Lower Bounds for Nonrelativistic Atomic Energies.

A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H-) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.

Comparison of an improved self-consistent lower bound theory with Lehmann's method for low-lying eigenvalues.

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple's lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple's results. In this paper, we further improve the SCLBT and compare its quality with Lehmann's theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann's theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann's lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann's theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein's and Temple's methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann's theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann's lower bounds.

Upper and lower bounds for tunneling splittings in a symmetric double-well potential.

The accurate determination of tunneling splittings in chemistry and physics is an ongoing challenge. However, the widely used variational methods only provide upper bounds for the energy levels, and thus do not give bounds on the gap between them. Here, we show how the self-consistent lower bound theory developed previously can be applied to provide upper and lower bounds for tunneling splitting between symmetric and antisymmetric doublets in a symmetric double-well potential. The tight bounds are due to the very high accuracy of the lower bounds obtained for the energy levels, using the self-consistent lower bound theory. The accuracy of the lower bounds is comparable to that of the Ritz upper bounds. This is the first time that any theory gave upper and lower bounds to tunneling splittings.

A two-layer approach to the coupled coherent states method.

In this paper, a two-layer scheme is outlined for the coupled coherent states (CCS) method, dubbed two-layer CCS (2L-CCS). The theoretical framework is motivated by that of the multiconfigurational Ehrenfest method, where different dynamical descriptions are used for different subsystems of a quantum mechanical system. This leads to a flexible representation of the wavefunction, making the method particularly suited to the study of composite systems. It was tested on a 20-dimensional asymmetric system-bath tunnelling problem, with results compared to a benchmark calculation, as well as existing CCS, matching-pursuit/split-operator Fourier transform, and configuration interaction expansion methods. The two-layer method was found to lead to improved short and long term propagation over standard CCS, alongside improved numerical efficiency and parallel scalability. These promising results provide impetus for future development of the method for on-the-fly direct dynamics calculations.

Numerical implementation and test of the modified variational multiconfigurational Gaussian method for high-dimensional quantum dynamics.

In this paper, a new numerical implementation and a test of the modified variational multiconfigurational Gaussian (vMCG) equations are presented. In vMCG, the wave function is represented as a superposition of trajectory guided Gaussian coherent states, and the time derivatives of the wave function parameters are found from a system of linear equations, which in turn follows from the variational principle applied simultaneously to all wave function parameters. In the original formulation of vMCG, the corresponding matrix was not well-behaved and needed regularization, which required matrix inversion. The new implementation of the modified vMCG equations seems to have improved the method, which now enables straightforward solution of the linear system without matrix inversion, thus achieving greater efficiency, stability and robustness. Here the new version of the vMCG approach is tested against a number of benchmarks, which previously have been studied by split-operator, multiconfigurational time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH) techniques. The accuracy and efficiency of the new implementation of vMCG is directly compared with the method of coupled coherent states (CCS), another technique that uses trajectory guided grids. More generally we demonstrate that trajectory guided Gaussian based methods are capable of simulating quantum systems with tens or even hundreds of degrees of freedom previously accessible only for MCTDH and ML-MCTDH.