First Hyperpolarizability of Collagen Using the Point Dipole

Letter pubs.acs.org/JPCL

First Hyperpolarizability of Collagen Using the Point Dipole Approximation Ignat Harczuk, Olav Vahtras, and Hans Ågren* School of Biotechnology, Division of Theoretical Chemistry and Biology, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden S Supporting Information *

ABSTRACT: The application of localized hyperpolarizabilities to predict a total protein hyperpolarizability is presented for the first time, using rat-tail collagen as a demonstration example. We employ a model comprising the quadratic Applequist point-dipole approach, the so-called LoProp transformation, and a procedure with molecular fractionation using conjugate caps to determine the atomic and bond contributions to the net β tensor of the collagen [(PPG)10]3 triple-helix. By using Tholes exponential damping modification to the dyadic tensor in the Applequist equations, a correct qualitative agreement with experiment is found. The intensity of the βHRS signal and the depolarization ratios are best reproduced by decomposing the LoProp properties into the atomic positions and using Tholes exponential damping with the original damping parameter. Some ramifications of the model for general protein property optimization are briefly discussed.

C

stacking of the peptide bonds between the (hydroxy)-proline and glycine residues. The specific rigidness caused by the [PPG] subunit has been used to explain the buildup of the tensor component in the longitudinal direction of the chains, something which could not be found when modeling the [GGG] trimer. The point dipole model was originally proposed by Silberstein16,17 and later used by Applequist et al.18 to predict molecular polarizabilities. Several others generalized Applequist’s formulation to higher orders, with applications to intermolecular potentials,19 the second-order susceptibility χ(2) of DAN,20 and the third-order susceptibility χ(3) for benzene and naphthalene.21 The calculations of χ(2) and χ(3) in these works would be formulated with properties residing at the molecular center-of-mass, or distributed to the conjugated ring center-of-masses in the case of benzene and naphthalene, using the Lorentz local field tensor to account for the crystal surrounding. It has furthermore recently been shown that the distribution of molecular properties based on QTAIM22 could be used for the calculation of χ(2) in crystal structures.23 In the present work, it is however, to the best of our knowledge, the first time the nonlinear extension of Applequist’s equations is applied to covalently bonded fragments. We have previously shown how the second-order induced dipoles could be obtained with the aim of using LoProp24 hyperpolarizabilities of atoms or bonds to calculate the equivalent induced hyperpolarizability of interacting molecules.25 Here, the same formulation is extended with Tholes

alculations of nonlinear optical properties of large molecules have progressively become more accessible and attended during recent years. However, the inherent requirement of high electron correlation in quantum mechanical methods to determine higher-order nonlinear optical properties such as the first hyperpolarizability1,2 still sets rather severe limits on the size of the systems that can be investigated at high levels of accuracy. Finding approximate methods which could give the nonlinear optical characteristics of large molecular ensembles is thus highly desirable and would have wide ramifications for technical applications. One such scheme with accompanying approximations was recently presented by the present authors and was shown to have the potential of evaluating the first hyperpolarizability of extended systems. Furthermore, using the localized polarizabilities, this method was applied to accurately calculate the Rayleigh scattering of extended water clusters with coadsorbed organic molecules.3 We here further demonstrate the capability of this method, now in the area of biomolecular sensing, and choose for the purpose the hyperpolarizability of a protein of topical interest, namely, the collagen triple helix, which in recent years has drawn attention due to its high first-order nonlinear response. Collagen is the most abundant structural protein in the human body and has a long history as the object for second harmonic generation (SHG) imaging,4−8 and also for use as a biosensor. One important application is the detection of collagen fibrils in the extracellular matrix that can lead to diagnosis of heart disease,9−11 where the strong directionality of SHG has important implications. Previous measurements of the hyper-Rayleigh scattering in collagen12 together with additive approximate models,13,14 and more advanced models as the socalled ONIOM model,15 have related the large value of β to the © XXXX American Chemical Society

Received: March 31, 2016 Accepted: May 20, 2016

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DOI: 10.1021/acs.jpclett.6b00721 J. Phys. Chem. Lett. 2016, 7, 2132−2138

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The Journal of Physical Chemistry Letters exponential damping.26−28 The LoProp transformation is used for the atom and bond decomposition of the molecular properties of each residue, in combination with the so-called molecular fractionation with conjugate caps (MFCC)29,30 procedure for cutting at the peptide bonds, to derive the properties of the collagen triple helix model,31 which accounts for the properties of the helix without any long-range interactions between the residues. The second-order quadratic point-dipole model is then used to calculate the inter- and intrachain hyperpolarization between and inside the chains. The total hyper-Rayleigh scattering intensity and depolarization ratio, which are rotationally invariant and measurable in hyperRayleigh scattering experiments, are then computed from the resulting βCollagen tensor and compared to previous experiments and calculations. The localized properties are obtained by a transformation which diagonalizes the atomic overlap matrix.24 In its general form, the LoProp decomposition assigns molecular properties to bonds between atoms and to the atoms themselves. Here, both the methods of including the bond and atomic properties, as well as decomposing the bond polarizabilities to their constituent atoms, are investigated. These are denoted as the bond and atomic models, respectively. The comparison is relevant in that similar point-dipole models, such as the one formulated by Applequist, use atomic properties only, whereas LoProp thus gives both atomic and bond polarizabilities. The obvious advantage of using the midpoints between atoms is the additional degree of freedom and the possibly better overall description of the molecular potential. This observation dates back to early work of Karlström.32 The exact derivation of the Applequist equations for the hyperpolarizability together with the derivation of localized analytic atomic polarizabilities and hyperpolarizabilities can be found in previously published work.25,33 The discrete points are represented by their localized multipole expansions, polarizabilities, and hyperpolarizabilities; the total dipole-moment of each localized point can be written as pi = pi0 + αi·Ei +

1 β : E iE i 2 i

βm =

ij

R̃ ij = (1δij − αĩ ·Tij)−1 ·αj̃

δ pi = αĩ ·δ Ei + βi : δ Eiδ Ei

α̃j = αj + βj ·Ej

fV = 1 −

⎛1 ⎞ −v ⎜ v + 1⎟e ⎝2 ⎠

fE = fV −

⎛ 1 2 1 ⎞ −v ⎜ v + v ⎟e ⎝2 2 ⎠

(10)

fT = fE −

1 3 −v ve 6

(11)

Tij =

(9)

3rijrijfT − fE rij21 rij5

(12)

As an illustration of the damped model, Figure 1 shows the Ex and Ey electric field components in the xy-plane stemming from a positron at the origin, with (left column) and without (right column) damping. In this work, Tholes exponential damping is used and compared to the native Applequist approach, denoted as “point” in the figure legends. We also include results from the additive scheme, in which all the components of the dipolecoupling tensor in eq 4 are set to zero, and the Applequist properties are the sums of the localized atomic or bond polarizabilities and hyperpolarizabilities, respectively. In Figure 2, the βHRS ∞ intensity is plotted as a function of the amount of residues of each chain. The model peptide [(PPG)10]3 is approximately obtained with 29 residues in each chain. This collagen model has 87 residues in total, because the X-ray structure has one glycine missing in the Cterminal of each chain. The experimental result measured to βHRS 790nm = 11460 ± 690, was extrapolated to static limit by the

(2)

(3)

3rijrij − rij21 rij5

(7)

where v = au, and a is an arbitrary damping parameter. The parameter derived most accurately using the largest set of data27 is a = 2.1304, and this is the one also used in this work. In the damped model, the dyadic tensor now reads

(1)

where the dipole−field coupling tensor is Tij =

(6)

which gives the following damping parameters for the potential, field, and field-gradient, respectively

∑ Tij·pj j≠i

(5)

The points in the above notation can represent atoms or bonds, and if the localized hyperpolarizabilities are set to zero, the classical Applequist model is obtained, which gives rise only to the molecular polarizability. Due to very large molecular polarizabilities caused by the induced dipoles at short interatomic distances, the original Applequist18,34 equations were modified by Thole.26 The main difference in Tholes model derives from the introduction of the so-called damping parameters, which depend on the scaled interatomic distances. The equation used to calculate the damping reads rij u= iso iso 1/6 (αi αj ) (8)

The local fields are determined by all the other induced dipoles. Ei = Fi +

kl

and where the modified induced polarizabilities α̃ j due to the static field created by the localized multipoles reads

δ pi = αi·δ Ei + βi : Eiδ Ei 2

kl

where the relay matrix R̃ is obtained by the original Applequist equations

Assuming that an instantaneous induction of the point-dipoles occurs with respect to an external uniform electric field, the first- and second-order change in the dipoles can be expressed as

2

∑ R̃ ij·βj : (1 + ∑ Tjk ·R̃ kl)(1 + ∑ Tjk ·R̃ kl)

(4)

From the first- and second-order induction of the localized dipole moments, the total polarizability and hyperpolarizability, respectively, are obtained. The explicit equations to compute the total hyperpolarizability, which is the computed property of the collagen helix in this paper, reads 2133


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denominator in eq 8 gives less damping, overall increasing the total electric field at all sites. Another trend seen in Figure 2 is that the scattering intensity grows in a slight sigmoidal shape for all the models, whereas the additive scattering, i.e., just taking the sum of the localized hyperpolarizabilities, grows linearly. Adding more residues would thus not necessarily produce a growth in intensity but might converge with respect to the chain length. It is furthermore noteworthy that when damping is included, the intensity is not as sensitive to the computational method used to obtain the properties. Figure 3 shows all the methods tested and all the models used for the depolarization ratio. For the full system, i.e., when the x-axis is at 29, the ranking is that TDHF outperforms TDCAMB3LYP, followed by TD-B3LYP. It can also be seen that the depolarization oscillates more wildly than the scattering intensity. The experimental value of 8.4 hints to the fact that the collagen helix is highly dipolar, with no significant octupole character. The reason why the quadratic Applequist model underestimates the dipolar nature of this collagen model might be due to some components, such as βxxx and βyyy in the collagen tensor, are overinduced. They should be close to zero, because they are in the cross section of the collagen helix, but in fact are quite large for all quantum mechanics (QM) methods, which may contribute to a larger value of ⟨β2XZZ⟩ in comparison 2 ⟩, leading to a lower depolarization ratio. It is to ⟨βZZZ interesting that TD-B3LYP under-performs with respect to TDHF and TD-CAMB3LYP. For TDHF and TD-CAMB3LYP, βxxz and βyyz are close in magnitude, whereas for TD-B3LYP, the value of βxxz is quite small in comparison to βyyz. This might be attributed to the long-range exchange being important for the calculation of the nondiagonal hyperpolarizability components using the quadratic Applequist equations. All the QM methods show qualitative experimental agreement in the depolarization ratio for the pure additive summation of the localized hyperpolarizability tensors. This is a natural consequence of that the cross-sectional components βxxx and βyyy cancel each other out when extending the chain in the zdirection, giving only large contributions to the βxxz, βyyz, and βzzz components, maximizing the dipolar nature of the collagen. Previous calculations elucidated that the successive stacking of the π-character peptide bonds was the main reason behind the large second-order response in collagen.13,14 In Figure 4, we

Figure 1. Ex and Ey components due to an ideal point charge at at the origin with q = 1 au. The right column illustrates how Tholes exponential damping gradually decreases the electric field near the charge source.

two-state approximation, see the Supporting Information of the work by de Wergifosse et al.15 The atomic decomposition stemming from TD-DFT/B3LYP using Tholes damping shows the best agreement with experiment with a value of 7224 au. The corresponding values are 6785 and 8949 for TDHF and TD-DFT/CAMB3LYP, respectively. Because the originally derived exponential damping parameter of 2.1304 is used, which was parametrized without taking bond polarizabilities into account, the obtained result confirms the expectation that the atomic decomposition should give the most physically accurate picture. Decomposing the molecular properties with the LoProp bond polarizabilities between atoms gives a βHRS ∞ larger than that for the purely atomic properties. This is the case for both using no damping and with the damping. This originates from the nature of the LoProp decomposition, which assigns the larger portion of the molecular polarizability between atoms rather than on the atoms themselves. The use of bond polarizabilities also increases the total amount of points which are polarized by the Applequist equations, which could be the reason why the total βHRS ∞ is larger than for the atomic decomposition. Moreover, because the isotropic LoProp polarizabilities on average are smaller for each site, the

Figure 2. Total static scattering intensity βHRS for a growing collagen triple helix. For 29 residues, the rat-tail collagen is obtained. 2134


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Figure 3. Depolarization ratio, ρ, for the growing collagen triple helix. For 29 residues, the rat-tail collagen is obtained.

Figure 4. Sum total for αLoProp (left), and βLoProp (right) for all different atom types in the PDB file, after the LoProp/MFCC procedure had been zz zzz performed. The H1 and HCO types exist only in the N-terminal prolines and C-terminal prolines, respectively, and are added for completeness.

find that indeed the largest contribution to the polarizability component αzz and the negative hyperpolarizability component βzzz, which are the largest two components of the polarizability and hyperpolarizability for the total collagen model, respectively, originates from the peptide bond atoms. The only two atoms with a large response not directly participating in the peptide stacking are the CD and HD2 atom types. Because CD is directly bonded to the nitrogen, it will, however, have a high decomposition of the total hyperpolarizability, because it is close to the peptide bonded nitrogen. For the properties at atoms using only the Thole damping, the 27 residue containing [(PPG)3]3 system, which has been studied by full QM calculations, the results are 1707, 1780, and

2179 au for TDHF, TD-B3LYP, and TD-CAMB3LYP, respectively. In the reference calculations, the corresponding value of this tripeptide was calculated to 517 and 472 au using TDHF/CPHF and TDDFT/LC-BLYP, respectively. The proposed method thus gives results of an order 3.5 larger than the tested QM methods for the smaller oligomer structure. The presented calculations show that using atomic properties obtained using TDHF with the Applequist equation, modified with the exponential damping scheme of Tholes, the HRSscattering intensities obtained are close to the experimental ones, extrapolated to the static limit. It is intriguing that properties of different amino acids, as seen in Figure 4, will show up in different total collagen properties, plotted in Figures 2135


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The Journal of Physical Chemistry Letters 2 and 3. As the composition of collagen fibrils consisting of type I and II have been found to be correlated with hypersensitive heart disease,10 the signal intensity could help discern the composition of the fibril network and aid in medical diagnosis. Our code deals with static external fields only, and direct calculations of the dynamical properties is not possible at the moment. However, the extension to the dynamical contribution βHRS(−2ω; ω, ω) is indeed inherent in our model. The effort to implement the dynamical response into our model is limited only by code implementation for the postprocessing script, because the quadratic response part is already implemented. The ambition of the present work was to calculate the net hyperpolarizability of collagen by considering the interaction of its separate residues. We calculated the first hyperpolarizability for the amino acid constituents for a model system of the collagen triple-helix, also denoted as rat-rail collagen, using quadratic response theory, employing time-dependent Hartree−Fock and density functional theory. The localized polarizabilities and hyperpolarizabilities of the atoms and bonds were obtained for this collagen model by using the LoProp algorithm. The influence of the dipole-coupling tensor was studied in the Applequist equations, where the original formulation was tested alongside the modified Tholes damping interaction. The inclusion of damping, which avoids the overinduction inherent in the Applequist equations at small interatomic separations, was found to be crucial. It was shown that although the new formalism overestimates QM data for smaller structures, overall qualitative trends can be found with respect to the ONIOM model and other approximate methods. In the case of atomic properties with damping, good agreement with available experimental data is obtained for the scattering intensity at the static limit. Investigations on other systems for which accurate experimental data is available could help discern the proposed model’s validity and generality. For the total dipole−octupole ratio of collagen as measured by the depolarization ratio, the additive model gave the best agreements with experiments. Out of the Applequist interaction models, the TDHF properties fared best. The TDDFT/B3LYP properties gave the worst depolarization ratios. We related the computed first hyperpolarizability of a collagen model to isotropical averages measured in the liquid phase. Because water has a high dielectric constant, it will affect the hyperpolarizability of the collagen, and its long-range effect is difficult to elucidate when extrapolating theoretical predictions to experimental observables. Moreover, the discrepancy in the reported values of the first hyperpolarizability of water range from 10 au35 to 65 au,36 which makes the extrapolation from dynamical measurements to the static limit difficult. Furthermore, because direct comparison of the hyperpolarizability tensor components from theoretical calculations is possible only to experimental quantities for crystals of fixed periodicity, it is difficult to relate the measured intensities of unstructured materials to theoretical calculations. However, it must be noted that the strongest advantage in the presented model lies in the computational speed, because the total calculation time is limited only by the size of the MFCC fragments. After the quadratic response on each fragment has been performed, the LoProp decomposition in combination with the Applequist procedure can be calculated on a regular modern computer in the time scale of minutes. Further interesting considerations involves the properties of more complicated protein structures, such as those containing aromatic residues or proteins with β-sheet formations. The

investigation of the properties for such systems with respect to the size of the capping fragments could shed light on the importance of many-body effects when cutting across covalent bonds, which could be tested in many different ways, but that lies outside the scope of this Letter. A motivation behind this work can be found in the general ambition to derive models to efficiently predict properties of large biostructures like proteins, with an accuracy that allows the optimization of the biostructures for a particular property by structural modifications like substitutions or residual mutation. The sole application presented in this work, on the collagen triple helix, indicates that the advised quadratic Applequist point-dipole approach using localized property components is such a model with sufficient accuracy for optimizing protein hyperpolarizability and thereby the second harmonic generation used in medical diagnosis. Obviously, many more examples, studied under different conditions, must be explored to firmly prove that indication.

■

COMPUTATIONAL METHODS The collagen triple helix chain was obtained from the PDB structure with the code 1K6F.31 The first 3 chains from the file were used as a model for the miniature version of collagen. The pdb 2gmx script as part of the GROMACS37 package was used to add hydrogen atoms to the chains. All water molecules were removed, and the N- and C-terminal groups were set to the neutral HN and −CH groups, respectively, for all three chains. The geometry was optimized in the gas phase using Gaussian09,38 with the semiempirical PM6 method.39 The properties are obtained using linear and quadratic response theory in Dalton40 and transferred to the LoProp basis using a script.41 For the properties, the 6-31+G* basis set is used, which has been shown to give a good trade-off between accuracy and computational speed for molecular properties in molecules that have simple electronic configurations. The polarizabilities and hyperpolarizabilities were evaluated at static field, with no vibrational contributions. Because the experimental value is extrapolated to the static limit from 780 nm measurements using the two-state approximation, frequency dispersion is omitted. For the static value of the hyperpolarizability, the vibrational contributions are assumed to be negligible. The script used for obtaining the MFCC29,30 fragments is called pdbreader.py and is part of a larger software suite publicly available on GitHub.42 In the case of the bond inclusive force fields, the properties of the bond midpoint between separate residues are evenly distributed to the C and N atom. To avoid issues with overinduction of the point dipoles due to the short interatomic distance between two neighboring residues, the properties of the peptide bond atoms were evenly distributed to their nearest neighboring atoms. For the C atoms, the properties were distributed evenly to the O and CA atoms, and for the N atoms, the properties were distributed evenly to the HN and CA atom. For the atoms and bonds in each separate residue, only the local static field and field caused by the induced dipoles from other residues is calculated. This is because the properties of each residue are already accounted for from the QM calculations. The hyper-Rayleigh scattering intensity is calculated as βHRS =

2 2 ⟨βXXZ ⟩+⟨βZZZ ⟩

(13)

and the depolarization ratio as 2136


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2 ⟨βZZZ ⟩ 2 ⟨βXXZ ⟩

(11) Porter, K. E.; Turner, N. A. Cardiac Fibroblasts: at the Heart of Myocardial Remodeling. Pharmacol. Ther. 2009, 123, 255−278. (12) Deniset-Besseau, A.; Duboisset, J.; Benichou, E.; Hache, F.; Brevet, P.-F.; Schanne-Klein, M.-C. Measurement of the Second-order Hyperpolarizability of the Collagen Triple Helix and Determination of its Physical Origin. J. Phys. Chem. B 2009, 113, 13437−13445. (13) Loison, C.; Simon, D. Additive Model for the Second Harmonic Generation Hyperpolarizability Applied to a Collagen-mimicking Peptide (Pro-Pro-Gly) 10. J. Phys. Chem. A 2010, 114, 7769−7779. (14) Tuer, A. E.; Krouglov, S.; Prent, N.; Cisek, R.; Sandkuijl, D.; Yasufuku, K.; Wilson, B. C.; Barzda, V. Nonlinear Optical Properties of Type I Collagen Fibers Studied by Polarization Dependent Second Harmonic Generation Microscopy. J. Phys. Chem. B 2011, 115, 12759−12769. (15) de Wergifosse, M.; de Ruyck, J.; Champagne, B. How the Second-Order Nonlinear Optical Response of the Collagen Triple Helix Appears: A Theoretical Investigation. J. Phys. Chem. C 2014, 118, 8595−8602. (16) Silberstein, L., VII. Molecular Refractivity and Atomic Interaction. Philos. Mag. 1917, 33, 92−128. (17) Silberstein, L. L. Molecular Refractivity and Atomic Interaction. II. Philos. Mag. 1917, 33, 521−533. (18) Applequist, J.; Carl, J. R.; Fung, K.-K. An Atom Dipole Interaction Model for Molecular Polarizability. Application to Polyatomic Molecules and Determination of Atom Polarizabilities. J. Am. Chem. Soc. 1972, 94 (2960), 2952. (19) Dykstra, C. E. Efficient Calculation of Electrically Based Intermolecular Potentials of Weakly Bonded Clusters. J. Comput. Chem. 1988, 9, 476−487. (20) Munn, R. Theory of Molecular Optoelectronics. IX. Calculation of χ (2) from Theoretical Hyperpolarizabilities for DAN. Int. J. Quantum Chem. 1992, 43, 159−169. (21) Reis, H.; Papadopoulos, M. G.; Calaminici, P.; Jug, K.; Köster, A. Calculation of Macroscopic Linear and Nonlinear Optical Susceptibilities for the Naphthalene, Anthracene and Meta-Nitroaniline Crystals. Chem. Phys. 2000, 261, 359−371. (22) Bader, R.; Nguyen-Dang, T. Quantum Theory of Atoms in Molecules-Dalton Revisited. Adv. Quantum Chem. 1981, 14, 63−124. (23) Seidler, T.; Krawczuk, A.; Champagne, B.; Stadnicka, K. QTAIM-Based Scheme for Describing the Linear and Nonlinear Optical Susceptibilities of Molecular Crystals Composed of Molecules with Complex Shapes. J. Phys. Chem. C 2016, 120, 4481−4494. (24) Gagliardi, L.; Lindh, R.; Karlström, G. Local Properties of Quantum Chemical Systems: the LoProp Approach. J. Chem. Phys. 2004, 121, 4494−500. (25) Harczuk, I.; Vahtras, O.; Ågren, H. Hyperpolarizabilities of Extended Molecular Mechanical Systems. Phys. Chem. Chem. Phys. 2016, 18, 8710−8722. (26) Thole, B. Molecular Polarizabilities Calculated With a Modified Dipole Interaction. Chem. Phys. 1981, 59, 341−350. (27) Van Duijnen, P. T.; Swart, M. Molecular and Atomic Polarizabilities: Thole’s model Revisited. J. Phys. Chem. A 1998, 102, 2399−2407. (28) Swart, M.; van Duijnen, P. T. DRF90: a Polarizable Force Field. Mol. Simul. 2006, 32, 471−484. (29) Zhang, D. W.; Zhang, J. Z. H. Molecular Fractionation with Conjugate Caps for Full Quantum Mechanical Calculation of ProteinMolecule Interaction Energy. J. Chem. Phys. 2003, 119, 3599. (30) Söderhjelm, P.; Ryde, U. How Accurate Can a Force Field Become? A Polarizable Multipole Model Combined with Fragmentwise Quantum-Mechanical Calculations. J. Phys. Chem. A 2009, 113, 617−627. (31) Berisio, R.; Vitagliano, L.; Mazzarella, L.; Zagari, A. Crystal Structure of the Collagen Triple Helix Model [(Pro-Pro-Gly) 10] 3. Protein Sci. 2002, 11, 262−270. (32) Karlström, G. Local Polarizabilities in Molecules, Based on ab initio Hartree-Fock Calculations. Theor. Chim. Acta 1982, 60, 535− 541.

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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.jpclett.6b00721. Detailed MFCC explanation, example coordinate files containing MFCC fragments, full LoProp atomic and bond properties averaged over residues, and computed components of the collagen model PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Morten Nørby Pedersen for providing the basis set. The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputing Center (NSC) and High Performance Computing Centre North (HPC2N), project “Design of Force Fields for Theoretical Spectroscopy”, SNIC2015-1-230.

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REFERENCES

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