Expansion methods for Adams–Gilbert equations. I. Modified Adams–Gilbert equation and common and fluctuating basis sets

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Abstract
Adams–Gilbert (AG) equation for nonorthogonal localized orbitals of a single‐determinant wavefunction has been modified so as to enable one to compute wavefunctions of large polyatomic systems by the expansion method. This equation is named as modified Adams–Gilbert (MAG) equation. One solves the AG or the MAG equation by each subsystem and, collecting all the orbitals obtained, one constructs wavefunction of the system. It is shown that when one employs the expansion method, one must actually use basis functions common to all the subsystems (common basis set) to solve the AG equation, while one can employ, by each subsystem, different basis functions appropriate to the subsystem (fluctuating basis set) to solve the MAG equation. An expansion method suitable for solving the AG and the MAG equations has been presented. Application of the method to HF, H2O, and CH4 has revealed that (1) the method proposed is workable, (2) actually so many basis functions are not needed for describing some subsystems, especially for core electrons, and (3) it is necessary to orthogonalize approximately, not necessarily rigorously, the orbitals in the system.