Theor Chem Acc (2015) 134:151
1 3
In NOF theory [ 31 ], the solution is established optimizing the energy functional with respect to the occupation
numbers and to the natural orbitals, separately. If PNOF5
is employed, the occupancies can be expressed through
auxiliary variables in order to enforce automatically the
N -representability bounds on the 1-RDM, so the variation
can be performed without constraints. A self-consistent
procedure proposed in Ref. [ 32 ] yields the natural orbitals.
This scheme requires computational times that scale as n
4 ,
n being the number of basis functions, like in the Hartree–
Fock (HF) approximation. However, our implementation
in the molecular basis set requires also four-index transformation of the electron repulsion integrals, which is the
time-consuming step though a parallel implementation of
this part of the code has substantially improved the performance of our program. Besides the matrix scaling carried
out in [ 32 ], the direct inversion in the iterative subspace
(DIIS) method has recently been implemented in our code,
but new techniques should be considered to reduce the
number of iterations required to achieve the convergence.
In spite of its promising performance, the computational
cost of PNOF5 calculations prevents them for a wider use.
In this sense, any strategy to reduce calculation times is
highly interesting, as it would expand the range of applicability of NOF theory. The fragmentation technique has
allowed to span the range of systems attainable by wavefunction-based and density-functional theories, by means
of a dividing approach that allows the calculation of the
whole system fragmented in subsystems. In the present
paper, we use such an approach, the so-called thermodynamic fragment energy method (FEM) [ 33 ], to assess the
performance of PNOF5 in the context of fragment energy
calculations.
The aim of this work is to determine how PNOF5-FEM
energies converge to the exact PNOF5 values for selected
oligomers, namely the polyalkene chains C n H 2 n +2 and planar zigzag (FH) n clusters. The size-consistency property,
and the fact that the functional tends to localize spatially
the natural orbitals, makes this functional an exceptional
candidate for fragment calculations. We demonstrate that
the convergence of fragment energy calculations is very
fast, especially in those cases where the interaction between
fragments is small.
2 Methods
2.1 The functional
At the beginning [ 4 ], PNOF5 was formulated as an orbitalpairing approach that involves coupling each orbital g ,
below the Fermi level ( g ≤ F = N /2), with only one orbital
above F ( N g = 1). This model was further improved by
a better description of the electron pair in the so-called
extended PNOF5 [ 5 ], in which each orbital g was coupled with N g > 1 orbitals above F . This pairing condition
is refl ected in the following sum rule for the occupation
numbers:
where p is the running index referring to the spatial part
of natural spin orbitals and n p their occupation numbers.
Notice that for spin-compensated systems the spatial part
of α and β natural spin orbitals are the same, so that the
total occupation number for a given natural spatial orbital
is n occ
p = n α
p + n
β
p = 2 × n p and therefore can take values
between [0, 2].
In Eq. ( 1 ), g is the subspace containing the orbital g
and its N g coupled orbitals. It is worth to note that these
subspaces are mutually disjoint
g 1 ∩ g 2 = ∅
, i.e.,
each orbital belongs only to one subspace g . The PNOF5
energy for a singlet state of an N -electron system can be
cast as
The fi rst term of the energy ( 2 ) draws the system as independent F electron pairs described by
where H pp denotes for the one-particle matrix elements of
the core Hamiltonian. J pq = pq|pq and K pq = 횿pq|qp횿
are the usual direct and exchange integrals, respectively.
L pq = 횿pp|qq횿 is the exchange and time-inversion integral
[ 34 ], which only differs in phases of the natural orbitals
with respect to the exchange integrals, so L pq = K pq for
real orbitals. The interaction energy E int
pq is given by
Accordingly, the last term of Eq. ( 2 ) contains the contribution to the HF mean fi eld of the electrons belonging to
different pairs. It is clear that the weaknesses of this Ansatz
is the absence of the interpair electron correlation. Recently
[ 35 ], a new functional PNOF6, which includes interpair
correlations, has been developed. The latter is able to treat
orbital delocalization in aromatic systems such as in benzene, a key aspect in radical stabilization. In this work, we
study systems in which such effects are not present, so it
seems proper to use the PNOF5 approach. Moreover, we
(1)
p∈ g
n p = 1; g = 1, F
(2)
E =
F
g=1
E g +
F
g 1 =g 2
p∈ g 1
q∈ g 2
E
int
pq
(3)
E g =
p∈ g
n p
2H pp + J pp
+
p,q∈ g ,p =q
n p , n q
L qp
(4)
n p , n q
=
−
√
n p n q , p = g or q = g
√ n p n q , p, q > F
(5)
E
int
pq = n q n p
2J pq − K pq
182
Reprinted from the journal
1 3
In NOF theory [ 31 ], the solution is established optimizing the energy functional with respect to the occupation
numbers and to the natural orbitals, separately. If PNOF5
is employed, the occupancies can be expressed through
auxiliary variables in order to enforce automatically the
N -representability bounds on the 1-RDM, so the variation
can be performed without constraints. A self-consistent
procedure proposed in Ref. [ 32 ] yields the natural orbitals.
This scheme requires computational times that scale as n
4 ,
n being the number of basis functions, like in the Hartree–
Fock (HF) approximation. However, our implementation
in the molecular basis set requires also four-index transformation of the electron repulsion integrals, which is the
time-consuming step though a parallel implementation of
this part of the code has substantially improved the performance of our program. Besides the matrix scaling carried
out in [ 32 ], the direct inversion in the iterative subspace
(DIIS) method has recently been implemented in our code,
but new techniques should be considered to reduce the
number of iterations required to achieve the convergence.
In spite of its promising performance, the computational
cost of PNOF5 calculations prevents them for a wider use.
In this sense, any strategy to reduce calculation times is
highly interesting, as it would expand the range of applicability of NOF theory. The fragmentation technique has
allowed to span the range of systems attainable by wavefunction-based and density-functional theories, by means
of a dividing approach that allows the calculation of the
whole system fragmented in subsystems. In the present
paper, we use such an approach, the so-called thermodynamic fragment energy method (FEM) [ 33 ], to assess the
performance of PNOF5 in the context of fragment energy
calculations.
The aim of this work is to determine how PNOF5-FEM
energies converge to the exact PNOF5 values for selected
oligomers, namely the polyalkene chains C n H 2 n +2 and planar zigzag (FH) n clusters. The size-consistency property,
and the fact that the functional tends to localize spatially
the natural orbitals, makes this functional an exceptional
candidate for fragment calculations. We demonstrate that
the convergence of fragment energy calculations is very
fast, especially in those cases where the interaction between
fragments is small.
2 Methods
2.1 The functional
At the beginning [ 4 ], PNOF5 was formulated as an orbitalpairing approach that involves coupling each orbital g ,
below the Fermi level ( g ≤ F = N /2), with only one orbital
above F ( N g = 1). This model was further improved by
a better description of the electron pair in the so-called
extended PNOF5 [ 5 ], in which each orbital g was coupled with N g > 1 orbitals above F . This pairing condition
is refl ected in the following sum rule for the occupation
numbers:
where p is the running index referring to the spatial part
of natural spin orbitals and n p their occupation numbers.
Notice that for spin-compensated systems the spatial part
of α and β natural spin orbitals are the same, so that the
total occupation number for a given natural spatial orbital
is n occ
p = n α
p + n
β
p = 2 × n p and therefore can take values
between [0, 2].
In Eq. ( 1 ), g is the subspace containing the orbital g
and its N g coupled orbitals. It is worth to note that these
subspaces are mutually disjoint
g 1 ∩ g 2 = ∅
, i.e.,
each orbital belongs only to one subspace g . The PNOF5
energy for a singlet state of an N -electron system can be
cast as
The fi rst term of the energy ( 2 ) draws the system as independent F electron pairs described by
where H pp denotes for the one-particle matrix elements of
the core Hamiltonian. J pq = pq|pq and K pq = 횿pq|qp횿
are the usual direct and exchange integrals, respectively.
L pq = 횿pp|qq횿 is the exchange and time-inversion integral
[ 34 ], which only differs in phases of the natural orbitals
with respect to the exchange integrals, so L pq = K pq for
real orbitals. The interaction energy E int
pq is given by
Accordingly, the last term of Eq. ( 2 ) contains the contribution to the HF mean fi eld of the electrons belonging to
different pairs. It is clear that the weaknesses of this Ansatz
is the absence of the interpair electron correlation. Recently
[ 35 ], a new functional PNOF6, which includes interpair
correlations, has been developed. The latter is able to treat
orbital delocalization in aromatic systems such as in benzene, a key aspect in radical stabilization. In this work, we
study systems in which such effects are not present, so it
seems proper to use the PNOF5 approach. Moreover, we
(1)
p∈ g
n p = 1; g = 1, F
(2)
E =
F
g=1
E g +
F
g 1 =g 2
p∈ g 1
q∈ g 2
E
int
pq
(3)
E g =
p∈ g
n p
2H pp + J pp
+
p,q∈ g ,p =q
n p , n q
L qp
(4)
n p , n q
=
−
√
n p n q , p = g or q = g
√ n p n q , p, q > F
(5)
E
int
pq = n q n p
2J pq − K pq
182
Reprinted from the journal