7.10.6 Regularised Excited States with BWs-ADC

A disadvantage of the standard, Møller-Plesset based (MP-)ADC scheme is that excitation energies and properties start to diverge if the HOMO/LUMO gap approaches zero. This is caused by divergence of the first order $t_2$-amplitudes, which enter the secular matrix. This issue can be avoided by choosing a regular partitioning of the Hamiltonian for the perturbative expansion of the polarisation propagator. Here, we employ the size-consistent Brillouin-Wigner partitioning by Carter-Fenk and Head-Gordon. ¹⁹⁵ Carter-Fenk K., Head-Gordon M.
J. Chem. Phys.
(2023), 158, pp. 234108. Link ^{, 201} Carter-Fenk K., Shee J., Head-Gordon M.
J. Chem. Phys.
(2023), 159, pp. 171104. Link While it was originally proposed as a repartitioning to make second order Brillouin-Wigner theory size-consistent and size-extensive, it is best to instead view BWs-$n$ as a regularised Rayleigh-Schrödinger perturbation theory, whose unperturbed Hamiltonian and perturbation are given by:

Here, $\widehat{W}$ is a block-diagonal one-electron operator whose properties are extensively explored in Ref. 335 Dittmer L. B., Head-Gordon M.
J. Chem. Phys.
(2025), 162, pp. 054109. Link Within the context of BWs-ADC, we recommend using the $A_0$ and $B_0$ parameters of MPR-BWs2 (see also 6.9):

A key aspect of BW-s2 theory is that the combined zeroth order Hamiltonian still constitutes a one-particle operator and can thus be diagonalised self-consistently to yield the dressed orbitals ${\varphi }_p^{\prime }(r,\sigma )$ with corresponding orbital energies ${\epsilon }_p^{\prime }$:

If the BWs-ADC equations (both the secular matrix-vector product as well as densities required for ISR property calculations) are expressed in terms of dressed orbitals, the algebraic structure of the BWs-ADC equations is almost completely equivalent to that of the corresponding MP-ADC expression. This implies that the computational cost of a BWs-ADC calculation is nearly identical to its MP-ADC equivalent. Two additional required steps increase the cost slightly: First, the BW-s2 ground state is iterative. Second, the second order ground state doubles amplitudes ($t_{ij}^{ab}{}^{(2)}$) require a one-shot correction of $𝒪(N^5)$ cost due to the presence of $\widehat{W}$ in ${\widehat{H}}^{(1)}$.

In practice, a BWs-ADC(2) calculation only takes $1-5\%$ longer than an equivalent MP-ADC(2) calculation on the same machine. For a BWs-ADC(3) calculation, this difference is typically even less as all tensor contractions with a computational complexity of $𝒪(N^6)$ are identical between BWs-ADC(3) and MP-ADC(3).

So far, BWs-ADC is only implemented for particle number preserving excited states (i. e. those accessed by the keyword EE_STATES). Below we provide an overview over the properties of each implemented BWs-ADC method.

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  BWs-ADC(0) is implemented for the sake of completeness. Since all zeroth order ADC schemes yield spin-adapted Koopman excitations as excitation vectors and (dressed) orbital energy differences as the corresponding excitation energies, this method is not recommended for practical use with any parameter choice.

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  BWs-ADC(1) is exactly equal to MP-ADC(1) for excitation energies. If a BWs-ADC(1) calculation is attempted, Q-Chem will crash and refer to a regular ADC(1) calculation instead.

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  BWs-ADC(2)-s has been found to consistently outperform MP-ADC(2) as well as CC2, CIS(D) and CIS(D${}_{\mathrm{\infty }}$). Note that it performs best for $A_0=3.8$ and $B_0=0$ with minimal dependence on $A_0$, which is extremely close to the optimal parameter choice for the ground state ($A_0=4$, $B_0=0$). Also note that any choice of $B_0\ne 0$ drastically reduces the excitation energy quality.

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  BWs-ADC(2)-x is implemented for the sake of completeness but has not been benchmarked. Due to the ad hoc inclusion of first order terms in the 2p2h-2p2h block of the secular matrix, only the second order contribution to the ph-ph block carries any dependence on the regularisation parameters. Its performance is therefore expected to mimic that of MP-ADC(2)-x.

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  BWs-ADC(3) was found to perform best for $A_0=-1$ and $B_0=0$, though similar parameter choices with the constraint of $A_0-B_0=-1$ exhibit almost the same errors. Note that this parameter choice is deregularising, i. e. it cannot describe systems with a vanishing or small HOMO/LUMO gap. For states with weak static correlation and small double excitation character, BWs-ADC(3) strongly outperforms BWs-ADC(2). At the same time, excited states whose proper description requires triples which exhibit significant static correlation are comparatively poorly described, leading to a similar overall MAE to BWs-ADC(2) across benchmarks.

An overview over the methods and recommended parameters and settings is given in table 7.4.

BWs-ADC calculations are executed in a two-step process inside a single Q-Chem job. First, a RI-BW-s2 calculation inside the libgmbpt module is run. Second, the converged dressed orbitals and orbital energies are handed over to adcman for the subsequent ADC calculation. For this reason, BWs-ADC is only available with the Resolution of Identity approximation.

To run a BWs-ADC calculation, include the following in your input file:

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  In the REM section, set METHOD to RIBWS2.

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  In the REM section, specify both BASIS and either AUX_BASIS or AUX_BASIS_CORR

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  Choose a partitioning with the METHOD keyword in the $bws2 section. We recommend:

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    Set METHOD to MPR-BWS2 in the $bws2 section.

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    Set MPR_A0 to the desired $A_0$ value in the $bws2 section.

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    Set MPR_B0 to the desired $B_0$ value in the $bws2 section.

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  In the $bws2 section, use the EXC_METHOD keyword to determine the BWs-ADC variant:

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    For BWs-ADC(0), set EXC_METHOD to ADC(0) or ADC0.

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    For BWs-ADC(2)-s, set EXC_METHOD to any of ADC(2), ADC2, ADC(2)-S or ADC2-S.

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    For BWs-ADC(2)-x, set EXC_METHOD to ADC(2)-X or ADC2-X.

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    For BWs-ADC(3), set EXC_METHOD to ADC(3) or ADC3.

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  For a restricted reference, specify the number of excited states using the EE_STATES or the EE_SINGLETS and EE_TRIPLETS keywords in the REM section.

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  For an unrestricted reference, only use the EE_STATES keyword.

All keywords used for Job Control of RI-BW-s2 calculations (see section 6.8.2) can be used to specify details about the ground state calculation in a BWs-ADC calculation. Additionally, all keywords used for ADC Job Control (see section 7.10.13) can be used to adjust the ADC part of a BWs-ADC calculation.