PyOrbb—Automated Analyses of Orbital‐Interaction Mechanisms

ABSTRACT

We present PyOrbb, an open‐source, user‐friendly Python‐based program that automates the analysis and visualization of orbital interaction mechanisms between molecular fragments or reactants, based on quantum‐chemical computations with the Amsterdam Density Functional (ADF) program. PyOrbb extracts the essential mixing patterns, presents them in an interactively adjustable orbital interaction diagram, and summarizes all relevant quantitative data in a convenient Excel spreadsheet for further analysis. The tool enables efficient interpretation of chemical bonding across a broad range of systems, including examples from organic, inorganic, and supramolecular chemistry. The application runs on macOS and Windows operating systems. PyOrbb is shown to accurately and rapidly visualize relevant bonding mechanisms from the often‐overwhelming output of density functional computations.

PyOrbb is an open‐source program that automates fragment‐based orbital interaction analysis from ADF calculations. It identifies the chemically most significant attractive and repulsive interactions and presents them as publication‐ready MO diagrams.

1. Introduction

Molecular orbital (MO) theory is a central tenet in chemistry and an essential tool for understanding and predicting chemical bonding and reactivity [1, 2]. The analysis of how orbitals of different molecular fragments interact with each other to form the molecular orbitals has provided deep insight into various chemical phenomena [3, 4, 5, 6]. As a parallel, valence bond (VB) theory has also proven useful in this pursuit [7, 8]. With the advent of Kohn–Sham DFT, MO theory has become a quantitative model for revealing accurate causal relationships between the electronic structure of molecules and their ability to form bonds or to react [9, 10]. The activation strain model (ASM) [11, 12, 13, 14] is a comprehensive framework for the analysis of bonding and reactivity with MO theory at its core. ASM has proven to be a versatile tool for generating useful frameworks to rationalize chemical reactivity, for example, cycloadditions [15, 16, 17, 18, 19, 20, 21, 22], metalorganic catalysis [23, 24], substitution reactions [25, 26, 27], and many more.

The primary bottleneck in applying MO theory is deconvoluting the decisive, dominant MO interaction mechanism or pattern in the complex output of modern Kohn–Sham DFT programs. For instance, one MO can arise from interactions and mixing between many different fragment MOs (FMOs). However, in most cases, only a few orbital interactions and mixing patterns are crucial, that is, they make the dominant contribution to the strength of a chemical bond or an intermolecular interaction. Textbooks focus on these leading orbital interactions in an MO diagram to explain chemical phenomena [1].

While the graphical user interfaces (GUI) accompanying modern quantum chemical software can easily generate an MO diagram showing any MO mixing occurring in a fragmented system, it is still a laborious, time‐consuming task for chemists to identify key FMO interactions. In molecular orbital theory [1], an interaction between two FMOs results in a stabilized bonding and a destabilized antibonding combination. In the case of a doubly occupied and an unoccupied FMO, the interaction will be net stabilizing as the electrons in the bonding combination will be stabilized relative to their original FMOs and the destabilized antibonding combination remains unoccupied (Figure 1a). Between two doubly occupied FMOs, the interaction will be destabilizing (Pauli repulsion, Figure 1b) as the destabilization of the higher‐lying orbital is larger than the stabilization of the lower‐lying orbital. Additionally, an MO diagram must be closed, that is, the number of FMOs should be equal to the number of the resulting overall MOs. For example, two FMOs must form two overall MOs. Lastly, the number of electrons in the FMOs must be equal to the number of electrons in the MOs. Generating diagrams based only on, for example, Mulliken contributions generally does not yield a satisfactory description of the bonding mechanism. An additional point of interest is the interpretation of orbital interactions involving charged fragments. In this case, the second fragment perturbs the FMO energies, necessitating a change in the orbital interaction analysis. For such cases, it is preferable to use the effective energies, which are adjusted for the presence of the second fragment (Section S1) [28, 29, 30, 31].

FIGURE 1.

Two MO diagrams showing interactions between two FMOs. (a) depicts the interaction between an occupied (${\psi }_1$) and an unoccupied (${\psi }_2$) orbital, which causes a net stabilization of the occupied MOs, whereas (b) shows the interaction between two occupied FMOs (${\psi }_1$ and ${\psi }_2$), which results in net destabilization of the occupied MOs. The ${\epsilon }^{\left(2\right)}$ assigned to the stabilization and destabilization is the second order perturbative correction to the orbital energies.

To resolve these challenges, we have developed PyOrbb, a user‐friendly MO analysis tool and Python code that serves as a wrap‐around to the output of the Amsterdam Density Functional (ADF) module of the Amsterdam Modeling Suite (AMS) [32, 33, 34]. PyOrbb summarizes the key bonding and antibonding MO interactions between FMOs, allows for choice between energy types and assists in interpreting the complex numerical output of modern quantum chemical software. We first describe the fundamental principles and workflow of PyOrbb and then provide a suite of examples of bonding motifs in molecules and bimolecular systems that cover much of chemical space.

2. Fragment Approach to Bonding Analysis

One of the core goals of chemical research is to understand the mechanism behind the strength of a chemical interaction. This includes the formation of a bond within a molecule and the formation and breaking of bonds from reactants to products along a reaction path or intermolecular interactions. One needs to establish a causal relationship between the electronic structures of atoms, molecular fragments, or molecules and the bonding or reactivity of these species. The conceptual approach based on the analysis of fragment‐based MOs provides a direct route to establish causal relationships in terms of the interactions between the orbitals of the molecular fragments that are involved in bond formation (e.g., two methyl radicals forming the C–C bond in ethane) or in a chemical reaction (e.g., chloride and chloromethane reacting via S_N2 substitution) [1, 2, 3, 9]. Therefore, the ability to generate a correct and transparent MO interaction scheme based on accurate KS–DFT computations is essential.

The ASM [11, 12, 13, 14] is a generalization of this fragment‐based approach to describe any bond formation or reaction not only in equilibrium geometries but also at saddle points and non‐stationary points along a reaction's potential energy surface. The bonding energy (∆E(ζ)) can be decomposed into the strain energy (ΔE _strain(ζ)) that is associated with the geometrical deformation of the individual reactants as the process takes place and the actual interaction energy (ΔE _int(ζ)) between the deformed reactants (Equation 1).

$$\mathrm{\Delta }E\left(\zeta \right)=\mathrm{\Delta }{E}_{\text{strain}}\left(\zeta \right)+\mathrm{\Delta }{E}_{\mathrm{int}}\left(\zeta \right).$$ (1)

The interaction energy (ΔE _int(ζ)) between the deformed reactants is further analyzed in the conceptual framework provided by the Kohn–Sham molecular orbital (KS–MO) model [9, 10], using a quantitative energy decomposition analysis scheme (EDA, Equation 2) [35, 36]. In this scheme, the electrostatic interaction (ΔV _elstat(ζ)) corresponds to the classical Coulomb interaction between the unperturbed charge distributions of the deformed, but still separate reactants, the Pauli repulsion energy (ΔE _Pauli(ζ)) comprises the destabilizing interactions between occupied orbitals on the respective reactants, the orbital interaction energy (ΔE _oi(ζ)) accounts for charge transfer (interaction between occupied orbitals on one fragment with unoccupied orbitals on the other fragment) and polarization (empty‐occupied orbital mixing on one fragment due to the presence of another fragment) and electron‐pair interactions (e. g., SOMO–SOMO interactions). Furthermore, a dispersion term (ΔE _disp(ζ)) can be added as an empirical correction, for example, based on Grimme's approach for long‐range dispersion interactions between the fragments [37, 38, 39].

$$\mathrm{\Delta }{E}_{\mathrm{int}}\left(\zeta \right)=\mathrm{\Delta }{V}_{\text{elstat}}\left(\zeta \right)+\mathrm{\Delta }{E}_{\text{Pauli}}\left(\zeta \right)+\mathrm{\Delta }{E}_{\mathrm{oi}}\left(\zeta \right)+\mathrm{\Delta }{E}_{\text{disp}}\left(\zeta \right)$$ (2)

These quantitative energy terms can provide insight into chemical bonding by quantifying the various features in the bonding mechanism. The latter, that is, the actual physical mechanism behind bond formation, is obtained by analyzing the MO interactions that determine the relevant interaction patterns. To obtain these mechanisms in a user‐friendly, automated manner, we have developed the program PyOrbb. PyOrbb analyzes and extracts the most relevant destabilizing and stabilizing orbital interaction patterns associated respectively with ΔE _Pauli, and ΔE _oi. It requires the user to supply the output generated by a fragment calculation as implemented in AMS [32, 33, 34] or by the PyFrag 2019 program [40, 41], which is specially designed to perform fragment calculations along a reaction coordinate. Once analyzed, PyOrbb provides the results in a clear, automated, and informative way.

3. Description of the Program

The core of the program consists of three parts (see Figure 2). First, our Python backend reads and summarizes all relevant data and performs a preliminary FMO‐based Mulliken analysis. The obtained data are then used to generate the mixing patterns that are determined to be most relevant in the bonding mechanism of the studied chemical system. Furthermore, PyOrbb provides a GUI that allows users to focus on specific aspects of the data, resulting in clear, concise overviews of the results. All of PyOrbb's functionality is also accessible as a Python module [42, 43]. A step‐by‐step case study for the PyOrbb program is provided in Section S2, illustrated with the H₃B ← NH₃ Lewis adduct. PyOrbb will warn the user whenever suspicious data are encountered or when the user has potentially made an error in setting up the calculations. An overview of these warnings and errors along with potential solutions is provided online (https://theochem‐vu.github.io/PyOrbb/).

FIGURE 2.

A flowchart describing the PyOrbb program, including the five‐step algorithm used to generate FMO mixing patterns described in Section 3.2. A full step‐by‐step case study is described in Section S2.

3.1. Reading and Preparing Data

The result file from an ADF calculation is the input for the PyOrbb program. The rkf‐file contains information about MOs, FMOs, occupation numbers, and information such as coefficients, Fock and overlap matrices. The PyOrbb program performs preparatory Mulliken analyses that will be used to generate the key mixing patterns. We use Equation (3) to construct the Mulliken contribution matrix:

$$\mathit{M}=\mathit{c}\odot \left(\mathit{c}·\mathit{S}\right),$$ (3)

where $\mathit{c}\in {ℝ}^{n\times n}$ (for $n$ MOs) is the coefficient matrix ($c_{\mathit{ki}}=⟨{\psi }_i|{\mathrm{\Psi }}_k⟩$), $\mathit{S}\in {ℝ}^{n\times n}$ is the overlap matrix ($S_{\mathit{ij}}=⟨{\psi }_i|{\psi }_j⟩$), $\odot$ is the Hadamard (elementwise product, e.g., ${\left(\mathrm{A}\odot \mathrm{B}\right)}_{\mathit{ij}}={\mathrm{A}}_{\mathit{ij}}{\mathrm{B}}_{\mathit{ij}}$) product, and $·$ is the inner product (e.g., ${\left(\mathrm{A}·\mathrm{B}\right)}_{\mathit{ij}}=\sum _k{\mathrm{A}}_{\mathit{ik}}{\mathrm{B}}_{\mathit{kj}}$). We then follow Equation (4) to construct the gross Mulliken population:

$$\mathit{p}=\mathit{o}·\mathit{M},$$ (4)

where $\mathit{o}\mathit{\in }{ℝ}^{\mathit{n}}$ is a vector containing occupation numbers for the MOs.

3.2. Generating FMO Mixing Patterns

After reading and preparing orbital data, PyOrbb executes a five‐step algorithm to determine the leading mixing patterns, that is, the FMO interaction patterns that exhibit the strongest mixing. The algorithm is designed to fulfill three key requirements of closed interactions: (1) the number of used FMOs must be equal to the number of resulting MOs in the mixing diagram; (2) the number of electrons in the FMOs must be equal to the number of electrons in the resulting MOs; and (3) interacting FMO pairs must form a bonding and antibonding MO. The goal of the algorithm is to obtain mixing patterns that meet the three requirements and are most likely relevant to the bonding mechanism.

3.2.1. Detection of FMO Interaction Patterns

The first step of the algorithm is to identify and rank important FMO interactions. Internally, we distinguish two types of interactions: stabilizing orbital interactions (oi) between an occupied and a virtual FMO, and destabilizing Pauli‐repulsive (pr) interactions between two occupied FMOs. The strength of these interactions can be estimated using a second‐order perturbative approximation [1, 2], and leads to the following approximate expression for nondegenerate orbital interactions:

$$\mathrm{\Delta }{E}_{ij}^{\mathrm{oi}}\propto -\frac{{⟨{\psi }_i|{\psi }_j⟩}^2}{\left|{\epsilon }_i-{\epsilon }_j\right|},$$ (5)

where $⟨{\mathit{\psi }}_{\mathit{i}}|{\mathit{\psi }}_{\mathit{j}}⟩$ is the overlap between two FMOs ${\psi }_i$ and ${\psi }_j$, and ${\epsilon }_i$ is the orbital energy of ${\psi }_i$.

For electron‐pair bonds between degenerate FMOs $\mathrm{\Delta }{E}_{\mathit{ij}}^{\mathrm{oi}}$ is approximately proportional to the overlap [1, 2]:

$$\mathrm{\Delta }{E}_{\mathit{ij}}^{\mathrm{oi}}\propto ⟨{\psi }_i|{\psi }_j⟩.$$ (6)

The perturbative approximation also leads to the following approximate expression for the strengths of destabilizing Pauli‐repulsive interactions:

$$\mathrm{\Delta }{E}_{\mathit{ij}}^{\mathrm{pr}}\propto {⟨{\psi }_i|{\psi }_j⟩}^2.$$ (7)

3.2.1.1. Quantifying Orbital Interactions

To rank the orbital interactions between FMOs ${\psi }_i$ and ${\psi }_j$ we extend on Equation (7). The gain or depletion of electrons to or from FMO ${\psi }_i$ is taken into account to quantify the “activity” of the FMOs. This is captured by $\mathrm{\Delta }{p}_i$ as the deviation of the Mulliken population $p_i$ (Equation 4) from the initial occupation (0, 1, or 2 electrons):

$$\mathrm{\Delta }{p}_i=\left\{\begin{array}{c}2-p_i,\text{if}{\psi }_i\text{was initially doubly occupied}\\ \left|1-p_i\right|,\text{if}{\psi }_i\text{was initially singly occupied}\\ p_i,\text{if}{\psi }_i\text{was initially unoccupied}\end{array}\right..$$ (8)

FMOs with a large $\mathrm{\Delta }p$ tend to participate in stronger orbital mixing, while those with a small $\mathrm{\Delta }p$ are considered relatively inert, unimportant, or nonbonding.

To quantify one‐, two‐, and three‐electron two‐center orbital interactions, the Mulliken populations ($p_i$) are used to determine the number of electrons in the resulting stabilizing bonding combination ($p_{\text{bond},\mathit{ij}}$) and the number of left‐over electrons that occupy the destabilizing antibonding combination ($p_{\text{anti},\mathit{ij}}$):

$$p_{\text{bond},\mathit{ij}}=\min \left(p_i+p_j,p_{\max }\right),$$ (9)

$$p_{\text{anti},\mathit{ij}}=\max \left(p_i+p_j-p_{\max },0\right),$$ (10)

where ${\mathit{p}}_{\max }$ refers to the maximum number of electrons per orbital for either unrestricted or restricted DFT, depending on the method used

$$p_{\max }=\left\{\begin{array}{c}1e^{-}\text{if unrestricted}\\ 2e^{-}\text{if restricted}\end{array}\right..$$ (11)

The stabilization of a chemical system due to an orbital interaction is strengthened when the number of bonding electrons increases and weakened when there are more antibonding electrons. As such, we include the term $\left(p_{\text{bond},\mathit{ij}}-p_{\text{anti},\mathit{ij}}\right)$. This term will be largest when the combined population, $p_i+p_j$, equals $p_{\max }$ electrons, will be weaker when the populations deviate from $p_{\max }$, and will approach zero when 0 or 4 electrons are involved. The use of populations also allows for interpretation of systems where the fragment‐ and molecular electron configurations do not correspond (see Section S3).

Combining the above, the orbital interactions between FMOs ${\psi }_i$ and ${\psi }_j$ are ranked according to Equation (12):

$$R_{\mathit{ij}}^{\mathrm{oi}}=\mathrm{\Delta }{p}_i\mathrm{\Delta }{p}_j\left(p_{\text{bond},\mathit{ij}}-p_{\text{anti},\mathit{ij}}\right)\frac{{⟨{\psi }_i|{\psi }_j⟩}^2}{\left|{\epsilon }_i-{\epsilon }_j\right|}.$$ (12)

Please note that the ranking values calculated by PyOrbb only serve to provide an indication of the relative strengths of the interactions between pairs of FMOs and are not directly related to the energy terms of the canonical EDA procedure (Equation 2).

3.2.1.2. Quantifying Pauli Repulsions

The quantification of Pauli repulsive interactions between FMOs ${\psi }_i$ and ${\psi }_j$ is based on Equation (7), which relates the overlap of occupied FMOs to the Pauli repulsion term. The sum of the initial occupations, $o_i+o_j$, is used to define the number of electrons in the antibonding combination, $o_{\text{anti},\mathit{ij}}$:

$$o_{\text{anti},\mathit{ij}}=\max \left(o_i+o_j-o_{\max },0\right),$$ (13)

where $o_{\max }$ refers to the maximum electron occupancy per orbital for either unrestricted or restricted DFT, depending on the method used:

$$o_{\max }=\left\{\begin{array}{c}1e^{-}\text{if unrestricted}\\ 2e^{-}\text{if restricted}\end{array}\right..$$ (14)

The occupation of the antibonding combination allows for four‐ and three‐electron two‐center interactions to be included in ranking $R_{\mathit{ij}}^{\mathrm{pr}}$. Whereas FMO combinations with lower occupation numbers are omitted:

$$R_{\mathit{ij}}^{\mathrm{pr}}=o_{\text{anti},\mathit{ij}}{⟨{\psi }_i|{\psi }_j⟩}^2.$$ (15)

3.2.2. Filtering Interaction Patterns

The first step, in principle, ranks all FMO interactions in the chemical system. In the second step, PyOrbb filters the interactions to ensure only relevant interactions are kept. For orbital and Pauli repulsive interactions, we compare the values for $R_{\mathit{ij}}^{\mathrm{oi}}$ and $R_{\mathit{ij}}^{\mathrm{pr}}$ to thresholds ${\tau }_{\mathrm{oi}}$ and ${\tau }_{\mathrm{pr}}$, respectively:

$$R_{\mathit{ij}}^{\mathrm{oi}}\ge {\tau }_{\mathrm{oi}},$$ (16)

$$R_{\mathit{ij}}^{\mathrm{pr}}\ge {\tau }_{\mathrm{pr}},$$ (17)

If the values calculated using Equations (12) and (15) are below the thresholds ${\tau }_{\mathrm{oi}}$ and ${\tau }_{\mathrm{pr}}$, set by the user, the interaction is omitted. PyOrbb provides an interactive orbital mixing diagram that gives the user an easy way to change ${\tau }_{\mathrm{oi}}$ and ${\tau }_{\mathrm{pr}}$ by means of a slider.

3.2.3. Matching MO and FMO Pairs

The third step finds MO pairs that correspond to the FMO pairs that were found in Step 3.2.2 based on the Mulliken contributions. First, the Mulliken contributions $M_{\mathit{ik}}$ (contribution of ${\psi }_i$ to ${\mathrm{\Psi }}_k$) are normalized to reduce the effect of a well‐known artifact of the Mulliken analysis in combination with large basis sets [44, 45, 46]. Especially in cases where the orbital overlap is large, there may be negative (or greater than $p_{\max }$) populations and contributions. PyOrbb warns the user of this by displaying a warning whenever the sum of the absolute Mulliken contributions is larger than 130% for a given orbital (MO or FMO).

By definition, the sum of the contributions from an FMO or to an MO is always equal to 1, any negative contributions induce higher positive contributions, leading to qualitatively incorrect MO selection. To alleviate this problem, we discard negative contributions and normalize all positive Mulliken contributions. The normalized Mulliken contributions (${\mu }_{\mathit{ik}}$) are obtained by dividing $M_{\mathit{ik}}$ by the average of the sum over all positive Mulliken contributions to ${\psi }_i$ and ${\mathrm{\Psi }}_k$:

$${\mu }_{\mathit{ik}}=\frac{\max \left(M_{\mathit{ik}},0\right)}{\frac{1}{2}\left[{\sum }_j\max \left(M_{\mathit{jk}},0\right)+{\sum }_l\max \left(M_{\mathit{il}},0\right)\right]}.$$ (18)

This renormalization ensures that there are no negative contributions and, conveniently, the contributions are again normalized to 100%. To match FMO and MO pairs we take the products of the normalized Mulliken contributions from the FMO pairs to all occupied and virtual MOs. Note that for Pauli repulsive pairs, only the normalized Mulliken contributions to occupied MOs are considered. PyOrbb matches a pair of FMOs (${\psi }_i,{\psi }_j$) to a pair of MOs (${\mathrm{\Psi }}_k,{\mathrm{\Psi }}_l$) by calculating the product of the normalized Mulliken contributions from the FMOs to each MO:

$$\left({\mathrm{\Psi }}_k,{\mathrm{\Psi }}_l\right)=\underset{\left(k,l\right),k\ne l}{\text{argmax}}\left(\left({\mu }_{\mathit{ik}}{\mu }_{\mathit{jk}}\right)\left({\mu }_{\mathit{il}}{\mu }_{\mathit{jl}}\right)\right)$$ (19)

The product is large only when both FMOs contribute to both MOs, ensuring a complete and meaningful closed interaction.

3.2.4. Obtaining N‐FMO Mixing Patterns

So far, we have treated interactions involving two FMOs. However, many key interactions involve more than two FMOs. In the fourth step, PyOrbb combines the two‐FMO mixing patterns to generate N‐FMO mixing patterns by constructing a graph that includes all FMOs and MOs identified in the third step. This graph is separated into its connected subgraphs, which then constitute the isolated N‐FMO mixing patterns.

3.2.5. Ensuring Closed Interaction Patterns

The established N‐FMO mixing patterns do not necessarily meet the three requirements for closed interactions. The fifth step ensures that these requirements are met. PyOrbb checks each N‐FMO mixing pattern and checks the number of FMOs, MOs, and electrons involved in the interaction. If they do not match, PyOrbb adds new virtual or occupied MOs or FMOs based on the Mulliken gross populations already present in the N‐FMO mixing patterns. It chooses the new MOs or FMOs based on the normalized Mulliken contributions (Equation 18) to the known MOs and FMOs. When all three requirements are met (Section 3.2), the final MO diagram describing the key orbital interactions is drawn in the interactive PyOrbb interface.

3.3. PyOrbb Graphical User Interface

PyOrbb provides a user‐friendly GUI that displays the results of the MO analysis it performs. The GUI contains five main elements that show information to the user or modify the analysis shown (Figure 3).

FIGURE 3.

Example output of the graphical user interface of PyOrbb applied to a calculation describing the donor–acceptor interaction between NH₃ and BH₃. The elements of the GUI window have been numbered and are indicated by red boxes.

The main element in the app window (Element 1) displays the most important orbital interaction diagram extracted by PyOrbb for the provided calculation. It considers the options set by the user in Elements 2 and 3. The orbital diagram is divided into columns that show the orbitals of the involved fragments and the overall MOs. The lines between the orbitals indicate whether there is interaction and mixing between them, and their transparency indicates the magnitude of the mixing. For example, the BH₃(3A₁) orbital contributes strongly to the highest MO (the LUMO of the complex), and the NH₃(3A₁) orbital contributes weakly to the highest MO. The use of colors in the diagram indicates the type of interaction; a green‐colored interaction indicates a stabilizing orbital interaction, a red interaction indicates a destabilizing Pauli repulsive interaction, a gray interaction indicates both orbital interaction and Pauli repulsive character, and a purple interaction indicates an interaction that was added during the fifth step of the orbital analysis algorithm.

Element 2 is a control panel that allows for modification of the FMO interaction filtering step of the algorithm. The checkboxes are used to either show or hide the orbital and Pauli‐repulsive interactions. The sliders adjust the thresholds used to filter mixing situations during the second step of the algorithm.

To give an indication of the relative importance of a two‐FMO Pauli repulsive or orbital interaction, we calculate the relative ranking value as a fraction ($f_{\mathit{ij}}$) of the sum of the ranking values:

$$f_{\mathit{ij}}=\frac{R_{\mathit{ij}}}{{\sum }_{\mathit{ab}}{R}_{\mathit{ab}}},$$ (20)

where $R_{\mathit{ij}}$ is either the ranking value for an orbital interaction ($R_{\mathit{ij}}^{\mathrm{oi}}$, Equation 12) or for a Pauli repulsive interaction ($R_{\mathit{ij}}^{\mathrm{pr}}$, Equation 15). We emphasize again that both the ranking values ($R_{\mathit{ij}}^{\mathrm{oi}}$ and $R_{\mathit{ij}}^{\mathrm{pr}}$) and the fractions ($f_{\mathit{ij}}$) do not directly map to the orbital interaction and Pauli repulsion energy terms from the canonical EDA method. We also obtain the cumulative sum of fractions over the set ($T=\left\{\left(i,j\right):R_{\mathit{ij}}\ge \tau \right\}$) including all two‐FMO interactions with ranking values larger than the threshold set by the user:

$$F_T=\sum _{\mathit{ij}\in T}{f}_{\mathit{ij}}.$$ (21)

The cumulative sums of the included orbital interactions ($F_T^{\mathrm{oi}}$) and Pauli repulsive interactions ($F_T^{\mathrm{pr}}$) are shown in the PyOrbb GUI next to the associated slider (i.e., the interaction between BH₃(3A1) and NH₃(3A1) represents 95.2% of the total orbital interaction ranking values for this system). The default slider position for the orbital interaction is chosen such that $F_T^{\mathrm{oi}}$ is at least 70%. Similarly, for the Pauli repulsive slider, the default is chosen such that $F_T^{\mathrm{pr}}$ is at least 10%. These defaults ensure that in most systems only the relevant interactions are shown to the user. The sliders also have a minimum and a maximum depending on the ranking values of the interactions that have been detected.

Element 3 is a control panel that provides the user with options to filter the FMOs and MOs that are part of the interaction analysis. This allows the user to select which orbitals are included in the analysis, with quick filtering based on spin, irreducible representations, or specific orbitals of interest. There is also an option to choose the FMO energy type between regular orbital energies (i.e., FMO energies as eigenvalues of the separate fragments), effective orbital energies (recommended for charged fragments), and approximate effective energies (for more information about orbital energy types, see Section S1) [28, 29, 30, 31]. Furthermore, when one or more orbital levels are selected, the “Draw Orbital” and “Filter” buttons are activated. The former calculates and draws orbital iso‐surfaces of the selected orbitals, while the latter applies a filter such that the selected orbitals are always included in the analysis. Pressing the “Filter” button again disables the filter.

Element 4 provides the option to create a comprehensive Excel sheet containing relevant calculation data (vide infra, Section 3.4) and to save the generated MO diagram as a PNG file.

Element 5 summarizes context‐specific information. This box appears when the user clicks an FMO, MO, or interaction line (Figure 4). If an FMO is selected, the information box provides the name of the FMO, its name in terms of its energy position relative to the HOMO or LUMO (e.g., HOMO, HOMO–1, LUMO, LUMO+1, etc.), and the same within its irreducible representation (e.g., A₁ HOMO, A₁ HOMO–1, etc.). Additionally, general information is provided, such as the fragment the FMO belongs to, its orbital energy, its gross Mulliken population in the complex, its gross Mulliken spin population in the complex, its spin, and the irrep it belongs to. Information related to its interactions with other FMOs, such as the overlap (S) and the orbital energy gap (∆ε), is provided below the general information, as well as information about its contribution to MOs, such as the Mulliken contribution of the FMO into the MOs and the respective coefficients. Similar information can be obtained when the user selects an MO. If an interaction line is selected, the information box displays the involved FMO and MO and provides useful information, such as the Mulliken contribution and the FMO coefficient for that MO. The information box also shows the other FMOs that interact with the selected FMO and MO and determines whether the interaction is bonding or antibonding.

FIGURE 4.

Context‐specific information box in Element 5 of the PyOrbb GUI (see Figure 3). Information shown when the user clicks (a) a fragment orbital, (b) a molecular orbital, (c) an interaction line, or (d) two interaction fragment orbitals.

Clicking on an FMO or MO opens an interactive window, where the orbital is displayed graphically. The viewer also provides options to customize how orbitals and molecules are visualized (see Figure 5), as well as to export high‐quality images.

FIGURE 5.

Interactive PyOrbb Viewer screen showing the 3A₁ FMO of the BH₃ fragment.

3.4. PyOrbb Excel File

PyOrbb provides a user‐friendly spreadsheet for summarizing the most important orbital data for detailed orbital analysis (see Figure 6). The spreadsheet contains a sheet providing general information, such as coordinates of the molecules, the number of orbitals, and energy decomposition analysis terms. PyOrbb also adds sheets that provide information about MOs and FMOs, including overlaps, energy gaps, coefficients, and Mulliken contributions. A summary of the most important two‐orbital mixing patterns, both charge‐transfer and Pauli‐repulsive, is also provided. The spreadsheet provides the user with a complete and comprehensive overview of all data found in the fragment calculation.

FIGURE 6.

Example of a spreadsheet generated by PyOrbb for a calculation describing the donor–acceptor interaction between NH₃ and BH₃, showing the MO coefficients originating from the BH₃ FMOs.

4. Applications

To showcase PyOrbb's ability to generate correct MO diagrams that describe bonding mechanisms, we explore seven diverse chemical systems that have been studied extensively in the past, thereby providing data to verify PyOrbb's outcomes. Numerical data for each example is provided in Section S5 (Tables S1–S10). The chemical systems include the analysis of a Lewis acid–base pair H₃B ← NH₃, the C–H bond activation by Fe(CO)₄, the interaction of the reactants in the transition state of a Diels–Alder reaction, the Lewis‐acid catalysis of a Diels–Alder reaction, the hydrogen‐bonding in a guanine‐cytosine Watson‐Crick base pair, the analysis of the pair‐bond formation in ethane and orbital analysis of its radical fragment H₃C^•, and, lastly, the homolytic and heterolytic bond cleavage of fluoromethane. These systems span a wide range of chemical bonding motifs commonly explored by computational chemists. All calculations were performed at the level of density functional theory used in the respective publications, with an all‐electron basis set.

4.1. Donor–Acceptor Bonding: H₃B ← NH₃ Lewis Adduct

The H₃B ← NH₃ complex is a prototypical example of a Lewis adduct and has been previously studied using the KS–MO approach [47]. An ADF input file for this system is provided in Section S4. Upon approach of the Lewis pair, a dative bond is formed by donation of a filled lone‐pair orbital on NH₃ into a virtual orbital of BH₃. PyOrbb successfully ranks this interaction as the main interaction and includes several other important interactions (Figure 7). Despite expecting a relatively simple orbital interaction mechanism, detecting the main interactions from all possible interactions is non‐trivial (Figure 7a). We observe an excess of irrelevant mixing patterns, and importantly, non‐closed mixing patterns.

FIGURE 7.

Orbital interaction diagrams generated by PyOrbb for the H₃B ← NH₃ Lewis adduct showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, and (c) a schematic orbital interaction for the H₃B ← NH₃ Lewis adduct showing the most important interactions identified by PyOrbb. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐BLYP‐D3(BJ)/TZ2P.

The most important interaction in H₃B ← NH₃ (Figure 7) identified by PyOrbb is the main donor‐acceptor orbital interaction where electrons are donated from NH₃(3A₁) into BH₃(3A₁). Additionally, a strong Pauli repulsive interaction is observed between the filled NH₃(3A₁) and BH₃(2A₁) FMOs.

4.2. Oxidative Addition: C–H Bond Activation by Fe(CO)₄

Catalytic reactions are indispensable tools in modern synthetic chemistry [48, 49]. The oxidative addition step is often the first and rate‐determining step in transition metal catalysis [50, 51, 52, 53]. In this example, we analyze the transition state of the oxidative addition of an iron complex into the methane C–H bond [54]. PyOrbb correctly identified the key orbital interaction mechanism (Figure 8). We see two major orbital interaction patterns, namely, the interaction between CH₄(5A) and Fe(CO)₄(42A), which corresponds to the σ‐donation from the σ _C‐H orbital of CH₄ to the d _σ orbital of Fe(CO)₄. PyOrbb also finds the π‐backdonation interaction between CH₄(6A) and Fe(CO)₄(41A), which involves the donation of electrons from the d_π orbital of Fe(CO)₄ to the σ*_C‐H orbital of CH₄.

FIGURE 8.

Orbital interaction diagrams generated by PyOrbb for the transition state of the oxidative insertion of Fe(CO)₄ into the methane C–H bond, showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, and (c) a schematic orbital interaction diagram showing the most important interactions identified by PyOrbb. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐BLYP/TZ2P.

4.3. Diels–Alder Transition State

Diels–Alder cycloadditions are a crucial class of reactions in heterocyclic chemistry [55], materials chemistry [56, 57, 58], and total synthesis [59, 60], and have been extensively studied computationally [61, 62, 63, 64, 65, 66]. In this example, we study the bonding mechanism between the diene, buta‐1,3‐diene, and the dienophile, ethylene, in the Diels–Alder transition state. Generally, in pericyclic transition states, two interactions play an important role, namely the normal electron demand (NED) interaction between a filled orbital on the diene and a virtual orbital of the dienophile, and the inverse electron demand (IED), where electrons are donated from a filled orbital on the dienophile to a virtual orbital on the diene [15, 16, 17, 18, 19, 20, 21, 22]. PyOrbb correctly finds both the NED and IED interactions (Figure 9). Additionally, it identifies the Pauli repulsive interaction between the π‐HOMO‐1 of the diene and the π‐HOMO of the dienophile, which contributes significantly to the Pauli repulsion term in the EDA computation.

FIGURE 9.

Orbital interaction diagrams generated by PyOrbb for the Diels–Alder transition state, showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, and (c) a schematic orbital interaction diagram showing the most important interactions identified by PyOrbb. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐BLYP/TZ2P.

4.4. Lewis‐Acid Catalysis: Diels–Alder

The Diels–Alder reaction can be catalyzed by Lewis acids (LA) [67, 68]. Conventionally, LA catalysis has been explained as a consequence of enhanced reactivity resulting from stronger orbital interactions (viz. LUMO‐lowering catalysis) [2, 69, 70, 71, 72, 73, 74, 75]. In 2020, Vermeeren et al. used ASM and EDA methodologies to point out that the enhanced reactivity upon LA‐catalysis of the Diels–Alder reaction between isoprene and methyl acrylate was due to the reduced four‐electron (Pauli) repulsion (viz. Pauli‐lowering catalysis) [76, 77]. They demonstrated that the LA pulls density away from the dienophile, resulting in a smaller amplitude of occupied π‐orbitals on the C–C double bond of the dienophile. This, in turn, results in less occupied–occupied overlap with the π‐system of the diene and thus less Pauli repulsion.

These insights can be obtained conveniently by applying the PyOrbb program to both the uncatalyzed and LA‐catalyzed reaction (Figure 10). To focus on the Pauli lowering effect, we hide the orbital interactions, leaving only the four‐electron (Pauli) repulsive interactions in the interactive MO diagram. For both systems, the key interaction was identified to be between the occupied diene (π‐HOMO_diene–1), which is the in‐phase combination of the π‐system of the diene, and the π‐bonding orbital in the ester (π‐HOMO_dienophile for both the uncatalyzed and the LA‐catalyzed reaction). As reported in the literature, the lowering of the overlap between these two π‐systems upon LA coordination causes the reduced Pauli repulsion.

FIGURE 10.

Orbital interaction diagrams generated by PyOrbb for the transition state of the Diels–Alder reaction between a diene and ester with (b, d) and without (a, c) AlCl₃ as a Lewis‐acid catalyst showing (a, b) all interactions, (c, d) key Pauli repulsive interactions with their overlap (S). (e) To summarize, a schematic Pauli repulsive interaction diagram for the uncatalyzed (gray lines) and the AlCl₃‐catalyzed reaction (green lines) is shown. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐M06‐2X/TZ2P.

4.5. Hydrogen‐Bonding: Guanine–Cytosine Base Pair

Hydrogen bonds play an important role in biochemical processes [78, 79], as well as areas such as materials science [80, 81], rational drug design [82, 83], and nanotechnology [84, 85, 86]. Therefore, developing a sound physical model of the hydrogen‐bonding mechanism is a task frequently encountered [87]. To illustrate, we investigate the guanine‐cytosine (GC) Watson–Crick base pair, which plays a key role in determining the structure and stability of DNA. Guanine and cytosine form a structurally and electronically complementary base pair through attractive electrostatic and orbital interactions, as revealed by EDA [88]. Because of the importance of the orbital interactions term in the EDA, we inspect the orbital interactions between the two bases. This reveals a complex pattern of donor–acceptor orbital interactions in the σ‐electronic system. Because weak interactions can consist of a large number of orbital interactions with small ranking values $R_{\mathit{ij}}^{\mathrm{oi}}$, it can be valuable to look beyond the initial 70% cutoff of $F_T^{\mathrm{oi}}$. Based on the Mulliken populations, orbital overlap, and energy gap (Equation 12), PyOrbb efficiently generates a reasonable and clear FMO interaction pattern, as illustrated in Figure 11. To understand why these interactions were chosen, refer to the ranking sheet in the Excel file produced by PyOrbb, which quantifies each term included in Equation (12).

FIGURE 11.

Orbital interaction diagrams generated by PyOrbb for the guanine‐cytosine hydrogen‐bond pair showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, and (c) the orbitals that are identified as most significant by PyOrbb. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐BLYP/TZ2P.

The PyOrbb program identified the key orbital interactions in the A′ symmetry (σ‐in plane) as reported by Fonseca Guerra et al. [88]. The strongest orbital mixing and interaction occurs between the cytosine HOMO–1 and the LUMO of guanine, which corresponds to an occupied FMO with lone‐pair character on the cytosine nitrogen atom and an unoccupied FMO with antibonding character on the frontier N–H bonds. Correspondingly, the unoccupied FMOs that PyOrbb identified as central in this hydrogen‐bonded complex show significant N–H antibonding character. Together with the occupied lone‐pair orbitals on a nitrogen or an oxygen atom, they form the σ _LP–σ*_N–H donor–acceptor interactions. Additionally, due to Step 5 of the algorithm, PyOrbb successfully identifies the polarization interaction between the HOMO and HOMO–1 of guanine, which mix to form the overall MOs HOMO and HOMO–1.

A well‐known component of hydrogen bonding is resonance assistance, in which the presence of the hydrogen‐bonding partner induces polarization of the π system through mixing of occupied and unoccupied π orbitals within each base. This effect is predominantly intrafragment in character and is substantially smaller than the direct interfragment donor–acceptor interactions in the A′‐symmetric σ‐electron system of the interacting DNA bases; consequently, it is less prominent in the present FMO interaction analysis [88].

4.6. Homolytic Bond Cleavage: Ethane and the Methyl Radical

The chemical bonding of molecules can be studied using Kohn–Sham MO analysis. In this section, we see that PyOrbb can assess the key interactions for the C–C bond formation and can readily isolate the orbital interactions within the pyramidalized H₃C^• fragment.

To investigate the C–C bond formation in ethane, the radical fragments were treated within the spin‐unrestricted formalism. In Figure 12a, the MO interaction scheme from the ADF‐GUI is presented, for which the most important interactions were identified by PyOrbb and are summarized in Figure 12b and schematically shown in Figure 12c. The key orbital interactions, including the pair bond interaction between the 3A1 orbitals and the Pauli repulsive interaction between the 2A₁, 1E₁ ¹, and 1E₁ ² orbitals on each respective fragment, are correctly described as reported by Rodrigues Silva et al. [89].

FIGURE 12.

Orbital interaction diagrams generated by PyOrbb for the homolytic bond cleavage of ethane, showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, and (c) schematic orbital mixing diagram with α‐spin orbitals. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐UOLYP/TZ2P.

To discuss the formation of the H₃C^• fragment, we fragment the pyramidal H₃C^• into the bare carbon atom, C^••••, and the three hydrogen atoms, 3 H^•. In Figure 13 we see that the pair bond formation occurs between the C (2s) and 1A₁′, C (2p _x ) and the H₃(1E₁ ¹′), and C (2p _y) and the H₃(1E₁ ²′), respectively (for clarity, PyOrbb can separate the orbital mechanisms into the α‐ and β‐spin mechanisms, which are treated separately in the unrestricted formalism). The C (2p _z) is correctly excluded, as this is the H₃C^•(3A₁) singly occupied orbital which partakes in pair bond formation in ethane, and, with a Mulliken contribution of 80.5%, is primarily formed by the C (2p _z) orbital.

FIGURE 13.

Orbital interaction diagrams generated by PyOrbb for the fragmentation of the methyl radical into the carbon atom and hydrogen atoms, showing (a) all interactions, (b) key orbital interactions extracted by PyOrbb, (c) the α‐spin orbital interactions, (d) the β‐spin orbital interactions, and (e) a schematic orbital mixing diagram showing the most important orbital interactions. Orbital isosurfaces drawn at isovalues of ±0.03. Computed at ZORA‐UOLYP/TZ2P.

4.7. Hetero/Homolytic Bond Cleavage: Fluoromethane

In the previous example, we investigated the homolytic bond cleavage of ethane and, subsequently, the fragmentation of the methyl radical. Interestingly, the same bond can be formed in more than one way, that is, from more than just the pair of fragments that is associated with the lowest‐energy dissociation. This issue is addressed in the next example, which showcases the changes in orbital interaction mechanisms associated with the homolytic and heterolytic bond cleavage of fluoromethane using PyOrbb, following the work of Vermeeren and Bickelhaupt [90]. The carbon‐fluorine bond can be cleaved in three ways: homolytically, heterolytically with fluorine taking both bonding electrons, and heterolytically with CH₃ taking both bonding electrons. PyOrbb correctly identifies the orbital interaction mechanism for all three bond cleavage pathways.

In this example, we investigate charged species, which requires using the effective energies of the FMOs (Figure 14a–c) [28, 29, 30, 31]. The effective energies refer to the FMO energies fully relaxed in the field of the molecule, as opposed to the orbital energies obtained from ${\psi }_{\mathrm{A}}$ and ${\psi }_{\mathrm{B}}$ of the isolated fragments (see Section S1). At this point, the FMO energies shift due to the inclusion of electrostatic potential between the fragments. We find a large shift in FMO energies when the effective energies are used for the two heterolytically cleaved systems (Figure 14b,e,c,f), indicating a strong stabilization of the FMOs of the anion due to the field of the cationic fragment, and in the same way, a strong destabilization of the FMOs of the cation. Conversely, the energies of the FMOs of the homolytically cleaved carbon‐fluorine bond are negligibly affected by the field of the uncharged second fragment. The impact of energy choice is illustrated by the heterolytic cleavage of fluoromethane into CH₃ ⁺ and F ⁻ (Figure 14b,e). In Figure 14b, using effective orbital energies corrects the orbital interaction selection: the energy gap between the CH₃ ⁺ (4A′) and F ⁻ (2A′) FMOs (Figure 14e) is only 0.1 eV with regular orbital energies but 17.2 eV with effective energies, making the interaction appear much stronger under regular orbital energies.

FIGURE 14.

Orbital interaction diagrams generated by PyOrbb for the C–F bond of fluoromethane into (a, d) CH₃ ^• + F^•, (b, e) CH₃ ⁺ + F⁻, and (c, f) CH₃ ⁻ + F⁺, showing the most important interactions extracted by PyOrbb using (a, b, c) effective energies for FMOs, that is, the Fock matrix elements after full orbital relaxation (see Section S1) and (d, e, f) orbital energies of the isolated fragments. Computed at ZORA‐(U)BLYP‐D3(BJ)/TZ2P.

5. Conclusions

In this work, we present PyOrbb, an open‐source program that automates the analysis of Kohn–Sham molecular orbital bonding mechanisms of molecules and reactions. This software tool facilitates the analysis of key molecular orbital interactions and efficiently extracts useful information from fragment calculations. PyOrbb (i) reads and prepares raw orbital data from ADF calculations for direct use in analysis, (ii) obtains and presents the most important mixing patterns in an interactive orbital interaction diagram, (iii) conveniently summarizes all key information, including the essential mixing patterns, effective FMO energies, FMO coefficients, and Mulliken analyses in an easily interpretable Excel file, and (iv) exposes all extracted data and provides various analysis tools in an easy‐to‐use Python module.

We showcase the program's effectiveness in determining the key orbital interaction mechanisms for a wide range of chemical interactions and reactions, including donor–acceptor interactions, bond activation by a transition metal catalyst, organic transition states, Lewis‐acid catalysis, hydrogen‐bonded base‐pairs and pair‐bond formation, homo‐ and heterolytic bond cleavage. We also show that the PyOrbb‐generated MO interaction diagrams in all our showcases agree with the literature.

PyOrbb streamlines the systematic analysis of complex bonding mechanisms for atoms and molecules. The user is freed from the formidable task of turning data into bonding mechanisms and can instead focus on interpreting these mechanisms. The open‐source PyOrbb code can be retrieved online along with extensive documentation on our GitHub repository (https://github.com/TheoChem‐VU/PyOrbb).

Funding

This work was supported by the China Scholarship Council (CSC), the Spanish Ministerio de Ciencia, Innovación y Universidades (PID2025‐170997NB‐I00, CEX2021‐001202‐M), the Netherlands Organization for Scientific Research (NWO), Microsoft Research, and the Open Life Science program.

Supporting information

Figure S1: FMO energies of Cl⁻(3p _σ ) (in green) and Na⁺ (3s) (in yellow) taken from isolated fragments (A + B) and in the potential of the NaCl molecule (AB) as implemented in ADF, as well as the approximate effective energies calculated at stage AB. Vertical gray lines indicate the orbital energy gap (ε _LUMO − ε _HOMO) between Na⁺ (3s) and Cl⁻(3p _σ ) calculated at each step of the procedure.

Figure S2: Frontier FMO energies of butadiene (diene, yellow) and ethylene (dienophile, green) taken from isolated fragments (A + B) and in the potential of the molecule (AB) as implemented in ADF, as well as the approximate effective energies calculated at stage AB for the transition state geometry of the (a) normal electron demand and (b) inverse electron demand Diels–Alder reaction. Vertical gray lines indicate the orbital energy gap (ε _LUMO − ε _HOMO) between the relevant occupied and virtual orbitals calculated at each step of the procedure.

Figure S3: FMO energies of F(1A″) (green) and CH₃(4A') (yellow) taken from isolated fragments (A + B) and in the potential of the molecule (AB) as implemented in ADF, as well as the approximate effective energies calculated at stage AB of the bond formation mechanism of CH₃F from (a) F· + CH₃·, (b) F⁻ + CH₃ ⁺, and (c) F⁺ + CH₃ ⁻.

Figure S4: (a) Overlap, (b) coefficient, (c) the Mulliken contribution, and (d) the energy gap matrices of the H₃B ← NH₃ Lewis adduct calculated at the BLYP‐D3(BJ)/SZ level of theory.

Figure S5: (a) Division of the initial populations of ${\psi }_i$ and ${\psi }_j$ over the bonding and antibonding MOs that are formed by their mixing, and (b) the scaling factor as a function of the total population ($o_i+o_j$).

Figure S6: The terms that go into the calculation of the ranking number (d). (a) The second‐order perturbative approximation of the stabilization due to the mixing of filled and unfilled orbitals, (b) the joint electronic activity of the interaction FMOs, and (c) the scaling factor related to the total initial occupations of the interacting FMOs.

Figure S7: The (a) $o_{\text{anti},\mathit{ij}}$ matrix, (b) $S_{\mathit{ij}}^2$ matrix, and (c) the Pauli repulsion ranking matrix for the H₃B ← NH₃ Lewis adduct.

Figure S8: Interaction between two FMOs ${\psi }_i$ and ${\psi }_j$ and the forming MOs ${\Psi }_k$ and ${\Psi }_l$ with the normalized Mulliken contributions on the edges connecting the FMOs to the MOs.

Figure S9: Process of finding matching MOs for (a, b, c) the orbital interaction between NH₃(3A₁) and BH₃(3A₁) and (d, e, f) the Pauli repulsive interaction between NH₃(3A₁) and BH₃(2A₁). Here (a, d) are the masks used for orbital interactions (MO pairs where there is one virtual and one occupied MO) and for Pauli repulsion (two different filled MOs). (b, e) are the $P_{\mathit{ij}}$ matrices of Mulliken contribution products. (c,f) are the masked Mulliken contribution product matrices from which the highest value will be selected.

Figure S10: The combination of the 2‐FMO mixing patterns between NH₃(3A₁) and BH₃(3A₁) (left) with NH₃(3A₁) and BH₃(2A₁) (middle) to form the 3‐FMO mixing pattern.

Figure S11: The main orbital interactions forming cyclobutadiene from two ethylene fragments prepared in their (a) ground state, and (b) in their triplet excited state where an electron from the 4A₁ orbital is excited to 3B₂ as obtained by PyOrbb. In this diagram $p$ refers to the Mulliken population of the SFOs summed over both alpha and beta spins.

Table S1: All quantitative data, as reported in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 7 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the normalized Mulliken contributions (μᵢₖ). Rows highlighted in green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S2: All quantitative data, as reported in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 8 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the normalized Mulliken contributions (μᵢₖ). Rows highlighted in green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S3: All quantitative data, as reported in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 9 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the normalized Mulliken contributions (μᵢₖ). Rows highlighted in green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S4: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 10 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S5: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 11 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S6: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 12 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S7: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 13 of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S8: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 14a,d of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S9: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 14b,e of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Table S10: All quantitative data, as generated in the Excel spreadsheet, for the highest ranked FMO pairs corresponding to the MO diagram in Figure 14c,f of the main article. Listed values include the FMOs (ψᵢ), occupations (oᵢ), gross Mulliken populations (pᵢ), MOs (Ψₖ), overlap (S), orbital energy gap (Δε), ranking values (R ^oi and R ^pr), Fraction and cumulative fraction of the total ranking values (Frac. and $\sum$ Frac.) and the Mulliken contributions (μᵢₖ). Rows highlighted green denote the FMO pairs displayed in the default MO diagram seen if the user loads in the rkf‐file into PyOrbb. Non‐highlighted entries represent the next‐highest‐ranked pairs included to demonstrate the selection cutoff.

Data Availability Statement

The source code for PyOrbb is freely available on GitHub at https://github.com/TheoChem‐VU/PyOrbb under the MIT license. Documentation, tutorials, installation instructions, rkf files, and spreadsheets for the examples described in the main text are provided inside the GitHub repository. The PyOrbb Python module requires Python 3.9 or higher to run. PyOrbb can analyze results from AMS ADF2019 or later (adf.rkf files). The PyOrbb GUI is available for macOS (ARM and intel) and Windows.