90
E.J. Brändas
tal particles are fermions like electrons or protons), subject to the Liouville equation
( briefly omitted)
i
∂ρ
∂t
= ˆ
Lρ
(4.7)
where the Liouvillian is defined as usual from the Hamiltonian H describing the
system, e.g. the molecular configuration under investigation, i.e.
ˆ
L = Hρ − ρH
† .
(4.8)
Normally H is self-adjoint, but to carry out analytic continuation a “dagger” is inserted in (4.8). Since the microscopic system might be very complex, not to mention
depicting biological structures and organization, it is convenient to start by introducing the N th particle fermionic (and its q-reduced) representable density matrix Γ (q)
as follows
Γ
(q)
x 1 · · · x q |x
1 · · · x
q
=
N
q
Ψ (x 1 · · · x q , x q+1 · · · x N )Ψ
∗
x
1 · · · x
q , x q+1 · · · x N
dx q+1 · · · dx N
(4.9)
defined in terms of the many body (normalized) wave function Ψ (here Γ represents
a pure state, with obvious extensions for an ensemble) in Eq. (4.9). In particular we
will discuss the 2-particle reduced density matrix below, since it is of fundamental
importance in connection with the all-embracing electronic correlation problem in
ab initio quantum chemistry, primarily in connection with strongly correlated structures [16, 17], like e.g. high-T C cuprates [18]. In the so-called extreme case [17] the
density matrix takes on a very simple form. For instance defining an m-dimensional
preferred real localized basis |h of geminals (two-particle functions) on suitable
sites of correct symmetry, one can show that the finite dimensional representation
essentially writes [13, 19]
Γ
(2)
= ρ =
m
k,l
|h k ρ kl h l |;
Tr{ρ} =
N
2
ρ kk = p;
ρ kl = p(1 − p);
k = l; p =
N
2m
.
(4.10)
In Eq. (4.10) N is the number of fermions or N/2 (quasi-)bosonic pairs and p is
“the probability to find” a pair in the state m. The number of possible states must
fulfill m ≥ N/2. The associated secular equation renders a non-degenerate large
eigenvalue λ L = mp − (m − 1)p 2 and a small (m − 1)-degenerate λ S = p 2 . Note
that for large m and p small (m N ), λ L ≈ N/2. Hence the density operator writes
Γ
(2)
= ρ = λ L |g 1 g 1 | + λ S
m
k,l=1
|h k
δ kl −
1
m
h l |.
(4.11)
E.J. Brändas
tal particles are fermions like electrons or protons), subject to the Liouville equation
( briefly omitted)
i
∂ρ
∂t
= ˆ
Lρ
(4.7)
where the Liouvillian is defined as usual from the Hamiltonian H describing the
system, e.g. the molecular configuration under investigation, i.e.
ˆ
L = Hρ − ρH
† .
(4.8)
Normally H is self-adjoint, but to carry out analytic continuation a “dagger” is inserted in (4.8). Since the microscopic system might be very complex, not to mention
depicting biological structures and organization, it is convenient to start by introducing the N th particle fermionic (and its q-reduced) representable density matrix Γ (q)
as follows
Γ
(q)
x 1 · · · x q |x
1 · · · x
q
=
N
q
Ψ (x 1 · · · x q , x q+1 · · · x N )Ψ
∗
x
1 · · · x
q , x q+1 · · · x N
dx q+1 · · · dx N
(4.9)
defined in terms of the many body (normalized) wave function Ψ (here Γ represents
a pure state, with obvious extensions for an ensemble) in Eq. (4.9). In particular we
will discuss the 2-particle reduced density matrix below, since it is of fundamental
importance in connection with the all-embracing electronic correlation problem in
ab initio quantum chemistry, primarily in connection with strongly correlated structures [16, 17], like e.g. high-T C cuprates [18]. In the so-called extreme case [17] the
density matrix takes on a very simple form. For instance defining an m-dimensional
preferred real localized basis |h of geminals (two-particle functions) on suitable
sites of correct symmetry, one can show that the finite dimensional representation
essentially writes [13, 19]
Γ
(2)
= ρ =
m
k,l
|h k ρ kl h l |;
Tr{ρ} =
N
2
ρ kk = p;
ρ kl = p(1 − p);
k = l; p =
N
2m
.
(4.10)
In Eq. (4.10) N is the number of fermions or N/2 (quasi-)bosonic pairs and p is
“the probability to find” a pair in the state m. The number of possible states must
fulfill m ≥ N/2. The associated secular equation renders a non-degenerate large
eigenvalue λ L = mp − (m − 1)p 2 and a small (m − 1)-degenerate λ S = p 2 . Note
that for large m and p small (m N ), λ L ≈ N/2. Hence the density operator writes
Γ
(2)
= ρ = λ L |g 1 g 1 | + λ S
m
k,l=1
|h k
δ kl −
1
m
h l |.
(4.11)