Maximization of linear independence of basis function products.

Basis sets consisting of functions that form linearly independent products (LIPs) have remarkable applications in quantum chemistry but are scarce because of mathematical limitations. We show how to linearly transform a given set of basis functions to maximize the linear independence of their products by maximizing the determinant of the appropriate Gram matrix. The proposed method enhances the utility of the LIP basis set technology and clarifies why canonical molecular orbitals form LIPs more readily than atomic orbitals.

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential () within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential that is equivalent to () within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials () defined on the infinite-dimensional Hilbert space, the potentials reconstructed from matrices of () within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between and () improves to the extent that the basis function products are appropriate as a basis for ().

Pomeron Eigenvalue at Three Loops in N=4 Supersymmetric Yang-Mills Theory.

We obtain an analytical expression for the next-to-next-to-leading order of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron eigenvalue in planar N=4 SYM using quantum spectral curve (QSC) integrability-based method. The result is verified with more than 60-digit precision using the numerical method developed by us in a previous paper [N. Gromov, F.

Exact slope and interpolating functions in N=6 supersymmetric Chern-Simons theory.

Using the quantum spectral curve approach we compute, exactly, an observable (called slope function) in the planar Aharony-Bergman-Jafferis-Maldacena theory in terms of an unknown interpolating function h(λ) which plays the role of the coupling in any integrability based calculation in this theory. We verified our results with known weak coupling expansion in the gauge theory and with the results of semiclassical string calculations. Quite surprisingly at strong coupling the result is given by an explicit rational function of h(λ) to all orders.