1 3
Theor Chem Acc (2015) 134:85
DOI 10.1007/s00214-015-1688-5
REGULAR ARTICLE
A study of the compactness of wave functions based on Shannon
entropy indices: a seniority number approach
Luis Lain
1 · Alicia Torre
1 · Diego R. Alcoba
2,3 · Ofelia B. Oña
4 ·
Gustavo E. Massaccesi
5
Received: 16 April 2015 / Accepted: 16 June 2015 / Published online: 4 July 2015
© Springer-Verlag Berlin Heidelberg 2015
function expansions, and to study their multiconfi gurational character.
Keywords Compactness of wave functions · Seniority
number · Shannon entropy indices · Optimization of
molecular orbital basis sets
1 Introduction
The confi guration interaction (CI) methods have played an
important role in describing N -electron systems, since they
demand a lower computational cost than that required for
determining the full confi guration interaction (FCI) expansions which provide the exact descriptions for a given Hilbert space. Consequently, there has been a considerable
interest in formulating N -electron wave functions in terms
of CI expansions providing a rapid convergence to the
FCI ones [ 1 – 4 ]. Traditionally, the CI wave functions have
been expanded by means of N -electron Slater determinants
selected according to their excitation degrees with respect
to a given reference determinant. However, more recently,
another selection criterion has also been proposed. This criterion is based on the seniority number of the Slater determinants used to construct the CI expansions [ 5 – 11 ]. Results
arising from both excitation- and seniority number-based
CI schemes show that the seniority number-based selection
procedure is particularly suitable to describe systems exhibiting strong (static) correlation [ 5 ], what has increased the
interest on this approach [ 9 – 11 ]. As is well known, the seniority number of a Slater determinant is defi ned as the number of singly occupied orbitals which possesses that determinant [ 12 , 13 ]. The seniority number concept has been
extended in Refs. [ 6 , 9 ] to N -electron wave functions which
describe electronic states of atomic and molecular systems,
Abstract This work reports the formulation of Shannon entropy indices in terms of seniority numbers of the
Slater determinants expanding an N -electron wave function. Numerical determinations of those indices prove that
they provide a suitable quantitative procedure to evaluate
compactness of wave functions and to describe their confi gurational structures. An analysis of the results, calculated
for full confi guration interaction wave functions in selected
atomic and molecular systems, allows one to compare
and to discuss the behavior of several types of molecular
orbital basis sets in order to achieve more compact wave
Dedicated to Prof. P. R. Surjan on occasion of his 60th birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Luis Lain
qfplapel@lg.ehu.es
1
Departamento de Química Física, Facultad de Ciencia
y Tecnología , Universidad del País Vasco , Apdo. 644 ,
48080 Bilbao , Spain
2
Departamento de Física, Facultad de Ciencias Exactas
y Naturales , Universidad de Buenos Aires , Ciudad
Universitaria , 1428 Buenos Aires , Argentina
3
Instituto de Física de Buenos Aires , Consejo Nacional de
Investigaciones Científi cas y Técnicas , Ciudad Universitaria ,
1428 Buenos Aires , Argentina
4
Instituto de Investigaciones Fisicoquímicas Teóricas y
Aplicadas , Universidad Nacional de la Plata, CCT La Plata,
Consejo Nacional de Investigaciones Científi cas y Técnicas ,
Diag. 113 y 64 (s/n), Sucursal 4, CC 16 , 1900 La Plata ,
Argentina
5
Departamento de Ciencias Exactas, Ciclo Básico Común ,
Universidad de Buenos Aires , Ciudad Universitaria ,
1428 Buenos Aires , Argentina
115
Reprinted from the journal
Theor Chem Acc (2015) 134:85
DOI 10.1007/s00214-015-1688-5
REGULAR ARTICLE
A study of the compactness of wave functions based on Shannon
entropy indices: a seniority number approach
Luis Lain
1 · Alicia Torre
1 · Diego R. Alcoba
2,3 · Ofelia B. Oña
4 ·
Gustavo E. Massaccesi
5
Received: 16 April 2015 / Accepted: 16 June 2015 / Published online: 4 July 2015
© Springer-Verlag Berlin Heidelberg 2015
function expansions, and to study their multiconfi gurational character.
Keywords Compactness of wave functions · Seniority
number · Shannon entropy indices · Optimization of
molecular orbital basis sets
1 Introduction
The confi guration interaction (CI) methods have played an
important role in describing N -electron systems, since they
demand a lower computational cost than that required for
determining the full confi guration interaction (FCI) expansions which provide the exact descriptions for a given Hilbert space. Consequently, there has been a considerable
interest in formulating N -electron wave functions in terms
of CI expansions providing a rapid convergence to the
FCI ones [ 1 – 4 ]. Traditionally, the CI wave functions have
been expanded by means of N -electron Slater determinants
selected according to their excitation degrees with respect
to a given reference determinant. However, more recently,
another selection criterion has also been proposed. This criterion is based on the seniority number of the Slater determinants used to construct the CI expansions [ 5 – 11 ]. Results
arising from both excitation- and seniority number-based
CI schemes show that the seniority number-based selection
procedure is particularly suitable to describe systems exhibiting strong (static) correlation [ 5 ], what has increased the
interest on this approach [ 9 – 11 ]. As is well known, the seniority number of a Slater determinant is defi ned as the number of singly occupied orbitals which possesses that determinant [ 12 , 13 ]. The seniority number concept has been
extended in Refs. [ 6 , 9 ] to N -electron wave functions which
describe electronic states of atomic and molecular systems,
Abstract This work reports the formulation of Shannon entropy indices in terms of seniority numbers of the
Slater determinants expanding an N -electron wave function. Numerical determinations of those indices prove that
they provide a suitable quantitative procedure to evaluate
compactness of wave functions and to describe their confi gurational structures. An analysis of the results, calculated
for full confi guration interaction wave functions in selected
atomic and molecular systems, allows one to compare
and to discuss the behavior of several types of molecular
orbital basis sets in order to achieve more compact wave
Dedicated to Prof. P. R. Surjan on occasion of his 60th birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Luis Lain
qfplapel@lg.ehu.es
1
Departamento de Química Física, Facultad de Ciencia
y Tecnología , Universidad del País Vasco , Apdo. 644 ,
48080 Bilbao , Spain
2
Departamento de Física, Facultad de Ciencias Exactas
y Naturales , Universidad de Buenos Aires , Ciudad
Universitaria , 1428 Buenos Aires , Argentina
3
Instituto de Física de Buenos Aires , Consejo Nacional de
Investigaciones Científi cas y Técnicas , Ciudad Universitaria ,
1428 Buenos Aires , Argentina
4
Instituto de Investigaciones Fisicoquímicas Teóricas y
Aplicadas , Universidad Nacional de la Plata, CCT La Plata,
Consejo Nacional de Investigaciones Científi cas y Técnicas ,
Diag. 113 y 64 (s/n), Sucursal 4, CC 16 , 1900 La Plata ,
Argentina
5
Departamento de Ciencias Exactas, Ciclo Básico Común ,
Universidad de Buenos Aires , Ciudad Universitaria ,
1428 Buenos Aires , Argentina
115
Reprinted from the journal