Why Projection-Based DMRG-in-DFT Cannot Be Exact, Even with the Exact Exchange-Correlation Functional

Abstract

We establish the theoretical foundations for embedding a correlated wave function in an environment formed by Kohn-Sham orbitals. We show that introducing an approximation which equates two, in principle distinct, kinetic-energy functionals yields an embedding functional identical to the projection-based wavefunction-in-DFT formulation of Miller and co-workers. We demonstrate that this functional is inherently nonvariational: its minimum is not guaranteed to coincide with the exact ground-state energy and remains bounded from above by it. Building on this formal framework, we analyze the dominant sources of error in projection-based DMRG-in-DFT embedding with approximate exchange-correlation (xc) functionals. Using molecules with dissociating covalent bonds as a diagnostic example, we demonstrate that the primary source of error is the nonadditive exchange-correlation energy describing the nonclassical coupling between the active subsystem and its environment. Eliminating the fractional-spin error by employing a pair-density xc functional (PDFT) instead of a semilocal GGA does not remedy this deficiency, because the inaccuracy stems from self-interaction effects at the subsystem-environment interface.

Figures (5)

• Figure 1: Model systems investigated in this work: (a) H$_{20}$ chain with a central active fragment (A) composed of four hydrogen atoms with stretched interatomic bonds of length R$_{\text{st}}$. The environment (B) consists of hydrogen dimers with an intra-dimer bond length of 1.0 Å. Dashed lines indicate the inter-dimer separation and the A--B distance, both equal to 1.4 Å. H$_{\text{emb}}$ denotes hydrogen atoms that remain coupled to the environment in the dissociation limit, $\text{R}_{\text{st}} \rightarrow \infty$. (b) Propionitrile (CH$_3$CH$_2$CN) molecule with the active fragment (A) defined as the $-$CN group with triple bond of the length R stretched. C$_{\text{emb}}$ denotes a carbon atom that remains coupled to the environment in the dissociation limit.
• Figure 2: Results for the H$_{20}$ chain with a four-atom active fragment using the cc-pVDZ basis set. (a) Absolute energies and (b) relative energies (in Ha) obtained with DMRG and DMRG-in-DFT. For the DFT part, the PBE functional was employed (see main text). Two variants are shown in which the nonadditive exchange–correlation energy (see Eq. (\ref{['Approx0']})) is computed using either the PBE or the PDFT functional, denoted as DMRG-in-DFT(PBE) and DMRG-in-DFT(PDFT), respectively.
• Figure 3: Results for the propionitrile molecule using the cc-pVDZ basis set. (a) Absolute energies and (b) relative energies (in Ha) obtained with DMRG and DMRG-in-DFT. For the DFT part, the PBE functional was employed (see main text). Two variants are shown in which the nonadditive exchange–correlation energy (see Eq. (\ref{['Approx0']})) is computed using either the PBE or the PDFT functional, denoted as DMRG-in-DFT(PBE) and DMRG-in-DFT(PDFT), respectively.
• Figure 4: Results for the H$_{20}$ chain with a 8-atom active fragment using the cc-pVDZ basis set. (a) Absolute energies and (b) relative energies (in Ha) obtained with DMRG and DMRG-in-DFT. For the DFT part, the PBE functional was employed (see main text). Two variants are shown in which the nonadditive exchange–correlation energy (see Eq. (\ref{['Approx0']})) is computed using either the PBE or the PDFT functional, denoted as DMRG-in-DFT(PBE) and DMRG-in-DFT(PDFT), respectively.
• Figure 5: Relative error energies (in mHa) for the H$_{20}$ chain with a 4-atom active fragment obtained with DMRG and DMRG-in-DFT using the cc-pVDZ basis set. Results are shown for three separations between the active fragment (A) and the environment (B): R$_{\text{AB}}$ = 1.4 Å, R$_{\text{AB}}$ = 1.8 Å and R$_{\text{AB}}$ = 2.6 Å. For each separation, two variants are presented in which the nonadditive exchange-correlation energy [see Eq. (\ref{['Approx0']})] is computed using either the PBE or the PDFT functional, denoted as DMRG-in-DFT(PBE) and DMRG-in-DFT(PDFT), respectively.