1 3
Theor Chem Acc (2016) 135:6
DOI 10.1007/s00214-015-1761-0
REGULAR ARTICLE
Effi cient calculation of the density response function
from generalized polarizabilities
Tomasz Janowski
1,2 · Krzysztof Wolinski
3 · Peter Pulay
1
Received: 26 June 2015 / Accepted: 7 November 2015 / Published online: 21 December 2015
© Springer-Verlag Berlin Heidelberg 2015
proportional to the product of the number occupied and
virtual orbitals, and therefore scaling quadratically with the
system size at constant basis set quality. The diagonal elements of the DRF (local polarizabilities) of water and butadiene in the molecular plane are displayed. An example of
non-diagonal DRF is presented for butadiene 0.77 Å above
the molecular plane, with one point fi xed over the midpoint
of a C=C bond. We obtain a fairly localized DRF, even in
this conjugated system. The values become quite small if
the distance of the two local perturbations exceeds a bond
length. By using frequency-dependent polarizabilities, one
could readily calculate the dynamic DRF.
Keywords Density response function · Polarization
propagator · Density response kernel · Generalized
polarizability · Polarizability · Conceptual density
functional theory · Random phase approximation
1 Introduction
The density response function (DRF), also known as polarization propagator or density response kernel, χ ( r , r ′, ω ),
[ 1 , 2 ] is an important quantity in electronic structure theory,
particularly in evaluating dispersion energies, in conceptual density functional theory (DFT) and in random phase
approximation (RPA) and GW (approximate Green’s function) methods. The static ( ω = 0) limit of this function is
defi ned as the second functional derivative of the molecular
energy E with respect to the external potential u at constant
electron number N . Its frequency-dependent form is widely
used, as its poles give the electronic excitation energies.
The zero-frequency limit [ 3 ] can be used to defi ne electric response in larger molecules and is useful in conceptual density functional theory [ 4 , 5 ]. The fi rst functional
Abstract We present a method to calculate the density response function (DRF), also known as the density
response kernel or the polarization propagator in the static
limit. Our method uses generalized polarizabilities (GPs)
which are second derivatives of the molecular energy with
respect to two arbitrary perturbations of the external electrostatic potential. They are generalizations of the common multipole polarizabilities. The latter use solid spherical harmonics as perturbing potentials, while GPs can
use any function. We use a sine function expansion of the
electrostatic potential. Generalized polarizabilities were
originally introduced for a different project, ultrafast quantum/molecular mechanics calculations. By transforming
the GPs to a (discretized) direct space representation, we
obtain the DRF in the static limit. The method has been
implemented for single-determinant (Hartree–Fock and
density functional theory) wavefunctions, but can be generalized to more accurate wavefunctions. The number of coupled-perturbed self-consistent fi eld (CP-SCF) calculations
in our method is proportional to the molecular volume at
a given spatial resolution, i.e., scales linearly with the system size, and is independent of the basis set size. The best
previous method has the number of CP-SCF calculations
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Peter Pulay
pulay@uark.edu
1
Department of Chemistry and Biochemistry , The University
of Arkansas , Fayetteville , AR 72701 , USA
2
Present Address: Department of Chemistry , Duke University ,
Durham , NC 27708 , USA
3
Department of Chemistry , Marie Curie-Sklodowska
University , 20-031 Lublin , Poland
9
Reprinted from the journal
Theor Chem Acc (2016) 135:6
DOI 10.1007/s00214-015-1761-0
REGULAR ARTICLE
Effi cient calculation of the density response function
from generalized polarizabilities
Tomasz Janowski
1,2 · Krzysztof Wolinski
3 · Peter Pulay
1
Received: 26 June 2015 / Accepted: 7 November 2015 / Published online: 21 December 2015
© Springer-Verlag Berlin Heidelberg 2015
proportional to the product of the number occupied and
virtual orbitals, and therefore scaling quadratically with the
system size at constant basis set quality. The diagonal elements of the DRF (local polarizabilities) of water and butadiene in the molecular plane are displayed. An example of
non-diagonal DRF is presented for butadiene 0.77 Å above
the molecular plane, with one point fi xed over the midpoint
of a C=C bond. We obtain a fairly localized DRF, even in
this conjugated system. The values become quite small if
the distance of the two local perturbations exceeds a bond
length. By using frequency-dependent polarizabilities, one
could readily calculate the dynamic DRF.
Keywords Density response function · Polarization
propagator · Density response kernel · Generalized
polarizability · Polarizability · Conceptual density
functional theory · Random phase approximation
1 Introduction
The density response function (DRF), also known as polarization propagator or density response kernel, χ ( r , r ′, ω ),
[ 1 , 2 ] is an important quantity in electronic structure theory,
particularly in evaluating dispersion energies, in conceptual density functional theory (DFT) and in random phase
approximation (RPA) and GW (approximate Green’s function) methods. The static ( ω = 0) limit of this function is
defi ned as the second functional derivative of the molecular
energy E with respect to the external potential u at constant
electron number N . Its frequency-dependent form is widely
used, as its poles give the electronic excitation energies.
The zero-frequency limit [ 3 ] can be used to defi ne electric response in larger molecules and is useful in conceptual density functional theory [ 4 , 5 ]. The fi rst functional
Abstract We present a method to calculate the density response function (DRF), also known as the density
response kernel or the polarization propagator in the static
limit. Our method uses generalized polarizabilities (GPs)
which are second derivatives of the molecular energy with
respect to two arbitrary perturbations of the external electrostatic potential. They are generalizations of the common multipole polarizabilities. The latter use solid spherical harmonics as perturbing potentials, while GPs can
use any function. We use a sine function expansion of the
electrostatic potential. Generalized polarizabilities were
originally introduced for a different project, ultrafast quantum/molecular mechanics calculations. By transforming
the GPs to a (discretized) direct space representation, we
obtain the DRF in the static limit. The method has been
implemented for single-determinant (Hartree–Fock and
density functional theory) wavefunctions, but can be generalized to more accurate wavefunctions. The number of coupled-perturbed self-consistent fi eld (CP-SCF) calculations
in our method is proportional to the molecular volume at
a given spatial resolution, i.e., scales linearly with the system size, and is independent of the basis set size. The best
previous method has the number of CP-SCF calculations
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Peter Pulay
pulay@uark.edu
1
Department of Chemistry and Biochemistry , The University
of Arkansas , Fayetteville , AR 72701 , USA
2
Present Address: Department of Chemistry , Duke University ,
Durham , NC 27708 , USA
3
Department of Chemistry , Marie Curie-Sklodowska
University , 20-031 Lublin , Poland
9
Reprinted from the journal