On the fallibility of variational calculations: Ab initio versus empirical
potential energy functions for HCN
Alice M. Smith”) and William Klemperer
Department of Chemistry, Harvard University, Cambridge, Massachusetts 01238
Kevin K. Lehmannb)
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
(Received 3 1 August 1990; accepted 4 December 1990)
HCN is one of the most advantageous candidates among the polyatomics for an empirical
determination of the anharmonic potential energy function because of the extensive list of
precise spectroscopic data available for a number of isotopomers. We have determined an
empirical stretching potential energy surface for HCN with a rms error of less than 3 cm - ’ for
vibrational energy levels of up to 18 400 cm - ‘. The best ab initio stretching potential has a rms
error of 10.5 cm - ’ for the same levels. However, using the ab initio dipole moment surface and
vibrational wave functions from the empirical vs ab initio potentials, we find that the predicted
intensities are remarkably different, a factor of 4 for the (0,0,6) band at 18 377 cm - ‘.
Furthermore, the predicted intensities from the theoretical potential agree significantly better
with the experimental values. The most natural explanation of these results is that the ab initio
potential is closer to reality than an empirical surface obtained from exhaustive spectroscopic
data.
I. INTRODUCTION
Characterization of the intramolecular potential energy
surface remains one of the principal goals of vibrational
spectroscopy. While harmonic force fields are known for a
large number of polyatomic molecules, for only a handful of
polyatomics does sufficient spectroscopic data exist for a
meaningful determination of the anharmonic potential function. Of these, hydrogen cyanide is one of the most advantageous candidates because of the extensive list of precise data
available for a number of isotopomers and because the spectrum is mostly free of erratic perturbations that exist in almost every other polyatomic molecule at high excitation energy. These perturbations make it extremely difficult to fit
the spectrum since the eigenstate character changes rapidly
with potential parameters.’
We have empirically determined a potential energy surface for HCN by fitting to a form of an expansion in the
Morse oscillator coordinate. The observations used included
40 vibrational energy levels of three isotopomers and 8 vibration-rotation interaction constants. The vibrational energy
levels were calculated variationally and the vibration-rotation constants via perturbation theory expressions. The rms
error in the fit of the vibrational energy levels is less than 3
cm - ’ for energy levels up to 18 400 cm - I.
Botschwina has calculated a vibrational potential using
the CEPA-1 ab initio method.* The surface is generated
from the ab initio points by a high order Dunham expansion
‘) Current Address: Institut fiir Physikalische und Theoretische Chemie.
Technische Universitiit Miinchen, Lichtenbergstrasse 4, D-8046 Garching, Bundesrepublik Deutschland.
b’Camile and Henry Dreyfus Teacher-Scholar.
5040
J. Chem. Phys. 94 (7), 1 April 1991
in bond coordinates. Based upon a comparison of the contour plots, his surface is in excellent agreement with our empirical one for energies up to 30 000 cm - ‘. After small empirical adjustment to match the known equilibrium bond
lengths and fundamentals, his predictions for the observed
vibrational energy levels have an rms error of 10.5 cm - ‘. He
also produced a dipole moment surface from his ab initio
points.
The question remains, which of the two surfaces is in
fact closer to reality, at least for the experimentally studied
region. Recently, we have measured the absolute intensity of
vibrational transitions from the first overtone of the C-H
stretch through the overtone and combination bands in the
visible.3-5 One expects the predicted vibrational energies to
be sensitive to first order to errors in the potential function,
unlike the second order sensitivity to errors in the vibrational wave function, yet work by Quack on CX3H has shown
that relative intensity data is sensitive to the degree of state
mixing and thus the character of the wave function in a way
qualitatively different than the vibrational energies.6 In general, accurate absolute overtone intensity data should provide a sensitive test of the accuracy of the potential energy
surface which is complementary to the frequency data. Relative intensities can be estimated from the overlap of a basis
state, such as a local mode excitation, if it is assumed to carry
the only optically bright state. But the accuracy of such approximations has not been established for realistic system
and in many cases, such as in HCN, many levels probably
obtain intensity from several basis states. To overcome such
limitations, we have used the theoretical dipole moment surface and predicted the overtone intensities for our empirical
potential energy surface. These will be compared with both
the experimental values and those reported earlier by
Botschwina.
0021-9606/91/075040-11$03.00
0 1991 American Institute of Physics
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5041
Smith, Klemperer, and Lehmann: The fallibility of variational calculations
II. AN EMPIRICAL STRETCHING POTENTIAL FOR HCN
A. Method
The basis for our approach is the assertion that a reasonable zeroth order description for the stretching potential of
HCN is uncoupled potentials for the CH and CN coordinates. This is a separate issue from whether the motion of the
molecule is best viewed as normal modes or as local
modes.7*8 For a molecule with only one hydrogen stretching
mode, the bond mode and normal mode motions are nearly
the same. In fact, because of kinetic coupling, the motion is
still coupled even with an uncoupled potential. For most XH bonds in polyatomics, a two parameter local mode expansion gives an excellent fit to the observed energy levels. This
has been taken as evidence that a simple Morse oscillator
provides an accurate zeroth order picture for the X-H potential. This was realized by 1975 and led to a resurgence of
interest in the mathematics of the Morse potential. For example, during this time, Sage’ developed analytic expressions for evaluating matrix elements of a number of operators of the Morse oscillator. We have made use of these
analytical forms to evaluate matrix elements of our Hamiltonian; analytical evaluation is much faster and more accurate
than numerical evaluation, especially for far off-diagonal
near-zero matrix elements. While a simple Morse potential
can describe the vibrational energy levels, earlier work on
C, H, has demonstrated that this form is too restricted to fit
the vibration-rotation
interaction as well as the anharmonic
shifts. Baggott and Law have used a simple Morse model to
fit the vibrational energy levels of HCN using an approximate perturbative method. Rotational constant information
was not included in their fit.
We model the HCN stretching potential by a Taylor
expansion in the Morse variable z, for each ith stretching
mode, and the interbond potential coupling as z, z3, dz3,
z, 4, etc. Throughout this work, the labeling is
(v, ttZ%j3) = (CN stretch, bend, CH stretch)
as consistent with Herzberg”
ii,n =ii,
+si,
+i3,,,
ii, = !&,jf + v,& + vi,z; + viiiiz;(,
213 =g,38,83 + v,,z,z,
zi = ( 1 - e - Q’,).
+ V,,,tiz,
+ ~133Zl~~
(1)
This form is flexible while still retaining many of the attractive features of a simple Morse potential including physical
dissociation behavior. It has been used earlier to model the
HCN potential and we shall refer to it as the generalized
Morse (GM) model.” Our measured intensities for II-E
(bend-stetch combination band or perpendicular band) and
II-II (bending hot band) transitions relative to those for xcC transitions demonstrate that bend-stretch interaction is of
minor importance for the states of HCN reached by direct
overtone excitation. II-B transitions have the same intensity
within experimental accuracy of the corresponding 2-I:
transitions with the same quanta of stretching excited, while
II-2 transitions have less than 0.5% of the intensity of the
corresponding 2-2 transitions.5
Bend-stretch interaction is included empirically in the
same manner as Botschwina. In a normal mode treatment,
to second order, the harmonic stretching energy levels are
shifted by
x,,(n,
+$I(&
+ 1) +x*,(4
-tj)(n2
+ 1)
(2)
due to bend-stretch interaction. In a local mode basis the
coordinates are rotated, which changes the diagonal terms
and introduces off-diagonal terms as well. (Actually, there
are also off-diagonal terms in the normal mode treatment
but they are neglected since they are only important when
there is Darling-Dennison
resonance.12 ) To include the
bend-stretch interaction terms in our local mode basis calculation, we have first expressed the normal mode number operators in terms of ladder operators, transformed the ladder
operators to a local mode basis, and then interpreted them as
Morse quantum numbers. In fitting the observed spectrum,
the quantum numbers used are in fact closer to Morse than
harmonic quantum numbers. This treatment requires
knowledge of the harmonic frequencies, the normal mode
x’s, and the L matrix, defined by r = ZQ. The x and w values
used for three HCN isotopomers were taken from Strey and
Mills.13 Due to the minor importance of bend-stretch interaction discussed above, we believe such an approximate
treatment is justified.
6. Model potential
refinement
Given this Hamiltonian model, one can calculate certain observables, e.g., vibrational eigenenergies, vibrationrotation constants, and dissociation energies. These calculated values are then compared to the observed values and
the sum of the square of their weighted differences is minimized. Since the observables do not in general depend linearly on the parameters, this problem requires nonlinear programming. We have used a nonlinear parameter estimation
(NLPE) and programming package from the program library of the IBM corporation.
Particularly when using more than one type of observable in the fit, the most appropriate weighting is not clear.
The errors in the experimental observables are much smaller
than the model errors and thus a traditional assignment of
weights based upon random error is not necessarily appropriate. In theory, however, if the model can fit well all of the
data simultaneously, then the fit should not be very sensitive
to the weights; alternatively, alarge sensitivity to the weights
signals that the model is not sufficient.
7. Vibrational
energies
The vibrational energies are calculated variationally.
The full Hamiltonian matrix, including bend-stretch interaction, is diagonalized in a Morse oscillator product basis to
obtain eigenenergies and eigenvectors. All basis states with a
diagonal energy below 50 000 cm - ’ are included. This
yields a Hamiltonian matrix of dimension 247 for the main
isotope with well-converged eigenvalues below 20 000
cm - ‘. The nonlinear least squares fit to the potential model
included 40 observed vibrational band origins for three isotopomers of HCN (see Table II). The energy levels were all
weighted equally since the model error is much larger than
the experimental errors. Specifically, they were all weighted
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
5042
with 2 = 1.O cm - 2 or roughly 0.0 1% of the absolute energy
values.
2. Vibration-rotation
interaction
constants
Rotational constants for excited vibrational states of
HCN with a fractional accuracy of 10e4 or better have been
determined through high resolution spectroscopy. Rotational constants and centrifugal distortion constants reported by spectroscopy are just the coefficients of a perturbation
expansion in powers of J( J -t 1). This is sufficient since the
convergence of the perturbation expansion is easily determined experimentally.’ In the traditional approach, the rotational constants for several vibrational states are combined
in a perturbation expansion in powers of [v + (d,/2) ] to
determine the a and y vibration-rotation constants.5 Here,
di is the degeneracy of the ith normal mode
B,=B,- i$,ai(ui+ 2) + ijj$, Y,(Vi
+2)
The a’s, which in turn are used to help determine the
cubic force field, are the first order perturbative corrections
to the rigid rotor and these alone have been shown to provide
a meaningful description of the interaction of vibration and
rotation. This formula fit AB, values of HCN to better than
0. 1%, even at six quanta of CH stretch excitation, e.g., for
the highest observed transition, the (006)-(000), its AR of
was fit with a residual of
- 67.18~10-~
cm-’
- 0.06 x 10 - 3 cm - ’ to this formula.5
The simplest method of including vibration-rotation interaction in a variational treatment is to directly fit the a’s
predicted from the cubic force field dictated by the model
potential parameters using the usual perturbation formulas.
A more complete procedure would be to calculate the rotational constants for each observed vibrational state using a
procedure similar to that developed earlier for the variational vibrational wave functions of C, H, .14 Such a procedure is
time consuming. Furthermore, the accuracy of the perturbation expansion at fitting the experimental data suggests that
such a procedure would not contribute substantially more
information to our fit.
Using formulas given by Hoy et ~1.” and Mills” for
calculating the a’s perturbatively for a linear molecule, we
extended the variational calculation to include a fit to the
two stretch-dependent a’s for each of HCN, H13CN,
HCt5N, and DCN. The a’s were all weighted with
&= 1.0~10-~ cm-* or a sigma of 30 MHz, which is
roughly 10% of a typical a value. Such a low weighting was
not intended and could partly explain our poor fits to the (r
values ( < 14% errors) in comparison to the traditional quartic force field fit of Strey and Mills ( ( 1% errors).13 In principle, the vibrational dependence of the D ‘s (centrifugal distortion constants) and q’s (Z-doubling constants) could also
be fit. These depend on up to quartic force constants. We
have not yet included these in the variational fit because of
the complexity of the perturbation formulas.”
3. Dissociation
energies
We fit the energies associated with two possible dissociation pathways for HCN, as exhibited in Table II. The
“relaxation” of the fragment diatomic should be considered
when calculating these energies, i.e., the CN bond in diatomic CN is longer than the CN bond length in HCN, and similarly for CH. Given our potential model, the CH dissociation
energy is found, by setting zj = 1 (r, = CO> and minimizing
the resulting potential with respect to z, . Similarly, the CN
dissociation is found by setting z, = 1 and minimizing the
resulting potential with respect to z,. Although the two dissociation energies for HCN were included in the variational
fit, they provided minimal additional constraints due to the
large errors of 15% (H f CN) and 8% (HC + N) in the
predicted dissociation energies.
C. Results
Our best fit potential constants are given in Table I(A)
and the residuals of the fit are listed in Table II. Including the
observed a’s in the fit changed the CH potential parameters
only slightly, but did make a significant change in the CN
parameters. The CN contribution to both stretching a’s
dominates over the CH contribution due to the fact that
dl /JQ, = 6.5 X Z/JQ, . Thus the a’s aid principally in determining the CN part of the potential, which makes them
actually quite complimentary to the energy level data, which
gives more attention to the CH dependence due to the experimental data set and the greater anharmonicity of the CH
bond. On the other hand, the domination of the a’s by the
CN dependence hinders our ability to use the a’s to break
correlations between the parameters and improve the full
molecular potential. The vibrational data contains a strong
correlation of the third and fourth derivatives of the potential, much like a diatomic molecule; we had hoped the vibration-rotation interaction constants would have broken that
correlation.
Employing the CEPA-1 method, Botschwina has presented a stretching potential and dipole moment surface for
HCN.2 His corrected potential includes correcting for slight
errors in the calculated equilibrium bond lengths, eliminating the linear dependence on the bond coordinates, and scaling the CN and CH diagonal force constants. In addition, he
empirically included stretch-bend interactions through use
of the experimentall anharmonicity constants xl2 = - 3.4
cm -‘andx,,
= - 18.9 cm - ‘. In order to obtain vibrational energies and wave functions, he diagonalized his vibrational Hamiltonian in a basis set of ~860 normal mode,
harmonic oscillator product functions. Since he worked with
a harmonic basis, the xlZ and xZ3 stretch-bend interaction
constants contributed only to the diagonal elements of his
Hamiltonian. The ab initio dipole moment function is parametrized as a 2 1 term Taylor series in Ar(CH)
and
AR (CN), including diagonal terms up to A/ and AR 5.
From the contour plots shown in Figs. 1 and 2, it is clear
that the CEPA- 1 ab initio potential and our empirical potential are very similar in the experimentally observed region
below 21 000 cm - ‘, which is approximately the energy of
the highest observed state, the (006), including zero point
energy. The largest differences occur on the inner wall,
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
TABLE I(A) Best fit potential constants for the generalized Morse (GM)
model. (B) Best fit potential constants for Baggott and Law’s simple Morse
(SM) model. (C) Best fit potential constants’ for the Carter, Handy, and
Mills (CHM) potential.
(A)
V,,, = 1.161 423 al
F’,,, -0.019998aJ
VII,, = 0.030 954 aJ
a, = 2.306 112 A ’
(B)
fi, = 18.809aJii
a1 = 2.363 A ’
’
V,, = 0.912 139 aJ
V,,, = 0.016 146 aJ
V )JJJ = 0.001 35 al
a, = 1.841393 A ’
V,, = -0.049287a.J
v,,, =0.586X 1o-4 a.l
v,,, = -0.151x10
SaJ
J;, =6.21aJ.i
a, = 1.12lW
f,, = -0.280 aJA* 2
CC)
(20)
(II)
(02)
(30)
(21)
(12)
(03)
(‘lo)*
(31);
a?)*
(13)*
c(M)*
.
B(v) (6,)
(50)*
(41)*
(ii)
0.424 841 1
- 0.010 835 2
0.201 300 0
-
0.000 843
o.ca 410
0.004 843
O.OCCI225
(32)*
(In)*
(14)*
(OS)*
6
0
1
3
(60)*
(51)*
(42)
(33)*
(24)
(Is)*
0.003 182 0
O.ooOl61 3
0.003 085 8
- 0.004 394 9
0.019 3115
2.244131/i
(06)*
’
a,
’
’
O.ooO 236 4
- O.M)2 612 9
-0.000011
8
-0.0004324
-0.0052121
OX042663
- 0.000 132 5
o.ooo 222 1
O.C@
- o.ooa 019 9
O.OOb
-0.005 1484
0.0050242
1.886 098 A
’
” k’(r) ,r, ) = Z,,,B(ij)Y, z”, where the z’s are defined as in Eq. ( 1) of the
text and 1 = (CN); 3 = (CH), as throughout this text. Force constants
marked * were refined in the least squares fit. Potential parameters with
(i + j = 2) and (i + j = 3) were constrained to agree exactly with the harmonic and cubic force constants of Strey and Mills (Ref. 13).
“Constrained to 0.0 in the fit.
which is most easily observed from the difference contours of
Fig. 2. Comparison of his theoretical energy and intensity
values to our experimental values for stretching states
between 6000 and 18 400 cm - ’ has been previously given.5
The energy residuals for the ab initio and empirical potentials are displayed together in Fig. 3. The agreement for intensity values from the fundamentals to very high overtone
bands is phenomenal, considering their dynamic range of lo9
and the fact that no adjustments of the calculation were
made to improve agreement with experiment [see Table
III I. The adequacy of a linear model for the stretching states
of HCN is clearly demonstrated.
III. INTENSITY CALCULATIONS
Absolute intensities covering a dynamic range of one
million to one have been recently determined for 22 stretch-
5043
ing bands of HCN through high resolution vibrational spectroscopy.3” In addition, average dipole moments for eight
vibrationally excited states have been determined.” Using
the ab initio CEPA-1 electric dipole moment function, we
have predicted absolute intensities of vibrational transitions
of HCN as well as average dipole moments from our empirically optimized vibrational wave functions. In addition, we
have performed the identical intensity calculation for the
empirically optimized vibrational wave functions of Baggott
and Law.2o We shall distinguish here the two sets of empirical functions with the labels GM (for our generalized
Morse) and SM (for the simple Morse of Baggott and Law,
whose model potential includes only the z2 Morse potential
terms). Table I(B) gives the parameters for the simple
Morse stretching potential of Baggott and Law. As far as
energy level predictions, both empirical potentials are superior to the ab initio potential. Fig. 3 compares energy residuals for the ab initio and empirical potentials. The rms errors
are 2.41,2.02, and 10.55 cm- i for the GM, SM, and CEPAI, respectively, including all observed stretching states of
H12C!r4N up to the 006.
Since the ub zizitio and empirical potential functions are
very similar, we expected to predict similar intensities with
perhaps ours being in even closer agreement with experiment, since we believed our potential form was somewhat
more accurate. Table III compares the intensities calculated
with the empirical potentials (GM and SM) to the ub initio
results, both using the CEPA- 1 dipole moment function. In
this table, we group together the intensity of bands known to
be in strong anharmonic resonance since their relative intensity is expected to be a hypersensitive function of the potential form. For the lowest transitions the agreement of the
three calculations is excellent; the predictions for the CH
fundamental, for example, are within 1%. Even the predictions for very weak CN fundamental agree remarkably well,
given its unusually small intensity which is due to almost
exactly fortuitous cancellation of the contributions from the
two bond stretches. As one moves up in energy, however,
larger discrepancies appear. Referring to Figs. 4 and 5 and
Table III, our GM potential does better than the CEPA-1
surface at predicting the lower overtones and combination
bands up to about 12 000 cm - ‘. In the region of four and five
quanta of CH stretch, the experimental values lie between
the empirical and CEPA-1 predicitions. For the highest observed levels, the CEPA-1 predictions are clearly superior.
All three calculations fail to predict the large ( 105) band
intensity, but the predictions of the empirical potentials are
considerably worse than that of the ub initio. For the highest
CH overtone transitions, the intensities of the two empirical
potentials are quite .close but substantially smaller (four
times for the 006-000 transition) than the CEPA-1 intensities.
In order to clarify these results, in particular the relative
effects of mode-mode coupling vs changes in the individual
mode potentials, we have looked at the predicted intensities
of a hypothetical C-H diatomic with the same potential and
dipole moment functions as the C-H fragment of the three
HCN potentials. First we have looked at the overlap of the
corresponding eigenstates for the three potentials. We see
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
5044
TABLE II. Residuals of empirical fit to generalized Morse PEF.
Description
Computed
1st order vibration-rotation
HCN
Q3
Q-1
DCN
a3
aI
H’%N
a3
al
HC15N
a3
al
Observed”
Computed - Observed
interaction constants in MHzh
270.2
288.8
313.0
298.9
- 42.8
- 10.1
290.3
174.6
321.8
178.2
- 31.4
- 3.58
248.1
280.1
288.3
287.0
- 40.2
- 7.23
259.5
277.2
302.0
292.0
- 42.55
- 14.80
MIS error = 23.6 MHz
Two dissociation energies’
HCN(g) -H(g)
+ CN(g)AF;H 0.990 851
HCN(g) -HC(g)
+ N(g)A;,OH 1.843 04
0.859 02
1.699 90
0.131 83
0.143 14
Vibrational transitions“ for HCN
100
001
200
101
300
002
201
102
301
003
202
103
(004 + 302 )/2
203
(104+402)/z
005
303
(502 + 204)/2
105
006
2 095.74
3 312.69
4 171.00
5 394.01
6 225.68
6 522.90
7 455.15
8 587.79
9 495.87
9 632.10
10 633.04
11 677.82
12 650.31
13 704.62
14 664.2 1
15 555.82
15 712.66
16 658.75
17 549.32
18 377.54
2 096.68
3 311.48
4 173.07
5 393.70
6 228.59
6 519.61
7 455.42
8 585.57
9 496.43
9 627.08
10 631.44
11 674.45
12 646.88
13 702.24
14 662.1
15 551.9
15 710.5
16 657.3
17 550.4
18 377.0
- 0.94
1.21
- 2.07
- 0.32
- 2.92
3.29
- 0.26
2.22
- 0.56
5.02
1.60
3.36
3.43
2.37
2.15
3.88
2.13
1.49
- 1.07
0.54
rms error = 2.41 cm
’
0.18
- 0.41
1.08
0.55
1.48
0.64
0.50
- 2.97
0.76
- 1.35
- 6.31
- 8.39
rms error = 3.24 cm
’
Vibrational transitions for H’%N
001
101
002
102
003
103
004
005
303
204
105
006
3 293.69
5 343.24
6 484.36
8 519.78
9 573.16
11 592.05
12 562.34
15 452.23
15 573.56
16 543.42
17 425.63
18 248.82
3 293.51
5 343.66
6 483.28
8 519.24
9 571.69
11 591.4
12 561.8
15 455.2
15 572.8
16 544.8
17 431.9
18 257.2
Vibrational transitions for HC15N
001
002
003
005
303
204
105
3 311.11
6 519.32
9 625.91
15 541.59
15 561.80
16 604.58
17 507.40
3 310.09
6 516.50
9 621.73
15 539.7
15 615.6
16 603.6
17 509.9
1.02
2.82
4.18
1.91
2.40
1.02
- 2.48
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
TABLE
5045
II. (continued).
Description
Computed
006
Computed - Observed
Observed”
18 357.73
18 359.9
- 2.14
rms error = 2.44 cm - ’
rms error for all 40 energy levels = 2.69 cm - ’
“Reference 5 and references within.
hThe convention for labeling the a’s is analogous to that of vibrational transitions, that is: Q, is the “CN”
normal mode and Q,, the “CH” normal mode.
‘Reference 18.
‘Units are cm ’. All transitions originate in the vibrational ground state (000).
that the respective wave functions are almost identical along
the C-H coordinate, especially for the two Morse-type potentials. For example, at six quanta of CH stretching excitation, the overlaps of the vibrational wave functions are
0.999 7 16 (SM/GM) and 0.98 1792 (GMKEPA);
at eight
quanta, these are 0.999 624 and 0.975 729.2’
Given these results, one would expect the matrix elements of the dipole moment operator to be very similar, yet
as Table IV shows, the intensities calculated from the same
dipole function show large discrepancies. By the time one
gets to eight quanta of C-H stretch, the predicted intensities
differ by a factor of 24 while the two wave functions have an
overlap of 0.97. When one examines the matrix elements of
x”, one finds that lower powers, which have the smallest
matrix elements, show the greatest divergence.
These result show the extreme sensitivity of the high
overtone intensity data to very small changes in the potential. This sensitivity is so great, that clearly no potential can
be known with sufficient accuracy to determine the dipole
1.50
1
EYPIFUCAL (GM) and AE INITIO POTENTLUS
I
I
moment function from a measurement of all the overtone
intensities, even if the ambiguities of matrix element sign can
be resolved. It is interesting to note that it is often remarked
that one can, in principle, determine the complete dipole
moment function from measurements of a single row of the
dipole moment operator matrix. While this is formally correct, the present results show the instability of any such determination. The fractional errors introduced by uncertainty in the potential appear to grow exponentially as one
examines transitions forbidden to higher order.
In order to compare this calculation to that for the twodimensional problem performed in a Morse oscillator basis
in the section above, the intensities for those transitions
which are derived from the pure CH stretching state are
summed. For example, the 005 and the 402 + 104 are the
three dominant states which share the 005 bond mode intensity. Table V shows that these independent calculations yield
the identical qualitative trend. What we learn by comparing
these calculations is that the assignment of all overtone in-
I
FIG. 1. Contour map of our empirical
(GM) potential (solid) overlapped with
that of the corrected CEPA-1 ab initio potential (dashed). The contours of constant energy begin at 3000 cm - ’ and continue in steps of $ 3000 to 2 1 Ooo cm - ‘,
which is approximately the energy of the
highest observed level including zero
point energy.
I
t
1.10
CH Bond
I
Length
1.30
in A
J. Chem. Phys., Vol. 94, No. 7,1 April 1991
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
5046
FIG. 2. As in Fig. 1, plus a contour
map of the difference [empirical
(GM)-& inifio] potential. The difference potential contours begin at
+ 500.0 cm - ’at the right and continue to a leftmost value of - 2500 cm - ’
in steps of - 100 cm -I; difference
contours with negative values are
dashed.
1.10
1.30
Bond Length
CH
in A
tensity in a given region as coming from a single “local
mode” basis function close in energy is clearly an excellent
approximation for all three model potentials.
IV. CONCLUSIONS
Our recent intensity experiments and the present work
was undertaken with the goal of determining the dipole moment function of HCN, beyond the simple dipole derivatives
commonly determined from infrared fundamental intensities. Our results are, however, disappointing to say the least.
It is clear that not one polyatomic molecule has its potential
energy surface known with sufficient accuracy to predict the
intensities of the high overtone bands, even when these
bands are used in a refinement of the potential energy surface. If a full set of harmonically allowed transitions could be
obtained between the highly excited states, such a determin-
3
2
0
Y
A
-10
A
.
.
A
-15
.
Aab initio
5' -20
;
-
.
+ empirical
A
1-25
h
I
2000
”
’ I
6000
”
’ I”’
II
10000
IL000
11
I
16000
band origin in cm-’
FIG. 3. Frequency residuals for Botschwina’s corrected CEPA-I ab initio
potential compared with those for our empirical (GM) potential.
ation may be possible, but it is clear at present that such an
experiment is not feasible in the near future. It thus appears
that spectroscopists will not practically be able to improve
dipole moment surfaces predicted by highly correlated electron structure calculations such as CEPA; at least beyond
improvements on the equilibrium and dipole derivatives.
One is tempted to try to turn this extreme sensitivity on the
potential around, i.e., use overtone intensity data to help
determine the shape of the potential. One requires a dipole
moment surface. It is not clear, at present, whether ab initio
dipole moment surfaces are of sufficient accuracy that this
can be done effectively. Iachello has found in recent studies
of HCN intensities using the vibron model that overtone
intensities are also extremely sensitive to changes in the dipole moment operator. However, Medvedev has shown that
the inner repulsive wall of the potential is the most important
determinant in overtone intensity falloff.
We must now critically examine the empirical potential
that we have determined in this study by a fit to nearly all the
available spectroscopic data on HCN and its isotopomers.
As spectroscopists, we would like to believe that the potential extracted from such a large amount of very precise data
should be the best potential available, at least in the region
sampled classically by the observed states. One could point
to the improved overtone energies for our empirical potential over the corrected CEPA potential to support this claim,
but that is perhaps misleading as we fit our potential to agree
with this data, unlike the CEPA potential. The empirical
potentials for HCN have predicted the position of unobserved vibrational bands of HCN with similar accuracy as
the fitted levels, but this is not really “independent” data.
Thus this is a necessary but not sufficient condition for trusting the results.
What we need is truly independent observations, and
that is what the observed intensities provide. While we are
forced to use the CEPA dipole moment function, there is no
J. Chem. Phys., Vol. 94, No. 7,l April 1991
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
TABLE
5047
III. Comparison of empirical and CEPA-1 vib. energy levels, intensities, and dipole moments using the CEPA-1 EDMF.
State
no.
Label
1
2
3
4
5
6
7
8
9
10
11
12
13
I4
IS
16
17+ 18
20
22 + 23
25
26
28 + 29
31
34
100
001
200
101
300
002
201
400
102
301
003
500
202
401
103
600
004 + 302
203
402-t 104
005
303
502 + 204
105
006
Energies (cm- ‘)
Empirical-CEPA- 1
SM
GM
- 2.33
1.41
- 4.82
- 0.58
- 1.44
6.15
- 2.84
- 10.20
4.27
- 5.07
12.22
- 13.02
2.46
- 1.39
10.50
- 15.87
18.99
9.06
15.59
23.49
7.99
12.72
21.46
28.09
- 3.95
2.14
- 10.30
- 3.18
- 19.03
9.63
- 10.79
- 30.13
3.09
- 20.73
20.7 1
- 43.52
- 5.52
- 33.05
13.17
- 59.14
17.93
3.86
- 3.33
49.14
- 6.98
- 28.27
39.59
65.91
Empirical
GM
Absolute intensities (cm/mol)
Empirical
CEPA- I
SM
15 018.9
58.34eQ5
170.2
2.487&4
22.78
1.13oeQ5
49.46
0.30
916.9
6.39
2101.6
0.046
1.17
0.922
83.73
0.0029
120.9
2.06
10.44
9.61
0.049
1.201
1.18
0.12
10 871.3
57.83dI5
20.83
2.868&t
16.15
1.343eO5
16.14
0.12
1536.1
16.96
3798.2
0.029
16.87
1.356
184.79
0.0024
285.04
5.19
28.92
28.20
0.205
1.568
4.53
2.88
Obs.
19 201.2
58.16eO5
418.2
2.79leO4
5.87
1.143eo5
45.30
0.21
1088.4
16.60
2285.
0.037
10.21
1.174
103.52
0.0057
140.23
2.75
13.39
11.68
0.11
0.545
I.60
0.93
< 2520.0
60.83~05
2.589eO4
23.21(79)
0.8499cQ5
20.92(86)
1010.0(3.0)
10.53(20)
2547.0( 12.0)
6.79( 10)
lOO.O(5.0)
166.11(1.03)
2.89(4)
17.75( 13)
17.5(4)
0.05(l)
2.1(l)
16.7(4)
2.4(2)
State
no.
Label
Changes in average dipole moments in D”
Observed
Empirical (GM)
CEPA-lh
GMKEPA-I
I
2
6
I1
I7
25
26
28 + 29
31
34
100
001
002
003
004
005
303
502 + 204
105
006
0.003
- 0.027
- 0.054
- 0.080
- 0.099
-0.1279
- 0.070
- 0.133
- 0.125
- 0.147
1.053
0.889
0.897
0.904
0.865
0.918
0.894
0.891
0.919
0.926
9
5
56
7
0
-
4
6
2
6
0.0037
0.0309
0.0608
0.0893
0.1145
0.1394
0.0788
0.1449
0.1362
0.1595
0.004 66
- 0.032 22
- 0.058 81
- 0.109 81
- 0.156 8
- 0.171 8
a Using the experimental p. = - 2.9852 D. The sign of the dipole moment is chosen by Botschwina to be negative at the equilibrium
hObtained through private correspondence with Botschwina.
geometry.
I
A
0
ab lnltio
f empIrical
NQul
u
B
-1
-2
-1
-L
+
I”‘I”‘I”I
2000
11
6000
10000
band
orlgm
lLOO0
“I
I
18000
in cm-l
FIG. 4. Calculated absolute band intensities for HCN 10n -00 transitions
where n = 1,2,3,4,5. The solid line passes through the measured intensities.
Both sets of calculated intensities use the CEPA-1 (ED) ab initio dipole
moment function, and vibrational wave functions from either the CEPA-1
ab initio potential or from our empirical (GM) potential.
2000
”
’
I
”
6000
’
I
10000
band
origin
1’
’
I
IL000
“‘I’
18000
m cm-’
FIG. 5. Calculated absolute band intensities for HCN OOn-OOO transitions
where n = 1,2,3,4,5,6. The solid line passes through the measured intensities. Both sets of calculated intensities use the CEPA-1 (ED) a6 inirio dipole moment function, and vibrational wave functions from either the
CEPA-1 ab initio potential or from our empirical (GM) potential.
J. Chem. Phys., Vol. 94, No. 7,1 April 1991
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
5048
TABLE IV. Comparison of empirical and CEPA-I CH bond energies, intensities, and dipole moments using
the CEPA-1 electric dipole moment function.
Excitation Energies (cm-‘)
Empirical-CEPA- 1
GM
SM
CH quanta
1
2
3
4
5
6
7
8
-
6.4
10.87
15.74
22.73
32.90
46.60
63.88
84.17
CH quanta
-
10.18
16.33
20.33
23.49
26.40
29.04
30.83
30.81
Empirical
GM
Absolute intensities (cm/mol )
Empirical
SM
CEPA- 1
5.51 x lob
1.27x 10’
2.42 x lo”
176.65
18.71
1.81
0.14
0.0066
5.48 x 10h
1.49x 10’
4.41 x 10’
407.72
52.43
7.21
1.04
0.158
Changes in average dipole moments in D
GM
CEPA-1
- 0.029 34
- 0.058 60
- 0.087 34
-0.11501
-0.14088
-0.16406
-0.183 39
- 0.197 49
- 0.204 61
- 0.032 66
- 0.064 65
- 0.095 48
- 0.124 56
-0.15120
-0.17453
- 0.193 47
- 0.206 64
- 0.212 37
5.5 1 x 10h
1.30x 10’
2.73 x 10’
212.6
23.59
2.49
0.23
0.0169
SM
- 0.029 87
- 0.059 51
- 0.088 47
-0.116 19
- 0.141 96
-00.16490
- 0.183 92
- 0.197 72
- 0.204 73
reason to assume that the errors in that function should favor the intensities predicted from either CEPA or an empirical potential. In making just such a comparison, the empirical potential does only slightly better then the CEPA
potential for the lower overtones, and significantly worse for
the highest observed states. In fact the two surfaces are extremely similar over the region sampled by experiment and
the ab initio points. Yet overall, it appears that the ab initio
potential for HCN is in fact the more reliable one, closer to
the potential that only nature knows for sure.
Following this work, Carter, Handy, and Mills (CHM)
have independently determined an empirical force field for
HCNz3 [see Table I(C) 3. They explicitly included all three
degress of freedom, though the basis was optimized for predicting states with mostly stretching excitation. Their method thus did not invoke the approximations for bend-stretch
coupling that we were forced to make. Their potential was
expanded in Morse functions for the stretching modes; thus
we can conveniently compare our method directly with
theirs to estimate the errors introduced by our approximate
treatment of the bending motion. We find that for the states
used in our fit, the rms difference is 4.0 cm - ‘, with the largest difference for the (006) state which we calculate at
18 388.8 cm - ’ and they at 18 376.8 (Botschwina at
18 348.8, experiment 18 377.1). The errors for states with
substantial Y, excitation are somewhat smaller and of opposite sign. In fitting to experimental data, it is clear that by a
very slight change in the stretching potential, most of this
error can be removed in an effective stretching potential. It is
interesting to note that the approximate treatment of the
bend causes the (006) state to be predicted too high by 12
cm-‘, and we would expect a similar error to apply to
Botschwina’s predictions as well. Since his prediction for the
(006) state is too low by 40 cm - ‘, we see that errors would
likely increase slightly by an explicit treatment of the bending mode. Recent work by Gazdy and Bowman has found
errors of similar size in a similar comparison using the Murrell-Carter-Halonen
potential which is quite different from
those considered at present.24
As illustrated by comparing Figs. 1 and 6 (or 2 and 7),
the CHM potential is significantly closer to the ab initio potential than either the SM or GM potentials discussed ear-
TABLE V. Comparison of 1D harmonic oscillator and 2D Morse oscillator basis results.
Intensity group
5CHor
(005 + 402
+ 104 + 303)
6CHor
(006 -I- 105)
GM PEF
HO basis
MO basis
SM PEF
HO basis
MO basis
CEPA- 1
1D HO talc
2D HO talc
18.71
20.62
23.59
25.18
52.43
57.32
1.81
1.90
2.49
2.53
7.21
7.41
J. Chem. Phys., Vol. 94, No. 7,1 April 1991
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Smith, Klemperer, and Lehmann: The fallibility of variational calculations
EUFTRICAL
(CI-ILM~
and
AB INITIO
5049
POTENTIALS
1.50
FIG. 6. Contour map of the empirical
(CHM ) potential (solid) overlapped
with that of the corrected CEPA-1 ab
initio potential (dashed). The contours of constant energy begin at 3GOO
cm-’ and continue in steps of + 3C00
to 21000 cm-‘, which is approximately the energy of the highest observed level including zero point energy.
I
I
I
I
I
0.90
1.10
1.30
1.50
1.70
CH Bond Length in A
lier. As might be expected, the CHM potential predicts intensities that are extremely close to those of the ab initio
potential, being only 4% larger for the (006) state and 9%
larger for the ( 105) state. For many of the weaker combination levels, the differences in intensity are larger, e.g., 16%
for the (202) band. Overall, the CHM does slightly worse
than the ab initio calculation at predicting the observed intensities and thus we cannot confidently conclude that it is
an overall improvement.
Thus it is possible to do better than our present results.
It is important, therefore, to contrast the two fits. In addition
to CHMs explicit treatment of the bending mode, two other
differences appear noteworthy. First, CHM constrained
quadratic and cubic terms in the potential expansion to give
EbmRICAL
(CH&Y)
-
AB INITIO
the same quadratic and cubic force constants that were determined earlier by Strey and Mills in a traditional anharmanic force field fit.13 The second difference is that CHM
used a much higher order expansion than we did (sixth order vs fourth order). The rigid constraints that they imposed
reduced the amount of parameter correlation inherent in
such an expansion. Their paper may leave the false impression that these constraints were used for reasons of computational economy; rather it was a deliberate and astute strategy
as demonstrated by comparison of our two results.25 Earlier
Lehmann et al. pointed out the importance of including vibration-rotation constants to supplement a fit to vibrational
energies, even when variational methods are used.14 The
present results demonstrate that it is likely more accurate to
POTENTIAL
1.50
'q
1.30
.c_
FIG. 7. As in Fig. 6, plus a contour
map of the difference [empirical
(CHM)-ab inirio] potential. The difference potential contours begin at
- 300.0 cm ’ (negative-valued contours are dashed) at the top and continue downwards to + 400 cm ’ in
steps of + 100 cm ‘.
f
P
f
2
P
5
1.x)
1.10
1.30
CH Bond Length in A
J. Chem. Phys., Vol. 94, No. 7,1 April 1991
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5050
Smith, Klemperer, and Lehmann: The fallibility of variational calculations
treat them as constraints than as data equivalent to the vibrational intervals.
Spectroscopy can still determine energy levels to far
higher accuracy than theory (lo-’ compared with 10h3)
but that is rarely the raison d ‘Cfre for spectroscopic study.
One hopes to determine molecular properties, such as structure, forces, and electronic moments, that influence physical
and chemical properties of matter. It appears that even when
extensive spectroscopic information is available, the present
“inversion” methods can, even in a careful study, lead to
errors larger than those of completely ab initio methods, at
least for simple first row molecules such as HCN. Even the
CHM potential, while more accurate than our empirical potential, does not appear to be obviously more accurate than
current ab initio calculations. If we look at molecules more
complicated than triatomics, we can not even calculate the
forward problem (the rovibrational energies given a potential) with even 1O-3 accuracy above 10 000 cm-’ without
approximations in the spirit of our treatment of the bending
mode to reduce the dimensionality of the dynamical space.
Thus our limited ability to predict and treat rovibrational
dynamics remains the weak link in the chain joining theory
with experiment despite the enormous attention that has
been given to this problem by the chemical physics community.
ACKNOWLEDGMENTS
K.K.L.
acknowledges support from the National
Science Foundation in the form of a Presidential Young Investigator Award. This work was supported by the National
Science Foundation.
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J. Chem. Phys., Vol. 94, No. 7, 1 April 1991
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