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If a tenant gives you a notice to quit, they should leave on the date they have specified in the notice. If they fail to leave on that date without your permission, then it is possible to charge them double the rent that
was due before the notice date expired. [s.18 Distress for Rent Act 1737].

s 18 Tenants holding after the time they notify for quitting, to pay double rent.

And whereas great inconveniences have happened and may happen to landlords whose tenants have power to determine their leases, by giving notice to quit the premisses by them holden, and yet refusing to deliver up the possession when the landlord hath agreed with another tenant for the same: from and after the said twenty fourth day of June one thousand seven hundred and thirty eight, in case any tenant or tenants shall give notice of his, her, or their intention to quit the premisses by him, her, or them holden, at a time mentioned in such notice, and shall not accordingly deliver up the possession thereof at the time in such notice contained, that then the said tenant or tenants, his, her, or their executors or administrators, shall from thenceforward pay to the landlord or landlords, lessor or lessors, double the rent or sum which he, she, or they should otherwise have paid, to be levied, sued for, and recovered at the same times and in the same manner as the single rent or sum, before the giving such notice, could be levied, sued for, or recovered; and such double rent or sum shall continue to be paid during all the time such tenant or tenants shall continue in possession as aforesaid.

There are however strict rules that must be followed before a claim for double rent may be made. These were decided in Oliver Ashworth (Holdings) Ltd. v. Ballard (Kent) Ltd. [1999] 3 W.L.R. 57.

Laws L.J. states:
"...the right to double rent conferred by section 18 of the Act of 1737 only arises where (a) the tenant holding over after his own notice to quit is in fact a trespasser (thus, the notice must be valid), and (b) the landlord treats him as such. Any other case departs from what I regard as the plain purpose of the section..."

Though s.18 and Laws L.J. both specifically state that the notice must be "valid" a landlord is entitled to accept an invalid notice to quit. It is suggested by the Guild that acceptance of an invalid notice may provide an argument that the notice was "valid").

One of the intentions of s.18 was to compensate the landlord should he have contracted a new tenant to take the property after the date specified in the notice, however there is no requirement to have a tenant in waiting to charge double rent.

Laws L.J. states
"...it is to my mind entirely clear that the legislature was concerned only to compensate landlords for the potential loss of rent arising where a tenant holds over against the landlord's insistence that he should comply with his own notice to quit. I do not say that the section applies only where the landlord has a new tenant ready and waiting; the recital gives the thrust, but not necessarily the focus, of the section's reach."

Presumably, as the requirement is that the landlord treats the tenant as a trespasser, any request for double rent should be made using the words "mesne profits" so as to ensure the landlords intention is that the occupier is a trespasser. In addition, a landlord would be well advised to immediately commence possession.

Seeking possession

No notice by the landlord will be required because the service by the tenant of a valid notice to quit has ended the tenancy. However, a possession order in the usual way will be required.

You should contact the Guild for advice on seeking possession but essentially, it is an almost identical procedure to the rent arrears procedure after the service of a section 8 notice. It is the same court forms with some minor modifications explaining the reason for possession is because of the tenants notice to quit. 

You should bear in mind that any claim for possession based on a tenants notice is extremely rare in the courts and therefore judges may be unfamiliar with what you are asking for. If an alternative method of possession is available (for example after service of a section 21) this maybe preferred.