  In this work, the composite control-variate stratified sampling (CCSS) method
is presented for calculation of MO integrals without transformation of AO
integrals. The central idea of this approach is to obtain the 2-electron MO
integrals by direct integration of 2-electron coordinates. This method does not
require or use pre-computed AO integrals and the value of the MOs at any point
in space is obtained directly from the linear combination of AOs. The
integration over the electronic coordinates was performed using stratified
sampling Monte Carlo method. This approach was implemented by dividing the
integration region into a set of non-overlapping segments and performing Monte
Carlo calculations on each segment. The Monte Carlo sampling points for each
segment were optimized to minimize the total variance of the sample mean.
Additional variance reduction of the overall calculations was achieved by
introducing control-variate in the stratified sampling scheme. The composite
aspect of the CCSS allows for simultaneous computation of multiple MO integrals
during the stratified sampling evaluation. The main advantage of the CCSS
method is that unlike rejection sampling Monte Carlo methods such as Metropolis
algorithm, the stratified sampling uses all instances of the calculated
functions for the evaluation of the sample mean. The CCSS method is designed to
be used for large systems where AO-to-MO transformation is computationally
prohibitive. Because it is based on numerical integration, the CCSS method can
be applied to a wide variety of integration kernels and does not require
\textit{a priori} knowledge of analytical integrals. In this work, the
developed CCSS method was applied for calculation of excitonic properties in
CdSe quantum dots using electron-hole explicitly correlated Hartree-Fock
(eh-XCHF) and geminal-screened electron-hole interaction kernel (GSIK) methods.
