J. Phys. Chem. 1995,99, 17544-17550
17544
Testing the Validity of Scaling the Quantum Mechanical Molecular Force Fields for Rotational Isomers Yurii N. Panchenko’ and George R. De Mad* Laboratoire de Chimie Physique Moliculaire, Faculti des Sciences CP.160/09, Universiti Libre de Bruxelles, 50, ave. F.-D. Roosevelt, B-1050 Brussels, Belgium
Vladimir I. Pupyshev Laboratory of Molecular Structure and Quantum Chemistry, Chair of Physical Chemistry, Department of Chemistry, Moscow State University, Moscow 11 9899, Russian Federation, C. I. S. Received: June 23, 1995; In Final Form: September 13, 1995@
The conditions under which the scaling method can be applied to quantum mechanical force fields to predict the fundamental vibrational frequencies of related molecules, with acceptable accuracy, are reinvestigated. Most of all, transferability of scale factors to the force fields of related organic molecules (especially molecules with heteroatoms) requires that the quantum mechanical calculation gives results which are rather close to the Hartree-Fock limit. Another requirement is singlet stability of the SCF wave function. The calculation of the vibrational frequencies of the rotational isomers of glyoxal, O%H-CH=O, is presented as an example. Thus when quantum mechanical force fields obtained with a basis set giving results near the Hartree-Fock limit (i.e., HF/6-31G*, HF/6-31G**) are used, transfer of the scale factors obtained for the light s-transglyoxal conformer to the force field of the s-cis rotamer gives a wholly satisfactory prediction of the vibrational frequencies of the latter. Comparison of the theoretical and experimental fundamental vibrational spectra of the s-trans and s-cis conformers of nitrous acid, HONO, shows that the above is probably true for inorganic molecules. Agreement between the calculated and experimental vibrational frequencies of rotational isomers is a more stringent test of the transferability of scale factors to related molecules than only agreement between the calculated and experimental frequencies of isotopomers.
1. Introduction have been developed for correction of Scaling the systematic overestimations of the quantum mechanical (QM) force constants obtained either by ab initio computations at the Hartree-Fock (HF) level or by semiempirical methods. The conditions which must be met to obtain transferable scale factors are reexamined in this paper. In one of the first variants of the scaling method, only the diagonal force constants were multiplied by fixed empirical factors.2 An improved method used the experimental vibrational frequencies in a fitting procedure to obtain the scale factors for the diagonal force constant^.^ (The off-diagonal force constants were not involved in the fitting pr~cedure.~) In another method, reported at about the same time, the diagonal force constants were grouped into several sets and all the off-diagonal ones were placed into one common set. A specific scale factor was obtained for each set of force constants by a fitting procedure using the experimental frequencie~.~ Scale factors for the offdiagonal force constants have also been calculated as the arithmetic means of the scale factors for the corresponding diagonal force constant^.^ Some computations are performed directly on the calculated ab initio vibrational frequencies without any modification of the harmonic QM force fields.6 However, with the exception of the simple multiplication of the computed frequencies by a single factor7(which is equivalent to a scaling procedure using a single scale factor for all the force constants), such procedures cannot be considered on the same level as the above-mentioned scaling methods.
* Author
to whom correspondence should be addressed. Telephone:
x-32-2-650.4088. Fax: x-32-2-650.4232. E-mail:
[email protected]. @Abstractpublished in Advance ACS Abstracts, November 15, 1995.
0022-365419512099-17544$09.00/0
The contemporary h l a y method (see review in ref 8) for scaling QM force fields consists of congruent transformation of the force constant matrix in internal coordinates. This corresponds to the following equation,
where F is the scaled QM (SQM) force field, 9 3 s the force constant matrix calculated at the HF level, and D is the diagonal matrix of scale factors. This method is used either with optimized scale factors9or with scale factors transferred, without further optimization, from related molecules.lO,llNote that the force fields obtained by both of these techniques have been used to compute fundamental vibrational frequencies for both optimizedi2-I4 and unoptimized (or “adjusted” experimental) molecular geometrical structures.2.s-10 Note also that sometimes when Pulay’s method8 is used, some off-diagonal force constants are scaled separately. In ref 15 this procedure was applied to the force field of hexatriene, to reproduce the splitting of the C=C stretching vibration. However this splitting can also be explained by vibronic coupling (see refs 13 and 16 and references cited therein). Pulay’s scaling method8 is the most widely used because it is invariant under certain transformations of the valence coordinates. A theoretical basis for the method is given in ref 17, where it was shown that the force constants determined near the HF limit are approximately linearly connected with the exact force constants (full CI approximation). The general conditions which must be fulfilled to obtain satisfactory, transferable scale factors for related molecules were considered in ref 17. A theoretical basis for the technique of transferring the scale factors to related molecules is presented in ref 18. Also, using ethylene 0 1995 American Chemical Society
Validity of Scaling Force Fields
J. Phys. Chem., Vol. 99, No. 49, 1995 17545
.3 3
'P
\o
W
Cr,
-
m
m + m
00 P
f 4
m
8
H4
H3
W
Figure 1. Atom numbering used throughout the text. Only the s-cis conformers are shown.
as an example, it has been shown that the anharmonicity of the experimental vibrational frequencies affects the values of the scale factors only slightly.'* Many SQM force fields have been computed at the standard, split-valence 6-3 1G basis set level of theory and gave satisfactory results (see examples in refs 12-14 and 18). However, when the scale factors obtained for the 6-31G QM force field of s-trans-glyoxal (see Figure 1) were transferred to that of the s-cis conformer,20 the difference between the calculated and experimental values of the fundamental frequency of the formal carbon-carbon single bond stretching mode was about 60 cm-I, which is markedly larger than the usually accepted deviations. Note that an even higher theoretical value of this vibrational frequency was obtained previously with a scaled semiempirical force field (the deviation was about 100 cm-I).I1 Such marked deviations may complicate the vibrational assignment dramatically in the case of larger molecules. These anomalies obviously require additional study of the force fields of the glyoxal rotamers to try to find an explanation. This paper presents such an attempt. To further test the validity of the scaling procedure and the transferability of the scale factors to rotational conformers, the theoretical and experimental vibrational spectra of the s-trans and s-cis conformers of the nitrous acid molecule (see Figure 1) are also investigated.
. .
e
m
0
e
!€
e
I
2. Methods
The procedure used for rotational isomers at a given theoretical level is the following. (i) Obtain completely optimized conformer geometries (preferably with "tight" optimization criteria). (ii) Verify the singlet stability of the wave functions. (iii) Compute the QM force fields and theoretical frequencies. (iv) Determine, by a fitting procedure using the experimental vibrational frequencies, the scale factors for the QM force field of the rotamer for which the fundamental frequencies are known most accurately. (v) Finally, transfer the scale factors obtained in step iv to the QM force field of the remaining rotamer and calculate its vibrational frequencies. Completely optimized geometrical parameters for s-trans- and s-cis-glyoxal as calculated using the standard gradient method at the HF/6-31G*, HF/6-31G**, and MP2/6-31G* levels of theory21,22 with the Gaussian series of programsz3were obtained p r e v i ~ u s l yor~ computed ~ ~ ~ ~ in this work. These geometrical parameters and some other literature data are compared in Table 1.
The QM force fields were computed for both the s-transand s-cis-glyoxal conformers at each theoretical level. The Cartesian force constants were transformed into the local (valence) symmetry coordinate^.^^ Scale factors (Table 2) for eq 1 were determined for the s-trans conformer using the experimental set of the vibrational frequencies of s-trans glyoxal-h2. The theoretical vibrational frequencies calculated using the three SQM force fields for the s-trans rotamer of glyoxal-hz are juxtaposed to the experimental data in Table 3. The force fields obtained for the s-cis rotamer were scaled by transferring the corresponding scale factors determined for the s-trans conformer. The theoretical vibrational frequencies are given in Table 4.
E
Ps
0)
=
17546 J. Phys. Chem., Vol. 99, No. 49, 1995
Panchenko et al.
TABLE 2: Scale Factors for the Glyoxal ConformersD theoretical level coordinate
HF16-31G ref 20
HF/6-31G* this work
HF/6-3 1G* * this work
MP2/6-31G* this work
C-C stretch C-0 stretch C-H stretch O=C-C in-plane bend C-C-H in-plane bend C-C torsion C-H wag
0.829 0.824 0.754 0.792 0.792 0.519 0.781
0.780 0.725 0.773 0.838 0.807 0.645 0.779
0.782 0.725 0.789 0.840 0.815 0.645 0.776
0.839 0.984 0.864 0.999 0.920 0.710 0.954
a The scale factors for the HF/6-31G*, HF/6-31G**, and MP2/6-31G* force fields of s-rrans-glyoxal were determined using the experimental frequencies of the s-trans-h2 isotopomer. At the MP2/6-31G* level, except for the 0-C-C angle coordinates (scale factor set at 0.9999, see text), the initial approximation of 0.93 was used for the scale factors of all the coordinates. The refined MP2/6-31G* scale factors were used as the initial approximation for the scale factors of the HF/6-31G** force field, and the refined HF/6-31G** scale factors were taken as the initial guess for determination of the HF/6-31G* scale factors. Note however that the starting approximation usually has no effect on the final refined values of the scale factors.
TABLE 3: Experimental and Calculated Vibrational Frequencies of s-trans-Glyoxal(cm-') calculated from scaled SQM force fields
exptl V
Sa
Ab
1 2 3 4 5 6 7 8 9 10 11 12
a,
R
a,
IR
b, b,
R IR
gas phase
HF16-31G ref 20
HF/6-31G* this work
HF/6-31G** this work
MP2/6-3 1G* this work
2843.2736 1744.1236 1352.6036 1065.8136 550.5337 801.363s 126.7039 1047.8136 2835.0738 1732.lo4',' 1312.3838 338.5538
2848 1764 1336 1074 564 803 127 1048 285 1 1719 1317 340
2840 1762 1347 1066 549 795 127 1057 2837 1716 1317 340
284 1 1762 1347 1066 550 794 127 1054 2838 1716 1317 340
284 1 1742 1348 1069 548 796 130 1052 2837 1733 1316 334
assignment v(C-H)str V(C==O)Stl d(C-H)bend v(C-C)str 6(o=c-c)bend X(C-H)W,, t(C=C)tors x(C-H)wa, V(C-H)str v(C=O)str d(C-H)bend 6(o=c-c)knd
a S, species symmetry. A: R and IR, active in Raman and in infrared, respectively. The extremely accurate value of 1732.099 12 f 0.OOO 23 cm-I for this mode, obtained in a Fourier transform study of the region 1650-1800 cm-I, was reported in ref 40.
TABLE 4: Experimental and Calculated Vibrational Frequencies of s-cb-Glyoxal (cm-') exptl v
Sa
Ab
assignment
1 2 3 4 5 6 7 8 9 10 11 12
al
IR, R
V(C-H),~ v(C=O),tI a(C-H)bend v(C-C)s,r 6(O=C-C)bend x(C-H),,, t(C-C)tors ~(C-H)wag V(C-H)str v(C=O)stl V(C-H)bend 6(o=c-c)bend
a
a2
R
b2 bl
IR,R IR, R
gas phase refs 25 and41 1746 827 284.5 1049 91
calculated from SQM force fields
Ar matrix ref42
N2 matrix ref 42
HF/6-31G ref 20
HF/6-31G* this work
HF/6-31G** this work
MP2/6-31G* this work
284 1 1736.6 1368.6 816.8
2853.7 1745.5 1374.1 839.8
1760.7
1762.2
2834 1739 1375 89 1 29 1 1066 89 175 2809 1752 1367 787
2812 1745 1382 837 288 1065 89 756 2782 1758 1367 808
2811 1744 1382 837 288 1062 87 755 278 1 1758 1367 809
2810 1724 1372 825 285 1052 117 749 2783 1760 1367 826
S, species symmetry. A: R and IR, active in Raman and infrared, respectively.
The calculation procedures for H O N O are exactly the same as those used for glyoxal. The results are given in Tables 5-8. The definition and numbering of internal coordinates and the force fields are given in Tables 9-12 as supporting information.
3. Discussion To obtain a set of scale factors which are suitable and transferable for the evaluation of vibrational molecular frequencies at high accuracy, it is necessary to fulfill some obvious conditions: (I) Experimental data. The harmonically corrected experimental vibrational frequencies which are to be used in the scaling procedure must be available. Note, this problem is not trivial, since the theory is not sufficiently advanced.
(11) Molecular state under investigation. (1) Molecular vibrations must be described by the theory of small vibrations in the adiabatic approximation. (2) The evaluated force constants must be rather characteristic. For fulfillment of condition 11.1, it is sufficient to require a relatively wide gap between the exact energies of the ground and first electronically excited states* (111) Hartree-Fock molecular model. (1) The calculations must be carried out With a rather complete A 0 basis Set. (2) The Hartree-Fock solution for the ground state must be singlet stable with respect to small variations of the MO's. (3) The System O f occupied ground state MO's must be well-localizable. These conditions d o not have equal status. Fulfillment of condition III.3, well-localizability of the electronic density
J. Phys. Chem., Vol. 99, No. 49, 1995 17547
Validity of Scaling Force Fields TABLE 5: Experimental and Optimized ab Initio Quantum Mechanical Geometrical Parameters and Energies for HONO (Bond Lengths, Bond Angles, and Energies in Picometers. Degrees, and Atomic Units)
parameter r(01-N2) r(O3aN2) r(H4-01) LO-N=O LH-0-N
energy r(01-N2) r(03-N2) r(H4-01) LO-N-0 LH-0-N
energy
exptl spectr ref 43
theoretical level HF/6-31G this work
HF/6-311G*
MPU6-311G**
this work
this work
s-trans 143.2 138.67 134.35 117.0 117.13 114.36 95.8 95.32 94.38 110.7 111.6 111.7 102.1 108.5 105.3 -204.52 1940 -204.700 158 s-cis 139.2 136.81 132.44 118.5 118.17 115.21 98.2 96.39 95.27 113.6 114.0 114.0 104.1 112.1 107.9 -204.5206 14 -204.70 1446
141.55 118.04 96.52 110.9 101.2 -205.274580 137.73 119.38 97.66 113.2 104.2 -205.27526 1
TABLE 6: Scale Factors for the HONO Conformer@ scale factors coordinate HF/6-31G HF/6-311G** MP2/6-311G** N-0 stretch 0.717 0.517 0.860 0.805 N=O stretch 0.696 0.9999 0-H stretch 0.799 0.750 0.874 0.637 0.557 0.860 O=N-0, in-plane N-0-H, in-plane 0.792 0.719 0.930 N-0 torsion 0.930 0.855 0.855 The scale factors were calculated using the experimental frequencies of light trans-nitrous acid. At the MP2/6-31lG* level, 0.93 was used as the initial approximation for all the scale factors except that for the N2-03 stretch coordinate, for which the scale factor was set and kept fixed equal to 0.9999. This procedure was used because the computed frequencies for the N2-03 stretch were lower than the experimental values (see Table 7). distributions, can be evaluated easily for chemical systems. Conditions I and I1 are usually assumed to be fulfilled for any ideal molecular model. However, fulfillment of condition III.1, completeness of the A 0 basis set, depends on the investigator’s choice. Fulfillment of condition I11.2, the singlet stability of the solutions, determines the reliability of the computational results. Thus it is necessary to examine these two criteria more carefully when choosing a molecular system for scaling of its QM force field by empirical factors. The general features of our investigations will be presented first of all using glyoxal as an example. The conclusions obtained are then confirmed by our results for HONO. 3.1. Glyoxal. General Consideration. From a formal point of view, the quality of the force constant estimations in the scaling method depends on the extent to which the equation
connecting the second derivatives of the Hartree-Fock (EHF) and exact (&I) energies with respect to the geometrical parameters q is fulfilled.17 (dq is the element in the matrix D corresponding to the parameter q (see eq l).) Of course, this fulfillment also depends on the degree to which the potential surface, defined at the adiabatic level by ECI, permits the evaluation of the nuclear motion characteristics within the framework of the small vibration approximation. As mentioned above, agreement of the final results with the experimental data also depends on the possibility of obtaining the “experimental” harmonic frequencies. In general, the values of dq (eq 2) are
functions of the characteristics of a particular molecule. When using a restricted A 0 basis set, there is persistent dependence of the EHF values on the positions of the A 0 centers which possess distinct nonlocal character. In order to exclude this dependence, the A 0 basis set must be adequately complete. If one uses the MP2 method to calculate the force constants, it is still necessary to scale them because the MP2 computation does not pick up the entire correlation energy.7 Nevertheless, perturbation corrections decrease the force constants and thereby the frequencies, bringing them closer to the experimental values. Thus, the MP2/6-31G* scale factors reported in Table 2 are closer to unity than those for the other computations. All of the results of MP2/6-31G computations for both the trans- and cis-glyoxal conformers are unsatisfactory. The optimized C=O bond lengths are too long (-126 pm) and the v2 and V I O frequencies calculated with the unscaled QM force field are lower than the experimental values. This feature can probably be connected with the incorrectness of the modeling of this wave function. It is difficult to separate the two factors: incompleteness of the A 0 basis set and incompleteness of the configurational expansion (i.e. electron correlation description). This is the reason why the corresponding MP2/ 6-31G data are not included in this paper. Stability Problem and Transferability of Scale Factors. The second derivative of the energy with respect to a given parameter is known to have two component^.'^ One of them is of Hellman-Feynman type and is determined by the second derivative of the Hamiltonian with respect to q. The corresponding operator has the form of a potential which decreases rapidly with increasing distance between the nuclei of the moiety involving the parameter q. Thus the localization of such a component of the force constant is evident. In general, however, the relaxation component of the force constant is not so local but it involves, as a partial, a factor A-I, where A is a matrix of the second derivatives of the energy functional at the HF level (or, what is close to this, the energy denominators in the expansion series of the perturbation theory). Therefore, the local character of the estimated force constant depends on the relative importance of the relaxation term.1795’ The transferability of the scale factors dq depends on the stability of the wave function and hence on the energy gap between the ground state and the excited states of the same multiplicity both in the exact problem (full CI level) and at the HF level. For s-trans-glyoxal the localizability of the MO system is beyond question. Therefore using the s-trans ts-cis changes, it is possible to study, together, the importance of the completeness of the A 0 system and the stability of the SCF solution because the local geometry changes are rather small (see Table 1). However there are some peculiarities which are important enough to be considered. Unfortunately, it is impossible to separate the problems of stability of the SCF solutions from the problems of the basis set completeness, since extension of the basis set usually results in the enhancement of the stability of the solution. Note in this connection that the scale factors corresponding to rather low frequencies (see Table 2) deviate most strongly from unity (thus the relaxation terms are certainly high for the torsional motions). The estimations of the eigenvalues of the second variations matrix show that the lowest eigenvalues at the HF/6-31G* level are equal to 0.007 au for the s-trans-glyoxal triplet stability matrix (B, symmetry) and 0.140 au for the singlet stability matrix (A, symmetry). For s-cis-glyoxal the corresponding values are -0.002 au (B2) and 0.135 au ( B I ) . The near zero values of the lowest eigenvalue of the triplet eigenvector in the
17548 J. Phys. Chem., Vol. 99, No. 49, 1995
Panchenko et al.
TABLE 7: Experimental and Calculated Vibrational Frequencies (cm-') of s-trans- and s-cis-Nitrous Acid, HONO (AU Modes Are Active in Both the Raman and Infrared) calculated"
exptl gas phase V
1 2 3 4 5 6
HF/6-31G
HF/6-31G**
symmetry
assignment
s-trans
s-cis
s-cisb
s-cisb
s-trans
s-cis
a'
v(O--H)s, v(N=O)sw 6(NOH)bend ~(N-O)str 6(ONO)knd S(N-O)tars'
3590.77OC 1699.760c 1263.2071' 790.117 l C 595.62f 543.88r
3426.196' 1640.519' 1261' 85 1.943lC 609.0' 639.5h
3439 1638 1296 838 577 65 1
3463 1655 1277 839 590 613
3591 1679d 1264 803 591 544
3418 163od 1292 880 617 668
a"
MP2/6-311G**
a Calculated using the SQM force fields. The scale factors derived for the force field of the s-trans conformer at each theoretical level were transferred to the force field of the s-cis conformer without any alteration. The calculated values for the s-trans conformer coincide completely with the corresponding experimental values and are not given here. See refs 44-46 and references cited therein. For the "N=O stretch" coordinates the scale factors were assumed to be 0.9999, since the calculated frequencies tumed out to be lower than the experimental ones. e See ref 47 (low-temperature matrix). f See ref 49. g Obviously the torsional vibration w, should be excluded from consideration: the definition of the torsional coordinate is valid only for axis-symmetrical molecules such as ethane. See ref 50.
TABLE 8: Evaluated Scale Factors, Calculated Frequencies (v, cm-l), and Their Deviations (Av, cm-') from Experimental Values for the Quantum Mechanical HF/6-311G** Force Fields of s-trans- and seis-HONO transition from s-cis to s-trans
from s-trans to s-cis V
symmetry
mode
scale factor
stability matrices of the restricted HF method mean that the wave function has not moved into an area of substantial changes and reproduces, at least qualitatively, the real density distribut i ~ n . ~ Note ' , ~ ~that the wave function is essentially single configurational. (The weight of the dominant configuration with closed shells at the CASSCF level is equal to about 0.998,53in accordance with the singlet stability of the SCF solutions.) However, the excited states are markedly multiconfigurational as usual. In general this results in somewhat higher errors in the force field scaling for the vibrations of the corresponding types of symmetry. A 0 Basis Set Completeness. For the glyoxal molecule, the inclusion of polarization functions on the two oxygen atoms seems rather important. Indeed, Table 1 shows that the inclusion of d-polarization functions on the heavy atoms decreases the total energy rather markedly. However, the transition from the 6-31G* basis set to the 6-31G** (p-functions on hydrogen) does not result in any marked decresae of the total energy. This shows the closeness of the obtained value to the HF limit. Note that, for s-cis-glyoxal, the degree of closeness of the calculated value of v4 to the experimental magnitude is apparently correlated with the transition to the HF limit: going from the HF/6-31G to the HF/6-31G* and to the HF/6-31G** levels of theory improves the agreement between the calculated and experimental v4 vibrational frequencies (see Table 4). Force Fields. Only a few of the off-diagonal force constants of the s-cis conformer either change their values significantly or change their signs on going from the HF/6-3 1G to the MP2/ 6-31G* computational level. The absolute values of these force constants are rather small compared to the diagonal force constants in the corresponding row or column in the matrix of SQM force constants. For s-cis-glyoxal, the values of v4 calculated with the HF/6-31G and MP2/6-31G* SQM force fields are 891 and 825 cm-I, respectively (see Table 4). It is tempting to correlate the lower MP2/6-31G* value with the decrease in the scaled force constant (0.7 mdyn.A-I) for the
V
Av
scale factor
v
Av
3463 1655 1277 839 590 673
+37 +14 +16 -11 -19 133
0.525 0.685 0.735 0.607 0.695 0.775
3554 1687 1246 802 615 518
-37 -13 - 17 112 +19 -26
C-C stretching coordinate as compared to that for the HF/631G SQM force field. However, approximately the same decrease in the C-C stretching force constant is also observed in the case of the s-trans rotamer, where the calculated values of v4 are at the same level of accuracy for all the theoretical levels under consideration. Thus the whole set of in-planeforce constants is responsible for the accuracy of the calculated v4 frequency for the s-cis rotamer. Changes in the SQM out-of-plane force constants for both conformers do not result in marked changes of the calculated frequencies; perhaps they are caused mainly by the variations of the optimized geometrical parameters on going from the HF/ 6-31G to the MP2/6-31G* computational level. The same is probably true, to a certain degree, for the in-plane force constants. The theoretical MP2/6-31G* analysis of the vibrational spectra of the isotopomers of the s-trans and s-cis conformers of glyoxal was performed in ref 14, where two experimental bands of s-cis-glyoxal-d2, v4 and YIZ, were reassigned. Note also that the prediction of the vibrational frequencies of isotopomers of the s-trans-glyoxal conformer was satisfactory even with the HF/6-31G SQM force field.20 Thus, ifthe scale factors obtained for the QM force field of one light rotamer can be transferred to the QM force field of another and give satisfactory vibrationalfrequencies, this is a more stringent test of the transferability of the scale factors than only the satisfactory reproduction of isotopomer frequencies. 3.2. Nitrous Acid. An analogous example is given by calculation of the vibrational frequencies of the inorganic molecule nitrous acid (HONO), for which the s-trans and s-cis conformers exist in the gas phase in thermodynamic equilibrium with HzO, NO, NOz, N203, N204, and nitric acid.56*57 The analysis of the HONO spectra is complicated further by the facts that the reported experimental values of v3 and v5 for the s-cis conformer were measured in low-temperature matrices47and that the computed intensities of these two bands are
Validity of Scaling Force Fields relatively Indeed, the authors of refs 49 and 54 state that the v3 and v5 fundamental bands for s-cis-HONO have not yet been observed. We shall proceed to show that the bands reported in the matrix studies are probably assigned correctly. All of the fundamental vibrational bands of s-trans-HONO are w e l l - k n ~ w n , and ~ - ~we ~ have ~ ~ ~determined the SQM force fields at each theoretical level used in this paper (see Table 12 in the supporting information). As mentioned in the Methods section, the scale factors reported in Table 6 for s-trans-HONO have been used to scale the corresponding QM force fields for the s-cis rotamer. Assuming that the reported experimental values for the v3 and vg fundamental bands are indeed correct, then the differences between the theoretical HF/6-3 1G and the experimental v3 and vg frequencies for the s-cis conformer are greater than the usually accepted deviations (see Table 7). However the calculation at the HF/6-311G** level, i.e. with extension of the split-valence representation and inclusion of the polarization functions, shifts the theoretical v3 and v5 frequencies to values within acceptable accuracy limits, thus allowing the prediction of the vibrational spectrum of the s-cis conformer. The possible difference in the anhmonicity corrections for the s-trans and s-cis rotamers also should be taken into account here. It is interesting that the scale factor for the torsional motion of the HONO molecule, while still less than unity, is markedly larger than the other scale factors (Table 6). This may reflect significantly larger conjugation effects in the case of glyoxal than in the case of nitrous acid. It is necessary to treat the MP2/6-311G* results on the calculated frequencies for s-cis-HONO separately. The computed theoretical values of v2 (V(N=O),~~) are lower than the experimental values for both conformers of nitrous acid (see Table 7). This testifies that the corresponding wave functions are simulated incorrectly at this theoretical level. This circumstance also results in overestimated values of the calculated v3 and v4 frequencies of the s-cis conformer. The stabilities of the HF/6-311G** wave functions for both the s-trans- and s-cis-HONO conformers have been checked, and they are both singlet and triplet stable; Le., the lowest eigenvalues of the singlet stability matrix are 0.150 (A”; s-trans) and 0.160 (A”; s-cis), and those of the triplet stability matrix are 0.008 (A’; s-trans) and 0.016 (A’; s-cis). This can be considered as a sufficient condition to more or less satisfactory transferability of the scale factors. This transferability is not directly relevant to the values of the scale factors. If we assume that all the frequencies for s-cis and s-trans rotamers of HONO are known, it is interesting to test our suggestions by an altemative procedure for determination of the scale factors for the force fields of these species. Indeed, for a sufficiently complete basis set and approximately the same level of singlet stability of the SCF solutions (see above), we can use the scale factors for the s-trans conformer force field to compute the frequencies of the s-cis rotamer or, vice versa, determine the scale factors for the s-cis rotamer and transfer them to the force field of the s-trans conformer. From the theoretical point of view both procedures are equivalent. This is supported by the results reported in Table 8. It is easy to see that for both procedures even the absolute values of deviations of estimated frequencies are practically the same. The only interesting detail is that the deviations for the same vibrations in both procedures have opposite signs. In both cases the difference in the scale factors is very small, except for the torsional and the N-0 stretching vibrations, where the differences are respectively 0.08 and 0.05, Le. -9%. (See also footnote g to Table 7.) For the other scale factors the deviations
J. Phys. Chem., Vol. 99, No. 49, 1995 17549 are much smaller and do not correlate directly with eventual errors in the experimental frequencies. The agreement obtained between the calculated HF/6-311G** and the experimental frequencies of s-cis-HONO is taken to examplify an approximate fulfillment of all the conditions for the scaling procedure in the case of inorganic molecules and makes it possible to apply the scaling method for molecules with polar or ionic bonds.
4. Conclusions The fulfillment of the conditions put forward in ref 17 for the accurate prediction of fundamental vibrational frequencies (using the SQM force field obtained by Pulay’s method8) is shown to be necessary. These conditions have been tested, and the following are found to be more stringent than the others: the quantum mechanical results should be close to the HartreeFock limit, and the wave function should be strongly singlet stable. For glyoxal the fulfillment of these conditions is achieved by including polarization functions on the heavy atoms, Le. on going from the 6-31G to the 6-31G* basis set. Thus the force field obtained with the latter, more complete basis set, when scaled using Pulay’s method,g yields a more accurate prediction of v4 (V(C-C),~)of s-cis-glyoxalthan the SQM force field at the 6-31G computational level. An analogous result is obtained for the v3 and v5 vibrations (6(N-0-H)be,d and d(O=N-O)&,d, respectively) of the inorganic molecule HONO. The agreement obtained between the calculated HF/6-311G** and the experimental frequencies of s-cis-HONO is taken to exemplify an approximate fulfillment of all the conditions for the scaling procedure in the case of inorganic molecules and makes it possible to apply the scaling method to the force fields of this molecular class. It is worth stressing that the necessity of fulfillment of the stringent conditions for the accurate prediction of fundamental vibrational frequencies (closeness to the Hartree-Fock limit and strongly singlet stable SCF solution) has been demonstrated in this work for the rotational isomers, rather than for isotopomers. Satisfactoryresults were obtained previously at the 6-3 1GZ0and 4-21G55computational levels using the same SQM force field for different isotopomers. This probably means that only scale factors which are transferable to rotational isomers can be considered as physically meaningful empirical factors correcting the Hartree-Fock force fields for electron correlation effects. In general it is worthwhile checking the stability of the wave function when the symmetry of the nuclear skeleton is important (see details in refs 50 and 51) and in all cases when SCF iterations converge slowly. Acknowledgment. The authors wish to thank Georges DestrCe of the U.L.B.N.U.B. Computing Center for his assistance with the GAUSSIAN 92 program package. Financial support by the Russian Foundation for Fundamental Investigations (Grant 93-03-18386) is gratefully acknowledged. Yu.N.P. also thanks the U.L.B. for an international scientific collaboration grant. Supporting Information Available: Tables of definition and numbering of intemal coordinates and force fields for glyoxal and HONO (4pages). Ordering information is given on any current masthead page.
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