Chapter 8
The Dissociation of Water Analysis of the CF1 Central Force Model of Water 1
Downloaded by UNIV OF GUELPH LIBRARY on October 26, 2012 | http://pubs.acs.org Publication Date: September 29, 1994 | doi: 10.1021/bk-1994-0568.ch008
Anna Nyberg and A. D. J. Haymet
School of Chemistry, University of Sydney, New South Wales 2006, Australia
The relative free energies of the solvated speciesH+(aq)andΗ O+(aq)are 3
calculated for the CF1 model of water. The calculations lead to an upperbound for the pH of CF1 water. Comparison is made with other calculations for the dissociation of water.
1
Recently we have calculated the pH of the CF1 central force model of water. The CF1 model is a slight modification of the central force model of Stillinger and 2
3
Rahman, designed to improve the pressure at 25 °C and 1.00 g c m " . For the CF1 model an upper bound to the pH is found to be 8.5±0.7. The model has a 1
dielectric constant of 69 ± 11. Our first calculation used classical mechanics, and predicted the equilibrium concentration of 'loosely solvated' (defined below) species sulting from the dissociation H 0 ( ^ ^ 2
and O H ^ , re 3
+ 0HJ" . Standard methods were Q(?)
used to calculate the relative fraction of dissociated species. Since the extent of hydration of H
+
and O H " in the CF1 model of water is not known (nor to our
knowledge, is it known for any other model), this calculation of the pH in the CF1 model established an upper bound. Further stabilisation of the 'loosely solvated' species would lead to an increase in the total equilibrium concentration of H ^ . Within the CF1 model, a hydrogen species is defined to be 'loosely solvated' if both (i) the distance to the nearest oxygen species is greater than 1.2 Â and (ii) the distance to the nearest hydrogen species is greater than 1.8 Â. 1
Corresponding author
0097-6156/94/0568-0110S08.00/0 © 1994 American Chemical Society
In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.
8. NYBERG AND HAYMET
The Dissociation of Water
111
Here we begin the calculation of the relative stability of tightly solvated species such as H ^ , H 0 ^ , ... , Η 0 ί ( ) and similar O H ^ species. For the 3
ç )
9
αί7
5
CF1 model, 'tightly solvated ions have all oxygen and hydrogen species connected by 'bonds', where an oxygen-hydrogen bond has a separation less than 1.2 Â, and a hydrogen-hydrogen bond has a separation less than 1.8 Â. Our ultimate goal is to calculate the total concentration of dissociated species. We begin with the H 0t Downloaded by UNIV OF GUELPH LIBRARY on October 26, 2012 | http://pubs.acs.org Publication Date: September 29, 1994 | doi: 10.1021/bk-1994-0568.ch008
3
ion, defined in the CF1 model to have all three O H distance less than 1.2
g )
Â, and all three intramolecular hydrogen-hydrogen distances less than 1.8 Â.
C F 1 M o d e l of W a t e r Despite the well-known role of pH on the structure of proteins and activity of en zymes, only modest interest has been shown in the dissociation of water, with the 4
5
6
notable exceptions of work by Stillinger, Warshel, and Bratos and co-workers. " The rigid models of water
10,11
9
used most frequently in computer simulations have
+
zero H ion concentration, since by construction they cannot address dissociation. 12
Some flexible models also do not permit dissociation. ""
15
2,16 17
The central force model of Stillinger and co-workers ' consists of three pair potentials acting between fractionally charged hydrogen and oxygen species. There is a single Hamiltonian which describes both intra- and inter-molecular degrees of freedom. The central force (CF) potential energies, as revised in 1978 2
by Stillinger and Rahman are:
V
o o ( r )
VWr) H
H
l
l
=
=
3
^ 0
3
} r
, V (r)
τ /
λ
ou
=
4
5
+
?6^^_ ^-4(r^,»_ 0
—
4^
i Α
1 + 40(r-2.05C ) e
72.269 6.23403 — + 3ΓΠ^ 1912
-...(r-«,»
-7.62177(r-1.45251)*
7 ί
0 - 2 f c
β
2
10 γ _j_ 4 0 ( r - 1 . 0 5 ) e
4 ^ - f 5.49305(r-2.2) > e
where the values of the constants are C\ = C2 = 1. The hydrogen species have a fraction charge of approximately one-third, which should be regarded as an effective value arising from integrating out the many contributions to the total potential energy omitted from a two-body prescription. At the temperature Τ = 3
25 °C and density 1.00 g c m " , this model has a pressure of 3,540 bar, thousands
In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.
112
STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION
18
1
of times the correct value. The CF1 model attempts to both preserve the useful properties of this model and correct the pressure. The same potential energies are used, but with the slightly different constants C\ = 0.9 and C2 = 1/1.025. With this change, the pressure for the CF1 model is 120 bar. r
Two high peaks in CF1 pair correlation functions gou( ) and gnn(r) cor respond to species within the same molecule, and at 25 °C, these peaks arise Downloaded by UNIV OF GUELPH LIBRARY on October 26, 2012 | http://pubs.acs.org Publication Date: September 29, 1994 | doi: 10.1021/bk-1994-0568.ch008
solely from intra-molecular correlations. The CF1 model has also been studied by integral equation methods.
19,20
The Free Energy Calculation +
+
The Helmholtz free energy of solvation for the species H , OH~ and H 0 , all of 3
importance in the dissociation of water, are calculated using gradual changes of the interaction potential between the species and the surrounding solvent. This method is called thermodynamic integration, and it has been used in calculations of the chemical potential of water
21,22
and free energy of hydration of molecules 22
and ions by Jorgensen, Kollman and others. "
25
A parameter λ describes the path chosen between the initial and final state, and the change in free energy is N
N
dH(p ,q ,\)
(2)
where Η is the Hamiltonian of a system of Ν particles, and q are coordinates and p the corresponding momenta. The angle brackets denote an average over phase space, which is approximated by a (relatively short) time average from a molecular dynamics simulation. The Helmholtz free energy corresponds to the canonical ensemble, a choice that is implemented easily in molecular dynamics simulations. The path is chosen so that no phase transitions are encountered. For example, the pressure remains positive throughout the simulation. The computational details are the same as those used earlier.
1
The change in free energy is calculated for the following processes: +
Process 1. Η ( ) —> M α
