In a recent study, we developed a kinetic-energy density functional that can be utilized in orbital-free quantum mechanical/molecular mechanical (OF-QM/MM) simulations. The functional includes the nonlocal term constructed from the response function of the reference system of the QM solute. The present work provides a method to combine the OF-QM/MM with a theory of solutions based on the energy representation to compute the solvation free energy of the QM solute in solution. The method is applied to the calculation of the solvation free energy Δμ of a QM water solute in an MM water solvent. It is demonstrated that Δμ is computed as −7.7 kcal/mol, in good agreement with an experimental value of −6.3 kcal/mol. We also develop a theory to map the free energy δμ due to electron density polarization onto the coordinate space of electrons. The free energy density obtained by the free-energy mapping for the QM water clarifies that each hydrogen atom makes a positive contribution (+34.7 kcal/mol) to δμ, and the oxygen atom gives the negative free energy (−71.7 kcal/mol). It is shown that the small polarization free energy −2.4 kcal/mol is generated as a result of the cancellation of these counteracting energies. These analyses are made possible by the OF-QM/MM approach combined with a statistical theory of solutions.
REFERENCES
1.
W.
Kohn
and L. J.
Sham
, “Self-consistent equations including exchange and correlation effects
,” Phys. Rev.
140
, A1133
–A1138
(1965
).2.
R. G.
Parr
and W.
Yang
, Density-Functional Theory of Atoms and Molecules
(Oxford University Press
, New York
, 1989
).3.
R. M.
Martin
, Electronic Structure, Basic Theory and Practical Methods
(Cambridge University Press
, Cambridge
, 2004
).4.
W.
Koch
and M. C.
Holthausen
, A Chemist’s Guide to Density Functional Theory
(Wiley VCH
, 2001
).5.
Recent Progress in Orbital: Free Density Functional Theory
, Recent Advances in Computational Chemistry
, edited by T. A.
Wesolowski
and Y. A.
Wang
(World Scientific
, Singapore
, 2013
).6.
L. H.
Thomas
, “The calculation of atomic fields
,” Proc. Cambridge Philos. Soc.
23
, 542
–548
(1927
).7.
E.
Fermi
, “A statistical method for the determination of some atomic properties and the application of this method to the theory of the periodic system of elements
,” Z. Phys.
48
, 73
–79
(1928
).8.
C. F.
von Weizsacker
, “Zur theorie dier kernmassen
,” Z. Phys.
96
, 431
–458
(1935
).9.
E.
Chacón
, J. E.
Alvarellos
, and P.
Tarazona
, “Nonlocal kinetic energy functional for nonhomogeneous electron systems
,” Phys. Rev. B
32
, 7868
–7877
(1985
).10.
L.-W.
Wang
and M. P.
Teter
, “Kinetic-energy functional of the electron density
,” Phys. Rev. B
45
, 13196
–13220
(1992
).11.
W. C.
Witt
, B. G.
Del Rio
, J. M.
Dieterich
, and E. A.
Carter
, “Orbital-free density functional theory for materials research
,” J. Mater. Res.
33
, 777
–795
(2018
).12.
Y. A.
Wang
, N.
Govind
, and E. A.
Carter
, “Orbital-free kinetic-energy functionals for the nearly free electron gas
,” Phys. Rev. B
58
, 13465
–13471
(1998
).13.
Y. A.
Wang
, N.
Govind
, and E. A.
Carter
, “Orbital-free kinetic-energy density functionals with a density-dependent kernel
,” Phys. Rev. B
60
, 16350
(1999
).14.
C.
Huang
and E. A.
Carter
, “Nonlocal orbital-free kinetic energy density functional for semiconductors
,” Phys. Rev. B
81
, 045206
(2010
).15.
W.
Mi
, A.
Genova
, and M.
Pavanello
, “Nonlocal kinetic energy functionals by functional integration
,” J. Chem. Phys.
148
, 184107
(2018
).16.
L. A.
Constantin
, E.
Fabiano
, and F.
Della Sala
, “Nonlocal kinetic energy functional from the jellium-with-gap model: Applications to orbital-free density functional theory
,” Phys. Rev. B
97
, 205137
(2018
).17.
W.
Mi
and M.
Pavanello
, “Orbital-free density functional theory correctly models quantum dots when asymptotics, nonlocality, and nonhomogeneity are accounted for
,” Phys. Rev. B
100
, 041105(R)
(2019
).18.
J.
Xia
, C.
Huang
, I.
Shin
, and E. A.
Carter
, “Can orbital-free density functional theory simulate molecules
,” J. Chem. Phys.
136
, 084102
(2012
).19.
J.
Xia
and E. A.
Carter
, “Density-decomposed orbital-free density functional theory for covalently bonded molecules and materials
,” Phys. Rev. B
86
, 235109
(2012
).20.
H.
Takahashi
, “Development of kinetic energy density functional using response function defined on the energy coordinate
,” Int. J. Quantum Chem.
122
, e26969
(2022
).21.
J.
Lindhard
, “On the properties of a gas of charged particles
,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd.
28
(8
), 1
(1954
).22.
H.
Takahashi
, “Density-functional theory based on the electron distribution on the energy coordinate
,” J. Phys. B: At., Mol. Opt. Phys.
51
, 055102
(2018
).23.
H.
Takahashi
, “Development of static correlation functional using electron distribution on the energy coordinate
,” J. Phys. B: At., Mol. Opt. Phys.
53
, 245101
(2020
).24.
H.
Takahashi
, “Development of nonlocal kinetic-energy density functional for the hybrid QM/MM interaction
,” J. Chem. Phys.
158
, 014102
(2023
).25.
A.
Warshel
and M.
Levitt
, “Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme
,” J. Mol. Biol.
103
, 227
–249
(1976
).26.
J.
Gao
and X.
Xia
, “A priori evaluation of aqueous polarization effects through Monte Carlo QM-MM simulations
,” Science
258
, 631
–635
(1992
).27.
M. F.
Ruiz-López
, “Combined QM/MM calculations in chemistry and biochemistry
,” J. Mol. Struct.: THEOCHEM
632
, 9
(2003
).28.
H.
Takahashi
, T.
Hori
, H.
Hashimoto
, and T.
Nitta
, “A hybrid QM/MM method employing real space grids for QM water in the TIP4P water solvents
,” J. Comput. Chem.
22
, 1252
–1261
(2001
).29.
H.
Takahashi
, D.
Suzuoka
, and A.
Morita
, “Why is benzene soluble in water? Role of OH/π interaction in solvation
,” J. Chem. Theory Comput.
11
, 1181
–1194
(2015
).30.
H.
Takahashi
, D.
Suzuoka
, S.
Sakuraba
, and A.
Morita
, “Role of the photosystem II as an environment in the oxidation free energy of the Mn cluster from S1 to S2
,” J. Phys. Chem. B
123
, 7081
–7091
(2019
).31.
N.
Matubayasi
and M.
Nakahara
, “Theory of solutions in the energetic representation. I. Formulation
,” J. Chem. Phys.
113
, 6070
–6081
(2000
).32.
H.
Takahashi
, N.
Matubayasi
, M.
Nakahara
, and T.
Nitta
, “A quantum chemical approach to the free energy calculations in condensed systems: The QM/MM method combined with the theory of energy representation
,” J. Chem. Phys.
121
, 3989
–3999
(2004
).33.
H.
Takahashi
, A.
Omi
, A.
Morita
, and N.
Matubayasi
, “Simple and exact approach to the electronic polarization effect on the solvation free energy: Formulation for quantum mechanical/molecular mechanical system and its applications to aqueous solutions
,” J. Chem. Phys.
136
, 214503
(2012
).34.
J.
Tomasi
, B.
Mennucci
, and R.
Cammi
, “Quantum mechanical continuum solvation models
,” Chem. Rev.
105
, 2999
–3094
(2005
).35.
D.
Frenkel
and B.
Smit
, Understanding Molecular Simulation
(Academic Press
, London
, 1996
).36.
H.
Takahashi
, W.
Satou
, T.
Hori
, and T.
Nitta
, “An application of the novel quantum mechanical/molecular mechanical method combined with the theory of energy representation: An ionic dissociation of a water molecule in the supercritical water
,” J. Chem. Phys.
122
, 044504
(2005
).37.
H.
Takahashi
, Y.
Kawashima
, and T.
Nitta
, “A novel quantum mechanical/molecular mechanical approach to the free energy calculation for isomerization of glycine in aqueous solution
,” J. Chem. Phys.
123
, 124504
(2005
).38.
H.
Takahashi
, S.
Umino
, Y.
Miki
, R.
Ishizuka
, S.
Maeda
, A.
Morita
, M.
Suzuki
, and N.
Matubayasi
, “Drastic compensation of electronic and solvation effects on ATP hydrolysis revealed through large-scale QM/MM simulations combined with a theory of solutions
,” J. Phys. Chem. B
121
, 2279
–2287
(2017
).39.
N.
Matubayasi
and M.
Nakahara
, “Theory of solutions in the energy representation. II. Functional for the chemical potential
,” J. Chem. Phys.
117
, 3605
–3616
(2002
).40.
Y.
Karino
, M. V.
Fedorov
, and N.
Matubayasi
, “End-point calculation of solvation free energy of amino-acid analogs by molecular theories of solution
,” Chem. Phys. Lett.
496
, 351
–355
(2010
).41.
M. J.
Frisch
, G. W.
Trucks
, and H. B.
Schlegel
, Gaussian 09, Revision C.01
, Gaussian, Inc.
, Wallingford, CT
, 2010
.42.
43.
A. D.
Becke
, “Density-functional exchange-energy approximation with correct asymptotic behavior
,” Phys. Rev. A
38
, 3098
–3100
(1988
).44.
A. D.
Becke
, “A new mixing of Hartree-Fock and local density-functional theories
,” J. Chem. Phys.
98
, 1372
–1377
(1993
).45.
C.
Lee
, W.
Yang
, and R. G.
Parr
, “Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,” Phys. Rev. B
37
, 785
–789
(1988
).46.
T. H.
Dunning
, “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
,” J. Chem. Phys.
90
, 1007
–1023
(1989
).47.
H.
Takahashi
, T.
Hori
, T.
Wakabayashi
, and T.
Nitta
, “A density functional study for hydrogen bond energy by employing real space grids
,” Chem. Lett.
29
, 222
–223
(2000
).48.
H.
Takahashi
, T.
Hori
, T.
Wakabayashi
, and T.
Nitta
, “Real space ab initio molecular dynamics simulations for the reactions of OH radical/OH anion with formaldehyde
,” J. Phys. Chem. A
105
, 4351
(2001
).49.
H.
Takahashi
, H.
Hashimoto
, and T.
Nitta
, “Quantum mechanical/molecular mechanical studies of a novel reaction catalyzed by proton transfers in ambient and supercritical states of water
,” J. Chem. Phys.
119
, 7964
–7971
(2003
).50.
J. R.
Chelikowsky
, N.
Troullier
, and Y.
Saad
, “Finite-difference-pseudopotential method: Electronic structure calculations without a basis
,” Phys. Rev. Lett.
72
, 1240
–1243
(1994
).51.
J. R.
Chelikowsky
, N.
Troullier
, K.
Wu
, and Y.
Saad
, “Higher-order finite-difference pseudopotential method: An application to diatomic molecules
,” Phys. Rev. B
50
, 11355
–11364
(1994
).52.
T.
Ono
and K.
Hirose
, “Timesaving double-grid method for real-space electronic-structure calculations
,” Phys. Rev. Lett.
82
, 5016
–5019
(1999
).53.
H.
Takahashi
, S.
Sakuraba
, and A.
Morita
, “Large-scale parallel implementation of Hartree-Fock exchange energy on real-space grids using 3D-parallel fast Fourier transform
,” J. Chem. Inf. Model.
60
, 1376
–1389
(2020
).54.
L.
Kleinman
and D. M.
Bylander
, “Efficacious form for model pseudopotentials
,” Phys. Rev. Lett.
48
, 1425
–1428
(1982
).55.
P.
Hohenberg
and W.
Kohn
, “Inhomogeneous electron gas
,” Phys. Rev.
136
, B864
–B871
(1964
).56.
D.
Suzuoka
, H.
Takahashi
, and A.
Morita
, “Computation of the free energy due to electron density fluctuation of a solute in solution: A QM/MM method with perturbation approach combined with a theory of solutions
,” J. Chem. Phys.
140
, 134111
(2014
).57.
H. J. C.
Berendsen
, J. R.
Grigera
, and T. P.
Straatsma
, “The missing term in effective pair potentials
,” J. Phys. Chem.
91
, 6269
–6271
(1987
).58.
W.
Wagner
, J. R.
Cooper
, A.
Dittmann
, J.
Kijima
, H.-J.
Kretzschmar
, A.
Kruse
, R.
Mareš
, K.
Oguchi
, H.
Sato
, I.
Stöcker
, O. O.
Šifner
, Y.
Takaishi
, I.
Tanishita
, J.
Trübenbach
, and T.
Willkommen
, “The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam
,” J. Engin. Gas Turb. Pow.
122
, 150
–184
(2000
).59.
C. H.
Bennett
, “Efficient estimation of free energy differences from Monte Carlo data
,” J. Comput. Phys.
22
, 245
–268
(1976
).60.
T. A.
Wesołowski
, “Embedding a multideterminantal wave function in an orbital-free environment
,” Phys. Rev. A
77
, 012504
(2008
).61.
T. A.
Wesolowski
and A.
Warshel
, “Frozen density functional approach for ab initio calculations of solvated molecules
,” J. Phys. Chem.
97
, 8050
–8053
(1993
).62.
H.
Takahashi
, S.
Umino
, and A.
Morita
, “Construction of exchange repulsion in terms of the wave functions at QM/MM boundary region
,” J. Chem. Phys.
143
, 084104
(2015
).63.
S.
Umino
, H.
Takahashi
, and A.
Morita
, “Condensed phase QM/MM simulations utilizing the exchange core functions to describe exchange repulsions at the QM boundary region
,” J. Chem. Phys.
145
, 084107
(2016
).64.
A. D.
Becke
, “A multicenter numerical integration scheme for polyatomic molecules
,” J. Chem. Phys.
88
, 2547
–2553
(1988
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.