Projection-based embedding offers a robust framework for modeling chemical processes in large molecular systems. It partitions computations into low-level treatments for the environmental subsystems and high-level treatments for the active subsystems. To address the computational bottleneck of applying high-accuracy wavefunction theory to the active regions, we introduce frozen natural orbitals (FNOs) approach. This technique efficiently compresses the virtual orbital space to generate a compact set of natural orbitals, which significantly accelerates the convergence of correlation energy recovery. We evaluated the FNO-based embedding approach in comparison with two alternative virtual space truncation approaches for a diverse range of molecular systems. Our results demonstrate that the FNO-embedded method exhibits superior performance by delivering accurate correlation energies with a substantially fewer number of virtual orbitals. Its potential for quantum computing is also discussed, as this reduction in orbitals can be directly translated to lower qubit requirements and thereby facilitate the quantum simulations on near-term quantum hardware.
Projection-based embedding offers a robust framework for modeling chemical processes in large molecular systems. It partitions computations into low-level treatments for the environmental subsystems and high-level treatments for the active subsystems. To address the computational bottleneck of applying high-accuracy wavefunction theory to the active regions, we introduce frozen natural orbitals (FNOs) approach. This technique efficiently compresses the virtual orbital space to generate a compact set of natural orbitals, which significantly accelerates the convergence of correlation energy recovery. We evaluated the FNO-based embedding approach in comparison with two alternative virtual space truncation approaches for a diverse range of molecular systems. Our results demonstrate that the FNO-embedded method exhibits superior performance by delivering accurate correlation energies with a substantially fewer number of virtual orbitals. Its potential for quantum computing is also discussed, as this reduction in orbitals can be directly translated to lower qubit requirements and thereby facilitate the quantum simulations on near-term quantum hardware.
We present an open-shell frozen natural orbital (FNO) approach, which utilizes the second-order Z-averaged perturbation theory (ZAPT2), to reduce the restricted open-shell Hartree-Fock virtual space size with controllable accuracy. Our ZAPT2 frozen natural orbital (ZAPT-FNO) selection scheme significantly outperforms the canonical molecular orbital virtual space truncation scheme based on Hartree-Fock orbital energies, especially when using large multiple-polarized and augmented basis sets. We demonstrate that the ZAPT-FNO-selected virtual orbitals lead to a systematic convergence of the correlation energies but, more importantly, to the singlet-triplet T1-S0 energy gaps with respect to the complete active space (CAS) [occupied + virtual] size. We confirm our findings by simulating T1-S0 gaps in H2O2 and O2 molecules using the traditional complete active space configuration interaction (CASCI) approach, as well as in stretched CH2, for which we also employed the iterative qubit coupled cluster (iQCC) method as a quantum eigensolver. Finally, we applied the iQCC method with a ZAPT-FNO-selected active space to the phosphorescent Ir(ppy)3 complex with 260 electrons, where extended basis sets are required to achieve chemical (∼1 mEh) accuracy. In this case, CASCI results are not available; however, the iQCC-computed T1-S0 gaps show robust convergence with an enlarging basis set and CAS size, approaching the experimental value. Thus, the ZAPT-FNO method is very promising for improving the accuracy of quantum chemical modeling in a resource-efficient manner and opens the door to simulating open-shell states of large materials within realistic active space sizes and without compromising on basis-set quality.
The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near-term noisy intermediate-scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present an orbital-optimized VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO-VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO-OO-VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO-OO-VQE and SVO-OO-VQE improve the variational accuracy while substantially reducing measurement cost.
The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum (NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation energies for the case of small molecules (i.e., BeH2, CH4, NH3, H2O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex catalyst that is utilized in industrial settings for the polymerization of α-olefins. We show that the MI-FNO approach provides a significant reduction in the quantum bit (qubit) requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices.
Quantum embedding methods offer a promising route to extend quantum chemical calculations to large multiscale systems by treating a chemically important subsystem at a high level of theory while describing its surrounding environment at an affordable level. The methods are also quite relevant for quantum computing approaches based on hardware with limited resources. Here, we present an iterative projection-based embedding framework combined with VQE, in which the environment density is allowed to respond self-consistently to the refined electronic structure of the embedded subsystem described by VQE. Unlike conventional one-shot approaches where the environment remains frozen after the initial orbital optimization, the proposed iterative scheme alternates between the VQE-level treatment of the subsystem and a mean-field-level refinement of the environment until mutual self-consistency is achieved. The convergence behavior of the scheme is first examined using several small test systems. Its practical applicability is then demonstrated with a composite system with a CH2NH molecule sandwiched by two benzene rings, with the C=N dihedral angle rotating from 0 to 90 deg. The iterative procedure consistently converges within ~10 iteration steps across all tested geometries, yielding energies below the conventional one-shot embedding results. The converged results well reproduce the fully correlated reference energy employing the same active space, and the resulting potential energy surface with respect to the dihedral rotation is also in good agreement with the reference one. These results demonstrate that our iterative embedding framework is numerically robust and physically sound, yielding a self-consistent and reliable treatment of inter-subsystem correlation. We expect that its formulation will be particularly compatible with the emerging paradigm of quantum-classical hybrid computing.
We present a unified frozen natural orbital (FNO) framework for periodic Gaussian-orbital post-Hartree-Fock calculations with explicit k-point sampling. The approach includes conventional ground-state FNOs together with state-specific (SS-FNO) and state-averaged (SA-FNO) variants constructed from perturbative one-particle density matrices. We implement these approximations for periodic Moller-Plesset perturbation theory, algebraic diagrammatic construction, and equation-of-motion coupled cluster methods in the PySCF software package and benchmark them for correlation energies, equations of state, fundamental band gaps, and quasiparticle band structures of semiconductors and insulators. FNO truncation reproduces canonical results with small errors while substantially reducing computational cost. SS-FNO enables accurate large-basis band-gap calculations, including quadruple-zeta results, whereas SA-FNO efficiently compresses the virtual space for multi-state band-structure calculations. Combined with basis-set and thermodynamic-limit extrapolations, the framework provides a practical route to high-accuracy correlated calculations in periodic systems.
Density functional theory with plane-wave basis sets is widely employed in computational materials science, including applications to isolated molecular systems. However, the inadequate description of electron correlation remains a fundamental limitation. Accurate correlation treatments based on many-body Hamiltonians require reliable representations of both occupied and virtual orbitals, yet virtual orbitals are often poorly described in conventional computational schemes, resulting in reduced accuracy. In this work, we introduce localized correlation-converged virtual orbitals (LCCVOs) as an efficient basis for constructing accurate many-body Hamiltonians in molecular systems. Using a substantially reduced number of orbitals, the LCCVO framework yields dissociation energies for singlet, doublet, and triplet molecules that are comparable to, and in many cases exceed, those obtained with high-level correlation-consistent basis sets such as cc-pVXZ (X = D, T, Q, 5). These results demonstrate the efficiency, scalability, and robustness of the LCCVO approach for high-accuracy quantum chemical calculations.