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  • DFT without eigenvectors

    Introduction

    Several methods have been proposed (Bekas 2005)

    Theory

    We define a set of orthonormal functions \(\left\{\varsigma(\vec{r})_j\right\}_j\) that span the same basis as the orbitals \(\left\{\varphi(\vec{r})_j\right\}_j\). The are related through a orthogonal transformation \({\overleftrightarrow{R}}\) such that \[\varphi_i(\vec{r}) = \sum_jR_{ij}\varsigma_j(\vec{r}) \ .\] \[\begin{aligned} n(\vec{r}) &= \sum_if_i\varphi_i^*(\vec{r})\varphi_i(\vec{r}) \\ &=\sum_{ijk}f_iR^*_{ij}\varsigma_j^*(\vec{r})R_{ik}\varsigma_k(\vec{r})\\ &=\sum_{jk}\varsigma_j^*(\vec{r})\left(\sum_iR^*_{ij}f_iR_{ik}\right)\varsigma_k(\vec{r})\\ &=\sum_{jk}\varsigma_j^*(\vec{r})F_{jk}\varsigma_k(\vec{r})\end{aligned}\]

    \[F_{jk}=\sum_iR^*_{ij}f_iR_{ik} \label{eq:occupationmatrix}\]

    \[\begin{aligned} \mathrm{Tr}({\overleftrightarrow{F}})&=\sum_jF_{jj}\\ &=\sum_{ij}R^*_{ij}f_iR_{ij}\\ &=\sum_{j}f_j\\ &=N\end{aligned}\]

    \[\begin{aligned} \sum_if_i\epsilon_i &= \sum_if_i\int\mathrm{d}\vec{r}\varphi_i^*(\vec{r})\int\mathrm{d}\vec{r}'\mathrm{H}(\vec{r},\vec{r}')\varphi_i(\vec{r}') \\ &=\sum_{ijk}f_i\int\mathrm{d}\vec{r}R^*_{ij}\varsigma_j^*(\vec{r})\int\mathrm{d}\vec{r}'\mathrm{H}(\vec{r},\vec{r}')R_{ik}\varsigma_k(\vec{r}')\\ &=\sum_{jk}F_{jk}\int\mathrm{d}\vec{r}\varsigma_j^*(\vec{r})\int\mathrm{d}\vec{r}'\mathrm{H}(\vec{r},\vec{r}')\varsigma_k(\vec{r}')\\ &=\sum_{jk}F_{jk}{\cal{E}}_{jk}\\ &=\mathrm{Tr}({\overleftrightarrow{F}}^\intercal {\overleftrightarrow{{\cal{E}}}})\end{aligned}\]

    \[{\cal{E}}_{jk}=\int\mathrm{d}\vec{r}\mathrm{d}\vec{r}'\varsigma_j^*(\vec{r})\mathrm{H}(\vec{r},\vec{r}')\varsigma_k(\vec{r}')\]

    In temperature-dependent DFT, the occupations are obtained from the Fermi-Dirac distribution \[f_i=\frac1{1+e^{(\epsilon_i-\mu)/kT}} \label{eq:fermidirac}\] where the chemical potential \(\mu\) is obtained from imposing the condition that the occupations must sum to the total number of electrons. This procedure establishes a relation between the set of KS eigenvalues \(\epsilon_i\) and the occupations \(f_i\). For our approach, we must find a relation between the subspace Hamiltonian \({\overleftrightarrow{h}}\) and the occupation matrix \({\overleftrightarrow{F}}\). Formally, we can diagonalize \({\overleftrightarrow{h}}\) to obtain both \({\overleftrightarrow{R}}\) and \(\epsilon_i\), from there we can then obtain \({\overleftrightarrow{F}}\) from eqs. \ref{eq:fermidirac} and \ref{eq:occupationmatrix} as \[F_{jk}=\sum_iR^*_{ij}\frac1{1+e^{(\epsilon_i-\mu)/kT}}R_{ik}\ . \label{eq:fermidiracdiagonal}\] Of course this involves the diagonalization of \({\overleftrightarrow{{\cal{E}}}}\) which is what we wanted to avoid from the beginning. But since \({\overleftrightarrow{R}}\) and \(\epsilon_i\) are the eigenvectors and eigenvalues of \({\overleftrightarrow{{\cal{E}}}}\) we can rewrite eq. \ref{eq:fermidiracdiagonal} as a matrix formula \[{\overleftrightarrow{F}}=\left[I + \exp\left(\frac1{kT}\left({\overleftrightarrow{{\cal{E}}}}-\mu\right)\right)\right]^{-1} \ . \label{eq:fermidiracmatrix}\] with the condition that the value of \(\mu\) must be such that \(\mathrm{Tr}({\overleftrightarrow{F}})=N\).

    We have converted the problem from the diagonalization of \({\cal{E}}\) to the evaluation of the function of a matrix for which several methods exists. So the question is now, can we find a method to evaluate \ref{eq:fermidiracmatrix} that is more efficient for large matrices than diagonalization.

    While most approaches are focused on sparse matrices (Sidje 2010)

    We can use a Padé approximation (Moler 2003) \[\exp\left({\overleftrightarrow{A}}\right)\sim{\overleftrightarrow{R}}_n({\overleftrightarrow{A}})=\left[{\overleftrightarrow{P}}_n({\overleftrightarrow{A}})\right]^{-1}\left[{\overleftrightarrow{Q}}_n({\overleftrightarrow{A}})\right]\] where \[P_n({\overleftrightarrow{A}})=\sum_{j=0}^n\frac{(2n-j)!\,n!}{(2n)!\,j!\,(n-j)!}{\overleftrightarrow{A}}^j\] and \[Q_n({\overleftrightarrow{A}})=\sum_{j=0}^n\left(-1\right)^j\frac{(2n-j)!\,n!}{(2n)!\,j!\,(n-j)!}{\overleftrightarrow{A}}^j\ .\]

    \[\frac{\partial{\overleftrightarrow{F}}}{\partial \mu} = \frac1T\exp\left(\frac1{T}\left({\overleftrightarrow{{\cal{E}}}}-\mu\right)\right){\overleftrightarrow{F}}^{-2}\]

    (CHECK ORDER, NON-COMMUTATIVE MULTIPLICATION)

    References

    1. Constantine Bekas, Yousef Saad, Murilo L. Tiago, James R. Chelikowsky. Computing charge densities with partially reorthogonalized Lanczos. Computer Physics Communications 171, 175–186 (2005). Link

    2. Roger B. Sidje, Yousef Saad. Rational approximation to the FermiDirac function with applications in density functional theory. Numer Algor 56, 455–479 (2010). Link

    3. Cleve Moler, Charles Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix Twenty-Five Years Later. SIAM Rev. 45, 3–49 (2003). Link