  Four large ring molecules composed by 15 nitrogen-substituted benzene rings,
named as "knot-isomers of Moebius cyclacene", i.e. non-Moebius cyclacenes
without a knot (0), Moebius cyclacenes with a knot (1), non-Moebius cyclacenes
with two knots (2), and Moebius cyclacenes with three knots (3), are
systematically studied for their structures and nonlinear optical properties.
The first hyperpolarizability (beta_0) values of these four knot-isomers
structures are 4693 (0) < 10484 (2) < 25419 (3) < 60846 au (1). The beta_0
values (60846 for 1, 10484 for 2 and 25419 au for 3) of the knot-isomers with
knot(s) are larger than that (4693 au for 0) of the knot-isomer without a knot.
It shows that the beta_0 value can be dramatically increases (13 times) by
introducing the knot(s) to the cyclacenes structures. It is found that
introducing knots to cyclacenes is a new means to enhance the first
hyperpolarizability.
