Gor'kov factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)〉〈\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉$, is made the basis of a generalized self-consistent-field (SCF) theory precisely analogous to that previously developed for Hartree factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1)\ensuremath{\psi}(4)〉〈({\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)〉$. The generalized SCF method is reviewed in the context of Gor'kov factorization. The corresponding simple SCF theory is developed to illustrate the method and is shown to give a simple Hamiltonian version of the Bardeen-Cooper-Schrieffer theory of super-conductivity and, more generally, of coherent pairing. The notion of an external pairing field is introduced and the corresponding response functions developed via a formalism like that of Kubo. General fluctuation-dissipation theorems are proved for the response functions. An equation of state is then obtained by a formulation analogous to the dielectric formulation. The insertion of the simple SCF approximation to the response function into the equation of state yields a generalization of the usual low-density (or short-range) evaluation of the grand potential to arbitrary temperatures. Screening the bare interaction with this approximate response function converts the former into the $t$ matrix of the independent-pair approximation. The entire analysis gives an underlying unity to the currently disparate treatments of superconductivity, of long-range correlations, and of short-range correlations.Gor'kov factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)〉〈\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉$, is made the basis of a generalized self-consistent-field (SCF) theory precisely analogous to that previously developed for Hartree factorization, $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1){\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)\ensuremath{\psi}(4)〉\ensuremath{\rightarrow}〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1)\ensuremath{\psi}(4)〉〈({\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\psi}(3)〉$. The generalized SCF method is reviewed in the context of Gor'kov factorization. The corresponding simple SCF theory is developed to illustrate the method and is shown to give a simple Hamiltonian version of the Bardeen-Cooper-Schrieffer theory of super-conductivity and, more generally, of coherent pairing. The notion of an external pairing field is introduced and the corresponding response functions developed via a formalism like that of Kubo. General fluctuation-dissipation theorems are proved for the response functions. An equation of state is then obtained by a formulation analogous to the dielectric formulation. The insertion of the simple SCF approximation to the response function into the equation of state yields a generalization of the usual low-density (or short-range) evaluation of the grand potential to arbitrary temperatures. Screening the bare interaction with this approximate response function converts the former into the $t$ matrix of the independent-pair approximation. The entire analysis gives an underlying unity to the currently disparate treatments of superconductivity, of long-range correlations, and of short-range correlations.

Physical ReviewJan 18, 19654 citations

Localized sample-based quantum diagonalization for strongly correlated chemistry

Qiaohong Wang, Kevin J. Sung, Ruhee D’Cunha +7 more

We develop a hybrid quantum-classical workflow combining sample-based quantum diagonalization (SQD) and the localized active space self-consistent field method (LASSCF) to solve for the ground states of transition-metal complexes, a longstanding challenge for both classical and quantum algorithms. The resulting approach, named LASSQD, integrates quantum sampling with fragment-based multireference theory to reduce the computational cost of solving strongly correlated active spaces. We test LASSQD on multiple iron-based complexes and demonstrate that it agrees with LASSCF within 1 kcal/mol, albeit at a much reduced computational cost. The cost reduction originates from the use of a sparse approximation of the exact and combinatorially large ground-state wavefunction, which also enables LASSQD to treat fragment sizes that are computationally inaccessible to LASSCF, as demonstrated by our computation of the spin gap of iron-porphyrin. These results establish that LASSQD is a scalable strategy for generating reliable multireference wave functions, providing a robust starting point for post-SCF correlation methods that recover dynamic correlation beyond the active space, and a promising pathway toward quantum-enhanced electronic structure calculations.We develop a hybrid quantum-classical workflow combining sample-based quantum diagonalization (SQD) and the localized active space self-consistent field method (LASSCF) to solve for the ground states of transition-metal complexes, a longstanding challenge for both classical and quantum algorithms. The resulting approach, named LASSQD, integrates quantum sampling with fragment-based multireference theory to reduce the computational cost of solving strongly correlated active spaces. We test LASSQD on multiple iron-based complexes and demonstrate that it agrees with LASSCF within 1 kcal/mol, albeit at a much reduced computational cost. The cost reduction originates from the use of a sparse approximation of the exact and combinatorially large ground-state wavefunction, which also enables LASSQD to treat fragment sizes that are computationally inaccessible to LASSCF, as demonstrated by our computation of the spin gap of iron-porphyrin. These results establish that LASSQD is a scalable strategy for generating reliable multireference wave functions, providing a robust starting point for post-SCF correlation methods that recover dynamic correlation beyond the active space, and a promising pathway toward quantum-enhanced electronic structure calculations.

Proceedings of the National Academy of SciencesJul 10, 20260 citations

Exact relations for the spin correlation functions

M. Okada, Tadashi Toyoda

AIP conference proceedingsJan 1, 19990 citations

SCF framework, HF stability and RPA correlation for Jordan-Wigner-transformed spin Hamiltonians on arbitrary coupling topologies

Shadan Ghassemi Tabrizi, Thomas M. Henderson, Thomas D. Kühne +1 more

Mapping spins to fermions via the Jordan-Wigner (JW) transformation can render mean-field (Hartree-Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering, because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-Algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytic orbital Hessian to diagnose HF stability and compute ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J1-J2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.Mapping spins to fermions via the Jordan-Wigner (JW) transformation can render mean-field (Hartree-Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering, because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-Algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytic orbital Hessian to diagnose HF stability and compute ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J1-J2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.

arXiv:2601.13078Jan 19, 20260 citations

t

matrix of the independent-pair approximation. The entire analysis gives an underlying unity to the currently disparate treatments of superconductivity, of long-range correlations, and of short-range correlations.