Molecular equilibrium structures from experimental rotational constants and calculated vibration–rotation interaction constants

(1)

Molecular equilibrium structures from experimental rotational constants and calculated vibration–rotation interaction constants

Filip Pawłowski, Poul Jørgensen, Jeppe Olsen, and Flemming Hegelund Department of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark

Trygve Helgaker

Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway Ju¨rgen Gauss

Institut fu¨r Physikalische Chemie, Universita¨t Mainz, D-55099 Mainz, Germany Keld L. Bak

UNI-C, Olof Palmes Alle´ 38, DK-8200 Aarhus N, Denmark John F. Stanton

Institute of Theoretical Chemistry, Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, Texas 78712

共

Received 26 October 2001; accepted 22 January 2002

兲

A detailed study is carried out of the accuracy of molecular equilibrium geometries obtained from least-squares fits involving experimental rotational constants B₀ and sums of ab initio vibration–

rotation interaction constants␣r

B. The vibration–rotation interaction constants have been calculated for 18 single-configuration dominated molecules containing hydrogen and first-row atoms at various standard levels of ab initio theory. Comparisons with the experimental data and tests for the internal consistency of the calculations show that the equilibrium structures generated using Hartree–Fock vibration–rotation interaction constants have an accuracy similar to that obtained by a direct minimization of the CCSD

共

T

兲

energy. The most accurate vibration–rotation interaction constants are those calculated at the CCSD

共

T

兲

/cc-pVQZ level. The equilibrium bond distances determined from these interaction constants have relative errors of 0.02%–0.06%, surpassing the accuracy obtainable either by purely experimental techniques

共

except for the smallest systems such as diatomics

兲

关

DOI: 10.1063/1.1459782

兴

I. INTRODUCTION

One of the most successful approaches to the determination of molecular equilibrium geometries is based on the expansion of the rotational constants B_v of the vth vibra- tional state in powers of v⫹¹2.^1–7 Such an expansion is obtained by applying perturbation theory to the molecular vibration–rotation Hamiltonian expressed in normal coordinates.^8,9To second order, terms linear in v⫹¹2 appear in the expansion of B_v. The corresponding vibration–

rotation interaction constants, which give the dependence of the rotational constants on the vibrational quantum numbers, are conventionally denoted by ␣r

B. Expressions for the ␣r B

constants in terms of the parameters that occur in the vibration–rotation Hamiltonian

共

e.g., the harmonic and cubic force constants

兲

may be found in Ref. 9.

To second order, the molecular rotational constant for the vibrational ground state B₀ is related to the equilibrium rotational constant B_eas

B₀⫽B_e⫺1

2

兺

_r ^␣^r^B^,

^共

¹

^兲

where the summation is over the 3N⫺6(5) vibrational modes, N being the number of nuclei. The rotational constant B_eis inversely proportional to the principal moment of iner-

tia I_B at the equilibrium geometry. Since, within the Born–

Oppenheimer approximation, the equilibrium geometry is the same for all isotopomers, we may determine B_e and hence the equilibrium geometry by a least-squares fit provided B₀ and

⌺

r␣r

B are known for a sufficiently large number of isotopomers. The purpose of the present study is to carry out a detailed investigation of the accuracy obtainable for molecular equilibrium geometries with this approach.

Whereas accurate experimental rotational constants are available for the vibrational ground state of most small molecules, accurate vibration–rotation interaction constants are more scarce. In principle, they can be determined from the rotational structure of vibrational spectra. However, for larger molecules, their determination becomes increasingly more difficult and is often complicated by Coriolis and Fermi resonances. Alternatively, accurate vibration–rotation interaction constants can be obtained from quantum-chemical calculations. Together with experimental rotational constants of the vibrational ground state, the calculated vibration–rotation constants may be used to determine high-quality equilibrium structures.^1–7We shall refer to these equilibrium structures as empirical, as distinct from purely experimental or purely the- oretical structures.

In the present paper, we examine the accuracy obtainable

6482

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for empirical equilibrium structures at various wave-function and basis-set levels using the vibration–rotation interaction constants obtained from a second-order treatment of the vibration–rotation Hamiltonian, starting from the rigid-rotor harmonic-oscillator Hamiltonian.⁹In particular, the accuracy of the empirical equilibrium structures is examined for four standard ab initio N-electron models: Hartree–Fock self- consistent-field

共

SCF

兲

theory, Møller–Plesset second-order perturbation theory

共

MP2

兲

,¹⁰ coupled-cluster singles-and- doubles

共

CCSD

兲

theory,¹¹and CCSD theory with the pertur- bative treatment of triple excitations

共

CCSD

共

T

兲兲

.¹²In the calculations, the correlation-consistent basis sets of Dunning and co-workers are used.^13–15 The calculations of the vibration–rotation interaction constants are performed for 18 single-configuration dominated molecules, constituting a representative set where the hydrogen and first-row atoms are joined by single, double, and triple bonds.

Although, for many of the molecules considered here, the vibration–rotation interaction constants have previously been reported at selected levels of theory,¹⁶a comprehensive and systematic investigation of the accuracy obtainable at the different levels of theory has not been presented. In this paper, we analyze the results for the 18 molecules statistically so as to establish the accuracy of the calculated vibration–

rotation interaction constants and of the empirical equilibrium geometries at the different levels of theory. The results may be used to predict the accuracy of calculations on molecules with bonds similar to those of the sample molecules but do not necessarily carry over to open-shell molecules and to molecules containing heavier elements.

As the errors in empirical equilibrium geometries arise mainly from errors in the calculated vibration–rotation interaction constants, we begin in Sec. II by analyzing how the errors in the vibration–rotation interaction constants propa- gate to errors in the equilibrium geometries. In Sec. III, some computational details are reported for the calculation of the vibration–rotation interaction constants. The errors in the calculated vibration–rotation interaction constants are then in Sec. IV established by comparing with the experimental constants and by examining the internal consistency of the calculated constants. Next, in Sec. V, we review the experimental rotational constants that are used to obtain the empirical equilibrium geometries. The statistical analysis of the empirical equilibrium geometries is presented in Sec. VI, where we also present the best empirical equilibrium geometries with uncertainties. Section VII contains some conclud- ing remarks.

II. UNCERTAINTIES IN THE EQUILIBRIUM GEOMETRIES

A. Errors in the least-squares fitting

As noted in the introduction, empirical equilibrium geometries can be determined when the experimental rotational constants B₀ and the sums of the calculated vibration–

rotation interaction constants

兺

r␣r

B are known for a sufficiently large set of isotopomers. The geometries may be obtained from least-squares fits based on Eq.

共

1

兲

since the rotational constants B_edepend only on the moments of iner-

tia at the equilibrium geometry, which in the Born–

Oppenheimer approximation is the same for all the isoto- pomers. The differences in B_e are thus only due to differences in isotope masses. In the general case of molecules with three rotational axis, there are three rotational constants A₀

⭓

B₀

⭓

C₀, leading to a set of 3n equations, where n is the number of isotopomers. In the following, we shall examine the accuracy of the empirical equilibrium geometries calculated in this manner.

The empirical equilibrium geometries contain errors arising from the least-squares procedure, as quantified by the residual

⌫⫽ 冑 ^兺

ⁱ

冋

^B^eⁱ^⫺

冉

^B⁰ⁱ^⫹¹²

^兺

^r ^␣^r^Bⁱ

冊册

²^,

^共

²

^兲

where i runs over the rotational constants included in the fit.

In general, the quality of the fitted geometries is expected to improve as the number of rotational constants increases relative to the number of geometrical parameters. However, since the residuals may also arise from inconsistencies in the experimental rotational constants, they do not represent a true measure of the quality of the fit or, for that matter, of the accuracy of the geometries obtained in the refinement. We also note that, when the number of rotational constants is equal to the number of independent geometrical parameters, the residual vanishes. In Sec. VI A, we shall see that the residuals of Eq.

共

2

兲

are small, especially compared with the errors in the vibration–rotation interaction constants. The uncertainties in the geometries arising from the limitations in the least-squares-fitting procedure can therefore be safely ignored.

A more fundamental source of error in the equilibrium geometries arises from the fact that the least-squares fits

共

as carried out in this work

兲

are all based on Eq.

共

1

兲

, which is a truncated expansion of B₀ in v⫹¹2 around B_e, with cubic and higher-order terms neglected. We shall here assume that the omitted terms are small and their effect negligible, but return briefly to this point in Sec. IV A.

Perhaps equally important are the contributions to the vibration–rotation interaction constants that arise from the deviation between the true electronic distribution and that assumed by subsumming the electron masses into the nuclear mass—see Flygare.¹⁷ In structure refinements, such effects have been accounted for in at least two cases—namely, for ketene⁴and for the cyclic isomers of SiC₃

共

Ref. 18

兲

—leading to a considerable improvement in the quality of the least- squares fit and also in the inertial defects calculated for the refined structures. For SiC₃, the differences between the equilibrium bond distances obtained by excluding and in- cluding these electronic contributions were of the order of 0.01 pm

共

with maximum change of 0.05 pm

兲

. In general, therefore, this effect can be neglected for the molecules considered by us.

Finally, there are two sources of error in Eq.

共

1

兲

that may affect the equilibrium geometries: the errors in the experimental rotational constants ␦^B0, and the errors in the sums of the calculated vibration–rotation interaction constants

兺

r␦␣r

B. As discussed in Sec. V, the errors in the experimen-

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tal rotational constants are small compared with the errors in the interaction constants and may be ignored,

␦^B0

⬇

0.

共

3

兲

In conclusion, subject to the above proviso regarding the truncation of the expansion and assuming that the experimental constants have been measured accurately, the only significant source of error in the empirical equilibrium structures are the errors in the calculated sums of the vibration–

rotation interaction constants

兺

r␦␣r

B. In the next sections, we examine how this error propagates into errors in the equilibrium geometries.

B. Relative errors in the equilibrium moments of inertia

From Eq.

共

1

兲

, we obtain

␦^Be

B_e

⬇

1 2

兺

r␦␣r

B

B₀⫹¹2

兺

p␣p

B,

共

4

兲

where we have also assumed the validity of Eq.

共

3

兲

. Since B₀ is large compared with the first-order term

兺

p␣p

B of Eq.

共

1

兲共

see Sec. IV A

兲

, the relative error in B_ebecomes

␦^Be

B_e

⬇ 兺

r␦␣r B

2B₀ .

共

5

兲

The equilibrium rotational constant B_e is a simple function of the principal moment of inertia I_B

B_e⫽ h 8␲²^cIB

,

共

6

兲

where h is Planck’s constant, c the velocity of light in vacuum, and B_e is given in cm^⫺¹. The errors ␦^Be and␦^IB

are therefore related as

␦^Be⫽

冏

^dB^dI^B^e

冏

^␦Î^B^⫽^Bê^␦ÎÎ^B^B^.

^共

⁷

^兲

Using Eq.

共

5

兲

, we then arrive at the following expression for the relative error in the principal moment of inertia

␦^IB

I_B ⫽␦^Be

B_e

⬇ 兺

r␦␣r B

2B₀ .

共

8

兲

We are now in a position to establish the relation between the error in the sum of the vibration–rotation interaction constants

兺

r␦␣r

B and the relative errors in the equilibrium geometry for various types of molecules.

C. Relative errors in bond lengths of X₂, XY, and linear YXY molecules

For the diatomics X₂

共

H₂, N₂, and F₂in the present set

兲

and XY

共

CO and HF

兲

, the moment of inertia I_B is a simple quadratic function of the equilibrium bond distance R_e

I_B⫽␮^Re

2,

共

9

兲

where ␮⫽m_Xm_Y/(m_X⫹m_Y) is the reduced mass of the atoms.¹⁹ The relation between the errors ␦^IB and ␦^Re is therefore given by

␦^IB⫽

冏

^dR^dI^B^e

冏

^␦^Rê^⫽²^␮^Rê^␦^Rê^⫽^2I^B^␦^R^Rêê

^共

¹⁰

^兲

and the relative error in the equilibrium bond distance

␦^Re/R_ebecomes

␦^Re

R_e ⫽␦^IB

2I_B.

共

11

兲

Likewise, for linear molecules of the type YXY, the moment of inertia is a quadratic function of the equilibrium bond distance

I_B⫽2m_YR_e².

共

12

兲

The relative error in the bond distance is therefore also in this case given by Eq.

共

11

兲

. Combining Eqs.

共

8

兲

and

共

11

兲

, we conclude that, for linear molecules of the type X₂, XY, and YXY, the relative error in the equilibrium bond distance may be expressed as

␦^Re

R_e

⬇ 兺

r␦␣r B

4B₀ .

共

13

兲

D. Errors in bond lengths and bond angles of larger molecules

For molecules containing more geometrical parameters

共

bond distances or bond angles

兲

, Eq.

共

11

兲

is in general not satisfied. Nevertheless, Eq.

共

11

兲

is still approximately satis- fied for linear molecules of the type YXXY (C₂H₂) and YXZ

共

HCN and HNC

兲

and for nonlinear molecules of the type YXY

共

H₂O and CH₂

兲

, provided that the relative errors in the interatomic distances are the same

共

see the Appendix

兲

. In the Appendix, it is also shown that, under the same assumption, the error in the bond angles for nonlinear YXY molecules becomes

关

see Eq.

共

A22

兲兴

␦␪

⬇ 兺

r␦␣r B

B₀ tan␪

2,

共

14

兲

where␪is the bond angle.

For larger molecules, the moments of inertia are more complicated functions of the equilibrium geometries. The analytical relations between the relative error in the moment of inertia and the errors in the bond distances and bond angles then become more complex. However, Eqs.

共

13

兲

and

共

14

兲

still give a rough idea of the accuracy of the fitted equilibrium structures.

E. Resonances in the summed vibration–rotation interaction constants

The determination of the empirical equilibrium geometries requires a knowledge of the sum of the vibration–

rotation interaction constants ⫺¹2

⌺

r␣r

B. The individual vibration–rotation interaction constants ␣r

B may be calculated using second-order perturbation theory, starting from the rigid-rotor harmonic-oscillator Hamiltonian,⁹

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␣r

B⫽⫺2B_e²

␻r

冋 ^兺

^␰ ³

^共

^4I^a^r^b^␰^␰

^兲

²^⫹

^兺

^s^⫽^r

^共

^␨^rs^b

^兲

²³^␻^␻^r²^r²^⫺^⫹^␻^␻^s²^s²

⫹␲

冉

^h^c

冊

^1/2

^兺

^s ^␾^rrs^a^s^bb

冉

^␻^␻^s^3/2^r

冊册

^.

^共

¹⁵

^兲

Here ␻r is the harmonic frequency of normal mode r, I_␰ is the ␰th principal component of the moment of inertia at the equilibrium geometry, a_r^abis the derivative of the moment of inertia or inertia product with respect to the normal coordi- nate r,␨rs

b is a Coriolis coupling constant, and␾rstis a cubic force constant. The first contribution to Eq.

共

15

兲

is a moment-of-inertia correction term, the second term is due to Coriolis interaction, and the last term is an anharmonic correction term.

The Coriolis coupling term has a resonance when ␻r

⬇

␻s. By contrast, as recognized by East, Johnson, and Allen,³ the summed Coriolis coupling term

⫺1

2

兺

_r ^␣^r^B

^共

^Coriolis

^兲⫽ 兺

_r _␻^B_r^e²

兺

_s_⫽_r

^共

^␨^rs^b

^兲

²³_␻^␻_r²^r²_⫺^⫹_␻^␻_s²^s²

⫽⫺s

兺

⬎r

B_e²

共

␨rs

b

兲

²

共

␻r⫺␻s

兲

²

␻r␻s

共

␻r⫹␻s

兲共

16

兲

does not have a resonance contribution. In cases of Coriolis resonance, therefore, the summed Coriolis contribution is un- affected by the resonance and may be safely calculated. Still, for strong Coriolis interactions, one has to be careful when comparing individual theoretical and experimental vibration–rotation interaction constants. Among the molecules studied here, only CH₂O exhibits a strong Coriolis resonance. Accordingly, this system is excluded from our direct comparison of individual values.

Another problem may arise due to Fermi resonances when ␻r

⬇

␻s⫹␻t. However, the calculated individual vibration–rotation interaction constants are free of contributions from Fermi resonances as no denominator of the type

␻r⫺(␻s⫹␻t) appears in the expression for the vibration–

rotation interaction constants Eq.

共

15

兲

. Thus, the sum

⫺¹2

⌺

r␣r

B is not affected by Fermi resonances either. By contrast, experimentally determined vibration–rotation interaction constants may include resonance contributions and must be carefully corrected for comparison with calculated values.

We conclude that the use of Eq.

共

1

兲

to determine equilibrium geometries depends on whether the vibration–

rotation interaction constants are determined experimentally or theoretically. With regard to Coriolis resonance contributions, the individual experimental and theoretical vibration–

rotation interaction constants contain such contributions but their sum in Eq.

共

1

兲

does not—see Eq.

共

16

兲

. With regard to Fermi resonance contributions, no such contributions appear in the theoretical constants, whereas the experimental constants may contain such contributions, which must be carefully eliminated before the constants can be used to determine the equilibrium geometries.

III. COMPUTATIONAL DETAILS

The vibration–rotation interaction constants have been calculated for the following 18 single-configuration dominated molecules containing hydrogen and first-row atoms:

H₂, N₂, F₂, HF, CO, CO₂, HCN, HNC, C₂H₂, CH₂, H₂O, HNO, HOF, N₂H₂, CH₂O, C₂H₄, NH₃, and CH₄ using the SCF, MP2, CCSD, and CCSD

共

T

兲

models. For each model, we have used the correlation-consistent polarized valence X-tuple zeta (cc-pVXZ),¹³ correlation-consistent polarized core–valence X-tuple zeta (cc-pCVXZ),¹⁴ and augmented cc-pVXZ (aug-cc-pVXZ)

共

Ref. 15

兲

basis sets of Dunning and co-workers. The calculations were carried out for the cardinal numbers 2

⭐

X

⭐

4 except that the aug-cc-pVTZ, aug-cc-pVQZ, and cc-pCVQZ calculations were not performed for CO₂, HCN, HNC, C₂H₂, HNO, HOF, N₂H₂, CH₂O, C₂H₄, NH₃, and CH₄.

In each case, the vibration–rotation interaction constants were calculated and the geometry was optimized at the same level of theory, correlating all electrons. Analytical second- derivative techniques²⁰were used to obtain the quadratic and cubic force fields, with the latter computed as described in Ref. 21. The calculations were carried out using a local ver- sion of the ACESII program.²² In the following, we shall refer to the quantities calculated at the CCSD

共

T

兲

/cc-pVQZ level as the best calculated values

共

the reference values

兲

. When necessary, we shall label the reference values with ( )^°.

IV. ACCURACY OF THE CALCULATED

VIBRATION–ROTATION INTERACTION CONSTANTS In this section, we examine the accuracy of the vibration–rotation interaction constants calculated at differ- ent levels of ab initio theory. The accuracy is established in two ways: first, by comparing the best calculated and experimental constants in Sec. IV A; next, by estimating

共

from the internal consistency of the calculations

兲

the wave-function/

basis-set limits of these constants in Sec. IV B.

A. Comparison of the CCSD„T…Õcc-pVQZ vibration–rotation interaction constants with the experimental values

In Sec. IV A 1, we examine the accuracy of the best calculated constants (␣r

B)^° by comparing with the experimental values; in Sec. IV A 2, the sums

⌺

r (␣r

B)^° are examined.

1. Comparison of the individual␣r

Bconstants

In Tables I and II, we have listed the best calculated and experimental vibration–rotation interaction constants ␣r

A,

␣r

B, and ␣r

C for all molecules, noting that complete sets of experimental constants are not available for NH₃, N₂H₂, C₂H₄, CH₄, and CH₂O. For the linear molecules, the experimental vibration–rotation interaction constants are usually larger than the calculated ones; for the nonlinear molecules, no such systematic trend can be discerned.

In Tables I and II, we have also presented the errors in the calculated vibration–rotation interaction constants, esti- mated as

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␦␣r B⫽

共

␣r

B

兲

^°⫺␣_r^B^exp^.

共

17

兲

The largest discrepancies are found for the A rotational con- stant of the planar molecules

共

H₂O, HNO, CH₂, and HOF

兲

, which are close to quasilinearity. The quasilinearity causes a strong coupling of the rotation about the a axis with the symmetrical bending mode; this coupling is neglected in our calculations. In general, for planar molecules, there is a near dependency among the equations for the three rotational con- stants. The equations for the A constants are therefore omit- ted from the least-squares fit for the planar molecules when the empirical equilibrium geometries are determined—see Sec. V. The large discrepancies observed for these constants will therefore not affect the empirical geometries; the re- maining discrepancies are mostly small compared with␣_r^B^exp^. The experimental vibration–rotation interaction constants become less accurate as the size of the molecule increases. To illustrate this point, we have, in the last columns of Tables I and II, given the relative discrepancies between the calculated and experimental constants

␦␣r B_relat

⫽

共

␣r

B

兲

^°⫺␣_r^B^exp

共

␣r

B

兲

^° .

共

18

兲

The smallest errors

共

less than 2%

兲

are found for the diatomics, for which the experimental vibration–rotation interaction constants have been determined to high accuracy. Likewise, for the linear triatomic molecules HCN, HNC, CO₂, the experimental results are of similar accuracy and the largest error is 6.7%

共

for the

⌸

g bending mode of CO₂

兲

. For the four-atomic linear molecule C₂H₂, the relative errors are larger, with an error of 28% for the

⌸

g bending mode

共

although the absolute error is small

兲

.

By contrast, for the bent triatomic molecules H₂O, HNO, CH₂, HOF

共

for which the A constant has been ex- cluded from the least-squares fit

兲

, we obtain relative errors for the B and C vibration–rotation interaction constants that in many cases are significantly larger than those for the linear molecules. For HNO, the absolute relative error is as large as 309%

共

for the interaction between the C rotational mode and the symmetrical stretching mode

兲

; moreover, the calculated and experimental vibration–rotation interaction constants sometimes differ in sign. The coupled-cluster model is size extensive and the calculated vibration–rotation interaction

constants are expected to have similar accuracy for small and large molecular systems.

In general, we have used␦␣r

B as a measure of the error in the calculated vibration–rotation interaction constants. For the diatomics and for the linear triatomics, this leads to small errors in the calculated interaction constants; for the nonlinear molecules, it gives larger errors. As␦␣r

Bcontains experimental errors,␦␣r

Bmay be viewed as a conservative estimate of the error in the calculated vibration–rotation interaction constants.

2. Comparison of⌺r␣r B

In Table III we have collected the best calculated and experimental values of the sum of the vibration–rotation interaction constants

⌺

r␣r

B. We have also included

兺

r ␦␣r

B⫽

兺

r

共

␣r

B

兲

^°⫺

兺

r ␣_r^B^exp

共

19

兲

as estimates of the error in the best calculated sums of vibration–rotation interaction constants, as well as the relative sums

⌺

r(␣r

B)^°/B₀ and the relative errors

⌺

r␦␣r B/B₀. Due to lack of experimental data, NH₃, N₂H₂, C₂H₄, CH₂O, and CH₄ are not included in this table.

The discrepancies of

⌺

r␣r

B are similar to those of the individual ␣r

B constants. Thus, the largest differences occur for the A rotational constant of the quasilinear triatomics. For H₂O, for example,

⌺

r␦␣r

A⫽0.5053 cm^⫺¹, which is significantly larger than

⌺

r␣r

Bfor many of the considered rotational constants. We recall, however, that the A rotational constants of the planar molecules are not needed to determine the empirical equilibrium geometries.

The values of

⌺

r(␣r

B)^°/B₀ in Table III confirm that the rotational constants B₀ are considerably larger than

⌺

r␣r

B, justifying the assumption that we made in going from Eq.

共

4

兲

to Eq.

共

5

兲

.

For diatomics, the constant␥e—that is, the cubic term in the expansion of B₀in Eq.

共

1

兲

—is of the same magnitude as or smaller than␦␣r

B. For HF, N₂, and CO, for example, ␥e

is equal to 0.0127, ⫺0.000 03, and 0.000 0005 cm^⫺¹, respectively.²³ Since ␥e is multiplied by (¹₂)² to obtain its contribution to the rotational constant, it appears well justi- fied to neglect the cubic terms in Eq.

共

1

兲

for the diatomics.

TABLE I. Experimental␣r B_exp

and calculated CCSD共T兲/cc-pVQZ (␣r

B)° vibration–rotation interaction constants of the diatomic molecules. In the last two columns, the absolute discrepancy␦␣r

Band relative discrepancy␦␣r

B_relat

are listed.

Molecule

Rot. const.

B0关cm^⫺¹兴 (␣r B)°

关cm^⫺¹兴 ␣r

B_exp

关cm^⫺¹兴 ␦␣r

B

关cm^⫺¹兴 ␦␣r

B_relat

关%兴

H2 59.334 3.020 3.062^a ⫺0.042 ⫺1.38

F2 0.882 95 0.012 57 0.012 595^b ⫺0.000 025 ⫺0.198

N2 1.989 59 0.016 95 0.017 29^c ⫺0.000 34 ⫺2.01

HF 20.559 0.787 0.798^a ⫺0.011 ⫺1.41

CO 1.922 53 0.017 24 0.017 50^a ⫺0.000 27 ⫺1.55

aK.P. Huber and G.H. Herzberg, Constants of Diatomic Molecules共Van Nostrand Reinhold, New York, 1979兲.

bR.Z. Martinez, D. Bermejo, J. Santos, and P. Cancio, J. Mol. Spectrosc. 168, 343共1994兲.

cJ. Bendtsen, J. Raman Spectrosc. 2, 133共1974兲.

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TABLE II. Experimental ␣r B_exp

and calculated CCSD共T兲/cc-pVQZ (␣r

B)° vibration–rotation interaction constants of polyatomic molecules. For a given rotational constant, the vibrational modes are listed in order of increasing energy. In the last two columns, the absolute discrepancy ␦␣r

B and relative discrepancy␦␣r

B_relat

are listed.

Rotational Rotational

constant Mode (␣r

B)° ␣_r^B^exp ␦␣r

B ␦␣r

B_relat

constant Mode (␣r

B)° ␣_r^B^exp ␦␣r

B ␦␣r

B_relat

关cm^⫺1兴 r 关cm^⫺1兴关cm^⫺1兴关cm^⫺1兴关%兴关cm^⫺1兴 r 关cm^⫺1兴关cm^⫺1兴关cm^⫺1兴关%兴

CO₂

B₀⫽0.3902 ⌸g ⫺0.000 75 ⫺0.0007^a ⫺0.000 05 6.67

⌺g⫹ 0.0012 0.0012^a 0.0000 0.00

⌺u⫹ 0.0031 0.0031^a 0.0000 0.00

HCN

B0⫽1.4782 ⌸ ⫺0.003 75 ⫺0.003 57^b ⫺0.000 18 4.80

⌺^⫹ 0.009 78 0.010 01^b ⫺0.000 23 ⫺2.36

⌺^⫹ 0.010 27 0.010 43^b ⫺0.000 16 ⫺1.55 HNC

B0⫽1.5121 ⌸ ⫺0.0054 ⫺0.0051^c ⫺0.0003 5.56

⌺^⫹ 0.0109 0.0112^c ⫺0.0002 ⫺2.24

⌺^⫹ 0.0100 0.0098^c 0.0002 2.04

C₂H₂

B0⫽1.1766 ⌸g ⫺0.001 885 ⫺0.001 354^d ⫺0.000 531 28.2

⌸u ⫺0.002 388 ⫺0.002 232^d ⫺0.000 156 6.53

⌺g⫹ 0.005 753 0.006 181^e ⫺0.000 429 ⫺7.46

⌺u⫺ 0.005 681 0.005 882^e ⫺0.000 201 ⫺3.54

⌺g⫹ 0.006 689 0.006 904^e ⫺0.000 215 ⫺3.21

H₂O

A0⫽27.8807 A1 ⫺2.607 067 ⫺3.247 630^f 0.640 563 ⫺24.57 A1 0.693 204 0.758 570^g ⫺0.065 366 ⫺9.43 B2 1.162 759 1.232 670^g ⫺0.069 911 ⫺6.01 B0⫽14.5217 A1 ⫺0.156 019 ⫺0.165 880^f 0.009 861 ⫺6.32 A1 0.218 836 0.216 980^g 0.001 856 0.85 B2 0.095 603 0.090 380^g 0.005 223 5.46 C0⫽9.2774 A1 0.148 081 0.148 320^f ⫺0.000 239 ⫺0.16 A1 0.177 715 0.172 950^g 0.004 765 2.68 B2 0.142 656 0.139 290^g 0.003 366 2.36

HNO

A0⫽18.4792 A⬘ ⫺0.3392 ⫺0.3455^h 0.0062 ⫺1.83 A⬘ ⫺0.1204 ⫺0.0975^h ⫺0.0228 18.97

A⬘ ^0.8691 ^0.8031^h ^0.0661 ^7.60

B0⫽1.4115 A⬘ ^0.0035 ^0.0014^h ^0.0021 ^60.58 A⬘ ^0.0098 ^0.0101^h ⫺0.0003 ⫺3.17 A⬘ ⫺0.0063 ⫺0.0078^h 0.0014 ⫺22.24 C0⫽1.3071 A⬘ ^0.0252 ^0.0149^h ^0.0103 ^40.73 A⬘ ⫺0.0035 0.0073^h ⫺0.0108 309.16 A⬘ ⫺0.0029 ⫺0.0005^h ⫺0.0024 82.15

CH₂

A0⫽20.1182 A1 ⫺1.5974 ⫺1.7082ⁱ 0.1108 ⫺6.94

A1 0.5022 0.2890ⁱ 0.2132 42.46

B2 0.8600 0.7942ⁱ 0.0658 7.65

B0⫽11.2050 A1 ⫺0.1274 ⫺0.1214ⁱ ⫺0.0060 4.69 A1 0.1523 0.1613ⁱ ⫺0.0090 ⫺5.90 B2 0.0534 0.0572ⁱ ⫺0.0038 ⫺7.13

C0⫽7.0686 A1 0.1255 0.1251ⁱ 0.0004 0.31

A1 0.1277 0.1231ⁱ 0.0046 3.59

B2 0.0931 0.0831ⁱ 0.0100 10.75

HOF

A0⫽19.5346 A⬘ ^0.0244 ^0.0193^j ^0.0051 ^20.82 A⬘ ⫺0.4757 ⫺0.4996^j 0.0239 ⫺5.01 A⬘ ^0.7350 ^0.7387^k ⫺0.0037 ⫺0.51 B0⫽0.8926 A⬘ ^0.0110 ^0.0112^j ⫺0.0002 ⫺1.90

A⬘ ^0.0038 ^0.0038^j ^0.0000 ^0.00

A⬘ ^0.0007 ^0.0004^k ^0.0003 ^41.71

C0⫽0.8509 A⬘ ^0.0108 ^0.0110^j ⫺0.0002 ⫺1.88 A⬘ ^0.0070 ^0.0072^j ⫺0.0002 ⫺3.04

A⬘ ^0.0018 ^0.0016^k ^0.0002 ^8.47

aJ.L. Teffo, O.N. Sulakshina, and V.I. Perevalov, J. Mol. Spectrosc. 156, 48共1992兲.

bA.G. Maki, G. ChMellan, S. Klee, M. Winnewisser, and W. Quapp, J. Mol. Spectrosc. 202, 67共2000兲.

cR.A. Creswell and A.G. Robiette, Mol. Phys. 36, 869共1978兲: first and third values from Table II footnote d and second value from Table II footnote e共R.G.

Woods et al., private communication兲.

dY. Kabbadj, M. Herman, G. Di Lonardo, L. Fusina, and J.W.C. Johns, J. Mol. Spectrosc. 150, 535共1991兲.

eM.A. Temsamani and M. Herman, J. Chem. Phys. 105, 1355共1996兲.

fC. Camy-Peyret and J.-M. Flaud, Mol. Phys. 32, 523共1976兲.

gJ.M. Flaud and C. Camy-Peyret, J. Mol. Spectrosc. 51, 142共1974兲with the use of A0, B0, C0from Table XI of this paper.

hJ.W.C. Johns, A.R.W. McKellar, and E. Weinberger, Can. J. Phys. 61, 1106共1983兲.

iH. Petek, D.J. Nesbitt, D.C. Darwin, P.R. Ogilby, C.B. Moor, and D.A. Ramsay, J. Chem. Phys. 91, 6566共1989兲.

jH. Bu¨rger, G. Pawelke, A. Rahner, E.H. Appelman, and J.M. Mills, J. Mol. Spectrosc. 28, 278共1988兲.

kH. Bu¨rger, G. Pawelke, A. Rahner, E.H. Appelman, and J.M. Mills, J. Mol. Spectrosc. 138, 346共1989兲.

The fact that the

⌺

r␣r

B/B₀ have similar magnitudes for all molecules suggests that the cubic terms in B_ein general can be neglected.

From Eqs.

共

13

兲

and

共

14

兲

, we see that

⌺

r␦␣r

B/B₀ propagates directly into the errors in the equilibrium geometries.

For the rotational constants of Table III that are included in the least-squares fits, this ratio does not exceed 0.23% and it is especially small for the diatomic and linear triatomic molecules. In addition, the ratio is small for bent triatomic HOF, for which we have found a surprisingly good agreement between the calculated and experimental vibration–rotation interaction constants. A conservative estimate of the relative error in the bond distances is 0.06%, obtained by inserting

the largest value of

⌺

r␦␣r

B/B₀ in Table III

共

0.23%

兲

into Eq.

共

13

兲

.

B. Statistical analysis of the accuracy of the calculated vibration–rotation interaction constants

In Sec. IV B 1, we carry out a statistical analysis to establish the accuracy of the ␣r

B constants that are calculated with the different N-electron models and basis sets. Next, in Sec. IV B 2, we examine the errors in

⌺

r␣r

Brelative to B₀, as this quantity essentially determines the accuracy of the empirical equilibrium structures.

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1. Statistical analysis of the errors in the calculated

␣r

B constants

To investigate the accuracy of the␣r

B constants that are calculated using the different standard models and basis sets, we examine the deviation of the calculated␣r

B from the best calculated value (␣r

B)^°:

⌬

␣r B⫽␣r

B⫺共␣r

B

兲

^°.

共

20

兲

In adopting this error measure, we are assuming that the errors in the best calculated values are negligible compared with the errors obtained at lower levels. In the statistical analysis, we consider the mean error

⌬

¯

共

Table IV

兲

, the standard deviation

⌬

std

共

Table V

兲

, the mean absolute error

⌬

¯

共

Table VI

兲

, and the maximum error

⌬

max

共

Table VII

兲

. abs

The errors in the cc-pVXZ and cc-pCVXZ basis sets are similar for all wave-function models, indicating that valence basis sets may safely be used for calculating the vibration–

rotation interaction constants. Larger deviations exist between the results obtained with the cc-pVXZ and

aug-cc-pVXZ basis sets. In particular, as we increase the cardinal number X, a monotonic change is usually observed for the different statistical error measures for the cc-pVXZ family; for the aug-cc-pVXZ basis sets, only the mean error changes monotonically.

The mean errors in Table IV decrease

共

become more negative

兲

with increasing cardinal number but increase with improvements in the wave-function description in the se- quence SCF, MP2, CCSD, CCSD

共

T

兲

. Except for the corre- lated models in the smallest basis sets, the mean and absolute mean deviations have the same magnitude but opposite signs, indicating that the errors in the calculated vibration–

rotation interaction constants behave in a systematic manner.

Looking at the trend as the cardinal number increases, it is clear that basis-set saturation has not been reached at the quadruple-zeta level. Still, the changes from triple-zeta to quadruple-zeta basis sets are rather small compared with the differences between the best calculated␣r

B constants and the experimental ones. In general, the uncertainty

共

mean abso-

TABLE III. Experimental⌺r␣r B_exp

and calculated CCSD共T兲/cc-pVQZ⌺r(␣r

B)° sums of the vibration–rotation interaction constants corresponding of the different rotational constants. For the nonlinear molecules, the rota- tional constants are listed in the order A0, B0, C0.

Molecule

Rot. const.

B0关cm^⫺1兴 ⌺r(␣r B)°

关cm^⫺1兴 ⌺r␣r B_exp

关cm^⫺1兴 ⌺r␦␣r B

关cm^⫺1兴 ⌺r(␣r B)°/B0

关%兴 ⌺r␦␣r

B/B0

关%兴

H2 59.334 3.0205 3.0620 ⫺0.0415 5.16 ⫺0.07

F2 0.8830 0.0126 0.0126 ⫺0.0000 1.46 ⫺0.0

N2 1.989 59 0.0170 0.0173 ⫺0.0003 0.87 ⫺0.02

HF 20.559 0.7869 0.7980 ⫺0.0111 3.88 ⫺0.05

CO 1.922 53 0.0172 0.0175 ⫺0.0003 0.91 ⫺0.01

CO2 0.3902 0.0027 0.0028 ⫺0.0001 0.72 ⫺0.02

HCN 1.4782 0.0126 0.0133 ⫺0.0007 0.90 ⫺0.05

HNC 1.5121 0.0102 0.0107 ⫺0.0005 0.71 ⫺0.03

C2H2 1.1766 0.0096 0.0118 ⫺0.0022 1.00 ⫺0.19

H2O 27.8807 ⫺0.7511 ⫺1.2564 0.5053 ⫺4.51 1.81

14.5217 0.1584 0.1415 0.0169 0.98 0.12

9.2774 0.4685 0.4606 0.0079 4.96 0.08

HNO 18.4792 0.4095 0.3601 0.0494 1.95 0.27

1.4115 0.0070 0.0038 0.0032 0.27 0.23

1.3071 0.0189 0.0218 ⫺0.0029 1.66 ⫺0.22

CH2 20.1182 ⫺0.2352 ⫺0.6250 0.3898 ⫺3.11 1.94

11.2050 0.0783 0.0971 ⫺0.0188 0.87 ⫺0.17

7.0686 0.3463 0.3313 0.0150 4.69 0.21

HOF 19.5346 0.2836 0.2584 0.0252 1.32 0.13

0.8926 0.0154 0.0153 0.0001 1.72 0.01

0.8509 0.0195 0.0198 ⫺0.0003 2.33 ⫺0.03

TABLE IV. Mean errors in the calculated vibration–rotation interaction constants共cm^⫺¹兲relative to the reference CCSD共T兲/cc-pVQZ level.

SCF MP2 CCSD CCSD共T兲

cc-pVDZ ⫺0.060 ⫺0.009 0.008 0.017

cc-pVTZ ⫺0.064 ⫺0.014 ⫺0.001 0.008

cc-pVQZ ⫺0.072 ⫺0.021 ⫺0.009 0.000

aug-cc-pVDZ ⫺0.069 ⫺0.014 ⫺0.000 0.009 aug-cc-pVTZ ⫺0.072 ⫺0.022 ⫺0.008 0.001 aug-cc-pVQZ ⫺0.108 ⫺0.040 ⫺0.017 ⫺0.007

cc-pCVDZ ⫺0.040 0.011 0.020 0.029

cc-pCVTZ ⫺0.054 ⫺0.007 ⫺0.003 0.007

cc-pCVQZ ⫺0.084 ⫺0.018 ⫺0.011 0.000

TABLE V. Standard deviations of errors in the calculated vibration–rotation interaction constants 共cm^⫺1兲 relative to the reference CCSD共T兲/cc-pVQZ level.

SCF MP2 CCSD CCSD共T兲

cc-pVDZ 0.119 0.113 0.086 0.097

cc-pVTZ 0.118 0.042 0.025 0.033

cc-pVQZ 0.134 0.045 0.020 0.000

aug-cc-pVDZ 0.149 0.088 0.058 0.064

aug-cc-pVTZ 0.132 0.035 0.031 0.023

aug-cc-pVQZ 0.172 0.057 0.037 0.016

cc-pCVDZ 0.077 0.056 0.062 0.076

cc-p-CVTZ 0.110 0.027 0.016 0.030

cc-pCVQZ 0.158 0.023 0.024 0.001

Molecular equilibrium structures from experimental rotational constants and calculated vibration–rotation interaction constants