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Stochastic perturbative triples

Stochastic formulation
Memoization
Importance sampling
Multiple buckets
Semi-stochastic algorithm
Test code
- Orbital energies
  - Occupied
  - Virtual
  - Code
- Uniform distribution
- Importance sampling
  - Preparation
  - Sampling
- Semi-stochastic
  - Preparation
  - Sampling
- Final plot
  - Water
  - 14 electrons
  - 38 electrons
  - HBN
Appendix
- Fortran code
  - Naive expression
  - Optimized functions
Plots
- Test 2
Results QP

Stochastic formulation

The perturbative correction is expressed as:

\[ E_{(T)} = \sum_{ijkabc} E_{ijk}^{abc} = \sum_{ijkabc} \frac{(4 W_{ijk}^{abc} + W_{ijk}^{bca} + W_{ijk}^{cab}) (V_{ijk}^{abc} - V_{ijk}^{cba})}{\epsilon_i + \epsilon_j + \epsilon_k - \epsilon_a - \epsilon_b - \epsilon_c} \]

If $E_{(T)}$ is computed as \[ E_{(T)} = \sum_{ijkabc} \frac{1}{N} \left( E_{ijk}^{abc} \times N \right) \] where $N$ is the total number of terms, it can be computed by drawing uniformly samples $E_{ijk}^{abc}$ and computing the statistical average.

The stochastic calculation is unbiased, but the rate of convergence of the statistical error is be governed by the variance of the samples. Moreover, the total computational time is proportional to the number of computed samples. The objective of this work is to propose a stochastic algorithm which gives an estimate of $E_{(T)}$ with a small enough statistical error in less time than what is required for the exact computation.

Memoization

The perturbative correction can be re-expressed as the sum of corrections indexed only by virtual orbitals:

\[ E_{(T)} = \sum_{a,b,c} E_{abc} \]

where

\[ E_{abc} = \sum_{ijk} E_{ijk}^{abc} \]

The Monte Carlo sampling can be made by drawing samples $E_{abc}$. The great advantage of this re-expression is that the number of triplets $(a,b,c)$ is $N_v^3$, which is expected to be small enough so that all contributions $E_{abc}$ can be stored in memory. Therefore, the first time a triplet $(a,b,c)$ is drawn, it is computed and stored. Every next time the same triplet is drawn, the value is fetched from memory. In this way, the $N_o^3 \times N_v$ cost of computing the sample is paid only once, and all other evaluations are negligible.

Hence, using an number of Monte Carlo samples large enough such that each contribution has been drawn at least once will have a computational cost larger only by a negligible amount than the cost of making the exact computation.

Importance sampling

To reduce the variance of the samples, the samples are drawn using the probability

\[ P(a,b,c) = \frac{1}{\mathcal{N}} \frac{1}{\epsilon_{\text{occ}} - \epsilon_a - \epsilon_b - \epsilon_c} \]

where $\mathcal{N}$ is such that $\sum P(a,b,c) = 1$, and $\epsilon_{\text{occ}}$ is the average of the energy contributions of the occupied orbitals: \[ \epsilon_{\text{occ}} = \frac{3}{N_o} \sum_i \epsilon_i. \]

The perturbative contribution is computed as

\[ \[ E_{(T)} = \mathcal{N} \sum_{abc} P(a,b,c) \, E_{abc} \, (\epsilon_{\text{occ}} - \epsilon_a - \epsilon_b - \epsilon_c) \, \]

This modification reduces the statistical error bars by a factor of two for the same computational cost for two reasons:

The estimator has less fluctuations
Some triplets are more likely to be drawn than others, so memoization is even more effective

Multiple buckets

We use the inverse transform sampling technique to draw the samples, so an array of pairs $( P(a,b,c), (a,b,c) )$ is stored. To reduce even more the variance of the samples, this array is sorted in decreasing order of $P(a,b,c)$, and the array is divided into buckets $B$ where the sum of $P(a,b,c)$ in the same in each bucket. As all buckets are equiprobable, one can safely generate samples which are averages of samples drawn from each bucket. If the values of $E_{abc}$ are skewed, this latest improvement reduces significantly the variance.

Semi-stochastic algorithm

The total perturbative contribution is the sum of contributions in the buckets:

\[ E_{(T)} = \sum_B E_B = \sum_B\sum_{(a,b,c) \in B} E_{abc} \]

When all the triplets of a bucket $B$ have been drawn at least once, the contribution $E_B$ can be computed. Then, there is no need to evaluate $E_B$ stochastically and we can separate the buckets into stochastic ($\mathcal{S}$) and deterministic ($\mathcal{D}$) buckets:

\[ E_{(T)} = \sum_{B \in \mathcal{D}} E_B + \frac{1}{|\mathcal{S}|} \sum_{B \in \mathcal{S}} \left \langle E^B_{abc} \times \frac{\epsilon_{\text{occ}} - \epsilon_a - \epsilon_b - \epsilon_c}{\mathcal{N}} \right \rangle_{P(a,b,c), (a,b,c) \in B} \]

All the buckets don't have the same size: the number of triplets per bucket is decreasing with the index of the bucket. So the first buckets enter into the deterministic set first, and the stochastic contribution becomes smaller and smaller. When all the triplets have been drawn, the exact value is obtained without any statistical error.

To accelerate the filling of the buckets, at every Monte Carlo iteration we trigger the computation of the first non-computed triplet. Hence, after $N$ drawings we are sure to have computed the exact contribution.

Test code

import numpy as np
import gzip
import time

ev_to_h = 0.03674930813664888

The gzipped input file h2o_qz.gz contains data for the computation of the valence perturbative triples correction of the water molecule in the cc-pVQZ basis set. Each line is composed of $a$, $b$, $c$, $(\epsilon_a + \epsilon_b + \epsilon_c)$ and $3\times E_{abc}$.

#TODO
#filetype=0
#filename = "h2o_qz.gz"

filetype=1
#filename = "UEG/nelec-14-norb-147.out.gz"
#filename = "UEG/nelec-38-norb-341.out.gz"
filename = "HBN/hbn.out.gz"
#filename = "HBN/hbn-36-176.out.gz"

denom = []
eabc  = []
triplets = []

with gzip.open(filename, 'r') as f:
  if filetype == 0:
      for line in f:
          a, b, c, d, e = line.split()
          triplets.append( [int(a),int(b),int(c)] )
          denom.append(float(d))
          eabc.append(float(e)/3.)

  else:
      for line in f:
          a, b, c, d, e = line.split()
          triplets.append( [int(a),int(b),int(c)] )
          denom.append(float(d)*ev_to_h)
          eabc.append(-float(e)*ev_to_h)
      
triplets = np.array(triplets)
denom = np.array(denom)
eabc  = np.array(eabc)

The exact contribution is the sum of all $E_{abc}$:

Nabc = int(len(eabc))
sum(eabc)

-0.03022743231272565

Orbital energies

Occupied

The energies of the occupied orbitals are required to compute $\epsilon_{\text{occ}}$:

#+NAME:orb_energies-h2o

1	-1.3495166001
2	-0.7144028522
3	-0.5819477116
4	-0.5081442470

#+NAME:orb_energies-14

1	-0.40729666
2	-0.18688829
3	-0.18688829
4	-0.18688829
5	-0.18688829
6	-0.18688829
7	-0.18688829

#+NAME:orb_energies-38

1	-0.40945392
2	-0.28354874
3	-0.28354874
4	-0.28354874
5	-0.28354874
6	-0.28354874
7	-0.28354874
8	-0.15943825
9	-0.15943825
10	-0.15943825
11	-0.15943825
12	-0.15943825
13	-0.15943825
14	-0.15943825
15	-0.15943825
16	-0.15943825
17	-0.15943825
18	-0.15943825
19	-0.15943825

#+NAME:orb_energies-hbn

1	-0.93953654
2	-0.92121891
3	-0.78505256
4	-0.77976882
5	-0.48653647
6	-0.47973909
7	-0.33857221
8	-0.30326521
9	-0.30034533
10	-0.22859891
11	-0.18754927
12	-0.18606588
13	-0.15297849
14	-0.15148893
15	-0.12146057
16	-0.084733892

#+NAME:orb_energies-hbn-36

1	-0.93323231
2	-0.91493460
3	-0.77878864
4	-0.77351063
5	-0.48043513
6	-0.47364310
7	-0.33220283
8	-0.29719087
9	-0.29425309
10	-0.22234047
11	-0.18186670
12	-0.18043512
13	-0.14726115
14	-0.14582101
15	-0.11539548
16	-0.078697018

Virtual

The energies of the virtual orbitals are required for the importance sampling function $\epsilon_{\text{vir}}$:

#+NAME:vir_energies-h2o

1	0.1170780927
2	0.1710190904
3	0.4494673807
4	0.4621202460
5	0.5016951241
6	0.5836694028
7	0.6051947963
8	0.6146697086
9	0.6565051177
10	0.7182063878
11	0.8516191972
12	0.9198243228
13	1.1120958387
14	1.1574553944
15	1.3462327164
16	1.4165525262
17	1.4800979932
18	1.4849525169
19	1.5786053722
20	1.6868289625
21	1.9114339643
22	2.0688122674
23	2.1955484330
24	2.2920067741
25	2.3622513733
26	2.4242696557
27	2.4882741718
28	2.5209123081
29	2.5830337542
30	2.5867616694
31	2.6542316478
32	2.6703494422
33	2.8415133159
34	2.8617578839
35	3.0383168770
36	3.1276003251
37	3.2948068036
38	3.3464975106
39	3.4589491034
40	3.6284221842
41	3.8256509147
42	4.0020447839
43	4.1256277268
44	4.1867673535
45	4.2141487650
46	4.4362998361
47	4.4947291066
48	4.7060661936
49	4.7521034075
50	4.7996510056
51	4.9280320715
52	5.3624272963
53	5.4026006683
54	5.9752399036
55	6.1056730507
56	6.2495153622
57	6.3165406593
58	6.7312795276
59	6.7921966285
60	7.0753837689
61	7.2750220790
62	7.3155097230
63	7.3689101210
64	7.4373090211
65	7.5364710135
66	7.5557420909
67	7.5687553410
68	7.9995654839
69	8.0806018979
70	8.0989771117
71	8.1437725027
72	8.1500310933
73	8.2750489171
74	8.3193045638
75	8.3573477176
76	8.4311162300
77	8.6177564754
78	8.8437328029
79	8.9093818035
80	8.9656095271
81	9.2370137178
82	9.3925685769
83	9.4140146116
84	9.9230302951
85	10.0156977150
86	10.2889761593
87	10.4424540894
88	10.6589001318
89	10.7573224755
90	10.8181869588
91	11.2934414670
92	11.4080819018
93	11.6045421043
94	11.6427826465
95	11.7038554829
96	11.8458216549
97	12.1936857310
98	12.3399716139
99	12.4248976056
100	12.4326416584
101	12.4615374955
102	13.6268899965
103	13.8184890674
104	14.2866145595
105	14.6487775661
106	14.6790346791
107	14.8758761065
108	16.4564806016
109	16.8432677225
110	44.3828489701

#+NAME:vir_energies-14

1	0.19320282
2	0.19320282
3	0.19320282
4	0.19320282
5	0.19320282
6	0.19320282
7	0.19320282
8	0.19320282
9	0.19320282
10	0.19320282
11	0.19320282
12	0.19320282
13	0.37219048
14	0.37219048
15	0.37219048
16	0.37219048
17	0.37219048
18	0.37219048
19	0.37219048
20	0.37219048
21	0.52219882
22	0.52219882
23	0.52219882
24	0.52219882
25	0.52219882
26	0.52219882
27	0.68533129
28	0.68533129
29	0.68533129
30	0.68533129
31	0.68533129
32	0.68533129
33	0.68533129
34	0.68533129
35	0.68533129
36	0.68533129
37	0.68533129
38	0.68533129
39	0.68533129
40	0.68533129
41	0.68533129
42	0.68533129
43	0.68533129
44	0.68533129
45	0.68533129
46	0.68533129
47	0.68533129
48	0.68533129
49	0.68533129
50	0.68533129
51	0.83869825
52	0.83869825
53	0.83869825
54	0.83869825
55	0.83869825
56	0.83869825
57	0.83869825
58	0.83869825
59	0.83869825
60	0.83869825
61	0.83869825
62	0.83869825
63	0.83869825
64	0.83869825
65	0.83869825
66	0.83869825
67	0.83869825
68	0.83869825
69	0.83869825
70	0.83869825
71	0.83869825
72	0.83869825
73	0.83869825
74	0.83869825
75	1.1378306
76	1.1378306
77	1.1378306
78	1.1378306
79	1.1378306
80	1.1378306
81	1.1378306
82	1.1378306
83	1.1378306
84	1.1378306
85	1.1378306
86	1.1378306
87	1.2852508
88	1.2852508
89	1.2852508
90	1.2852508
91	1.2852508
92	1.2852508
93	1.2860149
94	1.2860149
95	1.2860149
96	1.2860149
97	1.2860149
98	1.2860149
99	1.2860149
100	1.2860149
101	1.2860149
102	1.2860149
103	1.2860149
104	1.2860149
105	1.2860149
106	1.2860149
107	1.2860149
108	1.2860149
109	1.2860149
110	1.2860149
111	1.2860149
112	1.2860149
113	1.2860149
114	1.2860149
115	1.2860149
116	1.2860149
117	1.4331484
118	1.4331484
119	1.4331484
120	1.4331484
121	1.4331484
122	1.4331484
123	1.4331484
124	1.4331484
125	1.4331484
126	1.4331484
127	1.4331484
128	1.4331484
129	1.4331484
130	1.4331484
131	1.4331484
132	1.4331484
133	1.4331484
134	1.4331484
135	1.4331484
136	1.4331484
137	1.4331484
138	1.4331484
139	1.4331484
140	1.4331484

#+NAME:vir_energies-38

1	0.089587821
2	0.089587821
3	0.089587821
4	0.089587821
5	0.089587821
6	0.089587821
7	0.089587821
8	0.089587821
9	0.19634974
10	0.19634974
11	0.19634974
12	0.19634974
13	0.19634974
14	0.19634974
15	0.28787609
16	0.28787609
17	0.28787609
18	0.28787609
19	0.28787609
20	0.28787609
21	0.28787609
22	0.28787609
23	0.28787609
24	0.28787609
25	0.28787609
26	0.28787609
27	0.28787609
28	0.28787609
29	0.28787609
30	0.28787609
31	0.28787609
32	0.28787609
33	0.28787609
34	0.28787609
35	0.28787609
36	0.28787609
37	0.28787609
38	0.28787609
39	0.38006254
40	0.38006254
41	0.38006254
42	0.38006254
43	0.38006254
44	0.38006254
45	0.38006254
46	0.38006254
47	0.38006254
48	0.38006254
49	0.38006254
50	0.38006254
51	0.38006254
52	0.38006254
53	0.38006254
54	0.38006254
55	0.38006254
56	0.38006254
57	0.38006254
58	0.38006254
59	0.38006254
60	0.38006254
61	0.38006254
62	0.38006254
63	0.54632180
64	0.54632180
65	0.54632180
66	0.54632180
67	0.54632180
68	0.54632180
69	0.54632180
70	0.54632180
71	0.54632180
72	0.54632180
73	0.54632180
74	0.54632180
75	0.62788608
76	0.62788608
77	0.62788608
78	0.62788608
79	0.62788608
80	0.62788608
81	0.62788608
82	0.62788608
83	0.62788608
84	0.62788608
85	0.62788608
86	0.62788608
87	0.62788608
88	0.62788608
89	0.62788608
90	0.62788608
91	0.62788608
92	0.62788608
93	0.62788608
94	0.62788608
95	0.62788608
96	0.62788608
97	0.62788608
98	0.62788608
99	0.62856333
100	0.62856333
101	0.62856333
102	0.62856333
103	0.62856333
104	0.62856333
105	0.70752745
106	0.70752745
107	0.70752745
108	0.70752745
109	0.70752745
110	0.70752745
111	0.70752745
112	0.70752745
113	0.70752745
114	0.70752745
115	0.70752745
116	0.70752745
117	0.70752745
118	0.70752745
119	0.70752745
120	0.70752745
121	0.70752745
122	0.70752745
123	0.70752745
124	0.70752745
125	0.70752745
126	0.70752745
127	0.70752745
128	0.70752745
129	0.78589516
130	0.78589516
131	0.78589516
132	0.78589516
133	0.78589516
134	0.78589516
135	0.78589516
136	0.78589516
137	0.78589516
138	0.78589516
139	0.78589516
140	0.78589516
141	0.78589516
142	0.78589516
143	0.78589516
144	0.78589516
145	0.78589516
146	0.78589516
147	0.78589516
148	0.78589516
149	0.78589516
150	0.78589516
151	0.78589516
152	0.78589516
153	0.86364035
154	0.86364035
155	0.86364035
156	0.86364035
157	0.86364035
158	0.86364035
159	0.86364035
160	0.86364035
161	0.94081079
162	0.94081079
163	0.94081079
164	0.94081079
165	0.94081079
166	0.94081079
167	0.94081079
168	0.94081079
169	0.94081079
170	0.94081079
171	0.94081079
172	0.94081079
173	0.94081079
174	0.94081079
175	0.94081079
176	0.94081079
177	0.94081079
178	0.94081079
179	0.94081079
180	0.94081079
181	0.94081079
182	0.94081079
183	0.94081079
184	0.94081079
185	1.0177694
186	1.0177694
187	1.0177694
188	1.0177694
189	1.0177694
190	1.0177694
191	1.0177694
192	1.0177694
193	1.0177694
194	1.0177694
195	1.0177694
196	1.0177694
197	1.0177694
198	1.0177694
199	1.0177694
200	1.0177694
201	1.0177694
202	1.0177694
203	1.0177694
204	1.0177694
205	1.0177694
206	1.0177694
207	1.0177694
208	1.0177694
209	1.0177694
210	1.0177694
211	1.0177694
212	1.0177694
213	1.0177694
214	1.0177694
215	1.0177694
216	1.0177694
217	1.0177694
218	1.0177694
219	1.0177694
220	1.0177694
221	1.0177694
222	1.0177694
223	1.0177694
224	1.0177694
225	1.0177694
226	1.0177694
227	1.0177694
228	1.0177694
229	1.0177694
230	1.0177694
231	1.0177694
232	1.0177694
233	1.1708206
234	1.1708206
235	1.1708206
236	1.1708206
237	1.1708206
238	1.1708206
239	1.2468683
240	1.2468683
241	1.2468683
242	1.2468683
243	1.2468683
244	1.2468683
245	1.2468683
246	1.2468683
247	1.2468683
248	1.2468683
249	1.2468683
250	1.2468683
251	1.2468683
252	1.2468683
253	1.2468683
254	1.2468683
255	1.2468683
256	1.2468683
257	1.2468683
258	1.2468683
259	1.2468683
260	1.2468683
261	1.2468683
262	1.2468683
263	1.2469208
264	1.2469208
265	1.2469208
266	1.2469208
267	1.2469208
268	1.2469208
269	1.2469208
270	1.2469208
271	1.2469208
272	1.2469208
273	1.2469208
274	1.2469208
275	1.2469208
276	1.2469208
277	1.2469208
278	1.2469208
279	1.2469208
280	1.2469208
281	1.2469208
282	1.2469208
283	1.2469208
284	1.2469208
285	1.2469208
286	1.2469208
287	1.3228043
288	1.3228043
289	1.3228043
290	1.3228043
291	1.3228043
292	1.3228043
293	1.3228043
294	1.3228043
295	1.3228043
296	1.3228043
297	1.3228043
298	1.3228043
299	1.3228723
300	1.3228723
301	1.3228723
302	1.3228723
303	1.3228723
304	1.3228723
305	1.3228723
306	1.3228723
307	1.3228723
308	1.3228723
309	1.3228723
310	1.3228723
311	1.3228723
312	1.3228723
313	1.3228723
314	1.3228723
315	1.3228723
316	1.3228723
317	1.3228723
318	1.3228723
319	1.3228723
320	1.3228723
321	1.3228723
322	1.3228723

#+NAME:vir_energies-hbn

1	0.46523374
2	0.47190413
3	0.52766278
4	0.75894665
5	0.79372894
6	0.81533720
7	0.85502910
8	0.85996841
9	0.86164099
10	0.87081777
11	0.92274576
12	0.93485125
13	0.98156862
14	0.98258918
15	0.98352706
16	1.0139348
17	1.1023545
18	1.1149920
19	1.2342007
20	1.2570760
21	1.2905259
22	1.3642916
23	1.3661178
24	1.4402250
25	1.5383045
26	1.5553226
27	1.6069830
28	1.6114899
29	1.6612452
30	1.6830711
31	1.7813984
32	1.8183162
33	1.8190747
34	1.8305486
35	1.8460614
36	1.8723801
37	2.0206319
38	2.0432636
39	2.0757063
40	2.1004740
41	2.1054700
42	2.1063651
43	2.2238620
44	2.2446128
45	2.3007112
46	2.3328563
47	2.4020334
48	2.4188139
49	2.4204084
50	2.4650720
51	2.5651099
52	2.7104512
53	2.7373724
54	2.7437489
55	2.8547023
56	2.8580549
57	2.8726837
58	2.9212925
59	2.9333791
60	2.9595462
61	2.9712978
62	2.9781069
63	2.9805534
64	3.0321222
65	3.0844009
66	3.0880791
67	3.1729389
68	3.1765382
69	3.2249290
70	3.2617942
71	3.2988499
72	3.5313407
73	3.5900472
74	3.5924177
75	3.7169678
76	3.7567167
77	3.8425762
78	3.8658065
79	3.8664628
80	3.9456514

#+NAME:vir_energies-hbn-36

1	0.45821960
2	0.45955397
3	0.52417218
4	0.75814810
5	0.76944736
6	0.79918247
7	0.83833757
8	0.84234429
9	0.84624883
10	0.84773610
11	0.86555326
12	0.90752856
13	0.91381841
14	0.93078158
15	0.94927271
16	1.0019894
17	1.0516346
18	1.0603957
19	1.1622263
20	1.2124273
21	1.2251086
22	1.2669459
23	1.3119527
24	1.3236910
25	1.4020074
26	1.4180400
27	1.4247132
28	1.4252123
29	1.4343132
30	1.4943183
31	1.5538082
32	1.5739816
33	1.5797415
34	1.5878624
35	1.6135695
36	1.6136001
37	1.6199772
38	1.6589831
39	1.7186777
40	1.7815760
41	1.8076690
42	1.8148269
43	1.8396877
44	1.8671517
45	1.8879163
46	1.9024199
47	1.9776692
48	1.9806444
49	2.0328782
50	2.0402167
51	2.0535394
52	2.1583521
53	2.2173123
54	2.2298141
55	2.2445000
56	2.2944750
57	2.3297962
58	2.3424792
59	2.3443938
60	2.3605343
61	2.4364144
62	2.4587039
63	2.4595817
64	2.5735021
65	2.5857946
66	2.6315694
67	2.6660058
68	2.6670340
69	2.7380125
70	2.8020294
71	2.8441878
72	2.8727076
73	2.8830104
74	2.9157291
75	2.9276150
76	2.9350208
77	2.9790396
78	3.0588475
79	3.1333010
80	3.1380314
81	3.1675500
82	3.1780752
83	3.1878095
84	3.2247691
85	3.2446009
86	3.2626894
87	3.3304832
88	3.3478010
89	3.4550268
90	3.4889406
91	3.5369474
92	3.5465361
93	3.5513747
94	3.5521487
95	3.5986419
96	3.6403020
97	3.7330773
98	3.7426986
99	3.8073074
100	3.8372094
101	3.8411368
102	3.8413932
103	3.8451913
104	3.8725514
105	3.9332957
106	3.9460972
107	3.9506332
108	3.9697667
109	3.9830317
110	4.0843935
111	4.1546720
112	4.1606148
113	4.1872647
114	4.1948277
115	4.1983197
116	4.2076585
117	4.2141985
118	4.2384778
119	4.2905590
120	4.3597337
121	4.4268950
122	4.4429214
123	4.4447499
124	4.4831871
125	4.4875016
126	4.5399434
127	4.5508515
128	4.5755937
129	4.5866014
130	4.7271287
131	4.7297052
132	4.7502066
133	4.7600181
134	4.7745013
135	4.8402334
136	4.8438511
137	4.8754985
138	4.8769072
139	4.9093041
140	4.9869284
141	4.9905789
142	4.9981826
143	5.0028200
144	5.0579505
145	5.0811681
146	5.1274989
147	5.1731798
148	5.2753752
149	5.2894543
150	5.2915312
151	5.3884059
152	5.4042175
153	5.4225445
154	5.5326874
155	5.5802786
156	5.6484393
157	5.7163910
158	5.7426619
159	5.9004863
160	6.0031234

Code

epsilon = np.array(orb_energies)[:,1]
epsilon_v = np.array(vir_energies)[:,1]
e_occ = 3.*sum(epsilon)/len(epsilon)
e_occ

-1.208795827875

Uniform distribution

Below is a simple code which computes the stochastic average with a uniform sampling. Each unit of time corresponds to the computation of a $E_{abc}$ contribution. The output is a table containing the time, the running average and the statistical error.

The maximum running time for the computation of the exact quantity is $N_v^3$ =

len(eabc)

date = 0        # Current time: increments when a new triplet is computed
ET  = 0.        # Accumulator
ET2 = 0.        # Accumulator of E^2 for variance
computed = np.zeros(len(eabc))   # If true, the (a,b,c) triplet has already
                              # been computed
result = {}

isample = 0
time0 = time.time()
while date < Nabc*0.99:
 if time.time() - time0 > 60.: break
 isample += 1
 eta = np.random.uniform()   # Random number between 0 and 1
 abc = int(eta * Nabc)       # Triplet (a,b,c) drawn
 s = eabc[abc] * Nabc
 if computed[abc] == 0:
     computed[abc] = 1
     date += 1
 ET += s
 ET2 += s*s
 if date % int(Nabc/100) == 0:
     average  = ET/isample
     variance = ET2/isample - average*average
     result[date/Nabc] =  (average, np.sqrt(variance/(isample-1)))

[ ("%.3f"%k, v1, v2) for (k, (v1, v2)) in result.items() ]

#+NAME:data_uniform

0.010	-0.03437642645957338	0.0030753630637821525
0.020	-0.03196781983908248	0.001837822949804852
0.030	-0.03177580311852873	0.001720127332029026
0.040	-0.032298975146952	0.001428537377973462
0.050	-0.031126606689641206	0.001185117590763498
0.060	-0.030516572686857285	0.001022406320141158
0.070	-0.03031613476271244	0.0009134000426629288
0.080	-0.030158573217771642	0.0008258298216325343
0.090	-0.029933996997862877	0.0007575679186364425
0.100	-0.029939102663094793	0.0007449215086748057
0.110	-0.03027078211599003	0.0007190676006701886
0.120	-0.030412162274688476	0.000685566251636473
0.130	-0.030270750417570543	0.0006430858167234617
0.140	-0.030498507348269383	0.0006158540830116521
0.150	-0.030490507925363328	0.0005906102179915185
0.160	-0.030518954198717314	0.0005703431837634973
0.170	-0.030213640271687846	0.0005409517176894218
0.180	-0.030258679245643547	0.0005253652905405611
0.190	-0.030333274426206692	0.0005198998863964522
0.200	-0.0302733911551464	0.0005005950671897297
0.210	-0.03026559364033272	0.0004825220030277329
0.220	-0.03025586551613391	0.0004697550023708562
0.230	-0.030302109366235522	0.0004642042573853746
0.240	-0.030334648269316054	0.0004512365616056368
0.250	-0.030361980739061295	0.00043886764696707026
0.260	-0.030283750363813	0.0004275759671604114
0.270	-0.03019387833526731	0.00041502890927369997
0.280	-0.030136619610862777	0.00040250878507234827
0.290	-0.030268317789462457	0.0003982341164866337
0.300	-0.030181731817639737	0.0003874278989981271
0.310	-0.030301090062314424	0.0003846766981229691
0.320	-0.030181632844130373	0.0003735298694478283
0.330	-0.030346788080401393	0.00037055227897075124
0.340	-0.030321083505389944	0.0003625084242273785
0.350	-0.03031844120867735	0.000356806596366249
0.360	-0.03035148716708672	0.0003506711400078661
0.370	-0.030277830709765356	0.000342171531867949
0.380	-0.030378816651894357	0.00033986289765587766
0.390	-0.030365402689548193	0.00033692599619419145
0.400	-0.030354069178420165	0.0003304829598541888
0.410	-0.03035383639056811	0.0003244053095678985
0.420	-0.030471269175802472	0.00032212755052154384
0.430	-0.030418257107866987	0.00031449404856775384
0.440	-0.030474283193158074	0.00030961993631343425
0.450	-0.03040771869558633	0.0003028770206064099
0.460	-0.03043384154552774	0.0002986987882455902
0.470	-0.03039059595415329	0.0002931504370002957
0.480	-0.030399559731904162	0.0002914390291029102
0.490	-0.03040840624488255	0.0002871330399747792
0.500	-0.030433618775920112	0.0002831855505470479
0.510	-0.030391034199860004	0.0002784795162104784
0.520	-0.030423899086325607	0.0002743308344880197
0.530	-0.030444941603789376	0.00027245899521168617
0.540	-0.03041330750788333	0.00026863589194129245
0.550	-0.030443843453372817	0.0002647887384537327
0.559	-0.030395222302231326	0.0002601415325019499
0.569	-0.03038575991898126	0.0002560350009762799
0.579	-0.030397495421124035	0.0002527936986919491
0.589	-0.03035703357086474	0.00024921038282965305
0.599	-0.030367073503388766	0.00024530800378388815
0.609	-0.03034865292438221	0.00024057255779665634
0.619	-0.030347539456847126	0.00023759795719966657
0.629	-0.03036816530066652	0.00023386406758813552
0.639	-0.030309451544814934	0.0002309476317856454
0.649	-0.03028625725934087	0.0002285039730992457
0.659	-0.030283171113352016	0.00022505030956627053
0.669	-0.030216545004483557	0.00022075959106823286
0.679	-0.03018791803686861	0.000216409617148716
0.689	-0.030130871402653105	0.00021274698033747194
0.699	-0.030153535988059074	0.00020996270408764868
0.709	-0.030168044632427723	0.0002072153449956144
0.719	-0.03015132154087348	0.00020371937659283525
0.729	-0.03014144950971066	0.00020003627563828567
0.739	-0.030186887963037024	0.00019794215640513352
0.749	-0.030190200081583077	0.00019445302614139054
0.759	-0.03018461151070908	0.00019183973114613275
0.769	-0.03015529655509179	0.00018896773795239234
0.779	-0.030134327355261274	0.00018637601547252417
0.789	-0.03020981162451403	0.0001840077069555368
0.799	-0.030186994680259767	0.0001818700167991961
0.809	-0.030213376244598432	0.00018020714866842657
0.819	-0.030183390128740703	0.00017697198221723528
0.829	-0.030188624472397507	0.00017415838147679145
0.839	-0.030213048689140157	0.0001718757355807843
0.849	-0.030212269469307065	0.0001688540171202461
0.859	-0.03023028482655304	0.00016658369092061525
0.869	-0.03025162562379045	0.00016327923756701636
0.879	-0.030271262286699296	0.00016065941240476654
0.889	-0.03024066238325353	0.00015672825220353865
0.899	-0.030180757283723773	0.00015292568557125476
0.909	-0.030208761640158285	0.00015035198771780757
0.919	-0.030197429406441458	0.0001473411347540838
0.929	-0.03020283963451932	0.0001434342057913342
0.939	-0.030210225556278718	0.0001389552464182351
0.949	-0.030208953008535155	0.0001355572097081645
0.959	-0.030234393661962507	0.00013108868970967383
0.969	-0.03017826794775221	0.00012494937843904978
0.979	-0.03019828956553072	0.00011867981991075709
0.989	-0.03022660358379614	0.00010982186878635525

Importance sampling

Preparation

Sort the data according to $-1/\Delta \epsilon_o \times 1/\epsilon_a$:

def importance_function(i):
  ea, eb, ec = epsilon_v[triplets[i,:]-1]
#     return -1./denom[i]
#     return -1./denom[i]**2
#     return -1./denom[i] * 1./ea
  return -1./denom[i] * 1./ea**2
#     return -1./(ea*eb*ec)
#     return -1./(ea*eb*ec)**2
#     return -1./denom[i] * (
#                1./((ea-eb)**2-e_occ) + 1./((ea-ec)**2-e_occ) + 1./((eb-ec)**2-e_occ))
#     return 1.
 
to_sort = []
for i in range(Nabc):
 to_sort.append( (importance_function(i), eabc[i], denom[i]) ) 
to_sort.sort()

eabc_sorted = []
denom_sorted = []
for (_, y, z) in to_sort:
 eabc_sorted.append(y)
 denom_sorted.append(z)
denom_sorted = np.array(denom_sorted)
eabc_sorted = np.array(eabc_sorted)

We build the array waccu which contains $w(x) = \frac{\int_0^x P(y)\,dy}{\int_0^1 P(y)\, dy}$ for the inverse transform sampling.

w = []
for (x, _, _) in to_sort:
 w.append(-x)
w = np.array(w)
waccu = np.array(w)

sum_w= 0.
for i in range(len(w)):
 sum_w += w[i]
 waccu[i] = sum_w
waccu = waccu / sum_w
w = w / sum_w
w

0.000162384392

0.000161615667

0.000156174841

…

2.82313861e-07

2.80523353e-07

2.80412476e-07

We will draw uniform random numbers and find their index in the array waccu:

def find_sample(eta):
 return np.searchsorted(waccu, eta, side='left', sorter=None)

Sampling

ET  = 0.        # Accumulator
ET2 = 0.        # Accumulator of E^2 for variance
computed = np.zeros(Nabc)  # If true, the (a,b,c) triplet has already
                         # been computed
date = 0.        # Current time: increments when a new triplet is computed
dt = 1./Nabc
result = {}

isample = 0
prevdate = 0.
time0 = time.time()
while date < 0.99: 
  if time.time() - time0 > 60.: break
  isample += 1
  eta = np.random.uniform()   # Random number between 0 and 1
  abc = find_sample(eta)
  s = eabc_sorted[abc] / w[abc]
  if computed[abc] == 0:
      computed[abc] = 1
      date += dt
  ET += s
  ET2 += s*s
  if date - prevdate > 0.01:
      prevdate = date
      dtmp = 1./isample
      average  = ET*dtmp 
      variance = ET2*dtmp - average*average
      result[date] = ( average, np.sqrt(variance/(isample-1)) )

[ ("%.3f"%k, v1, v2) for (k, (v1, v2)) in result.items() ]

#+NAME:data_importance

0.010	-0.03027212822147209	0.0011742893979352206
0.020	-0.0313009364341866	0.0009252120899929328
0.030	-0.031057363530977423	0.0007281055373661676
0.040	-0.031249853375632717	0.0006402801404234216
0.050	-0.03122110948462111	0.0005750574146217905
0.060	-0.03128348204672714	0.0005275262000247139
0.070	-0.0308998979808618	0.0004726597407782536
0.080	-0.03082342928319695	0.00044136523641729995
0.090	-0.03061878595835833	0.00040625560663516984
0.100	-0.03039442587808384	0.00037445927958614157
0.110	-0.030414153392358843	0.0003555022536120971
0.120	-0.030433890781777325	0.0003386617754861883
0.130	-0.030412842186671795	0.00032521342707965395
0.140	-0.030135851295907408	0.0003054031156376609
0.150	-0.03017758373987041	0.00029160561167065966
0.160	-0.030071710241627626	0.00028285433322913355
0.170	-0.030187691754911785	0.00027362801402723217
0.180	-0.03017792919745682	0.00026256836577371164
0.190	-0.030066918230931893	0.0002520216317979708
0.200	-0.03012732632783178	0.0002442961255034912
0.210	-0.030138915946570068	0.00023654277814720348
0.220	-0.030135951124831976	0.00022882972065217942
0.230	-0.030189298295077573	0.00022288280644866787
0.240	-0.030179114235883018	0.0002165756153934681
0.250	-0.03020755367006979	0.00021062441315912225
0.260	-0.030150949451180773	0.00020297280220009072
0.270	-0.030113105404734098	0.0001968538782253076
0.280	-0.03001178610814453	0.000190510244582427
0.290	-0.030105460516270968	0.00018580175733023656
0.300	-0.030206578515471226	0.0001820686148935213
0.310	-0.030220596073402568	0.00017695300024148726
0.320	-0.030253362198926707	0.000173110033483545
0.330	-0.03025716666455987	0.00016884806376386556
0.340	-0.03024767268207987	0.00016474215718864923
0.350	-0.030245608121976592	0.0001610389817601625
0.360	-0.030228130070767308	0.00015681822177276763
0.370	-0.03019666434207847	0.00015286660948749957
0.380	-0.0302247108757732	0.00014956619859720995
0.390	-0.030277506060936692	0.0001473951646458563
0.400	-0.030311036212474794	0.00014525747787829678
0.410	-0.030319395740519034	0.0001423403120586929
0.420	-0.030328754540608446	0.00013953020416052482
0.430	-0.030335025610265645	0.00013717074592026099
0.440	-0.03032536551327421	0.00013420580225141168
0.450	-0.030317525082447477	0.00013090745868107453
0.460	-0.030331154567198596	0.0001280412784818255
0.470	-0.030346821228602428	0.0001254894342261729
0.480	-0.030341062669058966	0.0001227585525308749
0.490	-0.03032284123943016	0.0001199017169732116
0.500	-0.030324593697131128	0.00011716907194040494
0.510	-0.03036599520797054	0.0001149841429523499
0.520	-0.030373500023963107	0.00011257695085119413
0.530	-0.030398568034173238	0.00011033955037405243
0.540	-0.030379771640186178	0.00010826945179330241
0.550	-0.030392078925771775	0.00010591950473282571
0.560	-0.0304026454971627	0.00010358492676711116
0.570	-0.030389082287990102	0.0001011027020601961
0.580	-0.03042023586750155	9.908928944756498e-05
0.590	-0.03042662529206729	9.688702814501591e-05
0.600	-0.03044723783359543	9.482918893094575e-05
0.610	-0.03043219469351895	9.26971869085586e-05
0.620	-0.03041964871512123	9.045177316867549e-05
0.630	-0.03040801626098958	8.839118804599184e-05
0.640	-0.030423340462181085	8.667968332871836e-05
0.650	-0.03039653789830176	8.44014731598794e-05
0.660	-0.03038052592884054	8.248031393751431e-05
0.670	-0.030383047104467806	8.066997772997924e-05
0.680	-0.03037101495895282	7.865737348265977e-05
0.690	-0.030390109013467823	7.687785167133467e-05
0.700	-0.03039164278824196	7.530409352036397e-05
0.710	-0.030376353526121343	7.323158011397256e-05
0.720	-0.030343481470340126	7.126805565409954e-05
0.730	-0.03031777838164579	6.936365378349401e-05
0.740	-0.030329363559433456	6.769454854815683e-05
0.750	-0.030330011749874996	6.59686784438342e-05
0.760	-0.030329426846525974	6.427621173496454e-05
0.770	-0.030326548388485877	6.244765973081217e-05
0.780	-0.030329650883986148	6.082490018420884e-05
0.790	-0.030322073238055823	5.910514588411257e-05
0.800	-0.030312381215268632	5.7421583243115227e-05
0.810	-0.030299053653321258	5.5699918230207584e-05
0.820	-0.030311512280480676	5.4102331646809365e-05
0.830	-0.030325578671112484	5.2582898838102214e-05
0.840	-0.030321914469912063	5.089177178614602e-05
0.850	-0.030319285429001232	4.914371564960337e-05
0.860	-0.030313500328097136	4.7470383207069204e-05
0.870	-0.030305455309798633	4.582229909456817e-05
0.880	-0.03029270937963236	4.401899624643128e-05
0.890	-0.030287372422237557	4.221534263667604e-05
0.900	-0.030273826343971428	4.0445160816126796e-05
0.910	-0.030279937846679802	3.864802165815449e-05
0.920	-0.03027407657538685	3.680080289715885e-05
0.930	-0.030275647635292845	3.5030900270752445e-05
0.940	-0.030275268108168343	3.30505449660589e-05
0.950	-0.030255391500928076	3.1140646530112084e-05
0.960	-0.030249858250144413	2.9083139511987113e-05
0.970	-0.030243251806653753	2.6630322771430047e-05
0.980	-0.030257401812890404	2.3870299829544822e-05

Semi-stochastic

Preparation

We split the contributions into 40 buckets:

Nbuckets = 40 # Number of buckets

bucket = []
abc_old = 0
idx0 = 0
idx1 = 0
bounds = []
for i in range(Nbuckets):
  accu = 0
  xi = (i+1)/Nbuckets
  while idx1 < len(eabc) and waccu[idx1] <= xi:
      accu += 1
      idx1 += 1
  bucket.append(accu)
  bounds.append( (idx0, idx1) )
  idx0 = idx1

Contributions of each bucket:

#+name:E_B

result = []
for i in range(Nbuckets):
    result.append( (i+1, sum(eabc_sorted[bounds[i][0]:bounds[i][1]])) )
result

1	-0.0013677742901972176
2	-0.0011041914108332854
3	-0.0009275898609788052
4	-0.0008073200317293237
5	-0.0007790507659793898
6	-0.0006121615146329108
7	-0.0005770622533542697
8	-0.0005142604992375494
9	-0.00046382101884437956
10	-0.0004039453565831979
11	-0.000389650155472885
12	-0.00029544073304587146
13	-0.0003250768234388132
14	-0.00028341936089169763
15	-0.0002473003234040006
16	-0.0003442889080859639
17	-0.0003435619631738471
18	-0.0004858997139826501
19	-0.0009618635728221805
20	-0.001429989465699455
21	-0.0013470049374201519
22	-0.001176633686853677
23	-0.0009970812970662656
24	-0.0009399212374881834
25	-0.0009377932840110686
26	-0.0008548897786545894
27	-0.0007917116515238299
28	-0.0008377443725186978
29	-0.0008085988173851557
30	-0.0007679194358436122
31	-0.0006805914110088752
32	-0.0006632464581929114
33	-0.000653886276418067
34	-0.0006694575639034294
35	-0.0006153089357946734
36	-0.0008010751950592366
37	-0.0010914570803250902
38	-0.0011707797198706584
39	-0.0009276499869679824
40	-0.0008310131640315182

reset
set grid
set xlabel "Bucket index"
set ylabel "E_B"
plot data u 1:2 w lp title ""

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/E_B.png

Sampling

ET  = np.zeros(Nbuckets)  # Energy of the bucket
ET2 = np.zeros(Nbuckets)
computed = np.zeros(Nabc)        # If true, the (a,b,c) triplet has already
                               # been computed
bucket_comp = [ 0 ]*Nbuckets     # Number of computed elements in the bucket
bucket_sampled = [ 0 ]*Nbuckets  # Number of samples drawn in the bucket
imin = 0                         # First non-computed contribution
date = 0
dt = 1./Nabc
result = {}

isample = 0
prevdate = 0.
time0 = time.time()
while date < 0.99:
   if time.time() - time0 > 60.: break
   isample += 1
   # Force computation of 1st uncomputed triplet
   while imin < Nabc and computed[imin] == 1:
       imin += 1
   computed[imin] = 1
   date += dt
   for i in range(Nbuckets):
       if imin < bounds[i][1]:
           bucket_comp[i] += 1
           break

   # Random sampling
   eta = np.random.uniform(0.,1./Nbuckets)   # Random number between
                                             # 0 and 1/Nbuckets
#    if True:
#        i = np.random.randint(low=0,high=Nbuckets)
   for i in range(Nbuckets):
       eta_i = eta + i/Nbuckets
       abc = find_sample(eta_i)
       if computed[abc] == 0:
           computed[abc] = 1
           date += dt
       bucket_comp[i] += 1
       bucket_sampled[i] += 1
       s = eabc_sorted[abc] / w[abc] 
       ET[i] += s
       ET2[i] += s*s
       
   if date - prevdate > 0.001:
       prevdate = date
       sum_det = 0.
       average = 0.
       variance = 0.
       N = 0
       norm = 0.
       for i in range(Nbuckets):
           if bucket_comp[i] >= bucket[i]:
               sum_det += sum(eabc_sorted[bounds[i][0]:bounds[i][1]])
           else: 
               average  += ET [i] 
               variance += ET2[i] 
               N += bucket_sampled[i]
               norm += sum(w[bounds[i][0]:bounds[i][1]])
       if N == 0:
           break
       average  = average/N * norm 
       variance = variance/N * norm * norm - average*average
       result[date] = ( sum_det + average, np.sqrt(variance/(N-1)) )

[ ("%.3f"%k, v1, v2) for (k, (v1, v2)) in result.items() ]

#+NAME:data_semistoch

0.001	-0.03700252934055045	0.0039446587198956025
0.003	-0.03816535382359609	0.0038208208777224563
0.004	-0.03823708722776118	0.0028909427306171133
0.006	-0.035477058071137275	0.002247199750908954
0.007	-0.034245613400924205	0.001896709248931999
0.008	-0.03364206951878407	0.0016639509024875523
0.010	-0.03272048663758599	0.0014711824914169181
0.011	-0.03204661390178344	0.001322421630968049
0.012	-0.03247239543109624	0.0012886273407492139
0.014	-0.03175941391454308	0.00118359399240578
0.015	-0.03147754125129678	0.0011108993494206017
0.016	-0.031017678239190105	0.0010391294918667723
0.018	-0.031231735536857996	0.0009952425287539474
0.019	-0.030918409685504133	0.0009434169449510998
0.020	-0.03076948352418771	0.000903940058511309
0.021	-0.030731805827973045	0.000870986845165927
0.023	-0.030657131449181856	0.0008446635016691032
0.024	-0.030418434317601425	0.0008236052557121908
0.026	-0.030413475405211866	0.0008038247047447392
0.027	-0.0302634458743812	0.00077726661752534
0.028	-0.030099860020646314	0.0007493781343440913
0.030	-0.030031414584741206	0.0007247571482425683
0.031	-0.029861913390748302	0.0007034977703759773
0.032	-0.030059889998951636	0.000683003955292924
0.034	-0.029942110833037733	0.0006673144407280344
0.035	-0.03010585057798657	0.0006590408230506174
0.036	-0.030239317949261265	0.0006561827265267656
0.037	-0.030196049865866686	0.0006443243760346761
0.038	-0.030241106881002105	0.0006319770449670202
0.040	-0.03016811032728042	0.000617108033453784
0.041	-0.030350176642184847	0.0006165635992594986
0.042	-0.030240504055469575	0.0006027883414881043
0.044	-0.030243452024318145	0.000592125863157674
0.045	-0.030224876961973447	0.0005814798995328184
0.046	-0.030123772311770113	0.0005681146708359756
0.048	-0.030144622828395962	0.0005605907864994292
0.049	-0.030015693219271205	0.0005321448596201531
0.050	-0.03018458396335074	0.0005355348097130672
0.052	-0.03016822279138483	0.0005238345524062661
0.053	-0.03011562822223151	0.0005160375156707
0.054	-0.030195222853138512	0.0005103559989166096
0.055	-0.03010728198315345	0.0005014043081161718
0.056	-0.030248017288061903	0.0004970499573047812
0.058	-0.030191948307978143	0.000490481069666529
0.059	-0.030125170066960517	0.000483367721285067
0.060	-0.030154424033964577	0.0004800909060633886
0.061	-0.030133248378154606	0.00047395650514493845
0.062	-0.030107309050102435	0.0004680280913700501
0.063	-0.030272043639534847	0.00046749450189163664
0.065	-0.03031856589756985	0.00046210918278845266
0.066	-0.030277388584620448	0.00045599444321776407
0.067	-0.03036370316099435	0.00045280823138617787
0.069	-0.03046322299658464	0.00045512734443032383
0.070	-0.03035119702465893	0.0004486259793506022
0.071	-0.030419039108779976	0.00044817401080980614
0.073	-0.030395213559428674	0.0004398158499293913
0.074	-0.030341524748297603	0.00043307016376981794
0.075	-0.030283854097569864	0.00042836583969847286
0.076	-0.030253742572353238	0.00042354346970364465
0.077	-0.03027487341006264	0.00042009345507324294
0.078	-0.0302785611097896	0.000416736445723873
0.080	-0.030281274099255278	0.0004120406856983387
0.081	-0.030290494100580056	0.00040908692765803266
0.082	-0.030266893434727403	0.0004028018039494786
0.083	-0.030218347929600587	0.0003980635385584676
0.085	-0.030211381844881494	0.0003925339697622152
0.086	-0.03017150141441146	0.00038838228641229313
0.087	-0.030291812317871827	0.0003813236293705212
0.088	-0.03036709412699521	0.00037996636281608693
0.089	-0.030373880408704056	0.00037560712293775744
0.091	-0.030350177812443428	0.0003720575164884457
0.092	-0.03039761574920972	0.0003708143976083477
0.093	-0.030393689061324412	0.0003680000113594057
0.094	-0.030422938001869772	0.00036615246750459
0.095	-0.030403519598244076	0.00036306145436255764
0.096	-0.030372434859504083	0.0003592451269566279
0.098	-0.03042907201231523	0.0003585900840137507
0.099	-0.03047975612038179	0.0003583197208192352
0.100	-0.030471112401891765	0.0003555786225227624
0.101	-0.03043905824709334	0.0003528144664398024
0.102	-0.030478758561294655	0.00035116811688065225
0.103	-0.030414108061452134	0.0003478379830471161
0.105	-0.030376183345412592	0.00034484121606776153
0.106	-0.030380072599887526	0.0003424579946640505
0.107	-0.030387583043484854	0.00034024031802088996
0.108	-0.030374165523760457	0.000338044280965926
0.110	-0.03033849374729628	0.0003345066245276828
0.111	-0.03031727568567442	0.00033052853662265654
0.112	-0.030281967826368623	0.00032835256029365855
0.113	-0.030265615547090625	0.00032541475802995727
0.115	-0.03026128193460788	0.0003224163021125531
0.116	-0.03024607304019136	0.00031972980916718203
0.117	-0.03026073495328826	0.00031499577065609947
0.118	-0.030264954328951987	0.0003132423667434477
0.119	-0.03026289333742615	0.00031130745336599094
0.120	-0.030239630655707395	0.00030908035669914504
0.121	-0.030203261923097146	0.0003061335983241776
0.123	-0.030231088082779283	0.0003049587459237616
0.124	-0.0301730313875267	0.000302370517010408
0.125	-0.030202128014130414	0.00029943670500927897
0.126	-0.030201066947482796	0.0002977742191085441
0.128	-0.030188591868682564	0.0002968017991007076
0.129	-0.030144215932881073	0.00029421439605051367
0.130	-0.03018152969570565	0.0002945417297584062
0.131	-0.030151259521389383	0.00028932786786329504
0.132	-0.03016450051716881	0.0002871074159703193
0.133	-0.030181448983663456	0.0002865425972303796
0.135	-0.030216273138280916	0.00028528209181156853
0.136	-0.030199968626720503	0.00028344547928226357
0.137	-0.03023700286436646	0.0002825077893557387
0.138	-0.030247039925638375	0.00028090496168322763
0.139	-0.030162271861245007	0.0002738852550863887
0.140	-0.030122989681895636	0.00027167922190168874
0.141	-0.030134372467299327	0.0002709623098598168
0.143	-0.03009900538432069	0.0002686160040496558
0.144	-0.030104817136945974	0.0002670161498611978
0.145	-0.03010273925619643	0.0002652067452667331
0.146	-0.03016523992656411	0.0002657065615534412
0.147	-0.030152339781916723	0.00026429659274772327
0.149	-0.03024206341983937	0.00026314361970561996
0.150	-0.030251041086898616	0.0002616324774368503
0.151	-0.03025428715031608	0.00025996418879560806
0.152	-0.030236131124096156	0.00025822305799472797
0.153	-0.03020472235330434	0.00025601153096578696
0.155	-0.03021052502499859	0.00025393969531851024
0.156	-0.030205840596079156	0.00025245921447642394
0.157	-0.030189212489828428	0.00025086507253774997
0.158	-0.03016856465436066	0.0002493797170808302
0.160	-0.03017077252464674	0.00024826142272371855
0.161	-0.030179440380813403	0.00024715940312143564
0.162	-0.03017421152202646	0.0002459467210308922
0.163	-0.03021588473929045	0.000244533542649064
0.165	-0.03019969814884073	0.00024312163433322476
0.166	-0.030182431291430823	0.000241566612532491
0.167	-0.030166372702147552	0.00024012970477529813
0.168	-0.030167324192878413	0.00023831741434575136
0.169	-0.030188007750374248	0.00023700790923796895
0.170	-0.03016658130189743	0.00023581321732665228
0.171	-0.030167313931836377	0.0002342929913485219
0.173	-0.030162246540474277	0.00023415781118600666
0.174	-0.030175515504466413	0.00023315517828239484
0.175	-0.030189390713570383	0.00023256552253274665
0.176	-0.030190884632317724	0.0002319059741678459
0.177	-0.030206993469326898	0.00023136301138234518
0.178	-0.03023492859736461	0.00023097018486762993
0.179	-0.03023831938167204	0.00022957836072401704
0.180	-0.03024489933095384	0.00022835028505019408
0.182	-0.030224676992683885	0.00022695445375434762
0.183	-0.030230048575579947	0.00022627751043832235
0.184	-0.030263132732766488	0.00022464786433213155
0.185	-0.03023963860198528	0.00022338386655867352
0.186	-0.03022487618027306	0.00022254698029167922
0.187	-0.03022979657204305	0.0002217019852122815
0.189	-0.03021735319477277	0.00022067147008575323
0.190	-0.030241922654610794	0.0002202961176514565
0.191	-0.03023681639725889	0.00021921659157412307
0.192	-0.030241180359839026	0.0002172537127060832
0.193	-0.030278985445591787	0.0002164931612457535
0.194	-0.0302788256782817	0.00021560627893732405
0.196	-0.030279784650915537	0.00021487165743497446
0.197	-0.03028055746694189	0.0002138988361614016
0.198	-0.030290229580351036	0.0002131666606221079
0.199	-0.030294548917311023	0.00021226540029090218
0.200	-0.030303457424882607	0.0002095215441786332
0.201	-0.030303138557863693	0.00020873413209768877
0.202	-0.030289753696823837	0.0002076113972099128
0.204	-0.030290971287446206	0.00020667567733688569
0.205	-0.03027296050018466	0.0002059141325145798
0.206	-0.03025424279985809	0.0002047722719490207
0.207	-0.03028286778773851	0.00020440334657071855
0.208	-0.030266001433572068	0.00020160439032623553
0.209	-0.030284733845724953	0.0002014157986686664
0.210	-0.03025884701720464	0.00020033586458304418
0.211	-0.030246715220208702	0.00019963483650434137
0.213	-0.030239118794403014	0.0001988244731310209
0.214	-0.030221792219556884	0.00019773254538100075
0.215	-0.03021877538562136	0.00019683307800529948
0.216	-0.030260064746390695	0.00019497073331051684
0.217	-0.030250296900292073	0.0001941233772043744
0.219	-0.03023132549864826	0.00019337966678921128
0.220	-0.03024622513697771	0.00019271219087325722
0.221	-0.030249304064692	0.00019223921917964538
0.222	-0.03024150274025897	0.00019141060223654932
0.223	-0.030216225116226163	0.00019012377320775655
0.225	-0.03021327770246388	0.00018890809396165196
0.226	-0.030236133414685364	0.00018820408274914706
0.227	-0.030215112830770982	0.00018396680359866516
0.228	-0.030206390677198026	0.00018323203127174788
0.229	-0.030197548563893617	0.0001826354689151217
0.230	-0.03019872827733519	0.00018221826272274362
0.231	-0.030215655343314905	0.00018164929965094323
0.233	-0.030208999547148528	0.00018099769455358597
0.234	-0.030194511024504496	0.00017977379833222906
0.235	-0.030179442596690753	0.00017914537021458373
0.236	-0.030198505327008567	0.0001789769841347495
0.238	-0.030207049127088027	0.0001785334404204488
0.239	-0.030162208889488475	0.0001748691065269009
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0.846	-0.030213044359015773	8.11004180595196e-06
0.847	-0.030213244276805364	8.098481751376493e-06
0.848	-0.030212914230549053	8.069731197354463e-06
0.849	-0.030212953716411677	8.043986969019347e-06
0.850	-0.03021337904791928	8.052526081092125e-06
0.851	-0.03021253619709572	8.019288359006694e-06
0.852	-0.03021216948387092	7.992129530728743e-06
0.853	-0.030211891511483624	7.960756929300705e-06
0.854	-0.03021142179201084	7.929445023496794e-06
0.855	-0.030211836368956625	7.909213798763553e-06
0.856	-0.03021157505453853	7.879422529059699e-06
0.857	-0.03021193164627511	7.85791751968503e-06
0.858	-0.03021132442692887	7.83470242959927e-06
0.859	-0.030211287149203982	7.813621931912011e-06
0.860	-0.030211584071993525	7.80397160335191e-06
0.861	-0.030211285278577468	7.77654482554094e-06
0.862	-0.03021070343028317	7.745932597042503e-06
0.863	-0.030211016980507457	7.744047226813484e-06
0.864	-0.03021071484125383	7.716390287305682e-06
0.865	-0.030211632585633637	7.705545160475695e-06
0.866	-0.030211663711614573	7.682790041511805e-06
0.867	-0.030211254671414292	7.65550693394998e-06
0.868	-0.03021092567597769	7.626054483899228e-06
0.869	-0.030210264176316608	7.599758782715608e-06
0.870	-0.03021000909210467	7.571923607063591e-06
0.872	-0.030209337376001313	7.545823488032606e-06
0.873	-0.030209832543496615	7.53232886321725e-06
0.874	-0.030210172562503072	7.5195359229938765e-06
0.875	-0.030209880607499167	7.491189136978581e-06
0.876	-0.03021015086278947	7.4684975729141745e-06
0.877	-0.03021062663656375	7.447351195738599e-06
0.878	-0.030210761567348005	7.421476379596416e-06
0.879	-0.030211788583739314	7.408919410352043e-06
0.880	-0.030212088640987327	7.386235194380184e-06
0.881	-0.030211786771356528	7.358734158329978e-06
0.882	-0.03021107271687905	7.332152942587671e-06
0.883	-0.030210820700011933	7.309760714878662e-06
0.884	-0.03021032528245062	7.287552042071234e-06
0.885	-0.030210667617637382	7.272619550731226e-06
0.886	-0.030210534702831755	7.250939416238533e-06
0.887	-0.030210946730824674	7.2793252349112675e-06
0.888	-0.030211252139993497	7.2648090326639045e-06
0.889	-0.030210665082162854	7.240579121463229e-06
0.890	-0.03021087774424671	7.221960125950946e-06
0.891	-0.030211452716676957	7.211951374554893e-06
0.892	-0.030210694858355754	7.184074515170302e-06
0.893	-0.030210083062529838	7.158333262367399e-06
0.894	-0.030210977834800413	7.151539847082595e-06
0.895	-0.030210669641171445	7.12454694261228e-06
0.896	-0.030211331051111158	7.128299400843476e-06
0.897	-0.03021120935321755	7.104054172134005e-06
0.898	-0.03021140956618183	7.079676713170904e-06
0.899	-0.03021129189126742	7.062917562447002e-06
0.900	-0.0302117660429539	7.048190506182905e-06
0.901	-0.030212107465792854	7.035624120166649e-06
0.902	-0.030212355366550418	7.019206498291099e-06
0.903	-0.03021337302594268	7.017319386632158e-06
0.904	-0.03021369810945497	7.001908048869648e-06
0.905	-0.030213185303807757	6.9751250094704174e-06
0.906	-0.03021451380662918	7.03340512194063e-06
0.907	-0.030213729788559407	7.003593200684789e-06
0.908	-0.030213722778432706	6.98776083870591e-06
0.909	-0.030213117850751096	6.962538580572595e-06
0.910	-0.030213237683233737	6.939545272611981e-06
0.911	-0.03021314737188534	6.918589578781604e-06
0.912	-0.030213018975091734	6.893923822089388e-06
0.913	-0.03021330245351052	6.892247712272672e-06
0.914	-0.03021311587060586	6.866801502716889e-06
0.915	-0.030212959227322575	6.84181437274924e-06
0.916	-0.030212414092534588	6.816561394471185e-06
0.917	-0.030212136360792487	6.7960736338012995e-06
0.918	-0.030211732584037084	6.7747221438673855e-06
0.919	-0.03021210585476781	6.7697727461243385e-06
0.920	-0.0302116693858556	6.7435640414947464e-06
0.921	-0.030211483225436563	6.722243439601207e-06
0.922	-0.030211487960060443	6.697392227767628e-06
0.923	-0.030211120762383147	6.675194193865902e-06
0.924	-0.03021186881170036	6.663111583642154e-06
0.925	-0.030211387204286975	6.639492381564352e-06
0.926	-0.030211223994855024	6.616685809080732e-06
0.927	-0.030211709678181403	6.609004532495233e-06
0.928	-0.0302119772474608	6.588167128881656e-06
0.929	-0.030211614701277516	6.564996041769278e-06
0.930	-0.030212049809856197	6.547742505037693e-06
0.931	-0.030211521488944102	6.528313624966542e-06

System	(T) in mH	1mH absolute error	1% relative error
Water	-8.9892	0.1%	6%
HBN	-30.2274	1.8%	13%
UEG (14,147)	-37.8054	0.5%	2%
UEG (38,341)	-138.2314	x 2%	x 2%

Final plot

Water

reset
set grid
set style fill transparent solid 0.50 border
e_ref = e_ref + 0.
set ylabel "E_{(T)} (a.u.)"
set xlabel "Fraction of computed (a,b,c) triplets"
#set yrange [e_ref-.0005:e_ref+.0005]
plot data_uniform    u ($1):($2+$3):($2-$3) w filledcurves ls 1 title "Uniform", \
  data_uniform u ($1):($2) w l ls 1 title "", \
  data_importance u ($1):($2+$3):($2-$3) w filledcurves ls 3 title "Importance sampling", \
  data_importance u ($1):($2) w l ls 3 title "", \
  data_semistoch u ($1):($2+$3):($2-$3) w filledcurves ls 4 title "Semi-stochastic", \
  data_semistoch u ($1):($2) w l ls 4 title "", \
  e_ref title "Exact" lc black

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/convergence-h2o.png

reset
set grid
set log y
set style fill transparent solid 0.50 border
set ylabel "Error bar (a.u)"
set xlabel "Fraction of computed (a,b,c) triplets"
set format y "10^{%T}"
plot data_uniform    u 1:3 w l ls 1 title "Uniform", \
  data_importance u 1:3 w l ls 3 title "Importance sampling", \
  data_semistoch  u 1:3 w l ls 4 title "Semi-stochastic"

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/error-h2o.png

14 electrons

reset
set grid
set style fill transparent solid 0.50 border
e_ref = e_ref + 0.
set ylabel "E_{(T)} (a.u.)"
set xlabel "Fraction of computed (a,b,c) triplets"
#set yrange [e_ref-.0005:e_ref+.0005]
plot data_uniform    u ($1):($2+$3):($2-$3) w filledcurves ls 1 title "Uniform", \
  data_uniform u ($1):($2) w l ls 1 title "", \
  data_importance u ($1):($2+$3):($2-$3) w filledcurves ls 3 title "Importance sampling", \
  data_importance u ($1):($2) w l ls 3 title "", \
  data_semistoch u ($1):($2+$3):($2-$3) w filledcurves ls 4 title "Semi-stochastic", \
  data_semistoch u ($1):($2) w l ls 4 title "", \
  e_ref title "Exact" lc black

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/convergence-14.png

reset
set grid
set log y
set style fill transparent solid 0.50 border
set ylabel "Error bar (a.u)"
set xlabel "Fraction of computed (a,b,c) triplets"
set format y "10^{%T}"
plot data_uniform    u 1:3 w l ls 1 title "Uniform", \
  data_importance u 1:3 w l ls 3 title "Importance sampling", \
  data_semistoch  u 1:3 w l ls 4 title "Semi-stochastic"

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/error-14.png

38 electrons

reset
set grid
set style fill transparent solid 0.50 border
e_ref = e_ref + 0.
set ylabel "E_{(T)} (a.u.)"
set xlabel "Fraction of computed (a,b,c) triplets"
#set yrange [e_ref-.0005:e_ref+.0005]
plot data_uniform    u ($1):($2+$3):($2-$3) w filledcurves ls 1 title "Uniform", \
  data_uniform u ($1):($2) w l ls 1 title "", \
  data_importance u ($1):($2+$3):($2-$3) w filledcurves ls 3 title "Importance sampling", \
  data_importance u ($1):($2) w l ls 3 title "", \
  data_semistoch u ($1):($2+$3):($2-$3) w filledcurves ls 4 title "Semi-stochastic", \
  data_semistoch u ($1):($2) w l ls 4 title "", \
  e_ref title "Exact" lc black

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/convergence-38.png

reset
set grid
set log y
set style fill transparent solid 0.50 border
set ylabel "Error bar (a.u)"
set xlabel "Fraction of computed (a,b,c) triplets"
set format y "10^{%T}"
plot data_uniform    u 1:3 w l ls 1 title "Uniform", \
  data_importance u 1:3 w l ls 3 title "Importance sampling", \
  data_semistoch  u 1:3 w l ls 4 title "Semi-stochastic"

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/error-38.png

HBN

reset
set grid
set style fill transparent solid 0.50 border
e_ref = e_ref + 0.
set ylabel "E_{(T)} (a.u.)"
set xlabel "Fraction of computed (a,b,c) triplets"
#set yrange [e_ref-.0005:e_ref+.0005]
plot data_uniform    u ($1):($2+$3):($2-$3) w filledcurves ls 1 title "Uniform", \
  data_uniform u ($1):($2) w l ls 1 title "", \
  data_importance u ($1):($2+$3):($2-$3) w filledcurves ls 3 title "Importance sampling", \
  data_importance u ($1):($2) w l ls 3 title "", \
  data_semistoch u ($1):($2+$3):($2-$3) w filledcurves ls 4 title "Semi-stochastic", \
  data_semistoch u ($1):($2) w l ls 4 title "", \
  e_ref title "Exact" lc black

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/convergence-hbn.png

reset
set grid
set log y
set style fill transparent solid 0.50 border
set ylabel "Error bar (a.u)"
set xlabel "Fraction of computed (a,b,c) triplets"
set format y "10^{%T}"
plot data_uniform    u 1:3 w l ls 1 title "Uniform", \
  data_importance u 1:3 w l ls 3 title "Importance sampling", \
  data_semistoch  u 1:3 w l ls 4 title "Semi-stochastic"

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/error-hbn.png

Appendix

Fortran code

Naive expression

energy = 0d0
do a = 1, nV 
do b = 1, nV
  do c = 1, nV
    do k = 1, nO
      do j = 1, nO
        do i = 1, nO
          energy = energy + (4.d0 * W(i,j,k,a,b,c) + &
                             W(i,j,k,b,c,a) + &
                             W(i,j,k,c,a,b)) * &
                            (V(i,j,k,a,b,c) - V(i,j,k,c,b,a)) / &
                            (f_o(i) + f_o(j) + f_o(k) - f_v(a) - f_v(b) - f_v(c))
        enddo
      enddo
    enddo
  enddo
enddo
enddo

energy = energy / 3.d0
print*,"(T)", energy

Optimized functions

Here, we compute in the inner-most loop all possible permutations of $(a,b,c)$ with $a \ne b \ne c$. In the second loop, we compute the contributions where two indices are the same. When $a=b=c$, the contribution is zero.

e = 0d0
!$OMP DO SCHEDULE(dynamic)
do a = 1, nV
do b = a+1, nV
  do c = b+1, nV
    delta_abc = f_v(a) + f_v(b) + f_v(c)
    call form_w_abc(nO,nV,a,b,c,T_voov,T_oovv,X_vovv,X_ooov,W_abc,W_cba,W_bca,W_cab,W_bac,W_acb)
    call form_v_abc(nO,nV,a,b,c,t1,X_oovv,W_abc,V_abc,W_cba,V_cba,W_bca,V_bca,W_cab,V_cab,W_bac,V_bac,W_acb,V_acb)
    do k = 1, nO
      do j = 1, nO
        do i = 1, nO
          delta = 1.d0 / (f_o(i) + f_o(j) + f_o(k) - delta_abc)
          e = e + delta * ( &
             (4d0 * (W_abc(i,j,k) - W_cba(i,j,k)) + &
                     W_bca(i,j,k) - W_bac(i,j,k)  + &
                     W_cab(i,j,k) - W_acb(i,j,k)  ) * (V_abc(i,j,k) - V_cba(i,j,k)) + &
             (4d0 * (W_acb(i,j,k) - W_bca(i,j,k)) + &
                     W_cba(i,j,k) - W_cab(i,j,k)  + &
                     W_bac(i,j,k) - W_abc(i,j,k)  ) * (V_acb(i,j,k) - V_bca(i,j,k)) + &
             (4d0 * (W_bac(i,j,k) - W_cab(i,j,k)) + &
                     W_acb(i,j,k) - W_abc(i,j,k)  + &
                     W_cba(i,j,k) - W_bca(i,j,k)  ) * (V_bac(i,j,k) - V_cab(i,j,k)) )
        enddo
      enddo
    enddo
  enddo
enddo

c = a
do b = 1, nV
  if (b == c) cycle
  delta_abc = f_v(a) + f_v(b) + f_v(c)
  call form_w_abc(nO,nV,a,b,c,T_voov,T_oovv,X_vovv,X_ooov,W_abc,W_cba,W_bca,W_cab,W_bac,W_acb)
  call form_v_abc(nO,nV,a,b,c,t1,X_oovv,W_abc,V_abc,W_cba,V_cba,W_bca,V_bca,W_cab,V_cab,W_bac,V_bac,W_acb,V_acb)
  do k = 1, nO
    do j = 1, nO
      do i = 1, nO
        delta = 1.0d0 / (f_o(i) + f_o(j) + f_o(k) - delta_abc)
        e = e + delta * ( &
           (4d0 * W_abc(i,j,k) + W_bca(i,j,k) + W_cab(i,j,k)) * (V_abc(i,j,k) - V_cba(i,j,k)) + &
           (4d0 * W_acb(i,j,k) + W_cba(i,j,k) + W_bac(i,j,k)) * (V_acb(i,j,k) - V_bca(i,j,k)) + &
           (4d0 * W_bac(i,j,k) + W_acb(i,j,k) + W_cba(i,j,k)) * (V_bac(i,j,k) - V_cab(i,j,k)) )
      enddo
    enddo
  enddo
enddo
enddo
!$OMP END DO NOWAIT

!$OMP CRITICAL
energy = energy + e
!$OMP END CRITICAL

deallocate(W_abc, W_cab, W_bca, W_bac, W_cba, W_acb, &
         V_abc, V_cab, V_bca, V_bac, V_cba, V_acb )

!$OMP END PARALLEL

Plots

reset
binwidth=1
set boxwidth binwidth
bin(x,width)=width*floor(x/width)
plot 'data' using (bin($4,binwidth)):($7) smooth freq with boxes

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/tmp.png

import numpy as np
data = []
with open('data','r') as f:
 for line in f:
     l = line.split()
     a, b, c = int(l[0]), int(l[1]), int(l[2])
     ea, eb, ec = float(l[3]), float(l[4]), float(l[5])
     e = float(l[6])
#        data.append(e)
#        data.append(e * (ea + eb + ec))
#        data.append(e * (ea + eb + ec) * ea)
#        data.append(e * (ea + eb + ec) * eb)
#        data.append(e * (ea + eb + ec) * ec)
#        data.append(e * (ea + eb + ec) * ea*eb)
#        data.append(e * (ea + eb + ec) * ea*eb*ec)
#        data.append(e * (ea + eb + ec + 0.037)**2)
     data.append(e * (ea * eb * ec)**1)

data = np.array(data)
average = sum(data)/len(data)
variance = sum( (data-average)**3 ) /(len(data)-1)
error = np.sqrt(variance/len(data))
print(average, '+/-', error)
print(error/average)

0.0030955494327421676 +/- 4.131592930193792e-06
0.0013346880804077012

1/(ea*eb*ec) : 0.004730162427068331 1/(ea*eb) : 0.004730162427068331 1/ec : 0.006151067343303064 1/eb : 0.0056790423986561225 1/ea : 0.007281256366224999 denom : 0.008468893789021222 orig : 0.011634889545203767

reset
plot 'data' using :($7 * ($4*$5*$6)**1 ) w p

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/tmp.png

Test 2

with open('data2','w') as f:
 for i in range(Nabc):
     f.write("%e %e\n"%(w[i], eabc_sorted[i]))

reset
plot 'data2' using :($7 * ($4*$5*$6)**1 ) w p

Results QP

Benzene

12
Benzene
C      0.000000     1.392503     0.000000
C     -1.205943     0.696252     0.000000
C     -1.205943    -0.696252     0.000000
C      0.000000    -1.392503     0.000000
C      1.205943    -0.696252     0.000000
C      1.205943     0.696252     0.000000
H     -2.141717     1.236521     0.000000
H     -2.141717    -1.236521     0.000000
H      0.000000    -2.473042     0.000000
H      2.141717    -1.236521     0.000000
H      2.141717     1.236521     0.000000
H      0.000000     2.473042     0.000000

Benzene DZ

Intel(R) Core(TM) i7-1165G7 @ 2.80GHz, 4 cores

Gaussian16

CCSD(T,TWInCore)

Wavefunction amplitudes converged. E(Corr)= -231.54403350 CCSD(T)= -0.23157977364D+03 283.27 s

Stochastic algorithm

Time: 2.36414194107056 s

E(CCSD) = -231.544034027686 Ha Correlation = -0.821789081365 Ha Conv = 9.18E-07

Computing (T) correction…

E(CCSD(T))	Error	%
-231.58030122	0.4195E-03	19.66
-231.58021542	0.2290E-03	39.16
-231.57982156	0.1324E-03	58.61
-231.57980529	0.6177E-04	78.09
-231.57977003	0.6253E-05	97.53

Time: 5.18708086013794 s

E(CCSD(T)) = -231.579770030057 Ha E(T) = -0.035736002371 Ha Correlation = -0.857525083736 Ha

Benzene TZ

Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz, 32 cores

Gaussian16

Wavefunction amplitudes converged. E(Corr)= -231.75392219

Stochastic algorithm

Time: 90.0595779418945 s

Successful convergence after 9 iterations

E(CCSD) = -231.753922932570 Ha Correlation = -0.974220010793 Ha Conv = 6.97E-07

#+NAME:benzene_tz

E(CCSD(T))	Error	%
-231.80582073	0.1702E-02	0.43
-231.80568602	0.1182E-02	0.86
-231.80638784	0.9730E-03	1.28
-231.80583046	0.8243E-03	1.70
-231.80564026	0.7246E-03	2.13
-231.80549184	0.6546E-03	2.55
-231.80528021	0.5917E-03	2.97
-231.80533305	0.5513E-03	3.39
-231.80552546	0.5262E-03	3.81
-231.80531583	0.4945E-03	4.23
-231.80535472	0.4705E-03	4.66
-231.80557527	0.4565E-03	5.08
-231.80557303	0.4355E-03	5.50
-231.80548029	0.4202E-03	5.92
-231.80556655	0.4050E-03	6.34
-231.80563017	0.3936E-03	6.77
-231.80566114	0.3827E-03	7.19
-231.80558699	0.3701E-03	7.61
-231.80567134	0.3635E-03	8.03
-231.80560792	0.3524E-03	8.45
-231.80599656	0.2819E-03	8.87
-231.80600921	0.2748E-03	9.29
-231.80602737	0.2682E-03	9.71
-231.80602047	0.2619E-03	10.13
-231.80600268	0.2562E-03	10.55
-231.80605143	0.2532E-03	10.97
-231.80603473	0.2490E-03	11.39
-231.80606029	0.2452E-03	11.81
-231.80608868	0.2406E-03	12.23
-231.80608311	0.2367E-03	12.65
-231.80603067	0.2316E-03	13.07
-231.80602845	0.2276E-03	13.49
-231.80598091	0.2235E-03	13.91
-231.80592227	0.2203E-03	14.33
-231.80586276	0.2164E-03	14.75
-231.80585569	0.2125E-03	15.17
-231.80588786	0.2088E-03	15.59
-231.80580990	0.1836E-03	16.01
-231.80575077	0.1801E-03	16.43
-231.80575959	0.1777E-03	16.85
-231.80577139	0.1757E-03	17.27
-231.80573497	0.1730E-03	17.69
-231.80571827	0.1703E-03	18.11
-231.80570941	0.1679E-03	18.53
-231.80568515	0.1654E-03	18.95
-231.80565681	0.1628E-03	19.37
-231.80563140	0.1606E-03	19.79
-231.80564907	0.1587E-03	20.21
-231.80561730	0.1562E-03	20.63
-231.80558851	0.1539E-03	21.05
-231.80559825	0.1525E-03	21.48
-231.80563699	0.1512E-03	21.90
-231.80562532	0.1326E-03	22.32
-231.80558771	0.1310E-03	22.74
-231.80558484	0.1299E-03	23.16
-231.80559053	0.1284E-03	23.58
-231.80560391	0.1271E-03	24.00
-231.80561828	0.1261E-03	24.42
-231.80562601	0.1249E-03	24.84
-231.80562503	0.1235E-03	25.26
-231.80561025	0.1221E-03	25.68
-231.80561714	0.1209E-03	26.10
-231.80561468	0.1197E-03	26.52
-231.80562579	0.1185E-03	26.94
-231.80561713	0.1103E-03	27.36
-231.80560273	0.1091E-03	27.78
-231.80563253	0.1084E-03	28.20
-231.80564377	0.1075E-03	28.62
-231.80564167	0.1064E-03	29.04
-231.80563093	0.1054E-03	29.46
-231.80563660	0.1049E-03	29.88
-231.80563212	0.1040E-03	30.30
-231.80563092	0.1030E-03	30.72
-231.80563308	0.1021E-03	31.14
-231.80563106	0.1011E-03	31.56
-231.80567721	0.9500E-04	31.98
-231.80566323	0.9423E-04	32.40
-231.80565084	0.9338E-04	32.82
-231.80564784	0.9250E-04	33.24
-231.80565170	0.9174E-04	33.66
-231.80566332	0.9103E-04	34.08
-231.80566231	0.9026E-04	34.50
-231.80567450	0.8967E-04	34.92
-231.80567643	0.8886E-04	35.34
-231.80567236	0.8800E-04	35.76
-231.80570112	0.8340E-04	36.18
-231.80569928	0.8278E-04	36.60
-231.80570171	0.8228E-04	37.02
-231.80568327	0.8151E-04	37.44
-231.80568723	0.8078E-04	37.86
-231.80568265	0.8013E-04	38.28
-231.80569338	0.7963E-04	38.69
-231.80567324	0.7899E-04	39.11
-231.80563524	0.7452E-04	39.53
-231.80564097	0.7398E-04	39.95
-231.80565289	0.7357E-04	40.37
-231.80564818	0.7296E-04	40.79
-231.80565600	0.7251E-04	41.21
-231.80565207	0.7194E-04	41.63
-231.80566001	0.7143E-04	42.05
-231.80566004	0.7083E-04	42.47
-231.80564267	0.6744E-04	42.89
-231.80563415	0.6689E-04	43.31
-231.80563791	0.6641E-04	43.72
-231.80563744	0.6596E-04	44.14
-231.80563826	0.6551E-04	44.56
-231.80563215	0.6500E-04	44.98
-231.80563720	0.6456E-04	45.40
-231.80563619	0.6246E-04	45.82
-231.80563835	0.6198E-04	46.24
-231.80564409	0.6157E-04	46.66
-231.80563762	0.6119E-04	47.08
-231.80563979	0.6080E-04	47.50
-231.80565249	0.6042E-04	47.92
-231.80565287	0.6003E-04	48.34
-231.80566663	0.5788E-04	48.76
-231.80567701	0.5766E-04	49.18
-231.80567437	0.5721E-04	49.60
-231.80567603	0.5686E-04	50.02
-231.80567932	0.5655E-04	50.44
-231.80568558	0.5616E-04	50.86
-231.80568611	0.5583E-04	51.28
-231.80568801	0.5329E-04	51.70
-231.80569324	0.5295E-04	52.12
-231.80569806	0.5258E-04	52.53
-231.80569919	0.5219E-04	52.95
-231.80569441	0.5179E-04	53.37
-231.80569178	0.5145E-04	53.79
-231.80570274	0.4828E-04	54.21
-231.80571023	0.4804E-04	54.64
-231.80570955	0.4772E-04	55.05
-231.80571522	0.4742E-04	55.47
-231.80571110	0.4711E-04	55.89
-231.80569718	0.4400E-04	56.31
-231.80570477	0.4381E-04	56.73
-231.80569947	0.4350E-04	57.15
-231.80569318	0.4321E-04	57.56
-231.80569576	0.4290E-04	57.98
-231.80570782	0.4050E-04	58.40
-231.80570906	0.4027E-04	58.82
-231.80570963	0.4000E-04	59.24
-231.80570842	0.3974E-04	59.66
-231.80570764	0.3947E-04	60.08
-231.80571662	0.3712E-04	60.50
-231.80571952	0.3690E-04	60.92
-231.80572084	0.3668E-04	61.34
-231.80571920	0.3644E-04	61.76
-231.80572058	0.3620E-04	62.18
-231.80572094	0.3389E-04	62.59
-231.80572010	0.3366E-04	63.01
-231.80572148	0.3347E-04	63.43
-231.80572100	0.3327E-04	63.85
-231.80571891	0.3128E-04	64.27
-231.80571950	0.3106E-04	64.68
-231.80571796	0.3086E-04	65.10
-231.80571719	0.3064E-04	65.52
-231.80570329	0.2833E-04	65.94
-231.80570411	0.2819E-04	66.36
-231.80570353	0.2802E-04	66.78
-231.80570459	0.2786E-04	67.20
-231.80570001	0.2628E-04	67.61
-231.80570281	0.2612E-04	68.03
-231.80570545	0.2596E-04	68.45
-231.80570837	0.2583E-04	68.87
-231.80571420	0.2446E-04	69.29
-231.80571712	0.2430E-04	69.70
-231.80571366	0.2411E-04	70.12
-231.80572181	0.2295E-04	70.54
-231.80572527	0.2282E-04	70.96
-231.80572548	0.2266E-04	71.38
-231.80572717	0.2159E-04	71.79
-231.80572893	0.2147E-04	72.21
-231.80573143	0.2133E-04	72.63
-231.80573576	0.2039E-04	73.04
-231.80573648	0.2025E-04	73.46
-231.80573447	0.2010E-04	73.88
-231.80573276	0.1929E-04	74.30
-231.80573004	0.1914E-04	74.71
-231.80572435	0.1824E-04	75.13
-231.80572626	0.1812E-04	75.54
-231.80572718	0.1800E-04	75.96
-231.80573168	0.1727E-04	76.38
-231.80573042	0.1717E-04	76.80
-231.80572534	0.1637E-04	77.21
-231.80572609	0.1626E-04	77.63
-231.80572710	0.1562E-04	78.05
-231.80572948	0.1551E-04	78.47
-231.80572869	0.1540E-04	78.89
-231.80573529	0.1483E-04	79.30
-231.80573552	0.1473E-04	79.72
-231.80573496	0.1396E-04	80.14
-231.80573536	0.1339E-04	80.55
-231.80573698	0.1330E-04	80.97
-231.80573519	0.1268E-04	81.39
-231.80573370	0.1258E-04	81.81
-231.80573230	0.1201E-04	82.23
-231.80573221	0.1192E-04	82.64
-231.80572889	0.1129E-04	83.06
-231.80573070	0.1087E-04	83.48
-231.80573069	0.1079E-04	83.90
-231.80572791	0.1030E-04	84.32
-231.80572916	0.9864E-05	84.73
-231.80573030	0.9504E-05	85.15
-231.80573409	0.9133E-05	85.57
-231.80573446	0.9056E-05	85.99
-231.80573537	0.8694E-05	86.40
-231.80573086	0.8322E-05	86.82
-231.80573012	0.7984E-05	87.24
-231.80572517	0.7645E-05	87.65
-231.80572484	0.7265E-05	88.07
-231.80572223	0.6877E-05	88.49
-231.80572365	0.6538E-05	88.90
-231.80572196	0.6240E-05	89.32
-231.80572052	0.5931E-05	89.74
-231.80572150	0.5322E-05	90.16
-231.80572299	0.5070E-05	90.57
-231.80572435	0.4825E-05	90.99
-231.80572740	0.4575E-05	91.41
-231.80572812	0.4136E-05	91.83
-231.80572667	0.3904E-05	92.25
-231.80572641	0.3526E-05	92.66
-231.80572734	0.3352E-05	93.08
-231.80572923	0.2997E-05	93.50
-231.80572857	0.2716E-05	93.92
-231.80572984	0.2452E-05	94.33
-231.80572846	0.2219E-05	94.75
-231.80572965	0.2026E-05	95.17
-231.80572998	0.1864E-05	95.58
-231.80573034	0.1634E-05	96.00
-231.80573038	0.1455E-05	96.42
-231.80572994	0.1339E-05	96.84
-231.80573009	0.1191E-05	97.26
-231.80573022	0.1077E-05	97.68
-231.80573054	0.9794E-06	98.10
-231.80573088	0.8814E-06	98.52
-231.80573038	0.8139E-06	98.95
-231.80573031	0.4206E-06	99.38
-231.80572937	0.2185E-06	99.82

Time: 239.890999078751 s

E(CCSD(T)) = -231.805729365546 Ha E(T) = -0.051806432976 Ha Correlation = -1.026026443769 Ha

reset
set grid
set format y "%10.4f"
set ylabel "(T) (a.u)"
set xlabel "Wall clock time (s)"
plot data using :1:2 w errorlines notitle, -231.805729365546 notitle

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/benzene_tz.png

Streptocyanine QZ

Geometry

8
Streptocyanine-C1   
C       0.000000     0.000000     0.425929
N       0.000000     1.161139    -0.177701
N       0.000000    -1.161139    -0.177701
H       0.000000     0.000000     1.505176
H       0.000000     1.254035    -1.182034
H       0.000000    -1.254035    -1.182034
H       0.000000     2.007765     0.367245
H       0.000000    -2.007765     0.367245

315 MOs HF: -149.5404341190290

Stochastic algorithm

Time: 127.198801040649 s

Successful convergence after 10 iterations

E(CCSD) = -150.162036576177 Ha Correlation = -0.621602548836 Ha Conv = 6.91E-07

#+NAME:strepto_qz

E(CCSD(T))	Error	%
-150.18755289	0.3236E-03	1.33
-150.18749603	0.2240E-03	2.69
-150.18753707	0.1571E-03	4.05
-150.18764846	0.1373E-03	5.41
-150.18760591	0.1214E-03	6.77
-150.18754960	0.9845E-04	8.13
-150.18754334	0.9176E-04	9.49
-150.18755251	0.7994E-04	10.84
-150.18754178	0.7515E-04	12.20
-150.18754220	0.7023E-04	13.56
-150.18754476	0.6258E-04	14.92
-150.18755984	0.5938E-04	16.27
-150.18757249	0.5661E-04	17.63
-150.18756762	0.4951E-04	18.99
-150.18756197	0.4759E-04	20.35
-150.18757288	0.4598E-04	21.70
-150.18758115	0.4099E-04	23.06
-150.18758419	0.3965E-04	24.41
-150.18757470	0.3824E-04	25.77
-150.18757089	0.3491E-04	27.12
-150.18757876	0.3382E-04	28.48
-150.18760689	0.3078E-04	29.84
-150.18759859	0.2991E-04	31.19
-150.18760844	0.2776E-04	32.55
-150.18761368	0.2702E-04	33.90
-150.18762310	0.2486E-04	35.26
-150.18762591	0.2429E-04	36.62
-150.18762535	0.2228E-04	37.97
-150.18762385	0.2176E-04	39.33
-150.18763225	0.2007E-04	40.68
-150.18763223	0.1958E-04	42.04
-150.18763397	0.1798E-04	43.39
-150.18763340	0.1757E-04	44.75
-150.18764070	0.1621E-04	46.10
-150.18763986	0.1498E-04	47.46
-150.18763739	0.1464E-04	48.82
-150.18763883	0.1346E-04	50.17
-150.18764267	0.1255E-04	51.53
-150.18764269	0.1229E-04	52.88
-150.18764424	0.1142E-04	54.24
-150.18765046	0.1058E-04	55.60
-150.18765008	0.9804E-05	56.95
-150.18765061	0.9602E-05	58.31
-150.18764718	0.8941E-05	59.66
-150.18765089	0.8398E-05	61.02
-150.18764789	0.7875E-05	62.37
-150.18764931	0.7430E-05	63.72
-150.18764581	0.6968E-05	65.08
-150.18764195	0.6562E-05	66.43
-150.18764743	0.6188E-05	67.79
-150.18764632	0.5825E-05	69.14
-150.18764715	0.5539E-05	70.49
-150.18765097	0.5292E-05	71.84
-150.18765066	0.5036E-05	73.20
-150.18764911	0.4698E-05	74.55
-150.18765017	0.4478E-05	75.90
-150.18764929	0.4274E-05	77.25
-150.18764876	0.4021E-05	78.61
-150.18764747	0.3796E-05	79.96
-150.18764725	0.3571E-05	81.31
-150.18764619	0.3359E-05	82.66
-150.18764469	0.3151E-05	84.01
-150.18764433	0.2899E-05	85.36
-150.18764369	0.2633E-05	86.71
-150.18764453	0.2399E-05	88.06
-150.18764511	0.2151E-05	89.41
-150.18764428	0.1906E-05	90.76
-150.18764536	0.1634E-05	92.12
-150.18764554	0.1442E-05	93.47
-150.18764540	0.1139E-05	94.82
-150.18764707	0.7326E-06	96.18
-150.18764777	0.2304E-06	97.55
-150.18764760	NaN	98.92

Time: 74.8391861915588 s

E(CCSD(T)) = -150.187647603273 Ha E(T) = -0.025611027096 Ha Correlation = -0.647213575932 Ha

reset
set grid
set format y "%10.4f"
set ylabel "(T) (a.u)"
set xlabel "Wall clock time (s)"
plot data using :1:2 w errorlines notitle, -150.187647603273 notitle

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/strepto%20qz.png

Caffeine def2-svp

Geometry

H       -3.380413    -1.127237     0.573304
N        0.966830    -1.073743    -0.819823
C        0.056729     0.852720     0.392316
N       -1.375174    -1.021224    -0.057055
C       -1.261502     0.259071     0.523414
C       -0.306834    -1.683633    -0.716934
C        1.139424     0.187412    -0.270090
N        0.560263     2.083910     0.825159
O       -0.492680    -2.818056    -1.209473
C       -2.632807    -1.730396    -0.006095
O       -2.230134     0.798862     1.089973
H        2.549699     2.973498     0.622959
C        2.052743    -1.736089    -1.493128
H       -2.480772    -2.726953     0.488263
H       -3.008904    -1.902526    -1.049802
H        2.917610    -1.848152    -0.785787
H        2.378786    -1.121192    -2.374366
H        1.718988    -2.748992    -1.843921
C       -0.151845     3.097005     1.534835
C        1.893410     2.118125     0.419319
N        2.286125     0.996844    -0.244030
H       -0.168703     4.043656     0.930109
H        0.353532     3.297906     2.517775
H       -1.207450     2.753759     1.720305

HF: -675.7951033355195

Time: 125.921180963516 s

E(CCSD) = -677.937618736963 Ha Correlation = -2.142515403784 Ha Conv = 5.18E-07

#+NAME:caffeine_svp

E(CCSD(T))	Error	%
-678.02853922	0.5258E-02	0.10
-678.02976334	0.3787E-02	0.19
-678.02906265	0.3220E-02	0.27
-678.02888097	0.3069E-02	0.35
-678.02579694	0.2578E-02	0.43
-678.02539115	0.2311E-02	0.51
-678.02621342	0.2161E-02	0.59
-678.02699050	0.2054E-02	0.67
-678.02692044	0.1975E-02	0.75
-678.02637757	0.1821E-02	0.83
-678.02647872	0.1720E-02	0.92
-678.02562780	0.1627E-02	0.99
-678.02619592	0.1579E-02	1.07
-678.02595517	0.1507E-02	1.15
-678.02586955	0.1442E-02	1.23
-678.02637232	0.1414E-02	1.31
-678.02664314	0.1399E-02	1.39
-678.02689058	0.1373E-02	1.47
-678.02685870	0.1328E-02	1.55
-678.02652022	0.1284E-02	1.63
-678.02684814	0.1283E-02	1.71
-678.02660690	0.1240E-02	1.79
-678.02661689	0.1230E-02	1.87
-678.02656400	0.1202E-02	1.94
-678.02649762	0.1174E-02	2.03
-678.02622086	0.1144E-02	2.10
-678.02622715	0.1114E-02	2.18
-678.02626622	0.1091E-02	2.26
-678.02636527	0.1078E-02	2.34
-678.02631587	0.1058E-02	2.42
-678.02657845	0.1043E-02	2.50
-678.02687191	0.1048E-02	2.58
-678.02655328	0.1025E-02	2.65
-678.02667586	0.1013E-02	2.74
-678.02662550	0.9936E-03	2.81
-678.02673402	0.9801E-03	2.89
-678.02671217	0.9668E-03	2.97
-678.02654649	0.9498E-03	3.05
-678.02641864	0.9325E-03	3.13
-678.02619833	0.9146E-03	3.20
-678.02635933	0.9117E-03	3.28
-678.02638714	0.9020E-03	3.36
-678.02638762	0.8877E-03	3.44
-678.02620569	0.8725E-03	3.52
-678.02632108	0.8752E-03	3.60
-678.02624951	0.8666E-03	3.68
-678.02634128	0.8553E-03	3.76
-678.02629702	0.8438E-03	3.84
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-678.02618891	0.7384E-05	99.39
-678.02618188	0.5774E-05	99.46
-678.02618004	0.4525E-05	99.54
-678.02617763	0.3286E-05	99.62
-678.02617839	0.3286E-05	99.69
-678.02617826	0.1864E-05	99.77
-678.02617811	0.1849E-05	99.85
-678.02617949	0.1036E-05	99.92

Time: 1328.17061614990 s

E(CCSD(T)) = -678.026179485578 Ha E(T) = -0.088560748615 Ha Correlation = -2.231076152399 Ha

reset
set grid
set format y "%10.4f"
set ylabel "(T) (a.u)"
set xlabel "Wall clock time (s)"
set yrange[-678.027:-678.025]
plot data using :1:2 w errorlines notitle, -678.026179485578 notitle

/scemama/StochasticTriples/media/commit/4a7f2e462d34892aecad650093d577e7a270f15b/caffeine_svp.png

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Stochastic perturbative triples

Stochastic formulation

Memoization

Importance sampling

Multiple buckets

Semi-stochastic algorithm

Test code

Orbital energies

Occupied

Virtual

Code

Uniform distribution

Importance sampling

Preparation

Sampling

Semi-stochastic

Preparation

Sampling

Final plot

Water

14 electrons

38 electrons

HBN

Appendix

Fortran code

Naive expression

Optimized functions

Plots

Test 2

Results QP

Benzene

Benzene DZ

Gaussian16

Stochastic algorithm

Benzene TZ

Gaussian16

Stochastic algorithm

Streptocyanine QZ

Geometry

Stochastic algorithm

Caffeine def2-svp

Geometry

180 KiB

Raw Blame History