Theor Chem Acc (2015) 134:102
1 3
of GPUs in quantum chemistry has been recognized, and
since then, their merits in applications to evaluation of twoelectron repulsion integrals [ 13 – 17 ], semiempirical methods [ 18 , 19 ], Hartree–Fock calculations [ 19 , 20 ], DFT [ 21 ,
22 ], CIS [ 23 ], MP2 [ 24 – 26 ], CCSD and CCSD(T) [ 27 , 28 ]
and Quantum Monte Carlo calculations [ 29 ] were reported.
Nevertheless, we considered it expedient to note a possible use of our computational scheme also for post-Hartree–
Fock methods. We mention post-Hartree–Fock methods
on purpose, because for energy gradients our factorization
approach needs the coupled perturbed Hartree–Fock calculation for evaluation of derivatives of expansion coeffi cients
of molecular orbitals [in Eq. ( 15 )]. It is assumed that evaluation of the gradient at the correlated level is considerably
more time-consuming than the coupled perturbed Hartree–
Fock calculation. Provided that evaluation of derivatives
would be done on GPU in double-precision arithmetics and
with an optimized FT quadrature, the calculations may be
effective. Below, we indicate how it could work.
Consider an integral of the type [ p (1) q (1)| r (2) s (2)],
where p , q , r , s are molecular orbitals, constructed from a
Gaussian atomic basis set. Direct application of the Fourier
transform of 1/ r by means of Eqs. ( 1 ) and ( 2 ) leads to the
following expression
where ( pqk ) and ( krs ) are one-electron overlap integrals
that are easy to evaluate. Provided that we have good computational facilities at our disposal, and that GPUs can be
effectively applied, we can expect considerable time-saving
with respect to a standard evaluation, as we demonstrated
it with ( gk | gk ) integrals [ 1 ]. Obviously, the utility of this
approach cannot be taken for granted and it should be
tested against well-established quantum chemical methods,
known under specifi cations such as RI (resolution of identity), density fi tting, Cholesky decomposition or chain-ofspheres approach which employs factorization based on a
mixed analytical and numerical evaluation of the exchange
integrals [ 30 ]. They all use sort of a factorization, and for
most of them, the factorization of ( pq | rs ) integrals can be
expressed as
where
The ( pq | Q ) and ( Q | P ) are, respectively, three- and twoindex two-electron repulsion integrals. The overlap integrals ( pqk ) in our approach in Eq. ( 26 ) are easier to
(26)
(pq|rs) =
1
2π 2
t
√
ω t (pqk t )|(k t rs)
√ ω t ,
(27)
(pq|rs) =
P
B
P
pq B
P
rs ,
(28)
B
P
pq =
Q
(pq|Q)|(Q|P)
−1/2 .
evaluate, but this need not be a substantial advantage
because performance of factorization depends mainly
on effi cient coding the matrix multiplication operations.
Three-center overlap integrals in pure Gaussian basis were
also exploited in tensor hypercontraction density fi tting
applied perturbation approaches [ 31 ] and coupled clusters
method [ 32 ]. Recent experience with the development of
quantum chemical methods shows that when an ingenious use is made of GPUs, not depending on linear algebra
libraries only, unprecedented progress in the evaluation of
exchange integrals can be achieved [ 13 , 14 , 20 , 21 ]. Hence,
the factorization method presented in this paper for derivatives of exchange integrals is worth of further exploration,
if it is a viable alternative to RI-type methods.
6 Conclusions
We have shown that the factorization method, proposed in
our earlier paper [ 1 ] for exchange integrals of the ( gk | gk )
type, can also be applied to their derivatives. We found that
the Fourier transform of 1/ r alone does not bring an advantage of speedup. However, its merit is its ability to cast
mathematical expressions into a form which is suited ideally for use of GPUs. Net result of such a combination is
considerable computer time-saving. In this paper, the test
calculations were extended for evaluation of derivatives.
The calculations were done for a set of exchange integral
and their derivatives that were needed for a treatment of
electron scattering by cyclopropane, benzene and adamantane. The data on timing of calculations are listed in Table 1 .
The mere Fourier transform by means of Eqs. ( 1 ) and ( 2 )
did not bring any profi t. But, if on top of that the A*.A and
B*.A + A*.B multiplications in Eqs. ( 3 ) and ( 17 ) were performed on GPU, the CPU time dropped by more than an
order of magnitude. It is important to note that the present
speedup achieved for the derivatives of the ( gk | gk ) integrals
is smaller when comparing to speedup of the GPU implementation for the ( gk | gk ) integrals alone [ 1 ]. A reason for
this is due to very effi cient implementation of the nuclear
derivatives in the original rigorous method that uses tabulated Shavitt functions while the present algorithm scales
linearly with the number of nuclear degrees of freedom.
The factorization method in a form presented in this
paper is general and can be, at least in principle, applied
to any type of two-electron integrals. To become an alternative to existing established quantum chemical methods,
the computer code has to enable an effi cient use of GPU.
Although reorganization of the program for this purpose is
tedious and requires programming skill, the result may be
rewarding. The factorization method may also have a merit
in supporting the idea of Füsti-Molnar and Pulay [ 12 ] to
substitute Gaussians by plane-wave functions. So far, their
22
Reprinted from the journal
1 3
of GPUs in quantum chemistry has been recognized, and
since then, their merits in applications to evaluation of twoelectron repulsion integrals [ 13 – 17 ], semiempirical methods [ 18 , 19 ], Hartree–Fock calculations [ 19 , 20 ], DFT [ 21 ,
22 ], CIS [ 23 ], MP2 [ 24 – 26 ], CCSD and CCSD(T) [ 27 , 28 ]
and Quantum Monte Carlo calculations [ 29 ] were reported.
Nevertheless, we considered it expedient to note a possible use of our computational scheme also for post-Hartree–
Fock methods. We mention post-Hartree–Fock methods
on purpose, because for energy gradients our factorization
approach needs the coupled perturbed Hartree–Fock calculation for evaluation of derivatives of expansion coeffi cients
of molecular orbitals [in Eq. ( 15 )]. It is assumed that evaluation of the gradient at the correlated level is considerably
more time-consuming than the coupled perturbed Hartree–
Fock calculation. Provided that evaluation of derivatives
would be done on GPU in double-precision arithmetics and
with an optimized FT quadrature, the calculations may be
effective. Below, we indicate how it could work.
Consider an integral of the type [ p (1) q (1)| r (2) s (2)],
where p , q , r , s are molecular orbitals, constructed from a
Gaussian atomic basis set. Direct application of the Fourier
transform of 1/ r by means of Eqs. ( 1 ) and ( 2 ) leads to the
following expression
where ( pqk ) and ( krs ) are one-electron overlap integrals
that are easy to evaluate. Provided that we have good computational facilities at our disposal, and that GPUs can be
effectively applied, we can expect considerable time-saving
with respect to a standard evaluation, as we demonstrated
it with ( gk | gk ) integrals [ 1 ]. Obviously, the utility of this
approach cannot be taken for granted and it should be
tested against well-established quantum chemical methods,
known under specifi cations such as RI (resolution of identity), density fi tting, Cholesky decomposition or chain-ofspheres approach which employs factorization based on a
mixed analytical and numerical evaluation of the exchange
integrals [ 30 ]. They all use sort of a factorization, and for
most of them, the factorization of ( pq | rs ) integrals can be
expressed as
where
The ( pq | Q ) and ( Q | P ) are, respectively, three- and twoindex two-electron repulsion integrals. The overlap integrals ( pqk ) in our approach in Eq. ( 26 ) are easier to
(26)
(pq|rs) =
1
2π 2
t
√
ω t (pqk t )|(k t rs)
√ ω t ,
(27)
(pq|rs) =
P
B
P
pq B
P
rs ,
(28)
B
P
pq =
Q
(pq|Q)|(Q|P)
−1/2 .
evaluate, but this need not be a substantial advantage
because performance of factorization depends mainly
on effi cient coding the matrix multiplication operations.
Three-center overlap integrals in pure Gaussian basis were
also exploited in tensor hypercontraction density fi tting
applied perturbation approaches [ 31 ] and coupled clusters
method [ 32 ]. Recent experience with the development of
quantum chemical methods shows that when an ingenious use is made of GPUs, not depending on linear algebra
libraries only, unprecedented progress in the evaluation of
exchange integrals can be achieved [ 13 , 14 , 20 , 21 ]. Hence,
the factorization method presented in this paper for derivatives of exchange integrals is worth of further exploration,
if it is a viable alternative to RI-type methods.
6 Conclusions
We have shown that the factorization method, proposed in
our earlier paper [ 1 ] for exchange integrals of the ( gk | gk )
type, can also be applied to their derivatives. We found that
the Fourier transform of 1/ r alone does not bring an advantage of speedup. However, its merit is its ability to cast
mathematical expressions into a form which is suited ideally for use of GPUs. Net result of such a combination is
considerable computer time-saving. In this paper, the test
calculations were extended for evaluation of derivatives.
The calculations were done for a set of exchange integral
and their derivatives that were needed for a treatment of
electron scattering by cyclopropane, benzene and adamantane. The data on timing of calculations are listed in Table 1 .
The mere Fourier transform by means of Eqs. ( 1 ) and ( 2 )
did not bring any profi t. But, if on top of that the A*.A and
B*.A + A*.B multiplications in Eqs. ( 3 ) and ( 17 ) were performed on GPU, the CPU time dropped by more than an
order of magnitude. It is important to note that the present
speedup achieved for the derivatives of the ( gk | gk ) integrals
is smaller when comparing to speedup of the GPU implementation for the ( gk | gk ) integrals alone [ 1 ]. A reason for
this is due to very effi cient implementation of the nuclear
derivatives in the original rigorous method that uses tabulated Shavitt functions while the present algorithm scales
linearly with the number of nuclear degrees of freedom.
The factorization method in a form presented in this
paper is general and can be, at least in principle, applied
to any type of two-electron integrals. To become an alternative to existing established quantum chemical methods,
the computer code has to enable an effi cient use of GPU.
Although reorganization of the program for this purpose is
tedious and requires programming skill, the result may be
rewarding. The factorization method may also have a merit
in supporting the idea of Füsti-Molnar and Pulay [ 12 ] to
substitute Gaussians by plane-wave functions. So far, their
22
Reprinted from the journal