Exploring a spectral filtering approach to electronic structure calculations

ABSTRACT

We make an initial exploration of a spectral filtering approach to numerically approximating solutions to the time-independent Schrödinger equation for a model system of two spinless fermions with harmonic interactions. Fourier transformation of the time evolution of an arbitrary antisymmetrized wave packet results in resonance at frequencies proportional to eigenenergies of the system Hamiltonian, while the Fourier coefficient approximates the corresponding eigenfunction. Spectral filtering is explored using the two-particle model through both direct numerical integration of the Schrödinger equation and a semiclassical parametrisation. In the former, a discrete position basis describes the time-evolution operator, which is then repeatedly applied to an antisymmetrized wave packet to obtain the packet's time evolution. In the semiclassical parametrisation, equations of motion for the parameters are obtained from the Dirac-Frenkel-McLachlan functional. Integration of the parameter equations of motion describes the packet's time evolution. The semiclassical approach requires less memory and accurately obtains the eigenspectrum. In a step towards application of spectral filtering to realistic systems, a possible ansatz for use with Coulomb potentials is explored, using a simple model Hamiltonian.

GRAPHICAL ABSTRACT

Keywords:

• Spectral filtering
• filter-diagonalisation
• semiclassical
• time-dependent Hartree-Fock
• electronic structure theory

Acknowledgments

We thank Phil Kovac, University of Oregon, Anthony Dutoi, University of the Pacific, and Christiane Koch, Universität Kassel, for helpful conversations. We also thank Cathy Wong, University of Oregon, for suggesting the consideration of squarish wave packets. AJK thanks Eric Beyerle, University of Oregon, for helpful conversations on the topic of this report. This work was supported by NSF grant CHE-1565680 and benefited from access to the University of Oregon high performance computer, Talapas. This work was carried out by unionised labour, one member each of the Graduate Teaching Fellows Federation, AFT Local 3544, and the United Academics of the University of Oregon, AAUP/AFT Local 3209.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The requirement described in Equation (4(4) ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}\omega ⟨\widetilde{\text{ψ}}(\omega )|\widetilde{\text{ψ}}(\omega )⟩=1.$(4) ) leads to a window function that isn't unitless; as a result, $|\widetilde{\text{ψ}}(\omega )⟩$ has different units in Equation (3(3) $|\widetilde{\text{ψ}}(\omega )⟩\cong \sum _{k=-N_t}^{N_t}\delta t{\mathrm{e}}^{i\omega t_k}g(t_k)|\text{ψ}(t_k)⟩,$(3) ) than it does in Equation (1(1) $|\widetilde{\text{ψ}}(\omega )⟩\equiv {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}t{\mathrm{e}}^{i\omega t}|\text{ψ}(t)⟩.$(1) ), its definition. This leads to no problems in practice.

2 Another interpretation of this model could be that the particles are electrons, where the choice to make $⟨q_1,q_2|\text{ψ}(0)⟩$ antisymmetric in the orbital degrees of freedom results in triplet states. Singlet states would then be obtained by a separate calculation where the initial state is symmetric in the orbital degrees of freedom.

3 $\mathbf{\text{M}}$ is not necessarily a positive definite or positive semi-definite matrix. With the regularisation scheme used here, a negative eigenvalue could cause an unintentional regularisation. One possible solution is to instead make use of the equation $\dot{\mathit{\lambda }}=({\mathbf{\text{M}}}^T\mathbf{\text{M}}{)}^{-1}{\mathbf{\text{M}}}^T\cdot \mathit{\chi }$, which follows directly from Equation (22(22) $\dot{\mathit{\lambda }}={\mathbf{\text{M}}}^{-1}\cdot \mathit{\chi },$(22) ). The product ${\mathbf{\text{M}}}^T\mathbf{\text{M}}={\mathbf{\text{M}}}^2$ is positive semi-definite, so it removes the issue of negative eigenvalues and may be regularised and inverted in place of $\mathbf{\text{M}}$. However, it has a larger condition number than $\mathbf{\text{M}}$, which, if too large, could be a problem for accuracy due to the calculation of a matrix exponential. In each calculation presented here, we encounter only one negative eigenvalue, which is much smaller in magnitude than $\overline{ϵ}$. Therefore, ${\mathbf{\text{M}}}^2$ is never inverted in place of $\mathbf{\text{M}}$, and accurate eigenstates and eigenenergies are still obtained. The small magnitudes of the negative eigenvalues actually seen suggest the possibility that these eigenvalues are really zero, but appear negative due to round-off error. This possibility, however, remains to be demonstrated.

Additional information

Funding

This work was supported by National Science Foundation (NSF) grant CHE-1565680.