The construction of the self-energy operator is known to be a challenging task, which becomes drastically more difficult with increasing system size due to the cubic scaling of the number of integrals that must be summed with respect to the number of orbital basis functions. In this work, we demonstrate how the technique of stochastic enumeration combined with stratified sampling can be used to construct the self-energy operator obtained from the electron-propagator formulation using Monte-Carlo numerical integration techniques. In order to overcome the problematic scaling associated with constructing the self-energy, we use a stratified stochastic enumeration scheme which allows us to replace the sequential sums in both the 2p1h and 1p2h terms of the self-energy with a stochastic summation. The SESSMO method utilizes a dual stratified sampling scheme where stratified sampling is performed in the discrete integer space of the molecular orbitals and in the 3-D real space of the electronic coordinates. In addition to the implementation of this stratified stochastic enumeration scheme, we also use a control-variate term to reduce the numerical error in the Monte-Carlo evaluation of the two-electron integrals. In this work we obtain both single-shot and iterative solutions to the diagonal approximation of the Dyson equation. The SESSMO method was used to study semiconductor quantum dots and metallic nanowires. The results of this work show that the SESSMO method can be used to efficiently and accurately construct the self-energy operator and subsequently solve the Dyson equation.

Development of stochastically enumerated stratified sampling of molecular orbitals (SESSMO) method for the efficient construction of the frequency-dependent self-energy operator

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