score:25
Those seem like the mitre caps which would be part of grenadier uniform in many armies of the time, including imperial Russia. I don't have a good source there but here's an example of a very similar uniform from a Russian grenadier regiment.
Here are grenadiers of three British regiments from what appears to be the late 18th century:
The hats don't have any special function aside from making an impression - grenadiers were generally elite troops selected among other things for their height, and a tall hat visually makes the already tall men seem even taller. Another hat type associated with grenadiers is the tall bearskin hat that modern tourists might have seen in London worn by the Queen's Guard.
It's also worth noting that by the time of Battle of Borodino the standard grenadier uniform in Imperial Russia had already switched to the shako hat (the other type of hats in this painting), so this picture might even be used to limit the troopers to a specific regiment such as the Pavlovsky Guard Regiment (Russian: Павловский лейб-гвардии полк) which retained the mitre hats after this reform.
Upvote:3
The soldiers are from the Pavlovski Regiment. Allowed to keep the old 'mitre' cap (popular during the period 1700-1800) as a reward by Czar Nicholas after the uniform change of c. 1809 because the Pavloski or Pavlov Regiment acted bravely at the battle of Friedland (1807). For information, the 'mitre' cap was used by Grenadiers because they (originally, c.1700) threw grenades. Most troops 1700-1800 wore a three-corner hat, or tricorn, but you would hit your tricorn hat if you threw a grenade 'over arm' like a cricket ball!