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Related Concept Videos

Energy Diagrams - II01:10

Energy Diagrams - II

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Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the system.
The point in the energy diagram at which the system’s potential energy is the lowest is known as the local minima. The system tends to stay in this position indefinitely unless acted upon by a net force. The slope of the potential energy diagram at the local minima is zero, indicating that zero net force is acting on the system. The...
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Energy Diagrams - I01:14

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The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the interpretation of the dynamics of the system becomes even simpler.
Take the example of a skater on a parabolic ramp. The potential energy at different points along the ramp will be proportional to the height of the ramp, which varies quadratically with the horizontal position on the ramp. As the skater moves down the ramp from the highest position,...
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Energy Diagrams, Transition States, and Intermediates02:13

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Free-energy diagrams, or reaction coordinate diagrams, are graphs showing the energy changes that occur during a chemical reaction. The reaction coordinate represented on the horizontal axis shows how far the reaction has progressed structurally. Positions along the x-axis close to the reactants have structures resembling the reactants, while positions close to the products resemble the products.  Peaks on the energy diagram represent stable structures with measurable lifetimes, while...
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The Arrhenius equation relates the activation energy and the rate constant, k, for chemical reactions. In the Arrhenius equation, k = Ae−Ea/RT, R is the ideal gas constant, which has a value of 8.314 J/mol·K, T is the temperature on the kelvin scale, Ea is the activation energy in J/mole, e is the constant 2.7183, and A is a constant called the frequency factor, which is related to the frequency of collisions and the orientation of the reacting molecules.
The Arrhenius equation can be used...
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Path Between Thermodynamics States01:21

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Consider the two thermodynamic processes involving an ideal gas that are represented by paths AC and ABC in Figure 1:
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Free Energy Changes for Nonstandard States03:25

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The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; concentrations other than 1 M) is related to the standard free energy change according to this equation:
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Related Experiment Video

Updated: Apr 4, 2026

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method
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Energy error bars in direct configuration interaction iteration sequence.

Zsuzsanna Tóth1, Ágnes Szabados1

  • 1Laboratory of Theoretical Chemistry, Institute of Chemistry, Loránd Eötvös University, 1518 Budapest, P.O. Box 32, Hungary.

The Journal of Chemical Physics
|September 3, 2015
PubMed
Summary

This study presents a computational method for estimating lower bounds of eigenvalues using Löwdin

Area of Science:

  • Computational chemistry
  • Quantum mechanics
  • Theoretical physics

Background:

  • Accurate eigenvalue determination is crucial for understanding quantum systems.
  • Existing methods often provide only upper bounds (expectation values).
  • Löwdin's bracketing function offers a theoretical basis for lower bounds.

Purpose of the Study:

  • To develop and implement a computational scheme for approximate lower bounds to eigenvalues.
  • To integrate this scheme into the full configuration interaction (FCI) algorithm.
  • To establish an error bar for wavefunction characterization.

Main Methods:

  • Elaboration of a computational scheme based on Löwdin's bracketing function.
  • Implementation within a direct full configuration interaction (FCI) algorithm.

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  • Approximation of strict lower bound properties.

    • Main Results:

    • The developed method yields approximate lower bounds for eigenvalues.
    • These lower bounds are of the same order of magnitude as the usual upper bounds (expectation values).
    • The difference between upper and lower bounds provides a useful error bar.

    Conclusions:

    • The computational scheme provides valuable, albeit approximate, lower bounds for eigenvalues.
    • The method offers an error estimation for wavefunctions in iterative steps.
    • This approach enhances the characterization of quantum mechanical systems.