Rapid Calculation of the Three-body Interaction Energies in Water Clusters†

doi:10.7503/cjcu20190038

Abstract

Abstract:

A method was proposed to calculate the three-body interaction energies in water clusters. The O—H bond of a water molecule was regarded as a bond-dipole, and the three-body interaction energy were expressed as the sum of long-range induction and short-range correction. The long-range induction was treated by using Thole model, and the short-range correction was derived by considering both permutations among different water molecules and permutations between the two dipoles within each molecule. The parameters needed in the method were determined by fitting to the CCSD(T)/aug-cc-pVTZ three-body interaction energies of 12347 water trimers taken from the reference. The accuracy of the method is validated through comparing the three-body interaction energies of 67 water clusters ranging from trimer to 30-mer obtained by CCSD(T) method, MP2 method, and M06-2X method. The method can reproduce the CCSD(T)/aug-cc-pVDZ total three-body interaction energies, with the RMSD of 3.32 kJ/mol and the MRD of 2.43%, slightly better than the MP2 method, and outperforms the M06-2X method for large water clusters. The method is helpful in simulating the dynamic properties of water and other biomolecules.

Key words: Water cluster, Three-body interaction, Bond-dipole

CLC Number:

O641

TrendMD:

LI Xiaolei,WANG Changsheng. Rapid Calculation of the Three-body Interaction Energies in Water Clusters^†[J]. Chem. J. Chinese Universities, 2019, 40(5): 950.

Figures/Tables 8

Table 1 Calculated total three-body interaction energies(E3B) of 21 small water clustersa

No.	Cluster	Ref.	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	E_3B	Δ	δ(%)	$E 3 B b$	Δ	δ(%)	E_3B	Δ	δ(%)
1	3w-1a	-5.82^c	-5.77	0.05	0.86	-5.65	0.17	2.92	-5.77	0.05	0.86
2	3w-1b	4.52^c	4.35	-0.17	3.76	5.36	0.84	18.58	4.85	0.33	7.30
3	3w-1c	-10.13^c	-10.29	-0.16	1.58	-11.34	-1.21	11.94	-10.00	0.13	1.28
4	4w-1	-25.82^d	-25.94	-0.12	0.46	-26.53	-0.71	2.75	-25.27	0.55	2.13
5	4w-2	-24.73^d	-24.85	-0.12	0.49	-25.23	-0.50	2.02	-24.60	0.13	0.53
6	4w-3	-15.40^d	-16.11	-0.71	4.61	-19.04	-3.64	23.64	-14.64	0.76	4.94
7	5w-1	-37.95^d	-37.87	0.08	0.21	-36.90	1.05	2.77	-38.41	-0.46	1.21
8	5w-2	-31.46^d	-32.22	-0.76	2.42	-32.38	-0.92	2.92	-30.84	0.62	1.97
9	5w-3	-26.57^d	-27.82	-1.25	4.70	-29.71	-3.14	11.82	-27.07	-0.50	1.88
10	5w-4	-24.85^d	-25.94	-1.09	4.39	-27.95	-3.10	12.47	-24.27	0.58	2.33
11	5w-5	-24.18^d	-25.31	-1.13	4.67	-26.07	-1.89	7.82	-23.30	0.88	3.64
12	5w-6	-26.07^d	-26.53	-0.46	1.76	-26.32	-0.25	0.96	-24.48	1.59	6.10
13	5w-7	-25.94^d	-26.36	-0.42	1.62	-26.48	-0.54	2.08	-25.06	0.88	3.39
14	6w-prism	-36.40^d	-38.07	-1.67	4.59	-37.99	-1.59	4.37	-37.91	-1.51	4.15
15	6w-cage	-37.53^d	-38.99	-1.46	3.89	-40.29	-2.76	7.35	-38.07	-0.54	1.44
16	6w-book1	-43.43^d	-43.76	-0.33	0.76	-42.97	0.46	1.06	-43.22	0.21	0.48
17	6w-book2	-42.30^d	-42.84	-0.54	1.28	-41.21	1.09	2.58	-41.80	0.50	1.18
18	6w-bag	-43.30^d	-44.10	-0.80	1.85	-41.59	1.71	3.95	-41.59	1.71	3.95
19	6w-chair	-49.29^d	-49.08	0.21	0.43	-48.53	0.76	1.54	-51.13	-1.84	3.73
20	6w-boat1	-47.45^d	-47.36	0.09	0.19	-46.28	1.17	2.47	-49.08	-1.63	3.44
21	6w-boat2	-47.45^d	-47.40	0.05	0.11	-46.32	1.13	2.38	-49.41	-1.96	4.13
	RMSD			0.74			1.68			1.02
	MAD			1.67			3.64			1.96
	MRD(%)				2.12			6.11			2.86

Table 1 Calculated total three-body interaction energies(E3B) of 21 small water clustersa

No.	Cluster	Ref.	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	E_3B	Δ	δ(%)	$E 3 B b$	Δ	δ(%)	E_3B	Δ	δ(%)
1	3w-1a	-5.82^c	-5.77	0.05	0.86	-5.65	0.17	2.92	-5.77	0.05	0.86
2	3w-1b	4.52^c	4.35	-0.17	3.76	5.36	0.84	18.58	4.85	0.33	7.30
3	3w-1c	-10.13^c	-10.29	-0.16	1.58	-11.34	-1.21	11.94	-10.00	0.13	1.28
4	4w-1	-25.82^d	-25.94	-0.12	0.46	-26.53	-0.71	2.75	-25.27	0.55	2.13
5	4w-2	-24.73^d	-24.85	-0.12	0.49	-25.23	-0.50	2.02	-24.60	0.13	0.53
6	4w-3	-15.40^d	-16.11	-0.71	4.61	-19.04	-3.64	23.64	-14.64	0.76	4.94
7	5w-1	-37.95^d	-37.87	0.08	0.21	-36.90	1.05	2.77	-38.41	-0.46	1.21
8	5w-2	-31.46^d	-32.22	-0.76	2.42	-32.38	-0.92	2.92	-30.84	0.62	1.97
9	5w-3	-26.57^d	-27.82	-1.25	4.70	-29.71	-3.14	11.82	-27.07	-0.50	1.88
10	5w-4	-24.85^d	-25.94	-1.09	4.39	-27.95	-3.10	12.47	-24.27	0.58	2.33
11	5w-5	-24.18^d	-25.31	-1.13	4.67	-26.07	-1.89	7.82	-23.30	0.88	3.64
12	5w-6	-26.07^d	-26.53	-0.46	1.76	-26.32	-0.25	0.96	-24.48	1.59	6.10
13	5w-7	-25.94^d	-26.36	-0.42	1.62	-26.48	-0.54	2.08	-25.06	0.88	3.39
14	6w-prism	-36.40^d	-38.07	-1.67	4.59	-37.99	-1.59	4.37	-37.91	-1.51	4.15
15	6w-cage	-37.53^d	-38.99	-1.46	3.89	-40.29	-2.76	7.35	-38.07	-0.54	1.44
16	6w-book1	-43.43^d	-43.76	-0.33	0.76	-42.97	0.46	1.06	-43.22	0.21	0.48
17	6w-book2	-42.30^d	-42.84	-0.54	1.28	-41.21	1.09	2.58	-41.80	0.50	1.18
18	6w-bag	-43.30^d	-44.10	-0.80	1.85	-41.59	1.71	3.95	-41.59	1.71	3.95
19	6w-chair	-49.29^d	-49.08	0.21	0.43	-48.53	0.76	1.54	-51.13	-1.84	3.73
20	6w-boat1	-47.45^d	-47.36	0.09	0.19	-46.28	1.17	2.47	-49.08	-1.63	3.44
21	6w-boat2	-47.45^d	-47.40	0.05	0.11	-46.32	1.13	2.38	-49.41	-1.96	4.13
	RMSD			0.74			1.68			1.02
	MAD			1.67			3.64			1.96
	MRD(%)				2.12			6.11			2.86

Table 2 Calculated total three-body interaction energies of 23 medium-sized water clustersa

No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
1	10w-1	-70.58	-73.64	-3.06	4.34	-61.63	8.95	12.68	-70.63	-0.05	0.07
2	10w-2	-71.13	-74.52	-3.39	4.77	-65.81	5.32	7.48	-68.74	2.39	3.36
3	10w-3	-66.94	-70.84	-3.90	5.83	-61.09	5.85	8.74	-66.40	0.54	0.81
4	10w-4	-69.41	-72.97	-3.56	5.13	-62.76	6.65	9.58	-69.79	-0.38	0.55
5	10w-5	-70.84	-73.30	-2.46	3.47	-61.04	9.80	13.83	-75.02	-4.18	5.90
6	10w-6	-68.37	-73.01	-4.64	6.79	-64.43	3.94	5.76	-71.09	-2.72	3.98
No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
7	10w-7	-70.42	-72.59	-2.17	3.08	-61.67	8.75	12.43	-75.27	-4.85	6.89
8	10w-8	-70.08	-72.89	-2.81	4.01	-61.09	8.99	12.83	-73.47	-3.39	4.84
9	10w-9	-69.45	-71.84	-2.39	3.44	-59.20	10.25	14.76	-72.01	-2.56	3.69
10	10w-10	-66.57	-70.17	-3.60	5.41	-61.21	5.36	8.05	-66.44	0.13	0.20
11	11w-1	-79.54	-82.09	-2.55	3.21	-70.00	9.54	11.99	-82.01	-2.47	3.11
12	11w-2	-82.63	-84.89	-2.26	2.74	-73.43	9.20	11.13	-82.30	0.33	0.40
13	11w-3	-78.87	-81.80	-2.93	3.71	-69.54	9.33	11.83	-79.54	-0.67	0.85
14	11w-4	-82.26	-84.22	-1.96	2.38	-69.41	12.85	15.62	-81.92	0.34	0.41
15	11w-5	-81.76	-83.60	-1.84	2.25	-71.84	9.92	12.13	-82.09	-0.33	0.40
16	12w-1	-88.70	-92.47	-3.77	4.25	-77.78	10.92	12.31	-87.91	0.79	0.89
17	13w-1	-99.87	-103.51	-3.64	3.64	-85.77	14.10	14.12	-99.54	0.33	0.33
18	14w-1	-111.55	-115.48	-3.93	3.52	-95.10	16.45	14.75	-108.95	2.60	2.33
19	15w-1	-123.39	-127.90	-4.51	3.66	-101.96	21.43	17.37	-121.92	1.47	1.19
20	17w-1	-142.05	-148.36	-6.31	4.44	-117.32	24.73	17.41	-138.87	3.18	2.24
21	17w-2	-146.15	-150.92	-4.77	3.26	-120.75	25.40	17.38	-142.67	3.48	2.38
22	21w-1	-182.17	-190.54	-8.37	4.59	-141.46	40.71	22.35	-181.71	0.46	0.25
23	21w-2	-184.81	-191.42	-6.61	3.58	-136.86	47.95	25.95	-179.16	5.65	3.06
	RMSD			4.04			17.91			2.50
	MAD			8.37			47.95			5.65
	MRD(%)				3.98			13.50			2.09

Table 2 Calculated total three-body interaction energies of 23 medium-sized water clustersa

No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
1	10w-1	-70.58	-73.64	-3.06	4.34	-61.63	8.95	12.68	-70.63	-0.05	0.07
2	10w-2	-71.13	-74.52	-3.39	4.77	-65.81	5.32	7.48	-68.74	2.39	3.36
3	10w-3	-66.94	-70.84	-3.90	5.83	-61.09	5.85	8.74	-66.40	0.54	0.81
4	10w-4	-69.41	-72.97	-3.56	5.13	-62.76	6.65	9.58	-69.79	-0.38	0.55
5	10w-5	-70.84	-73.30	-2.46	3.47	-61.04	9.80	13.83	-75.02	-4.18	5.90
6	10w-6	-68.37	-73.01	-4.64	6.79	-64.43	3.94	5.76	-71.09	-2.72	3.98
No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
7	10w-7	-70.42	-72.59	-2.17	3.08	-61.67	8.75	12.43	-75.27	-4.85	6.89
8	10w-8	-70.08	-72.89	-2.81	4.01	-61.09	8.99	12.83	-73.47	-3.39	4.84
9	10w-9	-69.45	-71.84	-2.39	3.44	-59.20	10.25	14.76	-72.01	-2.56	3.69
10	10w-10	-66.57	-70.17	-3.60	5.41	-61.21	5.36	8.05	-66.44	0.13	0.20
11	11w-1	-79.54	-82.09	-2.55	3.21	-70.00	9.54	11.99	-82.01	-2.47	3.11
12	11w-2	-82.63	-84.89	-2.26	2.74	-73.43	9.20	11.13	-82.30	0.33	0.40
13	11w-3	-78.87	-81.80	-2.93	3.71	-69.54	9.33	11.83	-79.54	-0.67	0.85
14	11w-4	-82.26	-84.22	-1.96	2.38	-69.41	12.85	15.62	-81.92	0.34	0.41
15	11w-5	-81.76	-83.60	-1.84	2.25	-71.84	9.92	12.13	-82.09	-0.33	0.40
16	12w-1	-88.70	-92.47	-3.77	4.25	-77.78	10.92	12.31	-87.91	0.79	0.89
17	13w-1	-99.87	-103.51	-3.64	3.64	-85.77	14.10	14.12	-99.54	0.33	0.33
18	14w-1	-111.55	-115.48	-3.93	3.52	-95.10	16.45	14.75	-108.95	2.60	2.33
19	15w-1	-123.39	-127.90	-4.51	3.66	-101.96	21.43	17.37	-121.92	1.47	1.19
20	17w-1	-142.05	-148.36	-6.31	4.44	-117.32	24.73	17.41	-138.87	3.18	2.24
21	17w-2	-146.15	-150.92	-4.77	3.26	-120.75	25.40	17.38	-142.67	3.48	2.38
22	21w-1	-182.17	-190.54	-8.37	4.59	-141.46	40.71	22.35	-181.71	0.46	0.25
23	21w-2	-184.81	-191.42	-6.61	3.58	-136.86	47.95	25.95	-179.16	5.65	3.06
	RMSD			4.04			17.91			2.50
	MAD			8.37			47.95			5.65
	MRD(%)				3.98			13.50			2.09

Table 3 Calculated total three-body interaction energies of 23 water nanoparticlesa

No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
1	16w-1	-135.65	-141.75	-6.10	4.50	-120.16	15.49	11.42	-135.44	0.21	0.15
2	16w-2	-141.00	-147.70	-6.70	4.75	-125.69	15.31	10.86	-139.58	1.42	1.01
3	16w-3	-150.21	-154.60	-4.39	2.92	-129.45	20.76	13.82	-149.91	0.30	0.20
4	16w-4	-150.21	-154.60	-4.39	2.92	-130.46	19.75	13.15	-150.79	-0.58	0.39
5	16w-5	-152.17	-156.82	-4.65	3.06	-132.17	20.00	13.14	-149.83	2.34	1.54
6	17w-3	-132.59	-137.90	-5.31	4.00	-111.67	20.92	15.78	-128.24	4.35	3.28
7	17w-4	-140.88	-147.07	-6.19	4.39	-116.02	24.86	17.65	-139.49	1.39	0.99
8	18w-1	-147.07	-152.80	-5.73	3.90	-123.30	23.77	16.16	-141.21	5.86	3.98
9	19w-1	-164.89	-172.00	-7.11	4.31	-128.57	36.32	22.03	-155.98	8.91	5.40
10	20w-1	-173.05	-180.54	-7.49	4.33	-140.50	32.55	18.81	-179.24	-6.19	3.58
11	20w-2	-170.75	-178.45	-7.70	4.51	-144.01	26.74	15.66	-170.75	0	0
12	20w-3	-166.23	-173.22	-6.99	4.21	-140.25	25.98	15.63	-173.34	-7.11	4.28
13	20w-4	-167.99	-175.98	-7.99	4.76	-133.93	34.06	20.28	-173.76	-5.77	3.43
14	20w-5	-170.87	-178.49	-7.62	4.46	-143.30	27.57	16.14	-175.60	-4.73	2.77
15	20w-6	-166.23	-175.81	-9.58	5.76	-144.10	22.13	13.31	-172.09	-5.86	3.53
16	20w-7	-166.65	-175.39	-8.74	5.24	-136.02	30.63	18.38	-167.86	-1.21	0.73
17	20w-8	-166.86	-174.22	-7.36	4.41	-138.41	28.45	17.05	-171.25	-4.39	2.63
18	20w-9	-168.49	-176.19	-7.70	4.57	-133.97	34.52	20.49	-167.44	1.05	0.62
19	20w-10	-167.82	-175.77	-7.95	4.74	-137.44	30.38	18.10	-172.42	-4.60	2.74
20	22w-1	-164.81	-173.43	-8.62	5.23	-124.73	40.08	24.32	-154.64	10.17	6.17
21	30w-1	-209.87	-221.42	-11.55	5.50	-166.73	43.14	20.56	-217.36	-7.49	3.57
22	30w-2	-196.77	-211.88	-15.11	7.68	-157.19	39.58	20.11	-203.64	-6.87	3.49
23	30w-3	-193.34	-215.60	-22.26	11.51	-164.47	28.87	14.93	-193.30	0.04	0.02
	RMSD			8.98			28.91			4.98
	MAD			22.26			43.14			10.17
	MRD(%)				4.85			16.86			2.37

Table 3 Calculated total three-body interaction energies of 23 water nanoparticlesa

No.	Cluster	CCSD(T)	MP2/aug-cc-pVDZ			M06-2X/jul-cc-pVTZ			This work
No.	Cluster	$E 3 B b$	$E 3 B c$	Δ	δ(%)	$E 3 B c$	Δ	δ(%)	E_3B	Δ	δ(%)
1	16w-1	-135.65	-141.75	-6.10	4.50	-120.16	15.49	11.42	-135.44	0.21	0.15
2	16w-2	-141.00	-147.70	-6.70	4.75	-125.69	15.31	10.86	-139.58	1.42	1.01
3	16w-3	-150.21	-154.60	-4.39	2.92	-129.45	20.76	13.82	-149.91	0.30	0.20
4	16w-4	-150.21	-154.60	-4.39	2.92	-130.46	19.75	13.15	-150.79	-0.58	0.39
5	16w-5	-152.17	-156.82	-4.65	3.06	-132.17	20.00	13.14	-149.83	2.34	1.54
6	17w-3	-132.59	-137.90	-5.31	4.00	-111.67	20.92	15.78	-128.24	4.35	3.28
7	17w-4	-140.88	-147.07	-6.19	4.39	-116.02	24.86	17.65	-139.49	1.39	0.99
8	18w-1	-147.07	-152.80	-5.73	3.90	-123.30	23.77	16.16	-141.21	5.86	3.98
9	19w-1	-164.89	-172.00	-7.11	4.31	-128.57	36.32	22.03	-155.98	8.91	5.40
10	20w-1	-173.05	-180.54	-7.49	4.33	-140.50	32.55	18.81	-179.24	-6.19	3.58
11	20w-2	-170.75	-178.45	-7.70	4.51	-144.01	26.74	15.66	-170.75	0	0
12	20w-3	-166.23	-173.22	-6.99	4.21	-140.25	25.98	15.63	-173.34	-7.11	4.28
13	20w-4	-167.99	-175.98	-7.99	4.76	-133.93	34.06	20.28	-173.76	-5.77	3.43
14	20w-5	-170.87	-178.49	-7.62	4.46	-143.30	27.57	16.14	-175.60	-4.73	2.77
15	20w-6	-166.23	-175.81	-9.58	5.76	-144.10	22.13	13.31	-172.09	-5.86	3.53
16	20w-7	-166.65	-175.39	-8.74	5.24	-136.02	30.63	18.38	-167.86	-1.21	0.73
17	20w-8	-166.86	-174.22	-7.36	4.41	-138.41	28.45	17.05	-171.25	-4.39	2.63
18	20w-9	-168.49	-176.19	-7.70	4.57	-133.97	34.52	20.49	-167.44	1.05	0.62
19	20w-10	-167.82	-175.77	-7.95	4.74	-137.44	30.38	18.10	-172.42	-4.60	2.74
20	22w-1	-164.81	-173.43	-8.62	5.23	-124.73	40.08	24.32	-154.64	10.17	6.17
21	30w-1	-209.87	-221.42	-11.55	5.50	-166.73	43.14	20.56	-217.36	-7.49	3.57
22	30w-2	-196.77	-211.88	-15.11	7.68	-157.19	39.58	20.11	-203.64	-6.87	3.49
23	30w-3	-193.34	-215.60	-22.26	11.51	-164.47	28.87	14.93	-193.30	0.04	0.02
	RMSD			8.98			28.91			4.98
	MAD			22.26			43.14			10.17
	MRD(%)				4.85			16.86			2.37

Table 4 Calculated total three-body interaction energies of 18 water clusters

No.	Cluster	CCSD(T)	WHBB			This work
No.	Cluster	$E 3 B *$ /(kJ·mol^-1)	E_3B/(kJ·mol^-1)	Δ/(kJ·mol^-1)	δ(%)	E_3B/(kJ·mol^-1)	Δ/(kJ·mol^-1)	δ(%)
1	11w-1	-79.54	-81.17	-1.63	2.05	-82.01	-2.47	3.11
2	11w-2	-82.63	-82.84	-0.21	0.25	-82.30	0.33	0.40
3	11w-3	-78.87	-81.59	-2.72	3.45	-79.54	-0.67	0.85
4	11w-4	-82.26	-83.68	-1.42	1.73	-81.92	0.34	0.41
5	11w-5	-81.76	-83.26	-1.50	1.83	-82.09	-0.33	0.40
6	12w-1	-88.70	-93.30	-4.60	5.19	-87.91	0.79	0.89
7	13w-1	-99.87	-103.76	-3.89	3.90	-99.54	0.33	0.33
8	14w-1	-111.55	-115.06	-3.51	3.15	-108.95	2.60	2.33
9	15w-1	-123.39	-123.01	0.38	0.31	-121.92	1.47	1.19
10	17w-1	-142.05	-148.95	-6.90	4.86	-138.87	3.18	2.24
11	17w-2	-146.15	-144.77	1.38	0.94	-142.67	3.48	2.38
12	17w-3	-132.59	-140.16	-7.57	5.71	-128.24	4.35	3.28
13	17w-4	-140.88	-141.84	-0.96	0.68	-139.49	1.39	0.99
14	18w-1	-147.07	-153.97	-6.90	4.69	-141.21	5.86	3.98
15	19w-1	-164.89	-169.03	-4.14	2.51	-155.98	8.91	5.40
16	21w-1	-182.17	-193.30	-11.13	6.11	-181.71	0.46	0.25
17	21w-2	-184.81	-190.37	-5.56	3.01	-179.16	5.65	3.06
18	22w-1	-164.81	-190.37	-25.56	15.51	-154.64	10.17	6.17
	RMSD/(kJ·mol^-1)			7.61			4.14
	MAD/(kJ·mol^-1)			25.56			10.17
	MRD(%)				3.66			2.09

Table 4 Calculated total three-body interaction energies of 18 water clusters

No.	Cluster	CCSD(T)	WHBB			This work
No.	Cluster	$E 3 B *$ /(kJ·mol^-1)	E_3B/(kJ·mol^-1)	Δ/(kJ·mol^-1)	δ(%)	E_3B/(kJ·mol^-1)	Δ/(kJ·mol^-1)	δ(%)
1	11w-1	-79.54	-81.17	-1.63	2.05	-82.01	-2.47	3.11
2	11w-2	-82.63	-82.84	-0.21	0.25	-82.30	0.33	0.40
3	11w-3	-78.87	-81.59	-2.72	3.45	-79.54	-0.67	0.85
4	11w-4	-82.26	-83.68	-1.42	1.73	-81.92	0.34	0.41
5	11w-5	-81.76	-83.26	-1.50	1.83	-82.09	-0.33	0.40
6	12w-1	-88.70	-93.30	-4.60	5.19	-87.91	0.79	0.89
7	13w-1	-99.87	-103.76	-3.89	3.90	-99.54	0.33	0.33
8	14w-1	-111.55	-115.06	-3.51	3.15	-108.95	2.60	2.33
9	15w-1	-123.39	-123.01	0.38	0.31	-121.92	1.47	1.19
10	17w-1	-142.05	-148.95	-6.90	4.86	-138.87	3.18	2.24
11	17w-2	-146.15	-144.77	1.38	0.94	-142.67	3.48	2.38
12	17w-3	-132.59	-140.16	-7.57	5.71	-128.24	4.35	3.28
13	17w-4	-140.88	-141.84	-0.96	0.68	-139.49	1.39	0.99
14	18w-1	-147.07	-153.97	-6.90	4.69	-141.21	5.86	3.98
15	19w-1	-164.89	-169.03	-4.14	2.51	-155.98	8.91	5.40
16	21w-1	-182.17	-193.30	-11.13	6.11	-181.71	0.46	0.25
17	21w-2	-184.81	-190.37	-5.56	3.01	-179.16	5.65	3.06
18	22w-1	-164.81	-190.37	-25.56	15.51	-154.64	10.17	6.17
	RMSD/(kJ·mol^-1)			7.61			4.14
	MAD/(kJ·mol^-1)			25.56			10.17
	MRD(%)				3.66			2.09

Table 5 Deviations of different methods with respect to the CCSD(T) three-body interaction energies of 67 water clusters

Method	RMSD/(kJ·mol^-1)	MAD/(kJ·mol^-1)	MRD(%)
MP2	5.78	22.26	3.70
M06-2X	19.95	47.95	12.34
This work	3.32	10.17	2.43

Fig.1 Correlation plots for the total three-body interaction energies of the 67 water clusters (A) MP2/aug-cc-pVDZ; (B) M06-2X/jul-cc-pVTZ; (C) this work.

Fig.2 Three-body interaction energy curves with respect to the intermolecular distances(A—C) and angles(D—F)

Fig.3 Total three-body interaction energy curves of the prism hexamer

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