# H.2 Recursive Algorithm to Calculate Following Appendix H.1, the algorithm to calculate the electron density (diagonal elements of ) is discussed in terms of the DYSON equation for the lesser and the left-connected GREEN's functions. The solution to the matrix equation (H.12)

can be written as (H.13)

where and have been defined in (H.7) and (H.8), and and are readily identifiable from (H.12). Using the relation , (H.13) can be written as (H.14)

where (H.15)

The left-connected lesser GREEN's function is defined by the first blocks of (H.2) (H.16) is defined in a manner identical to except that the left-connected system is comprised of the first blocks of (H.2). In terms of (H.12), the equation governing follows by setting and . Using the DYSON equations for and , can be recursively obtained as (H.17)

which can be written in a more intuitive form as (H.18)

where . Equation (H.18) has the physical meaning that has contributions due to four injection functions: (i) an effective self-energy due to the left-connected structure that ends at , which is represented by , (ii) the diagonal self-energy component at grid point that enters (H.2), and (iii) the two off-diagonal self-energy components involving grid points and .

To express the full lesser GREEN's function in terms of the left-connected GREEN's function, one can consider (H.12) such that , and . Noting that the only non-zero block of is and using (H.14), one obtains (H.19)

Using (H.14), can be written in terms of and other known GREEN's functions as (H.20)

Substituting (H.20) in (H.19) and using (H.5), one obtains (H.21)

where (H.22)

The terms inside the square brackets of (H.21) are HERMITian conjugates of each other. In view of the above equations, the algorithm to compute the diagonal blocks of is given by the following steps:
• ,
• For , (H.18) is computed,
• For , (H.21) and (H.22) are computed.

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