Explicitly-Correlated Effective Hamiltonian

• Explicitly-correlated effective Hamiltonians embed electron correlation directly into the operator, addressing slow basis convergence and singularity issues.
• Multiple formulations exist—canonical transcorrelation, congruent transformation, and Jastrow-factorized methods—each balancing Hermiticity, operator rank, and computational efficiency.
• These methods have practical applications in quantum simulation, stochastic CI, and relativistic corrections, enabling enhanced accuracy and resource savings.

An explicitly-correlated effective Hamiltonian is a Hamiltonian representation in which correlation effects are transferred from the wave-function ansatz, or from eliminated sectors of the full problem, into the operator itself. In quantum chemistry this usually denotes transcorrelated, congruent-transformed, Jastrow-factorized, or explicitly correlated Gaussian formalisms that encode short-range electron correlation, regularize singular relativistic operators, or supply matrix representations of operators beyond the nonrelativistic electronic Hamiltonian; in correlated-electron and materials settings, closely related usage includes effective operators inferred from correlated many-body wave functions or from low-energy projections (Motta et al., 2020, Cohen et al., 2019, Rácsai et al., 2024, Chang et al., 2018).

1. Conceptual scope and terminological range

The term does not identify a single formalism. In the explicitly correlated quantum-chemical sense, the defining feature is the inclusion of correlation information at the Hamiltonian level through objects such as a geminal generator, a Jastrow factor, or an explicitly correlated Gaussian basis. The central motivation is the slow basis convergence of ordinary orbital expansions, which do not efficiently reproduce the electron–electron cusp or other short-range structures of the exact wave function (Motta et al., 2020, Cohen et al., 2019).

A second usage arises when an effective Hamiltonian is inferred from correlated many-body states rather than from mean-field states. In that setting, the Hamiltonian is “effective” because it acts in a reduced manifold, and “correlated” because its parameters are fitted to observables computed with correlated wave functions, as in fixed-node diffusion Monte Carlo downfolding of spin-orbit couplings (Chang et al., 2018). A third usage appears in relativistic explicitly correlated Gaussian work, where the emphasis is not on deriving a new model Hamiltonian, but on making matrix elements of effective correction operators—such as the Breit–Pauli Hamiltonian or $H^2$—numerically available in a correlated basis (Rácsai et al., 2024).

This variety of meanings is substantive rather than merely terminological. Some constructions are exact changes of representation followed by truncation, some are approximate similarity or congruent transformations, some are projected positive-energy relativistic operators, and some are regression-based low-energy models. A common thread is that part of the many-body problem is absorbed into the operator before, or instead of, being handled solely by the final variational or projector solver (Bayne et al., 2013, Ferenc et al., 2021).

2. Operator-level constructions in quantum chemistry

Three operator constructions are central in the explicitly correlated literature: canonical transcorrelation, congruent transformation, and Jastrow-factorized similarity transformation.

Construction	Defining operator	Characteristic structure
Canonical transcorrelated F12	$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$	Hermitian; one- and two-body only; free of electron-electron singularities
Congruent transformed Hamiltonian	$\tilde H = G^\dagger H G$	Hermitian; variationally evaluated; generates up to six-particle operators
Jastrow-factorized similarity transformation	$\tilde H = e^{-\tau} H e^\tau$	Non-Hermitian; induced two- and three-body operators

In canonical transcorrelation, the explicitly correlated operator is an anti-Hermitian two-body geminal generator,

$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$

with

$$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$

The practical Hamiltonian is obtained by truncating the Baker–Campbell–Hausdorff expansion after the double commutator and retaining only one- and two-body parts. The resulting operator is Hermitian, contains no more than two-particle interactions, and is stated to be free of electron-electron singularities (Motta et al., 2020). Closely related constructions extended this framework with generalized pair excitations and a one-body orbital-relaxation operator $\hat S'$ to improve excited states in minimal bases (Kumar et al., 2022).

The congruent transformed Hamiltonian uses an explicitly correlated operator

$G(1,\dots,N)=\sum_{i<j} g(r_{ij})$

to define

$$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$

Because the transformation is congruent rather than similarity-based, Hermiticity is preserved. The energy is evaluated variationally as

$$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$

The price is operator growth: even though $$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$0 contains only one- and two-particle terms, $$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$1 generates two-, three-, four-, five-, and six-particle operators. The resolution-of-identity congruent transformed Hamiltonian and its partial infinite-order summation variant address this by projecting the explicitly correlated operator into a finite basis and summing selected diagram classes to infinite order (Bayne et al., 2013).

In the Jastrow-factorized formulation, the wave function is written as

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$2

leading to

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$3

Because only the kinetic operator fails to commute with $$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$4, the expansion terminates exactly at second order. The transformed Hamiltonian is non-Hermitian and decomposes into induced two- and three-body operators,

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$5

which were then solved projectively with stochastic CI/FCIQMC methods (Cohen et al., 2019).

These constructions differ sharply in formal properties. Canonical transcorrelation retains a solver-friendly Hermitian one- plus two-body form at the cost of truncation; congruent transformation preserves Hermiticity and a variational principle at the cost of high particle rank; Jastrow factorization retains the exact transformed equation for a chosen $$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$6 but produces a non-Hermitian operator with explicit three-body terms (Motta et al., 2020, Bayne et al., 2013, Cohen et al., 2019).

3. Explicitly correlated Gaussian bases and operator technology

A distinct branch of the subject uses explicitly correlated Gaussians (ECGs), especially floating ECGs (fECGs), as the primary representation of the electronic wave function. For clamped-nuclei molecular problems the nonrelativistic reference equation is

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$7

with the wave function expanded as

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$8

and floating ECG spatial factors

$$\hat{H}^\prime \approx \hat{H} + [\hat{H},\hat{A}]_{1,2} + \frac{1}{2}[[\hat{F},\hat{A}]_{1,2},\hat{A}]_{1,2}$$9

The shift vector $\tilde H = G^\dagger H G$0 is essential for molecular efficiency because the electronic density is not naturally centered at the origin, but it also destroys many analytic simplifications available for zero-shift atomic ECGs (Rácsai et al., 2024).

The principal obstacle in this setting has been the evaluation of singular operators required for relativistic and other effective Hamiltonians. For the leading relativistic term in the $\tilde H = G^\dagger H G$1-expansion,

$\tilde H = G^\dagger H G$2

one has

$\tilde H = G^\dagger H G$3

where, for singlet or spin-averaged states,

$\tilde H = G^\dagger H G$4

Direct evaluation converges poorly with smooth fECGs because the basis does not satisfy the exact electron–electron or electron–nucleus cusp conditions (Rácsai et al., 2024).

The key advance was the extension of Drachman’s regularization to molecular fECGs. Singular operators are replaced by smoother expressions, for example

$\tilde H = G^\dagger H G$5

$\tilde H = G^\dagger H G$6

and

$\tilde H = G^\dagger H G$7

In matrix terms, the delta-function and mass-velocity contributions are therefore evaluated through matrix elements of $\tilde H = G^\dagger H G$8, $\tilde H = G^\dagger H G$9, and derivative couplings rather than through local cusp-sensitive operators (Rácsai et al., 2024).

For molecular fECGs, the missing matrix elements were products of Coulomb factors,

$\tilde H = e^{-\tau} H e^\tau$0

The solution was to approximate one Coulomb factor by a Gaussian expansion,

$\tilde H = e^{-\tau} H e^\tau$1

using the Beylkin–Monzón exponential-sum construction. This converts intractable $\tilde H = e^{-\tau} H e^\tau$2 operators into sums of tractable $\tilde H = e^{-\tau} H e^\tau$3 operators. The paper reports that a universal choice such as $\tilde H = e^{-\tau} H e^\tau$4 is already near machine precision in double precision for the applications considered, with $\tilde H = e^{-\tau} H e^\tau$5 used as a cross-check (Rácsai et al., 2024).

The significance for effective Hamiltonians is direct. Once these matrix elements are available, one can assemble accurate fECG matrix representations not only of $\tilde H = e^{-\tau} H e^\tau$6 but also of

$\tilde H = e^{-\tau} H e^\tau$7

whose construction requires products of Coulomb operators through $\tilde H = e^{-\tau} H e^\tau$8. The work therefore supplies the missing operator-evaluation machinery for geometry-dependent effective correction operators beyond the nonrelativistic Hamiltonian itself (Rácsai et al., 2024).

Benchmark calculations covered He, Li, Be, H$\tilde H = e^{-\tau} H e^\tau$9, H$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$0, H$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$1, H$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$2, HeH$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$3, He$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$4, and He$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$5. For the singular Breit–Pauli pieces, direct evaluation converged slowly, whereas numerical Drachman regularization yielded much faster and more stable convergence. The $$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$6 state of H$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$7 was also computed near its local minimum, indicating suitability for geometry-dependent scans and, by plausible implication, for effective Hamiltonians defined along potential curves and surfaces (Rácsai et al., 2024).

4. Relativistic and spectroscopic effective Hamiltonians in ECG and fECG frameworks

The explicitly correlated Gaussian framework also supports relativistic effective Hamiltonians in a narrower spectroscopic sense. For spin-dependent Breit–Pauli theory, the total correction is decomposed as

$$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$8

In the molecular fECG implementation of spin-dependent terms, the working spin-dependent Hamiltonian through order $$\hat{A} = \frac{1}{2} G^{\alpha\beta}_{ij}\left(\hat{E}^{\alpha\beta}_{ij}-\hat{E}^{ij}_{\alpha\beta}\right),$$9 is

$$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$0

The purpose is not to derive new NRQED operator content, but to make the known fine-structure operators numerically usable with high precision in a molecular fECG basis (Jeszenszki et al., 23 Jun 2025).

In this setting the effective-Hamiltonian object is often a reduced fine-structure matrix assembled from perturbative couplings between nonrelativistic electronic states. For the excited helium dimer, the explicitly correlated calculation produces coupling parameters such as $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$1, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$2, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$3, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$4, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$5, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$6, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$7, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$8, $$G^{\alpha\beta}_{ij} = \frac{3}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ij\rangle + \frac{1}{8}\langle \alpha\beta|\hat Q_{12}\hat F_{12}|ji\rangle, \qquad \hat F_{12}(r_{12})=-\frac{e^{-\gamma r_{12}}}{\gamma}.$$9, and $\hat S'$0, which define reduced matrix Hamiltonians over coupled triplet and singlet manifolds (Jeszenszki et al., 23 Jun 2025). For the $\hat S'$1 state, the dipolar spin–spin splitting is summarized by

$\hat S'$2

This is an explicitly correlated effective Hamiltonian in the spectroscopic sense: the fECG computation supplies the entries of the reduced operator (Jeszenszki et al., 23 Jun 2025).

A related relativistic construction is the positive-energy projected no-pair Dirac–Coulomb or Dirac–Coulomb–Breit operator represented in an ECG basis,

$\hat S'$3

Here $\hat S'$4 projects onto the positive-energy subspace of the external-field Dirac problem. The resulting operator is naturally interpreted as an effective Hamiltonian for the electronic positive-energy sector because the negative-energy electron–positron space has been removed by projection (Ferenc et al., 2021).

The Breit interaction enters as

$\hat S'$5

and was treated both perturbatively and variationally. The comparison with low-$\hat S'$6 Breit–Pauli benchmarks showed that the perturbative no-pair Dirac–Coulomb plus first-order Breit correction is much closer to the leading-order reference $\hat S'$7 than the fully variational no-pair Dirac–Coulomb–Breit energy for the tested low-$\hat S'$8 systems (Ferenc et al., 2021). This does not reduce the importance of the projected relativistic Hamiltonian; rather, it clarifies that its higher-order content differs from low-order Breit–Pauli effective theory.

5. Downfolding from correlated wave functions

Another established meaning of explicitly-correlated effective Hamiltonian is a low-energy model whose parameters are fitted to matrix elements or expectation values computed with correlated many-body wave functions. In the diffusion Monte Carlo spin-orbit framework, the trial state is of Slater–Jastrow form,

$\hat S'$9

with $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$0 including up to three-body terms. Within a degenerate or nearly degenerate manifold, the effective relativistic Hamiltonian is chosen as

$G(1,\dots,N)=\sum_{i<j} g(r_{ij})$1

and the parameter $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$2 is obtained by the linear relation

$G(1,\dots,N)=\sum_{i<j} g(r_{ij})$3

This is a regression-based downfolding of expectation values across a manifold of explicitly correlated many-body states rather than a transformation of the full Hamiltonian matrix (Chang et al., 2018).

The significance is methodological. Correlation is not inserted by an $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$4-dependent operator acting on the Hamiltonian; instead, it is carried by the wave functions used to determine the model. The approach therefore belongs to the broader explicitly correlated effective-Hamiltonian program in which correlated first-principles calculations act as the solver and the effective Hamiltonian is the compact low-energy representation (Chang et al., 2018). In monolayer WS$G(1,\dots,N)=\sum_{i<j} g(r_{ij})$5, the resulting valence-band spin-orbit splitting at $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$6 was reported as

$G(1,\dots,N)=\sum_{i<j} g(r_{ij})$7

statistically consistent with a reported PBE value of $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$8 eV and an experimental estimate of $G(1,\dots,N)=\sum_{i<j} g(r_{ij})$9 eV (Chang et al., 2018).

A formally different, but conceptually adjacent, construction represents the exact electronic Hamiltonian in terms of fluctuation operators between internally correlated fragment states. If $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$0 is the antisymmetrized supersystem basis built from fragment states, the exact Hamiltonian can be written as

$$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$1

After truncating the fragment state spaces, high-energy local arrangements of electrons are removed, yielding a systematically improvable correlated effective Hamiltonian. This is not explicitly correlated in the F12/R12 sense, but it is correlated because the retained fragment states may themselves be highly correlated many-electron states (Dutoi et al., 2017).

These two examples mark an important boundary. A Hamiltonian can be “explicitly correlated” because explicit correlation enters the operator; it can also be “correlated effective” because its parameters or basis states are derived from correlated wave functions. The literature contains both meanings (Chang et al., 2018, Dutoi et al., 2017).

6. Quantum simulation and stochastic CI applications

Explicitly correlated effective Hamiltonians have been adopted as a way to reduce basis requirements before applying expensive solvers. In canonical transcorrelation for quantum simulation, the transformed Hamiltonian is constructed classically and then passed to VQE with a standard q-UCCSD ansatz. For the molecules studied, explicitly correlated energies based on an underlying 6-31G basis had cc-pVTZ quality, and the reduction in hardware resources relative to directly simulating cc-pVTZ was reported as up to two orders of magnitude in CNOT count and about a factor of three in qubits (Motta et al., 2020). The direct improvement is Hamiltonian-side: at fixed orbital basis, the q-UCCSD circuit resources are unchanged, but the modified Hamiltonian yields small-basis energies comparable to much larger-basis calculations (Motta et al., 2020).

The transcorrelated strategy was later extended to balanced ground- and excited-state simulation. Using a minimal ANO-RCC-MB basis, an explicitly correlated two-body operator with generalized pair excitations and a one-body orbital-relaxation operator were combined in a Hermitian, one- and two-body transcorrelated Hamiltonian. Ground-state energies became comparable to much larger cc-pVTZ calculations, the number of required CNOT gates could be reduced by more than three orders of magnitude for the species studied, and qEOM excitation-energy errors were reduced by an order of magnitude (Kumar et al., 2022). The paper’s central lesson was that cusp removal alone is insufficient for accurate excited states; generalized pair excitations and the singles/CABS-like orbital-relaxation transformation are both needed (Kumar et al., 2022).

The same operator-preprocessing logic also proved useful in stochastic CI. In FCIQMC, canonical transcorrelation supplies a Hermitian one- and two-body Hamiltonian on which the stochastic projector acts directly. Across the G1 set, ordinary aug-cc-pVDZ FCIQMC had an average absolute error of $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$2 per correlated electron, whereas CT-FCIQMC reduced this to $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$3 per correlated electron; for atomization energies the median absolute error was reduced from $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$4 to $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$5, with the abstract summarizing the improvement as from about $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$6 to $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$7 (Kersten et al., 2016). For $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$8, convergence within $$\tilde H = G^\dagger H G, \qquad \tilde S = G^\dagger \mathbf 1\, G.$$9 required less than $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$0 walkers with CT-FCIQMC, whereas conventional FCIQMC required more than $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$1 walkers (Kersten et al., 2016).

These applications show a recurrent pattern. Explicit correlation is incorporated classically or analytically into the Hamiltonian, while the downstream solver—VQE, ADAPT-VQE, qEOM, or FCIQMC—remains structurally conventional. This suggests that one of the most robust roles of explicitly correlated effective Hamiltonians is as solver-agnostic operator preconditioning (Motta et al., 2020, Kumar et al., 2022, Kersten et al., 2016).

7. Formal boundaries, related usages, and limitations

Several neighboring literatures use “effective Hamiltonian” in ways that overlap conceptually but not technically with explicitly correlated quantum-chemical constructions. Adaptive VQE dressed-Hamiltonian methods based on products of linear combinations of excitation operators,

$$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$2

move correlation into the Hamiltonian and keep state-preparation circuits shallow, but they do not use explicit $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$3-dependence, geminals, or cusp-enforcing factors (Liu et al., 2022). Likewise, Krylov–Bogoliubov–Mitropolsky averaging for deriving the $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$4-$$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$5 and Kondo models, stochastic Hamiltonians for ring, ladder, parquet, and induced-interaction theories, and EFT-style perturbation theory for correlated effective interactions all concern correlated effective Hamiltonians, but not explicit correlation in the F12/Jastrow/ECG sense (Saiko, 2010, Green, 2019, Photopoulos et al., 2024).

The formal trade-offs are persistent. Canonical transcorrelation remains approximate because the BCH series is truncated, only one- and two-body commutator parts are retained, and the double commutator uses the Fock operator rather than the full Hamiltonian (Motta et al., 2020, Kumar et al., 2022, Kersten et al., 2016). Congruent transformation preserves Hermiticity and a variational principle, but finite RI truncation and partial infinite-order summation leave the final renormalization only partially systematic (Bayne et al., 2013). Jastrow-factorized similarity transformation admits highly flexible explicit correlation and very accurate small-basis energies, but the Hamiltonian is non-Hermitian and contains three-body interactions whose six-index matrix elements are the main computational bottleneck (Cohen et al., 2019). In fECG molecular regularization, the operator technology is now available for $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$6 and $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$7, but the method remains practically targeted at few-electron systems and still depends on high-quality optimized Gaussian bases (Rácsai et al., 2024, Jeszenszki et al., 23 Jun 2025).

A frequent misconception is that every “effective Hamiltonian with correlation” is an explicitly correlated Hamiltonian. The literature reviewed here suggests a sharper classification. In the narrow sense, explicitly correlated effective Hamiltonians incorporate correlation through $$E[\Psi_T,G] = \frac{\langle \Psi_T|\tilde H|\Psi_T\rangle}{\langle \Psi_T|\tilde S|\Psi_T\rangle}.$$8-dependent geminals, Jastrow factors, or explicitly correlated Gaussian representations of operators (Motta et al., 2020, Cohen et al., 2019, Rácsai et al., 2024). In a broader sense, the phrase can also refer to low-energy Hamiltonians inferred from correlated wave functions or projected correlated subspaces (Chang et al., 2018, Ferenc et al., 2021). The distinction matters because the operator algebra, the approximations, and the intended applications are different.

Taken together, these developments establish the topic as an operator-centered branch of many-body theory. Whether the goal is singularity removal, relativistic correction surfaces, stochastic or quantum simulation in compact bases, or low-energy model extraction from correlated wave functions, the defining move is the same: correlation that would otherwise reside entirely in the final wave function is encoded, at least in part, in the Hamiltonian itself (Motta et al., 2020, Rácsai et al., 2024, Chang et al., 2018).