The properties of the exciton in the surface layer of crystals influence the properties of crystals very remarkably. In recent years

a lot of authors studied the surface polaron and the surface exciton. The properties of the exciton have been studied by many theoretical method by many investigators.Sumi

et al

. studied self-trapping of excitons(or electrons)interacting with phonons via short-range potential and found that the self-trapping depends strongly on the degree of lattice freedom.Chen

et al

. calculated the ground state energy of the strongly bound exciton-phonon system by using a concise variational approach.The influence of the interaction between phonons of different wave vectors in the recoil process on the effective potential between electron and hole

the self-trapping energy

and the self-trapping condition of the exciton in polar crystals has been discussed by using a perturbation method by present authors.Using the nearest-neighbor tight-binding approximation Luban

et al

. calculated the lowest bound-state energy of the effective 1D electron Hamiltonian and the exciton binding energy is obtained.The results for the exciton binding energy are in very good agreement both with experiment and the results of other theoretical calculations.Kasapoglu

et al

. calculated the binding energy of the exciton in the symmetric and asymmetric quantum wells by using a variational approach.The third-order nonlinear optical absorptions in hyperbolic quantum wires are studied by Guo

et al

.

with most emphasis devoted to the influence of excitons on the third-order nonlinear optical absorptions. The analytic form for the third-order nonlinear optical absorption coefficient of this system is derived by means of density matrix treatment.A variational calculation of the ground-state energy of neutral excitons and of positively and negatively charged excitons(trions)confined in a single-quantum well is presented by Riva

et al

. They studied the dependence of the correlation energy and of the binding energy on the well width and on the hole mass.By using the method of few-body physics

the binding energy spectra of the second bound state of a negatively charged exciton X - in a GaAs quantum dot with a parabolic confinement have been calculated as a function of the electron-hole mass ratio and of the dot radius by Xie. The properties of the exciton have been investigated by many methods. Many of these mainly concentrated attention on the weak-and intermediate-coupling cases and on the ground state energy. However

the exciton in strong-coupling polar crystals and the excited state energy of the exciton has not been studied so far. Recently we study the internal excited state of the weak-coupling surface magnetopolaron by means of Huybrechts's linear combination operator method. In this paper

the properties of internal excited state of the strong-coupling surface exciton in polar crystals are investigated by using the linear combination operator and the unitary transformation method and the excited state energy

the excitation energy and the mean number of phonon of the strong-coupling surface exciton are calculated. The results show that the self-energy of ground state and excited state

the excitation energy and the mean number of phonon of the strong-coupling surface exciton could be written as a series in α

s

-1

the first term being proportion to

the coupling constant α

s

for Wannier surface exciton.