Second-Order Perturbation Theory with Spin-Symmetry-Projected Hartree–Fock

Abstract

We propose two different schemes for second-order perturbation theory with spin-projected Hartree-Fock. Both schemes employ the same ansatz for the first-order wave function, which is a linear combination of spin-projected configurations. The first scheme is based on the normal-ordered projected Hamiltonian, which is partitioned into the Fock-like component and the remaining two-particle-like contribution. In the second scheme, the generalized Fock operator is used to construct a spin-free zeroth-order Hamiltonian. To avoid the intruder state problem, we adopt the level-shift techniques frequently used in other multi-reference perturbation theories. We describe both real and imaginary shift schemes and compare their performances on small systems. Our results clearly demonstrate the superiority of the second perturbation scheme with an imaginary shift over other proposed approaches in various aspects, giving accurate potential energy curves, spectroscopic constants, and singlet-triplet splitting energies. We also apply these methods to the calculation of spin gaps of transition metal complexes as well as the potential energy curve of the chromium dimer.

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach

“…Our imaginary-shi ed approach employs the explicit complex-valued reference Hamiltonian Ĥ0 → Ĥ0 + i Q instead of the approximate forms used to retain real arithmetic elsewhere. 67,70 We nd that this complex implementation requires shi values that are an order of magnitude smaller than those used in approximate real-valued approaches. 78 As a result, the explicit imaginary shi can recover smooth potential energy surfaces with far less distortion away from intruder-state singularities.…”

“…e quadratic form (A9) is the nonorthogonal extension of the imaginary-shi ed approach introduced for the SUPT2. 67 Note that the use of a "projection-a er-excitation" formalism in SUPT2 means that the underlying basis is orthogonal and Q pq = Q pq = δ pq . In contrast, the presence of the inverse overlap matrix elements Q sp for the nonorthogonal functions ( 22) makes solving Eq.…”

“…70 Alternatively, the SUPT2 approach introduced a real-valued rst-order wave function equation by back-transforming the imaginary-shi ed amplitudes, although this modi ed expression becomes quadratic in M . 67 In the case of NOCI-PT2, both of these approaches require either the inverse overlap matrix or a diagonal approximation for the matrix elements between rst-order interacting determinants, as outlined in Appendix A. Our preliminary investigations indicate that these approximations lead to extremely poor iterative convergence.…”

“…63 e recently derived Spin-Symmetry-Projected HF Perturbation eory (SUPT2) also provides a similar nonorthogonal rst-order wave function equation. 67 However, in SUPT2, the rst-order wave function is expanded using symmetry-projections of the single and double excitation determinants through a contracted "projection-a er-excitation" style. 67 e underlying basis functions for the SUPT2 equations are therefore orthogonal excitations from the symmetry-broken reference determinant.…”

“…67 However, in SUPT2, the rst-order wave function is expanded using symmetry-projections of the single and double excitation determinants through a contracted "projection-a er-excitation" style. 67 e underlying basis functions for the SUPT2 equations are therefore orthogonal excitations from the symmetry-broken reference determinant. In contrast, our NOCI-PT2 approach considers excitations from each reference determinant individually, leading to fundamentally nonorthogonal perturbation equations.…”