Formally Exact and Practically Useful Analytic Solution of Harmonium.

Area of Science:

• Quantum Mechanics
• Computational Chemistry
• Atomic Physics

Background:

• Harmonium, a fundamental model system in quantum mechanics, describes interacting electrons.
• Solving the Schrödinger equation for harmonium, especially with strong Coulomb interactions, is computationally challenging.
• Existing methods often rely on approximations or basis-expansion techniques, limiting accuracy and efficiency.

Purpose of the Study:

• To develop a novel, exact analytic solution for the Schrödinger equation of harmonium.
• To provide accurate wave function representations for both ground and excited states across arbitrary Coulomb interaction strengths.
• To analyze the behavior of electron density and natural occupation numbers in the strong correlation limit.

Main Methods:

• A recently developed method for solving Schrödinger equations was employed.
• Three formally exact analytic representations of the wave function were compared.
• The efficiency of the new method was benchmarked against basis-expansion methods.

Main Results:

• An exact factorized form, including a noninteger power pre-exponential factor, exponential decay, and modulator function, was identified as the optimal representation for the ground state wave function.
• Additional factors were determined to be necessary for accurately representing excited states, incorporating nodal information.
• The proposed method demonstrated significantly higher efficiency compared to traditional basis-expansion techniques.

Conclusions:

• The novel analytic solution provides an accurate and efficient means to describe harmonium systems.
• The findings offer valuable insights into the physics of strong electron correlation.
• This method advances the computational treatment of quantum systems with arbitrary interaction strengths.