可快速计算水团簇三体作用强度的新方法

doi:10.7503/cjcu20190038

摘要/Abstract

摘要：

将水分子视为由2个O—H键偶极构成, 再将水分子间的三体作用视为长程诱导作用和短程校正之和, 使用Thole模型计算长程诱导作用, 通过同时考虑不同水分子间的置换和同一个水分子中2个键偶极间的置换计算短程校正, 从而提出了一个可快速计算水团簇三体作用强度的新方法. 根据已报道的12347个水三聚体的结构和CCSD(T)三体作用能, 确定了该方法所需参数. 将该方法和所确定的参数应用于67个水团簇体系, 计算这些体系的三体作用能, 并与CCSD(T), MP2, M06-2X方法的计算结果进行比较. 结果表明, 相对于CCSD(T)方法的总三体作用能, 本文方法的均方根偏差(RMSD)仅为3.32 kJ/mol, 平均相对偏差(MRD)仅为2.43%; 对较大水团簇体系, 该方法计算精度稍优于MP2方法, 明显优于M06-2X方法, 并且更快捷高效.

关键词: 水团簇, 三体作用, 键偶极

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

中图分类号:

O641

TrendMD:

李晓蕾, 王长生. 可快速计算水团簇三体作用强度的新方法. 高等学校化学学报, 2019, 40(5): 950.

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

图/表 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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