The computational cost of performing a configuration interaction (CI) calculation for treating electron-electron correlation is directly proportional to the number of terms in the CI expansion. In this work, we present a diagrammatic projection approach for a priori identification of non-contributing terms in a CI expansion. This method known as the geminal-projected configuration interaction (GP-CI) method is based on using a two-body R12 geminal operator for describing electron-electron correlation in a reference many-electron wave function. The diagrammatic projection procedure was performed by first deriving the Hugenholtz diagrams of the energy expression of the R12 reference wave function and then performing diagrammatic factorization of effective particle-hole creation operators. The projection operation, which is a functional of the geminal function, was defined and used for the construction of the geminal-projected particle-hole creation operators. The form of the two-body R12 geminal operator was derived analytically by imposing an approximate Kato cusp condition. A linear combination of the geminal-projected oneparticle one-hole and two-particle two-hole operators were used for the construction of the GP-CI wave function. The applicability and implementation of the diagrammatic projection method was demonstrated by performing proof-of-concept calculations on an isoelectronic series of 10 electron systems: CH 4 , NH 3 , H 2 O, HF, Ne. The results from the calculations show that, as compared to conventional CI calculations, the GP-CI method was able to substantially reduce the size of the CI space (by a factor of 6-9) while maintaining an accuracy of 10 −5 Hartrees for the ground state energies. These results demonstrate the ability of the diagrammatic projection procedure to identify non-contributing states using an analytical form of the R12 geminal correlator operator. The geminal-projection method was also applied to second order Moller-Plesset perturbation theory (GP-MP2) giving similar results to the GP-CI method in terms of reduction of the double excitation space and accuracy to the ground state energy. This work also extends the analytical derivation of the geminal-projected particle-hole creation operators that were used for the construction of the CI wave function to coupled-cluster theory (GP-CCSD). This general derivation can also be applied to other many-electron theories and multi-determinant quantum Monte Carlo calculations.

The relationship between structure and property is central to chemistry and enables the understanding of chemical phenomena and processes. Need for an efficient conformational sampling of chemical systems arises from the presence of solvents and the existence of non-zero temperatures. However, conformational sampling of structures to compute molecular quantum mechanical properties is computationally expensive because a large number of electronic structure calculations are required. In this work, the development and implementation of the effective stochastic potential (ESP) method is presented to perform efficient conformational sampling of molecules. The overarching goal of this work is to alleviate the computational bottleneck associated with performing a large number of electronic structure calculations required for conformational sampling. We introduce the concept of a deformation potential and demonstrate its existence by the proof-by-construction approach. A statistical description of the fluctuations in the deformation potential due to non-zero temperature was obtained using infinite-order moment expansion of the distribution. The formal mathematical definition of the ESP was derived using the functional minimization approach to match the infinite-order moment expansion for the deformation potential. Practical implementation of the ESP was obtained using the random-matrix theory method. The developed method was applied to two proof-of-concept calculations of the distribution of HOMO-LUMO gaps in water molecules and solvated CdSe clusters at 300 K. The need for large sample size to obtain statistically meaningful results was demonstrated by performing 10 ESP calculations. The results from these prototype calculations demonstrated the efficacy of the ESP method for performing efficient conformational sampling. We envision that the fundamental nature of this work will not only extend our knowledge of chemical systems at non-zero temperatures but also generate new insights for innovative technological applications.

We present the development of a real-space and projected congruent transformation method for treating electron correlation in chemical systems. This method uses an explicitly correlated function for performing congruent transformation on the electronic Hamiltonian. As a result of this transformation, the electronic Hamiltonian is transformed into a sum of two-, three-, four-, five-, and six-particle operators. Efficient computational implementation of these many-particle operators continues to be challenging for application of the congruent transformation approach for many-electron systems. In this work, we present a projected congruent transformed Hamiltonian (PCTH) approach to avoid computation of integrals involving operators that couple more than two particles. The projected congruent transformation becomes identical to the real-space congruent transformation in the limit of infinite basis size. However, for practical calculations, the projection is always performed on a finite-dimensional space. We show that after representing the contributing expressions of the PCTH in terms of diagrams, it is possible to identify a subset of diagrams that can be summed up to infinite order. This technique, denoted as partial infinite-order summation (PIOS), partly alleviates the limitation from the finite-basis representation of the PCTH method. The PCTH and PCTH-PIOS methods were applied to an isoelectronic series of 10-electron systems (Ne,HF,H 2 O,NH 3 ,CH 4 ) and results were compared with configuration interaction (CISD) calculations. The results indicate that the PCTH-PIOS method can treat electron-electron correlations while avoiding explicit construction and diagonalization of the Hamiltonian matrix.

Electron-hole or quasiparticle representation plays a central role in describing electronic excitations in many-electron systems. For charge-neutral excitation, the electron-hole interaction kernel is the quantity of interest for calculating important excitation properties such as optical gap, optical spectra, electron-hole recombination, and electron-hole binding energies. The electron-hole interaction kernel can be formally derived from the density-density correlation function using both Green's function and time-dependent density functional theory (TDDFT) formalism. The accurate determination of the electron-hole interaction kernel remains a significant challenge for precise calculations of optical properties in the GW+BSE formalism. From the TDDFT perspective, the electron-hole interaction kernel has been viewed as a path to systematic development of frequency-dependent exchange-correlation functionals. Traditional approaches, such as many-body perturbation theory formalism, use unoccupied states (which are defined with respect to Fermi vacuum) to construct the electron-hole interaction kernel. However, the inclusion of unoccupied states has long been recognized as the leading computational bottleneck that limits the application of this approach for larger finite systems. In this work, an alternative derivation that avoids using unoccupied states to construct the electron-hole interaction kernel is presented. The central idea of this approach is to use explicitly correlated geminal functions for treating electron-electron correlation for both ground and excited state wave functions. Using this ansatz, it is derived using both diagrammatic and algebraic techniques that the electron-hole interaction kernel can be expressed only in terms of linked closed-loop diagrams. It is proved that the cancellation of unlinked diagrams is a consequence of linked-cluster theorem in real-space representation. The electron-hole interaction kernel derived in this work was used to calculate excitation energies in many-electron systems, and results were found to be in good agreement with the EOM-CCSD and GW+BSE methods. The numerical results highlight the effectiveness of the developed method for overcoming the computational barrier of accurately determining the electron-hole interaction kernel to applications of large finite systems such as quantum dots and nanorods.