Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial

Area of Science:

• Materials Science
• Computational Chemistry
• Crystallography

Background:

• Löwenstein's rule prohibits Al-O-Al bridges in aluminosilicate zeolite frameworks.
• Graph-theoretical interpretations and vector methods have been used to analyze these frameworks.
• The concept of the 'independence ratio' quantifies the maximal Al/(Al+Si) ratio in a zeolite framework.

Purpose of the Study:

• To present a practical computational technique for calculating the independence ratio of periodic nets, specifically 2-periodic nets.
• To introduce and utilize the 'independence polynomial' for determining maximal independent sets and the independence ratio.
• To apply the method to determine the independence ratio of the 2-periodic net 'dhc'.

Main Methods:

• Application of graph theory and the vector method to periodic nets of aluminosilicate frameworks.
• Development of a calculation technique based on propositional calculus.
• Introduction and use of the multivariate 'independence polynomial' to find maximal independent sets.

Main Results:

• The independence polynomial can be computed automatically to yield the independence ratio.
• Properties of the independence polynomial are discussed.
• The independence ratio for the 2-periodic net 'dhc' was determined using this method.

Conclusions:

• The independence polynomial offers an efficient way to calculate the independence ratio for zeolite frameworks.
• This method facilitates the determination of site distributions that achieve the maximal Al/(Al+Si) ratio.
• The approach provides a robust tool for analyzing and designing aluminosilicate structures based on Löwenstein's rule.