Alchemical Derivatives of Atoms: A Walk Through the Periodic Table

Abstract

Exploring the Chemical Compound Space is at stake when looking for molecules with optimal properties. In order to guide experimentalists to navigate through this unimaginably huge space, theoreticians should look for efficient and cheap algorithms. One of the strategies put forward some years ago was to look for transmutation of molecular structures, thereby changing their nuclear charge content, for which alchemical derivatives are instrumental. A collection of well tested isolated atom alchemical derivatives would be a basic instrument in a navigation toolbox. In this work, isolated atom alchemical derivatives were evaluated with different techniques, from the more accurate numerical differentiation and Coupled Perturbed Kohn–Sham approaches to the $Z^{-1}$ energy expansion model which upon derivation with respect to Z yields the desired derivatives. For this third approach a systematic, computationally elegant, method is developed to routinely evaluate an optimal set of all expansion coefficients in the energy expansion for a given N. For the lighter elements, $Z=1-18$, the comparison between the three approaches shows that the order of magnitude and sequences in the different approaches are similar paving the way for a walk through the complete Periodic Table by combining the $Z^{-1}$ expansion approach with the National Institute of Standards and Technology (NIST) databank atomic energy values at various levels of LDA. A uniform decrease is retrieved not only for the alchemical potential (the electrostatic potential at the origin) but also for the alchemical hardness, with some minor exceptions. The latter values are relatively strongly influenced by relativistic effects for the heavy elements. The uniform decrease of the first derivative is evidenced and quantified. Periodicity shows up in some exploratory calculations on the third derivative (the hyperhardness) which turn out to be strongly basis set dependent. The Periodic Tables generated could be used in a first step in exploring Chemical Compound Space in a systematic, efficient and cheap way. Some possible refinements (atoms-in-molecules corrections) and extensions (inclusion of mixed Z and N derivatives) are touched upon.

Appendices

Appendices

20.A Solution of the System of Equations Using the NIST Data

The system of equations to be considered is

$$\begin{aligned} N^2 \varepsilon _0^{[N]} + N \varepsilon _1^{[N]} + N^{-1} \varepsilon _3^{[N]}&= E^{[Z=N,N]} ,\end{aligned}$$

(20.29a)

$$\begin{aligned} 2N \varepsilon _0^{[N]} + \varepsilon _1^{[N]} - N^{-2} \varepsilon _3^{[N]}&= V_{en}^{[Z=N,N]} / N ,\end{aligned}$$

(20.29b)

$$\begin{aligned} (N+1)^2 \varepsilon _0^{[N]} + (N+1) \varepsilon _1^{[N]} + \varepsilon _2^{[N]} + (N+1)^{-1} \varepsilon _3^{[N]}&= E^{[Z=N+1,N]} ,\end{aligned}$$

(20.29c)

$$\begin{aligned} 2(N+1) \varepsilon _0^{[N]} + \varepsilon _1^{[N]} - (N+1)^{-2} \varepsilon _3^{[N]}&= V_{en}^{[Z=N+1,N]} / (N+1) , \end{aligned}$$

(20.29d)

where $E^{[Z=N,N]}$, $V_{en}^{[Z=N,N]}$, and $\varepsilon _j^{[N]}$ mean $E^{[Z=N,N]}_{\mathrm {NIST}}$, $V_{en,\, \mathrm {NIST}}^{[Z=N,N]}$, and $\tilde{\varepsilon }_j^{[N;4]}$.

After some simple algebra, the solutions of Eq. (20.29) can be written as follows

$$\begin{aligned} \varepsilon _0^{[N]}&= V_{en}^{[Z=N+1,N]} + V_{en}^{[Z=N,N]} - (2N+1) \varDelta E^ + ,\end{aligned}$$

(20.30a)

$$\begin{aligned} \varepsilon _3^{[N]}&= N^2 (N+1)^2 \left( 2 \varDelta E^+ - \frac{1}{N} V_{en}^{[Z=N,N]} - \frac{1}{N+1} V_{en}^{[Z=N+1,N]} \right) , \end{aligned}$$

(20.30b)

$$\begin{aligned} \varepsilon _1^{[N]}&= \varDelta E^+ - (2N+1) \varepsilon _0^{[N]} + \frac{1}{N(N+1)} \varepsilon _3^{[N]} ,\end{aligned}$$

(20.30c)

$$\begin{aligned} \varepsilon _2^{[N]}&= E^{[Z=N,N]} - N^2 \varepsilon _0^{[N]} -N \varepsilon _1^{[N]} - N^{-1} \varepsilon _3^{[N]} , \end{aligned}$$

(20.30d)

with $\varDelta E^+ = E^{[Z=N+1,N]} - E^{[Z=N,N]}$.

Table 20.6 Alchemical hyperhardness, $\gamma _{\mathrm {al}}$, Eq. (20.31)

20.B Alchemical Hyperhardness Values for Li to Cl

The alchemical hyperhardness can be calculated numerically from Eq. (20.7) or as the second derivative of the alchemical potential, Eq. (20.1):

$$\begin{aligned} \gamma _{\mathrm {al}} [Z=N, N ] = \frac{\mu _{\mathrm {al}}[Z+\delta ,N] - 2 \mu _{\mathrm {al}}[Z,N] + \mu _{\mathrm {al}}[Z-\delta ,N]}{\delta ^2} . \end{aligned}$$

(20.31)

The results in Table 20.6 are calculated with aug-cc-pCVTZ basis set.