Symmetry-based perturbation theory for electronic structure calculations

Abstract

We develop a multi-reference perturbation theory for electronic structure calculations based on symmetries of the Hamiltonian. The reference Hamiltonian in the symmetry-based perturbation theory (SBPT) is chosen such that it possesses more symmetries than the original Hamiltonian, leading to a larger reduction of computational resources in terms of both the number of configurations in the configuration interaction expansion and the number of required qubits in quantum computing applications. We provide approximate, scalable solutions for the second-order correction, as well as an application to selected configuration interaction. We show that SBPT is an extension of other existing multi-reference perturbation theories and that it can give better results for some molecular systems in a robust way.

Figures (7)

• Figure 1: The matrix on the left is a $D \times D$ Hamiltonian matrix, $H_{IJ} = \left< \Phi_I \right| H \left| \Phi_J \right>$. The Hamiltonian can be reduced to a block diagonal form using the symmetries of $H$, where unshaded elements are all zero. Each block matrix is labeled by its irreducible representation $\theta$ and given by a $D_{\theta} \times D_{\theta}$ matrix, $H_{I_{\theta}J_{\theta}} = \left<\Phi_{I_{\theta}} \right| H \left| \Phi_{J_{\theta}} \right>$.
• Figure 2: A schematic orbital energy diagram, with each horizontal black line representing a spin orbital $\psi_p$. The orbitals in the solid box transform nontrivially under a symmetry operation in the abelian group. The number of such operations and the number of solid boxes in SBPT are greater than in FCI - the more the number of boxes, the smaller the computational cost to solve for the reference Hamiltonian.
• Figure 3: In this orbital energy diagram, the highest four spin orbitals couple weakly in FCI. In the middle figure, the standard MRPTs treat them as external orbitals and form a CAS with the rest of the orbitals. In other words $H_{\rm{ref}}$ has $\mathbb{Z}_2$ symmetries for each external spin orbital. The right figure corresponds to a situation where each spin orbital has $\mathbb{Z}_2$ symmetry, so that the HF state is the eigenstate of $H_{\rm{ref}}$. The computational cost decreases as we make more approximations.
• Figure 4: The four spin orbitals in $A_0$ couple nontrivially. Besides $A_4$ in standard MRPTs, SBPT can explore other groupings of $\mathbb{Z}_2$ symmetries, such as $A_1$, $A_2$, and $A_3$ shown here.
• Figure 5: Ground-state potential energy curve of $\mathrm{H_2O}$ in the STO-3G basis. (a) The dissociation curve for the deformed geometry, $\Delta r = 0.01$. (b) Schematic orbital energy diagrams for $r = 1.8$ with the groupings of the exact (solid) and approximate (dashed) $\mathbb{Z}_2$ symmetries in MRPTs. (c, d) NEVPT2(4,4) and SBPT2 results of the leading-order contributions and the 2nd order corrections: the uncontracted (UC), strongly-contracted (SC), and Epstein-Nesbet (EN) approximations are given in Eqs. \ref{['2nd_order']}, \ref{['sc']}, and \ref{['en']}, respectively. PySCF uses the Dyall Hamiltonian with the SC approximation. (e, f) The SCI-SBPT2 results using the cutoffs given in Eqs. \ref{['epsilon_1']} and \ref{['epsilon_2']}: varying $\epsilon_1$ with fixed $\epsilon_2=0$ and varying $\epsilon_2$ with fixed $\epsilon_1$, which gives the number of irreps, $N_{\theta} = 12$. The number in parenthesis is the number of selected configurations.