Is there a specified way to calculate the whitespace around a logo? I have searched a lot but all I found is something like using the first word of the brand name to determine the whitespace.
There is no formula for calculating what the clear space should be. It's entirely a matter of preference for whoever is designing the logo and/or style guide for the use of said logo.
For instance, if you take the GDSE logo, the clear space could be defined as the height of the text (which would be quite tight) or the height of the circle (which would be quite loose) or some other amount, all would be equally valid choices.
There is a certain amount of instinctive logic to thinking that something will look 'right' if it is the same distance from the page or box edge as it is tall, but it won't always look correct.
The reason that many people use the height of a letter or other part of the logo to define the clear space is that it is an easy way to distribute a logo with a built in measure for how to place it. There are other ways, including incorporating an invisible (transparent) box around the logo.
Decide what looks pleasant and balanced and how much space is needed to allow the logo to breath and then only use an aspect of the logo to define this measurement if it is close to the amount of space that you want anyway.
If you're struggling for inspiration then an image search for "clear space around logo" or similar will serve up lots of examples of different ways to do it.
I work for a publicly funded agency, which I won't name, but which has the letter "E" in its logo. Our guidance is: "To determine the clear space, measure the width of the 'E' in the logotype. The clear space ewill change depending on the scale."
Then, we try to leave enough space to fit that letter between our logo and the edge of the frame containing it or-- between the logo and any other neighboring element.
I would search for the clients Brand Guidelines, this information is normally available there. Or use the prominent object in the logo for spacing around the entire logo.