7. Constraining the WF center
The matrix manifold based view point of Wannierization provides a transparent way to apply constraint on WF centers, during both disentanglement and maximal localization. Instead of localizing the MV spread functional, we optimize
\[\Omega_r = \Omega + \lambda(\langle \bm{r} \rangle - \bm{r}_0)^{2},\]
where $\lambda$ is a Lagrange multiplier, and $r_0$ is the desired center.
In this tutorial, we will start from the s, p initial projections of silicon valence + conduction bands, add WF center penalty, to force the WFs to be centered at the bond centers, i.e., bonding and anti-bonding orbitals.
Outline
- construct a
Model, by reading thewin,amn,mmn, andeigfiles - disentangle, without WF center penalty
- disentangle, with WF center penalty
This is a HTML version of the tutorial, you can download corresponding
- Jupyter notebook:
tutorial.ipynb - Julia script:
tutorial.jl
Preparation
Load the package
using WannierModel generation
We will use the read_w90 function to read the win, amn, mmn, and eig files, and construct a Model that abstracts the calculation
model = read_w90("si2")lattice: Å
a1: 0.00000 2.71527 2.71527
a2: 2.71527 0.00000 2.71527
a3: 2.71527 2.71527 0.00000
atoms: fractional
Si: 0.00000 0.00000 0.00000
Si: 0.25000 0.25000 0.25000
n_bands: 16
n_wann : 8
kgrid : 4 4 4
n_kpts : 64
n_bvecs: 8
b-vectors:
[bx, by, bz] / Å⁻¹ weight
1 0.28925 -0.28925 0.28925 1.49401
2 -0.28925 -0.28925 -0.28925 1.49401
3 -0.28925 0.28925 -0.28925 1.49401
4 -0.28925 -0.28925 0.28925 1.49401
5 0.28925 0.28925 -0.28925 1.49401
6 -0.28925 0.28925 0.28925 1.49401
7 0.28925 -0.28925 -0.28925 1.49401
8 0.28925 0.28925 0.28925 1.49401Disentanglement
First let's disentangle the valence + conduction manifold, without WF center penalty
U = disentangle(model);[ Info: Initial spread
WF center [rx, ry, rz]/Šspread/Ų
1 -0.00000 0.00000 0.00000 1.33119
2 0.00001 0.00011 -0.00000 1.92293
3 -0.00000 0.00011 0.00011 1.92297
4 0.00001 -0.00000 0.00011 1.92303
5 1.35763 1.35763 1.35763 1.33119
6 1.35762 1.35752 1.35763 1.92293
7 1.35763 1.35752 1.35752 1.92297
8 1.35762 1.35763 1.35752 1.92303
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 9.69628
Ω̃ = 4.50396
ΩOD = 4.49427
ΩD = 0.00968
Ω = 14.20024
[ Info: Initial spread (with states freezed)
WF center [rx, ry, rz]/Šspread/Ų
1 0.00000 0.00000 0.00000 1.79953
2 0.00000 -0.00000 -0.00000 2.44042
3 0.00000 -0.00000 0.00000 2.44042
4 -0.00000 -0.00000 -0.00000 2.44042
5 1.35763 1.35763 1.35763 1.79953
6 1.35763 1.35763 1.35763 2.44042
7 1.35763 1.35763 1.35763 2.44042
8 1.35763 1.35763 1.35763 2.44042
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 12.82012
Ω̃ = 5.42148
ΩOD = 5.34421
ΩD = 0.07726
Ω = 18.24160
Iter Function value Gradient norm
0 1.824160e+01 6.135873e-01
* time: 0.004395961761474609
1 1.609863e+01 8.063143e-02
* time: 0.039566993713378906
2 1.571861e+01 4.479970e-02
* time: 0.06500411033630371
3 1.559753e+01 3.171032e-02
* time: 0.18116402626037598
4 1.551187e+01 1.252183e-02
* time: 0.2165989875793457
5 1.548086e+01 8.310836e-03
* time: 0.2519509792327881
6 1.547552e+01 4.120118e-03
* time: 0.2770700454711914
7 1.547423e+01 1.505832e-03
* time: 0.31289196014404297
8 1.547405e+01 5.299607e-04
* time: 0.34834909439086914
9 1.547403e+01 4.244924e-04
* time: 0.3841531276702881
10 1.547402e+01 2.528288e-04
* time: 0.4096529483795166
11 1.547402e+01 8.185063e-05
* time: 0.509821891784668
12 1.547402e+01 4.605264e-05
* time: 0.5460410118103027
[ Info: Final spread
WF center [rx, ry, rz]/Šspread/Ų
1 0.00000 0.00000 0.00000 1.44670
2 0.00000 0.00000 -0.00000 2.09677
3 -0.00000 -0.00000 0.00000 2.09677
4 -0.00000 -0.00000 -0.00000 2.09677
5 1.35763 1.35763 1.35763 1.44670
6 1.35763 1.35763 1.35763 2.09677
7 1.35763 1.35763 1.35763 2.09677
8 1.35763 1.35763 1.35763 2.09677
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 10.05117
Ω̃ = 5.42285
ΩOD = 5.36573
ΩD = 0.05711
Ω = 15.47402Now we arrive at s, p WFs centered at atom centers,
omega(model, U) WF center [rx, ry, rz]/Šspread/Ų
1 0.00000 0.00000 0.00000 1.44670
2 0.00000 0.00000 -0.00000 2.09677
3 -0.00000 -0.00000 0.00000 2.09677
4 -0.00000 -0.00000 -0.00000 2.09677
5 1.35763 1.35763 1.35763 1.44670
6 1.35763 1.35763 1.35763 2.09677
7 1.35763 1.35763 1.35763 2.09677
8 1.35763 1.35763 1.35763 2.09677
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 10.05117
Ω̃ = 5.42285
ΩOD = 5.36573
ΩD = 0.05711
Ω = 15.47402
Disentanglement with constraint
As has been done in the 1. Maximal localization of isolated manifold tutorial, we can use the find_nearests function to find the bond centers, which are
r₀ = [
0.67882 -0.67882 -0.67882 0.67882 0.67882 -0.67882 -0.67882 0.67882
-0.67882 -0.67882 0.67882 0.67882 -0.67882 -0.67882 0.67882 0.67882
-0.67882 0.67882 -0.67882 0.67882 -0.67882 0.67882 -0.67882 0.67882
]3×8 Matrix{Float64}:
0.67882 -0.67882 -0.67882 0.67882 0.67882 -0.67882 -0.67882 0.67882
-0.67882 -0.67882 0.67882 0.67882 -0.67882 -0.67882 0.67882 0.67882
-0.67882 0.67882 -0.67882 0.67882 -0.67882 0.67882 -0.67882 0.67882note each column is a target center.
We use 1.0 as the Lagrange multiplier,
λ = 1.01.0First, we calculate the initial spread with WF center penalty, now we need to use omega_center instead of omega,
Wannier.omega_center(model, r₀, λ) WF center [rx, ry, rz]/Å spread/Ų ω ωc ωt
1 -0.00000 0.00000 0.00000 1.33119 1.38239 2.71358
2 0.00001 0.00011 -0.00000 1.92293 1.38255 3.30549
3 -0.00000 0.00011 0.00011 1.92297 1.38239 3.30535
4 0.00001 -0.00000 0.00011 1.92303 1.38223 3.30526
5 1.35763 1.35763 1.35763 1.33119 8.75506 10.08625
6 1.35762 1.35752 1.35763 1.92293 8.75457 10.67751
7 1.35763 1.35752 1.35752 1.92297 8.75447 10.67744
8 1.35762 1.35763 1.35752 1.92303 1.38220 3.30523
Sum spread: Ωt = Ω + Ωc, Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
Ωt = 47.37610
Ωc = 33.17586
Ω = 14.20024
ΩI = 9.69628
ΩOD = 4.49427
ΩD = 0.00968
Ω̃ = 4.50396where the last three columns, ω, ωc, ωt are the MV spread, the penalty, and the total spread for each WF. As expected, there are large penalties on WFs due to there centers.
Then disentangle with disentangle_center function,
U1 = disentangle_center(model, r₀, λ);[ Info: Initial spread
WF center [rx, ry, rz]/Å spread/Ų ω ωc ωt
1 -0.00000 0.00000 0.00000 1.33119 1.38239 2.71358
2 0.00001 0.00011 -0.00000 1.92293 1.38255 3.30549
3 -0.00000 0.00011 0.00011 1.92297 1.38239 3.30535
4 0.00001 -0.00000 0.00011 1.92303 1.38223 3.30526
5 1.35763 1.35763 1.35763 1.33119 8.75506 10.08625
6 1.35762 1.35752 1.35763 1.92293 8.75457 10.67751
7 1.35763 1.35752 1.35752 1.92297 8.75447 10.67744
8 1.35762 1.35763 1.35752 1.92303 1.38220 3.30523
Sum spread: Ωt = Ω + Ωc, Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
Ωt = 47.37610
Ωc = 33.17586
Ω = 14.20024
ΩI = 9.69628
ΩOD = 4.49427
ΩD = 0.00968
Ω̃ = 4.50396
[ Info: Initial spread (with states freezed)
WF center [rx, ry, rz]/Å spread/Ų ω ωc ωt
1 0.00000 0.00000 0.00000 1.79953 1.38239 3.18192
2 0.00000 -0.00000 -0.00000 2.44042 1.38239 3.82281
3 0.00000 -0.00000 0.00000 2.44042 1.38239 3.82281
4 -0.00000 -0.00000 -0.00000 2.44042 1.38239 3.82281
5 1.35763 1.35763 1.35763 1.79953 8.75506 10.55459
6 1.35763 1.35763 1.35763 2.44042 8.75506 11.19549
7 1.35763 1.35763 1.35763 2.44042 8.75506 11.19549
8 1.35763 1.35763 1.35763 2.44042 1.38236 3.82278
Sum spread: Ωt = Ω + Ωc, Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
Ωt = 51.41871
Ωc = 33.17711
Ω = 18.24160
ΩI = 12.82012
ΩOD = 5.34421
ΩD = 0.07726
Ω̃ = 5.42148
Iter Function value Gradient norm
0 5.141871e+01 6.242602e-01
* time: 0.0044519901275634766
1 4.551119e+01 1.554105e-01
* time: 0.0418400764465332
2 4.177915e+01 1.775544e-01
* time: 0.0794670581817627
3 3.815452e+01 1.142556e-01
* time: 0.11723208427429199
4 3.527368e+01 8.740494e-02
* time: 0.15498709678649902
5 3.278466e+01 9.874413e-02
* time: 0.2578110694885254
6 3.027711e+01 1.128481e-01
* time: 0.29560399055480957
7 2.775961e+01 1.039067e-01
* time: 0.33287596702575684
8 2.522508e+01 1.084466e-01
* time: 0.37023210525512695
9 2.309199e+01 7.927632e-02
* time: 0.40793609619140625
10 2.137551e+01 7.746493e-02
* time: 0.44551897048950195
11 2.015442e+01 6.228587e-02
* time: 0.48319005966186523
12 1.910341e+01 6.407832e-02
* time: 0.520578145980835
13 1.826504e+01 6.152310e-02
* time: 0.6228151321411133
14 1.758323e+01 6.980949e-02
* time: 0.65997314453125
15 1.694671e+01 5.148097e-02
* time: 0.6972110271453857
16 1.643278e+01 3.637169e-02
* time: 0.7344491481781006
17 1.601539e+01 3.740355e-02
* time: 0.771726131439209
18 1.570069e+01 3.790137e-02
* time: 0.8090739250183105
19 1.547133e+01 2.547564e-02
* time: 0.8465070724487305
20 1.529566e+01 2.693237e-02
* time: 0.9495749473571777
21 1.516174e+01 2.033466e-02
* time: 0.9868021011352539
22 1.505334e+01 2.423032e-02
* time: 1.0240590572357178
23 1.496660e+01 2.330651e-02
* time: 1.0614590644836426
24 1.488270e+01 1.732872e-02
* time: 1.0989069938659668
25 1.480827e+01 1.512787e-02
* time: 1.136146068572998
26 1.474092e+01 1.478235e-02
* time: 1.173293113708496
27 1.467948e+01 1.548079e-02
* time: 1.2104599475860596
28 1.462497e+01 1.346388e-02
* time: 1.3053219318389893
29 1.457279e+01 1.213910e-02
* time: 1.3422510623931885
30 1.452001e+01 1.505024e-02
* time: 1.3792610168457031
31 1.447352e+01 1.392017e-02
* time: 1.4163451194763184
32 1.443694e+01 1.269613e-02
* time: 1.4534571170806885
33 1.441082e+01 9.530009e-03
* time: 1.4905710220336914
34 1.438903e+01 8.680185e-03
* time: 1.5277149677276611
35 1.437123e+01 8.084113e-03
* time: 1.6228950023651123
36 1.435696e+01 7.296080e-03
* time: 1.660187005996704
37 1.434533e+01 6.870232e-03
* time: 1.697632074356079
38 1.433581e+01 6.213121e-03
* time: 1.7350759506225586
39 1.432762e+01 4.560134e-03
* time: 1.7725629806518555
40 1.432042e+01 4.927508e-03
* time: 1.8100099563598633
41 1.431422e+01 4.703799e-03
* time: 1.8474020957946777
42 1.430837e+01 5.161186e-03
* time: 1.8858189582824707
43 1.430230e+01 4.375159e-03
* time: 1.9842700958251953
44 1.429671e+01 4.914855e-03
* time: 2.021631956100464
45 1.429206e+01 4.529336e-03
* time: 2.0589001178741455
46 1.428789e+01 4.718275e-03
* time: 2.096066951751709
47 1.428373e+01 4.728993e-03
* time: 2.1331441402435303
48 1.427936e+01 4.067328e-03
* time: 2.1700379848480225
49 1.427519e+01 4.025320e-03
* time: 2.2069880962371826
50 1.427080e+01 4.811489e-03
* time: 2.3017079830169678
51 1.426554e+01 5.206415e-03
* time: 2.3385660648345947
52 1.425989e+01 6.114556e-03
* time: 2.37540602684021
53 1.425420e+01 5.560564e-03
* time: 2.412285089492798
54 1.424849e+01 5.345485e-03
* time: 2.44931697845459
55 1.424263e+01 4.650235e-03
* time: 2.4862051010131836
56 1.423743e+01 4.783608e-03
* time: 2.523061990737915
57 1.423308e+01 4.892709e-03
* time: 2.6172850131988525
58 1.422953e+01 4.610459e-03
* time: 2.6544570922851562
59 1.422653e+01 3.552166e-03
* time: 2.691411018371582
60 1.422427e+01 2.950846e-03
* time: 2.7283289432525635
61 1.422269e+01 2.815676e-03
* time: 2.765259027481079
62 1.422136e+01 2.553205e-03
* time: 2.802204132080078
63 1.422031e+01 2.322051e-03
* time: 2.839195966720581
64 1.421948e+01 1.827253e-03
* time: 2.876189947128296
65 1.421880e+01 1.830575e-03
* time: 2.9712560176849365
66 1.421820e+01 1.481196e-03
* time: 3.0084190368652344
67 1.421773e+01 1.699133e-03
* time: 3.0457911491394043
68 1.421734e+01 1.221767e-03
* time: 3.0831949710845947
69 1.421699e+01 1.035086e-03
* time: 3.1204910278320312
70 1.421667e+01 1.138761e-03
* time: 3.1578140258789062
71 1.421637e+01 1.153531e-03
* time: 3.1950860023498535
72 1.421613e+01 1.050299e-03
* time: 3.290087938308716
73 1.421590e+01 1.189060e-03
* time: 3.3276119232177734
74 1.421571e+01 9.106610e-04
* time: 3.364629030227661
75 1.421555e+01 7.664420e-04
* time: 3.401447057723999
76 1.421541e+01 7.162366e-04
* time: 3.4383981227874756
77 1.421528e+01 7.303721e-04
* time: 3.475172996520996
78 1.421517e+01 8.672785e-04
* time: 3.511824131011963
79 1.421507e+01 8.299648e-04
* time: 3.548380136489868
80 1.421498e+01 6.495339e-04
* time: 3.6428000926971436
81 1.421490e+01 6.401547e-04
* time: 3.679219961166382
82 1.421482e+01 7.780928e-04
* time: 3.715916156768799
83 1.421475e+01 5.953102e-04
* time: 3.752671957015991
84 1.421467e+01 6.539914e-04
* time: 3.7894160747528076
85 1.421460e+01 6.605057e-04
* time: 3.826396942138672
86 1.421453e+01 7.601512e-04
* time: 3.863326072692871
87 1.421445e+01 7.227036e-04
* time: 3.9582040309906006
88 1.421436e+01 6.771225e-04
* time: 3.995337963104248
89 1.421428e+01 6.319613e-04
* time: 4.032376050949097
90 1.421421e+01 7.628183e-04
* time: 4.069455146789551
91 1.421412e+01 7.562003e-04
* time: 4.106621980667114
92 1.421402e+01 8.858775e-04
* time: 4.143702030181885
93 1.421389e+01 1.032017e-03
* time: 4.180773973464966
94 1.421370e+01 1.396906e-03
* time: 4.217836141586304
95 1.421336e+01 1.417632e-03
* time: 4.3128180503845215
96 1.421256e+01 2.626765e-03
* time: 4.349996089935303
97 1.421062e+01 2.508767e-03
* time: 4.387232065200806
98 1.420863e+01 4.077040e-03
* time: 4.413782119750977
99 1.420465e+01 8.115206e-03
* time: 4.440356016159058
100 1.419562e+01 9.638571e-03
* time: 4.4774720668792725
101 1.418284e+01 7.003445e-03
* time: 4.5145790576934814
102 1.416522e+01 1.282492e-02
* time: 4.541260004043579
103 1.413663e+01 7.319819e-03
* time: 4.625324964523315
104 1.409797e+01 1.137819e-02
* time: 4.6520750522613525
105 1.406908e+01 1.551680e-02
* time: 4.678635120391846
106 1.404043e+01 8.036903e-03
* time: 4.716172933578491
107 1.399818e+01 1.222842e-02
* time: 4.75330114364624
108 1.397809e+01 8.049092e-03
* time: 4.779739141464233
109 1.395343e+01 9.203412e-03
* time: 4.817017078399658
110 1.393863e+01 7.755720e-03
* time: 4.855158090591431
111 1.392722e+01 6.743063e-03
* time: 4.892483949661255
112 1.391865e+01 4.669041e-03
* time: 4.987519025802612
113 1.391287e+01 4.704973e-03
* time: 5.024906158447266
114 1.390922e+01 3.460858e-03
* time: 5.062174081802368
115 1.390697e+01 2.827598e-03
* time: 5.099524974822998
116 1.390539e+01 2.136470e-03
* time: 5.136837959289551
117 1.390419e+01 1.880080e-03
* time: 5.17409610748291
118 1.390325e+01 1.413695e-03
* time: 5.211395025253296
119 1.390242e+01 1.380716e-03
* time: 5.306957960128784
120 1.390173e+01 1.645893e-03
* time: 5.344264984130859
121 1.390111e+01 2.060189e-03
* time: 5.381724119186401
122 1.390047e+01 1.750806e-03
* time: 5.4190709590911865
123 1.389973e+01 1.424874e-03
* time: 5.45653510093689
124 1.389888e+01 1.616405e-03
* time: 5.493885040283203
125 1.389800e+01 1.930972e-03
* time: 5.531183958053589
126 1.389708e+01 1.985363e-03
* time: 5.568349123001099
127 1.389608e+01 2.369816e-03
* time: 5.663486003875732
128 1.389484e+01 2.014579e-03
* time: 5.700727939605713
129 1.389356e+01 2.168547e-03
* time: 5.737942934036255
130 1.389229e+01 1.915106e-03
* time: 5.775361061096191
131 1.389126e+01 2.102994e-03
* time: 5.813575029373169
132 1.389038e+01 1.936740e-03
* time: 5.851124048233032
133 1.388955e+01 1.829416e-03
* time: 5.890165090560913
134 1.388881e+01 1.381362e-03
* time: 5.985392093658447
135 1.388820e+01 1.321446e-03
* time: 6.022536039352417
136 1.388774e+01 1.110782e-03
* time: 6.0598509311676025
137 1.388743e+01 1.132308e-03
* time: 6.099574089050293
138 1.388720e+01 1.133313e-03
* time: 6.136857986450195
139 1.388700e+01 8.134191e-04
* time: 6.174065113067627
140 1.388682e+01 8.180784e-04
* time: 6.211137056350708
141 1.388665e+01 9.253950e-04
* time: 6.305774927139282
142 1.388652e+01 8.050521e-04
* time: 6.343440055847168
143 1.388641e+01 8.703993e-04
* time: 6.380488157272339
144 1.388629e+01 7.296819e-04
* time: 6.417686939239502
145 1.388619e+01 5.653899e-04
* time: 6.454920053482056
146 1.388610e+01 4.359365e-04
* time: 6.492016077041626
147 1.388601e+01 4.729001e-04
* time: 6.529036045074463
148 1.388595e+01 5.467996e-04
* time: 6.5659120082855225
149 1.388590e+01 4.984536e-04
* time: 6.66035795211792
150 1.388586e+01 3.919395e-04
* time: 6.697222948074341
151 1.388584e+01 3.194011e-04
* time: 6.7340309619903564
152 1.388582e+01 2.180319e-04
* time: 6.770916938781738
153 1.388581e+01 2.287536e-04
* time: 6.807768106460571
154 1.388580e+01 1.716130e-04
* time: 6.844694137573242
155 1.388579e+01 1.219654e-04
* time: 6.881693124771118
156 1.388579e+01 1.010460e-04
* time: 6.97870397567749
157 1.388579e+01 8.824319e-05
* time: 7.015944957733154
158 1.388578e+01 1.045123e-04
* time: 7.0529890060424805
159 1.388578e+01 1.275517e-04
* time: 7.0899059772491455
160 1.388578e+01 1.179329e-04
* time: 7.126917123794556
161 1.388577e+01 1.026798e-04
* time: 7.163928985595703
162 1.388577e+01 9.629964e-05
* time: 7.200922012329102
163 1.388577e+01 9.151837e-05
* time: 7.237774133682251
164 1.388576e+01 1.230071e-04
* time: 7.333606958389282
165 1.388576e+01 9.666684e-05
* time: 7.37045693397522
166 1.388576e+01 9.519450e-05
* time: 7.407377004623413
167 1.388576e+01 8.045397e-05
* time: 7.444282054901123
168 1.388575e+01 6.994162e-05
* time: 7.481281042098999
169 1.388575e+01 9.317564e-05
* time: 7.518265962600708
170 1.388575e+01 1.000114e-04
* time: 7.555211067199707
171 1.388575e+01 7.418015e-05
* time: 7.649691104888916
172 1.388575e+01 7.294361e-05
* time: 7.686767101287842
[ Info: Final spread
WF center [rx, ry, rz]/Å spread/Ų ω ωc ωt
1 0.67899 -0.67852 -0.67890 1.15678 0.00000 1.15678
2 -0.67957 -0.68012 0.67692 1.15682 0.00001 1.15683
3 -0.67667 0.67948 -0.68023 1.15688 0.00001 1.15689
4 0.67816 0.67815 0.68009 1.15686 0.00000 1.15686
5 0.67856 -0.67909 -0.67858 2.31459 0.00000 2.31459
6 -0.67822 -0.67775 0.68084 2.31458 0.00001 2.31459
7 -0.68081 0.67819 -0.67725 2.31466 0.00001 2.31466
8 0.67945 0.67943 0.67760 2.31456 0.00000 2.31456
Sum spread: Ωt = Ω + Ωc, Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
Ωt = 13.88575
Ωc = 0.00003
Ω = 13.88572
ΩI = 10.71900
ΩOD = 3.15284
ΩD = 0.01387
Ω̃ = 3.16672the final spreads are
Wannier.omega_center(model, U1, r₀, λ) WF center [rx, ry, rz]/Å spread/Ų ω ωc ωt
1 0.67899 -0.67852 -0.67890 1.15678 0.00000 1.15678
2 -0.67957 -0.68012 0.67692 1.15682 0.00001 1.15683
3 -0.67667 0.67948 -0.68023 1.15688 0.00001 1.15689
4 0.67816 0.67815 0.68009 1.15686 0.00000 1.15686
5 0.67856 -0.67909 -0.67858 2.31459 0.00000 2.31459
6 -0.67822 -0.67775 0.68084 2.31458 0.00001 2.31459
7 -0.68081 0.67819 -0.67725 2.31466 0.00001 2.31466
8 0.67945 0.67943 0.67760 2.31456 0.00000 2.31456
Sum spread: Ωt = Ω + Ωc, Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
Ωt = 13.88575
Ωc = 0.00003
Ω = 13.88572
ΩI = 10.71900
ΩOD = 3.15284
ΩD = 0.01387
Ω̃ = 3.16672Look, the the WF centers are positioned where we want, and now these are 4 degenerate bonding WFs, and 4 degenerate anti-bonding WFs!
Note now the total spread is smaller than the s, p disentanglement case, since they are the bonding/anti-bonding combinations. 🎉
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