A simple model potential energy surface for double proton transfer

I love simple models.

There is a very nice paper
Correlated double-proton transfer. I. Theory
Zorka Smedarchina, Willem Siebrand, and Antonio Fernández-Ramos

It considers an incredibly simple potential energy surface to describe double proton transfer.

x_1 is the (dimensionless) position of one proton relative to the middle of its donor and acceptor.
x_2 is the corresponding position for the second proton.
The first term describes a quartic potential with an energy barrier for transfer of the proton between the donor and acceptor.

The dimensionless parameter G describes the extent of correlation or coupling between the two hydrogen bonds. The coupling term is chosen to have the important property that it is symmetric in the two co-ordinates but sensitive to their sign. This is an important difference to earlier [rather nice] work by Benderskii et al. who considered competition between two dimensional quantum tunneling paths [instantons] associated with concerted and sequential transfer.

Three types of Potential Energy surface (PES) can occur, depending on the value of G.
The three cases shown below correspond to
a) 0 < G < 1/2
b) 1/2 < G < 1
c)  G > 1.

The coordinates x_a=1/2(x_1-x_2) and x_s=1/2(x_1+x_2). The horizontal line between the left and right minimum corresponds to a concerted path.

For the top PES a sequential proton transfer is possible via one of the two intermediate (INT) states.

These three cases correspond to the different types of PES's seen for double proton transfer for different molecules.