H.2 Recursive Algorithm to Calculate $G^<$

Following Appendix H.1, the algorithm to calculate the electron density (diagonal elements of $G^<$) is discussed in terms of the DYSON equation for the lesser and the left-connected GREEN's functions. The solution to the matrix equation

$$\left[ \begin{array}{cc} \ensuremath{{\underline{A}}}_{Z,Z} & \en... ...uremath{{\underline{G}}}^\mathrm{a}_{Z^\prime,Z^\prime} \end{array} \right] \ ,$$ (H.12)

can be written as

$$\begin{array}{lll} \ensuremath{{\underline{G}}}^< &=& \ensure... ...a}}}^< \ \ensuremath{{\underline{G}}}^\mathrm{a} \ ,\end{array}$$ (H.13)

where $\ensuremath{{\underline{G}}}^\mathrm{r0}$ and $\ensuremath{{\underline{U}}}$ have been defined in (H.7) and (H.8), and $\ensuremath{{\underline{G}}}^<$ and $\ensuremath{{\underline{G}}}^a$ are readily identifiable from (H.12). Using the relation $\ensuremath{{\underline{G}}}^\mathrm{a}={\ensuremath{{\underline{G}}}^\mathrm{... ...0} \ensuremath{{\underline{U}}}^\dagger \ensuremath{{\underline{G}}}^\mathrm{a}$, (H.13) can be written as

$$\begin{array}{lll} \ensuremath{{\underline{G}}}^< &=& \ensure... ...dagger\ \ensuremath{{\underline{G}}}^\mathrm{a0}\ , \end{array}$$ (H.14)

where

$$\begin{array}{lll}\displaystyle \ensuremath{{\underline{G}}}^... ...{\Sigma}}}^<\ \ensuremath{{\underline{G}}}^{a0} \ . \end{array}$$ (H.15)

The left-connected lesser GREEN's function $\ensuremath{{\underline{g}}}^<_{L_{q}}$ is defined by the first $q$ blocks of (H.2)

$$\begin{array}{l} \ensuremath{{\underline{A}}}_{_{1:q,1:q}} \ ... ...emath{{\underline{g}}}^\mathrm{a}_{L_{1:q,1:q}} \ . \end{array}$$ (H.16)

$\ensuremath{{\underline{g}}}^{<}_{L_{q+1}}$ is defined in a manner identical to $\ensuremath{{\underline{g}}}^{<}_{L_{q}}$ except that the left-connected system is comprised of the first $q+1$ blocks of (H.2). In terms of (H.12), the equation governing $\ensuremath{{\underline{g}}}^<_{L_{q+1}}$ follows by setting $Z=1:q$ and $Z^\prime=q+1$. Using the DYSON equations for $\ensuremath{{\underline{G}}}^\mathrm{r}$ and $\ensuremath{{\underline{G}}}^<$, $\ensuremath{{\underline{g}}}^{<}_{L_{q+1,q+1}}$ can be recursively obtained as [8]

$$\begin{array}{l} \ensuremath{{\underline{g}}}^<_{L_{q+1,q+1}}... ...emath{{\underline{g}}}^\mathrm{a}_{L_{q+1,q+1}} \ , \end{array}$$ (H.17)

which can be written in a more intuitive form as

$$\begin{array}{l} \ensuremath{{\underline{g}}}^<_{L_{q+1,q+1}}... ...emath{{\underline{g}}}^\mathrm{a}_{L_{q+1,q+1}} \ , \end{array}$$ (H.18)

where $\ensuremath{{\underline{\sigma}}}^<_{_{q+1}} = \ensuremath{{\underline{A}}}_{_... ...{\underline{g}}}^{<}_{L_{q,q}} \ensuremath{{\underline{A}}}^\dagger_{_{q,q+1}}$. Equation (H.18) has the physical meaning that $\ensuremath{{\underline{g}}}^<_{L_{q+1,q+1}}$ has contributions due to four injection functions: (i) an effective self-energy due to the left-connected structure that ends at $q$, which is represented by $\ensuremath{{\underline{\sigma}}}^<_{_{q+1}}$, (ii) the diagonal self-energy component at grid point $q+1$ that enters (H.2), and (iii) the two off-diagonal self-energy components involving grid points $q$ and $q+1$.

To express the full lesser GREEN's function in terms of the left-connected GREEN's function, one can consider (H.12) such that $\ensuremath{{\underline{A}}}_Z=\ensuremath{{\underline{A}}}_{_{1:q,1:q}}$, $\ensuremath{{\underline{A}}}_Z^\prime=\ensuremath{{\underline{A}}}_{_{q+1:N,q+1:N}}$ and $\ensuremath{{\underline{A}}}_{Z,Z^\prime}=\ensuremath{{\underline{A}}}_{_{1:q,q+1:N}}$. Noting that the only non-zero block of $\ensuremath{{\underline{A}}}_{_{1:q,q+1:N}}$ is $\ensuremath{{\underline{A}}}_{_{q,q+1}}$ and using (H.14), one obtains

$$\begin{array}{l} \ensuremath{{\underline{G}}}^<_{_{q,q}} = \e... ...,q+1}} \ensuremath{{\underline{G}}}^<_{_{q+1,q}}\ . \end{array}$$ (H.19)

Using (H.14), $\ensuremath{{\underline{G}}}^<_{_{q+1,q}}$ can be written in terms of $\ensuremath{{\underline{G}}}^<_{_{q+1,q+1}}$ and other known GREEN's functions as

$$\begin{array}{l} \ensuremath{{\underline{G}}}^<_{_{q+1,q}} = ... ...ensuremath{{\underline{g}}}^\mathrm{a}_{L_{q,q}} \ .\end{array}$$ (H.20)

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

$$\begin{array}{lll} \ensuremath{{\underline{G}}}^<_{_{q,q}}&=&... ...remath{{\underline{g}}}^{<0}_{_{q+1,q}} \right] \ , \end{array}$$ (H.21)

where

$$\begin{array}{lll} \ensuremath{{\underline{g}}}^{<0}_{_{q,q+1... ...nsuremath{{\underline{g}}}^\mathrm{a0}_{_{q,q}} \ . \end{array}$$ (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 $\ensuremath{{\underline{G}}}^<$ is given by the following steps:

• $\ensuremath{{\underline{g}}}^<_{L_{1,1}} = \ensuremath{{\underline{g}}}^\mathr... ...{{\underline{\Sigma}}}^<_L \ \ensuremath{{\underline{g}}}^\mathrm{a0}_{_{1,1}}$,
• For $q=1,~2,~\ldots,~N-1$, (H.18) is computed,
• For $q=~N-1,~N-2,~\ldots,~1$, (H.21) and (H.22) are computed.

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