Skip to content

Commit 1025016

Browse files
kcpearcekcpearce
committed
Fix missing picture objects in README
1 parent 86f2a30 commit 1025016

1 file changed

Lines changed: 145 additions & 29 deletions

File tree

‎Project#03/README.md‎

Lines changed: 145 additions & 29 deletions
Original file line numberDiff line numberDiff line change
@@ -26,19 +26,35 @@ Read the nuclear repulsion energy from the [enuc.dat](./input/h2o/STO-3G/enuc.da
2626

2727
Read the AO-basis [overlap](./input/h2o/STO-3G/s.dat)
2828

29-
<img src="./figures/overlap.png" height="50">
29+
<picture>
30+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/overlap.png">
31+
<source media="(prefers-color-scheme: light)" srcset="./figures/overlap.png">
32+
<img src="./figures/overlap.png" height="50">
33+
</picture>
3034

3135
[kinetic-energy](./input/h2o/STO-3G/t.dat)
3236

33-
<img src="./figures/kinetic-energy.png" height="50">
37+
<picture>
38+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/kinetic-energy.png">
39+
<source media="(prefers-color-scheme: light)" srcset="./figures/kinetic-energy.png">
40+
<img src="./figures/kinetic-energy.png" height="50">
41+
</picture>
3442

3543
[nuclear-attraction integrals](./input/h2o/STO-3G/v.dat)
3644

37-
<img src="./figures/nuclear-attraction.png" height="55">
45+
<picture>
46+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/nuclear-attraction.png">
47+
<source media="(prefers-color-scheme: light)" srcset="./figures/nuclear-attraction.png">
48+
<img src="./figures/nuclear-attraction.png" height="55">
49+
</picture>
3850

3951
and store them in appropriately constructed matrices. Then form the "core Hamiltonian":
4052

41-
<img src="./figures/core-hamiltonian.png" height="25">
53+
<picture>
54+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/core-hamiltonian.png">
55+
<source media="(prefers-color-scheme: light)" srcset="./figures/core-hamiltonian.png">
56+
<img src="./figures/core-hamiltonian.png" height="25">
57+
</picture>
4258

4359
Note that the one-electron integrals provided include only the *permutationally unique* integrals, but you should store the full matrices for convenience.
4460
Note also that the AO indices on the integrals in the files start with "1" rather than "0".
@@ -52,24 +68,44 @@ Read the two-electron repulsion integrals from the
5268
file.
5369
The integrals in this file are provided in Mulliken notation over real AO basis functions:
5470

55-
<img src="./figures/eri.png" height="45">
71+
<picture>
72+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/eri.png">
73+
<source media="(prefers-color-scheme: light)" srcset="./figures/eri.png">
74+
<img src="./figures/eri.png" height="45">
75+
</picture>
5676

5777
Hence, the integrals obey the eight-fold permutational symmetry relationships:
5878

59-
<img src="./figures/permutational-symmetry.png" height="20">
79+
<picture>
80+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/permutational-symmetry.png">
81+
<source media="(prefers-color-scheme: light)" srcset="./figures/permutational-symmetry.png">
82+
<img src="./figures/permutational-symmetry.png" height="20">
83+
</picture>
6084

6185
and only the permutationally unique integrals are provided in the file, with the restriction that, for each integral, the following relationships hold:
6286

63-
<img src="./figures/index-restrictions.png" height="20">
87+
<picture>
88+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/index-restrictions.png">
89+
<source media="(prefers-color-scheme: light)" srcset="./figures/index-restrictions.png">
90+
<img src="./figures/index-restrictions.png" height="20">
91+
</picture>
6492

6593
where
6694

67-
<img src="./figures/compound-index-restrictions.png" height="20">
95+
<picture>
96+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/compound-index-restrictions.png">
97+
<source media="(prefers-color-scheme: light)" srcset="./figures/compound-index-restrictions.png">
98+
<img src="./figures/compound-index-restrictions.png" height="20">
99+
</picture>
68100

69101
Note that the two-electron integrals may be stored efficiently in a one-dimensional array and the above relationship used to map between given
70102
&mu;, &nu;, &lambda;, and &sigma; indices and a "compound index" defined as:
71103

72-
<img src="./figures/compound-index-restrictions2.png" height="20">
104+
<picture>
105+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/compound-index-restrictions2.png">
106+
<source media="(prefers-color-scheme: light)" srcset="./figures/compound-index-restrictions2.png">
107+
<img src="./figures/compound-index-restrictions2.png" height="20">
108+
</picture>
73109

74110
* [Hint 1](./hints/hint3-1.md): Compound indices
75111
* [Hint 2](./hints/hint3-2.md): Pre-Computed Lookup Arrays
@@ -80,13 +116,21 @@ Note that the two-electron integrals may be stored efficiently in a one-dimensio
80116

81117
Diagonalize the overlap matrix:
82118

83-
<img src="./figures/diag-mw-hessian.png" height="20">
119+
<picture>
120+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/diag-mw-hessian.png">
121+
<source media="(prefers-color-scheme: light)" srcset="./figures/diag-mw-hessian.png">
122+
<img src="./figures/diag-mw-hessian.png" height="20">
123+
</picture>
84124

85125
where L<sub>S</sub> is the matrix of eigenvectors (columns) and &Lambda;<sub>S</sub> is the diagonal matrix of corresponding eigenvalues.
86126

87127
Build the symmetric orthogonalization matrix using:
88128

89-
<img src="./figures/symm-orthog-matrix.png" height="25">
129+
<picture>
130+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/symm-orthog-matrix.png">
131+
<source media="(prefers-color-scheme: light)" srcset="./figures/symm-orthog-matrix.png">
132+
<img src="./figures/symm-orthog-matrix.png" height="25">
133+
</picture>
90134

91135
where the tilde denotes the matrix transpose.
92136

@@ -97,21 +141,37 @@ where the tilde denotes the matrix transpose.
97141

98142
Form an initial (guess) Fock matrix in the orthonormal AO basis using the core Hamiltonian as a guess:
99143

100-
<img src="./figures/initial-fock.png" height="25">
144+
<picture>
145+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/initial-fock.png">
146+
<source media="(prefers-color-scheme: light)" srcset="./figures/initial-fock.png">
147+
<img src="./figures/initial-fock.png" height="25">
148+
</picture>
101149

102150
Diagonalize the Fock matrix:
103151

104-
<img src="./figures/diag-fock.png" height="20">
152+
<picture>
153+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/diag-fock.png">
154+
<source media="(prefers-color-scheme: light)" srcset="./figures/diag-fock.png">
155+
<img src="./figures/diag-fock.png" height="20">
156+
</picture>
105157

106158
Note that the &epsilon;<sub>0</sub> matrix contains the initial orbital energies.
107159

108160
Transform the eigenvectors into the original (non-orthogonal) AO basis:
109161

110-
<img src="./figures/transform-coeff.png" height="25">
162+
<picture>
163+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/transform-coeff.png">
164+
<source media="(prefers-color-scheme: light)" srcset="./figures/transform-coeff.png">
165+
<img src="./figures/transform-coeff.png" height="25">
166+
</picture>
111167

112168
Build the density matrix using the occupied MOs:
113169

114-
<img src="./figures/density-matrix.png" height="55">
170+
<picture>
171+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/density-matrix.png">
172+
<source media="(prefers-color-scheme: light)" srcset="./figures/density-matrix.png">
173+
<img src="./figures/density-matrix.png" height="55">
174+
</picture>
115175

116176
where *m* indexes the columns of the coefficient matrices, and the summation includes only the occupied spatial MOs.
117177

@@ -123,11 +183,19 @@ where *m* indexes the columns of the coefficient matrices, and the summation inc
123183

124184
The SCF electronic energy may be computed using the density matrix as:
125185

126-
<img src="./figures/initial-scf-energy.png" height="60">
186+
<picture>
187+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/initial-scf-energy.png">
188+
<source media="(prefers-color-scheme: light)" srcset="./figures/initial-scf-energy.png">
189+
<img src="./figures/initial-scf-energy.png" height="60">
190+
</picture>
127191

128192
The total energy is the sum of the electronic energy and the nuclear repulsion energy:
129193

130-
<img src="./figures/initial-total-energy.png" height="25">
194+
<picture>
195+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/initial-total-energy.png">
196+
<source media="(prefers-color-scheme: light)" srcset="./figures/initial-total-energy.png">
197+
<img src="./figures/initial-total-energy.png" height="25">
198+
</picture>
131199

132200
where *0* denotes the initial SCF energy.
133201

@@ -137,7 +205,11 @@ where *0* denotes the initial SCF energy.
137205

138206
Start the SCF iterative procedure by building a new Fock matrix using the previous iteration's density as:
139207

140-
<img src="./figures/new-fock.png" height="60">
208+
<picture>
209+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/new-fock.png">
210+
<source media="(prefers-color-scheme: light)" srcset="./figures/new-fock.png">
211+
<img src="./figures/new-fock.png" height="60">
212+
</picture>
141213

142214
where the double-summation runs over all the AOs and *i-1* denotes the density for the last iteration.
143215

@@ -150,34 +222,58 @@ Form the new density matrix following the same procedure as in Step #5 above:
150222

151223
Orthogonalize:
152224

153-
<img src="./figures/orthog-fock.png" height="20">
225+
<picture>
226+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/orthog-fock.png">
227+
<source media="(prefers-color-scheme: light)" srcset="./figures/orthog-fock.png">
228+
<img src="./figures/orthog-fock.png" height="20">
229+
</picture>
154230

155231
Diagonalize:
156232

157-
<img src="./figures/diag-new-fock.png" height="15">
233+
<picture>
234+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/diag-new-fock.png">
235+
<source media="(prefers-color-scheme: light)" srcset="./figures/diag-new-fock.png">
236+
<img src="./figures/diag-new-fock.png" height="15">
237+
</picture>
158238

159239
Back-transform:
160240

161-
<img src="./figures/back-transform-coeff.png" height="20">
241+
<picture>
242+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/back-transform-coeff.png">
243+
<source media="(prefers-color-scheme: light)" srcset="./figures/back-transform-coeff.png">
244+
<img src="./figures/back-transform-coeff.png" height="20">
245+
</picture>
162246

163247
Compute the density:
164248

165-
<img src="./figures/compute-density.png" height="55">
249+
<picture>
250+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/compute-density.png">
251+
<source media="(prefers-color-scheme: light)" srcset="./figures/compute-density.png">
252+
<img src="./figures/compute-density.png" height="55">
253+
</picture>
166254

167255
where *i* denotes the current iteration density.
168256

169257
## Step #9: Compute the New SCF Energy
170258

171259
Compute the new SCF energy as before:
172260

173-
<img src="./figures/compute-new-scf-energy.png" height="125">
261+
<picture>
262+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/compute-new-scf-energy.png">
263+
<source media="(prefers-color-scheme: light)" srcset="./figures/compute-new-scf-energy.png">
264+
<img src="./figures/compute-new-scf-energy.png" height="125">
265+
</picture>
174266

175267
where *i* denotes the SCF energy for the *i*th iteration.
176268

177269
## Step #10: Test for Convergence
178270
Test both the energy and the density for convergence:
179271

180-
<img src="./figures/convergence-test.png" height="125">
272+
<picture>
273+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/convergence-test.png">
274+
<source media="(prefers-color-scheme: light)" srcset="./figures/convergence-test.png">
275+
<img src="./figures/convergence-test.png" height="125">
276+
</picture>
181277

182278
If the difference in consecutive SCF energy and the root-mean-squared difference in consecutive densities do not fall below the prescribed thresholds, return to Step #7 and continue from there.
183279

@@ -187,22 +283,38 @@ If the difference in consecutive SCF energy and the root-mean-squared difference
187283
### The MO-Basis Fock Matrix
188284
At convergence, the canonical Hartree-Fock MOs are, by definition, eigenfunctions of the Fock operator, viz.
189285

190-
<img src="./figures/canonical-mos.png" height="25">
286+
<picture>
287+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/canonical-mos.png">
288+
<source media="(prefers-color-scheme: light)" srcset="./figures/canonical-mos.png">
289+
<img src="./figures/canonical-mos.png" height="25">
290+
</picture>
191291

192292
If we multiply on the left by an arbitrary MO and integrate, we obtain:
193293

194-
<img src="./figures/mo-fock-matrix-element.png" height="30">
294+
<picture>
295+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/mo-fock-matrix-element.png">
296+
<source media="(prefers-color-scheme: light)" srcset="./figures/mo-fock-matrix-element.png">
297+
<img src="./figures/mo-fock-matrix-element.png" height="30">
298+
</picture>
195299

196300
In other words, the Fock matrix should be diagonal in the MO basis, with the orbital energies as its diagonal elements. We can demonstrate this explicitly using the AO-basis Fock matrix by first re-writing the above expression using the LCAO-MO coefficients:
197301

198-
<img src="./figures/mo-fock-matrix.png" height="50">
302+
<picture>
303+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/mo-fock-matrix.png">
304+
<source media="(prefers-color-scheme: light)" srcset="./figures/mo-fock-matrix.png">
305+
<img src="./figures/mo-fock-matrix.png" height="50">
306+
</picture>
199307

200308
Use the above equation to transform the Fock matrix from the AO basis to the MO basis and demonstrate that it is indeed diagonal (to within the convergence limits of the SCF iterative procedure).
201309

202310
### One-Electron Properties
203311
As discussed in detail in Ch. 3 of the text by Szabo and Ostlund, the calculation of one-electron properties requires density matrix and the relevant property integrals. The electronic contribution to the electric-dipole moment may be computed using,
204312

205-
<img src="./figures/electric-dipole-moment.png" height="50">
313+
<picture>
314+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/electric-dipole-moment.png">
315+
<source media="(prefers-color-scheme: light)" srcset="./figures/electric-dipole-moment.png">
316+
<img src="./figures/electric-dipole-moment.png" height="50">
317+
</picture>
206318

207319
where the vector notation implies three sets of dipole-moment integrals -- one for each Cartesian component of the dipole operator.
208320

@@ -215,7 +327,11 @@ The test cases provided below include the structural information dipole integral
215327
### Population Analysis/Atomic Charges
216328
A Mulliken population analysis (also described in Szabo & Ostlund, Ch. 3) requires the overlap integrals and the electron density, in addition to information about the number of basis functions centered on each atom. The charge on atom *A* may be computed as:
217329

218-
<img src="./figures/atomic-charge.png" height="50">
330+
<picture>
331+
<source media="(prefers-color-scheme: dark)" srcset="./figures/dark/atomic-charge.png">
332+
<source media="(prefers-color-scheme: light)" srcset="./figures/atomic-charge.png">
333+
<img src="./figures/atomic-charge.png" height="50">
334+
</picture>
219335

220336
where the summation is limited to only those basis functions centered on atom *A*.
221337

0 commit comments

Comments
 (0)