Krylov subspace methods for approximating the action of the matrix exponential exp(A) on a vector v are analyzed with A Hermitian and negative semidefinite. Our approach is based on approximating the exponential with the commonly employed diagonal Pad´e and Chebyshev rational functions, which yield a system of equations with a polynomial coefficient matrix. We derive optimality properties and error bounds for the convergence of a Galerkin-type approximation and of a computationally feasible and extensively used alternative. As complementary results, we theoretically justify the use of a popular a posteriori error estimate, and we provide upper bounds for the components of the solution vector. Our theoretical and numerical results show that this methodology may provide an appropriate framework to devise new strategies such as more powerful acceleration schemes.