Forces on atoms are both fundamental and crucial physical quantities that give rise to the infinite diversity of matter and its properties. How do atomic forces arise? That is the origin of atomic force is therefore crucial for a comprehensive understanding of a rich variety of structural, mechanical, and dynamic properties as well as physical and chemical phenomena. In particular, the influence of electronic states on atomic forces via chemical bonding and crystal structures, however, remains elusive. Here we develop a computational method for calculating the quantum electronic force, which is decomposed into atomic orbitals and/or spins, implemented in first-principles calculations based on a linear-combination-of-pseudoatomic-orbital (LCPAO) method. We demonstrate this methodology for simple diatomic molecules and extend its application to hydrogen diffusion on a platinum catalyst and the nitrogen-vacancy center in diamond. We clarify that specific orbitals dominate the driving forces for hydrogen diffusion, while spin-dependent forces determine the stable nitrogen-vacancy center configuration in diamond by monitoring quantum electronic forces. Our approach provides electronic insight into structural and dynamic phenomena by visualizing how each electron induces atomic forces.

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• Electronic structure
• First-principles calculations
• Structural properties
• Molecules
• Nitrogen vacancy centers in diamond
• Density functional theory