** Introduction: **Mott-insulating antiferromagnets are ubiquitously found in the parent compounds of many strongly-correlated materials, including cuprates and 3

*d*transition metal mono-oxides. In order to describe such systems, there is growing interest in wavefunction methods, including configuration-interaction (CI) methods, such as full configuration interaction Quantum Monte Carlo, (

*i*)-FCIQMC [1], applied to periodic clusters. Whilst wavefunctions of strongly-correlated molecular systems primarily build on restricted one-particle bases, the general approach in condensed matter physics communities is to start from a qualitatively correct broken-symmetry solution. We therefore ask how the structure of the many-electron wavefunction of a typical Mott-insulator depends on the representation of the single-particle basis used to describe the configuration space of the system. On the face it, such a question appears to be more of a mathematical rather than a physical one. However, we will show that the choice of basis impacts the convergence of the exact full configuration interaction (FCI) expansion of the many-electron wavefunction in a dramatic and counterintuitive manner. Thus a broken-symmetry mean-field basis (UHF), which produces a qualitatively correct description of the antiferromagnet, is shown to provide an extremely poor basis to construct the exact FCI wavefunction: the CI coefficients decay very slowly with increasing particle-hole excitation of the reference, resulting in a highly complex wavefunction which is very difficult to approximate.

In contrast, we find that the qualitatively incorrect band-structure of spin-restricted mean-field theory (RHF) provides a rapidly convergent CI expansion, which is much more amenable to approximation. Natural orbitals provide yet more rapidly converging, compact, wavefunctions. Our results suggest that with the correct single-particle basis, such strongly-correlated systems may be amenable to powerful single-reference wavefunction methods, opening a new direction of theoretical research for these systems [2].

** Model:** The three-band (p–d) Hubbard Hamiltonian describes the dynamics of holes in a CuO

_{2}plane represented by a Cu 3

*d*

_{x2–y2}orbital and two O 2

*p*

_{σ}orbitals. Its Hamiltonian comprises kinetic energy,

*t*

_{ij}, and hole interaction terms,

*U*

_{ij},

where *a*_{i,σ}^{†} (*a*_{i,σ}) creates (annihilates) a hole with spin σ in the Cu 3*d* or O 2*p*_{σ} orbital at site *i*. We present results for a tilted lattice with 10 CuO_{2} units and *N* = 10 holes, an undoped system at half-filling with a full Hilbert space of 20.3×10^{9} determinants.

* Method:* In (

*i*)-FCIQMC [1] the ground state wavefunction |ψ>, expressed as an FCI expansion |ψ> = ∑

_{i}C

_{i}|D

_{i}>, is sampled by an ensemble of

*N*

_{w}signed walkers which stochastically evolve in a combinatorially large Hilbert space of

*N*-particle Slater determinants constructed from an orthonormal one-particle basis set of size

*M*, acording to

The discrete basis allows for effective cancellation algorithms, such that for sufficiently large walker populations *N*_{w}, the fermion sign problem, an exponential increase in noise, can be controlled and (*i*)-FCIQMC converges to the FCI limit within stochastic errors, with a reduced scaling compared to traditional FCI (exact diagonalization).

The choice of one-particle basis set affects the nature of the *C*_{i} coefficients and may lead to a compact and sparse FCI representation of |ψ> that is more amenable to treatment with configuration-based methods. A natural measure for the sparsity of a wavefunction is the *L*_{1}-norm, *L*_{1} = ∑_{i} |*C*_{i}| and we therefore seek a representation of |ψ> in which the *L*_{1}-norm and level of complexity is small. For this purpose, we investigate two widely available sources of single-particle orbitals, restricted and unrestricted Hartree-Fock spin orbitals (RHF, UHF) which we also compare with restricted and unrestricted natural orbitals (RNO, UNO) which diagonalize the exact one-particle density matrix (γ_{q}^{p} = <Ψ|*a*_{p}^{† }*a*_{q}|Ψ>) and are known to give rapidly converging FCI expansions. Whereas restricted HF and NO orbitals restrict the spatial distributions ψ_{i}(**r**) to be equal for α and β channels, unrestricted spin orbitals, are characterized by relaxation of this constraint, ψ_{i}^{α} (**r**) ≠ ψ_{i}^{β} (**r**) [3].

** Results and discussions:** Counterintuitively, we observe that working with RHF orbitals in

*i*-FCIQMC leads to a |ψ> with a smaller

*L*

_{1}-norm (

*L*

_{1}= 723.6) as opposed to UHF spin orbitals (

*L*

_{1}= 1059.5).

** Ground states:** In order to shed light on the reasons behind this, we examine the

*i*-FCIQMC, RHF and UHF ground states (

**Tab. 1**). Like previous studies, we find that the exact

*i*-FCIQMC ground-state establishes an antiferromagnetic long-range order across the copper sites. |ψ

_{RHF}> and |ψ

_{UHF}> constitute a clear example of the symmetry dilemma [3]. While the UHF basis provides a physically closer single-determinant description of the antiferromagnetic ground-state, it breaks spin symmetry by separating α and β orbitals on two sublattices thereby leading to an inherent spin contamination. In contrast, the spin symmetry conserving RHF basis yields a qualitatively incorrect metallic ground state, but is still found to be a much more effective basis for correlated calculations within

*i*-FCIQMC.

**Tab. 1:**Ground-state energies

*E*

_{0 }(eV/hole), energy of the lowest-energy determinant

*E*

_{D }(eV/hole), percentage of correlation energy

*p*

_{corr}= (

*E*

_{0 }–

*E*

_{D})/(

*E*

_{exact }–

*E*

_{D}) (%) captured by |Ψ

_{CAS}>, average hole densities per atom <

*n*

_{at}> (holes/atom), staggered magnetization <

*M*

^{2}> = N

^{–1}∑

_{ij}(–1)

^{(xi+yi)+(xj+yj)}<Ψ

_{FCI}|

**S**

_{i}·

**S**

_{j}|Ψ

_{FCI}>, square magnitude of spin <

*S*

^{2}>. Errors in the previous digit are presented in parentheses. Local spin-spin correlation function <Ψ

_{FCI}|

**S**

_{1}·

**S**

_{j}|Ψ

_{FCI}> of the correlated

*i*-FCIQMC wavefunction |Ψ

_{FCI}> in the metallic RHF one-particle basis.

**Tab. 1:**Ground-state energies

*E*

_{0 }(eV/hole), energy of the lowest-energy determinant

*E*

_{D }(eV/hole), percentage of correlation energy

*p*

_{corr}= (

*E*

_{0 }–

*E*

_{D})/(

*E*

_{exact }–

*E*

_{D}) (%) captured by |Ψ

_{CAS}>, average hole densities per atom <

*n*

_{at}> (holes/atom), staggered magnetization <

*M*

^{2}> = N

^{–1}∑

_{ij}(–1)

^{(xi+yi)+(xj+yj)}<Ψ

_{FCI}|

**S**

_{i}·

**S**

_{j}|Ψ

_{FCI}>, square magnitude of spin <

*S*

^{2}>. Errors in the previous digit are presented in parentheses. Local spin-spin correlation function <Ψ

_{FCI}|

**S**

_{1}·

**S**

_{j}|Ψ

_{FCI}> of the correlated

*i*-FCIQMC wavefunction |Ψ

_{FCI}> in the metallic RHF one-particle basis.

** Orbital occupations numbers:** A clue with the difficulty introduced by the UHF basis can be obtained by considering the orbital occupation numbers γ

_{p}

^{p }in the four bases (

**Fig. 1**). Whilst for the RHF, RNO and UNO bases the γ

_{p}

^{p }decay roughly monotonically with mean-field orbital energy ε

_{p}(Fig. 1), a sharp increase is observed for the

*N*highest UHF virtuals, which are also far higher in energy than those of any other basis. These virtuals correspond to spin-flipped counterparts of the occupied UHF orbitals which introduces anti-boding character and splits the Hubbard bands far apart.

**Fig. 1:**The magnitude of coefficients |

*C*

_{i}| ≥ 0.0001 in the FCI expansion |Ψ> = ∑

_{i }

*C*

_{i }|

*D*

_{i}> shown against the respective determinant energy

*E*

_{i}for RHF (upper left), RNO (lower left), UHF (upper middle) and UNO (lower middle) basis sets. The |

*C*

_{i}coefficients of the full space |Ψ> are depicted in the lower panels while those of the (10,10)-CAS space |Ψ

_{CAS}> are shown in the top panels. The colours distinguish the

*x*-fold excitations (

*x*∈ {1,2,...,

*N*}) of the reference. The orbital occupation numbers γ

_{p}

^{p}(upper right) and mean-field orbital energies ε

_{i}(diagonal elements of mean-field generalized Fock matrix) (lower right) [3]. The numbers indicate degeneracies which are exact for HF and approximate for NO spin orbitals.

By breaking spin symmetry, UHF theory leads to a set of single-particle states characterized by localized spatial distribution ψ_{i}^{σ}(**r**) which strongly differ in their extent. In contrast, the metallic RHF orbitals are very delocalized and hence exhibit similar ψ_{i}(**r**). This facilitates correlation of the single-particle states, thereby favouring a more rapidly converging CI expansion in comparison to UHF orbitals. Additionally, the RHF orbital and determinant energies cover a smaller energy range (Fig. 1), a characteristic also shared by both NO bases. A quantitative measure of the configurational mixing present in the |ψ> representations is provided by the correlation entropy, *S*_{CE} = –(N)^{–1} ∑_{p} γ_{p}^{p} ln γ_{p}^{p}. With *S*_{CE} = 0.6421 for RNOs and *S*_{CE} = 0.6115 for UNOs, the entanglement in |ψ> is smallest in NO bases, followed by the RHF (*S*_{CE} = 0.7635) and UHF (*S*_{CE} = 0.8846) orbitals.

** Subspace diagonalizations:** The FCIQMC descriptions of |ψ> (Fig. 1) are all highly multiconfigurational with many single- to

*N*-fold particle-hole excitations of the reference with |

*C*

_{i}| decaying exponentially with determinant energy. In particular, in the UHF expansion a plethora of 10-fold excitations contribute to |ψ>. By contrast, the RHF, and more so RNO and UNO bases, are both sparser and strongly weighted towards the low particle-hole excitations, which are more amenable to accurate treatment of correlations via a compact set of explicit configurations.

Remarkably, |ψ_{FCI}> in the RHF basis can be well approximated by simple subspace diagonalizations in a (10,10)-CAS space. This is also the case for RNO and UNO spaces where |ψ_{CAS}> captures a majority of the respective correlation energy (Tab. 1) and basic structure of |ψ>. However, in the UHF space barely any correlation energy is captured by this subspace. This is a consequence of the fact that determinant weight is entirely absent from high particle-hole excitation when compared to |ψ> despite the fact that many of the significant high-excitations determinants are included in the CAS space. This suggests that orbitals outside the CAS space, especially the *N* highest-energy virtuals, are essential for establishing the basic structure of |ψ> in the UHF basis.

** Conclusions:** We have analysed the FCI wavefunction representation in different single-particle basis sets and their amenability to accurate correlation treatments using

*i*-FCIQMC and the strongly-correlated three-band Hubbard model. Counterintuitively, the effectiveness of single-particle basis sets for rapidly converging CI expansions is not necessarily paralleled by their ability to reproduce the physics of the system within a single-determinant description. Whilst the UHF determinant represents qualitatively the correct insulating antiferromagnet, imposing spin symmetry in the RHF basis gives an RHF determinant describing a qualitatively incorrect metal. Yet, in this basis the FCI representation of |ψ> is sparser, and converges rapidly with particle-hole excitations of the reference. Our results therefore suggest that with an appropriate single-particle description, it may be possible to describe the many-electron wavefunction of strongly-correlated materials based on single-reference quantum chemical methodologies [2,3], which opens up a vast array of powerful many-body techniques for the study of such systems.