The MRChem multiresolution analysis code for molecular electronic structure calculations performance and scaling properties

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The MRChem multiresolution analysis code for

molecular electronic structure calculations:

performance and scaling properties

Peter Wind,∗,†Magnar Bjørgve,†Anders Brakestad,†Gabriel A. Gerez S.,†Stig

Rune Jensen,∗,†Roberto Di Remigio Eik˚as,‡and Luca Frediani∗,†

†Department of Chemistry, UiT - The Arctic University of Norway, N-9037 Tromsø,

Norway

‡Algorithmiq Ltd, Kanavakatu 3C, FI-00160 Helsinki, Finland

E-mail: peter.wind@uit.no; stig.r.jensen@uit.no; luca.frediani@uit.no

Abstract

MRChem is a code for molecular electronic structure calculations, based on a multi-

wavelet adaptive basis representation. We provide a description of our implementation

strategy and several benchmark calculations. Systems comprising more than a thou-

sand orbitals are investigated at the Hartree–Fock level of theory, with an emphasis

on scaling properties. With our design, terms which formally scale quadratically with

the system size, in eﬀect have a better scaling because of the implicit screening intro-

duced by the inherent adaptivity of the method: all operations are performed to the

requested precision, which serves the dual purpose of minimizing the computational

cost and controlling the ﬁnal error precisely. Comparisons with traditional Gaussian-

type orbitals based software, show that MRChem can be competitive with respect to

performance.

arXiv:2210.01011v2 [physics.chem-ph] 9 Nov 2022

1 Introduction

Gaussian-type orbitals (GTOs) and more generally Linear Combination of Atomic Or-

bitals (LCAOs)1are well established as a standard for ab initio molecular electronic structure

calculations. As their shape is closely related to the electronic structure of atoms, even very

small basis sets with only a few functions per Molecular Orbital (MO) can give reasonable

results for describing molecular properties. However, for extended systems the description of

each orbital still requires the contributions from the entire basis in order to ensure orthog-

onality. Without further precautions, even when using localized orbitals, a large proportion

of the coeﬃcients will be very small for those systems.

In a Multiresolution Analysis (MRA) framework like multiwavelets (MWs), the basis can

adapt according to the function described (for an in-depth review of the MRA method in

the ﬁeld of Quantum Chemistry, see Ref. 2). The available basis is in principle inﬁnite and

complete, and, in practice, it is dynamically adapted to the local shape of the function and the

required precision. This can require the real-space basis to comprise millions of elementary

functions for each orbital. In this sense, the method starts with a big handicap compared

to LCAO basis sets. On the other hand, the exponential growth of available computational

resources has in recent years enabled MRA calculations on systems comprising thousands of

atoms.3,4

Two main challenges need to be addressed in order to achieve adequate performance:

the large number of operations to perform and the large memory footprint. The former

is addressed by limiting the numerical operations to those that are strictly necessary to

achieve the requested precision: rigorous error control at each elementary operation is the

main strength of a MW approach, enabling eﬀective screening. The latter is achieved by

algorithmic design: beyond a certain system size, not all data can be stored in local (i.e.

fast access) memory. On modern computers, data access is generally more time consum-

ing than the mathematical operations, especially if the data is not available locally.aThe

aThe online interactive chart at https://colin-scott.github.io/personal_website/research/

computer implementation must be able to copy large amounts of data eﬃciently between

compute nodes and the algorithm must be devised to reuse the data already available on

the compute node or in cache memory when possible. In this article, we will present the

main implementation features of our MRA code, MRChem,5to tackle those challenges, thus

enabling calculations on large systems at the Hartree–Fock (HF) level.

Beyond the eﬀective thresholding that screens numerically negligible operations, the large

computational cost is addressed by parallelization either at the orbital level or through real

space partitioning. This dual strategy allows to minimize the relative cost of communication

and the overall computational costs. Further, the most intensive mathematical operations

are expressed as matrix multiplications, which allows for optimal eﬃciency. Using localized

orbitals, the adaptivity of the MW description will signiﬁcantly reduce the computational

eﬀort required to compute the terms involving remote orbitals. Within a MRA framework,

the operators will exhibit an intrinsic sparsity, even if the orbitals have contributions from

remote regions. This is achieved because the diﬀerent length scales are naturally separated

through the adaptive grid representation. This opens the way for a method that scales

linearly with system size (N), where the linearity arises naturally from the methodology,

rather than being obtained with ad hoc techniques, such as fast-multipole methods6for

Fock matrix construction, or puriﬁcation of the density matrix.7

The large memory requirement is addressed by storing the data in a distributed “memory

bank”, where orbitals can be sent and received independently by any CPU. The total size

of the memory bank is then only limited by the overall memory available on the entire

computer cluster. Benchmark calculations at the HF level show that MRChem is able

to describe systems with thousands of electrons. The implementation exhibits near-linear

scaling properties and it can also be competitive with state-of-the-art electronic structure

software based on LCAO methods.

interactive_latency.html (Accessed 2022-10-29) is useful to understand the relative performance of on-

chip vs. oﬀ-chip memory accesses. In particular, it shows how the evolution of memory chips has been

lagging behind that of CPUs.

2 Solving the Hartree–Fock equations with multiwavelets

We consider the Self-Consistent Field (SCF) equations of the HF method:

(T+V)ϕi=

Nocc

Fij ϕj,(1)

where the Fock matrix elements Fij are obtained as

Fij =hϕi|T+V|ϕji.(2)

In the above equations, T=−1

2∇2is the kinetic energy operator, and the potential Vcon-

tains the nuclear-electron attraction Vnuc, the Coulomb repulsion J, and the exact exchange

K. To solve the SCF equations within a MW framework, we rewrite the diﬀerential equation

(1) as an integral equation:

ϕi=−2Hµi?"V ϕi−

Nocc

j6=i

Fij ϕj#,(3)

where ?stands for the convolution product. This form is obtained by recalling that the

shifted kinetic energy operator T−has a closed-form Green’s function:8,9

(T−)−1= 2Hµ, Hµ=e−µr

r, µ =√−2, (4)

By setting i=Fii and µi=√−2Fii, the diagonal term is excluded from the summation in

Eq. (3). It can be shown that iterating on the integral equation corresponds to a precon-

ditioned steepest descent. Practical applications show that this approach is comparable in

eﬃciency (as measured in the rate of convergence of the SCF iterations) to more traditional

methods.10

A MW representation does not have a ﬁxed basis. It is therefore not possible to express

functions and operators as vectors and matrices of a predeﬁned size, and the virtual space is

to be regarded as inﬁnite. Still, each function has a ﬁnite, truncated representation deﬁned

by the chosen precision threshold, but this representation will in general be diﬀerent for

diﬀerent functions. It is therefore necessary to develop working equations which allow the

direct optimization of orbitals. On the other hand, only occupied orbitals are constructed

and the coupling through the Fock matrix in Eqs. (1) and (3) is therefore limited in size.

Another advantage is that, to within the requested precision, the result can be formally

considered exact, and exploiting formal results valid in a complete basis – most notably the

Hellmann–Feynman theorem – becomes straightforward.11

Diﬀerential operators such as the Laplacian pose a fundamental problem for function

representations which are not continuous, and a naive implementation leads to numerical

instabilities. Robust approximations are nowadays available,12 but for repeated iterations

avoiding the use of the Laplacian operator is still an advantage. This is the main reason for

using the integral form in Eq. (4) rather than the diﬀerential form.

3 Implementation details and parallelization strategy

MW calculations can be computationally demanding. Both the total amount of elemen-

tary operations, the large amount of memory required and the capacity of the network are

important bottlenecks.

In practice, a supercomputer is required for calculations on large systems. For example

the representation of one orbital can demand between 10 and 500MB of data, depending

on the kind of orbital and the precision requested (see Section 4.2.1 for details). Moreover,

several sets of functions are necessary in a single SCF iteration (orbitals at the previous

iterations, right-hand side of equations etc.). Large systems will eventually require more

memory than is locally available on a single compute node in a cluster, and distributed

data storage becomes mandatory. At the same time, the SCF equations will require pairs of

orbitals, i.e. all the data must at some point be accessible locally.

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TheMRChemmultiresolutionanalysiscodeformolecularelectronicstructurecalculations:performanceandscalingpropertiesPeterWind,,yMagnarBjrgve,yAndersBrakestad,yGabrielA.GerezS.,yStigRuneJensen,,yRobertoDiRemigioEikas,zandLucaFrediani,yyDepartmentofChemistry,UiT-TheArcticUniversityofNorway,N-9037Troms...

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