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An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum...
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According to valence bond theory, a covalent bond results when: (1) an orbital on one atom overlaps an orbital on a second atom, and (2) the single electrons in each orbital combine to form an electron pair. The strength of a covalent bond depends on the extent of overlap of the orbitals involved. Maximum overlap is possible when the orbitals overlap on a direct line between the two nuclei.
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sp3d and sp3d 2 Hybridization
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Van der Waals Equation01:10

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The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
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Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. However, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. 
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IR Spectroscopy: Hooke's Law Approximation of Molecular Vibration01:16

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A covalently bonded heteronuclear diatomic molecule can be modeled as two vibrating masses connected by a spring. The vibrational frequency of the bond can be expressed using an equation derived from Hooke's law, which describes how the force applied to stretch or compress a spring is proportional to the displacement of the spring. In this case, the atoms behave like masses, and the bond acts like a spring.
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Updated: Aug 15, 2025

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
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Accurate Hellmann-Feynman forces from density functional calculations with augmented Gaussian basis sets.

Shivesh Pathak1, Ignacio Ema López2, Alex J Lee3

  • 1Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87123, USA.

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The Hellmann-Feynman theorem enables efficient force calculations using machine learning electron densities. Suppressing Pulay forces with augmented basis sets allows accurate simulations for large systems.

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Area of Science:

  • Computational chemistry
  • Quantum mechanics
  • Materials science

Background:

  • The Hellmann-Feynman (HF) theorem offers a direct method for calculating forces from electron density.
  • Machine learning (ML) models for electron density promise efficient force calculations in large systems.
  • A significant barrier to HF theorem adoption with atom-centered basis sets is the Pulay force error.

Purpose of the Study:

  • To demonstrate a method for suppressing the Pulay force in HF calculations.
  • To enable accurate force calculations for large molecular systems using ML-derived electron densities.
  • To facilitate reliable geometry optimization and molecular dynamics simulations with HF forces.

Main Methods:

  • Utilizing density functional calculations with suitably augmented Gaussian basis sets.
  • Suppressing the Pulay force by careful selection of basis sets.
  • Comparing HF forces with analytical forces for accuracy.

Main Results:

  • The Pulay force can be effectively suppressed using augmented Gaussian basis sets.
  • HF forces computed with this method achieve accuracy comparable to state-of-the-art analytical forces.
  • Reliable geometry optimization and molecular dynamics are achievable with HF forces.

Conclusions:

  • Augmented basis sets overcome the Pulay force issue, enabling accurate HF force calculations.
  • This approach paves the way for efficient and accurate simulations of large systems using ML densities and the HF theorem.
  • The findings support the broader application of the HF theorem in computational chemistry and materials science.