Our implementation of the reference relaxation procedure applied in Ref. 1 to various flavors of the multireference driven similarity renormalization group (MR-DSRG) contained a minor implementation error.2 In addition, the procedure used to canonicalize complete-active-space self-consistent-field (CASSCF) frozen core orbitals was potentially ambiguous. In this erratum, we present revised results computed using the correct reference relaxation procedure and uniquely defined canonical frozen core orbitals (see Ref. 2 for details). In addition, we correct a minor typo found in Appendix B of Ref. 1.

In Ref. 1, we explored several procedures to relax the reference and concluded that the “relaxed” (r) and “fully relaxed” (fr) schemes overestimated relaxation effects. We also attributed the higher accuracy of the “partially relaxed” (pr) scheme to error cancellation. As shown in Fig. 3, the revised relaxed and fully relaxed energies do not overestimate reference relaxation effects and instead lie between the results of the unrelaxed and partially relaxed schemes. These corrections lead us to positively reconsider our assessment of the relaxed and fully relaxed schemes. For all cases, the relaxed variant yields results that are indistinguishable from those obtained with the fully relaxed approach. Our revisions also show that the partially relaxed scheme is a good approximation to the fully relaxed approach, especially for the DSRG-MRPT3 and MR-LDSRG(2) methods. Note that we have also updated the unrelaxed and partially relaxed data, but these new results are only affected by the change in canonicalization procedure (absolute energies are found to differ at most by 0.4 mEh).

FIG. 3.

Energy deviations of various multireference DSRG (s = 0.5 Eh2) methods for the ground-state potential energy curves of (a) F2, (b) H2O2, (c) C2H6, and (d) N2 relative to FCI, CCSDT(2)Q, CCSDT(2)Q, and FCI, respectively.

FIG. 3.

Energy deviations of various multireference DSRG (s = 0.5 Eh2) methods for the ground-state potential energy curves of (a) F2, (b) H2O2, (c) C2H6, and (d) N2 relative to FCI, CCSDT(2)Q, CCSDT(2)Q, and FCI, respectively.

Close modal

Table I reports the corrected error statistics for the potential energy curves of F2, H2O2, C2H6, and N2. Judging the quality of our updated data using the average nonparallelism error (NPE, reported in Table I), we conclude that all three reference relaxation approaches are more accurate than the unrelaxed scheme and they yield similar errors. For instance, in the case of the MR-LDSRG(2), the unrelaxed, partially relaxed, relaxed, and fully relaxed schemes yield an average NPE equal to 4.77, 2.84, 2.83, and 2.85 mEh, respectively.

TABLE I.

Maximum error (MAX) and nonparallelism error (NPE) for the ground-state potential energy curves of F—F, HO—OH, H3C—CH3, and N≡N computed with various methods (reported in units of mEh). All DSRG methods employ a value of the flow parameter s = 0.5 Eh2. The last column shows the average NPE.

F2H2O2C2H6N2Average NPE
MethodMAXNPEMAXNPEMAXNPEMAXNPE
u-DSRG-MRPT2 New 25.23 10.24 31.15 7.23 52.13 5.49 32.96 18.95 10.48 
 Old 25.23 10.24 31.15 7.23 52.14 5.50 32.91 18.93 10.47 
DSRG-MRPT2 New 20.33 5.34 29.38 5.47 51.06 4.42 32.39 18.41 8.41 
 Old 20.33 5.34 29.38 5.47 51.06 4.42 32.35 18.39 8.40 
r-DSRG-MRPT2 New 22.01 7.00 29.49 5.59 51.30 4.66 32.80 18.83 9.02 
 Old 17.36 3.03 28.96 7.78 50.72 4.08 32.14 18.24 8.28 
fr-DSRG-MRPT2 New 22.10 7.09 29.49 5.58 51.30 4.66 32.79 18.82 9.04 
 Old 17.39 3.50 28.96 8.22 50.72 4.08 32.14 18.24 8.51 
u-DSRG-MRPT3 New 9.46 5.03 15.51 6.05 19.15 2.73 15.21 6.06 4.97 
 Old 9.46 5.03 15.50 6.04 19.15 2.73 15.19 6.09 4.97 
DSRG-MRPT3 New 6.81 2.39 15.05 5.60 19.14 2.71 13.90 4.78 3.87 
 Old 6.81 2.39 15.04 5.59 19.13 2.71 13.88 4.80 3.87 
r-DSRG-MRPT3 New 6.89 2.48 15.10 5.67 19.14 2.71 14.56 5.45 4.08 
 Old 4.73 3.59 14.58 6.79 19.12 3.22 13.12 4.09 4.42 
fr-DSRG-MRPT3 New 6.89 2.48 15.10 5.67 19.14 2.71 14.56 5.46 4.08 
 Old 4.72 3.80 14.58 7.09 19.12 3.27 13.11 4.09 4.56 
u-MR-LDSRG(2) New 3.83 4.86 3.89 3.95 6.08 3.44 9.33 6.86 4.77 
 Old 3.83 4.86 3.89 3.95 6.08 3.43 9.30 6.50 4.68 
pr-MR-LDSRG(2) New −1.03 1.34 0.79 0.86 5.29 2.89 8.30 6.26 2.84 
 Old −1.03 1.34 0.78 0.86 5.28 2.87 8.25 5.90 2.74 
r-MR-LDSRG(2) New −1.05 1.55 0.97 1.07 5.23 2.76 8.15 5.94 2.83 
 Old −4.79 3.86 −3.53 3.42 4.94 3.13 7.62 5.57 3.99 
MR-LDSRG(2) New −1.05 1.60 1.00 1.10 5.24 2.77 8.14 5.93 2.85 
 Old −5.19 4.24 −3.87 3.75 4.92 3.12 7.61 5.56 4.17 
F2H2O2C2H6N2Average NPE
MethodMAXNPEMAXNPEMAXNPEMAXNPE
u-DSRG-MRPT2 New 25.23 10.24 31.15 7.23 52.13 5.49 32.96 18.95 10.48 
 Old 25.23 10.24 31.15 7.23 52.14 5.50 32.91 18.93 10.47 
DSRG-MRPT2 New 20.33 5.34 29.38 5.47 51.06 4.42 32.39 18.41 8.41 
 Old 20.33 5.34 29.38 5.47 51.06 4.42 32.35 18.39 8.40 
r-DSRG-MRPT2 New 22.01 7.00 29.49 5.59 51.30 4.66 32.80 18.83 9.02 
 Old 17.36 3.03 28.96 7.78 50.72 4.08 32.14 18.24 8.28 
fr-DSRG-MRPT2 New 22.10 7.09 29.49 5.58 51.30 4.66 32.79 18.82 9.04 
 Old 17.39 3.50 28.96 8.22 50.72 4.08 32.14 18.24 8.51 
u-DSRG-MRPT3 New 9.46 5.03 15.51 6.05 19.15 2.73 15.21 6.06 4.97 
 Old 9.46 5.03 15.50 6.04 19.15 2.73 15.19 6.09 4.97 
DSRG-MRPT3 New 6.81 2.39 15.05 5.60 19.14 2.71 13.90 4.78 3.87 
 Old 6.81 2.39 15.04 5.59 19.13 2.71 13.88 4.80 3.87 
r-DSRG-MRPT3 New 6.89 2.48 15.10 5.67 19.14 2.71 14.56 5.45 4.08 
 Old 4.73 3.59 14.58 6.79 19.12 3.22 13.12 4.09 4.42 
fr-DSRG-MRPT3 New 6.89 2.48 15.10 5.67 19.14 2.71 14.56 5.46 4.08 
 Old 4.72 3.80 14.58 7.09 19.12 3.27 13.11 4.09 4.56 
u-MR-LDSRG(2) New 3.83 4.86 3.89 3.95 6.08 3.44 9.33 6.86 4.77 
 Old 3.83 4.86 3.89 3.95 6.08 3.43 9.30 6.50 4.68 
pr-MR-LDSRG(2) New −1.03 1.34 0.79 0.86 5.29 2.89 8.30 6.26 2.84 
 Old −1.03 1.34 0.78 0.86 5.28 2.87 8.25 5.90 2.74 
r-MR-LDSRG(2) New −1.05 1.55 0.97 1.07 5.23 2.76 8.15 5.94 2.83 
 Old −4.79 3.86 −3.53 3.42 4.94 3.13 7.62 5.57 3.99 
MR-LDSRG(2) New −1.05 1.60 1.00 1.10 5.24 2.77 8.14 5.93 2.85 
 Old −5.19 4.24 −3.87 3.75 4.92 3.12 7.61 5.56 4.17 

We note that in the absence of any approximation, only the fully relaxed scheme can reproduce full configuration interaction (FCI). Since the fully relaxed scheme is both (in principle) exact and accurate (as suggested by our updated results), this is the method of choice for computations using nonperturbative approximations. In the case of perturbative methods, our conclusions are unchanged and we continue to recommend the partially relaxed procedure from a perspective of balancing the cost and accuracy.

In the supplementary material, we also report the updated absolute energies for the singlet and triplet states of 9,10-anthracyne, which were revised due to canonicalization of the frozen core orbitals. These new results do not change the singlet-triplet splittings for 9,10-anthracyne reported in Ref. 1.

We also identified a typo in Eq. (B5) of Appendix B: Reference relaxation. The correct version of Eq. (B5) is as follows:

cuv=ouv+mCwumvm=H¯uvxyAH¯uxvyγyx.
(B5)

See supplementary material for the corrected single-point energies of F2, H2O2, C2H6, N2, and 9,10-anthracyne.

1.
C.
Li
and
F. A.
Evangelista
,
J. Chem. Phys.
146
,
124132
(
2017
).
2.
C.
Li
and
F. A.
Evangelista
,
J. Chem. Phys.
148
,
079903
(
2018
).

Supplementary Material