Derivation of Hierarchically Correlated Orbital Functional Theory: The Role of Hypercomplex Orbitals

This work presents a detailed mathematical derivation of the hierarchically correlated orbital functional theory (HCOFT), a framework based on hypercomplex orbitals. Recent study [Phys. Rev. Lett. 133, 206402] has demonstrated that hypercomplex orbitals in a determinant are equivalent to a set of real-valued orbitals that allow fractional occupations, making them desirable fundamental descriptors for many-electron systems. The algebraic properties of Clifford algebra are rigorously applied to derive key quantities within HCOFT, addressing the complexities introduced by the hypercomplex representation. It is shown that, despite this added complexity, the resulting density and kinetic energy remain physically meaningful and satisfy essential properties, including the Pauli exclusion principle. To establish the uniqueness of HCOFT, alternative definitions of hypercomplex orbitals within Clifford algebra are explored. These alternatives can lead to the loss of physical meaning in fundamental quantities for many-electron systems. Overall, this work demonstrates that HCOFT not only preserves the desired physical properties but also provides a single-determinant framework capable of describing multi-reference systems.