O1NumHess: A Fast and Accurate Seminumerical Hessian Algorithm Using Only O(1) Gradients

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Theoretical Chemistry

Background:

• Calculating the Hessian of molecular systems is crucial for determining vibrational frequencies and thermodynamic properties.
• Conventional seminumerical Hessian algorithms require a large number of displaced geometries, leading to high computational cost.

Purpose of the Study:

• To introduce a novel algorithm, O1NumHess, for efficient Hessian calculation.
• To reduce the number of gradient evaluations required for Hessian computation.

Main Methods:

• Developed O1NumHess algorithm utilizing finite differentiation of gradients at O(1) displaced geometries.
• Leveraged the off-diagonal low-rank (ODLR) property of Hessians to reduce independent entries from O(N_atom^2) to O(N_atom).
• Implemented and tested the algorithm using the BDF program on various molecular systems.

Main Results:

• O1NumHess achieves accuracy comparable to conventional double-sided seminumerical Hessians for frequencies, zero-point energies, and free energies.
• The algorithm demonstrates significant speed improvements over conventional numerical and often analytic Hessian methods.
• Requires only approximately 100 gradients for large systems, a substantial reduction from traditional approaches.

Conclusions:

• O1NumHess provides an efficient and accurate alternative for calculating molecular Hessians.
• The method's speed and reduced computational demand make it suitable for large molecular systems.
• An open-source implementation is available, applicable beyond computational chemistry.