The correct answer is option 1: Rhizobia. Free-living nitrogen fixing microorganisms are organisms that have the ability to convert atmospheric nitrogen into a usable form for plants. Rhizobia are a type of bacteria that form a mutualistic relationship with leguminous plants, such as soybeans and clover. These bacteria infect the root nodules of the plants and convert atmospheric nitrogen into ammonia through a process called nitrogen fixation. The ammonia is then used by the plants for growth and development.

Option 2, soil fungi, are not typically classified as nitrogen-fixing microorganisms. Though they play important roles in soil nutrient cycling, they are not known for their ability to fix atmospheric nitrogen.

Option 3, vesicular arbuscular mycorrhizae, are a type of fungi that form symbiotic associations with the roots of most plants. They aid in nutrient uptake, but they do not directly fix atmospheric nitrogen.

Option 4, blue-green algae (cyanobacteria), are indeed capable of nitrogen fixation. However, they do not exclusively exist as free-living microorganisms and can also form symbiotic associations with plants and fungi.