Theor Chem Acc (2015) 134:85
1 3
as well as to N -electron spin-adapted Hilbert spaces. The
expectation value of the seniority number operator with
respect to an N -electron wave function is a weighted sum
of the seniority numbers of all determinants involved in the
expansion of that wave function. The weights that determine those expectation values depend on the molecular
orbital basis set used to express the wave function, and
consequently, this feature has been utilized to evaluate
the compactness of the FCI and CI expansions in several
molecular basis sets. Likewise, the seniority number value
with respect to a wave function allows one to analyze the
multiconfi gurational character of the N -electron expansion,
which is useful to describe the static and dynamic correlation of a determined state [ 14 – 17 ].
On the other hand, in Ref. [ 18 ], the extent of the multiconfi gurational character of an N -electron wave function
was evaluated by means of numerical determinations of
an index set formulated within the Shannon information
entropy approach [ 19 – 21 ]. This treatment provides a suitable information concerning the distribution of the wave
function among different confi gurations characterized by
the excitation degree of the Slater determinants. The aim
of this work is to extend this methodology to the senioritybased CI scheme and to report the corresponding Shannon
index numerical values in terms of the contributions of Slater
determinants classifi ed according to the seniority number criterion. More recently, in Ref. [ 6 ], we have proposed unitary
transformations which lead to the construction of basis sets
of molecular orbitals in which the expectation values of the
seniority number operator with respect to N -electron wave
functions reach minimum values. The results found using
this type of molecular orbitals show that the wave function
expansions present a more rapid convergence than those arising from the use of other molecular orbitals [ 9 , 11 ]. Another
aim of this work is to evaluate and compare quantitatively,
by means of the proposed Shannon entropy indices, the compactness of wave functions expressed in canonical molecular
orbital (CMO) basis sets, natural orbitals (NO), and those
mentioned orbitals M min , which minimize the expectation
value of the seniority number operator.
This article has been organized as follows. Section 2
summarizes the notation and formulation of the main concepts used in this work; it also reports the formulation
of the Shannon entropy indices in terms of the seniority
numbers of the Slater determinants. In Sect. 3 , we present
numerical values of those indices for wave functions of
selected atomic and molecular systems; these values allow
one to characterize the compactness of the wave function
expansions. The calculation level and the computational
details are also indicated in this section. An analysis and
discussion of these results are reported in Sect. 4 . Finally,
in the last section, we highlight the main conclusions and
perspectives of this work.
2 Theoretical framework
The K orbitals of an orthonormal basis set will be denoted
by i, j, k, l, . . . and their corresponding spin-orbitals by
i σ , j σ , . . . ( σ and σ mean the spin coordinates α or β ). The
spin-free version of the N -electron seniority number operator ˆ
has been formulated as [ 6 , 9 , 11 ]
where ˆ
E i
i =
σ a
†
i σ a i σ and ˆ
E ii
ii =
σ ,σ a
†
i σ a
†
i σ a i σ a i σ are
the spin-free fi rst- and second-order replacement operators,
respectively [ 22 – 25 ] and a
†
i σ and a i σ are the usual creation
and annihilation fermion operators [ 26 ].
Closing both sides of Eq. ( 1 ) by an N -electron Slater
determinant of S z spin projection quantum number, one
obtains
where, according to Eq. ( 1 ), the expectation value is the
difference
i E i
i −
i E ii
ii , which is the number of total
electrons N minus the number of electrons corresponding
to doubly occupied orbitals. The possible values for the
parameter are positive integers belonging to the sequence
= 2|S z |, 2(|S z | + 1), . . . , max (where max = N if
K ≥ N and max = 2K − N if K < N ); their meaning is the
number of non-repeated orbitals in each determinant. That
parameter allows one to classify the Slater determinants
of S z quantum number, according to the corresponding
seniority level, and they will be denoted hereafter by () .
Consequently, a FCI N -electron wave function with given
spin quantum numbers S and S z will be expressed by
where C (() stands for the coeffi cient corresponding to the Slater determinant () . Obviously, since
there is no contribution of Slater determinants with
< 2S to spin-adapted N -electron wave functions
( (( < 2S)|(N, S, S z ) = 0 ), the lowest integer in the
sum
is 2 S . If we truncate the series
in Eq. ( 3 ),
we obtain CI( ) wave function expansions involving only
Slater determinants belonging to the selected levels.
According to Eqs. ( 1 ) and ( 3 ), the expectation value
of the operator ˆ
with respect to the FCI wave function
(N, S, S z ) is [ 6 ]
(1)
ˆ
=
K
i=1
ˆ
E
i
i − ˆ
E
ii
ii
(2)
= =| ˆ
|
(3)
|(N, S, S z ) > =
max
=2S
()
C (() |(() >
(4)
ˆ
(N,S) = N −
max
=2S
()
C (()
2
K
i
(()
ˆ
E ii
ii
(()
116
Reprinted from the journal
1 3
as well as to N -electron spin-adapted Hilbert spaces. The
expectation value of the seniority number operator with
respect to an N -electron wave function is a weighted sum
of the seniority numbers of all determinants involved in the
expansion of that wave function. The weights that determine those expectation values depend on the molecular
orbital basis set used to express the wave function, and
consequently, this feature has been utilized to evaluate
the compactness of the FCI and CI expansions in several
molecular basis sets. Likewise, the seniority number value
with respect to a wave function allows one to analyze the
multiconfi gurational character of the N -electron expansion,
which is useful to describe the static and dynamic correlation of a determined state [ 14 – 17 ].
On the other hand, in Ref. [ 18 ], the extent of the multiconfi gurational character of an N -electron wave function
was evaluated by means of numerical determinations of
an index set formulated within the Shannon information
entropy approach [ 19 – 21 ]. This treatment provides a suitable information concerning the distribution of the wave
function among different confi gurations characterized by
the excitation degree of the Slater determinants. The aim
of this work is to extend this methodology to the senioritybased CI scheme and to report the corresponding Shannon
index numerical values in terms of the contributions of Slater
determinants classifi ed according to the seniority number criterion. More recently, in Ref. [ 6 ], we have proposed unitary
transformations which lead to the construction of basis sets
of molecular orbitals in which the expectation values of the
seniority number operator with respect to N -electron wave
functions reach minimum values. The results found using
this type of molecular orbitals show that the wave function
expansions present a more rapid convergence than those arising from the use of other molecular orbitals [ 9 , 11 ]. Another
aim of this work is to evaluate and compare quantitatively,
by means of the proposed Shannon entropy indices, the compactness of wave functions expressed in canonical molecular
orbital (CMO) basis sets, natural orbitals (NO), and those
mentioned orbitals M min , which minimize the expectation
value of the seniority number operator.
This article has been organized as follows. Section 2
summarizes the notation and formulation of the main concepts used in this work; it also reports the formulation
of the Shannon entropy indices in terms of the seniority
numbers of the Slater determinants. In Sect. 3 , we present
numerical values of those indices for wave functions of
selected atomic and molecular systems; these values allow
one to characterize the compactness of the wave function
expansions. The calculation level and the computational
details are also indicated in this section. An analysis and
discussion of these results are reported in Sect. 4 . Finally,
in the last section, we highlight the main conclusions and
perspectives of this work.
2 Theoretical framework
The K orbitals of an orthonormal basis set will be denoted
by i, j, k, l, . . . and their corresponding spin-orbitals by
i σ , j σ , . . . ( σ and σ mean the spin coordinates α or β ). The
spin-free version of the N -electron seniority number operator ˆ
has been formulated as [ 6 , 9 , 11 ]
where ˆ
E i
i =
σ a
†
i σ a i σ and ˆ
E ii
ii =
σ ,σ a
†
i σ a
†
i σ a i σ a i σ are
the spin-free fi rst- and second-order replacement operators,
respectively [ 22 – 25 ] and a
†
i σ and a i σ are the usual creation
and annihilation fermion operators [ 26 ].
Closing both sides of Eq. ( 1 ) by an N -electron Slater
determinant of S z spin projection quantum number, one
obtains
where, according to Eq. ( 1 ), the expectation value is the
difference
i E i
i −
i E ii
ii , which is the number of total
electrons N minus the number of electrons corresponding
to doubly occupied orbitals. The possible values for the
parameter are positive integers belonging to the sequence
= 2|S z |, 2(|S z | + 1), . . . , max (where max = N if
K ≥ N and max = 2K − N if K < N ); their meaning is the
number of non-repeated orbitals in each determinant. That
parameter allows one to classify the Slater determinants
of S z quantum number, according to the corresponding
seniority level, and they will be denoted hereafter by () .
Consequently, a FCI N -electron wave function with given
spin quantum numbers S and S z will be expressed by
where C (() stands for the coeffi cient corresponding to the Slater determinant () . Obviously, since
there is no contribution of Slater determinants with
< 2S to spin-adapted N -electron wave functions
( (( < 2S)|(N, S, S z ) = 0 ), the lowest integer in the
sum
is 2 S . If we truncate the series
in Eq. ( 3 ),
we obtain CI( ) wave function expansions involving only
Slater determinants belonging to the selected levels.
According to Eqs. ( 1 ) and ( 3 ), the expectation value
of the operator ˆ
with respect to the FCI wave function
(N, S, S z ) is [ 6 ]
(1)
ˆ
=
K
i=1
ˆ
E
i
i − ˆ
E
ii
ii
(2)
= =| ˆ
|
(3)
|(N, S, S z ) > =
max
=2S
()
C (() |(() >
(4)
ˆ
(N,S) = N −
max
=2S
()
C (()
2
K
i
(()
ˆ
E ii
ii
(()
116
Reprinted from the journal