1 3
Theor Chem Acc (2015) 134:102
DOI 10.1007/s00214-015-1701-z
REGULAR ARTICLE
Use of graphics processing units for effi cient evaluation
of derivatives of exchange integrals by means of Fourier
transform of the 1/ r operator and its numerical quadrature
Petr Čársky
1 · Roman Čurík
1
Received: 20 May 2015 / Accepted: 8 July 2015 / Published online: 7 August 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Use of graphical processing units ·
Derivatives of exchange integrals · Fourier transform of
1/ r · Electron scattering
1 Introduction
In our recent paper [ 1 ], we presented a way of evaluation
of exchange integrals that is based on the use of Fourier
transform of the 1/ r operator and its subsequent numerical quadrature. In this way, we succeeded [ 1 ] to reduce
the computational time for exchange integrals considerably. The stimulus for undertaking that study arose from
our urgent need to calculate effi ciently exchange integrals in electron–molecule scattering calculations. These
integrals are of the [ g (1) k (1)| g (2) k (2)] type, where g and
k symbols, respectively, refer to Gaussians and planewave functions. Lengthy evaluation of theses integrals
has been a bottleneck in the scattering theory [ 2 ], and it
hampers ab initio applications to larger polyatomic molecules. The use of the Fourier transform of the 1/ r operator itself
and its subsequent numerical quadrature
brought only a minor computer time-saving. However,
the merit of Fourier transformation is that it can express
exchange integrals for a Hartree–Fock exchange potential
V ex as a scalar product of two vectors with one-electron
elements
(1)
1
r 12
=
1
2π 2
1
k 2 e
−ik.r 1 e
ik.r 2 dk
(2)
1
r 12
=
1
2π 2
pj
ω p ω j e
−ik pj .r 1 e
ik pj .r 2
Abstract In this paper, we propose an effi cient way for
evaluation of derivatives of exchange integrals. We propose
an approach in which we factorize the non-local exchange
kernel into a sum of separable terms. We exploit a discretized Fourier transform for the 1/ r operator, and we devise
a method that allows us to employ a manageable number
of plane-wave functions in the Fourier expansion while still
keeping necessary accuracy. Resulting formulas are amenable for effi cient evaluation on graphics processing units
(GPU). We discuss the GPU implementation for derivatives of two-electron repulsion integrals of the ( gk | gk ) type
in the hybrid Gaussian and plane-wave basis. Derivatives
of such integrals are needed for computation of cross sections in vibrationally inelastic electron scattering by polyatomic molecules. Speedup and accuracy achieved are demonstrated for cross sections of selected vibrational modes
of cyclopropane, benzene and adamantane. The proposed
factorization method is general and may be applied to any
type of exchange integrals. We note briefl y on its possible
application to exchange integrals and their derivatives in
quantum chemical computational methods.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Roman Čurík
roman.curik@jh-inst.cas.cz ; curik@jh-inst.cas.cz
Petr Čársky
carsky@jh-inst.cas.cz
1
J. Heyrovský Institute of Physical Chemistry, v.v.i. ,
Academy of Sciences of the Czech Republic , Dolejškova 3 ,
18223 Prague 8 , Czech Republic
15
Reprinted from the journal
Theor Chem Acc (2015) 134:102
DOI 10.1007/s00214-015-1701-z
REGULAR ARTICLE
Use of graphics processing units for effi cient evaluation
of derivatives of exchange integrals by means of Fourier
transform of the 1/ r operator and its numerical quadrature
Petr Čársky
1 · Roman Čurík
1
Received: 20 May 2015 / Accepted: 8 July 2015 / Published online: 7 August 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Use of graphical processing units ·
Derivatives of exchange integrals · Fourier transform of
1/ r · Electron scattering
1 Introduction
In our recent paper [ 1 ], we presented a way of evaluation
of exchange integrals that is based on the use of Fourier
transform of the 1/ r operator and its subsequent numerical quadrature. In this way, we succeeded [ 1 ] to reduce
the computational time for exchange integrals considerably. The stimulus for undertaking that study arose from
our urgent need to calculate effi ciently exchange integrals in electron–molecule scattering calculations. These
integrals are of the [ g (1) k (1)| g (2) k (2)] type, where g and
k symbols, respectively, refer to Gaussians and planewave functions. Lengthy evaluation of theses integrals
has been a bottleneck in the scattering theory [ 2 ], and it
hampers ab initio applications to larger polyatomic molecules. The use of the Fourier transform of the 1/ r operator itself
and its subsequent numerical quadrature
brought only a minor computer time-saving. However,
the merit of Fourier transformation is that it can express
exchange integrals for a Hartree–Fock exchange potential
V ex as a scalar product of two vectors with one-electron
elements
(1)
1
r 12
=
1
2π 2
1
k 2 e
−ik.r 1 e
ik.r 2 dk
(2)
1
r 12
=
1
2π 2
pj
ω p ω j e
−ik pj .r 1 e
ik pj .r 2
Abstract In this paper, we propose an effi cient way for
evaluation of derivatives of exchange integrals. We propose
an approach in which we factorize the non-local exchange
kernel into a sum of separable terms. We exploit a discretized Fourier transform for the 1/ r operator, and we devise
a method that allows us to employ a manageable number
of plane-wave functions in the Fourier expansion while still
keeping necessary accuracy. Resulting formulas are amenable for effi cient evaluation on graphics processing units
(GPU). We discuss the GPU implementation for derivatives of two-electron repulsion integrals of the ( gk | gk ) type
in the hybrid Gaussian and plane-wave basis. Derivatives
of such integrals are needed for computation of cross sections in vibrationally inelastic electron scattering by polyatomic molecules. Speedup and accuracy achieved are demonstrated for cross sections of selected vibrational modes
of cyclopropane, benzene and adamantane. The proposed
factorization method is general and may be applied to any
type of exchange integrals. We note briefl y on its possible
application to exchange integrals and their derivatives in
quantum chemical computational methods.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Roman Čurík
roman.curik@jh-inst.cas.cz ; curik@jh-inst.cas.cz
Petr Čársky
carsky@jh-inst.cas.cz
1
J. Heyrovský Institute of Physical Chemistry, v.v.i. ,
Academy of Sciences of the Czech Republic , Dolejškova 3 ,
18223 Prague 8 , Czech Republic
15
Reprinted from the journal