A Reduced-Parameter Variational Strategy for Accelerated Evaluation of Thermal Density Matrices via Quantum Effective Harmonic Oscillators Approach

Accurate treatment of anharmonic and temperature-dependent quantum effects is crucial for the reliable evaluation of thermal density matrices and vibrational partition functions, which in turn affect the accuracy of thermodynamic and spectroscopic predictions. In this study, a McLachlan-type variational principle based on the Bloch equation is developed for the calculation of the thermal density matrices and the associated vibrational partition function. Quantum anharmonic effects are captured through the independent-particle Effective Harmonic Oscillator (EHO) approximation, in which vibrational wave functions are defined solely by two variational parameters: the centroids and widths of multidimensional Gaussian product functions. The proposed variational method optimizes the trace of the squared deviation between the derivatives of the exact and model density matrices with respect to the parameters of the model Hamiltonian. The accuracy and efficiency of this Bloch equation based thermal-EHO (TB-EHO) approach are evaluated by calculating the thermal properties of various molecular systems, and comparing the results with those from the thermal vibrational configuration interaction (T-VCI) and earlier proposed Bloch equation based thermal vibrational self-consistent field (TB-VSCF) methods as benchmarks. The TB-EHO method yields results that closely match the reference values across a wide temperature range (0 to 1000 K). The approach is further validated by computing the equilibrium constant for a cis-trans isomerization reaction, yielding results in good agreement with TB-VSCF and experimental data. Moreover, the TB-EHO method exhibits substantial computational efficiency, reducing CPU time by ~98% compared to the TB-VSCF, making it highly suitable for applications to large molecular systems.