Adaptive-Partitioning Multilayer Dynamics Simulations: 1. On-the-Fly Switch between Two Quantum Levels of Theory

Abstract

We propose to generalize the previously developed two-layer permuted adaptive-partitioning QM/MM, which reclassifies atoms as QM or MM on-the-fly in dynamics simulations, to multi-layer adaptive-partitioning algorithms that enable multiple levels of theory. In this work, we formulate two new algorithms that smoothly interpolate the energy between two QM (Q1 and Q2) levels of theory. The first “permuted adaptive-partitioning” scheme is based on the weighted many-body expansion of the potential, as in the adaptive-partitioning QM/MM. Unconventional and potentially more efficient, the second “interpolated adaptive-partitioning” method employs alchemical QM calculations with Q1/Q2-mixed basis sets, Fock matrices, and overlap matrices. To our knowledge, this is the first time that such alchemical calculations are performed in QM, although they are routinely done in MM. Test calculations on water-cluster models show that both new algorithms indeed yield smooth energy curves when water molecules shift between Q1 and Q2.

Graphical Abstract

1. Introduction

Adaptive quantum-mechanics/molecular-mechanics (QM/MM)^1–34 allows the boundary between the QM and MM regions to be relocated on-the-fly as needed. This is especially suitable for modeling diffusive systems such as a substrate moving into and binding at the active site of an enzyme, or an ion migrating through the membrane via a channel. Our focus is usually the given substrate or ion and its immediate surroundings. This is often achieved by introducing a buffer zone (typically of 1–2 Å thickness) between the QM and MM regions and then smoothing the energy and/or forces when molecules or functional groups travel in the buffer zone. Adaptive QM/MM enables the use of a small, mobile QM zone centered at the substrate or ion and follows it wherever it goes, with the contents of the QM zone updated on-the-fly as the trajectory propagates. This treatment removes the limits caused by a static QM/MM boundary, achieving better integration of the QM and MM descriptions. Adaptive QM/MM has been demonstrated to yield accurate structural and kinetic properties of simulated model systems when compared with conventional QM/MM treatments.^{17, 27, 29}

To further improve computational efficiency and accuracy, we have undertaken an endeavor to systematically generalize the adaptive QM/MM schemes for two layers (QM and MM) to multi-layer algorithms that permit two or more QM layers to be used. This will allow us to treat the solute’s 1^st solvation shell at a high-level QM (denoted Q1, e.g., density functional theory (DFT) with a large basis set) and 2^nd solvation shell at a low-level QM (denoted Q2, e.g., DFT with a small basis set, or simply semi-empirical QM). Consequently, an atom may be reclassified on-the-fly as Q1, Q2, or MM. Doing so leads to an adaptive-partitioning multi-layer (APML) Q1/Q2/MM model (Fig. 1). Because the 2^nd shell, while important, is generally less influential than the 1^st shell on a solute, treating it with a more approximate method lowers the computational cost without notably compromising accuracy. Moreover, when a semi-empirical QM or density functional based tight binding (DFTB) method is used for Q2, it may be feasible to extend the Q2 layer to include atoms beyond the 2^nd shell, allowing for a more accurate description of critical interactions over longer ranges when compared to the two-layer QM/MM model. The APML models will be beneficial to many applications, e.g., the study of ion channels where electron delocalization is prominent along the transport path.^{35, 36}

Figure 1.

Although not required, here all except the MM (white) zones are centered at the solute O (purple) in APML. Buffer1 (light green) is between Q1 (brown) and Q2 (yellow), and Buffer2 (light blue) between Q2 and MM zones. Groups are purple/red in Q1, orange in Buffer1, magenta in Q2, green in Buffer2, and blue in MM. Distance from O to a Buffer1 group R₁ satisfies R_min1 ≤ R₁ ≤ R_max1, and from O to a Buffer2 group R₂ satisfies R_min2 ≤ R₂ ≤ R_max2.

A critical component in the APML method is the adaptive treatments for the Q1/Q2 layer. Here, we report the preliminary results of two milestones in our new adventure: (1) generalizing a two-layer adaptive potential for QM/MM to Q1/Q2, where the potential is expressed in the form of many-body expansion with individual terms smoothly interpolated between the Q1 and Q2 levels, and (2) proposing a more efficient approach, the interpolated adaptive-partitioning (IAP) treatment, that smoothly switches between levels of theory through alchemical QM calculations. The complete formulation and demonstration of the APML algorithms for a Q1/Q2/MM three-layer model will be reported in a future paper.

2. Method

2.1. General Considerations

The new algorithms in this work are based on the permuted-AP (PAP) QM/MM method, which was first proposed by Heyden et al.³ and further developed by Lin and coworkers.^{6, 9, 15, 27, 29, 32} In PAP-QM/MM, the potential V of the system is defined through a many-body expansion where the many-body terms are scaled by smoothing functions that vary smoothly between 0 and 1 when the positions of involved buffer groups move from the MM into the QM zones. The gradients of these employed smoothing functions (called transition forces) are associated with the differences in chemical potentials for the buffer groups between the QM and MM levels of theory. Various schemes^{3, 4, 9, 17, 27} have been successfully developed to properly treat these gradients and avoid artifacts in simulations.

It is straightforward to generalize the PAP algorithm to two QM levels of theory. For simplicity, we present the algorithms for the cases where each buffer group is a whole molecule (e.g., a water molecule). In principle, extensions are straightforward when the boundary passes through covalent bonds: For the Q1/Q2 boundary, the fragments can be capped by H atoms with scaled bond-lengths as in the integrated molecular orbital and molecular orbital (IMOMO) method by Morokuma and coworkers.³⁷ Alternatively, one can use the hybrid orbital projection technique in the fragment molecular orbital method.³⁸ The methods are general and can in principle be applied to both mechanical- and electrostatic-embedding schemes; here we will only discuss the simpler situation, mechanical embedding. Furthermore, the subtractive formula for QM/QM energy³⁷ is adopted.

Without loss of generality, we assume that there are sequentially one Q1 group (the 1^st), N buffer groups (the 2^nd to (N+1)-th), and one Q2 group (the (N+2)-th), leading to a total of (N + 2) groups. The Q1 and Q2 characteristics of group i are P_i and (1–P_i), respectively, where P_i is set to the value of a smoothing function² for group i that varies from 0 at R_i = R_max to 1 at R_i = R_min, and R_i is the distance between the group and the Q1-zone center. The P_i is equivalent to the quantum weight λ_i in Ref.²² and fractional QM character σ_i in Ref.¹⁷ For the 1^st (pure-Q1) and (N + 2)-th (pure-Q2) groups, P₁ = 1 and P_M = 0, respectively.

2.2. PAP Q1/Q2

Following the subtractive formula, which has been used in both QM/QM³⁷ and QM/MM,³⁹ we define the PAP Q1/Q2 potential $V_{12\cdots (N+2)}^{\text{Q}1/\text{Q}2;\text{PAP}}$ as

$$V_{12\cdots (N+2)}^{\text{Q}1/\text{Q}2;\text{PAP}}=V_{12\cdots (N+2)}^{\text{Q}2}-V_{12\cdots (N+1)}^{\text{Q}2}+V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{PAP}}$$ (1)

In this work, unless otherwise noted, the superscripts indicate the levels of theory at which the groups listed in the subscripts are treated. The last two terms in the right side indicate that, for the cohort of Q1 and buffer groups, the single-level Q2 descriptions $V_{12\cdots (N+1)}^{\text{Q}2}$ are replaced by the mixed-Q1/Q2 descriptions $V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{PAP}}$. The interactions between this Q1-buffer cohort and the Q2 group as well as the interactions within the Q2 group remain at the Q2 level. The mixed-Q1/Q2 descriptions for the Q1-buffer cohort are a sum of many-body interaction terms,

$$V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{PAP}}=V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;1-\text{body}}+\Delta V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;2-\text{body}}+\Delta V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;3-\text{body}}+\cdots$$ (2)

where the many-body interactions terms are

$$V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;1-\text{body}}={\sum }_{i=1}^{N+1}\left(P_i{V}_i^{\text{Q}1}+\left(1-P_i\right)V_i^{\text{Q}2}\right)$$ (3)

$$\Delta V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;2-\text{ body }}={\sum }_{i=1}^N{\sum }_{j=i+1}^{N+1}\left(P_i{P}_j\Delta V_{ij}^{\text{Q}1}+\left(1-P_i{P}_j\right)\Delta V_{ij}^{\text{Q}2}\right)$$ (4)

$$\Delta V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;3-\text{body}}={\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N{\sum }_{k=j+1}^{N+1}\left(P_i{P}_j{P}_k\Delta V_{ijk}^{\text{Q}1}+\left(1-P_i{P}_j{P}_k\right)\Delta V_{ijk}^{\text{Q}2}\right)$$ (5)

In eq. 4 and eq. 5, ΔV_ij and ΔV_ijk are given by

$$\Delta V_{ij}=V_{ij}-\left(V_i+V_j\right)$$ (6)

$$\Delta V_{ijk}=V_{ijk}-\left(V_{ij}+V_{ik}+V_{jk}\right)+\left(V_i+V_j+V_k\right)$$ (7)

It is obvious that, for groups i and j, their two-body interactions will be $\Delta V_{ij}^{\text{Q}1}$ if both are Q1. If one group is Q1 and the other is buffer or if both groups are buffer, the interactions will be a mixture of $\Delta V_{ij}^{\text{Q}1}$ and $\Delta V_{ij}^{\text{Q}2}$, weighted by the respective scaling functions P_i and P_j. Similarly, for three groups i, j, and k, their three-body interactions will be $\Delta V_{ijk}^{\text{Q}1}$ if all groups are Q1 and a mixture of $\Delta V_{ijk}^{\text{Q}1}$ and $\Delta V_{ijk}^{\text{Q}2}$ if any of the groups is buffer. The zero of energy in the one-body term for a given group is arbitrary and often set to the energy at its equilibrium geometry at the designated Q1 or Q2 level for convenience.

We note that our scheme differs from ONIOM-extension solvation (ONIOM-XS)^{2, 40, 41} in that PAP yields a smooth potential energy surface when the number of buffer groups changes, while ONIOM-XS exhibits abrupt energy changes when the number of buffer groups varies.³

As previously found, truncating the many-body expansion in PAP QM/MM increases the efficiency without significant loss of accuracy.^{3, 6, 9, 15, 29, 32} It is likely that this is also true for PAP Q1/Q2 calculations; as demonstrated in the fragment molecular orbitals (FMO) studies by Kitaura and colleagues^{42, 43} and in the electrostatically embedded many-body expansions by Dahlke et al.,^{44, 45} many-body potentials truncated after two-body interactions can reach accuracy of ~2 kcal/mol against reference calculations.

2.3. IAP Q1/Q2

Another way to achieve a smooth Q1-Q2 transition is IAP, which uses interpolated parameters in special alchemical QM calculations. Similar to PAP, the IAP potential $V_{12\cdots (N+2)}^{\text{Q}1/\text{Q}2;\text{IAP}}$ is given by:

$$V_{12\cdots (N+2)}^{\text{Q}1/\text{Q}2;\text{IAP}}=V_{12\cdots (N+2)}^{\text{Q}2}-V_{12\cdots (N+1)}^{\text{Q}2}+V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{IAP}}$$ (8)

where the potential for the Q1-buffer cohort $V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{IAP}}$ is computed alchemically. To our knowledge, this is the first time such alchemical QM calculations are proposed for two different QM levels of theory, although a different ansatz based on scaling electron integrals had been used by Field²² for single-level QM calculations in adaptive QM/MM. To illustrate this concept, we examine the case of combining two different basis sets. If the basis sets are {ϕ^Q1} in Q1 and {ϕ^Q2} in Q2, one assigns an interpolated basis set to group i, in a similar way to a contracted basis set:

$${\varphi }_i^{\text{Q}1/\text{Q}2}=P_i{\varphi }_i^{\text{Q}1}+\left(1-P_i\right){\varphi }_i^{\text{Q}2}$$ (10)

We mandate that an atomic orbital centered at a given atoms inherits the smoothing function value of the group that contains the atom. For instance, if atomic orbital μ is centered at an atom belonging to group i, then P_μ = P_i. The interpolation indicated by eq. 10 may be accomplished in different ways, and here we combine the respective shells from each basis set before the SCF iterations. For instance, the 1s shell from Q1 is combined with the 1s shell from Q2, and so on. Whenever a basis function exists in Q1 but not in Q2 (assuming Q1 is the larger basis), the ${\varphi }_i^{Q2}$ coefficient is simply set to zero. (A piece of pseudo code is given in Chart S1 in the Supporting Information.) This ensures that the Fock matrix has a consistent size throughout the buffer region. Apart from the above-mentioned interpolation, no change was made to the computation of integrals or the Fock matrix. (Although we only present here the IAP equations for the Hartree-Fock method, the extension to DFT is straightforward, with the exchange-correlation numerical integration computed using the interpolated basis.)

With these interpolated basis sets, one can form the alchemical Fock matrices and alchemical overlap matrices at the Q1/Q2 level through interpolation. For example, the alchemical Fock matrix $f_{\mu v}^{\text{Q}1/\text{Q}2}$ is a hybrid of the Fock matrices at the $\text{Q}1\left(f_{\mu \nu }^{\text{Q}1}\right)$ and $\text{Q}2\left(f_{\mu \nu }^{\text{Q}2}\right)$ levels and a new Fock matrix constructed using the interpolated basis set:

$$f_{\mu \nu }^{\text{Q}1/\text{Q}2}=P_{\mu }{P}_v{f}_{\mu v}^{\text{Q}1}+\left(1-P_{\mu }\right)\left(1-P_v\right)f_{\mu v}^{\text{Q}2}+f_{\mu v}^{\text{Q}1+\text{Q}2}$$ (11)

$$f_{\mu v}^{\text{Q}1}=h_{\mu \nu }^{\text{Q}1}+g_{\mu v}^{\text{Q}1}=h_{\mu v}^{\text{Q}1}+{\sum }_{\lambda \sigma }{P}_{\lambda }{P}_{\sigma }{D}_{\lambda \sigma }\left([\mu v\mid \lambda \sigma ]-\frac{1}{2}[\mu \sigma \mid \lambda v]\right)$$ (12)

$$f_{\mu \nu }^{\text{Q}2}=h_{\mu \nu }^{\text{Q}2}+g_{\mu \nu }^{\text{Q}2}=h_{\mu \nu }^{\text{Q}2}+{\sum }_{\lambda \sigma }\left(1-P_{\lambda }\right)\left(1-P_{\sigma }\right)D_{\lambda \sigma }\left([\mu \nu \mid \lambda \sigma ]-\frac{1}{2}[\mu \sigma \mid \lambda \nu ]\right)$$ (13)

Here, μ and ν denote two atomic orbitals (AOs), P_μ is the scaling factor for the function ϕ_μ, ℎ_μν and g_μν are the one- and two-electron terms, respectively, and D_λσ the charge density matrix element for atomic orbitals λ and σ. Here, D_λσ is modified by weights P_λP_σ in $f_{\mu \nu }^{\text{Q}1}$ and (1 − P_λ)(1 – P_σ) in $f_{\mu \nu }^{\text{Q}2}$, respectively. Note that $f_{\mu \nu }^{\text{Q}1}$ and $f_{\mu \nu }^{\text{Q}2}$ are Fock matrices constructed only with the Q1 and Q2 basis sets, respectively, whereas $f_{\mu v}^{\text{Q}1+\text{Q}2}$ is the new Fock matrix that contains cross terms from the mixed basis set defined in eq. 10. This cross-term Fock matrix ensures that the energy is correct at the limits of P_i = 0 and P_i = 1.

A special case is that Q1 and Q2 use different methods but the same basis set, which should result in simply interpolated matrices. Another special case is that Q1 and Q2 use the same method and the same basis set, leading to the regular Fock matrix for the system.

The alchemical overlap matrix interpolation is straightforward

$$S_{\mu \nu }^{\text{Q}1/\text{Q}2}=P_{\mu }{P}_v{S}_{\mu \nu }^{\text{Q}1}+\left(1-P_{\mu }\right)\left(1-P_v\right)S_{\mu \nu }^{\text{Q}2}+S_{\mu \nu }^{\text{Q}1+\text{Q}2}$$ (14)

This IAP scheme is in principle very general but leads to larger matrices with interpolated basis sets for the buffer groups; however, unless the number of the buffer groups is very large, the increase in computational cost is expected to be modest. Moreover, when two different basis sets are employed, the Q2 basis is often smaller in practical applications, again, leading to only modest increase in computation cost for the buffer region. Some matrix elements may become near 0 when P_i is approaching 0 or 1, leading to near linear dependence in the basis sets, and care must be taken to eliminate numerical instabilities when handling these matrices.

For alchemical QM calculations, just like in conventional QM calculations, only the interaction energy is of real interest. The interaction energy of each term in the right side of eq. 8 is computed in the usual way by subtracting the energies of isolated individual groups from the total energy of at the corresponding level of theory. For example, for $V_{12\cdots (N+1)}^{\text{Q}1/\text{Q}2;\text{IAP}}$, one will compute the energy of the individual i-th (i = 1, 2, … N + 1) group in the gas phase at the mixed Q1/Q2 level of theory with the associated P_i value; because P₁ = 1, the 1^st groups is computed simply at the Q1 level. Because the size of a group is usually rather small, the additional costs are insignificant. One can also tabulate pre-computed zeros of energy for given groups at equilibrium geometries on a fine grid of P_i (e.g., with a step size of 0.01 or 0.001 in P_i) for quick access (or for interpolations that approximate the zero of energy) when needed.

Although alchemical QM calculations may sound radical, the idea of alchemical calculations are widely used in MM calculations, e.g., in the alchemical free-energy calculations, where the parameters change on-the-fly depending on an extent parameter that describes the changes.⁴⁶ The idea has been extended to QM/MM free-energy calculations, examples including the quantum mechanical free energy (QM-FE) approach,⁴⁷ the QM/MM free energy (QM/MM-FE) method,⁴⁸ the reference-potential⁴⁹ algorithm, and the quantum-mechanical thermodynamic cycle perturbation (QTCP)⁵⁰ scheme. In this work, the alchemical changes in IAP are in accord with the value of the smoothing function and are taken to be at the more fundamental level of electronic structures. We note that alchemical QM has been employed in a different context (exploration of chemical compound space) by von Lilienfeld and others.^{51, 52} Additionally, the IAP method may be considered in similar spirit to the dual-basis QM^53–56 methods with the following major difference: in the dual-basis QM methods, the small and large basis sets are successively applied to all atoms in different stages of a calculation, whereas IAP introduces a special (buffer) region where both basis sets coexist throughout the entire calculation. In general, PAP and IAP yield different energies. However, both PAP and IAP produce smooth energy changes for groups moving across the buffer zone.

3. Computation

As proof of concept, we calculate both the PAP and IAP energy curves for 6 simple models: 1 two-water cluster (Model a), 3 three-water clusters (Models b to d) and 2 four-water clusters (Models e and f), which have distinct arrangements of the water molecules, respectively (Fig. 2). In the two-water model, W1 is fixed in the Q1 zone, with its O atom (O1) serving as the center of the Q1 zone, and W2 is moving along the O1-O2 line between the Q1 and Q2 zones. The three-water models are similar to the two-water model, except that additionally W3 is fixed in the Q2 zone, ~5–6 Å away from W1. In the four-water models, W1, W2, and W4 are fixed in the Q1, buffer, and Q2 zones, respectively, while W3 travels along the O1-O3 line between Q1 and Q2.

Figure 2.

Water-cluster Models a to f for proof-of-concept calculations. Water O atom is colored in red/yellow/green if in the Q1/buffer/Q2 zone. W2 in Models a to d and W3 in Models e and f move between Q1 and Q2 crossing the buffer zone, while all other water molecules are held fixed.

Generally, the buffer-zone setup is system specific, but buffer zones of 1–2 Å thickness are commonly adopted for the compromise between accuracy and computational efficiency. We have tested various buffer zone thicknesses and found that the overall results are similar. Thus, we present in detail the data with the spherical-shell buffer zones of 2.0-Å thickness (3.0 Å ≤ R ≤ 5.0 Å) and discuss the results for other buffer zones briefly. To situate the buffer zones, the distance R is measured between the O1 and O atom (O2 or O3) of the migrating water molecule, as appropriate. To simplify data analysis and interpretation, we have selected the popular DFT functional B3LYP,^57–59 and we have tested two basis set combinations: MIDI/MINI⁶⁰ and 6–31G(d)^61–64/STO-3G,⁶⁵ with the larger basis set for Q1 and the smaller basis set for Q2. We note that the MINI and STO-3G basis sets are likely too small for real applications, but they serve well for the proof-of-concept calculations targeted here. Because the two basis set combinations essentially reveal the same picture, we will focus on the MIDI/MINI results in the main text, with the 6–31G(d)/STO-3G results given in the Supporting Information. As all water molecules have fixed internal geometries in these models, the zeros of energy in AP treatments are computed using the fixed geometry. Full single-level Q1 and Q2 energy curves are also computed for comparison. The calculations were performed using a local version of the GAMESS^66–68 program, where the IAP algorithms are implemented.

4. Results

Fig. 3 shows the scan of the total intermolecular interaction energy E as one of the water molecules travels from the Q1 to the Q2 zone through the buffer. As can be seen, both PAP and IAP generate smooth potential energy surfaces as the moving water travels across the buffer zone. First, let us look at Models a to d. As expected, the IAP and PAP curves do not match each other exactly within the buffer region, but the small (up to 1 kcal/mol) differences diminish when approaching the boundaries of the buffer zones. The PAP and IAP curves superimpose with each other outside the buffer region. Moreover, both curves match the Q2 curve when W2 moves into the Q2 zone, where all interactions between the fixed-geometry water molecules are described at the Q2 level in both PAP and IAP. Notably, for Model a, both IAP and PAP curves display smooth transitions between Q1 and Q2. As to Models e and f, the IAP and PAP curves are similar but do not match each other exactly even when the migrating water molecule W3 is not in the buffer zone, because another water molecule W2 remains in the buffer all the time. (The IAP curve in Model f is incidentally close to the Q1 curve but does not exactly superimpose with it.)

Figure 3:

Potential energy curves of Models a to f computed at the Q1 (orange squares), Q2 (red circles), PAP (green “×”), and IAP (blue “+”) levels of theory, respectively, as the moving water molecule migrates between the Q1 and Q2 zones through the buffer zone (3.0 Å ≤ R ≤ 5.0 Å, highlighted in yellow). The calculations are performed at the B3LYP level with basis set combination MIDI/MINI.

In the IAP calculations, we sometimes observe quick changes in the energy curves near the buffer-Q2 boundary. These changes are small (<0.5 kcal/mol) and are prominent for Models b and d in our test calculations. Intriguingly, the changes seem to be more noticeable for larger buffer zones. In Fig. 4, for Models b and d, the 2.0-Å buffer-zone energy curves are compared with those with smaller buffer zones of 3.0 Å ≤ R ≤ 4.5 Å and 3.0 Å ≤ R ≤ 4.0 Å. Although we have not yet fully understood the cause of these changes, we suspect that they are due to the presence of both the Q1 and Q2 basis sets (through interpolation) for the water molecule in the buffer zone, which together provide a very “large” basis set for the self-consistent field (SCF) calculations. However, when the buffer group moves near the buffer-Q2 boundary, GAMESS will discard some Q1 basis functions with very small contributions during the canonical orthogonalization, effectively leading to an increasingly smaller basis set in the SCF calculations. This process continues until the buffer group reaches the buffer-Q2 boundary, when all Q1 basis functions are gone and only the Q2 basis set is available. Such changes are more noticeable in our tests when a larger buffer zone is adopted, as the smoothing functions take more smaller values near the buffer-Q2 boundary. Similarly, the Q2 basis functions are gradually removed when the buffer group is approaching the Q1-buffer boundary; but because the basis functions in the larger Q1 basis set usually dominates over those in the smaller Q2 basis set, the effects due to such changes are negligible. Note that the energy plotted here is the interaction energy, not the absolute energy, and therefore in general these changes may be seen going down or up, or undetectable, depending on the model system (e.g., they are insignificant in other test models) and the employed levels of theory (e.g., they are overall less notable when using the 6–31G(d)/STO-3G combination, as revealed in Fig. S1 in the Supporting Information). Currently, we are exploring algorithms to eliminate/minimize these quick changes, such as projections between Q1 and Q2 basis sets.

Figure 4:

Comparisons of IAP potential for Models b (left panel) and d (right panel) with different buffer sizes: 3.0 Å ≤ R ≤ 4.0 Å (blue “+”), 3.0 Å ≤ R ≤ 4.5 Å (orange squares), and 3.0 Å ≤ R ≤ 5.0 Å (red circles). The calculations are performed at the B3LYP level with basis set combination MIDI/MINI.

5. Discussion

Although the energy curves display smooth transitions, one may ask if the alchemical QM process produces an utterly unphysical electronic structure. To address this question, we verify the integrity of the electronic structure in the IAP method. Two key features are examined, the canonical molecular orbitals (MO) and the Mulliken atomic charges.⁶⁹ We understand that the meanings of the MO in DFT calculations are debatable and that the Mulliken charge model can sometimes lead to unreasonable charges; nevertheless, when used with care, they can provide useful insights toward the validity of the electronic structure obtained by the IAP method against the Q1 and Q2 calculations. For example, the Mulliken charges are well-known to depend strongly on the basis set, making them sensitive tests on the alchemical mixing of basis sets in this study.

Taking Model a as an example, we show here the results computed by the buffer zone of 3.0 Å ≤ R ≤ 5.0 Å. First, we look at the orbital energies. Table 1 lists the energies of two MOs with prominent contributions from the atomic orbitals at the buffer water molecule W2 in this dimer model: One MO (MO_a) describes the σ bonds between the O and H atoms, and the other MO (MO_b) involves the lone pair at the O atom. It is apparent that, except for near the buffer-Q2 boundary (R ≥ 4.8 Å), these MOs are dominated by contributions of the Q1 basis functions in the interpolated basis set, as mentioned above. As W2 approaches the buffer-Q2 boundary, some Q1 basis functions are gradually removed, and the Q2 basis functions become increasingly important. Upon reaching the buffer-Q2 boundary, the Q1 basis functions are no longer available, and only Q2 basis functions contribute to the MO. The data for the 6–31G(d)/STO-3G basis set combination (Table S1 in Supporting Information) yielded the same qualitative picture, although the deviations away from Q1 towards Q2 are seen much earlier in the journey of W2 moving toward the Q2 zone.

Table 1.

Orbital energies (a.u.) of two selected molecular orbitals for Model a with buffer zone (3.0 Å ≤ R ≤ 5.0 Å) calculated at the B3LYP level with basis set combination MIDI/MINI.

	MOa			MOb
R (Å)	Q1	Q2	IAP	Q1	Q2	IAP
3.0	−0.525	−0.505	−0.525	−0.360	−0.322	−0.360
3.1	−0.524	−0.504	−0.524	−0.358	−0.321	−0.358
3.2	−0.523	−0.503	−0.523	−0.356	−0.320	−0.356
3.3	−0.522	−0.502	−0.521	−0.355	−0.319	−0.355
3.4	−0.521	−0.502	−0.520	−0.353	−0.318	−0.353
3.5	−0.519	−0.501	−0.519	−0.351	−0.317	−0.351
3.6	−0.518	−0.501	−0.518	−0.350	−0.316	−0.350
3.7	−0.517	−0.500	−0.517	−0.349	−0.316	−0.349
3.8	−0.516	−0.500	−0.516	−0.347	−0.315	−0.348
3.9	−0.515	−0.499	−0.515	−0.347	−0.315	−0.347
4.0	−0.514	−0.499	−0.514	−0.346	−0.314	−0.346
4.1	−0.514	−0.499	−0.514	−0.345	−0.314	−0.345
4.2	−0.513	−0.498	−0.513	−0.344	−0.314	−0.345
4.3	−0.513	−0.498	−0.513	−0.344	−0.313	−0.344
4.4	−0.513	−0.498	−0.512	−0.343	−0.313	−0.343
4.5	−0.512	−0.497	−0.512	−0.343	−0.313	−0.343
4.6	−0.512	−0.497	−0.512	−0.343	−0.313	−0.342
4.7	−0.511	−0.497	−0.511	−0.342	−0.312	−0.342
4.8	−0.511	−0.497	−0.508	−0.342	−0.312	−0.339
4.9	−0.511	−0.496	−0.496	−0.342	−0.311	−0.311
5.0	−0.511	−0.496	−0.496	−0.342	−0.311	−0.311

A similar story is told in Fig. 5 by the Mulliken charges of the atoms of the mobile water calculated at the Q1, Q2, and IAP levels of theory. The IAP charges of the O and H atoms (panels a and b) gradually deviate from the Q1 charges as the mobile water move into the buffer zone, reflecting the effects due to mixing with the Q2 charge density through the matrix interpolations, but they do not simply approach the Q2 charges monotonously. It should be emphasized that, however, all changes in these charges are rather small (< 0.03 e), and that overall, the charge values seem very reasonable, at least qualitatively. Moreover, the IAP total charge of the mobile water molecule is almost the same as the Q1 total charge (difference ~0.0001 e) along the way when the water molecule travels across the buffer zone (panel c). The results are qualitatively similar for the 6–31G(d)/STO-3G basis set combination (Fig. S2 in Supporting Information), although this time the atomic charges monotonously approach Q2. We note that the difference between the 6–31G(d) and STO-3G basis sets in the atomic Mulliken charges are 0.2 e-0.4 e. In contrast, this difference is <0.01 e for the MINI and MIDI basis sets. Again, the IAP total charge of W2 agree with Q1 very well (difference ≤ 0.004 e). The results imply that, although the Q1/Q2 mixing does perturb the electronic structure of the buffer groups, the impacts are relatively small and are likely acceptable in most applications.

Figure 5:

Mulliken charges of the (a) O and (b) H atoms of the mobile water W2 calculated for Model a. The charges of the two H atoms are essentially the same. Panel (c) plots the total charge of W2. The calculations are performed at the B3LYP level with basis set combination MIDI/MINI.

Although both PAP and IAP yield smooth potential energy surfaces, these two energy surfaces do not necessarily match each other, and neither is “more correct” in theory than the other, as they are just two different ways of potential interpolation. However, potentially, the IAP can be computationally much more efficient for large systems, although this depends on the actual setup of the calculations such as the number of buffer groups. In general, the computational costs of adaptive Q1/Q2 will be dominated by the number of high-level Q1 calculations and the number of atoms in these Q1 calculations. In PAP, a set of Q1 calculations are performed for the Q1 group and various buffer groups combined, and the number of such Q1 calculations scales factorially with the number of buffer groups, unless the many-body expansion of the potential is significantly truncated. In contrast, IAP requires only one large alchemical calculation for the Q1 and all buffer groups combined, which is modestly more expensive than the pure-Q1 calculations for the same system. For example, for three water molecules, when using the MIDI/MINI basis set combination, on average one SCF cycle in the alchemical calculations took 0.7 second, as opposed to those in the Q1 calculations (0.5 second) and Q2 calculations (0.3 second). For the 6–31G(d)/STO-3G basis set combination, where the difference in basis set sizes is more prominent, the average times of SCF/cycle were 1.1 second for the alchemical, 1.1 second for the Q1, and 0.5 second for the Q2 calculations, respectively. Admittedly, to get the zero of energy, IAP also requires alchemical calculations for each buffer group, but the number of such calculations scale linearly with the number of buffer groups. Moreover, these calculations on single buffer groups are inexpensive owing to the small size of individual buffer groups. For instance, on average the SCF/cycle times were about 0.07 second (for MIDI/MINI) and 0.14 second (for 6–31G(d)/STO-3G) for an alchemical water molecule.

Of course, the number of SCF cycles may vary between calculations, affecting the actual time needed for the calculations. This can be a concern for the unconventional alchemical calculations in IAP. For the model systems tested here, the SCF convergence turned out to be reasonable, and we have not encountered issues. To give readers a feeling on the timing, we benchmarked the total “wall o’clock times” for representative single-point PAP and IAP calculations on Model e with both W2 and W3 in the buffer zone: The times for PAP were 45 seconds for the MIDI/MINI and 76 seconds for the 6–31G(d)/STO-3G basis set combinations, respectively. The counterparts in IAP reduced to 34 and 48 seconds, respectively. The data demonstrate that IAP is computationally more efficient than PAP for this model, albeit the difference is less than a factor of 2. However, for model systems with larger numbers of Q1 and buffer groups, IAP can be more competitive.

Although only variational DFT methods with different-sized basis sets are tested in the present work, the IAP approach is general and can be expanded (such work is underway) to post-Hartree-Fock algorithms, such as IAP MP2⁷⁰/HF.⁷¹ For example, an IAP MP2 correction following the IAP SCF convergence could be implemented through the following tentative ansatz:

$$E_{\text{MP2 }}^{(2),\text{IAP }}=\frac{1}{4}{\sum }_{ij}^{\text{Occ}}{\sum }_{ab}^{\text{Vir}}\frac{\left[{(ia\mid jb)}^{*}-{(ib\mid ja)}^{*}\right]}{{ϵ}_i+{ϵ}_j-{ϵ}_b-{ϵ}_a}$$ (15)

$${(ia\mid jb)}^{*}={\sum }_{\mu \nu \sigma \lambda }^{\text{AO}}{P}_{\mu }{P}_{\nu }{P}_{\sigma }{P}_{\lambda }{C}_{\mu i}{C}_{vj}{C}_{\sigma a}{C}_{\lambda b}(\mu \nu \mid \sigma \lambda )$$ (16)

The equations take the form of the canonical MP2 expression, with modified two-electron integrals where weights are applied according to the groups that the atomic orbitals belong to, and the MO energies being those obtained in the IAP SCF calculations. Such a treatment is a natural extension of the variational IAP implementation and is also in line with the ansatz by Field.²²

6. Conclusions

In this contribution, we report the extension of the adaptive QM/MM algorithm to multilayer models. A key ingredient to this development is a new adaptive scheme that can produce smooth potential energy surfaces between two QM (Q1 and Q2) levels of theory. To this end, we formulate two new adaptive Q1/Q2 schemes. The first scheme, PAP Q1/Q2, is a straightforward extension of the PAP QM/MM potential based on smoothly interpolating the many-body contributions to the potential energy. The second and more radical method, IAP Q1/Q2 utilizes an unconventional alchemical QM approach, where the basis sets, Fock matrices, and overlap matrices of Q1 and Q2 are interpolated. Alchemical treatments are routinely used in MM simulations, but it is, to our knowledge, the first time that they are applied to QM calculations. Both approaches take existing partitioning schemes^{3, 6, 22} previously developed for QM/MM methods and adapt them to an entirely quantum system. Test calculations on water-cluster models show overall smooth transitions in the energy curves by both PAP and IAP treatments when water molecules move across the buffer zones. Although the IAP energy curves do not seem to be always perfectly smooth near the Q2-buffer boundary (Fig. 4) because GAMESS removes basis functions with very small contributions during the canonical orthogonalization, the observed changes between the adjacent data points of concern are rather small (<0.5 kcal/mol), and the smoothness may be improved in the future, e.g., by projections between the Q1 and Q2 basis sets and/or by using different smoothing techniques. The alchemical QM calculations maintain a reasonably high fidelity in the descriptions of the electronic structure of the buffer groups, which appears to largely resemble the Q1 calculations. Currently, implementations and test calculations are underway for three-layer Q1/Q2/MM treatments, which integrate the Q1/Q2 description with an additional MM layer to tackle much larger systems. In short, the two schemes presented here are simple but nevertheless offer a promising starting point for future development.