An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato’s theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.
REFERENCES
1.
R. G.
Parr
and W.
Yang
, “Density-functional theory of the electronic structure of molecules
,” Annu. Rev. Phys. Chem.
46
, 701
–728
(1995
).2.
H.
Chermette
, “Chemical reactivity indexes in density functional theory
,” J. Comput. Chem.
20
, 129
–154
(1999
).3.
P.
Geerlings
, F.
De Proft
, and W.
Langenaeker
, “Conceptual density functional theory
,” Chem. Rev.
103
, 1793
–1874
(2003
).4.
F.
De Proft
and P.
Geerlings
, “Conceptual and computational DFT in the study of aromaticity
,” Chem. Rev.
101
, 1451
–1464
(2001
).5.
P. W.
Ayers
, J. S. M.
Anderson
, and L. J.
Bartolotti
, “Perturbative perspectives on the chemical reaction prediction problem
,” Int. J. Quantum Chem.
101
, 520
–534
(2005
).6.
J. L.
Gázquez
, “Perspectives on the density functional theory of chemical reactivity
,” J. Mex. Chem. Soc.
52
, 3
–10
(2008
).7.
S. B.
Liu
, “Conceptual density functional theory and some recent developments
,” Acta Phys.-Chim. Sin.
25
, 590
–600
(2009
).8.
P.
Geerlings
, E.
Chamorro
, P. K.
Chattaraj
, F. De.
Proft
, J. L.
Gázquez
, S. B.
Liu
, C.
Morell
, A.
Toro-Labbé
, A.
Vela
, and P.
Ayers
, “Conceptual density functional theory: Status, prospects, issues
,” Theor. Chem. Acc.
139
, 36
(2020
).9.
N.
Sablon
, F.
De Proft
, and P.
Geerlings
, “The linear response kernel: Inductive and resonance effects quantified
,” J. Phys. Chem. Lett.
1
, 1228
–1234
(2010
).10.
S.
Fias
, Z.
Boisdenghien
, T.
Stuyver
, M.
Audiffred
, G.
Merino
, P.
Geerlings
, and F.
De Proft
, “Analysis of aromaticity in planar metal systems using the linear response kernel
,” J. Phys. Chem. A
117
, 3556
–3560
(2013
).11.
S.
Fias
, P.
Geerlings
, P.
Ayers
, and F.
De Proft
, “π, σ aromaticity and anti-aromaticity as retrieved by the linear response kernel
,” Phys. Chem. Chem. Phys.
15
, 2882
–2889
(2013
).12.
Z.
Boisdenghien
, C.
Van Alsenoy
, F.
De Proft
, and P.
Geerlings
, “Evaluating and interpreting the chemical relevance of the linear response kernel for atoms
,” J. Chem. Theory Comput.
9
, 1007
–1015
(2013
).13.
P.
Geerlings
, S.
Fias
, Z.
Boisdenghien
, and F.
De Proft
, “Conceptual DFT: Chemistry from the linear response function
,” Chem. Soc. Rev.
43
, 4989
–5008
(2014
).14.
P.
Geerlings
, Z.
Boisdenghien
, F.
De Proft
, and S.
Fias
, “The E = E[N, v] functional and the linear response function: A conceptual DFT viewpoint
,” Theor. Chem. Acc.
135
, 213
(2016
).15.
R. F.
Nalewajski
and R. G.
Parr
, “Legendre transforms and Maxwell relations in density functional theory
,” J. Chem. Phys.
77
, 399
–407
(1982
).16.
M.
Berkowitz
and R. G.
Parr
, “Molecular hardness and softness, local hardness and softness, hardness and softness kernels, and relations among these quantities
,” J. Chem. Phys.
88
, 2554
–2557
(1988
).17.
M. H.
Cohen
, M. V.
Ganduglia‐Pirovano
, and J.
Kudrnovský
, “Reactivity kernels, the normal modes of chemical reactivity, and the hardness and softness spectra
,” J. Chem. Phys.
103
, 3543
–3551
(1995
).18.
P.
Senet
, “Nonlinear electronic responses, Fukui functions and hardnesses as functionals of the ground-state electronic density
,” J. Chem. Phys.
105
, 6471
–6489
(1996
).19.
P.
Senet
, “Kohn-Sham orbital formulation of the chemical electronic responses, including the hardness
,” J. Chem. Phys.
107
, 2516
–2524
(1997
).20.
P. W.
Ayers
and R. G.
Parr
, “Variational principles for describing chemical reactions. Reactivity indices based on the external potential
,” J. Am. Chem. Soc.
123
, 2007
–2017
(2001
).21.
P. W.
Ayers
, “Strategies for computing chemical reactivity indices
,” Theor. Chem. Acc.
106
, 271
–279
(2001
).22.
S.
Liu
, T.
Li
, and P. W.
Ayers
, “Potentialphilicity and potentialphobicity: Reactivity indicators for external potential changes from density functional reactivity theory
,” J. Chem. Phys.
131
, 114106
(2009
).23.
E.
Echegaray
, S.
Rabi
, C.
Cárdenas
, F. H.
Zadeh
, N.
Rabi
, S.
Lee
, J. S. M.
Anderson
, A.
Toro-Labbe
, and P. W.
Ayers
, “In pursuit of negative Fukui functions: Molecules with very small band gaps
,” J. Mol. Model.
20
, 2162
(2014
).24.
P. W.
Ayers
, R. C.
Morrison
, and R. K.
Roy
, “Variational principles for describing chemical reactions: Condensed reactivity indices
,” J. Chem. Phys.
116
, 8731
–8744
(2002
).25.
P.
Bultinck
and R.
Carbó-Dorca
, “Negative and infinite Fukui functions: The role of diagonal dominance in the hardness matrix
,” J. Math. Chem.
34
, 67
–74
(2003
).26.
P. W.
Ayers
, “Can one oxidize an atom by reducing the molecule that contains it?
,” Phys. Chem. Chem. Phys.
8
, 3387
–3390
(2006
).27.
A.
Savin
, F.
Colonna
, and M.
Allavena
, “Analysis of the linear response function along the adiabatic connection from the Kohn–Sham to the correlated system
,” J. Chem. Phys.
115
, 6827
–6833
(2001
).28.
Z.
Boisdenghien
, S.
Fias
, C.
Van Alsenoy
, F.
De Proft
, and P.
Geerlings
, “Evaluating and interpreting the chemical relevance of the linear response kernel for atoms II: Open shell
,” Phys. Chem. Chem. Phys.
16
, 14614
–14624
(2014
).29.
S.
Fias
, Z.
Boisdenghien
, F.
De Proft
, and P.
Geerlings
, “The spin polarized linear response from density functional theory: Theory and application to atoms
,” J. Chem. Phys.
141
, 184107
(2014
).30.
Z.
Boisdenghien
, S.
Fias
, F.
Da Pieve
, F.
De Proft
, and P.
Geerlings
, “The polarisability of atoms and molecules: A comparison between a conceptual density functional theory approach and time-dependent density functional theory
,” Mol. Phys.
113
, 1890
–1898
(2015
).31.
P.
Geerlings
, S.
Fias
, T.
Stuyver
, P. W.
Ayers
, R.
Balawender
, and F.
De Proft
, “New insights and horizons from the linear response function in conceptual DFT
,” in Density Functional Theory
(IntechOpen
, 2019
), pp. 3
–29
.32.
P.
Geerlings
, “The linear response function
,” in Towards a New Chemical Reactivity Theory
, edited by S.
Liu
(Wiley-VCH
, 2022
).33.
W.
Yang
and R. G.
Parr
, “Hardness, softness, and the Fukui function in the electronic theory of metals and catalysis
,” Proc. Natl. Acad. Sci. U. S. A.
82
, 6723
–6726
(1985
).34.
F.
Heidar-Zadeh
, M.
Richer
, S.
Fias
, R. A.
Miranda-Quintana
, M.
Chan
, M.
Franco-Pérez
, C. E.
González-Espinoza
, T. D.
Kim
, C.
Lanssens
, A. H. G.
Patel
, X. D.
Yang
, E.
Vöhringer-Martinez
, C.
Cárdenas
, T.
Verstraelen
, and P. W.
Ayers
, “An explicit approach to conceptual density functional theory descriptors of arbitrary order
,” Chem. Phys. Lett.
660
, 307
–312
(2016
).35.
S.
Fias
, F.
Heidar-Zadeh
, J. S. M.
Anderson
, P. W.
Ayers
, and R. G.
Parr
, “A reference-free stockholder partitioning method based on the force on electrons
,” J. Comput. Chem.
39
, 1044
–1050
(2017
).36.
W.
Kohn
, “Density functional and density matrix method scaling linearly with the number of atoms
,” Phys. Rev. Lett.
76
, 3168
–3171
(1996
).37.
E.
Prodan
and W.
Kohn
, “Nearsightedness of electronic matter
,” Proc. Natl. Acad. Sci. U. S. A.
102
, 11635
–11638
(2005
).38.
S.
Fias
, F.
Heidar-Zadeh
, P.
Geerlings
, and P. W.
Ayers
, “Chemical transferability of functional groups follows from the nearsightedness of electronic matter
,” Proc. Natl. Acad. Sci. U. S. A.
114
, 11633
–11638
(2017
).39.
M.
Berkowitz
, S. K.
Ghosh
, and R. G.
Parr
, “On the concept of local hardness in chemistry
,” J. Am. Chem. Soc.
107
, 6811
–6814
(1985
).40.
S. K.
Ghosh
and M.
Berkowitz
, “A classical fluid-like approach to the density-functional formalism of many-electron systems
,” J. Chem. Phys.
83
, 2976
–2983
(1985
).41.
M. K.
Harbola
, P. K.
Chattaraj
, and R. G.
Parr
, “Aspects of the softness and hardness concepts of density-functional theory
,” Isr. J. Chem.
31
, 395
–402
(1991
).42.
P. K.
Chattaraj
, D. R.
Roy
, P.
Geerlings
, and M.
Torrent-Sucarrat
, Theor. Chem. Acc.
118
, 923
–930
(2007
).43.
P. W.
Ayers
and R. G.
Parr
, “Local hardness equalization: Exploiting the ambiguity
,” J. Chem. Phys.
128
, 184108
(2008
).44.
S.
Liu
and R. G.
Parr
, “Second-order density-functional description of molecules and chemical changes
,” J. Chem. Phys.
106
, 5578
–5586
(1997
).45.
P.
Geerlings
, F.
De Proft
, and S.
Fias
, “Analogies between density functional theory response kernels and derivatives of thermodynamic state functions
,” Acta Phys.-Chim. Sin.
34
, 699
–707
(2018
).46.
T.
Gál
, “Why the traditional concept of local hardness does not work
,” Theor. Chem. Acc.
131
, 1223
(2012
).47.
R.
Cuevas-Saavedra
, N.
Rabi
, and P. W.
Ayers
, “The unconstrained local hardness: An intriguing quantity, beset by problems
,” Phys. Chem. Chem. Phys.
13
, 19594
–19600
(2011
).48.
T.
Gál
, P.
Geerlings
, F.
De Proft
, and M.
Torrent-Sucarrat
, “A new approach to local hardness
,” Phys. Chem. Chem. Phys.
13
, 15003
–15015
(2011
).49.
M.
Torrent-Sucarrat
, F.
De Proft
, P. W.
Ayers
, and P.
Geerlings
, “On the applicability of local softness and hardness
,” Phys. Chem. Chem. Phys.
12
, 1072
–1080
(2010
).50.
M.
Torrent-Sucarrat
, F.
De Proft
, P.
Geerlings
, and P. W.
Ayers
, “Do the local softness and hardness indicate the softest and hardest regions of a molecule?
,” Chem.-Eur. J.
14
, 8652
–8660
(2008
).51.
C.
Cárdenas
, P. W.
Ayers
, and A.
Cedillo
, “Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory
,” J. Chem. Phys.
134
, 174103
(2011
).52.
P.
Bultinck
, C.
Cárdenas
, P.
Fuentealba
, P. A.
Johnson
, and P. W.
Ayers
, “Atomic charges and the electrostatic potential are ill-defined in degenerate ground states
,” J. Chem. Theory Comput.
9
, 4779
–4788
(2013
).53.
P.
Bultinck
, C.
Cárdenas
, P.
Fuentealba
, P. A.
Johnson
, and P. W.
Ayers
, “How to compute the fukui matrix and function for systems with (quasi-)degenerate states
,” J. Chem. Theory Comput.
10
, 202
–210
(2014
).54.
P.
Bultinck
, D.
Jayatilaka
, and C.
Cárdenas
, “A problematic issue for atoms in molecules: Impact of (quasi-)degenerate states on quantum theory atoms in molecules and Hirshfeld-I properties
,” Comput. Theor. Chem.
1053
, 106
–111
(2015
).55.
M.
Levy
, “Electron densities in search of Hamiltonians
,” Phys. Rev. A
26
, 1200
–1208
(1982
).56.
H.
Englisch
and R.
Englisch
, “Hohenberg–Kohn theorem and non-v-representable densities
,” Physica A
121
, 253
–268
(1983
).57.
E. H.
Lieb
, “Density functionals for Coulomb systems
,” Int. J. Quantum Chem.
24
, 243
–277
(1983
).58.
H.
Englisch
and R.
Englisch
, “Exact density functionals for ground-state energies. 1. General results
,” Phys. Status Solidi B
123
, 711
–721
(1984
).59.
H.
Englisch
and R.
Englisch
, “Exact density functionals for ground-state energies. 2. Details and remarks
,” Phys. Status Solidi B
124
, 373
–379
(1984
).60.
H.
Eschrig
, The Fundamentals of Density Functional Theory
(Teubner
, Stuttgart-Leipzig, Germany
, 1996
).61.
P. W.
Ayers
, “Axiomatic formulations of the Hohenberg–Kohn functional
,” Phys. Rev. A
73
, 012513
(2006
).62.
J. P.
Perdew
and M.
Levy
, “Extrema of the density functional for the energy: Excited states from the ground-state theory
,” Phys. Rev. B
31
, 6264
–6272
(1985
).63.
H.
Eschrig
, “T > 0 ensemble-state density functional theory via Legendre transform
,” Phys. Rev. B
82
, 205120
(2010
).64.
M.
Franco-Pérez
, P. W.
Ayers
, J. L.
Gázquez
, and A.
Vela
, “Local and linear chemical reactivity response functions at finite temperature in density functional theory
,” J. Chem. Phys.
143
, 244117
(2015
).65.
M.
Franco-Pérez
, P. W.
Ayers
, J. L.
Gázquez
, and A.
Vela
, “Thermodynamic responses of electronic systems
,” J. Chem. Phys.
147
, 094105
(2017
).66.
R. A.
Miranda-Quintana
and P. W.
Ayers
, “Fractional electron number, temperature, and perturbations in chemical reactions
,” Phys. Chem. Chem. Phys.
18
, 15070
–15080
(2016
).67.
J. P.
Perdew
, R. G.
Parr
, M.
Levy
, and J. L.
Balduz
, Jr., “Density-functional theory for fractional particle number: Derivative discontinuities of the energy
,” Phys. Rev. Lett.
49
, 1691
–1694
(1982
).68.
R. G.
Parr
and W.
Yang
, Density-Functional Theory of Atoms and Molecules
(Oxford University Press
, London; New York
, 1989
).69.
J. L.
Gázquez
, M.
Franco‐Pérez
, P. W.
Ayers
, and A.
Vela
, “Temperature-dependent approach to chemical reactivity concepts in density functional theory
,” Int. J. Quantum Chem.
119
, e25797
(2019
).70.
71.
R. P.
Feynman
, “Forces in molecules
,” Phys. Rev.
56
, 340
–343
(1939
).72.
D.
Hilbert
, “Grundzüge einer allgemeinen theorie der linearen integralgleichungen [erste Mitteilung]
,” Gottinger Nachr.
1904
, 49
–91
.73.
J.
Mercer
, “Functions of positive and negative type, and their connection with the theory of integral equations
,” Philos. Trans. R. Soc., A
209
, 415
–446
(1909
).74.
T.
Bromwich
, Quadratic Forms and Their Classification by Means of Invariant-Factors
, Cambridge Tracts in Mathematics and Mathematical Physics Vol. 3 (Cambridge University Press
, 1906
).75.
G.
Giorgio
, “Various proofs of the Sylvester criterion for quadratic forms
,” J. Math. Res.
9
, 55
–66
(2017
).76.
T.
Helgaker
, P.
Jorgensen
, and J.
Olsen
, Molecular Electronic-Structure Theory
(John Wiley & Sons
, 2000
), p. 404
.77.
T.
Helgaker
, P.
Jorgensen
, and J.
Olsen
, Molecular Electronic-Structure Theory
(John Wiley & Sons
, 2000
), p. 126
.78.
P. W.
Ayers
, “The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons
,” J. Math. Chem.
43
, 285
–303
(2008
).79.
D.
Peng
and W.
Yang
, “Fukui function and response function for nonlocal and fractional systems
,” J. Chem. Phys.
138
, 184108
(2013
).80.
T.
Kato
, “On the eigenfunctions of many-particle systems in quantum mechanics
,” Commun. Pure Appl. Math.
10
, 151
–177
(1957
).81.
R. T.
Pack
and W. B.
Brown
, “Cusp conditions for molecular wavefunctions
,” J. Chem. Phys.
45
, 556
–559
(1966
).82.
P. W.
Ayers
, “Density per particle as a descriptor of Coulombic systems
,” Proc. Natl. Acad. Sci. U. S. A.
97
, 1959
–1964
(2000
).83.
Á.
Nagy
and K. D.
Sen
, “Exact results on the curvature of the electron density at the cusp in certain highly excited states of atoms
,” Chem. Phys. Lett.
332
, 154
–158
(2000
).84.
Á.
Nagy
and K. D.
Sen
, “Higher-order cusp of the density in certain highly excited states of atoms and molecules
,” J. Phys. B: At., Mol. Opt. Phys.
33
, 1745
–1751
(2000
).85.
R.
Nalewajski
, “The hardness based molecular charge sensitivities and their use in the theory of chemical reactivity
,” in Chemical Hardness
, edited by K.
Sen
(Springer-Verlag
, 1993
), pp. 115
–186
.86.
R. F.
Nalewajski
, “Chemical reactivity concepts in charge sensitivity analysis
,” Int. J. Quant. Chem.
56
, 453
–476
(1995
).87.
P.
Senet
, “Variational principle for eigenmodes of reactivity in conceptual density functional theory
,” ACS Omega
5
, 25349
–25357
(2020
).88.
P. K.
Chattaraj
, A.
Cedillo
, and R. G.
Parr
, “Fukui function from a gradient expansion formula, and estimate of hardness and covalent radius for an atom
,” J. Chem. Phys.
103
, 10621
–10626
(1995
).89.
R. G.
Parr
and L. J.
Bartolotti
, “Some remarks on the density functional theory of few-electron systems
,” J. Phys. Chem.
87
, 2810
–2815
(1983
).90.
P. W.
Ayers
, F.
De Proft
, and P.
Geerlings
, “Comparison of the utility of the shape function and electron density for predicting periodic properties: Atomic ionization potentials
,” Phys. Rev. A
75
, 012508
(2007
).91.
M.
Levy
, “Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem
,” Proc. Natl. Acad. Sci. U. S. A.
76
, 6062
–6065
(1979
).92.
S. M.
Valone
, “A one-to-one mapping between one-particle densities and some n-particle ensembles
,” J. Chem. Phys.
73
, 4653
(1980
).93.
M.
Levy
, J. S. M.
Anderson
, F. H.
Zadeh
, and P. W.
Ayers
, “Kinetic and electron–electron energies for convex sums of ground state densities with degeneracies and fractional electron number
,” J. Chem. Phys.
140
, 18A538
(2014
).94.
W.
Langenaeker
, F.
de Proft
, and P.
Geerlings
, “Development of local hardness related reactivity indices: Their application in a study of the SE at monosubstituted benzenes within the HSAB context
,” J. Phys. Chem.
99
, 6424
–6431
(1995
).95.
W. T.
Yang
, A. J.
Cohen
, F.
De Proft
, and P.
Geerlings
, “Analytical evaluation of Fukui functions and real-space linear response function
,” J. Chem. Phys.
136
, 144110
(2012
).96.
R.
Seeger
and J. A.
Pople
, “Self-consistent molecular orbital methods. XVIII. Constraints and stability in Hartree–Fock theory
,” J. Chem. Phys.
66
, 3045
(1977
).97.
V.
Bach
, E. H.
Lieb
, M.
Loss
, and J. P.
Solovej
, “There are no unfilled shells in unrestricted Hartree–Fock theory
,” Phys. Rev. Lett.
72
, 2981
–2983
(1994
).98.
R.
Bauernschmitt
and R.
Ahlrichs
, “Stability analysis for solutions of the closed shell Kohn–Sham equation
,” J. Chem. Phys.
104
, 9047
(1996
).99.
P. D.
Robinson
, “: A problem in perturbation theory
,” Proc. Phys. Soc.
78
, 537
–548
(1961
).100.
A.
Ruzsinszky
, J. P.
Perdew
, G. I.
Csonka
, O. A.
Vydrov
, and G. E.
Scuseria
, “Density functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+
,” J. Chem. Phys.
126
, 104102
(2007
).101.
T.
Helgaker
, P.
Jorgensen
, and J.
Olsen
, Molecular Electronic-Structure Theory
(John Wiley & Sons
, 2000
), p. 115
.102.
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