The alchemical integral transform revisited

Area of Science:

• Quantum chemistry
• Computational physics
• Theoretical chemistry

Background:

• The Alchemical Integral Transform (AIT) was recently introduced to predict energy differences.
• An ansatz was previously proposed to parameterize the spatial change (r) during an alchemical transformation (λ).

Purpose of the Study:

• To provide a rigorous mathematical derivation of the kernel (K) for the Alchemical Integral Transform (AIT).
• To thoroughly discuss the parameterization of spatial coordinates (r) as a function of alchemical change (λ) in n-dimensional space.
• To explore the necessary conditions, mathematical freedoms, and additional constraints involved in obtaining the r(λ) parameterization.

Main Methods:

• Rigorous mathematical derivation of the AIT kernel (K).
• Analysis of n-dimensional parameterization of spatial coordinates r(λ).
• Derivation of analytical expressions for spectral and density changes.

Main Results:

• A rigorous derivation of the Alchemical Integral Transform (AIT) kernel (K) is presented.
• The parameterization of alchemical change r(λ) in n dimensions is discussed, including conditions and constraints.
• Analytical expressions for energy spectra and density changes are derived for various quantum systems.

Conclusions:

• The rigorous derivation and detailed discussion of AIT enhance its applicability for predicting energy differences.
• The presented methods and results are validated through applications to diverse quantum mechanical systems.
• This work provides a robust theoretical framework for alchemical transformations in quantum systems.