Cross correlations between hard and soft forces: A molecular dynamics
study
G. Sridhar, P. Vijayakumar, and B. L. Tembe
Citation: J. Chem. Phys. 99, 3147 (1993); doi: 10.1063/1.465172
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NOTES
Cross correlations between hard and soft forces: A molecular
dynamics study
G. Sridhar, P. Vijayakumaral , and B. L. Tembe
Department of Chemistry, Indian Institute of Technology Bombay-400076
(Received 5 February 1993; accepted 28 April 1993)
The time dependent response of the solvent molecules
to a charge redistribution among the two reacting ions
plays a major role in determining the rate of the charge
transfer process between these ions in solution. The response of the polar solvent to the motion of a charged
particle in it may be studied in terms of the friction experienced by the particle in the solvent medium. This friction
consists of two parts, one due to the Stokes term arising
from the collisions of the solvent molecules with the ions,
and the other due to the dielectric relaxation of the polar
medium. This has been extensively studied in terms of both
the continuum theories 1,2 and the molecular theories. 3- 5
The friction coefficient ~(t) may be expressed in terms of
the time correlation function of the random forces acting
on the ion. 3 For a fixed ion, the random force is the same
as the total force acting on the ion 3
1
~(t)=3kBT(F(t) ·F(O).
(1)
Allnatt and Rice6 separated the total force acting on
the ion into hard (H) and soft (8) forges. Accordingly,
~(t) can be written as the sum of four force-force correlation functions 3
F(t) =FH(t) +FS(t) ,
(2a)
(F(t) ·F(O»=(FH(t) ·FH(O» + (Fs(t) ·Fs(O)
+ <FH(t) ·FS(O) + (Fs(t) ·FH(O),
(2b)
(2c)
3
In his molecular theory, Wolynes neglected the cross
correlations between the hard and the soft forces in the
calculation of the friction coefficient. He identified the friction due hard-hard interactions ~H(t) as the Stokes friction and the soft term ~(t) as the dielectric friction. Iii. the
present note, these pure (or self) and cross correlations are
evaluated for ferrous and ferric ions in water. The details of
the contributions of the cross correlations to the friction
coefficient are presented in this note.
In his molecular theory for dipolar liquids, Bagchi5 has
discussed the role of separability of these forces into hard
and soft parts, which are distinct in nature, and their corresponding cross correlations. He has obtained an expression for the cross-correlation functions. He concludes that,
when the soft forces are dependent on the orientations of
the interaction sites, the cross-correlations may be neglected only in weakly dipolar liquids. The crosscorrelations are negative at short times and decay to zero
J. Chern. Phys. 99 (4), 15 August 1993
at long times. Berkowitz and Wan7 have also studied the
effect on the cross correlations in their simulations ofNa+
and Cl- in water. They obtained nonzero values for the
cross-correlation terms, since the relaxation times characteristic of the soft and the hard forces are not widely separated.
In our present study, we evaluate these cross correlations between the forces using molecular dynamics (MD)
simulations. The system contains a ferrous and a ferric ion
held fixed at a separation of 5 A.. in a cubic box of length
15.55 A.., containing 125 water molecules. Periodic boundary conditions were employed in these simulations. A molecule based spherical cutoff (radius 7.5 A..) was used to
truncate potentials. The Coulombic interactions were modified by using the technique of reaction field to correct the
truncated long range interactions. 8,9,10 This approach has
been shown to give results which closely agree with the
results obtained by using Ewald sums. 8,9 To check the sensitivity of the results with respect to the treatment of long
ranged forces, we have done a separate simulation using
the minimum image method for all the interactions. In this
case, the initial values of the TDF are lower by 40%, but
the time dependence is similar to the results reported in
this note. The water-water interaction potential used is the
flexible SPC model developed by Toukan and Rahman. ll
The ion-water potential employed was developed by Curtiss et al 12 This potential has 5 kinds of terms as shown in
Eq. (3). One set of terms is due to the electrostatic interactions and the other four terms are non-Coulombic. The
forces acting on the ion can therefore be separated into five
types. These forces cannot be separated strictly into hard
and soft ones, although the extent of softness and hardness
of each term in the total force may be discussed:
V =A exp( - BrpeO) - D/r~eo-E/r~eO-F/r~~
+ qoqpeirpeO + qHqPe/r~eH + qHqpeir;eH·
(3 )
Figure 1 shows the contribution from the Coulombic
(C), non-Coulombic (NC) and their cross (C-NC) terms
to the total friction ~(t) for the ferrous ion. The total
friction at t=O is 1280 pS-2: The contribution from different parts are 1880 (C), 2910 (NC), -3510 (C-NC)
(pS-2). The corresponding values for ferric ion (curves
not shown) are 2960 (total), 2700 (C), 5380 (NC) and
-5120 (C-NC). The non-Coulombic term is not entirely
hard as there are relatively softer terms in the potential.
The five terms contributing to the total potential can be
arranged in the order of increasing degree of hardness. The
one with the steepest slope at small distances (in the range
0021-9606/93/99(4)/3147/2/$6.00
© 1993 American Institute of Physics
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3147
Letters to the Editor
3148
4000.0
r----~-~~--'--------__r
"'~
2+
Fe
'\11a.
.~o
c
.2
~
2000·0
c
'"
"0
....... ..••..
C
'"
o'"
a.
0·0
;:
...." ........:...... ~.. .
.....•. '" -,'vI
, -... - -
'\
'\..,,;
---
'"
E
i=
-2000.0
-40 00·0 ~_ _-L.._ _--1._ _ _l...-_ _- ' -_ _.....J
0.00
0.20
0.40
0.60
O·BO
1.00
time
(ps)
FIG. 1. The Coulornbic, non-Coulornbic and their cross term in (;(t) for
Fe2 + ion in water: (a) non-Coulornbic; (b) Coulornbic; (e) cross.
1 to 2 .A) can be considered as the hardest. Figure 2 (for
the ferrous ion) shows that the r- 12 term is the hardest
while the Coulombic potential is the softest. The exponential term appearing in the potential is also relatively soft.
400.-----------------,
200
A detailed study of the correlations 'between the Coulombic, exponential and other terms in the potential shows
that the main contribution to the cross-correlations at
short times (t ..... 0) arise from the Coulombic-exponential
(-4600 Fe2+, -5800 Fe3+) and the Coulombic-r- 6 (930
Fe2 +, 550 Fe3+) cross-correlations. This study also shows
that the cross-correlations between the relatively softer
terms and the hardest r- 12 term are very small. But the
softer terms are more correlated with one another. Also,
the main contribution to the non-Coulombic selfcorrelation arises from the self-correlation of the exponential term (5060 Fe2 +, 6900 Fe3+) and its cross correlation
with thez- 6 term (- 2060 Fe2 +, - 1300 Fe3 + ). The softest non-Coulombic term (the exponential one at the relevant distance range) is least correlated with the hardest
non-Coulombic term (r- 12 ).
The relaxation times for all these terms are in the range
of 0.1 ps to 0.4 ps and the curves are similar to those shown
in Fig. 1, except for the initial values which are given in the
above paragraph. The results indicate that at short times
(t ..... O), (;(t) is a result of a strong superposition of the
different self and cross terms, with the cross terms often
dominating. While the separation of forces into C and NC
indicates that the contribution of these terms in Fe3+ are
nearly twice the corresponding values in Fe2 +, a more detailed separation brings out the dominance of the
exponential-C, and the exponential-r- 6 correlations in the
cross terms. The contributionfrom the exponential selfterm-is nearly twice the Coulombic self-term. The results
indicate that the softer terms contribute dominantly to the
magnitude of TDF, while the contribution from the hard
component of the force to the self or the cross terms is very
small. The results also support Bagchi's observation that
the spherically symmetric soft part of the force can contribute significantly to the cross-correlation function.
Present address: Department of Chemistry, The Pennsylvania State
University, University Park, Pennsylvania.
IT. W. Nee and R. Z. Zwanzig, J. Chern. Phys. 52, 6353 (1970).
2 J. Hubbard and L. Onsager, J. Chern. Phys. 67, 4850 (1977).
3p. G. Wolynes, J. Chern. Phys. 68, 473 (1978).
4p. Colornons and P. G. Wolynes, J. Chern. Phys. 71, 2644 (1979).
5B. Bagchi, J. Chern. Phys. 95, 467 (1991).
6S. A. Rice and A. R. Allnatt, J. Chern. Phys. 34, 2144 (1961).
1M. Berkowitz and W. Wan, J. Chern. Phys. 86, 376 (1987).
8M. Neumann, MoCPhys.50, 841 (1983).
9M. Neumann, O. Steinhauser, and G. S. Pawley, Mol. Phys. 52, 97
(1984).
lOp. Vijayakurnar and B. L. Ternbe, J. Chern. Phys. 97, 4356 (1992).
11K. Toukan and A. Rahman, Phys. Rev. B 31,2643 (1985).
12L. A. Curtiss, J. W. Halley, J. Hautrnan, and A. Rahman, J. Chern.
Phys. 86, 2319 (1987).
a)
-20
0.50
1.00
1.50
2.00
2.50
3.00
FIG. 2. The five terms contributing to the total potential experienced by
the Fe2 + ion due to a solvent water molecule along its C2v axis.
(a) exponential term; (b) r- 12; (c) r- 6 term; (d) r- 8 term;
(e) Coulornbic term.
J. Chern. Phys., Vol. 99, No.4, 15 August 1993
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