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.

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

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

• Environment (lower-level DFT, classical computer)

Projection-based embedding differs from other 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 Molecule Orbitals Use methods like Pipek-Mezey to obtain localized Molecular Orbitals (LMOs)

3. Partition the occupied space Divide occupied orbitals into:

   • Embedded region (A)

   • Environment region (B)

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 space orbitals are projected to be orthogonal to environment orbitals.

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 response of the environment to changes in the embedded region is included by default

• Despite embedding, the subspace 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

• Manby, F. R., Stella, M., Goodpaster, J. D., & Miller III, T. F. A projector-embedding approach for multiscale coupled-cluster-in-DFT calculations. J. Chem. Theory Comput. 8, 2564–2572 (2012)

• Goodpaster, J. D., Barnes, T. A., Manby, F. R., & Miller III, T. F. Density functional embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexes. arXiv:1211.6052

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

• Simon J. Bennie, Marc W. van der Kamp, Robert C. R. Pennifold, Martina Stella, Frederick R. Manby, and Adrian J. Mulholland Journal of Chemical Theory and Computation 2016 12 (6), 2689-2697

• Bennie, S. J., Stella, M., Miller III, T. F., & Manby, F. R. A projector-embedding approach for multiscale coupled-cluster-in-DFT calculations. J. Chem. Phys. 143, 024105 (2015)

• Graham, D. S., Wen, X., Chulhai, D. V., & Goodpaster, J. D. Robust, accurate, and efficient: quantum embedding using the Huzinaga level-shift projection operator for complex systems. J. Chem. Theory Comput. (2020)

• Lee, S. J. R., Welborn, M., Manby, F. R., & Miller III, T. F. Projection-based wavefunction-in-DFT embedding. Acc. Chem. Res. 52 (5), 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., Pavosevic, F., & Veis, L. Projection-based density matrix renormalization group in density functional theory embedding. arXiv:2210.16289 (2022)