Theor Chem Acc (2015) 134:148
1 3
Waals energy formulae. Our procedure produces an explicit
model, derived from the wave function of the system, for
the dynamical polarizabilities associated with the building
blocks, which are bonds, lone pairs and in general localized
electron pairs. Thus one arrives to the quantum chemical analogs of the “quantum harmonic oscillators” (QHO) appearing
in the semiclassical theory of dispersion forces, elaborated
within a RPA framework by Tkatchenko et al. [ 72 , 73 ].
A promising perspective of the projected oscillator orbital
approach concerns the description of dispersion forces in
plane wave calculations for solids. Much effort has been
spent recently in fi nding a compact representation of the
dielectric matrix in plane wave calculations [ 10 ], but it still
remains a bottleneck for a really fast non-empirical calculation of the long-range correlation energy. Projected oscillator orbitals may offer an opportunity to construct extremely
compact representations of a part of the conduction band,
which contributes the most to the local dipolar excitations at
the origin of van der Waals forces. However, this approach
is probably limited to relatively large gap semiconductors,
where the construction of the MLWF [ 74 , 75 ], which are
the solid-state analogs of the Boys localized orbitals, is a
convergent procedure. Such a methodology could become a
fully non-empirical variant of Silvestrelli’s extension of the
Tkatchenko–Scheffl er approach for MLWFs [ 46 – 50 ].
The main purpose of the present work has been to
describe the principal features of the formalism, without
extensive numerical applications. However, the fi rst rudimentary numerical tests on the molecular C 6 coeffi cients
have indicated that the results are quite plausible in spite
of the simplifying approximations and it is reasonable to
expect that more sophisticated variants of the method will
improve the quality of the model. In view of the modest
computational costs and the fully ab initio character of
the projected oscillator orbital approach applied at various
approximation levels of the RPA, which is able to describe
dispersion forces even beyond a pairwise additive scheme,
the full numerical implementation of the presently outlined
methodology has certainly a great potential for the treatment of London dispersion forces in the context of density
functional theory.
Acknowledgments J.G.A. thanks Prof. Péter Surján (Budapest), to
whom this article is dedicated, the fruitful discussions during an early
stage of this work.
Appendix 1: Dipolar oscillator orbitals in local
frame
Let R
i be the rotation matrix which transforms an arbitrary vector v from the laboratory frame to the vector v loc in
the local frame defi ned as the principal axes of the second
moment tensor of the charge distribution associated with
a given localized occupied orbital i , R
i
· v = v loc . The
expression in the local frame of a POO i α constructed from
the LMO i is then:
Appendix 2: Riccati equations in POO basis
The fi rst-order wave function Ψ (1) can be written in terms
of Slater determinants | . . . ab . . . | formed with LMOs
and canonical virtual orbitals a and b on the one hand,
and on the other hand in terms of Slater determinants
| . . . m α n β . . . | formed with LMOs and POOs m α and n β .
That is to say that:
The canonical virtual orbitals and the POOs in question
are related by (see Eq. 6 ):
This allows us to write:
and leads to the transformation rule between the amplitudes
in the VMO and in the POO basis:
Multiplication of the Riccati equations of Eq. 9 by V †
and V from the left and from the right, respectively, and
expressing the amplitudes in POOs using Eq. 43 leads to:
(39)
|i
loc
α =
ˆ
I −
occ
m
|mm|
R
i
·
r − D
i
α
|i
=
R
i
· r
α
|i −
occ
m
R
i
· ·m|r|i
α
|m
=
β
R
i
αβ r β |i −
occ
m
β
R
i
αβ m|r β |i|m
(40)
Ψ
(1) =
occ
ij
virt
ab
T
ij
ab | . . . ab . . . | ≈
occ
ij
POO
mα nβ
T
ij
mα nβ | . . . m α n β . . . |.
(41)
|m α =
virt
a
|a am α and |n β =
virt
b
|b bn β .
(42)
Ψ
(1)
=
occ
ij
virt
ab
T
ij
ab | . . . ab . . . |
≈
occ
ij
POO
m α n β
T
ij
m α n β
| . . . m α n β . . . |
=
occ
ij
POO
m α n β
virt
ab
V am α T
ij
m α n β
V
†
n β b | . . . ab . . . |,
(43)
T
ij
= VT
ij
POO V
† .
109
Reprinted from the journal
1 3
Waals energy formulae. Our procedure produces an explicit
model, derived from the wave function of the system, for
the dynamical polarizabilities associated with the building
blocks, which are bonds, lone pairs and in general localized
electron pairs. Thus one arrives to the quantum chemical analogs of the “quantum harmonic oscillators” (QHO) appearing
in the semiclassical theory of dispersion forces, elaborated
within a RPA framework by Tkatchenko et al. [ 72 , 73 ].
A promising perspective of the projected oscillator orbital
approach concerns the description of dispersion forces in
plane wave calculations for solids. Much effort has been
spent recently in fi nding a compact representation of the
dielectric matrix in plane wave calculations [ 10 ], but it still
remains a bottleneck for a really fast non-empirical calculation of the long-range correlation energy. Projected oscillator orbitals may offer an opportunity to construct extremely
compact representations of a part of the conduction band,
which contributes the most to the local dipolar excitations at
the origin of van der Waals forces. However, this approach
is probably limited to relatively large gap semiconductors,
where the construction of the MLWF [ 74 , 75 ], which are
the solid-state analogs of the Boys localized orbitals, is a
convergent procedure. Such a methodology could become a
fully non-empirical variant of Silvestrelli’s extension of the
Tkatchenko–Scheffl er approach for MLWFs [ 46 – 50 ].
The main purpose of the present work has been to
describe the principal features of the formalism, without
extensive numerical applications. However, the fi rst rudimentary numerical tests on the molecular C 6 coeffi cients
have indicated that the results are quite plausible in spite
of the simplifying approximations and it is reasonable to
expect that more sophisticated variants of the method will
improve the quality of the model. In view of the modest
computational costs and the fully ab initio character of
the projected oscillator orbital approach applied at various
approximation levels of the RPA, which is able to describe
dispersion forces even beyond a pairwise additive scheme,
the full numerical implementation of the presently outlined
methodology has certainly a great potential for the treatment of London dispersion forces in the context of density
functional theory.
Acknowledgments J.G.A. thanks Prof. Péter Surján (Budapest), to
whom this article is dedicated, the fruitful discussions during an early
stage of this work.
Appendix 1: Dipolar oscillator orbitals in local
frame
Let R
i be the rotation matrix which transforms an arbitrary vector v from the laboratory frame to the vector v loc in
the local frame defi ned as the principal axes of the second
moment tensor of the charge distribution associated with
a given localized occupied orbital i , R
i
· v = v loc . The
expression in the local frame of a POO i α constructed from
the LMO i is then:
Appendix 2: Riccati equations in POO basis
The fi rst-order wave function Ψ (1) can be written in terms
of Slater determinants | . . . ab . . . | formed with LMOs
and canonical virtual orbitals a and b on the one hand,
and on the other hand in terms of Slater determinants
| . . . m α n β . . . | formed with LMOs and POOs m α and n β .
That is to say that:
The canonical virtual orbitals and the POOs in question
are related by (see Eq. 6 ):
This allows us to write:
and leads to the transformation rule between the amplitudes
in the VMO and in the POO basis:
Multiplication of the Riccati equations of Eq. 9 by V †
and V from the left and from the right, respectively, and
expressing the amplitudes in POOs using Eq. 43 leads to:
(39)
|i
loc
α =
ˆ
I −
occ
m
|mm|
R
i
·
r − D
i
α
|i
=
R
i
· r
α
|i −
occ
m
R
i
· ·m|r|i
α
|m
=
β
R
i
αβ r β |i −
occ
m
β
R
i
αβ m|r β |i|m
(40)
Ψ
(1) =
occ
ij
virt
ab
T
ij
ab | . . . ab . . . | ≈
occ
ij
POO
mα nβ
T
ij
mα nβ | . . . m α n β . . . |.
(41)
|m α =
virt
a
|a am α and |n β =
virt
b
|b bn β .
(42)
Ψ
(1)
=
occ
ij
virt
ab
T
ij
ab | . . . ab . . . |
≈
occ
ij
POO
m α n β
T
ij
m α n β
| . . . m α n β . . . |
=
occ
ij
POO
m α n β
virt
ab
V am α T
ij
m α n β
V
†
n β b | . . . ab . . . |,
(43)
T
ij
= VT
ij
POO V
† .
109
Reprinted from the journal