Chapter 2 - Exploring chemical space with alchemical derivatives

Abstract

The concept, basic theory and implementation and the advantages of alchemical derivatives are sketched in the broad context of exploring chemical space. The Alchemical Derivatives approach is casted in the framework of Conceptual Density Functional Theory (CDFT) in which, for the case considered, the nuclear charges are playing a predominant role permitting to navigate between atomic and molecular systems with different nuclear constellation. The theoretical framework is set up starting from an energy functional (and its functional Taylor expansion) depending on the external potential (involving nuclear charges and geometrical variables), the number of electrons and the spin number. In the expansion alchemical derivatives are pinpointed as those in which at least a derivation with respect to one nuclear charge is involved. Their electronic and nuclear components are scrutinized and relationships with more traditional CDFT descriptors are highlighted at first, second and third order. The evaluation of the derivatives is discussed both analytically in a Coupled Perturbed approach and numerically with particular attention to the role of the basis-set and the accuracy of numerical differentiation in the latter case. The link with von Lilienfeld's Alchemical Coupling approach is established pinpointing a major advantage of the Alchemical Derivative approach, namely the need for only a single SCF calculation and its alchemical derivatives of the parent molecule. The applications are treated in ascending order of complexity, varying from annihilation of the simplest atom (H) (deprotonation energies) to changing a more complex central atom in a series of AX4 molecules, to mutating simultaneously two atoms, with N2 as a case study. Its conclusive results enabled a systematic study of the BN substitution of benzene and its 3D analogue C60, with a substitution pattern (CC-BN)n with n varying from 1 up to the fully transmuted system (BN)30. A number of simple “rules of thumb” governing the complete substitution process were derived. The constraint that the number of electrons should be preserved is dropped in the transmutation of atoms where now both the number of electrons and the nuclear charge are allowed to vary. Extremely promising results are obtained for transformation processes where a given atom is transmuted in one of its neutral neighbors with a precision reaching chemical accuracy, i.e. less than 1 kcal/mol.