=Paper=
{{Paper
|id=Vol-2843/shortpaper007
|storemode=property
|title=Modeling the adsorption process on graphic processors (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2843/shortpaper007.pdf
|volume=Vol-2843
|authors=Dmitry Vinokursky,Natalia Kononova,Yana Agakhanova,Dmitry Miroshnichenko,Nikita Shelestov
}}
==Modeling the adsorption process on graphic processors (short paper)==
https://ceur-ws.org/Vol-2843/shortpaper007.pdf
Modeling the adsorption process on graphic processors*
Dmitry Vinokursky1, Natalia Kononova1, Yana Agakhanova2, Dmitry
Miroshnichenko3 and Nikita Shelestov1
1
Federal State Autonomous Educational Institution of Higher Education North-Caucasus Fed-
eral University, Stavropol, 355009, Russia
2
Moscow Institute of Physics and Technology, Moscow, 117303, Russia
3
CRT-GROUP, Moscow, 117587, Russia
knv_fm@mail.ru
Abstract. This paper briefly describes the technology of non-specialized com-
puting on GPUs and the theory of density functional. The Quantum ESPRESSO
software package was used to simulate the adsorption of an aluminum atom by
the C60 fullerene. It is shown that the binding energy of aluminum and
fullerene atoms depends on the aluminum atom position. The density of states is
calculated for each case.
Keywords: Computing on graphics processing units, Density functional theory,
Adsorption.
1 Introduction
General Purpose GPU Computing (GPGPU) has exploded in the last few years.
Transferring calculations from a central processing unit (GPU) to a video chip (GPU)
can often significantly speed up the computing speed. For example, according to
Nvidia, GPU molecular dynamics simulations using CUDA technology provide a 24x
improvement in performance over the same calculations on a CPU. The use of
GPGPU technology in quantum chemical calculations made it possible to carry out
calculations that previously took days in a few hours.
2 Materials and methods
2.1 GPGPU Technology
GPGPU (General-purpose graphics processing units) is a technology that uses a
graphics processing unit (GPU), which typically processes computations only for
computer graphics, to perform computations that are traditionally performed by a
central processing unit (CPU) [1].
*
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0).
Originally developed in 2003 to improve shader programs, the GPGPUs have
shown to be well suited to scientific computing needs, and development has continued
in this direction since then.
GPGPU is a form of parallel processing of data between one or more GPUs and
CPUs that parse the data as if it were in the form of an image or some other graphical
form. Although GPUs run at lower frequencies, they usually have multiple cores and
a fundamentally different architecture (Figure 1). This allows GPUs to process much
more images and graphics data per second than CPUs. Transferring data in graphical
form, scanning it and analyzing it with a GPU can significantly speed up calculations.
Fig. 1. GPU and CPU device. CPU is on the left, GPU is on the right.
GPUs implement the Single instruction, multiple data method (SIMD). This means
that the GPU cores execute the same set of instructions for each data instance. Most
graphics algorithms use this approach as the most efficient method for graphics ren-
dering. The module that converts a stream of input data into an output is called the
kernel.
The fundamental difference between CPU and GPU architectures is that if the CPU
is designed to execute one stream of instructions sequentially, then the GPU assumes
the simultaneous execution of a large number of parallel threads [2].
CPU performance is determined by the number of tasks executed per clock cycle.
However, it is impossible to continue to increase the speed of execution of algorithms
at the expense of a proportional increase in the number of cores, while maintaining
the architecture of sequential execution of the instruction flow. The number of transis-
tors that need to be added for the corresponding performance gain will very quickly
become too large.
On the other hand, GPUs were designed to process large numbers of instruction
streams in parallel. Moreover, these threads are natively parallelized, and there is
simply no overhead of parallelizing instructions in the GPU. Due to this organization
of computations, the graphics processor uses many execution units, which, unlike the
sequential flow of instructions for the central processor, are easily loaded [3, 4].
In terms of organizing access to memory, the CPU and GPU also differ signifi-
cantly. If the CPU structure is focused on algorithms with random access to memory,
then the GPU reading and, accordingly, writing occurs sequentially, which means that
if there was an access to a memory cell, then the data that follows will be used [5].
The problem of data access latency in the CPU is solved using the caching technology
and code branch prediction. The solution to the same problem with the help of a GPU
looks completely different: if one of the simultaneously running processes has sus-
pended its execution pending access to memory, the video chip switches to another
process, which already has all the data necessary for further execution. The caching
mechanism is used in the GPU to reduce memory access latency. However, if in the
central processor the cache occupies an impressive part of the chip, then the graphics
processor allocates only 128-256 kilobytes for the needs of the cache, which, in turn,
is used to increase the bandwidth.
Multi-threading in GPUs is also implemented at the hardware level. It makes no
sense to increase the number of threads in the CPU, since each switching between
threads inevitably leads to significant delays of several hundred cycles, so the CPU
core can usually only execute several threads at the same time (8-12). The GPU, in
turn, is able to instantly switch between threads, and each core can handle over a
thousand threads [6,7].
Video card manufacturers saw a big promise in GPGPU in 2003 and soon both of
the largest GPU manufacturers presented their solutions for performing general com-
puting on GPUs: Nvidia - CUDA, AMD - ATI Stream.
2.2 Using GPUs for Quantum Chemical Calculations
In this work, all calculations were performed using Quantum ESPRESSO. Quantum
ESPRESSO is a software package based on the density functional theory (DFT) and
the projected augmented wave method (PAW method) [8]. It is a powerful tool for
energy calculations of multi electronic systems and is intended for simulation at the
quantum mechanical level. It is distributed under the GNU General Public License
and is completely open source [9].
What distinguishes Quantum ESPRESSO from its counterparts is the ability to per-
form calculations on GPUs that support CUDA technology, which can significantly
speed up calculations even on personal computers [10].
3 Results
In 1964, Hohenberg and Kohn published a paper laying the foundation for density
functional theory (DFT) [11]. The essence of DFT is to replace a complex and, there-
fore, difficult to calculate many-electron wave function, which contains 3N variables
(N is the number of electrons, and each electron has 3 spatial variables), with an elec-
tron density functional, which contains only 3 variables [12]. So in the new system,
we don't have to worry about the huge number of 3N variables, instead we only deal
with 3 variables that are much easier to deal with. Hohenberg and Kohn proposed
their first theorem, which indicates that the ground state energy is uniquely dependent
on the electron density, which means that it is a functional of the electron density.
Their second theorem proved that by minimizing the energy of the system according
to the electron density, one can obtain the energy of the ground state.
Despite the fact that the theorems of Hohenberg and Kohn confirm the truth of the
fact that there is a one-to-one correspondence between the electron density functional
and the properties of the system, they do not give any exact information what exactly
these interactions are.
So the most commonly used method instead of “minimizing system energy” is the
Kohn-Sham method. Kohn and Sham published the paper in 1965, just a year after the
important paper by Hohenberg and Kohn, and in this paper they simplified the many-
electron problem to the problem of noninteracting electrons in the effective potential.
This potential includes the external potential and the effects of Coulomb interactions
between electrons, for example, exchange and correlation interactions. Working with
the exchange and correlation interactions is the main difficulty in Kohn and Sham
density functional theory. There is still no rigorous way to solve the exchange and
correlation energy. However, the simplest approximation is the local density ap-
proximation (LDA). The LDA is based on the use of a uniform electron gas model to
obtain the exchange energy (the exact value can be obtained from the Thomas-Fermi
model) and to obtain the correlation energy from fits to a uniform electron gas [13].
DFT has become very popular for computations in solid state physics since the
1970s. Compared to other methods dealing with quantum mechanical problems of
multiple bodies, LDA gives satisfactory results with experimental data. But in the
field of quantum chemistry, DFT was not accurate until the 1990s, when approxima-
tion methods were significantly improved to better simulate exchange correlation
interactions [14]. DFT is currently the leading method for calculating electronic struc-
ture in many fields. However, it is still difficult to use DFT to work with highly corre-
lated systems, band gaps in semiconductors, and highly dispersive systems.
In the theory of the electron density functional, the total energy of a crystal is de-
termined by the following expression:
(1)
Here the first term is the kinetic energy of the electrons in the crystal; - popula-
tion of orbital ν, σ - spin variable (α - spin up, β - spin down); - wave function.
The second term in expression (1) is the energy of the electron-nuclear interaction. In
the equation - is an electric charge density operator. The third term in
formula (1) is the energy of the electron-electron interaction. The last term in this
formula describes the exchange-correlation interaction.
Otherwise, equation (1) can be represented as:
(2)
Where - zone energy, - Hartree potential, - potential of exchange-
correlation interaction, - matrix trace computation.
4 Discussion
In this work, the adsorption of an aluminum atom on the C60 fullerene was studied.
Due to the symmetry of the problem, only the following placement options are possi-
ble: aluminum on the carbon atom, aluminum above the carbon-carbon bond, and
aluminum in the center of the cell on the fullerene surface (Figure 2).
Fig. 2. The location of an aluminum atom above C60 molecule.
Figure 3 shows the density of states of pure fullerene and fullerene with an alumi-
num atom:
─ is the density of states for fullerene C60;
─ is the density of states when aluminum is positioned in the center of the cell on the
fullerene surface;
─ is the density of states when aluminum is positioned above the carbon-carbon
bond;
─ is the density of states at the position of aluminum on the carbon atom.
Fig. 3. Density of states.
Table 1 shows the calculated binding energies for various positions of aluminum.
Table 1. Binding energies for aluminum and fullerene atoms.
Aluminum atom over Aluminum atom between Aluminum atom in the cen-
the carbon atom, eV carbon atoms, eV ter, eV
54.14 54.12 54.35
All calculations were performed using the Quantum ESPRESSO software package.
5 Conclusion
The calculations were performed on the GPU cluster of the North-Caucasus Federal
University.
The paper shows the relationship between the binding energy and the position of
the aluminum atom on the fullerene surface.
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