Exploring Chemical Space with Alchemical Derivatives: BN-Simultaneous Substitution Patterns in C60

The use of collective variables and enhanced sampling in the simulations of existing and emerging microporous materials

Microporous materials, including zeolites, metal-organic frameworks, and cage compounds, offer diverse functionalities due to their unique dynamics and guest confinement properties. These materials play a significant role in separation, catalysis, and sensing, but their complexity hinders exploration using traditional atomistic simulations. This review explores collective variables (CVs) paired with enhanced sampling as a powerful approach to enable efficient investigation of key features in microporous materials. We highlight successful applications of CVs in studying adsorption, diffusion, phase transitions, and mechanical properties, demonstrating their crucial role in guiding material design and optimisation. The future of CVs lies in integration with techniques like machine learning, allowing for enhanced efficiency and accuracy. By tailoring CVs to specific materials and developing multi-scale approaches we can further unlock the intricacies of these fascinating materials. Simulations are a cornerstone in unravelling the complexities of microporous materials and are crucial for our future understanding.

Extending the definition of atomic basis sets to atoms with fractional nuclear charge

Alchemical transformations showed that perturbation theory can be applied also to changes in the atomic nuclear charges of a molecule. The alchemical path that connects two different chemical species involves the conceptualization of a non-physical system in which an atom possess a non-integer nuclear charge. A correct quantum mechanical treatment of these systems is limited by the fact that finite size atomic basis sets do not define exponents and contraction coefficients for fractional charge atoms. This paper proposes a solution to this problem and shows that a smooth interpolation of the atomic orbital coefficients and exponents across the periodic table is a convenient way to produce accurate alchemical predictions, even using small size basis sets.

AROFRAG─A Systematic Approach for Fragmentation of Aromatic Molecules

We present a new systematic fragmentation scheme of polycyclic aromatic hydrocarbons (PAHs), including fullerenes and nanotubes, based on an idea to treat a sextet ring as a single unbreakable unit so that the basic unit of aromaticity remains preserved upon fragmentation. In the approach, denoted as AROFRAG (from aromatic fragmentation), a set of predefined elementary subsystems, such as naphthalene and biphenyl in the first model and larger PAHs in the second and third models, is generated with appropriate weights with the aim of reproducing the structure of the original molecule. The energies of the molecules are approximated as weighted sums of the energies of these subsystems. For symmetric cases, e.g., fullerenes, the point-group symmetry is preserved during the decomposition. The AROFRAG is used in conjunction with the molecule-in-molecule (MIM) technique to obtain an accurate description of the electronic energies. The new approach has been applied for selected graphene structures and fullerene doped with boron and nitrogen atoms, for which isomerization energies were calculated, as well as for several nanotubes and regular fullerene molecules. The combination of the third AROFRAG model and the MIM approach leads to the reproduction of electronic energies with a few milli-hartree accuracy at a fraction of the computational cost of the original method.

Optimization of General Molecular Properties in the Equilibrium Geometry Using Quantum Alchemy: An Inverse Molecular Design Approach

Inverse molecular design allows the optimization of molecules in chemical space and is promising for accelerating the development of functional molecules and materials. To design realistic molecules, it is necessary to consider geometric stability during optimization. In this work, we introduce an inverse design method that optimizes molecular properties by changing the chemical composition in the equilibrium geometry. The optimization algorithm of our recently developed molecular design method has been modified to allow molecular design for general properties at a low computational cost. The proposed method is based on quantum alchemy and does not require empirical data. We demonstrate the applicability and limitations of the present method in the optimization of the electric dipole moment and atomization energy in small chemical spaces for (BF, CO), (N₂, CO), BN-doped benzene derivatives, and BN-doped butane derivatives. It was found that the optimality criteria scheme adopted for updating the molecular species yields faster convergence of the optimization and requires a less computational cost. Moreover, we also investigate and discuss the applicability of quantum alchemy to the electric dipole moment.

On the Prediction of Lattice Energy with the Fukui Potential: Some Supports on Hardness Maximization in Inorganic Solids

Using perturbation theory within the framework of conceptual density functional theory, we derive a lower bound for the lattice energy of the ionic solids. The main element of the lower bound is the Fukui potential in the nuclei of the molecule corresponding to the unit formula of the solid. Thus, we propose a model to calculate the lattice energy in terms of the Fukui potential. Our method, which is extremely simple, performs well as other methods using the crystal structure information of alkali halide solids. The method proposed here correlates surprisingly well with the experimental data on the lattice energy of a diverse series of solids having even a non-negligible covalent characteristic. Finally, the validity of the maximum hardness principle (MHP) is assessed, showing that in this case, the MHP is limited.

Alchemical geometry relaxation

We propose to relax geometries throughout chemical compound space (CCS) using alchemical perturbation density functional theory (APDFT). APDFT refers to perturbation theory involving changes in nuclear charges within approximate solutions to Schr\"odinger's equation. We give an analytical formula to calculate the mixed second order energy derivatives with respect to both, nuclear charges and nuclear positions (named "alchemical force"), within the restricted Hartree-Fock case.We have implemented and studied the formula for its use in geometry relaxation of various reference and target molecules.We have also analysed the convergence of the alchemical force perturbation series, as well as basis set effects.Interpolating alchemically predicted energies, forces, and Hessian to a Morse potential yields more accurate geometries and equilibrium energies than when performing a standard Newton Raphson step. Our numerical predictions for small molecules including BF, CO, N2, CH$_4$, NH$_3$, H$_2$O, and HF yield mean absolute errors of of equilibrium energies and bond lengths smaller than 10 mHa and 0.01 Bohr for 4$^\text{th}$ order APDFT predictions, respectively.Our alchemical geometry relaxation still preserves the combinatorial efficiency of APDFT: Based on a single coupled perturbed Hartree Fock derivative for benzene we provide numerical predictions of equilibrium energies and relaxed structures of all the 17 iso-electronic charge-netural BN-doped mutants with averaged absolute deviations of $\sim$27 mHa and $\sim$0.12 Bohr, respectively.

Arbitrarily accurate quantum alchemy

Doping compounds can be considered a perturbation to the nuclear charges in a molecular Hamiltonian. Expansions of this perturbation in a Taylor series, i.e., quantum alchemy, have been used in the literature to assess millions of derivative compounds at once rather than enumerating them in costly quantum chemistry calculations. So far, it was unclear whether this series even converges for small molecules, whether it can be used for geometry relaxation, and how strong this perturbation may be to still obtain convergent numbers. This work provides numerical evidence that this expansion converges and recovers the self-consistent energy of Hartree-Fock calculations. The convergence radius of this expansion is quantified for dimer examples and systematically evaluated for different basis sets, allowing for estimates of the chemical space that can be covered by perturbing one reference calculation alone. Besides electronic energy, convergence is shown for density matrix elements, molecular orbital energies, and density profiles, even for large changes in electronic structure, e.g., transforming He₃ into H₆. Subsequently, mixed alchemical and spatial derivatives are used to relax H₂ from the electronic structure of He alone, highlighting a path to spatially relaxed quantum alchemy. Finally, the underlying code that allows for arbitrarily accurate evaluation of restricted Hartree-Fock energies and arbitrary order derivatives is made available to support future method development.

Links among the Fukui potential, the alchemical hardness and the local hardness of an atom in a molecule

This paper presents a brief summary of the difficulty that resides in the definition of the elusive concept of local chemical hardness. We argue that a definition of local hardness should be useful to a reactivity principle and not just as a mere definition. We then continue with a formal discussion about the benefits and difficulties of using the Fukui potential, which is interpreted as an alchemical derivative (alchemical hardness), as descriptor of local hardness of molecules. Computational evidence shows that the alchemical hardness is at least as good a descriptor as the combination of other two well-stabilized descriptors of local hardness, such as the Fukui function and grand canonical local hardness. Although our results are auspicious for the alchemical hardness as descriptor of local hardness, we finish by calling the attention of the community on the importance of discussing the raison d'être of a local hardness function and its main characteristics. We suggest that an axiomatic construction of local hardness could be they way of constructing a local hardness which is both useful and free of arbitrariness.

Ab Initio Machine Learning in Chemical Compound Space

Chemical compound space (CCS), the set of all theoretically conceivable combinations of chemical elements and (meta-)stable geometries that make up matter, is colossal. The first-principles based virtual sampling of this space, for example, in search of novel molecules or materials which exhibit desirable properties, is therefore prohibitive for all but the smallest subsets and simplest properties. We review studies aimed at tackling this challenge using modern machine learning techniques based on (i) synthetic data, typically generated using quantum mechanics based methods, and (ii) model architectures inspired by quantum mechanics. Such Quantum mechanics based Machine Learning (QML) approaches combine the numerical efficiency of statistical surrogate models with an ab initio view on matter. They rigorously reflect the underlying physics in order to reach universality and transferability across CCS. While state-of-the-art approximations to quantum problems impose severe computational bottlenecks, recent QML based developments indicate the possibility of substantial acceleration without sacrificing the predictive power of quantum mechanics.

Quantifications and Applications of Relative Fisher Information in Density Functional Theory

Though density functional theory is widely accepted as one of the most successful developments in theoretical chemistry in the past few decades, the knowledge of how to apply this new electronic structure theory, to help us better understand chemical processes and transformations, is still an unaccomplished task. The information-theoretic approach is emerging as a viable option for that purpose in the recent literature, providing new insights about steric effect, cooperativity, electrophilicity, nucleophilicity, stereoselectivity, homochirality, etc. In this work, based on the result from a recent paper by one of us [ J. Chem. Phys, 2019, 151, 141103], we present two quantifications of the relative Fisher information and discuss their physiochemical properties and possible applications. To that end, their analytical properties have been elucidated. They have also been applied to six categories of systems to illustrate their applicability. A better descriptor to quantify the single bond rotation barrier has been obtained. The relative Fisher information can also simultaneously determine electrophilicity and nucleophilicity, and effectively describe helical structures with different homochiral and heterochiral propensities. As integral parts of the information-theoretic approach, these newly introduced quantities will provide us with more analytical tools toward the long-term goal of crafting a chemical reactivity theory in the density-based language.

Simplifying inverse materials design problems for fixed lattices with alchemical chirality

Brute-force compute campaigns relying on demanding ab initio calculations routinely search for previously unknown materials in chemical compound space (CCS), the vast set of all conceivable stable combinations of elements and structural configurations. Here, we demonstrate that four-dimensional chirality arising from antisymmetry of alchemical perturbations dissects CCS and defines approximate ranks, which reduce its formal dimensionality and break down its combinatorial scaling. The resulting "alchemical" enantiomers have the same electronic energy up to the third order, independent of respective covalent bond topology, imposing relevant constraints on chemical bonding. Alchemical chirality deepens our understanding of CCS and enables the establishment of trends without empiricism for any materials with fixed lattices. We demonstrate the efficacy for three cases: (i) new rules for electronic energy contributions to chemical bonding; (ii) analysis of the electron density of BN-doped benzene; and (iii) ranking over 2000 and 4 million BN-doped naphthalene and picene derivatives, respectively.

Effects of perturbation order and basis set on alchemical predictions

Alchemical perturbation density functional theory has been shown to be an efficient and computationally inexpensive way to explore chemical compound space. We investigate approximations made, in terms of atomic basis sets and the perturbation order, introduce an electron-density based estimate of errors of the alchemical prediction, and propose a correction for effects due to basis set incompleteness. Our numerical analysis of potential energy estimates, and resulting binding curves, is based on coupled-cluster single double (CCSD) reference results and is limited to all neutral diatomics with 14 electrons (AlH⋯NN). The method predicts binding energy, equilibrium distance, and vibrational frequencies of neighboring out-of-sample diatomics with near CCSD quality using perturbations up to the fifth order. We also discuss simultaneous alchemical mutations at multiple sites in benzene.

Predicting Deprotonation Sites Using Alchemical Derivatives

An alchemical transformation is any process, physical or fictitious, that connects two points in the chemical space. A particularly important transformation is the vanishing of a proton, whose energy can be linked to the proton dissociation enthalpy of acids. In this work we assess the reliability of alchemical derivatives in predicting the proton dissociation enthalpy of a diverse series of mono- and polyprotic molecules. Alchemical derivatives perform remarkably well in ranking the proton affinity of all molecules. Additionally, alchemical derivatives could be use also as a predictive tool because their predictions correlate quite well with calculations based on energy differences and experimental values. Although second-order alchemical derivatives underestimate the dissociation enthalpy, the deviation seems to be almost constant. This makes alchemical derivatives extremely accurate to evaluate the difference in proton affinity between two acid sites of polyprotic molecule. Finally, we show that the reason for the underestimation of the dissociation enthalpy is most likely the contribution of higher-order derivatives.

Rapid and accurate molecular deprotonation energies from quantum alchemy

We assess the applicability of alchemical perturbation density functional theory (APDFT) for quickly and accurately estimating deprotonation energies. We have considered all possible single and double deprotonations in one hundred small organic molecules drawn at random from QM9 [Ramakrishnan et al., JCTC, 2015]. Numerical evidence is presented for 5160 deprotonated species at both HF/def2-TZVP and CCSD/6-31G* levels of theory. We show that the perturbation expansion formalism of APDFT quickly converges to reliable results: using CCSD electron densities and derivatives, regular Hartree-Fock calculations are outperformed at the second or third order for ranking all possible doubly or singly deprotonated molecules, respectively. CCSD single deprotonation energies are reproduced within 1.4 kcal mol^-1 on average within third order APDFT. We introduce a hybrid approach where the computational cost of APDFT is reduced even further by mixing first order terms at a higher level of theory (CCSD) with higher order terms at a lower level of theory only (HF). We find that this approach reaches 2 kcal mol^-1 accuracy in absolute deprotonation energies compared to CCSD at 2% of the computational cost of third order APDFT.

Atoms in Molecules from Alchemical Perturbation Density Functional Theory

Based on thermodynamic integration, we introduce atoms in molecules (AIM) using the orbital-free framework of alchemical perturbation density functional theory (APDFT). Within APDFT, atomic energies and electron densities in molecules are arbitrary because any reference system and integration path can be selected as long as it meets the boundary conditions. We choose the uniform electron gas (jellium) as a reference and linearly scale up all nuclear charges, situated at any query molecule's atomic coordinates. Within the approximations made when calculating one-particle electron densities, this universal choice affords unambiguous and exact definitions of energies and electron densities of AIMs. Numerical results are presented for neutral small molecules (CO, N₂, BF, CO₂), various small molecules with different electronic hybridization states of carbon (CH₄, C₂H₆, C₂H₄, C₂H₂, HCN), and all of the possible BN-doped mutants connecting benzene to borazine (C_2nB_3-nN_3-nH₆, 0 ≤ n ≤ 3). Our results, as well as comparison to atomic energy estimates resulting from either DFT trained neural network models or atomic basis set overlap within CCSD, suggest that APDFT based AIMs enable meaningful, interesting, and counterintuitive interpretations of chemical bonding and molecular electron densities.

Search for Catalysts by Inverse Design: Artificial Intelligence, Mountain Climbers, and Alchemists

In silico catalyst design is a grand challenge of chemistry. Traditional computational approaches have been limited by the need to compute properties for an intractably large number of possible catalysts. Recently, inverse design methods have emerged, starting from a desired property and optimizing a corresponding chemical structure. Techniques used for exploring chemical space include gradient-based optimization, alchemical transformations, and machine learning. Though the application of these methods to catalysis is in its early stages, further development will allow for robust computational catalyst design. This review provides an overview of the evolution of inverse design approaches and their relevance to catalysis. The strengths and limitations of existing techniques are highlighted, and suggestions for future research are provided.

The chemical space of B, N-substituted polycyclic aromatic hydrocarbons: Combinatorial enumeration and high-throughput first-principles modeling

Combinatorial introduction of heteroatoms in the two-dimensional framework of aromatic hydrocarbons opens up possibilities to design compound libraries exhibiting desirable photovoltaic and photochemical properties. Exhaustive enumeration and first-principles characterization of this chemical space provide indispensable insights for rational compound design strategies. Here, for the smallest seventy-seven Kekulean-benzenoid polycyclic systems, we reveal combinatorial substitution of C atom pairs with the isosteric and isoelectronic B, N pairs to result in 7 453 041 547 842 (7.4 tera) unique molecules. We present comprehensive frequency distributions of this chemical space, analyze trends, and discuss a symmetry-controlled selectivity manifestable in synthesis product yield. Furthermore, by performing high-throughput ab initio density functional theory calculations of over thirty-three thousand (33k) representative molecules, we discuss quantitative trends in the structural stability and inter-property relationships across heteroarenes. Our results indicate a significant fraction of the 33k molecules to be electronically active in the 1.5-2.5 eV region, encompassing the most intense region of the solar spectrum, indicating their suitability as potential light-harvesting molecular components in photo-catalyzed solar cells.

A first step towards quantum energy potentials of electron pairs

A first step towards the construction of a quantum force field for electron pairs in direct space is taken. Making use of topological tools (Interacting Quantum Atoms and the Electron Localisation Function), we have analysed the dependency of electron pairs electrostatic, kinetic and exchange-correlation energies upon bond stretching. Simple correlations were found, and can be explained with elementary models such as the homogeneous electron gas. The resulting energy model is applicable to various bonding regimes: from homopolar to highly polarized and even to non-conventional bonds. Overall, this is a fresh approach for developing real space-based force fields including an exchange-correlation term. It provides the relative weight of each of the contributions, showing that, in common Lewis structures, the exchange correlation contribution between electron pairs is negligible. However, our results reveal that classical approximations progressively fail for delocalised electrons, including lone pairs. This theoretical framework justifies the success of the classic Bond Charge Model (BCM) approach in solid state systems and sets the basis of its limits. Finally, this approach opens the door towards the development of quantitative rigorous energy models based on the ELF topology.

Alchemical Normal Modes Unify Chemical Space

In silico design of new molecules and materials with desirable quantum properties by high-throughput screening is a major challenge due to the high dimensionality of chemical space. To facilitate its navigation, we present a unification of coordinate and composition space in terms of alchemical normal modes (ANMs) which result from second order perturbation theory. ANMs assume a predominantly smooth nature of chemical space and form a basis in which new compounds can be expanded and identified. We showcase the use of ANMs for the energetics of the isoelectronic series of diatomics with 14 electrons, BN doped benzene derivatives (C_{6-2 x}(BN) _xH₆ with x = 0,1,2,3), predictions for over 1.8 million BN doped coronene derivatives, and genetic energy optimizations in the entire BN-doped coronene space. Using Ge lattice scans as reference, the applicability of ANMs across the periodic table is demonstrated for III-V and IV-IV semiconductors Si, Sn, SiGe, SnGe, SiSn, as well as AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, and InSb. Analysis of our results indicates simple qualitative structure property rules for estimating energetic rankings among isomers. Useful quantitative estimates can also be obtained when few atoms are changed to neighboring or lower lying elements in the periodic table. The quality of the predictions often increases with the symmetry of system chosen as reference due to cancellation of odd order terms. Rooted in perturbation theory, the ANM approach promises to generally enable unbiased compound exploration campaigns at reduced computational cost.

Exploring chemical space with alchemical derivatives: alchemical transformations of H through Ar and their ions as a proof of concept

Alchemical derivatives have been used previously to obtain information about transformations in which the number of electrons is unchanged. Here an approach for combining changes in both the number of electrons and the nuclear charge is presented.