Variational Calculations of Vibrational Energies and IR Spectra of

Article pubs.acs.org/JPCA

Variational Calculations of Vibrational Energies and IR Spectra of trans- and cis-HOCO Using New ab Initio Potential Energy and Dipole Moment Surfaces Yimin Wang, Stuart Carter, and Joel M. Bowman* Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory University, Atlanta, Georgia 30322, United States ABSTRACT: We report ab initio potential energy and dipole moment surfaces that span the regions describing the minima of trans- and cisHOCO and the barrier separating them. We use the new potential in three types of variational calculations of the vibrational eigenstates, for zero total angular momentum. Two use the code MULTIMODE (MM) in the socalled single-reference and reaction path versions. The third uses the exact Hamiltonian in diatom−diatom Jacobi coordinates. The single-reference version of MM is limited to a description of states that are localized at each minimum separately, whereas the reaction-path version and the Jacobi approach describe localized and delocalized states. The vibrational IR spectrum for zero total angular momentum is also reported for the trans and cis fundamentals and selected overtone and combination states with significant oscillator strength.

I. INTRODUCTION The HOCO radical has been intensively studied over the past 20 years, in part because of the important role it plays in hydrocarbon combustion.1 Recently, the vibrational energies of the fundamentals of trans- and cis-HOCO have been the focus of a number of papers.2−4 Johnson et al., reported new bands of HOCO and DOCO using a combination of photodetachment spectroscopy of the anion and high-quality ab initio electronic structure calculations to develop quartic force fields (QFFs). These were used in second-order vibrational perturbation theory (VPT2) calculations of the energies of the fundamentals.2 Very shortly after that report appeared, Fortenberry et al. reported QFFs, also based on high-level ab initio electronic structure calculations, for cis- and trans-HOCO.3,4 Those force fields were used in variational as well as VPT2 calculations of the energies of fundamentals. In general, agreement between theory and experiment was quite good, with the exception of the cis-OH stretch, where experiment (in a matrix) reported 3316 cm−1 (ref 5) versus the latest theory results of 3447 cm−1 (ref 4) and 3458 cm−1 (ref 2). Virtually simultaneous with the reports of these new QFFs, a new ab initio global potential energy surface (PES) was reported.6 (This new PES has been used in several dynamical studies of the OH+CO chemical reaction,7,8 and those papers can be consulted for a review of previous dynamical studies using earlier PESs.) The PES does describe the trans- and cisHOCO minima as well as the barrier for trans−cis isomerization much more accurately than previous global PESs; however, below the level of accuracy needed for spectroscopic analysis. Here we report a semiglobal, ab initio PES that describes trans- and cis-HOCO and the barrier separating them, at near © XXXX American Chemical Society

spectroscopic accuracy. This new PES is used to study the vibrational energies of these two isomers of HOCO as well as eigenstates that sample both minima. In addition, we report a dipole moment surface (DMS) that also spans these regions. We use the new PES in three sets of variational calculations of the vibrational eigenstates. One is a method that cannot rigorously treat the large amplitude torsional motion. This is the same method used in the work of Fortenberry et al., mentioned briefly above. The two other methods do rigorously describe this motion. Comparisons of vibrational energies from these three methods are made, along with previous results using QFFs and experiments. Finally, IR transition intensities are reported for the fundamentals as well as a number of overtone and combination states, using the new DMS. The paper is organized as follows. Section II presents details of the calculations and properties of the potential and dipole moment surfaces. Section III gives a brief review of the theory of the vibrational methods. Details of the vibrational calculations are given in section IV, and the results and discussion are given in section V. A brief summary and conclusions are given in the final section. Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: October 7, 2012 Revised: November 13, 2012

A

dx.doi.org/10.1021/jp309911w | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

Article

II. POTENTIAL ENERGY AND DIPOLE MOMENT SURFACES A. Potential Energy Surface. A global potential energy surface (PES) for HOCO that describes reactants OH + CO, the HOCO complex, and H + CO2 products has recently been reported with two of us as coauthors.6 The present semiglobal PES was developed using the same electronic structure method used to develop the global PES.6 Here, the electronic energies, obtained with UCCSD(T)-F12/aug-cc-pVTZ)9−11 calculations, are concentrated in the regions of the trans- and cisHOCO minima and the saddle point separating these two minima. A total of 14 830 electronic energies were obtained with this method using MOLRPO version 2010.1.12 The fitting basis for the PES is invariant with respect to the permutation of the two O atoms. The present fit is represented in terms of primary and secondary invariant polynomials12 up to a total polynomial order of seven. Standard, linear least-squares fitting was employed to determine 918 free coefficients of the PES. The root-mean-square (RMS) fitting error is about 5 cm−1 for the energies up to the saddle point region and around 100 cm−1 for the energies up to 22 000 cm−1 above the trans minimum. The whole data set covers configurations of a fairly wide energy range, 123 000 cm−1 above the global minimum. Structures and energies, relative to the global minimum, of the three stationary points of interest were optimized using both the PES and direct UCCSD(T)-F12/aug-cc-pVTZ calculations. The comparison is shown in Table 1, along with

Figure 1. Relaxed torsional potential from the PES.

B. Dipole Moment Surface. The dipole moment surface (DMS) is represented in the form of a model of effective charges at the locations of the four nuclei as follows, 4

μ⃗ (X) =

i=1

energy mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 H−O C−O′ O−C ∠HOC ∠OCO′ τ

cis-HOCO

trans−cis saddle point

PES

ab initio

PES

ab initio

PES

ab initio

0 535 619 1090 1260 1900 3823 0.963 1.178 1.342 107.8 127.1 180.0

0 533 620 1087 1256 1897 3824 0.963 1.178 1.342 107.9 127.0 180.0

620 589 600 1082 1312 1859 3653 0.972 1.184 1.329 108.2 130.2 0.0

618 576 604 1082 1311 1858 3655 0.973 1.184 1.329 108.2 130.2 0.0

3245 575i 634 963 1123 1886 3801 0.963 1.177 1.362 109.0 129.6 85.7

3251 582i 646 997 1118 1872 3785 0.966 1.177 1.363 108.9 129.4 86.7

(1)

where X = (rO⃗ , rO⃗ , rC⃗ , rH⃗ ) denotes the Cartesian configuration of the molecule, qi(X), which depends on the entire configuration, X, is the effective charge on the location of the ith atom, and ri⃗ = (xi, yi, zi)T are the X,Y,Z-coordinates of the ith atom. Each of four effective charges is represented as a linear combination of the polynomial basis in terms of the same variables as the potential; however, the polynomial basis here is covariant under the permtuatation of two O atoms, i.e., effective charges of two O atoms should interchange as their coordinates permute. In the present fit, 1928 coefficients of covariant polynomials of maximum degree 6 are determined in a leastsquares fashion, under the constraint of zero total charge to reproduce the UCCSD(T)-F12/aug-cc-pVDZ dipole moments at the same configurations used to obtain the electronic energies for the PES fitting. More details on the covariant fitting of the DMS are given elsewhere.13,14 The RMS fitting error of the dipole moment is 0.002 au for configurations that have potential energy up to the saddle point region and around 0.01 au for all configurations with energies up to 22 000 cm−1 above the global minimum. Cuts of the DMS along the normal modes of the trans and cis minima are shown in Figure 2. Both minimum structures used for generating cuts are in the xy plane. Among all normal modes, only mode 1, which is the torsion-like normal mode, has only the out-of-plane motion and, as seen from the figure, is also the only normal mode that has a changing z-component of the dipole at both minima. Note, over the small range of the normal modes in these figures, the dipole moment is virtually a linear function of those modes. However, the plot of the dipole moment along the relaxed torsional path, shown in Figure 3, clearly displays its nonlinearity.

Table 1. Structures (Å, deg), Energies (cm−1), and Harmonic Frequencies (cm−1) of HOCO′ trans-HOCO

∑ qi(X) ri ⃗

the harmonic frequencies obtained using both methods. As seen, the maximum deviation of the PES optimized structures to the ab initio ones is 0.003 Å for bond lengths and 0.2° for angles. The PES energies for the saddle point and cis minimum are also in excellent agreement with the ab initio ones. As for the harmonic frequencies, the PES and the ab initio results agree to within 5 cm−1 at the trans global minimum and 15 cm−1 for the other two stationary points. A plot of the relaxed potential, from the PES, as a function of the standard dihedral angle, denoted τ, is shown in Figure 1, with corresponding structures of HOCO indicated. As seen, this relaxed potential, which is explicitly used in one set of variational calculations, varies smoothly with τ, as it should.

III. VIBRATIONAL CALCULATIONS As noted, three types of vibrational calculations are performed for the energies and wave functions of HOCO. The major sets of results are obtained with the reaction-path version of the code MULTIMODE (MM).15,16 This code, which is based on the reaction-path Hamiltonian,17 is designed to describe one large-amplitude torsional mode. It was recently applied to an B



Article

Figure 2. Cuts of the DMS along normal modes of trans (upper) and cis (lower).

obtained previously with MM-SR and VPT2 are generally in very good agreement with each other. However, there were differences for the fundamental of the torsion mode, 13 cm−1 for trans-HOCO and 26 cm−1 for cis-HOCO. These differences are surprisingly large, and further motivates us to investigate MM-SR calculations. Both sets of MM calculations make use of the efficient nmode representation of the potential; in the present case n equals 4. A separate level of mode coupling is used for the vibrational angular momentum terms, in the present case this is 3. These are described in more detail below. These levels of mode coupling are tested, as usual, to determine the accuracy. However, based on many previous calculations it should be accurate to within 1−2 cm−1. The third set of variational calculations is done using an exact Hamiltonian in Jacobi coordinates (describing the OH and CO diatom−diatom arrangement of HOCO). The calculations, which are exact in formulation, and done with no approximations to the potential, provide benchmark results, in principle. However, they are more computer intensive than the MM calculations and very tight convergence of the results is not obtained for all the states reported here. A brief review of the methodology of these variational methods is given next. A. Variational Calculations in Jacobi Coordinates. The specific approach we take is based on earlier work described in detail for C2H2, where diatom−diatom Jacobi coordinates were used.19 The Hamiltonian is well-known from scattering theory.20 The six internal variables are the two diatom internuclear distances, rHO and rCO, the distance between their centers of mass, R, and the three internal angles, θ1, θ2, and ϕ. The approach (which we term truncation/recoupling) begins by representing the three angles initially in a primitive basis of free-rotor total angular momentum states denoted |j1, j2, j12⟩. A 3d numerical basis is obtained from diagonalization of the Hamiltonian given by

Figure 3. Dipole moment along the torsional path.

accurate calculation of the line-list, ro-vibrational spectrum of H2O2.16 This version of MM, denoted MM-RP, is clearly well suited for a study of the torsional motion in HOCO. The second set of vibrational calculations is done using the single-reference version of MM,18 denoted MM-SR. This version is based on the well-known Watson Hamiltonian in mass-scaled, rectilinear normal modes of a specific reference configuration. In the present case, these are the trans and cis minima of HOCO. The variational calculations reported earlier for trans- and cis-HOCO, using corresponding QFFs.3,4 used this method and software. This approach is expected to be valid for vibrational energies that are below the isomerization barrier, which is roughly 3250 cm−1. In fact, for states that do not involve significant torsional excitation, the expectation is that many states with energies above this torsional barrier are localized in these two minima and so a single-reference approach should be accurate for these states. VPT2 calculations are generally accurate for vibrational fundamentals of semirigid molecules and so, as expected, the vibrational fundamentals C


The Journal of Physical Chemistry A ̂ (θ1 ,θ2 ,ϕ) = H3d

j12̂

2

2μR R e 2

+

j1̂

Article

2

2μ1r1e 2

+

j2̂

2

coordinates Q are dependent on s, that the multimode representation of the potential must maintain the coordinate s in each term of the expansion, thus

2μ2 r2e 2

+ V3d(θ1 ,θ2 ,ϕ;r1e ,r2e ,R e)

(2)

V (s ;Q 1 ,Q 2 ,...,Q 3N − 7)

The reference potential, V3d, uses fixed values of the Jacobi distances. The 3d numerical basis is then combined with a 1d numerical basis in R. These 1d numerical functions are eigenfunctions of 1d Hamiltonian at the reference geometry. The 4d Hamiltonian, in which the three angles and R vary, is represented in the direct-product basis of the 3d and 1d numerical functions and is diagonalized. Finally, the full 6d Hamiltonian is diagonalized in a 6d basis, which is a direct product of a subset of 4d eigenstates and numerical functions in rHO and rCO; these are obtained from solving reference 2d potentials using a direct product of 1d numerical bases in rHO and rCO. Matrix elements are evaluated using standard quadrature methods. There are two parity solutions depending on whether the sum j1 + j2 + j12 is even or odd,20−22 and the full Hamiltonian matrix is blocked in terms of these two parities. B. MULTIMODE Calculations. For the MM-RP approach, we follow the procedure recently described in the study of the energy levels of H2O2,16 which uses the exact Hamiltonian of Miller, Handy, and Adams17 in the form developed by Carter and Handy. 15 In this procedure, one large-amplitude coordinate, s, the arc length along the path, is treated as a special coordinate. For HOCO, as in H2O2, this path is taken as the torsional motion. (The torsional path in the present case was shown in Figure 1.) The potential along this coordinate is thus one that repeats once every 2π radians, which in turn suggests a choice of primitive basis in sin(nsτ) and cos(nsτ) for use in the J = 0 variational procedure. The basis for the remaining 3N − 7 coordinates orthogonal to the path are conventional harmonic-oscillator functions for normal coordinates. These normal coordinates are developed via the following two-step algorithm. The first step is to determine the points along the path commencing with the cis configuration and terminating with the trans configuration. We do this by following the minimumenergy potential from cis (τ = 0) to trans (τ = 2π) and minimize the energy with respect to the remaining five internal coordinates. We note that this is not the true definition of a reaction path, in which case the Eckart conditions between adjacent points s and s + 1 would be automatically observed. We therefore enforce this condition by rotating each structure Xi(s+1) through the Euler angles α, β, and γ such that

∑ mi X i(s) × X i(s+1) = 0 i

= V (0)(s) +

∑ Vi(1)(s; Q i) + ∑ Vij(2)(s;Q i , Q j) i

+

ij

(4) ∑ Vijk(3)(s;Q i ,Q j ,Q k) + ∑ Vijkl (s ;Q i ,Q j ,Q k ,Q l) ijk

+

∑

ijkl (5) V ijklm (s ;Q i ,Q j ,Q k ,Q l ,Q m)

+ ··· (4)

ijklm

In this expression the one-mode representation (1MR) of the potential contains only Vi(1)(s;Qi) terms, which can be interpreted as a cut through the hyperspace of normal coordinates at a point s along the path with just one coordinate varying at a time. The two-mode representation of the potential contains 1MR terms plus the V(2) ij (s;Qi,Qj) terms, where any pairs of normal modes vary, etc. Note that all n-mode potentials contain contributions from the pure path potential V(0)(s). With this representation of the potential, truncated at 5MR, which represents complete integration in the case of HOCO, the dimensionality of integrals involving V has a maximum value of 6 (because integration is always performed over the path variable s), for any number of normal coordinates. An important feature of MM-RP is the option to vary the n-mode representation of the potential to test the convergence of the vibrational energies with respect to n ≤ 3N − 7. It should also be mentioned that the inverse moment of inertia μ(s,Q) is also a function of s and the normal coordinates Q, and this is expressed in an identical multimodal form as V(s,Q). We refer to the n-mode representations of V(s,Q) and μ(s,Q) as nMR and nMC, respectively. As well as J = 0 energy calculations, we have also considered the infrared spectrum of HOCO. There are, of course, no allowed J = 0 → J = 0 infrared transitions, but an approximation to the rovibrational spectrum can be made by considering the vibrational transition dipole moment matrix Ri, given by R i = ⟨ψv|μi |ψv ′⟩

i = X, Y , Z

(5)

Here, ψv is a vibrational wave function and μi is a component of the molecule-fixed dipole moment. We have generated two sets of such matrices for transitions from the zero-point levels of trans- and cis-HOCO. To do this, we save the vibrational wave functions in the energy calculation, and then restart by replacing the potential energy by the dipole moment components. These are calculated in terms of the instantaneous principal axis system generated by the s and Q integration points, which allows us to integrate the quantity Ri directly for each of the X, Y, Z dipole moment components. In Cs symmetry (taking cis as reference), the X and Z components transform as A′, and the Y component transforms as A″. For the six modes of HOCO, only that of the torsional (path) motion transforms as A″; the remaining five normal modes transform as A′. It is therefore strightforward to compute the vibrational expansion sets (above) of A′ and A″ symmetries, whence the nonzero symmetry-allowed vibrational transition dipole moment matrices Ri can be evaluated by the same numerical integration procedure as used in the calculation of the vibrational energies themselves. The vibrational intensities are then the sum of the squares of the Ri.

(3)

Here, i labels the atoms and Xi(s) are the Cartesian coordinates at path point s. The second step is to determine the normal coordinates, Q, at each point s along the path. This is done by projecting out translations, rotations and torsional motion from the 3N × 3N force constant matrix F by standard means.15 We then use the 3N − 7 basis functions that are optimized at the true trans minimum for use at all remaining points s along the path. In the range τ = 0 to τ = 2π we thus have sets of normal coordinate vectors that describe displacements from each Cartesian structure Xi(s), separated from its neighbors by the step size in τ. Numerical integration points are taken to coincide with those from this set such that they are equivalent upon reflection about τ = π. It follows from this, and the fact that the normal D



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Table 2. Fundamentals and Zero-Point Energy (cm−1) of trans-HOCO′ ZPE ν1 ν2 ν3 ν4 ν5 ν6 a

description

MM-RP

Jacobi

MM-SR

CcCR/5MRa

VPT2a

VPT2b

expt

OH stretch CO′ stretch HOC bend OC stretch OCO′ bend torsion

4547.4 3639.9 1855.5 1210.0 1049.4 612.4 500.3

4546.2 3639.6 1855.5 1209.7 1049.7 612.7 502.4

4546.5 3639.5 1855.3 1209.2 1050.2 612.5 505.0

4549.8 3634.4 1862.2 1214.2 1052.8 616.9 484.6

4559.2 3640.7 1862.2 1213.7 1053.0 616.6 497.5

3641 1854 1217 1057 614 507

3635.7c 1852.6d 1194b 1048b 629b

Reference 3. bReference 2. cReference 35. dReference 36.

that the fundamentals are converged to within 0.3 cm−1 for a given basis using the 4MR. However, this does not necessarily mean that the MM-SR energies are accurate to within 0.3 cm−1 (for this PES) owing to double-well nature of the PES. Details of the MM-RP calculations are as follows. We construct a reaction path commencing at the cis configuration that has the coordinate origin at the center of mass, the Z-axis in the OC direction with the molecule lying in the XZ plane. The axes are then rotated such that X, Y, Z become the principal axes with rotational constant A corresponding to Z, B corresponding to X, the so-called Ir representation.26 This structure is kept as the initial guess in the potential minimization procedure for the next path point at τ = 0.5°. The six independent internal coordinates RHO, ROC, RCO, HÔ C, OĈ O, and dihedral angle τ are used in the minimization, with only τ constrained. This structure is converted to Cartesian coordinates, which are then rotated to obey the Eckart conditions in (1). This structure is now saved and the process successively repeated for τ = 1° etc. until the trans structure is processed at τ = 180°. Path points for τ = 180.5° to τ = 359.5° are then obtained by reflection. The remainder of the algorithm is identical to that for H2O2.15 Of these 720 points thus generated, we select 90 equally spaced integration points with equal weights which are equivalent for both of the cis−trans regions. These are used to integrate a primitive basis in a total of 56 sin(nsτ) and cos(nsτ) functions. These are contracted to 46 orthonormal basis functions for the variational procedure that follows. The remaining basis functions in the 5 orthogonal normal modes comprises 31 primitive harmonic-oscillator functions integrated by 46 Gauss-Hermite quadrature points, contracted to 10 basis functions at the trans minimum. These are integrated by 26 optimized HEG quadrature points.27 Both basis functions and integration points so obtained are used at all remaining τ integration points. These parameters give Hamiltonian matrices of order 36573 A′ and 35702 A″ in Cs symmetry. Finally, 3MR and 2MC multimodal representations of V(s,Q) and μ(s,Q), respectively, were used, which, as noted above, exclude the torsional path coordinate s. Thus, these are in total 4MR and 3MC, respectively. Some tests with a smaller basis were also run by comparing 5MR and 3MC with 4MR and 2MC and the results agree to within 1 cm−1 or less. For the calculations using diatom−diatom Jacobi coordinates, the trans global minimum was used as the reference structure. The maximum value of j1 and j2 are 31 and 71, respectively. (Other values were also used to test convergence.) The quadratures in θ1 and θ2 were done using 54 and 84 GaussLegendre quadrature points, respectively. The range of j12 is from |j1 − j2| to min(j1+j2,62). As for the quadrature in ϕ, 120 equally spaced grid points from 0 to 2π were employed. Depending on the parity, the size of the 3d Hamiltonian is 26 730 for the even parity and 24 846 for the odd parity. The first

The approach in MM-SR calculations differs from the MMRP approach in that all the coordinates are normal coordinates of a reference geometry and the relevant Hamiltonian is the exact Watson Hamiltonian. This approach has been reviewed in detail previously,18,23 and so we omit details here. The approach is clearly limited in the present case to either transor cis-HOCO, as reference geometries. The calculations are of interest because such calculations were recently reported for cisand trans-HOCO by Fortenberry et al.,3,4 as noted already. They are also of interest to investigate how accurately a singlereference approach (also taken in VPT2 calculations) can describe the vibrational energies for this double-well radical. In both types of MM calculations, the first step is a vibrational self-consisent field calculation24 of the ground vibrational state, followed by virtual-state configuration calculations25 (these are denoted VSCF/VCI). The scheme for determining the excitation space is somewhat involved and is described in detail elsewhere.18,23 Basically, excitations out of the ground state are controlled by the number of modes simultaneously excited. The point to note here is that for a given nMR of the potential (and vibrational angular momentum terms) convergence is monitored with respect to the size of the excitation space. Finally, we note that the n-mode representation is used in both MM approaches. This representation is exact in the present case if n = 6; however, experience with similar molecules and testing indicates that a smaller value does produce accurate results with much less computational effort. That will be tested (and verified) in the present case. In summary, MM-RP and Jacobi Hamiltonians and bases describe the torsional motion in HOCO fully. Those states that may be (and in fact are) localized in either HOCO well emerge from these calculations owing to the behavior of the wave functions. This is not the case in the MM-SR approach, which assumes that states are localized in either HOCO well.

IV. DETAILS OF THE CALCULATIONS First, we describe the MM-SR calculations. These were done in the standard way,18,23 using as the reference configuration the global minimum, trans-HOCO, and the local minimum, cisHOCO, in separate sets of calculations, each using the present semiglobal PES. Both sets of calculations were done using a 4MR of the potential and make use of the Cs symmetry to block the Hamiltonian matrix. Convergence studies were carried out by increasing the basis size, based on the scheme for choosing the excitation space briefly mentioned above. The final set of calculations resulted in H-matrices of order 19624 (A′) and 8022 (A″) for both trans-HOCO and cis-HOCO. These are not very large; however, tests with smaller matrices indicate that the MM-SR fundamentals are converged to within a wavenumber or less. Also, some tests using a 5MR of the potential indicate E



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Table 3. Fundamentals and Zero-Point Energy (cm−1) of cis-HOCO′ description ZPE

a

ν1

OH stretch

ν2 ν3 ν4 ν5 ν6

CO′ stretch HOC bend OC stretch OCO′ bend torsion

MM-RP

Jacobi

MM-SR

5101.1 4480.3 3438.4a 3444.7a 1817.6 1268.7 1037.6 594.5 552.5

5100.1 4481.3 3438.3a 3443.5a 1819.0 1271.4 1040.0 596.8 553.6

5099.8 4480.0 3436.5a 3442.4a 1819.0 1271.5 1040.0 596.8 554.3

CcCR/5MRb

VPT2b

VPT2c

4485.7 3452.3

4491.4 3450.8

3458

1824.1 1280.2 1042.4 601.2 540.2

1823.4 1284.4 1045.9 601.7 566.5

1815 1282 1042 596 545

exptc

1290 1040 605

Mixed states, see text for details. bReference 4. cReference 2.

approximately equal to the molecular eigenestates energy contribute to these eigenstates. Next, consider the comparison with experiment. For transHOCO agreement is within 2−4 cm−1 for three modes and 15 and 17 cm−1 for two other bands that have been recently measured using photodetachment spectroscopy of HOCO−.2 For cis-HOCO only three bands have been reported recently from these experiments.2 These are not high-resolution measurements; however, the level of disagreement with the present results, i.e., 20 and roughly 10 cm−1, is probably outside the experimental uncertainties. Finally, the comparison of the present calculations with the previous ones using QFFs and VPT2 generally show good agreement, with differences less than 10 cm−1, with the exception of the OH-stretch for cis-HOCO, where the present results are roughly 14−20 cm−1 lower. The comparison of the present MM-SR results with the previous MM-SR ones,3,4 show agreement to within 2−7 cm−1, with the exception of the torsion fundamental, which is below the present energy by roughly 20 cm−1 for trans-HOCO and 14 cm−1 for cis-HOCO. For these fundamentals the present results are closer to the VPT2 ones, especially those of ref 2. Table 4 contains energies of the six vibrational overtones of trans- and cis-HOCO from the three sets of vibrational

200 3d eigenfunctions were combined with 15 numerical functions for R to form a direct product basis to solve the 4d Hamiltonian. For the quadrature in R, we used 20 optimized quadrature points in the range 3.0−4.5 bohr. Once the 3000 × 3000 4d Hamiltonian was diagonalized, the first 300 eigenstates together with 30 numerical functions in 2d were then used to diagonalize the final (9000 × 9000) 6d Hamiltonian. The 2d numerical functions came from diagonalizing the 2d Hamiltonian expanded by a direct product basis of 5 numerical functions of rHO and 8 numerical functions of rCO. The quadrature was done with 12 optimized quadrature points in the range of 1.0−3.2 bohr for rHO and 15 optimized quadrature points in the range of 1.8−3.0 bohr for rCO. All of these calculations yield vibrational energies that are converged to within a few wavenumbers at most for the fundamentals. For higher excited states, convergence is less good and we comment on that when we present results.

V. RESULTS AND DISCUSSION A. Vibrational Fundamentals. Tables 2 and 3 contain the energies of the trans- and cis-HOCO fundamentals from the present MM-RP, Jacobi, and MM-SR calculations, relative to the corresponding zero-point energy (ZPE), which is also given in these tables. Note the cis-HOCO ZPE is relative to the transHOCO global minimum and also given relative to the cis minimum to compare to previous calculations, which are also given in these tables, along with results from experiment. Consider first the results from the present calculations. For trans-HOCO the ZPEs agree to within 1.2 cm−1. Agreement for the fundamentals 1−5 is also excellent. For the torsional mode there is more dispersion among the results, with the MM-RP result below the Jacobi coordinate one by 2.1 cm−1 and the MM-SR result above it by 2.6 cm−1. Similar agreement among the three sets of calculations is seen for the cis-HOCO ZPE and fundamentals. Overall, we see that MM-SR results are in very good agreement with the benchmark Jacobi results. As expected, the MM-RP energies are also in good agreement with the Jacobi ones. Note, the cis eigenstates labeled ν1 in Table 3 are mixed states. Each one indicated has a leading coefficient corresponding to the fundamental of the cis-OH stretch in a zero-order basis. However, in all three vibrational calculations, roughly 40−50% of the character of these eigenstates contains contributions from other zero-order states. The nature of the mixing with other states is, not surprisingly, different depending on whether the zero-order MM-SR, MM-RP, or Jacobi basis is used. However, in general zero-order states with combinations of the CO stretches and HOC bend that have energies

Table 4. Overtones (cm−1) of trans- and cis-HOCO trans-HOCO 2ν1 2ν2 2ν3 2ν4 2ν5 2ν6

cis-HOCO

MM-RP

Jacobi

MM-SR

3685.7 2400.6 2079.7 1227.4 970.8

7117.4 3685.3 2398.3 2080.2 1228.5 979.4

7118.7 3683.7 2398.2 2076.3 1228.0 999.2

MM-RP

Jacobi

MM-SR

3612.0 2523.8 2061.8 1190.8 1057.7

6645.6 3614.1 2528.6 2066.2 1194.9 1062.9

6644.9 3614.8 2528.5 2067.3 1194.7 1073.8

calculations. As seen, there is generally good agreement, i.e., within less than 4 cm−1, with the exception of the torsion, where the MM-SR result differs from the other two methods by 20−30 and 11−16 cm −1 for trans-HOCO and cis-HOCO, respectively. This is not an unexpected result for this singlereference method, which eventually will fail badly for highly excited torsional states. Note that for these states MM-RP and Jacobi differ by 8.6 and 9.2 cm −1, respectively. We also examined a variety of combination states for transand cis-HOCO from MM-SR and MM-RP calculations. For states that do not involve excitation of the torsional mode the energies agree to within 10 cm−1 or less. Differences as large as F



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Figure 4. Torsional wave functions.

15 cm−1 are seen for some states with the torsional mode excited. As noted by Fortenberry et al.,3,4 the VPT2 and MM-SR calculations done by those workers showed relatively large differences for the torsion fundamental. These observations are

not seen here for the three sets of variational calculations. Does this suggest that those VPT2 calculations of the torsion fundamental are suspect? We think probably not. The reason is those calculations for the torsion fundamental are closer to the present variational ones than are the previous MM-SR ones. Of G



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Figure 5. Torsional wave functions.

course, the present PES and those QFFs are not identical and so differences are to be expected; however, the overall level of agreement with the VPT2 and the previous MM-SR ones is quite good, with the exception of the MM-SR torsion fundamental. So, this does point to some issue with the MM-

SR calculations for the torsion fundamental. To explore this a bit, recall that the MM-SR calculations did not use the identical QFFs used in the VPT2 calculations. Instead, a transformation of the coordinates was made to make the transformed QFF compatible with the variational method used in MM. (Details H



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of this transformation can be be found in refs 3 and 4.) So perhaps the issue is with the transformed QFF. Indeed, this appears to be the case, according to the recent analysis of these differences, using the transformed QFF.28 In that work, vibrational calculations were done using the Jacobi-coordinate Hamiltonian used here. (However, in virtually every other aspect those calculations were different from the present ones.) Those calculations do produce results for the torsional fundamental closer (but not identical) to the VPT2 results than the previous MM-SR ones. The analysis focused on the several coupling terms in the transformed QFF that were found to be evidently responsible for the differences with the MM-SR calculations. As shown here, using a realistic and accurate description of the torsional motion, the MM-SR results are in good agreement with the MM-RP and Jacobi calculations. So this supports, indirectly to be sure, the conclusion that the issue with the previous calculations is the transformed QFFs, which are physically not correct, albeit not necessarily inaccurate, descriptions of the torsional motion of HOCO. Next, we investigate the torsional wave functions for increasing excitation of that mode to determine the threshold for delocalized states. B. Torsional Wave Functions. Wave functions of pure torsional states up to high levels of excitation were calculated using the MM-RP and Jacobi Hamiltonians. We examine these states here. First, however, it is useful to estimate the energy at which the first delocalized state occurs. As usual, a simple estimate is given by the energy of the vibrationally adiabatic ground-state (VAGS) barrier at the trans−cis saddle point. This is the sum of the potential and the harmonic zero-point energy of the real-frequency modes at the saddle point separating the trans- and cis-HOCO minima. From the relevant entries in Table 1, this barrier equals 7448 cm−1, which relative to the trans ZPE (4546 cm−1) is 2902 cm−1. This is the estimate of the energy of the first delocalized torsional state, relative to the ZPE of trans-HOCO. This number is a crude estimate not only because it uses the harmonic ZPE of the saddle point but also because it ignores the quantization of the torsional states energies, which could only accidentally match this barrier energy. Also, because there is no quantitative definition of “delocalized”, visualization of the wave functions is essential and that is done in Figures 4 and 5. These show the torsional wave functions from MM-RP calculations as a function of τ with the five normal modes orthogonal to the path fixed at zero. As seen, and as expected, there are a number of well-localized states in the trans- and cis-HOCO minima. (Recall that the trans−cis saddle point is at τ = 86°.) The first state with a hint of delocalization is the 4τcis state. The cis state at 2803 cm−1 is certainly delocalized. We examined the wave functions from the Jacobi-Hamiltonian calculations and have located this delocalized state, with an energy of 2806 cm−1, quite close to the MMRP energy. All higher-energy overtone torsional states above this energy are delocalized. Thus, the threshold for torsional states to become delocalized is roughly 100 cm−1 below the VAGS barrier, which does indicate the usefulness of that estimate. This threshold does not imply that all states with energies above 2803 cm−1 are delocalized. Clearly, from Table 4 the states above this energy that are accurately described by MM-SR are not delocalized. C. IR Intensities. Finally, consider the calculated IR transition intensities given in Table 5. The intensities of the fundamentals follow the slopes of the dipole moment

Table 5. Calculated MULTIMODE-Reaction Path Intensities (Atomic Units) trans-HOCO ν1 ν1 ν2 ν3 ν4 ν5 ν6 2ν2 2ν3 2ν4 2ν5 2ν6 ν1+ν6 ν2+ν6 ν3+ν6 ν4+ν6 ν5+ν6 ν4+ν2 ν4+ν3 ν4+2ν6 ν5+ν3 ν5+ν4 a

intensity

3639.9

6.47

1855.5 1210.0 1049.4 612.4 500.3 3685.7 2400.6 2079.7 1227.4 970.8 4127.4 2353.4 1703.9 1546.2 1114.5 2883.8 2248.3 2010.5 1818.4 1652.7

12.27 13.26 4.86 0.34 5.45 0.24 0.08 1.14 0.96 0.47 0.15 0.003 0.03 0.006 0.003 0.06 0.23 0.10 0.12 0.14

cis-HOCO a

3438.4 3444.7a 1817.6 1268.7 1037.6 594.5 552.5 3612.0 2523.8 2061.8 1190.8 1057.7 4013.0 2373.4 1816.6 1583.8 1146.1 2835.4 2283.5 2080.0 1858.8 1623.5

intensity 0.56a 0.47a 15.59 0.02 9.89 1.76 6.64 0.18 0.06 0.86 0.06 1.17 0.04 0.006 0.01 0.03 0.0004 0.11 0.10 0.16 1.69 0.12

Mixed states; see text for details.

components shown in Figure 2. The listed overtones and combinations bands are the most intense ones and are offered as predictions for possible future experiments. Similar calculations of intensities for the fundamentals of trans- and cis-HOCO using QFFs and a new DMS localized at these minima have just been done.29 Those results agree well with the present ones except for the two mixed fundamentals of the OH-stretch in cis-HOCO, which are not reported in those calculations.

VI. SUMMARY AND CONCLUSIONS Semiglobal, ab initio potential energy and dipole moment surfaces describing the trans- and cis-HOCO minima and the barrier separating them were reported. There are precise, permutationally invariant and covariant fits, respectively to high-level ab initio calculations. Three sets of variational calculations of vibrational states were done using this PES and vibrational IR intensities were calculated using the reactionpath version of MULTIMODE. The energies from these calculations agree well for the fundamentals. As expected, MMSR calculations deviate from the MM-RP and Jacobi energies for the overtone of the torsion. The computational effort in doing the two sets of MM calculations are roughly in the ratio 1:4 for MM-SR: MM-RP. This is basically due to the larger quadrature grid needed for the torsional motion in MM-RP. The Jacob calculations are roughly 100 times slower than MMRP ones, owing mainly to the potential matrix quadratures. Comparisons were also made with previous calculations of the vibrational fundamentals used separate quartic force fields for trans- and cis-HOCO minima and experiment. The threshold for delocalized pure torsional excited states was determined and shown to be in good agreement with the harmonic vibrationally adiabatic ground-state barrier. Further studies of isomerization dynamics, perhaps along the lines of those reported for HONO isomerization,30−33 can be I



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(29) Fortenberry, R. C.; Wang, Y.; Huang, X.; Francisco, J. S.; Crawford, T. D.; Bowman, J. M.; Lee, T. J. Phys. Chem. A 2012, in press. (30) Richter, F.; Hochlaf, M.; Rosmus, P.; Gatti, F.; Meyer, H. D. J. Chem. Phys. 2004, 120, 1306−1317. (31) Richter, F.; Rosmus, P.; Gatti, F.; Meyer, H. D. J. Chem. Phys. 2004, 120, 6072−6084. (32) Richter, F.; Gatti, F.; Leonard, C.; Le Quere, F.; Meyer, H. D. J. Chem. Phys. 2007, 127, 164315. (33) Sala, M.; Gatti, F.; Lauvergnat, D.; Meyer, H. D. Phys. Chem. Chem. Phys. 2012, 14, 3791−3801. (34) Ma, J.; Li, J.; Guo, H. Phys. Rev. Lett. 2012, 109, 063202. (35) Petty, J. T.; Moore, C. B. J. Mol. Spectrosc. 1993, 161, 149−156. (36) Sears, T. J.; Fawzy, W. M.; Johnson, P. M. J. Chem. Phys. 1992, 97, 3996−4007.

done with this new PES and DMS. In addition, stimulated by recent work,34 we plan to extend the present PES to the H + CO2 channel to permit a study of tunneling to those products.

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AUTHOR INFORMATION

Corresponding Author

*Electronic address: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the National Science Foundation (CHE-1145227) and Department of Energy (DE DFG0297ER14782) is gratefully acknowledged.

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REFERENCES

(1) Francisco, J. S.; Muckerman, J. T.; Yu, H. Acc. Chem. Res. 2010, 43, 1519−1526. (2) Johnson, C. J.; Harding, M. E.; Poad, B. L. J.; Stanton, J. F.; Continetti, R. E. J. Am. Chem. Soc. 2011, 133, 19606−19609. (3) Fortenberry, R. C.; Huang, X.; Francisco, J. S.; Crawford, T. D.; Lee, T. J. J. Chem. Phys. 2011, 135, 134301. (4) Fortenberry, R. C.; Huang, X.; Francisco, J. S.; Crawford, T. D.; Lee, T. J. J. Chem. Phys. 2011, 135, 214303. (5) Milligan, D. E.; Jacox, M. E. J. Chem. Phys. 1971, 54, 927−942. (6) Li, J.; Wang, Y.; Jiang, B.; Ma, J.; Dawes, R.; Xie, D.; Bowman, J. M.; Guo, H. J. Chem. Phys. 2012, 136, 041103. (7) Li, J.; Xie, C.; Ma, J.; Wang, Y.; Dawes, R.; Xie, D.; Bowman, J. M.; Guo, H. J. Phys. Chem. A 2012, 116, 5057−5067. (8) Xie, C.; Li, J.; Xie, D.; Guo, H. J. Chem. Phys. 2012, 137, 024308. (9) Knizia, G.; Adler, T. B.; Werner, H.-J. J. Chem. Phys. 2009, 130, 054104. (10) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007−1023. (11) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796−6806. (12) Werner, H.-J.; Knowles, P. J.; Manby, F. R.; Schütz, M.; Celani, P.; Knizia, G.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.; et al. MOLPRO, version 2010.1, a package of ab initio programs, 2010. (13) Braams, B. J.; Bowman, J. M. Int. Rev. Phys. Chem. 2009, 28, 577−606. (14) Wang, Y.; Huang, X.; Shepler, B. C.; Braams, B. J.; Bowman, J. M. J. Chem. Phys. 2011, 134, 094509. (15) Carter, S.; Handy, N. C. J. Chem. Phys. 2000, 113, 987−993. (16) Carter, S.; Sharma, A. R.; Bowman, J. M. J. Chem. Phys. 2011, 135, 014308. (17) Miller, W. H.; Handy, N. C.; Adams, J. E. J. Chem. Phys. 1980, 72, 99−112. (18) Bowman, J. M.; Carter, S.; Huang, X. C. Int. Rev. Phys. Chem. 2003, 22, 533−549. (19) Zou, S.; Bowman, J. M.; Brown, A. J. Chem. Phys. 2003, 118, 10012−10023. (20) Zhang, J. Z. H. Theory and Application of Quantum Molecular Dynamics; World Scientific: Singapore, 1999. (21) Launay, J. M. J. Phys. B 1977, 10, 3665. (22) Alexander, M. H.; DePristo, A. P. J. Chem. Phys. 1977, 66, 2166−2172. (23) Carter, S.; Bowman, J. M.; Handy, N. C. Theor. Chem. Acc. 1998, 100, 191−198. (24) Bowman, J. M. J. Chem. Phys. 1978, 68, 608−610. (25) Christoffel, K.; Bowman, J. Chem. Phys. Lett. 1982, 85, 220−224. (26) Flaud, J. M.; Camy-Peyret, C.; Johns, J. W. C.; Carli, B. J. Chem. Phys. 1989, 91, 1504−1510. (27) Harris, D.; Engerholm, G.; Gwinn, W. J. Chem. Phys. 1965, 43, 1515−1517. (28) Mladenovic, M. J. Chem. Phys. 2012, 137, 014306. J


Variational Calculations of Vibrational Energies and IR Spectra of

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