Extension of the Cluster Jastrow ( <i>k-</i> uCJ) Decomposition to Complex Hamiltonians: Applications to Relativistic Systems

We propose an extension of the <i>k</i>-uCJ decomposition for the treatment of complex Hamiltonians, specifically focusing on relativistic systems. The algorithm uses Takagi decomposition to initially provide a rank-one approximation of both the real and imaginary parts of the two-electron integral in the second quantization Hamiltonian, followed by optimization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math> matrices. We employ the algorithm to calculate the potential energy curves of molecules containing heavy atoms, particularly TlH and PoH₂, and compare the accuracy of the results with calculations using the relativistic self-consistent field complete active space (RelCASSCF) method. We show the difference in the results of the approximated equilibrium ground state energy, equilibrium bond distance, and dissociation energy with respect to the nonrelativistic CASSCF calculation. We further compare the number of decomposition terms (<i>k</i>) needed between the rank-one and the optimized approximation in the <i>k</i>-uCJ method to reproduce the RelCASSCF results within chemical precision. Finally, we considered employing a weighting function based on the two-electron integral elements in the convergence criteria during matrix optimization. Results show that, for the studied systems, the optimized <i>k-</i>uCJ decomposition allows a significant reduction of <i>k</i> compared to the rank-one approximation and that nonrelativistic calculations overestimate the equilibrium bonding distances and estimate much higher equilibrium and dissociation energies. The weighting function seems to improve energy convergence, which could allow further reduction in the number of decomposition terms.