Block effective Hamiltonian theory and its application to atomic and molecular systems

<sec><p indent="0mm">Over the past decades, the Hartree-Fock equation has been well known as a starting point for calculating the Schrödinger equation approximately. However, it ignores the electron correlation energy, which leads to errors in the computational results. Consequently, a series of post-HF methods, such as coupling cluster methods and perturbation methods, have been developed and applied widely in chemistry and chemical biology. </sec><sec> Effective Hamiltonian theory is one of the important quantum chemistry approaches used to describe the electronic structure of systems. Initially, Lowdin derives the effective Hamiltonian to solve secular equations with the partitioning technique, which is favorable to dealing with large-scale eigenvalue problems for one state each time. Recently, the block effective Hamiltonian theory (BEHT) was proposed with the basic idea of dividing the whole configuration interaction functions into <italic>P</italic>, <italic>Q</italic>, and <italic>R</italic> spaces, according to the importance of their contributions to the ground-state wavefunction. In our previous work, the block effective Hamiltonian is derived and its eigenvalues in the model space are computed iteratively. The BEHT method has demonstrated excellent performance with high accuracy and less computational cost. </sec><sec> In this paper, we derived the expression of four blocks effective Hamiltonians and the general formula of <italic>n</italic> blocks effective Hamiltonians using the partitioning technique. In addition, we exploit the methods of BEHT to calculate the ground-state energy of CH₄, the electron affinities of F and NH₂, the ionization potentials of HF and H₂O, and the single-triplet gap energies of CH₂ and NH₂⁺. Meanwhile, we analyzed the performance of the two approximate methods (BEHT3a and BEHT3b) and explored the impact of the size of model space on computational results. </sec><sec> The results show that the accuracy of BEHT3a is smaller than <sc>0.1 eV</sc> in calculating the ground-state energy of CH₄ and the electron affinity of F in larger model spaces. In contrast, BEHT3b exhibits excellent computational performance. The ground-state energy value of CH₄ is only <sc>0.04 eV</sc> lower than what is obtained by MRCI, and the accuracy of electron affinities for F and NH₂ is <sc>0.03 eV.</sc> Similarly, the calculated errors of ionization potentials for HF and H₂O are smaller than <sc>0.03 eV,</sc> and the accuracy of the single-triplet gap energies for CH₂ and NH₂⁺ is also smaller than <sc>0.10 eV.</sc> It is worth mentioning that the iteration numbers are less than 8 times in this work, which indicates a low computational cost for BEHT. </sec><sec> In conclusion, in comparison with BEHT3a, BEHT3b is a more reliable method with higher accuracy. Besides, its accuracy can be further improved with the increase of the model space. </sec>