Relative Energies and Geometries of the cis- and trans-HO3 Radicals

Article pubs.acs.org/JPCA

Relative Energies and Geometries of the cis- and trans-HO3 Radicals from the Parametric 2‑Electron Density Matrix Method Erik P. Hoy, Christine A. Schwerdtfeger, and David A. Mazziotti* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States S Supporting Information *

ABSTRACT: The parametric 2-electron reduced density matrix (2-RDM) method employing the M functional [Mazziotti, D. A. Phys. Rev. Lett. 2008, 101, 253002], also known as the 2-RDM(M) method, improves on the accuracy of coupled electron-pair theories including coupled cluster with single−double excitations at the computational cost of configuration interaction with single−double excitations. The cis- and trans-HO3 isomers along with their isomerization transition state were examined using the recent extension of 2-RDM(M) to nonsinglet open-shell states [Schwerdtfeger, C. A.; Mazziotti, D. A. J. Chem. Phys. 2012, 137, 034107] and several coupled cluster methods. We report the calculated energies, geometries, natural-orbital occupation numbers, and reaction barriers for the HO3 isomers. We find that the 2-RDM(M) method predicts that the trans isomer of HO3 is lower in energy than the cis isomer by 1.71 kcal/mol in the correlation-consistent polarized valence quadruple-ζ (cc-pVQZ) basis set and 1.84 kcal/mol in the augmented correlation-consistent polarized valence quadruple-ζ (aug-cc-pVQZ) basis set. Results include the harmonic zeropoint vibrational energies calculated in the correlation-consistent polarized valence double-ζ basis set. On the basis of the results of a geometry optimization in the augmented correlation consistent polarized valence triple-ζ basis set, the parametric 2RDM(M) method predicts a central oxygen−oxygen bond of 1.6187 Å. We compare these energies and geometries to those predicted by three single-reference coupled cluster methods and experimental results and find that the inclusion of multireference correlation is important to describe properly the relative energies of the cis- and trans-HO3 isomers and improve agreement with experimental geometries.

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INTRODUCTION In 1952 at a conference in Chalk River, A. John Coleman recognized that the ground-state energy and properties of an Nelectron atom or molecule could in principle be computed as a linear functional of the 2-electron reduced density matrix (2RDM) rather than the N-electron wave function.1 This observation was the beginning of a new field in electronic structurenamely, reduced-density-matrix mechanics2,3in which the 2-RDM rather than the many-electron wave function is the basic computational variable. Direct calculation of the 2-RDM requires that it be constrained by a set of conditions to derive from the integration of an N-electron density matrix. These constraints, called Nrepresentability conditions by Coleman in 1963,4 are necessary to ensure that the 2-RDM represents a realistic N-electron system.5−8 Three general approaches for the direct calculation of the 2-RDM include: (i) the minimization of the energy with the 2-RDM constrained explicitly by N-representability conditions, which leads to a problem in semidefinite programming,9−13 (ii) the minimization of the energy with the 2-RDM parametrized (or constrained implicitly) by Nrepresentability conditions,3,14−18 and (iii) the calculation of the 2-RDM from the solution of the contracted Schrödinger equation (or its anti-Hermitian part) with cumulant-based reconstruction of the higher RDMs.3,19−23 In this paper we focus on the second category of methods, the parametric 2© 2013 American Chemical Society

RDM methods, and their application to a problem representative of computing small energy differences in chemistry, the relative electronic energies of the HO3 radical’s two isomers. Weinhold and Wilson24,25 developed one of the earliest parametric 2-RDM methods. They parametrized the 2-RDM as a functional of the parameters of the configuration interaction doubles (CID) wave function with the aim of developing a more general parametrization beyond a specific wave function ansatz. More recently, Kollmar26 and one of the authors (Mazziotti)14,15 extended the work of Weinhold and Wilson to obtain 2-RDM parametrizations that are size extensive generalizations of the CID parametrization. These parametrizations can be shown to be a significant generalization of the traditional coupled electron-pair methods. Although the traditional CEPA methods are based solely on perturbative arguments, the parametric 2-RDM methods have parametrizations that consider the N-representability of the 2-RDM. Here we employ the parametric 2-RDM method with the M parametrization, which we denote as the 2-RDM(M) method.3,14,15 The 2-RDM(M) method has an accuracy approaching that of coupled cluster with single, double, and Received: October 24, 2012 Revised: January 24, 2013 Published: February 18, 2013 1817

dx.doi.org/10.1021/jp3105562 | J. Phys. Chem. A 2013, 117, 1817−1825

The Journal of Physical Chemistry A

Article

where 1K and 2V are the one and two-electron reduced Hamiltonian operators, 1D and 2D are the one- and twoelectron reduced density matrices (1- and 2-RDM), and pqkl are the indices for a finite number of spin orbitals. The 2-RDM can be expanded into zeroth-order (D0) and first-order (2T) and higher-order parts, and the 1-RDM can be expanded into second-order (1T) and higher-order parts:3

triple excitations, especially in the presence of some multireference correlation, at a computational cost that is less than coupled cluster with single−double excitations. Applications have been made to several closed-shell molecular systems3,14−18 including the calculation of (i) the small barrier separating oxywater from hydrogen peroxide,16 (ii) the relative energies of the cis−cis and cis−trans isomers of carbonic acid, and (iii) the rotational barrier separating cis- and trans-diazene.17 Recently the 2-RDM(M) method has been extended to open-shell systems,27 which often require a multiconfigurational reference to obtain chemically accurate energies and barriers.28 In this paper we apply the open-shell version of the 2-RDM(M) method to the isomers of the HO3 radical. The hydridotrioxygen molecule (HO3) has been the subject of many experimental and computational studies due to its potential importance in atmospheric, combustion, and organic/ biochemical oxidation processes.29−32 In recent years the molecule’s potential role as a reservoir for atmospheric OH has been particularly well examined.33−40 HO3 was first observed experimentally by Cacace et al. in 1999 using neutralization-reionization mass spectrometry.41 The following year Nelander et al. reported its presence in Ar and H2O ice matrices based on infrared spectroscopy results.42,43 The rotational spectrum of HO3 was measured by Suma et al. in 2005 using Fourier-transform microwave spectroscopy.74 On the basis of the experimental rotational constants and theoretical dipole moment calculations, Suma et al. determined that the trans isomer of HO3 was the source of this rotational spectrum. Subsequent experiments observed the fundamental HOO bend around 1250 cm−1 by irradiating H218O + O2 ice mixtures.44,45 The amount of energy required to dissociate HO3(X2A″) to OH(2Πi) + O2(3Σ−g ) was established to have an experimental upper limit between 5.31 and 6.12 kcal/mol by a series of studies in 2007 and 200833−36 and was given a value of 2.97 ± 0.07 kcal/mol (barrierless) by Le Picard et al. on the basis of kinetic studies.37,38 Additionally, Lester and co-workers confirmed that the trans isomer is the primary contributor to the HO3 rotational spectrum and suggested that cis-HO3 likely appears in the spectrum as broad, unstructured peak.30,33−36 The HO3 molecule has also received significant theoretical attention due to its potential importance in atmospheric processes and the notable difficulties in reconciling theoretical and experimental results.29,31,32,39,40,46−53,53−73 In this study we investigate two of these issues: matching the theoretical to the experimental geometric parameters for the trans-HO3 state (especially the unusually long central oxygen−oxygen bond)32,74 and determining whether the cis- or trans-HO3 isomer is more favorable energetically.31,35,36,39,48,73,74 Some disagreement exists as to the importance of multireference correlation and/or higher-order excitations to the HO 3 problem.31,32,39,73 The parametric 2-RDM(M) method’s results for the cis- and trans-HO3 states are compared to those from single-reference coupled cluster methods as well as previous experimental and theoretical results36,73,74 to better understand the role of correlation in the energies and geometries of the HO3 states.

1

D = 1D0 + 1T + O(3)

2

D = 2D0 + 2T + O(2)

1

D ≈ 1D0 + O(3)

q

q

pq

+

pq 2

∑ 2V kl pqkl

(4)

The 1- and 2-RDMs can then be rewritten as functionals of the first-order components of the 2-RDM, 2T, using contraction relations from the 3- and 4-RDM expansions originally derived to solve the contracted Schrodinger equation.2,3,15,76−79 2

D ≈ 2D( 2T )

(5)

The resulting 2-RDM scales linearly with system size (size extensive) and produces energies that are correct through the third order of renormalized perturbation theory.15 However, this parametrization of the 2-RDM is not Nrepresentable.14,15,26 To enforce approximate N-representability, we apply a subset of the N-representability conditions known as the Cauchy-Schwarz inequalities for the 2-particle (2D) and 2-hole (2Q) matrices, ij

ij

ab

(2Dab)2 ≤ 2Dij 2Dab ij

ij

(6) ab

( 2Q ab)2 ≤ 2Q ij 2Q ab

(7)

Applying these inequalities to the 2-RDM parametrization and averaging the result modifies the class of 2-RDM elements with two occupied {ij} and two unoccupied {ab} orbitals as shown in the equation (where {klcd} are arbitrary spin orbitals),3,14−16 2 ab Δij

ab

= 2T ij

1−

1 4

cd

∑ f ijklabcd | 2T kl |2

(8)

klcd

f abcd ijkl ,

known as the topological factor, is defined in Table 1 and enforces the parametrization on the 2-RDM elements.26,80,81 The parametrization that results from averaging the 2D and 2Q inequalities is referred to as the M parametrization, which is significantly different from traditional coupled electron-pair approximations. It is both size extensive and approximately Nrepresentable as well as applicable to both closed- and openTable 1. Topological Factors f nnov in Eq 8a

THEORY The energy of an N-electron system can be written without approximation as a functional of the 1- and 2-RDMs

∑ 1K p 1D p

(3)

The expanded 1-RDM can be simplified by rotating the Hartree−Fock (zero-order) orbitals to form Brueckner-like orbitals, which causes the 1T components to vanish from the expansion3,15,75

■

E=

(2)

method

0/0

1/0

2/0

0/1

0/2

1/1

2/1

1/2

2/2

D Q M

0 0 0

1 0 0

1 0 1

0 1 0

0 1 1

1 1 1

1 1 1

1 1 1

1 1 1

The factor in the table, f nnov , represents the topological factor, f abcd ijkl , where no is the number of occupied spin orbitals shared by the sets {i, j} and {k, l}, and nv is the number of virtual spin orbitals shared by the sets {a, b} and {c, d}.14,15. a

pq

Dkl

(1) 1818



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shell systems. Including single excitations (1T) in the parametrization explicitly modifies the above parametrization and yields the 2-RDM(M) method used for the calculations in this paper.

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most multireferenced with the transition state having slightly less multireference character. In previous studies of ammonia oxide and oxywater by Hoy et al. and Schwerdtfeger et al., small differences in multireference character were found to alter the relative energy differences between isomers noticeably, especially when the energy differences were small.16,18 The energies from all of the methods including coupled cluster are not invariant to orbital rotations between the occupied and virtual orbitals. Because the natural occupation numbers are not too far from 0 and 1, energy variations from changes in the reference determinant are likely not large. The parametric 2RDM method also has a slight invariance with respect to rotations between occupied orbitals and rotations between virtual orbitals. Calculations of this latter invariance, to be published elsewhere, show that typically it also has a small effect on the energy. Table 3 contains the total electronic energies before the addition of harmonic zero-point vibrational energies (harmonic ZPEs) for each of the states studied in the cc-pVQZ and augcc-pVQZ basis sets. In the cc-pVQZ basis set the parametric 2RDM(M) method recovers an amount of correlation energy between CCSD and CCSD(T), which is consistent with the results from previous studies of closed-shell molecules.16−18,92 In the aug-cc-pVQZ basis set the parametric 2-RDM(M) method yields correlation energies that are lower than those from CCSD(T) by 0−0.0024 H. Previous calculations with the 2-RDM method indicate that it recovers more correlation energy than CCSD(T) only when there are significant contributions from multireference effects.17 In conjunction with the natural-orbital occupation numbers, this finding suggests that the 2-RDM(M) method is capturing additional multireference correlation beyond that from the coupled cluster methods. The energy differences between the HO3 isomers before and after the addition of harmonic zero-point vibrational energies are given in Tables 4 and 5, respectively. Considering the uncorrected results first, we see that the 2-RDM(M) method predicts that the trans-HO3 isomer is lower in energy than the cis isomer in the cc-pVQZ basis set by 0.28 kcal/mol. The addition of augmented functions to the basis set lowers the trans further relative to the cis and thereby increases the predicted energy gap between the two isomers to 0.41 kcal/ mol. With the addition of harmonic zero-point vibrational energies (Table 5), the 2-RDM(M) predicts a trans−cis energy dif ference of −1.71 kcal/mol in the cc-pVQZ basis set and −1.84 kcal/mol in the aug-cc-pVQZ basis set because the cis state has a harmonic zero-point vibrational energy, which is 1.43 kcal/mol larger than the trans harmonic zero-point vibrational energy. The addition of harmonic zero-point vibrational energies or augmented functions does not noticeably change the energy difference between the transition state and the trans isomer for the 2-RDM(M) method which remains at ∼2.04 kcal/mol for all the cases studied. Without including harmonic zero-point vibrational energies (Table 4), all of the coupled cluster methods predict that the cis-HO3 isomer is lower in energy than the trans-HO3 isomer. Among the coupled cluster methods, the energetic favorability of the cis isomer is largest for the CCSD method, which predicts a 0.87 kcal/mol difference in favor of the cis isomer in the cc-pVQZ basis set and smallest for the CCSD(T) method, which predicts a 0.39 kcal/mol difference in favor of the cis isomer. In the aug-cc-pVQZ basis set, the CCSD method predicts a trans−cis gap of 0.69 kcal/mol whereas the

APPLICATIONS

COMPUTATIONAL METHODS We examined two isomers of the HO3 radical and their transition state, cis-HO3, trans-HO3, and their isomerization transition state, using the open-shell versions of 2-RDM(M), coupled cluster with single and double excitations [CCSD],82,83 completely renormalized coupled cluster exploiting left eigenstates [CR-CCSD(T)L or CR-CCL],82,83 and coupled cluster with single and double excitations plus perturbative triple excitations [CCSD(T)].84−89 The CCSD and CR-CCL calculations were performed with the GAMESS electronic structure package,90 and the CCSD(T) results were computed using Gaussian 09.91 The open-shell parametric 2-RDM calculations were performed using a GAMESS implementation of the method discussed in refs 3, 15, 16, 27. Using the numerical finite-difference optimization in GAMESS for the CCSD, CR-CCL, and 2-RDM(M) methods, we optimized the geometries of each molecule to a tolerance of 10−7 Hartrees/Å in the cc-pVTZ and aug-cc-pVTZ basis sets and then calculated single-point energies at these minimum-energy geometries in the cc-pVQZ and aug-cc-pVQZ basis sets. The core orbital of each oxygen was frozen for both the optimization and singlepoint calculations. The energy differences involving the two isomers and their transition state were calculated in the ccpVQZ and aug-cc-pVQZ basis sets both with and without the addition of harmonic zero-point vibrational energies. The harmonic zero-point energies were calculated in the cc-pVDZ basis set by numerical finite differences using the GAMESS electronic structure package for CCSD, CR-CCL, and 2RDM(M) and the Gaussian 09 package for CCSD(T).90,91

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RESULTS We calculated the natural-orbital occupation numbers for each isomer and their transition state using the 2-RDM(M) method in the cc-pVQZ and aug-cc-pVQZ basis sets. The results of these calculations can be found in Table 2. The deviations of Table 2. Natural-Orbital Occupation Numbers Calculated by the 2-RDM(M) Method in the cc-pVQZ and aug-cc-pVQZ Basis Sets HONO-1 molecule transHO3 cis-HO3 TS

HONO

LUNO

ccpVQZ

aug-ccpVQZ

ccpVQZ

aug-ccpVQZ

ccpVQZ

aug-ccpVQZ

0.9485

0.9410

0.4992

0.4954

0.0585

0.0674

0.9521 0.9486

0.9449 0.9401

0.4990 0.5012

0.4966 0.4996

0.0538 0.0542

0.0612 0.0645

the second highest occupied natural-orbital (HONO−1) occupation number from 1.0, the highest occupied naturalorbital (HONO) occupation number from 0.5, and the lowest unoccupied natural-orbital (LUNO) occupation number from 0.0 indicate increasing multireference character. Both the HONO−1 and LUNO numbers show that cis-HO3, transHO3, and their transition state have a small but noticeable amount of multireference character. The trans isomer is the 1819



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Table 3. Total Electronic Energies before Harmonic Zero-Point Vibrational Energies plus 225 Hartrees for the cc-pVQZ and aug-cc-pVQZ Basis Sets molecule

basis set

trans-HO3 cis-HO3 TS trans-HO3 cis-HO3 TS

cc-pVQZ

aug-cc-pVQZ

CCSD

2-RDM(M)

CR-CCL

CCSD(T)

−0.802560 −0.803949 −0.802261 −0.810248 −0.811341 −0.810000

−0.837671 −0.837227 −0.834424 −0.850797 −0.850146 −0.847562

−0.838718 −0.839737 −0.837823 −0.845735 −0.846471 −0.844740

−0.839630 −0.840244 −0.838569 −0.848349 −0.848597 −0.847217

CCSD(T) method predicts a difference of 0.16 kcal/mol in favor of the cis isomer. Additionally, the coupled cluster methods all predict that the trans state is less than 1 kcal/mol lower than the transition state before the addition of harmonic zero-point vibrational energies. Once harmonic zero-point vibrational energies are included, as shown in Table 5, the CCSD and CR-CCL methods in the aug-cc-pVQZ basis set predict that cis-HO3 remains lower than trans-HO3 by 0.42 and 0.21 kcal/mol, respectively. Of the coupled cluster methods, only the CCSD(T) method predicts that the relative ordering of the cis and trans states changes once harmonic zero-point vibrational energies are included. The CCSD(T) method predicts that the trans isomer is 0.02 kcal/mol higher than the cis isomer in the cc-pVQZ basis set but that the trans isomer is 0.21 kcal/mol lower than the cis isomer in the aug-cc-pVQZ basis. With the addition of harmonic zero-point energies to the CCSD energy results, the CCSD method predicts in both the cc-pVQZ and aug-ccpVQZ basis sets that the trans isomer is higher in energy than the transition state by less than 0.1 kcal/mol. The CCSD(T) method predicts that the addition of harmonic zero-point energies lowers the transition state to ∼0.2 kcal/mol above the trans state. The addition of harmonic zero-point vibrational energies to our total electronic energies does not noticeably change this energy difference in the CR-CCL method. Comparison of Tables 4 and 5 shows that the addition of the harmonic zero-point energies stabilizes the trans relative to the cis for all of the methods. The 2-RDM(M) method manifests the largest stabilization of ≈1.4 kcal/mol, which is approx-

Table 4. Energy Differences between States before the Addition of Harmonic Zero-Point Energies for cc-pVQZ and aug-cc-pVQZ Basis Setsa ΔEX−Y (kcal/mol) X−Y

basis set

CCSD

2RDM(M)

CCSD(T)

MRCI(Q)

trans− cis TS− trans trans− cis TS− trans

cc-pVQZ

0.87

−0.28

0.64

0.39

−0.05

0.19

2.04

0.56

0.65

0.69

−0.41

0.46

0.16

0.16

2.03

0.62

0.71

aug-ccpVQZ

CRCCL

−0.22

a For comparison, MRCI(Q) cc-pVQZ and aug-cc-pVQZ trans−cis energy differences calculated by Varandas in 2012 are provided.73

Table 5. Energy Barriers after the Inclusion of Harmonic Zero-Point Vibrational Energies for cc-pVQZ and aug-ccpVQZ Basis Sets ΔEX−Y (kcal/mol) X−Y

basis set

trans−cis TS−trans trans−cis TS−trans

cc-pVQZ aug-cc-pVQZ

CCSD

2RDM(M)

CRCCL

CCSD(T)

0.61 −0.04 0.42 −0.07

−1.71 2.04 −1.84 2.04

0.39 0.55 0.21 0.61

0.02 0.11 −0.21 0.17

Table 6. Harmonic Frequencies (cm−1) Calculated in the cc-pVDZ Basis Set for Each Molecule and Methoda molecule trans-HOOO

transition state

cis-HOOO

a

mode

CCSD

2-RDM(M)

CR-CC(T)

CCSD(T)

exp

ω1 ω2 ω3 ω4 ω5 ω6 ω2 ω3 ω4 ω5 ω6 ω1 ω2 ω3 ω4 ω5 ω6

106.67 466.75 681.05 1233.95 1369.38 3774.56 483.83 695.15 1217.50 1296.31 3778.68 230.79 494.19 723.56 1220.48 1423.75 3722.78

78.67 133.92 450.07 979.58 1424.94 3727.83 111.52 480.72 1015.18 1353.19 3843.99 352.46 414.63 779.94 1167.89 1452.61 3630.43

145.55 316.33 562.53 1178.43 1360.24 3737.22 345.60 627.59 1243.47 1336.75 3738.20 220.71 362.15 623.80 1217.89 1361.33 3690.99

120.92 179.96 475.62 1055.72 1455.06 3722.39 63.73 462.60 973.89 1421.32 3710.66 206.00 242.77 569.41 1213.71 1346.41 3686.54

122 244 482 998 3528

Experimental results from Derro et al. for the trans isomer are provided for comparison.36 1820



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Overall, we find that the 2-RDM(M)’s predicted results are closer to those predicted by multireference calculations than those from single-reference CCSD, especially in the energetic stability of the trans isomer relative to the cis isomer and the height of the barrier to isomerization (≈2 kcal/mol). Table 7 contains the oxygen bond lengths for each of the two isomers and their transition states calculated in the aug-cc-

imately 1 kcal/mol greater than that found in CCSD(T). The origin of this stabilization can be seen in the harmonic vibrational frequencies in Table 6. Although the 2-RDM(M)’s predicted frequencies for cis are close to those from CCSD, the predicted frequencies for trans are close to those from CCSD(T). This subtle difference between the frequencies for cis and trans is enough to account for the increased stabilization from 2-RDM(M). The parametric 2-RDM(M) method predicts that the trans isomer is more stable than the cis isomer both before and after harmonic zero-point vibrational energies are added whereas CCSD(T) only favors the trans in the aug-cc-pVQZ basis set after harmonic zero-point vibrational energies are added. Figure 1 graphically compares the relative energies with harmonic

Table 7. Oxygen Bond Lengths and Angles (Å or deg) Calculated in the aug-cc-pVTZ Basis Seta molecule

parameter

CCSD

2-RDM(M)

CR-CCL

CCSD(T)

trans-HO3

boo1 boo2 CE boo1 boo2 CE boo1 boo2 doooh CE

1.2489 1.4917 −0.7718 1.2611 1.4667 −0.7725 1.2558 1.4863 124.04 −0.7700

1.2446 1.6187 −0.8297 1.2652 1.5740 −0.8252 1.2527 1.6173 84.76 −0.8262

1.2387 1.5595 −0.8156 1.2576 1.5129 −0.8126 1.2505 1.5448 90.28 −0.8120

1.2304 1.6050 −0.8248 1.2520 1.5403 −0.8176 1.2416 1.5875 80.44 −0.8210

cis-HO3

TS

a

The total correlation energy (in Hartrees) recovered in the aug-ccpVQZ basis set for each molecule is given in the rows marked CE (correlation energy) below the relevant parameters. The experimental values for the trans-HO3 boo1 bond length are 1.225* Å74 and 1.235 Å,32 and the values for the boo2 bond length are 1.688 Å74 and 1.648 Å .32 *Fixed at the MRCI+Q value.

pVTZ basis set. We focus on the oxygen bond lengths here because they are typically underestimated by theoretical methods relative to experiment.32,74 For completeness, the full set of geometries for both the optimizations and the harmonic zero-point vibrational energy calculations are available in the Supporting Information. Comparing the 2RDM(M) results (Figure 2) to CCSD results for the trans-HO3

Figure 1. Relative energies of the two HO3 isomers and their transition state from the 2-RDM(M), CCSD, and CCSD(T) methods for the (a) cc-pVQZ and (b) aug-cc-pVQZ basis sets. MRCI(Q) calculations in the cc-pVQZ (a) and aug-cc-pVQZ (b) basis sets including harmonic vibrational zero-point energies extrapolated to the basis set limit from ref 73.

Figure 2. (a) cis- and (b) trans-HO3 geometries as calculated by the 2RDM(M) method in the aug-cc-pVTZ basis set.

zero-point energies included from the 2-RDM(M), CCSD, and CCSD(T) methods in the (a) cc-pVQZ and (b) aug-cc-pVQZ basis sets. When we compare these results to the multireference configuration (MRCI) results of Suma et al. and Varandas et al., we see some interesting parallels between the predictions of the MRCI and 2-RDM(M) method.31,49,73,74 Suma et al. optimized HO3 using the multireference single and double excitation configuration interaction method with the Davidson correction (MRCI+Q) in the aug-cc-pVTZ basis set, which predicted an energetic favoring of the trans isomer by 0.266 kcal/mol and an isomerization barrier of 0.921 kcal/mol. In a 2012 paper Varandas, using the MRCI and MRCI+Q methods in multiple correlation-consistent basis sets with and without extrapolation to the basis-set limit, predicts the trans−cis differences between 0.1 and 0.6 kcal/mol in favor of the trans isomer. The MRCI(Q) results from the Varandas paper in the cc-pVQZ and aug-cc-VQZ basis sets are given in Table 4 for comparison.

state, we find that the parametric 2-RDM(M) method predicts a geometry with longer oxygen−oxygen bonds than CCSD in both basis sets. Compared to the experimental results of McCarthy et al. (boo1 = 1.235 and boo2 = 1.684), 2-RDM(M) gives a longer bond length, 1.2446 Å, for the boo1 bond and a shorter one, 1.6187 Å, for the boo2 at the aug-cc-pVTZ level. Both the CCSD and CR-CCL methods predict much shorter bond lengths for the boo2 bond than the 2-RDM(M) method. The best coupled cluster method in terms of matching the experimental geometries, CCSD(T), predicts a boo1 bond length of 1.2304 in the aug-cc-pVQZ basis set; however, the boo2 bond length predicted by CCSD(T) is shorter (1.6050) than the one predicted by the 2-RDM(M) method and farther from the experimental value. Looking at the trends in the coupled cluster methods, we can see that in moving from 1821



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CCSD to CCSD(T) the boo1 bond length decreases while the boo2 bond length increases. Although not shown in Table 7, the 2-RDM(M) method displays a similar behavior when augmented functions are added to the basis set. Taken together, these results suggest that increasing the dynamic correlation energy recovered shortens the boo1 bond and lengthens the boo2 bond whereas the addition of multireference correlation lengthens both bonds. To help explain and support these results, we consider the correlation energy recovery of the 2-RDM(M) method as compared to the coupled cluster methods. For the HO3 isomers we find that the 2-RDM(M) method recovers 101−103% of the correlation energy added to CCSD by CR-CC(L) at the cc-pVQZ level. At the aug-cc-pVQZ level, this increases to 130−132%. These results are indicative of multireference character as the 2-RDM(M) method only recovered more correlation energy than CR-CC or CCSD(T) in the presence of multireference correlation such as seen in the transition state of diazene studied by Sand, Schwerdtfeger, and Mazziotti.17 Both natural-orbital occupation numbers and the total energy recovered suggest that multireference correlation plays a nontrivial role in describing the energetics of this system in line with some previous HO3 studies.31,48,73 The longer oxygen−oxygen bonds of the HO3 isomers, as predicted by 2RDM(M), are a likely consequence of the multireference correlation. A similar effect is caused by the addition of perturbative triples to CCSD [CCSD(T)]. Therefore, it is likely that a high level of both dynamic and multireference correlation is needed to describe the geometry of this system.

studies on the topic, which suggest that multireference correlation is necessary to describe this system.31,40,45,48,52,63,65,73,74,93 Most notably, we find that the CCSD method, the least multireferenced of the coupled cluster methods employed, showed the largest energetic favoring of the cis isomer before and after harmonic zero-point vibrational energies are added whereas the 2-RDM(M), which is capable of describing multireferenced systems, predicted the largest energy difference in favor of trans-HO3. After considering the geometry results, we found that the parametric 2-RDM(M) method predicted long oxygen−oxygen bonds overall compared to the coupled cluster methods. In the aug-cc-pVTZ basis set the 2-RDM(M) method predicted a central oxygen−oxygen bond length (boo2) for trans-HO3 of 1.6187 Å compared to the 1.4917 Å bond length predicted by CCSD and the 1.6050 Å bond length predicted by CCSD(T). Compared to the coupled clusters results, the 2-RDM(M) result was closer to but still shorter than the experimental bond length of 1.684/1.688 Å.32,74 In the same basis set the second oxygen−oxygen bond (boo1) was predicted by the 2-RDM(M) method to be 1.2446 Å long. The addition of augmented functions notably improved the geometry results, which depend heavily on correlation energy recovery. Other multireferenced methods from previous studies, particularly MRCI and MRCI(Q) results extrapolated to the basis-set limit, have been able to match the experimental central oxygen−oxygen bond length even more closely than 2-RDM(M).31,52,73,74 Considering this along with the lengthening of the central bond and shortening of the boo1 bond with basis set size, we speculate that a 2-RDM(M) geometry optimization in the aug-cc-pVQZ or larger basis set would potentially yield further improvements to the geometries and, hence, should be considered for future studies of this system. With the extension of the parametric 2-RDM(M) method to open-shell systems, we have a method that is capable of treating a large variety of systems with a high degree of accuracy.3,14−18,92 With a computational cost equivalent to configuration interaction with single and double excitations (CISD) and less than that of CCSD, the 2-RDM(M) method is capable of approaching the accuracy of coupled cluster with full triple excitations in single-reference cases as well as being able to treat some multireference cases. In this study, we found that the 2-RDM(M) method predicts that trans-HO3 is lower in energy than cis-HO3 in both the cc-pVQZ and aug-cc-pVQZ basis sets, in line with experimental results; furthermore, it improves on the predicted oxygen−oxygen bond lengths predicted by CCSD relative to experimental results. Both the energetic and geometric predictions are consistent with previous results from multireference methods. Even though the parametric 2-RDM method is a single-reference theory, the present results in addition to earlier calculations14−17 demonstrate its ability to capture multireference correlation effects. The open-shell 2-RDM(M) method, which performs similarly to its closed-shell counterpart, is applicable to a wide range of moderately correlated open-shell chemical systems.

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DISCUSSION AND CONCLUSIONS In this study we employed the parametric 2-RDM(M) method and several coupled cluster methods to study the HO3 molecular system in the cc-pVQZ and aug-cc-pVQZ basis sets. The study represents the first application of the parametric 2-RDM(M) method to an open-shell system of significant chemical interest. The two key features examined were (i) the relative orderings of the cis and trans isomers of HO3 and (ii) the experimental geometries of trans-HO3. Experimental results suggest that the trans-HO3 isomer is lower in energy than the cis state and possesses a long central oxygen−oxygen bond that is difficult for theoretical methods to match.30,32−36,74 The 2RDM(M) method predicted that the trans-HO3 isomer is lower in energy than the cis-HO3 isomer by between 1.71 and 1.81 kcal/mol once harmonic zero-point vibrational energies are added and between 0.28 and 0.41 kcal/mol without including the zero-point energies. The CCSD and CR-CCL methods both predict that cis-HO3 is lower in energy than trans-HO3. Of the coupled cluster methods, only the CCSD(T) method in the aug-cc-pVQZ basis set predicted that the trans isomer is lower in energy by 0.21 kcal/mol once harmonic zero-point vibrational energies are included. A key difference between the CCSD(T) and 2-RDM(M) results points to the ability of the 2-RDM(M) method, as seen in ref 17, to capture some electron correlation this is traditionally labeled multireference correlation. Without including harmonic zero-point vibrational energies in the results, only the 2-RDM(M) method predicts that trans-HO3 is electronically favored over the cis isomer. On the basis of the naturalorbital occupation numbers and correlation energy recovered, we find that the difference is based, at least in part, on the degree of multireference correlation captured by the 2RDM(M) method. This conclusion is consistent with previous

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ASSOCIATED CONTENT

S Supporting Information *

The complete set of geometric parameters for each calculation. This material is available free of charge via the Internet at http://pubs.acs.org/. 1822



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.A.M. gratefully acknowledges the NSF, the ARO, the HenryCamille Dreyfus Foundation, the David-Lucile Packard Foundation, the Keck Foundation, and the Microsoft Corporation for their support.

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Relative Energies and Geometries of the cis- and trans-HO3 Radicals

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