  Traditional solution approaches for problems in quantum mechanics scale as
$\mathcal O(M^3)$, where $M$ is the number of electrons. Various methods have
been proposed to address this issue and obtain linear scaling $\mathcal O(M)$.
One promising formulation is the direct minimization of energy. Such methods
take advantage of physical localization of the solution, namely that the
solution can be sought in terms of non-orthogonal orbitals with local support.
In this work a numerically efficient implementation of sparse parallel vectors
within the open-source finite element library deal.II is proposed. The main
algorithmic ingredient is the matrix-free evaluation of the Hamiltonian
operator by cell-wise quadrature. Based on an a-priori chosen support for each
vector we develop algorithms and data structures to perform (i) matrix-free
sparse matrix multivector products (SpMM), (ii) the projection of an operator
onto a sparse sub-space (inner products), and (iii) post-multiplication of a
sparse multivector with a square matrix. The node-level performance is analyzed
using a roofline model. Our matrix-free implementation of finite element
operators with sparse multivectors achieves the performance of 157 GFlop/s on
Intel Cascade Lake architecture. Strong and weak scaling results are reported
for a typical benchmark problem using quadratic and quartic finite element
bases.
