247.07b 2262.7d

2258.63

2265.71

2252.59 2264.06 2268.42 2272.40 2272.97 2272.42 2271.67 2260.73

2254.98 2260.11 2261.84 2270.89 2271.20

2261.58 2259.47 2263.10 2272.14 2272.45

ν 3 (N–N) VAR

Comparison of basis set extrapolation to R12

a

Exp

Theory

0.94 0.94 0.94 0.94 0.94

Fitting rms ×10−16

ADZ-R12 ATZ-R12 AQZ-R12 AQZ-R12+coreCa AQZ-R12+coreAa

QFFs

TABLE II. Least-squares fitting error (a.u.2 ), equilibrium geometry(Å), harmonic frequency (cm−1 ), vibrationally averaged rotational constants (cm−1 )/structure (Å/◦ ), and vibrational fundamentals (cm−1 ) computed by second-order perturbation theory (PT) or variationally (VAR by Multimode) on various N2 H+ QFFs. ACPF and MRCI corrections were computed with the aug-cc-pVQZ basis. See text for details of these QFFs.

244108-7 J. Chem. Phys. 133, 244108 (2010)

244108-8

Huang, Valeev, and Lee

J. Chem. Phys. 133, 244108 (2010)

TABLE III. Least-squares fitting error (a.u.2 ), equilibrium geometry (Å), harmonic frequency (cm−1 ), vibrationally averaged rotational constants (cm−1 )/structure (Å/◦ ), and vibrational fundamentals (cm−1 ) (computed with second-order perturbation theory) from various NO2 + QFFs. The anharmonic constant G for the degenerate linear bend is also included for comparison. Sum of rms(×10−16 )

Re

Be

ω1 Sym str

ω2 Linear

ω3 Asym str

ν1 Sym str

ν2 Linear

ν3 Asym str

R0

B0

G (cm−1 )

ADZ-R12 ATZ-R12 AQZ-R12 AQZR12+core/TQa

1.35 1.41 1.34 1.31

1.12130 1.12112 1.12066 1.11866

0.41912 0.41926 0.41960 0.42110

1401.30 1398.51 1401.24 1407.69

628.31 631.81 633.12 636.51

2414.96 2410.34 2413.15 2423.86

1393.43 1391.27 1393.57 1399.79

622.68 625.82 626.84 629.99

2354.77 2350.34 2352.47 2362.48

1.12418 1.12403 1.12358 1.12156

0.41722 0.41734 0.41767 0.41917

–1.109 –1.036 –1.012 –0.994

DZ-F12 TZ-F12 QZ-F12 QZF12+core/TQa

1.34 1.37 1.35 1.32

1.12232 1.12123 1.12082 1.11881

0.41836 0.41917 0.41948 0.42099

1395.78 1399.24 1400.23 1406.67

634.95 633.60 633.35 636.75

2405.15 2410.11 2410.94 2421.64

1388.33 1391.65 1392.66 1398.86

627.58 626.64 627.08 630.24

2345.23 2349.86 2350.34 2360.33

1.12527 1.12415 1.12375 1.12173

0.41642 0.41725 0.41755 0.41905

–0.855 –0.922 –1.009 –0.991

ATZ AQZ A5Z ATQ5 +core/TQa +rel +rel+core/TQ

1.42 1.33 1.34 1.37 1.34 1.40 1.37

1.12529 1.12225 1.12140 1.12077 1.11876 1.12085 1.11883

0.41616 0.41841 0.41905 0.41952 0.42103 0.41946 0.42097

1382.38 1395.60 1397.86 1398.44 1404.90 1396.34 1402.80

626.13 628.90 631.67 635.13 638.57 634.13 637.58

2389.91 2403.40 2406.71 2408.85 2419.61 2403.14 2413.91

1376.10 1387.98 1390.12 1390.79 1397.02 1388.82 1395.04

621.07 623.38 625.49 628.09 631.29 627.40 630.61

2329.99 2342.97 2346.11 2348.07 2358.12 2342.56 2352.57

1.12827 1.12518 1.12433 1.12373 1.12170 1.12381 1.12178

0.41421 0.41648 0.41711 0.41756 0.41907 0.41750 0.41901

–1.120 –1.106 –1.022 –0.904 –0.886 –0.934 –0.916

+rel+core/TQ +ACPF/ATZ +rel+core/TQ +MRCI/ATZ +rel+core/TQ +MRCIQ/ATZ

1.40

1.12027 0.41989 1394.86

628.87

2399.01

1385.74

622.19

2337.26

1.12326

0.41791 –0.960

1.40

1.11796 0.42163 1407.27

635.41

2421.52

1397.51

627.04

2359.09

1.12091

0.41966 –0.785

1.44

1.12016 0.41997 1395.74

629.47

2400.43

1386.11

621.11

2338.21

1.12315

0.41799 –0.788

QFF

Ref. 27

1.122(3)

Ref. 27 Exp [28] a

1375 /10 1384

618 /-7 625

2334 /-63 2344

1401.1

627.7

2376.5

The core/TQ correction is computed as E(2pt CCSD(T) with core) – E(2pt CCSD(T) no core) using the CT/QZ basis sets.

extrapolation of energies from the CTZ and CQZ basis sets— see the footnote to Table III for details. Examination of the AXZ-R12 results shows a mixed situation with regard to monotonic behavior on going from the X = D basis set to the X = Q basis set. The stretching harmonic and fundamental vibrational frequencies exhibit erratic behavior, similar to N2 H+ , but the bond distance (both equilibrium and vibrationally averaged) and the bending frequency (both harmonic and fundamental) exhibit monotonic behavior. The important point again, however, is that the range of values on going from X = D to X = Q is relatively small, as one would expect with a robust R12 approach. The XZ-F12 set of results mirrors the situation for N2 H+ —that is mostly monotonic convergence behavior is found, the bend fundamental being the only exception, and the range of values on going from X = D to X = Q is small. Comparison of the AQZ-R12+core/TQ and the QZ-F12+core/TQ data shows excellent agreement. For example, the range of differences for the fundamental vibrational frequencies is only –0.25 to 2.15 cm−1 . Comparing the R12 (AQZ-R12+core/TQ) and basis set extrapolation (ATQ5+core/TQ) results, we find slightly better agreement than found for N2 H+ . For example, the (AQZR12+core/TQ – ATQ5+core/TQ) differences are 2.8, –1.3,

and 4.4 cm−1 for the fundamental frequencies and 0.0001 cm−1 for B0 . All in all, the agreement is remarkable and very encouraging given the reduced computational cost of the R12 method relative to basis set extrapolation in which the 5Z basis set is needed. For NO2 + , we have computed scalar relativity and higher-order correlation corrections. Similar to N2 H+ , the higher-order correlation correction was computed with the ACPF, the MRCI, and the MRCIQ approaches (with the ATZ basis set). The results are presented in Table III. We again see that the MRCI approach underestimates the higher-order correlation correction, but that the MRCIQ and ACPF methods give very similar quantities. We take as our best results in the current study to be the row labeled +rel+core/TQ+ACPF/ATZ (in full, this is ATQ5+rel+core/TQ+ACPF/TZ), though for NO2 + we could have chosen AQZ-R12 rather than ATQ5 as the base method. Comparison of the molecular and spectroscopic parameters determined from this QFF with the best previous theoretical results27 shows reasonable agreement given that the previous theory did not include one-particle basis set extrapolation, core-correlation, scalar relativity, nor higher-order correlation corrections. A more detailed discussion concerning the

b

1.207558 1.205744 1.205373 1.201363 1.206460 1.205504 1.205224 1.201215 1.206719 1.203659 1.202898 1.201409 1.202099 1.202419 1.210208 1.206847 1.205841 1.204389 1.204785 1.205066 1.200373 1.200772 1.201055 1.207481 1.204060 1.200448 1.20272 1.2033 1.2024

1.064546 1.062776 1.062994 1.061180 1.063367 1.063083 1.062994 1.061180 1.063309 1.062093 1.061735 1.061197 1.061359 1.061465 1.063952 1.063553 1.063121 1.063270 1.062666 1.062572 1.061462 1.060848 1.060753 1.063625 1.062270 1.060845 1.06208 1.0605 1.0625

3503.18 3506.19 3504.32 3512.45 3505.84 3503.61 3504.16 3512.31 3505.61 3508.37 3509.85 3510.40 3511.42 3511.45 3502.47 3501.15 3502.59 3499.86 3504.13 3505.09 3507.61 3512.34 3513.40 3500.97 3507.07 3513.33 3506.9 3501.5

2005.51 2008.21 2008.85 2018.94 2005.87 2008.30 2009.29 2019.41 2009.25 2013.41 2015.25 2016.43 2017.18 2017.07 1995.20 2004.11 2006.68 2010.50 2009.38 2008.63 2020.50 2019.56 2018.85 2004.35 2011.20 2018.23 2011.2 2013.3

3410.73 3411.75 3412.15 3420.32 3412.90 3411.30 3412.35 3420.56 3414.65 3416.05 3417.60 3417.04 3419.24 3419.58 3394.35 3408.64 3410.47 3419.01 3412.40 3410.27 3426.99 3420.63 3418.55 3406.68 3414.07 3421.57 3414.4 3417.6

560.52 616.98 620.31 634.72 618.06 618.61 619.65 634.14 586.86 606.27 623.79 620.11 641.70 644.74 592.94 606.50 612.33 616.23 618.34 617.98 630.77 632.58 632.18 596.50 613.15 630.08 621.7 621.5

734.23 750.29 748.42 747.57 750.19 750.59 749.10 748.27 747.27 747.64 749.52 747.86 751.50 752.19 748.27 745.54 747.19 743.36 748.92 750.18 742.32 748.08 749.40 750.78 746.61 741.81 748.6 746.8

Core-correlation corrections were computed as E(2pt CCSD(T)/ACVTQZ+core) – E(2pt CCSD(T)/ACVTQZ no core). All theory and Exp reference values were taken from Tables I and V from Ref. 78.

0.39 0.28 0.22 0.20 0.29 0.27 0.30 0.32 0.30 0.28 0.28 0.28 0.29 0.29 0.28 0.28 0.35 0.29 0.49 0.57 0.30 0.51 0.59 0.30 0.29 0.29

Sum of rms ×10−16 Re (C–C) Re (C–H) ω1 (σ g ) ω2 (σ g ) ω3 (σ u ) ω4 (π g ) ω5 (π u ) 1.17398 1.17761 1.17804 1.18495 1.17624 1.17779 1.17825 1.18517 1.17589 1.18105 1.18238 1.18487 1.18376 1.18323 1.17045 1.17557 1.17728 1.17933 1.17908 1.17872 1.18626 1.18601 1.18565 1.17460 1.18037 1.18649 1.18245 1.1824

Be 1.16860 1.17165 1.17220 1.17931 1.17036 1.17186 1.17239 1.17942 1.17019 1.17532 1.17651 1.17912 1.17777 1.17720 1.16438 1.16980 1.17151 1.17378 1.17331 1.17289 1.18083 1.18036 1.17993 1.16866 1.17462 1.18096 1.17670 1.17665

B0 1.213581 1.211660 1.211234 1.207043 1.212347 1.211398 1.211084 1.206937 1.212705 1.209538 1.208733 1.207216 1.207890 1.208214 1.216292 1.212761 1.211694 1.210180 1.210574 1.210863 1.206024 1.206422 1.206713 1.213520 1.212499 1.206097 1.20857

1.057617 1.057558 1.057703 1.055994 1.058106 1.057888 1.057743 1.056070 1.057251 1.056426 1.056512 1.055796 1.056571 1.056773 1.058439 1.057949 1.057697 1.057587 1.057425 1.057428 1.055916 1.055738 1.055740 1.057970 1.057868 1.055470 1.05660

3370.18 3367.58 3359.11 3367.28 3366.69 3366.52 3368.42 3376.40 3369.80 3373.45 3373.07 3376.06 3372.68 3371.67 3362.00 3363.15 3374.51 3363.98 3386.65 3391.18 3371.70 3394.41 3398.91 3365.12 3369.85 3375.51 3371.0 3371.66

R0 (C–C) R0 (C–H) ν 1 (σ g ) −12.234 −13.846 −14.550 −13.037 −14.512 −14.588 −13.895 −12.582 −14.040 −13.270 −14.077 −12.754 −14.919 −15.441 −14.164 −16.073 −8.345 −17.252 −0.152 3.401 −15.870 −1.039 4.515 −14.583 −14.098 −13.238 −13.76 −12.979

X14 1970.48 1971.52 1961.27 1972.84 1969.52 1972.18 1973.50 1984.60 1972.21 1977.63 1979.07 1981.54 1980.57 1980.02 1958.21 1967.62 1970.10 1974.51 1972.82 1971.98 1985.51 1984.01 1983.18 1967.04 1975.13 1983.64 1975.1 1974.76

3287.06 3282.19 3302.47 3309.28 3283.18 3281.83 3284.04 3291.04 3286.61 3289.63 3288.59 3291.69 3287.57 3286.48 3279.63 3282.25 3280.91 3284.79 3279.97 3279.00 3291.80 3287.05 3286.05 3282.65 3286.38 3291.08 3286.9 3288.75

688.98 606.47 498.77 523.08 616.45 606.08 609.78 630.69 632.58 655.56 617.64 671.73 583.14 565.11 588.30 571.35 608.73 562.63 649.31 668.54 584.69 668.62 687.16 611.00 609.85 615.08 612.2 612.59

ν 2 (σ g ) ν 3 (σ u ) ν 4 (π g )

48.430 3.281 − 32.452 − 31.687 6.502 3.343 3.083 2.669 22.027 23.294 4.282 24.138 − 13.595 − 20.712 5.369 − 4.546 6.499 − 10.770 18.261 24.805 − 10.804 17.252 23.570 12.898 4.819 − 1.809 3.22 3.671

X44

738.68 729.76 686.58 693.39 731.55 729.07 732.95 739.95 736.22 736.37 734.19 736.49 732.07 730.90 723.04 726.86 724.26 730.84 722.93 721.91 737.57 729.71 728.65 726.91 731.36 738.61 732.0 730.36

ν 5 (π u )

Comparison of basis set extrapolation to R12

a

AVDZ-R12fix AVTZ-R12fix AVQZ-R12fix AVQZ-R12fix+corea VDZ-F12 VTZ-F12 VQZ-F12 VQZ-F12fix+corea CVTZ CVQZ CV5Z CVTQZ CVQ5Z CVTQ5Z AVTZ AVQZ AV5Z AVTQZ AVQ5Z AVTQ5Z AVTQZ+corea AVQ5Z+corea AVTQ5Z+corea ACVTZ ACVQZ ACVTQZ Theoryb Expb

QFFs

TABLE IV. Least-squares fitting error (a.u.2 ), equilibrium geometry (Å), harmonic frequency (cm−1 ), vibrationally averaged rotational constants (cm−1 ) / structure (Å), and vibrational fundamentals (cm−1 ) computed by 2nd-order perturbation theory (PT) on various C2 H2 QFFs. See text for details of these QFFs.

244108-9 J. Chem. Phys. 133, 244108 (2010)

244108-10

Huang, Valeev, and Lee

J. Chem. Phys. 133, 244108 (2010)

TABLE V. Spectroscopic constants and vibrationally averaged structures of N2 H+ isotopologues. Computed from the ACTQ5+rel+ACPF QFF using second-order perturbation theory and compared to available experiments. Units are cm–1 /Å unless specified. 14 N H+ 2

Expa

ω1 (N–H/D) ω2 (bend) ω3 (N–N) Be B0 B(ν 1 ) B(ν 2 ) B(ν 3 ) α1 α2 α3 Re (N–N) Re (N–H/D) R0 (N–N) R0 (N–H/D) ν 1 (N–H/D) ν 2 (bend) ν 3 (N–N) X11 X22 X33 X12 X13 X23 G22 −103 Q φ aaa (Hz) De (cm–1 ) τ aaaa (MHz) a

14 N H+ 2

3398.59 695.47 2297.74 1.562 43 1.553 97 1.541 36 1.557 47 1.542 97 0.012 92 − 0.003 35 0.011 00 1.033 59 1.092 77

3233.95 686.80 2257.87

− 23.663

2.93E–6

1.562 56 1.554 47 1.541 99 1.558 08 1.543 52 0.012 47 − 0.003 62 0.010 95 1.033 28 1.092 78 1.097 12 1.030 69

15 N H+ 2

3380.65 689.62 2229.27 1.480 12 1.472 55 1.461 42 1.475 66 1.462 35 0.011 14 − 0.003 10 0.010 19 — — 1.097 05 1.030 44

14 N15 NH+

3382.44 690.74 2268.67 1.529 54 1.521 64 1.509 96 1.524 90 1.510 96 0.011 68 − 0.003 27 0.010 67 — — 1.097 09 1.030 55

15 N14 NH+

14 N D+ 2

Expa

3396.54 694.36 2259.25

14 N D+ 2

2719.84 550.37 2065.92

1.513 67 1.505 92 1.494 00 1.509 37 1.495 45 0.011 92 − 0.003 45 0.010 47 — — 1.097 08 1.030 58

3232.10 692.94 2260.62

3214.97 687.30 2194.38

3216.85 688.35 2232.50

3229.95 691.89 2223.43

− 68.086 1.914 − 11.932 − 20.724 − 19.172 −3.674 3.922 8.272 0.0476 2.87E–6 –0.3440

− 68.913 1.925 − 11.432 − 19.926 − 15.859 − 4.096 3.916 7.485 0.0389 2.58E–6 –0.3099

− 68.554 1.944 − 11.796 − 20.051 − 16.856 − 4.150 3.877 7.980 0.0435 2.76E–6 –0.3304

− 68.495 1.893 − 11.585 − 20.585 − 18.037 − 3.629 3.962 7.775 0.0426 2.70E–6 –0.3233

1.291 66 1.286 05 1.274 51 1.290 60 1.278 07 0.012 36 − 0.004 15 0.007 97 1.033 59 1.092 77

2636.98 543.18 2024.04

− 18.207

2.03E–6

1.291 81 1.286 42 1.274 09 1.291 05 1.278 74 0.012 34 − 0.004 63 0.007 68 1.033 28 1.092 78 1.096 13 1.033 48 2634.84 547.72 2026.66

15 N D+ 2

2665.11 542.96 2032.52 1.233 55 1.228 47 1.217 13 1.232 61 1.221 37 0.011 34 − 0.004 14 0.007 10 — — 1.096 12 1.033 26 2580.25 540.48 1996.08

14 N15 ND+

2680.41 544.38 2058.80 1.273 54 1.268 22 1.256 42 1.272 54 1.260 75 0.011 80 − 0.004 32 0.007 47 — — 1.096 14 1.033 38 2596.44 541.85 2020.25

15 N14 ND+

2703.82 548.96 2040.97 1.252 39 1.247 24 1.235 36 1.251 69 1.239 94 0.011 88 − 0.004 44 0.007 30 — — 1.096 11 1.033 37 2618.00 546.37 2003.78

− 21.571 − 22.783 − 21.787 − 22.455 0.799 0.828 0.853 0.772 − 9.533 − 8.637 − 9.178 − 8.957 − 18.066 − 17.173 − 17.309 − 17.954 − 47.567 − 44.244 − 46.183 − 45.918 3.597 2.955 2.903 3.682 2.191 2.144 2.108 2.228 7.144 6.603 7.020 6.732 0.0349 0.0303 0.0334 0.0318 1.97E–6 1.79E–6 1.91E–6 1.85E–6 –0.2368 –0.2147 –0.2294 –0.2220

Reference 42 and references therein (fundamentals were taken from Ref. 43).

comparison between our best results and experiment28–31 is given later, but it is clear that there is some disagreement, especially for the antisymmetric stretch ν 3 . D. Comparison of R12 and basis set extrapolation: C2 H2

We decided to compare the R12 methods to basis set extrapolation for a larger molecule, but also one for which one-particle basis set deficiencies can be a problem. Acetylene, C2 H2 , was an excellent choice because it was established more than 20 years ago that molecules with C–C multiple bonds can exhibit unusual behavior for their harmonic and fundamental vibrational frequencies (for example, see Refs. 74 and 75). The vibrational modes affected most are bending frequencies and for acetylene this is especially the π g mode, but also to a lesser extent the π u mode. The fundamental vibrational frequencies of the other modes are affected through unusually large or small anharmonic constants that connect to the two π modes, especially the π g mode. This one-particle basis set deficiency has been investigated over the years and various solutions have been suggested—for example, see Refs. 76 and 77. The common theme is that there needs to be a balance between saturation in low angular mo-

mentum functions and inclusion of higher angular momentum basis functions. It is also true that this basis set deficiency is larger for some molecules relative to others and since C2 H2 exhibits one of the larger effects, it is an ideal test case for comparison of the R12 and basis set extrapolation approaches. Contained in Table IV are results for C2 H2 obtained with the R12 and conventional basis set approaches (including basis set extrapolation) investigated in the current study. Examination of the AXZ-R12 harmonic and fundamental frequencies shows that the variation on going from X = D to X = Q is much larger than found for H2 O, N2 H+ , or NO2 + , especially for ω4 and ν 4 , the π g mode. The XZ-F12 results exhibit less erratic behavior, especially for ω4, though the variation for ν 4 is still larger than found for H2 O, N2 H+ , or NO2 + . However, the basis set extrapolation results exhibit an even larger erratic behavior, especially when 5Z results are included in the extrapolation procedure. Examination of the ATQ5Z+core results shows that this erratic behavior extends to the symmetric C–H stretching fundamental ν 1 through the X14 anharmonic constant that is much too small in magnitude (see Table IV). Using simply CCSD(T) without including basis set extrapolation, such as found in Ref. 78, can lead to very good spectroscopic data for C2 H2 , but it is clear from the results in Table IV that extrapolation to the one-particle basis set limit is

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J. Chem. Phys. 133, 244108 (2010)

TABLE VI. Spectroscopic constants of NO2 + , computed from the ATQ5+rel+core+ACPF QFF using second-order perturbation theory. See text for details of the methods. Units are in cm−1 unless specified. 14 N16 O + 2

14 N16 O + 2

Expa

ω1 (sym) ω2 (bend) ω3 (asym)

2399.01 628.87 1394.86

14 N18 O + 2

2358.00 618.12 1314.91

15 N16 O + 2

2342.90 614.16 1394.86

15 N18 O + 2

2300.88 603.14 1314.91

Be B0

0.4176 0.417

0.419 89 0.417 91

0.373 13 0.371 44

0.419 89 0.417 93

0.373 13 0.371 46

R0 (N–O) / Å

1.123

1.123 26

1.123 15

1.123 23

1.123 12

ν 1 (sym) PT ν 2 (bend) PT ν 3 (asym) PTb

2376.5 627.7 1401.1

X11 X22 X33 b X12 X13 X23 G22 –103 Qc φ aaa (Hz) De (MHz) τ aaaa (MHz)

2337.26 622.19 1385.74

2298.54 611.71 1315.13

–16.740 1.692 –3.819 –15.918 –24.702 –5.669 –0.960 0.726 –7.7E–5 0.0046 –0.0183

(0.020d )

2283.90 607.76 1382.82

–16.283 1.639 –3.393 –15.511 –22.766 –5.270 –0.932 0.584 –5.4E–5 0.0036 –0.0144

2244.16 597.02 1309.49

–15.900 1.612 –3.819 –15.103 –24.203 –5.526 –0.914 0.744 –7.7E–5 0.0046 –0.0183

–15.443 1.558 –3.393 –14.696 –22.283 –5.132 –0.885 0.598 –5.4E–5 0.0036 –0.0144

14 N18 O16 O+

15 N18 O16 O+

2379.17 623.51 1354.45

2322.63 608.68 1354.38

0.396 110 0.394275 1.123 195 1.123 217 2318.53 616.97 1349.37 –16.545 1.662 –3.602 –15.709 –23.692 –5.463 –0.943 0.652 –6.6E–5 0.0041 –0.0163

0.396 090 0.394277 1.123 184 1.123 160 2264.72 602.42 1345.31 –15.708 1.582 –3.604 –14.894 –23.188 –5.322 –0.896 0.668 –6.7E–5 0.0041 –0.0163

a

Reference 28 and references therein. Affected by Fermi resonance ν 3 = 2ν 2 . c Rotational l-type doubling constant. d Reference 30. b

TABLE VII. Force constants and equilibrium structure (Å) of N2 H+ . See text for details of the methods. Based on V = (Fij · i j )/2 + (Fijk · i j k )/6 + (Fijkl · i j k l )/24, where unrestricted integer indices, 3≥i,j,k,l≥1, are used. The internal coordinate displacements were defined as: 1 = rN–N – rN–N (eq), 2 = rN–H – rN–H (eq), 3 = sin(θ HNH ) as θ HNH (eq) = 0◦ . Only unique force constants are given. Units of force constants are mdyn/Ån .radm appropriate for an energy unit of mdyn Å (1 mdyn Å ≡ 1 aJ). QZ-F12 rN–N (eq)/Å rNH (eq)/Å F11 F21 F22 F33 = F44

1.0942458 1.0341154 23.545922 –0.231 614 6.150225 0.232499

AQZ-R12 1.0942272 1.0339474 23.565638 –0.230762 6.163152 0.231408

AQZ-R12+coreAa 1.0920722 1.0330125 23.766667 –0.227646 6.180081 0.233631

ACTZ

ACQZ

AC5Z

ACTQ5

1.0962345 1.0342001

1.0936134 1.0331086

1.0926806 1.0331939

1.0918606 1.0335407

23.317025 –0.236349 6.147109 0.225169

23.580306 –0.226599 6.191756 0.232260

23.677766 –0.224371 6.173344 0.232655

23.766216 –0.223210 6.137527 0.231779

ACTQ5 +rel+ACPF 1.0927763 1.0332789 23.498881 –0.217617 6.138109 0.229391

F111 F211 F221 F222 F331 = F441 F332 = F442

–173.1497 0.0300 0.2672 –40.1156 –0.8791 –0.1491

–173.2918 0.0333 0.2710 –40.1718 –0.8786 –0.1389

–174.8647 0.0290 0.2714 –40.3048 –0.8791 –0.1399

–172.1764 0.02478 0.3117 –39.9573 –0.8851 –0.1319

–173.4850 0.0024 0.2275 –40.5525 –0.8801 –0.1395

–174.2279 0.0444 0.2610 –40.2802 –0.8727 –0.1357

–175.0449 0.1101 0.3260 –39.7646 –0.8634 –0.1289

–174.4337 0.0986 0.3138 –39.7680 –0.8604 –0.1252

F1111 F2111 F2211 F2221 F2222 F3311 F3321 F3322 F3333 F3344

1016.35 0.42 –0.66 –0.68 232.36 0.79 0.72 0.06 1.12 0.37

1015.61 0.44 –0.72 –0.69 230.14 0.85 0.75 0.01 1.20 0.40

1012.95 0.33 –0.80 –0.80 229.83 0.86 0.78 0.19 1.30 0.43

1024.76 1.02 0.08 –0.51 232.82 1.17 0.99 0.43 1.45 0.48

1015.15 0.49 –0.55 –0.51 232.08 0.85 0.70 0.01 1.16 0.38

1013.44 0.13 –0.70 –0.97 230.14 0.84 0.69 0.05 1.14 0.38

1012.92 –0.28 –0.80 –1.64 227.47 0.90 0.74 0.18 1.18 0.39

1012.80 –0.22 –0.78 –1.67 227.42 1.17 0.90 0.25 1.35 0.45

a

= F4411 = F4421 = F4422 = F4444

CoreA corrections are computed as E(3pt CCSD(T) with core) – E(3pt CCSD(T) no core) using ACT/Q/5Z basis sets.

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TABLE VIII. Force constants and equilibrium structure (Å) of NO2 + . See text for details of the methods. Based on V = (Fij · i j )/2 + (Fijk · i j k )/6 + (Fijkl · i j k l )/24, where unrestricted integer indices 3≥i,j,k,l≥1 are used. The internal coordinate displacements were defined as: 1/2 = r(N–O1/2 ) – rEQ (N–O), 3/4 = sin(θ ONO ) for the degenerate linear bend (only one component, 3 or 4 , should be included in the formula). Units of force constants are mdyn/Ån .radm appropriate for an energy unit of mdyn Å (1 mdyn Å ≡ 1 aJ). QZ-F12 RN–O (MIN)

1.1208 218

AQZ-R12

AQZ-R12 + core/TQa 1.120 664 3 1.118 657 9

ATZ 1.125 289 5

AQZ 1.122 250 9

A5Z 1.121 398 5

ATQ5+core/TQATQ5+core/TQa +rel +ACPF/TZ 1.120 773 5 1.118 759 9 1.120 273 9 ATQ5

F11 = F22 17.577 692 17.606 357 17.765 966 17.198 760 17.464 573 17.517 174 17.539 596 17.699 417 0.899 658 0.897 693 0.908 724 0.810 410 0.890 764 0.897 563 0.890 538 0.901 230 F21 0.722 944 0.722 220 0.727 350 0.712 199 0.714 640 0.719 851 0.726 956 0.732 210 F33 = F44 F111 = F222 –142.9680 –143.2729 –144.7260 –140.1416 –142.1876 –142.6490 –142.9170 –144.3805 –2.2457 –2.2782 –2.3138 –1.9779 –2.2823 –2.2686 –2.1867 –2.2204 F211 = F221 –1.6987 –1.6909 –1.7024 –1.6545 –1.6780 –1.6948 –1.7151 –1.7269 F331 = F332 = F441 = F442 F1111 = F2222 898.91 899.06 896.81 907.17 898.65 899.17 901.64 899.38 24.76 25.00 25.20 25.67 24.90 24.54 24.18 24.38 F2111 = F2221 –6.50 –6.50 –6.65 –6.47 –6.29 –6.46 –6.75 –6.89 F2211 F3311 = F3322 1.68 1.67 1.57 2.27 1.70 1.67 1.74 1.63 = F4411 = F4422 F3321 = F4421 4.46 4.41 4.50 4.69 4.35 4.38 4.49 4.57 5.21 5.18 5.14 5.48 5.41 5.23 4.98 4.95 F3333 = F4444 1.74 1.72 1.71 1.83 1.80 1.74 1.66 1.65 F3344 a

17.424 622 0.911 218 0.712 044 –142.9 709 –2.2300 –1.7369 897.31 23.74 –6.42 1.92 4.26 5.11 1.70

The core/TQ correction is computed as E(2pt CCSD(T) with core) – E(2pt CCSD(T) no core) using the CT/QZ basis sets.

fraught with difficulty due to the well known and documented one-particle basis set deficiency that accompanies the description of C–C multiple bonds. It is especially troubling in this case that extrapolation with the 5Z results leads to erratic behavior, since we have shown in many cases that in order to obtain accurate spectroscopic data, it is necessary to include 5Z energies in the extrapolation procedure.11, 13 As mentioned previously, however, C2 H2 exhibits a much larger effect than most systems with a C–C multiple bond, and we have used basis set extrapolation effectively in some cases—see Ref. 13, for example. Close examination of Table IV shows that the best results are obtained with the QZ-F12+core approach, making C2 H2 the one case in our study where use of the R12 method is clearly favored over basis set extrapolation. Given this example and the fact that the R12 results for H2 O, NH2 + , and NO2 + are all very close to the values obtained with basis set extrapolation, it is clear that current state-of-the-art R12 methods can yield very accurate quartic force fields, potential energy surfaces, and spectroscopic data. E. Spectroscopic constants of isotopologues: N2 H+ and NO2 +

As discussed in the Introduction, minor isotoplogues of molecules found in the ISM or in other astronomical observations are going to be more easily identified in coming years as the Herschel, SOFIA, JWST, and ALMA telescopes come on-line and begin yielding a significant amount of high precision data. Therefore, Table V contains a more complete list of spectroscopic data for the main isotopologue of N2 H+ as well as all deuterium and 15 N isotopologues. Similarly, Table VI contains a more complete list of spectroscopic data for the main isotopologue of NO2 + and all 15 N and 18 O isotopologues. Where possible, experimental data are in-

cluded for comparison (in parentheses).28–31, 35–44 For N2 H+ , the ACTQ5+rel+ACPF QFF was used to compute the spectroscopic data while for NO2 + the ATQ5+rel+core+ACPF QFF was used. Second-order perturbation theory was used to solve the nuclear Schrödinger equation. Examination of Table V shows that the agreement between experiment and theory is excellent for both 14 N2 H+ and 14 N2 D+ for the various rotational constants, B, and the vibration-rotation interaction constants, α. For 14 N2 H+ , the largest error for a fundamental vibrational frequency is 6.1 cm−1 for ν 2 , while for 14 N2 D+ the largest error of 4.5 cm−1 also occurs for the bending fundamental ν 2 . However, the error for the stretching vibrational fundamentals is less than 3 cm−1 in all cases. The excellent agreement across the board for both 14 N2 H+ and 14 N2 D+ suggests that the theoretical predictions for the other N2 H+ isotopologues should be just as good, and it is hoped these will be useful in the interpretation of future laboratory experiments and/or astronomical observations. For NO2 + , experimental data are only available for the main isotopologue. Agreement between experiment and theory for the rotational constant B0 is reasonable, but as mentioned previously, differences for the fundamental vibrational frequencies are larger than we expect for the residual error in the theoretical values. The largest (theory – experiment) difference, –39.2 cm−1 , occurs for the antisymmetric stretch, while the difference for the symmetric stretch is –15.7 cm−1 and for the bend it is –5.51 cm−1 . The difference for the bend is not outside possible errors in the theoretical methods, as found for N2 H+ , but for both stretching modes, the differences are well outside any remaining error in the theory. Hence we must conclude that the experimental values are in error. Support for this assertion is given by comparison of theory and experiment for the centrifugal distortion constant— agreement between experiment and theory for the magnitude

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Comparison of basis set extrapolation to R12

J. Chem. Phys. 133, 244108 (2010)

TABLE IX. Force constants and equilibrium structure (Å) of C2 H2 . See text for details of the methods. Based on V = (Fij · i j )/2 + (Fijk · i j k )/6 + (Fijkl · i j k l )/24, where unrestricted integer indices, 7≥i,j,k,l≥1, are used. The symmetry internal coordinate displacements were defined as in Ref. 78. Only unique force constants are given. Units of force constants are mdyn/Ån .radm appropriate for an energy unit of mdyn Å (1 mdyn Å ≡ 1 aJ).

rC–C (eq)/Å rC–H (eq)/Å F11 F21 F22 F33 F44 = F55 F66 = F77 F111 F211 F221 F222 F331 F332 F441 = F551 F442 = F552 F661 = F771 F662 = F772 F643 = F753 F1111 F2111 F2211 F2221 F2222 F3311 F3321 F3322 F3333 F4411 = F5511 F4421 = F5521 F4422 = F5522 F4433 = F5533 F6611 = F7711 F6621 = F7721 F6622 = F7722 F6633 = F7733 F6431 = F7531 F6432 = F7532 F4444 = F5555 F6666 = F7777 F6644 = F7755 F6655 = F7744 F5544 F7766 F7654 a

QZ-F12

AQZ-R12

AQZ-R12+ corea

ATZ

AQZ

A5Z

ATQ5

ATQ5 + corea

1.205 223 9 1.062 994 2 6.372 932 –0.156 103 16.332 921 6.378 620 0.347 346 0.156 936 –25.5283 0.2824 –0.1454 –95.4442 –25.5102 0.3291 –0.1360 –0.5547 –0.1442 –1.0619 –0.0904 92.80 –1.16 –1.41 –0.39 473.79 92.84 –1.16 –1.36 92.92 –0.05 0.12 0.21 0.01 –0.01 0.03 –0.01 –0.05 0.00 0.15 –0.21 0.44 –0.14 0.29 0.18 0.33 0.25

1.205 372 6 1.062 994 2 6.373 754 –0.156 217 16.325 310 6.377 853 0.344 672 0.157 282 –25.5084 0.2886 –0.1259 –95.2475 –25.4852 0.3390 –0.1382 –0.5554 –0.1375 –1.0643 –0.0886 92.26 –1.40 –3.07 –1.19 459.32 93.19 –0.21 –2.47 93.07 –1.58 –0.01 –2.30 1.01 –0.43 –1.07 –2.24 1.10 –1.09 0.78 –1.72 –1.40 –0.61 –0.13 –0.49 0.13 1.11

1.201 363 3 1.061 180 2 6.402 014 –0.152 930 16.492 863 6.408 457 0.344 746 0.163 977 –25.6115 0.3239 –0.0892 –96.1965 –25.5948 0.3714 –0.1250 –0.5261 –0.1300 –1.0394 –0.0858 91.46 –1.55 –3.14 –1.20 455.29 92.29 –0.58 –2.73 91.88 –1.66 –0.03 –2.12 1.01 –0.29 –0.95 –2.22 1.10 –1.11 1.43 1.66 1.44 –0.43 –0.02 –0.25 0.37 0.93

1.210 208 4 1.063 952 3 6.366 993 –0.173 612 16.105 220 6.311 482 0.347 197 0.144 190 –25.9822 0.1430 –0.1528 –94.3611 –25.8560 0.4518 –0.1547 –0.5891 –0.1264 –1.0677 –0.0905 95.47 –0.14 –0.86 –1.16 482.09 98.74 0.02 –0.38 109.60 –0.14 0.02 0.07 –0.10 –0.05 0.04 –0.03 0.55 0.02 0.08 –0.42 0.53 –0.14 0.13 0.08 0.26 0.15

1.206 847 0 1.063 552 6 6.364 920 –0.152 796 16.241 281 6.364 752 0.344 416 0.150 564 –25.4833 0.2867 –0.1204 –94.8310 –25.4559 0.3241 –0.1328 –0.5539 –0.1348 –1.0661 –0.0834 92.56 –1.37 –1.67 –0.71 472.79 92.66 –1.25 –1.43 93.12 –0.12 0.06 0.18 –0.03 –0.25 0.05 –0.12 0.06 –0.23 0.45 –0.07 0.05 –0.19 0.15 0.12 0.07 0.29

1.205 840 9 1.063 120 7 6.367 775 –0.156 903 16.289 175 6.371 584 0.345 660 0.153 317 –25.5284 0.2801 –0.1540 –95.1897 –25.4984 0.3354 –0.1356 –0.5509 –0.1398 –1.0492 –0.0841 93.33 –1.14 –1.04 –0.35 473.76 93.16 –1.29 –1.20 93.26 0.30 0.30 0.78 –0.10 0.28 –0.70 –0.24 –0.16 –0.55 0.14 –0.44 0.60 –0.30 0.26 –0.09 0.18 1.32

1.205 066 0 1.062 571 7 6.372 888 –0.166 927 16.331 573 6.370 845 0.348 075 0.155 987 –25.7001 0.2385 –0.2110 –95.6261 –25.6467 0.3758 –0.1451 –0.5552 –0.1450 –1.0234 –0.0852 95.09 –0.57 0.08 0.08 477.10 95.15 –1.08 –0.61 96.87 0.95 0.64 1.67 –0.19 1.12 –1.78 –0.37 –0.35 –0.96 –0.40 –1.00 1.53 –0.43 0.46 –0.37 0.40 2.79

1.201 054 9 1.060 752 9 6.401 976 –0.163 146 16.500 906 6.401 819 0.346 159 0.162 548 –25.8140 0.2591 –0.1875 –96.6493 –25.7577 0.4032 –0.1410 –0.5435 –0.1388 –1.0042 –0.0782 94.29 –0.72 0.02 0.08 473.07 94.25 –1.45 –0.87 95.69 0.88 0.62 1.85 –0.18 1.27 –1.65 –0.35 –0.35 –0.98 0.25 –0.92 1.51 –0.25 0.58 –0.12 0.64 2.61

Core-correlation corrections were computed as E(2pt CCSD(T)/ACTQZ+core) – E(2pt CCSD(T)/ACTQZ no core).

is very good, but different signs are obtained. It is hoped that the highly accurate data presented here for 14 N16 O2 + and all the other isotopologues of NO2 + will be useful in the interpretation of future laboratory experiments and astronomical observations. F. Quartic force fields: N2 H+ , NO2 + , and C2 H2

For completeness, some of the QFFs computed in this work for N2 H+ are collected in Table VII, and those for

NO2 + are given in Table VIII. For the most part, the AQZ-R12+coreA and ACTQ5 (or AQZ-R12+core/TQ and ATQ5+core) QFFs are in reasonable agreement, as one would expect from the comparisons of spectroscopic data. In most cases, differences begin to appear in about the third significant digit, except for the smaller quartic constants where differences sometimes appear even in the first significant digit. The QFFs that include extrapolation to the one-particle basis set limit, core-correlation, scalar relativity, and higherorder correlation should be the most accurate available for

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N2 H+ and NO2 + . For the interested reader, we also include various QFFs for C2 H2 obtained in the present work in Table IX. IV. CONCLUSIONS

Accurate CCSD(T) QFFs have been computed for the H2 O, N2 H+, , NO2 + , and C2 H2 molecules and spectroscopic data computed from these. QFFs have been computed using both one-particle basis set extrapolation and alternatively one of the new CCSD(T)-R12 methods, specifically the CCSD(T)R12 method of Valeev.18, 57 The best R12 spectroscopic data are generally in very good agreement with the best results obtained from basis set extrapolation usually with the minor exception of one or two molecular parameters. For H2 O and NO2 + , agreement between results obtained from R12 and basis set extrapolation is very good, for example, differences in fundamental vibrational frequencies of less than 5 cm−1 and usually much less. For these two molecules, it is not really clear which set of results is better. For N2 H+ , differences for the stretching fundamentals are larger, as much as 7 cm−1 for the N–H stretch (and 11 cm−1 for the harmonic frequency). In this case, one-particle basis set extrapolation is preferred, but the R12 QFF is still pretty accurate. For C2 H2 , QFFs obtained from basis set extrapolation suffer significantly from a known one-particle basis set deficiency for molecules that possess C–C multiple bonds. This is especially true for extrapolations that include the 5Z basis set, which is problematic since we have shown several times previously that in order to obtain the most accurate QFF and spectroscopic data from approaches that include one-particle basis set extrapolation, it is necessary to include the 5Z basis set.11, 13 For C2 H2 , there is no question that the R12 approach investigated here is far superior to one-particle basis set extrapolation for obtaining a reliable QFF and spectroscopic data, though we note that conventional CCSD(T) including core-correlation but without basis set extrapolation can yield a very good QFF and spectroscopic data for C2 H2 , provided the basis set is well balanced.78 While this latter comment suggests that there is a cancellation of errors, we also note that the common theme for resolving the one-particle basis set deficiency that has been known to exist for molecules with C–C multiple bonds for more than 20 years is to use a balanced basis set (balanced with respect to saturation of low angular momentum functions and inclusion of higher angular momentum basis functions). Overall, given the excellent quality of the R12 QFFs and spectroscopic data obtained in the present study, it is concluded that use of a robust R12 approach, as we have done here, will replace one-particle basis set extrapolation for high accuracy electronic structure applications, such as computing QFFs and spectroscopic data for comparison to high-resolution laboratory or astronomical observations in the near future. We note, however, that there are still significant advances to be made for R12 methods, such as development of a robust approach that includes core-correlation explicitly. The CCSD(T)R12 approach used in the present study incorporated a reformulation of the V intermediate which was found to yield more systematic (monotonic) basis set convergence. We also demonstrated that this method can robustly

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compute electronic energies with high precision which is absolutely needed for computing QFFs. We conclude that optimization of geminal amplitudes can lead to loss of numerical precision and thus the approach where the amplitudes are fixed from the first-order cusp condition is preferred. These details are discussed in depth in the supplementary material since the focus of the present paper was QFFs and spectroscopic data. Finally, for N2 H+ and NO2 + we have computed QFFs that include one-particle basis set extrapolation, corecorrelation, scalar relativity, and higher-order correlation and have used these to compute highly accurate spectroscopic data for all isotopologues of interest to the astronomy community. For the 14 N2 H+ and 14 N2 D+ isotopologues where highresolution experimental data are available, the agreement between experiment and theory is excellent. For 14 N16 O2 + , agreement with experiment for the rotational constant is good, but agreement for the two stretching fundamental vibrational frequencies is outside the expected remaining uncertainty in the theoretical calculations. Hence we conclude that the experimental values are in error. It is hoped that the highly accurate data presented for the other minor isotopologues of N2 H+ and NO2 + will be useful in the interpretation of future laboratory and astronomical observations.

ACKNOWLEDGMENTS

The authors gratefully acknowledge support from the NASA Herschel GO Program, Cycle 0 TR/LA PID 1022, and the NASA Grant No. 08-APRA08–0050. Dr. David Schwenke is gratefully acknowledged for helpful discussions. X.H. acknowledges the financial support by NASA/SETI Institute Cooperative Agreement NNX09AI49A. E.F.V. gratefully acknowledges the support of the American Chemical Society Petroleum Research Fund (Grant No. 46811-G6), the U.S. National Science Foundation (CAREER Award No. CHE-0847295 and CRIF:MU Award No. CHE-0741927), the Alfred P. Sloan Foundation, and the Camille & Henry Dreyfus Foundation. 1 T.

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