What does the index i or j in Eq. of local density of states (LDOS) stand for?

[berlin]

What is the mean of the i,j index in local density of states (LDOS):

from the notes in the end of this page:
http://www.quantumwise.com/documents/manuals/latest/ReferenceManual/index.html/ref.localdevicedensityofstates.html

Are these LCAO basis index?  Why is LDOS defined as this?

Moreover, the are both complex, but DOS should be real. What is wrong here?

And more, the LCAO is a kind of non-orthogonalization basis, there may have some overlaps bettween and . Does the overlaps are removed? 

Or any reference on LDOS ?

[Jess Wellendorff]

Yes, the i,j indices are the indices of the LCAO basis set orbitals, which in ATK are real functions, as mentioned just below the equation.

As explained here: http://www.quantumwise.com/documents/manuals/latest/ReferenceManual/index.html/ref.devicedensityofstates.html, the overlap matrix S is computed from the overlaps of the basis functions.

[berlin]

Even those LCAO basis are real, as you mentioned, but the must be complex matrix.
 How to get a real DOS function based on that Eq. ?

[Jess Wellendorff]

Yes, the density matrix is in general a complex matrix, but it is also hermitian. This means that the imaginary parts of \rho and \rho^\dagger cancel out when summing over all i's and j's, leaving only the real part of \rho (the density matrix) multiplied by the real LCAO basis functions. The DOS is therefore real (has no imaginary components).