doc: fixed documentation to reflect orthogonal implementation

Signed-off-by: Nick Papior <[email protected]>

README.md

@@ -164,12 +164,12 @@ To create the nearest neighbour tight-binding model for graphene you simply do
     # Create nearest-neighbour tight-binding
     # graphene lattice constant 1.42
     dR = ( 0.1 , 1.5 )
-    on = ( 0.  , 1.  )
-    nn = (-0.5 , 0.  )
+    on = 0. 
+    nn = -0.5
 
     # Ensure that graphene has supercell connections
     gr.sc.set_nsc([3,3,1])
-    tb = TightBinding(gr)
+    tb = Hamiltonian(gr)
     for ia in tb.geom:
         idx_a = tb.close(ia, dR=dR)
         tb[ia,idx_a[0]] = on
@@ -178,9 +178,9 @@ To create the nearest neighbour tight-binding model for graphene you simply do
 at this point you have the tight-binding model for graphene and you can easily create
 the Hamiltonian using this construct:
 
-    H, S = tb.tocsr(k=[0.,0.5,0])
+    Hk = tb.Hk(k=[0.,0.5,0])
 
-which returns the Hamiltonian and the overlap matrices in the `scipy.sparse.csr_matrix`
+which returns the Hamiltonian in the `scipy.sparse.csr_matrix`
 format. To calculate the dispersion you diagonalize and plot the eigenvalues
 
     import matplotlib.pyplot as plt
@@ -189,8 +189,8 @@ format. To calculate the dispersion you diagonalize and plot the eigenvalues
     for ik, k in enumerate(klist):
         eigs[ik,:] = tb.eigh(k, eigvals_only=True)
         # Or equivalently:
-        #   H, S = tb.tocsr(k)
-        #   eigs[ik,:] = sli.eigh(H.todense(), S.todense(), eigvals_only=True)
+        #   Hk = tb.Hk(k)
+        #   eigs[ik,:] = sli.eigh(Hk.todense(), eigvals_only=True)
      for i in range(tb.no):
         plt.plot(eigs[:,i])
 
@@ -207,6 +207,26 @@ which accomplishes the same thing, but at much faster execution. `iR` should be
 number such that `tb.geom.close(<any index>,dR = tb.geom.dR * iR)` is approximately
 1000 atoms.
 
+The above example is for the default orthogonal Hamiltonian. However, sisl is
+not limited to orthogonal basis functions. To construct the same example using
+explicit overlap matrix the following procedure is necessary:
+
+    # Create nearest-neighbour tight-binding
+    # graphene lattice constant 1.42
+    dR = ( 0.1 , 1.5 )
+    on = ( 0. , 1.)
+    nn = (-0.5, 0.) # still orthogonal (but with fake overlap)
+
+    tb = Hamiltonian(gr, ortho=False)
+    for ia in tb.geom:
+        idx_a = tb.close(ia, dR=dR)
+        tb[ia,idx_a[0]] = on
+        tb[ia,idx_a[1]] = nn
+    Hk = tb.Hk(k=[0.,0.5,0])
+    Sk = tb.Sk(k=[0.,0.5,0])
+    eigs = sli.eigh(Hk.todense(), Sk.todense(), eigvals_only=True)
+
+
 
 ## Downloading and installation ##