ARTICLE pubs.acs.org/JPCA
~rX ~ Transition Electronic Transition Moment for the 000 Band of the A in the Ethyl Peroxy Radical Dmitry Melnik, Phillip S. Thomas, and Terry A. Miller* Department of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, Ohio 43210, United States
bS Supporting Information ABSTRACT: The electronic transition moment for the G-conformer of ethyl peroxy was determined from the experimentally measured value of the peak absorption cross-section and the simulation of its rovibronic spectrum using the results of the high resolution spectroscopy of this molecule. The resulting value is |μeG| = 2.55(6) 102 Debye, which is compared to values from electronic structure calculations.
’ INTRODUCTION The chemistry and spectroscopy of the organic peroxy radicals, RO2, have been studied for over three decades due to the importance of these species in combustion and atmospheric processes.14 All organic peroxy radicals exhibit a ~rX ~ band in the UV, which, however, is diffuse and strong B cannot be used to differentiate one peroxy radical from ~rX ~ electronic transianother. The observation of a weak A tion in the near-infrared region (NIR) at about 1.3 μm allows one to discriminate among different peroxy species,5 as well as analyze the details of their molecular structure.68 Although ~ rX ~ absorption is over 2 orders the peak cross-section of the A ~rX ~ transition, the of magnitude smaller than that of the B recent development of the sensitive cavity ring-down spectroscopy (CRDS) technique allows one to use the NIR electronic transition as an analytical tool for quantitative measurements911 of the radical concentration. A key molecular parameter that is required to quantify the absorption spectrum is the peak absorption cross-section, σp, which can be obtained directly from experiment. However, the value of σp is pressure- and temperature-dependent and can be measured over only a limited range of conditions for a given laboratory experiment. Nevertheless, the square modulus of the electronic transition dipole moment (TDM), |μe|2, can be derived from σp for a given vibronic transition at a given pressure and temperature, as long as the structure of the energy levels is known. Knowledge of the TDM allows the NIR absorption to be used for analytical concentration determinations over a wide range of conditions. The experimental measurements of the transition moment are ~ r X ~ also important from a theoretical perspective. The A transition correlates with the triply forbidden 1Δg r 3Σ g transition in molecular oxygen. When a peroxy radical RO2 is • formed, the R adduct induces a perturbation that makes this transition weakly allowed. The magnitude of this perturbation ~ r X ~ electronic can be characterized by the TDM of the A transition. The value of μ can be calculated using modern r 2011 American Chemical Society
computational methods but is quite sensitive to the level of calculations12,13 and in this case, accurate determination is made more difficult due to its small magnitude. Therefore, the experimentally measured values of TDM serve as a key benchmark for quantum chemistry calculations and provide valuable information about the electronic structure of the species. In this work, we first discuss the relationship between the experimentally measured absorption cross-section and the TDM, which involves molecular parameters that can be obtained from a combination of vibrational and rotational spectroscopic studies and theoretical calculations. We then describe the analysis of the experimental results for σp and the resulting value of the TDM for the G-conformer of the ethyl peroxy radical, C2H5O2. Finally, we discuss theoretical calculations of μ and compare them to the value obtained from our analysis of the experiment.
’ THEORY General Expression for |μe|2. The electronic absorption spectrum of a sufficiently large molecule under ambient conditions consists of a large number of overlapping rotational lines belonging to vibrational bands of, generally, a few different conformers. Since the absorption of an individual rotational transition can be straightforwardly related to the magnitude of μe, the corresponding relationship for the entire spectrum can be obtained by adding up the profiles of individual lines with weights proportional to their line intensities. For practical use, such a relationship needs to be cast in terms of quantities that can be conveniently calculated using generic spectrum simulation software. To derive the value of the TDM, μe, from the experimentally measured σp for the observed band contour for a given electronic transition, we first express the absorption cross-section σc,v0 η0 ,v00 η00 (ν) Received: August 9, 2011 Revised: October 14, 2011 Published: October 17, 2011 13931
dx.doi.org/10.1021/jp207647h | J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
for an individual rotational transition of a conformer c as a function of the integrated line intensity S c,v0 η0 ,v00 η00 ,14 σc, v0 η0 , v00 η00 ðνÞ ¼ S c, v0 η0 , v00 η00 f c, v0 η0 , v00 η00 ðνÞ
σc, v0 η0 , v00 η00 ðνÞ ¼
ð1Þ
where vibrational and rotational quantum numbers are denoted as v and η, respectively, and the single and double primes denote quantum numbers of the upper and lower levels, respectively. On the right-hand side of eq 1, fc,v0 η0 ,v00 η00 (ν) is the line shape function for an individual rovibronic line whose maximum is at the center frequency, ν = ν0c,v0 η0 ,v00 η00 and which is normalized such that its integral over the frequency domain is equal to unity. In general, fc,v0 η0 ,v00 η00 (ν) includes the effects of Doppler, collisional, and natural broadening and depends on both pressure and temperature. S c,v0 η0 ,v00 η00 is not pressure-dependent and can be written as a product of a TDM-independent and a temperaturedependent Boltzmann factor and a temperature-independent, TDM-dependent factor Rc,v0 η0 ,v00 η00 , i.e.,15 F c, v0 η0 , v00 η00 ðTÞ 8π3 0 νc, v0 η0 , v00 η00 R c, v0 η0 , v00 η00 S c, v0 η0 , v00 η00 ¼ Q ðTÞ 3hc ð2Þ where is in cm . The factor Rc,v0 η0 ,v00 η00 in eq 2 denotes a weighted transition moment squared,15,14 defined as
∑q
Gv0 v00 jμcq j2 Sq ðη0 ; η00 Þg 1 η00
ð3Þ
where Gv0 v00 is the FranckCondon factor,14 Sq(η0 ;η00 ) is the HonlLondon factor,16 and μcq is one of the components (e.g., projections of the TDM on the molecule’s principal axes denoted by q) of the electronic TDM of conformer c. The HonlLondon factor depends on q implicitly since the type of the rotational transition imposes a constraint on the η0 and η00 quantum numbers for Sq(η0 ;η00 ) not to vanish. The factor of g1 η00 is included to cancel the corresponding factor in the definition of Sq(η0 ;η00 ) to allow the rotational degeneracy gη00 to be included in Fc,v0 η0 ,v00 η00 (T). In the remaining, bracketed term of eq 2, Q(T) is the total partition function, and Fc,v0 η0 ,v00 η00 (T) is the Boltzmann factor for a particular rovibronic transition. For NIR transitions, kT , hcν0c,v0 η0 ,v00 η00 , and all rotational levels of the excited electronic states are essentially unpopulated. Therefore, the Boltzman factor is independent of the parameters of the upper electronic state and can be written as Fc,v00 η00 , where Ec, v00 , η00 F c, v00 η00 ðTÞ ¼ g c g v00 gη00 exp kT Ec0, v00 , 0 Ec, 0, 0 ¼ g c exp g v00 exp kT kT " # c E0, v00 , η00 g η00 exp ð4Þ kT where E c,0,0 is the energy of the lowest rovibrational level of c 00 conformer c, E 0,v ,0 is the energy of the rotationless level of the v00 th vibrational level of conformer c, and Ec0,v 00 ,η00 is the rotational energy of the level η 00 relative to η00 = 0 for the v00 state; g c is the statistical weight of the conformer c, g v00 is the vibrational degeneracy, and g η00 is the rotational level degeneracy.
8π3 νc, v0 η0 , v00 η00 c 2 Ec, 0, 0 jμq j g c exp Q ðTÞ kT
∑q 3hc
Ec0, v00 , 0 0 00 g v00 Gv v exp kT 0
2
00
@Sq ðη0 ; η00 Þ exp4
Ec,0,v0, η00 kT
31 5Afc, v0 η0 , v00 η00 ðνÞ
ð5Þ
~ state At room temperature conditions, a large number of X rovibronic levels are populated, giving rise to a correspondingly large number of transitions that form a band contour. To obtain the absorption profile, σ(ν), of such a contour, eq 5 must be summed over all indexes c,v0 ,η0 ,v00 ,η00 . Assuming that the ν0c,v0 η0 ,v00 η00 are all approximately equal and that the band center peak frequency is νp, for a narrow absorption spectrum, we can write the expression for the frequency-dependent absorption cross-section σ(ν)
1
ν0c,v0 η0 ,v00 η00
R c, v0 η0 , v00 η00 ¼
By combining eqs 14, we obtain
σðνÞ ¼
8π3 νp 3hc Q ðTÞ
∑c jμce j2 A c ðνÞ
ð6Þ
The |μce|2 is the absolute value of the electronic transition moment for conformer c, i.e., |μce|2 = ∑q|μcq|2, and Ec, 0, 0 A c ðνÞ ¼ gc exp Ic, v0 , v00 Pc, v0 , v00 ðνÞ ð7Þ kT v0 , v00
∑
0 " #1 μc 2 Ec0, v00 , η00 q 0 00 @ Afc, v0 η0 , v00 η00 ðνÞ Pc, v0 , v00 ðνÞ ¼ c Sq ðη ; η Þ exp μe kT qη0 η00
∑
ð8Þ
I
c, v 0 , v 00
¼G
v 0 v 00
Ec0, v00 , 0 exp kT
ð9Þ
The quantity Pc,v0 ,v00 (ν) in eq 8 is the rotational contour of the individual v0 r v00 vibrational band of conformer c including all contributions to the rotational transitions for all values of q. The profiles of all vibrational bands of conformer c weighted by the intensity factors Ic,v0 ,v00 are summed resulting in the absorption profile A c (ν) of the specfied conformer. Similarly, summation over the |μce|2 of all conformers weighted by their absorption profiles and normalized by the total partition function Q(T) in eq 6 results in the complete spectrum of the molecule. Generally, the values of |μce|2 differ among the conformers contributing to the absorption spectrum, and therefore, to obtain their values, the absorption needs to be measured at several frequencies (the number of such independent measurements should be equal to or greater than the number of conformers). In some cases, however, only one conformer contributes significantly to the absorption at the peak absorption frequency. In such cases, the summation over c in eq 6 reduces to a single term, and the value of |μce|2 for the corresponding conformer c can be readily written as jμce j2 ¼ σp 13932
3hcQ ðTÞ 8π3 νp A p
ð10Þ
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
where A p = A c (ν0p). Equation 10 gives the value15 of the square of the transition moment in statC2 3 cm2 = erg 3 cm3 = 1036 Debye2 (1 D = 3.3356 1030 C 3 m in SI units). Equations 610 show that apart from the experimentally measured values of σp and νp, two additional quantities, A p and Q(T) are needed to to determine |μce|. Both Q(T) and A p ~ state. Additionally, depend on the energy level structure of the X ~ state, A p depends on the rovibrational level structure of the A the FranckCondon factors for the contributing vibrational bands and the orientation of the |μce| with respect to the principal axes of the molecule, as defined by the |μcq/μce| ratios. As eqs 79 indicate, to obtain A p , the knowledge of the vibrational energies, rotational constants of upper and lower vibrational levels, the FranckCondon factors of the contributing bands, and the orientation of μe are required. The precise calculation of the total partition function Q(T) in eq 10 is a nontrivial task. However, a few simplifying approximations can be made to render this task manageable. We first assume that the molecule exists in the form of the distinct conformers corresponding to the local minima of the potential energy surface. The total partition function includes contributions from electronic, conformational, vibrational, and rotational partition functions. Assuming the electronic partition function to be unity, we express Q(T) in the general form, Ec0, v00 , 0 r Ec, 0, 0 Q ðTÞ ¼ ∑ gc exp gv00 exp Qc, v00 ðTÞ kT ∑ kT c v00 ð11Þ where Qrc,v00 is the rotational partition function of the v00 vibrational level of the conformer c. The summations in eq 11 give the contributions to Q(T) from the various conformers and the vibrational levels populated. For a rigid asymmetric rotor without symmetry factors, in the high-temperature limit (kT . hcAc,v00 ), Qrc,v00 is given as a function of temperature as " #1=2 πðkTÞ3 r Qc, v00 ðTÞ ¼ ð12Þ ðhcÞ3 Ac, v00 Bc, v00 Cc, v00 where Ac,v00 , Bc,v00 , and Cc,v00 are the corresponding rotational constants in cm1. Although the rotational constants of a given conformer vary for different vibrational levels, for the purpose of the calculation of Qrc,v00 (T), this variation is small and can be ignored. We can therefore substitute Ac,v00 , Bc,v00 , and Cc,v00 with the values Ac, Bc, and Cc in the vibrational ground state of the given conformer, and replace Qrc,v00 (T) in eq 12 with a v00 independent function Qrc(T), " #1=2 πðkTÞ3 r ð13Þ Qc ðTÞ ¼ ðhcÞ3 Ac Bc Cc Then the last two terms in eq 11 separate, and the total partition function can be rewritten in the form Ec, 0, 0 vib gc exp ð14Þ Q ðTÞ ¼ Qc ðTÞQcr ðTÞ kT c
∑
where Qvib c (T) is the vibrational partition function of the conformer c. If the molecule is approximated as a system of noninteracting harmonic oscillators, then its vibrational partition
function is given as17 Qcvib ≈Qcvib, 0 ðTÞ
¼
Y
j
hvc, j 1 exp kT
1
ð15Þ
where j runs over all vibrational modes whose fundamental frequencies are νc,j, and the superscript 0 indicates the harmonic approximation. In eq 15, we dropped the ZPE term since the vibrational level energies in eq 6 and eq 11 are measured with respect to the vibrationless level. Potentials of realistic molecules exhibit anharmonicities, which affect the energies of the vibrational levels and therefore the values of the partition function. In general, calculation of the vibrational partition function of such potentials is a complicated task that requires the knowledge of the details of the molecular potential energy surface. In some cases, however, it is possible to account for the anharmonic effects by introduction of an adjustment factor qac to the harmonic partition function, eq 15, Qcν ðTÞ ¼ Qcv, 0 ðTÞqac ðTÞ
ð16Þ
The discussion of the anharmonic effects and derivation of qac for a moderately anharmonic potential are given in the Appendix from which it follows that fundamental frequencies νc,j and anharmonicities χc,j of the vibrational modes are required to calculate Qνc (T). Quantum Chemical Calculations. From the discussion above, it is clear that Q(T) and A p require considerable knowledge about the radical’s structure, which can be typically summarized as a set of the molecular parameters. Some of these parameters can be obtained directly from experiment. For others, we turn to the quantum chemical electronic structure calcula~ X ~ absorption spectrum tions. To aid in the simulation of the A of ethyl peroxy, a series of electronic structure calculations were performed at the UB3LYP/aug-cc-pVTZ level of theory. Equilibrium geometries, normal modes, and harmonic vibrational frequencies were obtained for each conformer in both electronic states. Fundamental vibrational frequencies were also calculated, where anharmonicity was included via second order vibrational perturbation theory. Since B3LYP is a ground state method, the ~ state electronic wave function was obtained by permuting the A HOMO and SOMO orbitals at the outset of an initial ROHF ~ state wave single point calculation; the resultant converged A function was used as an initial guess for a single point UHF calculation, and the converged UHF wave function served as an initial guess for all subsequent calculations at the UB3LYP level. ~ and A ~ state potential energy curve scans were also perX formed for the OOCC torsion vibration at the UB3LYP/aug-ccpVTZ level. Here, the OOCC dihedral angle was scanned in 5° intervals, and all remaining degrees of freedom were relaxed at each step. The time-independent Schr€odinger equation was solved numerically for each potential curve using a basis of 73 plane waves; resultant energy levels and wave functions are used to calculate Boltzmann weights and FranckCondon factors for all possible transitions within the range of the basis set. The reduced mass for each potential curve was chosen to minimize the root-mean-squared error for the 0 f 1 transitions of the two conformers generated from the surface relative to the 0 f 1 transitions obtained from the fundamental frequency calculations described above. All electronic structure calculations were performed using Gaussian 09.18 FranckCondon (FC) simulations were performed using the MolFC package,19 which calculates FranckCondon factors in 13933
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
Table 1. Calculated Values of the Harmonic Frequencies νG,j, in cm1, and Anharmonic Constants, χG,j, of the Normal ~ Electronic Modes of the G-Conformer of C2H5O2 in the X , State and the Level Energies with Single, E(1) G,j and Double, , Excitation in the Corresponding Vibrational Mode As E(2) G,j Obtained from the Anharmonic Calculations mode
νG,j
χG,j
E(1) G,j
E(2) G,j
Table 2. Calculated Values of the Harmonic Frequencies νT,j, in cm1, and Anharmonic Constants, χG,j, of the Normal ~ Electronic Modes of the T-Conformer of C2H5O2 in the X , State and the Level Energies with Single, E(1) T,j and Double, , Excitation in the Corresponding Vibrational Mode E(2) T,j Obtained from the Anharmonic Calculationsa mode
symmetry A0
νT,j
χT,j
E(1) c,j
E(2) c,j
ν1
3129
0.024
2980
5824
ν1
3108
0.023
2967
5872
ν2
3108
0.023
2963
5887
ν2
3057
0.027
2890
5808
ν3
3102
0.024
2955
5864
ν3
3043
0.023
2900
5850
ν4 ν5
3063 3041
0.020 0.019
2940 2925
5824 5851
ν4 ν5
1507 1494
0.008 0.009
1482 1468
2971 2948
ν6
1504
0.007
1482
2976
ν6
1418
0.009
1392
2766
ν7
1487
0.013
1447
2882
ν7
1379
0.011
1350
2689
ν8
1482
0.014
1441
2872
ν8
1192
0.010
1167
2321
ν9
1409
0.012
1375
2730
ν9
1132
0.012
1105
2206
ν10
1376
0.011
1346
2681
ν10
1012
0.015
982.6
1959
ν11
1307
0.013
1272
2537
ν11
837.6
0.012
818.3
1632
ν12 ν13
1204 1149
0.010 0.012
1181 1121
2353 2233
ν12 ν13
500.5 306.7
0.005 0.007
496 310.8
989.8 630.2
ν14
1096
0.011
1073
2140
ν14
3121
0.024
2972
5911
ν15
991.4
0.015
962.5
1917
ν15
3098
0.024
2949
5863
ν16
841.6
0.012
820.8
1639
ν16
1485
0.007
1463
2936
ν17
797.4
0.009
782.9
1568
ν17
1277
0.011
1249
2490
ν18
523.4
0.007
516.2
1031
ν18
1147.6
0.011
1123
2244
ν19
361.0
0.011
353.4
702.9
ν19
804.1
0.002
806.9
1624
ν20 ν21
225.4 109.3
0.046 0.092
246.2 89.16
505.1 165.1
ν20 ν21
219.0 78.71
0.088 0.021
257 75.44
529.0 140.0
A00
a
the harmonic approximation with inclusion of Duschinsky mixing; these used UB3LYP/aug-cc-pVTZ unscaled harmonic frequencies and normal modes as input. Excitations are considered ~ state to all A ~ state levels with from the vibrationless level of the X a maximum quantum number of ν = 2 for the seven modes of lowest frequency, excluding CCOO torsion, and ν = 1 for the remaining modes, including combination levels. Since the FC simulations only consider cold (0 K) absorption, additional hot (T = 298 K) sequence band (SB)13 structure was calculated ~ and A ~ state UB3LYP/augbased upon differences in computed X cc-pVTZ fundamental frequencies, where the weight of each ~ state level. All sequence transition is the Boltzmann factor for the X transitions out of levels for which the Boltzmann weight is g0.1% that of the vibrationless level are included, including combination levels. Nonsequence hot transitions are expected to be weak and are not included in the simulations. OOCC torsion transitions, already calculated from the respective potential curves (vide supra), are also omitted from the FC and SB simulations. The calculated values of harmonic frequencies of normal ~ state, νc,j, and the energies of the levels with modes in the X (1) (2) , Ec,j one and two quanta of excitation in each mode, Ec,j obtained in the anharmonic calculations described above, are given in Table 1 for the G-conformer and Table 2 for the T-conformer. Table 3 gives the predicted vibrational structure ~ rX ~ transition of the G-conformer of the origin region of the A of ethyl peroxy. The first column of Table 3 shows the vibrational bands whose absolute intensity Ic,v0 ,v00 at T = 298 K, defined in eq 9 and given in the last column of Table 3, is greater than 0.05 and which absorb in the observed11 origin region of the G-conformer. Two torsional bands, 2112 and 2123 originate from levels lying
The vibrational modes are arranged by the symmetry that is indicated in the second column.
close to the barrier to CCOO torsion and therefore split into two components, labeled (1) and (2). The second column of Table 3 shows the calulated shifts Δν of the band frequencies with respect to the frequency of the origin. For G-conformer bands, the shift is measured with respect to the previously obtained7 origin frequency at 7592 cm1, and for T-conformer bands, 7362 cm1.7 (The frequency shifts of several bands were subsequently adjusted to fit the experimental spectrum, and these values are given in the third column.)
’ EXPERIMENTAL RESULTS AND CALCULATION OF Q(T) AND A c (Ν) The above results from quantum chemistry calculations can be combined with molecular parameters from the high-resolution spectral analysis of C2H5O2 to produce a detailed energy level structure to enable an accurate calculation of Q(T) and A c (ν). For practical purposes, the spectra of the G and T conformers are well separated and except for minor correction (vide infra) can be treated independently. Ergo, the argument preceding eq 10 applies, and the previously measured11 peak absorption cross-section of the G-conformer of C2H5O2, σp σEP p , can be used to obtain its electronic TDM, which we, henceforth, refer to as μGe . (In the following, in the notation for all conformer-specific quantities, the generic conformer index c is replaced with the appropriate conformer designation, G or T, wherever a given conformer is specified.) To obtain the value of |μGe |2 from the 13934
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
Table 3. Frequencies and Intensities of the Vibrational Bands Used in the Simulation of the Contour of the Absorption ~rX ~ Electronic Transition of the Spectrum of the A G-Conformer of C2H5O2 in the Vicinity of the 000 Banda
Table 4. Rotational Constants (in cm1) of Vibrational ~ and A ~ Electronic States of the G-Conformer of Levels in the X C2H5O2 A
Δν band
calculated
Ic,v0 ,v00 adjusted
relative
level
C
calculated adjusted calculated adjusted calculated adjusted ~ State X
absolute
G-Conformer
B
00
0.60533
0.59099a
0.18241
0.18899a
000
0
0
1.0
0.427
211
0.61862
0.60397
0.18002
0.18651
0.15823
0.16200
2111
49.7
21.0
0.350
0.149
201
0.60382
0.58952
0.18258
0.18822b
0.15903
0.16201b
191
0.59902
0.58483
0.18377
0.18945b
0.15926
0.16224b
0.18125 0.17763
b
0.15866 0.15727
0.16163b 0.16102
2011
21.8
11.8
0.305
0.130
1911
25.7
22.3
0.182
0.078
2112(1)
27.6
27.6
0.178
0.076
181 212
2112(2)
29.3
29.3
0.162
0.069
213
2110 2101
140.9 91.1
140.9 77.1
0.166 0.143
0.071 0.061
2123(1)
39.5
39.5
0.118
0.051
2123(2)
54.0
54.0
0.092
0.039
20112111
27.9
27.9
0.106
0.045
2124
17.1
17.1
0.089b
0.038
2022 1811
61.0
64.0
0.084
0.036
64.5
64.5
0.083
0.035
0.61038 0.63191
0.62990c
b
0.16005c
0.60231
0.58804
0.18275
0.18840
0.15886
0.16183b
201211
0.61711
0.60249
0.18019
0.18669
0.15806
0.16183
0.15816
0.16239a
0.56558
0.55305a
0.18583
0.19250a
1
21
0.56868
0.55609
0.18542
0.19207
0.15811
0.16234
201 191
0.56920 0.56359
0.55660 0.55111
0.18486 0.18597
0.19246b 0.19360b
0.15779 0.15812
0.16283b 0.16316b
181
0.57036
0.55772
0.18500
0.19260b
0.15780
0.16283b
2
0.57179
0.55912
0.18501
0.19165
0.15806
00
21
118.5 184.5
88.2 184.5
0.373 0.143
0.181 0.069
213
21112011
50.9
50.9
0.185
0.090
1
experimentally measured value of σEP p , both Q(T) and A G (ν) need to be known. Although only the value of A G (νp) is required (see eq 10), simulation of the rotational contours in ~rX ~ origin in the vicinity of the vibrational structure of the A νp are necessary to ensure the correct analysis of the observed spectra and to avoid systematic errors. A few sequence bands of the T-conformer of ethyl peroxy also contribute slightly to the absorption to the red of the origin of the G-conformer and need to be included in the absorption simulation for completeness. Although generally the μce of the different conformers are not equal to each other, the calculations of the oscillator strengths imply13 that their values are similar for the G and T conformers of C2H5O2, and since the effect of the T-conformer transition to the G-conformer profile is quite small, for the purpose of these spectral simulations, we assume |μTe | ≈ |μGe | The rotational contours of the individual vibrational bands, Pc,v0 ,v00 (ν) depend on the rotational constants of the upper and lower vibrational levels and the orientation of the transition moment with respect to principal axes, |μcq/μce|. These parameters are available for the most significant contributor, the 000 band, from the recent analysis of the jet-cooled high resolution spectra of C2H5O2.7 The frequencies, as well as the intensity factors, Ic,v0 ,v00 , and FranckCondon factors of the considerably weaker vibrational sequence bands are obtained from the vibra~ and A ~ states. In addition, tional structure calculations of the X
0.18158c
202
2122 2133
The frequencies of the bands, Δν, (in cm1) are given with respect to the origin at 7592 cm1 of the G-conformer and with respect to the origin at 7362 cm1 of the T-conformer. The absolute intensities of the bands defined in eq 9 and their values relative to that of the G-conformer origin band are given in the fifth and fourth columns, respectively. b Calculated rotational profile of 2113 has been used to simulate this band.
0.18685 0.18404
0.16299a
~ State A
T-Conformer
a
0.59593 0.61694
0.15919
0.56214c
202 1
20 21
0.19124c
0.16229 0.16224c
0.57283
0.56015
0.18390
0.19145b
0.15743
0.16246b
0.57231
0.55964
0.18445
0.19107
0.15774
0.16197
a
Experimental values obtained in high resolution studies.7 b Rotational ~ state and by 1.005 constants are scaled by a factor of 0.995 in the X ~ state to match an experimentally observed band profile. in the A c Obtained from quadratic regression from the values of the 00, 211, and 212 levels.
Table 5. Rotational Constants (in cm1) of Vibrational ~ and A ~ Electronic States of the T-Conformer of Levels in the X C2H5O2 for the Levels Connected by the Bands Contributing to the Absorption in the Region of the G-Conformer A level
B
C
calculated adjusted calculated adjusted calculated adjusted ~ State X
00
1.08400
212
1.02690
1.10180
0.14600
1.04376
0.14622
1.05189
1.06915
0.14766a
0.13525
0.14788
0.13657
0.14799c
1.01474
213 201211
a
0.14594
0.13725a 0.13856 0.13972c
0.14759
0.13582
0.13774
~ State A 1.06429
1.06663a
0.14612
0.14844a
0.13488
0.13715a
2
21 213
1.00773
1.00995 1.01474
0.14623
0.14855 0.14799c
0.13572
0.13800 0.13972c
201211
1.03279
1.03506
0.14600
0.14832
0.13524
0.13751
00
a
Experimental values obtained in high resolution studies.7 c Obtained from quadratic regression from the values of the 00, 211, and 212 levels.
these electronic structure calculations can be used to scale the experimental rotational constants for the vibrationless state to correct values for the excited vibrational levels. 13935
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
The rotational constants of the various vibrational levels of the ~ and A ~ states, as obtained in the anharmonic calculations for G X and T conformers of ethyl peroxy, are given in Table 4 and Table 5, respectively. The results of the calculations are given in the second, fourth, and sixth columns of the tables. These constants were scaled to the experimentally measured values of the rotational constants in the ground levels of corresponding electronic state, exp
Bα ðvÞ ¼ Bcalc α ðvÞkα
Bα ð0Þ Bcalc α ð0Þ
ð17Þ
where Bcalc α (v) is the value of the rotational constant in vth level obtained in the quantum chemistry calculations, Bexp α (0) is the experimentally measured value of the rotational constant in the ground vibrational level,7 and kα is the small additional correction factor discussed below. By using the above information, the rotational profiles Pc,v0 ,v00 (ν) were calculated using the rotational Hamiltonian for an asymmetric rigid rotor using the SpecView20 program, assuming T = 298 K and including contributions from rotational levels with up to J = 100. For the G-conformer, all three types of the rotational transitions (a-, b-, and c-type) are allowed, with |μGa /μGe |2 = 0.229, |μGb /μGe |2 = 0.499, and |μGc /μeG|2 = 0.272, derived from the experimentally measured high-resolution spectrum.7 For the T-conformer, only the c-type transition is symmetry allowed, and experimentally observed; thereT fore, |μc /μeT| = 1. Inclusion of spin-rotational interaction and/or centrifugal distortion does not significantly affect the calculated absorption profiles; hence, these terms were ignored to expedite the calculations. For the individual rovibronic line shape, fc,v0 η0 ,v00 η00 (ν), the Doppler HWHM was calculated21 to be ΔνD = 0.006 cm1. The homogeneous broadening of the rotational lines for ethyl peroxy is the combination of the collisional broadening with the buffer gas (air at P = 300 Torr) and lifetime broadening due to the internal conversion. The latter has been measured at 1400 MHz FWHM in the collision-free high resolution free jet experiment, which corresponds to 0.023 cm1 HWHM. The pressure broadening coefficient for ethyl peroxy is not available; however, for the molecules of similar or smaller size, e.g., C2H6, its value falls in the range of 0.070.1 cm1/bar.15 For the purpose of the simulation, we assumed a J-independent HWHM homogeneous line width of 0.1 cm1 (3000 MHz). With the molecular parameters of Tables 15, it is possible to simulate relatively well the overall spectral profile in the vicinity ~ rX ~ origin transition. However, the preliminof the C2H5O2 A ary analysis showed that the rotational profiles generated using the rigid rotor model and rotational constants calculated using eq 17 with kα = 1 predict the absorption spectrum A G (ν) with multiple pronounced sharp features corresponding to Q-branches of all the sequence and other hot bands shown in Figure 1. Experimentally, such sharp features are observed only in the origin band and 2111 sequence band, while other bands exhibit a smoother rotational contour with no pronounced Q-branch or a head for the P- or R-branch. Such smooth band profiles can be obtained by small ~ and A ~ adjustments of the values of rotational constants in both the X electronic states. Even though the electronic structure calculations that were used may give quite accurate geometries and corresponding rotational constants,7 such adjustments may be necessary since the quantum chemistry calculations do not account for effects relating to the breakdown of the BornOppenheimer approximation, e.g., Coriolis interaction, which can give rise to effective
Figure 1. Experimentally observed (lower black trace) and simulated ~ rX ~ transition (upper red trace) spectrum of the origin region of the A of the G-conformer of C2H5O2. The details of the simulation are given in the text. The arrows indicate the positions of the vibrational bands that provide the major contributions, in addition to the 000 band, to the absorption in this region. The peak absorption cross-section, σEP p , in the experimental spectrum is measured11 to be 5.29(20) 1021 cm2 at the frequency of the head of the R-branch (νp = 7596 cm1) of the 000 band, which is indicated by the bold blue arrow. For the clarity of presentation, the simulated trace is vertically shifted by 1.5 1021 cm2 with respect to the experimental trace. Its corresponding vertical scale is shown in red.
rotational constants not entirely geometrically determined. The previously observed homogeneous broadening due to molecular dynamics in the collision-free environment may be ~ and X ~ electronic states, indicative of such coupling between the A which can potentially affect the vibrationally averaged values of the ~ state. The simularotational constants, particularly in the excited A tions showed that rotational profiles of the G-conformer are fairly sensitive to the values of B and C rotational constants. To obtain band profiles PG,v0 ,v00 (ν) that resulted in the satisfactory prediction of the absorption spectrum, the rotational constants of several vibrational levels were calculated using eq 17 for various values of kα. ~ Values of kα = 0.995 for B and C rotational constants of the X electronic state, kα = 1.005 for the B and C rotational constants of ~ state, and kα = 1 for all A rotational constants were found to be the A optimal. The resulting values of the rotational constants are given in the columns 3, 5, and 7 of Table 4. All rotational constants for the T-conformer were calculated using eq 17 without any further correction (i.e., kα = 1). The rotational constants of the levels with more than two quanta of excitation were obtained by quadratic regression using the rotational constants with levels of lower degrees of vibrational excitation. Because of the rapid changes of the rotational constants with the excitation of the torsional mode, ν21, the rotational constants for n21 = 4 were not calculated by the polynomial regression; instead, the rotational profile calculated for the 2113 hot band was used to simulate the 2124 hot band. With these adjustments to the rotational constants, the resulting simulation of the rotational profile A (ν) is shown in Figure 1. The black arrows indicate the vibrational bands, except the origin band, that provide substantial contribution to the absorption profile. The bold blue arrow indicates the feature and the frequency (νp = 7596 cm1) at which the peak absorption cross-section was measured. In Figure 1, the simulated spectral profile is compared to the experimental trace obtained in the preceding work11 to show the good agreement with the latter. 13936
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
Table 6. Vibrational Partition Functions, Anharmonic Corrections, and Rotational Partition Functions for the G and T Conformers of C2H5O2, Obtained Using Different Dataa Qv,0 G
data used νc,j,χc,j νec,j,
5.172
χec,j
5.399
qaG
QrG
Qv,0 T
1.263
3.92 10
4
1.210
3.92 10
4
7.344 6.517
qaT
QrT
0.940
3.55 10
4
1.222
3.55 10
4
0
νc,j, χc,j, En
Qvtotal
Qr
Q 6.78 105
18.18
3.82 10
5
7.03 105 6.95 105
The values of QGv,0 and QTv,0 are obtained using eq 15; the values of qGa and qTa are obtained from eq A.6; and the values of QGr and QTr are obtained using eq 13. The total partition function Q(T = 298 K) in the first two lines is obtained using eqs 14 and A.5. The total vibrational Qvtotal, the averaged rotational Q r partition functions, and the total partition function Q(T = 298 K) in the last line are obtained using eqs A.9 and A.10. The values of Ec,0,0 and gc used in these calculations are given in the text. a
The peak value of the simulated spectral profile obtained from eqs 79 using the specified parameters, is found to be A p = 1.79(4) 103 cm. We use eq 13 and eq A.5 from the Appendix to calculate the rotational and vibrational partition functions of the two conformers. To obtain the total partition function, we used eq 14 and the value of the energy gap between T- and G-conformers, ET,0,0 = 81 cm1, calculated previously,10 and EG,0,0 = 0, with gG = 2 and gT = 1. The results of the calculations of the partition functions are shown in Table 6. To calculate the vibrational partition functions, we used both the anharmonic oscillator approximation of eqs A.5 and A.6 and the direct calculation of the contribution from the CCOO torsional mode described by eqs A.9 and A.10. For the anharmonic oscillator approximation, we used the values of the harmonic frequencies νc,j calculated in this work. a The anharmonic parameters χc,j required for calculation of qc,j were obtained from the calculated energies of vibrational levels with a single quantum of excitation in each mode ð1Þ
χc, j ¼
Ec, j 2νc, j
ð18Þ
The values of νc,j and χc,j are given in columns 2 and 3 of Table 1 for the G-conformer and columns 3 and 4 in Table 2 for the T-conformer. It should be noted that eq 18 does not account for the higher order anharmonic terms, i.e., it implies that eq A.1 is rigorous, and eq 18, therefore, gives the correct value of the anharmonicity. This approximation may fail for the low-frequency highly anharmonic modes, whose energy level structure is sensitive to higher order anharmonic parameters. However, for the lowest few excited states (v = 1,2), this inaccuracy can be minimized by fitting the energy level structure, given in the last two columns of Tables 1 and 2, to eq A.1 and deducing the effective values of e e and χc,j νc,j ð1Þ
ð2Þ
νec, j ¼ 3Ec, j Ec, j χec, j ¼
1 ð1Þ ð2Þ ð2E Ec, j Þ 2νec, j c, j
ð19Þ ð20Þ
that describe the energies of the lowest few levels most accurately. While this approach still makes use of eq A.1 and the resulting expressions for the vibrational partition function, it implicitly accounts for the higher order anharmonic terms. If e e = νc,j and χc,j = χc,j, and therefore, the eq A.1 is rigorous, then νc,j e e and χc,j will be vibrational partition function, calculated with νc,j identical to that calculated with νc,j and χc,j. Therefore, the discrepancy in calculation of Qvc (T) using these two methods provides a measure of the inaccuracy associated with the
approximation of eqs A.1A.6. The results obtained using e e , χc,j for the νc,j, χc,j are consistent with those obtained using νc,j G-conformer but significantly inconsistent for the T-conformer. However, although the discrepancies in QTv,0 and qTa are relatively large, their product, the anharmonicity corrected partition function QTv, eq A.5 is more consistent, possibly due to the anticorrelation of QTv,0 and qTa. Moreover, the T-conformer makes only a quite minor contribution to σpEP. To calculate the partition function using eqs A.9 and A.10, we have solved the Schr€odinger equation for the torsional potential as described in the Quantum Chemical Calculations section and obtained the eigenvalues for the torsionally excited states. The 0 resulting energies En for up to 3000 cm1 are available in the Supporting Information. For the purpose of the calculation of the vibrational partition function, the energies of the 5 torsional ~ state were states localized in the T-conformer well of the X shifted up by 106.8 cm1 such that the lowest T-conformer level lies 81 cm1 above the doubly degenerate lowest level of the G-conformer.10 The resulting values of Qvtotal, Q r, defined in eqs A.8 and A.9 in the Appendix, respectively, and Q are given in the last row of Table 6.
’ RESULTS AND DISCUSSION For the determination of |μeG|, we use the average value of the total partition function obtained using the different methods described previously and obtained Q(T = 298 K) = Q = 6.92(10) 105. Using A p =1.79(4) 103 cm and σpEP=5.29(20) 3 1021 cm2, we obtain jμGe j2 ¼ 6:78ð32Þ 104 Debye2 jμGe j ¼ 2:55ð6Þ 102 Debye
ð21Þ
It is important to discuss the indicated errors associated with the value of |μe|. As eq 10 indicates, there are four sources of the statistical error of |μeG|2: νp, A p, σpEP, and Q(T). The peak frequency νp is measured to better than 0.01%, which for the purpose of this analysis constitutes a negligible error. The peak absorption of the spectral profile is calculated numerically; therefore, its uncertainty results from the error propagation from the input parameters. While some parameters such as rotational constants for the sequence bands were slightly adjusted to match the experimental profile using an approximately estimated parameter kα, the major contribution (about 85%) to the absorption at the peak value comes from the origin band, whose parameters are determined essentially exactly in the high-resolution studies.7 Therefore, the uncertainty in A p is primarily defined by the uncertainty of the FranckCondon factor calculations for the origin band of the G-conformer. 13937
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
Table 7. Comparison of the Results of the Quantum Chemistry Calculations with the Experimental Valuesa geometry calc
A (cm1)
B (cm1)
C (cm1)
MP2(Full)/6-31G(d)
0.59373 [0.46]
0.19055 [0.83]
0.16459 [0.98]
CCSD/6-31+G(d)
0.58430 [1.14]
0.18845 [0.29]
TDM calc
0.16231 [0.40]
UB3LYP/aug-cc-pVTZ
0.60628 [2.58]
0.18481[2.21]
0.16160 [0.85]
exp
0.59099
0.18899
0.16299
|μeG| (102 Debye)
ν (cm1)
fG (106)
UCIS/6-31G(d)
2.61
7707
2.49
UCIS/aug-cc-pVTZ
2.04
8500
1.67
EOM-CCSD/6-31G(d)
3.43
9081
5.05
EOM-CCSD/aug-cc-pVTZ
2.80
8796
3.26
UCIS/6-31G(d)
2.52
7091
2.12
UCIS/aug-cc-pVTZ
2.02
7906
1.52
EOM-CCSD/6-31G(d)
3.31
8567
4.44
UCIS/6-31G(d) UCIS/aug-cc-pVTZ
2.42 1.78
7590 8401
2.10 1.26
EOM-CCSD/6-31G(d)
3.17
8969
4.26
exp
2.55(6)
7592
2.34(9)
The method of calculation of the molecular geometry and the predicted rotational constants are given in the first 4 columns of the table. The numbers in the square brackets beside the calculated values of the rotational constants are the discrepancy (in %) with the experimental values. The right half of the table shows the results of the calculations of the TDM based on the pre-calculated geometry. The electronic transition frequency ν obtained in the calculation corresponds to the energy difference between the ground and the excited state without ZPE correction (i.e., vertical excitation energy) and is not to be compared directly to the origin frequency of the most stable (G) conformer. The experimental values of rotational constants and transition frequency are taken from the earlier high-resolution study.7 a
These uncertainties are difficult to estimate rigorously; therefore, we use semiquantitative guidelines for the evaluation of errors in FranckCondon calculations. There is a couple of sources of such errors. One occurs if the normal modes of the ground electronic state do not perfectly map onto the normal modes of the excited state. A quantitative measure of such a mode mapping is given by an estimate of this error and can be obtained from a metric determinant parameter calculated by MOLFC, which, in the case of the perfect mode mapping, is equal to unity. The value of the metric determinant for the G-conformer of C2H5O2 in the present calculations is found to be 0.998, which gives the corresponding error estimate of ∼0.2%, which we considered negligible. The second source of error in the FranckCondon calculations is the neglect of anharmonicity, whose contribution can be estimated as the relative difference in ZPE of the harmonic and anharmonic calculations, δaG0,0 δa G0, 0
1 ¼ 2
χG, j νG, j ∑ G, j νG, j ∑ G, j
ð22Þ
where the summation is performed over all vibrational modes, j, of the G-conformer. Using the values for the harmonic frequencies and anharmonicities from Table 1, we obtain a value for δaG0,0 ≈ 1.0%, which we take to be an estimate of the uncertainty of A p. The major sources of uncertainty of |μeG| are the experimental error in σpEP (∼3.8%) followed by the uncertainties in Q(T) (∼2.6%) (primarily due to the calculations of the vibrational partition function, as the data in Table 6 indicates) and the uncertainties in the value of A p of about 1.0% as discussed above. This results in the combined uncertainty of the experimental value of |μeG|2 of 4.7%, giving rise to errors indicated in eq 21 for |μeG|2 and |μeG|. To our knowledge, no other experimentally measured values ~rX ~ transition of organic peroxy radicals of the |μe| for the A have been reported. However, a number of computational studies has been reported. Weisman and Head-Gordon performed computational studies on the absorption properties of
various peroxy radicals12 and reported the oscillator strength for ~ rX ~ transition of all studied species to be zero to the the A accuracy of their calculations. Hirsch et. al22 performed extensive calculations on HO2 and obtained a value for the electronic transition moment of |μ| = 1.7 102 Debye. In the recent studies by our group,13 the oscillator strength f, related to the transition dipole moment,23 0
jμce j2 ¼ 00
3 ge pe2 c f 2 ge00 2πcνme
ð23Þ
0
where ge , ge are electronic degeneracies of the lower and upper state, has been reported for a number of peroxy radicals. It has been found that the predicted values of fc are fairly sensitive to the level of theory used in the calculations. Additionally, it has been shown that the results of the calculations depend strongly on the molecular geometry used as an input for the calculations of the transition dipole moment. Therefore, it is important to use computational methods calculating the molecular geometry most accurately. Just et. al7 compared the experimentally measured rotational constants obtained from high-resolution jetcooled CRDS measurements to the results of calculatons using several methods and found that the geometries obtained at MP2(Full)/6-31G(d) and CCSD/6-31+G(d) levels produce fairly accurate (with e1% discrepancy with the experiment) values of the rotational constants, with the corresponding discrepancy for the density functional calculations (B3LYP/6-31+G(d)) being 23 times larger. ~ rX ~ TDM of ethyl peroxy To calculate the value of the A ~ state radical, we first optimized the molecular geometry for the X using MP2(Full)/6-31G(d) and CCSD/6-31+G(d) methods. The molecular geometry optimized using the density functional calculations with a large basis set, UB3LYP/aug-cc-pVTZ, is also used to test the sensitivity of different methods of computation of TDM to the input geometry. The resulting optimized geometries were used as an input to calculate the transition dipole moment using the UCIS and EOM-CCSD methods with different basis sets. The value of the |μeG| was read from the output of the calculations in atomic units, ea0, and converted to Debye using 1 D = 0.393 ea0. In addition, the value of the oscillator strength 13938
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A fG was obtained using eq 23, and the calculated values of |μeG| and the transition frequency were derived from the energy gap ~ and A ~ states, obtained in the calculations of the between the X dipole moment. The results of these calculations are given in Table 7. The TDM calculations at the UCIS/6-31G(d) level of theory appear to give the most accurate prediction of the value of |μeG|, within or close to 1 σ from the experimental value for all three optimized geometry inputs. In these cases, the accuracy of the TDM prediction generally improves with the accuracy of the calculations of the molecular geometry, as estimated by the deviation of the predicted values of the rotational constants from the experimental values. Interestingly, the UCIS calculations with the larger basis set (aug-cc-pVTZ) tend to underestimate the magnitude of the TDM. The EOM-CCSD method predicts values of |μeG| that are systematically larger than the experimentally measured ones. Employing a larger basis set (aug-cc-pVTZ) at this level of theory results in a moderate reduction of the discrepancy with the experimental value at the expense of the significant (over a factor of 20) increase of the computational cost. It seems rather surprising that the least expensive calculations, UCIS/6-31G(d), produce the results that are in the best agreement with the experimental value of the TDM. To fully understand the effect of the basis set and the level of theory on the predicted values, systematic study of other peroxy radicals is needed. The predicted values of fG vary over a larger range due to the quadratic dependence on the transition dipole moment and ~ and A ~ states, dependence on the energy gap between the X which also varies with the basis set used. It should be noted, however, that the energy gap derived from a single-point calculation does not equal the frequency of the electronic (origin) transition; hence, the frequency-independent |μeG| value derived from the quantum chemical calculations is a more reliable quantity.
’ CONCLUSIONS In the present work, we report the first experimentally measured value, |μeG| = 2.55(6) 102 Debye, of the TDM ~rX ~ electronic transition of an organic peroxy radical, of an A namely, the G-conformer of C2H5O2. The value of the TDM is obtained from the previously reported11 peak absorption crosssection, which we measured at room temperature and a nearambient pressure environment. A consequence of using such an environment is a rather complicated rovibrational profile that requires a detailed analysis to extract a TDM. In this study, we show that these complicatons can be managed using both the experimental data for the rotational structure of the zero-point levels of the two electronic states involved and the results of a computational analysis of the vibrational level structure of both electronic states. The latter provides the means for predicting important quantities characterizing the spectrum, principally the frequencies and intensities of the various hot vibrational bands, and the total partition function. We have shown that the latter can be calculated reasonably accurately using the weakly anharmonic oscillator approximation even in the case where the details of the structure of the highly excited vibrational levels are not known. The results of the quantum chemistry calculations are generally consistent with the experimentally measured value of |μGe |. Predicted values of |μGe | are more consistent with each other and
ARTICLE
the experimental value, than the transition frequency-dependent values of fG, which vary more substantially among various computational methods. Both quantities provide important benchmarks for assessing the accuracy of various quantum chemistry calculational methods.
’ APPENDIX: VIBRATIONAL PARTITION FUNCTION OF A WEAKLY NONHARMONIC OSCILLATOR For a one-dimensional anharmonic oscillator, the vibrational energy of the level with v quanta of excitation in a given mode j can be generally expressed by a power series in v + 1/2, which we truncate at the quadratic term. The vibrational level energy measured from the ground level is therefore given as "
Ec, j ðvÞ ¼ h νc, j
# νc, j χc, j νc, j 1 1 2 νc, j χc, j v þ v þ h 2 2 2 4
¼ hνc, j ðv χc, j ðv2 þ vÞÞ
ðA:1Þ
It can be readily seen that eq A.1 does not correctly predict the energy level structure for highly excited levels, e.g., for v > (1 χe)/ 2χe as the predicted energies E(v) decrease with v. For the accurate prediction of the energies of such highly excited levels, higher order anharmonic terms are required. These terms are rather difficult to determine experimentally, and their calculation becomes increasingly resource-consuming for molecules with multiple vibrational modes. In many cases, however, the anharmonicity is relatively weak, such that the levels whose energies are not correctly predicted by eq A.1, are not populated and do not contribute significantlty to the partition function, therefore eq A.1 is still a useful approximation. For such a case, we substitute eq A.1 to the expression for the vibrational partition function, assuming that all vibrational levels are nondegenerate. We obtain Qc,vibj ðTÞ
¼
∑v
¼
∑v
¼
∑v
"
# Ec, j ðvÞ exp kT " # hνc, j v hνc, j χc, j ðv2 þ vÞ exp kT " # hνc, j χc, j ðv2 þ vÞ hνc, j v exp exp kT kT ðA:2Þ
We now specify the condition of weakness of the potential anharmonicity by requiring that the argument of the second exponent in the last equality in eq A.2 is small compared to unity for all levels whose population is significant. Then, we expand this exponent in a Taylor series in (ν2 + ν) and truncate the expansion at the linear term. Equation A.2 becomes Qc,vibj ðTÞ
¼
∑v
! hνc, j χc, j ðv2 þ vÞ hνc, j v exp 1 þ kT kT
ðA:3Þ 13939
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
The sum in eq A.3 can be evaluated in closed form and gives 1
0
Qc,vibj ðTÞ
¼
hνc, j 1 B hνc, j B B1 þ χc, j 1 exp @ kT kT
C 1 C C A hνc, j cosh 1 kT
0 a ¼ Qc,vib, j qc, j
ðA:4Þ is the partition function of the one-dimensional where Qvib,0 c,j harmonic oscillator, the term in the last parentheses, qac,j, is the anharmonic correction to the partition function, and cosh(x) = (1/2)(ex + ex) is the hyperbolic cosine function. To extend this treatment to the multimode case, we multiply the expressions of eq A.4 for all vibrational modes together and obtain, Y
Qcvib ðTÞ ¼
j
0 a vib, 0 Qc,vib, ðTÞqac ðTÞ j qc, j ¼ Qc
ðA:5Þ
where νc,j are the harmonic vibrational frequencies for j modes, Qvib,0(T) is the total vibrational partition function in the harmonic approximation given by eq 15, and qac (T) is given by 1 0 qac ðTÞ ¼
Y j
B hνc, j B B1 þ χc, j @ kT
C 1 C C A hνc, j cosh 1 kT
ðA:6Þ
Up to this point we have assumed that the molecule exists in the form of several conformers that are defined by the local minima of PES and do not interconvert to each other. At vibrational excitation energies above the barrier between the two conformers, the vibrational level structure deviates from that of the weakly anharmonic oscillator discussed above. For such levels, the vibrationally averaged structure of the molecule will be no longer characterized by any specific conformer geometry but will rather be conformationally averaged. In this case, the vibrational mode (e.g., CCOO torsion) correlating to the isomerization path can no longer be considered weakly anharmonic and requires a more detailed treatment. We assume that 00 the energies of vibrational levels En of such a unique mode can be calculated, e.g., by the solution of the vibrational Schr€odinger equation as we discussed in the Quantum Chemical Calculations section, and the mode’s contribution to the total vibrational partition function can be calculated directly. An obvious complication in this case is that the vibrational partition functions of isolated conformers in eqs 14, 15, and A.2A.5 are no longer well-defined if the levels above the barrier are substantially populated and the total vibrational partition function needs to be used instead. For this case, eq 14 can be further factorized as ! Ec, 0, 0 Qcvib ðTÞ vib gc exp Q r ðTÞ Q ðTÞ ¼ Qtotal ðTÞ vib kT ðTÞ c Qtotal c
∑
ðA:7Þ Qvib total(T)
where is the total vibrational partition function of the rotationless molecule, whose value for moderate temperatures is given as Ec, 0, 0 vib vib ðTÞ ¼ gc exp ðA:8Þ Qtotal Qc ðTÞ kT c
∑
As follows from eqs A.5 and A.6, Qvib c (T) is a function of the fundamental frequencies of the normal modes, which are very
similar among different conformers, except for the CCOO torsional mode, ν21, which provides the largest contribution to vib the Qvib c (T). If the values of Qc , eq A.5, do not differ dramatically among conformers, eqs A.7 and A.8 together yield ! 1 Ec, 0, 0 r vib Q ðTÞ ¼ Qtotal ðTÞ gc exp Qc ðTÞ Q c ðTÞ c kT
∑
vib ¼ Qtotal ðTÞQ̅ r ðTÞ
ðA:9Þ where Qc(T) = ∑cgc exp[Ec,0,0/kT] is the conformational partition function. The expression in parentheses is the conformationally averaged rotational partition function Q r(T). To obtain the expression for the vibrational partition function for the case of interconverting conformers, Qvib total(T), we modify eqs A.5 and A.6 by restricting multiplication to all but the lowest frequency torsional mode, j0 , and multiplying the result by the directly calculated contribution from the energies of the torsional levels ! " #! 0 Y En vib vib, 0 a Qtotal ðTÞ ¼ exp Qj q j ðA.10Þ kT j6¼ j0 n
∑
vib,0 a ≈ Qc,j and qaj ≈ qc,j for j 6¼ j0 for all conformers. where Qvib,0 j The total partition function is therefore obtained by the substitution of eq A.10 into eq A.9.
’ ASSOCIATED CONTENT
bS
Supporting Information. A figure showing a cut of the PES of C2H5O2 along a coordinate interconverting the T and G conformers and a table of the energies of the 50 lowest vibrational levels of the CCOO torsional mode, v21. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Phone: 614-292-2569. Fax: 614-292-1948. E-mail: tamiller@ chemistry.ohio-state.edu.
’ ACKNOWLEDGMENT We acknowledge the financial support of this work by the DOE via Grant DE-FG02-01ER14172 and a grant of computer time from the Ohio Supercomputer Center. ’ REFERENCES (1) Lightfoot, P. D.; Cox, R. A.; Crowley, J. N.; Destriau, M.; Hayman, G. D.; Jenkin, M. E.; Moortgat, G. K.; Zabel, F. Atmos. Environ. 1992, 26A, 1805. (2) Madronich, S.; Calvert, J. J. Geophys. Res. 1990, 95, 5697. (3) Wallington, T. J.; Dagaut, P.; Kurylo, M. J. Chem. Rev. 1992, 92, 667. (4) Atkinson, R. J. Phys. Chem. Ref. Data 1997, 26, 215. (5) Sharp, E. N.; Rupper, P.; Miller, T. A. Phys. Chem. Chem. Phys. 2008, 10, 3955–3981. (6) Wu, S.; Dupre, P.; Rupper, P.; Miller, T. A. J. Chem. Phys. 2007, 127, 224305. (7) Just, G. M. P.; Rupper, P.; Miller, T. A.; Meerts, W. L. J. Chem. Phys. 2009, 131, 184303. 13940
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
The Journal of Physical Chemistry A
ARTICLE
(8) Just, G. M. P.; Rupper, P.; Miller, T. A.; Meerts, W. L. Phys. Chem. Chem. Phys. 2010, 112, 4773. (9) Pushkarsky, M. B.; Zalyubovsky, S. J.; Miller, T. A. J. Chem. Phys. 2000, 112, 10695. (10) Rupper, P.; Sharp, E. N.; Tarczay, G.; Miller, T. A. J. Phys. Chem. A 2007, 111, 832. (11) Melnik, D.; Chantyal-Pun, R.; Miller, T. A. J. Phys. Chem. A 2010, 114, 11583. (12) Weisman, J. L.; Head-Gordon, M. J. Am. Chem. Soc. 2001, 123, 11686. (13) Thomas, P. S.; Chhantyal-Pun, R.; Miller, T. A. J. Phys. Chem. A 2010, 114, 218–231. (14) Bernath, P. F. Spectra of Atoms and Molecules; Oxford University Press: New York, 2005. (15) Rothman, L. S.; et al. J. Quant. Spectrosc. Radiat. Transfer 1998, 60, 665. (16) Zare, R. N. Angular Momentum. In Understanding Spatial Aspects in Chemistry and Physics; Wiley Interscience: New York, 1988. (17) McQuarrie, D. A. Statistical Mechanics; University Science: Sausalito, CA, 2000. (18) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, revision A.1; Gaussian, Inc.: Wallingford, CT, 2009. (19) Borrelli, R.; Peluso, A. J. Chem. Phys. 2003, 119, 8437. (20) Stakhursky, V.; Miller, T. A. Specview: Simulation of Fitting of Rotational Structure of Electronic and Vibronic Bands, 56th International Symposium on Molecular Spectroscopy, Columbus, Ohio, 2001. (21) Demtroeder, W. Laser Spectroscopy: Basic Concepts and Instrumentation; Springer: Berlin, 2003. (22) Osmann, G.; Bunker, P. R.; Jensen, P.; Buenker, R. J.; Gu., J.-P.; Hirsch, G. J. Mol. Spectrosc. 1999, 197, 262. (23) Hilborn, R. C. Am. J. Phys. 1982, 50, 982.
13941
dx.doi.org/10.1021/jp207647h |J. Phys. Chem. A 2011, 115, 13931–13941
