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Semiempirical scaled valence force fields for benzene
Journal of Molecular Structure (Theochem) , 306 (1994) 165- 115 0166-1280/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved
165
Semiempirical scaled valence force fields for benzene Goran BaranoviC*, Bernhard Schrader Institut fir
Physikalische
und Theoretische
Chemie, Universittit GH Essen, 45117 Essen, Germany
(Received 23 September 1993; accepted 1 October 1993) Abstract The performances vibrational properties tested. A set of the agreement to within substituted benzenes
of three semiempirical parameterization sets (MNDO, AMI and PM3) in describing the (harmonic frequencies, isotopic frequency shifts and Coriolis constants) of benzene were best (in the least-squares sense) scaling factors for the force constants, which improves 20 cm-‘, was designed with the idea of it being used for the more complex cases of monoand polycyclic aromatic hydrocarbons.
Introduction
The idea of transferability of empirical scaling factors from benzene to, for example, monosubstituted benzene or even polycyclic hydrocarbons [l] within semiempirical quantum chemistry is particularly appealing because it would make the vibrational analysis of these large molecules, which are unsuitable even for ab initio geometry optimization, rather straightforward. Scaling factors have to be introduced in order to compensate for errors in predicting the vibrational frequencies. This is necessary especially if one wants to use semiempirical data in solving a specific spectroscopic problem, as without additional corrections these results are not of great practical value. However, similarly as in an attempt to find the best values for an empirical force field, to obtain the best values of scaling factors one needs fully and unambiguously assigned vibrational spectra. The purpose of this paper is to assess to what extent semiempirical quantum chemical vibrational analysis provides a correct description of the vibrational properties of benzene and to calculate the scaling factors to be * Corresponding author - on leave from R. BoSkovi6 Institute, 41001 Zagreb, Croatia. SSDI 0166-1280(93)03605-7
used in the assignment of an unknown vibrational spectrum of a certain benzene derivative. This is possible because the benchmark harmonic force field for benzene has been reported [2]. The harmonic force field of benzene was computed by using MNDO, AM1 and PM3 Hamiltonians with version 6.0 of the MOPAC program package [3]. The results are compared with the values obtained by ab initio [2,4] and semiempirical CND0/2 [5] calculations and experiment. Force constants were scaled using the method described in Ref. 4, i.e. a multiplication factor ci is applied to the force constant Ai and (eiCi)*” to the interaction constant J7, unless the latter is scaled independently. As will be shown, a separate scaling parameter should be introduced for almost all force constants, in order to obtain good reproducibility of vibrational frequencies. This is not unexpected, knowing all the approximations involved in the semiempirical methods. Therefore, a reproducibility will be considered good or satisfactory if no absolute frequency difference is larger than about 20 cm-‘. Although this is an order of magnitude weaker criterion than the one proposed by Goodman et al. [2], we believe that with an estimation within 2Ocm-’ it is still possible to reach reliable interpretation of a vibrational
G. BaranoviC and B. Schrader/J. Mol. Struct. (Theochem) 306 (1994) 165-175
166
spectrum. Our aim was to achieve this while keeping the number of scaling parameters as low as possible. The best values in the least-squares sense were obtained iteratively by means of the Levenberg-Marquardt algorithm from the observed harmonic frequencies of benzene-h6 and benzene-d6.
The internal coordinates used are the same as those used in Ref. 4 (see Table Al in the Appendix). The in-plane symmetry coordinates used are those reported in Ref. 2, while the outof-plane coordinates are identical to those reported by Pulay et al. [4] (see Table A2 in the Appendix). The transformation of a force field from Cartesian to internal coordinates was performed according to the method described previously [8]. Minor differences between our coordinates and those taken from Ref. 2 were taken into account when the harmonic unscaled force fields for benzene in terms of internal symmetrized coordinates were compared with the benchmark force field given in Table 2 (see Table A3 in the Appendix). The unscaled and scaled values in terms of symmetry coordinates are given in Table 3. The corresponding scaling factors for different scaling schemes are given in Table 4. The quality of the semiempirical force fields was considered with respect to the frequency differences (Table 5), isotopic frequency shifts for C6D6 (Table 6) and Coriolis constants (Table 7). The observed frequencies are corrected for anharmonicity [2]. The corrections are important for the C-H stretching vibrations and are introduced here in order to minimize the influence of fitting to the anharmonic data. While MNDO overestimates, and PM3 underestimates the C-H/ C-D stretching frequencies, AM1 values are rather close to the harmonic observed frequencies. The mean values of the AM1 and measured
Results and discussion The theoretical geometries of benzene (Table 1) are used in calculating the force fields, not only because of consistency, but also because the C-C bond lengths obtained with AM1 and PM3 are rather close to the experimental values. The C-H bond length is overestimated by 1 %, and as a result the C-H bending frequencies will be additionally downshifted. The use of calculated and not experimental geometries may thus be a source of error. Therefore the frequencies were also calculated for the experimental geometry [6], which was also used by Goodman et al. [2]. However, not only was the overall agreement in frequencies slightly worse in all three cases, but also the influence of differences in geometry was found to be negligible when compared with the already existing frequency differences obtained by semiempirical methods. The semiempirical geometries are close enough to each other such that the main reasons for the differences among calculated frequencies are to be found in the force fields. Table 1 Experimental
and theoretical geometries (A) of benzene
Bond
Exp.
MNDO
AM1
PM3
4-21GC
6-31G*d
kc
1.397a 1.3920b
1.4066
1.3950
1.3911
1.3845
1.386
rCH
1.084a 1.0862b
1.0903
1.0996
1.0948
1.0721
1.076
a r0 structure from Ref. 6. b r,,, structure from Ref. 7. ’ Ref. 4. d Ref. 2.
G. BaranoviC and B. SchraderlJ.
Mol. Struct.
Table 2 Unscaled harmonic force field constants No.
(Theochem)
ti ti
f2
Si Si
A
919919
2
PiPi 920a920a
f6
ti t* + I
f7
ti ti+2
fs
ti h+3
Ex~.~
0.728 -0.419 0.383
1.145 -0.602 0.749
1.523 -0.483 0.819
1.441 -0.601 0.787
0.006 0.002 0.001
0.006 0.002 0.002
0.014 0.001 0.001
fll
sisi+3
0.007 0.008 -0.022
k
Pi P*Pit2 Pi+1
-0.012 0.008
f14
&Pi+3
-0.001
=
=
tjSi+l tiSi+S
fl7
tiS,+3
=
tiSi+4
fl8
t1920a
fi9
Sl919
f20
s142oa
f2l
liPi =
-tiPi+
f22
tiPi+2
=
-riPi+S
f23
liPi+
=
-tiPi+
PM3 9.190 5.165 1.178 0.455 1.226
sisi+2
tiSi
AM1 9.601 5.654 1.317 0.522 1.348
sisi+l
tiSj+Z
MNDO 8.613 6.453 1.163 0.547 1.236
f9
f16
167
6.616 5.547 1.284 0.523 1.257
flo
flS
165-175
of benzene in terms of internal valence coordinates
Type
fi
306 (1994)
-0.004 -0.007
-0.006
-0.003 -0.014
0.000
0.000
0.000
0.200 0.008 -0.130
0.413 0.011 -0.056
0.425 0.005 -0.055
0.211 0.002 -0.030
0.124
0.118
0.103
0.129
-0.135 -0.110
-0.153 -0.192
-0.147 -0.183
-0.073 -0.094
-0.153 0.015 -0.023
-0.269 0.004 -0.004
-0.281 0.006 -0.006
-0.201 -0.001 -0.002
0.048 0.019
0.000 0.003
0.000 0.003
0.008 0.001
f24
siPi+l
=
-siPi+
f25
siPi+
=
-SiPi+
f26
420a P2
0.075
0.018
0.037
0.026
f27
“/i ?;
0.447
0.48 1
0.454
0.424
f28
928928
f29
929a429a
0.394 0.312
0.259 0.245
0.274 0.246
0.269 0.225
f30
“ii%+1
f31
^/iYi+2
f32
Ti^li+3
f33
71928
f34
Y1929a
a Ref. 2. The coordinate pi differs by a factor of -J2 coordinate. Equivalencies among some out-of-plane
-0.069 0.002 -0.021
-0.033 -0.006 -0.016
-0.028 -0.006 -0.018
-0.034 -0.005 -0.010
-0.154 0.147
-0.098 0.105
-0.101 0.104
-0.092 0.097
from that in Ref. 2. There is also a sign difference in the definition of the yi potential constants: ylq29a = -2y2q29a = -2y3q2ga = y4q29a = -2y5q2ga = -2?6Y6q29a = -t2/d3h2q29b = +c2/d3h3q29b = +/t/3)%q29b = +@/d%6429b, Yjq29b = ,,‘4q29b = 0. Stretching bond CODstants are in units of mdyn A-‘, while mdyn A rad-2 are used for angle bendings, waggings and torsions.
G. BaranoviC and B. SchraderlJ. Mol. Struct. (Theochem) 306 (1994) 165-175
168
Table 3 Benzene harmonic force fields in terms of symmetry coordinates
Type
Exp.’
Unscaled MNDO
MNDO
AM1
PM3
F2,2
P
F3,3
0.515
0.525
0.500
0.421
0.503
0.510
0.487
1.284 -0.331 5.571
1.163 -0.376 6.443
1.317 -0.359 5.645
1.178 -0.179 5.138
1.288 -0.369 5.626
1.267 -0.352 5.626
1.265 -0.192 5.530
3.939 -0.228 0.486
3.830 -0.521 0.552
4.768 -0.539 0.523
4.318 -0.400 0.476
3.737 -0.189 0.528
3.751 -0.191 0.534
3.713 -0.240 0.551
1.257 -0.190 0.430 0.150 5.510 0.054 0.050 6.690 0.305 0.526
1.236 -0.333 0.409 0.062 6.445 0.335 -0.004 8.548 0.473 0.557
1.348 -0.316 0.610 0.075 5.648 0.361 -0.005 9.379 0.498 0.534
1.126 -0.163 0.446 0.051 5.151 0.177 0.012 9.137 0.352 0.472
1.216 -0.308 0.355 0.060 5.628 0.274 -0.004 6.680 0.286 0.533
1.182 -0.296 0.475 0.071 5.629 0.300 -0.005 6.818 0.281 0.545
1.222 -0.169 0.395 0.055 5.544 0.162 0.013 8.315 1.061 0.547
0.544 -0.160 -0.115 7.380 0.572 5.568
0.544 -0.274 -0.006 9.881 0.813 6.456
0.522 -0.287 -0.005 10.788 0.831 5.657
0.444 -0.198 -0.015 10.445 0.418 5.177
0.520 -0.174 -0.005 7.574 0.666 5.637
0.533 -0.184 -0.005 7.538 0.691 5.637
0.514 -0.118 -0.017 7.754 0.385 5.571
0.292
0.387
0.367
0.334
0.294
0.296
0.297
0.394 -0.377 0.610
0.259 -0.240 0.550
0.274 -0.246 0.516
0.268 -0.225 0.269
0.354 -0.339 0.577
0.363 -0.346 0.583
0.381 -0.359 0.589
0.396
0.470
0.450
0.405
0.403
0.413
0.404
0.312 0.254 0.493
0.244 0.182 0.504
0.246 0.179 0.470
0.226 0.168 0.453
0.335 0.257 0.501
0.325 0.252 0.503
0.319 0.268 0.516
blu Q
F12,12 F12,13
s
bzut
F13,13
F14,14 F14,15
P e2g0
F15,15
F6,6 F6,l F6,8 F6,9
s
Fl,7 F7,8 F7,9
t
F8,8 F8,9
P
F9,9
P
F18, 18 Fl8,19
t
F18,20 FI,,
19
F19, 20
a2”
PM3
s
FL, F1,2
eh
AM1
7.616 0.157 5.554
alg t
a2g
Scaled
s
F20,20
7
Fll,ll
bg 6
F4,4 F4,5
Y
F5,5
elg Y
F IO,10
e2u6
F16,
16
Fl6,17
Y
F 17,17
a Ref. 2. Symmetry coordinate
10.989 0.737 6.471
12.501 0.750 5.672
11.657 0.365 5.195
7.500 0.604 5.650
7.498 0.624 5.652
7.544 0.336 5.591
constants for ring deformations
and CH rockings and waggings are in mdyn A rade2
G. BaranoviC and B. SchraderlJ.
Table 4 Valence coordinate No.
c1
ti ti Si Si
306 (1994)
165-175
169
scale factors
Type
c2
Mol. Struct. (Theochem)
Set Ia
Set II
Set III
AM1
AM1
MNDO
AM1
0.683 0.996 0.959
0.694 0.997 0.962 0.877 1.021
0.787 1.076 1.074 0.997 1.157 0.378 1.333 1.049
PM3
c3
q19q19
0.702 0.996 0.969
c4
q20aq20a
c3
c5
PiPi
c3
:998
0.769 0.873 1.107 0.983 0.957
c6
ti ti+l
0.483
CI Q
ti ti+2
c3
0.435 0.506
0.548 0.835
ti ti +3
c6
c3
0.440
0.489 1.072 0.469
0.532 6.033 5.483
0.511 5.135 3.123
1.819 0.007 124.4 1.072 1.417
b
c9
tiPi
-tiPi+l
(CICS)
112
(Cl %I
112
Cl0
tiPi+
=
-tiPi+
(CIC5)
l/2
(ClC5)
l/2
CII
tiPi+3
=
-tiPi+
(ClC5)
l/2
(ClC5)
l/2
Cl2
Ti ^Ir
Cl3
m3q2s
0.900 1.221
0.900 1.221
0.928 1.370
0.996 1.323
Cl4
q29aq29a
Cl3
Cl3
Cl3
Cl3
Cl5
“/i^li+l
Cl2
Cl2
1.924
2.225
c16
^li ^li +2
Cl2
Cl2
Cl2
Cl2
Cl2
Cl7
YiYi+3
Cl2
Cl2
Cl2
Cl2
Cl2
=
Cl8
71 q28
(c12c13)l’2
(c12c13)l’2
1.413
1.405
Cl9
Yl q29a
(c12~13)l’2
(c12c13)l’2
Cl8
Cl8
24 23
19 18
9 9
9 9
Average error (cm-l)C
Cl3
1.965
1.595 Cl8
6 11
“,Sets I and II are used in Ref. 4. c4 = c3, i.e. qlgqlg and qZOaqZOa are scaled by the same factor. ‘The top value refers to C,H6, and the bottom to C,D,.
anharmonic corrections are 14 and 117 cm-‘, respectively, which gives the value of 131 cm-’ to be subtracted from any calculated AM1 C-H stretching frequency of an aromatic hydrocarbon in order to obtain the observed anharmonic frequencies. For other normal modes the same pattern is generally reproduced by all three methods before scaling, except for the v3, v7, v9, v12 and v14 modes where the calculated frequencies are shifted in the opposite direction (Table 5). The AM1 Hamiltonian is slightly superior to the other two. Not only does it give a qualitatively correct picture of the b2u modes (as does the PM3 Hamiltonian, while
MNDO does not), but also the average AM 1 errors are the smallest. The largest frequency differences are obtained for the vl, 4 and vrs modes, and the only explanation is that the C-C/C-C and C-C/ C-H interactions are described unrealistically. The same is true for the out-of-plane interactions because the calculated v4 and u5 frequencies do not deviate in the same direction. This means that any correction of the out-of-plane field has to include independent scaling of some off-diagonal constants. All three parameter sets predict the ars force constant solution which obviously corresponds to the physical alg force constant set. However, the
G. BaranoviC and B. SchraderlJ. Mol. Struct. (Theochem) 306 (1994) 165-175
170
Table 5 Observed harmonic frequencies and calculated frequency differences (cm-‘) for benzene Symmetry
ak
Wilson No.
Unscaled
Scaled
MNDO
AM1
PM3
-15 -283
106 -236
-1 4
-7 4
-2 2
125
8
I
32
5 -8 A
-10 4 -6 3
-3 -2 1
-4 2
-6 2
-1
MNDO
AM1
PM3
1
3191 993
-233 -204
%
3
1350
-22
e&
I 8 9 6
3174 1601 1178 608
-233 -117 -14 -5
-14 -166 -44 -40
114 -184 30 -12
13 12
3174 1010
-229 52
-11 -18
119 34
14 15
1309 1150
145 -128
-58 -30
7 -5
-23 A
-21 4
-18
eh
20 19 18
3181 1484 1037
-235 -71 -110
-15 -95 -109
108 -63 -32
-12 14 -5
-11 12 -4
-5 -8 I
b 2s
5 4
990 701
-84 109
-22 89
-34 88
12 32
18 21
%!
10
847
-70
-44
-2
-3
-8
a2"
11
674
-97
-70
-39
17 18
961
-10
398
29
-22 21
-11 42
116
60
IO
blu
e2”
2
Obs. freq.a
Average error
21
-10
2 -1
-19 9
5
-6 -15 9
1
4
-9
17 2 -1 2 -12 -1 7
a Ref. 2 and references therein.
off-diagonal alg constant is strongly overestimated (Table 3). The calculated a2, constant is surprisingly close to the physical one, except for its PM3 value. The bt, constants are again rather close to the experimental values, excluding again the off-diagonal PM3 value, i.e. semiempirical calculations choose among multiple experimental solutions the same set as ab initio calculations [2]. The b2u frequencies as obtained with the PM3 set are the closest to the experimental ones. The relative phases between carbon and hydrogen movement are those predicted by ab initio calculations. Among the e2, frequencies (Table 5) the
largest deviation is found for the vs mode, showing that in all semiempirical fields the stretching/stretching and stretching/bending interactions are rather poorly described. With regard to the b2s and e2” force constants, as ab initio calculations, all semiempirical results favour the same of the two experimental solutions. The two onedimensional blocks, a2” and elgr together with the yiyi diagonal force constant give three different values for the same scale factor, thus emphasizing the necessity of independent scaling of interaction force constants. Besides frequencies there are a few more
G. Ba~a~ovi~ and B. SckraderjJ.
Mol. Struct.
(Tkeo~k~mj
306 (1994)
171
165-I75
Table 6 Observed isotopic frequency shifts for benzene C6D6 and differences between the observed and calculated shifts Symmetry
Wilson No.
Isotopic shift obsa
Unscaled MNDO
Scaled AM1
PM3
MNDO
AMI
PM3
1
-14
-3 4
2 1
829 47
-65 -4
46 -16
-13
-11
ak
3
291
-14
-3
19
-7
-6
-1
ezs
7 8 9 6
842 43 311 28
-68 30 -47 6
-8 33 -36 0
46 19 -17 3
-7 9 -18 7
-8 7 -14 5
7 -6 -5 1
b lu
13 12
830 40
-78 6
-6 2
31 -1
-16 4
-15 3
-2
bzu
14 15
23 326
-56 40
3 -7
21 -10
-36 29
-33 27
-40 26
el,
20 19 18
82.5 148 223
-78 59 -65
-13 78 -77
35 76 -62
-21 21 -13
-21 19 -11
-7 21 -15
bzs
5 4
161 108
30 -8
-46 -16
59 -25
11 -3
17 -8
17 -12
ek
10
187
-17
-10
-1
-2
-2
-1
at,
11
178
-27
-20
-12
-1
-0
-1
e2,
17 16
180 53
5 -4
8 -5
-2
ais
14 -5
23 -6
24 -3
9
8
1
1
a Ref. 2 and references therein.
experimentally observed vibrational properties that are readily predictable from this kind of calculation. Among them are isotopic frequency shifts (Table 6). The differences between the observed and calculated shifts are reasonably small with the largest disagreement in the e2s and et, blocks for all three parameter sets. The Coriolis constants are all predicted with the correct sign, although Cls and
from zero. Thus, no scaling can provide a better description of the si,Oi+r interaction. Semiempirical calculations of the CND0/2 type have been applied previously to benzene [5]. The number of the scaling factors was three for the inplane force field, and two for the out-of-plane force field with the average errors in the frequencies being 19cm-’ and 17cm-‘, respectively. These results are comparable to those obtained by semiempirical calculations of the MNDO type, but with a greater number of scaling factors. Only for the in-plane force fields were the best values of the scaling parameters determined from the same set of experimental data. The CNDO force field is thus more suitable for scaling.
172
G. BaranoviC and B. Schrader/J.
Mol. Strut.
(Theochem)
306 (1994) 165-175
Table 7 Experimental and calculated Coriolis constants for benzene Isotopomer and symmetry
C6H6
eh
Ex~.~
Gs C-19 t0
e%!
<6
-0.6392 -0.2861 -0.088 0.5858
4-9
C8
-0.5797
c7
C6D6
eh
4-18 Cl9
-0.4195 -0.4344
c20
%
<6 c9 (8 c7
Ozkabak and Goodmana
Unscaled
Scaled
MNDO
AM1
PM3
MNDO
AM1
PM3
-0.637 -0.283 -0.080 0.583 0.033 -0.555 -0.061
-0.441 -0.554 -0.005 0.602 0.049 -0.650 -0.000
-0.331 -0.664 -0.005 0.561 0.081 -0.644 0.002
-0.286 -0.684 -0.030 0.577 0.068 -0.617 -0.028
-0.604 -0.388 -0.008 0.638 -0.034 -0.599 -0.004
-0.611 -0.382 -0.007 0.596 -0.016 -0.577 -0.003
-0.579 -0.392 -0.029 0.627 -0.005 -0.614 -0.009
-0.369 -0.497 -0.137 0.356 0.296 -0.508 -0.144
-0.269 -0.685 -0.046 0.456 0.164 -0.586 -0.035
-0.229 -0.706 -0.065 0.368 0.237 -0.561 -0.044
-0.221 -0.653 -0.127 0.393 0.213 -0.463 -0.143
-0.362 -0.590 -0.048 0.495 0.132 -0.582 -0.044
-0.365 -0.589 -0.046 0.435 0.170 -0.566 -0.039
-0.365 -0.545 -0.090 0.440 0.219 -0.616 -0.043
a Ref. 2 and references therein. A necessary condition for a successful scaling procedure is an agreement in signs of the interaction force constants of the valence force field. With due attention paid to the definition of internal valence coordinates, the sign difference is definitely found in four cases: SiSi+s, pipi+,, pipi+, and yi~iyi+z. The symmetry coordinate diagonal force constants are overestimated by semiempirical calculations, although not all to the same
sign. These differences can be traced to the Sipi+, interaction constant the theoretical values of which are not in agreement with the experimental ones. Among the numerous scaling schemes that were tried, the following deserve a special attention. (i) As expected, the simplest scheme, where only the scaling factors for the diagonal constants were introduced (four in-plane and two out-of-plane), while the off-diagonal constants were scaled by
extent. As a rule, the bending constants deviate much less than the stretching ones. There is no common trend when the three parametric sets are
the factor (cicj)1’2 , did not work well. For the MNDO and AM1 parameterization sets the disagreement for almost all normal vibrations
compared with the experimental one. For example, the empirical bzu CC stretching constant F14+, is slightly greater than the MNDO one, but lo-20% smaller relative to the other two. Neither of the three semiempirical fields was expected to be in good agreement with the benchmark force field or to be as good as an ab initio field. It is, however, particularly noteworthy that the large difference in the el, off-diagonal constant F,s,*s is common to the high-level ab initio and semiempirical fields. Another order-of-magnitude difference is also found for the constant F7,9 but, unlike the ab initio result, there is an agreement in
even for the out-of-plane modes, especially for v5, u4,vll and ~17, in contrast to the ab initio results of Pulay et al. [4] where no additional scaling was necessary. (ii) The sets I and II, as defined in Ref. 4 where additional scaling factors for independent scaling of some in-plane off-diagonal constants have been introduced, improved both the qualitative and quantitative agreement. With the set II factors only the differences for v3, v15 and v19 were substantially above 20 cm-‘. Independent scaling of some off-diagonal constants was considered
remained
high, i.e. above
20 cm-‘.
This was true
G. BaranoviC and B. SchraderjJ.
Mol. Struct.
(Theochem)
306 (1994)
necessary in order to take into account electron correlation effects. However, unlike the ab initio force field, the semiempirical force field needs further corrections. (iii) The final adopted scheme which gives the absolute values of AZ+, AZ+ and AU,,, Av4 and Av, for both isotopomers near to 20 cm-‘, contains eleven in-plane and four out-of-plane scaling parameters (Table 4). However, even this scheme does not produce a force field which might be given a significance due to its failure to reproduce correctly the relative magnitudes and signs of the Coriolis constants (Table 7). This fact that the same scheme cannot improve both an ab initio and a semiempirical force field is the consequence of the least-squares parameterization already implemented in the semiempirical Hamiltonians. A semiempirical potential energy surface seems to be deformed relative to an ab initio one in such a way that any prediction of molecular behaviour will contain rather nonsystematic errors. For example, ordering of the normal modes only according to their frequencies is very likely to establish an erroneous correspondence with the observed frequencies. This is even more true if the isotopic shifts or frequencies of an isotopomer are to be used in an explanation of the vibrational dynamics of a molecule.
165-175
173
empirical parameters. It is not clear whether a geometry of optimization of the ground state with all doubly excited configurations included would lead to any improvement in the force field. Moreover, it has been demonstrated that even the MP2 force constants do not represent an essential improvement over the HF ones [2]. Semiempirical force fields are not all of the same quality. The AM1 parameterization is in this case an improvement over MNDO, while PM3 is certainly no better than the other two. Recently, the capability of the three sets in predicting the rotational barrier of biphenyl has been tested [9], and the same conclusion has been reached. Therefore, AM 1 is recommended for further use, at least for compounds derived from benzene. Although from the beginning benzene was not expected to be suitable for simple scaling, with regard to the importance of its derivatives the complex scaling scheme proposed here is worthy of undertaking. Acknowledgement
One of us (G.B.) thanks the Heinrich Hertz Stiftung of the Ministerium fur Wissenschaft und Forschung des Landes Nordrhein-Westfalen for a scholarship. References
Conclusion
The vibrational potential surface of ground-state benzene was examined using the MNDO, AM1 and PM3 semiempirical parameterization sets. All three sets of results compare well with the ab initio results obtained with medium-sized basis sets and at the HF level, although in some cases they give qualitatively the same predictions as the most advanced ab initio calculations. However, they are not suitable for scaling because the errors are not systematic. Referring to the frequency and isotopic shift differences, electron correlation is taken into account, but in a rather erratic and unpredictable way. Only part of the electron correlation is contained in the values of the semi-
N. Rougeau, J.P. Flament, P. Youkharibache, H.P. Gervais and G. Berthier, J. Mol. Struct. (Theochem), 254 (1992) 405. L. Goodman, A.G. Ozkabak and S.N. Thakur, J. Phys. Chem., 95 (1991) 9044. J.J.P. Stewart, J. Comput. Aided Mol. Design, 4 (1990) 1. P. Pulay, G. Fogarasi and J.E. Boggs, J. Chem. Phys., 74 (1981) 3999. (a) F. Toriik, A. Hegedtis and P. Pulay, J. Mol. Struct., 32 (1976) 93. (b) G. Fogarasi and P. Pulay, J. Mol. Struct., 39 (1977) 275. (a) A. Langseth and B.P. Stoicheff, Can. J. Phys., 34 (1956) 350. (b) B.P. Stoicheff, Can. J. Phys., 32 (1954) 339. J. Pliva, J.W.C. Jones and L. Goodman, J. Mol. Spectrosc., 148 (1991) 427. P. Pulay, G. Fogarasi, G. Pongor, J.E. Boggs and A. Varga, J. Am. Chem. Sot., 105 (1983) 7037. C. Dehu, F. Meyers and J.L. Bredas, J. Am. Chem. Sot., 115 (1993) 6198.
G. BaranoviC and B. Schrader/J. Mol. Struct. (Theochem) 306 (1994) 165-175
174
Appendix: Supplementary material Table A 1 Internal coordinates No.
for benzenea Description
Coordinate
l-6 7-12 13-18 19 20 21 22-21 28 29 30
PI
=(l/JU(P1
a9
=
-d),...,Psb
(1/&)(a1
-
(~2 +
q20a= (l/J12)(2cq @Ob
=
CC stretch. CH stretch. CH ip bend. CCC bend. bl, CCC bend. e2s CCC bend. e2s CH op bend. CC tors. b2s CC tors. ezu CC tors. ezu
(1/2)(a2
a3
-
(~4 +
as
-
a6)
- ff2 - cx3+ 2cx4- (Y~- ck6)
-
a3
+
a5
-
6,
+
63 -
(y6)
71?...>^16 q28
=
(l/&)(61
-
429a
=
(1/2)(-6l
+
q29b
=
(l/,/12)(-61
‘53 +
64 +
s4 + 262
‘55 -
‘56)
64 +
265
66)
-
63 -
-
66)
a See Table 3 of Ref. 4. b Note the sign difference relative to Ref. 2, and agreement with Ref. 4.
Table A2 Symmetry coordinates for benzene redundant internal coordinates No.
Symm.
1 2
alg
3
as %
in terms
of nonTable A2 (continued) No.
Symm.
Coordinateb
t6
22
b2s
-/I -
s6
23
Coordinateb
t, + t2 + t3 + 14 + t5 + sl
PI
+
s2 +
+
s3 +
/?2 +
p3
s4 +
+
p4
s5 +
+
P5 +
P +
p6
t] - 2t2 + tj + t4 - 2t, + t6 2sl -
s2 -
s3 +
2s4
p5
p6
-
s5 -
14 15 16 17 18 19 20 21
P3
+
-
-q20b --z/3,
b lu
sl
-
+
@2 +
s2 +
&
s3 -
-
2@4 +
s4 +
s5 -
,&
+
s6
tj - t2 + tj - t4 + t5 - t6 PI
elu
p2
+
p3
-
@4 +
t, - t3 - t4 + t6 231f s2- s3 - 2x4 -p2
eh
-
-
P3 +
P5
+
p5
-
-
p6
s5 f
s6
p6
tl + 2t, + t3 - t4 - 21, - t6 ~72 + s3 - s5 - s6 2p1
+
p2
-
&
-
2p4
-
Ps
‘74 +
elg
2%
25
ek
72 +
73 -
75 -
76
26
a2”
71 +
72 +
73 +
74 +
e2”
2%
P6
29 30 b
419
b2u
-
24
21 28
t] - t3 + t4 - t6 .9 - s3 + s5 - s6
8 9 10 11 12 13
-
73
75
-
‘76
+
“12 -
73
-
274
-
75 +
76
s6
-420a p2
72 +
q28
+
p6
-
72 -
429a
e2u
Not normalized.
-h+73-75+76 q29b
73 +
274
75 +
-
76
7’5 -
76
G. BaranoviC and B. Schrader/J.
Table A3 Harmonic force field constants
Mol. Struct.
(Theochem)
306 (1994)
165-175
of benzene in terms of internal valence coordinates Unscaled
No.
Type
fl
ti ti
f2
si si
f3
419419 q20aq20a
;
PiPi
f6 f7 f8
Ex~.~
MNDO
Scaled AM1
PM3
MNDO
AM1
PM3
6.616 5.547 1.284 1.257 0.523
8.613 6.453 1.163 1.236 0.547
9.601 5.654 1.317 1.348 0.522
9.190 5.165 1.178 1.226 0.455
6.624 5.634 1.288 1.216 0.523
6.660 5.635 1.267 1.183 0.533
7.233 5.559 1.265 1.222 0.527
0.128 -0.419 0.383
1.145 -0.602 0.749
1.523 -0.483 0.819
1.441 -0.601 0.787
0.776 -0.503 0.392
0.744 -0.518 0.384
0.545 -0.802 0.826
0.007 0.008 -0.022
0.006 0.002 0.001
0.006 0.002 0.002
0.014 0.001 0.001
0.006 0.002 0.001
0.006 0.002 0.002
0.015 0.001 0.001
-0.007 -0.004 0.000
-0.006 -0.006 0.000
-0.014 -0.003 0.000
-0.006 -0.004 0.000
-0.006 -0.006 0.000
-0.016 -0.004 0.000
f9
sisi+l
fl0
sisi+2
fll
sisi+3
fl2
PiPi+l
f13
&Pi+2
h4
Pi&+3
0.008 -0.012 -0.001
f17
tiSi = tiSi+l tisi+2 = tisi+5 tisi+3 = tisi+4
0.200 0.008 -0.130
0.413 0.011 -0.056
0.425 0.005 -0.055
0.211 0.002 -0.030
0.338 0.009 -0.046
0.353 0.004 -0.046
0.192 0.002 -0.028
fl8
t142oa
0.124
0.118
0.103
0.129
0.178
0.137
0.114
fl5 f16
-0.135 -0.110
-0.153 -0.192
-0.147 -0.183
-0.073 -0.094
-0.151 -0.178
-0.144 -0.171
-0.079 -0.098
-0.153 0.015 -0.023
-0.269 0.004 -0.004
-0.281 0.006 -0.006
-0.201 -0.001 -0.002
-0.143 0.027 -0.022
-0.144 0.029 -0.018
-0.366 0.000 -0.247
f25
0.048 0.019
0.000 0.003
0.000 0.003
0.008 0.001
0.000 0.003
0.000 0.003
0.009 -0.001
f26
0.075
0.018
0.037
0.026
0.030
0.035
0.028
0.447 0.394 0.312
0.481 0.259 0.245
0.454 0.274 0.246
0.424 0.269 0.225
0.447 0.355 0.335
0.452 0.363 0.326
0.454 0.381 0.320
fl9 f20
f21 f22 f23
f24
f27
Ti ^li
f28
q28q28
f29
429aq29a
^ii^li+3
-0.069 0.002 -0.021
-0.033 -0.006 -0.016
-0.028 -0.006 -0.018
-0.034 -0.005 -0.010
-0.063 -0.006 -0.015
-0.063 -0.006 -0.018
-0.067 -0.006 -0.011
‘-!I 428 YI429a
-0.154 0.147
-0.098 0.105
-0.101 0.104
-0.092 0.097
-0.138 0.148
-0.141 0.145
-0.146 0.155
f30
YiYi+l
f31
“li^li+Z
f32
f33 f34
a Ref. 2.
165
Semiempirical scaled valence force fields for benzene Goran BaranoviC*, Bernhard Schrader Institut fir
Physikalische
und Theoretische
Chemie, Universittit GH Essen, 45117 Essen, Germany
(Received 23 September 1993; accepted 1 October 1993) Abstract The performances vibrational properties tested. A set of the agreement to within substituted benzenes
of three semiempirical parameterization sets (MNDO, AMI and PM3) in describing the (harmonic frequencies, isotopic frequency shifts and Coriolis constants) of benzene were best (in the least-squares sense) scaling factors for the force constants, which improves 20 cm-‘, was designed with the idea of it being used for the more complex cases of monoand polycyclic aromatic hydrocarbons.
Introduction
The idea of transferability of empirical scaling factors from benzene to, for example, monosubstituted benzene or even polycyclic hydrocarbons [l] within semiempirical quantum chemistry is particularly appealing because it would make the vibrational analysis of these large molecules, which are unsuitable even for ab initio geometry optimization, rather straightforward. Scaling factors have to be introduced in order to compensate for errors in predicting the vibrational frequencies. This is necessary especially if one wants to use semiempirical data in solving a specific spectroscopic problem, as without additional corrections these results are not of great practical value. However, similarly as in an attempt to find the best values for an empirical force field, to obtain the best values of scaling factors one needs fully and unambiguously assigned vibrational spectra. The purpose of this paper is to assess to what extent semiempirical quantum chemical vibrational analysis provides a correct description of the vibrational properties of benzene and to calculate the scaling factors to be * Corresponding author - on leave from R. BoSkovi6 Institute, 41001 Zagreb, Croatia. SSDI 0166-1280(93)03605-7
used in the assignment of an unknown vibrational spectrum of a certain benzene derivative. This is possible because the benchmark harmonic force field for benzene has been reported [2]. The harmonic force field of benzene was computed by using MNDO, AM1 and PM3 Hamiltonians with version 6.0 of the MOPAC program package [3]. The results are compared with the values obtained by ab initio [2,4] and semiempirical CND0/2 [5] calculations and experiment. Force constants were scaled using the method described in Ref. 4, i.e. a multiplication factor ci is applied to the force constant Ai and (eiCi)*” to the interaction constant J7, unless the latter is scaled independently. As will be shown, a separate scaling parameter should be introduced for almost all force constants, in order to obtain good reproducibility of vibrational frequencies. This is not unexpected, knowing all the approximations involved in the semiempirical methods. Therefore, a reproducibility will be considered good or satisfactory if no absolute frequency difference is larger than about 20 cm-‘. Although this is an order of magnitude weaker criterion than the one proposed by Goodman et al. [2], we believe that with an estimation within 2Ocm-’ it is still possible to reach reliable interpretation of a vibrational
G. BaranoviC and B. Schrader/J. Mol. Struct. (Theochem) 306 (1994) 165-175
166
spectrum. Our aim was to achieve this while keeping the number of scaling parameters as low as possible. The best values in the least-squares sense were obtained iteratively by means of the Levenberg-Marquardt algorithm from the observed harmonic frequencies of benzene-h6 and benzene-d6.
The internal coordinates used are the same as those used in Ref. 4 (see Table Al in the Appendix). The in-plane symmetry coordinates used are those reported in Ref. 2, while the outof-plane coordinates are identical to those reported by Pulay et al. [4] (see Table A2 in the Appendix). The transformation of a force field from Cartesian to internal coordinates was performed according to the method described previously [8]. Minor differences between our coordinates and those taken from Ref. 2 were taken into account when the harmonic unscaled force fields for benzene in terms of internal symmetrized coordinates were compared with the benchmark force field given in Table 2 (see Table A3 in the Appendix). The unscaled and scaled values in terms of symmetry coordinates are given in Table 3. The corresponding scaling factors for different scaling schemes are given in Table 4. The quality of the semiempirical force fields was considered with respect to the frequency differences (Table 5), isotopic frequency shifts for C6D6 (Table 6) and Coriolis constants (Table 7). The observed frequencies are corrected for anharmonicity [2]. The corrections are important for the C-H stretching vibrations and are introduced here in order to minimize the influence of fitting to the anharmonic data. While MNDO overestimates, and PM3 underestimates the C-H/ C-D stretching frequencies, AM1 values are rather close to the harmonic observed frequencies. The mean values of the AM1 and measured
Results and discussion The theoretical geometries of benzene (Table 1) are used in calculating the force fields, not only because of consistency, but also because the C-C bond lengths obtained with AM1 and PM3 are rather close to the experimental values. The C-H bond length is overestimated by 1 %, and as a result the C-H bending frequencies will be additionally downshifted. The use of calculated and not experimental geometries may thus be a source of error. Therefore the frequencies were also calculated for the experimental geometry [6], which was also used by Goodman et al. [2]. However, not only was the overall agreement in frequencies slightly worse in all three cases, but also the influence of differences in geometry was found to be negligible when compared with the already existing frequency differences obtained by semiempirical methods. The semiempirical geometries are close enough to each other such that the main reasons for the differences among calculated frequencies are to be found in the force fields. Table 1 Experimental
and theoretical geometries (A) of benzene
Bond
Exp.
MNDO
AM1
PM3
4-21GC
6-31G*d
kc
1.397a 1.3920b
1.4066
1.3950
1.3911
1.3845
1.386
rCH
1.084a 1.0862b
1.0903
1.0996
1.0948
1.0721
1.076
a r0 structure from Ref. 6. b r,,, structure from Ref. 7. ’ Ref. 4. d Ref. 2.
G. BaranoviC and B. SchraderlJ.
Mol. Struct.
Table 2 Unscaled harmonic force field constants No.
(Theochem)
ti ti
f2
Si Si
A
919919
2
PiPi 920a920a
f6
ti t* + I
f7
ti ti+2
fs
ti h+3
Ex~.~
0.728 -0.419 0.383
1.145 -0.602 0.749
1.523 -0.483 0.819
1.441 -0.601 0.787
0.006 0.002 0.001
0.006 0.002 0.002
0.014 0.001 0.001
fll
sisi+3
0.007 0.008 -0.022
k
Pi P*Pit2 Pi+1
-0.012 0.008
f14
&Pi+3
-0.001
=
=
tjSi+l tiSi+S
fl7
tiS,+3
=
tiSi+4
fl8
t1920a
fi9
Sl919
f20
s142oa
f2l
liPi =
-tiPi+
f22
tiPi+2
=
-riPi+S
f23
liPi+
=
-tiPi+
PM3 9.190 5.165 1.178 0.455 1.226
sisi+2
tiSi
AM1 9.601 5.654 1.317 0.522 1.348
sisi+l
tiSj+Z
MNDO 8.613 6.453 1.163 0.547 1.236
f9
f16
167
6.616 5.547 1.284 0.523 1.257
flo
flS
165-175
of benzene in terms of internal valence coordinates
Type
fi
306 (1994)
-0.004 -0.007
-0.006
-0.003 -0.014
0.000
0.000
0.000
0.200 0.008 -0.130
0.413 0.011 -0.056
0.425 0.005 -0.055
0.211 0.002 -0.030
0.124
0.118
0.103
0.129
-0.135 -0.110
-0.153 -0.192
-0.147 -0.183
-0.073 -0.094
-0.153 0.015 -0.023
-0.269 0.004 -0.004
-0.281 0.006 -0.006
-0.201 -0.001 -0.002
0.048 0.019
0.000 0.003
0.000 0.003
0.008 0.001
f24
siPi+l
=
-siPi+
f25
siPi+
=
-SiPi+
f26
420a P2
0.075
0.018
0.037
0.026
f27
“/i ?;
0.447
0.48 1
0.454
0.424
f28
928928
f29
929a429a
0.394 0.312
0.259 0.245
0.274 0.246
0.269 0.225
f30
“ii%+1
f31
^/iYi+2
f32
Ti^li+3
f33
71928
f34
Y1929a
a Ref. 2. The coordinate pi differs by a factor of -J2 coordinate. Equivalencies among some out-of-plane
-0.069 0.002 -0.021
-0.033 -0.006 -0.016
-0.028 -0.006 -0.018
-0.034 -0.005 -0.010
-0.154 0.147
-0.098 0.105
-0.101 0.104
-0.092 0.097
from that in Ref. 2. There is also a sign difference in the definition of the yi potential constants: ylq29a = -2y2q29a = -2y3q2ga = y4q29a = -2y5q2ga = -2?6Y6q29a = -t2/d3h2q29b = +c2/d3h3q29b = +/t/3)%q29b = +@/d%6429b, Yjq29b = ,,‘4q29b = 0. Stretching bond CODstants are in units of mdyn A-‘, while mdyn A rad-2 are used for angle bendings, waggings and torsions.
G. BaranoviC and B. SchraderlJ. Mol. Struct. (Theochem) 306 (1994) 165-175
168
Table 3 Benzene harmonic force fields in terms of symmetry coordinates
Type
Exp.’
Unscaled MNDO
MNDO
AM1
PM3
F2,2
P
F3,3
0.515
0.525
0.500
0.421
0.503
0.510
0.487
1.284 -0.331 5.571
1.163 -0.376 6.443
1.317 -0.359 5.645
1.178 -0.179 5.138
1.288 -0.369 5.626
1.267 -0.352 5.626
1.265 -0.192 5.530
3.939 -0.228 0.486
3.830 -0.521 0.552
4.768 -0.539 0.523
4.318 -0.400 0.476
3.737 -0.189 0.528
3.751 -0.191 0.534
3.713 -0.240 0.551
1.257 -0.190 0.430 0.150 5.510 0.054 0.050 6.690 0.305 0.526
1.236 -0.333 0.409 0.062 6.445 0.335 -0.004 8.548 0.473 0.557
1.348 -0.316 0.610 0.075 5.648 0.361 -0.005 9.379 0.498 0.534
1.126 -0.163 0.446 0.051 5.151 0.177 0.012 9.137 0.352 0.472
1.216 -0.308 0.355 0.060 5.628 0.274 -0.004 6.680 0.286 0.533
1.182 -0.296 0.475 0.071 5.629 0.300 -0.005 6.818 0.281 0.545
1.222 -0.169 0.395 0.055 5.544 0.162 0.013 8.315 1.061 0.547
0.544 -0.160 -0.115 7.380 0.572 5.568
0.544 -0.274 -0.006 9.881 0.813 6.456
0.522 -0.287 -0.005 10.788 0.831 5.657
0.444 -0.198 -0.015 10.445 0.418 5.177
0.520 -0.174 -0.005 7.574 0.666 5.637
0.533 -0.184 -0.005 7.538 0.691 5.637
0.514 -0.118 -0.017 7.754 0.385 5.571
0.292
0.387
0.367
0.334
0.294
0.296
0.297
0.394 -0.377 0.610
0.259 -0.240 0.550
0.274 -0.246 0.516
0.268 -0.225 0.269
0.354 -0.339 0.577
0.363 -0.346 0.583
0.381 -0.359 0.589
0.396
0.470
0.450
0.405
0.403
0.413
0.404
0.312 0.254 0.493
0.244 0.182 0.504
0.246 0.179 0.470
0.226 0.168 0.453
0.335 0.257 0.501
0.325 0.252 0.503
0.319 0.268 0.516
blu Q
F12,12 F12,13
s
bzut
F13,13
F14,14 F14,15
P e2g0
F15,15
F6,6 F6,l F6,8 F6,9
s
Fl,7 F7,8 F7,9
t
F8,8 F8,9
P
F9,9
P
F18, 18 Fl8,19
t
F18,20 FI,,
19
F19, 20
a2”
PM3
s
FL, F1,2
eh
AM1
7.616 0.157 5.554
alg t
a2g
Scaled
s
F20,20
7
Fll,ll
bg 6
F4,4 F4,5
Y
F5,5
elg Y
F IO,10
e2u6
F16,
16
Fl6,17
Y
F 17,17
a Ref. 2. Symmetry coordinate
10.989 0.737 6.471
12.501 0.750 5.672
11.657 0.365 5.195
7.500 0.604 5.650
7.498 0.624 5.652
7.544 0.336 5.591
constants for ring deformations
and CH rockings and waggings are in mdyn A rade2
G. BaranoviC and B. SchraderlJ.
Table 4 Valence coordinate No.
c1
ti ti Si Si
306 (1994)
165-175
169
scale factors
Type
c2
Mol. Struct. (Theochem)
Set Ia
Set II
Set III
AM1
AM1
MNDO
AM1
0.683 0.996 0.959
0.694 0.997 0.962 0.877 1.021
0.787 1.076 1.074 0.997 1.157 0.378 1.333 1.049
PM3
c3
q19q19
0.702 0.996 0.969
c4
q20aq20a
c3
c5
PiPi
c3
:998
0.769 0.873 1.107 0.983 0.957
c6
ti ti+l
0.483
CI Q
ti ti+2
c3
0.435 0.506
0.548 0.835
ti ti +3
c6
c3
0.440
0.489 1.072 0.469
0.532 6.033 5.483
0.511 5.135 3.123
1.819 0.007 124.4 1.072 1.417
b
c9
tiPi
-tiPi+l
(CICS)
112
(Cl %I
112
Cl0
tiPi+
=
-tiPi+
(CIC5)
l/2
(ClC5)
l/2
CII
tiPi+3
=
-tiPi+
(ClC5)
l/2
(ClC5)
l/2
Cl2
Ti ^Ir
Cl3
m3q2s
0.900 1.221
0.900 1.221
0.928 1.370
0.996 1.323
Cl4
q29aq29a
Cl3
Cl3
Cl3
Cl3
Cl5
“/i^li+l
Cl2
Cl2
1.924
2.225
c16
^li ^li +2
Cl2
Cl2
Cl2
Cl2
Cl2
Cl7
YiYi+3
Cl2
Cl2
Cl2
Cl2
Cl2
=
Cl8
71 q28
(c12c13)l’2
(c12c13)l’2
1.413
1.405
Cl9
Yl q29a
(c12~13)l’2
(c12c13)l’2
Cl8
Cl8
24 23
19 18
9 9
9 9
Average error (cm-l)C
Cl3
1.965
1.595 Cl8
6 11
“,Sets I and II are used in Ref. 4. c4 = c3, i.e. qlgqlg and qZOaqZOa are scaled by the same factor. ‘The top value refers to C,H6, and the bottom to C,D,.
anharmonic corrections are 14 and 117 cm-‘, respectively, which gives the value of 131 cm-’ to be subtracted from any calculated AM1 C-H stretching frequency of an aromatic hydrocarbon in order to obtain the observed anharmonic frequencies. For other normal modes the same pattern is generally reproduced by all three methods before scaling, except for the v3, v7, v9, v12 and v14 modes where the calculated frequencies are shifted in the opposite direction (Table 5). The AM1 Hamiltonian is slightly superior to the other two. Not only does it give a qualitatively correct picture of the b2u modes (as does the PM3 Hamiltonian, while
MNDO does not), but also the average AM 1 errors are the smallest. The largest frequency differences are obtained for the vl, 4 and vrs modes, and the only explanation is that the C-C/C-C and C-C/ C-H interactions are described unrealistically. The same is true for the out-of-plane interactions because the calculated v4 and u5 frequencies do not deviate in the same direction. This means that any correction of the out-of-plane field has to include independent scaling of some off-diagonal constants. All three parameter sets predict the ars force constant solution which obviously corresponds to the physical alg force constant set. However, the
G. BaranoviC and B. SchraderlJ. Mol. Struct. (Theochem) 306 (1994) 165-175
170
Table 5 Observed harmonic frequencies and calculated frequency differences (cm-‘) for benzene Symmetry
ak
Wilson No.
Unscaled
Scaled
MNDO
AM1
PM3
-15 -283
106 -236
-1 4
-7 4
-2 2
125
8
I
32
5 -8 A
-10 4 -6 3
-3 -2 1
-4 2
-6 2
-1
MNDO
AM1
PM3
1
3191 993
-233 -204
%
3
1350
-22
e&
I 8 9 6
3174 1601 1178 608
-233 -117 -14 -5
-14 -166 -44 -40
114 -184 30 -12
13 12
3174 1010
-229 52
-11 -18
119 34
14 15
1309 1150
145 -128
-58 -30
7 -5
-23 A
-21 4
-18
eh
20 19 18
3181 1484 1037
-235 -71 -110
-15 -95 -109
108 -63 -32
-12 14 -5
-11 12 -4
-5 -8 I
b 2s
5 4
990 701
-84 109
-22 89
-34 88
12 32
18 21
%!
10
847
-70
-44
-2
-3
-8
a2"
11
674
-97
-70
-39
17 18
961
-10
398
29
-22 21
-11 42
116
60
IO
blu
e2”
2
Obs. freq.a
Average error
21
-10
2 -1
-19 9
5
-6 -15 9
1
4
-9
17 2 -1 2 -12 -1 7
a Ref. 2 and references therein.
off-diagonal alg constant is strongly overestimated (Table 3). The calculated a2, constant is surprisingly close to the physical one, except for its PM3 value. The bt, constants are again rather close to the experimental values, excluding again the off-diagonal PM3 value, i.e. semiempirical calculations choose among multiple experimental solutions the same set as ab initio calculations [2]. The b2u frequencies as obtained with the PM3 set are the closest to the experimental ones. The relative phases between carbon and hydrogen movement are those predicted by ab initio calculations. Among the e2, frequencies (Table 5) the
largest deviation is found for the vs mode, showing that in all semiempirical fields the stretching/stretching and stretching/bending interactions are rather poorly described. With regard to the b2s and e2” force constants, as ab initio calculations, all semiempirical results favour the same of the two experimental solutions. The two onedimensional blocks, a2” and elgr together with the yiyi diagonal force constant give three different values for the same scale factor, thus emphasizing the necessity of independent scaling of interaction force constants. Besides frequencies there are a few more
G. Ba~a~ovi~ and B. SckraderjJ.
Mol. Struct.
(Tkeo~k~mj
306 (1994)
171
165-I75
Table 6 Observed isotopic frequency shifts for benzene C6D6 and differences between the observed and calculated shifts Symmetry
Wilson No.
Isotopic shift obsa
Unscaled MNDO
Scaled AM1
PM3
MNDO
AMI
PM3
1
-14
-3 4
2 1
829 47
-65 -4
46 -16
-13
-11
ak
3
291
-14
-3
19
-7
-6
-1
ezs
7 8 9 6
842 43 311 28
-68 30 -47 6
-8 33 -36 0
46 19 -17 3
-7 9 -18 7
-8 7 -14 5
7 -6 -5 1
b lu
13 12
830 40
-78 6
-6 2
31 -1
-16 4
-15 3
-2
bzu
14 15
23 326
-56 40
3 -7
21 -10
-36 29
-33 27
-40 26
el,
20 19 18
82.5 148 223
-78 59 -65
-13 78 -77
35 76 -62
-21 21 -13
-21 19 -11
-7 21 -15
bzs
5 4
161 108
30 -8
-46 -16
59 -25
11 -3
17 -8
17 -12
ek
10
187
-17
-10
-1
-2
-2
-1
at,
11
178
-27
-20
-12
-1
-0
-1
e2,
17 16
180 53
5 -4
8 -5
-2
ais
14 -5
23 -6
24 -3
9
8
1
1
a Ref. 2 and references therein.
experimentally observed vibrational properties that are readily predictable from this kind of calculation. Among them are isotopic frequency shifts (Table 6). The differences between the observed and calculated shifts are reasonably small with the largest disagreement in the e2s and et, blocks for all three parameter sets. The Coriolis constants are all predicted with the correct sign, although Cls and
from zero. Thus, no scaling can provide a better description of the si,Oi+r interaction. Semiempirical calculations of the CND0/2 type have been applied previously to benzene [5]. The number of the scaling factors was three for the inplane force field, and two for the out-of-plane force field with the average errors in the frequencies being 19cm-’ and 17cm-‘, respectively. These results are comparable to those obtained by semiempirical calculations of the MNDO type, but with a greater number of scaling factors. Only for the in-plane force fields were the best values of the scaling parameters determined from the same set of experimental data. The CNDO force field is thus more suitable for scaling.
172
G. BaranoviC and B. Schrader/J.
Mol. Strut.
(Theochem)
306 (1994) 165-175
Table 7 Experimental and calculated Coriolis constants for benzene Isotopomer and symmetry
C6H6
eh
Ex~.~
Gs C-19 t0
e%!
<6
-0.6392 -0.2861 -0.088 0.5858
4-9
C8
-0.5797
c7
C6D6
eh
4-18 Cl9
-0.4195 -0.4344
c20
%
<6 c9 (8 c7
Ozkabak and Goodmana
Unscaled
Scaled
MNDO
AM1
PM3
MNDO
AM1
PM3
-0.637 -0.283 -0.080 0.583 0.033 -0.555 -0.061
-0.441 -0.554 -0.005 0.602 0.049 -0.650 -0.000
-0.331 -0.664 -0.005 0.561 0.081 -0.644 0.002
-0.286 -0.684 -0.030 0.577 0.068 -0.617 -0.028
-0.604 -0.388 -0.008 0.638 -0.034 -0.599 -0.004
-0.611 -0.382 -0.007 0.596 -0.016 -0.577 -0.003
-0.579 -0.392 -0.029 0.627 -0.005 -0.614 -0.009
-0.369 -0.497 -0.137 0.356 0.296 -0.508 -0.144
-0.269 -0.685 -0.046 0.456 0.164 -0.586 -0.035
-0.229 -0.706 -0.065 0.368 0.237 -0.561 -0.044
-0.221 -0.653 -0.127 0.393 0.213 -0.463 -0.143
-0.362 -0.590 -0.048 0.495 0.132 -0.582 -0.044
-0.365 -0.589 -0.046 0.435 0.170 -0.566 -0.039
-0.365 -0.545 -0.090 0.440 0.219 -0.616 -0.043
a Ref. 2 and references therein. A necessary condition for a successful scaling procedure is an agreement in signs of the interaction force constants of the valence force field. With due attention paid to the definition of internal valence coordinates, the sign difference is definitely found in four cases: SiSi+s, pipi+,, pipi+, and yi~iyi+z. The symmetry coordinate diagonal force constants are overestimated by semiempirical calculations, although not all to the same
sign. These differences can be traced to the Sipi+, interaction constant the theoretical values of which are not in agreement with the experimental ones. Among the numerous scaling schemes that were tried, the following deserve a special attention. (i) As expected, the simplest scheme, where only the scaling factors for the diagonal constants were introduced (four in-plane and two out-of-plane), while the off-diagonal constants were scaled by
extent. As a rule, the bending constants deviate much less than the stretching ones. There is no common trend when the three parametric sets are
the factor (cicj)1’2 , did not work well. For the MNDO and AM1 parameterization sets the disagreement for almost all normal vibrations
compared with the experimental one. For example, the empirical bzu CC stretching constant F14+, is slightly greater than the MNDO one, but lo-20% smaller relative to the other two. Neither of the three semiempirical fields was expected to be in good agreement with the benchmark force field or to be as good as an ab initio field. It is, however, particularly noteworthy that the large difference in the el, off-diagonal constant F,s,*s is common to the high-level ab initio and semiempirical fields. Another order-of-magnitude difference is also found for the constant F7,9 but, unlike the ab initio result, there is an agreement in
even for the out-of-plane modes, especially for v5, u4,vll and ~17, in contrast to the ab initio results of Pulay et al. [4] where no additional scaling was necessary. (ii) The sets I and II, as defined in Ref. 4 where additional scaling factors for independent scaling of some in-plane off-diagonal constants have been introduced, improved both the qualitative and quantitative agreement. With the set II factors only the differences for v3, v15 and v19 were substantially above 20 cm-‘. Independent scaling of some off-diagonal constants was considered
remained
high, i.e. above
20 cm-‘.
This was true
G. BaranoviC and B. SchraderjJ.
Mol. Struct.
(Theochem)
306 (1994)
necessary in order to take into account electron correlation effects. However, unlike the ab initio force field, the semiempirical force field needs further corrections. (iii) The final adopted scheme which gives the absolute values of AZ+, AZ+ and AU,,, Av4 and Av, for both isotopomers near to 20 cm-‘, contains eleven in-plane and four out-of-plane scaling parameters (Table 4). However, even this scheme does not produce a force field which might be given a significance due to its failure to reproduce correctly the relative magnitudes and signs of the Coriolis constants (Table 7). This fact that the same scheme cannot improve both an ab initio and a semiempirical force field is the consequence of the least-squares parameterization already implemented in the semiempirical Hamiltonians. A semiempirical potential energy surface seems to be deformed relative to an ab initio one in such a way that any prediction of molecular behaviour will contain rather nonsystematic errors. For example, ordering of the normal modes only according to their frequencies is very likely to establish an erroneous correspondence with the observed frequencies. This is even more true if the isotopic shifts or frequencies of an isotopomer are to be used in an explanation of the vibrational dynamics of a molecule.
165-175
173
empirical parameters. It is not clear whether a geometry of optimization of the ground state with all doubly excited configurations included would lead to any improvement in the force field. Moreover, it has been demonstrated that even the MP2 force constants do not represent an essential improvement over the HF ones [2]. Semiempirical force fields are not all of the same quality. The AM1 parameterization is in this case an improvement over MNDO, while PM3 is certainly no better than the other two. Recently, the capability of the three sets in predicting the rotational barrier of biphenyl has been tested [9], and the same conclusion has been reached. Therefore, AM 1 is recommended for further use, at least for compounds derived from benzene. Although from the beginning benzene was not expected to be suitable for simple scaling, with regard to the importance of its derivatives the complex scaling scheme proposed here is worthy of undertaking. Acknowledgement
One of us (G.B.) thanks the Heinrich Hertz Stiftung of the Ministerium fur Wissenschaft und Forschung des Landes Nordrhein-Westfalen for a scholarship. References
Conclusion
The vibrational potential surface of ground-state benzene was examined using the MNDO, AM1 and PM3 semiempirical parameterization sets. All three sets of results compare well with the ab initio results obtained with medium-sized basis sets and at the HF level, although in some cases they give qualitatively the same predictions as the most advanced ab initio calculations. However, they are not suitable for scaling because the errors are not systematic. Referring to the frequency and isotopic shift differences, electron correlation is taken into account, but in a rather erratic and unpredictable way. Only part of the electron correlation is contained in the values of the semi-
N. Rougeau, J.P. Flament, P. Youkharibache, H.P. Gervais and G. Berthier, J. Mol. Struct. (Theochem), 254 (1992) 405. L. Goodman, A.G. Ozkabak and S.N. Thakur, J. Phys. Chem., 95 (1991) 9044. J.J.P. Stewart, J. Comput. Aided Mol. Design, 4 (1990) 1. P. Pulay, G. Fogarasi and J.E. Boggs, J. Chem. Phys., 74 (1981) 3999. (a) F. Toriik, A. Hegedtis and P. Pulay, J. Mol. Struct., 32 (1976) 93. (b) G. Fogarasi and P. Pulay, J. Mol. Struct., 39 (1977) 275. (a) A. Langseth and B.P. Stoicheff, Can. J. Phys., 34 (1956) 350. (b) B.P. Stoicheff, Can. J. Phys., 32 (1954) 339. J. Pliva, J.W.C. Jones and L. Goodman, J. Mol. Spectrosc., 148 (1991) 427. P. Pulay, G. Fogarasi, G. Pongor, J.E. Boggs and A. Varga, J. Am. Chem. Sot., 105 (1983) 7037. C. Dehu, F. Meyers and J.L. Bredas, J. Am. Chem. Sot., 115 (1993) 6198.
G. BaranoviC and B. Schrader/J. Mol. Struct. (Theochem) 306 (1994) 165-175
174
Appendix: Supplementary material Table A 1 Internal coordinates No.
for benzenea Description
Coordinate
l-6 7-12 13-18 19 20 21 22-21 28 29 30
PI
=(l/JU(P1
a9
=
-d),...,Psb
(1/&)(a1
-
(~2 +
q20a= (l/J12)(2cq @Ob
=
CC stretch. CH stretch. CH ip bend. CCC bend. bl, CCC bend. e2s CCC bend. e2s CH op bend. CC tors. b2s CC tors. ezu CC tors. ezu
(1/2)(a2
a3
-
(~4 +
as
-
a6)
- ff2 - cx3+ 2cx4- (Y~- ck6)
-
a3
+
a5
-
6,
+
63 -
(y6)
71?...>^16 q28
=
(l/&)(61
-
429a
=
(1/2)(-6l
+
q29b
=
(l/,/12)(-61
‘53 +
64 +
s4 + 262
‘55 -
‘56)
64 +
265
66)
-
63 -
-
66)
a See Table 3 of Ref. 4. b Note the sign difference relative to Ref. 2, and agreement with Ref. 4.
Table A2 Symmetry coordinates for benzene redundant internal coordinates No.
Symm.
1 2
alg
3
as %
in terms
of nonTable A2 (continued) No.
Symm.
Coordinateb
t6
22
b2s
-/I -
s6
23
Coordinateb
t, + t2 + t3 + 14 + t5 + sl
PI
+
s2 +
+
s3 +
/?2 +
p3
s4 +
+
p4
s5 +
+
P5 +
P +
p6
t] - 2t2 + tj + t4 - 2t, + t6 2sl -
s2 -
s3 +
2s4
p5
p6
-
s5 -
14 15 16 17 18 19 20 21
P3
+
-
-q20b --z/3,
b lu
sl
-
+
@2 +
s2 +
&
s3 -
-
2@4 +
s4 +
s5 -
,&
+
s6
tj - t2 + tj - t4 + t5 - t6 PI
elu
p2
+
p3
-
@4 +
t, - t3 - t4 + t6 231f s2- s3 - 2x4 -p2
eh
-
-
P3 +
P5
+
p5
-
-
p6
s5 f
s6
p6
tl + 2t, + t3 - t4 - 21, - t6 ~72 + s3 - s5 - s6 2p1
+
p2
-
&
-
2p4
-
Ps
‘74 +
elg
2%
25
ek
72 +
73 -
75 -
76
26
a2”
71 +
72 +
73 +
74 +
e2”
2%
P6
29 30 b
419
b2u
-
24
21 28
t] - t3 + t4 - t6 .9 - s3 + s5 - s6
8 9 10 11 12 13
-
73
75
-
‘76
+
“12 -
73
-
274
-
75 +
76
s6
-420a p2
72 +
q28
+
p6
-
72 -
429a
e2u
Not normalized.
-h+73-75+76 q29b
73 +
274
75 +
-
76
7’5 -
76
G. BaranoviC and B. Schrader/J.
Table A3 Harmonic force field constants
Mol. Struct.
(Theochem)
306 (1994)
165-175
of benzene in terms of internal valence coordinates Unscaled
No.
Type
fl
ti ti
f2
si si
f3
419419 q20aq20a
;
PiPi
f6 f7 f8
Ex~.~
MNDO
Scaled AM1
PM3
MNDO
AM1
PM3
6.616 5.547 1.284 1.257 0.523
8.613 6.453 1.163 1.236 0.547
9.601 5.654 1.317 1.348 0.522
9.190 5.165 1.178 1.226 0.455
6.624 5.634 1.288 1.216 0.523
6.660 5.635 1.267 1.183 0.533
7.233 5.559 1.265 1.222 0.527
0.128 -0.419 0.383
1.145 -0.602 0.749
1.523 -0.483 0.819
1.441 -0.601 0.787
0.776 -0.503 0.392
0.744 -0.518 0.384
0.545 -0.802 0.826
0.007 0.008 -0.022
0.006 0.002 0.001
0.006 0.002 0.002
0.014 0.001 0.001
0.006 0.002 0.001
0.006 0.002 0.002
0.015 0.001 0.001
-0.007 -0.004 0.000
-0.006 -0.006 0.000
-0.014 -0.003 0.000
-0.006 -0.004 0.000
-0.006 -0.006 0.000
-0.016 -0.004 0.000
f9
sisi+l
fl0
sisi+2
fll
sisi+3
fl2
PiPi+l
f13
&Pi+2
h4
Pi&+3
0.008 -0.012 -0.001
f17
tiSi = tiSi+l tisi+2 = tisi+5 tisi+3 = tisi+4
0.200 0.008 -0.130
0.413 0.011 -0.056
0.425 0.005 -0.055
0.211 0.002 -0.030
0.338 0.009 -0.046
0.353 0.004 -0.046
0.192 0.002 -0.028
fl8
t142oa
0.124
0.118
0.103
0.129
0.178
0.137
0.114
fl5 f16
-0.135 -0.110
-0.153 -0.192
-0.147 -0.183
-0.073 -0.094
-0.151 -0.178
-0.144 -0.171
-0.079 -0.098
-0.153 0.015 -0.023
-0.269 0.004 -0.004
-0.281 0.006 -0.006
-0.201 -0.001 -0.002
-0.143 0.027 -0.022
-0.144 0.029 -0.018
-0.366 0.000 -0.247
f25
0.048 0.019
0.000 0.003
0.000 0.003
0.008 0.001
0.000 0.003
0.000 0.003
0.009 -0.001
f26
0.075
0.018
0.037
0.026
0.030
0.035
0.028
0.447 0.394 0.312
0.481 0.259 0.245
0.454 0.274 0.246
0.424 0.269 0.225
0.447 0.355 0.335
0.452 0.363 0.326
0.454 0.381 0.320
fl9 f20
f21 f22 f23
f24
f27
Ti ^li
f28
q28q28
f29
429aq29a
^ii^li+3
-0.069 0.002 -0.021
-0.033 -0.006 -0.016
-0.028 -0.006 -0.018
-0.034 -0.005 -0.010
-0.063 -0.006 -0.015
-0.063 -0.006 -0.018
-0.067 -0.006 -0.011
‘-!I 428 YI429a
-0.154 0.147
-0.098 0.105
-0.101 0.104
-0.092 0.097
-0.138 0.148
-0.141 0.145
-0.146 0.155
f30
YiYi+l
f31
“li^li+Z
f32
f33 f34
a Ref. 2.