The computational modeling of molecules under high pressure is a growing research area that augments experimental high-pressure chemistry. Here, a new electronic structure method for modeling atoms and molecules under pressure, Gaussians On Surface Tesserae Simulate HYdrostatic Pressure (GOSTSHYP) approach, is introduced. In this method, a set of Gaussian potentials is distributed evenly on the van der Waals surface of the investigated chemical system, leading to a compression of the electron density and the atomic scaffold. Since no parameters other than pressure need to be specified, GOSTSHYP allows straightforward geometry optimizations and ab initio molecular dynamics simulations of chemical systems under pressure for nonexpert users. Calculated energies, bond lengths, and dipole moments under pressure fall within the range of established computational methods for high-pressure chemistry. A Diels-Alder reaction and the cyclotrimerization of acetylene showcase the ability of GOSTSHYP to model pressure-induced chemical reactions. The connection to mechanochemistry is pointed out.