This page gives a high level overview of the ideas underlying linear scaling codes, and Conquest in particular. Technical details of the Conquest code can be found in the CONQUEST References section of these web pages; a reprint giving details of recent Conquest developments is available here.

### Introduction

Normal DFT codes solve for the Kohn-Sham eigenstates ( $$\hat{H} \psi_n = \epsilon_n \psi_n$$ ); these eigenstates extend throughout the simulation cell. Additionally, the eigenstates must be kept orthogonal to each other, which involves integrals of the form $$\int \mathrm{d}\mathbf{r} \psi_n(\mathbf{r}) \psi_m(\mathbf{r})$$. These integrals contain three terms which scale with the simulation cell size, and hence with the number of atoms in the cell, giving $$\mathcal{O}(N^3)$$ scaling. This dominates as system size increases, and there is a practical limit of a thousand atoms or so, even on high performance computing (HPC) machines.

Linear scaling DFT codes, by contrast, take advantage of the well-known fact that electronic structure is local. Rather than solve for the eigenstates, most linear scaling codes solve for the density matrix $$\rho(\mathbf{r},\mathbf{r}^\prime)$$. This is forced to be local in space (i.e. setting $$\rho(\mathbf{r},\mathbf{r}^\prime) = 0, \vert\mathbf{r} - \mathbf{r}^\prime\vert > R_c$$ for some cutoff distance $R_c$). Then, provided that the implementation is performed carefully, the computer effort and memory will scale linearly with the number of atoms: $$\mathcal{O}(N)$$ scaling.

### Density Matrices

For practical implementation, the density matrix is assumed to be separable, and is written in terms of localised, atom-centred support functions: $\rho(\mathbf{r},\mathbf{r}^\prime) = \sum_{i\alpha j\beta} \phi_{i\alpha}(\mathbf{r}) K_{i\alpha j\beta} \phi_{j\beta}(\mathbf{r}^\prime)$

The support functions then define Hamiltonian and overlap matrices ($$H_{i\alpha j\beta}$$ and $$S_{i\alpha j\beta}$$). The ground state search now goes over the density matrix and the support functions, with the usual constraint on self-consistency between potential and charge density. Instead of requiring orthonormality between the eigenstates, the density matrix is required to be idempotent (i.e. $$\rho^2 = \rho$$). This is not an easy condition to impose, and there are various solutions. Conquest works by writing $$K = 3LSL - 2LSLSL$$, a transformation which has extrema at 0 and 1, which is due to McWeeny.

### Sparse matrices and parallel scaling

In order to have linear scaling both in computational time and memory, linear scaling codes must use different techniques to normal codes. In particular, the matrices used will be sparse (that is they will have a limited number of non-zero elements) but linear scaling will not happen unless the zero elements are eliminated. Conquest has specially designed sparse matrix techniques, which are also constructed to give good performance on massively parallel computers.

If a linear scaling code is to operate on thousands or millions of atoms, then it must operate on parallel machines. The spatial locality of linear scaling methods (all operations relating to a given atom are restricted to a local area of real space) makes them naturally parallelisable. In Conquest, each processor (or each core in the multi-core machines which are becoming common) is given responsibility for a different group of atoms (known as a bundle of atoms). The rows of matrices labelled by these atoms are stored on the core, and when matrix multiplications are performed the core is responsible for calculating the rows of the new matrix.

### Support functions

The support functions $$\phi_{i\alpha}(\mathbf{r})$$ are represented in terms of basis functions: $$\phi_{i\alpha}(\mathbf{r}) = \sum_s c_{i\alpha s} \chi_s(\mathbf{r})$$. In Conquest, there are two basis functions implemented:

• Blip functions or b-splines, which allow plane-wave accuracy to be achieved
• Pseudo-atomic orbitals (PAOs), which allow rapid, analytic calculations

If a limited set of PAOs is used, then Conquest operates as an ab initio tight binding (or tight binding DFT) code, either self-consistent or non-self-consistent. As the set of PAOs expands, or for blip function basis sets, then full DFT is recovered, and plane-wave accuracy can be achieved by increasing cutoffs on the support functions and the density matrix.

### Other technical details

The charge density is stored on a grid, which is also used for numerical integrals and FFTs. As with the atoms, each processor or core is given responsibility for a specific area of the integration grid. The operation of Conquest is most efficient if these areas overlap strongly and are compact.