Theor Chem Acc (2015) 134:85
1 3
distribution at equilibrium distances in the three molecular basis sets. Likewise, the molecular system descriptions
at the stretched geometries systematically present higher
values for the I C and I W indices (Table 1 ) than their counterparts at the equilibrium distances. This effect is interpreted in the framework of the progressive openness of the
chemical bonds until their complete dissociation, which is
refl ected in the values of both indices. However, the index
I C shows a more sensitive character than the I W one, and
consequently, its use must be favored in order to account
for the infl uence of the bond stretching on the wave function features.
5 Concluding remarks and perspectives
In this work, we have extended the formulation of the Shannon entropy indices, informational content ( I C ), cumulative
( I W ), and specifi c -subspace ( I ), within the framework of
the seniority number criterion for constructing N -electron
wave function expansions in terms of Slater determinants.
The quantitative evaluation of these indices has allowed us
to implement analyses of the wave function expansions,
determining their compactness in the well-known canonical
molecular orbital and natural orbital basis sets, as well as
in the recently proposed molecular orbital basis set which
minimizes the seniority number of a given wave function.
The results obtained for several atomic and molecular systems described at the FCI level show the suitability of the
seniority-based formulation of these indices to measure
quantitatively the wave function expansion compactness,
as well as to analyze their multiconfi gurational structure.
We have also studied the ability of these indices to provide
information on the evolution of the wave functions according to the stretching of the chemical bondings. We are currently working in our laboratories on the formulation of
unitary transformations of molecular basis sets leading to
the minimization of the Shannon entropy indices, in order
to achieve a higher improvement on the compactness of
wave function expansions.
Acknowledgments This work has been fi nancially supported
by the Grant Nos. GIU12/09 and UFI11/07 (Universidad del Pais
Vasco, Spain), UBACYT 20020100100197 (Universidad de Buenos Aires, Argentina), PIP No. 11220090100061, 11220090100369,
11220130100377CO, and 11220130100311CO (Consejo Nacional
de Investigaciones Científi cas y Técnicas, Argentina). We thank the
Universidad del País Vasco for allocation of computational resources.
References
1. Shavitt I (1998) Mol Phys 94:3
2. Sherrill CD, Schaefer HF III (1999) Adv Quantum Chem 34:143
and references therein
3. Bytautas L, Ivanic J, Ruedenberg K (2003) J Chem Phys
119:8217
4. Giesbertz KJH (2014) Chem Phys Lett 591:220
5. Bytautas L, Henderson TM, Jiménez-Hoyos CA, Ellis JK, Scuseria GE (2011) J Chem Phys 135:044119
6. Alcoba DR, Torre A, Lain L, Massaccesi GE, Oña OB (2013) J
Chem Phys 139:084103
7. Stein T, Henderson TM, Scuseria GE (2014) J Chem Phys
140:214113
8. Boguslawski K, Tecmer P, Limacher PA, Johnson PA, Ayers PW,
Bultinck P, De Baerdemacker S, Van Neck D (2014) J Chem
Phys 140:214114
9. Alcoba DR, Torre A, Lain L, Massaccesi GE, Oña OB (2014) J
Chem Phys 140:234103
10. Limacher PA, Kim TD, Ayers PW, Johnson PA, De Baerdemacker S, Van Neck D, Bultinck P (2014) Mol Phys
5–6:853
11. Alcoba DR, Torre A, Lain L, Oña OB, Capuzzi P, Van Raemdonck M, Bultinck P, Van Neck D (2014) J Chem Phys
141:244118
12. Ring P, Schuck P (1980) The nuclear many-body problem.
Springe, New York
13. Koltun DS, Eisenberg JM (1988) Quantum mechanics of many
degrees of freedom. Wiley, New York
14. Garza AJ, Jiménez-Hoyos CA, Scuseria GE (2013) J Chem Phys
138:134102
15. Jiménez-Hoyos CA, Rodríguez-Guzmán R, Scuseria GE (2013)
J Chem Phys 139:204102
16. Evangelista FA (2014) J Chem Phys 140:124114
17. Mentel LM, Van Meer R, Gritsenko OV, Baerends EJ (2014) J
Chem Phys 140:214105
18. Ivanov VV, Lyakh DI, Adamowicz L (2005) Mol Phys 103:2131
19. Kullback S (1959) Information theory and statistics. Wiley, New
York
20. Mathai AM, Tathie PN (1975) Basic concepts in information
theory and statistics. Wiley, New York
21. Pfeiffer PE (1978) Concepts of probability theory. Dover, New
York
22. Paldus J, Jeziorski B (1988) Theor Chim Acta 73:81
23. Lain L, Torre A, Karwowski J, Valdemoro C (1988) Phys Rev A
38:2721
24. Torre A, Lain L, Millan J (1993) Phys Rev A 47:923
25. Lain L, Torre A (1995) Phys Rev A 52:2446
26. Surjan PR (1989) Second quantized approach to quantum chemistry. Springer, Berlin
27. Subotnik JE, Shao Y, Liang W, Head-Gordon M (2004) J Chem
Phys 121:9220
28. Jonhson RD III (ed) (2006) Computational chemistry comparison and benchmark database. NIST Standard reference database
vol 101. http://www.srdata.nist.gov/cccbdb
29. Roos JB, Larson M, Larson A, Orel AE (2009) Phys Rev A
80:112501
30. Chakrabarti K, Tennyson J (2012) Eur Phys J 66:31
31. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA,
Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson
GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF,
Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K,
Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao
O, Nakai H, Vreven T, Montgomery JA Jr, Peralta JE, Ogliaro
F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov
VN, Kobayashi R, Normand J, Raghavachari K, Rendell A,
Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam MJ,
Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo
J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R,
Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski
VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels
120
Reprinted from the journal
1 3
distribution at equilibrium distances in the three molecular basis sets. Likewise, the molecular system descriptions
at the stretched geometries systematically present higher
values for the I C and I W indices (Table 1 ) than their counterparts at the equilibrium distances. This effect is interpreted in the framework of the progressive openness of the
chemical bonds until their complete dissociation, which is
refl ected in the values of both indices. However, the index
I C shows a more sensitive character than the I W one, and
consequently, its use must be favored in order to account
for the infl uence of the bond stretching on the wave function features.
5 Concluding remarks and perspectives
In this work, we have extended the formulation of the Shannon entropy indices, informational content ( I C ), cumulative
( I W ), and specifi c -subspace ( I ), within the framework of
the seniority number criterion for constructing N -electron
wave function expansions in terms of Slater determinants.
The quantitative evaluation of these indices has allowed us
to implement analyses of the wave function expansions,
determining their compactness in the well-known canonical
molecular orbital and natural orbital basis sets, as well as
in the recently proposed molecular orbital basis set which
minimizes the seniority number of a given wave function.
The results obtained for several atomic and molecular systems described at the FCI level show the suitability of the
seniority-based formulation of these indices to measure
quantitatively the wave function expansion compactness,
as well as to analyze their multiconfi gurational structure.
We have also studied the ability of these indices to provide
information on the evolution of the wave functions according to the stretching of the chemical bondings. We are currently working in our laboratories on the formulation of
unitary transformations of molecular basis sets leading to
the minimization of the Shannon entropy indices, in order
to achieve a higher improvement on the compactness of
wave function expansions.
Acknowledgments This work has been fi nancially supported
by the Grant Nos. GIU12/09 and UFI11/07 (Universidad del Pais
Vasco, Spain), UBACYT 20020100100197 (Universidad de Buenos Aires, Argentina), PIP No. 11220090100061, 11220090100369,
11220130100377CO, and 11220130100311CO (Consejo Nacional
de Investigaciones Científi cas y Técnicas, Argentina). We thank the
Universidad del País Vasco for allocation of computational resources.
References
1. Shavitt I (1998) Mol Phys 94:3
2. Sherrill CD, Schaefer HF III (1999) Adv Quantum Chem 34:143
and references therein
3. Bytautas L, Ivanic J, Ruedenberg K (2003) J Chem Phys
119:8217
4. Giesbertz KJH (2014) Chem Phys Lett 591:220
5. Bytautas L, Henderson TM, Jiménez-Hoyos CA, Ellis JK, Scuseria GE (2011) J Chem Phys 135:044119
6. Alcoba DR, Torre A, Lain L, Massaccesi GE, Oña OB (2013) J
Chem Phys 139:084103
7. Stein T, Henderson TM, Scuseria GE (2014) J Chem Phys
140:214113
8. Boguslawski K, Tecmer P, Limacher PA, Johnson PA, Ayers PW,
Bultinck P, De Baerdemacker S, Van Neck D (2014) J Chem
Phys 140:214114
9. Alcoba DR, Torre A, Lain L, Massaccesi GE, Oña OB (2014) J
Chem Phys 140:234103
10. Limacher PA, Kim TD, Ayers PW, Johnson PA, De Baerdemacker S, Van Neck D, Bultinck P (2014) Mol Phys
5–6:853
11. Alcoba DR, Torre A, Lain L, Oña OB, Capuzzi P, Van Raemdonck M, Bultinck P, Van Neck D (2014) J Chem Phys
141:244118
12. Ring P, Schuck P (1980) The nuclear many-body problem.
Springe, New York
13. Koltun DS, Eisenberg JM (1988) Quantum mechanics of many
degrees of freedom. Wiley, New York
14. Garza AJ, Jiménez-Hoyos CA, Scuseria GE (2013) J Chem Phys
138:134102
15. Jiménez-Hoyos CA, Rodríguez-Guzmán R, Scuseria GE (2013)
J Chem Phys 139:204102
16. Evangelista FA (2014) J Chem Phys 140:124114
17. Mentel LM, Van Meer R, Gritsenko OV, Baerends EJ (2014) J
Chem Phys 140:214105
18. Ivanov VV, Lyakh DI, Adamowicz L (2005) Mol Phys 103:2131
19. Kullback S (1959) Information theory and statistics. Wiley, New
York
20. Mathai AM, Tathie PN (1975) Basic concepts in information
theory and statistics. Wiley, New York
21. Pfeiffer PE (1978) Concepts of probability theory. Dover, New
York
22. Paldus J, Jeziorski B (1988) Theor Chim Acta 73:81
23. Lain L, Torre A, Karwowski J, Valdemoro C (1988) Phys Rev A
38:2721
24. Torre A, Lain L, Millan J (1993) Phys Rev A 47:923
25. Lain L, Torre A (1995) Phys Rev A 52:2446
26. Surjan PR (1989) Second quantized approach to quantum chemistry. Springer, Berlin
27. Subotnik JE, Shao Y, Liang W, Head-Gordon M (2004) J Chem
Phys 121:9220
28. Jonhson RD III (ed) (2006) Computational chemistry comparison and benchmark database. NIST Standard reference database
vol 101. http://www.srdata.nist.gov/cccbdb
29. Roos JB, Larson M, Larson A, Orel AE (2009) Phys Rev A
80:112501
30. Chakrabarti K, Tennyson J (2012) Eur Phys J 66:31
31. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA,
Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson
GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF,
Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K,
Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao
O, Nakai H, Vreven T, Montgomery JA Jr, Peralta JE, Ogliaro
F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov
VN, Kobayashi R, Normand J, Raghavachari K, Rendell A,
Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam MJ,
Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo
J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R,
Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski
VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels
120
Reprinted from the journal