Projection-Based Wavefunction-in-DFT Embedding

Overview

Projection-Based Wavefunction-in-DFT (WF-in-DFT) embedding is a quantum embedding technique that enables the accurate treatment of a chemically important region (the active subsystem or embedded region) using a high-level wavefunction (WF) method, while describing the remainder of the system (the environment) with a more affordable Kohn–Sham Density Functional Theory (DFT) approach.

The projection-based formulation enforces orthogonality between the active and environment subspaces, avoiding the double-counting of correlation and enabling seamless integration of WF and DFT descriptions.

In Kvantify Qrunch, the active region is treated on a quantum computer using an adaptive VQE algorithm (see What Is an Adaptive-VQE and FAST-VQE), making this approach a bridge between classical DFT and quantum wavefunction methods.

Context

Large molecular systems — such as enzymes, metal–ligand complexes, or solvated charge-transfer systems — often require high accuracy for only a small, localized region. Running a full high-level WF calculation for the entire system is computationally prohibitive and unnecessary.

Embedding methods address this by partitioning the system into two subsystems:

• Embedded region (high-level WF on quantum computer)

• Environment (lower-level DFT on a classical computer)

Projection-based embedding differs from other embedding approaches (e.g., ONIOM, FDE) by:

• Enforcing explicit orthogonality between WF and DFT orbitals

• Avoiding functional-specific approximations in the coupling

• Allowing reuse of standard WF methods (including VQE) on the embedded Hamiltonian

Method Summary

A high-level view of the algorithm:

1. Whole-system DFT calculation — Perform a Kohn–Sham DFT calculation for the entire system to obtain orbitals and electron density.

2. Localization of the occupied molecular orbitals — Use methods like Pipek-Mezey to obtain localized molecular orbitals (LMOs)

3. Partition the occupied space — Divide the LMOs into:

   • Embedded region

   • Environment region

4. Construct projection operators — Define a projection operator for the environment region. This is used to remove environment contributions from the embedded region. The embedded LMOs are projected to be orthogonal to environment LMOs.

5. Build embedded Hamiltonian — Construct the embedded Hamiltonian including:

   • One- and two-electron terms for the embedded region

   • An embedding potential from the environment’s DFT description

6. Solve the embedded problem with adaptive VQE — Use an adaptive VQE (see What Is an Adaptive-VQE and FAST-VQE) on a quantum computer to determine the ground state of the embedded Hamiltonian. Integrate the results back into the full system description.

Limitations and Challenges

• Requires a good orbital localization and subsystem partitioning

• Full-system DFT step can still be expensive

• No environment environment to changes in the embedded region is included by default

• Despite WF-in-DFT embedding, the embedded subsystem can still be too large for current quantum hardware and active space reduction techniques may be needed

See Also

• What Is an Adaptive-VQE and FAST-VQE - adaptive VQE algorithms in Kvantify Qrunch

• Gate Selection - gate selection in adaptive VQE algorithms

• Understanding and Using Kvantify Qrunch’s Fluent Builder Pattern - the builder pattern in Kvantify Qrunch

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References

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• Goodpaster, J. D., Barnes, T. A., Manby, F. R., & Miller III, T. F. Density functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexes. J. Chem. Phys. 137, 224113 (2012)

• Welborn, M., Manby, F. R., & Miller III, T. F. Even-handed subsystem selection in projection-based embedding. J. Chem. Phys. 149, 144101 (2018)

• Bennie, S. J., van der Kamp, M. W., Pennifold, R. C. R., Stella, M., Manby, F. R., Mulholland, A. J. A Projector-Embedding Approach for Multiscale Coupled-Cluster Calculations Applied to Citrate Synthase. J. Chem. Theory Comput. 12, 2689–2697 (2016)

• Bennie, S. J., Stella, M., Miller III, T. F., & Manby, F. R. Accelerating wavefunction in density-functional-theory embedding by truncating the active basis set. J. Chem. Phys. 143, 024105 (2015)

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• Lee, S. J. R., Welborn, M., Manby, F. R., & Miller III, T. F. Projection-Based Wavefunction-in-DFT Embedding. Acc. Chem. Res. 52, 1359-1368 (2019)

• Lee, S. J. R., Ding, F., Manby, F. R., & Miller III, T. F. Analytical gradients for projection-based wavefunction-in-DFT embedding. J. Chem. Phys. 151, 064112 (2019)

• Beran, P., Pernal, K., Pavošević, F., & Veis, L. Projection-Based Density Matrix Renormalization Group in Density Functional Theory Embedding. J. Phys. Chem. Lett. 14, 716–722 (2023)