Theor Chem Acc (2015) 134:138
1 3
Several topics deserve our attention as natural continuation of the present work to continue the understanding of
the reactivity phenomena from a rigorous point of view and
are being considered in our laboratory. Some of them are
related to obtain an equation that preserves the constant
rate of electron transfer between the open domains in the
molecular structure and provides DM evolution within this
context, the modeling of the interaction potential of different nature and the contraction mappings in Fock space for
coherent density matrices, among others.
Acknowledgments This report has been fi nancially supported by
Projects 20020130100226BA (Universidad de Buenos Aires) and PIP
No. 11220090100061 (Consejo Nacional de Investigaciones Científi -
cas y Técnicas, República Argentina).
References
1. Szabo A, Ostlund NS (1989) Modern quantum chemistry. Introduction to advanced electronic structure theory. Dover Publications Inc, Mineloa, New York
2. Helgaker T, Jörgensen P, Olsen J (2000) Molecular electronic
structure theory. Wiley Ltd, Chitester
3. Geerlings P, De Proft F, Langenaeker W (2003) Conceptual density functional theory. Chem Rev 103(5):1793–1873
4. Perdew JP, Parr RG, Levy M, Balduz J Jr (1982) Density-functional theory for fractional particle number: derivative discontinuities of the energy. Phys Rev Lett 49(23):1691–1694
5. Perdew JP (1985) In: Dreizler RM, da Providencia J (eds) Density functional theory methods in physics, NATO ASI series. Plenum, New York
6. Parr RG, Yang W (1989) Density-functional theory of atoms and
molecules. Oxford Univesity Press, New York
7. Iczkowski RP, Margrave JL (1961) Electronegativity. J Am
Chem Soc 83(17):3547–3551
8. Morales J, Martinez TJ (2001) Classical fl uctuating charge theories: the maximum entropy valence bond formalism and relationships to previous models. J Phys Chem A 105(12):2842–2850
9. Ayers PW, Parr RG (2008) Local hardness equalization: exploiting the ambiguity. J Chem Phys 128(18):184108
10. Parr RG, Pariser R (2013) In: Ghosh SK, Chattaraj PK (eds)
Concepts and methods in modern theoretical chemistry: electronic structure and reactivity. CRC Press, Boca Raton
11. Valone SM (2011) A concept of fragment hardness, independent
of net charge, from a wave-function perspective. J Phys Chem
Lett 2(20):2618–2622 and references
12. Bochicchio RC, Rial D (2012) Energy convexity and density
matrices in molecular systems. J Chem Phys 137(22):226101
13. Bochicchio RC, Miranda-Quintana RA, Rial D (2013) Reduced
density matrices in molecular systems: grand-canonical electron
states. J Chem Phys 139(19):199101
14. Nalewajski RF (1998) Kohn–Sham description of equilibria and charge transfer in reactive systems. Int J Quant Chem
69(4):591–605
15. Valone SM, Atlas SR (2006) Energy dependence on fractional
charge for strongly interacting subsystems. Phys Rev Lett
97(25):256402
16. Bochicchio RC, Lain L, Torre A (2003) Atomic valence in
molecular systems. Chem Phys Lett 375(1–2):45–53 and
references
17. Lain L, Torre A, Bochicchio RC (2004) Studies of population
analysis at the correlated level: determination of three-center
bond indices. J Phys Chem A 108(18):4132–4137 and references
18. Bader RFW (1994) Atoms in molecules: a quantum theory.
Oxford University Press, Oxford
19. Popelier P (1999) Atoms in molecules: an introduction. Pearson
Edu, London
20. Yang WT, Zhang YK, Ayers PW (2000) Degenerate ground states
and fractional number of electrons in density and reduced density matrix functional theory. Phys Rev Lett 84(22):5172–5175
21. Ayers PW (2008) The continuity of the energy and other molecular properties with respect to the number of electrons. J Math
Chem 43(1):285–303
22. Hirshfeld FL (1977) Bonded-atom fragments for describing
molecular charge densities. Theor Chim Acta 44(2):129–138
23. Valone SM (2011) Quantum mechanical origins of the Iczkowski–Margrave model of chemical potential. J Chem Theory
Comp 7(7):2253–2261
24. Cohen MH, Wasserman A (2007) On the foundations of chemical reactivity theory. J Chem Phys A 111(11):2229–2242
25. Ayers PW (2007) On the electronegativity nonlocality paradox.
Theor Chem Acc 118(2):371–381
26. Coleman AJ, Yukalov VI (2000) Reduced density matrices:
Coulson’s challenge. Springer, New York
27. Gatti C, Macchi P (eds) (2012) Modern charge-density analysis.
Springer, Dordrecht
28. Fradera X, Austen MA, Bader RFW (1999) The Lewis model
and beyond. J Phys Chem A 103(2):304–314
29. Alcoba DR, Torre A, Lain L, Bochicchio RC (2005) Energy
decompositions according to physical space partitioning
schemes. J Chem Phys 122(7):074102
30. Alcoba DR, Lain L, Torre A, Bochicchio RC (2005) A study
of the partitioning of the fi rst-order reduced density matrix
according to the theory of atoms in molecules. J Chem Phys
123(14):144113
31. Becke AD (1988) A multicenter numerical integration scheme
for polyatomic molecules. J Chem Phys 88(4):2547–2553
32. Torre A, Alcoba DR, Lain L, Bochicchio RC (2005) Determination of three-center bond indices from population analysis: a
fuzzy atom treatment. J Phys Chem A 109(29):6587–6591
33. Ter Haar D (1961) Theory and applications of the density matrix.
Rep Prog Phys 24(1):304–362
34. Blum K (1981) Density matrix theory and applications. Plenum,
New York
35. Emch GG (1972) Algebraic methods in statistical mechanics and
quantum fi eld theory. Wiley-Interscience, New York
36. Blaizot JP, Ripka G (2006) Quantum theory of fi nite systems.
The MIT Press, Cambridge
37. Rosina M (2001) In: Cioslowsky J (ed) Many-electron densities
and reduced density matrices. Kluwer, Dordrecht
38. Mazziotti DA (1998) Contracted Schrödinger equation: determining quantum energies and two-particle density matrices without wave functions. Phys Rev A 57(6):4219–4234
39. Hellgren M, Gross EKU (2012) Effect of discontinuities in
Kohn–Sham-based chemical reactivity theory. J Chem Phys
136(11):114102
40. McWeeny R (2001) Methods of molecular quantum mechanics.
Academic, San Diego
41. Kummer H (1967) n-Representability problem for reduced density matrices. J Math Phys 8(10):2063–2081
42. Paldus J (1976) In: Eyring H, Henderson DJ (eds) Theoretical
chemistry: advances and perspectives, vol 2. Academic Press,
New York
43. Surján PR (1989) Second quantized approach to quantum chemistry an elementary introduction. Springer-Verlag, Berlin
96
Reprinted from the journal
1 3
Several topics deserve our attention as natural continuation of the present work to continue the understanding of
the reactivity phenomena from a rigorous point of view and
are being considered in our laboratory. Some of them are
related to obtain an equation that preserves the constant
rate of electron transfer between the open domains in the
molecular structure and provides DM evolution within this
context, the modeling of the interaction potential of different nature and the contraction mappings in Fock space for
coherent density matrices, among others.
Acknowledgments This report has been fi nancially supported by
Projects 20020130100226BA (Universidad de Buenos Aires) and PIP
No. 11220090100061 (Consejo Nacional de Investigaciones Científi -
cas y Técnicas, República Argentina).
References
1. Szabo A, Ostlund NS (1989) Modern quantum chemistry. Introduction to advanced electronic structure theory. Dover Publications Inc, Mineloa, New York
2. Helgaker T, Jörgensen P, Olsen J (2000) Molecular electronic
structure theory. Wiley Ltd, Chitester
3. Geerlings P, De Proft F, Langenaeker W (2003) Conceptual density functional theory. Chem Rev 103(5):1793–1873
4. Perdew JP, Parr RG, Levy M, Balduz J Jr (1982) Density-functional theory for fractional particle number: derivative discontinuities of the energy. Phys Rev Lett 49(23):1691–1694
5. Perdew JP (1985) In: Dreizler RM, da Providencia J (eds) Density functional theory methods in physics, NATO ASI series. Plenum, New York
6. Parr RG, Yang W (1989) Density-functional theory of atoms and
molecules. Oxford Univesity Press, New York
7. Iczkowski RP, Margrave JL (1961) Electronegativity. J Am
Chem Soc 83(17):3547–3551
8. Morales J, Martinez TJ (2001) Classical fl uctuating charge theories: the maximum entropy valence bond formalism and relationships to previous models. J Phys Chem A 105(12):2842–2850
9. Ayers PW, Parr RG (2008) Local hardness equalization: exploiting the ambiguity. J Chem Phys 128(18):184108
10. Parr RG, Pariser R (2013) In: Ghosh SK, Chattaraj PK (eds)
Concepts and methods in modern theoretical chemistry: electronic structure and reactivity. CRC Press, Boca Raton
11. Valone SM (2011) A concept of fragment hardness, independent
of net charge, from a wave-function perspective. J Phys Chem
Lett 2(20):2618–2622 and references
12. Bochicchio RC, Rial D (2012) Energy convexity and density
matrices in molecular systems. J Chem Phys 137(22):226101
13. Bochicchio RC, Miranda-Quintana RA, Rial D (2013) Reduced
density matrices in molecular systems: grand-canonical electron
states. J Chem Phys 139(19):199101
14. Nalewajski RF (1998) Kohn–Sham description of equilibria and charge transfer in reactive systems. Int J Quant Chem
69(4):591–605
15. Valone SM, Atlas SR (2006) Energy dependence on fractional
charge for strongly interacting subsystems. Phys Rev Lett
97(25):256402
16. Bochicchio RC, Lain L, Torre A (2003) Atomic valence in
molecular systems. Chem Phys Lett 375(1–2):45–53 and
references
17. Lain L, Torre A, Bochicchio RC (2004) Studies of population
analysis at the correlated level: determination of three-center
bond indices. J Phys Chem A 108(18):4132–4137 and references
18. Bader RFW (1994) Atoms in molecules: a quantum theory.
Oxford University Press, Oxford
19. Popelier P (1999) Atoms in molecules: an introduction. Pearson
Edu, London
20. Yang WT, Zhang YK, Ayers PW (2000) Degenerate ground states
and fractional number of electrons in density and reduced density matrix functional theory. Phys Rev Lett 84(22):5172–5175
21. Ayers PW (2008) The continuity of the energy and other molecular properties with respect to the number of electrons. J Math
Chem 43(1):285–303
22. Hirshfeld FL (1977) Bonded-atom fragments for describing
molecular charge densities. Theor Chim Acta 44(2):129–138
23. Valone SM (2011) Quantum mechanical origins of the Iczkowski–Margrave model of chemical potential. J Chem Theory
Comp 7(7):2253–2261
24. Cohen MH, Wasserman A (2007) On the foundations of chemical reactivity theory. J Chem Phys A 111(11):2229–2242
25. Ayers PW (2007) On the electronegativity nonlocality paradox.
Theor Chem Acc 118(2):371–381
26. Coleman AJ, Yukalov VI (2000) Reduced density matrices:
Coulson’s challenge. Springer, New York
27. Gatti C, Macchi P (eds) (2012) Modern charge-density analysis.
Springer, Dordrecht
28. Fradera X, Austen MA, Bader RFW (1999) The Lewis model
and beyond. J Phys Chem A 103(2):304–314
29. Alcoba DR, Torre A, Lain L, Bochicchio RC (2005) Energy
decompositions according to physical space partitioning
schemes. J Chem Phys 122(7):074102
30. Alcoba DR, Lain L, Torre A, Bochicchio RC (2005) A study
of the partitioning of the fi rst-order reduced density matrix
according to the theory of atoms in molecules. J Chem Phys
123(14):144113
31. Becke AD (1988) A multicenter numerical integration scheme
for polyatomic molecules. J Chem Phys 88(4):2547–2553
32. Torre A, Alcoba DR, Lain L, Bochicchio RC (2005) Determination of three-center bond indices from population analysis: a
fuzzy atom treatment. J Phys Chem A 109(29):6587–6591
33. Ter Haar D (1961) Theory and applications of the density matrix.
Rep Prog Phys 24(1):304–362
34. Blum K (1981) Density matrix theory and applications. Plenum,
New York
35. Emch GG (1972) Algebraic methods in statistical mechanics and
quantum fi eld theory. Wiley-Interscience, New York
36. Blaizot JP, Ripka G (2006) Quantum theory of fi nite systems.
The MIT Press, Cambridge
37. Rosina M (2001) In: Cioslowsky J (ed) Many-electron densities
and reduced density matrices. Kluwer, Dordrecht
38. Mazziotti DA (1998) Contracted Schrödinger equation: determining quantum energies and two-particle density matrices without wave functions. Phys Rev A 57(6):4219–4234
39. Hellgren M, Gross EKU (2012) Effect of discontinuities in
Kohn–Sham-based chemical reactivity theory. J Chem Phys
136(11):114102
40. McWeeny R (2001) Methods of molecular quantum mechanics.
Academic, San Diego
41. Kummer H (1967) n-Representability problem for reduced density matrices. J Math Phys 8(10):2063–2081
42. Paldus J (1976) In: Eyring H, Henderson DJ (eds) Theoretical
chemistry: advances and perspectives, vol 2. Academic Press,
New York
43. Surján PR (1989) Second quantized approach to quantum chemistry an elementary introduction. Springer-Verlag, Berlin
96
Reprinted from the journal