Theor Chem Acc (2015) 134:102
1 3
Fig. 1 Two sets of 12 fi xed
orientations of the k 2 vectors
used for testing the accuracy
of derivatives ∂/∂ A x ( k 1 | V ex | k 2 )
for cyclopropane. The absolute
values of k 2 vectors were either
those shown in the fi gure or ten
times greater. All derivatives
were with respect to the C 2y
atomic coordinate, as shown on
the right side of the fi gure
Fig. 2 Error in derivatives ∂/∂ A x ( k 1 | V ex | k 2 ) with respect to values
obtained by regular calculations. The k 1 vectors are bound to lie in
the σ xy plane of cyclopropane, and its k 1x and k 1y components are varied with a grid of 0.01 a.u. The k 2 vector is kept fi xed at the position
(0; 10; 0). Left A smaller numerical quadrature with 3402 grid points
was used for Eq. ( 2 ). The largest error was in the imaginary part of
derivatives. Right A larger numerical quadrature with 31,516 grid
points was used for Eq. ( 2 ). For this quadrature, the largest error was
in the real part of derivatives
Fig. 3 Error in the real part of derivatives ∂/∂ A x ( k 1 | V ex | k 2 ) with
respect to values obtained by the regular calculations. The k 1 vectors
are bound to lie in the σ xy plane of cyclopropane, and its k 1x and k 1y
components varied with a grid of 0.1 a.u. The k 2 vector is kept fi xed
at the position (8.66, 5, 0). The left plot was obtained with a smaller
quadrature with 3402 grid points, whereas the right plot was obtained
with a larger quadrature with 31,516 grid points
20
Reprinted from the journal
1 3
Fig. 1 Two sets of 12 fi xed
orientations of the k 2 vectors
used for testing the accuracy
of derivatives ∂/∂ A x ( k 1 | V ex | k 2 )
for cyclopropane. The absolute
values of k 2 vectors were either
those shown in the fi gure or ten
times greater. All derivatives
were with respect to the C 2y
atomic coordinate, as shown on
the right side of the fi gure
Fig. 2 Error in derivatives ∂/∂ A x ( k 1 | V ex | k 2 ) with respect to values
obtained by regular calculations. The k 1 vectors are bound to lie in
the σ xy plane of cyclopropane, and its k 1x and k 1y components are varied with a grid of 0.01 a.u. The k 2 vector is kept fi xed at the position
(0; 10; 0). Left A smaller numerical quadrature with 3402 grid points
was used for Eq. ( 2 ). The largest error was in the imaginary part of
derivatives. Right A larger numerical quadrature with 31,516 grid
points was used for Eq. ( 2 ). For this quadrature, the largest error was
in the real part of derivatives
Fig. 3 Error in the real part of derivatives ∂/∂ A x ( k 1 | V ex | k 2 ) with
respect to values obtained by the regular calculations. The k 1 vectors
are bound to lie in the σ xy plane of cyclopropane, and its k 1x and k 1y
components varied with a grid of 0.1 a.u. The k 2 vector is kept fi xed
at the position (8.66, 5, 0). The left plot was obtained with a smaller
quadrature with 3402 grid points, whereas the right plot was obtained
with a larger quadrature with 31,516 grid points
20
Reprinted from the journal