1 3
Theor Chem Acc (2015) 134:151
DOI 10.1007/s00214-015-1756-x
REGULAR ARTICLE
PNOF5 calculations based on the “thermodynamic fragment
energy method”: C n H 2 n +2 ( n = 1, 10) and (FH) n ( n = 1, 8) as test
cases
Xabier Lopez
1,2 · Mario Piris
1,2,3
Received: 28 April 2015 / Accepted: 30 October 2015 / Published online: 20 November 2015
© The Author(s) 2015 . This article is published with open access at Springerlink.com
be comparable to accurate quantum chemistry methods in
many cases [ 4 – 18 ]. So far this is the only natural orbital
functional (NOF) that has been obtained by top-down
and bottom-up methods [ 19 ]. In the bottom-up method,
the functional was generated by progressive inclusion
of known necessary N -representability conditions on the
2-RDM, whereas the top-down method was used through
reducing the energy expression generated from an N -particle wavefunction to a functional of the occupation numbers
and natural orbitals [ 5 ]. In the case of PNOF5, this wavefunction is an antisymmetrized product of strongly orthogonal geminals (APSGs), with the expansion coeffi cients
explicitly expressed by the occupation numbers [ 20 , 21 ].
The idea of a function of this type dates back to the early
fi fties [ 22 , 23 ] and is inspired by the valence bond theory
[ 24 , 25 ]; in fact, PNOF5 can also be considered as a type of
GVB-PP method with fi xed signs for the expansion coeffi cients of the corresponding determinants. Many scientists
have worked actively in the fi eld of strongly orthogonal
geminals, and one of them is Professor Péter R. Surján to
whom is dedicated this Festschrift. Indeed, an excellent
review summarizing the evolution of the geminal theory
up to 1999 can be found at his work [ 26 ]. An overview of
geminal-based perturbative techniques for describing electron correlation was given recently [ 27 ].
Consequently, PNOF5 is an orbital-pairing approach
that takes into account most of the non-dynamical effects,
but also an important part of the dynamical electron correlation corresponding to the intrapair (intrageminal) interactions. The existence of a generating wavefunction confi rms
that PNOF5 is strictly N -representable, i.e., the 2-RDM is
derived from a function that is antisymmetric in N -particles
[ 28 ]. Moreover, it demonstrates the size extensivity and
size consistency of PNOF5, which is an inherent property
to the generating singlet-type APSG wavefunction [ 29 , 30 ].
Abstract The performance of the “thermodynamic fragment energy method” (FEM) in the context of natural
orbital functional theory (NOFT) in its PNOF5 implementation is assessed. Two test cases are considered: the linear
chains C n H 2 n +2 ( n = 1, 10) and the hydrogen-bonded (FH) n
( n = 1, 8) clusters. Calculations show a fast convergence
of the PNOF5-FEM method, which allows the treatment of
extended system at a fractional cost of the whole calculation. We show that this type of methodologies could expand
the range of systems achievable by NOFT due to the signifi cant reduction in the computational cost.
Keywords Natural orbital functional theory · Fragment
energy method · APSG wavefunction
1 Introduction
In the last decade, a series of functionals has been developed [ 1 , 2 ] using a reconstruction proposed by Piris [ 3 ] of
the two-particle reduced density matrix (2-RDM) in terms
of the one-particle RDM (1-RDM). In particular, the Piris
natural orbital functional 5 (PNOF5) [ 4 , 5 ] has proved to
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Xabier Lopez
xabier.lopez@ehu.es
1
Kimika Fakultatea , Euskal Herriko Unibertsitatea UPV/EHU ,
1072 , 20080 Donostia , Euskadi, Spain
2
Donostia International Physics Center (DIPC) ,
20018 Donostia , Euskadi, Spain
3
IKERBASQUE , Basque Foundation for Science ,
48013 Bilbao , Euskadi, Spain
181
Reprinted from the journal
Theor Chem Acc (2015) 134:151
DOI 10.1007/s00214-015-1756-x
REGULAR ARTICLE
PNOF5 calculations based on the “thermodynamic fragment
energy method”: C n H 2 n +2 ( n = 1, 10) and (FH) n ( n = 1, 8) as test
cases
Xabier Lopez
1,2 · Mario Piris
1,2,3
Received: 28 April 2015 / Accepted: 30 October 2015 / Published online: 20 November 2015
© The Author(s) 2015 . This article is published with open access at Springerlink.com
be comparable to accurate quantum chemistry methods in
many cases [ 4 – 18 ]. So far this is the only natural orbital
functional (NOF) that has been obtained by top-down
and bottom-up methods [ 19 ]. In the bottom-up method,
the functional was generated by progressive inclusion
of known necessary N -representability conditions on the
2-RDM, whereas the top-down method was used through
reducing the energy expression generated from an N -particle wavefunction to a functional of the occupation numbers
and natural orbitals [ 5 ]. In the case of PNOF5, this wavefunction is an antisymmetrized product of strongly orthogonal geminals (APSGs), with the expansion coeffi cients
explicitly expressed by the occupation numbers [ 20 , 21 ].
The idea of a function of this type dates back to the early
fi fties [ 22 , 23 ] and is inspired by the valence bond theory
[ 24 , 25 ]; in fact, PNOF5 can also be considered as a type of
GVB-PP method with fi xed signs for the expansion coeffi cients of the corresponding determinants. Many scientists
have worked actively in the fi eld of strongly orthogonal
geminals, and one of them is Professor Péter R. Surján to
whom is dedicated this Festschrift. Indeed, an excellent
review summarizing the evolution of the geminal theory
up to 1999 can be found at his work [ 26 ]. An overview of
geminal-based perturbative techniques for describing electron correlation was given recently [ 27 ].
Consequently, PNOF5 is an orbital-pairing approach
that takes into account most of the non-dynamical effects,
but also an important part of the dynamical electron correlation corresponding to the intrapair (intrageminal) interactions. The existence of a generating wavefunction confi rms
that PNOF5 is strictly N -representable, i.e., the 2-RDM is
derived from a function that is antisymmetric in N -particles
[ 28 ]. Moreover, it demonstrates the size extensivity and
size consistency of PNOF5, which is an inherent property
to the generating singlet-type APSG wavefunction [ 29 , 30 ].
Abstract The performance of the “thermodynamic fragment energy method” (FEM) in the context of natural
orbital functional theory (NOFT) in its PNOF5 implementation is assessed. Two test cases are considered: the linear
chains C n H 2 n +2 ( n = 1, 10) and the hydrogen-bonded (FH) n
( n = 1, 8) clusters. Calculations show a fast convergence
of the PNOF5-FEM method, which allows the treatment of
extended system at a fractional cost of the whole calculation. We show that this type of methodologies could expand
the range of systems achievable by NOFT due to the signifi cant reduction in the computational cost.
Keywords Natural orbital functional theory · Fragment
energy method · APSG wavefunction
1 Introduction
In the last decade, a series of functionals has been developed [ 1 , 2 ] using a reconstruction proposed by Piris [ 3 ] of
the two-particle reduced density matrix (2-RDM) in terms
of the one-particle RDM (1-RDM). In particular, the Piris
natural orbital functional 5 (PNOF5) [ 4 , 5 ] has proved to
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Xabier Lopez
xabier.lopez@ehu.es
1
Kimika Fakultatea , Euskal Herriko Unibertsitatea UPV/EHU ,
1072 , 20080 Donostia , Euskadi, Spain
2
Donostia International Physics Center (DIPC) ,
20018 Donostia , Euskadi, Spain
3
IKERBASQUE , Basque Foundation for Science ,
48013 Bilbao , Euskadi, Spain
181
Reprinted from the journal