A multiconfigurational adiabatic connection (AC) formalism is an attractive approach to compute the dynamic correlation within the complete active space self-consistent field and density matrix renormalization group (DMRG) models. Practical realizations of AC have been based on two approximations: (i) fixing one- and two-electron reduced density matrices (1- and 2-RDMs) at the zero-coupling constant limit and (ii) extended random phase approximation (ERPA). This work investigates the effect of removing the “fixed-RDM” approximation in AC. The analysis is carried out for two electronic Hamiltonian partitionings: the group product function- and the Dyall Hamiltonians. Exact reference AC integrands are generated from the DMRG full configuration interaction solver. Two AC models are investigated, employing either exact 1- and 2-RDMs or their second-order expansions in the coupling constant in the ERPA equations. Calculations for model molecules indicate that lifting the fixed-RDM approximation is a viable way toward improving the accuracy of existing AC approximations.

The biggest challenge of many-electron theories is to grasp the effect of electron correlation. Many-electron methods typically assume a model reference wavefunction Ψref and a pertinent reference energy Eref computed as the expectation value of the exact Hamiltonian Ĥ,
(1)
The electron correlation energy is then defined as the deviation of the model energy from the exact value:
(2)
This definition holds for the Hartree–Fock (HF) method and Kohn–Sham (KS) density functional theory (DFT), where the reference wavefunction takes the form of a single determinant, as well as for multiconfigurational (MC) wavefunction methods, where Ψref is given as a combination of Slater determinants. Adiabatic connection (AC) formalism, the subject of this work, enables computation of the correlation energy for a given reference. AC was first proposed in the KS-DFT framework,1–3 leading to the development of novel correlation energy functionals. Approximate AC methods were also formulated for a single determinantal Hartree–Fock reference wavefunction.4,5 Recently, AC has been extended to multiconfigurational wavefunctions.6–10 Approximations developed in the AC(MC) framework offer a lower computational cost compared to second-order multireference perturbation methods, at the same time rivaling them in terms of accuracy.11,12 Contrary to the AC(KS-DFT) and AC(HF) methods, which are limited to ground states of singlet spin symmetry, AC(MC) is applicable to both ground and excited states of arbitrary spin multiplicity.13–15 
The first step in the AC theory assumes choosing the model Hamiltonian Ĥ(0) under the requirement that the reference wavefunction Ψref is one of its eigenfunctions:
(3)
In the next step, the parameter-dependent adiabatic Hamiltonian Ĥα is constructed:
(4)
such that it is fixed at α = 0 and α = 1 to the model and the exact Hamiltonians, respectively:
(5)
(6)
Denoting by Ψνα the νth eigenfunction of Ĥα,
(7)
and by Ψα the particular eigenfunction, which at α = 0 coincides with the reference wavefunction
(8)
(9)
it is straightforward to show via the Hellmann–Feynman theorem that the correlation energy follows from the following integration:16 
(10)
(11)
where W̃α is the exact AC integrand.
In the KS-DFT theory, the KS determinant Ψref = ΦKS, by definition, yielding the exact electron density ρexact, is used as a reference. The corresponding Hamiltonian Ĥ(0) consists of the kinetic energy operator and the local Kohn–Sham potential. The adiabatic connection Hamiltonian satisfying the constraints of Eqs. (5) and (6) includes the kinetic energy operator, a linearly scaled electron interaction operator, and a local α-dependent potential, which fixes the density to the exact full-interacting density for each α, i.e., ĤKS-DFTα=T̂+αV̂ee+V̂locα, where V̂locα is such that α ρα = ρexact. Consequently, cf. Eq. (11), the AC(KS-DFT) integrand yielding the exact KS-DFT correlation energy takes the form
(12)
where EHX is a sum of the Hartree and exchange energies, i.e., EHX=ΦKS|V̂ee|ΦKS. The analysis of the exact integrand W̃KS-DFTα was conducted for a few model systems and paved the way for approximations to AC integrands, and, thus, the correlation energy functionals, ranging from simple interpolation schemes, see Ref. 17 and the references therein, to interaction-strength-interpolation models incorporating static correlation in KS-DFT.5 
The AC Hamiltonian Ĥα in the wavefunction theory is different from that in KS-DFT. In the case of the former, the reference function Ψref is, in general, multiconfigurational, and the external potential in the AC Hamiltonian is fixed, which leads to electron density varying with α. The Ĥ(0) Hamiltonian for multireference functions, which are based on partitioning the orbital set into inactive, active, and virtual orbitals, can be chosen either as a group-Hamiltonian7,18 or the familiar Dyall Hamiltonian.19 All existing approximations to the AC(MC) theory assume, in the first place, that the one-electron reduced density matrix (1-RDM), γ, and, therefore, the electron density, do not change with α,6,8,9,12
(13)
The AC approaches for strong correlation, based on multiconfigurational reference wavefunction, aim at capturing only the electron correlation not accounted for by Ψref, i.e., the dynamic correlation. Thus, it has been justified to adopt another approximation in those methods—the extended random phase approximation (ERPA), which is a single-excitation-operator theory.20–22 Encouraging results from the AC(MC) approximations applied to both, ground and excited states, with the complete active space self-consistent field (CASSCF) and density matrix renormalization group (DMRG) systems were obtained.11,12

Even though the approximation that RDMs are constant with the coupling parameter α is justified if Ψref involves a large active space, it is still one of the sources of inaccuracies of the AC(MC) methods. Thus, it is important to investigate possible ways of improving the AC models by lifting the fixed-RDM restriction. It is worth noticing that an initial study in this direction has recently been undertaken by Senjean et al.,10 who used the AC formalism to study second-order correlation corrections for the seniority-zero wavefunctions. In Ref. 10, the constraints on 1-RDM have been partially removed by keeping the natural occupation numbers constant and introducing α-dependence in the natural orbitals.

The goal of this work is twofold. First, we want to fill the gap between AC(KS-DFT) and AC(MC) theories. While the behavior of the AC integrand based on the KS reference has been extensively studied, exact AC solutions for CAS functions have only been obtained for the hydrogen molecule.7 We aim at investigating AC integrands of many-electron model systems in both ground and excited states. The second goal is to examine the effect of removing the fixed-RDM approximation, Eq. (13), on the accuracy of the AC(MC) methods, which employ ERPA. For that purpose, we used the exact α-dependent RDMs and their numerical second-order Taylor expansions. The question of finding practical approximations of the latter is left for future work.

Compared to the investigation of Ref. 10, employing exact α-dependent RDMs is equivalent to allowing both occupation numbers and natural orbitals to evolve with the coupling constant value. Thus, the remaining source of errors in the considered AC(MC) models is due to the quality of the transition density matrices obtained from ERPA. Alternatives to ERPA-based AC could be envisioned, for example, by replacing ERPA with interpolation schemes for the AC integrand. Although this would offer better computational efficiency, there does not seem to be a systematic way for improving such interpolating schemes to treat strongly correlated systems.

What follows pertains to reference wavefunctions constructed from inactive (doubly occupied) and active (fractionally occupied) orbitals (the remaining orbitals form a set of virtual orbitals), e.g., the complete active space (CAS) wavefunctions. Such wavefunctions belong to the family of group product functions (GPFs).18 Varying the coupling constant α between 0 and 1 in the AC Hamiltonian, Eq. (4), connects a model system described with Ψref, which includes only correlation within the space of active orbitals with the fully correlated limit. Presence of the inactive and virtual sets of orbitals leads to at least two possible ways of defining the Hamiltonian Ĥ(0) satisfying the condition in Eq. (3). One assumes the group product function Hamiltonian
(14)
where the group index I runs through 1, 2, 3 pertaining, to sets of inactive, active, and virtual orbitals, respectively. The effective one-electron Hamiltonian is given as a sum of the kinetic and external (electron–nuclear interaction) potential, hpq=φp|t̂+υ̂ext|φq, and the mean-field electron–electron interaction with electron assigned to groups other than I:
(15)
where ⟨pqrs⟩ denotes an antisymmetrized two-electron integral ⟨pqrs⟩ = ⟨pq|rs⟩ − ⟨pq|sr⟩, and γref is a one-electron reduced density matrix obtained from the reference wavefunction
(16)
Notice that all groups of orbitals are treated on equal footing in ĤGPF(0); in particular, the Hamiltonian includes two-particle interactions within each orbital group. Restricting the two-electron interaction operator to only active orbitals leads to the Dyall Hamiltonian,19 defined as
(17)
where Ip denotes a group that an orbital p belongs to, and it has been assumed that the set of spin orbitals is partitioned into two subsets: (i) active orbitals and (ii) inactive plus virtual orbitals:
(18)
The effective Hamiltonian in Eq. (17) differs from that in Eq. (15) by also including the mean field interaction between different inactive orbitals, namely,
(19)
The exact AC integrand, W̃α, Eq. (11), can be written solely in terms of one-electron functions, i.e., one-electron transition reduced density matrices (1-TRDMs) of the α-system, γα,ν,
(20)
where Ψα connects with Ψref; see Eq. (9), and Ψνα is the νth eigenfunction of the AC Hamiltonian Ĥα, cf. Eq. (7), and 1-RDM of the reference and α-dependent systems
(21)
This is possible by employing the relation connecting the two-electron reduced density matrix (2-RDM) with one-electron matrices23 
(22)
For both the GPF and Dyall Hamiltonian, one can write the W̃α function as a sum of the Wexactα term including transition density matrices and the Δexactα term
(23)
Δexactα is defined in such a way that it depends solely on 1-RDMs, and it vanishes if the fixed-1-RDM condition, Eq. (13), is imposed:
(24)
In general, the spectra of the GPF and Dyall Hamiltonians are not identical and only partially overlap (for example, the reference CASSCF wavefunction is an eigenfunction of both Hamiltonians). Consequently, the AC Hamiltonians, Eq. (4), based on GPF and Dyall partitionings lead to different AC integrands, for all values of α smaller than 1.
For the GPF Hamiltonian, explicit expressions for the functions Wexactα and Δexactα in terms of 1-RDMs and 1-TRDMs have already been presented in Ref. 7. By repeating the derivation for the Dyall Hamiltonian defined in Eq. (17), one arrives at the following expressions:
(25)
and
(26)
Notice that terms for which all indices pqrs belong to the set of active orbitals are excluded from the first term in Eqs. (25) and (26). Moreover, by inspection, it can be checked that the Δexactα term satisfies the condition in Eq. (24), as the last term in Eq. (26) vanishes for γα = γref due to the property IpIqγpqref=0.
In real systems, the 1-RDM obtained for α > 0 differs from its α=0γref limit and Δexactα0. If, however, the reference wavefunction Ψref is correlated (e.g., in the case of a large set of active space orbitals), then Δexactα is likely to stay close to Δexactα=0 for all values of α. This was the origin of the fixed-1-RDM approximation, Eq. (13), assumed in AC(MC) methods.6,8,11 These methods combine AC with the ERPA approximation for 1-TRDMs, γERPAα,ν. As a result, the AC integrand is given as
(27)
If the reference wavefunction is given as a HF determinant, AC(MC) based on ERPA reduces to the RPAx approximation.24 Notice that the α-dependent ERPA equation depends on the chosen AC Hamiltonian, Ĥα, and, in principle, it should be solved for α-dependent 1- and 2-RDMs corresponding to Ĥα. In practice, however, α-dependency enters the ERPA equation only via the Hamiltonian Ĥα matrix, while for the density matrices, the fixed-RDM approximation is used, i.e., γα = γref and Γα = Γref, leading to
(28)
The resulting AC method,6,8 which throughout the text will be called canonical AC, based on approximations defined in Eqs. (24), (27), and (28), recovers the correlation energy by integration of the approximate integrand WACα, i.e.,
(29)
(30)
where the expression for Wα is the same as that in Eq. (25). In another approximation, named AC0, the AC integrand WACα is expanded at α = 0 up to the first-order term in α, resulting in
(31)

Both GPF and Dyall partitioning of the AC Hamiltonian lead to the same working equations in approximate adiabatic connection methods, i.e., AC and AC0, introduced in Eqs. (29)(31). This is no longer the case if the fixed-RDM approximation is lifted in Eqs. (24) and (28).

The main goal of this work is to investigate if using α-dependent reduced density matrices in approximating AC improves the accuracy of the methods. For this purpose, highly accurate 1- and 2-RDMs will be found for either the GPF or Dyall AC Hamiltonians by means of the density matrix renormalization group (DMRG) method.25–29 Such obtained RDMs employed in the ERPA equations will give rise to α-dependent 1-TRDMs γERPAα,ν=γERPAα,νγα,Γα,Ĥα, which will be subsequently used to compute the correlation energy EcorrAC-RDM(α) as follows:
(32)
(33)
where Δexactα is given in Eq. (26). The AC-RDM(α) correlation energy expression, compared with its exact counterpart—Eqs. (10) and (11)—involves one approximation: the integrand Ψα|Ĥ|Ψα, which is expressed by α-dependent 1-RDM and 1-TRDMs via the exact relation (22), employs 1-TRDMs approximated at the ERPA level of theory. Notice that ERPA-based 1-TRDMs are determined by α-dependent 1- and 2-RDMs, but this dependence cannot be written explicitly; see Eqs. (22)–(31) in Ref. 8. We will also investigate a variant of the AC-RDM(α) approximation, with density matrices γα and Γα expanded at α = 0 up to second-order terms:
(34)
(35)
where
(36)
(similar expansion holds for ΓTaylorα). The expression for ΔAC-Taylorα follows from Eq. (26) upon inserting the expansion for 1-RDM shown in Eq. (36).
It is relevant for this work, to connect the AC approximation based on the random phase approximation with the Rayleigh–Schrödinger perturbation theory (PT). First, recalling that Eq. (11) was obtained by using the Hellmann–Feynman theorem, dEαdα=Ψα|Ĥ|Ψα, and the relation dEαdαα=0=Ψref|Ĥ|Ψref, it follows that a Taylor expansion of the function W̃α at α = 0 reads
(37)
Accordingly, the nth and (n + 1)th derivatives of the AC integrand and the energy, respectively, are equal at α = 0:
(38)
Thus, the slope of the AC integrand at α = 0 [set n = 1 in Eq. (38)] is equal to twice the second-order energy in the perturbation theory (PT), if the AC and PT theories are based on the same Hamiltonian partitioning. For a HF reference wavefunction and the Møller–Plesset (MP) Hamiltonian, the relation presented in Eq. (38) is already known.30 Notice that the first-order derivative of the exact AC integrand W̃α is consistent with the second-order energy in the MP theory only if the Dyall partitioning is employed, as opposed to the GPF Hamiltonian
(39)
since in the case of no active orbitals, only the Dyall Hamiltonian partitioning is equivalent to the partitioning used in the MP perturbation theory.

To examine the performance of different approaches within the adiabatic connection formalism, we have studied three different molecular systems of varying multireference character: the water molecule (H2O), the methylene biradical (CH2), and the nitrogen molecule (N2). All calculations were carried out in the cc-pVDZ basis.31 

First, we studied the two lowest singlet electronic states (S0, S1) and the first triplet state (T0) of the water molecule in close-to-equilibrium geometry with the H–O–H angle of 104° and the O–H bond length of 0.969 Å. For all three states, we employed uncorrelated, single configuration state functions as reference (a single HF determinant for the S0 state; combinations of two pertinent open-shell determinants for the S1 and T0 states).

The CH2 biradical represents a strongly correlated (multireference) system, in which the multireference character may be varied by changing the H–C–H angle.32 We studied two geometries: first one is close to the equilibrium structure of the S0 state, with the C–H bond length of 1.109 Å and the H–C–H angle of 101.89°; second one is close to linear, with the H–C–H angle of 170°. The near-degeneracy of the orbitals that occurs in the nearly linear arrangement results in a significantly larger electron correlation. We performed calculations for the S0 and T0 electronic states using state-specific CASSCF(2,2) reference wavefunctions.

Finally, as an example of a strongly correlated problem with a complex electronic structure, which is problematic for most of the single reference approaches,33 we examined the triple bond breaking in the N2 molecule. We compared two geometries with bond lengths of 1.090 and 10 Å. As a reference, we employed the S0 state CASSCF(6,6) wavefunction, optimized in the active space of six N 2p orbitals, which are involved in the bond breaking process.

The reference wavefunctions were computed using Orca program package.34 All AC calculations were performed with the GammCor program.35 

The exact α-dependent 1- and 2-RDMs that were used for computations of the reference exact AC correlation energies were obtained by means of the accurate DMRG calculations in the MOLMPS program,36 with the α-dependent AC Hamiltonian, Eq. (4), using either the GPF or Dyall partitioning, Eqs. (14) and (17), respectively. DMRG25–29 is a flexible polynomial scaling approximation to the full configuration interaction (FCI) method, which approximates the FCI coefficients by a tensor network called a Matrix Product State (MPS).37 Varying the dimensions of the contracted indices in the tensor network controls both the accuracy and the computational cost, allowing us to obtain nearly exact results even when FCI is computationally intractable.

The number of renormalized states (bond dimensions of MPS matrices) in the DMRG calculations was set to M = 2000, which resulted in truncation errors much smaller than 10−6. The calculations were warmed-up with the configuration interaction dynamically extended active space (CI-DEAS) procedure,28,38 and the initial DMRG orbital orderings were optimized with the Fiedler method.39 The accuracy of the DMRG-generated α-dependent RDMs was verified by comparison of the exact AC energies with the α = 1 DMRG energies. For the S1 state of the H2O molecule, we used the Harmonic Davidson procedure40 in order to track specifically this excited state and avoided state averaging, which would deteriorate the quality of the S1 MPS wavefunction. The sum over the 1-TRDMs appearing in the Wexactα definition in Eq. (25) was computed from the DMRG α-dependent 1- and 2-RDMs according to Eq. (22). This way, we were able to obtain the profiles of the exact integrands, Wexactα and Δexactα, which were then integrated by means of the Gauss–Legendre numerical quadrature with 30 points.

Next to exact AC, we present results of two approximate AC models introduced in Sec. II. First, the AC-RDM(α) approach in which ERPA equations are solved with the exact α-dependent RDMs obtained from DMRG calculations; see Eqs. (32) and (33). Second, the AC-Taylor model in which exact DMRG-derived RDMs are replaced by their numerical Taylor expansion at α = 0; see Eqs. (34)(36). Notice that the number of elements of 2-RDMs, which are not trivial, i.e., which cannot be expressed as a product of 1-RDM elements, grows from nactive4 at α = 0 for the CASSCF wavefunction reference to nbasis4, where nactive and nbasis are the number of active orbitals and basis set functions, respectively, .

To obtain the second-order Taylor expansion of the RDMs, we used the finite difference method, element-wise. We applied the 3 and 4-point forward difference scheme for the first and second derivative, respectively, which are exact for polynomials one order higher than the order of the derivative. We used the RDMs with α = 0.0, 0.04, 0.15, 0.32, which turned out to be a good compromise between avoiding the error due to the numerical noise and using a sufficiently small step in α in the numerical differentiation. However, varying the points (within the region where the numerical noise was not dominant) produced slight variations of the order of a few mhartrees in the final energy. As our goal was not to get numerically exact results with Taylor-expanded RDMs, but to check if approximating the α-dependence of the RDMs used in ERPA would be a viable approach, this error is not especially concerning.

To investigate the sensitivity of the AC models to the quality of the α-dependent RDMs, we constructed “noisy” 1- and 2-RDMs, by adding a random number, uniformly distributed from the interval (−10−3, 10−3), to each element of the density matrices, at a given value of α. These “noisy” RDMs were then passed to the procedure used to obtain the Wexactα and WAC-RDM(α) curves described earlier in this section. We denote the results obtained with these “noisy” RDMs as “Wexactα+RND” and “WAC-RDM(α) + RND” respectively.

In this section, we present numerical results of the individual AC approximations [AC0, AC, AC-RDM(α), and AC-Taylor] and their comparison to the exact reference. In Tables I and II, we present the correlation energies of the individual spin states and the respective singlet–singlet and singlet-triplet energy gaps. Table III contains correlation energies for the nitrogen molecule in the equilibrium and dissociation geometries. Figures 1, 3, and 4 show the W and Δ integrands corresponding to the energy gaps. Additionally, in Fig. 2, we show the W and Δ curves for the individual S0 state of water. The W and Δ profiles of the remaining electronic states can be found in the supplementary material.

TABLE I.

Correlation energy for the lowest singlet and triplet states of H2O molecule. The last two columns show energy differences between the ground S0 state and T0/S1 excited states. All values in mhartree.

(mhartree)S0T0S1S0-T0S0-S1
Ecorra −217.9 −191.6 −191.7 −26.3 −26.2 
AC0 −204.8 −171.1 −169.9 −33.7 −34.9 
AC −185.6 −161.2 −160.0 −24.4 −25.6 
GPF Hamiltonian 
AC-RDM(α−171.6 −163.8 −161.2 −10.4 −7.8 
AC-Taylor −172.0 −159.3 −158.6 −12.7 −13.4 
Dyall Hamiltonian 
AC-RDM(α−231.4 −203.7 −204.5 −27.6 −26.9 
AC-Taylor −225.5 −196.3 −196.3 −29.2 −29.2 
(mhartree)S0T0S1S0-T0S0-S1
Ecorra −217.9 −191.6 −191.7 −26.3 −26.2 
AC0 −204.8 −171.1 −169.9 −33.7 −34.9 
AC −185.6 −161.2 −160.0 −24.4 −25.6 
GPF Hamiltonian 
AC-RDM(α−171.6 −163.8 −161.2 −10.4 −7.8 
AC-Taylor −172.0 −159.3 −158.6 −12.7 −13.4 
Dyall Hamiltonian 
AC-RDM(α−231.4 −203.7 −204.5 −27.6 −26.9 
AC-Taylor −225.5 −196.3 −196.3 −29.2 −29.2 
a

We use correlation energy defined as Ecorr = EFCIEref, where Eref corresponds to a single configuration state (see text) energy.

TABLE II.

Correlation energy for the lowest singlet and triplet states of CH2 molecule for the equilibrium (θ = 102°) and the strong correlation (θ = 170°) geometries. The last two columns show energy differences between S0 and T0 states.

θ = 102°
(mhartree)S0T0S0-T0
Ecorra −122.4 −122.2 −0.2 
AC0 −97.3 −96.8 −0.5 
AC −101.2 −100.5 −0.7 
GPF Hamiltonian 
AC-RDM(α−96.7 −98.6 1.9 
AC-Taylor −95.5 −97.1 1.6 
Dyall Hamiltonian 
AC-RDM(α−129.4 −130.2 0.8 
AC-Taylor −119.1 −120.8 1.7 
θ = 102°
(mhartree)S0T0S0-T0
Ecorra −122.4 −122.2 −0.2 
AC0 −97.3 −96.8 −0.5 
AC −101.2 −100.5 −0.7 
GPF Hamiltonian 
AC-RDM(α−96.7 −98.6 1.9 
AC-Taylor −95.5 −97.1 1.6 
Dyall Hamiltonian 
AC-RDM(α−129.4 −130.2 0.8 
AC-Taylor −119.1 −120.8 1.7 
θ = 170°
(mhartree)S0T0S0-T0
Ecorra −129.1 −131.1 2.0 
AC0 −106.5 −103.3 −3.2 
AC −108.3 −106.9 −1.4 
GPF Hamiltonian 
AC-RDM(α−105.0 −106.6 1.6 
AC-Taylor −104.4 −105.5 1.1 
Dyall Hamiltonian 
AC-RDM(α−136.9 −140.4 3.5 
AC-Taylor −129.0 −130.6 1.6 
θ = 170°
(mhartree)S0T0S0-T0
Ecorra −129.1 −131.1 2.0 
AC0 −106.5 −103.3 −3.2 
AC −108.3 −106.9 −1.4 
GPF Hamiltonian 
AC-RDM(α−105.0 −106.6 1.6 
AC-Taylor −104.4 −105.5 1.1 
Dyall Hamiltonian 
AC-RDM(α−136.9 −140.4 3.5 
AC-Taylor −129.0 −130.6 1.6 
a

We use correlation energy defined as Ecorr = EFCIEref, where Eref corresponds to CASSCF energy.

TABLE III.

Correlation energy for the N2 molecule at the equilibrium geometry (Req = 1.090 Å) and in the dissociation limit (Rdiss = 10 Å). The last column shows energy differences between Rdiss and Req geometries.

(mhartree)ReqRdissΔE
Ecorra −190.4 −183.4 −7.0 
AC0 −155.4 −141.9 −13.5 
AC −159.2 −147.8 −11.4 
GPF Hamiltonian 
AC-RDM(α−166.1 −159.2 −6.9 
AC-Taylor −164.4 −158.0 −6.4 
Dyall Hamiltonian 
AC-RDM(α−188.5 −180.9 −7.6 
AC-Taylor −179.8 −175.0 −4.8 
(mhartree)ReqRdissΔE
Ecorra −190.4 −183.4 −7.0 
AC0 −155.4 −141.9 −13.5 
AC −159.2 −147.8 −11.4 
GPF Hamiltonian 
AC-RDM(α−166.1 −159.2 −6.9 
AC-Taylor −164.4 −158.0 −6.4 
Dyall Hamiltonian 
AC-RDM(α−188.5 −180.9 −7.6 
AC-Taylor −179.8 −175.0 −4.8 
a

We use correlation energy defined as Ecorr = EFCIEref, where Eref corresponds to CASSCF energy.

FIG. 1.

Plots of differences between integrands corresponding to the ground state (S0) and an excited state (S1 or T0) of the H2O molecule, obtained for either the GPF or the Dyall Hamiltonian.

FIG. 1.

Plots of differences between integrands corresponding to the ground state (S0) and an excited state (S1 or T0) of the H2O molecule, obtained for either the GPF or the Dyall Hamiltonian.

Close modal
FIG. 2.

Exact and approximate Wα integrands and the Δexactα function obtained for the S0 state of the water molecule, using either the GPF or the Dyall Hamiltonian.

FIG. 2.

Exact and approximate Wα integrands and the Δexactα function obtained for the S0 state of the water molecule, using either the GPF or the Dyall Hamiltonian.

Close modal

For the H2O molecule, the correlation energies and energy gaps are presented in Table I, while the W and Δ integrands are shown in Fig. 1 (see also Figs. S1–S2 and Table S1 in the supplementary material). As can be seen in Table I, the canonical AC method, assuming the fixed-reference RDM approximation, predicts both the singlet–singlet (S0-S1) and singlet–triplet (S0-T0) energy gaps in perfect agreement with the exact reference (errors of 0.6 and 1.9 mhartree, respectively). The AC0 approximation is less accurate and overestimates the S00 and S0-S1 values by 7.4 and 8.7 mhartree, respectively.

In Sec. II, we pointed out that in the fixed-RDM approximation, the ERPA-based AC methods (AC and AC0) give identical results, irrespective of the underlying Hamiltonian partitioning (GPF or Dyall). Thus, the WACα curves in the left and right panels of Fig. 1 are identical for a given energy gap, and they should be compared with the Wexactα+Δexactα plots. Interestingly, the result of integration of the WACα function leads to a good agreement, see Table I, with the result of integration of Wexactα+Δexactα for the GPF and Dyall Hamiltonian partitionings for two different reasons. In the case of the latter Hamiltonian, the agreement is due to the closeness of the WACα and Wexactα+Δexactα functions, describing correlation energy for energy gaps in the entire range of α. If the GPF Hamiltonian is employed, the curvatures of the functions are quite different, and yet, upon integration, they yield similar values of the energy gaps.

In contrast to the AC and AC0 methods, the performance of the α-dependent-RDM models relies heavily on the zeroth-order AC Hamiltonian. Both AC-RDM(α) and AC-Taylor methods based on the GPF reference Hamiltonian considerably underestimate the exact S0–T0 and S0-S1 gaps by about 13–18 mhartree. The excellent accuracy of AC and poor performance of AC-RDM(α) can be inferred from Fig. 1. The WAC curves for the energy gaps lie below Wexact for α smaller than ∼0.5 and above Wexact for the α ∈ (0.5, 1.0) range. As a result, in the canonical AC, the errors for both parts fortuitously cancel. Although both AC-RDM(α) and AC-Taylor methods better reflect the Wexact curvature for the individual states (see the left panel of Fig. 2 for the S0 state and Fig. S1 in supplementary material for all the states), they do not benefit from a similar error cancellation. Consequently, the α-dependent RDMs based on ĤGPF do not improve the AC/AC0 energy gaps.

The situation is different for the Dyall reference Hamiltonian. First , inspection of Fig. 2 shows that the curvature of the Wexactα function is reduced compared to the pertinent function obtained for the GPF Hamiltonian. Accounting for α-dependency of RDMs in the ERPA equations leads to WAC-RDM(α)α functions that are more curved than the almost linear functions WACα, but not sufficient to match the exact curve if the GPF Hamiltonian is used. Since with the Dyall partitioning the Wexactα function is more linear, WAC-RDM(α)α stays close to it over the whole range of the coupling constant α. This observation seems to be more general, i.e., the use of the Dyall Hamiltonian leads to Wexactα curves that are less bent and closer to the WACα curves for both ground and excited states (see plots in the supplementary material).

The close resemblance between Wexactα and WAC-RDM(α)α based on ĤDyall is reflected in the striking accuracy of the S0–T0 and S0-S1 energy gaps—the errors of AC-RDM(α) and AC-Taylor approximations do not exceed 3 mhartree. Specifically, the AC-RDM(α) has an error of only 1.3 and 0.7 mhartree for the two gaps, respectively. This is less than the errors of AC, which are already small due to fortunate error cancellation. Similarly, the AC-Taylor has only slightly larger errors of 2.9 and 3.0 mhartree, respectively.

Figures 1 and 2 show the shapes of the Δexactα functions, Eq. (26). The magnitude of the Δexactα term and its change between ground and excited states is smaller than that of Wexact for all considered references. The sign of the Δexact term changes when computed with different zeroth-order Hamiltonians: it is positive for the GPF partitioning and negative for the Dyall one (Fig. 2). This sign change has been observed for all studied systems (see the supplementary material). Recall that in the AC-Taylor model, the Δ terms are accounted for in an approximate manner, i.e., by employing a second-order Taylor expansion of 1-RDMs [cf. Eq. (34)]. For the water molecule, more accurate ΔTaylor terms are obtained with the Dyall partitioning (Table S1 in the supplementary material), which contributes to the overall good quality of the AC-Taylor results.

The results for the CH2 biradical are shown in Table II. As mentioned in Sec. III, we studied two geometries: the S0 equilibrium one and the almost linear one (θ = 170°). The former has a less pronounced multireference character, with the dominant determinant coefficients of 0.98 and −0.2 at the CASSCF(2,2) level, while the latter is strongly correlated with coefficients 0.73 and −0.68, respectively.

In the equilibrium geometry, the correlation contribution to the singlet–triplet (S0–T0) gap is small and amounts to −0.2 mhartree. In other words, the narrow gap (3 mhartree) is accurately described already at the reference CASSCF(2,2) level. Examination of the exact AC integrands in Fig. 3 shows that the small correlation contribution results from an almost perfect cancellation of positive and negative parts of Wexactα under the integration. This cancellation holds for both the GPF and the Dyall reference Hamiltonians.

FIG. 3.

Plots of differences between integrands corresponding to S0 and T0 states of CH2 molecule for the equilibrium (θ = 102°) and the strong correlation (θ = 170°) geometries obtained either with the GPF or Dyall Hamiltonian.

FIG. 3.

Plots of differences between integrands corresponding to S0 and T0 states of CH2 molecule for the equilibrium (θ = 102°) and the strong correlation (θ = 170°) geometries obtained either with the GPF or Dyall Hamiltonian.

Close modal

The WACα curves for CH2 (θ = 102°) differ from Wexactα (see top panel in Fig. 3). Since the WACα values are close to zero over the whole range of α, their contribution to the correlation energy remains small. In contrast to WACα, the shape of WRDM(α)α curves resembles the Wexactα reference. Nevertheless, WRDM(α)α does not achieve the same error cancellation as the Wexactα integrand. This leads to less accurate S0-T0 gaps at the AC-RDM(α) level of theory compared to the original AC0/AC approach. Still, AC-RDM(α) remains a sensible approximation—results obtained with the Dyall Hamiltonian deviate by no more than 2 mhartree from the exact values. The performance of AC-Taylor with the same Hamiltonian is only slightly worse (errors of ∼3 mhartree with respect to the benchmark).

A different story unfolds for the CH2 molecule with the bond angle of 170°. For this geometry, the WAC-RDM(α)α curve with the GPF reference Hamiltonian follows Wexactα almost perfectly (see bottom panel of Fig. 3). The similarity is worse, but not lost, by using the Taylor approximated RDMs. Although in the case of the Dyall Hamiltonian, WAC-RDM(α)α does not follow Wexactα for α > 0.5, the superiority over WACα is apparent.

The similarity between the approximate α-dependent curves and exact ones for CH2 (θ = 170°) translates into excellent results for the S0-T0 gap. Compared to the equilibrium geometry, the exact S0-T0 gap and out-of-CAS correlation contribution are larger (57 and 2 mhartree, respectively). AC with exact α-dependent RDMs gives errors of merely 0.4 and 1.5 mhartree for the GPF and Dyall Hamiltonians, respectively. Approximating the RDMs via the second-order Taylor expansion results in errors of 0.9 and 0.4 mhartree, respectively. Both the canonical AC and AC0 methods perform poorly—the correlation contribution has the wrong sign—which leads to larger errors than the uncorrected CASSCF value.

The results for the dissociation of the N2 molecule, which is the most complex system of our study, can be found in Table III. The AC-RDM(α) model based on the GPF reference Hamiltonian is the most accurate—the error in the correlation contribution to the dissociation energy is merely 0.1 mhartree. The same method with the ĤDyall reference performs slightly worse, with the error of 0.6 mhartree.

For the GPF Hamiltonian, the AC model with the second-order Taylor approximated α-dependent RDMs (AC-Taylor) differs only marginally from AC-RDM(α), with an error of 0.6 mhartree. The error obtained with the AC-Taylor based on the Dyall model is larger and amounts to 2.2 mhartree. A closer inspection of the results shows that the major source of this error is a poor description of the Δα term by the Taylor approximated 1-RDM (see Table S3 in the supplementary material). Unlike in H2O and CH2, in the nitrogen molecule, the contribution of the Δexactα term to the dissociation energy is crucial, slightly exceeding the Wexactα term (−3.7 and −3.3 mhartree, respectively, for the Dyall Hamiltonian).

The canonical AC and AC0 significantly overestimate the dynamical electron correlation contribution to the dissociation energy. This is confirmed by inspection of the W curves in Fig. 4. The WACα curve of the canonical AC formulation strongly deviates from the Wexactα reference, whereas both WAC-RDM(α)α and WAC-Taylorα closely match the benchmark (for both GPF and Dyall Hamiltonians).

FIG. 4.

Plots of differences between integrands corresponding to equilibrium (Req = 1.090 Å) and dissociation-limit (Rdiss = 10 Å) geometries for the N2 molecule obtained either for the GPF or Dyall Hamiltonian.

FIG. 4.

Plots of differences between integrands corresponding to equilibrium (Req = 1.090 Å) and dissociation-limit (Rdiss = 10 Å) geometries for the N2 molecule obtained either for the GPF or Dyall Hamiltonian.

Close modal

In Fig. 5, we plot the mean absolute errors (MAEs) of the computed energy gaps obtained by the individual methods for all systems. One can see that the Dyall reference Hamiltonian performs better than the GPF Hamiltonian for approximations with α-dependent RDMs, with AC-RDM(α) giving MAEs more than two times smaller than those of canonical AC. We would like to notice, however, that our statistics correspond to only five energy gaps studied in this work, and the conclusion should be confirmed by further, more extensive computational studies.

FIG. 5.

Mean absolute errors (MAEs) of energy gaps presented in Tables IIII for different AC approximations.

FIG. 5.

Mean absolute errors (MAEs) of energy gaps presented in Tables IIII for different AC approximations.

Close modal

It is of interest, at this point, to emphasize the observed immunity of the ERPA-based AC integrand to random inaccuracies in α-dependent RDMs employed in the ERPA equations, which is demonstrated in Fig. 6 (see computational details for the exact procedure). It is striking that the randomly perturbed α-dependent 1- and 2-RDMs used in the computation of the WAC-RDM(α)α term, cf. Eq. (33), lead to relatively smooth integrands, even if the amplitude of perturbations amounts to 10−3, see the “WAC-RDM(α)α + RND” curve in Fig. 6. This result stays in stark contrast with the exact AC integrand, Eq. (25), obtained via contracting the α-dependent 2-RDM with two-electron integrals, showing strong oscillations (the “Wexactα + RND” curve). As a consequence, the adiabatic connection method achieves the same accuracy even with RDMs of poor quality. This has already been demonstrated with DMRG,11 where RDMs from low bond dimension calculations gave nearly the same results as the more accurate ones. The insensitivity of the ERPA-based AC integrand to the quality of the input α-dependent RDMs suggests that approximate methods, for example, the Quantum Monte Carlo or selected CI, could be used to generate crude density matrices for AC-RDM(α).

FIG. 6.

The effect of adding random noise to the RDMs, demonstrating the remarkable stability of ERPA to inaccuracies in input RDMs.

FIG. 6.

The effect of adding random noise to the RDMs, demonstrating the remarkable stability of ERPA to inaccuracies in input RDMs.

Close modal

In this article, we have improved the accuracy of the multireference adiabatic connection methods by lifting the fixed-RDM restriction. We have tested ERPA-based AC models employing either exact α-dependent 1- and 2-RDMs or RDMs through second-order in α. Unlike AC methods, with fixed-reference RDMs (AC0, canonical AC), α-dependent AC models depend on the partitioning of the reference Hamiltonian. Both the GPF and Dyall Hamiltonians were studied in this work. The expression for the AC integrand in terms of the one-electron reduced functions for the Dyall Hamiltonian is presented for the first time. Numerical demonstration was carried out for several small molecules of varying multireference character.

A comparison of the exact adiabatic connecting integrands with their approximate counterparts has confirmed that the good performance of the canonical (fixed-RDM, ERPA-based) AC method, featuring the nearly linear AC integrand, is often a result of the cancellation of errors from the ERPA and fixed-RDM approximations.7 This error cancellation is lost if only one approximation, in our case the fixed-RDM one, is lifted. Fortunately, as is demonstrated in model systems, the accuracy of the AC-RDM(α) approach for the correlation energy and energy gaps is superior to that of the canonical AC. In particular, we have shown that in the case of truly multireference problems (the CH2 biradical in close-to-linear geometry and dissociation of the N2 molecule), the AC methods with α-dependent RDMs significantly outperform the fixed-RDM approximations. For these systems, the GPF reference Hamiltonian provided slightly more accurate energy gaps than the Dyall. In contrast, in the single-reference regime (the H2O molecule), the Dyall reference Hamiltonian showed much better performance. Overall, ĤDyall seems to be the reference Hamiltonian of choice for our approach, because it is capable of a balanced description of both strongly and weakly correlated molecular systems.

In most cases, the energy gaps provided by the second-order Taylor approximation of α-dependent RDMs were within 2 mhartree of the energy gaps computed by the AC method with the exact α-dependent RDMs. This provides a strong motivation for the development of new practical AC approximations without the fixed-RDM restriction, which will be the subject of our following work.

The supplementary material contains the numerical contributions of Wexactα, Δexactα, and ΔAC-Taylorα to the correlation energies of the studied molecules, profiles of all computed W and Δ curves, and plots demonstrating the effect of adding random noise to the RDMs of the N2 molecule.

This work was supported by the National Science Center of Poland, under Grant No. 2021/43/I/ST4/02250, the Czech Science Foundation (Grant No. 23-04302L), the Grant Scheme of Charles University in Prague (Grant No. CZ.02.2.69/0.0/0.0/19_073/0016935), and the Center for Scalable and Predictive methods for Excitation and Correlated phenomena (SPEC), which is funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, the Division of Chemical Sciences, Geosciences, and Biosciences.

Most of the computations were carried out on the Karolina supercomputer in Ostrava; the authors would, therefore, like to acknowledge the support by the Czech Ministry of Education, Youth and Sports, from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center-LM2015070.”

The authors have no conflicts to disclose.

Mikuláš Matoušek: Data curation (lead); Formal analysis (equal); Methodology (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Michał Hapka: Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Libor Veis: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Katarzyna Pernal: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material