JAGP Ansatz for Electron Correlation

• JAGP ansatz is a variational many-body wavefunction that merges geminal pairing with Jastrow correlators to recover both static and dynamic electron correlation in quantum systems.
• It utilizes a dense variational pairing matrix and polynomial-size Hilbert-space Jastrow factors to maintain size consistency and facilitate efficient Monte Carlo optimization.
• Advanced optimization techniques, including the Linear Method and fast determinant updates, enable accurate energy evaluations for both finite molecules and extended systems.

The Jastrow–Antisymmetrized–Geminal–Power (JAGP) ansatz is a class of multi-parameter, polynomial-scaling many-body wavefunctions that systematically merge the explicit electron-pairing structure of geminal (AGP) theories with Hilbert-space Jastrow correlators. JAGP addresses both strong (static) and weak (dynamic) electron correlation while maintaining size consistency, efficient optimization strategies, and robust nodal structures for projector methods. This ansatz is variational, size-consistent, applicable to a wide variety of finite and extended systems, and forms the basis for high-accuracy variational and diffusion Monte Carlo treatments of strongly correlated and multireference electronic problems.

1. Mathematical Structure of the JAGP Ansatz

JAGP is defined in an occupation-number (Fock) basis as: $$|\Psi_{\mathrm{JAGP}}\rangle = \exp(\hat{J})\,|\Psi_{\mathrm{AGP}}\rangle$$ where $\hat{J}$ is a polynomial order-2 (or higher) operator diagonal in occupation numbers, and $|\Psi_{\mathrm{AGP}}\rangle$ is an antisymmetrized geminal power (number-projected BCS state). Explicitly: $$|\Psi_{\mathrm{AGP}}\rangle = \left(G^\dagger\right)^{N/2}|0\rangle,\quad G^\dagger = \sum_{p\bar{q}} F_{p\bar{q}}\,a_p^\dagger a_{\bar{q}}^\dagger$$ where $N$ is the even total number of electrons, $F_{p\bar{q}}$ is a fully variational pairing matrix coupling an $\alpha$-spin orbital $p$ to a $\beta$-spin orbital $\bar{q}$, and the antisymmetry is guaranteed by the multiple application of $G^\dagger$.

The Hilbert-space Jastrow operator $\hat{J}$ is: $$\hat{J} = \sum_{p\leq q} J^{\alpha\alpha}_{pq} \hat{n}_p\hat{n}_q + \sum_{\bar{p}\leq \bar{q}} J^{\beta\beta}_{\bar{p}\bar{q}} \hat{n}_{\bar{p}}\hat{n}_{\bar{q}} + \sum_{p\,\bar{q}} J^{\alpha\beta}_{p\bar{q}} \hat{n}_p\hat{n}_{\bar{q}}$$ where $\hat{n}_p=a_p^\dagger a_p$ is the occupation operator in orbital $p$.

This construction allows a dense pairing matrix $F$ supporting all possible resonance structures, while $\hat{J}$ encodes both dynamic correlation and constraints (e.g., through partial number projection) that eliminate unphysical local charge fluctuations, enforcing proper electron counts on subsystems and restoring size consistency (Neuscamman, 2013).

2. Variational Energy Expression and Monte Carlo Evaluation

The variational ground-state energy is

$$E = \frac{\langle\Psi_{\mathrm{JAGP}}|\hat{H}|\Psi_{\mathrm{JAGP}}\rangle}{\langle\Psi_{\mathrm{JAGP}}|\Psi_{\mathrm{JAGP}}\rangle}$$

which, when sampled stochastically, adopts the usual Variational Monte Carlo (VMC) estimator: $$E = \left\langle E_L(n) \right\rangle_{|\langle n|\Psi\rangle|^2}$$ where $$E_L(n) = \langle n|\hat{H}|\Psi\rangle/\langle n|\Psi\rangle$$, and $|n\rangle$ are occupation number vectors. The presence of the exponential Jastrow factor is handled efficiently: in the occupation-number basis, both numerator and denominator simply acquire scalar weights $\exp[J(n)]$. The resulting AGP part reduces to determinants of appropriate sub-blocks of the matrix $F$.

Local energies in JAGP are "Jastrow-dressed," with one- and two-electron integrals contracted against the perturbed Fock configurations: $$E_L(n) = E_0(n) + \sum_{ia} t_{ia}^n (R\Theta)_{ai} + \frac{1}{2} \sum_{iajb} [ia|jb] ((R\Theta)_{ai}(R\Theta)_{bj} - (R\Theta)_{aj}(R\Theta)_{bi}) + \ldots$$ All bare integrals are replaced by Jastrow-dressed analogues, and contractions involve products or inverses of $F$ submatrices.

3. Optimization Algorithms and Computational Scaling

JAGP is optimized using the Linear Method, which operates by constructing the subspace spanned by $|\Psi\rangle$ and all first derivatives $\partial|\Psi\rangle/\partial \mu_x$ with respect to the variational parameters $\mu_x$, then solving the projected generalized eigenvalue problem: $$\sum_y \langle\Psi^x|H|\Psi^y\rangle c_y = \lambda \sum_y \langle\Psi^x|\Psi^y\rangle c_y$$ Parameter updates follow $\mu_x \leftarrow \mu_x + c_x/c_0$. Convergence of $\sim$10–20 iterations (to within a few m$E_h$ in energy) is typical (Neuscamman, 2013).

Sampling is performed via Metropolis–Hastings or continuous-time Monte Carlo, with each move involving rapid update (in $O(1)$ or $O(N^2)$ time) of determinant ratios (by the Sherman–Morrison or Pfaffian formula) and Jastrow ratios. The overall cost per VMC iteration, including local energy and parameter derivative accumulation, scales as $O(N^5)$ in a minimal implementation, dominated by double sums in two-electron integral contractions.

Recent advances exploiting screening, symmetry projection, and fast determinant update techniques enable efficient application to larger systems (Mahajan et al., 2019).

4. Recovery of Static and Dynamic Correlation

The AGP part of JAGP, by virtue of a non-block-diagonal $F$, automatically superposes all possible pairings, supporting multiple resonance structures and zero-seniority (paired) subspaces. This enables recovery of near-degeneracy (static) correlation characteristic of multireference systems, as in bond dissociation, diradical states, and molecular transition states (Neuscamman, 2013, Zen et al., 2014).

Hilbert-space Jastrow factors serve multiple roles: i) elimination of unphysical ionic components (wrong local electron count) by partial-number projection, ii) enforcing size consistency, iii) direct encoding of electron avoidance (dynamical, short-range correlation) and inter-pair correlation otherwise missing from pure AGP or perfect-pairing theories. The two-body Jastrow projects out unphysical valence-bond contributions and encodes dynamical (e.g., cusp) physics (Neuscamman, 2013, Neuscamman, 2012, Raghav et al., 2022).

Cluster-Jastrow generalizations (CJAGP) efficiently compress dynamic correlation via highly structured, low-parameter cluster operators while preserving the AGP’s static correlation flexibility (Neuscamman, 2016).

5. Numerical Performance and Benchmark Applications

JAGP’s balanced static–dynamic correlation capability is reflected across a range of benchmark systems:

System	JAGP NPE / max error	Best conventional reference	Comments
H$_4$ distortion (square)	$\sim$0.6 m$E_h$ ($\sim$0.4 kcal/mol)	FCI (exact, exponential cost)	JAGP exact within m$E_h$ (Jastrow essential for symmetry)
H$_2$O symmetric stretch	3.4 kcal/mol	CASPT2 NPE 1.0 kcal/mol	JAGP better than CCSD(T) NPE 5.1 kcal/mol
N$_2$ triple bond	$\sim$15 kcal/mol at dissociation	CASPT2(6,6) (exponential scaling)	JAGP curve smooth; CCSD(T) has nonphysical cusp/large error
Diradicals (ethylene, CH$_2$)	recovers $\sim$98% of reference barrier	MR-CISD+Q, CAS(12,12)	JAGP far superior to JSD (single determinant) on multirefs
G2 atomization set (55 mol.)	MAD 1.6 kcal/mol (LRDMC)	JSD MAD 3.2 kcal/mol	JsAGPs DMC achieves chemical accuracy ($\le$1 kcal/mol) for 26/55

In multi-reference transition-states (e.g., ethene), JAGP matches the accuracy of UCCSD(T) and MRCI+Q, overcoming qualitative breakdown observed in lower-level approximations (Neuscamman, 2013).

For size-consistency tests, JAGP with proper Jastrow projection yields exactly additive total energies for non-interacting fragments, unlike bare AGP (Neuscamman, 2012).

6. Extensions: Symmetry-Projection, Nonlinear Correlators, and Quantum Circuits

JAGP can be enhanced with explicit symmetry-projection operators restoring global symmetries (number, spin $S_z$, complex conjugation), yielding the symmetry-projected Jastrow-mean-field (SJMF) family. Optimization is performed in the symmetry-broken frame but projected observables are recovered through sampling (Mahajan et al., 2019).

Nonlinear generalizations (higher-body Jastrow or similarity-transformed formulations) can be constructed such that the similarity-transformed Hamiltonian is still exactly summable in the seniority-zero AGP algebra, allowing CC-like projective equations to be solved at $O(M^4)$ cost (Khamoshi et al., 2020).

On quantum hardware, AGP/JAGP can be exactly prepared as an elementary symmetric polynomial (ESP) state by a polynomial depth circuit, whose structure is a disentangled sequence of paired coupled-cluster (unitary pCCD) and Jastrow layers. This circuit natively generates multipartite entanglement not accessible to a product-Slater reference (Khamoshi et al., 2023).

7. Advantages, Limitations, and Outlook

Advantages:

• Variational, monotonic energy improvement.
• Strict size consistency via Hilbert-space Jastrow projections.
• Polynomial scaling ($O(N^5)$, potentially $O(N^4)$ with optimizations).
• Simultaneous recovery of strong (static) and weak (dynamic) correlation, beyond single-reference and standard CI approaches.
• Nodal structures of JAGP are robust indicators of qualitative electronic state, enabling high-accuracy diffusion Monte Carlo (DMC) projections.
• Amenable to symmetry-projection and cluster/Jastrow compressions for extended systems.

Limitations and Open Challenges:

• Residual dynamic correlation near equilibrium remains significant ($\sim$7–15 kcal/mol in extended bases without post-JAGP projection or perturbative correction) (Neuscamman, 2013, Neuscamman, 2016).
• Cluster-Jastrow and nonlinear correlators achieve balance with fewer parameters but cannot recover all dynamic details for absolute energies; most errors are well-balanced in energetics.
• Further gains could result from fixed-node projector Monte Carlo using JAGP as trial, real-space Jastrow with cusp enforcement, Pfaffian-genus geminals, or direct orbital basis optimization.
• Over-correlation near critical pairing strengths and the recoupling regime suggests a need for further refinement (e.g., hybridizing projective and variational strategies or including higher-order correlators).

JAGP thus provides a polynomial-cost, size-consistent, and systematically improvable framework for strong correlation, bridging the gap between mean-field, active-space, and coupled-cluster paradigms. Its versatility and accuracy establish it as a cornerstone ansatz for both ground and excited-state electronic-structure methods, particularly in regimes inaccessible to conventional single-reference theories.