1 3
Theor Chem Acc (2015) 134:148
DOI 10.1007/s00214-015-1751-2
REGULAR ARTICLE
Local random phase approximation with projected oscillator
orbitals
Bastien Mussard
1,2,3 · János G. Ángyán
4,5
Received: 31 July 2015 / Accepted: 22 October 2015 / Published online: 14 November 2015
© Springer-Verlag Berlin Heidelberg 2015
wave functions obtained by a range-separated hybrid
method reproduce experimental values with < 15 % error.
Keywords RPA · Oscillator orbitals · London dispersion
energy · Dispersion coeffi cient · Local correlation method
1 Introduction
According to the Perdew’s popular classifi cation of density
functional approximations (DFA) [ 1 ], the random phase
approximation (RPA) is situated on the highest, fi fth rung
of Jacob’s ladder, which leads from the simplest Hartree
level toward the “heaven” corresponding to the exact solution of the Schrödinger equation. When stepping upwards
on Jacob’s ladder, one uses more and more ingredients of
the Kohn–Sham single determinant. Starting from the lowest rungs, the density, its gradient, the full set of occupied
orbitals are successively necessary for the construction of
the functional. At the highest rung DFA is usually based on
many-body methods, which require the knowledge of the
complete set of occupied and virtual orbitals. Such methods have the drawback that the size of the virtual orbital
space can be very large even in a moderately sized atomic
orbital basis. In the case of plane wave calculations, the virtual space can become even prohibitively large. One solution to keep the size of matrices in reasonable limits makes
recourse to an auxiliary basis set to expand the occupiedvirtual product functions. Such approaches are known in
quantum chemistry as resolution of identity [ 2 ] or densityfi tting [ 3 , 4 ] methods. Similar advantages can be achieved
by Cholesky decomposition [ 5 ] techniques. In plane wave
calculations, the plane wave basis itself can be used to
expand the product states [ 6 ]. Further gain can be achieved
by projection methods, which avoid any explicit reference
Abstract An approximation to the many-body London dispersion energy in molecular systems is expressed
as a functional of the occupied orbitals only. The method
is based on the local-RPA theory. The occupied orbitals are localized molecular orbitals, and the virtual space
is described by projected oscillator orbitals, i.e., functions
obtained by multiplying occupied localized orbitals with
solid spherical harmonic polynomials having their origin at
the orbital centroids. Since we are interested in the longrange part of the correlation energy, responsible for dispersion forces, the electron repulsion is approximated by its
multipolar expansion. This procedure leads to a fully nonempirical long-range correlation energy expression. Molecular dispersion coeffi cients calculated from determinant
Dedicated to Prof. Péter Surján on the occasion of his 60th
birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* János G. Ángyán
janos.angyan@univ-lorraine.fr
Bastien Mussard
bastien.mussard@upmc.fr
1
Institut du Calcul et de la Simulation , Sorbonne Universités,
UPMC Univ Paris 06 , 75005 Paris , France
2
UMR 7616, Laboratoire de Chimie Théorique , Sorbonne
Universités, UPMC Univ Paris 06 , 75005 Paris , France
3
CNRS, UMR 7616, Laboratoire de Chimie Théorique ,
75005 Paris , France
4
Institut Jean Barriol, CRM2, UMR 7036 , Université de
Lorraine , 54506 Vandoeuvre-lès-Nancy , France
5
CNRS, Institut Jean Barriol, CRM2, UMR 7036 ,
54506 Vandoeuvre-lès-Nancy , France
99
Reprinted from the journal
Theor Chem Acc (2015) 134:148
DOI 10.1007/s00214-015-1751-2
REGULAR ARTICLE
Local random phase approximation with projected oscillator
orbitals
Bastien Mussard
1,2,3 · János G. Ángyán
4,5
Received: 31 July 2015 / Accepted: 22 October 2015 / Published online: 14 November 2015
© Springer-Verlag Berlin Heidelberg 2015
wave functions obtained by a range-separated hybrid
method reproduce experimental values with < 15 % error.
Keywords RPA · Oscillator orbitals · London dispersion
energy · Dispersion coeffi cient · Local correlation method
1 Introduction
According to the Perdew’s popular classifi cation of density
functional approximations (DFA) [ 1 ], the random phase
approximation (RPA) is situated on the highest, fi fth rung
of Jacob’s ladder, which leads from the simplest Hartree
level toward the “heaven” corresponding to the exact solution of the Schrödinger equation. When stepping upwards
on Jacob’s ladder, one uses more and more ingredients of
the Kohn–Sham single determinant. Starting from the lowest rungs, the density, its gradient, the full set of occupied
orbitals are successively necessary for the construction of
the functional. At the highest rung DFA is usually based on
many-body methods, which require the knowledge of the
complete set of occupied and virtual orbitals. Such methods have the drawback that the size of the virtual orbital
space can be very large even in a moderately sized atomic
orbital basis. In the case of plane wave calculations, the virtual space can become even prohibitively large. One solution to keep the size of matrices in reasonable limits makes
recourse to an auxiliary basis set to expand the occupiedvirtual product functions. Such approaches are known in
quantum chemistry as resolution of identity [ 2 ] or densityfi tting [ 3 , 4 ] methods. Similar advantages can be achieved
by Cholesky decomposition [ 5 ] techniques. In plane wave
calculations, the plane wave basis itself can be used to
expand the product states [ 6 ]. Further gain can be achieved
by projection methods, which avoid any explicit reference
Abstract An approximation to the many-body London dispersion energy in molecular systems is expressed
as a functional of the occupied orbitals only. The method
is based on the local-RPA theory. The occupied orbitals are localized molecular orbitals, and the virtual space
is described by projected oscillator orbitals, i.e., functions
obtained by multiplying occupied localized orbitals with
solid spherical harmonic polynomials having their origin at
the orbital centroids. Since we are interested in the longrange part of the correlation energy, responsible for dispersion forces, the electron repulsion is approximated by its
multipolar expansion. This procedure leads to a fully nonempirical long-range correlation energy expression. Molecular dispersion coeffi cients calculated from determinant
Dedicated to Prof. Péter Surján on the occasion of his 60th
birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* János G. Ángyán
janos.angyan@univ-lorraine.fr
Bastien Mussard
bastien.mussard@upmc.fr
1
Institut du Calcul et de la Simulation , Sorbonne Universités,
UPMC Univ Paris 06 , 75005 Paris , France
2
UMR 7616, Laboratoire de Chimie Théorique , Sorbonne
Universités, UPMC Univ Paris 06 , 75005 Paris , France
3
CNRS, UMR 7616, Laboratoire de Chimie Théorique ,
75005 Paris , France
4
Institut Jean Barriol, CRM2, UMR 7036 , Université de
Lorraine , 54506 Vandoeuvre-lès-Nancy , France
5
CNRS, Institut Jean Barriol, CRM2, UMR 7036 ,
54506 Vandoeuvre-lès-Nancy , France
99
Reprinted from the journal