Fermi's golden rule is expanded in Planck's constant to study quantum effects of nuclear baths on transitions between electronic states. For six approximations, the second-order expansion of the transition rates is compared with the exact expansion. The comparison provides the conditions for agreement between the exact and approximate rate constants in the second order. Two of the conditions show that the change in electronic states of the solute does not affect solvent structure much. The other concerns linearity of the reaction coordinate. Besides the comparison of analytical expressions, the expansion of approximations is numerically compared, using a simple model. Most approximations show similar values for a parameter set. The values approximate the value obtained by quantum molecular simulations.