Theor Chem Acc (2015) 134:148
1 3
to virtual orbitals. Such a technique, named projective dielectric eigenpotential (PDEP) method, has been successfully applied to construct the dielectric function in plane
wave RPA calculations [ 7 – 9 ]. Recently, Rocca succeeded
to reduce further the size of the problem by a prescreened
set of virtuals [ 10 ].
The purpose of the present work is to demonstrate that
an approximate variant of the RPA (and also of the MP2)
correlation energy can be expressed by using quantities
that are computable from occupied orbitals alone. In this
context, one should mention the beautiful result, which has
been obtained by Surján, who has shown that the MP2 correlation energy, exempt of any further approximation in a
given basis set, can be reformulated as a functional of the
Hartree-Fock density matrix, i.e., using exclusively the
occupied orbitals [ 11 ]. Our main interest is not to reproduce the full correlation energy with high numerical precision, but we focus our attention to a well-defi ned part of
it, namely on the long-range dynamical correlation energy,
which is usually taken responsible for the London dispersion forces.
It is now well-documented that most of the conventional density functional calculations in the Kohn–Sham
framework, unless special corrections are added to the
total energy, are unable to grasp the physics of these longrange forces. Various fairly successful dispersion correction schemes are known, but although most of them were
claimed to posses an essentially ab initio character, they
use, without exception, some “external” data, like atomic
polarizabilities, atomic radii, etc. [ 12 – 18 ]. Even if these
quantities are usually not taken from experimental sources,
and rather originate from ab initio computations, their presence deprives these theories of their self-contained character. Hence, although we do not pretend that the rather drastic approximations which are going to be implemented in
the following allow us to achieve results of a quality comparable to the precision attained by carefully fi ne-tuned
methodologies, we argue that the ingredients of the present
approach originate from a controlled series of approximations and do not use any “external” inputs. Moreover, in
contrast to the relatively costly and sophisticated methods,
like RPA, we do not need virtual orbitals, i.e., we stay on
the fourth rung of Jacob’s ladder.
Our approach relies on the use of localized occupied
molecular orbitals (LMOs) that can be obtained relatively
easily by a unitary transformation in the subspace of occupied orbitals, according to either an external or an internal
localization criterion [ 19 ]. The localization of virtual orbitals is much more diffi cult, since the usual localization criteria for the occupied orbitals lead often to divergent results.
It is to be noted that recently a signifi cant progress has been
reported [ 20 ] for the effi cient localization of virtual orbitals. However, we follow another strategy here and build
excited determinants using localized functions which are
able to span the essential part of the virtual space. One of
the most popular local correlation approaches in this spirit
consists in using the atomic orbital (AO) basis functions to
represent the virtual space. They are made orthogonal to the
occupied space by projection, leading to the projected AOs
(PAOs) techniques [ 21 – 23 ]. The locality of these functions
is guaranteed by construction, even if it may be somewhat
deteriorated by the projection procedure.
In the present work we are going to revisit and explore
a quite old idea of Foster and Boys from the early 60s
[ 24 – 26 ]. The main concern of these authors was to construct a set of virtual orbitals directly from the set of occupied LMOs by multiplying them with solid spherical harmonic functions centered on the barycenter of the LMO.
The orthogonality of these new functions can be ensured by
a projection procedure. Boys and Foster called these new
functions, obtained after multiplication, oscillator orbitals (OOs), and after removing the components of the OOs
in the space of the occupied orbitals they may be called
projected oscillator orbitals (POOs). Very few articles in
the literature mention Boys’ oscillator orbitals [ 27 ], probably because it had no particular numerical advantage in
high-precision confi guration interaction calculations and
its practical implementation raised a number of complications which could be avoided by more straightforward
algorithms, like the use of the full set of virtual molecular
orbitals (VMOs). We have found only a single, very recent
article, which referred to the notion of oscillator orbitals
[ 28 ] as a useful concept, but not as a practical computational tool. To the best of our knowledge, the mathematical
implications of using oscillator orbitals to defi ne the virtual
space has never been rigorously studied. Such an analysis
is beyond the scope of the present study: it is going to be
the subject of a forthcoming publication.
The POOs are non-orthogonal among each other, which
is at the origin one of the complications mentioned above.
This problem can be handled just like in the case of the
PAOs; therefore, a theory of electron correlation based on
POOs can follow a similar reasoning as local correlation
methods using PAOs [ 29 ]. In particular, in this paper, the
RPA method will be reformulated for a virtual space constructed from POOs.
It is worthwhile to mention that the projected oscillator orbitals bear some similarities to the trial perturbed
wave function in the variation-perturbational technique
associated with the names of Kirkwood [ 30 ], Pople and
Schofi eld [ 31 ] (KPS), to calculate multipole molecular
polarizabilities. The closely related Karplus–Kolker [ 32 ,
33 ] (KK) method and its variants [ 34 , 35 ] use a similar Ansatz for the perturbed orbitals. In these latter methods, which were formulated originally as simplifi ed perturbed Hartree–Fock theories, the fi rst-order perturbed
100
Reprinted from the journal
1 3
to virtual orbitals. Such a technique, named projective dielectric eigenpotential (PDEP) method, has been successfully applied to construct the dielectric function in plane
wave RPA calculations [ 7 – 9 ]. Recently, Rocca succeeded
to reduce further the size of the problem by a prescreened
set of virtuals [ 10 ].
The purpose of the present work is to demonstrate that
an approximate variant of the RPA (and also of the MP2)
correlation energy can be expressed by using quantities
that are computable from occupied orbitals alone. In this
context, one should mention the beautiful result, which has
been obtained by Surján, who has shown that the MP2 correlation energy, exempt of any further approximation in a
given basis set, can be reformulated as a functional of the
Hartree-Fock density matrix, i.e., using exclusively the
occupied orbitals [ 11 ]. Our main interest is not to reproduce the full correlation energy with high numerical precision, but we focus our attention to a well-defi ned part of
it, namely on the long-range dynamical correlation energy,
which is usually taken responsible for the London dispersion forces.
It is now well-documented that most of the conventional density functional calculations in the Kohn–Sham
framework, unless special corrections are added to the
total energy, are unable to grasp the physics of these longrange forces. Various fairly successful dispersion correction schemes are known, but although most of them were
claimed to posses an essentially ab initio character, they
use, without exception, some “external” data, like atomic
polarizabilities, atomic radii, etc. [ 12 – 18 ]. Even if these
quantities are usually not taken from experimental sources,
and rather originate from ab initio computations, their presence deprives these theories of their self-contained character. Hence, although we do not pretend that the rather drastic approximations which are going to be implemented in
the following allow us to achieve results of a quality comparable to the precision attained by carefully fi ne-tuned
methodologies, we argue that the ingredients of the present
approach originate from a controlled series of approximations and do not use any “external” inputs. Moreover, in
contrast to the relatively costly and sophisticated methods,
like RPA, we do not need virtual orbitals, i.e., we stay on
the fourth rung of Jacob’s ladder.
Our approach relies on the use of localized occupied
molecular orbitals (LMOs) that can be obtained relatively
easily by a unitary transformation in the subspace of occupied orbitals, according to either an external or an internal
localization criterion [ 19 ]. The localization of virtual orbitals is much more diffi cult, since the usual localization criteria for the occupied orbitals lead often to divergent results.
It is to be noted that recently a signifi cant progress has been
reported [ 20 ] for the effi cient localization of virtual orbitals. However, we follow another strategy here and build
excited determinants using localized functions which are
able to span the essential part of the virtual space. One of
the most popular local correlation approaches in this spirit
consists in using the atomic orbital (AO) basis functions to
represent the virtual space. They are made orthogonal to the
occupied space by projection, leading to the projected AOs
(PAOs) techniques [ 21 – 23 ]. The locality of these functions
is guaranteed by construction, even if it may be somewhat
deteriorated by the projection procedure.
In the present work we are going to revisit and explore
a quite old idea of Foster and Boys from the early 60s
[ 24 – 26 ]. The main concern of these authors was to construct a set of virtual orbitals directly from the set of occupied LMOs by multiplying them with solid spherical harmonic functions centered on the barycenter of the LMO.
The orthogonality of these new functions can be ensured by
a projection procedure. Boys and Foster called these new
functions, obtained after multiplication, oscillator orbitals (OOs), and after removing the components of the OOs
in the space of the occupied orbitals they may be called
projected oscillator orbitals (POOs). Very few articles in
the literature mention Boys’ oscillator orbitals [ 27 ], probably because it had no particular numerical advantage in
high-precision confi guration interaction calculations and
its practical implementation raised a number of complications which could be avoided by more straightforward
algorithms, like the use of the full set of virtual molecular
orbitals (VMOs). We have found only a single, very recent
article, which referred to the notion of oscillator orbitals
[ 28 ] as a useful concept, but not as a practical computational tool. To the best of our knowledge, the mathematical
implications of using oscillator orbitals to defi ne the virtual
space has never been rigorously studied. Such an analysis
is beyond the scope of the present study: it is going to be
the subject of a forthcoming publication.
The POOs are non-orthogonal among each other, which
is at the origin one of the complications mentioned above.
This problem can be handled just like in the case of the
PAOs; therefore, a theory of electron correlation based on
POOs can follow a similar reasoning as local correlation
methods using PAOs [ 29 ]. In particular, in this paper, the
RPA method will be reformulated for a virtual space constructed from POOs.
It is worthwhile to mention that the projected oscillator orbitals bear some similarities to the trial perturbed
wave function in the variation-perturbational technique
associated with the names of Kirkwood [ 30 ], Pople and
Schofi eld [ 31 ] (KPS), to calculate multipole molecular
polarizabilities. The closely related Karplus–Kolker [ 32 ,
33 ] (KK) method and its variants [ 34 , 35 ] use a similar Ansatz for the perturbed orbitals. In these latter methods, which were formulated originally as simplifi ed perturbed Hartree–Fock theories, the fi rst-order perturbed
100
Reprinted from the journal