  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, has been used in 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 He$_3$ into H$_6$. Subsequently, mixed alchemical and spatial
derivatives are used to relax H$_2$ from the electronic structure of He alone,
highlighting a path to spatially relaxed quantum alchemy. Finally, the
underlying code (APHF) which allows for arbitrary precision evaluation of
restricted Hartree-Fock energies and arbitrary-order derivatives is made
available to support future method development.
