Integrated QM/polarizable MM/continuum approaches to model chiroptical properties of strongly interacting solute–solvent systems

1.1 QM/classical solvation models for chiroptical properties

Continuum solvation models rooted in the PCM constitute the state-of-the-art of the computational approaches to chiroptical properties/spectroscopies of solvated system, and have become an integral part of the toolkit of scientists working in academia as well as industry. Nowadays, the large majority of the calculations performed to support the assignment of the chiral absolute configuration exploit such approaches, coupled to a QM description of the target system. QM/continuum solvation models for chiroptical properties have been recently reviewed: the interested reader can find a comprehensive discussion in recent literature.9, 18 However, as already mentioned in the Introduction, they suffer from serious limitations. Most important, they may fail dramatically to model strongly interacting molecule-environment couples, such as aqueous solutions, which are nonetheless the natural environment of most biomolecules.

Recent studies36, 37, 42, 44 have demonstrated that such a limitation can be successfully overcome by moving to QM/MM/continuum approaches,51, 52 such as the recently developed fully polarizable QM/FQ/PCM method with nonperiodic boundary conditions,45 which includes polarization effects in the classical MM portion by means of fluctuating charges (FQ)53 and resorts to a continuum PCM treatment of the outer solvation shell. In particular in this model the solvated molecular system is decomposed in three layers, in a multiscale fashion (Figure 1). The inner layer, treated at the QM level (density functional theory [DFT] and coupled-cluster-based methods have been exploited so far, however, the model can be extended to other families of Hamiltonians) is limited to the part of the system which is considered to determine a large part of the property under examination, within the general philosophy of focused models. Within the framework of solvation, such an inner core coincides with the solute. The second layer is constituted by the portion of the environment closer to the solute, and is treated at the atomistic MM level in terms of a set of FQs, which adjust to the QM density. In case of solvation studies, this portion is not covalently bonded to the inner core, however, the formulation of the model is general enough to allow for covalently bonded system to be treated in a ONIOM (our Own N-layer Integrated molecular orbital molecular mechanics)-like structure.54, 55 The QM/FQ system has a droplet-like shape (see Figure 1 for a schematic picture), of which the radius can be adjusted to the needs of the property/system under examination. A third layer, treated as a continuum PCM dielectric is used to account for bulk effects and assures the proper nonperiodic boundary conditions. In the study of solvation, the QM/FQ/PCM approach is used in combination with molecular dynamics (MD) simulations: a series of snapshots from the MD is extracted, which constitute the actual model system on which the QM/FQ/PCM calculations of the desired properties are performed and then averaged out to obtain the final results. Actually, the computational cost of the procedure is mainly determined by the cost of the single QM/FQ/PCM calculation, which basically depends on the QM level and the kind of property (first, second, third derivative, etc.) to be calculated, together with the number of representative snapshots necessary to allow the final results to converge to a sensible value.

The QM/FQ/PCM approach has recently been extended to treat the evaluation of analytical first and second derivatives,47 response equations,46 magnetic perturbations with gauge including atomic orbitals (GIAOs),48 excitation energies (at the TDDFT46 and EOM-CCSD49 levels of theory), and excitation energy gradients.50 Therefore, most chiroptical responses can be calculated.36, 37, 56 In this section, the physical and computational aspects of this model in connection with the calculation of chiroptical properties are briefly presented, whereas selected examples, showing its potentialities are reported in a forthcoming section.

1.1.1 The QM/FQ/PCM approach

The FQ approach45 is particularly suitable to be employed within a three layer QM/MM/(PCM) framework because of its connection both with quantum mechanics and classical electrostatics. In fact, the model is based on the concepts of atomic hardness and electronegativity, which can be rigorously defined within the DFT,57, 58 and the electronic distribution is represented in terms of a set of classical atomic charges depending on the electrostatic potential. The FQ model, possibly coupled to the PCM can be used to calculate molecular properties by means of response theory or analytical derivatives,46-49 however, it can be also exploited in MD simulations.59 In particular, if the QM/FQ(/PCM) is exploited in a time-independent fashion, the standard machinery of computational chemistry can be employed to calculate structural and spectroscopic properties.37, 46-49

The FQ approach represents the polarization of a classical, atomic system by endowing each atom with a FQ, of which the value depends on the environment53, 60, 61 according to the electronegativity equalization principle57, 62 The FQs are defined through the minimization of the following functional:

(1)

where α and β run on molecules and i, j on atoms and λ_α is a set of Lagrangian multipliers used to impose charge conservation constraints. Q_α is the total charge on molecule a. The vector collects the atomic electronegativities, and the J matrix collects the interaction kernel elements between the FQs. In our implementation, the Ohno functional form reference 63 is used:

(2)

where η_i is the hardness of the ith atom,

is the average of the atomic hardnesses of atoms i and j and is the distance between two MM atoms. Notice, however, that the implementation is general enough to allow other kernels to be exploited.

The stationarity conditions of the functional in Equation 1 are defined through the following equation45

(3)

where C_Q collects atomic electronegativities and total charge constraints, whereas charges and Lagrange multipliers are collected in and D includes the J matrix and the Lagrangian blocks.

The interaction between the FQs and the QM density of charge, ρ_QM can be defined as:

(4)

where is the electrostatic potential due to the QM density of charge at the ith FQ q_i placed at r_i.

At the self-consistent field (SCF) level of theory the global energy functional of the system then reads:

(5)

where h and G are the usual one- and two-electron matrices, and P is the QM density matrix.

By imposing the stationarity conditions taking into account the proper constraints, a modified set of Fock equations is obtained, defined in terms of the QM/FQ Fock matrix:

(6)

where μ, ν are atomic basis functions.

The FQs can therefore be obtained by solving:

(7)

The outer PCM layer can easily be taken into account by redefining the charges and the D matrix so as to include PCM contributions, that is, by solving the following set of equations:

(8)

where S and Ω represent the Coulomb interaction among the PCM charges and with the FQs, respectively, and accounts for the dielectric nature of the medium. In addition, a PCM contribution is to be added to in Equation 6. Further details on the coupling of the PCM with the FQ model can be found elsewhere.45, 46

1.1.2 Analytical derivatives and response equations

In the previous section, the physical fundamentals of the QM/FQ/PCM model is given, with the main focus being the definition of the energy. However, the calculation of energies is by definition completely insufficient to address chiroptical phenomena. In fact, to model the chiral response, any theoretical model has to account for the presence of the external polarized radiation. Also, by examining the literature treating the theoretical fundamentals of chirality,64, 65 the reader may be easily convinced that any chiroptical signal arises from the simultaneous interaction of the sample with both the electric and magnetic components of the electromagnetic field. This means that both fields have to be accounted for in the theoretical methodology presented above, and that is generally achieved by resorting to response theory. Also, any computational approach to spectroscopic quantities requires an accurate description of the sample. Among other requirements, this means that a way to determine the minimum energy structure is generally requested, and this is computationally achieved by calculating geometrical energy first (gradients) and second (vibrational frequencies) derivatives. All these quantities have been formulated for the QM/FQ/PCM approach, to enable it to effectively treat chiroptical phenomena. Therefore, this section and the rest of the theoretical part of the article give the technical details of such an extension, up to the formulation of OR and ECD spectra. The reader not interested in the technical details may directly jump to the next part of the article, where some selected examples of application of the method are discussed.

As it was mentioned in the previous paragraph, to discuss the extension of the QM/FQ/PCM method to chiral phenomena, it is worth briefly recalling the extension of the QM/FQ/PCM method to the evaluation of analytical first and second derivatives,47 response equations to electric46 and magnetic perturbations with GIAOs.48 In following treatment, only FQ terms will be considered: extension to PCM does not add any substantial difference, but only makes the notation more cumbersome. The reader interested in the details can find them in references 46 and 47.

Energy first derivative can be expressed by means of the chain rule: 66

(9)

where the last two terms vanish because of the stationarity conditions. The first term, that is, the derivative of the energy with respect to the position of a QM-described nucleus, reads

(10)

The term in Equation 9 involving the derivatives of the density matrix can be computed by resorting to the energy-weighted density matrix contribution

where the subscript oo denotes the occupied–occupied block of the matrix in the MO basis. Therefore:

(11)

In case of energy derivatives with respect to the position of MM atoms ξ, we obtain:

(12)

where is nothing else than the electric field produced by the QM density acting on the FQs, (see reference 47 for more details).

Second order response properties can be analytically derived by calculating the second derivatives of the variational QM/FQ energy. According to the philosophy of focused models, the extension of the QM/FQ/PCM approach to second derivatives has been done by assuming the external perturbation (i.e., the electric field or a nuclear displacement) only acting on the QM portion of the system, whereas indirect effects on the MM part through the perturbation on the QM density are considered. This approach cannot be applied when the QM and MM portions are covalently bonded, however, alternative approaches such as the ONIOM54, 55 extrapolation can be applied.

The second derivatives of the SCF/FQ energy with respect to two generic perturbations x, y (i.e., electric/magnetic field components, nuclear coordinates) acting on the QM portion of the system can be obtained by further differentiating Equation 11:47

(13)

where we have generally considered that the basis functions may depend on the perturbation.

Equation 13 involves FQ explicit contributions and response contributions, involving density matrix derivatives. The latter are obtained through a set of coupled perturbed Hartree Fock (CPHF) equations including FQ/PCM terms.46, 47 Therefore, electric, magnetic or vibrational properties require the evaluation of the perturbed density matrix and, possibly, of the perturbed FQs (and PCM charges).

Following reference 66, by differentiating once the SCF equations, working in the MO basis, projecting on the occupied-virtual block, using the density matrix idempotency and recalling that , we obtain the following equation:

(14)

The Fock matrix derivatives may be written as:

(15)

The derivatives of the FQ charges can be obtained by differentiating Equation 7

(16)

Substitution yields:

(17)

where:

By rearrangement of the terms, the CPHF equations are obtained:

(18)

By working in the MO basis and defining the following matrices:

(19)

(20)

where ϵ denotes orbital energies and are standard antisymmetrized bielectronic integrals.

After some algebra Equation 18 can be rewritten as

(21)

where and . The solution of Equations 21 and 16 yields the derivatives of the density matrix and the FQs with respect to QM nuclear positions, respectively, thus allowing the calculation of energy second derivatives.

The formulas for the full Hessian matrix, that is, including also the QM/MM and MM/MM blocks, can be found in reference 47.

1.1.3 Electric perturbations

In the presence of an external electric field E and assuming the FQ to be affected by the field only through the response of the QM molecule, an additional perturbation appears in the energy functional ( are dipole integrals):

(22)

Therefore, a contribution to the monoelectronic part of the Fock operator of the system arises:

(23)

The second derivatives of the energy with respect to the electric field, that is, the static polarizability, reads:

(24)

where there are not any explicit FQ contributions.

The right-hand side of the CPHF equations reads

(25)

Again, there is not any FQ contribution to this term. The right-hand side of the CPHF equations is real: therefore and the equations may be solved for X + Y. By summing up the CPHF equations, we obtain:

(26)

and , where involves an FQ contribution.

In the frequency dependent case (ω is the frequency of the oscillating field):

(27)

1.1.4 Magnetic perturbations

Let us assume a static magnetic field, given by the sum of a homogeneous magnetic field B and the field produced by the magnetic moment m_X of the nucleus X at position R_X is interacting with the molecular system. This can be accomplished using the minimal coupling principle67 which is valid also for nonlinear Hamiltonians.68 In this case, the following additional terms are to be included in the Fock operator describing the solute interacting with the magnetic field:69

(28)

(29)

(30)

(31)

In the previous equations, the Coulomb Gauge is exploited. h⁽⁰¹⁾ represents the first-order contribution to the interaction between the electronic motion and the external magnetic field; h⁽¹⁰⁾ is the corresponding first-order term which couples the nuclear magnetic moment to the electronic motion; h⁽²⁰⁾ and h⁽¹¹⁾ are second order terms connected with the diamagnetic contribution, respectively, to the magnetizability and to the magnetic shielding. Notice that in principle a further h⁽⁰²⁾ term should be present,69 that is, a second-order term relative to the interaction between the electrons and the nuclear magnetic moment, which is, however, not of interest to the present case.

The calculation of magnetic terms with finite basis sets suffers from a dependence on the choice of the origin. To circumvent the origin dependence of calculated magnetic and mixed electric-magnetic properties, two ways have been proposed in the literature, that is, the use of magnetic-field dependent basis sets (Gauge-including atomic orbitals GIAOs, also known as London Orbitals)70, 71 or the use of the velocity-gauge for dipole integrals. Here, the use of GIAOS is discussed, where a dependence of basis functions on the magnetic perturbation is assumed. Therefore, none of the terms in Equation 13 can be neglected. The right-hand side of the response equations for the insulated system reads

(32)

where

(33)

Q is purely imaginary, and hence , so that the response equations can be solved for X − Y. By subtracting the response equations

(34)

which can be used together with . Notice that in this case A − B does not include any FQ response contribution. The contribution to the right-hand side of the QM/FQ response equation reads

(35)

where

Again, the FQs contribute indirectly to the magnetic response.

1.2 Optical rotation

Once the (frequency-dependent) electric and magnetic CPHF are solved, the basic ingredients to calculate mixed electric–magnetic properties are obtained. The OR reads:6, 72-79

where

G_xy is the electric dipole–magnetic dipole polarizability tensor:

(36)

The integral in Equation 36 can be expressed as:

(37)

where

and, finally,

λ = E, B and the X and Y transition densities are obtained by solving the response equations for the (frequency dependent) electric and magnetic fields, respectively.

1.3 Electronic circular dichroism

The calculation of ECD spectra generally requires the evaluation of excited state energies. This can be carried out by applying a linear response approach to the QM/FQ/PCM model. The Casida approach to the time-dependent density functional theory (TD-DFT)80 has been extended to the QM/FQ model.46, 50 In particular, the form of the TD-DFT equations is

(38)

The eigenvalues Ω are the vertical excitation energies, while the amplitudes for the single particle excitation and de-excitation are contained in the eigenvectors X and Y, respectively. The response matrices A and B are defined as

(39)

where i, j are occupied orbitals, a, b are unoccupied ones and σ and τ are spin labels. The usual procedure for the diagonalization of Equation 38 involves the contraction of the combinations and with the (X + Y) and (X − Y) vectors, carried out in the atomic orbital basis.

The FQ contribution to the response matrices reads

(40)

where

(41)

The FQ contribution is purely electrostatic, and only contributes to the symmetric combination (A + B). Once the Casida equations are solved and excitation energies and transition amplitudes are known, the rotatory strength tensors can be computed in the velocity form, as described in reference 81 The integrals of the electric and magnetic multipoles are computed and stored in AO basis set, and then are traced with the transition densities.

2.1 The OR of (R)-methyloxirane

A well-known example of huge solvent effects on chiroptical properties is the OR of methyloxirane in aqueous solution.29, 30, 35, 82, 83 The application of the PCM approach has been amply reported, showing that continuum solvation is a viable tool to describe the OR of methyloxirane whenever solvent effects are dominated by bulk contributions, that is, for weakly interacting solute–solvent couples. Even in this case, however, to obtain reliable results, the basic PCM formulation is not completely adequate, and local field/nonequilibrium effects should be included in the modeling of the electronic property.26, 29 Also, vibrational corrections calculated with the inclusion of anharmonic terms give a substantial contribution to the final computed property, which permit to finally recover the experimental findings. On the contrary, in cases dominated by specific solute–solvent interactions, even this refined model is not able to correctly reproduce solvent effects, which can be large enough to change the sign of the OR value. The reason for such an effect can be understood from the data reported in Figure 2, where the results obtained by calculating with the QM/FQ/PCM approach the OR on 2000 snapshots taken from a MD simulation are given. The set of 2000 representative structures was obtained by cutting, for each snapshot, a sphere centered in the solute's geometrical center of 16 Å. A 1.5 Å larger radius was used to define the PCM spherical cavity representing the third layer (more details on the computational protocol can be found in reference 36). It is in particular worth to notice that the snapshots were extracted from a MD performed by keeping fixed the methyloxirane structure, that is, the snapshots only differ by a different arrangement of the water molecules around methyloxirane. Figure 2 clearly shows a large variability of the OR as a function of the particular geometrical arrangement of water around the target molecule. The OR can vary sensitively, not only in absolute value, but especially in sign, taking any value in the range −150/150 deg dm⁻¹ g⁻¹ cm³. This means that any static approach to such property, such as the PCM or simple cluster-based models, where the specific interaction is modeled by redefining the solute as the target molecule plus a limited number of explicitly treated solvent molecules, will most probably give a result falling in this range, however, a good reproduction of the average value is not guaranteed. This is more clearly explained in Figure 3, which reports the OR calculated values as obtained by using the PCM and the QM/FQ/PCM approach. The experimental value83 is also reported for comparison's sake. The solvent effect modeled by PCM goes in the right direction with respect to the value in vacuo, that is, a decrease in the absolute value is obtained. However, the account of bulk effects only is unable to correctly predict the OR sign, which is instead successfully recovered when the explicit presence of the water around methyloxirane is modeled by means of the QM/FQ/PCM approach. Notably, such an agreement is not limited to a single wavelength (see Figure 4), but the entire ORD curve is well reproduced by the hybrid explicit/implicit QM/FQ/PCM model, also as varying the choice of the density functional.84

It is also to notice that the analysis of the snapshots of MD simulations did not evidence any dominant solute–solvent configurations, that is, any persistent hydrogen bonding pattern around methyloxirane. This is probably the reason for the slow convergence of the property with respect to the simulation time, and the need of averaging over a huge (2000) number of snaphshots to allow the property to converge to a sensible value. Furthermore, such an absence of predominant patterns explains the failure of computational protocols based on clusters. Such approaches have been applied to this case30, 82 by re-defining the solute molecule in terms of a large number of clusters including only a limited number of solvent molecules, which were then treated at the QM level in the evaluation of the OR. A large effect of specific solute–solvent interactions in determining the sign of the calculated OR was obtained, however, a quantitative agreement with the experimental values was not achieved.

2.2 The ECD of (S)-nicotine

The example which has been reported in the previous section regards a rigid molecule, which has a single prevalent conformation. However, this is not the case of most molecules, which generally have conformational flexibility. As an example of nonrigid chiral molecules, in this section the ECD spectrum of (S)-nicotine is discussed.37, 85

Neutral nicotine (see inset in Figure 5 for its structure) exists in two possible conformations, denoted as A and B, which are defined through a rotation around the angle. Calculated populations of the two conformers by exploiting the PCM and the CAM-B3LYP/aug-ccpVDZ/level of theory show a prevalence of the A conformer (55.3% population) over the B one.

The two conformations show a very similar behavior in terms of the oscillator strengths of similar transitions, that is, very similar absorption spectra,37 however, the same is not true for rotatory strengths (R), determining ECD spectra. In fact (see Figure 5), in the region around 175 nm the sign of the rotatory strengths of the two conformers are opposite, with the A conformer showing a positive and large R value, whereas the B conformer has a low, negative value. Around 225 nm the sign of the R of the two conformers is the same, however, the B rotatory strength is generally higher compared with the other conformer. Therefore, despite the B conformation is predicted to be less populated by the PCM, it is expected to dominate the ECD spectrum in that region nonetheless. The final PCM ECD spectrum is therefore the result of the subtle interplay between the conformer populations and their relative R values. The resulting spectrum is shown in Figure 6, where the experimental spectrum37 is also reported for comparison's sake. Clearly, the PCM yields results that qualitatively agree with the experimental values, however, a quantitative reproduction of peak intensities is not achieved. Notice that the variation of the ECD spectrum on conformation (Figure 5) is not surprising. In fact, the close relationship between conformations and sign and magnitude of chiroptical properties has been reported in the literature,86, 87 and has been extensively investigated by Wiberg and coworkers (see reference 87 and references therein).

Because of its two basic lone pairs, nicotine is expected to undergo hydrogen bonding with water molecules, of which the effects can be evaluated by exploiting a MD simulation followed by QM/FQ/PCM calculations on selected snapshots (more details on the computational protocol can be found in reference 37).

The application of MD simulations to extract data on the conformational preference, shows that by exploiting the general Amber force field88 with atomic charges calculated by applying the multiconformation RESP (MultiRESP)89 procedure and water molecules modeled using the TIP4P/200590 parameter set, the B conformer is found to be the most stable, at variance with QM/PCM findings. The analysis of nitrogen-hydrogen radial distribution functions between nitrogen atoms in nicotine and hydrogen atoms in water molecules indicates the presence of first and second hydration shells separated from the bulk solvent. In particular, the aliphatic nitrogen (see inset in Figure 5) forms a very stable hydrogen bond with one water molecule, whereas the hydration structure of the aromatic nitrogen atom is much less defined compared with that of the aliphatic one. Also, in the case of the A conformer, the aliphatic nitrogen shows well-defined first and second hydration shells, while the aromatic nitrogen hydration structure is less defined. However, when conformer B is considered, the existence of a cooperative effect in which bridging water–water hydrogen bonds stabilize the B conformer, creating less disorder in the water liquid structure, can be found. The results of the MD simulations were also used to extract a set of snapshots to be exploited to compute ECD spectra of nicotine in the QM/FQ/PCM scheme. In particular, each snapshot was centered on the center of mass of the solute, around which a solvation sphere of radius 14 Å, containing an average of 366 water molecules, was cut. A 2.5 Å larger radius was used to define the PCM spherical cavity, used as boundary for the third layer.

The QM/FQ/PCM ECD spectrum was then obtained by averaging out the spectra of the single snapshots. The final result is shown in Figure 6; the sign pattern agrees with experiments, however, the relative intensities of the bands do not perfectly match the experimental spectrum. The discrepancy, however, is much lower than in case of QM/PCM results, thus demonstrating a significant improvement in the results with the inclusion of explicit hydrogen bonding effects, affecting both in TD-DFT calculations and the conformational sampling. It is also to notice that the convergence of the spectrum is achieved by averging 300 snapshots, that is, a number of an order of magnitude less than what is required to allow OR of methyloxirane to converge.