Note: An explicit solution of the optimal superposition and Eckart frame problems.

Area of Science:

• Computational physics
• Molecular dynamics
• Quantum chemistry

Background:

• The optimal superposition problem and the Eckart frame problem are fundamental in comparing molecular structures.
• Existing solutions often involve computationally intensive methods like matrix diagonalization or quaternion algebra.

Purpose of the Study:

• To introduce a simplified, explicit solution for both optimal superposition and Eckart frame problems.
• To demonstrate a method that avoids complex mathematical techniques.
• To explore the implications of using different forms of the solution matrix T.

Main Methods:

• Developed an explicit solution for the optimal superposition and Eckart frame problems.
• Introduced a single variable change in the solution matrix T for flexibility.
• Analyzed the selection of T for proper rotations versus general orthogonal transformations.

Main Results:

• An explicit solution is derived without matrix diagonalization or quaternion algebra.
• A single variable modification in the solution matrix T enables selection between proper rotations and general orthogonal transformations.
• The equivalence of the two problems and the implications of alternative selections are discussed.

Conclusions:

• The proposed method offers a computationally efficient alternative for solving optimal superposition and Eckart frame problems.
• The flexibility in selecting the transformation matrix T enhances the applicability of the solution.
• This approach simplifies structural comparisons in various scientific domains.