We have computed the correlated electronic structure of FeSe and its dependence on the A 1g mode versus compression. Using the self-consistent density functional theory -dynamical mean field theory (DFT-DMFT) with continuous time quantum Monte Carlo (CTQMC), we find that there is greatly enhanced coupling between some correlated electron states and the A 1g lattice distortion. Superconductivity in FeSe shows a very strong sensitivity to pressure, with an increase in Tc of almost a factor of 5 within a few GPa, followed by a drop, despite monotonic pressure dependence of almost all electronic properties. We find that the maximum A 1g deformation potential behaves similar to the experimental Tc. In contrast, the maximum deformation potential in DFT for this mode increases monotonically with increasing pressure.PACS numbers: 74.70. Xa, 74.25.Jb, 75.10.Lp So far there is no predictive theory for superconductivity in the cuprate and iron-superconductors, hence these superconductors can be classified as non-conventional superconductors, since there is a well-developed, predictive theory for electron-phonon superconductors whose normal state is well-represented by conventional density functional theory (DFT) [1][2][3]. The unconventional superconductors are very sensitive to applied pressure, so pressure provides a control to test theories and develop a better understanding [4][5][6]. Here we study superconductivity under applied pressure in pure FeSe; unlike cuprates, superconductivity in FeSe arises without doping. FeSe is an ideal system to study the electron pairing mechanism due to the simplicity of its crystal and electronic structure. It shows a very strong enhancement of T c upon application of modest pressure with dramatic increase of T c from 8K to ∼ 37K [4,7,8] and then decreases upon further application of pressure. Why does T c increase with pressure and then decrease for rather small lattice compression? This question was addressed in Ref. 8, where it was found that applied pressure (P) intensified antiferromagnetic spin fluctuations (SF). However, this did not explain the decrease in T c with further compression.The discovery of the iron superconductors showed that high T c is not specific to the cuprates, and suggests a wider field of potential high T c materials [9]. Although DFT gives many properties reasonably accurately for both cuprates [10][11][12] and Fe-superconductors [13][14][15], there is also significant indication of the importance of correlations and fluctuating local moments beyond DFT specially for the Fe-superconductors which are paramagnetic metals in room temperature, and Dynamical Mean Field Theory (DMFT) has proved to be a good approximation [16][17][18][19][20][21][22].While most studies suggest a spin fluctuation coupling mechanism for SC in Fe-superconductors similarly to cuprates [23][24][25], strong coupling phonons have also been proposed to play a role in both sets of materials [3,[26][27][28][29][30]. The study of strong electron-phonon coupling in correlated solids is in...