An effective electronic Hamiltonian for transition metal oxide compounds is presented. For Mnoxides, the Hamiltonian contains spin-2 "spins" and spin-3/2 "holes" as degrees of freedom. The model is constructed from the Kondo-lattice Hamiltonian for mobile eg electrons and localized t2g spins, in the limit of a large Hund's coupling. The effective electron bond hopping amplitude fluctuates in sign as the total spin of the bond changes. In the large spin limit, the hopping amplitude for electrons aligned with the core ions is complex and a Berry phase is accumulated when these electrons move in loops. The new model is compared with the standard double exchange Hamiltonian. Both have ferromagnetic ground states at finite hole density and low temperatures, but their critical temperatures could be substantially different due to the frustration effects induced by the Berry phase. 75.30.Et, 75.50.Cc The discovery of giant magnetoresistance effects in ferromagnetic metallic oxides R 1−x X x MnO 3 (where R = La, Pr, Nd; X = Sr, Ca, Ba, Pb) has triggered renewed attention into these compounds.[1] A decrease in resistivity of four orders of magnitude has been observed in thin films of Nd 0.7 Sr 0.3 MnO 3 at fields of ∼ 8T .[2] The phase diagram of La 1−x Ca x MnO 3 is very rich with ferromagnetic (metal and insulator) phases, as well as regions where charge-ordering is observed.[3] The magnetic and electronic properties of these manganese oxides are believed to arise, at least in part, from the strong coupling between correlated itinerant electrons and localized spins, both of 3d character. The Mn 3+ ions have three electrons in the t 2g state forming a local S=3/2 spin, and one electron in the e g state which hops between nearestneighbor Mn-ions, with double occupancy suppressed by Coulombic repulsion. The widely used Hamiltonian to describe manganese oxides is [4]where the first term is the e g electron transfer between nearest-neighbor Mn-ions at sites m, n, while the second term is the ferromagnetic Hund coupling between the S=3/2 localized spin S n and the mobile electron with spin σ n (J H > 0). A Coulombic repulsion to suppress double occupancy in the itinerant band is implicit. The on-site Hund coupling energy is larger than the conduction bandwidth favoring the alignment of the itinerant and localized spins. For Mn 3+ , the resulting spin is 2, while for Mn 4+ (vacant e g state) the spin is 3/2. Since the study of Hamiltonian Eq.(1) is a formidable task, simplifications have been introduced to analyze its properties. A familiar approach is the use of the doubleexchange Hamiltonian, [4][5][6][7][8][9] where the e g electrons move in the background of classical spins S cl n that approximate the S=3/2 almost localized t 2g electrons. The conduction electron effective hopping between sites m and n used in previous work is t ef f mn = t 1 + (S cl m · S cl n /S 2 ), where S is the magnitude of the classical spin. [4,6] Using this model, the ferromagnetic critical temperature, T c , was recently estimated.[9] Since t...