1 Ground states of Heisenberg spin clusters from a cluster -based p rojected Hartree -Fock approach

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Ground states of Heisenberg spin clusters from

a cluster-based projected Hartree-Fock approach

Shadan Ghassemi Tabrizi1,a,* and Carlos A. Jiménez-Hoyos1,†

1Department of Chemistry, Wesleyan University, Middletown, CT 06459, USA

aPresent address: Department of Chemistry, University of Potsdam,

Karl-Liebknecht-Str. 24-25, D-14476, Potsdam-Golm, Germany

*shadan_ghassemi@yahoo.com

†cjimenezhoyo@wesleyan.edu

Abstract. Recent work on approximating ground states of Heisenberg spin clusters by

projected Hartree-Fock theory (PHF) is extended to a cluster-based ansatz (cPHF). Whereas

PHF variationally optimizes a site-spin product state for the restoration of spin- and point-

group symmetry, cPHF groups sites into discrete clusters and uses a cluster-product state as

the broken-symmetry reference. Intracluster correlation is thus already included at the mean-

field level and intercluster correlation is introduced through symmetry projection. Variants of

cPHF differing in the broken and restored symmetries are evaluated for ground states and

singlet-triplet gaps of antiferromagnetic spin rings for various cluster sizes, where cPHF in

general affords a significant improvement over ordinary PHF, although the division into

clusters lowers the cyclical symmetry. On the other hand, certain two- or three-dimensional

spin arrangements permit cluster groupings compatible with the full spatial symmetry. We

accordingly demonstrate that cPHF yields approximate ground states with correct spin and

point-group quantum numbers for honeycomb lattice fragments and symmetric polyhedra.

1. Introduction

The calculation of magnetic properties of exchange-coupled spin clusters, e.g., molecules

with multiple open-shell transition-metal centers bridged by diamagnetic ligands [1,2], from

the Heisenberg model,

ˆˆˆ

ij i j



=

ss

, usually relies on approximations, because exact

diagonalization (ED) is only feasible for small systems. The ground state and perhaps a few

excited states are needed to interpret electron-paramagnetic resonance (EPR) or inelastic

neutron scattering (INS) spectra, or to assess other low-temperature properties [3]. The

density matrix renormalization group (DMRG [4]) is the most important variational method

for ground states of one-dimensional (1D) systems (rings or chains), but is less suitable for 2D

coupling topologies. The scarcity of computationally affordable and easily applicable

alternatives motivated our recent exploration of projected Hartree-Fock theory (PHF [5]) for

ground states of Heisenberg spin clusters [6]. PHF can be used in a black-box manner and has

a mean-field (HF) scaling, with a prefactor depending on the size of the symmetry-projection

grid. In finite spin systems, PHF restores spin (S) and point-group (PG) symmetry from a

general product state. For a collection of

sites, this broken-symmetry reference state is

simply a three-dimensional spin configuration [6]. PHF yields rather accurate ground-state

wave functions for symmetric rings with a moderate number of sites N and large local spin s

and predicts reasonably accurate singlet-triplet gaps. Limitations become evident for larger

rings, where the accuracy decreases [6]. PHF indeed recovers zero correlation energy per site

in the thermodynamic limit

N→

. In other words, the method is not size extensive [7]. To

ameliorate this problem and enable a more accurate treatment of larger systems by variational

symmetry-projection methods, one could either adopt a multi-component ansatz, where the

broken-symmetry reference is a linear combination of non-orthogonal mean-field states [8], or

work with a cluster basis that grants more flexibility than ordinary PHF, while still optimizing

just a single reference. We pursue the latter option, which we call cPHF. Note however that

both strategies could be combined into a multi-component cPHF ansatz, which may be

pursued in future work. For other correlated spin-cluster approaches (coupled-cluster and

many-body perturbation theory) and for further literature on related methods, see, e.g.,

Ref. [9].

2. Theory and Computations

PHF optimizes a broken-symmetry mean-field state



for the application of a symmetry

projector

[5]. In cPHF,



is a product of individual cluster states



 = 



, (1)

where Q is the total number of clusters. As an example, for a cluster comprising two

sites, the structure of



is given in Eq. (2).

,1 ,2 ,3 ,4i i i i i

c c c c =  +  +  + 

(2)

The



are independently optimized to minimize the variational energy, Eq. (3), of the

projected state

P = 

ˆ ˆ ˆ

H HP

   

 

. (3)

The site-permutation invariance [10] of spin Hamiltonians representing systems with spatial

symmetry (rings, symmetric polyhedra, etc.) is here referred to as point-group (PG)

symmetry. Each level is thus characterized by its total spin S and its PG-label



. To recover a

substantial fraction of the correlation energy for all but the smallest systems, it is mandatory

to combine S- with PG-projection in PHF [6,11]. In the PG-projector



of Eq. (4) (we

consider only one-dimensional irreducible representations



), h is the order of the group,

()

g



is the character of group element g, and

ˆg

is the respective symmetry operation [12].

ˆˆ

()

P g R





=

(4)

Multidimensional irreducible representations become relevant for projection onto

0S

sectors. The projector

ˆS

for spin S and magnetic quantum number m (the

ˆz

eigenvalue) is

expanded in terms of transfer operators

ˆS

ˆˆ

S S S

m m k mk

P f P =  = 



, (5)

which are conveniently parameterized by Euler angles [13],

ˆˆ

ˆsin( ) ( , , )

i S i S

mk mk

P d d d D e e e





      



−

−−

=

. (6)

For a given



, the coefficients

[Eq. (5)] correspond to the lowest-energy solution of the

generalized eigenvalue problem for the Hamiltonian

in the non-orthogonal basis spanned

 

ˆS

P

, 1,...,k S S S= − − + +

[11]. For combined S- and PG-projection, the projector is

a product,

ˆ ˆ ˆ

P P P



(spin rotations commute with site permutations). In the trivial case where

the projector is the identity,

ˆ1P=

, that is, if no symmetry projection is performed, cPHF is

equivalent to cHF, also called cluster mean-field theory [9,14]. If each cluster comprises just

one site, cPHF becomes equivalent to PHF, specifically, the “single fermion” variety of PHF

presented in Ref. [6]. Finally note that cPHF trivially yields the exact ground state in the

chosen symmetry sector

( , )S

if all sites are contained in a single cluster or if there are no

couplings between clusters.

In quantum-chemical terminology,



is of generalized HF type (GHF [15]) if it

completely breaks spin symmetry. An unrestricted HF (UHF) state also breaks total spin

symmetry (that is,



is not an eigenfunction of

), but conserves

ˆz

. In a UHF-type

reference, each cluster has a defined z-projection

. Different clusters may have different

, which add up to the total

ˆz

eigenvalue,

mm=

. Compared to complete spin-

symmetry breaking in GHF, the number of variational parameters is reduced in UHF. As an

example, a general

state of an

dimer is a superposition of only two basis states,

Eq. (7),

,1 ,2i i i

cc =  + 

. (7)

PHF variants that restore S- or PG-symmetry from a GHF- or UHF-type reference, are

called SGHF, PGSUHF, etc. In cPHF, the cluster size q may be appended, e.g., SGHF(2)

denotes a cluster-based SGHF calculation with dimers. For a given grouping, the lowest

variational energy is obtained when working with the largest symmetry group (PGSGHF). We

do not include complex-conjugation symmetry [5,16] in the cPHF scheme, because this

involves a more complicated formalism [6,17,18] and has comparatively small effects for

Heisenberg systems [6].

Self-consistent field (SCF [5,6,18,19]) and gradient-based optimization (Refs. [11,20], and

references cited therein) are two different strategies for the optimization of



. In the SCF

approach, the local cluster states



result from successively building and diagonalizing an

effective Fock matrix for each cluster. We found that reaching SCF convergence is often

challenging in cPHF and therefore prefer gradient-based optimization, where each



parameterized in terms of a Thouless rotation from an initial guess



. Details are provided

in the Supplemental Information (SI). With q sites of spin s, the number of real variational

parameters that define a general Thouless rotation for a single cluster is

var 2[(2 1) 1]

Ns= + −

leading to a total of

var 2 [(2 1) 1]

N Q s=  + −

for Q clusters (not counting the

coefficients

for S > 0, cf. Eq. (5)). Note however that the Thouless parameterization, though convenient, is

slightly redundant [6] with respect to S-projection from a GHF-type reference



, because

global spin rotations as well as certain gauge transformations of



leave the spin-projected

state unchanged [21].

The local cluster basis of dimension

(2 1)q

would have to be truncated for large clusters,

e.g., by considering a limited number of lowest levels of the intracluster Hamiltonian. As an

example, such a scheme could make a treatment of the Mn70 or Mn84 single-molecule magnets

(with N = 70 or N = 84 s = 2 sites) feasible in terms of a

7q=

division [22] but is beyond the

scope of this work.

As recommended previously [19], we discretized transfer operators, Eq. (6), with a

combined Lebedev-Laikov [23] and Trapezoid integration grid. For S-projection from a UHF-

type reference, the evaluation of integrals over Euler angles



and



is trivial [24] and

integration over



employs a Gauss-Legendre grid. A computational parallelization of the

summation over the grid is trivial [19]. The quality of S-projection can be checked by

computing

S

from a sum of spin-pair correlation functions (SPCFs), Eq. (8),

ˆˆˆ

( 1) 2 ij

Ns s



= + + 



S s s

. (8)

We ensured that

deviates by

10−



from the ideal value of

( 1)SS+

. Details on the

calculation of SPCFs are provided in SI.

Figure 1 illustrates that cluster formations are in general not fully compatible with spatial

symmetry, meaning that working with the cyclic

group (or the dihedral group

that

additionally includes vertical

axes) of spin rings with N sites would involve complicated

transformations between different cluster bases. A division in terms of q neighbors thus

reduces the cyclical symmetry of rings according to

/N N q

CC→

/N N q

DD→

. For example,

for

2q=

, sectors

and

( )mod

kN+

of group

[the crystal momentum k indicates the

eigenvalue

exp( 2 / )i k N



−

of the cyclic permutation

ˆN

] belong to the same sector of group

/2N

. Thus, a

0k=

state in

/2N

is generally a mixture of

0k=

(Mulliken label

A=

) and

( B)=

. This however does not imply that cPHF wave functions will

significantly break symmetry with respect to the full point group (see Results section).

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1GroundstatesofHeisenbergspinclustersfromacluster-basedprojectedHartree-FockapproachShadanGhassemiTabrizi1,a,*andCarlosA.Jiménez-Hoyos1,†1DepartmentofChemistry,WesleyanUniversity,Middletown,CT06459,USAaPresentaddress:DepartmentofChemistry,UniversityofPotsdam,Karl-Liebknecht-Str.24-25,D-14476,Potsdam...

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1 Ground states of Heisenberg spin clusters from a cluster -based p rojected Hartree -Fock approach

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