JOURNAL OF CHEMICAL PHYSICS
VOLUME 115, NUMBER 6
8 AUGUST 2001
Density functional theory predictions of anharmonicity and spectroscopic
constants for diatomic molecules
Mutasem Omar Sinnokrot and C. David Sherrill
Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry,
Georgia Institute of Technology, Atlanta, Georgia 30332-0400
共Received 19 April 2001; accepted 25 May 2001兲
The reliability of density functional theory and other electronic structure methods is examined for
anharmonicities and spectroscopic constants of the ground electronic states of several diatomic
molecules. The equilibrium bond length r e , harmonic vibrational frequency ␻ e , vibrational
anharmonicity ␻ e x e , rotational constant B e , centrifugal distortion constant D̄ e , and
vibration-rotation interaction constant ␣ e have been obtained theoretically for BF, CO, N2, CH⫹,
and H2. Predictions using Hartree–Fock, coupled-cluster singles and doubles 共CCSD兲, coupled
cluster singles and doubles with perturbative triples 关CCSD共T兲兴, and various density functional
methods 共S-VWN, BLYP, and B3LYP兲 have been made using the 6-31G* , aug-cc-pVDZ, and
aug-cc-pVTZ basis sets and compared to experimental values. Density functional theory predictions
of the spectroscopic constants are reliable 共particularly for B3LYP兲 and often perform as well as the
more expensive CCSD and CCSD共T兲 estimates. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1386412兴
I. INTRODUCTION
ity and vibration-rotation interaction have been very carefully studied for the Hartree–Fock 共HF兲 and configuration
interaction singles and doubles 共CISD兲 methods by Schaefer
and co-workers17,18 for a series of asymmetric top and linear
molecules. Although harmonic frequency predictions can
change significantly between the Hartree–Fock and CISD
methods, these workers found that Hartree–Fock with a polarized double-zeta basis 共DZP兲 was sufficient to provide
good agreement with experiment for the asymmetric top
molecules considered.17 The subsequent study of linear polyatomic molecules18 did not show as good agreement between
DZP HF anharmonic constants and experiment, presumably
because the linear molecules all contained multiple bonds,
making electron correlation effects more important.
Dunning and co-workers1–3 have studied the accuracy of
theoretical vibrational anharmonicities, vibration-rotation interaction constants, and other spectroscopic constants for
many diatomic molecules using Hartree–Fock, generalized
valence bond 共GVB兲, GVB CI, complete-active-space selfconsistent-field 共CASSCF兲, and CASSCF second-order configuration interaction 共SOCI兲 methods. These authors found
reasonable predictions at all levels of theory considered, but
basis sets of at least polarized triple-zeta quality and dynamic
electron correlation 共GVB-CI or CASSCF SOCI兲 were required to obtain high accuracy. Feller and Sordo6 have recently reported full coupled-cluster singles, doubles, and
triples 共CCSDT兲 spectroscopic constants for first row diatomic hydrides and find excellent agreement with experiment even without correcting for core-valence correlation or
relativistic effects; moreover, no significant differences were
found between predictions using full iterative triples or perturbative triples according to the CCSD共T兲 共Ref. 13兲 prescription. Additionally, full configuration interaction quartic
Predictions of the spectroscopic constants of diatomic
molecules 共e.g., r e , ␻ e , ␻ e x e , B e , D̄ e , and ␣ e 兲 are often
used to benchmark new theoretical methods or study electron
correlation or basis set effects.1– 8 Such predictions are also
useful for as yet unknown or poorly characterized electronic
states of diatomics.9,10 While the theoretical prediction of
equilibrium geometries and harmonic vibrational frequencies
for molecules is routine, predictions of properties related to
higher derivatives of the potential energy surface are relatively rare. Nevertheless, theoretical estimates of anharmonicity are critical for high accuracy predictions of experimentally observed fundamental vibrational frequencies;
harmonic computations often overestimate fundamental frequencies by a few (⬇1 – 3) percent. Likewise, anharmonic
corrections are necessary to transform vibrationally averaged
structures to equilibrium structures. Theoretically computed
anharmonic force fields are therefore valuable for obtaining
equilibrium geometries from, e.g., microwave and electron
diffraction techniques. Botschwina and co-workers11 have investigated the differences between equilibrium and ground
state moments of inertia at several levels of theory, and
Kochikov et al.12 have recently demonstrated that an accurate equilibrium structure for SF6 can be obtained by electron
diffraction and empirically scaled ab initio quadratic and cubic force constants. Gauss, Stanton, and co-workers have
computed the anharmonicity corrections to equilibrium rotational constants using coupled-cluster with single, double,
and perturbative triple substitutions 关CCSD共T兲兴 共Ref. 13兲 and
have used these corrections to obtain highly accurate equilibrium structures from experimentally observed, vibrationally averaged rotational constants for several molecules14
including cyclopropane15 and benzene.16
Electronic structure estimates of vibrational anharmonic0021-9606/2001/115(6)/2439/10/$18.00
2439
© 2001 American Institute of Physics
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2440
J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
M. O. Sinnokrot and C. D. Sherrill
force fields have been reported by Van Huis et al.19 for
NH⫹
2 .
Although density functional theory 共DFT兲 共Ref. 20兲 is
rapidly replacing more traditional correlated ab initio methods for use in many chemical problems, we are unaware of
any systematic study of DFT predictions of anharmonic
spectroscopic constants. However, recent work21,12 on benzene and SF6 suggests that the hybrid functional B3LYP
共Refs. 22, 23兲 can provide accurate anharmonic force fields.
Moreover, the study of benzene by Miani et al.21 indicates
that the B3LYP spectroscopic constants are significantly
more accurate than the Hartree–Fock results. In this study
we use several different DFT approaches in conjunction with
three different basis sets to determine the spectroscopic constants of BF, CO, N2, CH⫹, and H2. For comparison, spectroscopic constants were also predicted using Hartree–Fock,
coupled-cluster singles and doubles 共CCSD兲,24 and coupledcluster singles and doubles and perturbative triples
关CCSD共T兲兴;13 the results are compared to the experimental
values of Huber and Herzberg.25
II. THEORETICAL APPROACH
The vibrational term values G( v ) of a diatomic molecule are
G 共 v 兲 ⫽ ␻ e 共 v ⫹ 21 兲 ⫺ ␻ e x e 共 v ⫹ 21 兲 2 ⫹¯ ,
共1兲
while the rotational term values F v (J) are given by
F v 共 J 兲 ⫽B v J 共 J⫹1 兲 ⫺D̄ e J 2 共 J⫹1 兲 2 ⫹¯ .
共2兲
The effective rotational constant B v for vibrational level v
depends on the equilibrium rotational constant B e and the
vibration-rotation interaction terms ␣ e via
B v ⫽B e ⫺ ␣ e 共 v ⫹ 12 兲 ⫹¯ .
共3兲
Fundamental frequencies are thus related to harmonic frequencies via ␯ ⫽ ␻ e ⫺2 ␻ e x e ⫹¯ . The term ␣ e requires the
third derivative of the electronic energy, while ␻ e x e requires
up to the fourth derivative.26
In this study we determined the equilibrium bond length
r e , harmonic vibrational frequency ␻ e , vibrational anharmonicity ␻ e x e , rotational constant B e , centrifugal distortion
constant D̄ e , and vibration-rotation interaction constant ␣ e
for the ground states of BF, CO, N2, CH⫹, and H2. Spectroscopic constants were determined by a five-point method, in
which energies are very tightly converged 共typically to
10⫺12 hartree兲 for five bond lengths uniformly distributed
共with a spacing of 0.005 Å兲 around the equilibrium internuclear distance r e . The force constants up to the quintic
constant f rrrrr were used to determine these spectroscopic
constants as explained elsewhere.27
Computations were performed using three basis sets of
contracted Gaussian functions, namely 6-31G* , 28,29 aug-ccpVDZ, and aug-cc-pVTZ.30 The 6-31G* basis set is of
double zeta plus polarization quality and has a contraction
scheme of (10s4p1d)/关 3s2 p1d兴 for first-row elements. The
augmented correlation-consistent polarized valence doublezeta basis set, aug-cc-pVDZ, is formed from a contraction of
(9s4p1d) to 关 3s2p1d兴, and is augmented by a diffuse set of
functions (1s1 p1d), resulting in a contraction scheme of
(10s5 p2d)/关 4s3 p2d兴. The third basis set, the augmented
correlation-consistent polarized triple-zeta basis set, aug-ccpVTZ, is formed from contracting a set of (10s5p2d1 f )
primitives to 关 4s3 p2d1 f 兴, and is augmented by a diffuse set
of functions (1s1p1d1 f ), resulting in a contraction scheme
of (11s6 p3d2 f )/关 5s4 p3d2 f 兴. A set of six Cartesian Gaussian functions is used for 6-31G* , whereas the correlationconsistent basis sets employ pure angular momentum sets of
five d and seven f functions. The diffuse functions are included primarily to facilitate comparison to forthcoming
studies of excited electronic states. Based on the results of
Dunning and co-workers,1–3 we expect little difference between the cc-pVXZ and aug-cc-pVXZ predictions of the
spectroscopic constants of the ground states of the molecules
considered here, particularly for the triple-␨ basis sets.
Spectroscopic constants were predicted ab initio using
Hartree–Fock, coupled-cluster with single and double substitutions 共CCSD兲,24 and coupled-cluster with single, double,
and perturbative triple substitutions 关CCSD共T兲兴.13 Predictions were also made using the following Kohn–Sham density functional theory methods:20 the local spin density approximation with Slater exchange31 and the correlation
functional of Vosko, Wilk, and Nusair32 共denoted S-VWN兲;
the generalized gradient approximation 共GGA兲 methods
which pair Becke’s 1988 exchange functional33 with the correlation functionals of Lee, Yang, and Parr34 共BLYP兲; and a
hybrid22 gradient-corrected functional which mixes in
Hartree–Fock exchange, B3LYP.23,35 Geometries were optimized using analytic gradients for Hartree–Fock,36,37
CCSD,38 CCSD共T兲,39,40 and density functional theory. All
electrons were correlated in the coupled-cluster procedures.
All computations were performed using Q-CHEM 1.2 共Ref. 41兲
except the coupled-cluster results which used the ACES II
package.42
III. RESULTS AND DISCUSSION
Total electronic energies and spectroscopic constants are
presented in Tables I 共BF兲, II 共CO兲, III (N2), IV (CH⫹),
and V (H2). Errors versus experimental values from Huber
and Herzberg25 are displayed in Figs. 1 (r e ), 2 (B e ), 3 ( ␻ e ),
4 (D̄ e ), 5 ( ␣ e ), and 6 ( ␻ e x e ). Table VI summarizes the
average absolute relative errors for the spectroscopic constants considered at each level of theory.
A. Equilibrium bond lengths and rotational constants
The errors in bond lengths presented in Fig. 1 are consistent with the errors expected for these levels of theory
based on previous systematic studies of equilibrium
geometries.43– 46 Although the best geometry predictions for
Hartree–Fock are obtained with the aug-cc-pVDZ basis, for
all other theoretical methods this basis gives much poorer
geometries than 6-31G* or aug-cc-pVTZ. The coupledcluster methods usually improve the Hartree-Fock geometry
predictions substantially; with a 6-31G* basis, bond lengths
are slightly overestimated, and they are quite accurately predicted with the aug-cc-pVTZ basis. An unusually large error
in the BF bond length 共⫹0.0412 Å兲 makes our average ab-
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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
DFT for diatomic molecules
2441
TABLE I. Spectroscopic constants for the ground electronic state of BF.a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* CCSD共T兲
aug-cc-pVDZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
Experiment
a
Energy
re
␻e
␻ ex e
Be
D̄ e
␣e
⫺124.103 184
⫺124.115 039
⫺124.157 406
⫺124.356 725
⫺124.400 707
⫺124.524 602
⫺124.362 855
⫺124.408 519
⫺124.536 629
⫺123.732 509
⫺123.747 823
⫺123.793 468
⫺124.638 683
⫺124.659 056
⫺124.699 758
⫺124.605 678
⫺124.623 477
⫺124.664 131
1.2604
1.2757
1.2486
1.2807
1.3037
1.2629
1.2828
1.3078
1.2669
1.2697
1.2836
1.2609
1.2853
1.2994
1.2777
1.2737
1.2877
1.2654
1.2625
1471
1363
1493
1400
1262
1425
1390
1245
1406
1401
1304
1404
1347
1255
1342
1398
1301
1398
1402
10.5
9.4
11.8
10.0
8.6
11.5
10.1
8.7
11.7
8.4
7.7
11.6
8.9
9.5
11.3
9.2
8.9
11.7
11.8
1.522
1.486
1.551
1.474
1.423
1.516
1.470
1.414
1.507
1.500
1.468
1.521
1.464
1.432
1.482
1.491
1.459
1.510
1.507
6.52e⫺06
7.07e⫺06
6.70e⫺06
6.54e⫺06
7.24e⫺06
6.87e⫺06
6.57e⫺06
7.30e⫺06
6.92e⫺06
6.88e⫺06
7.44e⫺06
7.15e⫺06
6.92e⫺06
7.46e⫺06
7.22e⫺06
6.78e⫺06
7.32e⫺06
7.06e⫺06
7.60e⫺06
0.0174
0.0180
0.0182
0.0171
0.0173
0.0184
0.0173
0.0175
0.0186
0.0186
0.0189
0.0199
0.0181
0.0188
0.0196
0.0180
0.0186
0.0193
0.0198
Energies in hartrees, bond lengths in Å, and other quantities in cm⫺1. Core electrons correlated. Experimental data from Huber and Herzberg 共Ref. 25兲.
solute aug-cc-pVDZ CCSD bond length error 共0.0207 Å兲
noticeably larger than expected 共0.0096兲 from Helgaker’s
study;46 this error is even larger for aug-cc-pVDZ CCSD共T兲.
As one can see from Fig. 1 or Table VI, S-VWN and BLYP
DFT predictions do not improve over Hartree-Fock for the
bond lengths of these molecules except with the large augcc-pVTZ basis. However, the B3LYP DFT bond lengths are
very good with the 6-31G* and aug-cc-pVTZ basis sets,
with average absolute errors even smaller than CCSD or
CCSD共T兲.
The equilibrium rotational constants B e are simply related to the equilibrium bond lengths via B e ⫽h/(8 ␲ 2 ␮ r 2e ).
Hence, we expect the general trends observed for bond
length errors to persist for B e errors, although underestimates
of bond lengths lead to overestimates of B e and vice versa.
Consistent with this view, the B e errors in Fig. 2 are qualitatively like a mirror image 共through zero error兲 of the bond
length errors in Fig. 1. Most errors in B e are within ⫾6%,
with Hartree–Fock typically overestimating and other methods typically underestimating experiment. The BLYP and
B3LYP results are substantially improved over S-VWN and
rival or surpass the accuracy of the CCSD and CCSD共T兲 B e
predictions, depending on basis set. Just as in the bond
length predictions, the aug-cc-pVDZ basis set gives the least
accurate results. Our best results 关average absolute errors of
0.7% for B3LYP or CCSD共T兲 with the large basis兴 compare
very favorably with the 0.8% average absolute error found
by Peterson et al. using a much more intricate and expensive
TABLE II. Spectroscopic constants for the ground electronic state of CO.a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* CCSD共T兲
aug-cc-pVDZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
Experiment
a
Energy
re
␻e
␻ ex e
Be
D̄ e
␣e
⫺112.737 877
⫺112.755 481
⫺112.782 944
⫺113.033 762
⫺113.065 574
⫺113.173 946
⫺113.044 881
⫺113.078 371
⫺113.191 837
⫺112.414 653
⫺112.432 743
⫺112.466 892
⫺113.293 988
⫺113.316 439
⫺113.345 740
⫺113.257 211
⫺113.278 107
⫺113.306 642
1.1138
1.1108
1.1041
1.1412
1.1398
1.1241
1.1472
1.1465
1.1311
1.1419
1.1381
1.1287
1.1504
1.1463
1.1376
1.1381
1.1343
1.1260
1.1283
2439
2403
2421
2223
2175
2243
2159
2108
2173
2171
2151
2178
2109
2090
2113
2210
2187
2208
2170
11.1
11.2
11.2
12.0
12.0
12.2
13.1
13.1
13.2
13.6
13.5
13.6
14.2
13.9
13.9
12.9
12.7
12.9
13.3
1.982
1.993
2.017
1.888
1.893
1.946
1.868
1.870
1.922
1.886
1.898
1.930
1.858
1.871
1.900
1.898
1.911
1.939
1.931
5.24e⫺06
5.48e⫺06
5.60e⫺06
5.45e⫺06
5.73e⫺06
5.86e⫺06
5.59e⫺06
5.89e⫺06
6.01e⫺06
5.69e⫺06
5.91e⫺06
6.06e⫺06
5.77e⫺06
6.00e⫺06
6.15e⫺06
5.60e⫺06
5.84e⫺06
5.98e⫺06
6.12e⫺06
0.0145
0.0150
0.0150
0.0157
0.0162
0.0163
0.0167
0.0172
0.0173
0.0167
0.0170
0.0172
0.0171
0.0173
0.0176
0.0161
0.0165
0.0167
0.0175
Energies in hartrees, bond lengths in Å, and other quantities in cm⫺1. Core electrons correlated. Experimental data from Huber and Herzberg 共Ref. 25兲.
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2442
J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
M. O. Sinnokrot and C. D. Sherrill
TABLE III. Spectroscopic constants for the ground electronic state of N2 . a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* CCSD共T兲
aug-cc-pVDZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
Experiment
a
Energy
re
␻e
␻ ex e
Be
D̄ e
␣e
⫺108.943 950
⫺108.961 925
⫺108.987 796
⫺109.261 654
⫺109.285 417
⫺109.391 840
⫺109.274 461
⫺109.299 569
⫺109.411 366
⫺108.640 508
⫺108.656 410
⫺108.689 686
⫺109.510 660
⫺109.530 204
⫺109.558 855
⫺109.471 947
⫺109.490 688
⫺109.518 429
1.0784
1.0783
1.0700
1.1129
1.1137
1.0929
1.1190
1.1203
1.0999
1.1112
1.1098
1.0958
1.1182
1.1168
1.1030
1.1057
1.1046
1.0914
1.0977
2758
2736
2727
2416
2396
2449
2346
2323
2370
2403
2394
2399
2340
2328
2336
2458
2445
2448
2359
11.3
11.2
10.8
13.7
13.4
13.2
14.7
14.4
14.2
14.1
13.8
13.6
15.2
15.0
14.6
13.8
13.7
13.2
14.3
2.070
2.071
2.115
1.944
1.941
2.016
1.923
1.918
1.990
1.950
1.955
2.005
1.926
1.930
1.979
1.969
1.973
2.021
1.998
4.67e⫺06
4.75e⫺06
5.09e⫺06
5.04e⫺06
5.09e⫺06
5.47e⫺06
5.17e⫺06
5.23e⫺06
5.61e⫺06
5.14e⫺06
5.21e⫺06
5.60e⫺06
5.22e⫺06
5.31e⫺06
5.68e⫺06
5.06e⫺06
5.14e⫺06
5.51e⫺06
5.76e⫺06
0.0133
0.0136
0.0137
0.0158
0.0159
0.0161
0.0167
0.0168
0.0171
0.0159
0.0161
0.0162
0.0164
0.0166
0.0167
0.0154
0.0156
0.0157
0.0173
Energies in hartrees, bond lengths in Å, and other quantities in cm⫺1. Core electrons correlated. Experimental data from Huber and Herzberg 共Ref. 25兲.
multireference configuration interaction 共MRCI兲 procedure
in conjunction with a larger cc-pVQZ basis set.3
B. Harmonic vibrational frequencies
Figure 3 displays the relative errors in harmonic vibrational frequencies ␻ e . The Hartree–Fock predictions show a
typical 3%–17% overestimate in most cases, whereas the
CCSD frequencies are considerably improved, with most errors within 5%. Errors are further reduced with the CCSD共T兲
method. Our aug-cc-pVTZ CCSD average absolute error of
3.1% is quite similar to the TZ(2d f ,2pd) CCSD average
error of 3.7% in the systematic study of Thomas et al.44 The
DFT methods are generally as accurate as CCSD for the
current set of vibrational frequencies, with errors usually
within 5%; depending on the basis set, the DFT predictions
are almost as good or better than those of CCSD共T兲. S-VWN
and BLYP tend to underestimate, while B3LYP tends to overestimate harmonic frequencies. Basis set effects are generally
modest, and predictions are not always improved with the
larger basis set. However, we note that the aug-cc-pVDZ
basis substantially underestimates ␻ e with several of the
methods considered.
C. Centrifugal distortion constants
The centrifugal distortion constant D̄ e , which accounts
for the lengthening of the bond with higher angular momentum J, is given by D̄ e ⫽4B 3e / ␻ 2e . Errors in theoretical pre-
TABLE IV. Spectroscopic constants for the ground electronic state of CH⫹. a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* CCSD共T兲
aug-cc-pVDZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
Experiment
a
Energy
re
␻e
␻ ex e
Be
D̄ e
␣e
⫺37.895 537
⫺37.901 390
⫺37.907 902
⫺37.994 271
⫺38.003 773
⫺38.034 966
⫺37.995 823
⫺38.005 348
⫺38.037 437
⫺37.676 368
⫺37.678 046
⫺37.687 375
⫺38.069 081
⫺38.071 006
⫺38.080 060
⫺38.081 446
⫺38.062 280
⫺38.070 577
1.1046
1.1245
1.1133
1.1284
1.1456
1.1184
1.1291
1.1465
1.1194
1.1608
1.1758
1.1636
1.1496
1.1634
1.1475
1.1359
1.1516
1.1370
1.1309
3201
3071
3056
2931
2854
2920
2922
2845
2908
2699
2625
2622
2753
2672
2690
2882
2787
2795
2740
59.3
53.4
49.1
68.5
62.6
60.8
69.3
63.3
61.4
53.0
45.1
41.6
81.2
68.5
71.9
77.8
69.2
70.3
64.0
14.861
14.339
14.630
14.240
13.815
14.495
14.221
13.795
14.471
13.456
13.116
13.392
13.719
13.396
13.770
14.052
13.673
14.026
14.177
1.28e⫺03
1.25e⫺03
1.34e⫺03
1.35e⫺03
1.30e⫺03
1.43e⫺03
1.35e⫺03
1.30e⫺03
1.43e⫺03
1.34e⫺03
1.31e⫺03
1.40e⫺03
1.36e⫺03
1.35e⫺03
1.44e⫺03
1.33e⫺03
1.32e⫺03
1.41e⫺03
1.40e⫺03
0.459
0.441
0.433
0.525
0.498
0.524
0.529
0.502
0.529
0.521
0.494
0.493
0.551
0.547
0.551
0.513
0.512
0.509
0.492
Energies in hartrees, bond lengths in Å, and other quantities in cm⫺1. Core electrons correlated. Experimental data from Huber and Herzberg 共Ref. 25兲.
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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
DFT for diatomic molecules
2443
TABLE V. Spectroscopic constants for the ground electronic state of H2 . a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
Experiment
a
Energy
re
␻e
␻ ex e
Be
D̄ e
␣e
⫺1.126 828
⫺1.128 826
⫺1.133 056
⫺1.151 698
⫺1.164 899
⫺1.172 636
⫺1.133 046
⫺1.132 890
⫺1.137 281
⫺1.165 258
⫺1.162 882
⫺1.169 609
⫺1.168 718
⫺1.167 273
⫺1.173 268
0.7299
0.7481
0.7344
0.7462
0.7617
0.7430
0.7651
0.7811
0.7660
0.7480
0.7663
0.7467
0.7431
0.7612
0.7432
0.7414
4646
4559
4585
4367
4345
4402
4207
4145
4181
4374
4270
4344
4450
4353
4413
4401
121.0
106.4
109.3
141.9
118.3
127.4
115.8
106.7
112.0
127.7
113.1
116.4
123.1
110.0
115.6
121.3
62.783
59.770
62.019
60.080
57.662
60.602
57.149
54.828
57.015
59.788
56.963
59.994
60.582
57.741
60.569
60.853
4.59e⫺02
4.11e⫺02
4.54e⫺02
4.55e⫺02
4.06e⫺02
4.60e⫺02
4.22e⫺02
3.84e⫺02
4.24e⫺02
4.47e⫺02
4.05e⫺02
4.58e⫺02
4.49e⫺02
4.06e⫺02
4.56e⫺02
4.71e⫺02
2.953
2.564
2.742
3.364
2.804
3.120
2.910
2.591
2.779
3.059
2.739
3.009
3.017
2.678
2.929
3.062
Energies in hartrees, bond lengths in Å, and other quantities in cm⫺1. Experimental data from Huber and Herzberg 共Ref. 25兲.
dictions of D̄ e are presented in Fig. 4, which shows that
theory almost always underestimates D̄ e , by up to 20%. Estimates of D̄ e show a much larger improvement than the
previously discussed spectroscopic constants when the larger
aug-cc-pVTZ basis is used, errors being reduced by about a
factor of two 共except with Hartree–Fock兲. All three DFT
methods perform very well for D̄ e 关often as well as
CCSD共T兲兴, with errors within 4% for the aug-cc-pVTZ basis.
D. Vibration-rotation interaction constants
The vibration-rotation interaction constants ␣ e depend
on the third derivative of the potential energy and relate the
effective rotational constant B v for vibrational level v to the
equilibrium rotational constant B e via Eq. 共3兲. Figure 5 displays the errors in predicted values. Except for CH⫹, theory
usually underestimates ␣ e , by up to 23% for Hartree–Fock.
Most other methods are typically within 15%. Similarly to
FIG. 1. Error in theoretically predicted bond lengths.
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2444
J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
M. O. Sinnokrot and C. D. Sherrill
FIG. 2. Relative error in theoretically predicted rotational constants B e .
FIG. 3. Relative error in theoretically predicted harmonic vibrational frequencies.
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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
DFT for diatomic molecules
2445
FIG. 4. Relative error in theoretically predicted centrifugal distortion constants D̄ e .
FIG. 5. Relative error in theoretically predicted vibration rotation interaction constants ␣ e .
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2446
J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
M. O. Sinnokrot and C. D. Sherrill
FIG. 6. Relative error in theoretically predicted anharmonic constants ␻ e x e .
D̄ e , DFT methods are competitive with CCSD共T兲 and
slightly more reliable than CCSD, and the aug-cc-pVTZ predictions are substantially improved over those using the
smaller basis sets. With the larger basis set, the CCSD共T兲,
S-VWN, BLYP, and B3LYP average absolute errors in ␣ e
共see Table VI兲 are 3.6%, 3.6%, 3.8%, and 4.8%. These results compare well with the 1.5% average absolute error inferred from the much more expensive cc-pVQZ MRCI results of Peterson et al. for first-row homonuclear diatomics.3
The leading term in the difference between the 共zeropoint兲 vibrationally averaged rotational constant B 0 and the
equilibrium rotational constant B e is ⫺ ␣ e /2 关see Eq. 共3兲兴.
For the cases considered here, this difference is about 0.5%–
2.5% of B e . As noted above, we find average absolute errors
of 0.7% for aug-cc-pVTZ B3LYP or CCSD共T兲 predictions of
B e . Thus, the corrections for vibrational motion (B 0 ⫺B e )
can be larger than the errors in theoretical estimates of B e .
Given the very small errors in ␣ e predictions, the difference
TABLE VI. Average absolute relative errors 共percent兲 for spectroscopic constants 共bond length errors in Å, not relative兲.a
Method
6-31G* HF
aug-cc-pVDZ HF
aug-cc-pVTZ HF
6-31G* CCSD
aug-cc-pVDZ CCSD
aug-cc-pVTZ CCSD
6-31G* CCSD共T兲
aug-cc-pVDZ CCSD共T兲
aug-cc-pVTZ CCSD共T兲
6-31G* S-VWN
aug-cc-pVDZ S-VWN
aug-cc-pVTZ S-VWN
6-31G* BLYP
aug-cc-pVDZ BLYP
aug-cc-pVTZ BLYP
6-31G* B3LYP
aug-cc-pVDZ B3LYP
aug-cc-pVTZ B3LYP
a
re
␻e
␻ ex e
Be
D̄ e
␣e
0.0147
0.0126
0.0181
0.0107
0.0207
0.0047
0.0134
0.0244
0.0045
0.0176
0.0255
0.0122
0.0181
0.0263
0.0103
0.0071
0.0157
0.0039
11.3
9.0
9.9
2.5
3.5
3.1
1.9
4.1
1.4
1.6
3.9
2.3
1.7
4.2
2.2
2.5
2.9
1.6
11.2
17.3
14.7
10.7
9.6
5.7
8.8
6.4
2.3
10.8
16.2
10.3
14.0
8.5
5.4
10.3
10.1
5.2
3.0
2.2
3.7
1.8
3.7
1.0
2.2
4.3
0.7
3.3
4.8
2.6
3.1
4.7
1.7
1.1
2.8
0.7
11.7
11.7
8.0
8.9
8.7
4.6
7.9
7.6
3.5
8.4
8.0
3.9
6.4
5.8
2.5
8.2
7.7
3.5
12.5
14.3
13.1
9.8
7.5
5.9
7.6
5.3
3.6
5.9
6.0
3.6
5.6
6.4
3.8
6.8
7.6
4.8
Core electrons were correlated for coupled-cluster. Errors were computed relative to experimental data from Huber and Herzberg 共Ref. 25兲.
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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
(B 0 ⫺B e ) can be estimated with very high accuracy with any
of the theoretical methods considered here. Once the ⫺ ␣ e /2
vibrational correction is included, the majority of the error in
theoretical estimates of the vibrationally averaged rotational
constant B 0 again comes from the errors in the equilibrium
constant B e itself.
E. Vibrational anharmonic constants
Computations of the vibrational anharmonic constants
␻ e x e require up to the fourth derivative of the potential energy. Errors in the theoretical predictions for this constant are
presented in Fig. 6. This quantity appears to be somewhat
more difficult to compute accurately on a relative basis than
the other constants in this study; the estimates are usually
only good to about ⫾30%. Except for Hartree–Fock and
S-VWN, the larger aug-cc-pVTZ yields much better results
than the double-␨ basis sets. The aug-cc-pVTZ average absolute errors for CCSD, BLYP, and B3LYP are about 5%,
with the Hartree-Fock errors about three times as large and
the CCSD共T兲 average absolute error being 2.3%. Our best
results compare favorably with an average absolute error of
2.5% for much costlier cc-pVQZ MRCI predictions of first
row homonuclear diatomics3 and 3.0% for aug-cc-pVTZ
CCSDT predictions of first row diatomic hydrides.6 Since the
leading anharmonic correction to the harmonic vibrational
frequencies is just ⫺2 ␻ e x e , the observed relative errors in
␻ e x e translate into nearly the same percentage error in the
correction from harmonic frequencies ␻ e to experimentally
observed fundamental frequencies ␯ e . Given that the anharmonic correction is usually small anyway 共on the order of a
few percent兲, and considering the quite small errors in DFT
predictions of ␻ e , this suggests that density functional
theory is capable of providing rather accurate predictions of
fundamental frequencies ␯ e .
IV. CONCLUSIONS
Density functional theory provides reliable predictions
of the spectroscopic constants of several diatomic molecules,
including constants depending on anharmonicity such as
vibration-rotation interaction constants ␣ e and vibrational
anharmonicities ␻ e x e . While not as accurate as highly sophisticated multireference CI methods, DFT predictions are
frequently as good or better than more expensive CCSD results and are often competitive with CCSD共T兲. S-VWN performs the worst and B3LYP performs the best of the DFT
methods considered. By considering anharmonic corrections,
DFT appears capable of providing very accurate predictions
of vibrationally averaged rotational constants B 0 and fundamental vibrational frequencies ␯.
ACKNOWLEDGMENTS
This research was supported by the National Science
Foundation 共Grant No. CHE-0091380兲. One of the authors
共C.D.S.兲 acknowledges a Camille and Henry Dreyfus New
Faculty Award and an NSF CAREER Award 共Grant No.
CHE-0094088兲. The authors thank Alfred Park for assistance
DFT for diatomic molecules
2447
with the analysis. The Center for Computational Science and
Technology is funded through a Shared University Research
共SUR兲 grant from IBM and Georgia Tech.
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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001
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