Role of Torsion-Vibration Coupling in the Overtone Spectrum and

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

pubs.acs.org/JPCA

Role of Torsion-Vibration Coupling in the Overtone Spectrum and Vibrationally Mediated Photochemistry of CH3OOH and HOOH Published as part of The Journal of Physical Chemistry virtual special issue “Veronica Vaida Festschrift”. Laura C. Dzugan,† Jamie Matthews,‡ Amitabha Sinha,*,¶ and Anne B. McCoy*,§ †

Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States Analyst Research Laboratories, Ilan Ramon St. 2, Ness Ziona 7403635, Israel ¶ Department of Chemistry and Biochemistry, University of California, 9500 Gilman Drive, La Jolla, San Diego, California 92093, United States § Department of Chemistry, University of Washington, Seattle, Washington 98195, United States ‡

S Supporting Information *

ABSTRACT: The yield of vibrationally excited OH fragments resulting from the vibrationally mediated photodissociation of methyl hydroperoxide (CH3OOH) excited in the vicinity of its 2νOH and 3νOH stretching overtones is compared with that resulting from excitation of the molecule to states with three quanta in the CH stretches and to the state with two quanta in the OH stretch and one in the OOH bend (2νOH + νOOH). We find that the OH fragment vibrational state distribution depends strongly on the vibrational state of CH3OOH prior to photodissociation. Specifically, dissociation from the CH stretch overtones and the stretch/bend combination band involving the OH stretch and OOH bend produced significantly less vibrationally excited OH fragments compared to that produced following excitation of CH3OOH to an overtone in the OH stretch. While the absence of vibrationally excited OH photoproducts following excitation of the CH overtone is not surprising, the lack of vibrationally excited OH following excitation to the 2νOH+νOOH combination band is unexpected given that photodissociation following excitation to the lower-energy 2νOH state produces OH products in v = 1 as well as in its ground state. This trend persists even when the electronic photodissociation wavelength is changed from 532 to 355 nm and thus suggests that the observed disparity arises from differences in the nature of the initially populated vibrational states. This lack of vibrationally excited OH products likely reflects the enhanced intramolecular vibrational energy redistribution associated with the stretch/bend combination level compared to the pure OH stretch overtone. Consistent with this hypothesis, photodissociation from the stretch/bend combination level of the smaller HOOH molecule produces more vibrationally excited OH fragments compared to that resulting from the corresponding state of CH3OOH. These results are investigated using second-order vibrational perturbation theory based on an internal coordinate representation of the normal modes. Consistent with the observations, the first-order correction to the wave function shows stronger coupling of the 2νOH+νOOH state to states with torsion excitation compared to the other bands considered in this study.

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INTRODUCTION The study of intramolecular vibrational energy redistribution (IVR) is central to understanding the dynamics of vibrationally excited molecules and for providing insights into the pathways and time scales for energy flow in the energized system.1−3 In the frequency domain IVR manifests itself through vibrational state mixing, whereby the zeroth-order states that provided a useful description of the molecular vibrations at low energy become mixed as a result of terms neglected in the model vibrational Hamiltonian. This mixing becomes increasingly more important at higher energies.1,3 For systems where IVR is restricted or incomplete, bond and mode selective chemistry becomes possible.4−14 Vibrationally mediated photodissociation (VIMP), in which molecules excited to an initial vibrational state are subsequently photodissociated by promotion to a repulsive excited electronic © XXXX American Chemical Society

state (Figure 1), provides a powerful tool for probing state mixing and examining its influence on vibrational spectra as well as photodissociation dynamics.6−8,15−18 For example, one color VIMP experiments (λ1 = λ2 = 750 nm) by Crim and coworkers, in which the 4νOH stretch overtone state of HOOH is initially excited, exhibits the generation of substantially more OH fragments in the v = 1 vibrational state compared to amount of vibrationally excited OH products generated from direct single-photon dissociation (λ = 355 nm) at roughly the same total energy.15 This finding has been interpreted as suggesting that the eigenstates associated with the 4νOH level of HOOH are relatively pure and contain a significant amount of Received: October 2, 2017 Revised: November 7, 2017

A

DOI: 10.1021/acs.jpca.7b09778 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

molecular eigenstates (|n⟩) in which the red contributions to the horizontal lines indicate the contribution from |s⟩. Within vibrational perturbation theory we can explore these types of couplings through an analysis of the size of the coefficients of various terms in the first-order correction to the molecular eigenstate. This approach also allows us to compute the anharmonic transition frequencies and associated intensities. Finally, we can explore the roles of various types of resonance interactions in describing the molecular eigenstates. On the basis of this, the experimental findings are discussed and interpreted using second-order vibrational perturbation theory. Because high overtones are considered, the Hamiltonian is expanded in normal modes based on internal coordinates,19 rather than the usual rectilinear normal modes.20−22 The remainder of the paper is structured as follows. In the following section, we provide a brief description of the experiment. This is followed by a discussion of how vibrational perturbation theory has been implemented in the present study. The results of the VIMP studies on HOOH and CH3OOH are presented and discussed in light of the results of the vibrational perturbation theory calculations.

Figure 1. Schematic diagram illustrating IVR resulting from vibrational state mixing. The zeroth-order bright state |s⟩ carries the oscillator strength from the ground state |g⟩. Other isoenergetic states in the region, |l⟩, are dark states that do not carry any oscillator strength from the ground state. Coupling of the bright state with the dark states results in the formation of eigenstates |n⟩ and fractionation of the bright-state character. The oscillator strength associated with each eigenstate is dictated by the amount of bright-state character in the eigenstate, given by the square of the mixing coefficient, shown in red.

zeroth-order OH stretching character. Vibrationally mediated photodissociation experiments on H2O, where IVR is even more restricted than in HOOH, also demonstrated that the OH fragment vibrational state distribution depends intimately on the starting parent vibrational state.7,8 When H2O was excited to the third OH stretch overtone state in which all of the excitation was in one of the OH stretches (the |04⟩− state in a local mode representation), OH fragments are generated predominantly in the v = 0 level. While by contrast the excitation to the |13⟩− state, with the excitation distributed over both OH stretches, leads to generation of predominantly OH (v = 1).7,8 In the present work we explore the influence of vibrational state mixing on the OH fragment product state distribution resulting from state-selected photodissociation of CH3OOH, which represents a substantially larger system compared to H2O and HOOH. Whereas the earlier experiments on water and hydrogen peroxide compared fragment vibrational state distributions resulting from excitation of OH stretching states, here we explore changes in fragment vibrational state distribution arising from excitation of states that are close in energy to the OH stretch overtones, but which contain either excitation of the OOH bend along with the OH stretch or excitation of overtones and combination bands involving the CH stretches. In particular, we compare the relative yields of ground and vibrationally excited OH fragments resulting from vibrational state-selected photodissociation starting from two OHstretching overtones (2νOH and 3νOH) relative to that arising from photodissociation of a combination band involving the OH stretch and OOH bend (2νOH + νOOH) and of states with three quanta in the CH stretches, 3νCH. To elucidate the role played by vibrational state densities on the outcome, we compare the results from CH3OOH with analogous measurements on HOOH, a molecule that has an electronic chromophor similar to CH3OOH, but with a substantially lower vibrational state density. Our results for these two hydroperoxides demonstrate the rather slow rate of IVR associated with their OH-stretching overtone states and how the addition of a single quantum of bending excitation significantly increases the rate of IVR in both of these systems. The IVR model depicted in Figure 1 shows a strong analogy to vibrational perturbation theory. Specifically, in this model, a zero-order bright state (|s⟩) is directly coupled to a set of dark background states (depicted by the ket |l⟩), to form a set of

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EXPERIMENT We synthesize CH3OOH as outlined in previous studies23−25 and make concentrated H2O2 (>90%) by bubbling N2 gas through commercially available 30% H2O2 sample over a period of several days to remove excess water. Typically 60−80 mTorr of the reagent hydroperoxide is slowly introduced into a glass photolysis cell, and 1.4 Torr of N2 buffer gas is added to the cell to relax the rotational state distribution of the nascent OH product. The N2 buffer gas pressure, along with the time delay between the photolysis and probe lasers, are adjusted so that negligible relaxation of the vibrational degrees of freedom of excited parent molecule or the OH product occurs. This is possible because typically the rate for relaxation of the rotational degrees of freedom is much faster than that for vibration.26,27 Tunable IR radiation for vibrational excitation (λ1) is produced by a MOPO laser pumped by the third harmonic of a seeded Nd:Yag laser. The MOPO laser’s line width is 0.4 cm−1, and its idler pulse energy ranges from 6 to 10 mJ over the 6000−11 000 cm−1 spectral region. After vibrational excitation the hydroperoxide molecules are subsequently photodissociatated by promoting them to a repulsive electronic excited state using either 532 nm, or in some experiments 355 nm light, from a second Nd:Yag laser (λ2). The two beams λ1 and λ2 counter propagate relative to one another, and their temporal delay is fixed at ∼15 ns. Typical pulse energies for the 532 and 355 nm light (λ2) are, respectively, 50 and 15 mJ. The various vibrational levels of the fragment OH radical are probed via the A-X(0−0), A-X(1− 1), and A-X(2−2) diagonal transitions using the doubled output of a third Nd:Yag pumped dye laser operating at ∼310 nm (λ3). The probe laser light is greatly attenuated to prevent saturation of the OH transitions and photolysis of the peroxide molecules. The probe laser beam counter propagates relative to the IR light (λ1) with its temporal delay set at ∼1400 ns relative to λ1. This delay allows thermalization of the nascent OH fragment rotational state distribution without significant relaxation of its vibrations. The OH laser-induced fluorescence (LIF) signal is collected using an f/1 lens system and imaged onto an end-on photomultiplier. A color glass filter (Schott UG-11) and a set of 355 nm edge filters are used to reduce scattered laser light. The above scheme is illustrated in Figure 2. B


Article

The Journal of Physical Chemistry A ⎤ ℏωn ⎡ ∂ 2 2⎥ ⎢− Q + n ⎥⎦ 2 ⎢⎣ ∂Q n 2

3N − 6

H (0) =

∑ n=1

(4)

for the perturbation theory analysis described below. With H(0) defined, we define a normal mode . -matrix as 3N − 6

(3−1)i , m Gi , j(3−1)j , n

∑

.m , n =

(5)

i,j=1

where based on the above definition of 3 , . eq m , n = ωnδm , n

(6)

and, in general, the elements of the . matrix are functions of all 3N − 6 normal modes. With this definition of H(0), the firstand second-order corrections to the Hamiltonian become ⎡ ∂. ⎢− 1 n, m ∂ Q ∂ ⎢ 2 ∂Q o ∂Q m o ∂Q n m,n,o=1 ⎣ ⎤ 1 ∂ 3V Q mQ nQ o⎥ + ⎥⎦ 6 ∂Q m∂Q n∂Q o 3N − 6

H (1) =

Figure 2. Illustration of the three-laser experiment. Vibrational excitation, λ1 (purple), precedes photodissociation along the O−O bond, λ2 (red), before LIF is used to probe the resulting OH radicals, λ3 (blue).

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THEORY Nondegenerate Perturbation Theory. To explore the assignment of the bands in the spectrum of CH3OOH and HOOH in the 7000 cm−1 to 11 000 cm−1 region, we employ second-order perturbation theory. In these calculations, we expand the Hamiltonian in normal modes that are linear combinations of the internal coordinates, rather than linear combinations of Cartesian coordinates usually used in this type of analysis.20,21 Frequency Calculations. Following earlier studies,28−30 we first define our normal modes as linear combinations of internal coordinates starting from the harmonic Hamiltonian H=

1 2

eq [pG p + ΔrF i i , jΔrj] i i,j j

⎡ ∂ 2.n , m ∂ ∂ ⎢− 1 Q oQ p ⎢ 4 ∂Q o∂Q p ∂Q m ∂Q n m,n,o,p=1 ⎣ ⎤ ∂ 4V 1 + Q mQ nQ oQ p⎥ + V ′(Q e) ⎥⎦ 24 ∂Q m∂Q n∂Q o∂Q p 3N − 6

H (2) =

∂ 2V ∂ri ∂rj

22

The final term in H(2) is part of the kinetic energy and arises from the conversion from Cartesian to internal coordinates. The above expansion is constructed so that the terms in H(n) are of the order of ℏn/2+1. As such, the equilibrium contribution to V′, which is of order ℏ2, appears in H(2). As we are interested in energy differences (e.g., transition frequencies) as opposed to absolute energies, this term does not contribute to the final results. Therefore, it is not computed and not included in the discussion of the Hamiltonian that follows. With the Hamiltonian determined, the wave functions and energies are evaluated through second order, yielding the wellknown expressions:

evaluated at

(2)

and Δri represents the displacement of one of the internal coordinates from its equilibrium value. Unlike Cartesian coordinates, there is some flexibility in the choice of the 3N − 6 internal coordinates. In this study, we use the bond lengths, angles, and dihedral angles that define the Z-matrix as the internal coordinates. Diagonalizing the GF-matrix,22 we obtain the squares of harmonic frequencies, while the eigenvectors provide the transformation between the normal coordinates, Qn, and the Δri

En =

∑ n=1

⎛

∑ ℏωi⎝ni + ⎜

i

+

∑

3 i , nQ n

1 ⎞⎟ + ⟨ψn(0)|H (1)|ψn(0)⟩ 2⎠

⟨ψm(0)|H (1)|ψn(0)⟩

m≠n

3N − 6

Δri =

∑

(8)

(1)

i,j=1

where Gi,j is an element of the Wilson G-matrix, the equilibrium geometry of the molecule

Fi , j =

(7)

3N − 6

∑

∑

2

(0) (En(0) − Em )

+ ⟨ψn(0)|H (2)|ψn(0)⟩ (9)

and

(3)

Finally, we have defined the normal modes to be dimensionless (e.g., 3 i , n = Li , n / ωn where Li, n is an eigenvector of the GFmatrix). As such,

|Ψ(1) n ⟩ =

∑ m≠n

C

⟨ψm(0)|H (1)|ψn(0)⟩ (0) (En(0) − Em )

|ψm(0)⟩

(10)


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The Journal of Physical Chemistry A |Ψ(2) n ⟩

⟨ψm(0)|H (2)|ψn(0)⟩

∑

=

(0) (En(0) − Em )

m≠n

+∑

∑

⟨ψm(0)|H (1)|ψl(0)⟩⟨ψl(0)|H (1)|ψn(0)⟩ (0) (En(0) − E l(0))(En(0) − Em )

m≠n l≠n

H(n). In this procedure, we first calculate the frequencies and transition moments as described above, including only those terms in H(n) 0 in the evaluation of the energies in eq 9 and matrix elements of the dipole moment. Then the Hamiltonian matrix is constructed, recognizing that the only off-diagonal terms will come from matrix elements of H(n) deg. As the only offdiagonal terms in this matrix are ones that couple nearly degenerate states, this matrix is much smaller than the matrix that would be needed to solve the original Hamiltonian. This matrix is then diagonalized, and the eigenvectors are used to obtain the matrix elements of the dipole moment in this representation.31−33 Just as the diagonal terms in this smaller matrix are obtained by applying nondegenerate perturbation theory, the offdiagonal terms must also be corrected. Specifically, the matrix element that couples states m and n is given by

|ψm(0)⟩ |ψm(0)⟩

⎡ ⟨ψ (0)|H (1)|ψ (0)⟩⟨ψ (0)|H (1)|ψ (0)⟩ n n n − ∑ ⎢ m |ψm(0)⟩ (0) (0) 2 ⎢ (E n − E m ) m≠n ⎣ 2 ⎤ (0) (1) (0) H ⟨ ψ | | ψ ⟩ ⎥ m n 1 (0) + | ψ ⟩ ⎥ n (0) 2 2 (En(0) − Em ) ⎥ ⎦

(11)

where n represents the 3N − 6 vibrational quantum numbers for the state of interest, and |Ψ(0) n ⟩ is the product of 3N − 6 one-dimensional harmonic oscillator eigenstates, which are solutions to eq 4. The final term in the expression for |ψ(2) n ⟩ is included to ensure that the wave functions are orthonormal through second order. It is worth noting that with the definition of H(1) used in this study, the second term in eq 9 vanishes. Further, for reasons discussed below, we do not include the first term in eq 11 in the analysis. Intensity Calculations. The intensity of a transition requires the evaluation of

S0 → n ∝ ⟨ψn|μ ⃗ |ψ0⟩ 2

(1) (0) (1) (0) Hm , n = ⟨ψm |H |ψn ⟩ (2) (0) (2) (0) Hm , n = ⟨ψm |H |ψn ⟩ ⎛ ⟨ψ (0)|H (1)|ψ (0)⟩⟨ψ (0)|H (1)|ψ (0)⟩ 1 0 0 k k n + ∑ ⎜⎜ m 2 k ⎝ (En(0) − E k(0)) ⟨ψ (0)|H0(1)|ψk(0)⟩⟨ψk(0)|H0(1)|ψn(0)⟩ ⎞ ⎟ + m ⎟ (0) (E m − E k(0)) ⎠

(18)

(12)

(3) (0) (3) (0) Hm ψn ⟩ , n = ⟨ψm H

Applying perturbation theory to this calculation, we divide μ⃗ into its zeroth-, first-, and second-order contributions with 3N − 6

μ ⃗(0) = μe ⃗ +

∑ m=1

μ⃗

(1)

1 = 2

μ ⃗(2) =

1 6

3N − 6

∑ m,n=1

∂μ ⃗ Q ∂Q m m

∂ 2μ ⃗ Q Q ∂Q m∂Q n m n

3N − 6

∑ m,n,o=1

∂ 3μ ⃗ Q Q Q ∂Q m∂Q n∂Q o m n o

⎛ ⟨ψ (0) H (1) ψ (0)⟩⟨ψ (0) H (2) ψ (0)⟩ 0 k n ⎜ m 0 k +∑⎜ (0) (0) ⎜ Em − E k k ⎝

(

(13)

+ (14)

)

⟨ψm(0) H0(2) ψk(0)⟩⟨ψk(0) H0(1) ψn(0)⟩ ⎞⎟ ⎟⎟ (En(0) − E k(0)) ⎠

⎛ ⟨ψ (0) H (1) ψ (0)⟩⟨ψ (0) H (1) ψ (0)⟩⟨ψ (0) H (1) ψ (0)⟩ 0 0 l k k n ⎜ m 0 l + ∑⎜ (0) (0) (0) (0) ⎜ Em − E l )(En − E k l, k ⎝

(

(15)

With these expressions, the dipole moment matrix elements ⟨ψn|μ⃗ |ψ0⟩ are evaluated using the second-order expansion of the wave function provided in eqs 10 and 11. Degenerate Perturbation Theory. Perturbation theory is plagued by problems with small denominators that arise when (0) (0) E(0) n ≈ Em or El in eqs 9, 10 and 11. In particular, when any of the coefficients in eq 10 becomes large, the assumption that |ψ(1) n ⟩ provides a small correction to the zero-order wave function is no longer valid. This will be the case for couplings between states in CH3OOH in which one quantum in the symmetric CH stretch is replaced by two quanta in the HCH bend. To address this, we employ a degenerate form of perturbation theory. Specifically, the first- and second-order corrections to the Hamiltonian are expressed as harmonic oscillator raising and lowering operators and are further partitioned as (n) H (n) = H0(n) + Hdeg

(17)

)

(0) (1) (0) (0) (1) (0) (0) (1) (0) 1 ⟨ψm H0 ψl ⟩⟨ψl H0 ψk ⟩⟨ψk H0 ψn ⟩ − (0) 2 Em − E l(0) E k(0) − E l(0)

(

−

)(

)

(0) (1) (0) (0) (1) (0) (0) (1) (0) ⎞ 1 ⟨ψm H0 ψl ⟩⟨ψl H0 ψk ⟩⟨ψk H0 ψn ⟩ ⎟ ⎟⎟ 2 En(0) − E k(0) E l(0) − E k(0) ⎠

(

)(

)

(19)

While the first- and second-order corrections to the off-diagonal matrix elements are commonly included in degenerate perturbation theory treatments, we were concerned that, (1) while the quartic terms in H(2) m,n contain corrections from H0 , (1) there is not a similar correction to Hm,n. The reason is because such a correction does not come in until third order in the perturbation theory expressions.19 In a study of vibrational overtones in water using high-level perturbation theory, McCoy and Sibert19 found that the these third-order corrections decreased the Fermi resonance coupling term by a factor of 0.625. This led us to explore the importance of including the

(16)

H(n) deg

where includes only those terms that couple nearly degenerate states, while H(n) 0 contains the remaining terms in D


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The Journal of Physical Chemistry A Table 1. Coordinates and Equilibrium Geometries of HOOH and CH3OOH Used for This Study molecule HOOH

CH3OOH

a

coordinatea

value,b Å

coordinatea

value,b deg

coordinatea

value,b deg

rO1H1

0.9647

θH1O1O2

99.5591

ϕH1O1O2H2

121.5539

rO2O1

1.4500

θO1O2H2

99.5591

rO2H2

0.9647

rO1H1

0.9657

θH1O1O2

99.1645

ϕH1O1O2C

132.5390

rO2O1

1.4549

θO1O2C

104.9136

ϕO1O2CH2

177.2472

rCO2

1.4154

θO2CH2

104.4306

ϕO1O2CH3

58.9173

rH2C

1.0920

θO2CH3

110.7587

ϕO1O2CH4

−63.7528

rH3C

1.0938

θO2CH4

111.1926

rH4C

1.0942

See Figure 3 for the numbering of the atoms. bAll parameters were evaluated at the MP2/6-311++G(d,p) level of theory.

program package,37 in which the first and second derivatives of the Hessian with respect to the normal modes based on Cartesian coordinates are evaluated numerically. At the same time, the first and second derivatives of the Cartesian dipole derivatives are evaluated with respect to each of the normal mode coordinates. By contrast to the implementation of vibrational perturbation theory (VPT2) in most electronic structure theory packages, where the normal modes are defined as linear combinations of Cartesian displacements, we define the normal modes in terms of linear combinations of displaced bond lengths and angles. When defining the normal modes in terms of Cartesian displacements, one only needs to know the positions of the atoms in the equilibrium structure, and from there the normal modes are determined. Often the reference structure is generated in a center of mass frame, with the Cartesian axes corresponding to the principal axes of the molecule. In contrast, there is considerable flexibility when defining the internal coordinates. For the present study, for an N-atomic molecule, we use N − 1 bond distances, N − 2 angles, and N − 3 dihedral angles using a Z-matrix construction. For a stable molecule, the bond distances can be determined without ambiguity, as they will correspond to the covalent bond network. The angles and dihedrals require a bit more thought, and the convergence properties of the expanded potential will depend on this choice. The coordinates of HOOH and CH3OOH used for the present study, along with their equilibrium values, evaluated at the MP2/6-311++G(d,p) level of theory/basis set are provided in Table 1, and the minimum-energy geometries are shown in Figure 3 along with the labeling of the atoms. The evaluation of the force constants associated with these normal mode coordinates involves the application of the chain rule to convert from derivatives with respect to Cartesian coordinates and normal modes based on these coordinates, qcart m , to derivatives with respect to the internal coordinates and the associated normal modes, Qint n . While this is a nonlinear transformation, the two sets of normal modes were defined to int ensure that at the equilibrium geometry ∂qcart m /∂Qn = δm, n. The remaining derivatives are obtained using a finite-difference scheme,38 and a similar scheme is used to obtain the . -matrix elements and their derivatives as well as the dipole moment derivatives. In the case of the evaluation of changes in Cartesian coordinates when internal coordinates are displaced, an Eckart embedding of the Cartesian axis system is used to provide a unique definition of these coordinates.39,40 The parameters used for these calculations are provided in the Supporting Information.

third-order corrections to the off-diagonal matrix elements. Note that since H(3) m,n contains the fifth-order terms in the expansion of the Hamiltonian, by symmetry the third-order correction to the diagonal matrix elements is zero. As such, it does not need to be considered. Further, because we only expand the Hamiltonian through fourth order in Qn and ∂ , ∂Q n

H(3) = 0 in the present analysis. While this could be problematic for an expansion in normal modes based on Cartesian displacements, the fifth-order terms in the expansion of the Hamiltonian in the normal modes based on displacements of internal coordinates are expected to be small. A similar truncation was employed in the earlier study of water,19 and this choice is supported by the accuracy of higher-order perturbation theory on molecules, including water, formaldehyde, and methane,34−36 where a quartic expansion of the Hamiltonian was made in terms of a similar choice of coordinates. Expectation Values of Internal Coordinates. One way to explore the types of motions that are activated in the excited state of interest is to explore how the expectation value of the coordinate of interest is displaced from its equilibrium value with vibrational excitation. Expressions for such expectation values can be obtained by perturbation theory by first expanding the coordinates of interest in terms of the normal modes.20 Since the normal modes used in the present study are constructed from linear combinations of displacement coordinates 3N − 6

Δri ≡ Δri(1) =

∑

3 i , mQ m

(20)

m=1

the expression for the expectation value based on second-order perturbation theory becomes 3N − 6

⟨ψn|Δri|ψn⟩ = −

∑ m=1

⎡3 ⎢ i,m ⎢⎣ 8ωm

⎛ ∂ 3V × ⎜⎜ 2 ⎝ ∂Q l Q m

■

⎧ ⎨ (2nl + 1) ⎪ l=1 ⎩ ⎤ ∂.l , l ⎞⎫ ⎪ ⎟⎟⎬⎥ + ⎪ ⎥⎦ ∂Q m ⎠⎭ 3N − 6

∑

⎪

(21)

NUMERICAL DETAILS When we perform an anharmonic frequency (VPT2) calculation using an electronic structure theory package, all of the cubic and many of the quartic force constants are generated. In particular, for the present application we use the Gaussian 09 E


Article


surface with respect to the normal mode coordinates is also needed. While numerical first and second derivatives of the dipole surface with respect to the Cartesian coordinates are evaluated as part of the standard VPT2 calculations, we also needed to generate mixed third derivatives of the dipole surface with respect to the three CH stretches. This is achieved by a finite difference procedure using a step size of 0.2 in the dimensionless normal coordinates, and evaluating the dipole moment function by performing single-point calculations (MP2/6-311++G(d,p)) at the displaced geometries. From these values we calculated the third derivative of the dipole moment for the state with one quantum in each of the three unique CH stretches.

Figure 3. Equilibrium structures of HOOH and CH3OOH calculated at the MP2/6-311++G(d,p) level of theory/basis.

When implementing the transformation between qcart and Q , we noted that the cubic force constants did not display the expected symmetry, and the order in which the derivatives were taken affected the results (e.g., -ijk ≠ - jik ). We traced the problem to the fact that the cubic force constants were obtained by evaluating the derivatives of the analytically evaluated matrix elements of the Hesssian with respect to dimensionless normal coordinates numerically. In this representation, the same displacement of a normal mode coordinate with a small harmonic frequency will correspond to a much larger distortion of the structure of the molecule than an equal displacement of a normal mode coordinate with a larger harmonic frequency. As such, while this approach is fairly accurate for high-frequency modes, the numerical derivatives with respect to the lowfrequency normal modes are much less stable. On the basis of this, we evaluated the cubic and quartic force constants in such a way that we were able to minimize our use of terms in the expansion of the potential that required taking numerical derivatives of the Hessian with respect to lowest-frequency normal modes.38 In this study, we are interested in states with three quanta in the CH stretch. As such, in addition to requiring a quartic expansion of the potential, a cubic expansion of the dipole int

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RESULTS AND DISCUSSION

Experimental Spectra. The panels on the left side of Figure 4 display action spectra of vibrationally excited CH3OOH obtained by monitoring the OH(2Π3/2, N = 2, v) fragment rotational state associated, respectively, with its v = 0, 1, and 2 vibrational levels. In Figure 4a, for example, we see that monitoring the yield of OH fragments in the ground vibrational state, OH (N = 2, v = 0), while scanning the vibrational excitation laser λ1 over the range from ∼6500 to 10 800 cm−1 reveals spectral features of CH3OOH corresponding to the first overtone of the OH stretch (2νOH), the OH stretch/bend combination (2νOH + νOOH), the second overtone of the C−H stretch (3νCH, where three quanta are distributed among the three CH stretch vibrations), and finally the second OH stretching overtone (3νOH). To highlight the weaker spectral features, the intensity is magnified by a factor of 10 in the spectral region blue of ∼7800 cm−1. The spectra are normalized to the laser powers, and thus the observed intensity of 3νOH indicates that it is ∼20 times weaker than 2νOH. When the same

Figure 4. VIMP action spectra of CH3OOH (a−c) and HOOH (d−f). The detected OH radicals are in their vibrationless state (a,d), first excited state (b,e), and second excited state (c,f). The intensities to the right of the dashed gray line are magnified by a factor of 10 for CH3OOH and by a factor of 5 for HOOH. Panel (a) was reproduced from ref 24. Copyright 2009 American Chemical Society. F


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The Journal of Physical Chemistry A Table 2. OH(2Π3/2) Fragment Vibrational Product State Branching Ratio from Photodissociation of Hydrogen Peroxide and Methyl Hydroperoxide excitation banda

OH(v = 0)

OH(v = 1)

Table 3. Normal Modes and Frequencies for HOOH and CH3OOH Evaluated at the MP2/6-311++G(d,p) Level of Theory/Basis

OH(v = 2)

HOOH

CH3OOH 2ν1b (7025 cm−1) 2ν1 + ν8 (8290 cm−1) 3ν1 (10 260 cm−1)

99.2 (99.2)c 100 d 99.25 d

ν1 + ν5e (7050 cm−1) (ν1 + ν5) + ν6 (8300 cm−1) 2ν1 + ν5/3ν5 (10 290 cm−1)

89.8 (88.7)c 97.2 d 90.5 d

0.8 (0.76)c 0 (0)c 0.75 d

0 (0)c 0 (0)c 0 (0)c

9.7 11.0c 2.8 d 9.5 d

0.5 (0.3)c 0 (0)c 0 d

n 1 2 3 4 5 6

H2O2 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Values in parentheses correspond to λ2 = 355 nm. bIn CH3OOH, ν1 is the OH stretch and ν8 is the OOH bend. cObserved OH product resulting from photodissociation with λ2 = 355 nm. dValues could not be quantified due to generation of OH(v = 0) from direct singlephoton photodissociation at λ2 = 355 nm. eIn H2O2, ν1 and ν5 are the symmetric and antisymmetric OH stretches, and ν6 is the out-of-phase OOH bend. a

spectral region of CH3OOH is scanned monitoring OH (N = 2, v = 1), shown in Figure 4b, we find that, although vibrational features associated with the OH stretching overtones (2νOH and 3νOH) are still clearly visible, the OH stretch/bend combination band (2νOH + νOOH) and C−H stretching overtone features (3νCH) are missing. In essence, the spectra in Figure 4a,b suggest that substantially more vibrationally excited OH fragments in the v = 1 state are generated from excitation of the OH stretching overtone states (2νOH and 3νOH) compared with either the OH stretch/bend combination band or the C−H stretching overtone. Finally, when probing the OH(v = 2) fragments (Figure 4c), we do not observe any of the CH3OOH vibrational features over this same spectral region. This suggests that, within our detection limit, OH (v = 2) products are not formed from the photodissociation of methyl hydroperoxide vibrational states over this region. The measured values for the OH vibrational state branching ratios, resulting from dissociation of methyl hydroperoxide at the peak of its various vibrational bands, are summarized in upper half of Table 2. These values are obtained by converting the integrated OH fragment LIF intensities to relative population using available OH Einstein B-coefficients and Franck−Condon factors as well as normalizing for laser power. To explore the potential influence of parent vibrational state densities on the fragment vibrational state distribution, we also investigated the state-selected photodissociation of HOOH. Hydrogen peroxide is expected to have similar excited-state photochemistry as methyl hydroperoxide. However, the smaller number of atoms results in a smaller vibrational state density, and, as such, one can expect there to be fewer nominally dark background states coupled to the zeroth-order bright states in HOOH. VIMP spectra of HOOH have been reported over the 4νOH and 5νOH regions. While absorption data are available,41 to the best of our knowledge there are no published VIMP data covering the 2νOH−3νOH regions. The action spectra of HOOH obtained by monitoring the OH(2Π3/2, v = 0, 1, 2) fragment are

mode in-phase OH str in-phase OOH bend OO stretch torsion out-of-phase OH str out-of-phase OOH bend CH3OOH mode OH str out-of-phase sym CH str asym CH str in-phase sym CH str HCH bend CH wag CH3 umbrella OOH bend CH3 rock CH3 rock CO str OO str COO bend CH3 torsion HOOC torsion

νn, cm−1 3847.8 1457.6 921.3 395.3 3846.8 1300.5 νn, cm−1 3838.7 3190.8 3156.7 3067.6 1535.2 1485.8 1476.6 1377.5 1213.4 1193.5 1073.5 861.1 452.7 267.5 204.3

shown, respectively, in Figure 4d−f. In normal mode notation the 2νOH band at ∼7050 cm−1 is assigned to be the ν1 + ν5 combination band involving the symmetric (ν1) and antisymmetric (ν5) OH stretching normal modes (see Table 3 for descriptions of the normal modes and the corresponding frequencies). The much weaker band at ∼7350 cm−1 is assigned to the OH stretch/HOOH torsion combination mode (ν1 + ν5) + ν4. The band at ∼8280 cm−1 corresponds to a combination of two quanta in the OH stretch along with one quantum in the antisymmetric OOH bend (ν1 + ν5) + ν6, while the band at ∼8400 cm−1 corresponds to the combination band in which the symmetric bend is excited, (ν1 + ν5) + ν2. The 3νOH band, which involves either 2ν1 + ν5 or 3ν5, appears in the vicinity of ∼10 290 cm−1. As in the case of CH3OOH, one finds substantially more vibrationally excited OH fragments resulting from excitation of the pure OH stretching overtone states compared to the stretch/bend combination. In addition, for the first OH stretching overtone (2νOH), OH fragments in the v = 2 levels are observed. Excitation of the second OH stretching overtone (3νOH), however, does not produce any detectable amounts of OH(v = 2). The measured values for the OH vibrational state branching ratios resulting from the vibrational state-selected photodissociation of hydrogen peroxide are summarized in lower half of Table 2. Note that since both HOOH and CH3OOH contain one or more large-amplitude torsion, each of the levels mentioned above will reflect transitions between pairs of tunneling doublets. The tunneling splittings in both molecules have been studied previously, and while the ground-state tunneling splittings associated with the HOOX torsion are 11.4 (HOOH) and 15 cm−1 (CH3OOH) when there are two quanta in the OH stretch in CH3OOH, this splitting has decreased to 3.9 cm−1,24 while for the νOH level in HOOH it is 8.1 cm−1.42,43 G


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The Journal of Physical Chemistry A While these splittings have been determined from spectra recorded at higher resolution, they are treated as part of a vibrational feature in the present discussion. As an additional confirmation that the observed variation in the OH fragment vibrational state distributions is not due to differences in dissociation dynamics arising from accessing different regions of the electronically excited-state surface, we performed state-selected photodissociation experiments on both molecules using a different excitation wavelength for λ2. Specifically, we explored the effect of decreasing the value of λ2 from 532 to 355 nm. This change in λ2 should result in the molecule accessing a different portion of the excited electronic surface. As shown by the results in parentheses in Table 2 for the 2νOH level, the branching ratios are essentially unaffected by this change in wavelength. Because of the large OH background signal arising from direct single photon photolysis at 355 nm, we are only able to obtain information for the 2νOH vibrational level, as the OH background signal from direct 355 nm photolysis swamps the weaker signal associated with stateselected dissociation from the higher-energy the 3νOH and stretch−bend combination bands. The lack of variation in the OH vibrational state branching ratio with λ2 is consistent with our hypothesis that the observed OH fragment vibrational state branching ratios in these vibrational state-selected photodissociation experiments are determined predominantly by the nature of the initially prepared vibrational state of the parent hydroperoxide. Calculated Spectrum. To examine the nature of the initially prepared vibrational states in CH3OOH and HOOH and the extent of state mixing, we performed theoretical analyses of the vibrational states using second-order perturbation theory. In these calculations, we first obtained the optimized structure at the MP2/6-311++G(d,p) level of theory and basis shown in Figure 3. Using these optimized structures, we ran an anharmonic frequency calculation at the same level of theory using Gaussian 09.37 This allowed us to obtain most of the necessary force constants, the transition dipole moment derivatives, and the transformation matrix between the Cartesian coordinates and the normal modes defined as linear combinations of Cartesian coordinates. As noted above, only 2 the semidiagonal quartic force constants ∂4V/∂xi∂xj∂qcart and m 2 are third derivatives of the dipole moment vector ∂3μ⃗/∂xi∂qcart m evaluated within the VPT2 calculation as implemented in Gaussian 09. This information was then used to calculate the force constant derivatives, transition dipole moment derivatives, and the derivatives of the . -matrix elements in normal modes defined as linear combinations of internal coordinates, as described above. When nondegenerate perturbation theory is employed to evaluate transitions to states with up to two quanta of excitation, the missing terms are not needed. As we are interested in higher excited states, the third derivatives of the dipole moment with respect to the three CH stretching vibrations are obtained numerically. Figure 5 displays the results of perturbation when different treatments of resonance interactions are employed. Using nondegenerate perturbation theory, we calculated the anharmonic spectrum for CH3OOH in the 2νOH−3νOH region. The results of this analysis are plotted in Figure 5h, while the experimental spectrum is reproduced in the top panel of this figure. Comparing these spectra, we find that the position and relative intensities of the transitions associated with OH stretch overtones and combination bands involving the 2νOH overtone are generally in good agreement with the measured spectrum,

Figure 5. (a) Experimental spectrum of CH3OOH, when OH radicals in their vibrationless state are monitored, compared to the calculated spectra obtained using (h) non-degenerate perturbation theory and (b−g) degenerate perturbation theory with various terms in Hdeg included, as shown in eqs 17−19 and indicated in the panels. The xaxis range in (a) is shifted relative to the other panels to line up the 2νOH feature.

although the calculated OH stretch frequency appears to be higher in energy compared to the measured value. There is an extra peak in the calculated spectrum that lies to the red of the first overtone in the OH stretch, and the CH stretch frequencies span a broader frequency range than is seen in the recorded spectrum. Specifically, the transitions that are assigned as CH stretch overtones range in frequency from 7450 to 9000 cm−1. On the basis of the harmonic frequencies of the three CH stretch vibrations (3068, 3157, and 3191 cm−1) we would expect a much narrower range of overtone frequencies. After further investigation, we determined that the harmonic frequency of the symmetric CH stretch is only 3 cm−1 lower in energy than twice the harmonic frequency of the HCH bend, for example, at the harmonic level, 2ν5 − ν4 = 3 cm−1. Similarly small energy differences are found when an aug-cc-pVTZ basis set is used for the MP2 calculation, and the difference becomes 17 cm−1 in a CCSD(T) calculation and 8 cm−1 in a B3LYP calculation. As a result of this near degeneracy, the third term in eq 9 diverges, and states that have at least one quantum of excitation in the symmetric CH stretch are lowered in frequency by, in some cases, more than 1000 cm−1. H


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Hamiltonian matrix leads to the results plotted in Figure 5b− d. The differences between these three calculations involve which contributions from the cubic terms are included in the evaluation of the matrix elements that couple the states with three quanta in the CH stretch. As is seen the differences between these plots are small. The primary consequences of including the terms in eq 18 in Hdeg is to shift the CH stretch peaks to lower energies and to reduce the number of peaks that carry intensity. As is seen in eq 18, there are two contributions to the offdiagonal matrix elements in the Hamiltonian matrix at second order. The first contribution arises from terms in H(2) deg, while the remaining terms contain contributions from H(1) 0 . These two sets of contributions parallel the contributions to the Fermi resonance that come in at first and third order. In the case of the Fermi resonances, the contributions to this coupling term that come in at first- and third-order perturbation theory are comparable in size. On the basis of this, we became interested in the relative magnitudes of the two contributions to the off(2) diagonal terms in Hm,n . To this end, we explored the contributions of these two terms to the calculated spectrum. As is seen in the calculated spectra shown in Figure S1, when only the contribution from H(2) deg is included, the transitions involving three quanta in the CH stretch are shifted to higher frequency by several hundred cm−1, and the agreement with the measured spectrum becomes less good. This large effect was initially surprising, especially in light of the relatively small contributions of the terms in H(3) deg to the calculated spectrum. Focusing on the terms that are larger than 5 cm−1, we found that including only the contributions to H(2) m,n from H(2) 0 changed the signs of these coupling terms, while they had a small effect (

Role of Torsion-Vibration Coupling in the Overtone Spectrum and

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