BIRS Workshop Lecture Videos
Multiresolution spectral–element representations of electronic wave functions Valeev, Edward
One of the grand challenges of quantum chemistry is computing observables (energies, scattering cross–sections, response properties) and their confidence intervals with high precision. This requires fast methods, characterized by a low–order scaling of the computational cost with the system size and error. Despite significant advances in the area, we are still very far from this goal. This talk will address some of our recent work on fast methods, with the word fast referring to the scaling with precision (i.e. the discretization error) and with the system size. Specifically, I will discuss our efforts to compute two–particle wave functions in hierarchical (multiresolution) discontinuous spectral–element representations. [1, 2] Each volume element of the six–dimensional wave function is supported by an orthonormal basis set of tensor products of polynomials. To alleviate the severe (exponential) cost of finite–element representation in many dimensions, which makes precise representation and computation of correlated wave functions intractable, we use low–rank tensor approximations, namely the singular value decomposition SVD, and reformulate all operations necessary to solve the Schr ̈odinger equation in low–rank form.  The low-rank approximations alone are not sufficient; hence we use explicitly–correlated terms in the wave function to regularize the electron–electron Coulomb singularities of the Hamiltonian. Our approach does not assume any geometric symmetry, hence the method is tractable for molecules.  The method was used to compute the first–order Møller-Plesset wave function and the second-order energy of the helium atom with precision guaranteed by construction (our most precise value for the MP2 energy is −37.379 mEh). We will further highlight the strengths and weaknesses of the adaptive discretization strategy.  F. A. Bischoff, R. J. Harrison, and E. F. Valeev, J. Chem. Phys. 137, 104103 (2012).  F.A.BischoffandE.F.Valeev,inpreparation.  F. A. Bischoff and E. F. Valeev, J. Chem. Phys. 134, 104104 (2011).
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