A hierarchy of local Coupled Cluster Singles and Doubles response methods for
Ionization Potentials
Gero Wälz,1 Denis Usvyat,1 Tatiana Korona,2 and Martin Schütz1, ∗
1
Institute of Physical and Theoretical Chemistry, University of Regensburg
Regensburg, D-93040, Germany
2
Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
(Dated: February 8, 2016)
We present a hierarchy of local coupled cluster (CC) linear response (LR) methods to calculate
ionization potentials (IPs), i.e., excited states with one electron annihilated relative to a ground
state reference. The time-dependent perturbation operator V(t), as well as the operators related to
the first-order (w.r. to V(t)) amplitudes and multipliers thus are not number conserving and have
half integer particle rank m. Apart from calculating IPs of neutral molecules, the method offers also
the possibility to study neutral radicals as “excited states” of a reference closed-shell anion.
It turns out that IPs require a higher-order treatment than excitation energies; an IP-CC LR
method corresponding to CC2 LR or ADC(2) provides insufficient accuracy. We therefore systematically extended the order w.r. to the fluctuation potential of the IP-CC LR Jacobian up to
IP-CCSD LR, keeping the excitation space of the first-order (w.r. to V(t)) cluster operator restricted to the m = 12 ⊕ 32 subspace and the accuracy of the zero-order (ground-state) amplitudes
at the level of CC2 or MP2.
For the more expensive diagrams beyond the IP-CC2 LR Jacobian we employ local approximations. The implemented methods are capable of treating large molecular system with hundred atoms
or more.
I.
INTRODUCTION
During the past decade efficient implementations of
coupled cluster (CC) linear response theory for excitation energies and properties of excited states [1] became
available, which are applicable to large molecular systems
that previously could be treated only at the level of timedependent density functional theory (TD-DFT). These
CC response methods are essentially based on secondorder CC models like CC2 [2] (note that also the algebraic diagrammatic construction scheme through second
order, ADC(2) [3–7] can be considered as a CC linear
response method based on an unitary CC ansatz truncated at second order [8]). A particularly efficient implementation of CC2 linear response is available via the
TURBOMOLE program package; it relies on density fitting
(DF) and a partitioning of the eigenvalue problem [9, 10],
and also nuclear energy gradients are available [11–14].
These methods still employ spatially delocalized canonical molecular orbitals and therefore have a computational
cost scaling of O(N 5 ) with molecular size N . This scaling is reduced to O(N 4 ) in the tensor hypercontraction
based CC2 schemes of Martı́nez et al. [15, 16], which
are going beyond a DF based factorization of the fourindexed electron repulsion integrals (ERIs) by providing
a decomposition of the ERIs in terms of two- rather than
three-indexed intermediates.
In our group we have developed over the years an efficient CC2 linear response scheme based on localized orbitals and a rigorous exploitation of the locality of cor-
∗
[email protected]
relation effects. These local CC2 linear response methods enable calculations of properties of singlet and triplet
electronically excited states including excitation energies
[17, 18], orbital un-relaxed and relaxed first-order properties [19–22], and analytic gradients w.r. to nuclear displacements [23, 24]. By virtue of the latter it is possible
to perform geometry optimizations of molecular systems
like chlorophyllide a. DF is employed rigorously to factorize ERIs and related diagrams. Local approximations
are applied both to ground-state (or zeroth-order w.r.
to the time-dependent perturbation) amplitudes and Lagrange multipliers, and to the left and right eigenvectors of the Jacobian representing the individual excited
states, yet only to the doubly excited parts thereof, while
the singly excited parts remain un-truncated. Note that
in CC models like CC2 the singly excited amplitudes are
considered as zeroth-order parameters w.r. to the fluctuation potential, since they have to carry the burden
of describing orbital relaxation effects in the excitation
energies, i.e., in the positions of the poles of the linear
response function.
Local approximations rely on spatially localized basis
functions to span occupied- and virtual spaces, respectively, as specified by the Hartree-Fock reference determinant, in order to benefit from the short-range decay
behavior of dynamic electron correlation in non-metallic
systems. Usually, the localized molecular orbitals spanning the occupied space (LMOs) are obtained by unitary
transformation of the original occupied Hartree-Fock orbitals, with the unitary transformation matrix being determined according to a certain localization criterion like
Pipek-Mezey [25] or Boys [26]. Localized functions spanning the virtual space can be obtained by projection of
the atomic orbitals onto the virtual space (PAOs) [27],
2
but also alternative choices of virtuals, like orbital specific virtuals (OSVs) [28] or pair natural orbitals (PNOs)
[29–31] have been used [32–41].
Once having a description of occupied and virtual
space in terms of localized functions, say LMOs and
PAOs, it is possible to truncate a priori ground-state
amplitudes and multipliers on the basis of spatial locality criteria: the LMO pair list of doubles amplitudes and
multipliers can be restricted depending on the separation of the two LMOs, and pair-specific subspaces of the
virtual space (domains) can be specified for each pair of
the restricted pair list. This reduces the scaling of the
amount of relevant amplitudes and multipliers w.r. to
system size N from O(N 4 ) to O(N ). To apply local approximations to the eigenvectors of the Jacobian is less
straightforward and crucially depends on the nature of
the individual excited states represented by the related
eigenvectors. In Ref. 18 we introduced (in the context of
a multistate calculation) an adaptive scheme for specifying state-specific local approximations for each individual state during the iterative Davidson diagonalization
of the Jacobian by analyzing certain intermediates. Furthermore, by employing Almlöf’s Laplace trick [42, 43]
the LCC2 eigenvalue problem was partitioned such that
only an effective singles Jacobian has to be diagonalized,
as it is naturally possible in the canonical orbital basis
[9].
In the present work we extend our local CC linear response scheme to the calculation of ionization potentials
(IPs). To this end, the operators related to the time- or
frequency-dependent first-order amplitudes and multipliers in the quasienergy Lagrangian [1] are no longer number conserving, but contain one excess annihilator. This
implies that the reference state lives in the Fock subspace
F (M, N ), while the ionized states live in the Fock subspace F (M, N − 1), where M denotes the available spin
orbitals and N the number of electrons of the system.
Apart from the possibility of calculating ionization energies of neutral closed-shell systems such a method also
allows to study neutral doublet radicals by choosing as a
reference the related negatively charged closed-shell reference. This is a potentially very interesting application
area, in particular once also first-order properties and
gradients w.r. to nuclear displacements become available
for ionized states.
Similar methods for calculating IPs and properties of
ionized states have been presented before in the equationof-motion (EOM) CC framework [44–54], yet to our
knowledge so far only for non-local canonical or natural orbitals, and without a DF based factorization of the
Jacobian transforms. Apart from that, EOM-CC or CC
linear response programs for the calculation of excitation
energies can be exploited to obtain IPs by addition of a
very diffuse orbital to the space of the virtual orbitals
[48]. For canonical methods however, the computational
cost, i.e., the scaling thereof w.r. to system size N is
quite high, and the methods thus are limited to rather
small molecular systems. In contrast, the methods pre-
sented here exploit locality of dynamic correlation and
DF of ERIs such that molecular systems beyond hundred
atoms without symmetry can be treated.
The obvious alternative to linear response or EOM approaches of calculating IPs (or electron affinities) is the
∆ approach: here, the IPs (or EAs) are obtained as the
difference of the total energies of the N and the N ∓ 1
electron system. This ∆ approach is expected to provide
higher accuracy at a given level of the correlation treatment than the corresponding linear response or EOM
method, since it includes orbital relaxation explicitly for
both the N and the N ∓ 1 electron system (via optimized Hartree-Fock orbitals). On the other hand, the ∆
approach has clear limitations. One of these is that it
only permits the computation of the lowest ionized state
in each irreducible representation of the molecular point
group, i.e., only the lowest state for a molecule without
symmetry. Higher lying IPs may be of interest e.g. for
estimating excitation energies of radicals, which can be
accessed by the linear response or EOM methods by specifying an appropriate closed-shell anion as the (ground
state) reference. A further disadvantage of the ∆ approach is its higher computational cost: firstly, it is now
unavoidable to deal with an open shell system, and one
has to make sure also that the N and N − 1 electron systems are treated in a balanced way, otherwise potentially
large systematic errors can be introduced by subtracting
two similar numbers. Secondly, as discussed in section II
the diagrams that appear in the transforms of the trial
vector with the IP-CC Jacobian have a simpler structure than those appearing in the ground state amplitude
equations due to the lack of one particle line. In particular, no expensive four-external ladder diagrams appear in
the methods proposed in this work. It is therefore desirable to avoid these in the ground state calculation as well
and we restrict the ground state method in the following to either MP2 or CC2. It is shown that nevertheless
an accuracy close to that of the much more expensive
EOMIP-CCSD method is achieved.
II.
THEORY
In the following we briefly sketch the CC response theory for describing ionized states, following the review
about response theory for number-conserving perturbations in Ref. 1. As usually, we employ indices i, j, . . .
for occupied orbitals, i.e., LMOs, a, b, . . . for virtual orbitals, i.e., PAOs, p, q, . . . for general (occupied or virtual) molecular orbitals, and capital indices P, Q, . . . for
fitting functions.
A.
Response theory for ionized states
As the time-dependent perturbation we introduce a
formal non-physical (non-particle conserving) operator,
3
where
which destroys and creates a particle, i.e.,
n
X
V(t) =
(0)
exp(−iωk t)V(ωk ),
V(ωk ) =
Y (ωk )Y,
Y=
X
Yp apβ +
a†pβ
,
(0)
(0)
T2 = t(0)
µ2 τµ2
(1)
T 1 (t) = t(1)
µ 1 (t)τµ 1
2
(1)
T 3 (t) =
2
I
X h0|X|ĀihĀ|Y |0i
−
+
I¯
ω − ωĀ
XX
ij
+
ω − ωI¯
X h0|apβ |ĀihĀ|a†qβ |0i
Ā
=
h0|Y |ĀihĀ|X|0i
ω + (EĀ − E0 )
ω − (EĀ − E0 )
¯ I|a
¯ qβ |0i
X X h0|a†pβ |Iih
Ā
I¯
XX
ab
Ā
−
¯ I|a
¯ jβ |0i
h0|a†iβ |Iih
ω − ωI¯
−
ω + ωI¯
−
¯ I|a
¯ jβ |0i
h0|a†iβ |Iih
−
2
τi = aiβ ,
ω + ωI¯
h0|aaβ |ĀihĀ|a†bβ |0i
ω + ωĀ
(2)
¯ are the ionized eigenstates living in F (M, N −1)
where |Ii
with related energies EI¯, and |Āi the electron attached
states living in F (M, N + 1) with related energies EĀ .
Apart from a symmetrization hhX; Y iiω corresponds to
the one-electron Green’s function [55]. It has poles for
ionization energies ωI¯ and electron affinities ωĀ . In
the following, we are only interested in the first part of
hhX; Y iiω containing the poles for ωI¯. That part contains bra and kets of only ionized states (apart from the
ground state), but not electron attached states. A ket
ionized state is generated from the ground state ket by
an operator with one excess annihilator (cf. eq. (7)).
The time-dependent CC wavefunction ansatz after isolation of the phase can therefore be written as
f = exp(T(0) + T(1) (t) + . . . )|0i
|CCi
2
(3)
= tia τia ,
1
τ ab ,
= tij
2 ab ij
= ti (t)τi ,
(5)
2
t(1)
µ 3 (t)τµ 3
2
τia = a†aα aiα + a†aβ aiβ ,
ω + ωĀ
h0|aaβ |ĀihĀ|a†bβ |0i
ω − ωĀ
¯ I|a
¯ qβ |0i
h0|a†pβ |Iih
h0|apβ |ĀihĀ|a†qβ |0i
(4)
a
= tij
a (t)τij .
Note that Einstein convention is used above and in the
following, i.e., repeated indices are implicitly summed
up; summations are written explicitly only if it is helpful
for clarity.
In eqs. (4) and (5) we employ the particle rank m of
the related operator, i.e., the number of elementary operators of an operator string divided by two, as subscript
indices in the individual Tm operators. We truncate the
particle rank at m = 2 in the cluster operator. Furthermore, by virtue of the 2n + 1 rule it is sufficient to
consider amplitudes up to first order w.r. to V(t) (the
order is given by the superscripted numbers in parenthesis). From eq. (5) it is clear that zeroth-order amplitudes
with half-integer particle rank, as well as first-order amplitudes with integer particle rank are all zero.
The operators τm in eq. (5) are all spin-adapted, i.e.,
hhX; Y iiω
X h0|X|Iih
¯ I|X|0i
¯
¯ I|Y
¯ |0i h0|Y |Iih
−
=
ω − (EI¯ − E0 )
ω + (EI¯ − E0 )
¯
pq
2
T1 = t(0)
µ1 τµ1
with the (frequency-dependent) perturbation strengths
Y (ωk ), and elementary annihilators apβ and creators
a†pβ . This is somehow reminiscent of the response theory
for treating triplet excited states, where also an artificial
triplet coupled V(t) is introduced. We point out that by
treating ionization and electron attachment processes together Y still is Hermitian. With that, V(t) is Hermitian
under the assumption of the usual symmetry properties,
i.e., ω−k = −ωk , and ∗Y (ωk ) = Y (ω−k ). Furthermore,
since V(t) is unphysical anyway, we can as well set the
“integrals” Yp in the second quantized form of operator Y
to one, for simplicity. With operator X = Y the resulting
exact linear response function [1] for such a perturbation
can then be written as
=
(1)
2
and
p
Y
+
(1)
T(1) (t) = T 1 (t) + T 3 (t) + . . . ,
k=−n
X
(0)
T(0) = T1 + T2 + . . . ,
(1)
,
ab
τij
= τia τjb ,
a
τij
=
τia τj .
(6)
(7)
The operators in eq. (6) with integer particle rank are
spin-conserving or singlet-coupled excitation operators,
generating a singlet state when being applied to the
closed shell reference determinant |0i. On the other hand,
the operators in eq. (7) with half-integer rank produce a
doublet state with S = MS = 21 when being applied to
|0i. Note that the LMO pair list is triangular for zeroth3
order m = 2 amplitudes tij
ab , while it is not for the m = 2
ij
first-order amplitudes ta .
The contravariant bra functions (forming a biorthonormal set with the ket functions produced by applying the
operators in eqs. (6) and (7) to |0i) take the form
1
†
h0| (τia ) ,
2
1
ab
ab †
hµ̃2 | = hΦ̃ab
,
ij | = h0| 2τij + τji
6
†
hµ̃ 21 | = hΦ̃i | = h0| (τi ) ,
1
a
a †
hµ̃ 23 | = hΦ̃aij | = h0| 2τij
+ τji
.
3
hµ̃1 | = hΦ̃ai | =
(8)
Applying the derivation outlined in detail for number
conserving V(t) in Ref. 1 one finally arrives at the time-
4
averaged second-order quasienergy Lagrangian
"
n
i E
D h
X
(1)
2n+1 (2)
{
L (t)}T =
0 V(−ωk ), T 1 (ωk ) 0
Mµm νl = hµ̃m |τνl | 0i ,
(15)
2
k=−n
h
E
i
D
(1)
(1)
0 + λ(0)
µm µ̃m V(−ωk ), T 1 (ωk ) + T 3 (ωk ) CC
2
−
metric
and
2
(1)
λ(1)
µm (−ωk )ωk tνl (ωk ) hµ̃m
ξµYl =
|τνl | 0i
D 0 + λ(1)
µm (−ωk ) µ̃m V(ωk )
#
h
i E
(1)
(1)
(0)
+ H , T 1 (ωk ) + T 3 (ωk ) CC , (9)
2
2
where |CCi = exp(T(0) )|0i is the unperturbed CC wavefunction, hµ̃0m | = hµ̃m | exp(−T(0) ), and
(1)
T(1)
m (ωk ) = tµm (ωk )τµm , with
n
X
(1)
tµm (t) =
t(1)
µm (ωk ) exp(−iωk t)
∂ 2 {2n+1 L(2) (t)}T
(1)
∂λµm (−ω)∂Y (ω)
= hµ̃0m |Y| CCi ,
(16)
with m and l both half-integer indices. The rhs ξ Y of
eq. (13) is non-zero (due to the creator part of Y), and
tY(ω) has poles for the singular matrix A − ωM, which,
according to Eq. (11) consequently leads to poles for
these ω also in hhX; Y iiω . The eigenvalues of the CC
Jacobian A above hence correspond to the IPs of the
molecular system as described by the CC model.
(10)
B.
k=−n
The CC2 model
(1)
(and the λµm (ωk ) being analogously defined). Furthermore, the particle-rank index m in eq. (9) runs over
(0)
m = 1, 2 for zeroth-order multipliers λµm , and over
(1)
m = 12 , 32 for first-order multipliers λµm (ωk ) and ampli(1)
tudes and tµm (ωk ), respectively. Note that in eq. (9) no
(1)
terms containing products of Tm (ωk ) can occur. Such
terms correspond to diagrams with two hole lines terminating in single line vertices, which can neither be closed
by any number of operators with integer particle rank,
nor in combination with one operator with half-integer
particle rank. This implies that the second derivative of
{2n+1 L(2) (t)}T w.r. to the first-order amplitudes vanishes, which means a substantial simplification relative
to CC response theory for electronically excited states.
For the first part of the linear response function (having
poles for the IPs) we therefore obtain
hhX; Y ii0ω =
d2 {2n+1 L(2) (t)}T
= η X tY(ω) + η Y tX
(−ω) ,
dX (−ω)dY (ω)
(11)
with
ηµYl =
∂ 2 {2n+1 L(2) (t)}T
(1)
∂Y (−ω)∂tµl (ω)
D h
i E
0
= 0 Y, τµ 1 0 δl 12 + λ(0)
µm hµ̃m |[Y, τµl ]| CCi ,
2
(12)
and m integer and l half-integer.
the stationary conditions
tY(ω)
is obtained from
(13)
with the CC Jacobian A,
Aµm νl
H(0) = F[0](0) + W[1](0) ,
(17)
with F[0](0) representing the Fock operator, and W[1](0)
the fluctuation potential (we indicate the order w.r. to
the fluctuation potential, and w.r. to V(t) with superscript numbers in brackets, and parenthesis, respectively). In the following we drop, for convenience, the
superscripts specifying the orders for the individual parts
of the Hamiltonian. The zeroth-order amplitudes in T(0)
are determined by the amplitude equations[2]
i E
D h
(0) 0 = µ̃1 Ĥ + Ĥ, T2 0 ,
D h
i E
(0) (18)
0 = µ̃2 Ĥ + F, T2 0 ,
with the similarity-transformed Hamiltonian Ĥ =
(0)
(0)
exp(−T1 )H exp(T1 ). Due to this similarity-transform, dressed integrals
(pqˆ|rs) = (µν|ρσ)Λpµp Λhνq Λpρr Λhσs ,
(19)
do occur. The coefficient matrices Λp and Λh transforming from AO basis (indexed by Greek letters µ, ν, ...) to
MO basis are defined as [17]
Λpµa = Pµa − Lµi tia0 Sa0 a , Λpµi = Lµi ,
(A − ωM) tY(ω) + ξ Y = 0,
Aµm νl − ωMµm νl =
The CC2 model relies on the Møller-Plesset partitioning of the Hamiltonian,
Λhµa = Pµa ,
Λhµi = Lµi + Pµa tia .
(20)
(0)
∂ 2 {2n+1 L(2) (t)}T
(1)
(1)
∂λµm (−ω)∂tµl (ω)
h
D
i
E
0 (0)
= µ̃m H , τνl CC ,
,
(14)
and depend on the zeroth-order singles amplitudes tµ1 .
In eq. (20) L, P, and S denote the LMO and PAO coefficient matrices, and the overlap matrix of the PAOs,
respectively. Objects involving dressed integrals are all
decorated by a hat in the following. For example, after
5
decomposing the dressed integrals defined in eq. (19) by
DF one obtains
P
−1
(Qˆ|rs),
(pqˆ|rs) = (pqˆ|P )ĉP
rs , with ĉrs = J
PQ
and JP Q = (P |Q),
(21)
with J being the Coulomb metric of the auxiliary fitting
functions, which are indexed by P, Q. Furthermore, the
dressed Fock matrix is defined as
X
fˆpq = ĥpq +
2(iiˆ|pq) − (iqˆ|pi) .
(22)
i
In our local approach the zeroth-order doubles ampli(0)
tudes in T2 are confined by a restricted pair list P0 and
pair-specific domains [ij], i.e.,
(0)
T2
1 X X ij ab
tab τij ,
=
2
(23)
ij∈P0 ab∈[ij]
h
E D h
i
i E
(0) µ̃ 12 Ĥ, τν 3 0
Ĥ, τν 1 exp(T2 ) 0
2
2
D h
D h
i E
i E  .
3
3
µ̃ 2 F, τν 3 0
µ̃ 2 Ŵ, τν 1 0
2
vij
a
=
P
ĉP
ai B̂j
+
fab uij
b
−
Saa0 uik
a0 fkj
−
Saa0 ukj
a0 fki ,
(26)
(27)
with the intermediates
Zik = −(kc|P )VicP ,
fc
P
ki
P
ik P
W i = cP
kb ũb , Via = t̃ab ckb ,
(28)
In eqs. (26–28) amplitudes and trial vectors decorated
by a tilde correspond to contravariant bra functions as
defined in eq. (8), i.e.,
ij
ji
ij
ij
ji
t̃ij
ab = 2tab − tab , and ũa = 2ua − ua .
(29)
Furthermore, we have dropped the explicit dependence
of trial vectors U and products V on the related ionized
states for better readability.
As a sidenote, the right matrix trial-vector product
V(I) = AU (I) for the ADC(2) [3, 4] aka TD-UCC[2]-H
[8, 24] Jacobian A is very similar to eqs. (26–28); only
the singles part differs, reading instead
2
¯ are obThe IPs ωI¯ for the lowest few ionized states |Ii
tained by solving the right eigenvalue problem
¯
¯ = ωI¯MR(I).
AR(I)
P
fc
ˆ
vi = −uj fˆji + uk Zik + fˆjb ũji
b − W k (ki|P )
and B̂iP = −uk (kiˆ|P ).
with P0 and [ij] determined as usually for ground state
calculations on the basis of spatial locality (cf. section
III for details).
The CC2 Jacobian for ionized states is obtained
by differentiation of the time-averaged second-order
quasienergy Lagrangian for the CC2 model, yielding
!
A 12 12 A 12 23
A=
(24)
A 32 12 A 32 23
D
µ̃ 12
=
m = 12 subspace only by adapting the Davidson procedure described in detail in Ref. 18 accordingly. However,
already the straightforward approach of diagonalization
in the m = 12 ⊕ 23 subspace is very efficient (due to the
occurrence of just one virtual orbital index), such that
the Laplace based approach provides little gain, except
perhaps for extensively large systems.
The expressions for the right matrix trial-vector product V(I) = AU (I) were derived by employing diagrammatic techniques. The final working equations (after factorizing ERIs by DF) are
(25)
To this end we employ a Davidson diagonalization variant
for non-symmetric matrices [56, 57] such that only matrix
trial-vector products V(I) = AU (I) (rather than the full
matrix A) are needed. Note a subtlety in the nomenclature here: I¯ ∈ {1, . . . , NI¯ ≤ NDav } denotes a particular
ionized state (NI¯ is equal to the number of states treated
in the multistate calculation), while I ∈ {1, . . . , NDav }
denotes a certain basis vector of the Davidson subspace,
which, in turn, belongs to a certain I¯ (at each Davidson
refresh, we have NDav = NI¯).
No state-specific local approximations are invoked on
¯ at that stage;
the trial vectors U (I) and eigenvectors R(I)
only the truncation of the zeroth-order doubles amplitudes (vide supra) is exploited (in the computation of the
intermediate ViaP , cf. eq. (28)). Moreover, the locality
in the orbitals is exploited for prescreening in the evaluation of the diagrams given below. We also implemented
a Laplace based approach to solve eq. (25) by partitioning A and solve an effective eigenvalue problem in the
1
P
vi = −uj fji + uk (Zik + Zki ) − fc W k (ki|P )
2
(30)
and all integrals and Fock matrix elements are undressed
(0)
(since tµ1 = 0 for ADC(2)).
C.
Additional higher-order diagrams
1.
IP-CCSD[1]CC2
The CC2 model for ionized states as specified above in
section II B does by itself not provide IPs of satisfactory
accuracy; we use it to generate initial guesses for the right
¯ and to generate initial state-specific loeigenvectors R(I)
cal approximations (vide infra). Due to the generation of
the electron hole, orbital relaxation effects are expected
to be of greater importance for IPs than for electronic excitation energies, where the CC2 model already provides
acceptable accuracy for many applications. In order to
improve on the CC2 model we add higher-order diagrams
to the CC2 Jacobian, while still sticking to the m = 21 ⊕ 32
excitation subspace. From eq. (24) it is clear that the
A 21 12 and A 21 32 submatrices are already complete in the
6
sense that they contain all possible diagrams, whereas for
the submatrices A 23 12 and A 32 32 this is not the case. The
latter are only correct to first- and zeroth-order w.r. to
the fluctuation potential W, respectively. Increasing the
order of each of these two submatrices by one yields
D h
h
i
i E
(0)
(31)
A 32 21 = µ̃ 32 Ŵ + Ŵ, T2 , τν 1 0 ,
2
i E
D h
(32)
A 23 23 = µ̃ 32 F̂ + Ŵ, τν 3 0 .
2
This implies the addition of two related third-order terms
to the m = 32 amplitude equations in the time-averaged
second-order quasienergy CC2 Lagrangian, from which
the Jacobian is obtained by differentiation (note that the
(1)
(1)
lowest orders w.r. to W of T 1 and T 3 is 0 and 1,
2
with the new intermediates
ij
P
P
ˆ
ˆ
BaP = uk cP
ka , Ŷa = (ij |P )Ba , and Ŷabc = (ab|P )Bc .
(34)
In contrast to standard IP-CC2 or ADC(2) diagrams,
which (apart from the contraction of the ground state
amplitudes in eq. (28) outside the Davidson iterations)
scale nominally at most as O(N 4 ) with molecular size N ,
these additional diagrams scale nominally as O(N 5 ). We
will employ in the following local correlation techniques
to reduce this scaling.
2.
IP-CCSD[2]CC2
2
[0](1)
respectively, i.e., T 3
= 0 [58]). Evidently, the A 23 32
2
block of the Jacobian is no longer diagonal in the canonical basis, which precludes the partitioning of the eigenvalue problem to an effective m = 12 eigenvalue problem,
i.e., the application of the Laplace trick (cf. section II B).
For that reason, the Fock matrix in A 32 32 could as well
taken as dressed, including so further higher-order terms
at no additional cost. For further reference we denote
this method as IP-CCSD[1]CC2 . This acronym implies
that A used in the eigenvalue problem (25) is an approximation to the CCSD Jacobian, correct to first order in
W on top of a CC2 ground state calculation. Accordingly, the pure CC2 approach outlined in section II B
corresponds to the acronym IP-CCSD[0]CC2 . Extending
IP-CCSD[0]CC2 just according to eq. (32), i.e., without
adding the second order contribution to A 32 21 in eq. (31)
indeed leads to a method for which the IPs of eigenstates
with predominantly m = 32 character are treated formally
at first order w.r. to the fluctuation potential (rather
than at zeroth order as in ADC(2) or CC2). However,
test calculations performed in the context of this work
have shown that such a method is not really superior to
IP-CCSD[0]CC2 ; sometimes the ionization potentials are
better than those of IP-CCSD[0]CC2 , but often worse.
A similar behavior was also found for excitation energies of the ADC(2)-x method [59], where, analogously,
just the doubles-doubles block of the ADC(2) Jacobian
is augmented by the first order term hµ̃2 |[W, τν2 ]|0i [60].
IP-CCSD[1]CC2 , on the other hand, is clearly superior to
IP-CCSD[0]CC2 , as demonstrated in section III.
Due to the replacement of the undressed by the dressed
Fock operator in A 32 32 the undressed Fock matrix elements in eq. (27) have to be substituted by their dressed
counterparts. The inclusion of the two higher-order
terms (31) and (32) in the IP-CCSD[1]CC2 method entails
the addition of the following terms to the IP-CCSD[0]CC2
matrix trial-vector product given in eq. (27),
jk ki
[1]
ik kj
P P
0
∆vij
=
S
t
Ŷ
+
t
Ŷ
−
V
B̂
− Ŷabc tij
aa
a
ab b
ia j
ba b
bc
P
ˆ P
+ (aiˆ|P )fc W j + Saa0 ukl
a0 (ki|P )ĉlj
− ukj (abˆ|P )ĉP − uik (abˆ|P )ĉP ,
b
ki
b
kj
(33)
A further step up on the ladder towards full IPCCSDCC2 is the IP-CCSD[2]CC2 method, where an additional second-order term is added to A 23 32 , i.e.,
D h
h
i
i E
(0)
A 23 32 = µ̃ 32 F̂ + Ŵ + Ŵ, T2 , τν 3 0 .
(35)
2
The Jacobian of the IP-CCSD[2]CC2 method is identical
to that of the EOMIP-CCSD(2) method presented by
Stanton and Gauss (Ref. 46), yet the latter is based on a
MP2 rather than a CC2 ground state calculation. Consequently, EOMIP-CCSD(2) corresponds to the acronym
IP-CCSD[2]MP2 . The IP-CCSD[2]CC2 method adds the
following further terms to eqs. (27) and (33),
[2]
P
kj
ik
ij
P int
∆vij
Wj
a = Saa0 Zik ua + Zjk ua + Zac uc + Via
lj ik
jk
il kj
kl
+ (til
ab uc + tab uc − t̃ab uc )Kbc
ijkl kl
− tij
ua ,
ab Yb + Y
(36)
with the new intermediates
P
Zac = −Vka
(kc|P ),
int
P
W i = (kb|P )ũki
b ,
P
kl
Ya = (ka|P )int W k , Y ijkl = tij
bc Kbc ,
ij
and Kab
= (ia|P )cP
jb .
(37)
Note that the length of the operator list for the pairs
kl of the exchange operators Kkl in eq. (36) remains
essentially unaffected by the locality of ground state amplitudes and trial vectors. Hence it is governed by the
R−3 decay of the exchange integrals themselves.
3.
IP-CCSD[f ]CC2 =IP-CCSDCC2
The full IP-CCSDCC2 method includes a further
fourth-order term in A 32 21 , namely that involving the
dressing of the Fock operator (with second-order ground
state singles), hence
D h
h
i E
i
(0)
A 23 12 = µ̃ 32 Ŵ + F̂ + Ŵ, T2 , τν 1 0 .
(38)
2
7
of this on the ionization energies is presumably small. To
further improve the methods one therefore has to go to
m = 12 ⊕ 32 ⊕ 25 excitation spaces and beyond. All this is
not further considered in the context of the present work.
TABLE I. List of individual IP-CCSD[k]M methods and eventually existing synonyms. Furthermore, the correctness (w.r.
to the fluctuation potential) of the ionization energies with
dominant m = 21 and m = 32 character (according to the
analysis of Ref. [61]), and the correctness of the A 3 1 and
2 2
A 3 3 submatrices is given. The related nominal scaling of the
2 2
computational cost of the additional Jacobian × trial vector
diagrams w.r. to the number of occupied (no ), virtual (nv ),
and fitting (Nf ) functions is also provided.
Methods
Synonyms
IP-CCSD[k]M
k M
0
1
2
0
1
2
f
f
b
MP2
MP2
MP2
CC2
CC2
CC2
CC2
CCSD
IP-ADC(2)
a
Order
}|
z
Scaling
{
ω1 ω3 A3 1 A3 3
2
2
2
EOMIP-CCSD(2) 2
IP-CC2
2
2
2
2
EOMIP-CCSD
2
2
0
1
1
0
1
1
1
1
2 2
2 2
1
2
2
1
2
2
2
2
0
1
2
0
1
2
2
3
n2o nv Nf
n2o n3v
n3o n2v
n2o nv Nf
n2o n3v
n3o n2v
n2o n2v
n2o n3v
a) the scaling of ground state calculation is n2o n2v Nf for all
methods except EOMIP-CCSD, where it is n2o n4v
b) Jacobian is symmetrized according to eq. (30)
This leads to the additional term in the matrix trialvector product,
[f ]
D.
ij ˆ
k
∆vij
a = −Saa0 t̃a0 b fkb u .
(39)
Obviously, the additional computational effort necessary to go from IP-CCSD[2]CC2 to IP-CCSD[f]CC2 is
only minor. We note in passing that for MP2 ground
state amplitudes eq. (39) obviously is zero. Hence,
IP-CCSD[2]MP2 is already equivalent to the complete
method IP-CCSD[f]MP2 .
Table I compiles the hierarchy of IP-CCSD[k]M methods explored in this work; synonyms to our method
acronyms are also listed, if available. Furthermore, the
order of correctness w.r. to the fluctuation potential of
the ionization energies with dominant m = 12 and m = 32
character, as well as the correctness of the A 32 12 and A 23 32
submatrices is given: all methods apart from k = 0
describe ionization energies with dominant m = 12 and
m = 32 character correct through second- and first-order,
respectively. The highest nominal scaling of the computational cost of the Jacobian × trial vector diagrams,
with which the method is augmented by increasing k is
also given in Table I. Note however, that the nominal
scaling of the EOMIP-CCSD method is governed by the
ladder diagrams of the ground state CCSD calculation
and thus has an overall nominal scaling of n2o n4v .
Within the excitation space m = 12 ⊕ 32 the method
can only be further improved on going to higher-order
ground state methods, i.e., by inclusion of higher-order
[2](0)
[2](0)
doubles and triples amplitudes tµ2 , tµ3 . The effect
Local approximations
As shown in Table I, the additional terms of the methods beyond the pure CC2 model (aka IP-CCSD[0]CC2 )
exhibit a higher-order scaling of the computational cost
with molecular size, namely O(N 5 ) instead of O(N 4 ).
Consequently, these terms are much more expensive than
an initial IP-CCSD[0]CC2 calculation. Note that for the
CC2 ground state calculations local approximations according to eq. (23) are invoked.
In order to reduce the computational cost of these additional terms we introduce additional local approximations focussing on the m = 32 trial vectors uij
a (I) and related matrix trial-vector products vij
a (I): primarily, the
(non-triangular) pairs ij are restricted to the pair list PI¯.
To this end, after the initial IP-CCSD[0]CC2 calculation,
a subset of important pairs is determined by analyzing
their individual contribution
¯ = raij (I)S
¯ aa0 rij0 (I)
¯
dij (I)
a
(40)
to the norm of the m = 23 part of the present approxi¯ Note that in the
mations of the right eigenvectors raij (I).
previous equation the repeated LMO indices i, j are excluded from the implicit summation. The individual dij
are then normalized by their sum and sorted according
to decreasing size. The pair list PI¯ comprises the pairs
ij with largest dij until the cumulative dij reaches a certain specified threshold κe (of course, if a certain ij is
included in PI¯, then also all symmetry related pairs are
included in PI¯ as well).
The individual PI¯ are modified in the course of the
Davidson diagonalization such that an appropriate state
specific local approximation is attained for each ionized
¯ In the critical initial phase of the first few iterastate |Ii:
tions of the Davidson procedure where the lowest ionized
¯ have to be found, PI¯ is re-built in each iterastates |Ii
tion, i.e., for each additonal Davidson basis vector (cf.
section II.C in Ref. 18). Thereafter, the PI¯ are then further re-constructed only in each refresh of the Davidson
procedure [18].
Since only one virtual orbital index occurs in the trial
vectors uij
a (I) they can easily be stored on disk even without truncating ij according to PI¯. For pairs ij ∈
/ PI¯
vij
a (I) is not zero, but still calculated at the level of IPCCSD[0]CC2 , i.e., according to eq. (27) without further
terms. For pairs ij ∈ PI¯, on the other hand, vij
a (I) is
calculated according to eq. (27) and augmented by eqs.
(33), (36), (39), depending on the level of the method. In
such a setting, a “multi state” Davidson diagonalization
in the sense that multiple ionized states are calculated simultaneously, is straightforward, even though state spe-
8
cific local approximations apply and are exploited in the
calculation of the vij
a (I).
The much smaller set of the all internal three–index
ERIs (ijˆ|P ) can be reduced similarly as just described for
(abˆ|P ): (ijˆ|P ) decays exponentially w.r. to the distance
between the LMOs indexed by i and j. However, due
to the orthogonality of the LMOs, an overlap criterion
is obviously inappropriate. We use instead the product
of the Löwdin partial charges of the LMOs i and j to
reduce (ijˆ|P ), as suggested earlier by Kats (cf. Eq. (2)
in Ref. 62). In the following we refer to this threshold as
the ij-threshold.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
HF
D
S
∆CC
2
∆MP
CSD
CSD
2
C SD
CC2
_CC
[2 ]_
2
CC2
MP2
_MP
[1 ]_
[1 ]_
IP-C
EOM
IP-C
IP-C
CSD
CSD
CSD
The IP-CCSD[k]M methods introduced in the previous
section have been implemented in the MOLPRO program
package [63]. In this section we present vertical ionization
potentials for different molecules and ionized states. In
all calculations the cc-pVDZ AO basis set [64] was used,
together with the related MP2FIT [65] fitting basis and the
JKFIT [66] fitting basis related to at least the cc-pVTZ
AO basis (the latter for fitting in the construction of Fock
and dressed Fock matrices). The occupied orbitals were
localized by employing Pipek-Mezey localization [25].
IP-C
IP-C
IP-C
CC2
(2 )
TEST CALCULATIONS
1.0
0.9
0.8
ADC
III.
accuracy of the IP-CCSD[k]M hierarchy
In the following we explore the accuracy of the IPCCSD[k] hierarchy for a test set of small to medium
sized molecules by comparison to the ∆CCSD(T) reference. The individual molecules and states of the test set,
along with the corresponding detailed IPs so obtained,
are listed in Tab. S1 of the supplementary information
[67]. Most of the molecules have been used already in
previous work for testing our local CC linear response
approach [17–19, 22–24]; their individual geometries are
supplied as xyz-files as supplementary information [68].
“Multi state” Davidson diagonalizations were carried
out for seven states simultaneously (three for water), yet
we present results only for the 3-6 lowest states depending
on the availability of a related ∆CCSD(T) result. Fig. 1
displays the mean absolute errors (MAEs) of the IPs of
the various methods of sections II B and II C relative to
∆CCSD(T). The ∆MP2, ∆CCSD, and the ∆CCSD(T)
reference values are obtained as the difference between a
closed–shell calculation and a corresponding open–shell
calculation with one electron removed. By utilizing point
group symmetry in the open–shell calculations it is possible to obtain the ionization potential of the lowest lying ionized state in each irreducible representation of the
point group. Furthermore, also EOMIP-CCSD [45] results and IPs obtained as Hartree-Fock orbital energies
according to Koopmans’ theorem are included in Fig. 1
for comparison. The EOMIP-CCSD calculations were
performed with the CFOUR program package [69].
It was already mentioned in the introduction that the
accuracy of the ∆CC method is higher than that of
the corresponding IP-EOM-CC or linear response IP-CC
Mean Absolute Error [eV]
Apart from the truncations of ground state amplitudes
ij
tij
ab and trial vectors ua (I) as discussed above, we also
exploit locality in the three-index ERIs (abˆ|P ) and the
intermediate Ŷabc of eqs. (33) and (34): due to the exponential decay in the integral w.r. to the distance between the two centers of the PAOs indexed by a and b
the number of non-negligible integrals of this type scales
as O(N 2 ) with system size N , and the same holds also
for the intermediate Ŷabc . Hence, truncating PAO pairs
a, b according to an overlap criterion decreases substantially the size of the (abˆ|P ) integral distribution and the
Ŷabc object, and the computational cost of the related
terms in eq. (33). To this end we assign to every center
pair A, B the maximum value SAB of the corresponding
block of the PAO overlap matrix Sab , ∀a ∈ A, ∀b ∈ B.
A simple check of SAB against a threshold given in the
input (the AB-threshold) determines which center pairs
A, B are kept in (abˆ|P ) and Ŷabc . Furthermore, since
the intermediate Ŷabc is contracted with the ground state
doubles amplitudes tij
bc in eq. (33) it is also possible to
restrict the range bc of Ŷabc . This is achieved a priori by
setting up a list of non-vanishing center pairs B, C with
B ∈ [ij] ∧ C ∈ [ij], ∀ij ∈ P0 . This yields a diagonally
dominant B, C list since there is no ij including both B
and C with B being far from C, hence B cannot be far
from C.
A.
FIG. 1.
Mean absolute errors of the vertical IPs (in
[eV]) of the individual methods, i.e., IP-ADC(2), IP-CC2,
IP-CCSD[k]MP2 , IP-CCSD[k]CC2 , EOMIP-CCSD, ∆MP2,
∆CCSD, and Hartree-Fock Koopmans’ theorem relative to
the reference ∆CCSD(T). The MAE is calculated over all
ionized states of all molecules of the test set. The AB- and
ij-thresholds were both set to 10−8 , and the pair lists P0 and
PI¯ remained un-restricted.
1.00
0.50
methods are generally somewhat more accurate than the
corresponding IP-CCSD[k]MP2 methods, as is evident
from the RMS errors. Apparently, inclusion of ground
state singles is advisable.
0.00
B.
accuracy of local approximations
-0.50
-1.00
-1.50
FIG. 2. Mean (in red) and RMS (in blue) errors of the vertical IPs (in [eV]) of the individual methods, i.e., IP-ADC(2),
IP-CC2, IP-CCSD[k]MP2 , IP-CCSD[k]CC2 , EOMIP-CCSD,
∆MP2, and ∆CCSD, relative to the reference ∆CCSD(T).
The Mean and RMS errors are calculated over all ionized
states of all molecules of the test set. The AB- and ijthresholds were both set to 10−8 and the pair lists P0 and
PI¯ remained un-restricted.
0.030
ADC(2)
IP-CCSD[1]_CC2
IP-CCSD_CC2
0.025
CC2
IP-CCSD_MP2
IP-CCSD[1]_MP2
IP-CCSD[2]_CC2
0.020
0.015
0.010
0.005
0.000
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method. At the same time, the latter are less expensive
and provide simultaneous access to several, rather than
just the lowest ionized state. All the IP-CCSD[k]CC2 and
IP-CCSD[k]MP2 calculations referring to Fig. 1 were carried out without any local approximations and are thus
equivalent to their canonical counterpart.
As can be seen from Fig. 1, the simplest methods,
namely ADC(2) and CC2 (aka IP-CCSD[0]CC2 ) exhibit
large errors relative to ∆CCSD(T); the differences in the
ionization potentials range from 0.3 eV to more than one
eV; the MAEs are in the range of 0.6-0.7 eV, which is
only a slight improvement over pure Hartree-Fock orbital energies, i.e., using Koopmans’ theorem, where the
MAE amounts to about one eV. The situation improves
drastically on going from ADC(2) and CC2 to the IPCCSD[1]MP2 or IP-CCSD[1]CC2 methods, where differences mostly smaller than 0.2 eV are observed. The MAE
for these methods amounts to 0.13 eV.
IP-CCSD[2]CC2 exhibits a further slight improvement
compared to IP-CCSD[1]CC2 , the MAE improves to 0.10
eV. The differences between IP-CCSD[2]CC2 and full
IP-CCSDCC2 is very minor, both regarding accuracy
and computational cost. EOMIP-CCSD, which employs
CCSD rather than CC2 ground state amplitudes, is again
somewhat more accurate than IP-CCSDCC2 (MAE is
0.07 eV), but also considerably more costly, since fourexternal ladder diagrams have to be evaluated.
Fig. 2 shows the mean and RMS errors of the IPs for
the same methods as in Fig. 1 apart from Hartree-Fock
orbital energies, for which the mean±RMS error amounts
to -0.95±0.78 eV. It can be seen that ADC(2), IP-CC2,
as well as the k = 1 methods (IP-CCSD[1]MP2 and
IP-CCSD[1]CC2 ) generally underestimate the ∆CCSD(T)
reference value, while the k ≥ 2 methods also overestimate it in many cases. Furthermore, the IP-CCSD[k]CC2
Fig. 3 compiles, for each of the test molecules, the
MAE of the IPs of local vs. non-local (or canonical)
IP-CCSD[k]MP2 and IP-CCSD[k]CC2 over all computed
states of the molecule. The pair domains [ij] = [i] ∪ [j] of
the ground state amplitudes tij
ab in eq. (23) correspond to
the union of the orbital domains [i] and [j]. The latter are
Boughton-Pulay (BP) domains [70] obtained with a criterion of 0.98, and extended by the next nearest neighbour
atoms forming a covalent bond with any of the atoms
of the BP domain set (iext=1 option in MOLPRO). The
ground state pair list P0 is truncated according to a distance criterion R0 , i.e., only those pairs ij are included
in P0 for which the interorbital distance between i and
j (measured as the minimum distance between closest
atoms in the two respective BP domain sets) is smaller
than R0 . R0 is set to 10 bohr in all calculations. The
excited state pair list PI¯ was determined as described in
section II D, employing a threshold κe = 0.99. The ABthreshold employed to restrict the center pair list A, B of
the three-index ERIs (abˆ|P ) and intermediate Ŷabc was
set to 10−2 . Test calculations indicate that such a value
Mean Absolute Error [eV]
MeanValueandRMSdevia3ons[eV]
9
FIG. 3. Mean absolute local errors of the vertical IPs (in
[eV]) of the individual methods IP-ADC(2), IP-CC2, IPCCSD[k]MP2 , IP-CCSD[k]CC2 , i.e. MAEs of the differences
in the IPs between local and corresponding non-local (canonical) methods (the averaging is done over all ionized states of
all molecules of the test set). For the local calculations the
AB- and ij-threshold were set to 10−2 and 10−4 , respectively,
P0 truncated according to a distance criterion of of R0 = 10
bohr, and PI according to a threshold of κe = 0.99 (see text).
For the non-local calculations the AB- and ij-thresholds were
both set to 10−8 and the pair lists P0 and PI remained untruncated.
10
for the AB-threshold is a sensible setting for basis sets
without too diffuse functions; some related data is given
in Tables S5-S8 of the supplementary information [67].
Likewise, for the threshold restricting the pairs i, j in
(ijˆ|P ) on the basis of Löwdin partial charges a reasonably conservative value of 10−4 was used.
As can be seen from Fig. 3, the mean absolute local
errors are all below 0.025 eV. For ADC(2) and CC(2)
the PI¯ remains un-truncated, and the small local errors
are solely caused by the truncation of P0 and the groundstate pair domains [ij]. Interestingly, the IP-CCSD[1]M
methods exhibit significantly larger local errors than the
higher-order methods IP-CCSD[k]M with k ≥ 2. This
appears to be related to the tendency of the k = 1 methods to locate states with dominant or large m = 32 character (cf. section III C), which are not found by the
k ≥ 2 methods: for example, the largest errors of the
k = 1 methods were seen for the D2 and D3 states of
furan, i.e., 0.066 eV for M = CC2, and 0.071 eV for
M = MP2, respectively. Both the D2 state (for M=CC2)
and the D3 state (for M=MP2) are dominated by the
LUMO
same τHOMO τHOMO
ionization process with m = 32 character; the corresponding weight amounts to about 70 %.
The other methods, on the other hand, only find states
with small or insignificant m = 32 character, and the local errors are smaller, e.g., 0.005 eV for the D2 state of
furan when calculated with the IP-CCSD[2]CC2 method.
As it appears, states with dominant or substantial m = 32
are more sensitive to the local approximation as specified
by the present settings. Nevertheless, generally, the local
errors can be considered as acceptable on the scale of the
accuracy observed relative to ∆CCSD(T) (vide supra),
particularly so for the k ≥ 2 methods.
C.
excitation energies of radicals
As claimed in the introduction, the IP-CC response
methods can in principle also be used to compute excitation energies of radicals as differences of the corresponding IP and the IP of the lowest ionized state: an
excitation process into the singly occupied molecular orbital (SOMO) of the radical, e.g. SOMO-n → SOMO,
can be represented by a ionization process of the closedshell molecule in the m = 12 excitation manifold, e.g.
τ(HOMO−n) = a(HOMO−n)β (HOMO: highest occupied
molecular orbital in closed shell molecule). The related
excitation energy of the radical then corresponds to the
difference in the IPs related to τHOMO and τ(HOMO−n) ,
respectively. Such excitation energies can be expected
to be well described by our IP-CCSD[k] methods or by
EOMIP-CCSD. On the other hand, excitation processes
into the virtual space, e.g. SOMO → LUMO require
LUMO
a m = 32 ionization process, e.g. τHOMO τHOMO
. Such
3
m = 2 dominated ionization processes are expected to
be less well described by IP-CCSD[k] or EOMIP-CCSD
and an extension to the m = 52 excitation manifold may
be required for sufficient accuracy.
TABLE II. Excitation energies (in eV) of the lowest three
doublet states of H2 O+ and the acridine radical calculated
with CASSCF and CASPT, and via IP differences computed
with IP-CCSD[k]CC2 , k = 1 and k = f . The occupation string
of the dominant configuration state function in the CASSCF
wavefunction is also given. The CASSCF calculations were
state averaged over four roots; for H2 O+ and the acridine
radical a full valence active space, and a 9 electrons in 10
orbitals CAS space were used, respectively. No point group
symmetry was imposed.
dominant CSF CASSCF CASPT2 IP-CCSD[k]CC2
z
}|
{
k=1
D1
D2
D3
22a200
2a2200
2220a0
2202a0
D1
D2
D3
22220a0000
222200a000
222a200000
H2 O+
2.294
2.221
7.282
6.889
15.040
15.039
acridine radical
2.688
2.225
3.430
2.830
3.613
2.878
k=f
2.198 2.400
6.855 6.999
19.004 23.145
14.268
1.944
2.868
2.834
—
—
2.972
To explore this issue in the context of our IPCCSD[k]M methods we calculated excitation energies for
the H2 O+ and the acridine radical of Ref. [71]. Table II
compares excitation energies calculated via IP differences
from IP-CCSD[k]CC2 , k = 1 and k = f with CASCF and
CASPT2 excitation energies. The latter were computed
with the MOLPRO (H2 O+ ) and the MOLCAS v.7 (acridine
radical) program packages [63, 72]).
The D1 and D2 states of H2 O+ can be generated by
m = 12 ionization processes; the excitation energies calculated via the IP-CCSD[k]CC2 , k = 1 and k = f IPs
are, as expected, in good agreement with those obtained
through CASPT2. The much higher lying D3 state, on
the other hand, corresponds to the SOMO → LUMO excitation and requires a m = 23 ionization process. Here,
the agreement between CASPT2 and IP-CCSD[k]CC2 is
much worse. Interestingly, the error is much larger for
the full method than for the k = 1 method. Moreover,
the latter finds another ionized state corresponding to
LUMO
τ(HOMO−1) τ(HOMO−1)
(with an excitation energy of 14.3
eV), which is not found by the other methods. The
EOMIP-CCSD method of the CFOUR program package
[69] yields very similar results as our k = f method.
The computations on the acridine radical were performed in the same geometry as in Ref. [71]. Here,
the radical is neutral and the reference for the IPCCSD[k]CC2 calculations thus a closed-shell anion. The
two lowest lying excited states of the radical possess
mainly SOMO → LUMO and SOMO → (LUMO+1)
character, respectively, and are therefore dominated by
m = 23 ionization processes acting on the closed-shell anion. The D3 state, on the other hand, has (SOMO-1)
11
→ SOMO character and is well described by an m = 12
ionization process. As is evident from Table II, the k = 1
method yields excitation energies in good agreement with
CASPT2 for all three states. For the k = f method, on
the other hand, the D1 and D2 states are not found, even
when utilizing the final k = 1 eigenvectors as start vectors
in the Davidson diagonalization of the k = f calculation.
Generally, the k = 1 method has a much increased tendency to locate m = 32 dominated states than the more
complete k > 1 or the k = 0 methods. The origin of
this peculiar feature is the first-order correction in the
A 32 32 submatrix, eq. (32), since omission of the secondorder correction in the A 32 12 submatrix, eq. (31) does
not change the behavior of the k = 1 method in this respect. On the other hand, the higher-order terms of eq.
(35) apparently suppress this feature again in the k > 1
method.
D.
comprises 98 atoms and 262 correlated electrons. The
local approximations for the ground state amplitudes tij
ab
reduce the pair list P0 from 8646 to 2613 and the average
size of the virtual space from 759 to 129 functions, i.e.,
the number of amplitudes from 4981M to 48M by two
orders of magnitude. The size of the intermediate Ŷabc
reduces from 852M elements (6.6 Gbyte) to 160 M
elements by a factor of 5.3 by virtue of the restrictions
in the ab and bc ranges as discussed in section II D.
Note that all these truncations do not depend on the
individual ionized states.
calculations on D21L6
As an example for a bigger molecule we present
calculations on the 3-(5-(5-(4-(bis(4-(hexyloxy)phenyl)amino)phenyl)thiophene-2-yl)thiophene-2-yl)-2-cyanoacrylic acid (D21L6) dye shown in Fig. 4. This
molecule is utilized as an organic sensitizer for solar-cell
applications [73] and has already been used by us as
a test molecule in previous studies [21, 22]. D21L6
We computed the lowest seven ionized states of D21L6
with our local IP-CCSD[k]CC2 and IP-CCSD[k]MP2
methods. The results of these calculations are compiled
in Table III. As already seen above for the previous
test set of molecules, the k = 0 methods significantly
underestimate the IPs when taking IP-CCSD[f]CC2 as
the reference (by 0.65-0.95 eV; MAE is 0.79 eV for IPCCSD[k]CC2 , and 0.72 for IP-CCSD[k]MP2 ). For the
k = 1 methods, and this in contrast to all other methods, the D2 and D4 states pick up a large fraction of
m = 23 character (the ratio of the norms of the m = 32
and m = 21 parts of the related eigenvector amounts to
1.4 – 1.5). This behavior persists with full PI¯ lists (κe
set to 1.0) and is therefore not caused by the local approximation, but a genuine feature of the k = 1 method.
Enhanced m = 32 character when using the k = 1 methods was also observed for some of the molecules of the
test set, primarily for higher states outside those entering the statistics in section III A, and, of course, for the
acridine radical in section III C. Presently it is unclear if
the D2 and D4 states would also acquire a large fraction
of m = 23 when further improving the k > 1 methods
such that they include additionally the m = 52 excitation
FIG. 4. D21L6 dye molecule.
space in the Jacobian.
Limiting the statistics to states that can be clearly assigned the MAEs of IP-CCSD[1]CC2 and IP-CCSD[1]MP2
relative to IP-CCSD[f]CC2 come to 0.32 and 0.23 eV, respectively (on going back from a CC2 to a MP2 ground
state treatment, i.e., by essentially dropping ground state
singles, the IPs are generally shifted to somewhat larger
values, hence reducing the MAE of the latter method).
The k = 2 methods, on the other hand, provides results in good agreement with the reference, but also at a
comparable cost as IP-CCSD[f]CC2 itself: the first Davidson diagonalization at the k = 0 level (used to generate
a first guess of the local approximation) takes about 5
hours, the subsequent second Davidson diagonalization
about 15 hours. It is odd to see that the simpler k = 1
methods actually take more computation time than IPCCSD[f]CC2 itself: this again can be attributed to the
occurrence of enhanced m = 32 character in D2 and D4
and a slower convergence of the Davidson diagonalization
(in iteration 24 still four states are not yet converged for
the k = 1 methods , among them D2 and D4 , while for
the other methods just a single state has not yet converged at that step). This slow convergence makes the
12
TABLE III. Results for D21L6. The vertical ionization energies are given in [eV]. The timings for ground-state, the initial
k = 0 diagonalization, and the second k > 0 diagonalization with additional terms included are all given in [h]. In parenthesis
the number of Jacobian × trial vector products are given, which were computed during the iterative diagonalizations. The
convergence criteria for the individual states during the diagonalizations are the respective norm of the residual (10−4 ) and the
energy change in the eigenvalue, i.e., the IP (7.65 · 10−6 ). The calculations were performed in parallel mode on seven AMD
6180 SE cores @ 2.5 GHz.
ADC(2)
IP-CCSD[k]M
}|
z
k=0
M=CC2
k=1
k=2
M=MP2 M=CC2
M=MP2 M=CC2
IPs [eV]
D0
5.414
5.317
6.082
5.968
6.354
6.257
D1
6.516
6.400
6.935
6.817
7.285
7.186
a
a
D2
7.363
7.316
7.811
7.639
8.165
8.145
D3
7.533
7.451
7.888
7.847
8.305
8.239
D4
7.891
7.862
8.179b
8.021b
8.703
8.700
D5
8.237
8.146
8.436
8.340
8.860
8.793
D6
8.271
8.195
8.476
8.415
8.997
8.923
HF+M [h]
1.8
5.2
1.9
5.1
1.9
5.0
k = 0 step [h] 3.9 (143) 3.7 (134) 4.2 (143) 4.0 (134) 4.0 (143) 4.0 (134)
k > 0 step [h]
–
–
17.9 (170) 22.8 (172) 12.4 (136) 15.1 (137)
a) ionized state with dominant m =
b) ionized state with dominant m =
3
2
3
2
LUMO
LUMO
character: τHOMO τHOMO
, τ(HOMO−1) τHOMO
(LUMO+1)
LUMO
character: τHOMO τHOMO , τHOMO τHOMO
{
k=f
M=CC2
6.258
7.188
8.145
8.241
8.701
8.794
8.924
5.1
4.0 (134)
15.6 (137)
13
k = 1 methods computationally more expensive than the
k ≥ 2 methods.
In order to get a handle on the local error in this
molecule we also performed an IP-CCSD[f]CC2 calculation with full pair list PI¯ (κe set to 1.0), which is the most
critical local approximation in our local IP-CCSD[k]M
methods. Furthermore, the threshold restricting the A, B
center pairs in integrals (abˆ|P ) and intermediates Ŷabc on
the basis of the overlap, as well as the threshold restricting the LMO pairs in (ijˆ|P ) on the basis of Löwdin partial
charges (cf section II D) were each reduced by two orders
of magnitude relative to their default values. The effect
on the IPs is quite small, the largest deviation amounts to
0.045 eV, and the MAE over the seven states to 0.011 eV.
We thus can conclude that the local approximations are
uncritical also for such an extended molecule as D21L6.
IV.
CONCLUSIONS
In this paper we presented theory and implementation
of a hierarchy of coupled cluster linear response methods for calculating ionization potentials, i.e., “excitation
energies” of states with one electron being annihilated
relative to the ground state reference (unperturbed w.r.
to the time-dependent perturbation). Consequently, the
operators related to the first-order amplitudes and Lagrange multipliers (in the second-order quasi-energy Lagrangian) have non-integer particle rank. In the present
work the particle rank of the time-dependent cluster operators was restricted to the m = 12 ⊕ 32 excitation space.
Test calculations show that IP-CC LR methods equivalent to the electron excitation methods CC2 LR and
ADC(2) provide only rather poor accuracy for IPs; relative to the ∆CCSD(T) reference they exhibit errors up
to more than one eV. Apparently, compared to excitation
energies, IPs are more demanding w.r. to the order of the
electron correlation and orbital relaxation treatment. To
this end we have developed a hierarchy of IP-CC methods
ranging up to full IP-CCSD, which are denoted as IPCCSD[k]M , and are all based either on an M = CC2 or
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and eventually nuclear gradients.
ACKNOWLEDGMENTS
Financial support of the Deutsche Forschungsgemeinschaft (DFG), grants US-103/1-2 and SCHU 1456/9-1, is
gratefully acknowledged.
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