LDA energy bands, low-energy Hamiltonians, t', t'', t perpendicular to(k), and J perpendicular to.

We describe the LDA band structure of YBa2Cu3O7 in the epsilon(F) +/- 2 eV range using orbital projections and compare with YBa2Cu4O8. Then, the high-energy and chain-related degrees of freedom are integrated out and we arrive at two, nearest-neighbor, orthogonal, two-center, 8-band Hamiltonians H-8(+) and H-8(-) for respectively the even and odd bands of the bi-layer. Of the 8 orbitals Cu(x)2-(y)2. O2(x), O3(y) and Cu-s have sigma character and Cu-xz, Cu-yz, O2(z), and O3(z) have pi character. The roles of the Cu-s orbital, which has some Cu(3z)2-(1) character, and the four pi orbitais are as follows: Cu-s provides 2nd- and 3rd-nearest-neighbor (t' and t '') intra-plane hopping, as well as hopping between planes (t(perpendicular to)). The pi-orbitals are responsible for bifurcation of the saddle-points for dimpled planes. The 4-sigma-band Hamiltonian is generic for flat CuO2 planes and we use it for analytical studies The k(parallel to)-dependence is expressed as one on u = (cos bk(y) + cos ak(x))/2 and one on nu = (cos bk(y) - cos ak(x))/2. The latter arises solely through the influence of Cu-s. The reduction of the (sigma-Hamiltonian to 3- and 1-band Hamiltonians is explicitly discussed and we point out that, in addition to the hoppings commonly included in many-body calculations, the 3-band Hamiltonian should include hopping between all 2nd-nearest-neighbor oxygens and that the 1-band Hamiltonian should include 3rd-nearest-neighbor hoppings. We calculate the single-particle hopping between the planes of a bi-layer and show that it is generically: t(perpendicular to) (k(parallel to)) approximate to 0.25 eV .nu(2) (1 - 2ut'/t)(-2). The hopping through insulating spacers such as (BaO)Hg(BaO) is an order of magnitude smaller, but seems to have the same k(parallel to)-dependence. We show that the inclusion of t' is crucial for understanding ARPES for the anti-ferromagnetic insulator Sr2CuO2Cl2. Finally, we estimate the value of the inter-plane exchange constant J(perpendicular to) for an un-doped bi-layer in mean-field theory using different single-particle Hamiltonians, the LDA for YBa2Cu3O6, the eight- and four-band Hamiltonians, as well as an analytical calculation for the latter. We conclude that J perpendicular to similar to -20 meV.