# Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Computations

_{70}or Mn

_{84}single-molecule magnets (with N = 70 or N = 84 s = 2 sites) feasible in terms of a $q=7$ division [23] but is beyond the scope of this work.

_{18}ring (studied by INS in [27]) suggests a treatment in terms of six equivalent clusters, each hosting three neighboring $s={\scriptscriptstyle \frac{5}{2}}$ sites, which represent three chemically inequivalent Fe

^{III}ions. Furthermore, Mn

_{70}and Mn

_{84}tori, the largest single-molecule magnets known to date, have repeat units of 14 Mn

^{III}ions (s = 2), but the pattern of isotropic couplings in the Heisenberg model suggests a partitioning into two inequivalent types of clusters with seven sites each [23].

## 3. Results and Discussion

#### 3.1. Symmetric Rings

#### 3.2. Honeycomb Lattice Fragments

#### 3.3. Polyhedra

- (a)
- Icosahedron

- (b)
- Truncated Tetrahedron

_{2}or A

_{1}for half-integer or integer s, respectively).

- (c)
- Truncated Icosahedron

^{3+}, Cr

^{3+}, Fe

^{3+}, truncated icosahedra were not yet synthetically realized as magnetic molecules. The respective Heisenberg model was nevertheless addressed in a few works [29,43,44,45,46,47,48,49], some of which were motivated by gaining an understanding of the properties of buckminsterfullerene C

_{60}. However, electronic-structure calculations have shown that the Heisenberg model is at best of qualitative value for C

_{60}because this molecule possesses $\ll 60$ unpaired electrons [50] (but note that there is no unique measure for this number). It was eventually concluded that C

_{60}displays no significant strong correlation [51]. Ab initio GHF calculations [50] on C

_{60}still reproduce the exotic three-dimensional spin-density pattern of the classical Heisenberg ground state [29]. Geometry optimization on the GHF level-of-theory yields a perfect ${I}_{h}$ structure for C

_{60}[50], but the spatial inversion ${C}_{i}$ is the only obvious self-consistent symmetry of the GHF solution. We specify the hidden icosahedral symmetry I in Figure 14, where classical spin vectors are plotted in a Schlegel diagram. A uniform rotation was applied such that spins in the central pentagon in Figure 14 lie in the geometrical plane of that pentagon, with the spin on the first site pointing in the negative x-direction. This spin configuration is left unchanged by a combined spin rotation (R) by 144° and a five-fold site permutation (P). In the electronic-structure problem, the site permutation corresponds to a spatial rotation. These operations are performed about an axis through the coordinate origin (the center of the truncated icosahedron) and the center of the central pentagon (marked ${5}_{\mathrm{PR}}$ in Figure 14).

_{60}are up to six-fold degenerate: molecular orbitals span irreducible representations [52] of the double group ${I}_{h}^{*}$.

- (d)
- Truncated icosidodecahedron

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Optimization of the Broken-Symmetry Reference

**t**comprising matrix elements ${t}_{lm}$ is of dimension $(Q\cdot M)\times (Q\cdot M)$ but consists of Q blocks ${h}_{i}$, where ${h}_{i}$ is the Hamiltonian of the i-th isolated cluster. We similarly define reduced two-body integrals ${[kn|lm]}_{p{p}^{\prime}}$ (for $p,{p}^{\prime}=1,2,...,q$) for the interaction of site p of an arbitrary cluster with site ${p}^{\prime}$ of another arbitrary cluster. These integrals are generally zero for most combinations $p,{p}^{\prime}$ because only specific site pairs interact. Local spin matrices ${s}_{p,\alpha}$ $(p=1,2,...,q;$ $\alpha =x,y,z)$ are of dimension $M\times M$. The number Z of nonzero entries in ${s}_{p,\alpha}$ is independent of p (for $s={\scriptscriptstyle \frac{1}{2}}$, $Z=M$ for all $\alpha $). For a given $\alpha $, there are thus ${Z}^{2}$ combinations of nonzero entries in ${s}_{p,\alpha}$ and ${s}_{{p}^{\prime},\alpha}$, each combination yielding a nonvanishing integral ${[kn|lm]}_{p{p}^{\prime}}$. The value of ${[kn|lm]}_{p{p}^{\prime}}$ is the product of the respective nonzero entries in ${s}_{p,\alpha}$ and ${s}_{{p}^{\prime},\alpha}$ (all couplings are assumed to have the same strength, $J=1$); (k, n) and (l, m) are the (row, column) indices of the nonzero entries in ${s}_{p,\alpha}$ and ${s}_{{p}^{\prime},\alpha}$, respectively.

**Z**.

**G**with elements ${G}_{vo}$ $(o=1)$, defined in Equation (A14):

**1**is $1\times 1$ because there is only one occupied MO per site. For spin projection, the grid weights $t(\mathrm{\Omega})$ (an Euler-angle triplet $\mathrm{\Omega}=(\alpha ,\beta ,\gamma )$ defines a grid point) are combined with Wigner D-matrix elements for all combinations of magnetic quantum numbers m and k [20]:

**Figure A1.**The cyclic point-group operation ${\widehat{C}}_{3}$ keeps the numbering of spins in the clusters intact (

**a**), but a vertical rotation causes internal permutations (

**b**).

**G**are passed to the fminunc function in Matlab. The separation into real and imaginary parts (Equation (A14)) is required for fminunc, which optimizes with respect to a set of real variables.

## Appendix B. Spin-Pair Correlation Functions (SPCFs)

## Appendix C. Reference Energies for Spin Rings

**Table A1.**Singlet and triplet energies and the gap $\mathrm{\Delta}{E}_{\mathrm{ST}}$ in $s={\scriptscriptstyle \frac{1}{2}}$ rings with N sites and cluster size q. SUHF predictions are compared to exact results (from Table V.1 in [32]).

N | ||||||
---|---|---|---|---|---|---|

q | 6 | 12 | 18 | 24 | 30 | |

2 | ${E}_{\mathrm{S}}$ | −2.6514 | −4.8770 | −7.1335 | −9.3914 | −11.6485 |

${E}_{\mathrm{T}}$ | −1.8956 | −4.3114 | −6.6414 | −8.9434 | −11.2237 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.756 | 0.566 | 0.492 | 0.448 | 0.425 | |

6 | ${E}_{\mathrm{S}}$ | −2.8028 | −5.3482 | −7.7332 | −10.2071 | −12.6941 |

${E}_{\mathrm{T}}$ | −2.1180 | −4.7874 | −7.3492 | −9.8772 | −12.3954 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.561 | 0.384 | 0.330 | 0.299 | |

Exact | ${E}_{\mathrm{S}}$ | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 |

Exact | $\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |

**Table A2.**Singlet and triplet energies and the gap $\mathrm{\Delta}{E}_{\mathrm{ST}}$ in $s={\scriptscriptstyle \frac{1}{2}}$ rings with N sites and cluster size q. SGHF predictions are compared to exact results (from Table V.1 in [32]).

N | ||||||
---|---|---|---|---|---|---|

q | 6 | 12 | 18 | 24 | 30 | |

2 | ${E}_{\mathrm{S}}$ | −2.8028 | −5.0625 | −7.3603 | −9.6589 | −11.9416 |

${E}_{\mathrm{T}}$ | −2.1180 | −4.5485 | −6.8696 | −9.1446 | −11.4453 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.514 | 0.491 | 0.514 | 0.496 | |

6 | ${E}_{\mathrm{S}}$ | −2.8028 | −5.3768 | −7.9641 | −10.4728 | −12.9231 |

${E}_{\mathrm{T}}$ | −2.1180 | −5.0090 | −7.6042 | −10.1411 | −12.6090 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.368 | 0.360 | 0.332 | 0.314 | |

Exact | ${E}_{\mathrm{S}}$ | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 |

Exact | $\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |

**Table A3.**Singlet and triplet energies and the gap $\mathrm{\Delta}{E}_{\mathrm{ST}}$ in $s={\scriptscriptstyle \frac{1}{2}}$ rings with N sites and cluster size q. ${D}_{Q}\mathrm{SGHF}$ predictions $(Q=N/q)$ are compared to exact results (from Table V.1 in [32]).

N | ||||||
---|---|---|---|---|---|---|

q | 6 | 12 | 18 | 24 | 30 | |

2 | ${E}_{\mathrm{S}}$ | −2.8028 | −5.3710 | −7.8905 | −10.3945 | −12.8677 |

${E}_{\mathrm{T}}$ | −2.1180 | −5.0104 | −7.5544 | −10.0287 | −12.4834 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.361 | 0.336 | 0.366 | 0.384 | |

6 | ${E}_{\mathrm{S}}$ | −2.8028 | −5.3874 | −8.0224 | −10.6501 | −13.2762 |

${E}_{\mathrm{T}}$ | −2.1180 | −5.0315 | −7.7782 | −10.4356 | −13.0870 | |

$\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.356 | 0.244 | 0.215 | 0.189 | |

Exact | ${E}_{\mathrm{S}}$ | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 |

Exact | $\mathrm{\Delta}{E}_{\mathrm{ST}}$ | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |

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**Figure 1.**Two cluster groupings in an $N=6$ ring. Dihedral symmetry is broken by dimerization $({D}_{6}\to {D}_{3})$ or trimerization $({D}_{6}\to {D}_{2})$. The respective reduced symmetry groups can be employed in cPHF.

**Figure 2.**Dimers of diametrically opposite sites conserve the symmetry of rings (group ${D}_{6}$ in this example). A cyclic permutation interchanges sites in the last cluster (cluster 3, yellow box).

**Figure 4.**Relative correlation energies p (Equation (9)) from cluster variants of UHF, SUHF, SGHF and ${D}_{Q}\mathrm{SGHF}$ $(Q=N/q)$ in antiferromagnetic spin rings with $N=6,12,18,24,30$ sites. Data points are connected by dotted (q = 2), dashed (q = 3) or solid lines (q = 6).

**Figure 5.**Relative singlet-triplet gaps from cluster variants of SUHF, SGHF and ${D}_{Q}\mathrm{SGHF}$ $(Q=N/q)$ in antiferromagnetic spin rings with $N=6,12,18,24,30$ sites. Data points are connected by dotted (cluster size q = 2), dashed (q = 3) or solid lines (q = 6).

**Figure 6.**Cluster groupings in triphenylene $(N=18)$ conserving the full ${D}_{3}$ symmetry. In (

**a**–

**c**), clusters are shown in different colors; in (

**d**,

**e**), all dimer clusters $(q=2)$ are shown in the same color for simplicity.

**Figure 7.**Cluster groupings in triphenylene (N = 24) with symmetries ${C}_{6}$ (

**a**), ${D}_{6}$ (

**b**,

**f**–

**h**), and ${D}_{3}$ (

**c**–

**e**). In (

**e**,

**f**), all dimers (q = 2) are shown in the same color for simplicity. In (

**g**), three clusters comprise two separate trimers each. In (

**h**), two clusters comprise three separate dimers, and six disconnected sites (red) constitute another cluster.

**Figure 8.**Cluster groupings ((

**a**), $q=6)$ and ((

**b**), $q=7)$ in hexabenzocoronene (N = 42) conserving the full ${D}_{6}$ symmetry.

**Figure 9.**Cluster groupings in hexa-cata-hexabenzocoronene (N = 48) have symmetries ${D}_{6}$ (

**a**,

**c**) or ${C}_{6}$ (

**b**). In (

**a**), six disconnected sites (pink) constitute one cluster.

**Figure 10.**Cluster groupings in kekulene (N = 48) with symmetries ${D}_{3}$ (

**a**), ${D}_{6}$ (

**b**,

**c**), and ${C}_{6}$ (

**d**). In (

**a**,

**b**), two clusters (grey and pink) each comprise three separate dimers (

**a**) or six separate sites (

**b**).

**Figure 11.**Icosahedron (

**a**), truncated tetrahedron (

**b**), truncated icosahedron (

**c**), and truncated icosidodecahedron (

**d**).

**Figure 12.**Three cluster groupings in the icosahedron conserve ${I}_{h}$ ((

**a**), $q=2$), ${D}_{2h}$ ((

**b**), $q=2$), or ${D}_{5d}$ symmetry ((

**c**), $q=6$). In the planar projections (Schlegel diagrams), interacting sites belonging to the same or to different clusters are connected by blue or pink lines, respectively. In (

**a**), the color of sites assigns them to one of six clusters. Sites forming symmetry-inequivalent pairs with site 1 are numbered in (

**a**).

**Figure 13.**Dimers (

**a**) and trimers (

**b**) respect the full ${T}_{d}$ symmetry of the truncated tetrahedron. For more details, see caption to Figure 12.

**Figure 14.**Classical three-dimensional spin configuration of the truncated icosahedron. Green spin vectors lie in the xy-plane (paper plane); red/blue vectors point in the negative/positive z-direction. The first vector on the central pentagon points in the (horizontal) negative x-direction. Symmetry elements (${3}_{\mathrm{P}}\times {3}_{\mathrm{R}}={3}_{\mathrm{PR}}$, and ${5}_{\mathrm{PR}}$, see main text for details) are defined with respect to an axis through the coordinate origin and the center of the respective pentagon or hexagon.

**Figure 16.**Cluster groupings compatible with ${I}_{h}$ symmetry in the truncated icosidodecahedron. The midpoints of the squares ((

**a**), $q=4$), hexagons ((

**b**), $q=6$), and decagons ((

**c**), q = 10) form an icosidodecahedron (Q = 30), dodecahedron (Q = 20), or icosahedron (Q = 12), respectively.

**Figure 17.**Numbering of centers forming inequivalent pairs with site 1 in the bicolorable (red, blue) truncated icosidodecahedron lattice. Sites without a number are white but still belong to one of the two sublattices.

**Figure 18.**Magnitude of SPCFs (left y-axis; correlations are positive/negative for pairs in same/different lattices) and bond distances from reference site 1 (right y-axis) in the truncated icosidodecahedron. The site numbering follows Figure 17.

**Table 1.**Mulliken labels

^{a}of ground states in sectors $S=0$ and $S=1$ of even N antiferromagnetic $s={\scriptscriptstyle \frac{1}{2}}$ rings in the reduced dihedral group ${D}_{Q}$ $(Q=N/q)$.

$\mathit{N}=4\mathit{n}+2$ | $\mathit{N}=4\mathit{n}$ | ||||||
---|---|---|---|---|---|---|---|

$\mathit{S}=0$ | $\mathit{S}=1$ | $\mathit{S}=0$ | $\mathit{S}=1$ | ||||

q even | q odd | q even | q odd | q even | q odd | q even | q odd |

A_{2} | B_{1} | A_{1} | A_{1} | A_{1} | A_{1} | A_{2} | B_{1} |

^{a}Label A(B) and subscript 1(2) respectively denote symmetry(antisymmetry) under the cyclic permutation ${\widehat{C}}_{Q}$ or the vertical ${\widehat{C}}_{2}$ operation that exchanges all sites pairwise.

**Table 2.**Ground-state energy estimates for honeycomb-lattice fragments from cluster variants of UHF, SGHF, PGSGHF, and PT2. The lowest energy for each grouping/method is given in bold type.

System | Grouping (q, Bonds) ^{a} | UHF | SGHF | $\mathbf{PGSGHF}\text{}(\mathrm{\Gamma})$ | PT2 | Exact |
---|---|---|---|---|---|---|

Triphenylene | a (3, 12) | −7.2753 | −8.2062 | −8.6556 (A_{2}) | −8.6445 | −8.7697 |

b (6, 15) | −7.5804 | −8.4865 | −8.7342 (A_{2}) | −7.8205 | ||

c (6, 18) | −8.4083 | −8.7556 | −8.7696 (A_{2}) | −8.6640 | ||

d (2, 9) | −7.0229 | −8.0255 | −8.5364 (A_{2}) | −8.2068 | ||

e (2, 9) | −6.9584 | −7.8672 | −8.4093 (A_{2}) | −8.1914 | ||

Coronene | a (4, 24) | −10.2764 | −11.2733 | −11.7966 (A) | −11.4628 | −11.9755 |

b (4, 18) | −9.7660 | −10.7447 | −11.6399 (A_{1}) | −11.3781 | ||

c (6, 21) | −10.5736 | −11.3457 | −11.8103 (A_{1}) | −11.3740 | ||

d (8, 24) | −10.8304 | −11.6961 | −11.9459 (A_{1}) | −11.4992 | ||

e (2, 12) | −9.5702 | −10.6676 | −11.2997 (A_{1}) | −11.3109 | ||

f (2, 12) | −9.6055 | −10.7084 | −11.6190 (A_{1}) | −11.3968 | ||

g (6, 18) | −9.9867 | −10.9182 | −11.6693 (A_{1}) | −11.1837 | ||

h (6, 12) | −9.4117 | −11.1230 | −11.8635 (A_{1}) | −11.1287 | ||

Hexabenzo- coronene | a (6, 42) | −19.8044 | −20.4297 | −21.0044 (B_{1}) | −20.7082 | – ^{b} |

b (7, 42) | −19.6337 | −20.4824 | −21.0786 (B_{1}) | −20.7724 | ||

Hexa-cata-hexabenzo-coronene | a (6, 42) | −21.5999 | −23.0983 | −23.8296 (A_{1}) | −23.1753 | – ^{b} |

b (8, 48) | −22.4271 | −23.3169 | −23.7980 (A) | −23.4848 | ||

c (8, 42) | −21.1057 | −21.9885 | −23.0109 (A_{1}) | −23.0836 | ||

Kekulene | a (6, 48) | −21.4983 | −22.2249 | −22.7387 (A_{1}) | −22.9403 | – ^{b} |

b (6, 36) | −19.8349 | −20.9791 | −22.4499 (A_{1}) | −22.2943 | ||

c (8, 48) | −21.4735 | −22.2083 | −23.0747 (A_{1}) | −22.9187 | ||

d (8, 48) | −22.1306 | −22.7459 | −23.3603 (A) | −23.2404 |

**Table 3.**SPCFs $\langle {\widehat{s}}_{i}\cdot {\widehat{s}}_{j}\rangle $ in coronene from PGSGHF, compared against exact results. Letters (a, b, c, d) identifying cluster groupings, and the site numbers (first column) are defined in Figure 7

^{a}.

i–j | ${{C}}_{6}SGHF(4)\left(\mathbf{a}\right)$ | ${\mathit{D}}_{6}SGHF(4)\left(\mathbf{b}\right)$ | ${\mathit{D}}_{3}SGHF(6)\left(\mathbf{c}\right)$ | ${\mathit{D}}_{3}SGHF(8)\left(\mathbf{d}\right)$ | Exact |
---|---|---|---|---|---|

1–2 | −0.33729 | −0.35059 | −0.39063 | −0.36850 | −0.35875 |

2–3 | −0.33729 | −0.35059 | −0.38706 | −0.33627 | −0.35875 |

1–4 | −0.40404 | −0.35322 | −0.30621 | −0.37926 | −0.37507 |

4–5 | −0.37034 | −0.40476 | −0.37196 | −0.37095 | −0.36665 |

5–6 | −0.54911 | −0.42665 | −0.52422 | −0.52685 | −0.52881 |

6–7 | −0.30533 | −0.40476 | −0.37196 | −0.37095 | −0.36665 |

7–8 | −0.37034 | −0.40476 | −0.41287 | −0.36483 | −0.36665 |

8–9 | −0.54911 | −0.42665 | −0.45278 | −0.52027 | −0.52881 |

(1–3) | 0.17744 | 0.19472 | 0.19503 | 0.16792 | 0.16641 |

(1–5) | 0.18265 | 0.20098 | 0.16410 | 0.17971 | 0.17661 |

(2–6) | 0.18009 | 0.20098 | 0.16410 | 0.17971 | 0.17661 |

(4–6) | 0.20777 | 0.18967 | 0.19643 | 0.19440 | 0.19291 |

(5–7) | 0.17515 | 0.18967 | 0.19643 | 0.19440 | 0.19291 |

^{a}SPCFs for all distinct NN pairs i–j (first column) are given, including pairs that are equivalent in the full symmetry group but inequivalent in some of the PGSGHF wave functions. The NNN set (pairs i–j in parentheses) is not complete, except for (b), which maintains the full ${D}_{6}$ symmetry. Bold type is used for SPCFs of pairs that belong to the same cluster.

**Table 4.**Comparison of ${I}_{h}\mathrm{SGHF}\left(2\right)$ predictions of SPCFs in the ground state of the icosahedron with ${\scriptscriptstyle \frac{1}{2}}\le s\le 2$ against exact values. Site numbers are defined in Figure 12a.

s | $\langle {\widehat{s}}_{1}\cdot {\widehat{s}}_{2}\rangle $ | $\langle {\widehat{s}}_{1}\cdot {\widehat{s}}_{3}\rangle $ | $\langle {\widehat{s}}_{1}\cdot {\widehat{s}}_{4}\rangle $ | |
---|---|---|---|---|

1/2 | Exact | −0.2063 | 0.0841 | −0.1397 |

PHF | −0.2063 | 0.0841 | −0.1397 | |

1 | Exact | −0.6187 | 0.3680 | −0.7463 |

PHF | −0.6187 | 0.3680 | −0.7464 | |

3/2 | Exact | −1.2580 | 0.9060 | −1.9899 |

PHF | −1.2580 | 0.9062 | −1.9910 | |

2 | Exact | −2.1237 | 1.6616 | −3.6897 |

PHF | −2.1236 | 1.6621 | −3.6926 |

**Table 5.**GHF and PHF estimates of ground-state energies of the antiferromagnetic icosahedron with ${\scriptscriptstyle \frac{1}{2}}\le s\le 2$ for two different $q=2$ groupings (Figure 12a,b) with ${I}_{h}$ or ${D}_{2h}$ symmetry.

s | Grouping | GHF | SGHF | PGSGHF | $\mathbf{Exact}\text{}(\mathrm{\Gamma})$ |
---|---|---|---|---|---|

1/2 | ${D}_{2h}$ | −4.5000 | −5.3224 | −6.1717 | −6.1879 (Au) |

${I}_{h}$ | −3.3541 | −5.7644 | −6.1879 ^{a} | ||

1 | ${D}_{2h}$ | −14.3025 | −17.4565 | −18.1678 | −18.5611 (Ag) |

${I}_{h}$ | −13.4164 | −18.2225 | −18.5609 | ||

3/2 | ${D}_{2h}$ | −31.4256 | −36.2633 | −37.3073 | −37.7412 (Au) |

${I}_{h}$ | −30.1869 | −37.3842 | −37.7396 | ||

2 | ${D}_{2h}$ | −55.2658 | −61.7751 | −63.1481 | −63.7104 (Ag) |

${I}_{h}$ | −53.6656 | −63.2529 | −63.7075 |

^{a}Exact ground-state energy within numerical double precision.

**Table 6.**GHF and PHF estimates of ground-state energies of the antiferromagnetic ${\scriptscriptstyle \frac{1}{2}}\le s\le 2$ truncated tetrahedron for two different groupings (Figure 13).

s | q | GHF | SGHF | T_{d}SGHF | $\mathbf{Exact}\text{}(\mathrm{\Gamma})$ |
---|---|---|---|---|---|

1/2 | 2 | −4.5000 | −5.2700 | −5.7009 ^{a} | −5.7009 (A _{2}) |

3 | −3.8881 | −4.8147 | −5.7009 ^{a} | ||

1 | 2 | −14.0173 | −16.0342 | −17.1649 | −17.1955 (A _{1}) |

3 | −13.8696 | −15.7195 | −17.1775 | ||

3/2 | 2 | −29.7756 | −32.8938 | −34.4456 | −34.6402 (A _{2}) |

3 | −29.6977 | −32.5614 | −34.4796 | ||

2 | 2 | −51.5616 | −55.7815 | −57.7827 | −58.1140 (A _{1}) |

3 | −51.5327 | −55.3924 | −57.8181 |

^{a}Exact ground-state energy within numerical double precision.

**Table 7.**cGHF and cPHF estimates of ground-state energies of the $s={\scriptscriptstyle \frac{1}{2}}$ truncated icosahedron.

q | GHF | SGHF | PGSGHF ^{a} |
---|---|---|---|

2 | −24.2705 | −25.5486 | −27.8429 |

5 | −25.8525 | −26.6072 | −28.5653 |

10 | −28.6199 ^{b} | −29.2195 | −29.9842 |

^{a}Projection onto $S=0$, $\mathrm{\Gamma}={\mathrm{A}}_{\mathrm{g}}$.

^{b}All clusters assume their local singlet ground state.

q = 2 | q = 5 | ||||
---|---|---|---|---|---|

E | −25.5486 | −27.8429 | −26.6072 | −28.5653 | −30.69 |

j | SGHF | ${I}_{h}\mathrm{SGHF}$ | SGHF | ${I}_{h}\mathrm{SGHF}$ | VMC |

2 | −0.562 | −0.610 | −0.186 | −0.277 | −0.529 |

3 | −0.145 | −0.159 | −0.351 | −0.337 | −0.247 |

4 | 0.051 | 0.051 | 0.076 | 0.073 | 0.030 |

5 | 0.136 | 0.137 | 0.142 | 0.154 | 0.141 |

6 | −0.145 | −0.154 | −0.151 | −0.154 | −0.142 |

7 | −0.056 | −0.054 | −0.059 | −0.061 | −0.023 |

8 | −0.090 | −0.080 | −0.094 | −0.093 | −0.038 |

9 | 0.084 | 0.070 | 0.087 | 0.083 | 0.031 |

10 | −0.002 | 0.001 | −0.003 | −0.001 | 0.001 |

11 | 0.051 | 0.049 | 0.052 | 0.051 | 0.027 |

12 | −0.090 | −0.072 | −0.094 | −0.088 | −0.026 |

13 | −0.090 | −0.042 | −0.094 | −0.084 | −0.002 |

14 | 0.051 | 0.017 | 0.052 | 0.046 | −0.001 |

15 | −0.002 | −0.002 | −0.003 | −0.003 | −0.004 |

16 | 0.084 | 0.037 | 0.087 | 0.078 | 0.001 |

17 | −0.090 | −0.036 | −0.094 | −0.081 | 0.002 |

18 | −0.056 | −0.018 | −0.059 | −0.051 | 0.013 |

19 | −0.145 | −0.042 | −0.151 | −0.129 | 0.000 |

20 | 0.136 | 0.039 | 0.142 | 0.124 | −0.002 |

21 | 0.051 | 0.016 | 0.053 | 0.046 | −0.030 |

22 | −0.145 | −0.040 | −0.150 | −0.128 | 0.007 |

23 | −0.179 | −0.046 | −0.186 | −0.158 | 0.016 |

24 | 0.168 | 0.044 | 0.176 | 0.152 | −0.008 |

**Table 9.**Variational estimates from GHF; SGHF; ${I}_{h}\mathrm{SGHF}$ (projection onto $S=0$, $\mathrm{\Gamma}={\mathrm{A}}_{\mathrm{g}}$), a singlet-product on fused hexagons $(q=10)$; and PT2 for the ground state of the truncated icosahedron with ${\scriptscriptstyle \frac{1}{2}}\le s\le 2$.

s | GHF (2) | SGHF (2) | I_{h}SGHF (2) | $\mathit{q}=10,\text{}{\mathit{s}}_{\mathit{i}}=0$ | PT2 (1) | PT2 (2) |
---|---|---|---|---|---|---|

1/2 | −24.2705 | −25.5486 | −27.8429 | −28.6199 | −31.0543 | −29.1216 |

1 | −85.6371 | −87.7764 | −90.1147 | −89.8943 | −96.7113 | −96.9910 |

3/2 | −186.6961 | −189.5706 | −192.4293 | −183.7630 | −202.2428 | −203.7112 |

2 | −327.0802 | −330.4438 | −343.1173 | −310.5941 | −347.1526 | −349.5852 |

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**MDPI and ACS Style**

Ghassemi Tabrizi, S.; Jiménez-Hoyos, C.A.
Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach. *Condens. Matter* **2023**, *8*, 18.
https://doi.org/10.3390/condmat8010018

**AMA Style**

Ghassemi Tabrizi S, Jiménez-Hoyos CA.
Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach. *Condensed Matter*. 2023; 8(1):18.
https://doi.org/10.3390/condmat8010018

**Chicago/Turabian Style**

Ghassemi Tabrizi, Shadan, and Carlos A. Jiménez-Hoyos.
2023. "Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach" *Condensed Matter* 8, no. 1: 18.
https://doi.org/10.3390/condmat8010018