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Related Concept Videos

Coefficient of Correlation01:12

Coefficient of Correlation

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the strength of the linear...
Calculating and Interpreting the Linear Correlation Coefficient01:11

Calculating and Interpreting the Linear Correlation Coefficient

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable, x, and the dependent variable, y. Hence, it is also known as the Pearson product-moment correlation coefficient. It can be calculated using the following equation:
Correlation of Experimental Data01:23

Correlation of Experimental Data

Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
For example, a spherical particle moving through a viscous fluid experiences drag. Dimensional analysis shows that the drag force depends on the particle's diameter, velocity, and...
Correlations02:20

Correlations

Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
Correlation and Regression00:53

Correlation and Regression

In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a negative...
Correlation01:09

Correlation

In statistics, two variables are said to be correlated if the values of one variable are associated with the other variable. Depending on the relationship between two variables, correlation can be of three types– positive correlation, negative correlation, and zero correlation.
Two variables, for example, a and b, are said to be positively correlated if both variables move in the same direction. In other words, a positive correlation exists between two variables, a and b, if:

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Framework for constructing generic Jastrow correlation factors.

P López Ríos1, P Seth, N D Drummond

  • 1Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2012
PubMed
Summary
This summary is machine-generated.

Researchers created a flexible framework for Jastrow factors, improving quantum Monte Carlo calculations. Optimized factors recover over 90% of correlation energy for various systems.

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Area of Science:

  • Computational Chemistry
  • Quantum Many-Body Physics

Background:

  • Jastrow factors are crucial for improving the accuracy of quantum Monte Carlo (QMC) methods.
  • Accurate representation of electron correlation is essential for reliable predictions in computational chemistry and physics.

Purpose of the Study:

  • To develop a flexible framework for constructing Jastrow factors with arbitrary numbers of particles.
  • To investigate the impact of various multi-body Jastrow terms on QMC calculations.
  • To assess the performance of optimized Jastrow factors across diverse chemical and physical systems.

Main Methods:

  • Development of a flexible framework for Jastrow factor construction.
  • Inclusion of three- and four-body Jastrow terms, including van der Waals-like and anisotropic terms.
  • Application in variational Monte Carlo (VMC) and fixed-node diffusion Monte Carlo (FN-DMC) calculations.

Main Results:

  • Successfully implemented a flexible framework for Jastrow factors.
  • Demonstrated the effectiveness of three- and four-body terms in improving accuracy.
  • Achieved over 90% retrieval of FN-DMC correlation energy using VMC with optimized Jastrow factors for homogeneous electron gases, atoms, and molecules.

Conclusions:

  • The developed framework offers a versatile approach to enhance Jastrow factor construction.
  • Optimized multi-body Jastrow factors significantly improve the accuracy of variational Monte Carlo.
  • This methodology provides a robust tool for accurate electronic structure calculations.