Hi,
I'm a postdoc.
Improper dihedral angles -- hereafter referred to as "improper
dihedrals" -- have always been difficult for me to understand. They
are, and continue to be, my Achilles heel. Could you please help
me?
Dihedral angles ("dihedrals") are, in
general, angles between two planes, in molecular dynamics (MD)
simulations. For PROPER dihedrals, the atoms involved are connected by
bonds in order. If we have a proper dihedral defined by atoms i, j, k, and
l bonded in series, then the proper dihedral is i-j-k-l. The proper
dihedral is the angle between the plane i-j-k and the plane j-k-l, and the
torsion is about bond j-k.
On the other hand, for
IMproper dihedrals, the atoms involved do not follow bond order. Yet, the
improper dihedral is still an angle between two planes: namely, plane (i,j,k)
and plane (j,k,l); that's how the manual of the MD package Gromacs defines
improper dihedrals. Here's a snapshot from the Gromacs
manual:
https://i.imgur.com/k2JMgHM.jpg
As you
can see from the snapshot at the above link, the Gromacs developers are quite
clear that they define the improper dihedral angle, Greek letter xi, as the
angle between planes (i,j,k) and (j,k,l). Gromacs's default type of
improper dihedral is the harmonic type with the form given in Eq. 5.175 in the
image at this link:
https://i.imgur.com/G8NVHWp.jpg
Gromacs
also includes a periodic improper dihedral type, a Fourier form (Eq. 5.180),
shown in the image at this link:
https://i.imgur.com/B0VF674.jpg
If
you'd like to see the complete discussion in the Gromacs manual, it's
available around pages 372-376 at the following
link:
Now,
I'd like to specify improper dihedrals in Gromacs for alkylimidazolium
cations according to the force field developed by Canongia Lopes and Padua et
al. For reference, the citation and link to that paper is as
follows:
Canongia Lopes, J.; Deschamps, J.; Padua,
Agilio A. H. J. Phys. Chem. B 2004, 108, 2038-2047.
Note
there is a correction to this paper with citation as follows. But the
correction only affects a few of the numbers, and does not affect any of the
qualitative forms.
Canongia Lopes, J.; Deschamps, J.; Padua, Agilio A. H. J. Phys. Chem. B
2004, 108, 11250.
I'm
looking at the dihedrals section of Table 3 on page 2041. (Again, some of
the actual numbers are corrected in the correction, but none of the qualitative
forms is affected.) I have posted a screenshot of that part of Table 3 at
the following link (yellow highlighting and annotations are
mine):
https://i.imgur.com/wvWlPbB.jpg
That
table lists a number of dihedral types for
alkylimidazolium cations in Canongia Lopes & Padua's force field for
those cations. (Their force field is based on the OPLS force field of
Jorgensen et al.) However, just two of those
dihedral types are improper dihedral types: the last two: X-NA-X-X and
X-C_{W/R}-X-X, as indicated by the superscript italicized "c".
The potential energy coefficients listed there (V1, V2, and V3) correspond to
the Fourier-type dihedral form used in the OPLS force
field.
Meanwhile, the entries in that table with
headings C1, C2, C4, C6, C12, and C18 refer to the number of occurrences of each
term in an alkylmethylimidazolium cation with that number of carbon atoms in the
longer side chain. For example, C2 refers to the cation
1-ethyl-3-methylimidazolium (EMI+), whose structure I have posted in the image
here:
Table
3 (
https://i.imgur.com/wvWlPbB.jpg ) lists the number of
occurrences of the improper dihedrals for the EMI+ cation (column C2).
Specifically, there are 4 occurrences of X-NA-X-X and 6 occurrences of
X-C_{W/R}-X-X in EMI+. Could you please help me to understand where these
occurrences are and how I can correctly specify them? Here are some
specific questions:
(1) Does "number of
occurrences" in Table 3 (
https://i.imgur.com/wvWlPbB.jpg ) refer to the number of
dihedral TYPES or the total number of DIHEDRALS? I was thinking it was the
latter, but I'm not certain.
(2) The symmetries of
the potential energy forms for X-NA-X-X and X-C_{W/R}-X-X are the same, but
their barrier heights are different (V2 = 8.37 vs. 9.2 kJ/mol). Since atom
type NA is directly adjacent to both atom types CR and CW, it seems to me that
there could be overlap ("intersection") between the two types of
improper dihedrals,
X-NA-X-X and X-C_{W/R}-X-X. An improper dihedral X-NA-CW-X, for example,
has both NA and CW in the "2" position, if one defines the
"2" position as one away from the "end" of the
quadruplet. How do I deal with this? Do I include BOTH the
X-NA-X-X and X-C_{W/R}-X-X potentials, or just
one?
(3) What specifically are the occurrences of
X-NA-X-X and X-C_{W/R}-X-X? In the image at the link below, I have
enumerated two sets of quadruplets, red and blue, with the red being X-NA-X-X
improper dihedrals and the blue being X-C_{W/R}-X-X improper
dihedrals:
I have identified
8 "red" and 8 "blue," and each "blue" seems to be
a duplicate of a "red."
In the analysis for
(3), however, I have neglected the hydrogens HA bonded to each of CR, CW, and
CW.
(4) As a final attempt, I have considered the
hydrogens. In the image at the following link, it now seems I can come up
with four
X-NA-X-X improper dihedrals and five (not six)
X-C_{W/R}-X-X improper dihedrals:
Are those
correct? Have I missed any? What is the sixth X-C_{W/R}-X-X improper
dihedral? Again, Gromacs improper dihedrals are specified i-j-k-l, where
(i,j,k) and (j,k,l) are the two planes.
Thanks so much
for your time, and for any guidance you can provide,
Andrew DeYoung, PhD
Department of
Chemistry
Carnegie Mellon University