Determination of molecular properties by the Method of Moments. I Abstract: The method of moments appearst to have some significant advantages over other variational methods of quantum chemistry if complicated molecular systems are investigated and elaborate variational wave functions are applied. The main advantages are (a) the possibility of avoiding the worst difficulties of integration of reducing the number of integrals (b) the possibility of improving the accuacy of the wave function in physically important regions of the configurational space and (c) the possibility of obtaining some "random sampling type" error estimates. The aim of the series of papers starting with the present one is to collect experience in applying the method of moments to problems of quantum chemistry.