Interpolation of property-values between electron numbers is inconsistent with ensemble averaging

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Models studying ground state energy (E) variation with electron number (N) are crucial in quantum chemistry.
• Grand-canonical (GC) ensemble formulations are often used to describe systems with variable particle numbers.
• Interpolating electronic properties between integer electron numbers is a common practice in theoretical models.

Purpose of the Study:

• To investigate the physical foundations of E vs. N models using grand-canonical ensemble formulations.
• To analyze the mathematical and physical consistency of interpolating electronic properties between integer electron numbers.
• To clarify the relationship between smooth energy interpolation and ensemble averaging in electronic structure theory.

Main Methods:

• Analysis of grand-canonical ensemble formulations for electronic systems.
• Mathematical examination of energy interpolation schemes between integer electron numbers.
• Theoretical investigation of differentiability properties of energy functions in relation to ensemble averaging.

Main Results:

• Demonstration that energy interpolation consistent with a GC ensemble is non-differentiable.
• Proof that smooth energy interpolation cannot be formulated as any GC ensemble.
• Conclusion that interpolating electronic properties between integer electron numbers is fundamentally inconsistent with ensemble averaging.

Conclusions:

• Derivative discontinuities are essential for accurate electronic structure calculations involving variable electron numbers.
• The choice of ensemble formulation critically impacts the interpretation of interpolated electronic properties.
• The interaction with the subsystem's surroundings plays a vital role in determining its electronic properties, especially at fractional electron numbers.