Article pubs.acs.org/JPCA
Calculated Resonance Vibrational Raman Optical Activity Spectra of Naproxen and Ibuprofen Florian Krausbeck,† Jochen Autschbach,*,‡ and Markus Reiher*,† †
ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
‡
S Supporting Information *
ABSTRACT: Determining the absolute configuration of a molecule is possible by means of various chiroptical spectroscopic methods, one of them being vibrational Raman optical activity (VROA). By adopting a laser excitation wavelength in resonance with an electronic transition, the weak spectral signals can be selectively enhanced. We implement a Kohn−Sham methodology for the calculation of resonance and offresonance VROA spectra to (S)-naproxen and (S)-ibuprofen and discuss their band patterns. The resonance enhancement of the VROA spectrum of (S)-naproxen at an incident wavelength of 514.5 nm caused by an absorption tail as well as the typical offresonance behavior of the VROA spectrum of (S)-ibuprofen at the same incident wavelength can be well reproduced. VROA spectra are also predicted under full resonance conditions.
spectroscopy. Even coordination compounds24,43,44 and proteins45,46 are now in reach for computational methods. The VROA intensities are about a thousand times weaker than the already weak Raman intensities. It is therefore desirable to enhance the VROA and Raman signals selectively, by adopting a laser excitation wavelength in resonance with an electronic transition. This allows for the enhancement of the Raman signals of, for example, amide bonds in proteins or aromatic side chains.47−51 To calculate such spectra in resonance, several theoretical developments were accomplished in the past decade. Nafie52 developed the resonance VROA (RVROA) theory in the limit of a resonance with one excited electronic state. This RVROA theory was later extended by Jensen et al.38 to include multiple electronic states, via solving for complex linear response tensors by inclusion of a common excited-states damping, which was subsequently implemented in the Amsterdam Density Functional (ADF) program package53 in a Kohn−Sham (KS) density functional theory (DFT) framework. We54 studied the interference between several excited electronic states for RVROA spectra of naproxen and naproxen-OCD3 with a sum-over-states (SOS) approach and showed that the consideration of a second excited electronic state may change the RVROA intensities significantly. Experimental studies showing the RVROA effect for naproxen and ibuprofen were carried out by Vargek et al.,55 and other examples were also reported; see the RVROA study of single-walled carbon nanotube enantiomers by Magg and
1. INTRODUCTION Vibrational Raman optical activity (VROA) spectroscopy measures the difference of the Raman scattering intensity of right and left circularly polarized light. It has become one of the predominant techniques for the determination of the absolute configuration of a molecule1−4 and for the study of conformational changes of chiral biomolecules5,6 besides other chiroptical techniques such as electronic and vibrational circular dichroism (CD) and optical rotation.7,8 While measuring a VROA spectrum is now a routine task, the interpretation of its band structure poses a challenge due to a complex relation between the molecular structure and its spectral pattern. Here theoretical studies of VROA can be essential for the assignment of observed spectral features to a certain molecular vibration and structure. (For instance, we investigated spectroscopic signatures in proteins;9−13 for reviews, see refs 8 and 14−19). Prominent examples where quantum-chemical calculations were combined with experiment to assign an absolute configuration are chirally deuterated neopentane20 and methyloxirane in gas and liquid phases21 (see also ref 22) as well as the prototypical asymmetric carbon in the molecule CHFClBr.23 Methodological developments such as density fitting for the calculation of property tensors,24 the tensor transfer method by Bouř, Keiderling, and coworkers,25−27 the decomposition into gradient components,28,29 the interpretation of spectra in terms of localized vibrational modes,30−32 the provision of new graphical tools,33,34 the access to coupled-cluster data,35 London orbitals and gauge-origin independent spectra,36−38 surface-enhanced VROA,39,40 and investigations into how anharmonic effects contribute to a VROA spectrum41,42 contributed to the rapid advance of the off-resonance variant of VROA in computational © XXXX American Chemical Society
Received: October 2, 2016 Revised: November 16, 2016
A
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The Journal of Physical Chemistry A coworkers.56 Experimental RVROA spectra involving two electronically excited states for a chiral transition metal complex were reported by Merten et al.57 Very recently, Crassous and coworkers58 conducted an experimental study where among other techniques VROA spectroscopy was employed to examine modifications upon redox changes to helicene-based mono- and bis-FeII complexes. The spectra were obtained with a laser wavelength of 532 nm and evidently under resonance, as they were monosignate and followed the intensity patterns of the corresponding resonance Raman spectra. In a very recent computational study, Vidal et al.59 showed that a RVROA spectrum is not necessarily characterized by a purely monosignate character but is also able to feature sign alternations including intensity enhancement. It is obvious that computational methods for RVROA intensities should consider many electronically excited states. Here we describe such an implementation that is based on the short-time approximation to the time-dependent theory of Raman scattering, as formulated by Heller et al.60−62 In the offresonance case, it is identical to the standard theory introduced by Placzek, while it becomes a Placzek-like polarizability theory in the resonance case. If only short-time dynamics is involved, that is, the excited states dephase rapidly compared with the time scale of molecular vibrations, then this theory is accurate. We apply a (semi)numerical differentiation scheme for the evaluation of the spectroscopic intensities as numerical first derivates of the respective property tensors, which makes our implementation easily adaptable to new electronic structure methods that can deliver such property tensors (whereas their analytic derivatives might be more difficult to derive and implement). The numerical errors introduced are usually negligible in the framework of the double harmonic approximation.63,64 Hence, this approach has two advantages: First, one needs only analytic geometry gradients and molecular property tensors, and not analytic Hessian and property gradients, respectively, for the electronic structure method of choice, and second, it parallelizes trivially and makes large molecules easily accessible. We use this methodology to study the resonance and offresonance VROA spectra of (S)-naproxen and (S)-ibuprofen. Bands of the experimental spectra of these molecules are assigned. Experimental observations are interpreted in terms of our calculated results. This work is organized as follows: In Section 2, we describe the theory and the implementation in MoViPac and NWChem, followed by the computational details in Section 3. We then study the resonance and off-resonance VROA spectra of (S)-naproxen and (S)-ibuprofen in Section 4.
̂ = −1 Θαβ 2
∑ (3ri ri
α β
− δαβri2)
(3)
i
respectively. The summations run over the number of electrons, and the length representation of the electric dipole and quadrupole operators is employed. The position and momentum operators of electron i are given by ri with Cartesian components riα, riβ, riγ and by pi with its components piα, piβ, and piγ, respectively. Further, εα,β,γ is the third-rank antisymmetric tensor. In SI-based atomic units, the nonrelativistic form of the magnetic moment operator is given in terms of the orbital angular momentum liα by eq 2. The components of the complex electric dipole−dipole polarizability ααβ, the electric dipole−magnetic dipole polarizability Gαβ ′ , and the electric dipole−quadrupole polarizability Aγδβ tensors are given by αα , β = ααR, β + iααI , β
(4)
Gα′ , β = Gα′ R, β + iGα′ I, β
(5)
and Aα , βγ = AαR, βγ + iAαI , βγ
(6)
in terms of their real (R) and imaginary (I) parts. Adopting a SOS formulation, for convenience of notation, the real and imaginary components of the polarizability tensors can be defined in terms of the multipole operators of eqs 1−3 as ααR, β =
∑ f + (wn0 , ω , Γ)Re[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|μβ̂ |Ψ0⟩] (7)
n≠0
ααI , β =
∑ g −(wn0 , ω , Γ)Re[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|μβ̂ |Ψ0⟩] (8)
n≠0
Gα′ R, β =
∑ f − (wn0 , ω , Γ)Im[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|m̂β |Ψ0⟩] (9)
n≠0
Gα′ I, β =
∑ g +(wn0 , ω , Γ)Im[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|m̂β |Ψ0⟩] (10)
n≠0
AαR, βγ =
∑ f + (wn0 , ω , Γ)Re[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|Θ̂βγ |Ψ0⟩] n≠0
(11)
AαI , βγ
=
∑g
(wn0 , ω , Γ)Re[⟨Ψ0|μα̂ |Ψn⟩⟨Ψn|Θ̂βγ |Ψ0⟩]
−
(12)
n≠0 ±
The dispersion line-shape functions f (ωn0, ω, Γ) and absorption line-shape functions g±(ωn0, ω, Γ) related to finite lifetime broadening are given by65,66 ωn0 − ω ωn0 + ω f ± (ωn0 , ω , Γ) = ± 2 2 (ωn0 − ω) + Γ (ωn0 + ω)2 + Γ 2
2. BRIEF THEORY OVERVIEW For a self-contained presentation, we briefly review the theory of resonance and off-resonance VROA following the presentation of Jensen et al. in ref 38. The calculation of RVROA of a molecular system requires linear response functions involving the electric dipole, magnetic dipole, and electric quadrupole moment operators which are given by μ̂α = −∑ riα i
m̂ α = −∑ i
1 1 liα = − ∑ εαβγ riβpi γ 2 2 i,β ,γ
(13)
g ±(ωn0 , ω , Γ) =
Γ Γ ± 2 2 (ωn0 − ω) + Γ (ωn0 + ω)2 + Γ 2
(1)
(14)
(2)
To account for the broadening of the excited electronic states in the calculation of the polarizability tensors, a damping parameter Γ is included phenomenologically65,67−69 describing the relaxation and dephasing of the excited states (see also refs 70−74). In the equations above, Ψ0 is the electronic ground-
and B
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The Journal of Physical Chemistry A state wave function and Ψn is the nth excited-state wave function. Furthermore, ω is the angular frequency of the incident light, ℏωn0 = En − E0 is the excitation energy, and 1/Γ is the lifetime of the excitation, which equals the full-width at half-maximum (fwhm) of the absorption band. In the SOS formulation, the broadening Γ may be different for different electronic states. The KS-based linear response implementation used for the present study is based on a common broadening parameter, as in the approach of Jensen et al.38 For a given oneparticle basis set, the contributions from all electronic singlet excited states that can be formed contribute implicitly to the response functions; that is, these are formally equivalent to a full SOS within a given basis. Only states with relatively low energy correspond to electronically excited states of the system, but the full set is needed to exploit the full flexibility of the oneparticle basis in the response calculations. Furthermore, the usual approximations of KS theory apply, namely, the use of approximate functionals for the ground state and approximate adiabatic linear response kernels and the assumption that the perturbed KS density matrix provides not only accurate electric but also magnetic−electric polarizabilities. For theoretical VROA spectroscopy, the Placzek approximation63,75 is adopted to calculate the polarizability transition tensors from the corresponding molecular property tensors. This theory can be extended to a Placzek-like polarizability theory76 for both on- and off-resonance cases. Following this approach, the transition polarizability tensors are expanded into a Taylor series around the equilibrium geometry. In each expansion, the first-order term accounts for Rayleigh scattering, while the second-order term accounts for Raman/VROA scattering when adopting the harmonic approximation. Taking the derivative of the molecular property tensors with respect to the normal mode Qp of the pth vibration, those second terms are defined as follows ⎛ ∂α ⎞ ⎛ ∂α ⎞ αβ ⎟ ⎜ αβ ⎟ p p ααβ ααβ = ⟨0|ααβ|1p ⟩⟨1p |ααβ|0⟩ = ⎜⎜ ⎟ ⎜ ⎟ ⎝ ∂Q p ⎠0 ⎝ ∂Q p ⎠0 ⎛ ∂α ⎞ ⎛ ∂G′ ⎞ αβ ⎟ ⎜ αβ ⎟ p ′ p = ⟨0|ααβ|1p ⟩⟨1p |Gαβ ′ |0⟩ = ⎜⎜ Gαβ ααβ ⎟ ⎜ ∂Q ⎟ Q ∂ ⎝ p⎠ ⎝ p⎠ 0 0
(21)
⎛ 1 p p *⎞ (αG′)2p = Im⎜i ααα G′ββ ⎟ ⎝9 ⎠
(22)
αp2 =
1 R ,p R ,p I ,p I ,p (ααα αββ + ααα αββ ) 9
β(α)2p =
(23)
1 R ,p R ,p I ,p I ,p R ,p R ,p I ,p I ,p (3(ααβ ααβ + ααβ ααβ ) − ααα αββ − ααα αββ ) 2 (24)
1 R ,p R ,p I ,p I ,p R ,p R ,p I ,p I ,p ′ + ααβ ′ ) − ααα β(G′)2p = (3(ααβ Gαβ Gαβ G′ββ − ααα G′ββ ) 2 (25)
β(A)2p =
1 R ,p ,p I ,p p (ωααβ εαγδA γR, δβ + ωααβ εαγδA γI ,,δβ ) 2
(26)
(αG′)2p =
1 R ,p R ,p I ,p I ,p (ααα G′ββ + ααα G′ββ ) 9
(27)
In practice, RVROA measurements can be performed by employing different experimental setups, each with a different relation between the molecular properties and the measured intensities. In this work, we choose a backscattering geometry where the RVROA intensities are given by78
(15)
I R(180°) − I L(180°) = Δ (16)
dσ 4 (180°) ∝ (24β(G′)2p + 8β(A)2p ) dΩ c (28)
Here c is the speed of light.
3. COMPUTATIONAL DETAILS We implemented the theory described above in our MoViPac program package79 with which all calculations of the VROA and RVROA intensities were performed. (For a validation of our implementation at the well-investigated examples of methyloxirane and hydrogen peroxide, see the Supporting Information.) Structure optimizations and response calculations were performed at the KS DFT level with the BLYP density functional and the def2-TZVP Gaussian-type basis set, employing a locally modified version of the NWChem program.80 The modifications of NWChem entail, first, the interface with MoViPac to provide the calculated complex response tensors of eqs 4−6. These tensors were calculated with KS DFT using a complex response module implemented by one of us.81,82 The NWChem KS response module supports nonhybrid, global hybrid, and range-separated hybrid functionals, and the G′ tensor can be calculated with frequencydependent gauge-including atomic orbital (GIAO) basis
(17)
These derivatives may be calculated by an n-point central difference formula77 by performing n − 1 distortions of every Cartesian nuclear coordinate of the molecule. The transition polarizability tensors are then contracted to different isotropic and anisotropic invariants as follows
⎛ 3α p α p * − α p α p * ⎞ αα ββ αβ αβ ⎟ = Re⎜⎜ ⎟ 2 ⎝ ⎠
⎛1 ⎞ p β(A)2p = Re⎜ ωααβ εαγδA γp,*δβ ⎟ ⎝2 ⎠
where is the anisotropic invariant of the transition tensor “×”, and Re(×) and Im(×) denote the real and imaginary components of the invariants, respectively. In all equations above, the Einstein summation convention is applied for repeated Greek indices. A Greek subscript denotes x, y, or z in Cartesian coordinates. Substituting the complex expressions for the different property tensors given by eqs 4−6 yields
⎛ ∂A p ⎞ ⎛ ∂α ⎞ αβ ⎟ p ⎜ γ , δβ ⎟ ααβ εαγδA γp, δβ = ⟨0|ααβ|1p ⟩⟨1p |εαγδA γp, δβ |0⟩ = ⎜⎜ ε ⎟ ⎟ αγδ ⎜ ⎝ ∂Q p ⎠0 ⎝ ∂Q p ⎠0
β(α)2p
(20)
β(×)2p
and
⎛ 1 p p *⎞ αp2 = Re⎜ ααα αββ ⎟ ⎝9 ⎠
⎛ 3α p G′ p * − α p G′ p * ⎞ αα ββ αβ αβ ⎟ β(G′)2p = Im⎜⎜i ⎟ 2 ⎝ ⎠
(18)
(19) C
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The Journal of Physical Chemistry A
Furthermore, if a GIAO basis is not used, then only the first term on the right-hand side of eq 29 contributes to the results, with the m(α) μν simply being AO matrix elements of the magnetic moment operator of eq 2. After test calculations showed that the use of a GIAO basis is not vital as long as the molecules are not very large, as is the case here, and centered at the coordinate origin, the dipolelength gauge was chosen as it entails computational savings. The BLYP nonhybrid functional was chosen because it gave accurate results in RVROA benchmark calculations in the past.38 Single-point calculations were considered converged when the total electronic-energy difference between two iteration steps was
