Method and Implementation of Projected Hybrid Orbitals for Treating Multiple Covalent Bonds in Combined QM/MM Calculations

Abstract

The projected hybrid orbital (PHO) method for treatment of multiple boundary atoms introduces a novel solution for handling the covalent connection between quantum mechanical (QM) and molecular mechanical (MM) regions in QM/MM calculations. By projecting the QM basis, typically adequately large for computational accuracy, onto a secondary minimal basis set on the boundary atom, it preserves electronic interactions between the two regions without system-specific parameters. Applicable in both ab initio wave function theory and density functional theory, PHO has been shown to maintain structural and electronic integrity across various systems. It offers key advantages over traditional QM/MM methods, such as avoiding modification of MM charge distribution and energy corrections, while being flexible for biochemical systems and potentially broader QM/QM embedding frameworks.

1. INTRODUCTION

Combined quantum mechanical and molecular mechanical (QM/MM) approaches are widely used for studying chemical reactions in solutions and enzymes.^1–14 By combining a quantum mechanical treatment of the reactive core with molecular mechanics for the surrounding environment, QM/MM methods offer a powerful balance between computational efficiency and accuracy.¹⁵ However, one persistent challenge lies in properly defining the boundary between the QM and MM regions, particularly when covalent bonds span this transition.^16–24 In this work, we extend the projected hybrid orbital (PHO) technique to handle multiple covalent bonds across the QM/MM boundary.²⁴ This approach is system-agnostic and adaptable to any basis set in ab initio wave function theory or density functional theory (DFT), eliminating the need for system-specific parametrization. The PHO method addresses several shortcomings of traditional boundary treatments, providing a more universal and effective solution.

Historically, numerous strategies have been devised to manage the QM/MM boundary, which can be broadly categorized into two classes that either alter or preserve the degrees of freedom at the boundary.^25,26 In the first category is a common approach, known as link-atom methods,^27,28 which introduces an auxiliary atom (typically hydrogen) or an effective bond to cap the QM region at the boundary.^29–43 While straightforward, this method can disturb the local electrostatic environment and introduce artifacts, such as rebalanced atomic charges of the local MM group and additional degrees of freedom that do not naturally belong to the system.¹⁶ In contrast, more sophisticated techniques in the second category preserve the system’s integrity by situating the QM/MM boundary at an atomic site,^{22,24,44–58} allowing the boundary atom to interact with both regions without compromising electrostatic balance or introducing unnatural distortions. Recent developments saw embedding approaches using density matrix to optimize link orbitals,^59–61 though a drawback is that the link orbitals are constructed from a QM treatment of the full molecule. Other methods include projection-based embedding^62–64 and frozen density embedding.^65–68

The key innovation of the present multiple boundary PHO method lies in its ability to project the core and valence electrons of the boundary atoms in the QM region onto a secondary minimal basis set. The latter is then transformed into hybrid orbitals.^22,49 These secondary orbitals are optimized within the self-consistent field (SCF) procedure, ensuring a smooth and balanced transition between the QM and MM regions.²⁴ Unlike previous methods that require charge redistribution or the parametrization of atomic integrals, the PHO technique retains the essential characteristics of the basis set used to represent the QM region by the secondary orbitals in a least-squares sense. Thus, the projection method ensures a consistent treatment of the boundary across diverse systems.

For any QM/MM boundary method to be effective, it also need to meet certain key criteria that guarantee a seamless transition between the QM and MM regions:^24,49

1. Electronegativity Continuity: The electronegativity of the boundary atom must closely match that of the QM region. This continuity prevents artificial changes in reactivity or charge distribution, ensuring smooth electron density transitions.

2. Structural Integrity: The method must maintain the overall structure of the molecular system, avoiding artificial constraints or alterations to its degrees of freedom. Electrostatic interactions between the QM and MM regions should be preserved to reflect the true nature of the system.

3. Consistency in Molecular Properties: The hybrid method should yield molecular geometries and energies that are in agreement with fully quantum or fully classical treatments of the system. This consistency ensures that the QM/MM approach accurately models the system’s behavior without introducing significant errors.

In this article, we describe an extension of the projected hybrid orbital approach to treat multiple covalent boundaries in combined QM/MM calculations. Furthermore, we introduce a procedure to accelerate the convergence of self-consistent-field optimization of QM/MM energies and the evaluation of analytical gradients. This paper is organized as follows. First, we present a detailed theoretical and technical description of the multiple boundary PHO method. This is followed by a series of examples, designed to examine the effects of boundary atoms on geometry optimization, electrostatic effects, placement of boundary atoms and stability of molecular dynamics. Finally, the paper is concluded with a summary of future perspectives.

2. THEORY

2.1. Hybrid Orbitals.

A conceptually intuitive approach to saturate the valency of the “QM” region at a covalent connection across the QM and MM boundary is to introduce a local orbital on each atom directly bonded to the “MM” region.^44–49 Then, the boundary atom is used both in QM calculations and in MM force-field terms. In this study, we focus on the selection of sp³ carbon atoms as the boundary atoms, although generalization to other atom types and hybridization forms such as an sp³ oxygen or a sp² carbon atom is straightforward.

Depicted in Figure 1 are two sp³-carbon boundary atoms that separate the full system into a molecular system treated by a QM model and two MM fragments. Each boundary sp³ carbon (${\mathrm{C}}_B$), where $B=1,\dots ,n_B$ and $n_B$ is the number of boundary atoms, has three bonding terms to the MM region ($M_{B2},M_{B3}$, and $M_{B4}$) plus one covalent bond pointing toward QM atoms $C_Q$. The use of a valence sp³ hybridization implies that the boundary atom ${\mathrm{C}}_B$ also possesses a 1s core orbital, denoted as ${\eta }_0^B$. Of the four valence hybrid orbitals, three, ${\eta }_2^B,{\eta }_3^B$, and ${\eta }_4^B$, are constructed roughly in the direction of the three connecting MM atoms and one $\left({\eta }_1^B\right)$ in the direction of $C_Q$. The essential idea of the present multiple boundary PHO approach is that the core orbital ${\eta }_0^B$ and the hybrid orbitals $\left\{{\eta }_1^B\right.;\left.B=1,\dots ,n_B\right\}$ on the boundary atoms ${\mathrm{C}}_B$, together with all atomic basis orbitals in the QM region, are relaxed and fully optimized in the self-consistent-field (SCF) procedure. These orbitals are called active orbitals. The remaining three hybrid orbitals, ${\eta }_{\lambda }^B(\lambda =2,3,4)$ on each boundary atom ${\mathrm{C}}_B$ are excluded from the SCF optimization, but their electronic integrals are included in forming the Fock matrix and their hybridation components (i.e., the mixing of 2s and 2p orbitals) are dynamically varied according to the local geometry of the boundary atoms.^33,49 These orbitals are called auxiliary orbitals.

Figure 1.

Illustration depicting two boundary atoms (${\mathrm{C}}_b$) represented by projected hybrid orbitals (PHO). Orbitals in brown are actively included in the self-consistent-filed (SCF) optimization of the “QM” subsystem in circle, and orbitals in blue are auxiliary orbitals that vary following geometry changes but are not optimized in the SCF procedure.

A natural choice for mapping the local atomic hybrid orbitals is to use an auxiliary, minimum basis set. We have adopted the standard Pople style Gaussian orbitals,⁶⁹ i.e., $\mathrm{S}\mathrm{T}\mathrm{O}–n\mathrm{G}$, such as STO-3G. The minimal basis orbitals are transformed into 5 hybrid orbitals according to the geometry of the boundary atom and the four atoms directly bonded it.²⁴ Its definition and procedure have been detailed in the generalized hybrid orbital approach.^33,49 In short, the relationship between the hybrid orbitals and the auxiliary basis orbitals, denoted as {1s, 2s, $2{\mathrm{p}}_x,2{\mathrm{p}}_y,2{\mathrm{p}}_z$} for a single boundary atom ${\mathrm{C}}_1$, is given below.

$$\left(\begin{array}{l}{\eta }_0^1\\ {\eta }_1^1\\ {\eta }_2^1\\ {\eta }_3^1\\ {\eta }_4^1\end{array}\right)={\mathbf{T}}_1^{†}\left(\begin{array}{l}1\mathrm{s}\\ 2\mathrm{s}\\ 2{\mathrm{p}}_x\\ 2{\mathrm{p}}_y\\ 2{\mathrm{p}}_z\end{array}\right)$$ (1)

where ${\mathbf{T}}_1^{†}$ is the transformation coefficients determined by the coordinates of ${\mathrm{C}}_1$ and its bonding atoms.⁴⁹ Orthogonality among the hybrid orbitals ${\eta }_1^1,{\eta }_2^1,{\eta }_3^1$ and ${\eta }_4^1$ is automatically enforced in the transformation.

It follows that the hybridization transformation matrix for a group of $n_B$ boundary atoms from the secondary auxiliary basis, which is of dimension $5n_B$, is given as

$${\mathbf{T}}_{\mathrm{B}}=\left(\begin{array}{lll}{\mathbf{T}}_1& \cdots & \mathbf{0}\\ ⋮& \ddots & ⋮\\ \mathbf{0}& \cdots & {\mathbf{T}}_{n_B}\end{array}\right)$$ (2)

By including the primary basis functions in the QM region, the total transformation matrix of dimension $N_{\text{mix}}$ is

$${\mathbf{T}}_{\mathrm{H}}=\left(\begin{array}{ll}\mathbf{I}& \mathbf{0}\\ \mathbf{0}& {\mathbf{T}}_{\mathrm{B}}\end{array}\right)$$ (3)

where $\mathbf{I}$ is a unit matrix having a dimension equal to the number of basis orbitals, $N_{\text{tot}}$, of the QM fragment.

2.2. Basis Set Projection.

As can be seen, there are two basis sets involved in a PHO calculation: (a) the primary basis set both on atoms in the QM region ($N$) and on the boundary atoms ($n_B{N}_B$) of a total size $N_{\text{tot}}=N+n_B{N}_B$, and (b) a set of secondary, mixed atomic and hybrid basis orbitals ($N_{\text{hyb}}=5n_B$), together at the dimension of $N_{\text{mix}}=N+N_{\text{hyb}}$. The main objective of the PHO method is to construct a set of valence hybrid orbitals along with the 1s core orbital on each boundary atom from the secondary basis orbitals to have the maximum likelihood with respect to the primary basis set.²⁴ Thus, these hybrid orbitals would preserve the properties of the general (primary) basis set to the greatest extent without introducing additional parameters. This can be accomplished by projection of the primary basis on to the secondary basis.

Let $\left\{{\chi }_{\mu };\mu =1,\dots ,N_{\text{tot}}\right\}$ be the primary basis set, typically an adequately high-quality basis set (e.g., aug-cc-pVTZ) depending on the specific application, and $\left\{{\eta }_{B\lambda };B=1,\dots ,n_B;\lambda =0,\dots \right.,4\}$ be the secondary hybrid orbitals, constructed from a minimal basis set. In the present study, the STO-3G basis is used in the test cases to form the hybrid orbitals (though any $\mathrm{S}\mathrm{T}\mathrm{O}–n\mathrm{G}$ can be used). We denote the square overlap matrix between the primary basis orbitals on boundary atoms $A$ and $B$ as ${\mathbf{S}}^{AB}$ whose matrix element is given by

$${\mathbf{S}}_{\mu \nu }^{AB}=⟨{\chi }_{\mu }^A\mid {\chi }_{\nu }^B⟩$$ (4)

The rectangular overlap matrix ${\mathbf{R}}^{AB}$ between the primary and secondary basis orbitals and boundary atoms $A$ and $B$ is defined as

$${\mathbf{R}}_{\mu \lambda }^{AB}=⟨{\chi }_{\mu }^A\mid {\eta }_{\lambda }^B⟩$$ (5)

Then, for a single boundary atom ($A=B$), the projection matrix from the primary basis functions $\left\{{\chi }_{\mu }^1\right\}$ to its secondary basis orbitals $\left\{{\eta }_{\lambda }^1\right\}$ is

$${\mathbf{P}}_B={\left({\mathbf{S}}^{11}\right)}^{–1}{\mathbf{R}}^{11}$$ (6)

This projection leads approximately to the condition

$${[{\mathbf{P}}_B^{†}{\mathbf{s}}^{11}{\mathbf{P}}_B]}_{\lambda \sigma }\approx ⟨{\eta }_{\lambda }^1\mid {\eta }_{\sigma }^1⟩$$ (7)

In general, for $n_B$ boundary atoms, the projection matrix from the primary basis set $\left\{{\chi }_{\mu }^B\right\}$ to the secondary basis $\left\{{\eta }_{\lambda }^B\right\}$ on these atoms is then given as follows.

$${\mathbf{P}}_B={\mathbf{S}}_B^{–1}{\mathbf{R}}_B$$ (8)

where

$${\mathbf{S}}_B=\left(\begin{array}{lll}{\mathbf{S}}^{11}& \dots & {\mathbf{S}}^{1n_B}\\ ⋮& \ddots & ⋮\\ {\mathbf{S}}^{n_{B}1}& \dots & {\mathbf{S}}^{n_B{n}_B}\end{array}\right);{\mathbf{R}}_B=\left(\begin{array}{lll}{\mathbf{R}}^{11}& \cdots & {\mathbf{R}}^{1n_B}\\ ⋮& \ddots & ⋮\\ {\mathbf{R}}^{n_{B}1}& \cdots & {\mathbf{R}}^{n_B{n}_B}\end{array}\right)$$ (9)

Recall that ${\mathbf{S}}^{AB}$ and ${\mathbf{R}}^{AB}$ are the overlap matrices, respectively, between the primary basis orbitals and between the primary and secondary orbitals residing on atoms $A$ and $B$.

A small complication of the projection procedure defined in eq 8 is that the boundary orbitals on different atoms are nonorthogonal. However, at this stage, these hybrid orbital overlaps can be neglected since the boundary atoms $\left\{{\mathrm{C}}_B\right\}$ should be far away from each other. Otherwise, if different boundary atoms have strong interactions with nonnegligible overlap, such a QM/MM calculation would not have been ideal and it is advised that a different choice of the QM region be considered. In this regard, the projection matrix for multiple boundary projected orbitals for practical purposes can be approximated in the block-diagonal form

$${\mathbf{P}}_B\approx \left(\begin{array}{lll}{\mathbf{P}}_1& \cdots & \mathbf{0}\\ ⋮& \ddots & ⋮\\ \mathbf{0}& \cdots & {\mathbf{P}}_{n_B}\end{array}\right)$$ (10)

Finally, the projection matrix from the full $N_{\text{tot}}$ primary basis functions to the mixed $N_{\text{mix}}$ basis orbitals, including both atoms in the QM region and the boundary atoms, is

$${\mathbf{T}}_{\mathrm{P}}=\left(\begin{array}{ll}\mathbf{I}& \mathbf{0}\\ \mathbf{0}& {\mathbf{P}}_B\end{array}\right)$$ (11)

2.3. Orthogonalization.

With the hybridization transformation ${\mathbf{T}}_{\mathrm{H}}$ and orbital projection ${\mathbf{T}}_{\mathrm{P}}$, we combine both transformations into the mixed hybrid orbital (HO) space (of dimension $N+5n_B$) for QM/MM electronic structure calculations.

First, we express the overlap matrix of the mixed orbitals as follows

$$\mathbf{Q}={\left({\mathbf{T}}_{\mathrm{p}}{\mathbf{T}}_{\mathrm{H}}\right)}^{†}\mathbf{S}\left({\mathbf{T}}_{\mathrm{p}}{\mathbf{T}}_{\mathrm{H}}\right)$$ (12)

Second, we project out the three auxiliary hybrid orbitals on each boundary atom, pointing toward the three MM atoms, since their charge densities are not further optimized.

$$\hat{P}=1–\sum _{i=1}^{n_{\mathrm{B}}}\left(\left|{\eta }_2^i\right.⟩\left.⟨{\eta }_2^i\right|+\left|{\eta }_3^i\right.⟩\left.⟨{\eta }_3^i\right|+\left|{\eta }_4^i\right.⟩\left.⟨{\eta }_4^i\right|\right)$$ (13)

where ${\eta }_2^i,{\eta }_3^i$, and ${\eta }_4^i$ are the auxiliary (inactive) hybrid orbitals of boundary atom $i$. Specifically, for a basis function ${\chi }_{\mu }^{\mathrm{H}\mathrm{O}}$ in the HO space, the net projection result is

$$\hat{P}\left|{\chi }_{\mu }^{\mathrm{H}\mathrm{O}}\right.⟩=c_{\mu }\left(1–\sum _{i=1}^{n_B}\left(Q_{\mu 2}^i\left|{\eta }_2^i\right.⟩+Q_{\mu 3}^i\left|{\eta }_3^i\right.⟩+Q_{\mu 4}^i\left|{\eta }_4^i\right.⟩\right)\right)$$ (14)

where the coefficient $c_{\mu }$ is introduced as a normalization constant. Further, eq 14 is derived with the assumption that different boundary atoms are far away such that their overlap integral is negligible (eq 10). Thus, as noted in Subsection 2.1, each boundary atom in the PHO method is recognized both as QM and MM in character.^24,49,50 In all, this leaves a total of $N+2n_B$ basis functions, actively involved in the SCF procedure.

Finally, the transformation matrix that orthogonalizes all orbitals in the hybrid space is given below.

$${\mathbf{T}}_{\mathrm{O}}=\left(\begin{array}{lllllllllllll}{c}_1& \cdots & 0& 0& 0& 0& 0& 0& \cdots & 0& 0& 0& 0\\ ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \cdots & 0& 0& 0& 0\\ 0& \cdots & c_N& 0& 0& 0& 0& 0& \cdots & 0& 0& 0& 0\\ 0& \cdots & 0& 1& 0& 0& 0& 0& \cdots & 0& 0& 0& 0\\ 0& \cdots & 0& 0& 1& 0& 0& 0& \cdots & 0& 0& 0& 0\\ –c_1{Q}_{12}^1& \cdots & –c_N{Q}_{N2}^1& 0& 0& 1& 0& 0& \cdots & 0& 0& 0& 0\\ –c_1{Q}_{13}^1& \cdots & –c_N{Q}_{N3}^1& 0& 0& 0& 1& 0& \cdots & 0& 0& 0& 0\\ –c_1{Q}_{14}^1& \cdots & –c_N{Q}_{N4}^1& 0& 0& 0& 0& 1& \cdots & 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0& 0& 0& 0& 0& \cdots & 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \cdots & 0& 1& 0& 0\\ –c_1{Q}_{12}^{n_B}& ⋮& –c_N{Q}_{N2}^{n_B}& 0& 0& 0& 0& 0& \cdots & 0& 0& 1& 0\\ –c_1{Q}_{13}^{n_B}& ⋮& –c_N{Q}_{N3}^{n_B}& 0& 0& 0& 0& 0& \cdots & 0& 0& 0& 1\\ –c_1{Q}_{14}^{n_B}& ⋮& –c_N{S}_{N4}^{n_B}& 0& 0& 0& 0& 0& \cdots & 0& 0& 0& 0\end{array}\right)$$ (15)

2.4. PHO-SCF Energy.

The QM/MM energy calculation is performed using the mixed atomic and hybrid orbitals. The latter is obtained by (a) projecting the primary basis set (typically high-quality) to the secondary minimal basis orbitals, (b) forming a set of hybrid orbitals using the minimal basis, and (c) orthogonalizing the mixed atomic and hybrid orbitals. This process, from the primary basis set to PHO space, is given by

$${\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}={\mathbf{T}}_{\mathrm{P}}{\mathbf{T}}_{\mathrm{H}}{\mathbf{T}}_{\mathrm{O}}$$ (16)

SCF optimization of the PHO method is carried out using the mixed atomic and hybrid orbitals, excluding three auxiliary orbitals on each boundary atom, but all electronic integrals and Fock matrix formation are done using the standard, primary atomic orbitals. This minimizes the need to make major changes in the standard SCF procedure. Consequently, matrix transformations between the mixed atomic and hybrid orbital basis and the primary atomic basis are required as detailed in the following steps:

1. We first construct the Fock F and overlap matrix $\mathbf{S}$ in the primary basis set as done in standard electronic structure calculations, and then, transform $\mathbf{F}$ and $\mathbf{S}$ (dimension: $N_{\text{tot}}$) into the PHO basis (dimension: $N_{\text{mix}}=N+5n_B$)

   $${\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}={\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}{\mathbf{F}\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}$$ (17)

   $${\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}={\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}{\mathbf{S}\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}$$ (18)

   Here, the mixed hybrid basis in the PHO method and its dimension have been indicated as subscripts and superscripts.

2. Next, the columns and rows corresponding to the auxiliary orbitals in ${\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}$ and ${\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}$ are discarded to obtain ${\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$ and ${\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$. The auxiliary orbitals do not participate in bonding explicitly in QM calculations since they are the orbitals that would otherwise form chemical bonds with atoms in the MM region. Furthermore, they are orthogonal to all other orbitals in the mixed basis representation, and thus, can be removed from the Fock matrix diagonalization, i.e., without being explicitly optimized. The Roothaan equation is solved to yield a set of orbital coefficients in the $N+2n_B$ mixed orbital basis

   $${\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}{\mathbf{C}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}={\mathit{ϵ}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}{\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}{\mathbf{C}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$$ (19)
3. The density matrix ${\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$ is computed using the orbital coefficients obtained at the iteration above ${\mathbf{C}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$. Before we construct the Fock matrix for the next SCF iteration, ${\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$ is expanded to the full mixed orbital dimension ($N_{\text{mix}}$), which is accomplished by adding the charge densities of the boundary atoms, consisting of both the single electron in the auxiliary orbital and the partial charges on the boundary atom—a reflection of the bond polarization in the MM force field.

   $$\begin{gathered} {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}= \\ \left(\begin{array}{llllllll}{\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \cdots & \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& 1–\frac{q^1}{3}& 0& 0& \cdots & 0& 0& 0\\ \mathbf{0}& 0& 1–\frac{q^1}{3}& 0& \cdots & 0& 0& 0\\ \mathbf{0}& 0& 0& 1–\frac{q^1}{3}& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ \mathbf{0}& 0& 0& 0& \cdots & 1–\frac{q^{n_B}}{3}& 0& 0\\ \mathbf{0}& 0& 0& 0& \cdots & 0& 1–\frac{q^{n_B}}{3}& 0\\ \mathbf{0}& 0& 0& 0& \cdots & 0& 0& 1–\frac{q^{n_B}}{3}\end{array}\right) \end{gathered}$$ (20)

   where $q^i\left(i=1,\dots ,n_B\right)$ are the partial atomic charges on corresponding the boundary atoms of the MM force field. The denominator (3) assumes an equal partition of the partial charge to the three auxiliary bonds on each boundary atom.
4. We transform the density matrix from the mixed hybrid orbital basis back to the primary atomic orbital basis set

   $$\mathbf{P}={\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}{\mathbf{P}}^{\mathrm{P}\mathrm{H}\mathrm{O},N+5n_{\mathrm{B}}}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}$$ (21)

   Then, the total electronic energy is determined using the primary atomic orbital basis. If the convergence criteria on density and energy are met, the SCF procedure is accomplished. Otherwise, we go back to step 1 for the next iteration.

The total $\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}$ energy consists of both the electronic energy and standard force field terms,^5,6,15,24 which is given by

$$\begin{gathered} E=E_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}+\sum _{A=1,A<B}^{n_{\mathrm{Q}}}\frac{Z_A{Z}_B}{r_{AB}}+\sum _{A=1}^{n_{\mathrm{Q}}}\sum _{M=1}^{n_{\mathrm{M}}}\frac{Z_A{q}_M}{r_{AM}}+E_{\mathrm{M}\mathrm{M}} \\ +E_{\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}} \end{gathered}$$ (22)

where $E_{\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}}$ is the van der Waals interaction energy between $\mathrm{Q}\mathrm{M}$ and $\mathrm{M}\mathrm{M}$ atoms and the electronic energy contains also $\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}$ electrostatic interaction terms

$$\begin{gathered} E_{\text{elec}}=\sum _{\mu \nu }{P}_{\mu \nu }(H_{\mu \nu }^{\mathrm{Q}\mathrm{M}}+H_{\mu \nu }^{\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}}) \\ +\frac{1}{2}\sum _{\mu \nu }\sum _{\lambda \sigma }{P}_{\mu \nu }{P}_{\lambda \sigma }\left((\mu \nu \mid \lambda \sigma )–\frac{1}{2}(\mu \sigma \mid \lambda \nu )\right) \end{gathered}$$ (23)

2.5. Acceleration of SCF Convergence.

A second complication of the PHO method is in SCF convergence using standard method in the presence of multiple orbital transformations. Unfortunately, the commutator direct inverse of iterative space (CDIIS) of Pulay⁷⁰ cannot be directly applied since the commutator is no longer the orbital rotation gradient. To this end, we employed an energy-based DIIS (EDIIS) procedure,⁷¹ which is successful. In particular, we solve the following optimization problem using the Fock and density matrices in the PHO space, making use of information from the last $k$ iterations.

$$ϵ=\mathrm{i}\mathrm{n}\mathrm{f}\{F(\mathbf{c}),c_I\ge 0,\sum _{I=1}^k{c}_I=1\}$$ (24)

where

$$\begin{gathered} F(\mathbf{c})=\sum _{I=1}^k{c}_I{E}_{\text{elec}}^I–\frac{1}{2}\sum _{I=1}^k\sum _{J=1}^k{c}_I{c}_J \\ \mathrm{t}\mathrm{r}([{\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B,(I)}–{\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B,(J)}][{\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B,\left(I\right)} \\ –{\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B,\left(J\right)}]) \end{gathered}$$ (25)

The new Fock matrix at the $k+1$ iteration is constructed as $\sum _{I=1}^k{c}_i{\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B,\left(I\right)}$ for the next SCF cycle.

The convexity constraint $c_I\ge 0$ makes direct optimization of $F\left(\mathbf{c}\right)$ rather difficult. Inspired by ref 72, we adopted a variable substitution $c_I=t_I^2/\sum _{J=1}^k{t}_J^2$ to meet both conditions of $c_I\ge 0$ and $\sum _{I=1}^k{c}_I=1$.⁷² This becomes an unconstrained optimization of $F\left(\mathbf{t}\right)$ that can be solved in standard ways.

2.6. Analytic Gradient.

For energy gradients, we focus on the electronic term $E_{\text{elec}}$ since all other terms in eq 23 are trivial.³³

$$\begin{gathered} \frac{\partial E_{\text{elec}}}{\partial X}=\sum _{\mu \nu }{P}_{\mu \nu } \\ \frac{\partial (H_{\mu \nu }^{\mathrm{Q}\mathrm{M}}+H_{\mu \nu }^{\mathrm{Q}\mathrm{M}/\mathrm{M}\mathrm{M}})}{\partial X}+ \\ \frac{1}{2}\sum _{\mu \nu }\sum _{\lambda \sigma }{P}_{\mu \nu }{P}_{\lambda \sigma }\frac{\partial }{\partial X}\left((\mu \nu \mid \lambda \sigma )–\frac{1}{2}(\mu \sigma \mid \lambda \nu )\right)+\sum _{\mu \nu }\frac{\partial P_{\mu \nu }}{\partial X}{F}_{\mu \nu } \end{gathered}$$ (26)

The only additional terms to be considered are the derivatives with respect to the orbital projection and hybridization transformation, which come in from the gradients of the density matrix

$$\begin{gathered} \frac{\partial \mathbf{P}}{\partial X}=\frac{\partial {\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}}{\partial X}{\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}+{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}\frac{\partial {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}}{\partial \mathrm{X}}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†} \\ +{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}{\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}\frac{\partial {\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}}{\partial X} \end{gathered}$$ (27)

Note that the derivatives of ${\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}$ elements corresponding to auxiliary orbitals are always 0, thus we just need to calculate the gradient of ${\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$. The contribution from the second term of eq 27 to $\partial E_{\text{elec}}/\partial X$ is

$$\begin{gathered} \mathrm{t}\mathrm{r}\left({\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}\frac{\partial {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}}{\partial X}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}\mathbf{F}\right)=\mathrm{t}\mathrm{r}\left(\frac{\partial {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}}{\partial X}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}{\mathbf{F}\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}\right) \\ =\mathrm{t}\mathrm{r}\left(\frac{\partial {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}}{\partial X}{\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}\right)=\mathrm{t}\mathrm{r}\left(\frac{\partial {\mathbf{P}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}}{\partial X}{\mathbf{F}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}\right) \\ =–\mathrm{t}\mathrm{r}\left(\frac{\partial {\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}}{\partial X}{\mathbf{W}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}\right) \end{gathered}$$ (28)

where ${\mathbf{W}}_{\text{PHO}}^{N+2n_B}$ is the energy-weighted density matrix. The derivative of ${\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+2n_B}$ can be calculated by picking up elements from

$$\begin{gathered} \frac{\partial {\mathbf{S}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{N+5n_B}}{\partial X}=\frac{\partial {\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}}{\partial X}{\mathbf{S}\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}+{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}\frac{\partial \mathbf{S}}{\partial \mathrm{X}}{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}+{\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}^{†}\mathbf{S} \\ \frac{\partial {\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}}{\partial X} \end{gathered}$$ (29)

The last remaining term is

$$\frac{\partial {\mathbf{T}}_{\mathrm{P}\mathrm{H}\mathrm{O}}}{\partial X}=\frac{\partial {\mathbf{T}}_{\mathrm{P}}}{\partial X}{\mathbf{T}}_{\mathrm{H}}{\mathbf{T}}_{\mathrm{O}}+{\mathbf{T}}_{\mathrm{P}}\frac{\partial {\mathbf{T}}_{\mathrm{H}}}{\partial X}{\mathbf{T}}_{\mathrm{O}}+{\mathbf{T}}_{\mathrm{P}}{\mathbf{T}}_{\mathrm{H}}\frac{\partial {\mathbf{T}}_{\mathrm{O}}}{\partial X}$$ (30)

Now, $\partial {\mathrm{T}}_{\mathrm{p}}/\partial X=\mathbf{0}$ since all integrals are single-centered; $\partial {\mathrm{T}}_{\mathrm{P}}/\partial X$ has been given in ref $33;\partial {\mathrm{T}}_{\mathrm{O}}/\partial X$ can be calculated straightforwardly when the above quantities are available.

2.7. Tetrahedral Restraint.

For each boundary atom ${\mathrm{C}}_B(B=1,\dots ,n_B$) in the PHO method, only two orbitals (the 1s core and the hybrid orbital in the direction of the QM connecting atom $C_Q$) are included in the SCF optimization. The three auxiliary hybrid orbitals (${\eta }_{\lambda }^B,\lambda =\mathrm{2,3},4$), although used in constructing the Fock matrix, do not participate in forming chemical bonds explicitly in QM calculations (they are treated as MM terms). Each of these auxiliary orbitals carries an electron density equivalent to 1 electron plus one-third of the partial charge of the boundary atom. The lack of full delocalization with the QM part of the system and the partial representation of the auxiliary bonds (half-bond equivalence with just 1 electron density) in the MM region affects electronpair interactions. The consequence is a slightly longer bond distance to the boundary atom (from the QM atom) and larger bond angles about each boundary atom.^24,49 One possible way of alleviating this problem is to use the restrained hybridization (RH) approach introduced by Jung and Ten-no for partition of the hybrid orbitals according to a perfect tetrahedral geometry,⁵³ or to use different MM charge distributions by Eckard and Exner.⁵⁴ Here, to compensate this artifact (unavoidable in any QM and MM partition from a full QM system), we introduce empirical terms to preserve the tetrahedral geometry of the boundary atom

$$\begin{gathered} E_{\mathrm{B}}=\sum _{i=1}^{n_B}[K_{\mathrm{B}}{\left\{r^B\left({\mathrm{Q}}_i{\mathrm{C}}_i\right)–r^0\left({\mathrm{Q}}_i{\mathrm{C}}_i\right)\right\}}^2 \\ +\sum _{k=1}^3{K}_{\mathrm{A}}{\left\{{\theta }_k^B\left({\mathrm{M}\mathrm{C}}_i\mathrm{M}\right)–{\theta }^0\right\}}^2] \end{gathered}$$ (31)

where $r^0\left({\mathrm{Q}}_i{\mathrm{C}}_i\right)$ and ${\theta }^0$ are, respectively, the equilibrium bond distance, from the corresponding force field terms, between boundary atom ${\mathrm{C}}_i$ and its QM connecting atom, and the standard tetrahedral angle (109.47°) of the three chemical bonds in the MM region about the boundary atom ${\mathrm{C}}_i$. The force constants $K_{\mathrm{B}}$ and $K_{\mathrm{A}}$ are assigned empirical values of 500 kcal/mol/Ų and 300 kcal/mol/rad², respectively.

3. COMPUTATIONAL DETAILS

We have implemented the projected hybrid orbital (PHO) method for multiple boundary atoms in combined QM/MM calculations into the program QBICS (https://qbics.info). QBICS is a general purpose simulation program using potential energy functions based on electronic structure theory and multistate density functional theory, force fields and combined QM/MM methods. Single-point energy calculation, geometry optimization and MD simulation can be performed using standard techniques and multistate density functional theory. In particular, a combined QM/MM potential can be constructed with wave function and density functional theory for the QM representation along with a molecular mechanical force field^73,74 for which the CHARMM36 force field is used in the present study.⁷⁵ To illustrate the capabilities of the PHO method as implemented in Qbics, we use Hartree–Fock theory and Kohn–Sham density functional theory along with different basis functions as indicated in specific examples.

4. RESULTS AND DISCUSSION

The performance of the PHO method are investigated. We first examine optimized geometries and MD simulations of ethane performed with bond and tetrahedral restraints, with only tetrahedral terms and without any restraints on boundary carbon atoms. The results reveal directly the reason why a tetrahedral restraining term is added. The performance of the multiple PHO boundary method is further tested on the first and second deprotonation energies of dicarboxylic acid with respect to the placement of multiple boundaries. Furthermore, we investigate the performance on proton transfer energies between the zwitterionic and neutral forms of amino acids and the binding energies between the thiocyanate group of a tripeptide and a water molecule.

4.1. Ethane.

Ethane is perhaps one of the simplest model for a combined QM/MM treatment. Although not of practical interest, it in fact is an ideal system for testing methods that treat QM/MM covalent bond partition.^22,49 Here, one methyl group is assigned to the QM region, the carbon atom of the second methyl group is chosen as the boundary atom, and the remaining three hydrogen atoms are grouped in the MM region. This partition has been adopted in testing the original hybrid orbital methods, including the generalized hybrid orbital (GHO) developed for semiempirical, HF and DFT QM models, and an initial description of the projected hybrid orbital (PHO) approach.^{22,24,49,50,52}

Several changes have been made in the present version of the PHO method in comparison with that introduced in ref 24, including the key definition of the projection operator (eq 8) and introduction of a bond restraining term (the first term in eq 31) to improve the accuracy for the bond distance between the boundary atom and its QM connection. In this work, we recall this example and test it in terms of geometry optimization and stability in MD simulations. We use both Hartree–Fock (HF) theory and KS-DFT with the B3LYP functional in combination with 4 basis sets, including 6–31G(d), 6–311+G(d,p), def2-SVP and def2-TZVP to illustrate the performance of the PHO method.

We begin with an evaluation of the effects of the boundary atom on electronic structure calculations. The “NR” column in Table 1 shows results of a straightforward application of the PHO method without employing restraining terms (eq 31). This highlights the consequence of orbital projection from the larger, primary basis set to the minimal basis orbitals and the use of a hybrid covalent orbital that is determined by the geometry of the four bonding atoms (three MM atoms and one QM atom). Table 1 shows that across different basis functions used, direct application of the PHO orbitals makes the bond length between the boundary atom and QM atom somewhat longer than the corresponding DFT value by ca. 0.05 Å, and the bond angles involving a ${\mathrm{H}}_M$ atom too large by 12 to 15°. The origin of the large discrepancy in bond angle has been noted before,^22,24 largely due to the incomplete representation of the bond densities of the ${\mathrm{C}}_B–{\mathrm{H}}_M$, having only an equivalence of one electron to directly interact with the full chemical bonds in the QM calculation.^22,24 Thus, the electrostatic interactions among the three auxiliary bonds are much too week in comparison with the full chemical bond in the QM fragment.^53,54 This leads to an imbalance in valence bond electrostatic repulsion (two electrons in the ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond vs ${\mathrm{C}}_B–{\mathrm{H}}_M$). These terms are unavoidable in any QM/MM partition if one wishes to treat the potential energy surface and dynamics of the system properly.

Table 1.

Mean Unsigned Errors (MUE) in Bond Length (A) and Bond Angle (deg) of Ethane Optimized Using the Projected Hybrid Orbital (PHO) Method with B3LYP Density Functional Theory (DFT) for One Methyl Group and the CHARMM36 Force Field for the Second Methyl Group in Comparison with DFT Calculations for the FULL System^a

	def2-SVP			def2-TZVP			6–31G(d)			6–311+G(d,p)
	PHO	Tet	NR	PHO	Tet	NR	PHO	Tet	NR	PHO	Tet	NR
CQ–HQ	0.012	0.010	0.010	0.009	0.008	0.008	0.012	0.010	0.011	0.009	0.008	0.008
CQ–CB	0.034	0.096	0.064	0.028	0.077	0.048	0.029	0.088	0.057	0.025	0.074	0.044
CB–HM	0.030	0.028	0.026	0.038	0.036	0.035	0.037	0.034	0.033	0.035	0.033	0.032
HQ–CQ–HQ	3.3	2.3	2.7	2.7	1.8	2.2	3.0	2.0	2.4	2.6	1.8	2.2
HQ–CQ–CB	2.9	2.0	2.4	2.4	1.7	2.0	2.7	1.8	2.2	2.3	1.6	2.0
CQ–CB–HM	2.9	2.7	11.9	3.1	3.0	12.4	3.0	2.9	12.2	3.1	2.9	12.4
HM–CB–HM	3.2	3.0	14.8	3.5	3.3	15.4	3.4	3.2	15.0	3.4	3.3	15.4

^aThe carbon atom of the “MM” methyl group is designated as the boundary atom. Four basis sets are used, including def2-SVP, def2-TZVP, 6–31G(d), and 6–311+G(d,p), and the calculations were performed without local restraining terms (NR), with a tetrahedral restraint (Tet) and the full PHO model.

Having understood the origin of the valence bonding interactions, we use tetrahedral angle-restraining terms to maintain the sp³ hybridization at the boundary atom, and indeed, the mean unsigned errors (MUE) decreased significantly to an acceptable range (Table 1, Tet columns). However, the deviation in the ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond length becomes larger, due to the change in hybridization ratio of 2s and 2p orbitals. To correct this deviation, we also include a harmonic bonding term in eq 31, which becomes the final PHO model (Table 1). We have also test a larger (1000 kcal/mol/Ų) and smaller (300 kcal/mol/Ų) force constants, and recommend to adopt a value of 500 kcal/mol/Ų. These force-field terms involving any MM atoms are included in QM/MM calculations anyhow in original implementations in CHARMM, though MM terms for ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bonds are not included in the present PHO method.^22,76 Results obtained using Hartree–Fock theory generally show good performance and they are provided in Supporting Information. Here, the parameters are considered to be generally applicable to all situations with the use of carbon sp³ boundary atoms, and the overall results in Table 1 are adequate for these mixed computational models.

Table 2 lists the average ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond length and the root-mean-square deviations (RMSD) with respect to the minimized geometry from a 10 ps molecular dynamics simulation of an ethane molecule under the NVE ensemble at 300 K. The deviations in average bond lengths from MD simulation follow the same trends as those of geometry optimization. The bond length fluctuation, indicated by RMSE, however, vary greatly without or with partial tetrahedral restraints. Nevertheless, the RMSDs on calculations using the final PHO model are in good agreement with the full QM results both using the HF and DFT/B3LYP methods. The average ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond length from MD simulations at the HF/def2-SVP level of theory is shown in Figure 2. We find that the ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond can fluctuate up to 1.78–1.80 Å without the bond restraint term. On the other hand, the QM/MM bond fluctuation drops to the range between 1.49 to 1.61 Å using the final PHO model. This is in accord with the fluctuation range of 1.46 to 1.60 Å using the full QM method. Interesting, bond fluctuations increase with the addition of a tetrahedral angle restraint (Tet vs NR).

Table 2.

Average Bond Lengths (Å) and the Root-Mean-Square Deviations (RMSD) for the Carbon–Carbon Bond from Molecular Dynamic Simulations of Ethane at 300 K^a

	HF		PHO		Tet		NR
	average	RMSD	average	RMSD	average	RMSD	average	RMSD
def2-SVP	1.531	0.036	1.548	0.026	1.644	0.078	1.605	0.051
6–31G(d)	1.534	0.012	1.548	0.029	1.632	0.051	1.604	0.064
6–311+G(d,p)	1.531	0.023	1.544	0.028	1.615	0.059	1.593	0.079
def2-TZVP	1.528	0.038	1.544	0.028	1.616	0.59	1.591	0.067
	DFT		PHO		Tet		NR
	average	RMSD	average	RMSD	average	RMSD	average	RMSD
def2-SVP	1.532	0.016	1.546	0.030	1.640	0.074	1.612	0.072
6–31G(d)	1.536	0.023	1.546	0.023	1.637	0.073	1.604	0.065
6–311+G(d,p)	1.537	0.023	1.543	0.024	1.621	0.069	1.585	0.049
def2-TZVP	1.532	0.017	1.540	0.024	1.616	0.060	1.591	0.054

^aCalculations were performed with the Hartree–Fock theory (HF) and density functional theory (DFT) for the full system, and QM/MM potentials with the hybrid project hybrid orbital (PHO) model, the tetrahedral restraining potential only (Tet) and without any molecular mechanics restraints (NR).

Figure 2.

Histogram of the ${\mathrm{C}}_Q–{\mathrm{C}}_B$ bond fluctuations of ethane from molecular dynamic simulations at 300 K using Hartree–Fock theory with the 6–311+G(d,p) basis set (full QM in black), and hybrid QM/MM potential using the projected hybrid orbitals model with the local force field restraints (PHO in red), with the tetrahedral angles restraints only (Tet in yellow) and without restraints (NR in green).

4.2. Atomic Charges.

In hybrid QM/MM calculations, a critical requirement is preserving the electronic structure of the full QM fragment. While Mulliken population charges may not be the most precise for molecular simulations, they effectively capture key features of charge polarization. For this reason, they are used here to examine the charge distribution in the QM fragment. The focus is on the overall charge behavior rather than individual partial atomic charges.

Mulliken population charges were computed for ethane and butane using Hartree–Fock theory and DFT/B3LYP, with two basis sets: 6–31G(d) and 6–311G(d,p). For ethane, the molecule was divided at the covalent bond between the two methyl groups, assigning one methyl group to the QM region and treating the other as part of the MM region, with the boundary atom placed on the carbon of the MM methyl group. In butane, in its all-trans conformation, the central CH₂CH₂ group was included in the QM region, while the two terminal methyl groups were treated as MM. We placed boundary atoms on the carbon atoms of the MM methyl groups. Note that in the CHARMM force field, hydrogen atoms are assigned a partial charge of +0.09 |e| units and the carbon of a methyl group carries −0.27 |e| units to keep it neutral. The central question here is whether the PHO method induces any significant perturbations in the charge distribution near the QM/MM boundary. This concern rises from the possibility that the hybrid orbitals generated by PHO, which mimic the primary basis set, could introduce deviations in electro-negativity. Ideally, if the minimal basis set representation accurately reflects the electron-withdrawing power of the primary basis, the total partial charge on the methyl group in ethane should be zero.

The results in Table 3 show that the methyl group containing the boundary atom does exhibit nonzero charges, with the magnitude depending on the computational method and basis set used. For the 6–31G(d) basis set, the PHO method displays a slightly stronger electron-withdrawing effect, resulting in more negative charges on the methyl groups of the MM region (carrying the boundary atoms). In contrast, the larger 6–311G(d,p) basis set yields a less electronegative boundary atom, producing a positive partial charge on the methyl group. This trend is also observed in butane, where the charge behavior of the methyl groups mirrors that of ethane. Interestingly, a small charge reversal between the two basis sets is noted in butane, which is also present in full QM calculations using the same methods (Table 3). This suggests that the observed behavior is consistent with the intrinsic properties of the basis sets.

Table 3.

Mulliken Population Charges (Atomic Units) for Ethane and trans-Butane^a

atom/methyl group	6–31G(d)		6–311G(d,p)
	HF	B3LYP	HF	B3LYP
	Ethane
C1	−0.31	−0.26	−0.08	−0.15
Cb	−0.28	−0.32	−0.16	−0.20
(CbH3)MM	−0.01	−0.05	0.10	0.07
	trans-Butane
C2	−0.16	−0.11	−0.05	−0.09
Cb1	−0.30	−0.33	−0.16	−0.21
(CH2)QM	0.03	0.06	−0.11	−0.06
(Cb1H3)MM	−0.03	−0.06	0.11	0.06
	(−0.01)	(−0.02)	(0.02)	0.03

^aQM and MM fragments are indicated by subscripts, and C_b denotes boundary atoms assigned to methyl carbon atoms. Values in parentheses are group charges from full QM calculations.

Overall, despite minor deviations in charge, the fluctuations are reasonably small and centered around zero, depending on the basis set employed. These findings indicate that the PHO method effectively models the electron-withdrawing effects and maintains a balanced electronegativity between the QM fragment and the boundary atom. As a result, the PHO method is well-suited for general QM/MM applications, with minimal disruption to the charge distribution at the QM/MM boundary.

4.3. Deprotonation and Proton Transfer Energies.

To evaluate the influence of the positions where boundary atoms are placed and of different sizes of the QM region as well as the proximity of boundary atoms, we determined the first and second deprotonation energies on a series of dicarboxylic acids (Figure 3). We further examined the energies for the proton transfer between the neutral and zwitterionic forms of amino acids separated by five (5) to eight (8) methylene groups (Figure 3). The main purpose here is not on the accuracy of the theoretical models in comparison with experimental data, but we aim at an understanding of the ability to reproduce the corresponding full QM results using the PHO hybrid approach. Thus, Hartree–Fock theory is sufficient for this purpose, and we adopted the 6–311+G(d,p) basis functions in the calculations. Molecular geometries for all compounds were optimized in the all trans configuration, also using HF/6–311+G(d,p), and QM/MM energies are determined via single-point energy calculations at these geometries.

Figure 3.

Molecular structures of a series of amino acids (a) and dicarboxylic acids separated by different number of methylene groups (b), and same length dicarboxylic acids but different QM regions (c). The fragments bearing functional groups are treated by Hartree–Fock theory and the methylene groups in parentheses are modeled as molecular mechanics models. The carbon atoms labeled by ${\mathrm{C}}_b$ indicate the position where the boundary atoms are located, treated by the projected hybrid orbital method.

We first examine the effect of separation length between two boundary atoms on the computed energy change for proton transfer from the zwitterionic form of various amino acids to the corresponding neutral configuration (3a), and deprotonation energies of dicarboxylic acids shown in Figure 3b. In each case, the boundary atoms are placed at the methylene carbon that are two covalent bonds away from the functional groups (carboxylic group or amino group). Thus, while the number of atoms in the QM region remains the same, the number of linker methylene group ranges from 1 to 4 units. Although the neutral form of the amino acids is lower in energy than the zwitterion, the zwitterion geometries were optimized without collapsing since the hydrocarbon linker is in the all-trans conformation and there is no viable proton transfer path without fully ionizing it.

Tables 4 and 5 reveal that the computational errors in combined QM/MM calculations with the present PHO boundary method are reasonably small, generally below 2 kcal/mol for the proton transfer energies and the second deprotonation energies. However, significant difference (more than 5 kcal/mol) exists on the first deprotonation energy. There is minimal effect due to the separation distance between two QM fragments, exhibiting small deviations from the full QM results as the functional groups are farther away in all energy terms. The larger deviation in the first deprotonation energy in Table 5 may originate from the proximity of two boundary atoms to the functional groups, but it could be due to delocalization effects, e.g., greater size for the QM fragments. This is indeed confirmed next in Table 6. Nevertheless, only when four (4) methylene groups, −CH₂−CH₂−CH₂−CH₂−, are placed between two boundary atoms, can we obtain computational accuracy below 1 kcal/mol. The good news, however, is that the accuracy in the two deprotonation energies of the dicarboxylic acids are not influenced by using two boundary atoms; in fact, the second deprotonation energies have less errors than the first deprotonation energy (Tables 5 and 6).

Table 4.

Computed Proton Transfer Energies (kcal/mol) Using HF/6–311+G(d,p) and Signed Errors from PHO-QM/MM Calculations for the All-Trans Amino Acids^a

$\begin{array}{c}{}^{-}{\mathrm{O}}_2{\mathrm{C}\mathrm{C}\mathrm{H}}_2{\mathrm{C}}_{B1}\left[{\mathrm{H}}_2{\left({\mathrm{C}\mathrm{H}}_2\right)}_n{\mathrm{H}}_2\right]\\ {\mathrm{C}}_{B2}{\mathrm{C}\mathrm{H}}_2{\mathrm{N}\mathrm{H}}_3^{+}\end{array}$	HF (full QM)	PHO (QM/MM)
n = 1	−81.0	1.9
n = 2	−87.2	−1.3
n = 3	−91.7	−1.0
n = 4	−95.6	−0.8

^aHydrogen atoms and methylene groups in square parentheses are modeled by the CHARMM force field. The energy change for the proton transfer process is defined by $\mathrm{\Delta }{E}_{\mathrm{P}\mathrm{T}\mathrm{E}}=E\left(\mathrm{n}\mathrm{e}\mathrm{u}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\right)-E\left(\mathrm{z}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{n}\right)$.

Table 5.

Computed First and Second Deprotonation Energies (kcal/mol) Using HF/6–311+G(d,p) and Signed Errors from the PHO-QM/MM Calculation on Dicarboxylic Acids in the All-Trans Conformation^a

HO2CCH2CB1[H2(CH2)nH2]CB2CH2CO2H	ΔE1		ΔE2
	HF	PHO	HF	PHO
n = 1	−364.7	−5.6	−402.7	− 1.5
n = 2	−357.1	−5.2	−395.3	0.7
n = 3	−356.5	−5.0	−392.5	−0.3
n = 4	−357.8	−5.0	−388.0	−0.9

^aHydrogen atoms and methylene groups in square parentheses are modeled by the CHARMM force field. Deprotonation energies are defined as follows, where R indicates different lengths of methylene groups. $\begin{array}{l}\mathrm{\Delta }{E}_1=E\left({\mathrm{H}\mathrm{O}}_2\mathrm{C}\left[R\right]{\mathrm{C}\mathrm{O}}_2^{-}\right)-E\left({\mathrm{H}\mathrm{O}}_2\mathrm{C}\left[R\right]{\mathrm{C}\mathrm{O}}_2\mathrm{H}\right)\\ \mathrm{\Delta }{E}_2=E{(}^{-}{\mathrm{O}}_2\mathrm{C}\left[R\right]{\mathrm{C}\mathrm{O}}_2^{-})-E\left({\mathrm{H}\mathrm{O}}_2\mathrm{C}\left[R\right]{\mathrm{C}\mathrm{O}}_2^{-}\right)\end{array}$

Table 6.

Computed First and Second Deprotonation Energies (kcal/mol) Using HF/6–311+G(d,p) and Signed Errors from the PHO-QM/MM Calculation for Ridecanedioic Acid^a

	ΔE1		ΔE2
	HF	PHO	HF	PHO
R1CB1[H2(CH2)7H2]CB2R1	−359.7	−3.8	−382.5	−2.3
R2CB1[H2(CH2)5H2]CB2R2		−3.6		−1.4
R2CB1[H2(CH2)3H2]CB2R3		−2.3		−0.7
R4CB1[H2(CH2)H2]CB2R4		−1.0		0.1

^aaBoundary atoms placement and the short-hand notations (R_n; n = 1,..., 4) may be referenced in Figure 3c. Deproton energies are defined as that in Table 5.

We further examined the effect of the boundary atoms placed at different positions away from the functional groups in the tridecanedioic acid, HO₂C(CH₂)₁₁CO₂H, (Figure 3c). Table 6 shows that as the QM/MM boundary atoms are moved farther away from the carboxyl functional groups, the MUE decreases gradually from 3.8 and 2.3 kcal/mol, respectively, for the first and second deprotonation energies when ${\mathrm{C}}_b$ atoms are placed at $\beta$-carbon atom, to less than 1 kcal/mol for both deprotonation energies when the boundary atoms are 5 covalent bonds away from the carboxylic groups. Clearly, there is benefit for having a larger QM region in these calculations. Importantly, as far as the multiple boundary method is concerned, the present findings show that there is little interaction between boundary atoms.

4.4. Potential Energy Curves of Nitrile–Water Complex.

The cyano (−CN) group of organic nitriles and the closely related organo thiocyanates are widely used as probes to the local electrostatic environment of proteins based on vibrational Stark shifts.^77–80 Their infrared (IR) frequencies are sensitive to the fluctuating electric field and the observed spectral shifts reflect the specific orientation and strength of hydrogen bonding interactions. Combined QM/MM simulations have been used to model theses processes and the study of nuclear quantum effects,^81,82 and we chose a cysteinyl tripeptide system as a model to examine the performance of the present PHO multiple boundary approach.

To this end, we designed three QM/MM models making use of the bimolecular complex structure between cyanylated tripeptide and water by including zero, one, and two PHO boundary atoms as shown in Figure 4. Again, the main purpose here is to evaluate the possible errors in computed binding energies introduced in the presence of PHO boundary atoms with respect to the results from the corresponding QM/MM calculations without covalent boundary. In the first model in Figure 4a, the entire tripeptide is included in the QM region. In model (b), one boundary atom is placed at the ${\mathrm{C}}_{\alpha }$ position of cysteine (the second peptide). In model (c), two boundary atoms are anchored, respectively, at the ${\mathrm{C}}_{\alpha }$ position of residues 1 and 3. For simplicity, we used optimized monomer geometry for cyanylated tripeptide at B3LYP/6–31G(d) level and experimental geometry data are used for water.⁸³

Figure 4.

Geometries for the cyano group capped cysteine tripeptide and water complex. QM regions are indicated by circles in combined QM/MM calculations with zero (a), one (b) and two (c) projected hybrid orbital boundary atoms (black) which are indicated by arrows. Atom colors are coded as oxygen in red, nitrogen in blue, carbon in cyan, sulfur in yellow, and hydrogen in white.

It was known that the frequency-shifts of the −SCN probe stem from the effects of both the linear $\sigma$-type and the $\pi$-type hydrogen-bonding interactions,⁸⁰ for which we scanned the potential energy curves for the complex with water in Figure 4 at fixed C···N···H angles, respectively, for the two types of hydrogen-bonding interactions, from previous optimizations of the MeSCN−H₂O complex using MP2/aug-cc-pVTZ.⁸¹ These potential energy curves were determined as a function of the C···H_w distance along the directions defined by the C···N···H angles using B3LYP/6–31G(d) for the QM region and the TIP3P water for water and CHARMM36 force field for the peptides.^75,83

Figure 5 shows that although the optimal distances for both hydrogen bonding configurations (1.9 Å for $\sigma$-type interaction and at 2.1 Å for $\pi$-type complex) are the same across all three QM/MM constructs, the computed binding energies can have noticeable errors when the boundary atom is placed on the $\alpha$-carbon of the cysteine residue of the probe group. On the other hand, the agreement between the QM/MM model in which all peptide atoms included in the QM region and that of the two-boundary model that encapsulate two peptide bonds is quite good. This suggests that the effects of placing QM boundary atoms need to be carefully examined in QM/MM calculations. It is interesting to note that the hydrogen bonding energy is overestimated by 0.24 kcal/mol in the $\sigma$-complex, but it is weaker by 0.14 kcal/mol for the $\pi$-type structure when one PHO boundary is used and only the side chain of cysteine is included in the QM region (Figure 4b). The difference between PHO with two boundary atoms and QM/MM is only about 0.1 kcal/mol.

Figure 5.

Rigid potential energy scan along the bond vector between nitrogen of the cyanylated tripeptide and the donor hydrogen of water.

5. CONCLUSION

The projected hybrid orbital (PHO) method for multiple boundary atoms offers a novel approach for treating the covalent boundary between quantum mechanical (QM) and molecular mechanical (MM) subsystems in combined QM/MM calculations. It extends the generalized hybrid orbital (GHO) method by introducing a secondary, minimal basis set on the boundary atom, where QM and MM regions meet. This allows for the projection of the primary QM basis onto the secondary set using a new orbital projection approach, creating hybrid orbitals that preserve electronic interactions between the two regions. Notably, this method can be applied in ab initio wave function theory (WFT) and density functional theory (DFT) with any basis set without system-dependent parameters, making it widely applicable.

The PHO method has been tested on various molecules and properties using Hartree–Fock theory and density functional theory with several basis sets, demonstrating that it maintains structural and electronic integrity in the QM region. For example, in simulations of molecular structure, proton transfer and deprotonation energies and hydrogen bonding interactions, the results closely match full QM calculations or QM/MM results without covalent boundary. The dependence on the placement of the boundary atoms have been examined, showing that direct interactions between projected boundary orbitals are minimal. In energy calculations, accuracy improves as the boundary atoms are moved further from the functional group and further apart, with errors dropping to less than 1 kcal/mol when the boundary is at least three covalent bonds away.

A key advantage of the PHO method is its ability to overcome challenges faced by traditional QM/MM techniques, such as arbitrary energy corrections seen in the link-atom approach. Additionally, the method’s general, transferable nature eliminates the need for system-specific parametrization, making it superior to methods that make use of density matrix embedding data. The latter is a major limitation of this class of approaches that is generally system-dependent and requires electronic structure calculations on the entire system. The PHO method also holds potential for broader applications beyond QM/MM calculations, particularly in QM/QM embedding frameworks,⁸⁴ expanding the method’s utility in density matrix and density functional embedding schemes. Finally, we note that although PHO is a well-defined method, it is an approximate method, mimicking the basis set of the primary (QM) system by a minimal basis set. This makes the method generally applicable to any basis sets, but one still needs to validate boundary methods before applications.

Supplementary Material

Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c01326.

Optimized structures used in this article (ZIP)

Deviations from full QM calculations for ethane using different methods and basis sets (XLSX)