We present an ab initio O(N) method that combines an accurate optimized-orbital solution of the electronic structure problem with an efficient Green's function technique for evaluating the quantum conductance. As an important illustrative example, we investigate carbon nanotube-metal contacts and explain the anomalously large contact resistance observed in nanotube devices as due to the spatial separation of their conductance eigenchannels. The results for various contact geometries and strategies for improving device performance are discussed. The study of the electrical properties of nanostructures has seen intense activity in the last decade, due to the prom- ise of novel technological applications for nanoscale quan- tum electronic devices. The theoretical study of quantum conductance in such structures has thus become of primary interest and has been addressed by a variety of techniques. 1 Due to the complexity of describing an ''open'' system of a nanoscale device in contact with effectively infinite leads, most of the current approaches rely on phenomenological tight binding models, which for many systems may not pro- vide a sufficiently reliable and accurate description. There are only few examples of ab initio calculations of quantum conductance and the field is still in a critical phase of development. The existing methods are based on the so- lution of the quantum scattering problem for the electronic wave functions through the conductor using a number of related techniques: Lippman-Schwinger and perturbative Green's function methods have been used to study conduc- tance in metallic nanowires and recently in small molecular nanocontacts; 2,3 conduction in nanowires, junctions, and nanotube systems has been addressed using local 4 or nonlocal 5,6 pseudopotential methods and through the solution of the coupled-channel equations in a scattering-theoretic approach. 7-9 These methods are based on a plane wave rep- resentation of the electronic wave functions, which imposes severe restrictions on the size of the system because of the large number of basis functions necessary for an accurate description of the electron transmission process. Therefore, structureless jellium leads, which do not provide a micro- scopic description of the conductor-metal contact, had to be assumed in most cases for computational reasons. Only re- cently have real-space approaches been considered for a more efficient solution of the electronic transport problem. They are based on the use of linear combination of atomic orbitals 10 ~LCAO! or Gaussian 11 orbital bases. These are combined with either a scattering state solution for the transmission 10 or Green's function-based techniques. 11 In this paper we present an approach based on a real- space optimized-orbital solution of the electronic structure problem, combined with an efficient Green's function-based technique for the evaluation of the electron transmission probability. Both the ab initio and the transport algorithms scale essentially linearly with the size of the system, thus extending greatly the range of applicability of our method. This method has been already successfully applied to de- scribe quantum conductance in ideal and defective carbon nanotubes. 12 Following a brief overview of the methodology, we address the problem of contacts in a metal-carbon nano- tube assembly, which is very important in the design of effi- cient nanotube-based devices. Contact resistances of the or- der of MV are typically observed in most of the prototypical nanotube-based devices realized so far, 13-16 whereas from simple band structure arguments one would expect resis- tances of the order of a few tenths of kV, 17 because the fundamental resistance of a single ballistic channel is 12.9 kV. The results of our calculations provide an explanation for this pathologically high contact resistance and suggest strategies to improve the performance of nanotube-metal contacts. It is important to stress that this problem requires self-consistent ab initio methodology, in order to accurately describe the highly inhomogeneous environment of a nanowire-metal junction and to account for the charge trans- fer occurring at the interface between the two dissimilar ma- terials.