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THE JOURNAL OF CHEMICAL PHYSICS 144, 204103 (2016)
First time combination of frozen density embedding theorywith
the algebraic diagrammatic construction schemefor the polarization
propagator of second order
Stefan Prager,1,a) Alexander Zech,2,b) Francesco Aquilante,3,c)
Andreas Dreuw,1,d)and Tomasz A. Wesolowski2,e)1Interdisciplinary
Center for Scientific Computing, Ruprecht-Karls University, Im
Neuenheimer Feld 368,69120 Heidelberg, Germany2Dèpartement de
Chimie Physique, Université de Genève, 30 Quai Ernest-Ansermet,1211
Genève 4, Switzerland3Dipartimento di Chimica “G. Ciamician,”
Università di Bologna, Via Selmi 2, 40126 Bologna, Italy
(Received 24 February 2016; accepted 25 April 2016; published
online 24 May 2016)
The combination of Frozen Density Embedding Theory (FDET) and
the Algebraic DiagrammaticConstruction (ADC) scheme for the
polarization propagator for describing environmental effectson
electronically excited states is presented. Two different ways of
interfacing and expressingthe so-called embedding operator are
introduced. The resulting excited states are compared
withsupermolecular calculations of the total system at the ADC(2)
level of theory. Molecular test systemswere chosen to investigate
molecule–environment interactions of varying strength from
dispersioninteraction up to multiple hydrogen bonds. The overall
difference between the supermolecular andthe FDE-ADC calculations
in excitation energies is lower than 0.09 eV (max) and 0.032 eV
inaverage, which is well below the intrinsic error of the ADC(2)
method itself. Published by AIPPublishing.
[http://dx.doi.org/10.1063/1.4948741]
I. INTRODUCTION
Almost all chemical reactions occur in liquid phaseand also
almost all chemical properties are measured incondensed phases. In
these cases, the investigated moleculeinteracts with its
environment, leading to a change of itschemical properties.
Electronically excited states, especially,can be strongly
influenced by the environment. A largevariety of molecules and
solvents exhibit this effect ofsolvatochromism.1 Hence, including
the environment iscrucial for comparable calculations and reliable
predictionsusing quantum chemical methods. However, including
allenvironment molecules in quantum chemical calculationsexceeds
nowadays computational limits. Different methodsfor considering the
environment have been proposed, rangingfrom an implicit treatment
as in polarizable continuum models(PCM)2 to explicit models
treating the environment at amolecular mechanics level of theory
(QM/MM).3 It is knownthat implicit approaches can only model
dispersion andCoulombic interaction but may fail in describing
specificsolvent interactions, for instance hydrogen bonds.4–6 In
thesecases, an explicit model for the environment is needed.
Density embedding methods7,8 constitute such an explicitmodel,
and have received increasing attention in recent years(for the
latest reviews see Refs. 9 and 10). Frozen-DensityEmbedding Theory
(FDET)7,11 in particular provides a formal
a)Electronic mail:
[email protected])Electronic mail:
[email protected])Electronic mail:
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framework in which the whole system is described by meansof two
independent quantities: the embedded wavefunction(interacting or
not) and the density associated with theenvironment. The latter can
in principle be taken from lowerlevel quantum mechanical methods,
statistical theories forensembles, or even from experiment.
While the original work used DFT12–14 methods todescribe the
embedded species, also FDET variants based onwavefunction methods
have been developed.11,15–17 Carter andcollaborators were the first
to combine the FDET embeddingpotential with wavefunction based
methods in their approachin which the independent variables are the
total density andthe density of the embedded species.8,18 This
combination isespecially useful for the calculation of excited
states whereDFT methods have known limitations.19–21
The embedding is accomplished by means of anembedding potential
which depends on the density ofthe embedded species ρA(r⃗). In the
canonical form ofFDET, herein referred to as conventional FDET, an
iterativescheme is applied to obtain a wavefunction which is
self-consistent with respect to the embedding potential.
Theiterative scheme requires reconstruction of the
embeddingpotential with an updated density taken from the
previousiteration and solving the many-body problem with the
newlygenerated embedding potential.16,17,22,23 This
circumstancemakes high-level ab initio methods unfavorable.
Furthermore,the ρA-dependency of the embedding potential leads to
non-orthogonal wavefunctions within conventional FDET. In orderto
overcome ρA-dependency and reduce the computationalcost of
conventional FDET, an approximation is usuallymade in which the
embedding potential is evaluated at some
0021-9606/2016/144(20)/204103/14/$30.00 144, 204103-1 Published
by AIP Publishing.
http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741http://dx.doi.org/10.1063/1.4948741mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]
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204103-2 Prager et al. J. Chem. Phys. 144, 204103 (2016)
fixed density ρrefA(r⃗).16,23–26 Recently an alternative
method,
linearized FDET, has been proposed, which is based on a
fixeddensity ρref
A(r⃗) for the embedding potential, but still provides
consistency between wavefunction and energy.27,28
The algebraic diagrammatic construction (ADC) for
thepolarization propagator is a robust and accurate method for
thecalculation of excited states.29 Additionally, its
mathematicalstructure is well suited to include external
one-particleoperators since it is based on a non-iterative ground
statemethod, making it straight forward to include
environmentalmodels in ADC. ADC is size-consistent, Hermitian
andsystematically improvable.30 So far, including
environmentaleffects into an ADC calculation has been accomplished
bymeans of a polarizable continuum model (PCM)6,31 andthrough QM/MM
calculations.32
In this article, we report a multiscale variant using FDETin
combination with ADC. After giving a brief introductionto ADC and
FDET in Sec. II, we outline the computationaldetails in Sec. III.
The results of our test calculations arepresented in Sec. IV. A
discussion and conclusion follow inSec. V.
II. THEORETICAL METHODOLOGY
A. Algebraic diagrammatic construction schemefor the
polarization propagator of secondorder (ADC(2))
A full derivation of the algebraic diagrammatic construc-tion
scheme for the polarization propagator (ADC) isgiven in the
literature.30,33,34 ADC can be derived usingthe intermediate state
(IS) formalism. In this formalism, acomplete set of orthonormalized
intermediate states
�Ψ̃I
�is
constructed by applying creation and annihilation operatorsĈI
=∈
ĉ†aĉi; ĉ
†aĉ†bĉiĉj; . . .
onto the exact ground state
wavefunction followed by Gram-Schmidt orthogonalization.In this
intermediate state representation (ISR), the excited
statewavefunction can be expressed in the basis of the
intermediatestates according to
Ψn =J
XnJ�Ψ̃J
�, (1)
with XnJ as the eigenvectors of the ADC matrix, which isbuilt as
a matrix representation of the Hamiltonian shifted bythe exact
ground state energy E0,
MI J =
Ψ̃I
�Ĥ − E0
�Ψ̃J
�. (2)
This leads to the corresponding Hermitian eigenvalue
problem,
MX = ΩX, X†X = 1, (3)
with X as the eigenvector matrix andΩ as the diagonal matrixof
the eigenvalues which correspond to the exact
excitationenergies,
Ωn = En − E0. (4)
This eigenvalue problem can be solved by diagonalizationof the
ADC matrix M using typical iterative diagonalizationschemes like
the Davidson algorithm.35 Since neither theexact ground state
wavefunction nor the exact ground state
energy are known, they are replaced by the correlated
Møller-Plesset wavefunction and energy, respectively. Applying the
ISformalism to 2nd order perturbation theory (MP(2)) generatesthe
2nd order ADC equations (ADC(2)).
B. Frozen density embedding theory (FDET)
This section will deal with the key features ofFrozen Density
Embedding Theory (FDET) with respect towavefunction embedding,
while a more detailed view on theFDET formalism used in this study
is given in Refs. 11 and 28.
In FDET, the supersystem is divided into the embeddedspecies (A)
and the environment (B). Although FDET holds forall possible
choices for ρA and ρB, the practical FDET basedmethods, which hinge
on approximate density functionalfor the embedding potential, are
designed for systems withnon-covalently bound environments like
systems amenablefor QM/MM type of embedding methods. If charge
transferbetween subsystems cannot be excluded then modeling
suchshould be made by means of neither FDET nor QM/MM typeof
methods. In this particular variant of FDET, a wavefunctionis used
to describe the electronic state of the embeddedspecies while the
electron density ρB(r⃗) is the descriptor forthe environment. Thus,
the total energy of the supersystem(AB) takes the form of a
functional EEWF
AB[ΨA, ρB] depending
on the embedded wavefunction (EWF) and the environmentdensity.
Stationary wavefunctions are determined through theEuler-Lagrange
equation
δEEWFAB
[ΨA, ρB]δΨ
A
− λΨA = 0, (5)
where λ is the Lagrange multiplier referring to
thenormalization. While Eq. (5) holds rigorously for theelectronic
ground state, other than ground-state solutionscan be associated
with excited states on the basis of theLevy-Perdew theorem36 as
noticed by Khait and Hoffmann.37
The total energy of the supersystem follows as
EEWFAB�ΨA, ρB
�=
ΨA
�ĤA
�ΨA
�+ V nucB
�ρA
�
+ Jint�ρA, ρB
�+ Enadxc,T
�ρA, ρB
�
+ EHKvB�ρB
�+ V nucA
�ρB
�, (6)
where the following terms arise from the interaction of thetwo
subsystems:
Jint�ρA, ρB
�=
ρA(r⃗)ρB(r⃗ ′)�
r⃗ − r⃗ ′� dr⃗dr⃗ ′, (7)
V nucA�ρB
�=
ρB(r⃗)vA(r⃗)dr⃗ , (8)
V nucB�ρA
�=
ρA(r⃗)vB(r⃗)dr⃗ , (9)
Enadxc,T�ρA, ρB
�= Enadxc
�ρA, ρB
�+ Tnads
�ρA, ρB
�
+∆FMD�ρA
�. (10)
The first three terms describe classical electrostatic
interac-tions while Enadxc,T
�ρA, ρB
�comprises terms representing non-
additivity of the exchange-correlation and kinetic
densityfunctionals and a ρA dependent functional ∆FMD
�ρA
�
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204103-3 Prager et al. J. Chem. Phys. 144, 204103 (2016)
(usually neglected in practice) depending on the form ofthe
wavefunction used to solve Eq. (5). The non-additiveenergy
bifunctional is generally defined as
Enad�ρA, ρB
�= E
�ρA + ρB
�− E
�ρA
�− E
�ρB
�. (11)
The necessary condition for the stationary wavefunction
tosatisfy is the following Schrödinger-like equation:
�ĤA + v̂emb
�ΨA = ϵΨA, (12)
where v̂emb is the embedding operator describing the effectsof
the environment. This operator is determined uniquely bythree
charge densities (ρA(r⃗), ρB(r⃗), and that generated bythe nuclei
of the environment (equivalently represented by theelectrostatic
potential vB(r⃗))). It is thus the functional of thesethree
functions. It reads:
vemb[ρA, ρB, vB](r⃗)
= vB(r⃗) +
ρB(r⃗ ′)�r⃗ − r⃗ ′
�dr⃗ ′ +δEnadxc,T[ρA, ρB]
δρA
. (13)
This potential is obtained as the functional derivative of
thecorresponding terms in the total energy functional with
respectto ρA(r⃗).
In this study, a particular version of FDET is usedwhere the
non-additive energy functional Enadxc,T[ρA, ρB] isapproximated by a
functional which is linear in ρA(r⃗). Theapproximation is
constructed as a Taylor-expansion of thenon-additive energy
functional at a reference density ρref
A(r⃗)
with the series being truncated after the linear term,
Enadxc,T[ρA, ρB] ≈ Enadxc,T[ρrefA , ρB]
+
(ρA(r⃗) − ρrefA (r⃗)
) δEnadxc,T[ρrefA , ρB]δρ
refA(r⃗) dr⃗
= Enadxc,T[ρrefA , ρB] + ∆lin[ρA, ρrefA , ρB]. (14)From here on,
we will refer to this variant of FDET
as linearized FDET. The expression for the total energy
inlinearized FDET is
EL−FDETAB [ΨA, ρB, ρrefA ] =
ΨA
�ĤA
�ΨA
�+ V nucB
�ρA
�
+ Jint�ρA, ρB
�+ Enadxc,T[ρrefA , ρB]
+∆lin[ρA, ρrefA , ρB]+ EHKvB
�ρB
�+ V nucA
�ρB
�. (15)
The embedding potential obtained as functional derivative ofthe
corresponding terms in the total energy functional givenin Eq. (15)
reads
vemb[ρrefA , ρB, vB](r⃗)
= vB(r⃗) +
ρB(r⃗ ′)�r⃗ − r⃗ ′
�dr⃗ ′ +δEnadxc,T[ρrefA , ρB]
δρrefA
. (16)
The potential will be denoted as v linemb(r⃗) = vemb[ρref
A, ρB, vB](r⃗). Note that although both the potentials that
are given in Eqs. (13) and (16) are functionals of three
chargedensities, the latter one does not depend on the electronic
stateof the embedded system.
Since the embedding potential is added to the
1-particleHamiltonian, all electrostatic and part of the
non-electrostatic
interaction terms are already included when evaluating
theexpectation value of the Hamiltonian. In order to arrive at
thesame expression as Eq. (15), only constant,
state-independentterms and the DFT energy of system B need to be
added,
⟨ΨA|v̂emb|ΨA⟩ = V nucB�ρA
�+ Jint
�ρA, ρB
�
+
ρA(r⃗)
δEnadxc,T[ρrefA , ρB]δρ
refA(r⃗) dr⃗ . (17)
Therefore, the energy difference between two states I and J
isevaluated as the difference of two self-consistent expressionsfor
the total energy given in Eq. (15) and reads (for the
fullderivation see Zech et al.):28
∆EI J = ⟨ΨJA|ĤA + v̂ linemb|ΨJA⟩ − ⟨ΨIA|ĤA + v̂ linemb|ΨIA⟩.
(18)Benchmark calculations on linearized FDET show that
thisapproximation yields only negligible errors in
excitationenergies compared to conventional FDET. Even in states
withlarge differences in electron densities between ground
andexcited state as in charge-transfer states, the approximation
isvalid.28
III. COMPUTATIONAL METHODOLOGY
In general, the implementation of the FDE-ADC methodused in this
work consists of four separate steps: the individualcalculations of
the density matrices of A and B, the building ofthe embedding
potential, and the final calculation applying theembedding
potential on system A. In our implementation, thecore system (A) is
calculated with Q-Chem38 using Møller-Plesset perturbation
theory39,40 of second order (MP(2)) whilethe environment (B) is
calculated either in Q-Chem or inMolcas41,42 using Hartree-Fock
(HF)43 or density-functional-theory (DFT).12–14 Both density
matrices of A and B areused to construct the embedding potential
within Molcas.The embedding potential subsequently serves as input
forQ-Chem and is added to the 1-particle Hamiltonian used inthe HF
calculation,
Õ1 = O1 + vemb[ρrefA , ρB]. (19)The orbitals and orbital
energies obtained from the modifiedHF calculation include the
influence of the environment andserve as input for a subsequent
ADC(2) calculation. Theresulting excited states are influenced by
the environmentby virtue of the modified orbitals and orbital
energies.For applying this method, no modifications to the
originalADC equations need to be made. Furthermore, the
excitationenergies are a direct result (cf. Eq. (18)), i.e., no
furthercorrections have to be applied. The combination of thesetwo
steps, the HF calculation including the embeddingpotential and the
following ADC(2) calculation, are furtherreferred to as FDE-ADC(2)
calculation. Currently, twodifferent approaches are implemented to
perform FDE-ADC calculations, which we call supermolecular
expansion(SE) and reassembling of density matrix (RADM),
whichdistinguish the different numerical procedures to perform
thecontraction of the required density matrices.
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204103-4 Prager et al. J. Chem. Phys. 144, 204103 (2016)
In SE, the calculations of both systems A and B andthe final
FDE-ADC(2) calculation are performed in the basisfunctions of A +
B. Although this is not very efficient, itprovides an easy way to
analyze this method in detail as aproof of principle, since there
is no basis-set superpositionerror (BSSE) included.
In our newly developed approach called RADM, thecalculation of A
is split. A is calculated in the monomerbasis on MP(2) level of
theory and additionally in thesupermolecular basis on HF level of
theory, obtaining twoseparate density matrices. A new density
matrix in the basisof A and B is generated by adding the
HF-to-MP(2) differencedensity matrix in the basis of A to the AA
block of theHF density matrix in the basis of A + B. The
embeddingpotential is preliminary calculated in the basis of A + B
andtruncated afterwards to the basis of A. This approximation canbe
made because the off-diagonal blocks AB and BA in theembedding
potential matrix are almost zero and the BB blockhardly contributes
to A since the density of A is vanishinglysmall on the basis of B.
The final FDE-ADC(2) calculationincluding the embedding potential,
i.e., the computationallymost demanding step, is performed in the
basis of A only(Fig. 1).
For the FDE-ADC calculations, a development versionof Q-Chem
based on version 4.3 and a developmentversion of Molcas based on
version 8.0 have been used.Molecular pictures were captured using
Avogadro 1.1.0.44
Unless otherwise indicated, all calculations have been
carriedout at MP(2)/cc-pVDZ45 and ADC(2)/cc-pVDZ level oftheory for
system A and HF/cc-pVDZ level of theory forsystem B.
IV. RESULTS
FDE-ADC(2) was tested for four different systems withvarious
environments of different interaction strengths. Inall benchmarks,
the supersystem is optimized at MP(2)/cc-pVDZ level of theory and
the five lowest electronicallyexcited states are calculated at
ADC(2)/cc-pVDZ level of
theory. Additionally, the five lowest excited states of
theisolated system A were calculated at ADC(2)/cc-pVDZ levelof
theory without re-optimizing the geometry. In this way,the
electronic interactions between the core system andenvironment can
be analyzed directly without the influenceof geometry changes due
to environment interactions. TheFDE-ADC(2) calculations were
carried out using the sameoptimized geometries for system A and B
as in the supersystemcalculation. For all investigations, the
singlet multiplicityand neutral charge were conserved. The test
systems werechosen with respect to various kinds of
system-environmentinteractions. Benzene with hydrogen fluoride
(Sec. IV A 1)in two different orientations was chosen as a system
withonly weak dispersion interactions and polarization of
anaromatic π-system. Benzaldehyde with two water molecules(Sec. IV
A 2) forming a water dimer is an example forsystems with a hydrogen
bond between embedded speciesand environment in addition to a
polarization of the π-system.The last system, uracil with five
water molecules (Sec. IV A 3),shows the strongest interactions
because all water moleculesare involved in hydrogen bonds with the
embedded species.Transitions involving the electron lone pairs are
expected tobe more strongly shifted than typical π → π∗ states due
to theenvironment interactions.
A. Results of FDE-ADC(2) usingthe supermolecular expansion
In the first test, the supermolecular expansion was used.This
approach is the mathematically exact implementation ofFDE-ADC
without any further approximation and withoutrestrictions to the
basis set. Although no benefit incomputational cost with respect to
the supersystem calculationcan be achieved, this serves as a
benchmark for furtherapproximate FDE-ADC approaches.
1. Benzene with one hydrogen fluoride molecule
As a first test system, a hydrogen fluoride molecule isplaced
apical to the benzene ring, almost along the C6-axis
FIG. 1. Schematic representation ofthe reassembling of density
matrix(RADM) method. In the upper line, theisolated system A is
calculated and inthe lower line, the isolated system B.In system A,
the HF (grey) density ma-trix on the basis functions of A and Bis
combined with the HF/MP(2) (red)difference density matrix on the
basisfunctions of A to build the RADM ma-trix. In combination, the
embedding po-tential (yellow) is created and truncatedto the
elements in the basis of A. Fi-nally, FDE-ADC(2) (blue)
calculationsare performed in the basis of A only.
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204103-5 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 2. Molecular structures of the two benzene-hydrogen
fluoride supersys-tems. Benzene with apical hydrogen fluoride
(left) and side-on (right).
with the hydrogen pointing towards the benzene (Fig. 2 left).The
distance between the plane of the benzene ring and thehydrogen atom
is 2.161 Å.
The frontier orbitals of benzene, i.e., the highest
occupiedmolecular orbital (HOMO), HOMO-1, the lowest
unoccupiedmolecular orbital (LUMO) and LUMO+1 are the typical
πorbitals. The lower lying occupied orbitals (HOMO-2 andHOMO-3)
show σ-character while the higher unoccupiedorbitals (LUMO+2 and
LUMO+3) can be described asRydberg-orbitals. The analysis of the
five lowest electronicallyexcited states of isolated benzene shows
four locally excitedstates characterized by π → π∗ transitions and
an energeticallyhigher lying Rydberg state. The five lowest excited
states andtheir character are given in Table I.
In the supermolecule calculations of benzene togetherwith
hydrogen fluoride, the energetic order of the frontierorbitals as
well as the energetic order and the characterof the excited states
are changed due to the influence of
the environment and as the fifth excited state a σ →
π∗transition arises. The HOMO and HOMO-1 switch theirenergetic
order but since they are almost degenerate, it haspractically no
influence. However, this change leads to adifferent orbital
transition pattern in the characterization ofthe excited states. A
HOMO → LUMO transition in theisolated benzene corresponds now to a
HOMO-1 → LUMOtransition in the C6H6-HF supermolecular complex.
Thefive lowest excited states are listed and characterized inTable
I.
In the FDE-ADC calculation, the results of thesupersystem
calculation are nicely reproduced. The lowestfour excited states
are in almost perfect agreement with thecalculation of the
supersystem regarding excitation energies,oscillator strengths and
orbital transitions. The 3rd excitedstate is marginally shifted to
higher excitation energies insteadof slightly lower excitation
energies, as the supersystemcalculation would infer. The 5th and
6th excited state changedthe energetic order in FDE-ADC compared to
the supersystemcalculation because of the small energy difference
betweenthese excited states of only 0.02 eV (Table I). For the
sakeof comparison, the S6 is used and labeled as S5 according tothe
energetic order of this specific state in the
supersystemcalculation.
The difference of the excitation energies and the
oscillatorstrengths of the isolated system and the FDE-ADC
calculationto the supersystem calculation is shown in Fig. 3. With
amaximum deviation of 0.03 eV in excitation energies for
thissystem, the polarization of the π-system is reproduced
almostquantitatively.
TABLE I. Excitation energies, oscillator strengths, and orbital
transitions for the five energetically lowest electronically
excited singlet states of isolatedbenzene, the supersystem benzene
with apical hydrogen fluoride, and the FDE-ADC calculations in the
supermolecular expansion of benzene (A) and hydrogenfluoride
(B).
Excitation energies (eV) Oscillator strengthsOrb. trans.
Weight (%)
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC H+x → L+ya
Isol. Supersys. FDE-ADC
S1 5.287 5.323 5.328 0.0000 0.0001 0.0001 0 → 0 (π → π∗) 8.3
45.2 45.10 → 1 (π → π∗) 36.6 . . . . . .−1 → 0 (π → π∗) 36.6 . . .
. . .−1 → 1 (π → π∗) 8.2 43.6 43.5
S2 6.677 6.692 6.704 0.0000 0.0000 0.0000 0 → 0 (π → π∗) 37.9 .
. . . . .0 → 1 (π → π∗) 8.5 45.0 45.6−1 → 0 (π → π∗) 8.2 46.2
45.4−1 → 1 (π → π∗) 37.2 . . . . . .
S3 7.520 7.500 7.530 0.7155 0.6810 0.7000 0 → 0 (π → π∗) 18.3 .
. . . . .0 → 1 (π → π∗) 25.8 44.1 43.4−1 → 0 (π → π∗) 25.7 43.0
43.6−1 → 1 (π → π∗) 18.5 . . . . . .
S4 7.522 7.527 7.548 0.7162 0.6867 0.7057 0 → 0 (π → π∗) 25.5
42.6 42.60 → 1 (π → π∗) 18.3 . . . . . .−1 → 0 (π → π∗) 18.4 . . .
. . .−1 → 1 (π → π∗) 26.0 44.3 44.3
S5 8.131 8.298 8.317 0.0000 0.0005 0.0003 0 → 2 (π → R) 90.2 . .
. . . .−2 → 0 (σ → π∗) . . . 69.4 64.7b−3 → 1 (σ → π∗) . . . 14.3
18.3b
aH=HOMO, L=LUMO, R=Rydberg-type orbital.bOccurs as S6 in the
original FDE-ADC(2) calculation.
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204103-6 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 3. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated benzene (blue) to the
supersystem C6H6–HFapical andthe FDE-ADC(2) calculation (red) to
the supersystem. The blue bars cor-respond to the influence of the
environment on the excitation energies andoscillator strengths
(solvatochromic shift) while the red bars indicate theaccuracy of
the FDE-ADC(2) calculation in comparison with the ADC(2)calculation
of the full system.
In a second geometrical arrangement, the hydrogenfluoride is
placed side-on and in plane with the benzene(Fig. 2 right). The
geometry of benzene is slightly distortedby the hydrogen fluoride
and differs therefore marginallyfrom the geometry used in the
system above. The Hbenzene–Fdistance is 2.074 Å. However, the
frontier orbitals are visuallyindistinguishable to the orbitals of
the unperturbed benzeneor the orbitals of benzene of the previous
system due tothe weak system-environment interaction. The five
lowest
electronically excited states consist of four π → π∗
transitionsand one Rydberg state as the S5 state.
Calculating the supersystem, the frontier orbitals ofbenzene are
almost identical to the frontier orbitals of theisolated benzene
and only slightly distorted by the hydrogenfluoride. Only the
LUMO+2, which is a Rydberg orbital in thecase of the isolated
benzene, is now localized on the hydrogenfluoride and the original
Rydberg orbital located at the benzeneis the LUMO+3. The excited
states are more influenced bythe hydrogen fluoride in plane with
the benzene ring thanperpendicular to it. The S1 to S4 states are
still characterizedas local π → π∗ transitions on the benzene while
the S5 state isa mixed Rydberg state delocalized over benzene and
hydrogenfluoride.
In the FDE-ADC calculation, a higher lying virtualRydberg
orbital is lowered and becomes the LUMO, but thisorbital does not
contribute to any local π → π∗ transitions.Since the energy of the
higher lying unoccupied virtualorbitals (LUMO+1 to LUMO+10) is
practically identicalto the supersystem calculation, the character
of the excitedstates remains the same even if the number of the
electronaccepting orbital is increased by 1. The excited states S3
andS4 are, as in the supersystem calculation, degenerate.
Theexcited states of the supersystem and the isolated benzene
aswell as the FDE-ADC calculations are characterized in detailin
Table II.
The influence of the hydrogen fluoride molecule onto thebenzene
while it is located in-plane with the benzene-ringis reproduced
almost exactly by the FDE-ADC calculations.The largest deviation in
the excitation energies is lower than0.02 eV for the Rydberg state
and lower than 0.005 for theπ → π∗ transitions. The differences in
the excitation energiesand oscillator strengths between isolated
benzene and the
TABLE II. Excitation energies, oscillator strengths, and orbital
transitions for the five energetically lowest electronically
excited singlet states of isolatedbenzene, the supersystem benzene
with side-on hydrogen fluoride, and the FDE-ADC calculations in the
supermolecular expansion of benzene (A) and hydrogenfluoride
(B).
Excitation energies (eV) Oscillator strengthsOrb. trans.
Weight (%)
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC H+x → L+ya
Isol. Supersys. FDE-ADC
S1 5.287 5.435 5.435 0.0000 0.0001 0.0001 0 → 0 (π → π∗) 44.8
45.9 45.9b−1 → 1 (π → π∗) 44.8 43.8 43.8b
S2 6.677 6.791 6.792 0.0000 0.0000 0.0000 0 → 1 (π → π∗) 46.0
46.3 46.1b−1 → 0 (π → π∗) 46.0 45.9 46.2b
S3 7.521 7.659 7.660 0.7155 0.7447 0.7295 0 → 0 (π → π∗) 7.5 . .
. 43.2b0 → 1 (π → π∗) 36.7 44.0 . . .−1 → 0 (π → π∗) 36.7 44.3 . .
.−1 → 1 (π → π∗) 7.5 . . . 45.3b
S4 7.522 7.659 7.664 0.7162 0.7262 0.7200 0 → 0 (π → π∗) 36.7
43.1 . . .0 → 1 (π → π∗) 7.5 . . . 44.3b−1 → 0 (π → π∗) 7.5 . . .
44.2b−1 → 1 (π → π∗) 36.8 45.2 . . .
S5 8.131 8.177 8.163 0.0000 0.0004 0.0003 0 → 2 (π → R) 90.7
26.5 82.8b0 → 3 (π → R) . . . 63.6 . . .0 → 0 (π → π∗) . . . . . .
7.5
aH=HOMO, L=LUMO, R=Rydberg-type orbital.bThe low-lying Rydberg
orbital is neglected in counting, i.e., LUMO indices reduced by
1.
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204103-7 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 4. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated benzene (blue) to the
supersystem C6H6–HFside-onand the FDE-ADC(2) calculation (red) to
the supersystem. The blue barscorrespond to the influence of the
environment on the excitation energiesand oscillator strengths
(solvatochromic shift) while the red bars indicate theaccuracy of
the FDE-ADC(2) calculation in comparison with the ADC(2)calculation
of the full system.
FDE-ADC calculation to the supersystem calculation areshown in
Fig. 4.
2. Benzaldehyde with two water molecules
Benzaldehyde is able to form hydrogen bonds with a polarprotic
solvent. Because of this ability to act as a hydrogenacceptor, it
was chosen together with two water molecules asa model system (Fig.
5). Because of the hydrogen bond withinthe water dimer, the
strength of the hydrogen bond from thewater dimer to benzaldehyde
is increased.
FIG. 5. Molecular structure of the test system of benzaldehyde
with twowater molecules.
The frontier orbitals of isolated benzaldehyde optimizedas
supersystem are characterized as follows: HOMO-3: π,HOMO-2: n,
HOMO-1: π, HOMO: π, LUMO: π∗, LUMO+1:π∗, LUMO+2: Rydberg, LUMO+3:
Rydberg, LUMO+4: π∗.Analysis of the five energetically lowest
electronically excitedstates shows two states with n → π∗ character
(S1 and S4) and3 π → π∗ transitions. A detailed characterization is
given inTable III.
Going to the supersystem, the frontier orbitals are
hardlychanged but the HOMO and HOMO-1 change their energeticorder.
Since these two orbitals are almost degenerate it canbe considered
as negligible. However, this change has to beconsidered in the
characterization of the excited states. Incontrast, the excited
states are influenced significantly by thewater environment. As
expected, the hydrogen-bonded watermolecule stabilizes the
n-orbitals leading to a large increase
TABLE III. Excitation energies, oscillator strengths, and
orbital transitions for the five energetically lowest
electronically excited singlet states of isolatedbenzaldehyde, the
supersystem benzaldehyde with a hydrogen-bonded water dimer, and
the FDE-ADC calculations in the supermolecular expansion
ofbenzaldehyde (A) and the water dimer (B).
Excitation energies (eV) Oscillator strengthsOrb. trans.
Weight (%)
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC H+x → L+ya
Isol. Supersys. FDE-ADC
S1 3.692 3.897 3.891 0.0001 0.0001 0.0001 −2 → 0 (n → π∗) 73.9
74.3 73.9−2 → 4 (n → π∗) 13.2 9.1 . . .−2 → 5 (n → R) . . . . . .
6.9
S2 4.933 4.813 4.837 0.0104 0.0197 0.0174 0 → 0 (π → π∗) 18.5
61.9 59.10 → 1 (π → π∗) 20.5 . . . . . .−1 → 0 (π → π∗) 41.0 4.3
6.0−1 → 1 (π → π∗) 7.9 19.8 19.1
S3 5.771 5.554 5.595 0.3258 0.3444 0.3519 0 → 0 (π → π∗) 60.9
7.3 9.8−1 → 0 (π → π∗) 22.2 76.1 73.5
S4 6.272 6.751 6.742 0.0000 0.0059 0.0001 −2 → 1 (n → π∗) 83.6
82.1 81.4S5 6.884 6.753 6.784 0.3684 0.2942 0.2908 0 → 0 (π → π∗)
5.3 17.7 18.4
0 → 1 (π → π∗) 41.9 5.6 7.1−1 → 0 (π → π∗) 23.5 . . . . . .−1 →
1 (π → π∗) 12.4 50.3 47.7
0 → 4 (π → π∗) . . . 6.0 . . .aH=HOMO, L=LUMO, R=Rydberg-type
orbital.
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204103-8 Prager et al. J. Chem. Phys. 144, 204103 (2016)
of the excitation energies of the corresponding n → π∗
states.Simultaneously, the excitation energies of the π → π∗
statesare reduced. However, the energetic order and the characterof
the states are not changed by the environment (Table III).
In the FDE-ADC calculation, the interactions of thehydrogen
bonds are simulated almost quantitatively. Thedegeneracy of the
HOMO and HOMO-1 is retained andalso the n-orbitals are correctly
described. The excited statesin the FDE-ADC calculation show the
same influence of theenvironment on the n → π∗ as well as on the π
→ π∗ states asseen in the supersystem calculation. The full
characterizationis given in Table III.
In this example, the largest deviation in the excitationenergies
is lower than 0.01 eV for the n → π∗ states andlower than 0.05 eV
for the π → π∗ transitions. Especiallythe reproduction of the
influence of the hydrogen bonds onthe excited states is remarkable,
since hydrogen bonds havelarger orbital interactions compared to
dispersion interactionor polarization. The difference in excitation
energies andoscillator strengths of the isolated benzaldehyde and
theFDE-ADC(2) calculation to the supersystem calculation
aresummarized in Fig. 6.
Additionally, the benzaldehyde system was used as anexample
system for testing the basis set dependence ofFDE-ADC(2). The same
calculations as described above havebeen carried out using the
cc-pVTZ basis set. Generally,the excitation energies are lowered
due to the larger basis set.This is observed consistently in the
supersystem calculation aswell as in the FDE-ADC calculation. Both
in the supersystemcalculation and the FDE-ADC calculation, the S4
and S5state change their energetic order with respect to the
isolatedbenzaldehyde. This change is reproduced very nicely in
theFDE approach (see Table IV).
Calculating the difference of the deviation in
excitationenergies obtained with FDE-ADC and the supersystem
FIG. 6. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated benzaldehyde (blue) to
the supersystem of benzalde-hyde with a water dimer and the
FDE-ADC(2) calculation (red) to the super-system. The blue bars
correspond to the influence of the environment on theexcitation
energies and oscillator strengths (solvatochromic shift) while
thered bars indicate the accuracy of the FDE-ADC(2) calculation in
comparisonwith the ADC(2) calculation of the full system.
TABLE IV. Excitation energies and oscillator strengths for the
FDE-ADC(2)/cc-pVTZ calculation in the SE approach using
benzaldehyde assystem A and two water molecules as system B.
Exc. energies (eV) Osc. strength
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC
S1 3.623 3.827 3.820 0.0001 0.0002 0.0001S2 4.810 4.685 4.715
0.0127 0.0220 0.0188S3 5.549 5.350 5.399 0.3260 0.3378 0.3474S4
6.129 6.530 6.565 0.0000 0.2975 0.2876S5 6.645 6.599 6.587 0.3583
0.0001 0.0001
calculation in the two basis sets according to
∆Ωbasis =���(Ω
cc-pVDZFDE-ADC −Ω
cc-pVDZsupersystem
)−(Ω
cc-pVTZFDE-ADC −Ω
cc-pVTZsupersystem
) ��� , (20)with ٠as the excitation energy exhibits an error
smallerthan 0.008 eV in all investigated electronically excited
states.Hence, the deviation of FDE-ADC(2) from the
supersystemcalculation does not vary with the larger basis set. It
occursthat the error of the FDE-ADC(2) method compared to
thesupersystem calculation is largely independent of the basis
setsize. Certainly, the results of the ADC(2) and
FDE-ADC(2)calculations in general are affected using a larger basis
set.
3. Uracil with five water molecules
Uracil was chosen as a test system, because it has anaromatic π
system and in addition two hydrogen bond donors(B and D in Fig. 7)
and two oxygen atoms as hydrogenbond acceptors (A and C), thus
providing a large variety ofenvironment interactions. However,
uracil has also a moderate
FIG. 7. Geometrical arrangement of the supersystem of uracil and
five watermolecules.
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204103-9 Prager et al. J. Chem. Phys. 144, 204103 (2016)
size to allow benchmark calculations of the supersystemin
reasonable time. The geometrical arrangement of thesupersystem is
shown in Fig. 7.
The frontier orbitals of uracil, optimized in thesupersystem,
show π character for the HOMO-1 and HOMO,while the HOMO-2 and
HOMO-3 are n-orbitals and localizedmainly at the oxygen atom OA and
OC, respectively. TheLUMO and LUMO+2 exhibit π∗ character, the
LUMO+1 isa Rydberg orbital. Analyzing the five lowest
electronicallyexcited states of the isolated uracil, optimized in
thesupersystem structure, n → π∗ transitions are observed asthe S1,
S3 and S5 states. The S2 and S4 states are local π → π∗transitions.
The detailed characterization of the five lowestexcited states is
given in Table V.
In the supersystem structure, the water molecules
interactstrongly with the uracil forming hydrogen bonds. Two
watermolecules act as hydrogen bond donors for the OA, one asan
acceptor for NB, forming a hydrogen bond chain overone of the
previously mentioned water molecules to OA.Another hydrogen bond
chain is formed from ND over twowater molecules to OC. These
hydrogen bonds will influencethe n-orbitals and hence also the n →
π∗ excitations butwill do so differently for each state since the
interactionis more pronounced at the OA than at the OC. The rest
ofthe frontier orbitals are qualitatively unchanged except forthe
LUMO+1 and LUMO+2, which change their energeticorder. Analyzing the
five lowest electronically excited statesof the supersystem, the
excitation energy of the S1 state is, asexpected, largely increased
due to the stabilizing effects of theenvironment on the n-orbitals.
Accordingly, the influenceon the two other n → π∗ transitions is
weaker but still
significant. However, some orbitals delocalized over uraciland
one or more water molecules contribute only slightly tothe
excitation. A detailed description of the excited states isgiven in
Table V.
In the FDE-ADC calculations, the uracil was consideredas system
A while all 5 water molecules are used asthe environment B. As in
the supersystem calculation, theenergetic order of the LUMO+1 and
LUMO+2 is changedcompared to the isolated uracil calculation. The
remainingfrontier orbitals are essentially identical to the
orbitals ofisolated uracil, but higher lying virtual orbitals are
largelydistorted accounting for the effect of the environment.
Theresults of the FDE-ADC calculations are in very goodagreement
with the supersystem benchmark. The largest shiftin excitation
energy (S1) and the largest shift in oscillatorstrength (S4) are
well reproduced. Also, the different influenceof the hydrogen bonds
on the excited states is included inthe FDE-ADC calculation. The
characterization of the excitedstates is given in Table V.
For this system, the largest error is 0.09 eV for theexcitation
energies and 0.02 for the oscillator strength,which is slightly
higher than in the previous systems. Still,considering the strong
interaction between uracil and the waterenvironment, this error is
acceptable for this approximatetreatment of the environment. The
slightly larger error arisesbecause some orbitals, localized on the
water molecules ofthe environment, contribute slightly to the
excitations inthe supersystem but are not considered in the
FDE-ADCcalculation. Comparison of the FDE-ADC calculation and
theisolated uracil calculation to the supersystem is presented
inFig. 8.
TABLE V. Excitation energies, oscillator strengths, and orbital
transitions for the five energetically lowest electronically
excited singlet states of isolated uracil,the supersystem uracil
with five hydrogen-bonded water molecules, and the FDE-ADC
calculations in the supermolecular expansion of uracil (A) and the
fivewater molecules (B).
Excitation energies (eV) Oscillator strengthsOrb. trans.
Weight (%)
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC H+x → L+ya
Isol. Supersys. FDE-ADC
S1 4.440 5.143 5.108 0.0001 0.0016 0.0007 −2 → 0 (n → π∗) 77.2
72.2 72.9−3 → 0 (n → π∗) . . . . . . 5.8−4 → 0 (n → π∗) . . . 4.6 .
. .
S2 5.439 5.215 5.291 0.2261 0.2371 0.2162 0 → 0 (π → π∗) 81.3
80.7 80.1S3 5.866 6.211 6.221 0.0000 0.0001 0.0002 −2 → 0 (n → π∗)
4.6 . . . 5.8
−2 → 1 (n → π∗) 35.7b 20.0 . . .−3 → 0 (n → π∗) 28.3 17.4 31.0−3
→ 1 (n → π∗) 16.4b 10.8 . . .−4 → 0 (n → π∗) . . . 20.2 . . .−4 → 1
(n → π∗) . . . 5.6 . . .−2 → 6 (n → R) . . . . . . 24.8−3 → 6 (n →
R) . . . . . . 13.6
S4 6.194 6.292 6.320 0.0311 0.0963 0.0850 −1 → 0 (π → π∗) 85.9
84.4 84.1S5 6.468 6.657 6.749 0.0001 0.0005 0.0021 −2 → 1 (n → π∗)
38.0b 22.8 . . .
−3 → 0 (n → π∗) 44.6 15.3 45.9−3 → 1 (n → π∗) . . . 12.4 . . .−4
→ 0 (n → π∗) . . . 20.4 . . .−4 → 1 (n → π∗) . . . 6.6 . . .−2 → 6
(n → R) . . . . . . 19.1−3 → 6 (n → R) . . . . . . 9.2
aH=HOMO, L=LUMO, R=Rydberg-type orbital.bOrbital index for
LUMO+1 and LUMO+2 interchanged.
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204103-10 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 8. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated uracil (blue) to the
supersystem of uracil with five watermolecules and the FDE-ADC(2)
calculation (red) to the supersystem. Theblue bars correspond to
the influence of the environment on the excitationenergies and
oscillator strengths (solvatochromic shift) while the red
barsindicate the accuracy of the FDE-ADC(2) calculation in
comparison with theADC(2) calculation of the full system.
B. Results of FDE-ADC(2) using the reassemblingof density matrix
approach
In this chapter, the newly developed reassembling ofdensity
matrix (RADM) approach was used within the FDE-ADC calculations.
Using RADM, the density matrix forthe FDE-ADC(2) calculation is
conducted within the basisfunctions of A only leading to a
substantial decrease of thecomputational costs. The neglect of the
embedding potentialon the basis functions on B induces, however, an
additionalerror, which is also discussed in this section.
As in the previous chapters, the FDE-ADC calculationis tested
against the supersystem calculation and the isolatedsystem A. The
systems benzene with one hydrogen fluorideside-on, benzaldehyde
with two water molecules and uracilwith five water molecules were
chosen as test systems.
1. Benzene with one hydrogen fluoridemolecule side-on
The FDE-ADC calculations on benzene with onehydrogen fluoride
show almost identical orbitals comparedto the isolated benzene.
Only the LUMO+2 Rydberg orbitalshows a distorted surface in the
region pointing towardsthe hydogenfluoride molecule. In contrast to
using thesupermolecular expansion for FDE-ADC (SE-FDE-ADC) nohigher
lying virtual orbital is lowered to become the LUMO.HOMO-1, HOMO,
LUMO and LUMO+1 exhibit π andπ∗-character, respectively. The
character of the five lowestelectronically excited states is the
same as in the supersystemcalculation. The S1 to S4 are local π →
π∗ transitions, the S5 isa Rydberg state. As in the supersystem,
the S3 and S4 states aredegenerate. In this case, the state
corresponding to the S4 ofthe supersystem calculation shows a
slightly lower excitationenergy and becomes the S3 state (Table
VI).
Using RADM, the excited states are in very goodagreement with
the supersystem calculation and almostidentical to the SE-results.
The differences of RADM-FDE-ADC to the supersystem are below 0.006
eV for the localπ → π∗ transitions and below 0.03 eV for the
Rydberg state.In comparison to the SE-FDE-ADC(2) results, the error
ofthe RADM approximation alone is smaller than 0.04 eVfor the
excitation energy of the Rydberg state and almostzero (0.0001 eV)
for the local π → π∗ excitation energies.In summary, the
differences of SE-FDE-ADC, RADM-FDE-ADC and the isolated benzene
calculations to the supersystemcalculation are collected in Fig.
9.
2. Benzaldehyde with two water molecules
Analogous to the SE calculations, benzaldehyde wasused as system
A. The orbitals obtained in the RADM-FDE-ADC calculation are very
similar to the orbitals ofthe isolated benzaldehyde and only
slightly distorted. Thefrontier orbitals HOMO-1 to LUMO+1 and
LUMO+4 showπ and π∗-symmetry, respectively. The HOMO-2 is an
n-orbitallocalized at the oxygen and the LUMO+2 and LUMO+3 are
TABLE VI. Excitation energies, oscillator strengths, and orbital
transitions for the FDE-ADC calculation usingthe RADM approach for
benzene as system A and one hydrogen fluoride molecule in plane as
system B.
Excitation energies (eV) Oscillator strengthsOrb. trans.
RADM ∆Ωa ∆Ωb f ∆ f a ∆ f b H+x → L+y Weight (%)
S1 5.436 0.001 0.001 0.0001 0.0000 0.0000 0 → 0 (π → π∗) 46.0−1
→ 1 (π → π∗) 43.8
S2 6.792 0.001 0.000 0.0000 0.0000 0.0000 −1 → 0 (π → π∗) 46.20
→ 1 (π → π∗) 46.0
S3 7.661 0.002 0.001 0.7298 −0.0149 0.0003 0 → 0 (π → π∗) 43.2−1
→ 1 (π → π∗) 45.3
S4 7.665 0.006 0.001 0.7184 −0.0079 −0.0016 0 → 1 (π → π∗)
44.4−1 → 0 (π → π∗) 44.1
S5 8.199 0.022 0.036 0.0006 0.0001 0.0003 0 → 2 (π → R)
90.3aReferenced to the supersystem calculation.bReferenced to the
SE-FDE-ADC(2) calculation.
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204103-11 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 9. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated benzene (blue) to the
supersystem C6H6–HFside-on, theSE-FDE-ADC(2) calculation (red) to
the supersystem, and the RADM-FDE-ADC(2) calculation (green) to the
supersystem. The blue bars correspond tothe influence of the
environment on the excitation energies and oscillatorstrengths
(solvatochromic shift) while the red and green bars indicate
theaccuracy of the SE-FDE-ADC(2) and RADM-FDE-ADC(2) calculations
incomparison with the ADC(2) calculation of the full system. The
SE-FDE-ADC(2) results are taken from Sec. IV A 1.
Rydberg orbitals. The five lowest electronically excited
statesare closely related to the excited states of the
supersystemexhibiting a local π → π∗ transition in the excited
states S2,S3, and S5 and an n → π∗ transition in the excited states
S1and S4.
Also for this test, the agreement of the RADM-FDE-ADC
calculation with the supersystem calculation is verygood, but
shows, as expected, slightly larger differences thanthe SE
calculation. The difference in the excitation energiescompared to
the supersystem calculation is below 0.08 eV forthe calculated
excited states. The error induced by the RADM
FIG. 10. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated benzaldehyde (blue) to
the supersystem of benzalde-hyde with a water dimer, the
SE-FDE-ADC(2) calculation (red) to the super-system, and the
RADM-FDE-ADC(2) calculation (green) to the supersystem.The blue
bars correspond to the influence of the environment on the
excitationenergies and oscillator strengths (solvatochromic shift)
while the red andgreen bars indicate the accuracy of the
SE-FDE-ADC(2) and RADM-FDE-ADC(2) calculations in comparison with
the ADC(2) calculation of the fullsystem. The SE-FDE-ADC(2) results
are taken from Sec. IV A 2.
approximation alone compared to the SE-FDE-ADC(2) resultsis
smaller than 0.04 eV in excitation energies. A graphicalcomparison
is presented in Fig. 10.
Additionally, this system was benchmarked using thediffuse basis
set aug-cc-pVDZ in combination with theRADM-FDE-ADC(2) method.
Diffuse basis functions canbetter describe the peripheral regions
of the systems, in whichthe densities overlap. This results in a
better descriptionof the embedding potential and therefore in even
smallererrors in excitation energies. Since diffuse basis functions
areincluded, the orbitals as well as the orbital transitions of
theexcited states differ both for the supersystem calculation
and
TABLE VII. Excitation energies, oscillator strengths and orbital
transitions for the FDE-ADC calculation in theRADM approach using
benzaldehyde as system A and two water molecules as system B.
Excitation energies (eV) Oscillator strengthsOrb. trans.
RADM ∆Ωa ∆Ωb f ∆ f a ∆ f b H+x → L+y Weight (%)
S1 3.877 −0.020 0.014 0.0001 0.0000 0.0000 −2 → 0 (n → π∗)
74.1−2 → 4 (n → π∗) 12.2
S2 4.851 0.038 0.014 0.0164 −0.0033 −0.0010 0 → 0 (π → π∗)
58.0−1 → 0 (π → π∗) 6.6−1 → 1 (π → π∗) 20.4
S3 5.628 0.075 0.033 0.3490 0.0046 −0.0029 0 → 0 (π → π∗) 10.8−1
→ 0 (π → π∗) 72.6
S4 6.731 −0.020 −0.011 0.0001 −0.0058 0.0000 −2 → 1 (n → π∗)
83.9S5 6.803 0.050 0.019 0.2979 0.0037 0.0071 0 → 0 (π → π∗)
18.6
0 → 1 (π → π∗) 8.1−1 → 1 (π → π∗) 48.4
aReferenced to the supersystem calculation.bReferenced to the
SE-FDE-ADC(2) calculation.
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204103-12 Prager et al. J. Chem. Phys. 144, 204103 (2016)
TABLE VIII. Excitation energies and oscillator strengths for the
FDE-ADC(2)/aug-cc-pVDZ calculation in the RADM approach using
benzalde-hyde as system A and two water molecules as system B.
Exc. energies (eV) Osc. strength
State Isol. Supersys. FDE-ADC Isol. Supersys. FDE-ADC
S1 3.562 3.765 3.761 0.0001 0.0002 0.0002S2 4.812 4.682 4.716
0.0132 0.0234 0.0194S3 5.486 5.282 5.338 0.3308 0.3393 0.3534S4
5.943 6.351 6.350 0.0001 0.0154 0.0019S5 6.042 6.403 6.397 0.0115
0.0002 0.0001
the RADM-FDE-ADC(2) calculation compared to the resultsobtained
with the cc-pVDZ basis set. Applying the diffusebasis set
aug-cc-pVDZ, the error of the RADM-FDE-ADC(2)calculation compared
to the supersystem calculation is smallerfor all excitation
energies in this system compared to theresults obtained with the
cc-pVDZ basis set. (See Tables VIIand VIII.)
3. Uracil with five water molecules
In the strongly interacting system of uracil with five
watermolecules, the orbitals are hardly perturbed using
RADM-FDE-ADC. HOMO-2 and HOMO-3 exhibit n-type symmetrywhile the
frontier orbitals from HOMO-1 to LUMO+1 areπ and π∗ type orbitals,
respectively. The five energeticallylowest excited states are
investigated. Their character isconserved compared to the
supersystem calculation. Localπ → π∗ transitions can be observed in
the S2 and S4 whilethe remaining excited states show n → π∗
transition character.(See Table IX.)
As in the SE-FDE-ADC calculation, the difference to
thesupersystem calculation is slightly larger than in the
previousexamples due to the neglected contributions localized onthe
environment, i.e., the five water molecules. The largestdifference
in excitation energies is about 0.1 eV. The errorinduced by the
RADM approximation alone compared to the
FIG. 11. Excitation energy difference (left) and oscillator
strength difference(right) of the isolated uracil (blue) to the
supersystem of uracil with five watermolecules, the SE-FDE-ADC(2)
calculation (red) to the supersystem, and theRADM-FDE-ADC(2)
calculation (green) to the supersystem. The blue barscorrespond to
the influence of the environment on the excitation energiesand
oscillator strengths (solvatochromic shift) while the red and green
barsindicate the accuracy of the SE-FDE-ADC(2) and
RADM-FDE-ADC(2)calculations in comparison with the ADC(2)
calculation of the full system.The SE-FDE-ADC(2) results are taken
from Sec. IV A 2.
SE-FDE-ADC(2) results is smaller than 0.07 eV in
excitationenergies. Considering the large shift of the excitation
energieswith up to 0.7 eV due to the environment, this difference
is inexcellent agreement with the reference calculation (Fig.
11).
For this system, the computation time was alsoinvestigated. The
calculation of the supersystem takes about73 h CPU time on one core
keeping all data for the ADCcalculation in memory (Intel Xeon
E7-4870v2 2.3 GHz,software compiled using Intel C++ and Fortran
compilers46
v15.0 in combination with the MKL library47). In contrast,
thecomplete FDE-ADC calculations using the RADM approachwhich
consist of the MP(2) calculation on system A in thebasis functions
of A, the HF calculation of system A in thebasis functions of A and
B, the HF calculation of system
TABLE IX. Excitation energies, oscillator strengths, and orbital
transitions for the FDE-ADC calculation usingthe RADM approach for
uracil as system A and five water molecules as system B.
Excitation energies (eV) Oscillator strengthsOrb. trans.
RADM ∆Ωa ∆Ωb f ∆ f a ∆ f b H+x → L+y Weight (%)
S1 5.043 −0.100 −0.065 0.0004 −0.0011 −0.0003 −2 → 0 (n → π∗)
73.8−3 → 0 (n → π∗) 5.9
S2 5.319 0.104 0.028 0.2092 −0.0279 0.0070 0 → 0 (π → π∗) 81.4S3
6.198 −0.013 −0.023 0.0001 0.0000 −0.0001 −2 → 0 (n → π∗) 6.0
−2 → 1 (n → π∗) 31.8−3 → 0 (n → π∗) 30.2−3 → 1 (n → π∗) 16.9
S4 6.331 0.039 0.011 0.0720 −0.0243 0.0130 −1 → 0 (π → π∗)
85.7S5 6.763 0.106 0.014 0.0010 0.0005 −0.0011 −2 → 1 (n → π∗)
24.6
−3 → 0 (n → π∗) 48.4−3 → 1 (n → π∗) 10.0
aReferenced to the supersystem calculation.bReferenced to the
SE-FDE-ADC(2) calculation.
-
204103-13 Prager et al. J. Chem. Phys. 144, 204103 (2016)
FIG. 12. Comparison of the computational cost in CPU time of a
supersys-tem calculation and the RADM-FDE-ADC method.
B in the basis functions of A and B, the creation of
theembedding potential, and the final FDE-ADC(2) calculationsin the
basis function of A takes about 3 h on the samemachine. This
amounts to a saving of 70 h or more than95% of the time needed for
the supersystem calculation. Forlarger systems with even more
environment molecules, thepercentage of time saved will be
drastically larger due tothe formal O(N5)-scaling of the ADC(2)
calculation for thesupersystem (with N being the number of basis
functions).
A comparison of the computational time is shown inFig. 12. The
size of the environment is successively increasedfrom 1 to 5 water
molecules. The computational cost of thesupersystem calculation
increases dramatically already for asmall environment due to the
scaling of ADC(2). Hence, thecomputational cost is dominated by the
calculation of theembedded species for a small environment. Of
course, thistrend holds only for environments up to the same size
assystem A. The calculation of the environment scales formallywith
O(N3), which will become the most time-consuming stepfor large
environments (e.g., more than 100 water molecules).But for such
large systems, a full ADC calculation is no longerfeasible.
V. DISCUSSION AND CONCLUSION
In this work we have introduced a wavefunctionembedding approach
based on the combination of ADCand linearized FDET. Three molecular
model systems werestudied using two different FDE-ADC techniques in
whichthe environment consisted of up to five water molecules.
In the case of the supermolecular expansion (SE-FDE-ADC), we
also tested different spatial arrangements andanalyzed the basis
set dependence. Although polarizationof the environment is
neglected when using a frozen density,the results from this study
show that FDE-ADC is accurateenough to be employed in practical
calculations for larger
systems. Even large effects on the excited state due to
stronginteractions with the environment were described
correctly.Also large changes in oscillator strength due to the
effect ofthe environment were reproduced with satisfactory
accuracyby both FDE-ADC methods. The error of FDE-ADC islarger when
orbitals of the environment contribute to theexcitation. This
cannot be captured by FDET. In thesecases, the influence of the
environment can both be over-or underestimated depending on the
effect of the missingenvironmental orbital.
While SE-FDE-ADC yields excitation energies very closeto the
supersystem results (MAE = 0.025 eV), it providesno computational
advantage over an ADC calculation ofthe supersystem. On the other
hand, SE-FDE-ADC is wellsuited for an analysis of the FDE-ADC
method since apartfrom linearization of the non-additive energy
functionalEnadxc,T[ρA, ρB] no further approximations are
introduced.
With the second variant, RADM, it is possible toapproach a
system size of practical relevance due to thetruncation of the
embedding potential to the basis functionsof system A. Although
accompanied by the introduction ofnew approximations, the RADM
variant performs comparablywell (MAE = 0.040 eV). Computation of
the reassembleddensity matrix requires an additional Hartree-Fock
calculationof the supersystem and therefore increases the scaling
bythe term O(N3
A+B). Using the current implementation, also
investigations of chromophores in solution at FDE-ADC(3)level of
theory should be feasible using the existing ADC(3)implementation
in Q-Chem.48 Also, state-of-the-art tools forsystematic analysis of
electronic excitations can be combinedwith the FDE-ADC approach for
a detailed investigation ofthe influence of the environment on the
character of excitedstates.49,50
Note that the evaluation of the total energy was not neededin
the present work because of the relation for the excitationenergy
given in Eq. (18) holding for linearized FDET. Also,the evaluation
of the total energy (see Eq. (15)) involves notonly expectation
values of some quantum operators but alsoexplicit integration over
the real space.
A future implementation of FDE-ADC should includethe so called
Monomer Expansion (ME). In this approach, thecalculation of the
subsystem involves only the respective basisfunctions of one
system, which reduces the computational costeven further without
inducing an error due to a truncation asin the RADM approach.
In total, the FDE-ADC method is a promising approachfor
considering environmental effects on electronically excitedstates.
The error of this method is lower than the intrinsicerror of the
used ADC method and using the RADMapproximation explicit treatment
of extended environments isalready feasible. In further studies,
charge-transfer complexesare going to be investigated in polar
solvents using the RADM-FDE-ADC method.
ACKNOWLEDGMENTS
This research was supported by grants from the SwissNational
Science Foundation (No. 200021_152779). S.P.thanks the Heidelberg
Graduate School “Mathematical and
-
204103-14 Prager et al. J. Chem. Phys. 144, 204103 (2016)
Computational Methods for the Sciences” (HGS MathComp)for
financial support.
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