  For electron spin qubits in quantum dots, reducing charge noise sensitivity
is a critical step in achieving fault tolerant two-qubit gates mediated by the
exchange interaction. This work explores how the physical device geometry
affects the sensitivity of exchange to fluctuations in applied gate voltage and
interdot bias due to charge noise. We present a modified linear combination of
harmonic orbitals configuration interaction (LCHO-CI) method for calculating
exchange energies that is applicable to general quantum dot networks. In the
modified LCHO-CI approach, an orthogonal set of harmonic orbitals formed at the
center of the dot network is used to approximate the many-electron states. This
choice of basis significantly reduces the computation time of the full CI
calculation by enabling a pre-calculated library of matrix elements to be used
in evaluating the Coulomb integrals. The resultant many-electron spectra are
mapped onto a Heisenberg Hamiltonian to determine the individual pairwise
electronic exchange interaction strengths, $J_{ij}$. The accuracy of the
modified LCHO-CI method is further improved by optimizing the choice of
harmonic orbitals without significantly lengthening the calculation time. The
modified LCHO-CI method is used to calculate $J$ for a silicon MOSFET double
quantum dot occupied by two electrons. Two-dimensional potential landscapes are
calculated from a 3D device structure, including both the Si/SiO$_2$
heterostructure and metal gate electrodes. The computational efficiency of the
modified LCHO-CI method enables systematic tuning of the device parameters to
determine their impact on the sensitivity of $J$ to charge noise, including
plunger gate size, tunnel gate width, SiO$_2$ thickness and dot eccentricity.
Generally, we find that geometries with larger dot charging energies, smaller
plunger gate lever arms, and symmetric dots are less sensitive to noise.
