## Stochastic *GW*

We have recently developed a formalism to calculate quasiparticle
energies within the *GW* approximation using stochastic orbitals.
The *GW* approximation offeres a highly accurate theory for calculatiing
quasiparticle energies in a semiconducting materials. Practical use, however,
is limited by the theoretical fourth-order scaling with respect to
system size.

We have recently developed a stochastic, orbital-free formalism which effectively
circumvents the issues with scaling. In our method, the occupied and virtual orbitals
of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate
the Green function *G*, the polarization potential *W*, and thereby the
*GW* self-energy. The method enables *linear scaling* calculations, breaking
the theoretical scaling limit.

We illustrated the utility of the method for silicon nanoclusters of various sizes,
the largest of which have over 3000 electrons, and various organic fullerenes.

## Stochastic DFT

In the standard implementation of Kohn-Sham DFT, the electron density is

\(\rho(r) = 2[|\psi_1(r)|^2+|\psi_2(r)|^2+\ldots+|\psi_{\text{HOMO}}(r)|^2 \)

The occupied states are in turn obtained by diagonalzing the density-dependent Kohn-Sham matrix.
Formally, matrix diagonalization scales as N

^{3}_{}.
In stochastic DFT, on the other hand, the electron density is obtained as the average over densities of filtered stochastic orbitals:

\(\rho(r) = <|\xi_{filtered}(r)|^2>_{\xi}\)

The number of stochastic orbitals required is typically independent of the number of electrons resulting in an overall linear or sometimes sublinear scaling with this method.
We also have developed embedded fragment and overlapped embedded fragment stochastic DFT which allows us to study covalently bonded systems more accurately.

## Stochastic TDDFT

In conventional Kohn-SHam TDDFT, all occupied orbitals will be propagated individually. This makes TDDFT expensive for larger systems. In stochastic TDDFT, instead of propagating all occupied orbitals, we instead propagate the filtered stochastic orbitals. Each such orbital is a linear combination of all occupied states and thus carries the information of the entire system.

## Correlation Energies

Local and semi-local corrlation functionals commonly used within
DFT fail to describe long-range van der Waals interactions and other types
of dynamical screening effects. Post-DFT corrections to the correlation
energy, such as MP2 and RPA, offer a straightforward method for improving estimates
of various observables. The main hurdle with higher-order correlation energies
is computational complexity. Similar to the research above, we can use stochastic
orbitals to replace large sums over occuopied and virtual KS orbitals.

## Density functional theory

We also develop and implement improved time-dependent density functional theory (TDDFT) approaches
suitable for nanoscale device modeling.
These include:

- Orbital-free DFT methods
- Hydrodynamic tensor DFT
- Quantum Drude friction