The rate that a freelance cinematographer charges for a corporate project can vary depending on a number of factors, including the scope of the project, the location of the shoot, the duration of the project, and the experience and expertise of the cinematographer.

Here are a few things to consider when determining a rate for a corporate project:

Day rate: Many freelance cinematographers charge a day rate for their services. This is a flat rate that covers a full day of shooting, regardless of how many hours are actually worked. Day rates can vary widely depending on the experience and reputation of the cinematographer.
Half-day rate: Similar to the day rate, but it's for half the amount of hours, usually around 4 hours.
Project rate: Some cinematographers may charge a flat rate for the entire project, which can be beneficial for clients who want a more predictable cost.
Equipment rental: Some cinematographers may charge an additional fee for equipment rental, this can include cameras, lenses, lighting equipment and other specialized gear.
Travel: If the project requires travel, the cinematographer may charge additional fees for transportation, lodging, and per diem expenses.
Expediting: Some projects may require expediting the delivery of final footage, that can include additional costs for the freelancer.

It's worth noting that the prices can vary greatly depending on the location, the level of the production and the nature of the project, for example a short corporate video for a small business might cost less than a long-form commercial for a big brand.

It's important for the client and the freelancer to have a clear understanding of the scope of the project and the rate before work begins. It's also worth to have a contract that outlines the terms of the agreement and to avoid any misunderstandings.