A detailed theoretical study is presented for the vibrational population distribution of polyatomic molecules which results from electronic excitation from a thermal ground state. If the vibrational frequencies of the excited state are lower than the ground-state frequencies and if position shifts are not too large, then there exist excitation frequencies for which the excited-state vibrational distribution will be cooled in comparison to the ground state. An analytic theory for the vibrational distribution in the excited state is obtained by noting that the fast dephasing of a polyatomic molecule after excitation allows for the development of a Gaussian approximation for the excitation process. We show that the equilibrium energy distribution of a polyatomic molecule as well as the nascent distribution after excitation are well approximated as Gaussian. The average energy in the excited state is then found to be a quadratic function of the excitation frequency. If cooling takes place, it will usually be maximal for an excitation frequency which is to the red of the ground electronic state to ground electronic state excitation frequency. Cooling is not necessarily a quantum effect, it may also be found in the classical limit, in which one ignores quantization of the vibrational levels. The generality of the Gaussian approximation opens the way for theoretical treatment of anharmonic polyatomic molecules, using quantum Monte Carlo techniques.