Theor Chem Acc (2015) 134:102
1 3
which is an ideal task for general-purpose computation on
graphics processing units (GPU). The indices p and j in
Eq. ( 2 ) refer to radial and angular numerical quadrature,
respectively, ω p and ω j are weights of the grid points, N k 1
and N k 2 in Eq. ( 3 ) are normalization constants for planewave functions k 1 and k 2 , the fi rst summation in Eq. ( 3 )
is over occupied molecular orbitals, and q is a collective
index for numerical quadrature
Vectors A in Eq. ( 3 ) were obtained as
where c iμ stands for expansion coeffi cients of the i th
molecular orbital, and
where the notation ( μkk q ) denotes the overlap integral between the gaussian μ and a plane-wave function
exp[i( k + k q )]. By substituting the A terms in Eq. ( 3 ) from
Eqs. ( 5 ) and ( 6 ), we obtain the working form of Eq. ( 3 )
which is ready for differentiation. P μν is a density matrix
element.
In our original code [ 1 , 3 ], the ( gk | gk ) integrals were evaluated by means of complex Shavitt functions F n ( z ). When
the Fourier transform of the 1/ r operator was used together
with graphics processing units, the calculation of exchange
integrals for adamantane could be accomplished in 2 min,
whereas it lasted previously about 4 days and half on the
same computer. Our primary interest has been calculations
of cross sections for vibrationally inelastic electron scattering, for which, in contrast to elastic electron scattering, also
derivatives of exchange integrals with respect to atomic coordinates has to be evaluated [ 3 ]. Their evaluation by Fourier
transform and the use of GPU are the subject of this paper.
The paper is organized as follows. In Sect. 2 , we present derivation of formulas for derivatives of ( gk | gk ) integrals suitable for computer coding. Sections 3 and 4 are
dealing with test calculations on the molecules of cyclopropane, benzene and adamantane. By its size, adamantane is
a molecule considerably larger than molecules for which
calculations on vibrationally inelastic scattering have been
(3)
N k 1 N k 2 (k 1 |V ex |k 2 ) =
occ
i
q
(A
k 1
iq )
∗
(A
k 2
iq ),
(4)
q ≡ pj
(5)
A
k
iq =
μ
c iμ a
k
μq ,
(6)
a
k
μq = N k
ω p ω j
2π 2 (μkk q ),
(7)
N k 1 N k 2 (k 1 |V ex |k 2 )
= N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j (μk 1 k p,j ) ∗ (νk 2 k p,j )P μν ,
reported. In 2004, Itikawa [ 4 ] characterized the situation
as that “a lot of problems are still to be solved” and that
“compared to the large number of theoretical studies of
vibrational excitation of diatomic molecules, the number
of theoretical works for polyatomic molecules is very limited.” Hence, in this paper we wish to show that the use of
Fourier transform of the 1/ r operator and the use of GPU
open a way to treatments of polyatomic molecules considerably larger than it was possible so far. Section 3 comprises technical details of calculations, and in Sect. 4 , we
check performance and accuracy of the proposed computational scheme. The technique presented in this paper is general and can be also applied to exchange integrals of other
types. In Sect. 5 , we note on its possible use for integrals
and their derivatives in quantum chemical methods.
2 Derivatives of exchange integrals
Our task is to differentiate Eq. ( 7 ) with respect to a nuclear
coordinate denoted as variable y ,
The fi rst two terms on the r.h.s. of Eq. ( 8 ), representing the
Hellmann–Feynman contribution, can be easily factorized.
Substituting ( 5 ) and ( 6 ) into ( 8 ) and defi ning
we obtain
Factorization of the last term on the r.h.s. of Eq. ( 8 ) is
harder, but it is facilitated by availability of quantum chemical software. From the coupled perturbed Hartree–Fock
part of quantum chemical programs, it is possible to extract
the fi rst-order transformation matrix P
(1) which is needed
for evaluation of derivatives of density matrix elements.
(8)
N k 1 N k 2
∂
∂y
(k 1 |V ex |k 2 )
= N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j
∂
∂y
(μk 1 k p,j )
∗
(νk 2 k p,j )P μν
+ (μk 1 k p,j )
∗
∂
∂y
(νk 2 k p,j )
P μν + (μk 1 k p,j )
∗ (νk 2 k p,j )
∂
∂y
P μν
(9)
D
k,y
i,q =
μ
∂
∂y
a
k
μ,q
c iμ ,
(10)
N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j
∂
∂y
(μk 1 k p,j )
∗
(νk 2 k p,j )P μν
+ (μk 1 k p,j )
∗
∂
∂y
(νk 2 k p,j )
P μν
= 2
occ
i
q
[(D
k 1 ,y
i,q )
∗ A
k 2
i,q + (A
k 1
i,q )
∗ D
k 2 ,y
i,q ].
16
Reprinted from the journal
1 3
which is an ideal task for general-purpose computation on
graphics processing units (GPU). The indices p and j in
Eq. ( 2 ) refer to radial and angular numerical quadrature,
respectively, ω p and ω j are weights of the grid points, N k 1
and N k 2 in Eq. ( 3 ) are normalization constants for planewave functions k 1 and k 2 , the fi rst summation in Eq. ( 3 )
is over occupied molecular orbitals, and q is a collective
index for numerical quadrature
Vectors A in Eq. ( 3 ) were obtained as
where c iμ stands for expansion coeffi cients of the i th
molecular orbital, and
where the notation ( μkk q ) denotes the overlap integral between the gaussian μ and a plane-wave function
exp[i( k + k q )]. By substituting the A terms in Eq. ( 3 ) from
Eqs. ( 5 ) and ( 6 ), we obtain the working form of Eq. ( 3 )
which is ready for differentiation. P μν is a density matrix
element.
In our original code [ 1 , 3 ], the ( gk | gk ) integrals were evaluated by means of complex Shavitt functions F n ( z ). When
the Fourier transform of the 1/ r operator was used together
with graphics processing units, the calculation of exchange
integrals for adamantane could be accomplished in 2 min,
whereas it lasted previously about 4 days and half on the
same computer. Our primary interest has been calculations
of cross sections for vibrationally inelastic electron scattering, for which, in contrast to elastic electron scattering, also
derivatives of exchange integrals with respect to atomic coordinates has to be evaluated [ 3 ]. Their evaluation by Fourier
transform and the use of GPU are the subject of this paper.
The paper is organized as follows. In Sect. 2 , we present derivation of formulas for derivatives of ( gk | gk ) integrals suitable for computer coding. Sections 3 and 4 are
dealing with test calculations on the molecules of cyclopropane, benzene and adamantane. By its size, adamantane is
a molecule considerably larger than molecules for which
calculations on vibrationally inelastic scattering have been
(3)
N k 1 N k 2 (k 1 |V ex |k 2 ) =
occ
i
q
(A
k 1
iq )
∗
(A
k 2
iq ),
(4)
q ≡ pj
(5)
A
k
iq =
μ
c iμ a
k
μq ,
(6)
a
k
μq = N k
ω p ω j
2π 2 (μkk q ),
(7)
N k 1 N k 2 (k 1 |V ex |k 2 )
= N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j (μk 1 k p,j ) ∗ (νk 2 k p,j )P μν ,
reported. In 2004, Itikawa [ 4 ] characterized the situation
as that “a lot of problems are still to be solved” and that
“compared to the large number of theoretical studies of
vibrational excitation of diatomic molecules, the number
of theoretical works for polyatomic molecules is very limited.” Hence, in this paper we wish to show that the use of
Fourier transform of the 1/ r operator and the use of GPU
open a way to treatments of polyatomic molecules considerably larger than it was possible so far. Section 3 comprises technical details of calculations, and in Sect. 4 , we
check performance and accuracy of the proposed computational scheme. The technique presented in this paper is general and can be also applied to exchange integrals of other
types. In Sect. 5 , we note on its possible use for integrals
and their derivatives in quantum chemical methods.
2 Derivatives of exchange integrals
Our task is to differentiate Eq. ( 7 ) with respect to a nuclear
coordinate denoted as variable y ,
The fi rst two terms on the r.h.s. of Eq. ( 8 ), representing the
Hellmann–Feynman contribution, can be easily factorized.
Substituting ( 5 ) and ( 6 ) into ( 8 ) and defi ning
we obtain
Factorization of the last term on the r.h.s. of Eq. ( 8 ) is
harder, but it is facilitated by availability of quantum chemical software. From the coupled perturbed Hartree–Fock
part of quantum chemical programs, it is possible to extract
the fi rst-order transformation matrix P
(1) which is needed
for evaluation of derivatives of density matrix elements.
(8)
N k 1 N k 2
∂
∂y
(k 1 |V ex |k 2 )
= N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j
∂
∂y
(μk 1 k p,j )
∗
(νk 2 k p,j )P μν
+ (μk 1 k p,j )
∗
∂
∂y
(νk 2 k p,j )
P μν + (μk 1 k p,j )
∗ (νk 2 k p,j )
∂
∂y
P μν
(9)
D
k,y
i,q =
μ
∂
∂y
a
k
μ,q
c iμ ,
(10)
N k 1 N k 2
1
2π 2
μ
ν
p,j
ω p ω j
∂
∂y
(μk 1 k p,j )
∗
(νk 2 k p,j )P μν
+ (μk 1 k p,j )
∗
∂
∂y
(νk 2 k p,j )
P μν
= 2
occ
i
q
[(D
k 1 ,y
i,q )
∗ A
k 2
i,q + (A
k 1
i,q )
∗ D
k 2 ,y
i,q ].
16
Reprinted from the journal