Raman Optical Activity Spectra for Large Molecules through Molecules

Article pubs.acs.org/JCTC

Raman Optical Activity Spectra for Large Molecules through Molecules-in-Molecules Fragment-Based Approach K. V. Jovan Jose and Krishnan Raghavachari† Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States S Supporting Information *

ABSTRACT: We present an efficient method for the calculation of the Raman optical activity (ROA) spectra for large molecules through the molecules-inmolecules (MIM) fragment-based method. The relevant higher energy derivatives from smaller fragments are used to build the property tensors of the parent molecule to enable the extension of the MIM method for evaluating ROA spectra (MIM-ROA). Two factors were found to be particularly important in yielding accurate results. First, the link-atom tensor components are projected back onto the corresponding host and supporting atoms through the Jacobian projection method, yielding a mathematically rigorous method. Second, the longrange interactions between fragments are taken into account by using a less computationally expensive lower level of theory. The performance of the MIMROA model is calibrated on the enantiomeric pairs of 10 carbohydrate benchmark molecules, with strong intramolecular interactions. The vibrational frequencies and ROA intensities are accurately reproduced relative to the full, unfragmented, results for these systems. In addition, the MIM-ROA method is employed to predict the ROA spectra of Dmaltose, α-D-cyclodextrin, and cryptophane-A, yielding spectra in excellent agreement with experiment. The accuracy and performance of the benchmark systems validate the MIM-ROA model for exploring ROA spectra of large molecules.

1. INTRODUCTION Vibrational circular dichroism (VCD) and Raman optical activity (ROA) are widely used spectroscopic techniques to determine the absolute configurations of chiral molecules.1−6 ROA characterizes the differential inelastic scattering of the Stokes lines in the Raman spectra of a chiral molecule, whereas VCD characterizes the differential absorption of the left and right circularly polarized infrared (IR) radiation. Due to the differences in the associated mechanisms, ROA spectroscopy yields complementary information to that in VCD. In both spectra, the two enantiomers exhibit spectral intensities of the same magnitude, but opposite signs. The advantage of ROA relative to VCD is the ability to obtain lower frequency vibrational spectroscopic information. Moreover, ROA spectroscopy opens the possibility of exploring structural aspects of chiral molecules in their natural aqueous environment. These characteristics make ROA spectroscopy very unique, when compared to other vibrational spectroscopic techniques. ROA is an intrinsically weak phenomenon, the differential circular scattering being several orders of magnitude weaker than the corresponding vibrational Raman scattering. 7 Although instrumental advances allow ROA measurements for complex molecules, the unambiguous interpretation of experimental ROA spectra is not straightforward.8 As molecules get larger, the vibrational spectra become cluttered and the ROA spectral assignments become cumbersome. However, the combination of accurate quantum chemical calculations and experimental ROA measurements offers a robust approach for © XXXX American Chemical Society

the determination of the absolute configurations of chiral molecules.9,10 There have been several approaches for predicting the ROA spectra of molecules.11−18 Some key developments include the use of London atomic orbitals19 and techniques for the evaluation of the different mixed polarizability tensors through response equations at the frequency of the laser radiation.20−27 More recent developments in the evaluation of the analytic derivatives of these higher polarizability tensors have taken this area of research to wider applications.28 Indeed, there are a number of systems that have been explored theoretically and experimentally.29−32 However, the computation of ROA spectra can be computationally demanding, particularly since fairly large basis sets including diffuse and polarization functions are needed to assign experimental ROA spectra reliably.33 The high scaling of these methods can limit the applicability of direct ab initio ROA spectral calculations for large systems.33,34 Considerable progress in the quantum mechanical studies of large molecules in the past decade has come from the development of fragment-based methods.35 These methods are based on the general idea of cutting the large molecule into a set of small fragments and overlapping subsystems and assembling their energies in a systematic way to construct the total energy of the large molecule. Though there are many such methods for evaluating the energies, the corresponding Received: November 28, 2015

A

DOI: 10.1021/acs.jctc.5b01127 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

polarizability tensors are given in the sum-over-states representation as ωj0 2 ααβ = ∑ Re[⟨0|μα |j⟩·⟨j|θβ|0⟩] 2 ℏ j ≠ 0 ωj0 − ω 2 (4)

derivative expressions needed for the calculation of higher order molecular properties such as ROA are not available in most fragment-based methods. Recently, we have proposed a novel hybrid extrapolation method called molecules-in-molecules (MIM).36−39 This method has a generalized hybrid energy expression, similar in spirit to ONIOM (vide inf ra). MIM employs a multilayer partitioning technique with multiple levels of theory, making it an efficient extrapolation method compared to many other fragment-based methods that use a single layer. The performance of the MIM model has previously been validated for accurate energy evaluations and infrared,37 Raman,38 and VCD39 spectra of large molecules. Because the long-range interactions are accounted for at a low level of theory, the fragment size at the high level of theory can be kept reasonably small. In the present work, we have extended this method to evaluate the Hessian and the higher order polarizability derivative terms for the precise evaluation of ROA spectra of large molecules. The following sections of the current paper are organized as follows: Section 2 describes the theoretical background for ROA calculations and our procedure involved for evaluating MIM energies and their higher derivatives. Section 3 presents a benchmark analysis of MIM-ROA spectra on 10 carbohydrate molecules as well as a comparison with experiment for two larger molecules, α-D-cyclodextrin and cryptophane-A. Section 4 compares our approach with previous approaches used for large molecules and section 5 provides a summary and conclusions.

′ = Gαβ

Aαβγ =

(1)

mα = mα0 − ω−1G′βαFα̇ + ...

(2)

θαβ =

0 θαβ

+ 1/3Aαβγ Fγ + ...

∑

2 ℏ

∑

j≠0

j≠0

ω Im[⟨0|μα |j⟩·⟨j|θβ|0⟩] ωj0 2 − ω 2 ωj0 2

ωj0 − ω 2

(5)

Re[⟨0|μα |j⟩·⟨j|θβγ |0⟩] (6)

Here, ω is the frequency of the incident radiation, and ωj0 is the transition frequency from the ground state |0⟩ to an excited state |j⟩, related to the energy separation between these two states.6,17,40 Note that normal coordinate derivative of ααβ contributes to the Raman spectral intensity, whereas Gαβ ′ contributes to the optical rotation spectral intensity. The contribution of Aαβγ is of the same order as G′αβ tensor in the ROA spectrum; hence this gives additional richness to ROA spectrum over other spectroscopic methods. The evaluation of the vibrational ROA spectral intensities requires the evaluation of the derivatives of polarizabilities in eqs 4−6 with respect to the normal coordinates, (∂ααβ/∂Qi), (∂Gαβ ′ /∂Qi), and (∂Aαβγ/∂Qi). In practice, these higher order polarizability derivatives are evaluated in Cartesian coordinates, (∂ααβ/∂XA), (∂G′αβ/∂XA), and (∂Aαβγ/∂XA). The first of these, λ ααβ , is the electric dipole−electric dipole polarizability derivative and is already needed for the evaluation of Raman vibrational scattering. It represents the derivative of the molecular electric polarizability with respect to the Cartesian displacement of the λth atom, at the equilibrium geometry, Re as

2. COMPUTATIONAL METHODS AND THEORY ROA is a measure of the differential inelastic scattering of left and right circularly polarized electromagnetic radiation in the visible region. Unlike the Raman scattering spectra where the band intensities are always positive, the ROA band intensity can be either positive or negative depending on the difference in the scattering intensities. Enantiomers have ROA bands that are equal in magnitude but opposite in sign. Hence, the ROA spectra can be used to determine the absolute configuration of chiral molecules. The theory and implementation aspects of ROA spectral evaluations have been extensively discussed previously.12,13 The ROA spectral intensities are constructed under the far from resonance (FFR) approximation with the dynamic molecular properties being obtained from time dependent perturbation theory.14 Briefly, the electric and magnetic field vectors of the light wave (Fβ, Bβ) and the electric field gradient tensor (Fβγ ′) induce an oscillating electric dipole (μα), magnetic dipole (mα) and electric quadrupole moments (θαβ), as in the following equations: ′ Bβ̇ + ... μα = μα0 + ααβFβ + 1/3Aαβγ F′βγ + ω−1Gαβ

−2 ℏ

⎛ ∂α G ⎞ ⎡ ⎤ ∂ 3E αβ λ ⎟⎟ = ⎢ ⎥ = ⎜⎜ ααβ ⎢⎣ ∂Xλα ∂Fα ∂Fβ ⎥⎦ ⎝ ∂Xλα ⎠ R R e

e

(7)

The initial Cartesian coordinate derivatives are converted into the normal coordinate using the transformation matrix, S. The transformation matrix between the normal coordinate and the Cartesian displacement coordinate can be expressed as, Xλα = ∑iSλα,iQi. The electric dipole−electric quadrupole and electric dipole−magnetic dipole terms can be evaluated in a similar manner. The ROA calculations are performed using gauge invariant atomic orbitals to ensure that the resulting intensities are origin-independent. The Cartesian polarizability derivative tensors are first converted to a set of “invariants” as denoted in eqs 8−12. Raman and ROA invariants are combinations of products of tensors that are invariant to the orientation of the molecular coordinate frame relative to the laboratory coordinate frame.13 These Raman and ROA invariants are dependent on the higher order polarizability normal coordinate derivatives as ⎡ ⎤ 1 ⎢⎛ ∂ααα ⎞ ⎛ ∂αββ ⎞ ⎥ ⎟⎟ ⎜⎜ ⎟⎟ ai = ⎢⎜⎜ 9 ⎝ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎥ ⎣ Re Re⎦

(8)

⎡ ⎛ ∂α ⎞ ⎛ ∂αββ ⎞ ⎤ 1 ⎢ ⎛ ∂ααβ ⎞ ⎛ ∂ααβ ⎞ ⎟⎟ ⎜⎜ ⎟⎟ − ⎜⎜ αα ⎟⎟ ⎜⎜ ⎟⎟ ⎥ βi = ⎢3⎜⎜ 2 ⎝ ∂Q i ⎠ ⎝ ∂Q i ⎠ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎥ ⎝ ⎣ Re Re Re Re⎦

(9)

2

(3)

Here, the Greek subscripts denote the vector or tensor components in x, y, or z directions. ααβ is the electric dipole−electric dipole polarizability tensor, Gαβ ′ is the electric dipole−magnetic dipole polarizability tensor, and Aαβγ is the electric dipole−electric quadrupole polarizability tensor. The

2

B


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Journal of Chemical Theory and Computation ⎡ ⎤ 1 ⎢⎛ ∂ααα ⎞ ⎛ ∂G′ββ ⎞ ⎥ ⎟⎟ ⎜⎜ ⎟⎟ aGi′ = ⎢⎜⎜ 9 ⎝ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎥ ⎣ Re Re⎦

βG

2

i

2. In the two-layer MIM (MIM2), the total energy can be constructed from the subsystem energies through a generalized extrapolation derived from the two-layer ONIOM energy expression

(10)

⎡ ⎛ ∂α ⎞ ⎛ ∂G′ββ ⎞ ⎤ ′ ⎞ 1 ⎢ ⎛ ∂ααβ ⎞ ⎛ ∂Gαβ ⎟⎟ ⎜⎜ ⎟⎟ − ⎜⎜ αα ⎟⎟ ⎜⎜ ⎟⎟ ⎥ = ⎢3⎜⎜ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎥ 2 ⎝ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎝ ⎣ Re Re Re Re⎦

E Total = Erl − Eml + Emh

Here, Eml and Emh are the energies of the model region calculated at the low and high levels of theory, and Erl is the energy of the real (i.e., full) system calculated at the low level. However, MIM involves overlapping subsystems that can each be considered as a model system. It is convenient to redefine the energies of model-high (Emh) and model-low (Eml) levels of theory as a sum over all the subsystems as follows:

(11)

βA

2 i

⎡ ⎤ ω ⎢⎛ ∂ααβ ⎞ ⎛ εαγδ ∂A γ , δβ ⎞ ⎥ ⎟⎟ ⎜⎜ ⎟⎟ = ⎢⎜⎜ 2 ⎝ ∂Q i ⎠ ⎝ ∂Q i ⎠ ⎥ ⎣ Re Re⎦

(13)

(12)

Here, ai2 and βi2 are the Raman invariants and aGi′, βGi2, and βAi2 are the ROA invariants. The electric dipole−electric dipole polarizability derivatives are used to form two Raman tensor invariants whereas all three higher order polarizability derivatives are required to construct the ROA invariants. There are several experimental configurations possible for ROA measurements depending on the choice of polarization state of the incident and scattered radiation as well as the scattering angle.17 Hence, the intensity of the Raman or ROA band can be expressed as a linear combination of some or all of the invariants.14 The more specific equation for different scattering geometry and the invariants can be found in other publications.10 Overall, the prediction of ROA spectra within the harmonic approximation requires (a) evalutation of the second derivatives of the energy with respect to the nuclear displacement coordinates, i.e., the Hessian matrix that yields the harmonic vibrational frequencies, and (b) calculation of the ROA intensity through the evaluation of the higher oder polarizabilities and polarizability derivative terms. The first-principles evaluation of these terms can be the bottleneck for the evaluation of the ROA spectra for large molecules, particularly when used in conjunction with large basis sets that are known to be required to calculate ROA spectra accurately. This computational bottleneck in evaluating these higher order energy derivatives can be overcome by the MIM fragmentbased method. We now outline the computational procedure as well as the method used for the accurate evaluation of the electronic energy, molecular geometry, Hessian matrix, and polarizability derivatives through the MIM procedure. All the actual and MIM calculations are carried out using the Gaussian program suite.41 MIM fragmentation schemes and ROA evaluations are implemented through an external Perl script interface. Our fragment-based strategy for the efficient evaluation of ROA spectra for large molecules is as follows. 1. The MIM procedure36 for the evaluation of ROA spectra begins with the generation of the fragments by cutting single bonds in the large molecule (vide inf ra). The primary subsystems are then assembled from the interacting fragments through a number based scheme (vide inf ra). The overcounting of the overlapping regions of the primary subsystems is taken into account through derivative subsystems, constructed by means of the inclusion−exclusion principle. All the dangling bonds in the subsystems are saturated with hydrogen link atoms. The link atoms are placed along the vector connecting the supporting and host atoms at a distance defined using a scale factor as in the ONIOM method.42

Eml / mh =

∑ Eli/h − ∑ Eli/∩hj + ... + (−1)n− 1 i

∑

ij

Eli/∩hj ∩ k... ∩ n

(14)

n

i Here, El/h represents the energy of the ith primary subsystem (at low or high level of theory) and the other terms involving the overlapping terms represent the derivative subsystems. It is important to note that the redefined “model-high” and “modellow” levels in MIM extend through the whole molecule via summation over the subsystems 3. The link-atom force components are projected back onto the supporting and host atoms within the MIM2 model through a general Jacobian projection method, as

Fa =

∂E Total ∂Xa

=

∂Erl − ∂Xa

M

∑ L=1

∂Eml J(R 2 ;R1 ,R3) + ∂Xa

M

∑ L=1

∂Emh J(R 2 ;R1,R3) ∂Xa (15)

Here, the atomic forces with index “a” correspond to the Cartesian components in the X, Y, or Z directions. These atomic forces can be used to carry out a fragment-based geometry optimization procedure, as already described in our previous work.37 4. The mass-weighted Hessian matrix elements involving the link atom are projected back onto the supporting and host atoms through the general expression (at the optimized geometry, Re) ⎡ ∂ 2E ⎤ Total ⎥ Hab = ⎢ ⎢⎣ ∂Xα ∂Xβ ⎥⎦ R ∂ 2Erl = − ∂Xα ∂Xβ

e

M

∑ JT L=1

∂ 2Eml J+ ∂Xα ∂Xβ

M

∑ JT L=1

∂ 2Emh J ∂Xα ∂Xβ (16)

More details are given in our previous paper on the evaluation of infrared37 and Raman38 spectra with MIM2. The transformation matrix S from Cartesian to normal modes is then available for the full molecule. 5. In MIM2, the individual electric dipole−electric dipole polarizability derivative contributions for evaluating the ROA intensities are constructed from the separate fragment calculations through the general extrapolation expression C


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Figure 1. Set of L- and D-molecules employed for benchmarking the MIM-ROA method. A detailed analyses of the energies and ROA spectra of structures I−X are reported in Table 1. Optimized geometries are furnished in the Supporting Information. M λ [ααβ ,Total ]R e

=

λ ααβ , rl

electric dipole, electric dipole−magnetic dipole, and electric dipole−electric quadrupole polarizability derivatives from eqs 17−19 are employed to evaluate the intensity of the ROA intensities. The implementation of the MIM-ROA algorithm is general and will work with any arbitrary fragmentation scheme at the primary level and any combination of theoretical methods for which the necessary energy derivatives are available. For benchmarking the MIM-ROA method, we have taken into account 10 prototype carbohydrate molecules in the present work. There have been several recent calculations on the ROA spectra of sugars, especially carbohydrate molecules.43−45 Most of these molecules are conformationally flexible, though we have included only the most stable conformer in this benchmark study. The initial nonoverlapping fragments are constructed by breaking the backbone C−C and C−O single bonds of the benchmark molecules. The interacting fragments are combined, employing a number-based criterion (typically using trimer units) based on the connectivity information, for constructing the overlapping primary subsystems. Then the inclusion−exclusion principle is employed to construct the derivative subsystems from the primary subsystems. Finally, the above-mentioned MIM procedure for assembling the property tensors and evaluating the ROA spectra can be carried out for any level of theory (e.g., combinations of different density functionals and basis sets) for which the component property evaluations are available.

∑ ααβλ ,ml J(R 2 ;R1,R3)

−

L=1 M

+

∑ ααβλ ,mh J(R 2 ;R1,R3) L=1

(17)

The individual expression for the atomic polarizability derivative tensor is defined as in eq 7. 6. The derivative of the electric dipole−magnetic dipole tensor is constructed from a similar general expression from the individual fragment calculations, M λ λ [Gαβ ,Total ]R e = Gαβ , rl −

∑ Gαβλ ,ml J(R 2 ;R1,R3) L=1

M

+

∑ Gαβλ ,mh J(R 2 ;R1,R3) L=1

(18)

7. Similarly, the derivative of the electric dipole−electric quadrupole moment tensor is constructed through the general extrapolation expression M λ [Aαβγ ,Total ]R e

=

λ Aαβγ , rl

−

λ ∑ Aαβγ , ml J (R 2 ;R1 ,R3) L=1

M

+

λ ∑ Aαβγ , mh J (R 2 ;R1 ,R3) L=1

(19)

3. RESULTS AND DISCUSSION The carbohydrate molecules with multiple chiral centers show specific characteristics in the ROA spectrum, making them attractive benchmark candidates. To assess the accuracy and applicability of MIM-ROA, a careful benchmark study is performed for the enantiomeric pairs of 10 medium to large carbohydrate molecules. The agreement between MIM-ROA

All these individual derivative terms are evaluated in Cartesian coordinates and are transformed into normal coordinates through the transformation matrix, S. To summarize, the procedure starts with the initial determination of the optimized molecular geometry using the fragment-based energy gradients. The mass-weighted Hessian is then diagonalized to evaluate the vibrational normal-mode frequencies, and the electric dipole− D


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Table 1. Structure Number, Name, Number of Basis Functions (NBasis), Actual Energies (hartree) at MPW1PW91/aug-ccpVDZ and Mean Absolute Deviation (MAD) in Energies (kcal/mol), Frequencies (cm−1), and ROA Intensities (au 104) for the Test Systems Depicted in Figure 1a energy no.

system

NBasis

I II III IV V VI VII VIII IX X

arabinose galactose glucose lyxose xylose isomaltulose lactose maltose sucrose cellobiose average

320 384 384 320 320 727 727 727 727 727

Actual

E

−572.60638 −687.13290 −687.12576 −572.60644 −572.60598 −1297.82132 −1297.84292 −1297.84802 −1297.84692 −1297.83479

MAD @ MIM1 ΔE

ΔE

−1.903 −1.917 −1.853 −0.196 −1.135 9.017 18.269 18.703 21.891 17.126 9.201

1.013 0.777 0.894 0.417 0.667 1.266 −0.099 −0.823 −2.774 −1.103 0.983

MIM1

MIM2

MAD @ MIM2

Δν

Δ(0°)

Δ(90°)

Δ(180°)

Δν

Δ(0°)

Δ(90°)

Δ(180°)

3.500 2.461 3.361 2.370 2.977 2.458 2.587 3.096 9.573 3.291 3.567

2.439 2.108 2.011 2.383 3.217 3.272 3.101 2.807 4.193 3.229 2.876

0.683 0.592 0.725 0.992 0.897 1.096 1.148 0.813 1.360 1.040 0.935

1.367 1.184 1.451 1.985 1.794 2.191 2.296 1.627 6.331 2.393 2.262

1.183 0.988 1.164 0.643 0.771 0.844 0.857 1.018 1.208 1.462 1.014

1.322 0.950 1.103 2.169 1.648 1.204 1.412 1.473 1.874 1.730 1.489

0.426 0.290 0.397 0.633 0.520 0.383 0.523 0.471 0.697 0.513 0.485

0.852 0.581 0.794 1.265 1.040 0.765 1.046 0.942 1.393 1.027 0.971

a

The ROA spectrum is evaluated at an external frequency of 488 nm, except for galactose at 514.5 nm. The last row reports the average of entries in each column.

molecules are exactly the same, and the ROA spectral features have opposite signs. Hence, for convenience, we have presented the energies and spectral analysis on only one enantiomer (D form) for each carbohydrate in Table 1. The mean absolute deviations (MAD) at MIM1 and MIM2 are 9.21 and 0.98 kcal/ mol, respectively, when compared to the actual energies. This corresponds to a 89.3% improvement in accuracy at MIM2 compared to MIM1. The latter improvement in accuracy demonstrates the importance of the second layer of theory in accounting for the long-range intramolecular interactions in these benchmark molecules. The final columns in Table 1 report the MAD of each full spectrum, including all the normal modes between 0 and 2000 cm−1 at MIM1 and MIM2, relative to the actual spectrum. The scattering angle can be varied, the most important scattering directions being forward (0°), right-angle (90°), and backward (180°). The MAD at MIM2 shows ∼71.6% improvement over MIM1 in frequencies and ∼48.2, 48.1, and 57.1% improvement in ROA intensities calculated for scattering angles of 0°, 90°, and 180°, respectively. This is consistent with the significant improvement in the corresponding total energy. A pictorial representation of these improvements for the three relevant ROA invariants aGi′, βGi2, and βAi2 (see eqs 10−12) is illustrated in the Supporting Information (Figure S1). In this figure, the ROA invariants are denoted as AlphaG, Gamma2, and Delta2, respectively. Figure S1 depicts the percentage improvement in the corresponding invariants at MIM2 over MIM1. The percentage improvement for aGi′ (blue) is in the range 25− 66%, βGi2 (orange) is 38−67%, and βAi2 (gray) is 39−67%. To illustrate the strength of our proposed method, comparisons of the MIM-ROA spectra for D-maltose between 400 and 1600 cm−1 and the directly evaluated (actual) spectra are shown in Figure 2. A corresponding comparison between MIM-Raman and actual spectra are shown in the Supporting Information (Figure S2). Both figures show the MIM1 and MIM2 spectra using primary subsystems composed of trimer units (i.e., three interacting fragments in each subsystem). The actual spectrum is shown in black, whereas the MIM1 spectrum is shown in red, and MIM2 in blue. A better overlap between the black and the blue lines in the ROA (depicted in Figure 2B) and Raman spectra (depicted in Figure S2(B)), indicates a very good agreement between the full calculation and MIM2.

and directly evaluated (i.e., unfragmented) ROA spectra is discussed for these systems in subsection 3.1. The MIM-ROA spectrum of a chiral cyclic oligosaccharide system, α-Dcyclodextrin, is compared with the backscattered experimental spectrum and with previous work in subsection 3.2. The final subsection, 3.3, presents a comparative analysis of MIM-ROA spectrum with the backscattered experimental spectrum for a large cage molecule, cryptophane-A.46 3.1. Comparison of MIM-ROA with Full Calculations. All the enantiomeric forms of the 10 benchmark carbohydrate molecules are depicted in Figure 1. The first five molecules are simple monosaccharides and were calculated in their cyclic forms for benchmarking purposes, whereas molecules six through ten are more complicated and were calculated in their most stable geometries. These calibration studies are carried out using the MPW1PW91 density functional, previously shown to perform well for the calculation of chirooptical properties with basis sets such as aug-cc-pVDZ.13 For MIM2, we have employed a second layer at the MPW1PW91/ 6-31G(d) level of theory for correcting the molecular properties. These results are compared with the actual calculations performed at MPW1PW91/aug-cc-pVDZ levels of theory. The entries in the bracket following “MIM” give the level(s) of theory and basis set(s) employed for the calculations. The geometry optimizations are performed on the test systems (as described in the methodology section) at two different combinations of levels of theory: (1) MIM1[MPW1PW91/aug-cc-pVDZ] and (2) MIM2[MPW1PW91/ aug-cc-pVDZ:MPW1PW91/6-31G(d)]. For simplicity, they will be denoted as MIM1 and MIM2 in this subsection. The ROA spectra evaluated at these levels of theory are compared with the corresponding full unfragmented calculations (referred to as “actual”) at MPW1PW91/aug-cc-pVDZ. In previous work, we have shown that MIM reproduces the geometrical parameters of the actual high calculations very accurately with a correlation factor of ∼1.0.37 In the case of the 10 benchmark systems, all models work very well, and we do not present an analysis of the actual and MIM geometrical parameters of these systems. A comparison of MIM1 and MIM2 energies relative to the full “actual” calculations at MPW1PW91/aug-cc-pVDZ level in presented in Table 1 for the benchmark molecules depicted in Figure 1. The energies of D and L for each of the 10 benchmark E


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Figure 2. Comparison of actual [MPW1PW91/aug-cc-pVDZ] (in black) backscattered ROA spectrum with (A) MIM1[MPW1PW91/ aug-cc-pVDZ] (in red) and (B) MIM2[MPW1PW91/aug-ccpVDZ:MPW1PW91/6-31G(d)] (in blue) of D-maltose. Both actual and MIM spectra are evaluated at an external frequency of 488.0 nm.

To illustrate the performance of MIM-ROA and MIMRaman, comparisons of the D-Maltose MIM2 vibrational spectra between 400 and 1600 cm−1 with the corresponding backscattered experimental spectra47 are shown in Figure 3. The calculated normal modes are scaled with a scaling factor of 0.958 and fitted with Lorentzian functions with width 10.5 cm−1. The frequency-dependent polarizability derivatives are evaluated at an external frequency of 488.0 nm, the same as that used in the experiment. Figure 3A depicts the backscattered experimental ROA spectrum whereas Figure 3B shows the corresponding MIM2-ROA spectrum. The most intense ROA bands are at 1108 (−578.06) corresponding to the C−O stretching mode, and at 1297.75 (−318.02) corresponding to the C−H wagging modes. Similarly, Figure 3 also depicts a comparison of (C) backscattered experimental and (D) MIM2Raman spectra. The most intense peaks are at 1106.0 (118.08) and 1041.5 (104.92) and correspond to C−O stretching modes. The overall agreement between the backscattered experimental and theoretical spectra is excellent. The ROA and Raman intensities are evaluated from the polarizability derivatives and these derivatives obey certain translational and rotational sum rules.48 Here, we explicitly analyze the sum rules for the translational and rotational degrees of freedom for D-maltose. For the evaluation of Raman and ROA intensity evaluations, six components of these derivatives are needed. Hence, for ααβ and G′αβ we analyze the six components in xx, xy + yx, xz + zx, yy, yz + zy, and zz directions, and for Aαβγ, we consider the components in xyz, yzx, zxy, yzy − zyy + zxx − xzx, yzz − zyz + xyx − yxx, and zxz − 9 + xyy − yxy. The sum rule is verified by comparing the maximum deviation in the left-hand side (lhs) and right-hand side (rhs) of these derivative expressions48 relative to the translational and rotational degrees of freedom. The maximum deviations are reported for the full (i.e., actual) calculations outside the bracket and that for MIM2 inside the bracket as Actual (MIM2). The maximum deviation for the translational dααβ, dG′αβ and dAαβγ are 0.0 (0.0), 0.018 (0.019), and 0.0 (0.0). The corresponding maximum deviations for rotational components are 0.061 (0.094), 0.830 (0.469), and 0.750 (0.300). The maximum deviations at MIM2 also show very similar values as in the corresponding full calculations. These

Figure 3. Comparison of D-maltose (A) experimental backscattered ROA spectrum (in black) with (B) MIM2[MPW1PW91/aug-ccpVDZ:MPW1PW91/6-31G(d)] (in blue) and (C) experimental backscattered Raman spectrum (in black) with (D) MIM2[MPW1PW91/aug-cc-pVDZ:MPW1PW91/6-31G(d)] (in blue). Both experimental and theoretical spectra are at an external frequency of 488.0 nm.

results show that MIM2 obeys the sum rules to a similar extent as in the full calculation. 3.2. Comparison of α-Cyclodextrin MIM-ROA and MIM-Raman with Experiment. To validate the applicability of MIM-ROA in predicting experimental spectra, we have made an explicit comparison of calculated MIM-ROA spectra with experiment for two larger test systems. For the first test system, we have chosen the α-D-cyclodextrin, shown in Figure 4A. Cyclodextrins are cyclic oligosaccharides consisting of six,

Figure 4. Optimized geometries of (A) α-D-cyclodextrin at MIM2[B3LYP/6-311++G(d,p):HF/3-21G] and (B) the T1T1T1 isomer of cryptophane-A at MIM2[MPW1PW91/6-31+G(d,p):MPW1PW91/631G]. F


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functional with a large 6-311++G(d,p) basis set at the high level and a smaller 3-21G at the low level, MIM2 [B3LYP/6-311+ +G(d,p):HF/3-21G]. The frequency-dependent polarizability derivatives are evaluated at an external frequency of 488.0 nm, the same as that used in the backscattered experiment.47,50 The total energy and the most intense ROA vibrational modes are reported in Table 2. All the 3N − 6 (scaled) normal modes and intensities are reported in Table S2 in the Supporting Information. Each normal mode is scaled with 0.952 and fitted with a Lorentzian function of width 14.5 cm−1 (Figure 5). As seen in Figure 5, the most intense peak in the experimental ROA spectrum is a couplet at 922 cm−1. This is in agreement with a strong calculated couplet involving a motion of the glycosidic bond along with some contribution from C−O−H deformation. This couplet has the same sign, though significantly red-shifted, with a higher intensity when compared to the corresponding couplet in the D-maltose ROA spectrum depicted in Figure 3A,B. Although some deviations are seen between theory and experiment, it should be noted that the backscattered experimental spectra for α-cyclodextrin have been measured in water. Despite the fact that hydrogen bonding interactions with water are missing in our calculations, the overall calculated ROA spectrum shows a very reasonable agreement with experiment. 3.3. Comparison of Cryptophane-A MIM-ROA with Experiment. Cryptophanes are an examples of synthetic “container molecules” that have been well studied theoretically and experimentally.51 The chiroptical properties of enantiopure cryptophanes and the molecular recognition of chiral or achiral guest molecules have been investigated by polarimetry, electronic circular dichroism (ECD), VCD, and ROA.52 Hence, as a final system to demonstrate the applicability of our method, we have evaluated the MIM-ROA spectra for the stereoisomers of cryptophane-A. Cryptophane cages comprise two cyclotriveratrylene bowls connected by three aliphatic O−CH2−CH2−O bridge groups, and optically pure cryptophane-A isomers have been synthesized recently.53 Among the many nonequivalent possible conformers of cryptophane-A, previous work has shown that the high symmetry D3 isomer with a dihedral angle close to 180° (denoted as T1T1T1), provides a very good agreement with experimental ROA. Thus, we have evaluated the MIM-ROA spectrum for this isomer, shown in Figure 4B. From the theoretical point of view, evaluating the ROA and Raman spectrum of cryptophane-A (C54H54O12) can be computationally expensive, particularly when used with large basis sets. We have performed MIM2 calculations using MPW1PW91/6-31+G(d,p) as the high layer and MPW1PW91/6-31G as the low layer. These calculations on cryptophane-A with 120 atoms at MIM2 [MPW1PW91/631+G(d,p):HF/6-31G] required 1524 basis functions and 2490

seven, or eight D-glucopyranose residues, interconnected with glycosidic bonds. We have chosen the smallest of this toroidal structure with six moieties that are interconnected with intramolecular hydrogen bonds. A previous analysis of αcyclodextrin has shown that all six rings in this molecule preferentially adopt a chair form over the other conformations.49 This stereoisomer is the most stable conformer, and hence we have taken into account only this most stable conformer. This makes a theory-to-experiment comparison relatively straightforward. A comparison of the backscattered ROA experimental spectrum of α-D-cyclodextrin50 with the calculated MIM-ROA spectrum between 400 and 1600 cm−1 is depicted in Figure 5A,B. In the current comparison, we have employed the B3LYP

Figure 5. Comparison of α-D-cyclodextrin (A) experimental backscattered ROA spectrum (in black) with (B) MIM2[B3LYP/6-311+ +G(d,p):HF/3-21G] (in blue) and (C) experimental backscattered Raman spectrum (in black) with (D) MIM2[B3LYP/6-311++G(d,p):HF/3-21G] (in blue). Both experimental and theoretical spectra are at an external frequency of 488.0 nm.

Table 2. Structure Number, Name, Basis Functions (NBasis), Energies, and Most Intense Peaks in the ROA Spectrum for the Cryptophane-A and α-Cyclodextrina ROA frequency [intensity] no.

system

NBasis

electronic energy

ν[IMIM2 max ]

ν[IMIM2 max ]

ν[IMIM2 max ]

XIID XIIL

cryptophane-A α-cyclodextrin

1524 1872

−2992.25458 −3665.68071

1602.75 [17221.49] 934.50 [−2557.39]

1306.25 [−11446.91] 1345.50 [−2339.05]

990.25 [−7993.51] 899.50 [1845.82]

The ROA frequencies (cm−1) and intensities (au 104). The backscattered ROA spectrum of cryptophane-A is evaluated at MIM2[MPW1PW91/631+G(d,p):MPW1PW91/6-31G] with an external frequency of 532.0 nm, and that of α-cyclodextrin at MIM2[B3LYP/6-311++G(d,p):HF/3-21G] with 488.0 nm. Refer to the text for further details.

a

G


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However, this method requires a rotation and translation of the different fragments to match the positions of a given set of atoms with the equivalent atoms in other fragments of the composite system. In addition, London orbitals were not used in this approach, making the results not strictly gauge-invariant. Our MIM approach keeps all the subsystems in the same global coordinate frame, and rigorous cancellation of the overlapping subsystems including link-atom projection makes it mathematically rigorous. Additionally, the inclusion of long-range interactions via an inexpensive second layer makes our results quantitatively accurate relative to the full calculation. The major bottleneck in the MIM-ROA calculation involves the evaluation of the higher energy derivative matrices. MIM speeds up all the higher energy derivative calculations, including the Hessian matrix (second-order derivatives) and the polarizability derivative matrices (third-order derivatives). The primary subsystem derivative evaluations are the rate-limiting steps in MIM. The number of primary subsystems increases linearly with the size of the large molecule whereas the subsystem size is independent of the parent molecule, and the computational time for the second layer calculation, if carefully selected, can be made small. Hence, the MIM-ROA method asymptotically has a linear scaling with increasing system size, though it will depend on the model selected for the second layer. Our results in section 3 suggest that the MIM approach is useful to reproduce the experimental spectra for large molecules. However, we have not addressed the fundamental question of selecting the best level of theory and basis functions to reproduce the experimental spectrum. This question has been addressed by Cheeseman et al.13 for a set of small molecules. The main thrust of the current manuscript is to enable calculations on large molecules through MIM, to accurately reproduce the full ROA spectrum. Our comparison in section 3.1 suggests that the MIM fragmentation introduces little error relative to the actual calculations at the same level of theory. Thus, any discrepancy from experiment is related to the level of theory used as well as other factors such as such as solvent effects or conformational sampling that have not been considered in this paper. The MIM model will also be able to speed up the explicit solvated calculations as well as conformational sampling problems. Our research group is actively working on these research problems that will be explored in future studies.

primitive Gaussian functions. Optimizations and Raman and ROA spectral evaluations are all carried out uniformly at the same level of theory. The gas phase theoretical spectra are compared with the experiment conducted in 0.1 M chloroform solvent. It has previously been confirmed that the solvent molecules do not alter the ROA spectrum of the parent T1T1T1 isomers.54 The ROA spectrum in the 850−1700 cm−1 region as seen in the experiment using a backscattering geometry has been analyzed in the present work. Figure 6 shows the comparison of

Figure 6. Comparison of T1T1T1 isomer of cryptophane-A (A) experimental backscattered ROA spectrum (in black) with (B) MIM2 [MPW1PW91/6-31+G(d,p):MPW1PW91/6-31G] (in blue). Both experimental and theoretical spectra are at an external frequency of 532.0 nm.

experimental spectrum for cryptophane-A (top) with the MIM2 spectrum for the gas phase molecule (bottom) using frequencies scaled by 0.952 and using a Lorentzian line shape with a width of 7.5 cm−1. The frequency dependent polarizability derivatives are evaluated at an external frequency of 532.0 nm, as used in the experiment. The total energy and the most intense ROA vibrational modes are reported in Table 2. The full set of frequencies and ROA intensities are listed in Table S3 in the Supporting Information. The overall agreement between the MIM-ROA and the backscattered experimental spectra is very good. In this spectrum, the normal mode at 1607 cm−1 is assigned to the CC stretching vibration of the benzene rings.54 The wagging and twisting of the CH2 groups, in the COC linkers and bowls give rise to the three bands between 1350 and 1250 cm−1, with the same sign. The band at 1211 cm−1 can be assigned to the asymmetric stretching vibration of COC linkers, coupled with the twisting mode of the linker −CH2 groups. The two bands observed at 1002 and 982 cm−1 belong to the symmetric stretching vibration of COC groups. The region below 1200 cm−1 is more complex because the observed bands correspond to coupled modes involving several vibrations. Overall, the MIM2-ROA spectrum shows a good agreement with experiment for the T1T1T1 stereoisomer. This suggests that MIM-ROA can be a powerful tool to assign ROA spectra for other large molecules in the future.

5. CONCLUSIONS ROA is a powerful method for the determination of the absolute configurations and the structures of biomolecules in their native state. However, accurate evaluation of ab initio ROA spectra for large molecules can be computationally expensive, particularly in conjunction with large basis sets. In this work, we have implemented and calibrated the MIM fragment-based method for evaluating ROA spectra of large molecules. The incorporation of the Jacobian link-atom projection method as well as the employment of multiple layers of theory in accounting for long-range weak interactions are both important factors that contribute to the accuracy of the method. MIM-ROA is carefully benchmarked on a set of 10 pairs of carbohydrate enantiomers. The frequencies and ROA intensities show excellent agreement with the actual calculations. Larger systems such as α-D-cyclodextrin and cryptophane-A have also been investigated, and the MIM-ROA spectra show excellent correlation with the corresponding

4. GENERAL ASPECTS There has been one previous fragment-based approach by Thorvaldsen et al.55 for the calculation of ROA tensors. H


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experimental spectra. Overall, our MIM-ROA fragment-based approach significantly reduces the computational requirements and opens up a wide range of systems that can be studied through accurate ab initio methods.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b01127. A pictorial representation of the improvement in the ROA invariants for MIM2 over MIM1 is shown in Figure S1. A comparative analysis of the MIM Raman spectra of D-maltose with the full calculation is depicted in Figure S2. Coordinates, energies, vibrational frequencies, and ROA intensities for the actual calculations of D-maltose (Table S1), α-cyclodextrin (Table S2), and (+)-T1T1T1 cryptophane-A (Table S3) are also listed. (PDF)

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AUTHOR INFORMATION

Corresponding Author

†K. Raghavachari. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Professor Prasad Polavarapu and Dr. James R. Cheeseman for valuable discussions. This work was supported by funding from NSF Grant No. CHE-1266154 at Indiana University. The authors also thank the Indiana University Big Red II supercomputing facility for computing time.

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Raman Optical Activity Spectra for Large Molecules through Molecules

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