Directory UMM :wiley:Public:journals:jcc:suppmat:18:
Supplementary material:
Title:
A critical analysis of electronic density functionals for structural, energetic, dynamic and magnetic properties of hydrogen
uoride clusters
Authors with Aliation:
Ch. Maerker and P. v. R. Schleyer
Institut fur Organische Chemie der
Friedrich-Alexander Universitat Erlangen-Nurnberg, Henkestrasse 42
D-91054 Erlangen, Germany
K. R. Liedl
Institut fur Allgemeine, Anorganische und Theoretische Chemie
Leopold-Franzens-Universitat Innsbruck, Innrain 52a
A-6020 Innsbruck, Austria
T.-K. Ha, M. Quack, and M. A. Suhm
Laboratorium fur Physikalische Chemie der ETH Zurich (Zentrum),
CH-8092 Zurich, Switzerland
Description:
hfndft.ps
This contains the complete table material (tables 1{16), including table 8 which is omitted in the
printed paper and tables 1{7 and 9{13 which are given in the printed paper in an abbreviated
form.
method
dft
realization
rHF /pm
D-LDA
94.0
6-311++G**-LDA [86]
93.0
6-311++G**-B [86]
92.8
D-BP
94.0
6-311++G**-BP [86]
93.1
D-BLYP
94.3
6-311+G**-BLYP
93.3
6-311++G**-BLYP [86]
93.3
aug-cc-pV(T/Q)Z-BLYP
93.4
D95**-BHHLYP
91.6
6-311+G**-BHHLYP
90.9
6-311++G**-BHHLYP [86]
91.0
6-311+G**-B3LYP
92.2
6-311++G**-B3LYP [86]
92.2
6-311++G(3df,3pd)-B3LYP
92.2
aug-cc-pVDZ-B3LYP
92.6
aug-cc-pVTZ-B3LYP
92.4
aug-cc-pV(T/Q)Z-B3LYP
92.3
aug-cc-pVQZ-B3LYP
92.2
aug-cc-pV(T/Q)Z-B3'LYP
92.1
aug-cc-pV(T/Q)Z-B3"LYP
92.0
cqc
6-311++G(3df,3pd)(SCF)
89.7
aug-cc-pV5Z(SCF) [43]
89.7
aug-cc-pV5Z+MP2 [43]
91.8
aug-cc-pV5Z+CCSD(T) [43]
91.7
aug-cc-pV5Z+aug(F)+CCSD(T) [158]
91.8
TZ2P(f,d)+CCSD(T) [44]
91.8
DZP+MP2 [17]
91.9
6-311++G(3df,3pd)+MP2
91.7
aug-cc-pVDZ+MP2 [43]
92.5
aug-cc-pVTZ+MP2 [43]
92.2
aug-cc-pV(T/Q)Z+MP2
92.0
aug-cc-pVQZ+MP2 [43]
91.9
[8s6p2d/6s3p]+MP2 [17]
91.8
[3s2p1d/3s1p]+ACPF [40]
91.9
[8s6p2d/4s1p]+CPF [37]
91.9
Experiment [151,200,201]
91.7
a this work
/D !/cm,1
2.11
2.02
1.94
2.04
1.96
2.03
1.96
1.96
1.78
1.97
2.00
2.01
1.98
1.98
1.83
1.80
1.81
1.81
1.81
1.82
1.82
1.91
1.88
1.81
1.80
1.82
1.99
1.82
1.82
1.81
1.81
1.81
1.82
1.77
1.80
3942
4012
3989
3919
3978
3872
3941
3942
3914
4303
4291
4285
4099
4101
4093
4064
4076
4070
4077
4105
4110
4481
4473
4136
4142
4143
4157
4221
4173
4085a
4120a
4121
4135a
4143
4182
4135
4138
S /(km mol,1 )
99
140
102
80
110
69
106
106
91
124
161
161
130
130
106
111
111
111
111
115
116
159
102.5
110
117
116a
121a
119
122a
120
88
90
102
Table 1: Monomer equilibrium distance rHF , permanent electric dipole moment , harmonic wavenumber ! and integrated molar IR band strength S (S is given within the double-harmonic approximation
for convenient comparison to literature data, it can be converted to the more fundamental squared
transition dipole moment h01i2 and the integrated absorption cross section G via 41.624 h01i2 =D2=
G/pm2=16.6054 (S /(kmmol,1))/(! /cm,1 ) [156]) for di erent approaches (cqc=conventional quantum
chemistry), as discussed in the text. Experimental data are corrected for anharmonic zero point vibration e ects and therefore directly comparable to the predictions (this is not always the case [202]).
1
potential surface
D-LDA
6-311++G**-LDA [86]
LDA [89]
6-311++G**-B [86]
aug-cc-pV(T/Q)Z-B
D-BP
6-311++G**-BP [86]
BP [89]
D-BLYP
6-31++G**-BLYP [90]
6-311+G**-BLYP
6-311++G**-BLYP [86]
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311++G**-BHHLYP [86]
6-31++G**-B3P [90]
6-31+G**-B3LYP [87]
6-31++G**-B3LYP [90]
6-311+G**-B3LYP
6-311++G**-B3LYP [86]
6-311++G(3df,3pd)-B3LYP
aug-cc-pVDZ-B3LYP
aug-cc-pVTZ-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
aug-cc-pVQZ-B3LYP
aug-cc-pV(T/Q)Z-B3"LYP
6-311++G(3df,3pd)+MP2
aug-cc-pVDZ+MP2 [43]
aug-cc-pVTZ+MP2 [43]
aug-cc-pV(T/Q)Z+MP2
aug-cc-pVQZ+MP2 [43]
best ab initio [43,44,53]
Exp. [1]
Exp. [3]
best empirical [12,53,160]
y typographical error in [86]?
rHF /pm
95.3
94.5
95.3
y93.3
93.4
94.9
94.0
94.7
95.1
94.7
94.0
94.1
94.2
92.2
91.5
91.6
93.3
93.5
92.9
92.9
92.9
93.3
93.2
93.1
93.0
92.8
92.3
93.1
92.8
92.6
92.5
92.2-3
0 /pm
rHF
94.5
93.6
RFF /pm
=
259
11
256
10
255
8
290
8
287
6
271
9
274
9
269
6
272
12
276
8
278
8
277
8
275
6
267
10
272
8
272
8
270
8
273
8
273
8
275
8
275
9
273
5
273
6
273
6
273
6
273
6
272
6
274
5
275
7
275
6
275
6
274
6
273-4
7
272(3) 10(6)
7(3)
273-4
7-8
93.1
93.1
94.4
93.4
94.7
94.2
93.6
93.6
93.7
91.9
91.3
91.3
92.8
93.1
92.5
92.5
92.5
92.9
92.7
92.6
92.5
92.3
92.0
92.8
92.5
92.3
92.2
92.0-1
92.2-3 91.9-92.0
0 =
76
73
62
68
73
68
77
68
65
66
71
71
61
60
68
67
67
64
65
67
70
68
68
69
68
65
70
69
68
68
68-70
63(6)
60(2)
65-69
Table 2: Planar (HF)2 minimum geometry for di erent approaches. RF F is the distance between the
F atoms. Monomer bond lengths are denoted rHF , bond angles of the monomer with the FF axis .
Primed quantities refer to the non-bonded HF in the dimer.
2
potential surface
D-LDA
6-311++G**-LDA [86]
LDA [89]
6-311++G**-B [86]
aug-cc-pV(T/Q)Z-B
D-BP
6-311++G**-BP [86]
BP [89]
D-BLYP
6-31++G**-BLYP [90]
6-311+G**-BLYP
6-311++G**-BLYP [86]
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311++G**-BHHLYP [86]
6-31++G**-B3P [90]
6-31+G**-B3LYP [87]
6-31++G**-B3LYP [90]
6-311+G**-B3LYP
6-311++G**-B3LYP [86]
6-311++G(3df,3pd)-B3LYP
aug-cc-pVDZ-B3LYP
aug-cc-pVTZ-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
aug-cc-pVQZ-B3LYP
aug-cc-pV(T/Q)Z-B3"LYP
6-311++G(3df,3pd)+MP2
aug-cc-pVDZ+MP2 [43]
aug-cc-pVTZ+MP2 [43]
aug-cc-pV(T/Q)Z+MP2
aug-cc-pVQZ+MP2 [43]
best ab initio [43]/ [44]
best empirical [12,53,160]
experimental D0
De/(kJ mol,1) D0h /(kJ mol,1)
41.8
37.4
38.9
14.0
11.5
23.8
18.9
20.5
25.7
20.5 (18.4)
19.7
19.8
17.4
27.2
22.7
22.8
21.3 (19.7)
21.3
21.8 (20.0)
21.0
21.1
20.2
19.3
18.8
18.8
18.9
19.1
20.7 (17.0)
19.6 (16.8)
19.7 (17.7)
18.8 (17.8)
19.4 (18.3)
19.2/19.8
18.9(2),19.1(2)
33.5
16.0
17.3
12.5
10.0
19.1
15.2
13.7
12.3
11.9
11.4
11.3
11.5
12.4
12.2
12.1
11.4
11.9
{/12.3
12.0(2),11.7(2)
12.70(1) [13]
Table 3: Electronic (De ) and harmonically corrected (D0h ) dimer dissociation energies with respect
to separated monomers compared to MP2 benchmarks and best estimates from ab initio theory as
well as anharmonic empirically adjusted potential energy surfaces which reproduce the experimental
(anharmonic) dissociation energy D0 of ref. [13]. Values in parentheses include BSSE correction.
3
potential surface
D-LDA
!1
3892
(156)
6-311++G**-LDA [86]
3947
(181)
6-311++G**-B [86]
3959
(117)
D-BP
3893
(122)
6-311++G**-BP [86]
3935
(136)
D-BLYP
3841
(111)
6-311+G**-BLYP
3903
(127)
6-311++G**-BLYP [86]
3902
(129)
aug-cc-pV(T/Q)Z-BLYP
3872
(114)
D95**-BHHLYP
4262
(161)
6-311++G**-BHHLYP [86]
4245
(166)
6-311+G**-BHHLYP
4245
(166)
6-311++G**-B3LYP [86]
4058
(150)
6-311+G**-B3LYP
4059
(148)
6-311++G(3df,3pd)-B3LYP
4050
(127)
aug-cc-pVDZ-B3LYP
4019
(136)
aug-cc-pVTZ-B3LYP
4032
(133)
aug-cc-pV(T/Q)Z-B3LYP
4028
(132)
aug-cc-pV(T/Q)Z-B3"LYP
4064
(137)
6-311++G(3df,3pd)+MP2
4132
(126)
aug-cc-pVDZ+MP2
4038
(136)
aug-cc-pVTZ+MP2
4081
(137)
aug-cc-pV(T/Q)Z+MP2
4079
(134)
aug-cc-pVQZ+MP2
4094
(139)
best ab initio [44]
4119
(119)
empirical SQSBDE/SNB [12,160]
4100
with adjustment based on [25,162]
4090
new empirical [53]
4100
anharmonic fundamental (see [53,160]) 3931
!2
!3
3702
744
(583) (175)
3718
730
(728) (178)
3879
507
(359) (222)
3746
633
(446) (189)
3795
602
(486) (195)
3730
657
(392) (169)
3790
569
(459) (209)
3787
577
(455) (203)
3735
588
(492) (134)
4164
622
(492) (194)
4165
571
(524) (251)
4165
570
(525) (250)
3958
581
(481) (219)
3960
574
(485) (224)
3934
599
(499) (163)
3884
593
(511) (149)
3907
582
(505) (152)
3906
587
(511) (152)
3945
591
(516) (155)
4046
609
(476) (168)
3938
577
(466) (154)
3986
580
(475) (152)
3989
568
(472) (156)
3997
578
(481) (153)
4050
567
(427) (160)
4050
4030
4030
550
3868 480
!5
!6
179
(151)
203
(94)
120
(7)
141
(134)
162
(14)
177
(74)
155
(26)
157
(20)
165
(4)
182
(34)
166
(37)
165
(41)
160
(31)
159
(34)
169
(7)
167
(12)
164
(5)
168
(9)
170
(9)
166
(3)
159
(12)
160
(6)
159
(7)
163
(11)
157
(25)
150
523
(272)
550
(292)
415
(248)
483
(255)
471
(245)
466
(268)
446
(241)
452
(245)
472
(154)
479
(256)
464
(274)
462
(274)
461
(262)
457
(259)
505
(174)
482
(166)
481
(169)
480
(168)
483
(171)
547
(174)
473
(175)
477
(173)
468
(170)
473
(173)
458
(188)
410
210
155
465
160 125 420
Table 4: Caption see next page!
4
!4
302
(117)
292
(137)
194
(149)
244
(81)
239
(172)
277
(172)
215
(142)
222
(156)
233
(151)
254
(180)
222
(135)
220
(132)
221
(152)
217
(142)
236
(146)
225
(146)
233
(152)
231
(148)
233
(150)
238
(144)
217
(148)
222
(155)
218
(147)
220
(150)
210
(141)
210
or 380
,!1 ,!2
50
240
65
294
29
109
26
173
43
183
31
142
38
151
40
155
42
179
40
138
39
120
46
126
43
143
41
139
43
159
45
180
45
170
42
164
46
165
41
127
47
147
39
134
42
131
41
138
38
107
38
48
38
31
88
108
108
93
Table 4: Predicted harmonic wavenumbers ! /cm,1 and vibrational shifts ,!1 2 relative to the
monomer for (HF)2 compared to rounded best empirical (conservatively 20 cm,1 ) and ab initio estimates as well as MP2 benchmarks. In parentheses, integrated molar IR band strengths S /(kmmol,1)
are given in the double-harmonic approximation. Experimentally, the band strength enhancement of
1 (2 ) over the monomer is approximately 20% (300%) [164]. The fundamental 6 has been observed
only in the K = 0 ! 1 and K = 1 ! 2 transitions [203]. From this, two di erent extrapolations to
the rotationless fundamentals have been made, depending upon treatment of Coriolis e ects in various
potentials (380cm,1 and 420cm,1 [12,53]), with corresponding di erences for !6 (410 or 465cm,1 ).
i
;
i
5
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
D95**-B3LYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
6-311++G(3df,3pd)+MP2 [45]
aug-cc-pVTZ+MP2
best estimate
a planar saddle
1
94.0
94.0
94.3
93.3
93.4
91.6
90.9
92.2
92.2
92.3
91.7
92.2
91.7
3
98.7
96.7
96.7
95.4
4
104.
98.0
97.8
96.7
93.5 94.9
92.4
94.0 95.1
94.2 95.7
93.3 94.6
93.3 94.4
5
6
108. 109.
98.9 99.1
98.3 98.5
97.3
98.3
95.4 95.5a
93.6
98.3
95.5 95.7
96.3
96.4
95.0
95.7
94.8 94.9
Table 5: Size dependence of the HF bond length rHF in the minimum geometry of (HF)n clusters.
For n=3-5, the minimum geometry has Cnh symmetry, for n=6, a slightly puckered S6 -symmetric
minimum is typically found. The limit for an in nite (chain or ring) cluster can be estimated around
95-97 pm [169, 170] and in a supersonic jet expansion dominated by pentamers and hexamers, rHF is
increased by 3-5 pm [58] relative to the monomer. The best estimates are mostly derived from ab initio
calculations [17,40] and shifts relative to the monomer are expected to have an uncertainty of less than
20%.
potential surface
n=
2
D-LDA
259
D-BP
271
D-BLYP
272
6-311+G**-BLYP
278
aug-cc-pV(T/Q)Z-BLYP
275
D95**-BHHLYP
267
6-311+G**-BHHLYP
272
6-311+G**-B3LYP
275
6-311++G(3df,3pd)-B3LYP
273
aug-cc-pV(T/Q)Z-B3LYP
273
6-311++G(3df,3pd)+MP2 [45]
274
aug-cc-pVTZ+MP2
275
best estimate
273.5(1.0)
a planar saddle
3
243
258
260
263
4
234
253
255
255
253 246
261
262 254
259 250
261 252
259 251
5
6
231 230
248 247
252 251
252
248
243 242a
252
251 250
247
247
248
248
248 247
Table 6: Size dependence of the FF distance RF F /pm in the minimum geometry of (HF)n clusters. See
also table 5. Experimental data from pentamer/hexamer supersonic jets (253pm [58]) and the extended
solid (248-251pm [169,204]) have to be corrected for vibrational averaging e ects. Best estimates derive
from spectroscopic data for n=2,3 and ab initio calculations for the larger clusters [17, 40]. Their
estimated absolute error is about (n , 1) pm, but the size trend is more accurate.
6
potential surface
n=
2
D-LDA
11
D-BP
9
D-BLYP
12
6-311+G**-BLYP
8
aug-cc-pV(T/Q)Z-BLYP
6
D95**-BHHLYP
10
6-311+G**-BHHLYP
8
6-311+G**-B3LYP
8
6-311++G(3df,3pd)-B3LYP
5
aug-cc-pV(T/Q)Z-B3LYP
6
6-311++G(3df,3pd)+MP2 [45] 5
aug-cc-pVTZ+MP2
6
best estimate
7-8
a planar saddle
3 4 5 6
21 9 3 2
22 11 5 2
22 11 5 3
23 11 5
3
23 11 5 2a
25
7
24 12 6 2
21 9 4
2
22 9 4
4
24 12 6 3
Table 7: Size dependence of the hydrogen bond angle HF F / in the minimum geometry of (HF)n
clusters. See also table 5. Best estimates derive from ab initio calculations [17,40].
potential surface
D-LDA
n=
aug-cc-pV(T/Q)Z-BLYP
2
3
41.8 147.7
[67]
23.8 83.7
[67]
25.7 87.6
[65]
19.7 63.6
[62]
17.4
4
256.3
[117]
147.7
[118]
152.5
[113]
119.1
[115]
D95**-BHHLYP
27.2
6-311+G**-BHHLYP
22.7
161.5
[113]
6-311+G**-B3LYP
21.0
D-BP
D-BLYP
6-311+G**-BLYP
6-311++G(3df,3pd)-B3LYP 20.2
aug-cc-pV(T/Q)Z-B3LYP
18.8
6-311++G(3df,3pd)+MP2
aug-cc-pVTZ+MP2
best estimate
a planar saddle
20.7
19.7
19.1
91.7
[64]
69.0
[58]
66.0
[60]
66.3
[63]
121.8
[111]
125.6
[119]
64.7 121.7
63
117
5
6
345.2 422.5
[158] [193]
198.9 243.5
[160] [195]
203.9 249.2
[152] [185]
164.2
[159]
163.8
[180]
217.9 265.5a
[153] [186]
171.7
[144]
168.1 207.6
[153] [189]
173.5
[164]
167.5
[170]
168.1
165.7
161
199
Table 8: Size dependence of the electronic dissociation energy of (HF)n clusters with respect to
fragmentation into separate monomers De /(kJ/mol). Values in brackets are obtained by linear scaling
to the best dimer value of 19.1(2)kJ/mol. Best estimates derive from ab initio calculations [17, 40, 53]
and experiment [13, 15, 22, 171], connected by quantum Monte Carlo calculations [12, 171], as well as
thermodynamic modelling [23]. The error bars are approximately n kJ/mol for n > 2 and 0.2 kJ/mol
for n=2.
7
potential surface
D-LDA
n=
2
3
33.5 125.2
[44]
D-BP
16.0 61.5
[45]
D-BLYP
17.3 65.1
[44]
6-311+G**-BLYP
12.5 43.4
[41]
D95**-BHHLYP
19.1 69.0
[42]
6-311+G**-BHHLYP
15.2 48.4
[37]
6-311+G**-B3LYP
13.7 45.6
[39]
6-311++G(3df,3pd)-B3LYP 12.3 45.1
[43]
6-311++G(3df,3pd)+MP2 12.4 43.7
best D0h estimate
11.7
41
best D0 estimate
12.7 43.0
aug-cc-pVTZ+MP2 De
19.7
best De estimate
19.1
63
a planar saddle
4
229.1
[80]
117.2
[86]
121.4
[82]
89.6
[84]
128.3
[79]
91.4
[78]
93.5
[89]
88.3
83
84.5
117
5
6
316.2 388.8
[110] [136]
162.5 199.8
[119] [146]
167.1 204.8
[113] [139]
128.5
[120]
176.6 218.2a
[108] [134]
132.9
[102]
130.7
[112]
133.0
[127]
125.2
116
145
117
146
165.7
161
199
Table 9: Size dependence of the harmonically corrected dissociation energy of (HF)n clusters with
respect to fragmentation into separate monomers Dh0 /(kJ/mol). Values in brackets are obtained by
linear scaling to the dimer estimate. Best estimates derive from ab initio calculations [17, 40] and
experiment [15, 22, 171], connected by quantum Monte Carlo calculations [12, 171]. The error bars are
approximately n kJ/mol for n > 2 and 0.2 kJ/mol for n=2. For more dimer results, see also table 3.
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
DZP+MP2 [17]
6-311++G(3df,3pd)+MP2
anh. experiment
a planar saddle
3
708/1005
411/570
360/506
346/472
338/480
250/342
297/406
344/475
247/359
280/396
249/|
4
1382/2010
668/922
581/806
602/812
640/884
5
1752/2315
876/1180
729/976
736/961
802/1065
553/710
660/854
816/1068
599/796
686/895
661/|
529/710
660/896
472/659
554/755
516/|
6
7
1979(1373)/2435
979(715)/1263
810(590)/1032
873/1111a
660(474)/834
695/843
716/|
746/|
Table 10: Harmonic IR-active/totally symmetric HF stretching wavenumber shift ,! /cm,1 as a
function of cluster size. For the (HF)6 S6 structure, the puckering-induced weak IR satellite band is
given in parentheses.
8
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
6-311++G(3df,3pd)+MP2
best harmonic
anh. experiment n=5{6
a planar saddle
3
4
5
6
878/892 1250/1034 1336/1025 930{1353
714/750
945/812 1053/816 744{1160
705/729
914/781
991/768 676{1004
619/714
866/787
950/775
678/756
952/869 1053/877 1068/830a
558/699
893/772
589/707
841/787
934/780
633/733
942/855 1040/865
581/719
903/859 1007/873
600/700
850/800
950/800 700{1000
600{900 [23]
Table 11: Harmonic IR-active in plane/out-of-plane HF libration wavenumbers !lib =cm,1 as a function
of cluster size. For the non-planar (HF)6 S6 structure, the range of IR-active librations is given.
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
6-311++G(3df,3pd)+MP2 [45]
best harmonic
anh. experiment n=4!6
a planar saddle
3
314/317
246/273
242/275
200/220
235/259
195/229
198/223
205/228
190/215
200/220
4
491/298
299/222
292/227
277/209
324/247
5
6
529/264
516/241
295/180
275/167
276/179
256/170
272/179
323/217 289/185a
256/181
271/210 267/180
299/224 300/197
278/210 279/186
280/210 270/190
250/175
260!230 [23]/ 185!150 [22]
Table 12: Harmonic IR-active/totally symmetric FF stretching wavenumber !F F =cm,1 as a function
of cluster size n. For the (HF)6 S6 structure, only the dominant component is given. Anharmonic
experimental values for n = 4 ! 6 are approximate and below the best harmonic estimate, which is
derived from ab initio and analytical potential energy surfaces [17].
9
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
2
117.9
206
6-311++G(3df,3pd)+MP2 [45]
118.2
206
[8s6p2d/6s3p]+MP2
ACPF [40]
117.8
205
extended basis [40]
QCISD(T)//MP2 [51]
D-LDA
120.5
207
D-BP
121.5
209
D-BLYP
122.1
210
D95**-BHHLYP
117.2
204
6-311+G**-BHHLYP
117.4
204
6-311+G**-B3LYP
119.3
207
6-311++G(3df,3pd)-B3LYP
119.1
207
best theoretical estimate
118
206
F HF=
6
EB
EB
121 174.6 164.3
122 167.4 158.6
174 164
121 185
173
186.4 176.3
118 104.6 95.1
119 140.0 130.0
119 145.5 134.7
170.3 159.5
121 200.5 188.1
120 172.2 160.7
121 157.8 148.4
122 170 160
!i
2306
2262
1868
1999
2030
2387
2468
2241
2203
Table 13: a)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
3
114.9
223
6-311++G(3df,3pd)+MP2 [45]
115.1
224
CCSD(T) [50]
ACPF [40]
114.7
223
QCISD(T)//MP2 [51]
D-LDA
117.2
226
D-BP
118.3
229
D-BLYP
118.8
230
D95**-BHHLYP
114.4
222
6-311+G**-BHHLYP
114.4
222
6-311+G**-B3LYP
116.1
225
MP2//6-311+G**-B3LYP
MP4//6-311+G**-B3LYP
QCISD(T)//6-311+G**-B3LYP
CCSD(T)//6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
116.1
226
best theoretical estimate
115
224
F HF=
6
152
154
153
150
150
150
152
152
153
153
EBe
EBh
!i0
84.2
11.2
43.3
51.5
61.7
90.9
68.5
1155
1448
1522
1799
1917
1705
87.7
78.4
75
86.6
95.4
21.4
55.0
64.0
75.8
104.8
81.1
95.6
98.3
101.6
102.0
69.6
80
75.0 1773
66.8 1733
61
EBe
EBh
57.2 1653
65
Table 13: b)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
4
113.6
226
6-311++G(3df,3pd)+MP2 [45]
113.8
226
ACPF [40]
113.5
225
D-LDA
115.5
229
D-BP
116.7
232
D-BLYP
117.1
233
D95**-BHHLYP
113.1
225
6-311+G**-B3LYP
114.7
228
MP2//6-311+G**-B3LYP
MP4//6-311+G**-B3LYP
QCISD(T)//6-311+G**-B3LYP
CCSD(T)//6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
114.7
228
best theoretical estimate
113.5
226
Table 13: c)
10
6
F HF=
166
168
167
165
166
166
61.1 42.5
53.3 33.9
61.9
2.9 ,6.1
29.3 14.3
39.0 22.8
48.4 28.4
167 54.9 36.5
69.3
72.4
75.4
75.8
168 43.6 25.0
167 55
35
!i0
1526
1498
692
1183
1284
1515
1454
1403
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
5
113.0
226
6-311++G(3df,3pd)+MP2 [45]
113.2
226
aug-cc-pVTZ+MP2
113.5
227
D-LDA
113.0
226
D-BP
115.9
232
D-BLYP
116.5
233
6-311+G**-BLYP
115.5
231
aug-cc-pV(T/Q)Z-BLYP
115.8
232
D95**-BHHLYP
112.5
225
6-311+G**-BHHLYP
112.5
225
D95**-B3LYP
113.9
228
6-311+G**-B3LYP
114.0
228
6-311++G(3df,3pd)-B3LYP
114.1
228
aug-cc-pV(T/Q)Z-B3LYP
114.3
228
aug-cc-pV(T/Q)Z-B3'LYP
113.9
228
aug-cc-pV(T/Q)Z-B3"LYP
113.9
228
best theoretical estimate
113
226
6
F HF=
174
177
177
175
176
175
175
177
174
175
175
176
176
176
176
175
EBe
EBh
58.1 34.7
52.6 27.1
45.7
0.4 ,5.5
23.8 7.2
34.1 15.7
40.0 17.9
30.6
44.3 19.0
74.9 48.0
22.6 2.6
51.6 27.6
40.8 17.0
41.3
44.5
44.0
50
25
!i0
1436
1431
431
1086
1209
1418
1583
1132
1368
1318
Table 13: d)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
6
112.7
225
ACPF [40]
112.7
225
D-LDA
114.3
229
D-BP
115.6
231
D-BLYP
116.1
232
D95**-BHHLYP
112.2
224
6-311+G**-B3LYP
113.7
227
best theoretical estimate
112.5
225
6
F HF=
179
179
179
179
177
EBe
EBh
!i0
65.5 37.0 1431
69.5
3.1 ,3.5 327
28.3 7.2 1081
40.3 16.7 1201
48.9 19.5 1398
179 57.1
179 60
30
Table 13: e)
Table 13: Structure and energetics of the concerted hydrogen exchange transition state in cyclic planar
HF clusters ( a) dimer, b) trimer, c) tetramer, d) pentamer, e)hexamer (here, the planar structure
often corresponds to a 4th order saddle, which is however very close to the puckered 1st order saddle)).
Electronic (EBe ) and harmonically corrected (EBh ) threshold energies are given in kJ/mol. The imaginary
wavenumber along the reaction coordinate !i0 =!i /cm,1 is a measure of the reaction pro le curvature.
Single point calculations at higher level with the same basis set (where given) are indicated by //. Best
energy estimates remain quite uncertain ((5-10)kJ/mol) in the absence of experimental constraints
and large basis set MP2, QCISD(T), and CCSD(T) reference calculations.
11
compound
(HF)1 (exp) C1v
HF
C1v
(HF)2
Cs , Hb
Cs
C2h
D2h
C3h
D3h
C4h
D4h
C5h
D5h
S6
C6h
D6h
(HF)3
(HF)4
(HF)5
(HF)6
a
b
c
d
F
H
IGLO/BII
386.6(0.0) 28.0(0.0)
397.8(-11.1)
388.5(-1.8)
388.5(-1.9)
268.2(118.5)
387.6(-1.0)
346.3(40.4)
387.9(-1.3)
354.4(32.3)
389.0(-2.3)
359.8(26.9)
390.0(-3.4)
390.2(-3.6)
362.7(23.9)
25.7(2.3)
26.7(1.3)
17.0(1.0)
11.1(16.9)
24.1(4.0)
11.2(16.8)
21.6(6.4)
10.8(17.2)
20.5(7.5)
10.6(17.4)
20.1(7.9)
20.1(7.9)
10.6(17.5)
F
H
419.70.3a 29.20.5a
IGLO-DFPT/BIII
405.5(0.0) 29.7(0.0)
408.0b
29.2b
c
418.1
29.1c
d
418.6
29.2d
406.1(-0.6) 27.2(2.5)
395.4(10.1) 28.3(1.3)
392.4(13.1) 28.6(1.1)
248.2(157.4) 14.2(15.4)
377.7(27.9) 25.2(4.5)
316.4(89.2) 13.6(16.1)
374.0(31.6) 22.8(6.9)
324.3(81.3) 13.2(16.5)
374.4(31.2) 21.8(7.9)
331.0(74.6) 13.1(16.6)
376.2(29.3) 21.5(8.2)
376.4(29.1) 21.5(8.2)
335.8(69.8) 13.1(16.6)
ref. [182]
employing the Perdew-Wang exchange-correlation functional (PW91)
GIAO-CCSD values from [138]
GIAO-CCSD(T) values from [138]
Table 14: Computed 19F and 1 H NMR isotropic equilibrium shielding constants e (ppm, all IGLODFPT/BIII calculations employed the Perdew exchange-correlation functional, if not noted otherwise).
Relative chemical shifts (ppm) of the HF oligomers with respect to monomeric HF given in parentheses.
12
species
11
22
33 anis a
HF
C1v -10.58 -10.58 -10.21
0.37
0.61c
0.52d
0.52e
0.54f
dia
-8.39
(HF)2
Cs -20.81 -20.76 -21.11
-0.33 -16.85
C2h -20.59 -20.99 -21.05
-0.26 -16.87
2.93 -17.86
D2h -21.07 -20.01 -17.61
C1v -21.18 -21.18 -20.44
0.74 -16.85
(HF)3 C3h -31.07 -31.08 -31.24
-0.17 -25.50
D3h -30.52 -30.52 -29.64
0.88 -26.24
C1v -31.73 -31.73 -30.67
1.06 -25.34
(HF)4 C4h -41.33 -41.33 -41.36
-0.03 -34.09
D4h -40.70 -40.70 -39.35
1.35 -34.74
C1v -42.26 -42.26 -40.90
1.36 -33.84
(HF)5 C5h -51.61 -51.61 -51.51
0.01 -42.64
D5h -50.87 -50.87 -49.17
1.70 -43.34
C1v -52.78 -52.78 -51.13
1.65 -42.35
(HF)6
S6 -61.90 -61.89 -61.74
0.15 -51.16
C6h -61.92 -61.90 -61.70
0.21 -51.16
D6h -61.04 -61.06 -58.91
2.14 -51.97
a anis = 33 , 0:5( 11 + 22)
b = m ((HF )n) , n m (HF ) Positive sign means reduced
para
nl
-3.48
-3.56
-2.18
-3.50
-5.00
-4.00
-5.24
-6.54
-5.47
-6.96
-8.12
-6.93
-8.67
-9.74
-9.74
-8.36
-0.57
-0.55
0.48
-0.58
-0.63
0.01
-0.81
-0.71
-0.03
-1.01
-0.82
-0.03
-1.21
-0.95
-0.95
0.0
-1.75 -0.32
m
b
0.00
-10.46
-11.22c
-10.91d
-10.40e
-10.78f
liquid -8.6g
-20.90 0.02
-20.88 0.04
-19.56 1.36
-20.93 -0.01
-31.13 0.25
-30.23 1.15
-31.38 0.00
-41.34 0.50
-40.25 1.59
-41.81 0.03
-51.58 0.72
-50.30 2.00
-52.23 0.07
-61.85 0.91
-61.84 0.92
-60.33 2.43
diamagnetic behaviour
see caption
c LDA with exchange only [135], converted from magnetizabilities,
d LDA with exchange-correlation functional [135]
e SCF values from [185]
f MP2 values from [185]
g reference [186]
Table 15: Analysis of IGLO/BII//B3LYP/6-311+G** computed molar magnetic susceptibilities m of
(HF)n clusters (n = 1 , 6). Shown are the diagonal components , the anisotropy anis a , diamagnetic ( dia ), paramagnetic ( para) and non-local ( nl ) components, the average value m and the diamagnetic exaltation b . All quantities given in the table are expressed in the irrational system (mir)=(ppm
cm3 mol,1). To convert to SI units, 1 ppm cm3 mol,1= 1 \ppm cgs/mol" = 4 10,12 m3 mol,1.
The magnetizability = m=B [135], which is the magnetic equivalent to the electric polarizability, can
be obtained in the SI system via = m ,0 1 NA,1 , where 0 is the vacuum permeability and NA the
Avogadro number. Hence (mir) /ppm cm3 mol,1 1:66054 10,29=(J T,2 ) 0:2104=(e2a20=me ) [156].
The gauge origin for the non-IGLO results was at the center of mass (c.m.) if not noted otherwise.
n
Cnh
Dnh
(HF)n (HF)n (Hn)q a
3
{2.8
{5.6
{21.7
4
{1.1
{1.7
n.a.
5
{0.6
{0.8
{22.7
6 {0.4 (-0.35b)
{0.5
{24.2c
a Data for Dnh symmetric H+ , H, , and H6 , respectively, at
3 5
GIAO-SCF/6-31+G*//B3LYP/6-311++G(d,3pd)
b S6 symmetric ground state
c Data taken from reference [187]
Table 16: NICS values (ppm, GIAO-SCF/6-31+G*//B3LYP/6-311+G**) for (HF)n (n=3{6) and
related species.
13
Title:
A critical analysis of electronic density functionals for structural, energetic, dynamic and magnetic properties of hydrogen
uoride clusters
Authors with Aliation:
Ch. Maerker and P. v. R. Schleyer
Institut fur Organische Chemie der
Friedrich-Alexander Universitat Erlangen-Nurnberg, Henkestrasse 42
D-91054 Erlangen, Germany
K. R. Liedl
Institut fur Allgemeine, Anorganische und Theoretische Chemie
Leopold-Franzens-Universitat Innsbruck, Innrain 52a
A-6020 Innsbruck, Austria
T.-K. Ha, M. Quack, and M. A. Suhm
Laboratorium fur Physikalische Chemie der ETH Zurich (Zentrum),
CH-8092 Zurich, Switzerland
Description:
hfndft.ps
This contains the complete table material (tables 1{16), including table 8 which is omitted in the
printed paper and tables 1{7 and 9{13 which are given in the printed paper in an abbreviated
form.
method
dft
realization
rHF /pm
D-LDA
94.0
6-311++G**-LDA [86]
93.0
6-311++G**-B [86]
92.8
D-BP
94.0
6-311++G**-BP [86]
93.1
D-BLYP
94.3
6-311+G**-BLYP
93.3
6-311++G**-BLYP [86]
93.3
aug-cc-pV(T/Q)Z-BLYP
93.4
D95**-BHHLYP
91.6
6-311+G**-BHHLYP
90.9
6-311++G**-BHHLYP [86]
91.0
6-311+G**-B3LYP
92.2
6-311++G**-B3LYP [86]
92.2
6-311++G(3df,3pd)-B3LYP
92.2
aug-cc-pVDZ-B3LYP
92.6
aug-cc-pVTZ-B3LYP
92.4
aug-cc-pV(T/Q)Z-B3LYP
92.3
aug-cc-pVQZ-B3LYP
92.2
aug-cc-pV(T/Q)Z-B3'LYP
92.1
aug-cc-pV(T/Q)Z-B3"LYP
92.0
cqc
6-311++G(3df,3pd)(SCF)
89.7
aug-cc-pV5Z(SCF) [43]
89.7
aug-cc-pV5Z+MP2 [43]
91.8
aug-cc-pV5Z+CCSD(T) [43]
91.7
aug-cc-pV5Z+aug(F)+CCSD(T) [158]
91.8
TZ2P(f,d)+CCSD(T) [44]
91.8
DZP+MP2 [17]
91.9
6-311++G(3df,3pd)+MP2
91.7
aug-cc-pVDZ+MP2 [43]
92.5
aug-cc-pVTZ+MP2 [43]
92.2
aug-cc-pV(T/Q)Z+MP2
92.0
aug-cc-pVQZ+MP2 [43]
91.9
[8s6p2d/6s3p]+MP2 [17]
91.8
[3s2p1d/3s1p]+ACPF [40]
91.9
[8s6p2d/4s1p]+CPF [37]
91.9
Experiment [151,200,201]
91.7
a this work
/D !/cm,1
2.11
2.02
1.94
2.04
1.96
2.03
1.96
1.96
1.78
1.97
2.00
2.01
1.98
1.98
1.83
1.80
1.81
1.81
1.81
1.82
1.82
1.91
1.88
1.81
1.80
1.82
1.99
1.82
1.82
1.81
1.81
1.81
1.82
1.77
1.80
3942
4012
3989
3919
3978
3872
3941
3942
3914
4303
4291
4285
4099
4101
4093
4064
4076
4070
4077
4105
4110
4481
4473
4136
4142
4143
4157
4221
4173
4085a
4120a
4121
4135a
4143
4182
4135
4138
S /(km mol,1 )
99
140
102
80
110
69
106
106
91
124
161
161
130
130
106
111
111
111
111
115
116
159
102.5
110
117
116a
121a
119
122a
120
88
90
102
Table 1: Monomer equilibrium distance rHF , permanent electric dipole moment , harmonic wavenumber ! and integrated molar IR band strength S (S is given within the double-harmonic approximation
for convenient comparison to literature data, it can be converted to the more fundamental squared
transition dipole moment h01i2 and the integrated absorption cross section G via 41.624 h01i2 =D2=
G/pm2=16.6054 (S /(kmmol,1))/(! /cm,1 ) [156]) for di erent approaches (cqc=conventional quantum
chemistry), as discussed in the text. Experimental data are corrected for anharmonic zero point vibration e ects and therefore directly comparable to the predictions (this is not always the case [202]).
1
potential surface
D-LDA
6-311++G**-LDA [86]
LDA [89]
6-311++G**-B [86]
aug-cc-pV(T/Q)Z-B
D-BP
6-311++G**-BP [86]
BP [89]
D-BLYP
6-31++G**-BLYP [90]
6-311+G**-BLYP
6-311++G**-BLYP [86]
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311++G**-BHHLYP [86]
6-31++G**-B3P [90]
6-31+G**-B3LYP [87]
6-31++G**-B3LYP [90]
6-311+G**-B3LYP
6-311++G**-B3LYP [86]
6-311++G(3df,3pd)-B3LYP
aug-cc-pVDZ-B3LYP
aug-cc-pVTZ-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
aug-cc-pVQZ-B3LYP
aug-cc-pV(T/Q)Z-B3"LYP
6-311++G(3df,3pd)+MP2
aug-cc-pVDZ+MP2 [43]
aug-cc-pVTZ+MP2 [43]
aug-cc-pV(T/Q)Z+MP2
aug-cc-pVQZ+MP2 [43]
best ab initio [43,44,53]
Exp. [1]
Exp. [3]
best empirical [12,53,160]
y typographical error in [86]?
rHF /pm
95.3
94.5
95.3
y93.3
93.4
94.9
94.0
94.7
95.1
94.7
94.0
94.1
94.2
92.2
91.5
91.6
93.3
93.5
92.9
92.9
92.9
93.3
93.2
93.1
93.0
92.8
92.3
93.1
92.8
92.6
92.5
92.2-3
0 /pm
rHF
94.5
93.6
RFF /pm
=
259
11
256
10
255
8
290
8
287
6
271
9
274
9
269
6
272
12
276
8
278
8
277
8
275
6
267
10
272
8
272
8
270
8
273
8
273
8
275
8
275
9
273
5
273
6
273
6
273
6
273
6
272
6
274
5
275
7
275
6
275
6
274
6
273-4
7
272(3) 10(6)
7(3)
273-4
7-8
93.1
93.1
94.4
93.4
94.7
94.2
93.6
93.6
93.7
91.9
91.3
91.3
92.8
93.1
92.5
92.5
92.5
92.9
92.7
92.6
92.5
92.3
92.0
92.8
92.5
92.3
92.2
92.0-1
92.2-3 91.9-92.0
0 =
76
73
62
68
73
68
77
68
65
66
71
71
61
60
68
67
67
64
65
67
70
68
68
69
68
65
70
69
68
68
68-70
63(6)
60(2)
65-69
Table 2: Planar (HF)2 minimum geometry for di erent approaches. RF F is the distance between the
F atoms. Monomer bond lengths are denoted rHF , bond angles of the monomer with the FF axis .
Primed quantities refer to the non-bonded HF in the dimer.
2
potential surface
D-LDA
6-311++G**-LDA [86]
LDA [89]
6-311++G**-B [86]
aug-cc-pV(T/Q)Z-B
D-BP
6-311++G**-BP [86]
BP [89]
D-BLYP
6-31++G**-BLYP [90]
6-311+G**-BLYP
6-311++G**-BLYP [86]
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311++G**-BHHLYP [86]
6-31++G**-B3P [90]
6-31+G**-B3LYP [87]
6-31++G**-B3LYP [90]
6-311+G**-B3LYP
6-311++G**-B3LYP [86]
6-311++G(3df,3pd)-B3LYP
aug-cc-pVDZ-B3LYP
aug-cc-pVTZ-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
aug-cc-pVQZ-B3LYP
aug-cc-pV(T/Q)Z-B3"LYP
6-311++G(3df,3pd)+MP2
aug-cc-pVDZ+MP2 [43]
aug-cc-pVTZ+MP2 [43]
aug-cc-pV(T/Q)Z+MP2
aug-cc-pVQZ+MP2 [43]
best ab initio [43]/ [44]
best empirical [12,53,160]
experimental D0
De/(kJ mol,1) D0h /(kJ mol,1)
41.8
37.4
38.9
14.0
11.5
23.8
18.9
20.5
25.7
20.5 (18.4)
19.7
19.8
17.4
27.2
22.7
22.8
21.3 (19.7)
21.3
21.8 (20.0)
21.0
21.1
20.2
19.3
18.8
18.8
18.9
19.1
20.7 (17.0)
19.6 (16.8)
19.7 (17.7)
18.8 (17.8)
19.4 (18.3)
19.2/19.8
18.9(2),19.1(2)
33.5
16.0
17.3
12.5
10.0
19.1
15.2
13.7
12.3
11.9
11.4
11.3
11.5
12.4
12.2
12.1
11.4
11.9
{/12.3
12.0(2),11.7(2)
12.70(1) [13]
Table 3: Electronic (De ) and harmonically corrected (D0h ) dimer dissociation energies with respect
to separated monomers compared to MP2 benchmarks and best estimates from ab initio theory as
well as anharmonic empirically adjusted potential energy surfaces which reproduce the experimental
(anharmonic) dissociation energy D0 of ref. [13]. Values in parentheses include BSSE correction.
3
potential surface
D-LDA
!1
3892
(156)
6-311++G**-LDA [86]
3947
(181)
6-311++G**-B [86]
3959
(117)
D-BP
3893
(122)
6-311++G**-BP [86]
3935
(136)
D-BLYP
3841
(111)
6-311+G**-BLYP
3903
(127)
6-311++G**-BLYP [86]
3902
(129)
aug-cc-pV(T/Q)Z-BLYP
3872
(114)
D95**-BHHLYP
4262
(161)
6-311++G**-BHHLYP [86]
4245
(166)
6-311+G**-BHHLYP
4245
(166)
6-311++G**-B3LYP [86]
4058
(150)
6-311+G**-B3LYP
4059
(148)
6-311++G(3df,3pd)-B3LYP
4050
(127)
aug-cc-pVDZ-B3LYP
4019
(136)
aug-cc-pVTZ-B3LYP
4032
(133)
aug-cc-pV(T/Q)Z-B3LYP
4028
(132)
aug-cc-pV(T/Q)Z-B3"LYP
4064
(137)
6-311++G(3df,3pd)+MP2
4132
(126)
aug-cc-pVDZ+MP2
4038
(136)
aug-cc-pVTZ+MP2
4081
(137)
aug-cc-pV(T/Q)Z+MP2
4079
(134)
aug-cc-pVQZ+MP2
4094
(139)
best ab initio [44]
4119
(119)
empirical SQSBDE/SNB [12,160]
4100
with adjustment based on [25,162]
4090
new empirical [53]
4100
anharmonic fundamental (see [53,160]) 3931
!2
!3
3702
744
(583) (175)
3718
730
(728) (178)
3879
507
(359) (222)
3746
633
(446) (189)
3795
602
(486) (195)
3730
657
(392) (169)
3790
569
(459) (209)
3787
577
(455) (203)
3735
588
(492) (134)
4164
622
(492) (194)
4165
571
(524) (251)
4165
570
(525) (250)
3958
581
(481) (219)
3960
574
(485) (224)
3934
599
(499) (163)
3884
593
(511) (149)
3907
582
(505) (152)
3906
587
(511) (152)
3945
591
(516) (155)
4046
609
(476) (168)
3938
577
(466) (154)
3986
580
(475) (152)
3989
568
(472) (156)
3997
578
(481) (153)
4050
567
(427) (160)
4050
4030
4030
550
3868 480
!5
!6
179
(151)
203
(94)
120
(7)
141
(134)
162
(14)
177
(74)
155
(26)
157
(20)
165
(4)
182
(34)
166
(37)
165
(41)
160
(31)
159
(34)
169
(7)
167
(12)
164
(5)
168
(9)
170
(9)
166
(3)
159
(12)
160
(6)
159
(7)
163
(11)
157
(25)
150
523
(272)
550
(292)
415
(248)
483
(255)
471
(245)
466
(268)
446
(241)
452
(245)
472
(154)
479
(256)
464
(274)
462
(274)
461
(262)
457
(259)
505
(174)
482
(166)
481
(169)
480
(168)
483
(171)
547
(174)
473
(175)
477
(173)
468
(170)
473
(173)
458
(188)
410
210
155
465
160 125 420
Table 4: Caption see next page!
4
!4
302
(117)
292
(137)
194
(149)
244
(81)
239
(172)
277
(172)
215
(142)
222
(156)
233
(151)
254
(180)
222
(135)
220
(132)
221
(152)
217
(142)
236
(146)
225
(146)
233
(152)
231
(148)
233
(150)
238
(144)
217
(148)
222
(155)
218
(147)
220
(150)
210
(141)
210
or 380
,!1 ,!2
50
240
65
294
29
109
26
173
43
183
31
142
38
151
40
155
42
179
40
138
39
120
46
126
43
143
41
139
43
159
45
180
45
170
42
164
46
165
41
127
47
147
39
134
42
131
41
138
38
107
38
48
38
31
88
108
108
93
Table 4: Predicted harmonic wavenumbers ! /cm,1 and vibrational shifts ,!1 2 relative to the
monomer for (HF)2 compared to rounded best empirical (conservatively 20 cm,1 ) and ab initio estimates as well as MP2 benchmarks. In parentheses, integrated molar IR band strengths S /(kmmol,1)
are given in the double-harmonic approximation. Experimentally, the band strength enhancement of
1 (2 ) over the monomer is approximately 20% (300%) [164]. The fundamental 6 has been observed
only in the K = 0 ! 1 and K = 1 ! 2 transitions [203]. From this, two di erent extrapolations to
the rotationless fundamentals have been made, depending upon treatment of Coriolis e ects in various
potentials (380cm,1 and 420cm,1 [12,53]), with corresponding di erences for !6 (410 or 465cm,1 ).
i
;
i
5
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
aug-cc-pV(T/Q)Z-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
D95**-B3LYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
aug-cc-pV(T/Q)Z-B3LYP
6-311++G(3df,3pd)+MP2 [45]
aug-cc-pVTZ+MP2
best estimate
a planar saddle
1
94.0
94.0
94.3
93.3
93.4
91.6
90.9
92.2
92.2
92.3
91.7
92.2
91.7
3
98.7
96.7
96.7
95.4
4
104.
98.0
97.8
96.7
93.5 94.9
92.4
94.0 95.1
94.2 95.7
93.3 94.6
93.3 94.4
5
6
108. 109.
98.9 99.1
98.3 98.5
97.3
98.3
95.4 95.5a
93.6
98.3
95.5 95.7
96.3
96.4
95.0
95.7
94.8 94.9
Table 5: Size dependence of the HF bond length rHF in the minimum geometry of (HF)n clusters.
For n=3-5, the minimum geometry has Cnh symmetry, for n=6, a slightly puckered S6 -symmetric
minimum is typically found. The limit for an in nite (chain or ring) cluster can be estimated around
95-97 pm [169, 170] and in a supersonic jet expansion dominated by pentamers and hexamers, rHF is
increased by 3-5 pm [58] relative to the monomer. The best estimates are mostly derived from ab initio
calculations [17,40] and shifts relative to the monomer are expected to have an uncertainty of less than
20%.
potential surface
n=
2
D-LDA
259
D-BP
271
D-BLYP
272
6-311+G**-BLYP
278
aug-cc-pV(T/Q)Z-BLYP
275
D95**-BHHLYP
267
6-311+G**-BHHLYP
272
6-311+G**-B3LYP
275
6-311++G(3df,3pd)-B3LYP
273
aug-cc-pV(T/Q)Z-B3LYP
273
6-311++G(3df,3pd)+MP2 [45]
274
aug-cc-pVTZ+MP2
275
best estimate
273.5(1.0)
a planar saddle
3
243
258
260
263
4
234
253
255
255
253 246
261
262 254
259 250
261 252
259 251
5
6
231 230
248 247
252 251
252
248
243 242a
252
251 250
247
247
248
248
248 247
Table 6: Size dependence of the FF distance RF F /pm in the minimum geometry of (HF)n clusters. See
also table 5. Experimental data from pentamer/hexamer supersonic jets (253pm [58]) and the extended
solid (248-251pm [169,204]) have to be corrected for vibrational averaging e ects. Best estimates derive
from spectroscopic data for n=2,3 and ab initio calculations for the larger clusters [17, 40]. Their
estimated absolute error is about (n , 1) pm, but the size trend is more accurate.
6
potential surface
n=
2
D-LDA
11
D-BP
9
D-BLYP
12
6-311+G**-BLYP
8
aug-cc-pV(T/Q)Z-BLYP
6
D95**-BHHLYP
10
6-311+G**-BHHLYP
8
6-311+G**-B3LYP
8
6-311++G(3df,3pd)-B3LYP
5
aug-cc-pV(T/Q)Z-B3LYP
6
6-311++G(3df,3pd)+MP2 [45] 5
aug-cc-pVTZ+MP2
6
best estimate
7-8
a planar saddle
3 4 5 6
21 9 3 2
22 11 5 2
22 11 5 3
23 11 5
3
23 11 5 2a
25
7
24 12 6 2
21 9 4
2
22 9 4
4
24 12 6 3
Table 7: Size dependence of the hydrogen bond angle HF F / in the minimum geometry of (HF)n
clusters. See also table 5. Best estimates derive from ab initio calculations [17,40].
potential surface
D-LDA
n=
aug-cc-pV(T/Q)Z-BLYP
2
3
41.8 147.7
[67]
23.8 83.7
[67]
25.7 87.6
[65]
19.7 63.6
[62]
17.4
4
256.3
[117]
147.7
[118]
152.5
[113]
119.1
[115]
D95**-BHHLYP
27.2
6-311+G**-BHHLYP
22.7
161.5
[113]
6-311+G**-B3LYP
21.0
D-BP
D-BLYP
6-311+G**-BLYP
6-311++G(3df,3pd)-B3LYP 20.2
aug-cc-pV(T/Q)Z-B3LYP
18.8
6-311++G(3df,3pd)+MP2
aug-cc-pVTZ+MP2
best estimate
a planar saddle
20.7
19.7
19.1
91.7
[64]
69.0
[58]
66.0
[60]
66.3
[63]
121.8
[111]
125.6
[119]
64.7 121.7
63
117
5
6
345.2 422.5
[158] [193]
198.9 243.5
[160] [195]
203.9 249.2
[152] [185]
164.2
[159]
163.8
[180]
217.9 265.5a
[153] [186]
171.7
[144]
168.1 207.6
[153] [189]
173.5
[164]
167.5
[170]
168.1
165.7
161
199
Table 8: Size dependence of the electronic dissociation energy of (HF)n clusters with respect to
fragmentation into separate monomers De /(kJ/mol). Values in brackets are obtained by linear scaling
to the best dimer value of 19.1(2)kJ/mol. Best estimates derive from ab initio calculations [17, 40, 53]
and experiment [13, 15, 22, 171], connected by quantum Monte Carlo calculations [12, 171], as well as
thermodynamic modelling [23]. The error bars are approximately n kJ/mol for n > 2 and 0.2 kJ/mol
for n=2.
7
potential surface
D-LDA
n=
2
3
33.5 125.2
[44]
D-BP
16.0 61.5
[45]
D-BLYP
17.3 65.1
[44]
6-311+G**-BLYP
12.5 43.4
[41]
D95**-BHHLYP
19.1 69.0
[42]
6-311+G**-BHHLYP
15.2 48.4
[37]
6-311+G**-B3LYP
13.7 45.6
[39]
6-311++G(3df,3pd)-B3LYP 12.3 45.1
[43]
6-311++G(3df,3pd)+MP2 12.4 43.7
best D0h estimate
11.7
41
best D0 estimate
12.7 43.0
aug-cc-pVTZ+MP2 De
19.7
best De estimate
19.1
63
a planar saddle
4
229.1
[80]
117.2
[86]
121.4
[82]
89.6
[84]
128.3
[79]
91.4
[78]
93.5
[89]
88.3
83
84.5
117
5
6
316.2 388.8
[110] [136]
162.5 199.8
[119] [146]
167.1 204.8
[113] [139]
128.5
[120]
176.6 218.2a
[108] [134]
132.9
[102]
130.7
[112]
133.0
[127]
125.2
116
145
117
146
165.7
161
199
Table 9: Size dependence of the harmonically corrected dissociation energy of (HF)n clusters with
respect to fragmentation into separate monomers Dh0 /(kJ/mol). Values in brackets are obtained by
linear scaling to the dimer estimate. Best estimates derive from ab initio calculations [17, 40] and
experiment [15, 22, 171], connected by quantum Monte Carlo calculations [12, 171]. The error bars are
approximately n kJ/mol for n > 2 and 0.2 kJ/mol for n=2. For more dimer results, see also table 3.
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
DZP+MP2 [17]
6-311++G(3df,3pd)+MP2
anh. experiment
a planar saddle
3
708/1005
411/570
360/506
346/472
338/480
250/342
297/406
344/475
247/359
280/396
249/|
4
1382/2010
668/922
581/806
602/812
640/884
5
1752/2315
876/1180
729/976
736/961
802/1065
553/710
660/854
816/1068
599/796
686/895
661/|
529/710
660/896
472/659
554/755
516/|
6
7
1979(1373)/2435
979(715)/1263
810(590)/1032
873/1111a
660(474)/834
695/843
716/|
746/|
Table 10: Harmonic IR-active/totally symmetric HF stretching wavenumber shift ,! /cm,1 as a
function of cluster size. For the (HF)6 S6 structure, the puckering-induced weak IR satellite band is
given in parentheses.
8
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
6-311++G(3df,3pd)+MP2
best harmonic
anh. experiment n=5{6
a planar saddle
3
4
5
6
878/892 1250/1034 1336/1025 930{1353
714/750
945/812 1053/816 744{1160
705/729
914/781
991/768 676{1004
619/714
866/787
950/775
678/756
952/869 1053/877 1068/830a
558/699
893/772
589/707
841/787
934/780
633/733
942/855 1040/865
581/719
903/859 1007/873
600/700
850/800
950/800 700{1000
600{900 [23]
Table 11: Harmonic IR-active in plane/out-of-plane HF libration wavenumbers !lib =cm,1 as a function
of cluster size. For the non-planar (HF)6 S6 structure, the range of IR-active librations is given.
potential surface
n=
D-LDA
D-BP
D-BLYP
6-311+G**-BLYP
D95**-BHHLYP
6-311+G**-BHHLYP
6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
6-311++G(3df,3pd)+MP2 [45]
best harmonic
anh. experiment n=4!6
a planar saddle
3
314/317
246/273
242/275
200/220
235/259
195/229
198/223
205/228
190/215
200/220
4
491/298
299/222
292/227
277/209
324/247
5
6
529/264
516/241
295/180
275/167
276/179
256/170
272/179
323/217 289/185a
256/181
271/210 267/180
299/224 300/197
278/210 279/186
280/210 270/190
250/175
260!230 [23]/ 185!150 [22]
Table 12: Harmonic IR-active/totally symmetric FF stretching wavenumber !F F =cm,1 as a function
of cluster size n. For the (HF)6 S6 structure, only the dominant component is given. Anharmonic
experimental values for n = 4 ! 6 are approximate and below the best harmonic estimate, which is
derived from ab initio and analytical potential energy surfaces [17].
9
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
2
117.9
206
6-311++G(3df,3pd)+MP2 [45]
118.2
206
[8s6p2d/6s3p]+MP2
ACPF [40]
117.8
205
extended basis [40]
QCISD(T)//MP2 [51]
D-LDA
120.5
207
D-BP
121.5
209
D-BLYP
122.1
210
D95**-BHHLYP
117.2
204
6-311+G**-BHHLYP
117.4
204
6-311+G**-B3LYP
119.3
207
6-311++G(3df,3pd)-B3LYP
119.1
207
best theoretical estimate
118
206
F HF=
6
EB
EB
121 174.6 164.3
122 167.4 158.6
174 164
121 185
173
186.4 176.3
118 104.6 95.1
119 140.0 130.0
119 145.5 134.7
170.3 159.5
121 200.5 188.1
120 172.2 160.7
121 157.8 148.4
122 170 160
!i
2306
2262
1868
1999
2030
2387
2468
2241
2203
Table 13: a)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
3
114.9
223
6-311++G(3df,3pd)+MP2 [45]
115.1
224
CCSD(T) [50]
ACPF [40]
114.7
223
QCISD(T)//MP2 [51]
D-LDA
117.2
226
D-BP
118.3
229
D-BLYP
118.8
230
D95**-BHHLYP
114.4
222
6-311+G**-BHHLYP
114.4
222
6-311+G**-B3LYP
116.1
225
MP2//6-311+G**-B3LYP
MP4//6-311+G**-B3LYP
QCISD(T)//6-311+G**-B3LYP
CCSD(T)//6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
116.1
226
best theoretical estimate
115
224
F HF=
6
152
154
153
150
150
150
152
152
153
153
EBe
EBh
!i0
84.2
11.2
43.3
51.5
61.7
90.9
68.5
1155
1448
1522
1799
1917
1705
87.7
78.4
75
86.6
95.4
21.4
55.0
64.0
75.8
104.8
81.1
95.6
98.3
101.6
102.0
69.6
80
75.0 1773
66.8 1733
61
EBe
EBh
57.2 1653
65
Table 13: b)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
4
113.6
226
6-311++G(3df,3pd)+MP2 [45]
113.8
226
ACPF [40]
113.5
225
D-LDA
115.5
229
D-BP
116.7
232
D-BLYP
117.1
233
D95**-BHHLYP
113.1
225
6-311+G**-B3LYP
114.7
228
MP2//6-311+G**-B3LYP
MP4//6-311+G**-B3LYP
QCISD(T)//6-311+G**-B3LYP
CCSD(T)//6-311+G**-B3LYP
6-311++G(3df,3pd)-B3LYP
114.7
228
best theoretical estimate
113.5
226
Table 13: c)
10
6
F HF=
166
168
167
165
166
166
61.1 42.5
53.3 33.9
61.9
2.9 ,6.1
29.3 14.3
39.0 22.8
48.4 28.4
167 54.9 36.5
69.3
72.4
75.4
75.8
168 43.6 25.0
167 55
35
!i0
1526
1498
692
1183
1284
1515
1454
1403
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
5
113.0
226
6-311++G(3df,3pd)+MP2 [45]
113.2
226
aug-cc-pVTZ+MP2
113.5
227
D-LDA
113.0
226
D-BP
115.9
232
D-BLYP
116.5
233
6-311+G**-BLYP
115.5
231
aug-cc-pV(T/Q)Z-BLYP
115.8
232
D95**-BHHLYP
112.5
225
6-311+G**-BHHLYP
112.5
225
D95**-B3LYP
113.9
228
6-311+G**-B3LYP
114.0
228
6-311++G(3df,3pd)-B3LYP
114.1
228
aug-cc-pV(T/Q)Z-B3LYP
114.3
228
aug-cc-pV(T/Q)Z-B3'LYP
113.9
228
aug-cc-pV(T/Q)Z-B3"LYP
113.9
228
best theoretical estimate
113
226
6
F HF=
174
177
177
175
176
175
175
177
174
175
175
176
176
176
176
175
EBe
EBh
58.1 34.7
52.6 27.1
45.7
0.4 ,5.5
23.8 7.2
34.1 15.7
40.0 17.9
30.6
44.3 19.0
74.9 48.0
22.6 2.6
51.6 27.6
40.8 17.0
41.3
44.5
44.0
50
25
!i0
1436
1431
431
1086
1209
1418
1583
1132
1368
1318
Table 13: d)
potential surface
n rHF /pm RFF /pm
DZP+MP2 [17]
6
112.7
225
ACPF [40]
112.7
225
D-LDA
114.3
229
D-BP
115.6
231
D-BLYP
116.1
232
D95**-BHHLYP
112.2
224
6-311+G**-B3LYP
113.7
227
best theoretical estimate
112.5
225
6
F HF=
179
179
179
179
177
EBe
EBh
!i0
65.5 37.0 1431
69.5
3.1 ,3.5 327
28.3 7.2 1081
40.3 16.7 1201
48.9 19.5 1398
179 57.1
179 60
30
Table 13: e)
Table 13: Structure and energetics of the concerted hydrogen exchange transition state in cyclic planar
HF clusters ( a) dimer, b) trimer, c) tetramer, d) pentamer, e)hexamer (here, the planar structure
often corresponds to a 4th order saddle, which is however very close to the puckered 1st order saddle)).
Electronic (EBe ) and harmonically corrected (EBh ) threshold energies are given in kJ/mol. The imaginary
wavenumber along the reaction coordinate !i0 =!i /cm,1 is a measure of the reaction pro le curvature.
Single point calculations at higher level with the same basis set (where given) are indicated by //. Best
energy estimates remain quite uncertain ((5-10)kJ/mol) in the absence of experimental constraints
and large basis set MP2, QCISD(T), and CCSD(T) reference calculations.
11
compound
(HF)1 (exp) C1v
HF
C1v
(HF)2
Cs , Hb
Cs
C2h
D2h
C3h
D3h
C4h
D4h
C5h
D5h
S6
C6h
D6h
(HF)3
(HF)4
(HF)5
(HF)6
a
b
c
d
F
H
IGLO/BII
386.6(0.0) 28.0(0.0)
397.8(-11.1)
388.5(-1.8)
388.5(-1.9)
268.2(118.5)
387.6(-1.0)
346.3(40.4)
387.9(-1.3)
354.4(32.3)
389.0(-2.3)
359.8(26.9)
390.0(-3.4)
390.2(-3.6)
362.7(23.9)
25.7(2.3)
26.7(1.3)
17.0(1.0)
11.1(16.9)
24.1(4.0)
11.2(16.8)
21.6(6.4)
10.8(17.2)
20.5(7.5)
10.6(17.4)
20.1(7.9)
20.1(7.9)
10.6(17.5)
F
H
419.70.3a 29.20.5a
IGLO-DFPT/BIII
405.5(0.0) 29.7(0.0)
408.0b
29.2b
c
418.1
29.1c
d
418.6
29.2d
406.1(-0.6) 27.2(2.5)
395.4(10.1) 28.3(1.3)
392.4(13.1) 28.6(1.1)
248.2(157.4) 14.2(15.4)
377.7(27.9) 25.2(4.5)
316.4(89.2) 13.6(16.1)
374.0(31.6) 22.8(6.9)
324.3(81.3) 13.2(16.5)
374.4(31.2) 21.8(7.9)
331.0(74.6) 13.1(16.6)
376.2(29.3) 21.5(8.2)
376.4(29.1) 21.5(8.2)
335.8(69.8) 13.1(16.6)
ref. [182]
employing the Perdew-Wang exchange-correlation functional (PW91)
GIAO-CCSD values from [138]
GIAO-CCSD(T) values from [138]
Table 14: Computed 19F and 1 H NMR isotropic equilibrium shielding constants e (ppm, all IGLODFPT/BIII calculations employed the Perdew exchange-correlation functional, if not noted otherwise).
Relative chemical shifts (ppm) of the HF oligomers with respect to monomeric HF given in parentheses.
12
species
11
22
33 anis a
HF
C1v -10.58 -10.58 -10.21
0.37
0.61c
0.52d
0.52e
0.54f
dia
-8.39
(HF)2
Cs -20.81 -20.76 -21.11
-0.33 -16.85
C2h -20.59 -20.99 -21.05
-0.26 -16.87
2.93 -17.86
D2h -21.07 -20.01 -17.61
C1v -21.18 -21.18 -20.44
0.74 -16.85
(HF)3 C3h -31.07 -31.08 -31.24
-0.17 -25.50
D3h -30.52 -30.52 -29.64
0.88 -26.24
C1v -31.73 -31.73 -30.67
1.06 -25.34
(HF)4 C4h -41.33 -41.33 -41.36
-0.03 -34.09
D4h -40.70 -40.70 -39.35
1.35 -34.74
C1v -42.26 -42.26 -40.90
1.36 -33.84
(HF)5 C5h -51.61 -51.61 -51.51
0.01 -42.64
D5h -50.87 -50.87 -49.17
1.70 -43.34
C1v -52.78 -52.78 -51.13
1.65 -42.35
(HF)6
S6 -61.90 -61.89 -61.74
0.15 -51.16
C6h -61.92 -61.90 -61.70
0.21 -51.16
D6h -61.04 -61.06 -58.91
2.14 -51.97
a anis = 33 , 0:5( 11 + 22)
b = m ((HF )n) , n m (HF ) Positive sign means reduced
para
nl
-3.48
-3.56
-2.18
-3.50
-5.00
-4.00
-5.24
-6.54
-5.47
-6.96
-8.12
-6.93
-8.67
-9.74
-9.74
-8.36
-0.57
-0.55
0.48
-0.58
-0.63
0.01
-0.81
-0.71
-0.03
-1.01
-0.82
-0.03
-1.21
-0.95
-0.95
0.0
-1.75 -0.32
m
b
0.00
-10.46
-11.22c
-10.91d
-10.40e
-10.78f
liquid -8.6g
-20.90 0.02
-20.88 0.04
-19.56 1.36
-20.93 -0.01
-31.13 0.25
-30.23 1.15
-31.38 0.00
-41.34 0.50
-40.25 1.59
-41.81 0.03
-51.58 0.72
-50.30 2.00
-52.23 0.07
-61.85 0.91
-61.84 0.92
-60.33 2.43
diamagnetic behaviour
see caption
c LDA with exchange only [135], converted from magnetizabilities,
d LDA with exchange-correlation functional [135]
e SCF values from [185]
f MP2 values from [185]
g reference [186]
Table 15: Analysis of IGLO/BII//B3LYP/6-311+G** computed molar magnetic susceptibilities m of
(HF)n clusters (n = 1 , 6). Shown are the diagonal components , the anisotropy anis a , diamagnetic ( dia ), paramagnetic ( para) and non-local ( nl ) components, the average value m and the diamagnetic exaltation b . All quantities given in the table are expressed in the irrational system (mir)=(ppm
cm3 mol,1). To convert to SI units, 1 ppm cm3 mol,1= 1 \ppm cgs/mol" = 4 10,12 m3 mol,1.
The magnetizability = m=B [135], which is the magnetic equivalent to the electric polarizability, can
be obtained in the SI system via = m ,0 1 NA,1 , where 0 is the vacuum permeability and NA the
Avogadro number. Hence (mir) /ppm cm3 mol,1 1:66054 10,29=(J T,2 ) 0:2104=(e2a20=me ) [156].
The gauge origin for the non-IGLO results was at the center of mass (c.m.) if not noted otherwise.
n
Cnh
Dnh
(HF)n (HF)n (Hn)q a
3
{2.8
{5.6
{21.7
4
{1.1
{1.7
n.a.
5
{0.6
{0.8
{22.7
6 {0.4 (-0.35b)
{0.5
{24.2c
a Data for Dnh symmetric H+ , H, , and H6 , respectively, at
3 5
GIAO-SCF/6-31+G*//B3LYP/6-311++G(d,3pd)
b S6 symmetric ground state
c Data taken from reference [187]
Table 16: NICS values (ppm, GIAO-SCF/6-31+G*//B3LYP/6-311+G**) for (HF)n (n=3{6) and
related species.
13