p-DMRG
Perturbatively corrected DMRG by Sheng Guo, Zhendong Li, and Garnet Chan (in 2018).
Motivation: DMRG scales poorly with respect to the number of basis functions. the above paper says that DMRG's cost is $\mathcal{O}\left(M^3D^3\right)$ for $M$ basis functions and a bond dimension of $D$, and that $D$ often has to scale as $\mathcal{O}\left(M\right)$, making the DMRG cost scale quite steeply with the number of basis functions: $\mathcal{O}\left(M^6\right)$. Basically: DMRG is very powerful for studying 50 electrons with 50 basis functions, but would require too mcuh RAM even for studying 6 electrons in 500 basis functions. It is excellent for treating static correlation of highly multi-reference systems, but poor for treating the remaining dynamic correlation.
Description: Partition the Hamiltonian into $H_0+V$ where $H_0$ is solved "exactly" with standard DMRG, and the resulting MPS $|\psi_0\rangle$ is used to treat $V$ perturbatively: but the partition is done such that $H_0$ requires a much smaller bond dimension $D_0$ than you would need if you were to treat the whole Hamiltonian with standard DMRG; and while treatment of $V$ requires a large bond dimension $D_1$, the treatment is done via minimization of a Hylerraas functional and a sum of MPSs, which is far less expensive than a standard DMRG treatment, and this allows $D_1$ to be roughly as big as the $D$ in standard DMRG would be if one were to try to treat the entire Hamiltonian with similar accuracy in comparison with p-DMRG. Much thought went into how best to define $H_0$ in the first paper, but a second paper posted on arXiv 8 days later settled on an Epstein-Nesbet partitioning, and also introduced an even more efficient way to treat the perturbative correction by using a stochastic method.
Cost: The more efficient stochastic algorithm (2nd paper) has two cost contributions: $\mathcal{O} \left(M^3D_1^2D_0 \right)$ and $\mathcal{O} \left(N_s N^2 K^3 D_0^2 \right)$ for $N_s$ samples in the stochastic sampling.
Implementations: The two aforementioned papers don't mention any software packages in which the method is implemented, and I see no mention of it in the documentation for PySCF (of which all three of the authors of the aforementioned papers, are very involved). Since the method was introduced 2 years ago, it may be in early stages and only implemented in an in-house code by the original authors, though I would not be surprised if it were to be implemented in PySCF shortly.
Remarks: The original authors emphasize that p-DMRG is different from DMRG-CASPT2 or DMRG-NEVPT2 where there is a CAS/non-CAS partition rather than an Epstein-Nesbet partition. p-DMRG targets quantitative accuracy for systems with more basis functions than standard DMRG can handle, but far fewer basis functions than what one would have in a qualitative treatment of dynamic correlation in DMRG-CASPT2. Also, there is a parameter $\lambda$ which can be tuned for example to avoid intruder states in $|\psi_0\rangle$: They found $\lambda=1$ to be more prone to the intruder state problem, so they mainlused $\lambda=0$ and $\lambda=1/2$.
