High-Resolution Rovibrational Spectroscopy of Carbon Monoxide

Article pubs.acs.org/JPCA

High-Resolution Rovibrational Spectroscopy of Carbon Monoxide Isotopologues Isolated in Solid Parahydrogen Mario E. Fajardo* Air Force Research Laboratory, Munitions Directorate, Ordnance Division, Energetic Materials Branch, AFRL/RWME, 2306 Perimeter Road, Eglin Air Force Base, Florida 32542-5910, United States ABSTRACT: We report high-resolution infrared absorption spectra of six different CO isotopologues isolated in cryogenic parahydrogen (pH2) solids. These data provide a stringent test for theories of nearly free molecular rotors in crystalline solids, such as crystal field theory, rotation−translation coupling theory, and the pseudorotating cage model. A gas-phase molecule rotates about its center-of-mass (C.M.); a trapped molecule instead rotates about its “center of interaction” (C.I.) with the trapping cage, which may differ from the C.M. for heteronuclear diatomics like CO. Isotopic manipulation of CO allows the systematic variation of the C.M. relative to the C.I. We report remarkably good straight line correlation plots between the observed matrix effects and C.M. locations. Extrapolation of these lines to the limit of vanishing matrix effects yields an “experimental prediction” of the C.I. in excellent (fortuitous?) agreement with the C.I. calculated using a linear pH2−CO−pH2 toy model.

fit CFT parameters in the subsequent discussions. Instead, we adopt a simple and well-known scheme for calculating effective rotational constants and vibrational band origins using only the observed CO/pH2 R(0) and P(1) transitions. We then develop a novel approach for analyzing these data, inspired by two sensible extensions to CFT, the rotation−translation coupling (RTC)5,6 and pseudorotating cage (PC)7 models. This approach yields remarkably good straight line correlation plots between the observed matrix effects and the location of the center-of-mass (C.M.) within the CO molecules, which varies with isotopic substitution. Extrapolation of these straight line correlations to the limit of vanishing matrix effects yields an “experimental prediction” of the so-called “center-of-interaction” (C.I.),5,6 around which a CO molecule is supposed to rotate in the pH2 solid. The excellent agreement of this prediction with the C.I. calculated using a linear pH2−CO−pH2 toy model is striking but potentially spurious. Thus, these results are presented with the caveat that while the experimental data and the calculations of effective spectroscopic constants appear trustworthy, the significance of the extrapolation-based prediction of the CO/pH 2 C.I. remains questionable.

1. INTRODUCTION We previously reported high-resolution infrared (IR) absorption spectra of the ν = 1 ← 0 vibrational fundamental of 12C16O molecules isolated in cryogenic parahydrogen (pH2) solids,1 as part of an ongoing effort to illuminate the rotational dynamics of small molecules in quantum solids and liquids. That study employed a modified crystal field theory (CFT) analysis, in which the molecular spectroscopic constants are taken as adjustable parameters, to make good spectroscopic assignments for all of the observed rovibrational features. However, the CFT model neglects important physics in its description of the CO/ pH2 system; therefore, physical interpretation of some of the resulting “best-fit” CFT parameters remains problematic. Because of these limitations of CFT, we chose not to include in ref 1 experimental data on pH2 solids doped with other CO isotopologues, which is the subject of the present article. In the interim, continued interest in CO rotational dynamics in CO/pH2 solids is evinced by two recent publications. One describes an experimental study employing high-resolution IR diode laser spectroscopy to measure spectral line shapes versus sample temperature,2 and the other describes path-integral molecular dynamics (PIMD) simulations of a CO molecule in a large pH2 cluster.3 Both papers cite CO/pH2 solids as candidates for realizing molecular quantum bits (qubits) in solid-state quantum computational devices, highlighting a practical application for understanding the dephasing processes that drive decoherence of the qubits. This interest motivated the current article, which also revisits (and refines) a highly speculative analysis of our unpublished CO/pH2 data, originally presented years ago at the Ohio State University International Symposium on Molecular Spectroscopy.4 In what follows, we adopt the spectroscopic assignments from the CFT analysis in ref 1, but we do not rely on the bestThis article not subject to U.S. Copyright. Published 2013 by the American Chemical Society

2. EXPERIMENTAL SECTION These experiments were carried out at Edwards AFB, CA using the apparatus described in detail in refs 1 and 8−10, and references therein. Briefly, CO-doped pH2 solids are produced Special Issue: Terry A. Miller Festschrift Received: July 22, 2013 Revised: September 25, 2013 Published: September 26, 2013 13504

dx.doi.org/10.1021/jp407267u | J. Phys. Chem. A 2013, 117, 13504−13512

The Journal of Physical Chemistry A

Article

by the rapid vapor deposition (RVD)8 method, in which independent flows of isotopically manipulated CO gas and precooled pH2 gas from an ortho/para hydrogen converter9 operating at T ≈ 15 K are co-deposited onto a BaF2 substratein-vacuum cooled to T ≈ 2 K by a liquid helium bath cryostat. Individual sample preparation details are given in the figure captions. High-resolution Fourier transform IR (FTIR) absorption spectra are recorded using a Bruker IFS120HR spectrometer. The FTIR beam propagates through the sample and BaF2 substrate with the Poynting vector11 parallel to the substrate surface normal. As discussed before,1,8,10 as-deposited RVD pH2 solids exhibit a mixed hexagonal close packed/face-centered cubic (hcp/fcc) microstructure. Both structures have the same bulk density and the same number (12) of nearest-neighbor pH2 molecules surrounding their single substitutional trapping sites; however, the nearest neighbors in fcc regions are arranged in Oh symmetry, whereas those in hcp regions show a lower D3h symmetry. Warming to T ≈ 4.5 K for several minutes anneals these samples to a nearly pure hcp microstructure and also results in a high degree of alignment of the hcp crystallites’ c axes with the substrate surface normal. Figure 1. Absorption spectra of the ν = 1 ← 0 vibrational fundamental for four CO isotopologues in solid pH2, shifted by subtracting the wavenumber of the corresponding gas-phase R(0) line. The observed peak positions and assignments are summarized in Table 1. Each sample was deposited onto a BaF2 substrate held at T = 2.4 K, annealed for 1 h at T = 4.8 K, and then recooled to T = 2.4 K. Trace (a) is for a 2.9 mm thick, 13 ppm 12C16O/pH2 sample; the spectrum is shifted by 2147.0811 cm−1. Trace (b) is for a 3.1 mm thick, 10 ppm 13 16 C O/pH2 sample; the spectrum is shifted by 2099.7101 cm−1. Trace (c) is for a 3.1 mm thick, 10 ppm 12C18O/pH2 sample; the spectrum is shifted by 2095.7511 cm−1. Trace (d) is for a 3.0 mm thick, 3 ppm 13 18 C O/pH2 sample that also contains 27 ppm 13C16O; this spectrum is rescaled vertically by a multiplicative factor of 2 and shifted by 2047.1544 cm−1. The spectral resolution for trace (a) is 0.0075 cm−1, and it is 0.005 cm−1 for traces (b−d). The traces have been truncated for ΔA10 > 2.0 and displaced vertically for ease of presentation.

3. RESULTS AND DISCUSSION Figure 1 shows IR absorption spectra of the vibrational fundamental region for four CO isotopologues (12C16O, 13 16 C O, 12C18O, and 13C18O) isolated in annealed pH2 solids; the spectra are shifted for convenience by subtracting the wavenumber of the corresponding gas-phase R(0) line.12 Similar, but much weaker, spectra were observed for 12C17O and 13C17O. Not shown are spectra of the as-deposited CO/ pH2 samples that include features due to CO molecules trapped in metastable fcc regions. The observed peak positions and their assignments based on the CFT analysis in ref 1 are collected in Table 1. Most of the following discussion focuses on transitions between CO/pH2 states with total angular momentum J = 0 and 1, with a brief mention at the end of data involving the J′ = 2 and J″ = 2 levels. In the Edwards AFB apparatus, the electric field vectors of the unpolarized FTIR beam are oriented (primarily) perpendicular to the BaF2 substrate surface normal and hence to the annealed solid pH2 hcp crystallites’ c axes. This results in enhanced intensities for the ⊥-polarized (ΔM = ±1) IR transitions, greatly reduced intensities for the ∥-polarized (ΔM = 0) IR transitions, and nonobservation of the P(1)∥ peak in the already weak spectrum of 12C17O/pH2. However, an estimated value of P(1)∥ for 12C17O can be obtained by interpolating the trend in the M-dependent splittings between P(1)∥ and P(1)⊥ features for 12C16O and 12C18O ΔP(1)hcp = P(1)⊥ − P(1)

and 1) first rotational constant, Bavg(gas), can be obtained from the observed gas-phase R(0) and P(1) lines Bavg (gas) ≡

[R(0)gas − P(1)gas ] 1 (Bν = 0 + Bν = 1) ≈ 4 2

(2)

where Bν=0 and Bν=1 are the rotational constants for CO in the ν = 0 and 1 states, respectively.13 For example, for 12C16O, (1/ 2)(Bν=0 + Bν=1) = 1.91378 cm−1 and [R(0)gas − P(1)gas]/4 = 1.91375 cm−1. Table 2 includes similar values for the other gasphase CO isotopologues.12 This definition of Bavg(gas) is consistent with taking the vibrational origin as the midpoint between the R(0) and P(1) lines 1 ν0(gas) ≡ [R(0)gas + P(1)gas ] (3) 2 Values of ν0(gas) calculated in this manner are listed in Table 3. The same definitions extend readily to CO isotopologues in Oh symmetry sites in fcc solid pH2 regions, for which the R(0)fcc and P(1)fcc features are single peaks

(1)

From the values in Table 1, this splitting is 0.866 cm−1 for 12 16 C O and 0.868 cm−1 for 12C18O; therefore, the position of the unobserved 12C17O P(1) ∥ feature is estimated by subtracting 0.87 cm−1 (rounded to the nearest 0.01 cm−1) from the observed 12C17O P(1)⊥ peak position. This value and all subsequently derived quantities are marked by asterisks in the tables and by shaded symbols in the figures and are excluded from all fits. As discussed before,1 because CO is such a stiff rotor, an excellent approximation to a vibrationally averaged (over ν = 0

Bavg (fcc) ≡ 13505

[R(0)fcc − P(1)fcc ] 4

(4)



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Table 1. Peak Positions of Assigned ν = 1 ← 0 CO/pH2 Transitions (cm−1)a 12 16

C O

2134.551 2135.204 2136.741 2137.26 2137.607 2140.268 2140.39 2140.46 2142.953 2143.325 2143.818 2145.282 2145.48 2145.931 2146.14 w 2146.50 2147.12 2148.615 a

12 17

13 16

C O

2109.93*b 2110.80

2115.89 2116.76 w

2118.80 w

12 18

C O

2087.552 2088.15 2089.666 2090.18 2090.524 2093.129 2093.25 w 2093.32 2095.760 2096.123 2096.618 2098.054 2098.25 2098.643 2098.910 2099.27 2099.91 2101.27

C O

2083.83 2084.63 2085.914 2086.43 2086.782 2089.197 2089.30 2089.38 2091.624 2092.009 2092.500 2093.712 2093.92 2094.495

13 17

13 18

C O

C O

2035.612 2036.34 2037.605 2038.12 2038.464 2040.833 2040.94 w 2041.01 2043.221 2043.588 2044.080 2045.269 2045.46 2045.997 2046.12 sh 2046.54 2047.14 2048.385

2062.22 w 2063.07 2065.56 w

2068.06 2068.92 2070.23 2070.88

2094.94 2095.59 2096.93

assignment

(J′,M′) ← (J″,M″)

polarization

P(2) hcp P(2) hcp P(1) hcp P(1) fcc P(1) hcp Q(1) hcp Q fcc Q fcc R(0) hcp R(0) fcc R(0) hcp R(1) hcp R(1) fcc R(1) hcp R(1) hcp R(1) fcc R(2) hcp S(0) hcp

(1,1) ← (2,2) (1,0) ← (2,1) (0,0) ← (1,0)

⊥ ⊥ ∥

(0,0) ← (1,1) (1,1) ← (1,1)

⊥ ⊥

(1,1) ← (0,0)

⊥

(1,0) ← (0,0) (2,1) ← (1,0)

∥ ⊥

(2,2) ← (1,1) (2,1) ← (1,1)

⊥ ∥

(3,2) ← (2,1) (2,2) ← (0,0)

⊥ ⊥

w = weak; sh = shoulder. bEstimated; see the text.

Table 2. Gas-Phase CO Peak Positions12 and Vibrationally Averaged Rotational Constants (cm−1), Averaged Rotational Constants in fcc Solid pH2 (cm−1), M-Weighted Centroid Band Positions in hcp Solid pH2 (cm−1), Averaged Rotational Constants in hcp Solid pH2 (cm−1), and Reduced (pH2/Gas Phase) Rotational Constants isotopologue 12 16

C O 12 17 C O 13 16 C O 12 18 C O 13 17 C O 13 18 C O a

P(1)gas

R(0)gas

Bavg(gas)

Bavg(fcc)

P(1)cen

R(0)cen

Bavg(hcp)

Bavg(hcp)/ Bavg(gas)

Bavg(fcc)/ Bavg(gas)

2139.4261 2112.5471 2092.3910 2088.4597 2064.8733 2040.1991

2147.0811 2120.0092 2099.7101 2095.7511 2071.9994 2047.1544

1.9138 1.8655 1.8298 1.8229 1.7815 1.7388

1.5162

2137.3183 2110.51*a 2090.2380 2086.4927 2062.787 2038.1777

2143.2413 2116.18 2096.0460 2091.9160 2068.347 2043.5073

1.4808 1.418*a 1.4520 1.3558 1.3900 1.3324

0.7737 0.756*a 0.7935 0.7438 0.7802 0.7663

0.7923

1.4858 1.3948 1.3670

0.8120 0.7651 0.7862

Calculated using the estimated P(1)∥ band position.

Table 3. Vibrational Band Origins in the Gas Phase and in fcc and hcp Solid pH2 (cm−1), Solid−Gas Shifts (cm−1), and Normalized Shifts isotopologue 12 16

C O 12 17 C O 13 16 C O 12 18 C O 13 17 C O 13 18 C O a

ν0(gas)

ν0(fcc)

ν0(hcp)

ν0(hcp) − ν0(gas)

ν0(fcc) − ν0(gas)

Δν0(hcp)/ν0(gas)

Δν0(fcc)/ν0(gas)

2143.2536 2116.2782 2096.0506 2092.1054 2068.4364 2043.6768

2140.293

2140.280 2113.345*a 2093.142 2089.204 2065.567 2040.843

−2.974 −2.933*a −2.909 −2.901 −2.870 −2.834

−2.961

−0.0013875 −0.001386*a −0.0013876 −0.0013867 −0.0013874 −0.0013868

−0.0013816

2093.152 2089.220 2040.854

−2.899 −2.886 −2.823

−0.0013831 −0.0013794 −0.0013812

Calculated using the estimated P(1)∥ band position.

ν0(fcc) ≡

1 [R(0)fcc + P(1)fcc ] 2

before calculating the rotational constants and vibrational band origins

(5)

However, as illustrated in Figure 2, for CO molecules in D3h symmetry sites in hcp solid pH2 regions, we first calculate the M-weighted centroids of the crystal field split R(0) and P(1) features to account for the two-fold degeneracy of the M′ = 1 and M″ = 1 levels P(1)cen =

R(0)cen =

ν0(hcp) ≡

(6)

[R(0) + 2R(0)⊥ ] 3

(8)

1 [R(0)cen + P(1)cen ] 2

(9)

The averaged rotational constants are collected in Table 2 along with the normalized or “reduced” Bavg(pH2)/Bavg(gas) values. Figure 3 shows correlation plots of the observed gas-tomatrix shifts

[P(1) + 2P(1)⊥ ] 3

[R(0)cen − P(1)cen ] 4

Bavg (hcp) ≡

Δν0 = ν0(pH2) − ν0(gas)

(7) 13506

(10)



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because the flexible nature of the solvent cavity minimizes repulsive interactions.14 Similar excellent linear Δν0 versus ν0 correlations were reported for CO2 molecules trapped in solid pH2, except that the data for the 12C and 13C isotopologues formed two different parallel straight lines.15 These observations support the notion that the highly compressible pH2 solid easily accommodates CO molecules, whereas the larger CO2 molecules exist in more crowded trapping sites in which repulsive CO2−pH2 interactions are not negligible, making these shifts sensitive to the internal vibrational dynamics of the CO2 molecule. Figure 4 illustrates an early attempt to understand the reduced rotational constants listed in Table 2, based on a

Figure 2. Detail of the 12C16O/pH2 spectrum depicted above in Figure 1a, showing the locations of the M-weighted centroids of the P(1) and R(0) bands and the vibrational origin.

Figure 4. Plots of reduced rotational constants for CO isotopologues in pH2 solids and of the positions of the molecular C.M. versus the mass differences between the O and C atoms. The two vertical axes share the same numerical scale. Values of Bavg(fcc)/Bavg(gas) are shown as the open blue squares, and those of Bavg(hcp)/Bavg(gas) are shown as the open black circles. The estimated value for the 12C17O isotopologue is included as the shaded circle. Values of R(C.M.avg) are shown as the open green diamonds. The solid lines are least-squares fits to each data set.

Figure 3. Plots of gas-to-matrix shifts of the vibrational band origins of CO isotopologues in pH2 solids versus the gas-phase vibrational origins. Values for CO isolated in fcc regions in as-deposited CO/pH2 samples are shown as the open blue squares. Values for CO isolated in hcp regions in annealed CO/pH2 samples are shown as the open black circles. The estimated value for the 12C17O isotopologue is included as the shaded circle. The slope of the least-squares-fit straight line to the fcc data is −0.00139(7), and the slope of the line fit to the hcp data (excluding 12C17O) is −0.00140(2); the values in parentheses are standard errors.

suggestion16 by Prof. G. Scoles that the strength of the CO− pH2 interactions should increase with the mass asymmetry in the CO isotopologues Δm = mO − mC

(11)

in which mO and mC are the masses of the O and C atoms, respectively. The data do indeed follow this trend, albeit with two “kinks” for Δm ≈ 4 and 5 amu. The success of this preliminary analysis led to more a careful consideration of the details of the RTC model, which warrants a brief digression. Whereas a gas-phase molecule rotates about its C.M., in the RTC model,5,6 the trapped molecule is supposed to rotate about an interior point called the C.I. At equilibrium, the C.I. coincides with the center of the trapping cage and therefore is determined by the specific molecule−host interactions. For a homonuclear diatomic molecule (e.g., N2) in a highly symmetric trapping site, the C.I. and C.M. will coincide by symmetry. For a heteronuclear diatomic molecule (e.g., CO), the C.I. lies on the internuclear axis, but it need not coincide

versus gas-phase vibrational origins, ν0(gas), for the data from Table 3. The slopes of the least-squares-fit straight lines to the fcc and hcp data are the same (Δν0/ν0(gas) ≈ −0.0014) within experimental error, and the two lines are offset vertically by ν0(fcc) − ν0(hcp) ≈ 0.013 cm−1. These slopes compare well with the simple averages (excluding 12C17O) of the normalized shifts listed in Table 3, Δν0(fcc)/ν0(gas) = −0.0013813(8) and Δν0(hcp)/ν0(gas) = −0.0013872(2), for which the values in parentheses are standard errors of the mean in the last digit. Constant Δν0/ν0 is expected for liquid solvents, in which the shifts are dominated by electrostatic guest−host interactions 13507



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Table 4. Gas-Phase Rotational Constants, Internuclear Separations, and Positions (Relative to the C Atom Nuclei) of the Vibrationally Averaged C.M.’s for CO Isotopologs in the ν = 0 and 1 States and Positions of Combined C.M. Averages over ν = 0 and 1 States isotopologue

Bν=0 (cm−1)

R0 (Å)

R(C.M.ν=0) (Å)

Bν=1(cm−1)

R1(Å)

R(C.M.ν=1) (Å)

R(C.M.avg) (Å)

1.92253 1.87396 1.83797 1.83098 1.78940 1.74641

1.13450 1.13446 1.13444 1.13443 1.13440 1.13437

0.64820 0.66502 0.62574 0.68065 0.64274 0.65858

1.90503 1.85712 1.82161 1.81472 1.77368 1.73126

1.13970 1.13960 1.13952 1.13950 1.13942 1.13932

0.65117 0.66803 0.62854 0.68369 0.64558 0.66146

0.64968 0.66652 0.62714 0.68217 0.64416 0.66002

12 16

C O C O 13 16 C O 12 18 C O 13 17 C O 13 18 C O 12 17

For consistency with the definition of Bavg in eq 2, we take the position of the vibrationally averaged (over ν = 0 and 1) C.M. as

with the C.M. In this case, the molecular C.M. is forced to rotate (and translate) about the center of the trapping cage, introducing additional angular momenta that couple to the molecular rotations. Relaxing the rigid trapping cage assumption in the RTC model leads to rotor−phonon coupling,6 which is pictured in the PC model7 as the synchronous displacement of the host nearest neighbors by the rotating−translating molecule. The strengths of the RTC effect and the rotor−phonon coupling depend on the magnitude of the separation between the positions of the C.M. and C.I. within the rotor a = |R(C.M.) − R(C.I.)|

R(C.M.avg ) =

1 (R(C.M.ν = 0) + R(C.M.ν = 1)) 2

(17)

Table 4 summarizes the calculations of R(C.M.avg) for the various CO isotopologues; these results are also plotted in the bottom portion of Figure 4. Note the fairly good linear correlation between R(C.M.avg) and Δm and especially the two kinks for Δm ≈ 4 and 5 amu, which suggest the improved correlation plots of reduced rotational constants versus R(C.M.avg) shown in Figure 5. Figure 5 shows the “remarkably good straight line correlation plots” promised above in the Introduction. Note the ∼3× vertical scale expansion compared with Figure 4 and the virtual elimination of the Δm ≈ 4 and 5 amu “kinks”. The slopes of the straight line fits to the fcc and hcp data are significantly different; therefore, they will eventually intersect for some value

(12)

and a is often used as a perturbation expansion parameter. By symmetry, terms with odd powers of a vanish in the RTC and rotor−phonon perturbation energies,5,6 leaving a2-dependent leading terms. Similarly, the PC model predicts corrections that depend on the squares of the displacements of the nearest neighbors, which (plausibly) suggests an a2-dependent contribution. Thus, we see that isotopic manipulation of CO allows for the systematic variation of R(C.M.) and hence of the value of a. Quantitative evaluation of R(C.M.) requires taking a little care in the definitions of the CO molecular “dimensions”. The reduced masses of the CO isotopologues are calculated as mCmO μCO = (mC + mO) (13) using m(12C) = 12.0000, m(13C) = 13.0034, m(16O) = 15.9949, m(17O) = 16.9991, and m(18O) = 17.9992 amu.17 For each vibrational state, the rotational constant, Bν, and the vibrational average of the reciprocal of the square of the CO internuclear separation, ⟨1/R2⟩ ν, are related by13

Bν =

h⟨1/R2⟩ν (8π 2cμ)

(14)

where h is Planck’s constant and c is the speed of light. Solving eq 14 for ⟨1/R2⟩ν and taking the square root of the reciprocal provides a meaningful measure of the average CO internuclear separation in a given vibrational state Rν ≡

1 R2

−1/2 ν

⎤ ⎡ h ⎥ =⎢ 2 ⎣ (8π cμBν ) ⎦

Figure 5. Plots of reduced rotational constants for CO isotopologues in pH2 solids versus R(C.M.avg). Values of Bavg(fcc)/Bavg(gas) are shown as the open blue squares, and those of Bavg(hcp)/Bavg(gas) are shown as the open black circles. The estimated value for the 12C17O isotopologue is included as the shaded circle. The parameters of the least-squares-fit straight line (y = a0 + a1x) to the fcc data are a0 = 1.340(16) and a1 = −0.842(25) Å−1. Those for the hcp data (excluding 12C17O) are a0 = 1.362(20) and a1 = −0.905(31) Å−1. The least-squares-fit quadratic (y = a0 + a1x + a2x2) to the hcp data (excluding 12C17O) is shown as the dashed red curve, with parameters a0 = 0.34964, a1 = 2.18708 Å−1, and a2 = −2.35883 Å−2.

1/2

(15)

The position of the CO molecular C.M. relative to the C atom nucleus is thus R(C.M.ν ) =

R νmO (mC + mO)

(16) 13508



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of R(C.M.avg) closer to the C atom. Because the RTC model predicts an a2 dependence, Figure 5 also includes a quadratic fit to the hcp data. This yields a slight improvement over the linear fit as the error variance drops from 1.8 × 10−6 to 1.1 × 10−6. However, the incremental F statistic is only Finc = 3.1, too small to conclude that the quadratic model is preferred over the linear fit even at the undemanding 10% level of significance [i.e., Finc < F0.9(1,2) = 8.53].18 Figure 5 is unprecedented for two reasons. First, historically, ref 1. and this article are the only reports to date of rotationally resolved and completely assigned rovibrational spectra for any nonhydride molecule in the solid phase. The development of the CFT and RTC models was largely motivated and fueled by experimental data from HX-doped rare gas matrixes. However, isotopic manipulation of HX molecules is not effective at varying R(C.M.avg) in a uniform manner, for example, the H ↔ D substitution makes for a large change, but the X = 35Cl ↔ 37 Cl or 79Br ↔ 81Br substitutions yield only closely spaced, and hence practically redundant, C.M. positions. Second, the unusually high precision of our IR spectroscopic measurements (σν ≈ 0.001 cm−1), yielding calculated reduced rotational constants and vibrational origin shifts with four significant digits, enables more critical evaluation of the various models and modes of data presentation. For example, it is required to reveal the improvement obtained upon plotting the reduced rotational constants versus R(C.M.avg) instead of Δm. Indeed, it is this “remarkably good” quality of the straight line fits that encourages the following highly speculative attempt to extrapolate these lines toward their intersection and to draw some physical significance by comparison with a theoretical prediction of the CO/pH2 C.I. While the calculation of the CO molecular C.M. is fairly straightforward, making a theoretical estimate of the CO/pH2 C.I. is much more involved, requiring knowledge of the CO− pH2 interaction potential and of likely trapping site structures. Fortunately, a high-quality ab initio four degree of freedom CO−H2 potential energy surface (PES),19 in which both CO and H2 are treated as rigid rotors with bond lengths fixed at the average values in their ground rovibrational states, was available at the start of our project, and those authors kindly shared their computer code for evaluating this PES.20 Averaging the 4-D PES over spherical H2 orientations yields a 2-D CO−pH2 diatom−pseudoatom PES, with the CO−pH2 separation calculated relative to R(C.M.ν=0) for the 12C16O isotopologue. Figure 6 shows cuts through the resulting 2-D surface for collinear approach of the pH2 pseudoatom to the C and O ends of the CO diatomic; clearly, the potential is more repulsive for approach toward the C atom. Instead of attempting large-scale many-atom simulations of CO in “realistic” pH2 trapping sites, we use a simple linear pH2−CO−pH2 toy model to estimate the location of the CO/ pH2 C.I. This simple approach is relevant due to strong cancellation effects between opposing groups of “off-axis” pH2 nearest neighbors, resulting from the high symmetries of the Oh and D3h trapping sites. In the linear pH2−CO−pH2 toy model, the total potential energy is taken as the sum of the pH2−CO and CO-pH2 interactions, i.e., the pH2−pH2 interaction is ignored. The two pH2 molecules begin fixed at some distance apart (twice the candidate nearest-neighbor distance, Rnn), and the CO molecule is moved along the pH2−pH2 axis until the total potential energy is minimized. The C.I. is taken as the position of the midpoint of the pH2−pH2 line segment (i.e., the center of the ersatz trapping cage) within the CO molecule and

Figure 6. Plots of collinear pH2−CO intermolecular potentials for pH2 approaching the O atom (solid curve) and the C atom (dashed curve). The vertical red line indicates the Rnn = 3.789 Å nearest-neighbor separation in solid pH2.

tabulated along with the value of the minimized total potential energy. The separation between the two pH2 molecules is then adjusted to a new candidate Rnn value, and the energy minimization process repeated and so forth. Figure 7 shows a plot of the minimized potential energies versus locations of the C.I. within the CO molecule [temporarily labeled as XC.I. in the coordinate system centered at R(C.M.ν=0) for 12C16O] for 3.6 < Rnn < 4.2 Å. In particular, two values are highlighted by red crosses, XC.I. = −0.237 Å for Rnn = 3.79 Å (the nearest-neighbor separation in pure solid

Figure 7. Plot of minimized total potential energy of a linear pH2− CO-pH2 construct versus XC.I.. The average positions of the C and O nuclei for 12C16O in the ν = 0, J = 0 state are indicated by the small crosses. For reference, the positions of the various C.M.avg for the different CO isotopologues are indicated by the small circles. 13509



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pH2) and XC.I. = −0.268 Å for Rnn = 4.10 Å, which is very near the global energy minimum for the toy model. Because the toy model contains no additional surrounding pH2 molecules constraining the expansion of the CO trapping site, we expect that the Rnn = 4.10 Å value overestimates the actual trapping site size. Indeed, the PIMD simulations in ref 3 show the first peak in the CO−pH2 radial distribution function at 4.0 Å (with a 0.65 Å full width at half-maximum) for the relaxed trapping site structure. Therefore, adding R(C.M.ν=0) = 0.64820 Å to XC.I. to convert back to the 12C nucleus origin coordinate system, we interpret the toy model calculations as predicting 0.38 < R(C.I.) < 0.41 Å. This predicted range of theoretical R(C.I.) is depicted in Figure 8 by the hashed green box. Recall that in the RTC

pH2 with an effective reduced rotational constant of Beff/Bgas = 0.90(2).21 Because N2 is similar in size and electronic structure to CO, this suggests that either (1) our arbitrary choice of Bavg(pH2)/Bavg(gas) = 1.0 to represent vanishing matrix effects in Figure 8 may be incorrect or (2) the prediction of vanishing matrix effects for a = 0 is incorrect and points to a problem with the RTC5 and rotor−phonon coupling6 formalisms. This problem as a → 0 does not necessarily apply to the PC model;7 its picture of a rotating “ellipsoidal” molecule deforming the trapping cage simply by virtue of its anisotropic guest−host interactions seems compatible with nonvanishing matrix effects even for a = 0. Incidentally, because polynomial extrapolations are notoriously unreliable, we hesitate to even mention that very minor adjustments to the quadratic fit cause the parabola in Figure 8 to pass through the range of theoretically predicted R(C.I.)’s very near to the reduced rotational constant value of 0.9. Finally, Figure 9 shows a correlation plot for the reduced rotational constants calculated from the P(2) and R(1) hcp

Figure 8. Extrapolations of the straight line and quadratic fits shown in Figure 5. The value of 1.0 for the reduced rotational constant is indicated by the horizontal red line. The value of R(C.I.) = 0.402 Å is indicated by the vertical red line. The region 0.38 < R(C.I.) < 0.41 Å is shown as the hashed green box. Figure 9. Plot of reduced rotational constants calculated from the hcp CO/pH2 P(2) and R(1) features versus R(C.M.avg). The parameters of the least-squares-fit straight line are a0 = 1.233(16) and a1 = −0.818(25) Å−1.

model,5 the perturbations to free molecular rotation vanish for a = 0, that is, for coincidence of the C.I. with the C.M. We take this “vanishing of matrix effects” to correspond to Bavg(pH2)/ Bavg(gas) = 1.0, represented in Figure 8 by the horizontal red line. The extrapolated straight line fits to the fcc and hcp data in Figure 5 intersect this horizontal line at R(C.I.) = 0.404 and 0.400 Å, respectively; the vertical red line marks the central value of R(C.I.) = 0.402 Å. The 95% confidence limits on these calculated intersections18 are 0.367 < R(C.I.) < 0.433 Å for the fcc line and 0.369 < R(C.I.) < 0.425 Å for the hcp line, which still shows strong overlap with the 0.38 < R(C.I.) < 0.41 Å theoretically predicted range. Thus, Figure 8 appears to portray excellent agreement between the experimentally determined and theoretically estimated values of the CO/pH2 C.I. Even a more critical comparison, including the uncertainty estimates for the intersections, does not invalidate this agreement. However, two important objections are readily apparent. First, the possibility that the data actually do fall on a parabola instead of a straight line would make the long linear extrapolations meaningless. Second, while the RTC model predicts vanishing matrix effects as a → 0, we are aware that N2, a homonuclear diatomic molecule for which a = 0 by symmetry, rotates in solid

CO/pH2 peak positions in Table 1, using the following definitions 1 P(2)cen = [P(2)M ″= 2 + P(2)M ″= 1] (18) 2 R(1)cen =

1 [R(1)M ′= 1 + R(1)M ′= 2 ] 2

Bavg (J = 2) Bavg (gas)

=

[R(1)cen − P(2)cen ] [R(1)gas − P(2)gas ]

(19)

(20)

Again, the points are well-fit by a straight line; however, the reduction in rotational constant is much stronger than that calculated using only the J = 0 and 1 levels. This same issue plagued the modified CFT calculations in ref 1, resulting in a ∼4000× increase in the centrifugal distortion constant, De, relative to the gas-phase value. Extrapolation of this straight line back toward the C atom yields an intersection with Bavg(pH2)/ 13510



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Bavg(gas) = 1.0 at R(C.I.) = 0.285 Å, with 95% confidence limit 0.229 < R(C.I.) < 0.328 Å, which clearly misses the intersections shown above in Figure 8. Thus, the spectroscopic data involving J = 2 levels highlight the limitations of both the CFT model and the present analysis.

The well-worn simple approach of extracting vibrational band origins, ν0(pH2), and effective rotational constants, Bavg(pH2), using only the observed R(0) and P(1) transitions, appears to be very well suited to the CO/pH2 system. A consistent definition of similar quantities for gas-phase CO molecules provides reference values for calculating gas-to-matrix vibrational origin shifts, Δν0, and reduced rotational constants, Bavg(pH2)/Bavg(gas), that yield exceptionally strong linear correlations with easily calculated intrinsic properties of the gas-phase CO molecules. In particular, plotting Bavg(pH2)/ Bavg(gas) versus R(C.M.avg) for the different CO isotopologues constitutes a novel and improved method for presenting and understanding these data. Hopefully, these data will motivate a serious re-examination of the RTC and PC models to (1) establish a better definition of what is meant by “vanishing matrix effects” in the limit of coincident C.M. and C.I. (i.e., as a → 0) and (2) re-evaluate the previously discarded possibility of matrix effects that depend linearly on the separation between C.M. and C.I. Additional quantum simulations of CO isotopologues in solid pH2 and liquid helium, keyed toward calculation of observables related to the changes in R(C.M.avg) due to isotopic substitution, could yield valuable insight into the relative importance of RTC (cage moves molecule) and PC (molecule moves cage) physical pictures. Finally, we hope that our CO/pH2 experimental results and data analysis/presentation scheme will also help advance our understanding of CO/(H2)n clusters,3,19,22−24 CO/(He)n clusters,25,26 and CO in helium nanodroplets.27 In particular, our CO/pH2 data illustrate the physical limit of complete solvation in a highly isotropic crystalline environment, albeit one in which the large zero-point motion of the pH2 nearest neighbors leads to very accommodating “liquid-like” surroundings.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Fajardo, M. E.; Lindsay, C. M.; Momose, T. Crystal Field Theory Analysis of Rovibrational Spectra of Carbon Monoxide Monomers Isolated in Solid Parahydrogen. J. Chem. Phys. 2009, 130, 244508. (2) Toda, N.; Mizoguchi, A.; Kanamori, H. Spectral Line Shape Profile of Rovibrational Transitions of CO Embedded in p-H2 Crystals Studied by High Resolution IR Diode Laser Spectroscopy. J. Chem. Phys. 2010, 132, 234504. (3) Mizumoto, Y.; Ohtsuki, Y. Path Integral Molecular Dynamics Simulation of Quasi-Free Rotational Motion of CO Doped in a Large Para-Hydrogen Cluster. Chem. Phys. Lett. 2011, 501, 304−307. (4) Fajardo, M. E. Anomalous Rotation−Translation Coupling Effects in Carbon Monoxide Isotopomers Trapped in Solid Parahydrogen. Presented at the 60th International Symposium on Molecular Spectroscopy, Talk RG04, Columbus, OH, 20−24 June 2005. Available online: https:// molspect.chemistry.ohio-state.edu/symposium_60/symposium/ Program/RG.html#RG04. (5) Friedmann, H.; Kimel, S. Theory of Shifts of Vibration−Rotation Lines of Diatomic Molecules in Noble-Gas Matrices. Intermolecular Forces in Crystals. J. Chem. Phys. 1965, 43, 3925−3939. (6) Mannheim, P. D.; Friedmann, H. Theory of Optical Absorption by Diatomic Molecules Embedded in Rare Gas Crystals. Phys. Status Solidi 1970, 39, 409−420. (7) Manz, J. Rotating Molecules Trapped in Pseudorotating Cages. J. Am. Chem. Soc. 1980, 102, 1801−1806. (8) Fajardo, M. E.; Tam, S. Rapid Vapor Deposition of Millimeters Thick Optically Transparent Parahydrogen Solids for Matrix Isolation Spectroscopy. J. Chem. Phys. 1998, 108, 4237−4241. (9) Tam, S.; Fajardo, M. E. Ortho/Para Hydrogen Converter for Rapid Deposition Matrix Isolation Spectroscopy. Rev. Sci. Instrum. 1999, 70, 1926−1932. (10) Fajardo, M. E. Matrix Isolation Spectroscopy in Solid Parahydrogen: A Primer. In Physics and Chemistry at Low Temperatures; Khriachtchev, L., Ed.; Pan Stanford Publishing: Singapore, 2011. (11) Born, M.; Wolf, E. Principles of Optics, 6th (corr.) ed.; Pergamon Press: Oxford, U.K., 1987. (12) Coxon, J. A.; Hajigeorgiou, P. G. Direct Potential Fit Analysis of the X1Σ+ Ground State of CO. J. Chem. Phys. 2004, 121, 2992−3008 and EPAPS Document No. E-JCPSA6-121-009429.. (13) Herzberg, G. Molecular Spectra and Molecular Structure I, Spectra of Diatomic Molecules; Krieger: Malabar, FL, 1991. (14) Barnes, A. J. Theoretical Treatment of Matrix Effects. In Vibrational Spectroscopy of Trapped Species; Hallam, H. E., Ed.; Wiley: London, 1973. (15) Du, J. H.; Wan, L.; Wu, L.; Xu, G.; Deng, W. P.; Liu, A. W.; Chen, Y.; Hu, S. M. CO2 in Solid Para-Hydrogen: Spectral Splitting and the CO2···(o-H2)n Clusters. J. Phys. Chem. A 2011, 115, 1040− 1046. (16) Scoles, G. Private Communication, 2000. (17) Holden, N. E. Table of the Isotopes. In CRC Handbook of Chemistry and Physics, 84th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2003. (18) Mandel, J. The Statistical Analysis of Experimental Data; Dover: New York, 1984. (19) Jankowski, P.; Szalewicz, K. Ab-Initio Potential Energy Surface and Infrared Spectra of H2-CO and D2-CO van der Waals Complexes. J. Chem. Phys. 1998, 108, 3554−3565. (20) Jankowski, P. Fortran subroutine potH2CO_V04.f. Private Communication, 2004. (21) Hinde, R. J.; Anderson, D. T.; Tam, S.; Fajardo, M. E. Probing Quantum Solvation with Infrared Spectroscopy: Infrared Activity Induced in Solid Parahydrogen by N2 and Ar Dopants. Chem. Phys. Lett. 2002, 356, 355−360. (22) Moroni, S.; Botti, M.; DePalo, S.; McKellar, A. R. W. Small paraHydrogen Clusters Doped with Carbon Monoxide: Quantum Monte Carlo Simulations and Observed Infrared Spectra. J. Chem. Phys. 2005, 122, 094314.

4. CONCLUSIONS

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ACKNOWLEDGMENTS

M.E.F. thanks Mr. S. Tam for his assistance in acquiring the CO/pH2 spectroscopic data, Prof. G. Scoles for suggesting the correlation plot versus mass asymmetry used in Figure 4, Profs. P. Jankowski and K. Szalwicz for sharing their H2−CO PES, his coauthors on the previous CO/pH2 and N2/pH2 manuscripts, Dr. C. M. Lindsay and Profs. T. Momose, D. T. Anderson, and R. J Hinde for teaching him about molecular spectroscopy in solid pH2, and Dr. C. D. Molek for a critical reading of this manuscript. 13511



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(23) Jankowski, P.; McKellar, A. R. W.; Szalewicz, K. Theory Untangles the High-Resolution Infrared Spectrum of the ortho-H2-CO van der Waals Complex. Science 2012, 336, 1147. (24) Raston, P. L.; Jaeger, W.; Li, H.; LeRoy, R. J.; Roy, P. N. Persistent Molecular Superfluid Response in Doped para-Hydrogen Clusters. Phys. Rev. Lett. 2012, 108, 253402. (25) McKellar, A. R. W. Helium Clusters Seeded with CO Molecules: New Results for HeN−13C18O and the Approach to the Nanodroplet Limit. J. Chem. Phys. 2006, 125, 164328. (26) Raston, P. L.; Xu, Y.; Jaeger, W.; Potapov, A. V.; Surin, L. A.; Dumesh, B. S.; Schlemmer, S. Rotational Study of Carbon Monoxide Isotopologues in Small 4He Clusters. Phys. Chem. Chem. Phys. 2010, 12, 8260. (27) von Haeften, K.; Rudolph, S.; Simanovski, I.; Havenith, M.; Zillich, R. E.; Whaley, K. B. Probing Phonon-Rotation Coupling in Helium Nanodroplets: Infrared Spectroscopy of CO and Its Isotopomers. Phys. Rev. B 2006, 73, 054502.

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High-Resolution Rovibrational Spectroscopy of Carbon Monoxide

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