New Strategies in Modeling Electronic Structures and Properties …
145
s i = −
4
α=1
ω α,i ln ω α,i ,
(52)
where ω α,i are the eigenvalues of the ith orbital RDM and the sum runs over all
four possible occupations of a spatial orbitals. The single-orbital entropy reaches a
maximum value of ln(4).
The entanglement entropy between an orbital pair i j and the remaining orbitals
is obtained in a similar way. For our second case, the environment states are defined
as
e
=
k 1 , k 2 , . . . , k i−1 , k i+1 , . . . , k j−1 , k j+1 , . . . , k L
, while the quantum state for
this orbital partitioning reads
Ψ
i, j,e
el
=
k 1 ,k 2 ,...,k L
˜
c k 1 ,k 2 ,...,k L
k i , k j
⊗
e
.
(53)
The matrix elements of the two-orbital RDM are determined in a similar way and
have the elements
ρ (i, j),(i , j ) =
e
e
k i , k j
Ψ
i, j,e
el
Ψ
i, j,e
el
k i , k j
e
.
(54)
The indices i and j encode all possible occupations of orbitals i and j in the twoorbital Fock space that is spanned by 16 states (for spatial orbitals): (− −), (↑ −),
(↓ −), (− ↑), . . . , (↑↓↑↓). Thus, the two-orbital RDM can be expressed as a 16×16
matrix and determines the two-orbital entropy s i, j [18]. Specifically, the two-orbital
entropy quantifies the entanglement between the environment orbitals and a particular
orbital pair i j and is given by
s i, j = −
16
α=1
ω α,i, j ln ω α,i, j
(55)
where ω α,i, j are the eigenvalues of the two-orbital RDM. Given the one- and twoorbital RDMs, we can calculate the so-called mutual information between any orbital
pair i j,
I i| j = s i + s j − s i, j .
(56)
Most importantly, the mutual information is a measure of correlation and describes
both classical and quantum correlations. I i| j takes values in the range of [0, ln 16],
where 0 is obtained for uncorrelated wave functions such as a single Slater (or the
HF) determinant. We should note that evaluating the one- and two-orbital RDMs
using the general (51) and (54) might be cumbersome due to the phase factors that
have to be accounted for in (50) and (53). For practical calculations, the one- and twoorbital RDMs can be expressed in terms of conventional N -particle reduced density
matrices [15, 16]. Specifically, ρ i,i requires only the 1- and 2-particle RDMs, while
ρ (i, j),(i , j ) requires in addition some elements of the 3- and 4-particle RDMs. In
145
s i = −
4
α=1
ω α,i ln ω α,i ,
(52)
where ω α,i are the eigenvalues of the ith orbital RDM and the sum runs over all
four possible occupations of a spatial orbitals. The single-orbital entropy reaches a
maximum value of ln(4).
The entanglement entropy between an orbital pair i j and the remaining orbitals
is obtained in a similar way. For our second case, the environment states are defined
as
e
=
k 1 , k 2 , . . . , k i−1 , k i+1 , . . . , k j−1 , k j+1 , . . . , k L
, while the quantum state for
this orbital partitioning reads
Ψ
i, j,e
el
=
k 1 ,k 2 ,...,k L
˜
c k 1 ,k 2 ,...,k L
k i , k j
⊗
e
.
(53)
The matrix elements of the two-orbital RDM are determined in a similar way and
have the elements
ρ (i, j),(i , j ) =
e
e
k i , k j
Ψ
i, j,e
el
Ψ
i, j,e
el
k i , k j
e
.
(54)
The indices i and j encode all possible occupations of orbitals i and j in the twoorbital Fock space that is spanned by 16 states (for spatial orbitals): (− −), (↑ −),
(↓ −), (− ↑), . . . , (↑↓↑↓). Thus, the two-orbital RDM can be expressed as a 16×16
matrix and determines the two-orbital entropy s i, j [18]. Specifically, the two-orbital
entropy quantifies the entanglement between the environment orbitals and a particular
orbital pair i j and is given by
s i, j = −
16
α=1
ω α,i, j ln ω α,i, j
(55)
where ω α,i, j are the eigenvalues of the two-orbital RDM. Given the one- and twoorbital RDMs, we can calculate the so-called mutual information between any orbital
pair i j,
I i| j = s i + s j − s i, j .
(56)
Most importantly, the mutual information is a measure of correlation and describes
both classical and quantum correlations. I i| j takes values in the range of [0, ln 16],
where 0 is obtained for uncorrelated wave functions such as a single Slater (or the
HF) determinant. We should note that evaluating the one- and two-orbital RDMs
using the general (51) and (54) might be cumbersome due to the phase factors that
have to be accounted for in (50) and (53). For practical calculations, the one- and twoorbital RDMs can be expressed in terms of conventional N -particle reduced density
matrices [15, 16]. Specifically, ρ i,i requires only the 1- and 2-particle RDMs, while
ρ (i, j),(i , j ) requires in addition some elements of the 3- and 4-particle RDMs. In