Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach
Abstract
:1. Introduction
2. Theory and Computations
3. Results and Discussion
3.1. Symmetric Rings
3.2. Honeycomb Lattice Fragments
3.3. Polyhedra
- (a)
- Icosahedron
- (b)
- Truncated Tetrahedron
- (c)
- Truncated Icosahedron
- (d)
- Truncated icosidodecahedron
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Optimization of the Broken-Symmetry Reference
Appendix B. Spin-Pair Correlation Functions (SPCFs)
Appendix C. Reference Energies for Spin Rings
N | ||||||
---|---|---|---|---|---|---|
q | 6 | 12 | 18 | 24 | 30 | |
2 | −2.6514 | −4.8770 | −7.1335 | −9.3914 | −11.6485 | |
−1.8956 | −4.3114 | −6.6414 | −8.9434 | −11.2237 | ||
0.756 | 0.566 | 0.492 | 0.448 | 0.425 | ||
6 | −2.8028 | −5.3482 | −7.7332 | −10.2071 | −12.6941 | |
−2.1180 | −4.7874 | −7.3492 | −9.8772 | −12.3954 | ||
0.685 | 0.561 | 0.384 | 0.330 | 0.299 | ||
Exact | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 | |
Exact | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |
N | ||||||
---|---|---|---|---|---|---|
q | 6 | 12 | 18 | 24 | 30 | |
2 | −2.8028 | −5.0625 | −7.3603 | −9.6589 | −11.9416 | |
−2.1180 | −4.5485 | −6.8696 | −9.1446 | −11.4453 | ||
0.685 | 0.514 | 0.491 | 0.514 | 0.496 | ||
6 | −2.8028 | −5.3768 | −7.9641 | −10.4728 | −12.9231 | |
−2.1180 | −5.0090 | −7.6042 | −10.1411 | −12.6090 | ||
0.685 | 0.368 | 0.360 | 0.332 | 0.314 | ||
Exact | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 | |
Exact | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |
N | ||||||
---|---|---|---|---|---|---|
q | 6 | 12 | 18 | 24 | 30 | |
2 | −2.8028 | −5.3710 | −7.8905 | −10.3945 | −12.8677 | |
−2.1180 | −5.0104 | −7.5544 | −10.0287 | −12.4834 | ||
0.685 | 0.361 | 0.336 | 0.366 | 0.384 | ||
6 | −2.8028 | −5.3874 | −8.0224 | −10.6501 | −13.2762 | |
−2.1180 | −5.0315 | −7.7782 | −10.4356 | −13.0870 | ||
0.685 | 0.356 | 0.244 | 0.215 | 0.189 | ||
Exact | −2.803 | −5.387 | −8.023 | −10.670 | −13.322 | |
Exact | 0.685 | 0.356 | 0.241 | 0.183 | 0.147 |
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q even | q odd | q even | q odd | q even | q odd | q even | q odd |
A2 | B1 | A1 | A1 | A1 | A1 | A2 | B1 |
System | Grouping (q, Bonds) a | UHF | SGHF | PT2 | Exact | |
---|---|---|---|---|---|---|
Triphenylene | a (3, 12) | −7.2753 | −8.2062 | −8.6556 (A2) | −8.6445 | −8.7697 |
b (6, 15) | −7.5804 | −8.4865 | −8.7342 (A2) | −7.8205 | ||
c (6, 18) | −8.4083 | −8.7556 | −8.7696 (A2) | −8.6640 | ||
d (2, 9) | −7.0229 | −8.0255 | −8.5364 (A2) | −8.2068 | ||
e (2, 9) | −6.9584 | −7.8672 | −8.4093 (A2) | −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 (A1) | −11.3781 | ||
c (6, 21) | −10.5736 | −11.3457 | −11.8103 (A1) | −11.3740 | ||
d (8, 24) | −10.8304 | −11.6961 | −11.9459 (A1) | −11.4992 | ||
e (2, 12) | −9.5702 | −10.6676 | −11.2997 (A1) | −11.3109 | ||
f (2, 12) | −9.6055 | −10.7084 | −11.6190 (A1) | −11.3968 | ||
g (6, 18) | −9.9867 | −10.9182 | −11.6693 (A1) | −11.1837 | ||
h (6, 12) | −9.4117 | −11.1230 | −11.8635 (A1) | −11.1287 | ||
Hexabenzo- coronene | a (6, 42) | −19.8044 | −20.4297 | −21.0044 (B1) | −20.7082 | – b |
b (7, 42) | −19.6337 | −20.4824 | −21.0786 (B1) | −20.7724 | ||
Hexa-cata-hexabenzo-coronene | a (6, 42) | −21.5999 | −23.0983 | −23.8296 (A1) | −23.1753 | – b |
b (8, 48) | −22.4271 | −23.3169 | −23.7980 (A) | −23.4848 | ||
c (8, 42) | −21.1057 | −21.9885 | −23.0109 (A1) | −23.0836 | ||
Kekulene | a (6, 48) | −21.4983 | −22.2249 | −22.7387 (A1) | −22.9403 | – b |
b (6, 36) | −19.8349 | −20.9791 | −22.4499 (A1) | −22.2943 | ||
c (8, 48) | −21.4735 | −22.2083 | −23.0747 (A1) | −22.9187 | ||
d (8, 48) | −22.1306 | −22.7459 | −23.3603 (A) | −23.2404 |
i–j | 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 |
s | ||||
---|---|---|---|---|
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 |
s | Grouping | GHF | SGHF | PGSGHF | |
---|---|---|---|---|---|
1/2 | −4.5000 | −5.3224 | −6.1717 | −6.1879 (Au) | |
−3.3541 | −5.7644 | −6.1879 a | |||
1 | −14.3025 | −17.4565 | −18.1678 | −18.5611 (Ag) | |
−13.4164 | −18.2225 | −18.5609 | |||
3/2 | −31.4256 | −36.2633 | −37.3073 | −37.7412 (Au) | |
−30.1869 | −37.3842 | −37.7396 | |||
2 | −55.2658 | −61.7751 | −63.1481 | −63.7104 (Ag) | |
−53.6656 | −63.2529 | −63.7075 |
s | q | GHF | SGHF | TdSGHF | |
---|---|---|---|---|---|
1/2 | 2 | −4.5000 | −5.2700 | −5.7009 a | −5.7009 (A2) |
3 | −3.8881 | −4.8147 | −5.7009 a | ||
1 | 2 | −14.0173 | −16.0342 | −17.1649 | −17.1955 (A1) |
3 | −13.8696 | −15.7195 | −17.1775 | ||
3/2 | 2 | −29.7756 | −32.8938 | −34.4456 | −34.6402 (A2) |
3 | −29.6977 | −32.5614 | −34.4796 | ||
2 | 2 | −51.5616 | −55.7815 | −57.7827 | −58.1140 (A1) |
3 | −51.5327 | −55.3924 | −57.8181 |
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 |
q = 2 | q = 5 | ||||
---|---|---|---|---|---|
E | −25.5486 | −27.8429 | −26.6072 | −28.5653 | −30.69 |
j | SGHF | 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 |
s | GHF (2) | SGHF (2) | IhSGHF (2) | 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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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
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 StyleGhassemi 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
APA StyleGhassemi Tabrizi, S., & Jiménez-Hoyos, C. A. (2023). Ground States of Heisenberg Spin Clusters from a Cluster-Based Projected Hartree–Fock Approach. Condensed Matter, 8(1), 18. https://doi.org/10.3390/condmat8010018