local cc2 response method based on the laplace transform
TRANSCRIPT
Local CC2 response method based on the Laplace transform: Analytic energy
gradients for ground and excited states
Katrin Ledermuller1 and Martin Schutz1, ∗
1Institute of Physical and Theoretical Chemistry, University of Regensburg,
Universitatsstraße 31, D-93040 Regensburg, Germany
A multistate local CC2 response method for the calculation of analytic energy gradients withrespect to nuclear displacements is presented for ground and electronically excited states. Thegradient enables the search for equilibrium geometries of extended molecular systems. Laplacetransform is used to partition the eigenvalue problem in order to obtain an effective singles eigenvalueproblem and adaptive, state-specific local approximations. This leads to an approximation in theenergy Lagrangian, which however is shown (by comparison with the corresponding gradient methodwithout Laplace transform) to be of no concern for geometry optimizations. The accuracy of thelocal approximation is tested and the efficiency of the new code is demonstrated by applicationcalculations devoted to a photocatalytic decarboxylation process of present interest.
I. INTRODUCTION
Equilibrium and transition structures of molecules, whichare stationary points on potential energy hypersurfaces,are of great interest in chemistry and physics. Knowl-edge about the energy hypersurfaces of the electronicallyground and excited states is the basis for understandingor predicting photophysical processes, which are in thefocus of various fields like material science or biochem-istry.The gradient for geometry optimizations, i.e., the deriva-tive of the energy with respect to nuclear displacements,can be calculated numerically or analytically, but numer-ical calculations are only applicable to small molecules.The pioneering work of Pulay for SCF calculations [1–3]was followed by the development of analytic ground stategradients for a variety of ab initio methods, amongst oth-ers configuration interaction (CI) [4, 5], multiconfigura-tional SCF (MCSCF) [6, 7], Møller-Plesset (MP) pertur-bation theory [8, 9] and Coupled Cluster theory (CC)[10]. Also gradients for local ground state methods havebeen presented, e.g. for MP2 [11, 12] and quadratic CI[13]. Ground state methods are well-established nowa-days, while theoretical studies of electronically excitedstates at a reliable level of ab initio theory are still verychallenging.Within time-dependent (TD) response theory, a widely-used framework for calculating excitation energies andproperties of excited states, excitation energies are ob-tained as a property of the electronic ground state,namely as the poles of the frequency-dependent polar-izability (FDP). The use of TD response theory is well-established for various wavefunction approaches, e.g. inthe context of Hartree-Fock (TD-HF), density functional(TD-DFT) [14, 15], or Coupled Cluster theory (TD-CC) [16–19]. For traditional CC wavefunction ansatzean appropriate time-averaged quasienergy Lagrangianhas to be specified as the basis of the related TD re-
sponse theory [20–22], in contrast to variational CC re-sponse approaches discussed recently [23, 24]. Fromthe time-averaged quasienergy Lagrangian the linear re-sponse function, i.e., the FDP, is then obtained by differ-entiation.The equation-of-motion Coupled Cluster (EOM-CC)method [25–29] approaches excited states from the CIperspective, but has close relationships to TD-CC re-sponse. Excitation energies and densities of TD-CC re-sponse and EOM-CC are equivalent. Analytic energygradients for excited states have been developed both forEOM-CC [30–32] and TD-CC [17, 33]. They competeagainst analytic TD-DFT gradients [34–36], which arecomputationally cheaper, but often unreliable. If chargetransfer (CT) states, Rydberg states or excitations of ex-tended π systems are involved, TD-DFT methods oftenfail qualitatively [15, 37, 38].The computationally cheapest CC model including dy-namical correlation effects is the CC2 model [39]: ampli-tudes related to double substitutions are correct only tofirst order (w.r. to a Møller-Plesset (MP) partitioning ofthe Hamiltonian), yet the full exp (T1) part of the CCansatz is retained to provide partial orbital relaxation.The CC2 model provides rather accurate results for ex-cited states, provided that they are dominated by singlessubstitutions. Analytic CC2 energy gradients have beendeveloped for the ground state [40–42] and for excitedstates [43].Compared to TD-DFT methods canonical TD-CC2, al-though being one of the cheapest CC models, is com-putationally rather expensive because of the steep scal-ing of the computational cost with molecular size N ofO(N 5). In order to reduce the computational cost ofTD-CC the density fitting approximation (DF) can beemployed, which factorizes the electron repulsion inte-grals [44–46]. There are highly efficient CC2 and scaledopposite-spin (SOS) CC2 implementations using this ap-proach for properties and analytic gradients of excitedstates [43, 47–51]. However, DF reduces only the prefac-tor, but not the scaling: DF-CC2 still scales as O(N 5).For a further reduction of the scaling local correlation
2
methods have been proposed [52–57]. Local methodsuse a basis of spatially localized orbitals, e.g. localizedmolecular orbitals (LMOs), and projected atomic orbitals(PAOs) to span occupied and virtual space, respectively,to benefit from the short-range nature of dynamic cor-relation in nonmetallic systems [58, 59]. The LMO pairlist and the (pair specific) virtual spaces can then berestricted, the latter to subspaces of PAOs (domains).The a priori specification of such restrictions is ratherstraightforward for ground state amplitudes, but more in-tricate for eigenvectors of excited states, which can haveRydberg or charge transfer character [52, 54, 60, 61].In the local CC2 response method based on the Laplacetransform (LT), denoted as LT-DF-LCC2, just an ef-fective eigenvalue problem in the space of the untrun-cated singles determinants has to be solved (as in thecanonical case), the doubles part does not enter theDavidson diagonalization explicitly [54–57]. State spe-cific restricted pair-lists and PAO domains for the dou-bles are determined by analysis of the untruncated dou-bles part of the actual approximation to the eigenvectorrelated to diagonal pairs. The approximations are re-specified in every Davidson-refresh, thus they allow theeigenvectors to change their character during the David-son process. LT-DF-LCC2 excitation energies, transi-tion moments and first-order properties without orbitalrelaxation were implemented into the MOLPRO programpackage [62, 63] earlier and enable calculations for ex-tended molecular systems consisting of hundred or moreatoms [54–56]. This publication continues the workabout orbital-relaxed first-order properties, which waspresented recently [57]. In this work, analytic energygradients with respect to nuclear displacements were de-veloped for ground and excited states on the basis of theLT-DF-LCC2 method.The DF-LCC2 method without LT, which was presentedearlier [52, 53], does not feature state-specific and adap-tive local approximations, but determines them by ana priori analysis of the untruncated CIS (configurationinteraction singles) wavefunction. Moreover it has thedisadvantage, that also the doubles eigenvalue equa-tions have to be solved explicitly. Nevertheless, con-trary to LT-DF-LCC2, where an approximated energyLagrangian is used, the method without LT is based onthe proper Lagrangian [57, 64]. For first-order proper-ties it has been shown, that the use of the approximatedLagrangian in the LT-DF-LCC2 method does not causeproblems. For geometry optimizations the effects of theapproximation are expected to be larger, thus this aspectis also explored in this work.The outline of the paper is as follows: First the workingequations for the implementation of the gradients are de-rived for the ground state and singlet and triplet excitedstates (sec. II). The accuracy of the local approxima-tions is then explored. Finally, as an illustrative appli-cation example, excited state geometry optimizations oftwo molecules were carried out, which are of relevancefor a photocatalytic decarboxylation reaction of present
interest and consist of more than fifty atoms.
II. THEORY
Einstein convention is employed in the following, i.e., re-peated indices are implicitly summed up; summationsare only written explicitly, if we consider it to be neces-sary for clarity. As in earlier publications the formalismis derived for an orthonormal basis of localized occupiedand canonical virtual molecular orbitals (MO) and thetransformation to the basis of nonorthogonal PAOs isperformed a posteriori [12, 57]. The MOs are expandedin an AO-basis χµ with metric SAO
µν = 〈χµ|χν〉,
φp = χµCµp. (1)
The composite coefficient matrix C = (L|Cv) concate-nates the LMO coefficient matrix L and the coefficientmatrix of the canonical virtuals Cv. For LMOs andcanonical virtuals we use indices i, j, ..., and a, b, ..., re-spectively. General molecular orbitals are indexed bym,n, ..., and PAOs by r, s, ... The coefficient matrix P
for the PAOs is given by
Pµr = [CvCv†SAO]µr = [CvQ]µr , (2)
implicitly defining the matrix Q, which transforms fromcanonical to PAO basis. For the metric S of the PAOsone then obtains
S = P†SAOP = Q†Q . (3)
In order to reduce the computational cost density fitting[44–46] is employed to decompose the four-index integralsinto three-index objects, i.e.,
(mn|pq) ≈ (mn|P )cPpq , cPpq =(
J−1)
PQ(Q|pq) ,(4)
with capital letters P,Q indexing auxiliary fitting func-tions. JPQ = (P |Q) is an element of the Coulomb metricof the fitting functions.The work presented in this publication is based on thepreliminary work about orbital-relaxed properties in thecontext of the LT-DF-LCC2 method, which can be foundin Ref. 57. We will refer to that article in the followingwhenever it would be too lengthy to reiterate the detailsin the present contribution.
A. Ground state
1. Lagrangian
The gradient for the local CC2 ground state energy con-tains terms from the underlying HF calculation, whichare obtained starting from the Lagrangian for the HFenergy EHF
0 ,
LHF0 = EHF
0 − 2fij(
(C†SAOC)ij − δij)
. (5)
3
The second term of the Lagrangian containsthe orthonormality condition for the coefficients((C†SAOC)ij = δij) with the Fock matrix elementsfij as corresponding Lagrange multipliers. The termsresulting from the derivative of LHF
0 with respect tonuclear displacements are added a posteriori to the CC2correlation contributions to obtain the full gradient forthe local CC2 ground state energy.The CC2 model, which was proposed by Christiansen et
al. [39], is an approximation to the CCSD model. TheCC2 correlation energy is calculated as
ECC20 = 〈0 |exp(−T)H exp(T)| 0〉
=⟨
0∣
∣
∣H+ [H,T2]
∣
∣
∣0⟩
, (6)
where |0〉 is the Hartree-Fock reference determinant, Hthe normal-ordered Hamiltonian, and T the cluster op-erator containing single and double excitations, i.e.,
T = T1 +T2 = tµ1τµ1
+ tµ2τµ2
= tiaτai + tijabτ
abij , (7)
with excitation operators τ and related amplitudes t.For singlet substitutions, as they occur for the electronicground state and singlet excited states, τ is defined as
τai = a†aαaiα + a†aβaiβ ,
τabij =1
2(a†aαaiα + a†aβaiβ)(a
†bαajα + a†bβajβ) , (8)
in terms of the elementary second quantization creationand annihilation operators a† and a (the index iα im-plies a spin orbital related to a spatial LMO i timesspin function α, etc.). Operators decorated by a hatrepresent operators similarity transformed with the ex-ponential of the singles cluster operator T1, e.g. H =exp(−T1)H exp(T1). A consequence of the similaritytransformed operators is the occurrence of dressed inte-grals,
(mn|pq) = (µν|ρσ)ΛpµmΛh
νnΛpρpΛ
hσq , (9)
with the coefficient matrices Λp and Λh in LMO/PAO-basis defined as
Λpµr = Pµr − Lµit
ir′Sr′r , Λp
µi = Lµi ,
Λhµr = Pµr , Λh
µi = Lµi + Pµrtir .(10)
The CC2 amplitudes are determined by the equations
Ωµ1= 〈µ1|H+ [H,T2]|0〉 = 0 ,
Ωµ2= 〈µ2|H+ [F,T2]|0 〉 = 0 . (11)
〈µ1| and 〈µ2| are contravariant configuration state func-tions projecting onto the singles and doubles manifold[65] and F is the Fock operator.The local CC2 Lagrangian for the ground state correla-tion energy ECC2
0 is defined as
LCC20 = ECC2
0 + λ0µiΩµi
+ zloc,0ij rij
+z0aifai + x0pq
[
C†SAOC− 1]
pq. (12)
It includes the amplitude equations (Ωµi= 0), the lo-
calization conditions (rij = 0), the Brillouin condition(fai = 0), and the orthonormality condition (C†SAOC =
1). The related multipliers are λ0µi, zloc,0ij , z0ai, and x0
pq,respectively. The tilde denotes contravariant Lagrangemultipliers, i.e.,
λir =
1
2λir, and λij
rs = 2λijrs − λji
rs . (13)
By choosing Pipek-Mezey localization [66] the conditionsrij become
rij =∑
A
[SAii − SA
jj ]SAij = 0 for all i > j, (14)
with the matrix SA being defined as
SAkl =
∑
µ∈A
∑
ν
[LµkSAOµν Lνl + LµlS
AOµν Lνk] . (15)
The summation over µ is restricted to basis functionscentered on atom A.The Lagrangian is required to be stationary with respectto all parameters. Differentiation of LCC2
0 w.r. to the CC
amplitudes t yields the equations for the multipliers λ0,
−ηνj= λ0
µiAµiνj
, (16)
with the Jacobian
Aµiνj=
∂Ωµi
∂tνj
, (17)
and
ηνj=
∂ECC20
∂tνj
. (18)
For the working equations we refer to Ref. 55.Differentiation of LCC2
0 w.r. to orbital variations yieldsthe orbital z-vector equations [67], from which the mul-tipliers z0, zloc,0, and x0 are obtained [12, 57, 68]. Thevariations of the orbitals in the presence of the perturba-tion V0 are described by the coefficient matrix
Cµp(V0) = Cµq(0)Oqp(V0), (19)
where C(0) are the coefficients of the optimized orbitalswithout perturbation and the matrix O(V0) describesthe rotation of the orbitals caused by V0, with O(0) = 1.The derivative of the Lagrangian with respect to the vari-ation can be partitioned into four contributions,(
∂LCC20
∂Opq
)
V0=0
= [B0 + B(z0) + b(zloc,0) + 2x0]pq ,
(20)
with
[B0]pq =
(
∂
∂Opq
(ECC20 + λ0
µiΩµi
)
)
V0=0
,
[B(z0)]pq =
(
∂
∂Opq
z0aifai
)
V0=0
,
[b(zloc,0)]pi =
(
∂
∂Opi
zloc,0kl rkl
)
V0=0
. (21)
4
The stationarity of LCC20 w.r. to orbital variations, and
the relation x0 = x0† are exploited to obtain the linearz-vector equations,
(1− Ppq)[B0 + B(z0) + b(zloc,0)]pq = 0 , (22)
from which z0 and zloc,0 are obtained. The permuta-tion operator Ppq permutes the orbital indices p and q.As shown in Ref. 12, the z-vector equations can be fur-ther decoupled into the Z-CPL (coupled perturbed local-ization), and the Z-CPHF (coupled perturbed Hartree-Fock) equations. The former have to be solved first, sincethe Lagrange multipliers zloc,0, which are the solutionsof the Z-CPL equations, do appear in the Z-CPHF equa-tions, which, in turn, determine the multipliers z0. Themultipliers x0 can then be expressed as
x0pq = −
1
4(1 + Ppq)[B
0 + B(z0) + b(zloc,0)]pq . (23)
The quantities B(z0) and b(zloc,0) are identical to the
quantities A, and a(zloc) given explicitly in Eqs. (29)and (39), respectively, of Ref. 12. For the working equa-tions for B0 we refer to Ref. 57, Eq. (33) and Eqs. (A5-A7). Please note, that there is an error in Eq. (A8)of Ref. 57, which defines the intermediates for B0. Thedensity d in this equation is correctly defined as
d = d(s) +Dξ(λ0) + d′ , (24)
with
d(s)µν = 2tiaCµiCνa , (25)
and Dξ(λ0) and d′ as defined there.
2. Gradient
The LCC2 ground state gradient Lq0 is obtained by differ-
entiating the Lagrangian L0 with respect to the nucleardisplacement q,
Lq0 =
(
∂L0
∂q
)
q=0
=
(
∂(LHF0 + LCC2
0 )
∂q
)
q=0
. (26)
Employing the definitions of the undressed and dressedfock matrices,
fµν = hµν + 2LρkLσk[(µν|ρσ)− 0.5(µρ|σν)] ,
fµν = hµν + 2ΛpρkΛ
hσk[(µν|ρσ)− 0.5(µρ|σν)] , (27)
the gradient Lq0 can be written in terms of the derivative
AO integrals hqµν , (µν|ρσ)
q and Sqµν . Sorting the result-
ing terms according to the derivative AO integrals yieldsthe working equation for the gradient in LMO/PAO ba-sis,
Lq0 = hq
µνD0µν + Sq
µν
(
∂rij∂SAO
µν
)
zlocij +X0µν
+(µν|P )qY0µνP − Jq
PQZ0PQ, (28)
where the density fitting (DF) approximation,
(µν|ρσ)q = (µν|P )qcPρσ + cPµν(P |ρσ)q
−JqPQc
Pµνc
Qρσ , (29)
has been applied, cf. Eq. (4). The densities dHF and D0
are defined as
dHFµν = 2LµiLνi ,
D0µν = C loc
µp (D0 + z0)pqC
locνq + Λp
µpD0pqΛ
hνq ,
D0pq = 2(δpiδqj + δprδqit
ir) +Dξ
pq(λ0) ,
D0pq = Dξ
pq(λ0) ,
Dξij(λ) = −2λjk
rs′Sr′rtikr′sSs′s ,
Dξrs(λ) = 2λkl
st′St′ttklrt ,
Dξri(λ) = Dξ
ir(λ) = 0 ,
Dξir(λ) = λi
r ,
Dξri(λ) = λk
s′Ss′stikrs ,
Dξrs(λ) = Dξ
ij(λ) = 0, (30)
with the composite coefficient matrix Cloc = (L|P).For a detailed discussion about D0 we refer to sectionIID in our publication about orbital-relaxed properties(Ref. 57).Detailed expressions for the quantities Y0
µνP and Z0PQ,
which are contracted with the derivative integrals(µν|P )q and Jq
PQ, respectively, are given in Eqs. (A2)
and (A3) of appendix A. The intermediate X0 in Eq.(28) contains terms resulting from the the orthonormal-ity conditions in LHF
0 and LCC20 , and terms resulting from
the dependency of the transformation matrix Q on theAO overlap matrix. Details and the working equationscan be found in appendix B. Moreover the derivatives ofthe localization criterion rij ,
(
∂rij∂SAO
µν
)
= (1− Pij)∑
A
[
2LµiLνiSAij
+SAii (LµiLνj + LµjLνi)
]
δµ∈A , (31)
are contracted with the derivative AO overlap integralsSqµν . δµ∈A restricts the index µ to AOs on atom A.
B. Singlet excited states
1. Lagrangian
The local orbital-relaxed CC2 Lagrangian Lf ′ for the sin-glet excited state f ′ is the sum of the Lagrangian L0 forthe ground state energy and the Lagrangian Lf for theexcitation energy,
Lf ′ = L0 + Lf ,
Lf = LfARf + λfµiΩµi
− ωf [LfMRf − 1]
+zloc,fij rij + zfaifai + xfpq
[
C†SAOC− 1]
pq.(32)
5
To obtain the excitation energy ωf = LfARf , with the
contravariant left eigenvector Lf and the covariant righteigenvector Rf , the left and right eigenvalue equationsfor the Jacobian A,
ARf = ωfMRf and LfA = ωf LfM , (33)
have to be solved (M is the metric of contra- and covari-ant CSFs). The CC2 Jacobian for singlet excited statestakes the form
Aµiνj=
(
〈µ1|[H, τν1](1 +T2)|0〉 〈µ1|[H, τν2
]|0〉
〈µ2|[H, τν1]|0〉 〈µ2|[F, τν2
]|0〉
)
.
(34)
The second term of Lf in Eq. (32) is the condition forthe ground state amplitudes. The third term enforcesthe orthogonality of left and right eigenvector. The re-maining terms represent the localization, Brillouin andorbital-orthogonality conditions, respectively. For con-ciseness the state index f is omitted for L, R, and ω inthe following. Differentiation of the Lagrangian Lf w.r.to the amplitudes t yields the equation for the multipliersλf , for the working equations we refer to Ref. 55. Anal-ogously to the ground state, stationarity of Lf w.r. toorbital variations, i.e.,
0 =( ∂
∂Opq
[
LAR+ λfµiΩµi
− ω[LMR− 1] + zloc,fij rij
+zfaifai + xfpq
[
C†SC− 1]
pq
])
V0=0, (35)
yields the orbital z-vector equations,
0 = (1− Ppq)[Bf + B(zf ) + b(zloc,f )]pq, (36)
which correspond to Eq. (22) for the ground state, anda set of equations for the multipliers xf ,
xfpq = −
1
4(1 + Ppq)[B
f + B(zf ) + b(zloc,f )]pq , (37)
corresponding to Eq. (23). Eq. (36) again decouplesinto the Z-CPL equations determining zloc,f , and the Z-CPHF equations determining zf . Apart from a differentright hand side these equations are equivalent to those ofthe ground state. The quantities B(zf ) and b(zloc,f ) aredefined according to Eq. (21), and Bf as
[Bf ]pq =
(
∂
∂Opq
(LAR+ λfµiΩµi − ω(LMR− 1))
)
V0=0
.
(38)
The working equations for Bf can be found in Ref. 57,Eq. (41) and Eqs. (B3-B5), respectively.
2. Gradient
Analogously to the ground state the differentiation ofthe Lagrangian Lf ′ with respect to nuclear displacements
employing Eqs. (27) and (29) yields the expression forthe excited state gradient Lq
f ′ . It is the sum of the ground
state gradient Lq0 (Eq.(28)) and the gradient for the dif-
ference between ground and excited state Lqf ,
Lqf ′ = Lq
0 + Lqf . (39)
For the latter, we obtain by sorting the terms accordingto the AO derivative integrals the working equations inLMO/PAO basis as
Lqf = hq
µνDfµν + Sq
µν
(
∂rij∂SAO
µν
)
zlocij +Xfµν
+(µν|P )q(13YfµνP + 1Yf
µνP )
−JqPQ(
13ZfPQ + 1Zf
PQ), (40)
For compactness, we split the intermediates, which arecontracted with the derivative integrals (µν|P )q and Jq
PQ
into two parts: 13YfµνP and 13Zf
PQ collect those termswhich appear both in singlet and triplet excited states,
while 1YfµνP and 1Zf
PQ comprise the terms only appear-ing for singlet excited states. Detailed expressions forthese quantities are given in Eqs. (A7-A12) of appendixA. The quantity Xf is discussed in appendix B, and theexcited state density Df
µν is defined as
Dfµν = C loc
µp (Dξ(λf ) +Dη(LR) + zf )pqC
locνq
+Λpµp(D
ξ(λf ) + Dη(LR))pqΛhνq ,
Dηij = −2Srr′L
ikr′s′Ss′sR
jkrs ,
Dηrs = 2Lij
stStt′Rijrt′ ,
Dηir = Dη
ri = 0 ,
Dηij = −Li
r′Sr′rRjr ,
Dηrs = Li
sRir,
Dηir = 0 ,
Dηri = Lj
s′Ss′sRjisr . (41)
C. Triplet excited states
1. Lagrangian
Triplet states for canonical CC2 response were introducedin Ref. 49. They were also discussed in the context of theLT-DF-LCC2 method [56, 57]. For triplet substitutionsthe excitation operators τ are defined as
τai = a†aαaiα − a†aβaiβ ,
τabij = (a†aαaiα − a†aβaiβ)(a†bαajα + a†bβajβ) . (42)
The triplet double substitution operators τabij are linearlydependent and to get rid of these redundancies sym-
6
metrized operators of the form
(+)
τabij = τabij + τ baji , ∀ a > b, i > j and
(−)
τabij = τabij − τ baji , ∀ (ai) > (bj) (43)
are introduced. Thus symmetrized doubly excited ket
and bra CSFs for triplet states are defined as
|(+)
Φabij 〉 =
(+)
τabij |0〉 , |(−)
Φabij 〉 =
(−)
τabij |0〉 ,
〈(+)
Φabij | =
1
8〈0|(
(+)
τabij )† , 〈
(−)
Φabij | =
1
8〈0|(
(−)
τabij )† , (44)
and triplet singles- and doubles cluster operators U1 andU2 as
U1 =∑
ia
uiaτ
ai ,
U2 =∑
a>b,i>j
(+)
U ijab
(+)
τabij +∑
(ai)>(bj)
(−)
U ijab
(−)
τabij . (45)
The Jacobian A for triplet excited states takes the form
Aµiνj=
〈µ1|[H, τν1](1 +T2)|0〉 〈µ1|[H,
(+)τν2
]|0〉 〈µ1|[H,(−)τν2
]|0〉
〈(+)
µ2 |[H, τν1]|0〉 〈
(+)
µ2 |[F,(+)τν2
]|0〉 0
〈(−)
µ2 |[H, τν1]|0〉 0 〈
(−)
µ2 |[F,(−)τν2
]|0〉
. (46)
Solving the eigenvalue problems of Eq. (33) with thisJacobian yields the excitation energies and left and righteigenvectors for triplet excited states. Recall that thecluster operator T refers to the ground state and there-fore contains singlet excitation operators.
2. Gradient
As for the singlet excited state gradient, Lqf ′ for the ex-
cited triplet state f ′ is equal to the sum of the groundstate gradient Lq
0 and the gradient for the triplet exci-tation energy Lq
f . The expression for Lqf obtained in
the triplet case is formally analogous to that of the sin-glet state, Eq. (40), but with the singlet specific quan-
tities 1YfµνP ,
1ZfPQ being replaced by those specific for
the triplet state, i.e., 3YfµνP , and
3ZfPQ (cf. Eqs. (A7-
A12) of appendix A). Furthermore, the density matrix
Df in AO basis appearing in the quantities 13YfµνP , and
13ZfPQ, calculated according to Eq. (41), is now differ-
ent; the densities Dηpq and Dη
pq required for it are defined
for the triplet case as
Dηij = −Srr′Sss′(
(+)
Likr′s′
(+)
Rjkrs + 2
(−)
Likr′s′
(−)
Rjkrs) ,
Dηrs = Stt′(
(+)
Lijst′
(+)
Rijrt + 2
(−)
Lijst′
(−)
Rijrt) ,
Dηir = Dη
ri = 0 ,
Dηij = −Li
r′Sr′rRjr ,
Dηrs = Li
sRir ,
Dηir = 0 ,
Dηri = Lj
s′Sss′Rjisr. (47)
The triplet specific quantity Xf , which is contracted inEq. (40) with the AO derivative overlap integral matrixSq, is discussed in appendix B.
D. Hybrid method (LT-)DF-LCC2
So far no distinction was made, whether Laplace trans-form was used for solving the eigenvalue equations ornot. The derived equations are valid for both the DF-LCC2 and the LT-DF-LCC2 method. For details aboutthe two methods we refer to Refs. 52–55. In the caseof DF-LCC2 the Lagrangians L0 and Lf are the properenergy Lagrangians. But as discussed in detail in Ref. 64for the LT-LMP2 method, they are only approximationsto the exact energy Lagrangians, if Laplace transforma-
7
tion is employed. Nevertheless they are used, becausethe proper LT-DF-LCC2 Lagrangians are impractical (cf.Eq. (27) in Ref. 64 and the related discussion).Yet the errors introduced by the use of these approximateLagrangians turned out to be negligible for the calcula-tion of excitation energies and first-order properties. Forgeometry optimizations the effect of the approximate La-grangians might be more problematic.For investigation of this aspect also a hybrid methodwas implemented, which will in the following be called(LT-)DF-LCC2. The basic idea is to combine the ex-act energy Lagrangian of the DF-LCC2 method withthe local approximations obtained from the LT-DF-LCC2method, which are in many cases more appropriate thanthe pair lists and domains of the DF-LCC2 method. Thefirst step of an (LT-)DF-LCC2 calculation is the David-son diagonalization for the right eigenvalue problem em-ploying the LT-DF-LCC2 code. The converged LT-DF-LCC2 eigenvectors are used as starting guess for a DF-LCC2 calculation without LT. DF-LCC2 is not state-specific, thus the local approximations, which were ob-tained from the LT-DF-LCC2 step for the state of inter-est, are used for each of the excited states in the DF-LCC2 part.The methods DF-LCC2 and (LT-)DF-LCC2 have onlybeen implemented for singlet excited states.
III. TEST CALCULATIONS
The energy gradients for the ground state and excitedstates have been implemented in the MOLPRO programpackage [62, 63] and most of the relevant routines wereparallelized based on a shared file approach, i.e., thescratch files containing the amplitudes, integrals, etc. re-side on two file systems, which are common to all parallelthreads. Input/output is organized such, that both filesystems are concurrently used.The underlying HF reference was computed employingthe density fitting approximation [69]. In all LT-DF-LCC2 calculations three Laplace quadrature points wereused, exemplary calculations with five points showed nosignificant improvement of the results. The cc-pVDZ AObasis set [70] was employed together with the related fit-ting basis set optimized for DF-MP2 [71].The geometry optimizations were performed using thequadratic steepest descent algorithm [72–74] in combina-tion with the model Hessian proposed by Lindh [75].The correctness of the code was verified by comparing theresults of our program using untruncated pair lists andfull domains to the corresponding canonical results ob-tained with the RI-CC2 gradient code of the TURBOMOLE
program [43, 48, 50, 76].
A. Accuracy of the local methods
The local approximation implies truncations (i) in thelist of LMO pairs, and (ii) in the individual pair-specificvirtual spaces. The latter, denoted as pair domains, areconstructed as the union of the two orbital domains re-lated to the two LMOs making up the pair.For the electronic ground state the orbital domains areconstructed according to the Boughton Pulay (BP) pro-cedure [77], employing a criterion of 0.98. The final or-bital domains are then obtained by augmenting the BPcore domains by further centers separated by not morethan one bond from the closest atom in the original BPdomain (iext=1 option in MOLPRO). The ground state pairlist comprises all pairs of LMOs with a respective LMOinterorbital distance up to 10 bohr. LMO interorbitaldistances are always measured as the closest distance be-tween the two sets of nuclei related to the relevant BPdomains.For excited states the construction of the truncated pairlists and pair domains is less straightforward, and thecharacter of the individual state should be taken intoaccount. In LT-DF-LCC2 calculations the local approx-imation is set up by an adaptive scheme as explained indetail in section IIC of Ref. 54: essentially, for diagonalpairs, the un-truncated doubles part Uii of the actual ap-proximation to the eigenvector for each individual stateis analysed in each Davidson refresh, and a set of impor-tant LMOs is determined according to a certain threshold(κe = 0.999). State-specific truncated LMO pair lists arethen determined by including (i) all pairs of importantLMOs, (ii) all pairs between two LMOs with interorbitaldistance up to 5 bohr, and (iii) merging the resulting listwith the LMO pair list of the ground state. The orbitaldomains for the individual excited states are determinedaccording to the procedure described in section IIC ofRef. 54: for not important LMOs just the ground stateorbital domains are adopted, while for important LMOsthe ground state orbital domains are augmented by fur-ther atoms taken from an ordered list, which is obtainedfrom a Lowdin like analysis of Uii. Here, a threshold of0.98 was employed. Since the domains for the excitedstates include the ground state domains, extending thelatter with iext=1 (vide supra) also enlarges the excitedstate domains. The pair lists and domains of each excitedstate are re-specified in every Davidson-refresh.In DF-LCC2 calculations, on the other hand, the lo-cal approximations are not adaptive and state-specific.They are obtained from analysis of the CIS wavefunc-tion of the selected excited state as explained in detailin sections IIIA of Ref. 60 and IIB of Ref. 52: the im-portant orbitals, which determine the pair list, are ob-tained from the CIS coefficients (specified by thresholdκe = 0.995) and the domains are constructed by apply-ing the Boughton Pulay (BP) procedure with a criterionof 0.98 to the excited natural orbitals (defined in Eq.(13) of Ref. 60), which contain the CIS coefficients anddescribe for a given excited state the whole excitation
8
from the respective LMO. The final excited state pairlists and domains are obtained analogously to the LT-DF-LCC2 case, i.e., by either adopting/augmenting theground state pair lists and orbital domains.In the hybrid method (LT-)DF-LCC2 the local approx-imations are obtained by an initial LT-DF-LCC2 step,yet the geometry optimization is carried out by the DF-LCC2 method with the doubles quantities of all statesbeing restricted to the LT-DF-LCC2 lists and domainsof that state, for which the geometry optimization is car-ried out.The local approximations including the number of redun-dant functions in each pair domain, are kept fixed duringthe optimization process in order to avoid discontinuitiesin the potential energy surface. In the following, the ac-curacy of the local approximations as specified above isinvestigated by comparing the local to canonical refer-ence results, the latter calculated with the TURBOMOLE
program package [78].During a geometry optimization the energetical order ofthe relevant low-lying excited states may change. In ourprogram the character of the eigenvectors is analysed ineach iteration of the Davidson process by calculating theoverlap with the vectors from the preceding iteration.Thus, by default, the geometry optimization follows aparticular state even if the order of the states changes.Yet for our test calculations comparing local and canoni-cal results this feature was switched off, since TURBOMOLEis lacking this option.Table I compiles canonical adiabatic excitation energiesof several molecules and states and the deviations of thelocal results. Moreover, for the local methods, the root-mean-square (rms) deviation σrms in atomic positions Ri
from the canonical reference is listed, which is calculatedas
σrms =
√
√
√
√|N∑
i
(Rloci −Rcan
i )2|/N , (48)
where N denotes the number of atoms in the molecule.For measuring bond lengths and angles, as well as for cal-culating σrms and the preceding alignment of the struc-tures we used the VMD program [79].Evidently, neither the adiabatic excitation energies, norσrms show a noticeable difference in accuracy between theindividual local methods. Moreover, also the convergencebehaviour of the geometry optimization is very similar forall three methods, as can be seen in Table I (all excitedstate geometry optimizations were started from the re-spective ground state geometry, while all ground stategeometry optimizations were started from the respectivegeometries used originally in Ref. 60). This implies thatthe approximate Lagrangian poses no problems in geom-etry optimizations using the LT-DF-LCC2 method. Con-sequently, there is no need to use the DF-LCC2 methodor the hybrid (LT-)DF-LCC2 method, which are compu-tationally much more expensive, because the eigenvalueproblem cannot be reduced to an effective singles prob-
lem as for the LT-DF-LCC2 method. For example, thegeometry optimization of the S1 state of DMABN tookthree to four times longer for the (LT-)DF-LCC2 andDF-LCC2 calculations than for the LT-DF-LCC2 calcu-lation.The deviations of the adiabatic excitation energies arenot larger than those of the vertical excitation energies(cf. Table 1 in Refs. 54 and 56), generally below 0.05eV.Furthermore, σrms lies in all of the cases clearly below0.1A. To understand the higher values for HPA and ty-rosine, their structures have to be considered. HPA andtyrosine consist of an aromatic ring and a side chain.Thus a deviation in one of the angles at the connectionof the two parts can cause a larger σrms. E.g., in the op-timized ground state geometries of HPA the maximumdeviation from local to canonical dihedral angles, whichdescribe the position of the side chain relative to the aro-matic ring, is 2.6. This leads to a bad alignment forone half of the structure and a high σrms, although eachof the two parts of the structure considered separatly isvery similar to the canonical one. Also energetically, thegeometry of the local calculation is very close to that ofthe canonical calculation, e.g. for the tyrosine S1 stateit lies less than 0.05 kcal/mol above the canonical mini-mum. Additionally, in Table II the maximum deviationsin the bond lengths, bond angles, and dihedral angles aregiven separately for each molecule and state. For bondlengths the maximum deviation amounts to 0.003A. Thedeviations of the bond angles are in most of the casesclearly smaller than 0.5, the maximum deviation of 0.6
is observed for the T2 state of tyrosine. For dihedralangles deviations up to 2.9 are observed (again in tyro-sine), but in most of the cases they are clearly smaller.One should also mention in that context, that part ofthe reason for discrepancies between canonical and lo-cal methods is basis set superposition error (BSSE): thecontamination by BSSE is much smaller in local than incanonical methods [80], see also discussion about effectof BSSE on local properties in Ref. [57]. For example,for the tyrosine molecule the maximum deviation in thebond angles occurs between the three atoms linking thecarboxyl group to the ring, the latter hovering above theπ-ring system. In the canonical geometry the bond an-gle is slightly smaller (by 0.4-0.6), moving the carboxylgroup slightly closer to the π-ring system, as expectedfor a BSSE effect.
B. Application
As an illustrative example for the efficiency and appli-cability of the new code we present results from calcu-lations on the molecules 1 and 2, which are shown infigure 1. Molecule 2 is obtained from 1 via protona-tion of the phthalimide moiety. Molecule 1 comprises55 atoms, 162 correlated electrons, and 554 basis func-tions in the cc-pVDZ basis, and molecule 2 56 atoms,162 correlated electrons, and 559 basis functions. Sim-
9
ilar but smaller molecules were recently studied experi-mentally and theoretically in the context of the synthesisof 9,10-Dihydrophenanthrenes via photocatalytic decar-boxylation [81].As displayed in Fig.1, two possible reaction pathwayslead to the cationic biradical intermediate 4, which inturn initiates the subsequent decarboxylation reaction:(i) an initial protonation step transforming 1 to 2, withsubsequent intramolecular electron transfer (IET) lead-ing to 4; or (ii) an initial IET leading to 3, followed byprotonation of 3 yielding 4.Low lying charge transfer (CT) states indicate IET.Hence, in a first step the lowest lying excited states of1 and 2 at their corresponding relaxed electronic groundstate geometries were calculated. The resulting excita-tion energies and orbital-relaxed dipole moment changesare compiled in Table III. Notably, for the protonatedmolecule 2, there are several low lying singlet and tripletCT states featuring large changes in their dipole moment,whereas for the unprotonated molecule 1 the lowest ex-cited states are all local excitations with the sole excep-tion of the (relatively high lying) S3 state. This picture isentirely analogous to that obtained in the previous studyon the smaller system [81], where only for the protonatedmolecule (corresponding to 2) low lying CT states couldbe observed in the canonical CC2 calculations. More-over, the excitation energies of 2 are clearly lower thanthose of 1, with the latter lying above the range acces-sible to the sensitizer used in the experiments ([Ir(dtb-bpy)(ppy)2]PF6) [82]. Hence, we can conclude, that thecationic biradical intermediate 4 is formed by first pro-tonation of 1 with subsequent IET, which is in line withthe conclusion drawn in Ref. 81 for a similar system.Figure 2 displays the orbital-relaxed density differencesto the ground state for the states S1 and S3 of molecule1, and for the states S1 and T1 of molecule 2. The CTcharacter of the S1 and T1 states of molecule 2 shiftingelectron density from the phenyl to the phthalimide moi-ety is clearly visible. On the other hand, the S1 state of1 is entirely localized on the phthalimide moiety. The S3
state of 1 (having an excitation energy of 4.57eV) finallyalso has CT character, but is shifting charge from the car-bonyl group rather than the phenyl ring to phthalimidemoiety.In a second step, geometry optimizations for the lowestexcited states of the molecules 1 and 2 were carried out.The changes of the molecular geometries during these op-timizations are shown in figure 3. While for the S1 stateof molecule 1 no substantial geometry changes are ob-served relative to the ground state structure, this is notso for the S1 state of molecule 2: here the geometry doesnot converge and approaches a conical intersection withthe ground state. During the first iterations the lengthof the bond between the nitrogen atom of the phthalim-ide moiety and the oxygen atom connecting it to the restof the molecule rapidly increases from 1.36A to 1.41A.Simultaneously, the length of the bond between this oxy-gen and the carbon atom of the carbonyl group decreases
from 1.45A to 1.40A, while the angle between the twooxygen atoms increases slightly. The same behaviour isalso observed for the lowest triplet state of molecule 2.These findings are again in agreement with those of theprevious study on a similar system [81] and match theproposed mechanism, in which the subsequent step (af-ter formation of 4) is the elimination of phthalimide andCO2 from 4. Moreover, structural changes within thephenyl and phthalimide moieties indicate the proposedIET. In contrast to the system studied in Ref. 81 there isa carbonyl group next to the phenyl moiety, which seemsto play a role in the stabilization of molecule 4.The ratios local vs. canonical in the number of uniqueelements of the doubles vector in table III lie between7.6 and 22.7% for the individual excited states of 1 and2. These small ratios indicate substantial computationalsavings due to the local approximations.The calculations were run in parallel mode, e.g. the op-timization of the S1 state of molecule 1 was run onseven Intel(R) Xeon(R) X5660 2.80GHz cores and theoptimization of the S1 state of molecule 2 on seven AMDOpteron 6180 SE 2.50 GHz cores. The optimization ofmolecule 1 converged within 17 iterations with a thresh-old of 10−6 for the energy and 10−3 for the gradient. Theright eigenvalue equation was solved for the three lowestlying states, while the left eigenvector, Lagrange multi-pliers, densities and gradient were calculated only for theground and the first excited state. The optimization wasfinished after 11 days. About a day was needed for theinitial step, in which also the local approximations aredetermined, while one optimization step took less than13 hours. For the protonated molecule 2 the initial steptook about 1.5 days and one iteration less than 15 hoursdue to the larger domains (cf. the doubles ratios in ta-ble III). The optimization did not converge for that casedue to the conical intersection with the ground state, asdiscussed above.The timings for finding the left and right eigenvectors ofthe Jacobian and for the calculation of properties werediscussed in detail in earlier publications [54, 56]. TheDavidson diagonalization starts from the converged vec-tors of the preceding optimization step, thus in the firstoptimization steps it converges slower than in later opti-mization steps, where only little changes in the vectorsoccur. In the optimization steps 1 and 10 the right eigen-vectors are for 1 obtained within 5.3 and 3.5 hours, andfor 2 within 8.3 and 2.6 hours, respectively. The lefteigenvector is obtained starting from the right eigenvec-tor within several iterations. Thus (i) the effect of thelarger domains in 2 is not as distinct as for the righteigenvector and (ii) the duration of this step is quite con-stant during the optimization, i.e. about 1.5 hours.As discussed recently in Ref. 57, most of the time forthe calculation of the Lagrange multipliers z, zloc andx is needed for the intermediates for the linear z-vector
equations, i.e., Bµi, Bµr, Brµ, while solving the linearz-vector equations takes only a few minutes (the latteralmost entirely for the Z-CPHF equations, while the Z-
10
CPL equations take virtually no time). For 1 and 2 thelinear z-vector equations are solved within less than 5minutes, and the intermediates are calculated within abit less than one hour for 1 and about 75 minutes for 2.In the assembly of the final gradient according to Eqs.(28) and (40), the construction of the intermediate quan-tities for the contractions with the derivative integralshqµν , (µν|P )q, Sq
µν and JqPQ is dominating this step of
the calculation: for the two molecules 1 and 2 the overalltimes for assembling the gradient (including both groundand excited state parts) were 60, and 80 minutes, respec-tively.
IV. CONCLUSIONS
Formalism, implementation, and test calculations for ex-cited state gradients in the context of the local CC2response method LT-DF-LCC2 are reported. The newmethod enables geometry optimizations for the groundand excited states of extended molecular systems. It isdemonstrated, that the Laplace transformation can alsobe utilized in the context of local CC2 gradients and ge-ometry optimizations. Therefore, also for such calcula-tions state-specific local approximations can be employedfor all the states taken into account, rather than solely asingle local approximation suitable only for the targetedstate in the geometry optimization. The present workshows that the deviations from the canonical referencein geometries, adiabatic excitation energies, as well asthe convergence in the geometry optimizations are vir-tually identical for the (much slower) DF-LCC2 method,where the proper Lagrangian can be used, and the LT-DF-LCC2 method, where the true Lagrangian has tobe approximated. The approximated Lagrangian henceposes in practice no problems, neither for LT-DF-LCC2first-order properties as shown before [57], nor for geom-etry optimizations as demonstrated here.The deviations of the local adiabatic excitation ener-gies from the canonical reference values are smaller than0.05eV and thus as small as the deviations of the lo-cal vertical excitation energies reported in earlier papers.The equilibrium structures are in all of our test casesvery similar to the canonical ones. The maximum devia-tion in bond lengths as observed in our test calculationsamounts to 0.003A, the deviation in bond angles is inmost cases clearly smaller than 0.5. Deviations in di-
hedral angles are usually somewhat larger, the observedmaximum deviation in our test set amounts to 2.9.As an illustrative application example we also presentexcited state geometry optimizations for two molecules(each comprising more than fifty atoms), which occur ina photocatalytic decarboxylation reaction that is of in-terest presently in our group in the context of an applica-tion project. In agreement with the results for a similarsystem we find a clear indication that the first reactionstep has to be the protonation of the phthalimide moiety,which is followed by an intramolecular electron transferstep. For systems of this size a geometry optimization ofan excited state geometry is possible within several daysto weeks on a standard workstation, depending on theconvergence behaviour. For the studied system compris-ing 56 atoms a single optimization step took less than13 hours on one of our workstations, and in total elevendays until convergence was reached.
ACKNOWLEDGMENTS
The authors thank Thomas Merz for stimulating dis-cussions concerning the application calculations. Thiswork has been financially supported by the DeutscheForschungsgemeinschaft DFG (Schu 1456/9-1). K.L.gratefully acknowledges a PhD fellowship from the Ger-man National Academic Foundation.
Appendix A: Detailed gradient expressions
Ground state
The gradient expression in LMO/PAO basis for the elec-tronic ground state can be written as
Lq0 = hq
µνD0µν + Sq
µν
(
∂rij∂SAO
µν
)
zlocij +X0µν
+(µν|P )qY µνP0 − Jq
PQZPQ0 , (A1)
with the quantities Y0µνP , Z0
PQ, which are contracted
with the derivative integrals (µν|P )q and JqPQ, respec-
tively, being defined as
11
Y0µνP = (D0
µν −1
2dHFµν )HF bP + D0
bP dHFµν − 2(D0
µρ −1
2dHFµρ )cPiρLνi
+LµiPνr[2VPir + V P
ir + 4bP tir + 2bPX(λ0T )ir + 2λ0
bP tir + 2X(λ0T )bP tir
−2cPjitjr − cPjiX(λ0T )jr −
X(λ0T )cPkitkr ]
+ ΛpµrΛ
hνi[2
ˆV Pir + 2bP λi,0
r ] + LµiΛhνj [−λj,0
r Srr′VPir′ −
λ0tcPji] + ΛpµrPνs[λ
i,0r V P
is − cPkjtks λ
j,0r ],(A2)
Z0PQ = cPir[V
Qir + V Q
ir + 2bQtir + 2bQX(λ0T )ir + 2λ0
bQtir − tjr cQji −
X(λ0T )cQjitjr] + cPri
ˆV Qir − cPji
λ0tcQij
+ D0
bP HF bQ − (D0µν −
1
2dHFµν )cPµic
Qiν . (A3)
The densities dHF and D0 appearing in Eqs. (A1-A3) aregiven in Eq. (30). The other intermediates appearing inEqs. (A2) and (A3) are defined as
V Qir = tijrsc
jsQ , ˆV Q
ir = λij,0rs csjQ ,
V Qir = tijrs(λ
j,0t cPts − Sss′ λ
k,0s′ cPjk) ,
X(λ0T )ir = λj,0s Sss′ t
jis′r ,
HF bQ = cQµνdHFµν ,
D0
bQ = cQµν(D0µν −
1
2dHFµν ) ,
bQ = cQirtir , cQij = cQirt
jr ,
λ0
bQ = cQriλi,0r , λ0
cQij = cQrj λi,0r ,
X(λ0T )bQ = cQirX(λ0T )ir , X(λ0T )cQij = cQirX(λ0T )jr ,
λ0T cQij = λi,0s cQsrt
jr . (A4)
Excited singlet and triplet states
The gradient expression in LMO/PAO basis for excitedsinglet states can be written as
Lqf = hq
µνDfµν + Sq
µν
(
∂rij∂SAO
µν
)
zlocij +Xfµν
+(µν|P )q(13YfµνP + 1Yf
µνP )
−JqPQ(
13ZfPQ + 1Zf
PQ), (A5)
while for excited triplet states we have instead
Lqf = hq
µνDfµν + Sq
µν
(
∂rij∂SAO
µν
)
zlocij +Xfµν
+(µν|P )q(13YfµνP + 3Yf
µνP )
−JqPQ(
13ZfPQ + 3Zf
PQ). (A6)
For compactness we have splitted the intermediates,which are contracted with the derivative integrals(µν|P )q and Jq
PQ into parts which occur both in sin-
glet and triplet excited states (13YfµνP and 13Zf
PQ), and
parts which exclusively either occur in the singlet (1YfµνP ,
1ZfPQ), or the triplet (
3YfµνP ,
3ZfPQ) cases. They are de-
fined as
12
13YfµνP = Df
µνHF bP + Df
bP dHFµν − 2 Df
µρcPiρLνi
+ LµiPνr[2(λfLbP + λf
bP + LRbP + dbP )tir + 2bP dλfL
ir − LtcPjiRjr − (dcPji +
λfLcPji)tjr − cPjid
λfLjr
+V Pir + LRV P
ir − V Pis Sss′L
ks′R
kr + V P
jrdLij ]
+ ΛpµrΛ
hνi[2
ˆV Pir + LWP
ir + 2bP λi,fr ]
+LµiΛhνj [−
LRcPji −λf tcPji + 2bP dLij − cPkid
Lkj − V P
ir Srr′ λj,fr′ − RV P
ir Srr′Ljr′ −
LV PjrSrr′R
ir′ ]
+ΛpµrPνs[λ
i,fr V P
is + LirRV P
is + LV Pir R
is − cPikL
krR
is − cPikλ
k,fr tis + 2bP dLrs −
RcPkjtks L
jr], (A7)
1YfµνP = LµiPνr[2(
LbP + X(LT )bP )Rir + 2RbPX(LT )ir −
X(LT )cPjiRjr −
RcPjiX(LT )jr] + ΛpµrΛ
hνi2
RbP Lir, (A8)
3YfµνP = LµiPνr[−
X′(LT )cPjiRjr −
RcPjiX′(LT )jr] (A9)
13ZfPQ = cPir[V
Qir + LRV Q
ir + 2(λf
bQ + λfLbQ + LRbQ)tir −LtcQjiR
jr − (dcQji +
λfLcQji)tjr]
+cPri[ˆV Qir + LWQ
ir ] + cPij [−LRcQji −
λf tcQji + 2bQdLij ] +Df
bP HF bQ −Dfµνc
Pµic
Qiν , (A10)
1ZfPQ = cPir[2(
LbQ + X(LT )bQ)Rir −
X(LT )cQjiRjr], (A11)
3ZfPQ = −cPir
X′(LT )cQjiRjr, (A12)
For singlet states the excited state density matrix Dfµν is
given in Eq. (41). The other intermediates appearing inEqs. (A7-A12) are defined as
X(LT )ir = LjsSss′ t
jis′r ,
dLij = −LjsSss′R
is′ , dLrs = Lk
rRks ,
dλfL
ir = λj,fs Sss′ t
jis′r + Lj
sSss′Rjis′r
Df
bQ = cQµνDfµν ,
RbQ = cQirRir , RcQij = cQirR
jr ,
X(LT )bQ = cQirX(LT )ir , X(LT )cQij = cQirX(LT )jr ,
λfLbQ = cQirdλfLir , λfLcQij = cQird
λfLjr ,
dbQ = cQijdLij , dcQij = cQikd
Ljk ,
LRbQ = cQrsdLrs , LRcQij = Li
scQsrR
jr ,
LbQ = cQriLir , LtcQij = Li
scQsrt
jr ,
λf
bQ = cQriλi,fr , λf tcQij = λi,f
s cQsrtjr ,
RV Qir = Rij
rscQjs , LV Q
ir = 2Lijrsc
Qsj ,
V Qir = tijrs(
ˆBQjs +
ˆB′Qjs ) ,
ˆV Qir = λij,f
rs cQsj ,
ˆBQir = λi,f
s cQsr − Srr′λk,fr′ cQik ,
ˆB′Qir = dLkic
Qkr − Srr′d
Lr′sc
Qis ,
LWQir = 2Lij
rs(Rjt c
Qst − Sss′R
ks′ c
Qkj) ,
LRV Qir = Rij
rs(Ljt c
Qts − Sss′L
ks′ c
Qjk). (A13)
For triplet states, some of these intermediates are differ-ent and have to be redefined as
Rijrs = 2(
(+)
Rijrs +
(−)
Rijrs) , Lij
rs = 2((+)
Lijrs +
(−)
Lijrs) ,
RV Qir = Rij
rscQjs , LV Q
ir =1
2Lijrsc
Qsj ,
LWQir =
1
2Ljisr(R
jt c
Qst − Sss′R
ks′ c
Qkj) ,
LRV Qir = Rji
sr(Ljt c
Qts − Sss′L
ks′ c
Qjk) ,
X ′(LT )ir = −LjsSss′t
ijs′r , X′(LT )cQij = cQirX
′(LT )jr ,
dλfL
ir = λj,fs Sss′ t
jis′r + Lj
sSss′Rjis′r. (A14)
Furthermore, the excited state density matrix Dfµν is still
calculated according to Eq. (41), yet different triplet
specific densities Dηpq and Dη
pq as defined in Eq. (47)have to be used.
Appendix B: Derivatives with respect to the AOoverlap matrix
Ground state
X0 in Eq. (28) contains terms originating from the or-thonormality condition in LHF
0 , from the orthonormal-ity condition in LCC2
0 , and from the dependency of thetransformation matrix Q on the AO overlap matrix. Theterms of the latter are in the following collected in XQ,0,
XQ,0µν =
(
∂L0
∂λij,0ab
)(
∂λij,0ab
∂SAOµν
)
+
(
∂L0
∂tijab
)(
∂tijab∂SAO
µν
)
,
X0µν = Cµp(−2ǫiδij + x0
pq +XQ,0pq )C†
qν . (B1)
The derivatives have to be calculated for the doublesparts of amplitudes and multipliers only, which are re-stricted to pair lists and domains in the local basis.
13
The derivatives of the Lagrangian in local basis are zerowithin the pair domains, but not outside, i.e., they arenon-zero in the canonical basis. The singles parts, on theother hand, are unrestricted, and the derivatives of theLagrangian w.r. to singles amplitudes or multipliers incanonical basis are zero.The working equations for XQ,0 are obtained by firstapplying the relation Qar = Cv†
aµSAOµν δνr, as shown in the
following for the first term (cf. derivation of the LMP2gradient, Appendix C in Ref. 12),
(
∂L0
∂λij,0ab
)(
∂λij,0ab
∂SAOµν
)
=
(
∂L0
∂λij,0ab
)(
∂(Qarλij,0rs Q†
sb)
∂SAOµν
)
=
(
∂L0
∂λij,0ab
)
(
Cv†aµδνrλ
ij,0rs Q†
sb +Qarλij,0rs δsµC
vνb
)
,
(B2)
and then the relation 1 = LL†SAO+CvCv†SAO, yielding
Cvµa
(
∂L0
∂λij,0ab
)
Qbsλji,0sr δrρS
AOρσ (LσkL
†kν + Cv
σcCv†cν )
+(LµkL†kσ + Cv
µcCv†cσ)S
AOσρ δρsλ
ji,0sr Q†
ra
(
∂L0
∂λij,0ab
)
Cv†bν
= Cvµa
(
∂L0
∂λij,0ab
)
Qbsλji,0sr δrρS
AOρσ LσkL
†kν
+Cvµa
(
∂L0
∂λij,0ab
)
λji,0bc Cv†
cν
+LµkL†kσS
AOσρ δρsλ
ji,0sr Q†
ra
(
∂L0
∂λij,0ab
)
Cv†bν
+Cvµcλ
ji,0ca
(
∂L0
∂λij,0ab
)
Cv†bν . (B3)
Note that we have used in the previous two equations theshort-hand notation Cv†
aµ for [Cv†]aµ for conciseness.Analogous terms to those of Eq. (B3) are obtained forthe derivative w.r. to the amplitudes. Eq. (A4) in Ref. 57implicitly defines the relation
1
2B0
rνCvνa = tjirtQ
†tb
(
∂(ECC20 + λ0
µiΩµi
)
∂tijab
)
+λji,0rt Q†
tb
(
∂(ECC20 + λ0
µiΩµi
)
∂λij,0ab
)
. (B4)
The derivatives of (ECC20 + λ0
µiΩµi
) with respect to the
amplitudes tijab and multipliers λij,0ab are equivalent to the
derivatives of LCC20 given in Eq. (12), because in LCC2
0
only these terms depend on tijab and λij,0ab . Thus Eq. (B4)
can be used together with the result of Eq. (B3) to ex-press XQ,0
µν as
XQ,0µν = Cv
µa
[(
∂L0
∂λij,0ab
)
Qbsλji,0sr
+
(
∂L0
∂tijab
)
Qbstjisr
]
δrρSAOρσ LσkL
†kν
+Cvµa
[(
∂L0
∂λij,0ab
)
Qbrλji,0rs
+
(
∂L0
∂tijab
)
Qbrtjirs
]
Q†scC
v†cν
+LµkL†kσS
AOσρ δρs
[
λji,0sr Q†
ra
(
∂L0
∂λij,0ab
)
+tjisrQ†ra
(
∂L0
∂tijab
)]
Cv†bν
+CvµcQcr
[
λji,0rs Q†
sa
(
∂L0
∂λij,0ab
)
+tjirsQ†sa
(
∂L0
∂tijab
)]
Cv†bν
=1
2
Cvµa[C
v†aρ(B
0rρ)
†δrσSAOσκ Lκi]L
†iν
+Cvµa[C
v†aσ(B
0rσ)
†Q†rb]C
v†bν
+Lµi[L†iσS
AOσρ δρrB
0rκC
vκa]C
v†aν
+Cvµa[QarB
0rσC
vσb]C
v†bν
. (B5)
Since X0AO is traced with the symmetric derivative
overlap matrix in the expression of the gradient (Eq.(28)) only the symmetric part of X0
AO can contribute.From Eq. (B5) it can be seen (i) that there is nointernal-internal contribution from XQ,0, and (ii) thatthe external-external contribution is already symmet-ric. Furthermore, the multipliers x0
pq are already sym-metrized by construction, cf. Eq. (23). Therefore, theX0
ij and X0ab are obtained as
X0ab = x0
ab +XQ,0ab ,
X0ij = x0
ij − 2ǫiδij . (B6)
Due to the above mentioned symmetry of X0AO only the
upper triangular off-diagonal block, i.e., the external-internal part, needs to be considered (with a factor oftwo), while the internal-external part can be dropped(see also appendix C in Ref. 12). Thus we have
X0ai = x0
ai +XQ,0ai + (x0
ia +XQ,0ia )†
= 2x0ai +XQ,0
ai + (XQ,0ia )† ,
X0ia = 0 . (B7)
Finally employing Eq. (B5) for XQ,0pq , Eq. (23) for x0
pq
and Eq. (33) from Ref. 57, which defines the quantity
14
B0 needed for x0pq as
B0pq = CµpB
0µi + CµpB
0µrQra + CµpS
AOµρ δρrB
0rνC
vνa ,
(B8)
the working equations for X0µν = CµpX
0pqCνq are
X0ab =
1
2Cv†
aµ(−B0µr + (B0
rµ)†)Q†
rb ,
X0ij = −
1
4L†iµB
0µj −
1
4(B0
µi)†Lµj
−g(z0)ij −1
2b(zloc,0)ij − 2ǫiδij ,
X0ai = −Qar(B
0µr)
†Lµi − [z0f ]ai ,
X0ia = 0, (B9)
with
g(z0)pq = ((pq|mn)− 0.5(pn|mq))z0mn ,
z0 = z0 + z0†. (B10)
Note that in the final expression for X0ab in Eq. (B9)
the symmetry of this quantity is no longer obvious due
to cancellation of terms between x0ab and XQ,0
ab . Never-theless, the symmetry is still there, as it must be, sincethis quantity is obtained as the sum of two symmetricmatrices.The quantities B0 and b(zloc,0) are obtained from thederivative of the Lagrangian with respect to the orbitalvariation as defined in Eq. (21), the working equationsfor b(zloc,0) can be found in Ref. 12, Eq. (A2). Forthe working equations for B0
rµ, B0µi and B0
µr we refer toRef. 57, Eqs. (A5-A7) respectively.
Singlet excited states
Also the excited state gradient, Eq. (40), contains deriva-tives w.r. to the AO overlap matrix SAO, which have to
be contracted with the derivative overlap integrals col-lected in matrix Sq. In analogy to the ground state, i.e.,quantity X0, the quantity Xf collects the terms originat-ing from the orthogonality condition in Lf , and from thedependency of the transformation matrix Q on the AOoverlap matrix,
XQ,fµν =
(
∂Lf
∂λij,fab
)(
∂λij,fab
∂SAOµν
)
+
(
∂Lf
∂tijab
)(
∂tijab∂SAO
µν
)
+
(
∂Lf
∂Lijab
)(
∂Lijab
∂SAOµν
)
+
(
∂Lf
∂Rijab
)(
∂Rijab
∂SAOµν
)
,
Xfµν = Cµp(x
fpq +XQ,f
pq )C†qν . (B11)
Following the same strategy as outlined above for theground state, the working equations for Xf are obtainedas
Xfµν = CµpX
fpqCνq ,
Xfab =
1
2Cv†
aµ(−Bfµr + (Bf
rµ)†)Q†
rb ,
Xfij = −
1
4L†iµB
fµj −
1
4(Bf
µi)†Lµj − g(zf )ij −
1
2b(zloc,f )ij ,
Xfai = −Qar(B
fµr)
†Lµi − [zf f ]ai ,
Xfia = 0 . (B12)
For the working equations for Bfrµ, B
fµi and Bf
µr we refer
to Ref. 57, Eqs. (B3-B5) respectively.
Triplet excited states
For triplet excited states, the corresponding quantity Xf
takes the same form as for singlet excited states, butfor triplet states the plus and minus combination of the
left and right doubles eigenvectors,(+)
R ,(−)
R ,(+)
L and(−)
L ,have to be taken into account in XQ,f . Hence for tripletexcited states Xf is specified according to Eq. (B12),
yet the Bfrµ, B
fµi, and Bf
µr entering the equation are thecorresponding triplet quantities, which can be found inRef. 57, Eqs. (C4-C6), respectively.
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16
TABLE I. Canonical adiabatic excitation energies (in eV) are listed in column ω. For the local methods the deviations of theenergies ∆ω (local-canonical, in eV), the rms deviation σrms in atomic positions (in A) and the number of iterations of thegeometry optimization Nit are shown.
DF-LCC2 (LT-)DF-LCC2 LT-DF-LCC2
State ω ∆ω σrms Nit ∆ω σrms Nit ∆ω σrms Nit
DMABN S0 0.003 4 0.003 4
S1 4.251 0.011 0.003 6 0.000 0.003 6 -0.004 0.003 8
S2 4.640 -0.001 0.004 6 -0.007 0.004 6 -0.010 0.003 6
HPA S0 0.039 14 0.039 14
T1 3.777 -0.001 0.033 27
T2 4.287 -0.001 0.020 15
p-cresol S0 0.002 3 0.002 3
S1 4.708 0.010 0.002 5 -0.004 0.002 5 -0.011 0.002 5
S2 6.024 0.000 0.003 7 -0.002 0.003 7 -0.008 0.003 10
T1 3.755 -0.007 0.003 9
T2 4.283 -0.003 0.002 7
1-phenylpyrrole S0 0.008 12 0.011 12
S1 4.732 0.015 0.005 16 0.006 0.006 16 0.007 0.007 13
T1 3.736 0.000 0.008 16
Tyrosine S0 0.035 19 0.032 18
S1 4.717 0.018 0.038 17 0.002 0.041 16 -0.003 0.036 19
T1 3.823 0.000 0.040 25
T2 4.324 0.002 0.060 21
trans-urocanic S0 0.008 4 0.008 4
acid S1 3.862 0.018 0.004 8 0.011 0.004 8 0.004 0.006 7
S2 4.803 0.018 0.006 6 0.006 0.006 6 0.007 0.007 6
T1 2.795 -0.002 0.004 7
T2 3.734 -0.007 0.004 7
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17
TABLE II. The maximum deviations of the local bond lengths r (in A), bond angles α and dihedral angles τ (in ) are shown(absolute values).
DF-LCC2 (LT-)DF-LCC2 LT-DF-LCC2
State ∆r ∆α ∆τ ∆r ∆α ∆τ ∆r ∆α ∆τ
DMABN S0 0.002 0.1 <0.1 0.002 0.1 <0.1
S1 0.002 <0.1 <0.1 0.001 <0.1 <0.1 0.002 <0.1 <0.1
S2 0.002 0.2 <0.1 0.003 0.2 <0.1 0.001 0.2 <0.1
HPA S0 0.003 0.2 2.6 0.003 0.2 2.6
T1 0.003 0.2 2.6
T2 0.003 0.3 1.1
p-cresol S0 0.002 <0.1 <0.1 0.002 <0.1 <0.1
S1 0.002 0.1 <0.1 0.002 0.1 <0.1 0.002 <0.1 <0.1
S2 0.002 0.1 <0.1 0.002 0.1 <0.1 0.002 <0.1 <0.1
T1 0.003 <0.1 <0.1
T2 0.002 <0.1 <0.1
1-phenylpyrrole S0 0.002 <0.1 0.7 0.002 <0.1 0.9
S1 0.002 0.2 0.5 0.002 0.1 0.3 0.002 0.2 0.8
T1 0.002 0.1 0.5
Tyrosine S0 0.004 0.3 2.2 0.003 0.3 2.0
S1 0.003 0.5 2.1 0.003 0.5 2.3 0.003 0.4 2.0
T1 0.003 0.3 2.5
T2 0.003 0.6 2.9
trans-urocanic S0 0.003 0.4 <0.1 0.001 0.4 <0.1
acid S1 0.004 0.4 <0.1 0.002 0.3 <0.1 0.002 0.2 <0.1
S2 0.002 0.3 <0.1 0.002 0.4 <0.1 0.002 0.5 <0.1
T1 <0.001 0.2 <0.1
T2 0.002 0.1 <0.1
18
TABLE III. Vertical excitation energies ω (in eV) and the norm of the dipole moment vector describing the change fromground to excited state |µf | (in a.u.) are shown for the lowest excited states of molecules 1 and 2 at the ground state geometry.Moreover the ratio (local vs. canonical) of the number of unique elements of the doubles quantities is listed in %.
molecule 1 molecule 2
State character ω |µf | doubles ratio character ω |µf | doubles ratio
S1 n → π∗ 4.13 1.29 9.6 CT 2.38 6.49 20.5
S2 n → π∗ 4.48 1.04 11.1 CT 2.99 5.90 20.3
S3 CT 4.57 9.16 22.1 CT 3.29 5.55 22.3
S4 π → π∗ 4.73 0.48 22.1 π → π∗ 3.41 0.40 16.9
S5 π → π∗ 4.86 0.07 14.8 CT 3.80 5.39 19.5
T1 π → π∗ 3.82 0.16 10.9 CT 2.38 6.47 20.6
T2 n → π∗ 3.87 1.21 10.9 π → π∗ 2.91 2.24 19.9
T3 π → π∗ 3.99 0.13 7.6 CT 2.98 4.55 19.9
T4 π → π∗ 4.30 1.40 16.0 CT 3.27 5.13 22.7
T5 π → π∗ 4.51 0.93 8.4 n → π∗ 3.67 2.36 18.4
FIG. 1. Two possible pathways for the reaction of molecule 1 to molecule 4, both including a protonation step and an IET, cf.Scheme 4 in Ref. 81. An initial IET leads to intermediate 3, an initial protonation step to intermediate 2.
19
FIG. 2. Orbital-relaxed density differences relative to ground state density of the singlet excited states S1 and S3, and of thelowest triplet state T1 for the molecules 1 and 2 at the relaxed ground state geometry. The yellow (bright) and dark greyiso-surfaces represent a value of +0.003 and −0.003, respectively.
1 S1, ω=4.13eV 2 S1, ω=2.38eV
1 S3, ω=4.57eV 2 T1, ω=2.38eV
FIG. 3. Change of the geometry of the molecules 1 and 2 during the optimization starting from the ground state geometry(red) and the corresponding excitation energies at the beginning and at the end of the optimization (i.e. for molecule 2 50optimization steps without convergence). The optimization steps are indicated by color (from red to blue).
1 ω=4.13eV, ω=2.76eV 2 ω=2.38eV, ω=0.06eV