Steven R. White

Gausslets are one of the few basis constructions for electronic structure that combine locality, orthonormality, variable resolution, and an accurate diagonal approximation for the electron–electron interaction, but the original construction is tied to one dimension. Radial gausslets extended this idea to atoms while leaving the angular degrees of freedom in spherical harmonics, so the atomic interaction remained only partially diagonal in the combined basis. Here, we introduce generalized gausslets on the sphere and combine them shell by shell with radial gausslets to form an atom-centered basis in which the electron–electron interaction takes a two-index integral-diagonal form. The angular basis starts from localized spherical Gaussians and uses injection to make a low-ℓ spherical-harmonic subspace exact. Tests of the kinetic spectrum, low-ℓ Coulomb matrix elements, spherium, first-row Hartree–Fock calculations, and He exact diagonalization show systematic convergence with increasing angular resolution. We also develop density matrix renormalization group (DMRG) methods for this basis, including compact matrix-product operators, correlated small-space starting states, Givens-rotation transfers between nearby angular sizes, and embedded sampled variance extrapolation. We show that this combination of ingredients can be used to solve the Be atom, with extrapolations in the number of angular functions but with fixed radial resolution, to within about 0.1 mH of the complete-basis-set limit exact energy. This shows that DMRG calculations of first-row atoms that include both static and accurate dynamic correlation on the same footing are feasible.

Gausslets are one of the few examples of basis sets for electronic structure that allow for two-index/diagonal electron–electron interaction terms. A weakness of gausslets is that, because of their 1D origin, they have been tied to Cartesian coordinates. Here, we generalize the gausslet construction for the radial coordinate in three dimensions for atomic basis sets. These radial gausslets make a very compact radial basis with a relatively modest number of functions, with diagonal interaction terms. We illustrate the accuracy of this construction with Hartree–Fock and exact diagonalization on atomic systems.

We introduce a simple and efficient variation of the tangent-space excitation Ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the S=1 S = 1 Heisenberg chain, our method—site basis excitation Ansatz (SBEA)—efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition—leaving the basis nonorthogonal—is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.

The phase diagram of the S=1/2 easy-axis triangular-lattice J_{1}-J_{2} model is investigated using the density-matrix renormalization group and analytical insights. We find a significant spin-liquid region extending from the Heisenberg limit and residing between the Y phase-known as the magnetic analogue of the "supersolid"-and collinear stripe phase. The order parameters of the supersolid are analyzed and an understanding of its lack of a ferromagnetic moment is suggested.

High-energy contributions to the spectra of approximate ground states are a universal feature of matrix product states.