Block-orthogonalized orbitals and their applications in post-valence bond self-consistent field calculations

In modern quantum chemistry, valence bond (VB) theory remains a valuable tool for understanding chemical bonding, providing intuitive insights into electronic structures through its electron-pair bonding model. While molecular orbital (MO) theory dominates electronic structure calculations, VB theory continues to be significant for its ability to offer clear chemical interpretations and an intuitive understanding of molecular bonding, electronic effects, and chemical reaction mechanics. Among the ab initio approaches in VB theory, the valence bond self-consistent field (VBSCF) method is foundational. By expressing the wave function as a linear combination of VB structures, the VBSCF method is inherently multi-configurational. This characteristic allows VBSCF to effectively incorporate static correlation, which is crucial for accurately describing strongly correlated systems or processes, such as bond dissociation and singlet diradicals and is also applicable to excited-state calculations. However, while the VBSCF method accounts for static correlation, it does not incorporate dynamic correlation. One often used and computationally efficient improvement is to include the contribution from virtual orbitals to account for dynamic correlation. This is typically achieved through treatments such as configuration interaction (CI) or perturbation theory (PT). This study introduces a novel orbital type, termed block-orthogonalized orbitals (BOOs), within the framework of ab initio VB theory. BOOs are designed to tackle the long-standing challenge of balancing chemical intuitiveness with the requirement for orbital orthogonality in post-VBSCF methods. Traditional post-VBSCF methods, such as valence bond configuration interaction (VBCI) and valence bond perturbation theory (VBPT2), often concern a trade-off between maintaining localized orbital representations and achieving computational efficiency. To address this challenge, the BOOs are constructed as follows: first, the delocalized virtual orbitals are obtained by projecting out the occupied orbitals. These virtual orbitals are then projected onto each fragment of the studied molecule, and subsequently orthogonalized for the occupied orbitals. Thus, the proposed BOOs provide a solution by combining both internal orthogonality and orbital locality, enhancing the efficiency of post-VBSCF calculations without sacrificing chemical interpretability. Moreover, the BOO-based VBCI method improves computational accuracy, particularly for bond dissociation energies (BDEs) and resonance energies (REs) in both diatomic and polyatomic molecules. Computational results for molecules including LiH, HF, HCl, F₂, Cl₂, and several polyatomic systems such as ethylene and cyclopropane, are presented in this study. Two strategies for constructing BOOs with different types of locality are compared, and it is concluded that more localized BOOs are particularly advantageous for post-VBSCF methods. The absolute energies, BDEs and REs of the studied molecules are presented, indicating that the BOO-based VBCI method yields more accurate results compared to the traditional method utilizing hybrid atomic orbitals (HAOs), mainly due to the slight delocalization of BOOs. For instance, the BDEs calculated using the BOO-based VBCI for HCl, F₂, and Cl₂ are closer to experimental values than those obtained from HAO-based VBCI. Additionally, the BOO-based approach also provides reasonable REs for the studied molecules. Finally, the BOO-based method addresses the issue where part of the inactive HAOs are not strictly localized on a single atom, thereby successfully extending the applicability of VBCI to polyatomic molecules. In conclusion, the introduction of BOOs represents a significant advancement in post-VBSCF methods, offering a balanced approach to orbital localization and orthogonality. This new orbital construction scheme preserves the chemical intuitiveness inherent in VB theory while enhancing both computational efficiency and accuracy, thereby establishing the foundation for the further development of efficient post-VBSCF methods and algorithms.