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Within computational quantum alchemy, observables of one system $B$, such asthe energy $E_B$, can be obtained by perturbing the external potential of adifferent iso-electronic system $A$, simply by Taylor expansion of $E_A$ in analchemical change $\lambda$. We recently introduced the Alchemical IntegralTransform (AIT) enabling the effective prediction of $E_B$ but requiring us toparametrize space $\pmb{r}$ in terms of $\lambda$ for which we used an Ansatzfound by trial and error. The kernel $\mathcal{K}$ of AIT also requiredpre-computed solutions of certain Diophantine equations which increasedcomputational complexity and cost. Here, we present the exact one-dimensionalsolution to $\pmb{r}(\lambda)$, together with the full one-dimensional kernel$\mathcal{K}$. Numerical evidence and a proof are presented in form of thequantum harmonic oscillator (QHO) and the Morse potential. Furthermore, wederive an analytical expression of $\mathcal{K}$ for the free atom whichamounts to proof that the energy varies quadratically exactly, notapproximately, in nuclear charge $Z_B$ for the entire iso-electronic series ofall the possible charged ions, i.e. $E_B(Z, N_e)\propto-\frac{1}{2}Z_B^2$.Comparison to Levy's averaging formula for relative energies of iso-electronicsystems also enables us to derive his residual error term. Finally, we rewrite$\mathcal{K}$ in a simple, compact and easy to implement form for arbitrarychanges in external potentials $$E_B-E_A=\int_{\mathbb{R}^n}d\pmb{y}_A\,\rho_A(\pmb{y}_A)\,\mathcal{K}[v_A,v_B](\pmb{y}_A)=\int_{\mathbb{R}^n}d\pmb{y}_A\,\rho_A(\pmb{y}_A)\frac{v_B(\pmb{y}_A)-v_A(\pmb{y}_A)}{(2\pi)^n}\int_{\mathbb{R}^n}d\pmb{\phi}\int_{\mathbb{R}^n}d\pmb{r}_A\int_0^1d\lambda\,e^{i\pmb{\phi}\cdot\left(\pmb{y}_A-\pmb{r}(\lambda)\right)}$$Here, $v_A,v_B$ are initial and final external potential, $\pmb{r}(\lambda)$the $n$-dimensional parametrization.

Generalized Alchemical Integral Transform and the multi-electron atom energy