Expressivity of Determinantal Ansatzes for Neural Network Wave Functions.

Area of Science:

• Quantum mechanics
• Computational chemistry
• Artificial intelligence in science

Background:

• Neural network wave functions are increasingly used for the many-body quantum problem.
• Common methods involve determinants or sums of determinants for antisymmetrization.
• Projection onto fixed-spin states is typical but limited to spin-independent operators.

Purpose of the Study:

• To investigate the relationship between different wave function types for quantum problems.
• To determine the applicability of these wave functions to spin-dependent Hamiltonians.
• To establish strict upper bounds between various wave function representations.

Main Methods:

• Developing and analyzing neural network wave function formalisms.
• Investigating properties of Hartree-Fock-like determinants, full spinor wave functions, and full determinant wave functions.
• Performing numerical experiments on the H3 molecule and 2D homogeneous electron gas.

Main Results:

• A strict upper bound property is demonstrated for spin-independent Hamiltonians.
• The relationship between spinor and full determinant wave functions is clarified through projection.
• Full determinant wave functions are shown to be inapplicable to spin-dependent Hamiltonians due to lack of antisymmetry.

Conclusions:

• The established bounds provide a theoretical framework for understanding neural network wave function behavior.
• The findings highlight limitations of fixed-spin projections for spin-dependent quantum systems.
• Numerical validation confirms the theoretical bounds for molecular and condensed matter systems.