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The Uncertainty Principle

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...

Uncertainty: Overview

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.

Propagation of Uncertainty from Systematic Error

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...

Propagation of Uncertainty from Random Error

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

The Quantum-Mechanical Model of an Atom

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Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.

Uncertainty in Measurement: Accuracy and Precision

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Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or the accepted value. Precise values agree with each other; accurate values agree with a true value.

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Updated: Oct 26, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

A framework for quantifying uncertainty in DFT energy corrections.

Amanda Wang¹, Ryan Kingsbury¹, Matthew McDermott^1,2

¹Department of Materials Science and Engineering, University of California, Berkeley, CA, 94720, USA.

Scientific Reports

|July 30, 2021

Summary

This summary is machine-generated.

Quantifying uncertainty in density functional theory (DFT) energy corrections reveals that many chemical systems have unstable polymorphs predicted as stable. This method improves compound stability assessments using computational thermodynamics.

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Area of Science:

Computational chemistry
Materials science
Chemical physics

Background:

Density functional theory (DFT) is crucial for predicting material properties.
Empirical corrections enhance DFT accuracy for enthalpies of formation and phase stability.
Quantifying uncertainty in these corrections is vital for reliable predictions.

Purpose of the Study:

To develop a method for quantifying uncertainty in DFT energy corrections.
To integrate this uncertainty quantification into a novel DFT correction scheme.
To reassess the stability of chemical systems and polymorphs considering uncertainty.

Main Methods:

Developed a method to quantify uncertainty in empirical DFT energy corrections.
Created a new DFT energy correction scheme combining oxidation-state and composition-dependent corrections.
Applied the method to analyze the stability of various chemical systems and polymorphs.

Main Results:

Demonstrated that many chemical systems contain unstable polymorphs incorrectly predicted as stable without uncertainty quantification.
Showcased the integration of uncertainty into a new DFT energy correction scheme.
Quantified the probability of compound stability on compositional phase diagrams.

Conclusions:

Uncertainty quantification is essential for accurate computational predictions of compound stability.
The developed method enables more reliable assessments of phase stability and polymorph stability.
This approach facilitates better-informed materials discovery and design.