74
5 Appendix 1: Angle-Dependent Integral Equation Theory
into N grid points (r i = iδr, i=0, 1, …, N−1; δr = r L /N; δr = 0.01d S ; N = 4096),
and all the projections are represented by their values on these points. The numerical solution is performed using the robust, highly efficient algorithm developed by
Kinoshita and coworkers [14, 15]. The hydration entropy of the solute, S, is evaluated via the temperature derivative of hydration free energy μ calculated using the
Morita-Hiroike formula extended to a molecular liquid (V is the system volume) []:
S = −(∂μ/∂ T ) V ∼ −{μ(T + T ) − μ(T + T )}/(2T ), ,T = 5K. (5.4)
In general, the dielectric constant of water is a good measure of the validity of a
theory. The ADIE theory gives a value of ~83 that is in very close agreement with
the experimental value (~78).
In the three-dimensional reference interaction site model (3D-RISM) theory [16–
19], another IET for hydration of a polyatomic solute, the mathematical complications are avoided using an approximation in which the dependence of a correlation
function on the orientations of water molecules is not explicitly taken into account.
Due to this approximation, the OZ equation as well as the closure equation is not exact
[6]. The most serious drawback of the 3D-RISM theory is that it completely fails
to elucidate the hydrophobic hydration (i.e., process 1) [6]. For a nonpolar solute,
even when a realistic molecular model is employed for water, the water behaves
like a hard-sphere solvent (the number density and the particle diameter are equal to
those pertinent to water). As a consequence, the 3D-RISM theory predicts that the
hydrophobicity is strengthened at low temperatures. This prediction is opposite to the
experimental evidence that the hydrophobicity is weakened at low temperatures [5,
6, 9]. Therefore, it is incapable of elucidating such subjects as the cold denaturation
of a protein. It also gives too high a hydration free energy of a nonpolar solute and
too high a density profile of water near a nonpolar solute [6]. However, the 3D-RISM
theory can suitably be applied to an analysis on process 2 (see Chap. 6).
References
1. Hansen J-P, McDonald LR (2006) Theory of simple liquids, 3rd edn. Academic, London
2. Kusalik PG, Patey GN (1988) J Chem Phys 88:7715
3. Kusalik PG, Patey GN (1988) Mol Phys 65:1105
4. Cann NM, Patey GN (1997) J Chem Phys 106:8165
5. Kinoshita M (2008) J Chem Phys 128:024507
6. Hayashi T, Oshima H, Harano Y, Kinoshita M (2016) J Phys: Condens Matter 28:344003
7. Hikiri S, Hayashi T, Inoue M, Ekimoto T, Ikeguchi M, Kinoshita M (2019) J. Chem. Phys.
150:175101
8. Yamada T, Hayashi T, Hikiri S, Kobayashi N, Yanagawa H, Ikeguchi M, Katahira M, Nagata
T, Kinoshita M (2019) J Chem Inf Model 59:3533
9. Inoue M, Hayashi T, Hikiri S, Ikeguchi M, Kinoshita M (2020) J Mol Liq 317:114129
10. Inoue M, Hayashi T, Hikiri S, Ikeguchi M, Kinoshita M (2020) J Chem Phys 152:065103
11. Roth R, Harano Y, Kinoshita M (2006) Phys Rev Lett 97:078101
12. Oshima H, Kinoshita M (2015) J Chem Phys 142:145103
5 Appendix 1: Angle-Dependent Integral Equation Theory
into N grid points (r i = iδr, i=0, 1, …, N−1; δr = r L /N; δr = 0.01d S ; N = 4096),
and all the projections are represented by their values on these points. The numerical solution is performed using the robust, highly efficient algorithm developed by
Kinoshita and coworkers [14, 15]. The hydration entropy of the solute, S, is evaluated via the temperature derivative of hydration free energy μ calculated using the
Morita-Hiroike formula extended to a molecular liquid (V is the system volume) []:
S = −(∂μ/∂ T ) V ∼ −{μ(T + T ) − μ(T + T )}/(2T ), ,T = 5K. (5.4)
In general, the dielectric constant of water is a good measure of the validity of a
theory. The ADIE theory gives a value of ~83 that is in very close agreement with
the experimental value (~78).
In the three-dimensional reference interaction site model (3D-RISM) theory [16–
19], another IET for hydration of a polyatomic solute, the mathematical complications are avoided using an approximation in which the dependence of a correlation
function on the orientations of water molecules is not explicitly taken into account.
Due to this approximation, the OZ equation as well as the closure equation is not exact
[6]. The most serious drawback of the 3D-RISM theory is that it completely fails
to elucidate the hydrophobic hydration (i.e., process 1) [6]. For a nonpolar solute,
even when a realistic molecular model is employed for water, the water behaves
like a hard-sphere solvent (the number density and the particle diameter are equal to
those pertinent to water). As a consequence, the 3D-RISM theory predicts that the
hydrophobicity is strengthened at low temperatures. This prediction is opposite to the
experimental evidence that the hydrophobicity is weakened at low temperatures [5,
6, 9]. Therefore, it is incapable of elucidating such subjects as the cold denaturation
of a protein. It also gives too high a hydration free energy of a nonpolar solute and
too high a density profile of water near a nonpolar solute [6]. However, the 3D-RISM
theory can suitably be applied to an analysis on process 2 (see Chap. 6).
References
1. Hansen J-P, McDonald LR (2006) Theory of simple liquids, 3rd edn. Academic, London
2. Kusalik PG, Patey GN (1988) J Chem Phys 88:7715
3. Kusalik PG, Patey GN (1988) Mol Phys 65:1105
4. Cann NM, Patey GN (1997) J Chem Phys 106:8165
5. Kinoshita M (2008) J Chem Phys 128:024507
6. Hayashi T, Oshima H, Harano Y, Kinoshita M (2016) J Phys: Condens Matter 28:344003
7. Hikiri S, Hayashi T, Inoue M, Ekimoto T, Ikeguchi M, Kinoshita M (2019) J. Chem. Phys.
150:175101
8. Yamada T, Hayashi T, Hikiri S, Kobayashi N, Yanagawa H, Ikeguchi M, Katahira M, Nagata
T, Kinoshita M (2019) J Chem Inf Model 59:3533
9. Inoue M, Hayashi T, Hikiri S, Ikeguchi M, Kinoshita M (2020) J Mol Liq 317:114129
10. Inoue M, Hayashi T, Hikiri S, Ikeguchi M, Kinoshita M (2020) J Chem Phys 152:065103
11. Roth R, Harano Y, Kinoshita M (2006) Phys Rev Lett 97:078101
12. Oshima H, Kinoshita M (2015) J Chem Phys 142:145103