Local CC2 excitation energies

The CC2 excitation energies are identical to the eigenvalues of the CC2 Jacobian, which is given by the following formula [21],

Aµiνj = whereτνxis the covariant excitation operator, ˜µxis the contravariant config-uration function, andxdenotes the the order of excitation. For the modified CC2 method (CC2-b) the Fock operator occurring in the doubles-doubles blockAµ2ν2 is replaced by ˆF. The right-hand eigenvalue problem,

AU^f =ω^fU^f (2.17)

is solved iteratively using a Davidson method [60] generalized to non-symmetric matrices [61]. The right eigenvector U^f is constructed from the singles and doubles amplitudesuⁱ_a andU^ij_ab(from now on we drop the index f, denoting the number of the excited state). The corresponding linear wave operator U = U1 + U2 in its spin-adapted form is defined analogically to theToperator, i.e.

U₁ = X In each iteration the product of the Jacobian with an approximate vectoru, vAuis required, in order to compute the residual vector as

0 = X

where ωis an approximate value of the CC2 excitation energy in a given iteration. Having the residual vector R, the next expansion vector of the small Davidson subspace is computed by first-order perturbation theory, as described in Ref. 44.

Working expressions for the vectorv Auare again conveniently de-rived from eq. (2.16) by virtue of diagrammatic techniques. After invoking the DF approximation one obtains for the singles part ofv,

vⁱ

a = fˆacuⁱ_c+Saa^′

U˜_a^ik′cfˆkc−u^k_a′fˆki

−cˆ^P_kiYˆ^ak_P (2.20) + 2(ˆc^P_ai+V^P_ia′Sa^′a)bP+W^P_ic(acˆ|P)−Saa^′W^P_ka′(kiˆ|P)

− S_aa^′

u^k_a′V_ic^P(kc|P)+V_ka^P′Y^ki_P +T˜^ik_a′cx_kc and for the doubles part,

V^ij

ab = G^ij_ab+G_ba^ji +Saa^′U^ij_a′cfbc + facU_cb^ij_′Sb^′b

− Saa^′

U^ik_a′b^′fkj+U_a^kj_′b′fki

Sb^′b, (2.21)

with the new intermediates

Y^ij_P (P|ic)u_c^j, Yˆ^ai_P (Pˆ|ac)uⁱ_c, bP (P|kc)u^k_c, xia c^ka_PY^ik_P,

W^P_ia U˜^ik_acc^kc_P,

G^ij_ab Yˆ_P^ai−Saa^′u^k_a′(kiˆ|P) ˆ

c^{b j}_P, (2.22)

and ˜U_ab^ij being defined according to eq. (2.6). The summation over repeated indices (Einstein convention) is assumed in eqs. (2.20-2.22) to improve the visibility of the formulas. The intermediate V_ia^P has already been defined in eq. (2.13). For CC2-b the elements of the Fock matrix in eq. (2.21) again are replaced by their dressed counterparts.

Similarly as for the ground state we introduce local restrictions only on the double part of the excitation space, by allowing double excitations only to pair-specific domains. The determination of such ‘excited state’

domains is a non-trivial task. For example, for a charge-transfer eigenstate the relevant virtual space for a given LMO pair is not necessarily located in the spatial vicinity of these LMOs, but may in fact lie far from them. In this work we follow the approach proposed in Ref. 44: a set of ‘important’

LMOs is determined by an analysis of the CIS wave function. All LMOs are ordered according to decreasing values of their weight

wi X

a

cⁱ_a² (2.23)

in the CIS wave function (cⁱ_a denotes the CIS coefficients). Going sequen-tially through this ordered list and adding up their weights all LMOs are

considered as important until a certain thresholdκeis reached. Whenκeis set to 1, all LMOs are considered as important, due to the normalization of the CIS wave function. For each important LMO φi its ‘excited state’

orbital domain [i]^∗is constructed as the union of the original ‘ground state’

orbital domain [i] and an additional domain obtained by employing the Boughton-Pulay procedure [56] to the orbitalφ^∗_i defined as

φ^∗_i X

a

cⁱ_aφa, (2.24)

which describes for a given excited state the whole excitation from the LMO i. ‘Excited state’ orbital domains for unimportant orbitals, on the other hand, just comprise the related ‘ground state’ orbital domain, i.e., [i]^∗ =[i]. From these ‘excited state’ orbital domains the ‘excited state’ pair domains [i j]^∗ then are formed as the unions of the corresponding orbital domains, in an entire analogy to the ‘ground state’ pair domains [i j]. Fi-nally, restrictions on the ‘excited state’ pair list based on the classification into important and unimportant orbitals can be imposed. For example, the ‘excited state’ pair list may be restricted such that it includes all pairs between important orbitals, but only pairs up to a certain inter-domain distance if at least one unimportant orbital is involved. It should be noted that the inter-domain distance is defined as a minimum distance between atoms contributing to an ‘excited state’ orbital domain. It is usually dif-ferent (smaller) from the inter-orbital distance, used for the ground-state calculations. The latter is defined as a minimum distance between atoms belonging to LMOs. Domains and restricted pair lists are the cornerstone of all computational savings in local methods. Different ways to restrict the ‘excited state’ pair list and their effect on accuracy and efficiency of the method are discussed in the next section.

At this stage we performed only single-point calculations. However, it should be mentioned here that local methods have also been applied successfully to potential energy surface (PES) calculations [62,63]. In order to avoid PES discontinuities the same domains and pair lists are used along the whole reaction coordinate. These domains and pair lists are usually somewhat enlarged in comparison to a corresponding single-point calculation and include all configurations, which become important along the reaction coordinate. The very same approach can easily be adopted to the DF–LCC2 method and excited state surfaces.

Once the configuration space is selected, we look for the eigenvector, which is the most similar to the CIS wave function (in a sense that the overlap of LCC2 and CIS vectors is the largest), using the root-homing

procedure built in our Davidson algorithm. This assures that the calcula-tion always converges to the correct vector (i.e. having the largest overlap with the CIS vector), independent on possible order changes in CIS and CC2 spectra (root-switching). If the CIS vector appears to be inadequate, e.g., by comparison with the un-truncated CC2 singles vector at a certain iteration, it is of course also possible to re-specify the domains based on the actual CC2 singles vector. We should emphasize here that our approach is state-specific, i.e. in order to obtain the CC2 spectrum we repeat the whole procedure (i.e. a domain selection and looking for the eigenvector) for every CIS vector. Alternatively one could merge domains and pair lists obtained from several CIS vectors and calculate the corresponding CC2 en-ergies simultaneously. This approach will lead, however, to an increased cost of the calculation per state, since each state will be calculated in a larger configuration space than in the state-specific case.

The analysis of eqs. (2.20,2.21) shows that the most expensive steps of the product formation of the Jacobian with an approximate vector are the evaluations of the intermediates W_ia^P and G^ij_ab in eq. (2.22). The cost of calculation of those quantities has a nominal scaling of ∝ n_Pn²_AOn_FIT = O(N⁵), the same as the ground state intermediateV_ia^P defined in eq. (2.13).

Also here, the local approximation reduces this scaling toO(N²), and with the local fitting even to O(N). Of course, without any truncations of the singles vector the overall nominal scaling with respect to molecular size remainsO(N⁴). In contrast to canonical methods with de-localized orbitals, pre-screening may reduce this scaling to some extend. In any case, for the calculations presented here the singles part are not yet dominating the overall cost of the calculation.