MATCH Commun. Math. Comput. Chem. 56 (2006) 97-111
MATCH
Communications in Mathematical
and in Computer Chemistry
ISSN 0340 - 6253
Topological Research on Lattice Energies for Inorganic Compounds
Lailong Mu*, Changjun Feng
Department of Chemistry, Xuzhou Normal University, Xuzhou 221116, P. R. China
Hongmei He
XuZhou College of Industrial Technology, Xuzhou, 221006, P. R. China
MULL@XZNU.EDU.CN
(Received November 3, 2005)
A novel connectivity index mG and its converse index mG’ based on adjacency matrix of
molecular graphs are proposed as follows: mG = Σ (gi.gj.gk.…)0.5, mG’ = Σ (gi.gj.gk.…)-0.5. The gi
element of adjacency matrix is defined as gi=(1+mi1.7).Xi0.3/(1+ri), where mi,Xi,ri, are the
charge number, the electronegativity and Goldschmidt ionic radius of atom i respectively. By
using the connectivity index mG and its converse index mG’, the multiple linear regression and
multiple nonlinear regression can provide high-quality QSPR models for the lattice energies
of 318 inorganic compounds. The results imply that the lattice energies may be expressed as a
linear or nonlinear combination of the connectivity indices 0G,1G,0G’ and 1G’. For the
nonlinear model, the correlation coefficient r and the standard error s are 0.9998 and 217.6
kJ.mol-1, respectively. The cross-validation by using the leave-one-out method demonstrates
that the models are highly reliable from the point of view of statistics.
1. INTRODUCTION
The quantitative structure-property/activity relationship (QSPR/QSAR) studies of organic
compounds have been a focus of great attention by the scientists for a long time1-3. Large
numbers of QSPR/QSAR models have been developed by using various model parameters to
describe and predict the physical properties and biological activities of organic compounds
from their molecular structures. However, there are only a few papers4-11 about the QSPR/
*
Tel:+86-516-3403163;Fax: +86-516-3403164
E-mail address: mull@xznu.edu.cn
- 98 -
QSAR studies of inorganic compounds, due to the complexity of inorganic-compound
composition.
The lattice energy of an inorganic ionic crystal is a dominant term in the thermodynamic
analysis of the existence and stability of an ionic crystal. It is one of the most important
quantities in indicating the structure, character, and behavior (reactivity) of solids. Although
lattice energy can be determined experimentally from the Born-Haber-Fajans thermochemical
cycle, direct experimental determination is generally not possible since, in practice, the
crystalline solid dissociates into atoms but not gaseous ions, as it is required in the lattice
energy evaluation. Therefore, its indirect experimental determination, computation, or
estimation is of considerable interest in modern materials science. Theoretical studies on
lattice energy have been carried out almost since the beginning of 1900s, and a variety of
estimation methods for lattice energies were available.12~29
In recent years, Feng et al.7,9-11 and Xu et al.8 etc., by using various model parameters,
have developed some QSPR/QSAR models to estimate the lattice energies of inorganic
compounds. In this study, the ionic polarization force parameter gi is defined based on ionic
radii, electronegativity, ionic charge. A now topological index mG and its converse index mG’
are proposed based on the randic’s index and gi. The connectivity index mG together with mG’
has good correlations for lattice energies of inorganic ionic crystals.
2. METHODS
Ionic crystals are made up of positive and negative ions, and the strongest interactions
result from the nearest oppositely charged ions. The lattice energy contribution originates
mainly from these nearest interactions, and the sum of the lattice energies of these interacting
ion pairs is affected by the ionic radius, electronegativity, ionic charge. As mentioned above
the polarization force parameter gi is defined as:
gi=(1+mi1.7).Xi0.3/(1+ri)
(1)
where mi, Xi , ri are the charge number, the electronegativity30 and Goldschmidt31 ionic radius
of atom i respectively. For ions of different atoms and ions of the same atom with different
charges, the gi are different. The gi of 78 ions are calculated and listed in Table 1.
Table 1. Ionic polarization force parameters
No
1
2
3
4
Ionic
Ag+
Al3+
As3+
Au3+
ri
1.13
0.55
0.69
1.37
Xi
1.93
1.61
2.18
2.54
mi
1
3
3
3
gi
1.1437
5.5618
5.5866
1.1162
No
40
41
42
43
Ionic
Nd3+
Ni2+
P5+
Pb2+
ri
1.15
0.78
0.17
1.32
Xi
1.14
1.91
2.19
2.33
mi
3
2
5
2
gi
3.6152
2.8985
17.7613
2.3605
- 99 -
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
B3+
Ba2+
Be2+
Bi3+
Ca2+
Cd2+
Ce3+
Co2+
Co3+
Cr2+
Cr3+
Cs+
Cu+
Cu2+
Dy3+
Er3+
Fe2+
Fe3+
Ga3+
Gd3+
Ge2+
Hg2+
Ho3+
In3+
K+
La3+
Li+
Mg2+
Mn2+
Mn3+
Mn4+
Mo3+
Mo4+
Mo6+
Na+
0.20
1.38
0.34
1.20
1.05
0.99
1.19
0.82
0.65
0.89
0.65
1.70
0.96
0.72
1.07
1.04
0.83
0.67
0.62
1.11
0.65
1.12
1.05
0.92
1.33
1.15
0.70
0.75
0.91
0.62
0.52
0.69
0.65
0.41
0.98
2.04
0.89
1.57
2.02
1.00
1.69
1.12
1.88
1.88
1.66
1.66
0.79
1.90
1.90
1.22
1.24
1.83
1.83
1.81
1.20
2.01
2.00
1.23
1.78
0.82
1.10
0.98
1.31
1.55
1.55
1.55
2.16
2.16
2.16
0.93
3
2
2
3
2
2
3
2
3
2
3
1
1
2
3
3
2
3
3
3
2
2
3
3
1
3
1
2
2
3
4
3
4
6
1
7.7126
1.7240
3.6304
4.1945
2.0727
2.4992
3.5303
2.8214
5.4734
2.6173
5.2728
0.6902
1.2371
2.9949
3.8321
3.9074
2.7834
5.3643
5.5117
3.7408
3.1751
2.4675
3.8789
4.6272
0.8088
3.5766
1.1694
2.6329
2.5372
5.2611
8.6709
5.5712
8.8239
19.6856
0.9883
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
Pb4+
Pd2+
Pr3+
Rb+
Rh3+
Sb3+
Sb5+
Sc3+
Si4+
Sm3+
Sn2+
Sn4+
Sr2+
Ti2+
Ti3+
Ti4+
Tl+
Tl3+
Tm3+
V2+
V3+
V5+
Y3+
Zn2+
Zr4+
BrClFHIN3O2S2Se2Te2-
0.84
0.80
1.12
1.49
0.75
0.90
0.62
0.83
0.40
1.13
1.02
0.74
1.18
0.76
0.70
0.64
1.49
1.05
1.04
0.88
0.75
0.36
0.95
0.83
0.80
1.96
1.81
1.33
2.08
2.20
1.71
1.35
1.82
1.93
2.12
2.33
2.20
1.13
0.82
2.28
2.05
2.05
1.36
1.90
1.17
1.96
1.96
0.95
1.54
1.54
1.54
2.04
2.04
1.25
1.63
1.63
1.63
1.22
1.65
1.33
2.96
3.16
3.98
2.20
2.66
3.04
3.44
2.58
2.55
2.10
4
2
3
1
3
3
5
3
4
3
2
4
2
2
3
4
1
3
3
2
3
5
3
2
4
1
1
1
1
1
3
2
2
2
2
8.0947
2.9905
3.6566
0.7568
5.4681
4.8783
12.5759
4.4782
10.0071
3.6777
2.5740
8.1271
1.9193
2.7481
5.0038
8.0209
0.9948
4.5147
3.9169
2.6169
4.9444
13.9844
4.0679
2.6982
6.9935
0.9357
1.0051
1.2991
0.8226
0.8382
3.8494
2.6193
2.0023
1.9204
1.7014
Based on adjacency matrix of molecular graphs, the novel connectivity index mG and its
converse index mG’ can be defined as follows:
G=Σ(gi.gj.gk.…)0.5
m
(2)
G’=Σ(gi.gj.gk.…)-0.5
m
(3)
0
where, m is the order of the molecular connectivity index. The G, 1G, 0G’, and 1G’ are
defined as follows:
0
G=Σ(gi)0.5
(4)
G=Σ(gi.gj)0.5
(5)
G’=Σ(gi)-0.5
(6)
1
0
1
G’=Σ(gi.gj)
-0.5
(7)
- 100 -
In expressions (4) and (6), the “Σ” is the sum of atomic of inorganic compound chemical
formula. In expressions (5) and (7), the “Σ” is the sum of chemical single bonds of chemical
formula. For example, the 0G, 1G, 0G’, and 1G’ of Ba4NaSb3O12 are calculated as follows:
For Ba4NaSb3O12, it can be treated as consisting of four Ba2+, one Na+, three Sb5+ and
twelve O2-, it contains eight Ba-O single bonds, one Na-O single bonds and fifteen Sb-O
single bonds, so:
0
G=4×1.72400.5+0.98830.5+3×12.57590.5+12×2.61930.5= 36.3060
1
G=8×(1.7240×2.6193)0.5+(0.9883×2.6193)0.5+15×(12.5759×2.6193)0.5=104.6993
0
G’=4×1.7240-0.5+0.9883-0.5+3×12.5759-0.5+12×2.6193-0.5=12.3129
1
G’=8×(1.7240×2.6193)-0.5+(0.9883×2.6193)-0.5+15×(12.5759×2.6193)-0.5=6.9998
The 0G, 1G, 0G’, and 1G’ of 318 inorganic compounds are calculated and listed in Table 2.
Table2. The calculated and Experimental Lattice Energies of 318inorganic compounds (kJ.mol-1)
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Substance
LiF
NaF
KF
RbF
CsF
LiCl
NaCl
KCl
RbCl
CsCl
LiBr
NaBr
KBr
RbBr
CsBr
LiI
NaI
KI
RbI
CsI
BeF2
MgF2
CaF2
SrF2
BaF2
BeCl2
MgCl2
CaCl2
SrCl2
BaCl2
BeBr2
MgBr2
CaBr2
SrBr2
BaBr2
0
G
2.2211
2.1339
2.0391
2.0097
1.9705
2.0839
1.9967
1.9019
1.8725
1.8333
2.0487
1.9615
1.8666
1.8372
1.7981
1.9969
1.9097
1.8148
1.7855
1.7463
4.1849
3.9022
3.7192
3.6649
3.5925
3.9105
3.6278
3.4448
3.3905
3.3181
3.8400
3.5572
3.3743
3.3200
3.2476
1
G
1.2325
1.1331
1.0250
0.9915
0.9469
1.0841
0.9967
0.9016
0.8722
0.8329
1.0460
0.9617
0.8699
0.8415
0.8036
0.9900
0.9102
0.8233
0.7965
0.7606
4.3434
3.6988
3.2818
3.1581
2.9930
3.8205
3.2536
2.8868
2.7779
2.6327
3.6861
3.1391
2.7852
2.6802
2.5401
0
G’
1.8021
1.8832
1.9893
2.0269
2.0811
1.9222
2.0033
2.1094
2.1469
2.2011
1.9586
2.0397
2.1458
2.1833
2.2375
2.0170
2.0981
2.2042
2.2418
2.2960
2.2796
2.3710
2.4493
2.4765
2.5163
2.5197
2.6112
2.6895
2.7167
2.7565
2.5924
2.6839
2.7622
2.7894
2.8292
1
G’
0.8113
0.8825
0.9756
1.0085
1.0561
0.9224
1.0033
1.1091
1.1466
1.2006
0.9560
1.0399
1.1496
1.1884
1.2444
1.0101
1.0987
1.2146
1.2556
1.3148
0.9209
1.0814
1.2188
1.2666
1.3364
1.0470
1.2294
1.3856
1.4399
1.5193
1.0852
1.2742
1.4362
1.4924
1.5747
U(ref)
1049
930
829
795
759
864
790
720
695
670
820
754
691
668
647
764
705
650
632
613
3526
2978
2651
2513
2373
3033
2540
2271
2170
2069
2914
2451
2134
2040
1995
U(pred)
1025
941
848
818
778
897
817
725
695
654
863
783
691
661
620
813
733
640
610
567
3505
3012
2678
2575
2435
3141
2681
2363
2264
2128
3047
2595
2280
2182
2046
Er.
24
-11
-19
-23
-19
-33
-27
-5
0
16
-43
-29
0
7
27
-49
-28
10
22
46
21
-34
-27
-62
-62
-108
-141
-92
-94
-59
-133
-144
-146
-142
-51
ref
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
9
9
32
- 101 -
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
BeI2
MgI2
CaI2
SrI2
BaI2
BeO
MgO
CaO
SrO
BaO
BeS
MgS
CaS
SrS
BaS
CuCl
AgCl
Al2O3
AgF
AgBr
AgI
TiF2
VF2
CrF2
MnF2
FeF2
CoF2
NiF2
CuF2
ZnF2
TiCl2
VCl2
CrCl2
MnCl2
FeCl2
CoCl2
NiCl2
CuCl2
TiI2
VI2
CrI2
MnI2
FeI2
CoI2
NiI2
CuI2
TlF
TlCl
TlBr
TlI
Tl2O
Tl2S
Tl2Se
Tl2Te
TlF3
TlCl3
TlBr3
3.7364
3.4537
3.2707
3.2165
3.1441
3.5238
3.2410
3.0581
3.0038
2.9314
3.3204
3.0376
2.8547
2.8004
2.7280
2.1148
2.0720
9.5720
2.2092
2.0367
1.9850
3.9373
3.8972
3.8974
3.8724
3.9479
3.9593
3.9821
4.0101
3.9222
3.6629
3.6228
3.6230
3.5980
3.6735
3.6848
3.7076
3.7357
3.4888
3.4487
3.4489
3.4239
3.4994
3.5108
3.5336
3.5616
2.1372
1.9999
1.9647
1.9129
3.6132
3.4098
3.3805
3.2991
5.5441
5.1325
5.0267
3.4888
2.9711
2.6361
2.5367
2.4042
6.1674
5.2522
4.6601
4.4843
4.2500
5.3922
4.5921
4.0744
3.9207
3.7158
1.1151
1.0722
22.9008
1.2189
1.0345
0.9791
3.7789
3.6876
3.6879
3.6310
3.8031
3.8290
3.8810
3.9450
3.7445
3.3240
3.2437
3.2439
3.1939
3.3453
3.3680
3.4138
3.4701
3.0354
2.9621
2.9623
2.9166
3.0548
3.0756
3.1174
3.1688
1.1368
0.9999
0.9648
0.9131
3.2284
2.8226
2.7643
2.6019
7.2653
6.3907
6.1659
2.7094
2.8008
2.8791
2.9063
2.9461
1.1427
1.2342
1.3125
1.3397
1.3795
1.2315
1.3230
1.4013
1.4285
1.4683
1.8965
1.9325
2.7017
1.8124
1.9689
2.0273
2.3580
2.3729
2.3728
2.3825
2.3541
2.3501
2.3421
2.3326
2.3635
2.5981
2.6130
2.6130
2.6227
2.5943
2.5902
2.5822
2.5727
2.7878
2.8027
2.8026
2.8123
2.7839
2.7799
2.7719
2.7624
1.8800
2.0001
2.0364
2.0949
2.6231
2.7120
2.7269
2.7719
3.1027
3.4629
3.5720
1.1465
1.3463
1.5174
1.5768
1.6638
0.6486
0.7616
0.8584
0.8920
0.9412
0.7418
0.8711
0.9817
1.0202
1.0765
0.8968
0.9327
1.5720
0.8204
0.9667
1.0213
1.0585
1.0847
1.0846
1.1016
1.0518
1.0447
1.0307
1.0140
1.0682
1.2034
1.2332
1.2331
1.2524
1.1957
1.1876
1.1717
1.1527
1.3178
1.3504
1.3503
1.3715
1.3094
1.3005
1.2831
1.2623
0.8797
1.0001
1.0365
1.0951
1.2390
1.4171
1.4470
1.5373
1.2388
1.4083
1.4596
2813
2340
2087
1976
1890
4443
3791
3401
3223
3054
3770
3238
2966
2779
2643
996
918
15111
974
905
892
2757
2770
2939
2803
2967
3042
3089
3102
3053
2514
2593
2601
2551
2641
2706
2786
2824
2342
2470
2440
2388
2491
2561
2637
2694
850
751
734
710
2575
2258
2326
2172
5431
5278
5146
2909
2467
2156
2058
1922
4359
3738
3328
3204
3038
3842
3280
2906
2793
2640
925
886
15361
1013
852
802
3075
3003
3004
2959
3093
3113
3154
3203
3048
2740
2672
2673
2630
2757
2776
2814
2860
2524
2459
2459
2418
2541
2559
2596
2640
944
820
786
736
2661
2306
2253
2099
5704
5113
4962
-96
-127
-69
-82
-32
84
53
73
19
16
-72
-42
60
-14
3
71
32
-250
-39
53
90
-318
-233
-65
-156
-126
-71
-65
-101
5
-226
-79
-72
-79
-116
-70
-28
-36
-182
11
-19
-30
-50
2
41
54
-94
-69
-52
-26
-86
-48
73
73
-273
165
184
32
32
32
32
32
32
32
32
32
32
9
9
9
9
9
32
32
9
32
32
32
9
9
32
9
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
9
32
32
32
9
32
32
32
32
32
32
9
32
9
32
9
- 102 -
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
TiO2
PbF2
PbCl2
PbBr2
PbI2
PbO
PbS
PbSe
PbTe
PbF4
PbO2
AlF3
AlCl3
AlBr3
AlI3
TiBr2
TiO
TiS
TiF3
TiCl3
TiBr3
TiI3
Ti2O3
TiF4
TiCl4
TiBr4
TiI4
CuF
CuBr
CuI
Cu2O
Cu2S
Cu2Se
CdF2
HgF2
VO
MnO
FeO
CoO
NiO
CuO
ZnO
CdO
HgO
GeO
SnO
PdO
MnBr2
MnS
MnSe
MnTe
MnF3
MnCl3
MnO2
MgSe
MgTe
ScBr3
6.0690
3.8159
3.5415
3.4710
3.3675
3.1548
2.9514
2.9222
2.8408
7.4042
6.0820
5.7777
5.3660
5.2602
5.1049
3.5923
3.2762
3.0728
5.6563
5.2446
5.1388
4.9835
9.3291
7.3912
6.8424
6.7013
6.4942
2.2520
2.0795
2.0278
3.8429
3.6395
3.6103
3.8604
3.8504
3.2361
3.2113
3.2868
3.2981
3.3209
3.3490
3.2611
3.1993
3.1893
3.4003
3.2228
3.3477
3.5275
3.0079
2.9786
2.8972
5.7130
5.3014
6.1815
3.0084
2.9270
5.0181
18.3343
3.5023
3.0807
2.9723
2.8132
4.9731
4.3481
4.2582
4.0080
12.9712
18.4185
8.0639
7.0932
6.8437
6.4774
3.2071
5.3659
4.6915
7.6488
6.7280
6.4914
6.1439
21.7218
12.9119
11.3576
10.9581
10.3715
1.2677
1.0759
1.0183
3.6002
3.1477
3.0826
3.6037
3.5808
5.2362
5.1559
5.4002
5.4370
5.5108
5.6017
5.3170
5.1171
5.0846
5.7677
5.1931
5.5975
3.0816
4.5078
4.4147
4.1553
7.8430
6.8988
19.0628
4.4971
4.2330
6.1410
1.5889
2.4056
2.6457
2.7185
2.8354
1.2688
1.3576
1.3725
1.4175
3.8609
1.5872
3.0561
3.4163
3.5254
3.7008
2.6708
1.2211
1.3099
3.0791
3.4394
3.5484
3.7238
2.7477
3.8626
4.3428
4.4883
4.7222
1.7764
1.9329
1.9913
2.4160
2.5049
2.5198
2.3873
2.3913
1.2361
1.2457
1.2173
1.2132
1.2053
1.1957
1.2267
1.2504
1.2545
1.1791
1.2412
1.1962
2.6954
1.3345
1.3494
1.3945
3.0681
3.4283
1.5754
1.3379
1.3829
3.5740
0.8727
1.1421
1.2984
1.3457
1.4219
0.8043
0.9200
0.9394
0.9980
1.2335
0.8687
1.1161
1.2688
1.3151
1.3895
1.2472
0.7455
0.8526
1.1767
1.3377
1.3865
1.4649
1.6573
1.2392
1.4088
1.4601
1.5427
0.7888
0.9295
0.9820
1.1111
1.2708
1.2976
1.1100
1.1171
0.7639
0.7758
0.7407
0.7357
0.7259
0.7141
0.7523
0.7817
0.7867
0.6935
0.7702
0.7146
1.2980
0.8873
0.9061
0.9626
1.1475
1.3046
0.8393
0.8895
0.9450
1.4656
12054
2543
2282
2230
2177
3520
3161
3144
3039
9468
11682
6252
5513
5360
5227
2430
3811
3357
5665
5153
5023
4971
14487
9908
9129
9059
8918
1088
978
966
3189
2865
2936
2830
2780
3863
3745
3865
3910
4010
4050
3971
3806
3907
3919
3652
3736
2482
3415
3330
3207
6012
5556
13150
3365
3202
4761
11923
2857
2534
2449
2324
3546
3105
3041
2860
9727
11973
6269
5638
5476
5240
2653
3816
3351
5977
5367
5211
4982
14617
9688
8734
8491
8137
1054
891
840
2939
2570
2515
2937
2919
3727
3672
3839
3864
3915
3977
3782
3645
3623
4089
3697
3974
2545
3220
3154
2969
6114
5494
12355
3213
3026
4942
131
-314
-252
-219
-147
-26
56
103
179
-259
-291
-17
-125
-116
-13
-223
-5
6
-312
-214
-188
-11
-130
220
395
568
781
34
87
126
250
295
421
-107
-139
136
73
26
46
95
73
189
161
284
-170
-45
-238
-63
195
176
238
-102
62
795
152
176
-181
9
32
32
32
32
9
9
9
9
9
9
32
32
32
32
32
32
9
9
32
32
9
25
32
9
32
32
9
32
32
32
32
9
32
9
32
32
32
32
32
32
32
9
9
9
9
9
32
9
9
9
9
9
9
9
9
32
- 103 -
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
YBr3
LaBr3
CeBr3
CdCl2
CdBr2
CdI2
CdS
CdSe
CoCl2
CoBr2
CoI2
CoS
CoSe
ZnCl2
HgCl2
SnCl2
FeBr2
NiBr2
CuBr2
ZnBr2
HgBr2
SnBr2
ZnI2
HgI2
SnI2
FeS
CoS
NiS
CuS
ZnS
SnS
Ag2O
Ag2S
Ag2Te
AlH3
AsBr3
AsI3
AuBr
AuCl
AuH
AuI
BaH2
BeH2
BiCl3
CaH2
CeCl3
CeI3
Cr2O3
CrBr2
CrCl3
CrF3
CrI3
CrN
Cs2O
Cs2S
CsH
Cu2Te
4.9188
4.7931
4.7808
3.5860
3.5155
3.4119
2.9959
2.9667
3.6848
3.6143
3.5108
3.0947
3.0655
3.6478
3.5760
3.6095
3.6030
3.6371
3.6652
3.5772
3.5054
3.5390
3.4737
3.4019
3.4354
3.0834
3.0947
3.1175
3.1456
3.0577
3.0194
3.7573
3.5539
3.4433
5.0793
5.2655
5.1102
2.0238
2.0591
1.9635
1.9720
3.1270
3.7193
5.0558
3.2537
4.8866
4.6255
9.4478
3.5524
5.3040
5.7156
5.0429
4.2582
3.2800
3.0765
1.7378
3.5289
5.8529
5.4881
5.4525
3.1699
3.0584
2.8947
4.4740
4.3815
3.3680
3.2496
3.0756
4.7536
4.6554
3.2937
3.1497
3.2170
3.2276
3.2937
3.3480
3.1779
3.0389
3.1038
3.0078
2.8763
2.9377
4.7215
4.7536
4.8182
4.8976
4.6487
4.5404
3.4616
3.0266
2.7899
6.4170
6.8589
6.4918
1.0219
1.0592
0.9582
0.9672
2.3818
3.4563
6.1599
2.6116
5.6512
5.1606
22.2981
3.1298
6.9065
7.8517
6.3069
13.5157
2.6891
2.3511
0.7535
2.9015
3.5972
3.6302
3.6336
2.6274
2.7002
2.8171
1.3393
1.3542
2.5902
2.6629
2.7799
1.3020
1.3170
2.6037
2.6315
2.6182
2.6670
2.6550
2.6454
2.6764
2.7042
2.6909
2.7933
2.8211
2.8078
1.3061
1.3020
1.2941
1.2845
1.3155
1.3300
2.4880
2.5768
2.6368
3.7317
3.5245
3.6999
1.9803
1.9440
2.0491
2.0388
2.9667
2.7299
3.4806
2.8997
3.5245
3.8090
2.7246
2.6857
3.4278
3.0676
3.7123
0.9452
3.0253
3.1141
2.3063
2.5648
1.5377
1.6399
1.6506
1.2619
1.3079
1.3818
0.8941
0.9129
1.1876
1.2309
1.3005
0.8415
0.8592
1.2144
1.2699
1.2434
1.2393
1.2144
1.1947
1.2587
1.3162
1.2887
1.3299
1.3907
1.3616
0.8472
0.8415
0.8302
0.8167
0.8605
0.8810
1.1555
1.3216
1.4337
1.4025
1.3122
1.3864
0.9785
0.9441
1.0436
1.0339
1.6794
1.1573
1.4611
1.5317
1.5926
1.7440
1.6145
1.2780
1.3031
1.1463
1.4270
0.6659
1.4875
1.7013
1.3271
1.3786
4435
4280
4418
2565
2517
2455
3460
3310
2695
2643
2531
3653
3611
2748
2664
2310
2515
2721
2774
2689
2639
2256
2619
2624
2206
3506
3653
3694
3795
3674
3201
2910
2677
2600
5969
5365
5295
1059
1066
1108
1070
2133
3306
4707
2406
4348
4061
14957
2536
5529
6065
5294
8358
2063
1850
653
2683
4716
4421
4391
2610
2525
2398
3196
3130
2776
2689
2559
3395
3326
2714
2593
2650
2670
2726
2771
2628
2508
2564
2500
2382
2437
3372
3395
3441
3496
3320
3243
2836
2473
2263
5201
5487
5251
841
874
782
790
1901
2887
4936
2136
4535
4179
14981
2587
5500
6120
5109
8779
2247
1901
558
2358
-281
-141
27
-45
-8
57
264
180
-81
-46
-28
258
285
34
71
-340
-155
-5
3
61
131
-308
119
242
-231
134
258
253
299
354
-42
74
204
337
768
-122
44
218
192
326
280
232
419
-229
270
-187
-118
-24
-51
29
-55
185
-421
-184
-51
95
325
9
9
9
32
32
32
9
9
9
32
9
9
9
32
32
32
9
32
32
32
32
32
32
32
32
9
9
9
9
9
9
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
- 104 -
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
CuH
DyCl3
ErCl3
Fe2O3
FeCl3
Ga2O3
GaBr3
GaCl3
GaF3
GaI3
GdCl3
HoCl3
InBr3
InCl3
InI3
K2O
K2S
KH
LaCl3
LaH3
LaI3
LaN
Li2O
Li2S
LiH
MgH2
Mn2O3
MoBr4
MoCl3
MoCl4
ZrBr4
ZrCl4
ZrF4
ZrI4
Na2O
Na2S
Na2Te
NaH
NdCl3
PdBr2
PdCl2
PdI2
PrCl3
PrI3
Rb2O
Rb2S
RbH
RhCl3
SbBr3
SbCl3
SbF3
SbI3
Sc2O3
ScCl3
ScF3
ScN
SmCl3
2.0192
4.9653
4.9844
9.4875
5.3238
9.5507
5.2496
5.3554
5.7670
5.0943
4.9418
4.9772
5.0530
5.1588
4.8977
3.4170
3.2136
1.8063
4.8989
4.6122
4.6378
3.8532
3.7812
3.5778
1.9884
3.4366
9.4427
6.8397
5.3680
6.9808
6.5137
6.6548
7.2036
6.3066
3.6067
3.4033
3.2927
1.9011
4.9091
3.6639
3.7344
3.5604
4.9199
4.6588
3.3583
3.1549
1.7769
5.3461
5.1106
5.2164
5.6280
4.9553
9.0877
5.1239
5.5355
4.0782
4.9254
1.0088
5.8878
5.9454
22.4907
6.9662
22.7975
6.8128
7.0612
8.0275
6.4482
5.8173
5.9237
6.2423
6.4699
5.9082
2.9109
2.5451
0.8157
5.6882
5.1459
5.1943
11.1315
3.5002
3.0603
0.9808
2.9434
22.2733
11.4935
7.0992
11.9126
10.2322
10.6053
12.0566
9.6845
3.2179
2.8135
2.5935
0.9017
5.7187
3.3455
3.4675
3.1665
5.7515
5.2521
2.8159
2.4620
0.7890
7.0332
6.4094
6.6431
7.5522
6.0664
20.5493
6.3649
7.2359
12.4557
5.7680
2.0016
3.5031
3.4982
2.7172
3.4241
2.7055
3.5274
3.4183
3.0580
3.7027
3.5093
3.5001
3.5663
3.4572
3.7417
2.8418
2.9306
2.2145
3.5211
3.8364
3.8056
1.0385
2.4674
2.5562
2.0273
2.8214
2.7256
4.4718
3.4160
4.3264
4.5133
4.3679
3.8876
4.7472
2.6296
2.7185
2.7784
2.1084
3.5182
2.6459
2.5731
2.7628
3.5153
3.7997
2.9169
3.0057
2.2521
3.4200
3.5542
3.4451
3.0849
3.7296
2.7987
3.4649
3.1046
0.9822
3.5138
0.9913
1.5286
1.5138
1.6007
1.2920
1.5791
1.3210
1.2746
1.1211
1.3957
1.5471
1.5193
1.4418
1.3911
1.5233
1.3741
1.5717
1.2260
1.5822
1.7490
1.7327
0.8085
1.1428
1.3071
1.0196
1.3590
1.6163
1.3921
1.2678
1.3431
1.5637
1.5087
1.3271
1.6521
1.2430
1.4217
1.5423
1.1090
1.5738
1.1956
1.1536
1.2632
1.5648
1.7136
1.4205
1.6247
1.2674
1.2796
1.4042
1.3548
1.1917
1.4836
1.7519
1.4140
1.2438
0.7226
1.5603
1254
4529
4591
14774
5436
15220
5569
5665
6238
5496
4495
4572
5117
5183
5001
2232
1979
713
4242
4493
3986
6793
2814
2472
918
2718
15035
9500
5253
9603
7984
8144
8971
7801
2478
2203
2095
807
4415
2751
2818
2760
4387
4101
2161
1949
684
5665
4776
4857
5324
4692
13708
4901
5540
7506
4450
831
4724
4769
15103
5544
15296
5453
5614
6244
5217
4668
4752
5021
5173
4796
2420
2073
632
4564
4172
4208
7322
2864
2500
804
2446
14966
8862
5642
9113
7980
8212
9121
7641
2653
2299
2092
725
4589
2769
2858
2638
4615
4257
2347
2000
601
5594
5149
5304
5909
4921
13867
5094
5682
8135
4628
423
-195
-178
-329
-108
-76
116
51
-6
279
-173
-180
96
10
205
-188
-94
81
-322
321
-222
-529
-50
-28
114
272
69
638
-389
490
4
-68
-150
160
-175
-96
3
82
-174
-18
-40
122
-228
-156
-186
-51
83
71
-373
-447
-585
-229
-159
-193
-142
-629
-178
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
25
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
- 105 -
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
SnBr4
SnCl4
SrH2
TiH2
TiN
TmCl3
TmI3
V2O3
VBr2
VBr3
VCl3
VI3
VN
YCl3
YH3
YI3
BN
AlN
CaSe
MgAl2O4
FeAl2O4
CaMoO4
SrMoO4
ZrSiO4
YVO4
Ce2O2S
YPO4
LaFeO3
LaMnO3
LaCrO3
LaCoO3
LaAlO3
NdFeO3
CaSiO3
MgSiO3
Ba2SiO4
Ca2SiO4
Mg2SiO4
YAlO3
ZnFe2O4
PrMnO3
Y3Al5O12
Y3Fe5O12
Ba2In2O5
CaMgSiO4
Ca2Fe2O5
Al2SiO5
CaMgSi2O6
LiAlSi2O6
NaAlSi2O6
CaAl2Si2O8
KAlSi3O8
Ba4NaSb3O12
Ba4LiSb3O12
Sr4NaSb3O12
6.7200
6.8611
3.1994
3.4717
4.1989
4.9868
4.7257
9.3025
3.5523
5.1255
5.2313
4.9702
4.1856
5.0246
4.7379
4.7635
4.7391
4.3203
2.8255
12.8130
12.8587
12.3502
12.2960
12.2816
12.2302
8.4097
12.7050
9.0626
9.0402
9.0428
9.0860
9.1048
9.0727
9.4584
9.6413
12.2631
12.5165
12.8823
9.2305
12.7485
9.0612
37.2635
37.0523
15.0203
12.6994
15.6037
15.9722
19.0997
19.4771
19.3899
25.4306
25.6953
36.3060
36.3933
36.5956
11.0304
11.4326
2.5131
3.0071
13.1664
5.9526
5.4358
21.5924
3.1296
6.4527
6.6879
6.1073
13.0880
6.0663
5.4879
5.5396
16.3462
13.8810
3.9901
28.1530
28.3010
47.7444
47.5687
37.5988
40.0538
17.4810
43.8963
20.4276
20.3190
20.3313
20.5414
20.6327
20.4770
25.1390
25.7311
28.9789
29.7990
30.9833
21.2430
27.8076
20.4211
86.6299
85.6045
29.3884
30.3912
31.8108
43.3798
50.8701
54.1584
54.0173
68.5187
74.3427
104.6993
104.8404
105.6366
4.4860
4.3405
2.9269
2.8083
0.9567
3.4976
3.7821
2.7531
2.6858
3.5511
3.4420
3.7265
0.9594
3.4881
3.8035
3.7726
0.8698
0.9337
1.4162
3.9359
3.9190
3.3915
3.4187
3.1658
3.2348
3.0069
3.2046
2.8142
2.8184
2.8179
2.8098
2.8064
2.8113
2.8644
2.7861
4.3109
4.1768
4.0202
2.7735
3.9438
2.8126
11.0222
11.0608
5.5424
4.0985
5.3421
4.2536
5.6504
5.6883
5.7694
7.1179
7.4274
12.3129
12.2318
12.1537
1.4505
1.3995
1.5917
1.3302
0.6836
1.5119
1.6557
1.6672
1.2781
1.3948
1.3457
1.4736
0.6877
1.4836
1.6400
1.6247
0.5506
0.6484
1.0025
2.3336
2.3127
1.6939
1.7276
1.7159
1.7452
2.0676
1.6521
1.7805
1.7883
1.7874
1.7725
1.7661
1.7752
1.6396
1.5429
2.6636
2.4980
2.3045
1.7051
2.3530
1.7775
6.6872
6.7588
3.6058
2.4012
3.3174
2.3533
3.1825
2.9200
2.9701
3.9929
3.8169
6.9998
6.9496
6.8030
8852
8930
2265
2864
8033
4608
4340
14520
2534
5224
5329
5136
8233
4524
4910
4258
10911
8405
2862
19128
19216
29521
29403
24058
25846
12131
27207
13364
13487
13678
13805
13856
13854
16439
16725
19300
19831
20697
14570
18972
14084
58006
56504
20065
20364
21811
28687
33324
34815
35020
45005
47626
64996
65181
65481
8542
8785
2038
2503
8567
4775
4412
14534
2586
5182
5337
4953
8520
4864
4462
4498
10471
8999
2844
19133
19240
29240
29121
24022
25256
11839
27416
13781
13712
13720
13854
13912
13813
16531
16932
19195
19825
20691
14308
18914
13778
57449
56636
19861
20260
21664
28331
33192
35301
35166
44714
48036
64918
65146
66027
310
145
227
361
-534
-167
-72
-14
-52
42
-8
183
-287
-340
448
-240
440
-594
18
-5
-24
281
282
36
590
292
-209
-417
-225
-42
-49
-56
41
-92
-207
105
6
6
262
58
306
557
-132
204
104
147
356
132
-486
-146
291
-410
78
35
-546
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
9
9
32
9
9
9
9
9
9
9
9
25
25
25
25
25
25
25
25
25
25
25
28
28
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
- 106 -
3. REGRESSION ANALYSIS
The experimental values of 318 inorganic-compounds lattice energies (U) are taken from
Ref.9, Ref.25, Ref.28 and Ref.32, and listed in Table 2. Only multiple regression analysis
(MRA) is used to construct the prediction models in this study, because the results obtained
by using approach MRA are good enough.
Linear regression analysis is carried by SPSS. The results of the best subsets regression
of U versus 0G, 1G, 0G’, and 1G’ show: if only the index 1G is used, a good correlation for the
U of the 318 can be obtained, and the correlation coefficient r is 0.9987. When the index 0G is
included, the model can be significantly improved (r = 0.9993); whereas the results obtained
by using three-parameters, 1G, 0G, and 1G’, as well as four-parameters, 1G, 0G, 1G’, and 0G’
are better than the results of single-parameter and two-parameters (r = 0.9995 and 0.9996,
respectively). Four models are shown as follows:
U=641.688+633.3101G
(8)
r = 0.9987, r2 = 0.9974, s = 503.60, F = 118971.0, N = 318
U=-88.043+525.4261G+ 325.2660G
(9)
r = 0.9993, r2 =0.9986, s = 365.16, F = 113280.1, N = 318
U=185.578+487.0761G+ 558.9610G-829.6991G’
(10)
r = 0.9995, r2 =0.9991, s = 293.46, F = 116993.6, N = 318
U=153.238+502.9371G+ 51.1400G-1076.5051G’+177.442 0G’ (11)
r = 0.9996, r2 =0.9992, s =280.98, F = 95718.9, N = 318
If the nonlinear regression analysis is carried by SPSS, the best nonlinear model by using
four parameters is shown as follows:
U=-315.953+831.2161G0.896+ 44.5760G1.644-140.7121G’2.150+110.0960G’1.474
(12)
2
r = 0.9998, r =0.9995, s =217.6, F = 159603.7, N = 318
The results of t-test show that these variables in these models are significant. Obviously,
five models are excellent QSPR models without exception. The model (12) explains more
than 99.95% of the variance in the experimental values of U for these inorganic compounds.
The calculated results from Eq. (12) for 318 inorganic compounds show in Table 2. that the
average absolute deviation is 156.78 kJ. mol-1. The calculated U versus experimental data is
shown in Figure 1. Fig.1 shows that the model (12) is quite excellent.
- 107 -
Figure 1. A plot of calculated versus experimental U for 318 inorganic compounds.
Finally, the models above generated for 318 inorganic compounds are verified by the
cross-validation using leave-one-out method, and the correlation coefficients rcv and standard
deviations scv together with the normal r and s are shown in Table 3. This table reveals that the
results of the cross-validations for each model are very close to the normal results of the
models, which means that the models constructed in this work are stable.
Table3. Statistics of MLR and Leave-One-Out Cross-Validation for the Five Models
properties
Model (8)
Model (9)
Model (10)
Model (11)
Model (12)
r
0.9987
0.9993
0.9995
0.9996
0.9998
r2
0.9974
0.9986
0.9991
0.9992
0.9995
s
503.60
365.16
293.46
280.98
217.63
rcv
0.9986
0.9992
0.9995
0.9995
0.9997
rcv2
0.9972
0.9985
0.9990
0.9991
0.9995
scv
521.99
381.29
303.36
295.53
226.15
4. RESULTS AND DISCUSSION
The gi defined in this paper are different for ions of different elements due to their
different electronegativities. Even for the ions of the different valence states of the same
element, the gi are also different. Therefore, the gi can reflect the characteristics of different
ions.
- 108 -
In inorganic compounds, the lattice energy is positively correlated with the number of
ions of chemical formula. The topological indices 0G and 0G’ are proportional to the number
of ions of chemical formula, so the lattice energy is positively correlated with 0G and 0G’. The
more chemical single bonds there are in inorganic compound, the greater lattice energy the
compound has, so the lattice energy is positively correlated with 1G and 1G’ too. This fact is
in agreement with the results of the regression analysis. Among the 1G, 1G’, 0G , and 0G’, the
index 1G has the best correlation with the lattice energy, r=0.9987, When the index 0G is
included, the double-parameter model can be significantly improved (r = 0.9993). In the two
models, the coefficients of 0G and 1G are all positive. However in the three-parameter model,
1
G , 0G, and 1G’,(r=0.9995), as well as four-parameters model, 1G,0G,, 1G’, and 0G’(r=0.9996),
the coefficients of both 1G are negative. The results show that the infection factors contained
in index 1G’ are different from indices 1G, 0G and 0G’. Index 1G together with 0G, 1G’and 0G’
reflects the essential influencing factor of lattice energy.
The estimated lattice energies of 318 inorganic compounds containing 282 binary ionic
crystals and 36 complex ionic crystals are listed in Table 2. As shown in Table 2, our
calculated values agree well with the available experimental values. The average error of 318
crystals is 4.28 %( 5.18% for U<5000 kJ mol-1, 2.32% for U>5000 kJ mol-1). The average
error of 36 complex ionic crystals is only 0.82%. If the regression is carried only for the
experimental values of 36 complex ionic crystals, the best linear and nonlinear threeparameter models are shown as:
U=486.290+505.2211G+622.6750G-886.780G’
(13)
2
r=0.9999, r =0.9997, s=292.12, F=36722.6, N=36
U=-96.116+650.8061G0.955+138.9160G1.322-31.3950G’ 2.114
(14)
2
r=0.9999, r =0.9998, s=218.23, F=65807.1, N=36
The average errors of the estimated lattice energies of 36 complex ionic crystals are 1.08%
and 0.67% for linear model (13) and nonlinear model (14), respectively. The results show that
the current method is more effective than literature methods7-11,23,25-26,28 for complex ionic
crystals.
In addition, the current method is very simple comparing with literature methods23,25-26,28,
for the following reasons:
(a) In current method, the gi of an ion can be easily calculated from the charge number,
the electronegativity and Goldschmidt ionic radius of atom i. The electronegativity and
Goldschmidt ionic radii for any atomic can obtained from the handbooks.
- 109 -
(b) The calculations of 0G, 1G, 0G’, and 1G’ of inorganic compounds are simple too. The
0
G, and 0G’ are the summation of all the atoms of the chemical formula. The 1G, 1G’ are the
summation of all chemical single bonds. For every ion, the numbers of chemical single bonds
bonding to other ions equal to the charge numbers.
(c) The models can be used for calculating the lattice energies of binary ionic crystals
and complex crystals.
5. CONCLUSION
A novel connectivity index mG and its converse index mG’ are proposed for predicting the
lattice energies of inorganic ionic compounds. The excellent QSPR models for the lattice
energies can be constructed by using
0
G, 1G, 0G’,and 1G’. For the nonlinear model, the
correlation coefficient r, the standard error, and the average error of the nonlinear model are
0.9998, 217.6 kJ.mol-1 and 4.28%, respectively, for the 318 inorganic ionic crystals. However,
the average error of 36 complex ionic crystals is only 0.82%. The cross-validation by using
the leave-one-out method demonstrates that the models are highly reliable from the point of
view of statistics. The results show that the current method is not only more effective, but also
simpler than literature methods for estimating the lattice energies of inorganic ionic crystals,
especially for estimating the lattice energies of complex ionic crystals.
6. ACKNOWLEDGEMENT
This work is supported by the university natural science foundation of Jiangsu Province
in china (contract grant number:04KJD150195). The authors express our gratitude to the
referees for their value comments.
REFERENCES AND NOTES
(1) Trinajstic, N. Chemical Graph Theory, 2nd ed., CRC Press, Boca Raron, FL.1992.
(2) Balaban, A. T. Applications of Graph in Chemistry. J. Chem. Inf. Comput. Sci. 1985,25,334-343.
(3) Balaban, A. T. Chemical Graphs: looking Back and Glimpsing Ahead. J. Chem. Inf. Comput. Sci.
1995,35,339-350.
(4) Yang, F.; Wang Z. D.; Huang, Y. P.; Ding X. R.. Modification of Wiener Index and Its Application. J.
Chem. Inf. Comput. Sci. 2003, 43, 753-756.
(5) Yang, F.; Wang Z. D.; Huang, Y. P.. Modification of Wiener Index .2 . J Chem. Inf. Comput. Sci.
2003, 43, 1337-1341.
- 110 -
(6). Mu L. L.; Feng C. J.. Topological Research on Standard Entropies of Alkali and Alkaline Earth
Metal Compounds. Chin. J. Chem. Phys. 2003, 16(1), 19-24
(7). Feng C. J.. The connection index of average energy level for valence electron and its applications.
Chin. J. Chem. Phys. 2000, 13(1), 66-70
(8). Xu S. Y.; Tang J. H.. The topological index mF electronegativity and its calculation of physical and
chemical properties of inorganic compounds. Chin. J. Chem. Phys. 2003, 16(3), 203-208
(9). Feng C. J.. studies on the relationships between the quantitative structure and physico-chemical
properties for the inorganic compounds. Chin. J. Anal. Sci. 2002, 18(1), 27-32
(10). Feng C. J.. Estimating and Predicting lattice energies of inorganic compounds with connectivity
indices. Chin. J. Mol. Sci. 1999, 15(4), 223-228
(11). Feng C. J.; Chen Y.. A novel structural parameter for QSPR study of halides. Chin. J. Mol. Sci.
2000, 16(3), 177-183
(12). Altshuller A. P.. Lattice Energies and Related Thermodynamic Properties of the Alkali Metal
Borohydrides and of the Borohydride Ion. J. Am. Chem. Soc. 1955, 77(21), 5455-5457.
(13). Stokes R. H.. Crystal Lattice Energies and the Electrostatic Self Energies of Gaseous Ions. J. Am.
Chem. Soc. 1964, 86(6), 982-986.
(14). Gardner P. J. ; Finch A... Lattice Energies of Ionic Crystals. II. Some Group I-A and II-A Salts J.
Phys. Chem. 1965, 69(2), 384-389.
(15). D'Orazio L. A.; Wood R. H.. Thermodynamics of the Higher Oxides. I. The Heats of Formation
and Lattice Energies of the Superoxides of Potassium, Rubidium, and Cesium. J. Phys. Chem. 1965, 69(8),
2550-2557.
(16). D'Orazio L. A.; Wood R. H.. Thermodynamics of the Higher Oxides. II. Lattice Energies of the
Alkali and Alkaline Earth Peroxides and the Double Electron Affinity of the Oxygen Molecule. J. Phys.
Chem. 1965, 69(8), 2558-2561.
(17). Brackett T. E., Brackett E. B.. The Lattice Energies of the Alkaline Earth Halides. J. Phys. Chem.
1965, 69(10), 3611-3614.
(18). Jenkins H. D. B. Group 3 tetrahalide salts, lattice energies, and thermodynamic parameters.
Tetrachloroaluminate(1-) and tetrachlorogallate(1-). Inorg. Chem. 1976, 15(1), 241-243.
(19). Stevenson G. R.; Wiedrich C. R.. Anion-radical crystal lattice energies. J. Am. Chem. Soc. 1979,
101(17), 5092-5094.
(20). Voogd J.; Derissen J. L., Duijneveldt F. B. V. Calculation of proton-transfer energies and
electrostatic lattice energies of various amino acids and peptides using CNDO/2 and ab initio SCF methods.
J. Am. Chem. Soc. 1981, 103(26), 7701-7706.
(21). Stevenson G. R., Wiedrich C. R., Clark G. Crystal lattice energies of solid anion radical salts. J.
Phys. Chem. 1981, 85(4), 374-378.
(22) Glasser, L. Lattice Energies of Crystals with Multiple Ions: A Generalized Kapustinskii Equation
Inorg. Chem. 1995, 34, 4935-4936.
- 111 -
(23) Jenkins, H. D. B.; Roobottom. H. K.; Passmore, J.; Glasser, L. Relationships among Ionic Lattice
Energies, Molecular (Formula Unit) Volumes, and Thermochemical Radii Inorg.Chem. 1999, 38, 36093620.
(24). Abramov, Y. A.; Volkov, A.; Wu, G.; Coppens, P. Use of X-ray Charge Densities in the
Calculation of Intermolecular Interactions and Lattice Energies: Application to Glycylglycine, DLHistidine, and DL-Proline and Comparison with Theory. J. Phys. Chem. 2000, 104(9), 2183-2188.
(25). Glasser, L.; Jenkins, H. D. B. Lattice Energies and Unit Cell Volumes of Complex Ionic Solids. J.
Am. Chem. Soc. 2000, 122, 632-638.
(26). Jenkins, H. D. B.; Tudela, D.; Glasser, L. Lattice Potential Energy Estimation for Complex Ionic
Salts from Density Measurements. Inorg. Chem. 2002, 41,2364-2367.
(27). C. Mellot-Draznieks; S. Girard; G. Ferey. Novel Inorganic Frameworks Constructed from DoubleFour-Ring (D4R) Units: Computational Design, Structures, and Lattice Energies of Silicate,
Aluminophosphate, and Gallophosphate Candidates. J. Am. Chem. Soc. 2002, 124(51),15326-15335.
(28). Liu D. T.; Zhang S. Y.; Wu Z. Lattice Energy Estimation for Inorganic Ionic Crystals. Inorg.
Chem. 2003, 42, 2465-2469
(29). S. Beaucamp; D. Mathieu; V. Agafonov. Parametrization of Semiempirical Models against ab
Initio Crystal Data: Evaluation of Lattice Energies of Nitrate Salts. J. Phys. Chem. B. 2005,
109(34),16469-16473.
(30). David R. L.; Grace B.; Lev I. B.; Robert N. G.; Henry V. K.; Kozo K.; Gerd R.; Dana L. R.; Daniel
Z. CRC Handbook of Chemistry and Physics. 85th ed; CRC Press; Boca Raton, FL,2002;9-74
(31). Pan D. K.; Zhao D. C.; Zheng Z. X. Material Structure. 2nd ed., higher education Press, 1989; A8.(Chinese)
(32). David R. L.; Grace B.; Lev I. B.; Robert N. G.; Henry V. K.; Kozo K.; Gerd R.; Dana L. R.; Daniel
Z. CRC Handbook of Chemistry and Physics. 85th ed; CRC Press; Boca Raton, FL,2002;12-23