DOI: 10.1039/D4FD00054D
(Paper)
Faraday Discuss., 2024, Advance Article

Yang Guo*^{a} and
Katarzyna Pernal*^{b}
^{a}Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier Chemistry, School of Chemistry and Chemical Engineering, Shandong University, Qingdao, Shandong 266237, China. E-mail: yang.guo@sdu.edu.cn
^{b}Institute of Physics, Lodz University of Technology, ul. Wolczanska 217/221, 93-005 Lodz, Poland. E-mail: pernalk@gmail.com

Received
9th March 2024
, Accepted 2nd April 2024

First published on 3rd April 2024

The adiabatic connection (AC) approximation, along with its linearized variant AC0, was introduced as a method of obtaining dynamic correlation energy. When using a complete active space self-consistent field (CASSCF) wave function as a reference, the AC0 approximation is considered one of the most efficient multi-reference perturbation theories. It only involves the use of 1st- and 2nd-order reduced density matrices. However, some numerical results have indicated that the excitation energies predicted by AC0 are not as reliable as those from the second-order N-electron valence state perturbation theory (NEVPT2). In this study, we develop a spinless formulation of AC0 based on the Dyall Hamiltonian and provide a detailed comparison between AC0 and NEVPT2 approaches. We demonstrate the components within the correlation energy expressions that are common to both methods and those unique to either AC0 or NEVPT2. We investigate the role of the terms exclusive to NEVPT2 and explore the possibility of enhancing AC0’s performance in this regard.

To extend CASSCF/CASCI methods to systems with more strongly correlated electrons or orbitals, various approximate FCI methods have been developed. In the 1970s, Malrieu and Peyerimhoff introduced their own selected CI (sCI) methods,^{12–14} while Goddard and colleagues developed the generalized valence bond (GVB) method.^{15} In the 1980s, Roos and colleagues extended CASSCF to constrained CAS and restricted active space SCF (RASSCF) methods.^{16,17} After 2000, Olsen and Gagliardi proposed the generalized active space (GAS) method,^{18–20} and Ivanic developed the occupation-restricted-multiple-active-space (ORMAS) method.^{21} In recent decades, sCI has been revived by various research groups worldwide, including Head-Gordon,^{22,23} Evangelista,^{24–26} Neese,^{27,28} Liu,^{29–34} and Sharma,^{35} among others.^{36–39} The fundamental idea behind sCI, RAS, GAS, and other similar methods is to reduce the dimension of the FCI space by introducing various approximations. Thus, these methods could be collectively referred to as reduced CI (RCI) methods. Alternatively, FCI quantum Monte Carlo (FCIQMC)^{40–44} and density matrix renormalization group (DMRG) theories^{45–51} have been developed as approximate FCI methods as well. All these methods have been used to address static correlations in systems with about one hundred strongly correlated electrons.

To capture the missing dynamic correlations of electrons, multi-reference (MR) perturbation theory (PT), MRCI, and MR coupled cluster (CC) methods based on RCI, FCIQMC, or DMRG wave functions have been developed. Using the DMRG reference, Yanai and colleagues developed CASPT2,^{52–54} MRCISD,^{55} as well as MRCC^{56} methods for strongly correlated systems requiring large active spaces. The CASPT2 methods based on RCI^{57,58} and DMRG^{59–61} are also reported by various groups. As one of the most popular MRPT2 methods, n-electron valence state perturbation theories (NEVPT2)^{62,63} based on DMRG,^{64–68} RCI,^{69} as well as FCIQMC,^{70} have been developed by many groups. Various MRCI and MRCC methods based on these approximate FCI wave functions have been developed by many groups as well.^{71,72} In the fully internally contracted (FIC) MRCISD or NEVPT2 method,^{62,63} the 5th-order reduced density matrices (RDMs) are required in principle. To reduce computational and storage costs, the rank reduction trick could be employed to reduce the requirement of 5th-order RDMs to 4th-order, provided that CASSCF or CASCI references are used.^{73} However, when utilizing approximate CAS references, the rank reduction trick could introduce unexpected intruder states.^{74} To achieve a stable FIC NEVPT2 or MRCISD algorithm with approximate CAS references, the 5th-order RDMs must be evaluated without additional approximations.^{75} Although the storage bottleneck of the 5th-order RDM could be circumvented by defining intermediates,^{68,76} their computational costs remain demanding.

To extend FIC MR dynamic correlation methods to systems with large active spaces, it is imperative to develop methods that require only low-order RDMs. Employing approximate cumulant expansion to lower the maximum rank of RDMs entering NEVPT2 equations leads to the appearance of false intruder states, which severely deteriorates the performance of the method.^{77} Evangelista and colleagues have developed various MR driven similarity renormalization group (DSRG) theories, including DSRG-PT and DSRG-CC, which only necessitate up to 3rd-order RDMs.^{78} Their extensions to systems with large active spaces have been explored as well.^{79,80} In 2018, Pernal proposed the adiabatic connection (AC) theory for MR.^{81,82} Based on transition density matrices from the extended random phase approximation (ERPA), AC theory can be considered a FIC MR method that only requires 1st- and 2nd-order RDMs. The linearized AC approximation (AC0) method could be considered as an MRPT2 method, which reverts to MP2 with a Hartree–Fock reference.^{82}

Many applications of AC and AC0 indicate that they can deliver potential energy curves of ground states as accurately as CASPT2 and NEVPT2.^{83–85} However, for excited states, the AC0 approximation tends to overestimate the excitation energies.^{83} This is because, in the present AC0, the transition density matrices are computed from the quasiparticle ERPA, which lacks negative excitation transition density matrices. To address this issue without solving more complicated ERPA equations, the AC0D model has been developed to recover the missing correlation energies due to negative excitation transition density matrices for excited states.^{86} Most recently, Pernal and Veis extended AC0 to DMRG^{87} and explored further improvements in its accuracy by removing fixed-density matrix approximation.^{88}

Although the theory of AC0 has been reported elsewhere, its relationship with conventional MRPT2, like NEVPT2, has not been discussed. In the present work, the equations of AC0 employing the Dyall Hamiltonian^{62} as the 0th-order Hamiltonian are presented and compared with NEVPT2 in detail. Like NEVPT2, the correlation energies computed by AC0 are decomposed into eight different subspaces. Detailed comparisons of correlation energies between AC0 and NEVPT2 are provided, and the importance of D corrections to AC0 for excited states is also discussed.

Ĥ = Ĥ_{0} + Ĥ_{0} + .
| (1) |

_{Sijab} = Ê^{b}_{j}Ê^{a}_{i}
| (2) |

_{Sija} = Ê^{t}_{j}Ê^{a}_{i}
| (3) |

_{Siab} = Ê^{b}_{t}Ê^{a}_{i}
| (4) |

_{Sij} = Ê^{u}_{j}Ê^{t}_{i}
| (5) |

_{Sab} = Ê^{b}_{u}Ê^{a}_{t}
| (6) |

_{Si} = Ê^{w}_{u}Ê^{t}_{i}
| (7) |

_{Sa} = Ê^{w}_{u}Ê^{a}_{t}
| (8) |

_{Sia} = Ê^{u}_{t}Ê^{a}_{i}, Ê^{a}_{u}Ê^{t}_{i}.
| (9) |

In NEVPT2, the Dyall Hamiltonian is employed as the 0th-order Hamiltonian,^{62}

(10) |

(11) |

ê^{pr}_{qs} = Ê^{p}_{q}Ê^{r}_{s} − δ_{qr}Ê^{p}_{s},
| (12) |

(13) |

F_{ii}, defined as

(14) |

(15) |

Once the 1st-order wave function of NEVPT2 is obtained, the correlation energies could be computed as (from now on the acronym NEV will be used interchangeably with NEVPT2 for brevity)

(16) |

|Ψ_{Sk}〉 = _{Sk}|Ψ_{0}〉.
| (17) |

The amplitudes T^{Sk} are usually computed by solving the projected 1st-order equations within each subspace S_{k},

〈_{Sk}|(Ĥ_{0} − E_{0})|Ψ_{Sk}〉T^{Sk} + 〈_{Sk}||Ψ_{0}〉 = 0,
| (18) |

〈_{Sk}|(Ĥ_{0} − E_{0})|Ψ_{Sk}〉T^{Sk} + 〈_{Sk}|Ĥ|Ψ_{0}〉 = 0.
| (19) |

Furthermore, with CASCI or CASSCF reference, the rank reduction trick can be used to reduce one order of RDMs,^{73}

〈_{Sk}|[Ĥ_{0},_{Sk}]|Ψ_{0}〉T^{Sk} + 〈_{Sk}|Ĥ|Ψ_{0}〉 = 0.
| (20) |

Consequently, RDMs up to 4th-order are needed for CAS reference functions. In particular, for the subspaces S_{ij}, S_{ab}, S_{ia}, two active indices are present in the 1st-order wave function and eqn (20) involves up to 3rd-order RDMs in the active space. Equations for the subspaces S_{i} and S_{a} are most demanding and they require up to 4th-order RDMs. The other subspaces S_{k} do not involve higher than 2nd-order RDMs in the active space. The NEVPT2 equations derived using eqn (20) have been reported elsewhere.^{73,90} Note that when using an approximate CASCI reference, the rank-reduction trick does not hold^{75} and the original equation, eqn (19), should be employed. Thus, in the S_{i} and S_{a} subspaces, the 5th-order RDMs are required.

Since, as it has been shown, in NEVPT2 there is no coupling between different subspaces, the amplitude equations of different subspaces could be solved independently. Therefore, the total correlation energy of NEVPT2 can be computed individually within each subspace and subsequently summed up

(21) |

(22) |

2.2.1 Extended random phase approximation. In AC, the generalized eigenvalue equation of ERPA reads,

where A^{α}, B^{α} and M are defined as,

corresponding, respectively, to α = 0 and α = 1 limits. ERPA has been derived from Rowe’s equation of motion.^{92} By assuming that an excitation operator is truncated to single excitations

the μth eigenstate of Ĥ_{α} could be generated from a ground state |Ψ_{0}^{α}〉,

the matrices in the resulting equation of motion are expressible in terms of α-dependent RDMs up to the 2nd-order. In ERPA, see eqn (23), the RDMs from the reference wave function are employed. In equations of motion derived for the excitation operator as in eqn (27), the eigenvectors are directly related to one-electron transition density matrices

and in the ERPA model one finds

respectively. Various matrices in the above equations are defined as,

In eqn (47), A^{occ–vir} is diagonal, and M^{occ–vir} represents a unit matrix. Consequently, there is no need to solve the generalized eigenvalue (GE) equation in this block. For eqn (48) and (49), it suffices to solve the GE problems involving solely the active indexes,

are the same as the Koopmans’ matrices of S_{ija} and S_{iab} subspace in NEVPT2,^{73} the eigenvalues and corresponding eigenvectors of the above two GE equations are exactly the same as the GE equations in NEVPT2 for the S_{ija} and S_{iab} subspaces,

the superscripts 0 and 1 for act–act matrices are omitted in the rest of this work. To address eqn (37), a transformation is required,

where

and the vectors (^{act–act}_{μ} + Ȳ^{act–act}_{μ}) are computed as

(23) |

(24) |

(25) |

The Ĥ_{α} in eqn (24) is the linearly interpolated Hamiltonian between Ĥ_{0} and the exact Hamiltonian Ĥ,

Ĥ_{α} = Ĥ_{0} + αĤ′ = Ĥ_{0} + α(Ĥ − Ĥ_{0}),
| (26) |

(27) |

|Ψ_{μ}^{α}〉 = Ô_{μ}^{α}|Ψ_{0}^{α}〉.
| (28) |

By assuming the consistency (killer) condition,

〈Ψ_{0}^{α}|Ô_{μ}^{α} = 0,
| (29) |

(30) |

γ_{μ}^{α} = MX_{μ}^{α} + MY_{μ}^{α}.
| (31) |

In the present work, the Dyall Hamiltonian, see eqn (10), is chosen as Ĥ_{0}, which is different from the group Hamiltonian^{93} adopted in previous AC studies. Despite this difference in Ĥ_{0} selection, the explicit matrix elements engaged in AC0 remain exactly the same. The 1st- and 2nd-order RDMs of CASSCF involved in eqn (23) are symmetrized,

Γ_{tu} = 〈Ψ_{0}|Ê^{t}_{u}|Ψ_{0}〉, (_{tu} = 2δ_{tu} − Γ_{tu}),
| (32) |

(33) |

In eqn (32), the ^{t}_{u} is the 1st-order hole density matrix.

Only two types of elements, A^{0} (B^{0}) and A^{1} (B^{1}), corresponding to α = 0 and α = 1 limits are evaluated in the AC0 method, respectively. At α = 0, the EPRA equation displays a block-diagonal structure depending on the single excitation types. This allows for its decoupling into four blocks: occupied–virtual (occ–vir), occupied–active (occ–act), active-virtual (act–vir), and active–active (act–act):

(34) |

(35) |

(36) |

(37) |

A^{0,occ–vir}_{iajb} = δ_{ij}δ_{ab}(F_{aa} − F_{ii}),
| (38) |

M^{occ–vir}_{iajb} = δ_{ij}δ_{ab},
| (39) |

A^{0,occ–act}_{itju} = δ_{ij}(K^{occ–act}_{tu} − _{tu}F_{ii}),
| (40) |

M^{occ–act}_{itju} = δ_{ij}(2δ_{tu} − Γ_{tu}) = δ_{ij}(_{tu}),
| (41) |

A^{0,act–vir}_{taub} = δ_{ab}(K^{act–vir}_{tu} + Γ_{tu}F_{aa}),
| (42) |

M^{act–vir}_{taub} = δ_{ab}(Γ_{tu}),
| (43) |

(44) |

B^{0,act–act}_{tuwv} = A^{0,act–act}_{tuvw},
| (45) |

(46) |

Since there is no off-diagonal block in eqn (34)–(36), they could be further simplified to,

A^{0,occ–vir}X^{occ–vir}_{μ} = w^{occ–vir}_{μ}M^{occ–vir}X^{occ–vir}_{μ}, Y^{occ–vir}_{μ} = 0,
| (47) |

A^{0,occ–act}X^{occ–act}_{μ} = w^{occ–act}_{μ}M^{occ–act}X^{occ–act}_{μ}, Y^{occ–act}_{μ} = 0,
| (48) |

A^{0,act–vir}X^{act–vir}_{μ} = w^{act–vir}_{μ}M^{act–vir}X^{act–vir}_{μ}, Y^{act–vir}_{μ} = 0.
| (49) |

K^{occ–act}C^{occ–act}_{μ} = ε^{occ–act}_{μ}C^{occ–act}_{μ},
| (50) |

K^{act–vir}C^{act–vir}_{μ} = ε^{act–vir}_{μ}ΓC^{act–vir}_{μ}.
| (51) |

Since the matrices

(52) |

(53) |

ε^{occ–act}_{μ} = ε^{EA1}_{μ},
| (54) |

ε^{act–vir}_{μ} = ε^{IP1}_{μ}.
| (55) |

The final eigenvalues and eigenvectors of eqn (48) and (49) can be restored for each doubly occupied MO i or virtual MO a,

(56) |

(57) |

Since A^{0,act–act} (B^{0,act–act}) is exactly the same as A^{1,act–act} (B^{1,act–act}), i.e.

A^{1,act–act}_{tuvw} = B^{1,act–act}_{tuwv} = A^{0,act–act}_{tuvw} = B^{0,act–act}_{tuwv}
| (58) |

(59) |

(60) |

(61) |

(62) |

(63) |

Then, the transformed eqn 59 can be solved as in conventional time-dependent HF,

(Ā^{act–act} + ^{act–act})(Ā^{act–act} − ^{act–act})(^{act–act}_{μ} − Ȳ^{act–act}_{μ}) = (w^{act–act}_{μ})^{2}(^{act–act}_{μ} − Ȳ^{act–act}_{μ})
| (64) |

(65) |

Assuming a properly-selected active space, all eigenvalues [w^{act–act}_{μ}]^{2} correspond to squared excitation energies in the active space and they should be positive for ground state calculations. However, when dealing with excited states, a few eigenvalues may turn negative or approach zero. In this work, if any [w^{act–act}_{μ}]^{2} is smaller than 10^{−6}, the eigenvalue and its corresponding eigenvector are disregarded in the computations.

2.2.2 Energy expressions of AC0. The derivation of AC0 energy expression has been reported elsewhere.^{81,82} To compare it with NEVPT2, a new derivation based on the transition density matrices is given below. Without loss of generality, Ĥ′ = Ĥ − Ĥ_{0} is assumed to consist of one- and two-electron operators

can be expressed in terms of 0th- and 1st-order one-electron transition density matrices,

and

respectively. In eqn (67), Ψ^{(1)}_{0} is the 1st-order correction to the ground state wave function.

one obtains a resolution of identity

satisfied by 0th-order states and a null projector

where the 0th- and 1st-order RDMs of the ground state are defined as,

The spin-free ERPA transition density matrices, eqn (31), are given by the eigenvectors [X^{(0)}_{μ}, Y^{(0)}_{μ}] corresponding to α = 0 and the 1st -order ones [X^{(1)}_{μ}, Y^{(1)}_{μ}] of ERPA, respectively. The latter follows from a standard perturbation theory treatment^{94} applied to GE in eqn (23), which, ultimately, yields the working formula for AC0 reading

where A^{(1)} and B^{(1)} are the 1st-order perturbing matrices simply obtained as A^{(1)} = A^{1} − A^{0}, similarly for B^{(1)}. The vectors Z^{+} and Z^{−} in eqn (79) are 0th-order (α = 0) ERPA eigenvectors. Taking into account the decoupling of ERPA equations at α = 0, cf. eqn (34)–(37), the Z^{±} vectors could be factorized as,

where matrices O^{AC0}_{μλ} defined in eqn (79) are given in the Appendix and the eigenvalues ω_{μ} take forms as shown in eqn (83), depending on the type of μ block,

(66) |

The 2nd-order in α correction to the ground state energy,

E^{(2)}_{0} = 〈Ψ^{(0)}_{0}|Ĥ′|Ψ^{(1)}_{0}〉,
| (67) |

[γ^{(0)}_{μ}]_{pq} = 〈Ψ^{(0)}_{0}|Ê^{p}_{q}|Ψ^{(0)}_{μ}〉,
| (68) |

[γ^{(1)}_{μ}]_{pq} = 〈Ψ^{(0)}_{0}|Ê^{p}_{q}|Ψ^{(1)}_{μ}〉 + 〈Ψ^{(1)}_{0}|Ê^{p}_{q}|Ψ^{(0)}_{μ}〉,
| (69) |

The formal derivation begins with noticing that from the completeness of eigenstates of Ĥ_{α} at any value of α

(70) |

(71) |

(72) |

By inserting eqn (71) into the two-body term of eqn (66), the expression for the 2nd-order correlation energy reads,

(73) |

By adding , the final correlation energy could be computed from transition density matrices defined in eqn (68) and (69),

(74) |

The above expression could be further re-written as,

(75) |

[γ^{(0)}]_{pq} = 〈Ψ^{(0)}_{0}|Ê^{p}_{q}|Ψ^{(0)}_{0}〉,
| (76) |

[γ^{(1)}]_{pq} = 〈Ψ^{(0)}_{0}|Ê^{p}_{q}|Ψ^{(1)}_{0}〉 + 〈Ψ^{(1)}_{0}|Ê^{p}_{q}|Ψ^{(0)}_{0}〉.
| (77) |

Eqn (75) turns into the AC0 energy expression after assuming that the 1-RDM is fixed (γ^{(1)} = 0) and the transition density matrices in the 0th- and 1st-order are obtained from the ERPA equation of motion, eqn (23),

(78) |

(79) |

(80) |

(81) |

Depending on the μ and λ indices, the AC0 correlation energies can be classified into ten distinct components, see Table 1, which allows one to write the total AC0 energy as

(82) |

(83) |

When both μ and λ are in the act–act block, the final correlation energy is always zero. This is due to the fact that,

(84) |

Among the remaining nine components (μ, λ), two of them, (occ–act, act–vir) and (occ–vir, act–act), can be regarded as belonging to S_{ia} subspace. Consequently, akin to NEVPT2, the correlation energy of AC0 can be computed across eight different subspaces, presented in Table 1. The important result of this section is that by showing that the correlation energy in the 2nd-order as given in eqn (67), is equivalent to the formula in eqn (75) expressed by the 1st-order TRDMs. This justifies a comparison of E_{Sk} terms contributing to AC0 with their counterpart contributing to NEVPT2 energy. The explicit energy expressions of E_{Sk}^{AC0} are given subspace by subspace in the Appendix. Remarkably, the energy expressions of the S_{ijab}, S_{ija} and S_{iab} subspaces are exactly the same as those of FIC-NEVPT2 (ref. 73)

(85) |

(86) |

(87) |

This means that for subspaces S_{k} involving only one active index, contributions to NEVPT2 and AC0 energies are identical.

2.2.3 D correction. It has been found that the AC0 method can produce ground state energies as accurate as NEVPT2. However, it often overestimates the excitation energies, even for low-lying excited states. Recently, one of us proposed the D correction to recover the missing correlation energies in AC0 computed for an excited state by considering the negative-transition density matrices.^{86} In this subsection, the explicit expression of the D correction is recapitulated, and the final D correction is separated into three S_{k} subspaces.

(see the Appendix for explicit expression of the matrices O′^{AC0} and O^{AC0}). Only the S_{ia}, S_{i}, and S_{a} subspaces, involving excitations between states I and J in the active space, w^{act–act}_{λ} of state I, contribute to the D correction of state J. The final correlation energy of AC0D for state J is computed as follows:

For a given excited state J, computation of the AC0 correlation energy proceeds according to the same protocol as described in Sections 2.2.1 and 2.2.2, the difference being that the ERPA GEs, eqn (34)–(36), are solved for Ψ_{J} CAS reference wave function, instead of that of ground state Ψ_{0}. It is known that all of the eigenvalues w^{act–act}, corresponding to excitations within the active orbitals space, are positive in the quasiparticle ERPA, see eqn (37). Thus, only the positive-energy transitions are considered in the calculation of excited states. However, by enlarging the manifold of the excitation operators in ERPA, there should be, as well, negative eigenvalues w^{act–act} corresponding to transitions from state J to lower states, at least to the ground state. To account for the missing correlation energies due to the de-excitation in the AC0 calculation for state J, we could evaluate the correction from all states I(I < J) with lower excitation energies instead. In the AC0 calculations of state I, there should be a one-to-one correspondence between w^{act–act} and E^{CAS}_{K} − E^{CAS}_{I}, where K is a singly excited state higher than I. In principle, each state I with lower energy than state J should contribute D corrections to the correlation energies of state J. For state J, the D correction from state I is computed as follows:

(88) |

(89) |

Although there is a one-to-one correspondence between w^{act–act} of state I and CASSCF energy differences of states J and I, E^{CAS}_{J} − E^{CAS}_{I}, the assignment is not straightforward, especially when there are many states with energies close to state J. To assist in the assignment, for molecules with spatial symmetries, the irreducible representation (irrep) of each w^{act–act}_{λ} is determined from the irreps of orbital pairs with non-zero elements in Y^{act–act}_{λ}.

For the calculations of excited states of organic molecules, we adopted the active spaces reported by Thiel and coworkers, which are detailed in Table 2. Specifically, for furan, pyrrole, benzene, and octatetraene, only the π electrons and orbitals are included in the active space. In the case of imidazole, in addition to the π electrons and orbitals, the lone-pair orbital of nitrogen is also included. We also adopted the computational procedure suggested by Thiel and coworkers.^{99} The ground state energy using the state-specific (SS) CASSCF reference was computed to calculate the excitation energies of states with different symmetries compared to the ground state. For the D correction, it is necessary to determine the energy order of excited states at the CASSCF level beforehand. In some cases, the order of excited states obtained from state-averaged (SA) CASSCF calculations may differ from that obtained from SS-CASSCF calculations. The different orders of excited states could lead to slightly different D corrections. In this work, the order from SA-CASSCF calculations for states with the same spin symmetry was used, which is consistent with a previous work published by one of the authors.^{86} The D corrections were computed using the same SA-CASSCF reference as well.

Molecule (active space) | State | S_{ab} |
S_{ij} |
S_{ia} |
S_{a} |
S_{i} |
Total correlation energy | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AC0 | NEV | AC0 | NEV | AC0 | AC0D | NEV | AC0 | AC0D | NEV | AC0 | AC0D | NEV | AC0 | AC0D | NEV | ||

Furan CAS(6,5) | 1^{1}A_{1}(SS) |
−0.14 | −1.05 | −0.01 | −0.05 | 0.21 | 0.21 | −2.99 | 0.04 | 0.04 | −0.34 | 0.00 | 0.00 | 0.00 | 0.09 | 0.09 | −23.87 |

1^{1}A_{1}(SA) |
−0.15 | −1.12 | −0.01 | −0.05 | 0.23 | 0.23 | −2.77 | 0.07 | 0.07 | −0.51 | 0.00 | 0.00 | 0.00 | 0.13 | 0.13 | −24.27 | |

2^{1}A_{1} |
−0.11 | −0.99 | −0.01 | −0.04 | 0.57 | 0.39 | −2.52 | 0.14 | 0.11 | −0.50 | 0.00 | 0.00 | 0.00 | 0.59 | 0.39 | −24.13 | |

3^{1}A_{1} |
−0.12 | −1.16 | −0.01 | −0.07 | 2.24 | 1.01 | −3.96 | 0.57 | 0.28 | −0.85 | 0.00 | 0.00 | 0.00 | 2.68 | 1.16 | −26.16 | |

1^{1}B_{2} |
−0.17 | −1.30 | −0.01 | −0.07 | 1.23 | 0.54 | −3.64 | 0.34 | 0.11 | −0.92 | 0.00 | 0.00 | 0.00 | 1.39 | 0.47 | −25.95 | |

1^{3}A_{1} |
−0.10 | −0.95 | 0.00 | −0.04 | 0.26 | 0.07 | −2.30 | 0.03 | −0.02 | −0.38 | 0.00 | 0.00 | 0.00 | 0.18 | −0.06 | −23.66 | |

1^{3}B_{2} |
−0.12 | −0.97 | −0.01 | −0.04 | 0.18 | 0.18 | −2.26 | 0.01 | 0.01 | −0.27 | 0.00 | 0.00 | 0.00 | 0.06 | 0.06 | −23.43 | |

Pyrrole CAS(6,5) | 1^{1}A_{1}(SS) |
−0.15 | −1.06 | −0.01 | −0.05 | 0.21 | 0.21 | −3.00 | 0.07 | 0.07 | −0.42 | 0.00 | 0.00 | 0.00 | 0.12 | 0.12 | −23.30 |

1^{1}A_{1}(SA) |
−0.16 | −1.12 | −0.01 | −0.05 | 0.21 | 0.21 | −2.73 | 0.06 | 0.06 | −0.50 | 0.00 | 0.00 | 0.00 | 0.10 | 0.10 | −23.53 | |

2^{1}A_{1} |
−0.11 | −1.02 | 0.00 | −0.04 | 0.54 | 0.35 | −2.43 | 0.17 | 0.13 | −0.52 | 0.00 | 0.00 | 0.00 | 0.59 | 0.37 | −23.44 | |

3^{1}A_{1} |
−0.12 | −1.14 | −0.01 | −0.06 | 1.80 | 0.82 | −3.46 | 0.54 | 0.30 | −0.80 | 0.00 | 0.00 | 0.00 | 2.21 | 0.99 | −24.96 | |

1^{1}B_{2} |
−0.16 | −1.25 | −0.01 | −0.06 | 0.86 | 0.14 | −3.04 | 0.25 | 0.02 | −0.80 | 0.00 | 0.00 | 0.00 | 0.93 | −0.01 | −24.64 | |

1^{3}A_{1} |
−0.11 | −0.99 | 0.00 | −0.03 | 0.29 | 0.05 | −2.32 | 0.05 | −0.02 | −0.47 | 0.00 | 0.00 | 0.00 | 0.23 | −0.09 | −23.19 | |

1^{3}B_{2} |
−0.13 | −0.99 | −0.01 | −0.04 | 0.17 | 0.17 | −2.21 | 0.01 | 0.01 | −0.29 | 0.00 | 0.00 | 0.00 | 0.05 | 0.05 | −22.78 | |

Imidazole CAS(8,7) | 1^{1}A′(SS) |
−0.22 | −1.74 | −0.01 | −0.04 | 0.20 | 0.20 | −3.71 | 0.08 | 0.08 | −0.76 | 0.00 | 0.00 | −0.01 | 0.04 | 0.04 | −23.69 |

1^{1}A′(SA) |
−0.25 | −1.89 | −0.01 | −0.05 | 0.23 | 0.23 | −3.42 | 0.10 | 0.10 | −0.89 | 0.00 | 0.00 | −0.01 | 0.07 | 0.07 | −24.12 | |

2^{1}A′ |
−0.21 | −1.81 | −0.01 | −0.04 | 0.60 | 0.21 | −3.25 | 0.18 | 0.07 | −0.91 | 0.00 | 0.00 | −0.01 | 0.57 | 0.07 | −24.12 | |

3^{1}A′ |
−0.26 | −2.04 | −0.01 | −0.06 | 1.12 | 0.65 | −3.90 | 0.41 | 0.23 | −1.16 | 0.00 | −0.01 | −0.01 | 1.25 | 0.59 | −25.31 | |

4^{1}A′ |
−0.22 | −1.90 | −0.01 | −0.05 | 1.49 | 0.51 | −3.95 | 0.48 | 0.07 | −1.13 | 0.00 | −0.01 | −0.01 | 1.74 | 0.35 | −25.22 | |

1^{1}A′′ |
−0.23 | −1.84 | 0.00 | −0.03 | 0.30 | 0.27 | −3.29 | 0.06 | −0.06 | −0.75 | 0.03 | 0.02 | −0.09 | 0.15 | 0.00 | −23.82 | |

2^{1}A′′ |
−0.23 | −1.86 | −0.01 | −0.03 | 0.35 | 0.35 | −3.41 | 0.06 | 0.04 | −0.83 | 0.04 | 0.03 | −0.11 | 0.21 | 0.18 | −24.16 | |

1^{3}A′ |
−0.20 | −1.71 | −0.01 | −0.03 | 0.32 | 0.32 | −3.19 | 0.08 | 0.08 | −0.77 | 0.00 | 0.00 | −0.01 | 0.20 | 0.20 | −23.65 | |

2^{3}A′ |
−0.20 | −1.75 | −0.01 | −0.04 | 0.36 | 0.33 | −3.13 | 0.12 | 0.11 | −0.82 | 0.00 | 0.00 | −0.01 | 0.28 | 0.23 | −23.76 | |

3^{3}A′ |
−0.23 | −1.87 | −0.01 | −0.04 | 0.50 | 0.29 | −3.41 | 0.21 | 0.15 | −1.06 | 0.00 | 0.00 | −0.01 | 0.47 | 0.20 | −24.46 | |

4^{3}A′ |
−0.21 | −1.77 | 0.00 | −0.03 | 0.77 | 0.15 | −3.62 | 0.35 | 0.01 | −1.11 | 0.00 | −0.01 | −0.01 | 0.91 | −0.07 | −24.67 | |

1^{3}A′′ |
−0.22 | −1.81 | −0.01 | −0.03 | 0.31 | 0.29 | −3.30 | 0.09 | −0.01 | −0.80 | 0.02 | 0.01 | −0.08 | 0.18 | 0.06 | −23.81 | |

2^{3}A′′ |
−0.22 | −1.78 | 0.00 | −0.03 | 0.36 | 0.34 | −3.45 | 0.10 | 0.02 | −0.89 | 0.03 | 0.03 | −0.11 | 0.26 | 0.17 | −24.15 | |

Benzene CAS(6,6) | 1^{1}A_{g} |
−0.10 | −0.73 | −0.02 | −0.09 | 0.25 | 0.25 | −3.64 | 0.04 | 0.04 | −0.28 | 0.00 | 0.00 | 0.00 | 0.17 | 0.17 | −25.49 |

1^{1}E_{2g} |
−0.06 | −0.57 | −0.01 | −0.07 | 0.41 | −0.59 | −2.79 | 0.06 | −0.09 | −0.29 | 0.00 | 0.00 | 0.00 | 0.41 | −0.75 | −25.15 | |

1^{1}B_{2u} |
−0.07 | −0.64 | −0.01 | −0.09 | 0.34 | 0.29 | −2.99 | 0.01 | 0.00 | −0.27 | 0.00 | 0.00 | 0.00 | 0.26 | 0.21 | −25.25 | |

1^{1}B_{1u} |
−0.14 | −1.07 | −0.02 | −0.12 | 1.22 | 0.65 | −3.92 | 0.24 | 0.16 | −0.71 | 0.00 | 0.00 | 0.00 | 1.30 | 0.65 | −27.29 | |

1^{1}E_{1u} |
−0.11 | −1.00 | −0.02 | −0.12 | 1.69 | 0.46 | −4.42 | 0.35 | 0.14 | −0.83 | 0.00 | 0.00 | 0.00 | 1.92 | 0.47 | −27.90 | |

1^{3}E_{2g} |
−0.05 | −0.55 | −0.01 | −0.07 | 0.34 | −0.39 | −2.75 | 0.05 | −0.09 | −0.28 | 0.00 | 0.00 | 0.00 | 0.33 | −0.54 | −25.06 | |

1^{3}E_{1u} |
−0.07 | −0.70 | −0.01 | −0.08 | 0.38 | 0.03 | −3.11 | 0.11 | 0.03 | −0.47 | 0.00 | 0.00 | 0.00 | 0.40 | −0.02 | −25.67 | |

1^{3}B_{2u} |
−0.10 | −0.95 | −0.02 | −0.11 | 0.89 | 0.11 | −4.23 | 0.21 | 0.06 | −0.88 | 0.00 | 0.00 | 0.00 | 0.98 | 0.05 | −27.50 | |

1^{3}B_{1u} |
−0.06 | −0.59 | −0.01 | −0.08 | 0.26 | 0.26 | −2.93 | 0.02 | 0.02 | −0.23 | 0.00 | 0.00 | 0.00 | 0.21 | 0.21 | −25.02 | |

Octatetraene CAS(8,8) | 1^{1}A_{g}(SS) |
−0.13 | −0.87 | −0.03 | −0.11 | 0.31 | 0.31 | −4.41 | 0.01 | 0.01 | −0.22 | 0.00 | 0.00 | 0.00 | 0.16 | 0.16 | −34.34 |

1^{1}A_{g}(SA) |
−0.14 | −0.91 | −0.02 | −0.12 | 0.30 | 0.30 | −4.18 | 0.00 | 0.00 | −0.38 | 0.00 | 0.00 | 0.00 | 0.15 | 0.15 | −34.67 | |

2^{1}A_{g} |
−0.09 | −0.81 | −0.02 | −0.11 | 0.41 | 0.07 | −4.07 | 0.03 | −0.01 | −0.40 | 0.00 | 0.00 | 0.00 | 0.34 | −0.05 | −34.64 | |

3^{1}A_{g} |
−0.13 | −1.20 | −0.02 | −0.16 | 1.98 | −0.07 | −5.92 | 0.66 | 0.37 | −1.21 | 0.00 | 0.00 | 0.00 | 2.49 | 0.15 | −37.87 | |

4^{1}A_{g} |
−0.08 | −0.77 | −0.01 | −0.10 | 0.47 | −0.04 | −3.85 | 0.17 | 0.08 | −0.49 | 0.00 | 0.00 | 0.00 | 0.55 | −0.05 | −34.62 | |

1^{1}B_{u} |
−0.14 | −1.21 | −0.02 | −0.15 | 1.32 | 0.61 | −5.66 | 0.41 | 0.31 | −1.09 | 0.00 | 0.00 | 0.00 | 1.57 | 0.75 | −37.44 | |

2^{1}B_{u} |
−0.08 | −0.79 | −0.01 | −0.10 | 0.44 | 0.06 | −3.89 | 0.08 | 0.02 | −0.46 | 0.00 | 0.00 | 0.00 | 0.43 | −0.01 | −34.64 | |

3^{1}B_{u} |
−0.06 | −0.73 | −0.01 | −0.10 | 0.54 | −0.98 | −3.62 | 0.16 | −0.08 | −0.49 | 0.00 | 0.00 | 0.00 | 0.62 | −1.14 | −34.49 | |

1^{3}A_{g} |
−0.09 | −0.79 | −0.02 | −0.10 | 0.35 | 0.29 | −4.05 | 0.04 | 0.02 | −0.34 | 0.00 | 0.00 | 0.00 | 0.28 | 0.21 | −34.40 | |

1^{3}B_{u} |
−0.10 | −0.82 | −0.02 | −0.11 | 0.33 | 0.33 | −4.28 | 0.03 | 0.03 | −0.32 | 0.00 | 0.00 | 0.00 | 0.24 | 0.24 | −34.48 | |

ME | −0.14 | — | −0.01 | — | 0.58 | 0.22 | — | 0.15 | 0.06 | — | 0.00 | 0.00 | — | 0.58 | 0.12 | — | |

RMSE | 0.15 | — | 0.01 | — | 0.75 | 0.41 | — | 0.21 | 0.10 | — | 0.01 | 0.01 | — | 0.84 | 0.42 | — |

The energy deviation of AC0 with respect to NEVPT2 for the N_{2} molecule is depicted in Fig. 1. Energies from both methods have been obtained with CAS(10,8) wave function. Along the entire PES, the correlation energies of the S_{ab} subspace are significantly overestimated by AC0, while those of the S_{a} subspace are underestimated. The errors from the S_{ab} and S_{a} subspaces slightly cancel each other out. Since there are only two doubly occupied MOs in the calculations, the correlation energies of the S_{ij}, S_{i}, and S_{ia} subspaces are not very significant. The overestimated energies of S_{ab} in AC0 can be attributed to the violation of Pauli’s exclusion principle, as discussed in the literature.^{101} In the AC0 wave function of the S_{ab} subspace, one active electron is allowed to be excited to the same virtual molecular orbital (MO) twice. Throughout the PES of N_{2}, AC0 overestimates the correlation energy by an average of 0.25 eV. Although the binding energies predicted by AC0 are close to those by NEVPT2, there is a hump at the bond dissociation region (1.6 Å). This could be attributed to the missing high-order particle-hole contribution in the S_{a} subspace of AC0, which is important for the strongly correlated region.

The AC0 correlation energies of Cr_{2} with respect to those of NEVPT2, utilizing CAS(12,12) as reference, are shown in Fig. 2. Similar to N_{2}, the AC0 correlation energies of the S_{ab} subspace are overestimated by about 25 mEh throughout the curve. Moreover, since there are more doubly occupied MOs, the correlation energies from the S_{ij} subspace are also overestimated. In contrast, the AC0 energies of the S_{ia}, S_{i}, and S_{a} subspaces, which involve w^{act–act}, are underestimated with respect to those of NEVPT2. There is a hump located at the equilibrium region for S_{a}. It was reported by Roos and coworkers that CASPT2 suffers from the intruder state problem around the equilibrium bond length of Cr_{2}.^{102} For Cr_{2}, the equilibrium bond distance region might be more strongly correlated. The difference of S_{a} between AC0 and NEVPT2 could be an indicator of strong correlations and the need to go beyond single excitations in the active space (or even double excitations, as in this region also NEVPT2 struggles) in the excitation operator in ERPA. The PES computed by AC0 and NEVPT2 is shown in Fig. 3. Due to the hump introduced by the S_{a} subspace, the binding energy of Cr_{2} is underestimated by AC0 compared to NEVPT2.

Fig. 3 The potential energy curves of Cr_{2} computed by AC0 and NEVPT2. The newly fitted experimental results are given for comparison.^{103} |

The excited states of five organic molecules selected from Thiel’s test set^{99} are computed using AC0(D) and NEVPT2. The contributions to correlation energies from S_{k} subspaces and the total correlation energies of NEVPT2 together with the corresponding deviations between AC0(D) and NEVPT2 are presented in Table 2. For almost all states, similar to the calculations of diatomic molecules, the AC0 correlation energies of S_{ab} and S_{ij} subspaces are overestimated, whereas the energies of the remaining three subspaces are underestimated, using NEVPT2 results as the reference. For S_{ab} and S_{ij} subspaces, the errors of AC0 between ground states and excited states are insignificant. AC0 and NEVPT2 correlation energies of S_{i} subspaces are practically the same. However, for the other two subspaces involving w^{act–act}, S_{ia} and S_{a}, the AC0 approximation significantly underestimates the correlation energies of excited states. The higher the excitation energies, the greater the deviations. These observations confirm the important role of the negative transitions in the active space, which ERPA lacks. The errors between NEVPT2 and AC0 are mainly due to the S_{ia} subspace. For the S_{ia} subspace, the mean error (ME) and root mean squared error (RMSE) of AC0 with respect to NEVPT2 are 0.58 eV and 0.75 eV, respectively. These errors are close to the overall errors between NEVPT2 and AC0. By adding the D corrections to the subspace, the ME of S_{ia} with respect to the NEVPT2 results is reduced to 0.22 eV. For the S_{a} space the ME error is reduced to 0.06 eV. Significant reductions of the AC0 correlation energy errors in S_{ia} and S_{a} spaces after adding the D correction, proves that the lowering of the error is not coincidental. It is a consequence of effectively adding contributions to the correlation energies of excited states from negative transitions in the active space by the correction.

The results of excitation energies are summarized in Table 3. The AC0 approximation tends to overestimate the excitation energies compared to NEVPT2, which is a consequence of underestimating the correlation energies of S_{ia} and S_{a} subspaces, as already discussed. By including the D correction the errors in these subspaces are reduced and the results are significantly improved for most low-lying excited states. The RMSE of excitation energies computed by the AC0D is reduced to 0.42 eV, which is close to that of NEVPT2, employing CC3 results as reference. For excited states with excitation energies higher than 6.0 eV, the AC0D corrections may not always improve the results systematically. For example, AC0 delivers more accurate excitation energies for the 1^{3}B_{2u} state of benzene than that of NEVPT2. By including the D correction, the excitation energy is reduced by about 0.93 eV. To systematically improve the accuracy of AC0 for excited states, one should construct more accurate transition densities from high-order ERPA, using projection spaces beyond the one-particle-one-hole space. Note that for the 1^{1}B_{u} state of octatetraene, the NEVPT2 method underestimates the excitation energies significantly, by 1.19 eV, due to the mixing of Rydberg and valence states. Previous studies using triple–ζ basis sets also showed underestimations of about 0.9 eV.^{104,105} To achieve more accurate results for the 1^{1}B_{u} state, it may be necessary to increase the size of the active space.^{106}

Molecule | State | AC0 | AC0D | NEVPT2 | CC3 |
---|---|---|---|---|---|

Furan | 2^{1}A_{1} |
0.55 | 0.35 | 0.10 | 6.62 |

3^{1}A_{1} |
2.20 | 0.46 | 0.68 | 8.53 | |

1^{1}B_{2} |
0.90 | −0.02 | −0.40 | 6.60 | |

1^{3}A_{1} |
0.23 | −0.02 | 0.13 | 5.48 | |

1^{3}B_{2} |
0.12 | 0.12 | 0.15 | 4.17 | |

Pyrrole | 2^{1}A_{1} |
0.62 | 0.40 | 0.14 | 6.40 |

3^{1}A_{1} |
1.99 | 0.77 | −0.12 | 8.17 | |

1^{1}B_{2} |
0.71 | −0.24 | −0.10 | 6.71 | |

1^{3}A_{1} |
0.25 | −0.07 | 0.14 | 5.51 | |

1^{3}B_{2} |
0.18 | 0.18 | 0.25 | 4.48 | |

Imidazole | 2^{1}A′ |
0.70 | 0.20 | 0.21 | 6.58 |

3^{1}A′ |
0.81 | 0.15 | −0.37 | 7.10 | |

4^{1}A′ |
1.68 | 0.29 | 0.02 | 8.45 | |

1^{1}A′′ |
0.12 | −0.03 | 0.02 | 6.82 | |

2^{1}A′′ |
0.08 | 0.05 | −0.08 | 7.93 | |

1^{3}A′ |
0.23 | 0.23 | 0.07 | 4.69 | |

2^{3}A′ |
0.29 | 0.25 | 0.06 | 5.79 | |

3^{3}A′ |
0.35 | 0.07 | −0.08 | 6.55 | |

4^{3}A′ |
0.47 | −0.51 | −0.39 | 7.42 | |

1^{3}A′′ |
0.12 | 0.01 | −0.01 | 6.37 | |

2^{3}A′′ |
0.14 | 0.05 | −0.08 | 7.51 | |

Benzene | 1^{1}E_{2g} |
0.20 | −0.96 | −0.03 | 8.43 |

1^{1}B_{2u} |
0.24 | 0.18 | 0.15 | 5.07 | |

1^{1}B_{1u} |
0.57 | −0.08 | −0.56 | 6.68 | |

1^{1}E_{1u} |
1.14 | −0.31 | −0.61 | 7.45 | |

1^{3}E_{2g} |
0.27 | −0.60 | 0.11 | 7.49 | |

1^{3}E_{1u} |
0.22 | −0.20 | −0.01 | 4.90 | |

1^{3}B_{2u} |
−0.05 | −0.98 | −0.86 | 6.04 | |

1^{3}B_{1u} |
0.25 | 0.25 | 0.22 | 4.12 | |

Octatetraene | 2^{1}A_{g} |
−0.05 | −0.44 | −0.25 | 4.97 |

3^{1}A_{g} |
1.58 | −0.76 | −0.76 | 6.50 | |

4^{1}A_{g} |
0.28 | −0.32 | −0.12 | 6.81 | |

1^{1}B_{u} |
0.21 | −0.60 | −1.19 | 4.94 | |

2^{1}B_{u} |
0.04 | −0.41 | −0.23 | 6.06 | |

3^{1}B_{u} |
0.89 | −0.87 | 0.43 | 7.91 | |

1^{3}A_{g} |
0.17 | 0.10 | 0.05 | 3.67 | |

1^{3}B_{u} |
0.09 | 0.09 | 0.01 | 2.30 | |

ME | 0.51 | −0.08 | −0.12 | — | |

MUE | 0.51 | 0.32 | 0.24 | — | |

RMSE | 0.75 | 0.42 | 0.35 | — |

S_{ijab}:

O_{ijab} = (ia|jb)[2(ia|jb) − (ib|ja)]
| (A.1) |

(A.2) |

(A.3) |

O_{ija} = (ia|jμ)[2(ia|jμ) − (iμ|ja)]
| (A.4) |

(A.5) |

(A.6) |

O_{iab} = (ia|μb)[2(ia|μb) − (ib|μa)]
| (A.7) |

(A.8) |

(A.9) |

(A.10) |

O_{ij} = (iμ|jλ)N_{iμ,jλ}
| (A.11) |

(A.12) |

(A.13) |

(A.14) |

O_{ab} = (μa|λb)N_{μa,λb}
| (A.15) |

(A.16) |

(A.17) |

(A.18) |

O_{i} = (iμ|λ)N_{iμ,λ}
| (A.19) |

(A.20) |

(A.21) |

(A.22) |

O_{a} = (μa|λ)N_{μa,λ}
| (A.23) |

(A.24) |

(A.25) |

(A.26) |

(A.27) |

(A.28) |

O_{ia} = (iμ|λa)N_{iμ,λa}
| (A.29) |

(A.30) |

(A.31) |

(A.32) |

In the above expressions, the A^{1} and B^{1} matrices are closely related to the orbital Hessian of the reference wave function,

(A.33) |

G_{pq} = Γ_{pr}F^{c}_{qr} + Γ_{pr,sp′}(qr|sp′),
| (A.34) |

(A.35) |

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