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Harmonic force constants of formic acid: ab initio results and the uniqueness problem of force fields derived from vibrational data
Volume 59, number 1
CHEMICAL PHYSICS LETTERS
1 November 1978
H!ARMONK FORCE CONSTANTS OF FORMIC ACID: AR INITIO RESULTS AND THE UNIQUENESS PROBLEM OF FORCE FIELDS DERIVED FROM VIBRATIONAL Tae-Kyu HA, R. MEYER
DATA
and Hs- H. CmRD
Lubomtony of Physicd chernishy. Swiss Fe&ml Insritute of Technology. CH-8092 Zurich. swinertand
Received4 July 1978 A complete set of harmonic force constants of formic acid (HCOOH) is caIcula:ed using a 6-31G basis set. It is used to determine a modified valence force field consistent with both ab initio results and vibrational datz
The unusualIy large amount of assigned formic acid fimdamentals published recentIy by Redington [l] has prompted us to compute a complete harmonic valence force field ab initio for comparison with observations. In the present work, special attention is paid to the problem of comparing calculated with empirical force constants since the latter cannot be determined uniqueIy from vibrationaI data for polyatomic molecules. Some lack of uniqueness remains even in the case of fo,mic acid where 201 observed fundamentaIs stiU do not seem to determine all of the 3 1 quadratic force constants. The present ab initio force field is in good overall agreement with Redington’s data [l] . However, there are a number of discrepancies which motivated us to investigate the question whether a force field compatible both with ab initio results and the observed frequencies can be found. Such a procedure is aimed at restricting the manifold of soiutions of the normal coordinate probiem. Biom et al. 121 have derived a set of scale factors which should allow the correction of ab initio force constants in order to produce empirical force constants for series of similar molecules. These authors suggest the use of scale factors common to related molecules for predictive calculation of normal frequencies. In a quantum chemical study by Schlegel et al. [3] predicted force constants were found to agree remarkably well with data derived from vibrational spectra_ Based on the result the authors suggest the reexamination of the empirical force fields.
In the present paper use is made of ab initio constants, as well as of Redington’s vibrational dats to determine an alternative force field of formic acid, which represents a compromise between the two approaches_ The ab initio SCF calculations have been carried out using an extended basis set of contracted gaussian functions (6-31G) [4]. The force constants of formic acid have been calculated from ab initio SCF energies. The re structure reported by Kwei and Curl [S] has been used as an initial structure and the inte,tis of bond lengths, bond angles and torsional angles were chosen in such a way that the increase in the SCF energy due to each displacement amounts to about 1000 cm-’ compared to that of the initial structure. Owing to the anharmorzcity of the ab initio energy surface the results for the force constants obtained by interpolation will obviousIy depend on the spacing of the sample points. With an optimal choice of these spacings the interpolation process should be similar to the one involved in the evaluation of empirical force constants by a normal coordinate treatment of the spectra_ On the other hand, the intervals should be small enough, especially for bond lengths, to avoid excessive overestimation of stretching force constants, which is inherent in the Hartree-Fock type SCF calculations_ The above criterion was used in an attempt to define suitable intewals in the order of magnitude of the vibrational amplitudes in the lowest states. Similar considerations were applied previously in 2 calculation of force constants of methanol [6] and formi7
Volmne 59, number 1
C.XJWIGW PHYSICS LETTERS
1 November 1978
Tabfe 1 Valence force I’ieldof formic acid monomer
co-m
7.237 C20) - 0.029(19) -0.099 (49) 0.18F(94) 0.20 (13) -0.097 (55) -0.028 (19)
7.242(18) -0.0346(42) -0.090 (11) 0.175(21) 0.260(31) -0.102 (12) -0.0351(42)
8.03 -0.035 -0_090 3.174 0.261 -0.102 -0.035
4.705 (22) 0.26 (11)
4.756 (15) 0.263(33) 0132 (16) 0.0367(46) -0.137
5.62 0.299 0.132 0.038 -0_153 -0.027
13.35 (63) 0.91 (51) 0.47:(78) 1.00 (34) 0.721(82)
13.05 (40) 1.21 (38) 0.031 (13) 0.14 (16) 0.40 (11)
13.34 (16) 1.52 (14) 0.0243(29) 0.336(43) 0.360(38)
15.37 1.313 0.024 0.367 0.345
6.00 (35) 0.092(6.5) 0.004(111) -0.404 02)
6.1C (21) 0.184(37) 0.170(80) -0.347 (16)
6.226 (64) 0.21703) 0.211(24) -0.344 (12)
7.67 0.350 0.203 -0.389
0.696(11) -0.099 (25) -0.044 (13)
0.6453(81) -0X32(81) -0.0606 (83)
0.6454(49) -0.1452(45) -0.0604(36)
0.760 -0.164 -0.058
F 67
1.297
1.248 (39) 0.007s(92)
I.204 (13) 0_0179(22)
1.340 0.018
SC-m
F?7
0.624(8)
0_6142(28)
0.6162(25)
0.750
WC--H)
53 F89
0.470(2) 0.092(4)
0.4698(28) 0.0922<41)
0.4704(28) 0.093Oi42)
0.580 0.095
F99
0.168(1)
0.16750(82)
O-16766(87)
0.190
0.46%
058%
0.60%
v
7.188(17)C)
Fll PI2 Fl3 F14 FIS fi6 Fl7
V (C-H)
4.666 (14)
Fa2 F23 F24
0.116 0.020 -0.024 -0.025
F25 F26 F27 F33 54 F35 F36 57 P(C-0)
F44 5s F46 F47
6KOH)
FSS Fs6 Fs7
~Gxm~
r(c-o) OWV),
a) F-a&ix B from ref. [I 1.
F66
(69) (20) (61) (14)
b) See text
c) U&S are mdyn/A for stretching and mdyn A/wd2 for bending coordinates Numbers in bracketsgive oflitsofhstdigit
18
deviationsin
Volume 59, nmber
1
CHEMICAL PHYSICS LETTERS
anhydride ]7] _Using the internal coordinates chosen by Redingron [I], diagonal force constants were evaluated by 3 point interpolation while off-diagonal elements were evaluated from the SCF energies calculated on a 32 point grid in the two-dimensional subspaces spanned by the respective pairs of internal coordinates. The results are shown in the last column of table 1. Bosi et al. [8] have previously reported 5 ab initio diagonal force constants of formic acid monomer in connection with their dynamical charge transfer study of the cyclic dimer. The total SCF energy of their basis set (9/4 basis set) amounted to - 188.6689 au for the equilibrium geometry, which is comparable to - 188.6627 au for the empirical structure [S] obtained from the 6-3 1G basis set of this work. The force constants reported by Bosi et al. [S] are: Flr(O-H) = 8.16,F22(C-H) = 6.16, F33(C=O) = 13.44, F.++(C-0) = 8.05 and FsS(COH) = 0.81 in mdyn/A. Except for the F33(C=O) value all these diagonal force constants are higher than those of the present work (see table 1). By -comparing columns I and IV of table 1 one notes that the calculated diagonal force constants are higher than the empirical ones throughout with deviations ranging up to 28%. For the off-diagonal elements there is fair overall agreement. However, discrepancies significantly exceed experimental errors stated for the interaction constants of C=O stretching with bending coordinates and of C-O stretching with COH and OCO bending. In discussing the two sets of force constants we have to consider the possibility of errors in the empirical set as well as in the quantum chemical one. It is well known in vibrational analysis that empirical force fields are usually ill determined owing to limitations of the harmonic approximation and of the experimental accuracy. Furthermore the particular set of observed data will determine only part of the force field but will leave unknown other properties of the potential. As a consequence, the analysis of experimental data allows for multiple solutions, and a specific least squares solution arrived at may have been affected by hidden bias dictated by the actual supply of experimental information and by details of the computational method used- Contrary to common belief, Redington’s work [l] indicates that
1 November 1978
this situation does not seem to be evitabie by a large excess of isotopic data with respect to the number of unknown force field parameters. In fact, if we calculate the HCOOH fundamentals for the full set of calculated force constants and for the set obtained after discarding the interaction constants involving O-H and C-H stretching the frequencies change by less than 0.3% except for v2 whose change amounts to O.% (= 30 cm-‘). Hence there is little hope of determining these coupling constants from observed fundamentals, and the quantum chemical results may provide the only estimates available_ It is conceivable that othe: cross terms are similarly ill determined, and the question arises whether apparent conflicts between empirical and quantum chemical results can be eased or resolved. We may therefore involve the quantum chemical force field with appropriate weight in adjusting calculated to observed fundamentals_ That is, we deliberately apply a known bias intended to offset the hidden bias mentioned above, which is less controllable_ The resulting set of force constants Ffj will then represent a “compromise” between prediction and observation. In the present attempt to defme such a compromise the least squares adjustment is based on the equations* [(&/afij)LL?$j
- (vJbs - VJ] lr(yS) = e(v,),
and
.Z2 s
(v,)
+
g$(Fij)
i
= min.
The quantities r(vs) and r-(&7) are standard residuals allowed for frequencies and force constants, respectively_ Tbey define the weight factors, W(Vs> = l/r2 (us) ,
w(Q)
= 1/r2(Fii)
.
starting out from the empirical force field (cohunn I of table 1) caiculations were made wit-b r(ys) =Z O.O1v,Obsand two different assumptions for r(Fi& one attributing little weight to the quantum chemical force field,
r(Fg)= I FvQcl * we are grateful to Rofessor 1.M hSUsfor suggesting the technique in a private discussion. 19
Volume 59, number I
CHEWCAL
PHYSICS
and one with stronger bias, The resultingforce fields are shown in columns II and III of tabIe 1, respectively. In both cases the rms relative error for the 201 frequencies adjusted ~2s not significantly larger than the 0.46% error obtained by Redington [Il. In both solutions the diagonal force ccastants are closer to the empirical than the quantum chemical values which seem to be overestimated by 10 to 20%. On the other hand, the off&agonal constants wz closer to the quantum chemical values. This is obvious for the C-H and O-H coupling constants which are the least estimable from the observed fundamentals. But also the other interaction constants are found to agree better with the quantum chemical values than with the empiricA ones, particularly in those cases where the latter seem to be offin order of magnitude, One such case concerns the constant J;35 coupling c--O stre”cchingwith CC?H bending, whose empirical value is about 20 times the quantum chemical value and exceeds the interaction constant F& of C-O stretching and COH bending. me small quantum chemical value for F& and the F& value which is intermediate between the empirical and the quantum chemical result, are both found to be consistent with ex~erirnentaldata. In conclusion one may note that only relatively minor corrections to the ribinitio force field are
20
LJXTERS
1 November 1978
required to reach agreement with.experimentaJ data_There stilI remains the probIem of appropriate& choosing the rekitiveweights in the least squares adjustment, accounting for the systematic errors of the ab initio results.Neverthelessthe present treatment shows that ab initio results may help restricting ambiguities of empirical force fields, Further work linking predicted and experimental vibrational data of small molecules is underway. Support of this work by the Swiss National Science Foundation (Project Nr,2.712-0.77) is gratefully acknowledged. Furthermore. we thank the ETHZ computation center for a generous grant of free computer time.
References R-L_ Redington,I. Mot. Spectry.65 (1977) 171. CE- Morn, Dissertation, University of Leaden(19763, H-B_ S&k&, S, Wolfe and F. Bernardi. J. C&m. Phys_ 67 (1977) 4181. R. Ditchfield, 1V.J. Hehre and J.A. Pople, J_ Chem. Phys. 54 (1971) 724; 56 (1972) 2257. G-H. Kweiand RX. Cud Jr., J. Qem. Wys. 32 (1960) 1.592. T--K iia, R. Meyer and Hs- H. Giinthard, C&em. Phys. Letters 22 (1973) 68_ ii. Kiihne, T.-K. Ha, R. *Meyer and E&-H. G&xthard, J. hfol. Spemy. <1978), submitted for publication. P. Bosi. G. Zerbi and E Ckmenti, J. C&em_ Phys_ 66 (1977) 3376.
CHEMICAL PHYSICS LETTERS
1 November 1978
H!ARMONK FORCE CONSTANTS OF FORMIC ACID: AR INITIO RESULTS AND THE UNIQUENESS PROBLEM OF FORCE FIELDS DERIVED FROM VIBRATIONAL Tae-Kyu HA, R. MEYER
DATA
and Hs- H. CmRD
Lubomtony of Physicd chernishy. Swiss Fe&ml Insritute of Technology. CH-8092 Zurich. swinertand
Received4 July 1978 A complete set of harmonic force constants of formic acid (HCOOH) is caIcula:ed using a 6-31G basis set. It is used to determine a modified valence force field consistent with both ab initio results and vibrational datz
The unusualIy large amount of assigned formic acid fimdamentals published recentIy by Redington [l] has prompted us to compute a complete harmonic valence force field ab initio for comparison with observations. In the present work, special attention is paid to the problem of comparing calculated with empirical force constants since the latter cannot be determined uniqueIy from vibrationaI data for polyatomic molecules. Some lack of uniqueness remains even in the case of fo,mic acid where 201 observed fundamentaIs stiU do not seem to determine all of the 3 1 quadratic force constants. The present ab initio force field is in good overall agreement with Redington’s data [l] . However, there are a number of discrepancies which motivated us to investigate the question whether a force field compatible both with ab initio results and the observed frequencies can be found. Such a procedure is aimed at restricting the manifold of soiutions of the normal coordinate probiem. Biom et al. 121 have derived a set of scale factors which should allow the correction of ab initio force constants in order to produce empirical force constants for series of similar molecules. These authors suggest the use of scale factors common to related molecules for predictive calculation of normal frequencies. In a quantum chemical study by Schlegel et al. [3] predicted force constants were found to agree remarkably well with data derived from vibrational spectra_ Based on the result the authors suggest the reexamination of the empirical force fields.
In the present paper use is made of ab initio constants, as well as of Redington’s vibrational dats to determine an alternative force field of formic acid, which represents a compromise between the two approaches_ The ab initio SCF calculations have been carried out using an extended basis set of contracted gaussian functions (6-31G) [4]. The force constants of formic acid have been calculated from ab initio SCF energies. The re structure reported by Kwei and Curl [S] has been used as an initial structure and the inte,tis of bond lengths, bond angles and torsional angles were chosen in such a way that the increase in the SCF energy due to each displacement amounts to about 1000 cm-’ compared to that of the initial structure. Owing to the anharmorzcity of the ab initio energy surface the results for the force constants obtained by interpolation will obviousIy depend on the spacing of the sample points. With an optimal choice of these spacings the interpolation process should be similar to the one involved in the evaluation of empirical force constants by a normal coordinate treatment of the spectra_ On the other hand, the intervals should be small enough, especially for bond lengths, to avoid excessive overestimation of stretching force constants, which is inherent in the Hartree-Fock type SCF calculations_ The above criterion was used in an attempt to define suitable intewals in the order of magnitude of the vibrational amplitudes in the lowest states. Similar considerations were applied previously in 2 calculation of force constants of methanol [6] and formi7
Volmne 59, number 1
C.XJWIGW PHYSICS LETTERS
1 November 1978
Tabfe 1 Valence force I’ieldof formic acid monomer
co-m
7.237 C20) - 0.029(19) -0.099 (49) 0.18F(94) 0.20 (13) -0.097 (55) -0.028 (19)
7.242(18) -0.0346(42) -0.090 (11) 0.175(21) 0.260(31) -0.102 (12) -0.0351(42)
8.03 -0.035 -0_090 3.174 0.261 -0.102 -0.035
4.705 (22) 0.26 (11)
4.756 (15) 0.263(33) 0132 (16) 0.0367(46) -0.137
5.62 0.299 0.132 0.038 -0_153 -0.027
13.35 (63) 0.91 (51) 0.47:(78) 1.00 (34) 0.721(82)
13.05 (40) 1.21 (38) 0.031 (13) 0.14 (16) 0.40 (11)
13.34 (16) 1.52 (14) 0.0243(29) 0.336(43) 0.360(38)
15.37 1.313 0.024 0.367 0.345
6.00 (35) 0.092(6.5) 0.004(111) -0.404 02)
6.1C (21) 0.184(37) 0.170(80) -0.347 (16)
6.226 (64) 0.21703) 0.211(24) -0.344 (12)
7.67 0.350 0.203 -0.389
0.696(11) -0.099 (25) -0.044 (13)
0.6453(81) -0X32(81) -0.0606 (83)
0.6454(49) -0.1452(45) -0.0604(36)
0.760 -0.164 -0.058
F 67
1.297
1.248 (39) 0.007s(92)
I.204 (13) 0_0179(22)
1.340 0.018
SC-m
F?7
0.624(8)
0_6142(28)
0.6162(25)
0.750
WC--H)
53 F89
0.470(2) 0.092(4)
0.4698(28) 0.0922<41)
0.4704(28) 0.093Oi42)
0.580 0.095
F99
0.168(1)
0.16750(82)
O-16766(87)
0.190
0.46%
058%
0.60%
v
7.188(17)C)
Fll PI2 Fl3 F14 FIS fi6 Fl7
V (C-H)
4.666 (14)
Fa2 F23 F24
0.116 0.020 -0.024 -0.025
F25 F26 F27 F33 54 F35 F36 57 P(C-0)
F44 5s F46 F47
6KOH)
FSS Fs6 Fs7
~Gxm~
r(c-o) OWV),
a) F-a&ix B from ref. [I 1.
F66
(69) (20) (61) (14)
b) See text
c) U&S are mdyn/A for stretching and mdyn A/wd2 for bending coordinates Numbers in bracketsgive oflitsofhstdigit
18
deviationsin
Volume 59, nmber
1
CHEMICAL PHYSICS LETTERS
anhydride ]7] _Using the internal coordinates chosen by Redingron [I], diagonal force constants were evaluated by 3 point interpolation while off-diagonal elements were evaluated from the SCF energies calculated on a 32 point grid in the two-dimensional subspaces spanned by the respective pairs of internal coordinates. The results are shown in the last column of table 1. Bosi et al. [8] have previously reported 5 ab initio diagonal force constants of formic acid monomer in connection with their dynamical charge transfer study of the cyclic dimer. The total SCF energy of their basis set (9/4 basis set) amounted to - 188.6689 au for the equilibrium geometry, which is comparable to - 188.6627 au for the empirical structure [S] obtained from the 6-3 1G basis set of this work. The force constants reported by Bosi et al. [S] are: Flr(O-H) = 8.16,F22(C-H) = 6.16, F33(C=O) = 13.44, F.++(C-0) = 8.05 and FsS(COH) = 0.81 in mdyn/A. Except for the F33(C=O) value all these diagonal force constants are higher than those of the present work (see table 1). By -comparing columns I and IV of table 1 one notes that the calculated diagonal force constants are higher than the empirical ones throughout with deviations ranging up to 28%. For the off-diagonal elements there is fair overall agreement. However, discrepancies significantly exceed experimental errors stated for the interaction constants of C=O stretching with bending coordinates and of C-O stretching with COH and OCO bending. In discussing the two sets of force constants we have to consider the possibility of errors in the empirical set as well as in the quantum chemical one. It is well known in vibrational analysis that empirical force fields are usually ill determined owing to limitations of the harmonic approximation and of the experimental accuracy. Furthermore the particular set of observed data will determine only part of the force field but will leave unknown other properties of the potential. As a consequence, the analysis of experimental data allows for multiple solutions, and a specific least squares solution arrived at may have been affected by hidden bias dictated by the actual supply of experimental information and by details of the computational method used- Contrary to common belief, Redington’s work [l] indicates that
1 November 1978
this situation does not seem to be evitabie by a large excess of isotopic data with respect to the number of unknown force field parameters. In fact, if we calculate the HCOOH fundamentals for the full set of calculated force constants and for the set obtained after discarding the interaction constants involving O-H and C-H stretching the frequencies change by less than 0.3% except for v2 whose change amounts to O.% (= 30 cm-‘). Hence there is little hope of determining these coupling constants from observed fundamentals, and the quantum chemical results may provide the only estimates available_ It is conceivable that othe: cross terms are similarly ill determined, and the question arises whether apparent conflicts between empirical and quantum chemical results can be eased or resolved. We may therefore involve the quantum chemical force field with appropriate weight in adjusting calculated to observed fundamentals_ That is, we deliberately apply a known bias intended to offset the hidden bias mentioned above, which is less controllable_ The resulting set of force constants Ffj will then represent a “compromise” between prediction and observation. In the present attempt to defme such a compromise the least squares adjustment is based on the equations* [(&/afij)LL?$j
- (vJbs - VJ] lr(yS) = e(v,),
and
.Z2 s
(v,)
+
g$(Fij)
i
= min.
The quantities r(vs) and r-(&7) are standard residuals allowed for frequencies and force constants, respectively_ Tbey define the weight factors, W(Vs> = l/r2 (us) ,
w(Q)
= 1/r2(Fii)
.
starting out from the empirical force field (cohunn I of table 1) caiculations were made wit-b r(ys) =Z O.O1v,Obsand two different assumptions for r(Fi& one attributing little weight to the quantum chemical force field,
r(Fg)= I FvQcl * we are grateful to Rofessor 1.M hSUsfor suggesting the technique in a private discussion. 19
Volume 59, number I
CHEWCAL
PHYSICS
and one with stronger bias, The resultingforce fields are shown in columns II and III of tabIe 1, respectively. In both cases the rms relative error for the 201 frequencies adjusted ~2s not significantly larger than the 0.46% error obtained by Redington [Il. In both solutions the diagonal force ccastants are closer to the empirical than the quantum chemical values which seem to be overestimated by 10 to 20%. On the other hand, the off&agonal constants wz closer to the quantum chemical values. This is obvious for the C-H and O-H coupling constants which are the least estimable from the observed fundamentals. But also the other interaction constants are found to agree better with the quantum chemical values than with the empiricA ones, particularly in those cases where the latter seem to be offin order of magnitude, One such case concerns the constant J;35 coupling c--O stre”cchingwith CC?H bending, whose empirical value is about 20 times the quantum chemical value and exceeds the interaction constant F& of C-O stretching and COH bending. me small quantum chemical value for F& and the F& value which is intermediate between the empirical and the quantum chemical result, are both found to be consistent with ex~erirnentaldata. In conclusion one may note that only relatively minor corrections to the ribinitio force field are
20
LJXTERS
1 November 1978
required to reach agreement with.experimentaJ data_There stilI remains the probIem of appropriate& choosing the rekitiveweights in the least squares adjustment, accounting for the systematic errors of the ab initio results.Neverthelessthe present treatment shows that ab initio results may help restricting ambiguities of empirical force fields, Further work linking predicted and experimental vibrational data of small molecules is underway. Support of this work by the Swiss National Science Foundation (Project Nr,2.712-0.77) is gratefully acknowledged. Furthermore. we thank the ETHZ computation center for a generous grant of free computer time.
References R-L_ Redington,I. Mot. Spectry.65 (1977) 171. CE- Morn, Dissertation, University of Leaden(19763, H-B_ S&k&, S, Wolfe and F. Bernardi. J. C&m. Phys_ 67 (1977) 4181. R. Ditchfield, 1V.J. Hehre and J.A. Pople, J_ Chem. Phys. 54 (1971) 724; 56 (1972) 2257. G-H. Kweiand RX. Cud Jr., J. Qem. Wys. 32 (1960) 1.592. T--K iia, R. Meyer and Hs- H. Giinthard, C&em. Phys. Letters 22 (1973) 68_ ii. Kiihne, T.-K. Ha, R. *Meyer and E&-H. G&xthard, J. hfol. Spemy. <1978), submitted for publication. P. Bosi. G. Zerbi and E Ckmenti, J. C&em_ Phys_ 66 (1977) 3376.