Energy Landscape of State-Specific Electronic Structure Theory

Area of Science:

• Quantum Chemistry
• Theoretical Chemistry
• Computational Chemistry

Background:

• State-specific approximations improve electronic excitation calculations by allowing electron density relaxation.
• The global structure of the exact electronic energy landscape is not well understood.
• Approximate wave functions are known local minima or saddle points of approximate energy.

Purpose of the Study:

• Introduce a geometric perspective on the exact electronic energy landscape.
• Investigate the connectivity of exact stationary points.
• Explore how the exact energy landscape influences state-specific approximations.

Main Methods:

• Geometric analysis of the exact electronic energy landscape.
• Constraining stationary points to a hypersphere surface.
• Analyzing the relationship between energy gradient and Hamiltonian variance.
• Using minimal basis Hartree-Fock and excited-state mean-field for H2 (STO-3G).

Main Results:

• Ground and excited states are stationary points on the exact energy landscape with increasing Hessian index.
• The square-magnitude of the exact energy gradient is proportional to the Hamiltonian variance.
• Approximate excited states correspond to constrained stationary points on the exact energy landscape.
• Hessian index increases for higher energy approximate states.

Conclusions:

• The geometric structure of the exact energy landscape dictates the properties of state-specific approximations.
• Challenges in variance optimization algorithms stem from the landscape's properties, including unphysical saddle points.