Disentangling the Galactic Halo with APOGEE. I. Chemical and Kinematical Investigation of Distinct Metal-poor Populations

National Science Foundation [AST-1312863, AST-1616636]; Spanish MINECO [AYA-2014-56359-P]; PAPIIT of Mxico [IN103014]; CONACyT of Mexico [241732]; PAPIIT of Mexico [IG100115, IA101215, IA101517]; MINECO of Spain [AYA2015-65205-P]; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE); US National Science Foundation; Chilean BASAL Centro de Excelencia en Astrofisica y Tecnologias Afines (CATA) grant [PFB-06/2007]; Ramon y Cajal fellowship [RYC-2013-14182]; Spanish Ministry of Economy and Competitiveness (MINECO) [AYA-2014-58082-P]; Chilean Ministry of Economy, Development, and Tourism's Millennium Science Initiative [IC120009]; STFC/Newton Fund [ST/M007995/1]; CONICYT/Newton Fund [DPI20140114]; Alfred P. Sloan Foundation; U.S. Department of Energy Office of Science; Center for High-Performance Computing at the University of Utah; Brazilian Participation Group; Carnegie Institution for Science; Carnegie Mellon University; Chilean Participation Group; French Participation Group; Harvard-Smithsonian Center for Astrophysics; Instituto de Astrofisica de Canarias; Johns Hopkins University; Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo; Lawrence Berkeley National Laboratory; Leibniz Institut fur Astrophysik Potsdam (AIP); Max-Planck-Institut fur Astronomie (MPIA Heidelberg); Max-Planck-Institut fur Astrophysik (MPA Garching); Max-Planck-Institut fur Extraterrestrische Physik (MPE); National Astronomical Observatory of China; New Mexico State University; New York University; University of Notre Dame; Observatario Nacional / MCTI; Ohio State University; Pennsylvania State University; Shanghai Astronomical Observatory; United Kingdom Participation Group; Universidad Nacional Autonoma de Mexico; University of Arizona; University of Colorado Boulder; University of Oxford; University of Portsmouth; University of Utah; University of Virginia; University of Washington; University of Wisconsin; Vanderbilt University; Yale University; [PHY 14-30152]