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Intermediate state representation

Derivation via intermediate states

As the name suggest, the intermediate state representation (ISR) approach consists of constructing the ADC matrix with the help of intermediate states ∣Ψ~I⟩\ket{\tilde{\Psi}_I}, obtained by applying excitation operators to the ground state ∣0⟩\ket{0}. In second quantization, the excitation operator is written as

C^I={a^a†a^i;a^a†a^b†a^ja^i,a<b,i<j;...}\hat{C}_I=\{ \hat{a}_a^\dagger\hat{a}_i;\hat{a}_a^\dagger\hat{a}_b^\dagger\hat{a}_j\hat{a}_i, a<b, i<j;... \}
(1)

where the indices a,b...a,b... refer to unoccupied orbitals, while i,j...i,j... represent occupied orbitals Schirmer (1991)Mertins & Schirmer (1996)Schirmer & Trofimov (2004). Schematic representations of single and double excitations, which are the only two excitation classes that are needed for ADC up to third order in perturbation theory, are depicted, with (a) single excitations, (b) double excitations, and (c) the structure of the ADC(2) matrix. The numbers in parenthesis indicate the highest order of perturbation theory used to describe each particular block.

The intermediate states ∣Ψ~I⟩\ket{\tilde{\Psi}_I} are obtained by first applying C^I\hat{C}_I to the many-body ground state

∣ΨI0⟩=C^I∣Ψ0⟩ \ket{\Psi_I^{0}}=\hat{C}_I\ket{\Psi_0} \,
(2)

and then performing a Gram--Schmidt orthogonalization procedure with respect to lower excitation classes (including the ground state) to obtain precursor states ∣ΨI#⟩\ket{\Psi_I^{\#}}, which can then be orthonormalized symmetrically according to Wenzel (2016)

∣Ψ~I⟩=∑J∣ΨJ#⟩SIJ−1/2 %:label: eq:ISdefinition \ket{\tilde{\Psi}_I}=\sum_J\ket{\Psi_J^{\#}}S_{IJ}^{-1/2}\,
(3)

where SIJ=⟨ΨI#∣ΨJ#⟩S_{IJ}=\braket{\Psi_I^{\#}|\Psi_J^{\#}} are overlap integrals of the precursor states.

The elements of the ADC matrix M\mathbf{M} are obtained as matrix elements of the shifted Hamiltonian in the basis of the intermediate states

MIJ=⟨Ψ~I∣H^−E0∣Ψ~J⟩=∑K,LSIK−1/2⟨ΨK#∣H^−E0∣ΨL#⟩SLJ−1/2 %:label: eq:Mdef M_{IJ}=\bra{\tilde{\Psi}_I}\hat{H}-E_0\ket{\tilde{\Psi}_J}=\sum_{K,L}S_{IK}^{-1/2}\bra{ \Psi_K^{\#}}\hat{H}-E_0\ket{\Psi_L^{\#}}S_{LJ}^{-1/2} \,
(4)

where E0E_0 is the ground-state energy. This representation of the (shifted) Hamiltonian leads to a Hermitian eigenvalue equation

MY=YΩ ,Y†Y=1 %:label: eq:adc_eigenvalue_eq \mathbf{MY} = \mathbf{Y\Omega} \, , \quad \mathbf{Y}^\dagger \mathbf{Y} = \mathbf{1} \,
(5)

the solution of which yields vertical excitation energies (ωn=En−E0\omega_n = E_n - E_0) as eigenvalues, collected in the diagonal matrix Ω\mathbf{\Omega}, and the corresponding excitation vectors as eigenvectors Yn\mathbf{Y}_n, collected in the columns of Y\mathbf{Y}.

Having obtained an expression for the ADC matrix, we return to the series expansion of the polarization propagator. In the same way as the propagator is expanded in series, also the matrix elements can be written in terms of orders of perturbation Wenzel (2016)

MIJ(k+l+m)=∑K,L(SIK−1/2)(k)(⟨ΨK#∣H^−E0∣ΨL#⟩)(l)(SLJ−1/2)(m) %:label: eq:Mseries M_{IJ}^{(k+l+m)} = \sum_{K,L}\left(S_{IK}^{-1/2}\right)^{(k)}\left(\bra{\Psi_K^{\#}}\hat{H}-E_0\ket{\Psi_L^{\#}}\right)^{(l)}\left(S_{LJ}^{-1/2}\right)^{(m)}\,
(6)

where kk and mm are the orders of perturbation theory used for the overlap matrices SIKS_{IK} and SLJS_{LJ}, ll is the order used for the matrix elements of the shifted Hamiltonian in the basis of precursor states, and the sum k+l+m=nk + l + m = n represents the order of the contribution to the ADC matrix M\mathbf{M}. In order to get all contributions of a given order nn, one needs to sum k,l,mk,l,m over all terms for which k+l+m=nk+l+m = n.

The effective transition amplitudes f\mathbf{f} can analogously be obtained in the ISR via

fI,pq=⟨Ψ~I∣a^p†a^q∣Ψ0⟩f_{I,pq} = \bra{\tilde{\Psi}_{I}} \hat{a}_p^\dagger \hat{a}_q \ket{\Psi_0}
(7)

and the transition density matrices or transition amplitudes x\mathbf{x} by contracting with the eigenvectors, x=Y†f\mathbf{x} = \mathbf{Y}^\dagger \mathbf{f}. Note that since f\mathbf{f} is a transition amplitude and not an expectation value, it is not symmetric: fI,pq≠fI,qpf_{I,pq} \neq f_{I,qp}.

Structure of the ADC matrix

As is shown here, the ADC eigenvalue equation in the space of single (S) and double (D) excitations can be written as

MeffXS=[MSS+MSD(ω1−MDD)−1MDS]XS=ωXS \mathbf{M}_{\text{eff}} \mathbf{X}_{\text{S}} = \big[ \mathbf{M}_{\text{SS}} + \mathbf{M}_{\text{SD}} (\omega \mathbf{1} - \mathbf{M}_{\text{DD}})^{-1} \mathbf{M}_{\text{DS}} \big] \mathbf{X}_{\text{S}} = \omega \mathbf{X}_{\text{S}} \,
(8)

where the subscript denote the corresponding subblocks of the ADC matrix and XS\mathbf{X}_{\text{S}} is the singles part of the eigenvector. From this, the order structure of the ADC matrix can be explained as follows. The leading contributions of MSS\mathbf{M}_{\text{SS}} and MDD\mathbf{M}_{\text{DD}}, and thus also of (ω1−MDD)−1(\omega \mathbf{1} - \mathbf{M}_{\text{DD}})^{-1}, are of zeroth order, whereas the leading contributions of MSD\mathbf{M}_{\text{SD}} and MDS\mathbf{M}_{\text{DS}} are of first order. Hence, if excited states dominated by single excitations are desired through, say, second order, then Meff\mathbf{M}_{\text{eff}} needs to be correct through this order, meaning that MSS\mathbf{M}_{\text{SS}} needs to be correct through second order, MSD\mathbf{M}_{\text{SD}} and MDS\mathbf{M}_{\text{DS}} need to be correct through first order, whereas MDD\mathbf{M}_{\text{DD}} only needs to be correct through zeroth order. Expanding MDD\mathbf{M}_{\text{DD}} through first order would lead to a third-order term since the coupling blocks are at least of first order. Equivalently, expanding the coupling blocks through second order would lead to third-order terms, which are neglected in a second-order scheme. A consistent third-order method requires each block one order higher in perturbation theory, i.e. the singles-singles block through third oder, the coupling blocks through second order and the doubles-doubles block through first order. These findings are depicted schematically below for ADC(nn) schemes up to third order. ADC(2)-x represents an ad hoc extension of ADC(2), where the first-order terms in the doubles-doubles block from ADC(3) are taken into account in order to improve the description of doubly excited states. However, this generally leads to an imbalanced description of singly and doubly excited states Dreuw & Wormit (2015). The structure of the ADC(nn) matrix at different orders nn is illustrated below. The numbers indicate the orders of perturbation theory used to expand the corresponding block.

Explicit expressions for ADC(2)

Using above expansion of MIJM_{IJ} in combination with specific classes of excitation operators and truncating the series at the desired order, various levels of ADC theory are obtained. One aspect to note is that the excitation classes needed to construct a specific ADC level are directly connected to the order of perturbation theory. This can be easily seen in this illustration of the polarization propagator, where the zeroth and first order terms are related to single excitations (only one particle-hole pair is involved), while second order terms involve double excitations (two particle-hole pairs are involved). To illustrate this further, we list the explicit expressions for the ADC matrix M\mathbf{M} in spin-orbitals up to second order Wenzel (2016)Wormit (2009):

Mia,jb(0)=(εa−εi)δabδijMia,jb(1)=−⟨ja∣∣ib⟩Mia,jb(2)=14δij∑c,k,l[tackl⟨kl∣∣bc⟩+tbckl∗⟨ac∣∣kl⟩]+14δab∑c,d,k[tcdik⟨jk∣∣cd⟩+tjkcd∗⟨cd∣∣ik⟩]−12∑c,k[tacik⟨jk∣∣bc⟩+tjkbc∗⟨ac∣∣ik⟩]Mia,klcd(1)=⟨kl∣∣id⟩δac−⟨kl∣∣ic⟩δad−⟨al∣∣cd⟩δik+⟨ak∣∣cd⟩δilMijab,kc(1)=⟨kb∣∣ij⟩δac−⟨ka∣∣ij⟩δbc−⟨ab∣∣cj⟩δik+⟨ab∣∣ci⟩δjkMijab,klcd(0)=(εa+εb−εi−εj) δacδbdδikδjl\begin{align} M_{ia,jb}^{(0)} &= (\varepsilon_a - \varepsilon_i) \delta_{ab} \delta_{ij} \\ M_{ia,jb}^{(1)} &= -\braket{ja||ib} \\ %M_{ia,jb}^{(2)} &= \frac{1}{4} \delta_{ij} \sum_{c,k,l} \left[ \frac{\braket{ac||kl} \braket{kl||bc}}{\epsilon_a + \epsilon_c - \epsilon_k - \epsilon_l} + \frac{\braket{ac||kl} \braket{kl||bc}}{\epsilon_b + \epsilon_c - \epsilon_k - \epsilon_l} \right] \nonumber \\ %&\quad + \frac{1}{4} \delta_{ab} \sum_{c,d,k} \left[ \frac{\braket{cd||ik} \braket{jk||cd}}{\epsilon_c + \epsilon_d - \epsilon_i - \epsilon_k} + \frac{\braket{cd||ik} \braket{jk||cd}}{\epsilon_c + \epsilon_d - \epsilon_j - \epsilon_k} \right] \nonumber \\ %&\quad - \frac{1}{2}\sum_{c,k} \left[ \frac{\braket{ac||ik} \braket{jk||bc}}{\epsilon_a + \epsilon_c - \epsilon_i - \epsilon_k} + \frac{\braket{ac||ik} \braket{jk||bc}}{\epsilon_b + \epsilon_c - \epsilon_j - \epsilon_k} \right] \\ M_{ia,jb}^{(2)} &= \frac{1}{4} \delta_{ij} \sum_{c,k,l} \left[ t_{ackl} \braket{kl||bc} + t_{bckl}^* \braket{ac||kl} \right] \nonumber \\ &\quad + \frac{1}{4} \delta_{ab} \sum_{c,d,k} \left[ t_{cdik} \braket{jk||cd} + t_{jkcd}^* \braket{cd||ik} \right] \nonumber \\ &\quad - \frac{1}{2}\sum_{c,k} \left[ t_{acik} \braket{jk||bc} + t_{jkbc}^* \braket{ac||ik} \right] \\ M_{ia,klcd}^{(1)} &= \braket{kl||id}\delta_{ac} - \braket{kl||ic}\delta_{ad} - \braket{al||cd}\delta_{ik} + \braket{ak||cd}\delta_{il} \\ M_{ijab,kc}^{(1)} &= \braket{kb||ij}\delta_{ac} - \braket{ka||ij}\delta_{bc} - \braket{ab||cj}\delta_{ik} + \braket{ab||ci}\delta_{jk} \\ M_{ijab,klcd}^{(0)} &= (\varepsilon_a + \varepsilon_b - \varepsilon_i - \varepsilon_j)\, \delta_{ac} \delta_{bd} \delta_{ik} \delta_{jl} \end{align}
(9)

where εp\varepsilon_p are Hartree--Fock (HF) orbital energies, ⟨pq∣∣rs⟩\braket{pq||rs} are anti-symmetrized two-electron integrals in physicists’ notation Szabo & Ostlund (2012), δpq\delta_{pq} is the Kronecker delta, and the tt-amplitudes are tijab=⟨ab∣∣ij⟩εa+εb−εi−εjt_{ijab} = \frac{\langle ab || ij \rangle}{\varepsilon_{a} + \varepsilon_{b} - \varepsilon_{i} - \varepsilon_{j}}.

Explicit expressions for the effective transition amplitudes f\mathbf{f} can be derived in an analogous manner. The necessary blocks through second order in perturbation theory are given as

fjb,ai(0)=δijδabfjb,ia(1)=−tijabfka,ij(2)=δkjεa−εi[12∑jbctijbc⟨ja∣∣bc⟩+12∑jkbtjkab⟨jk∣∣ib⟩]fklab,ij(1)=−δkitjlabfic,ab(2)=−δacεb−εi[12∑jcdtijcd⟨jb∣∣cd⟩+12∑jkctjkbc⟨jk∣∣ic⟩]fijcd,ab(1)=δactijbdfjb,ai(2)=12∑kctikac∗tjkbc−12δijγab(2)+12δabγij(2)fjb,ia(2)=−1εa+εb−εi−εj[(1−P^ij)(1−P^ab)[∑kctikac⟨kb∣∣jc⟩]−12∑cdtijcd⟨ab∣∣cd⟩−12∑kltklab⟨kl∣∣ij⟩] \begin{align*} f_{jb,ai}^{(0)} &= \delta_{ij} \delta_{ab} \\ f_{jb,ia}^{(1)} &= - t_{ijab} \\ f_{ka,ij}^{(2)} &= \frac{\delta_{kj}}{\varepsilon_a - \varepsilon_i} \Big[ \frac12 \sum_{jbc} t_{ijbc} \braket{ja||bc} + \frac12 \sum_{jkb} t_{jkab} \braket{jk||ib} \Big] \\ f_{klab,ij}^{(1)} &= - \delta_{ki} t_{jlab} \\ f_{ic,ab}^{(2)} &= \frac{- \delta_{ac}}{\varepsilon_b - \varepsilon_i} \Big[ \frac12 \sum_{jcd} t_{ijcd} \braket{jb||cd} + \frac12 \sum_{jkc} t_{jkbc} \braket{jk||ic} \Big] \\ f_{ijcd,ab}^{(1)} &= \delta_{ac} t_{ijbd} \\ f_{jb,ai}^{(2)} &= \frac12 \sum_{kc} t_{ikac}^* t_{jkbc} - \frac12 \delta_{ij} \gamma_{ab}^{(2)} + \frac12 \delta_{ab} \gamma_{ij}^{(2)} \\ %- \delta_{ij} \frac14 \sum_{klc} t_{klac}^* t_{klbc} - \delta_{ab} \frac14 \sum_{kcd} t_{ikcd}^* t_{jkcd} \\ f_{jb,ia}^{(2)} &= - \frac{1}{\varepsilon_a + \varepsilon_b - \varepsilon_i - \varepsilon_j} \Big[ (1 - \hat{P}_{ij})(1 - \hat{P}_{ab}) \big[ \sum_{kc} t_{ikac} \braket{kb||jc} \big] \\ &- \frac12 \sum_{cd} t_{ijcd} \braket{ab||cd} - \frac12 \sum_{kl} t_{klab} \braket{kl||ij} \Big] \, \end{align*}
(10)

where the operator P^pq\hat{P}_{pq} permutes the indices pp and qq in the following expression, and the second-order corrections to the ground-state density matrix γab(2)\gamma_{ab}^{(2)} and γij(2)\gamma_{ij}^{(2)} were defined here.

The structure of the ADC(2) matrix is depicted in panel (c) above. In principle, the ADC(2) matrix contains all the possible single and double excitations which can be constructed for the system of interest (using a particular basis set). However, to calculate all these excitations would be computationally very expensive or practically impossible for all but the smallest of systems. In practice, therefore, only the lowest nn excited states are ever calculated by means of iterative diagonalization algorithms, where nn is the number of states requested by the user. This means that the space of valence excitations is easily accessible, but makes the space of core excitations impossible to reach, except for molecules with very few electrons. An approach to overcome this problem will be discussed in more detail in the next section.

The ADC scheme is related to approaches such as configuration interaction (CI) or coupled cluster (CC) Helgaker et al. (2014). The CI matrix H\mathbf{H} corresponds to a representation of the Hamiltonian H^\hat{H} within the basis of excited HF determinants, ∣ΦI⟩=C^I∣ΦHF⟩\ket{\Phi_I} = \hat{C}_I \ket{\Phi_{\text{HF}}}, where C^I\hat{C}_I are the excitation operators used also in the precursor expression, and ∣ΦHF⟩\ket{\Phi_{\text{HF}}} is the HF ground state. The elements of H\mathbf{H} are thus given by

HIJ=⟨ΦI∣H^∣ΦJ⟩ H_{IJ} = \bra{\Phi_I} \hat{H} \ket{\Phi_J} \,
(11)

The secular matrix used in CC excited-state methods Schirmer & Mertins (2010), on the other hand, can be regarded as the representation of a non-Hermitian, similarity-transformed Hamiltonian Hˉ=e−T^H^eT^\bar{H} = e^{-\hat{T}} \hat{H} e^{\hat{T}} within the basis of excited HF determinants

HˉIJ=⟨ΦI∣e−T^H^eT^∣ΦJ⟩ \bar{H}_{IJ} = \bra{\Phi_I} e^{-\hat{T}} \hat{H} e^{\hat{T}} \ket{\Phi_J} \,
(12)

where T^\hat{T} is the cluster operator that generates a linear combination of singly, doubly, …\ldots excited determinants Helgaker et al. (2014). While truncated CI methods suffer from the size-consistency problem, neither ADC nor CC excited-state schemes have this issue. On the other hand, while the secular matrices of both ADC and CI are symmetric, this is not true for CC schemes because the similarity-transformed Hamiltonian Hˉ\bar{H} is not Hermitian, which complicates the calculation of excited-state and transition properties Schirmer & Mertins (2010).

Another aspect is the so-called compactness property Mertins & Schirmer (1996), which means that can use smaller explicit configuration spaces than comparable CI treatments. Both ADC and CC methods are somewhat more compact than CI, and ADC schemes are more compact than CC approaches in odd orders of perturbation theory.

Furthermore, it can be shown that the unitary coupled-cluster (UCC) approach shares the same properties as the ADC/ISR such as size consistency and compactness, and it is in fact equivalent to ADC in low orders of perturbation theory, but differences will occur in higher orders Hodecker et al. (2022).

Excited-state properties

A distinct advantage of the ISR over the classical propagator approach is that it gives direct access to excited-state wave functions by expanding it in the intermediate-state basis as

∣Ψn⟩=∑JYJn∣Ψ~J⟩ \ket{\Psi_n} = \sum_{J} Y_{Jn} \ket{\tilde{\Psi}_J} \,
(13)

where the elements of the eigenvectors are the expansion coefficients, YJn=⟨Ψ~J∣Ψn⟩Y_{Jn} = \braket{\tilde{\Psi}_J | \Psi_n}. This immediately offers the opportunity to calculate physical properties DnD_n of electronically excited state nn via Schirmer & Trofimov (2004)

Dn=⟨Ψn∣D^∣Ψn⟩=Yn† D~ Yn=∑IJYIn∗ D~IJ YJn D_n = \bra{\Psi_n} \hat{D} \ket{\Psi_n} = \mathbf{Y}_n^\dagger \, \tilde{\mathbf{D}} \, \mathbf{Y}_n = \sum_{IJ} Y_{In}^* \, \tilde{D}_{IJ} \, Y_{Jn} \,
(14)

where D~\tilde{\mathbf{D}} is the representation of the operator D^\hat{D} corresponding to the observable in the intermediate state basis

D~IJ=⟨Ψ~I∣D^∣Ψ~J⟩ \tilde{D}_{IJ} = \bra{\tilde{\Psi}_I} \hat{D} \ket{\tilde{\Psi}_J} \,
(15)

The matrix D~\tilde{\mathbf{D}} has a perturbation expansion analogous to that of M\mathbf{M} Schirmer & Trofimov (2004). It should be noted that properties calculated in this manner generally differ from those calculated as derivatives of the energy EnE_n Hodecker et al. (2019). Transition moments between two different excited states (m≠nm \neq n) can be obtained in a completely analogous manner as

Tmn=⟨Ψm∣D^∣Ψn⟩=Ym† D~ Yn T_{mn} = \bra{\Psi_m} \hat{D} \ket{\Psi_n} = \mathbf{Y}_m^\dagger \, \tilde{\mathbf{D}} \, \mathbf{Y}_n \,
(16)

While explicit expressions for the ADC matrix M\mathbf{M} are available through third order, the effective transition moments f\mathbf{f} and property matrix D~\tilde{\mathbf{D}} are only available through second order. Combining the eigenvectors Y\mathbf{Y} (and eigenvalues) of the ADC(3) matrix together with the second-order descriptions of f\mathbf{f} and D~\tilde{\mathbf{D}} yields transition moments, oscillator strengths and other properties at a level referred to as “ADC(3/2)” Schirmer & Trofimov (2004).

Size consistency

As was discussed in previous sections, truncated CI schemes suffer from the size-consistency problem, while MP2 does not. For excited-state methods, we need to define size consistency somewhat differently. Imagine again two systems that are very far apart, such that they do not interact. For the ground-state energy, a size-consistent method has to give the same result for the composite system as the sum of the individual fragments. Concerning excitation energies, the result for one of the fragments should be the same when we apply the method to the individual fragment or to the composite system and look at the local excitations on the respective fragment (to be more specific, this is referred to as size intensivity). We will check this in the following for a system consisting of two small molecules for ADC(1) and ADC(2).

import veloxchem as vlx
import gator
from gator.adconedriver import AdcOneDriver
from gator.adctwodriver import AdcTwoDriver
import numpy as np

np.set_printoptions(precision=5, suppress=True)
# LiH molecule
lih_xyz="""2

Li  0.000000   0.000000   0.000000
H   0.000000   0.000000   1.000000
"""
lih = vlx.Molecule.read_xyz_string(lih_xyz)

# Water molecule
h2o_xyz = """3

O    0.000000000000        0.000000000000        0.000000000000
H    0.000000000000        0.740848095288        0.582094932012
H    0.000000000000       -0.740848095288        0.582094932012
"""
h2o = vlx.Molecule.read_xyz_string(h2o_xyz)

# Basis set
basis_set_label = "6-31g"
basis_lih = vlx.MolecularBasis.read(lih, basis_set_label)
basis_h2o = vlx.MolecularBasis.read(h2o, basis_set_label)
Loading...

Calculate HF reference states

# SCF will be run by VeloxChem through Gator
scf_lih = gator.run_scf(lih, basis_lih, conv_thresh=1e-10, verbose=False)
scf_h2o = gator.run_scf(h2o, basis_h2o, conv_thresh=1e-10, verbose=False)
e_scf_lih = scf_lih.get_scf_energy()
e_scf_h2o = scf_h2o.get_scf_energy()

Excitation energies of individual systems

# Number of excited states for H2O, twice as many for LiH, thrice for composite system
nexc = 3

# Calculate ADC(1) and ADC(2) excitation energies with Gator
adc1_drv = AdcOneDriver(scf_lih.comm)
adc1_drv.conv_thresh = 1e-5
adc1_results_h2o = adc1_drv.compute(h2o, basis_h2o, scf_h2o.scf_tensors)
adc1_drv.nstates = 2 * nexc
adc1_results_lih = adc1_drv.compute(lih, basis_lih, scf_lih.scf_tensors)

adc2_drv = AdcTwoDriver(scf_lih.comm, scf_lih.ostream)
adc2_drv.conv_thresh = 1e-5
adc2_results_h2o = adc2_drv.compute(h2o, basis_h2o, scf_h2o.scf_tensors)
adc2_drv.nstates = 2 * nexc
adc2_results_lih = adc2_drv.compute(lih, basis_lih, scf_lih.scf_tensors)
Output
print("LiH ADC(1) excitation energies:\n", adc1_results_lih['eigenvalues'])
print("\nH2O ADC(1) excitation energies:\n", adc1_results_h2o['eigenvalues'])
print("\nLiH ADC(2) excitation energies:\n", adc2_results_lih['eigenvalues'])
print("\nH2O ADC(2) excitation energies:\n", adc2_results_h2o['eigenvalues'])
LiH ADC(1) excitation energies:
 [0.1587  0.20598 0.20598 0.28257 0.33253 0.33253]

H2O ADC(1) excitation energies:
 [0.3528  0.42544 0.44361]

LiH ADC(2) excitation energies:
 [0.14168 0.18637 0.18637 0.26695 0.31894 0.31993]

H2O ADC(2) excitation energies:
 [0.31186 0.39817 0.40289]

Composite system

# LiH and H2O molecules 100 Å apart
lih_h2o_xyz="""5

Li  0.000000             0.000000          0.000000
H   0.000000             0.000000          1.000000
O   100.000000000000     0.000000000000    0.000000000000                         
H   100.000000000000     0.740848095288    0.582094932012                         
H   100.000000000000    -0.740848095288    0.582094932012
"""
lih_h2o = vlx.Molecule.read_xyz_string(lih_h2o_xyz)
basis_lih_h2o = vlx.MolecularBasis.read(lih_h2o, basis_set_label)
# Run SCF of composite system and compare to sum of individual SCF energies
scf_lih_h2o = gator.run_scf(lih_h2o, basis_lih_h2o, conv_thresh=1e-10, verbose=False)
print("Sum of SCF:  ", e_scf_lih + e_scf_h2o, "au")
Source
print("Sum of SCF:  ", e_scf_lih + e_scf_h2o, "au")
Sum of SCF:   -83.85469422644601 au

adc1_drv.nstates = 3 * nexc
adc1_results_lih_h2o = adc1_drv.compute(lih_h2o, basis_lih_h2o, scf_lih_h2o.scf_tensors)
adc2_drv.nstates = 3 * nexc
adc2_results_lih_h2o = adc2_drv.compute(lih_h2o, basis_lih_h2o, scf_lih_h2o.scf_tensors)
Source
print("LiH ADC(1) excitation energies:\n", adc1_results_lih['eigenvalues'])
print("\nH2O ADC(1) excitation energies:\n", adc1_results_h2o['eigenvalues'])
print("\nLiH and H2O ADC(1) excitation energies:\n", adc1_results_lih_h2o['eigenvalues'])
print("\nLiH ADC(2) excitation energies:\n", adc2_results_lih['eigenvalues'])
print("\nH2O ADC(2) excitation energies:\n", adc2_results_h2o['eigenvalues'])
print("\nLiH and H2O ADC(2) excitation energies:\n", adc2_results_lih_h2o['eigenvalues'])
LiH ADC(1) excitation energies:
 [0.1587  0.20598 0.20598 0.28257 0.33253 0.33253]

H2O ADC(1) excitation energies:
 [0.3528  0.42544 0.44361]

LiH and H2O ADC(1) excitation energies:
 [0.1587  0.20598 0.20598 0.28257 0.33253 0.33253 0.33477 0.3528  0.44361]

LiH ADC(2) excitation energies:
 [0.14168 0.18637 0.18637 0.26695 0.31894 0.31993]

H2O ADC(2) excitation energies:
 [0.31186 0.39817 0.40289]

LiH and H2O ADC(2) excitation energies:
 [0.14169 0.18637 0.18637 0.26695 0.31186 0.31894 0.31993 0.31993 0.39817]

As we can see, the excitation energies of the individual LiH molecule occur also in the composite system, both at the ADC(1) and ADC(2) levels. At higher excitation energies, also the ones from water occur again. This means that the ADC(nn) approaches are indeed size consistent (or size intensive). The interested reader can confirm the same finding also for the transition moments or oscillator strengths.

References
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Spectra and properties
Polarization propagator
Spectra and properties
Core-valence separation