Reconstruction of Local Potentials from Matrix Representations in Arbitrary Basis Sets.

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Applied Mathematics

Background:

• Multiplicative potential operators are frequently represented as matrices within finite basis sets.
• Reconstructing the real-space potential from these matrices is typically an ill-posed problem.
• This difficulty arises from inherent linear dependencies among products of basis functions.

Purpose of the Study:

• To develop a robust method for recovering real-space potentials from matrix representations.
• To overcome the ill-posed nature of potential reconstruction in finite basis sets.
• To provide a solution applicable to arbitrarily large basis sets and various potential representations.

Main Methods:

• Replaced conventional truncation and regularization with importance-based selection of basis function products.
• Iteratively identified a subset of linearly independent and important products for well-conditioned reconstruction.
• Utilized underlying linear dependencies to recover the remaining matrix elements.

Main Results:

• Demonstrated a robust solution to the ill-posed problem of real-space potential reconstruction.
• Achieved exact reproduction of matrix elements for the selected important products.
• Showed convergence to the exact real-space potential in the complete-basis-set limit.

Conclusions:

• The proposed importance-based method offers a stable and accurate approach to potential reconstruction.
• This method effectively handles linear dependencies, a key challenge in the field.
• The technique is versatile, applicable to various representations and large basis sets, advancing computational quantum mechanics.