Effects of perturbation order and basis set on alchemical predictions

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.

Transferability of atomic energies from alchemical decomposition

We study alchemical atomic energy partitioning as a method to estimate atomization energies from atomic contributions, which are defined in physically rigorous and general ways through the use of the uniform electron gas as a joint reference. We analyze quantitatively the relation between atomic energies and their local environment using a dataset of 1325 organic molecules. The atomic energies are transferable across various molecules, enabling the prediction of atomization energies with a mean absolute error of 23 kcal/mol, comparable to simple statistical estimates but potentially more robust given their grounding in the physics-based decomposition scheme. A comparative analysis with other decomposition methods highlights its sensitivity to electrostatic variations, underlining its potential as a representation of the environment as well as in studying processes like diffusion in solids characterized by significant electrostatic shifts.

Molecular Hessian matrices from a machine learning random forest regression algorithm

In this article, we present a machine learning model to obtain fast and accurate estimates of the molecular Hessian matrix. In this model, based on a random forest, the second derivatives of the energy with respect to redundant internal coordinates are learned individually. The internal coordinates together with their specific representation guarantee rotational and translational invariance. The model is trained on a subset of the QM7 dataset but is shown to be applicable to larger molecules picked from the QM9 dataset. From the predicted Hessian, it is also possible to obtain reasonable estimates of the vibrational frequencies, normal modes, and zero point energies of the molecules.

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.

Relative energies without electronic perturbations via alchemical integral transform

We show that the energy of a perturbed system can be fully recovered from the unperturbed system’s electron density. We derive an alchemical integral transform by parametrizing space in terms of transmutations, the chain rule, and integration by parts. Within the radius of convergence, the zeroth order yields the energy expansion at all orders, restricting the textbook statement by Wigner that the p-th order wave function derivative is necessary to describe the (2 p + 1)-th energy derivative. Without the need for derivatives of the electron density, this allows us to cover entire chemical neighborhoods from just one quantum calculation instead of single systems one by one. Numerical evidence presented indicates that predictive accuracy is achieved in the range of mHa for the harmonic oscillator or the Morse potential and in the range of machine accuracy for hydrogen-like atoms. Considering isoelectronic nuclear charge variations by one proton in all multi-electron atoms from He to Ne, alchemical integral transform based estimates of the relative energy deviate by only few mHa from corresponding Hartree–Fock reference numbers.

Exploration of Chemical Space for Designing Functional Molecules Accounting for Geometric Stability

The design of functional molecules is regarded as searching for molecules with desired functionalities in chemical space populated by candidate molecules. Considering the geometric stability of molecules during the search process is crucial for designing realistic molecules. Here, we propose a method for designing functional molecules by exploring chemical space while explicitly accounting for geometric stability based on computational quantum alchemy. The proposed design method allows the simultaneous prediction of functional molecule in the equilibrium structure and its target desired property in an inverse design fashion without preparing the molecular geometries and performing brute-force screening. The applicability of the design method is proven by obtaining molecules with the desired atomization and electronic energies in various chemical spaces: (BF, CO), (CH₄, NH₃), 18 BN-doped benzene derivatives, and 3.1 × 10⁵ BN-doped phenanthrene derivatives.

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.

Accurate acid dissociation constant (pKa) calculation for the sulfachloropyridazine and similar molecules

Accurate calculation of the acid dissociation constant (pK_a) has fundamental importance for the description of molecular systems with pharmacological activities. The search for a more appropriate procedure for its determination is always welcome and has aroused increasing interest from the scientific community. In this sense, this work presents a computational study involving the combination of ten DFT functionals (M062X, M06L, B3LYP, BLYP, PBEPBE, BP86, LC-BLYP, SPBE, CAM-B3LYP, LC-PBEPBE) and HF method, eight basis set functions (6-311G, 6-311 + G, 6-311G(d,p), 6-311 + G(d,p), 6-311+ +G(d,p), 6-311(2d,2p), 6-311+ +G(2d,2p), and aug-cc-pVDZ), and three solvation models (SMD, PCM, and CPCM) for an accurate sulfachloropyridazine (SCR) pK_a determination. It was found that the smallest deviation (0.02 unit of pK_a) between the current study and experimental result was achieved with the BLYP/6-311 + G(d,p)/PCM combination. Therefore, this combination was extended to calculate the pK_a of six SCR similar molecules selected through the eletroshape similarity method. For all these molecules, the difference between the obtained results and experimental data ranged between 0.14 and 0.69 units of pK_a. This feature suggests that the obtained combination can determine pK_a with experimental precision for complexes that are formed by sulfonamide functional group (SO₂NHR). Graphical Abstract A computational study involving the combination of different levels of theory, basis sets and solvation models for an accurate sulfanamide pK_a determination.