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Effective potential function of internal rotation in the nonrigid approximation
Journal of
MOLECULAR STRUCTURE ELSEVIER
Journal of Molecular Structure 376 (1996) 183-194
Effective potential function of internal rotation nonrigid approximation’ A.V. Abramenkov”,
Ch.W. Bockb, G.R. De MarF*,
in the
Yu.N. Panchenkod
‘Laboratory of Molecular Structure and Quantum Mechanics, Choir of Physicul Chrmistry. Department of Chrmistry, M.V. Lomonosov Moscow State lhriversity, Moscow 119899, Rmsiaw Federution bChen~istry Department. Philadelphia College of Textiles and Science, Philadelphiu, PA19144. WA ‘Lnborotoire de Chimw Physique Moldculaire, FacultC des Sciences, CP 160/09. UkiversitC Libre de Brrr.relles, Av F.-D. RomewIt 50, B-1050 Brussels. Belgium
Received 21 July 1995; accepted 4 September 1995
Abstract The feasibility of the construction of potential functions of internal rotation in the framework of semi- and nonrigid one-dimensional models is considered. The nonrigid approach under discussion is based on using ab initio quantum mechanical calculations to obtain informalion on molecular geometry relaxation during internal rotation and to find a physically meaningful starting approximation of the potential. Furthermore. the theoretical potential obtained is refined according to the criterion of the best agreement between the calculated and experimental frequencies of the torsional vibrations. During this latter optimization, some additional conditions may be imposed which support the maximal closeness of the results to the starting approximation. The refined potential is sensitive to the accuracy of determination of molecular geometrical parameters. For this reason, the quantum mechanical calculation should be performed at a rather high level of theory (accounting for electron correlation) to obtain reliable results. The comparison of semi- and
nonrigid models of internal rotation is performed using nitrobenzene, fluoroacryloyl fluoride, 1Jbutadiene fluoride as examples.
1. Introduction When studying intramolecular motions with large amplitudes such as hindered internal rotation. pseudorotation of cyclic molecules or inversion and ring puckering, the molecular models used involve, as dynamic variables, only the internal * Corresponding author ’Dedicated to Professor James E. Boggs on the occasion of his 75th birthday.
and acryloyl
coordinates corresponding to these motions and the angular variables characterizing the overall rotation. The simplest case in point is the so called one-dimensional model (one internal degree of freedom plus overall rotation). Many practically important problems can be successfully solved in the framework of the one-dimensional approximation. This is true, in particular, for the description of torsional vibratidns and internal rotatidn of molecules with one top or several weakly interacting
0022-2860/96/$15.00 IQ 1996 Elsevier Science B.V. All rights reserved SSDI 0022-2860(95)09098-3
tops.
184
A. V. Abrammkm
er ul.~/ounml
of Moleculm
Sixty years ago, the semirigid model of internal rotation [l-3] was developed, which considers a molecule as consisting of two rigid moieties (referred to as top and frame) rotating around their connecting axis. In spite of its simplicity, such a model provides a means for the correct description of qualitative peculiarities of the motions under consideration. Moreover, in the framework of this approximation, it is often possible to achieve quantitative agreement between the calculated and experimental torsional vibrational spectra. A more general approach was developed at the end of the 1960s with major contributions being reported in papers by Meyer and co-workers [446], Gtinthard [7] and Laane and co-workers [8-l I]. Some general expressions were obtained for the model Hamiltonians for any arbitrary kind of motion with large amplitude, with due regard to the interaction of internal degrees of freedom with the overall rotation. In particular, a model for the torsional vibrations and internal rotation was suggested, which was called “flexible” model by the authors [5,6]. We prefer the term “nonrigid model” used by Giinthard [7]. It is assumed that the top and frame may be distorted during internal rotation, i.e. all the geometrical parameters of such a molecular model depend on the angle of internal rotation p. If was also pointed out [6] that such an approach, remaining within the framework of the one-dimensional approximation, is a sort of alternative for more complicated (or “combined” [6]) multi-dimensional models accounting for other vibrational degrees of freedom at the lcvcl of small vibrations. Therefore, the nonrigid model was introduced as a means of accounting (at least partially) for interactions between the considered motion with large amplitude and other molecular vibrations. If it is necessary to take into account the interactions of several motions with large amplitude (for example, interaction of rotations of two tops or rotation of top and inversion), the nonrigid onedimensional model cannot provide a means for solution of this problem. A multi-dimensional model should be used here. constructed within the framework of the same formalism as a one-dimensional model [lO,ll]. For example, a
Srrrrcturc 376 (1996 i 183-194
two-dimensional model was used in Ref. [12] for describing the rotation of the methyl moieties in acetone and trans-dimethylglyoxal. An analogous approach was applied to analysis of the interaction between internal rotation of the methyl top and inversion of the aldehyde moiety in the ground and lower excited electronic states of acetaldehyde [13.14] and thioacetaldehyde [15]. Note that multidimensional models of motions with large amplitude can be constructed using either rigid or nonrigid approximations. In the latter case, the gcomctrical parameters of the moving moieties are assumed to depend on several variables. In the present paper. we shall restrict our consideration to a one-dimensional nonrigid model of hindered internal rotation and discuss some problems related to the practical construction of potential functions of internal rotation in this approximation.
2. The model The general forms of the Hamiltonians for rigid and nonrigid models of internal rotation are essentially the same. The relevant equations are well known [4-6,8,16-191. They are given here without any further comments, viz. 2
where cp is the internal rotation F = (t1/2)G,, and g is the determinant matrix G:
G=
-I21
122
-I23
x2,
41
x3q
angle, of the
(2) -41
-42
qd
Kp2
xp3
y.pp
Here IV are the elements of the usual inertia tensor; XI,- = X,+,;= cj mj[rl x (8rj/jla(p)]l are the elements corresponding to interaction between internal and overall molecular rotations; YVP = c, m,(iAj/c3p) (i3rj/dp); m, and ri are the mass and radius vector,
A. C’. .4hramenkov et al.lJournai of Molecular Structure 376 (1996,
respectively, of the,jth atom in the molecular frame of reference. The g and F values are functions of the atomic coordinates and masses and do not depend on momenta. The second and third terms in the kinetic energy operator, which are primarily responsible for the interaction of internal and overall rotation of a molecule, are therefore usually combined with the potential function. This gives the effective potential VefF:
The distinction between rigid and nonrigid models manifests itself in the character of the dependence of the G matrix elements on the internal rotation angle ‘p. In the case of the semirigid model, such a dependence is due to the change in the position of the molecular center of mass during rotation of the top with respect to the frame. This results in changes of radius vectors of all the atoms in the center of mass coordinate system. If the top is a symmetric one, i.e. its center of mass is situated on the axis of internal rotation, then, in the framework of the semirigid model, the G matrix does not depend on ‘p, and F in the kinetic energy operator becomes merely a constant (internal rotation constant). For the nonrigid model, along with the change of position of the center of mass, there is a parametric dependence on 9 for all the internal coordinates (internuclear distances. valence and dihedral angles). Because of this, in general, the G matrix in the nonrigid approximation will be a function of p even in the case of a symmetric top. There is a further feature distinguishing the nonrigid from a semirigid model. For the internal coordinate which is relevant to the angle ‘p, the latter is usually determined by the axis of internal rotation and a pair of atoms which are not aligned with this axis. One of these two atoms must belong to the top and the other to the frame. In the semirigid model, the different choices of dihedral angles differ from one another only by an additive constant (phase), i.c. the determination of the angle p is essentially unambiguous.
183-194
185
The value of the angle cp in nonrigid models depends on the reference atoms selected in the top and frame: selection of different reference atoms corresponds to nonequivalent definitions of cp which are related by some nonlinear transformation. This should be taken into consideration, e.g. in comparing potential functions obtained in different ways (for example, by different choice of coordinates). According to the results of numerous quantum mechanical calculations, internal rotations usually have less effect on the structural parameters of heavy (skeletal) atoms. It is therefore reasonable to use such reference atoms for the definition of the internal rotation angle. to minimize possible discrepancies related to angle definition. For the practical determination of the function F(p) there are at least two approaches. In the first, F is calculated as an ordinary rotational constant using the reduced inertia moment of internal rotation 1, [3], viz. F=-
h hC1,
Z, being explicitly expressed in terms of instant atomic coordinates and masses. This is achieved by using an implicit coordinate transformation depending on y and diagonalizing the G matrix
POI. The second approach is based on the direct calculation of the G matrix elements, the necessary derivatives ar,/&c being calculated by means of numerical differentiation [6,10,11]. Both methods are equally applicable to semiand nonrigid models, because they make it possible to introduce easily a parametric dependence of top and frame geometries on the internal rotation angle cp. In principle, the second technique can be used to analyze interactions between internal and overall molecular rotations by calculating cross terms X,, and estimating their contribution to F(p) and u(y). If the first technique is used, these terms are implicitly redistributed between the principal momenta of inertia and the reduced moment of inertia of internal rotation. However, in practice, the first technique is applied more frequently, probably because it is simpler computationally. Because of its conventional character, this is the technique used in our calculations.
186
A. V. Abramenkov
3. Computational
et al.lJournnl
of Molecular
details
The model potential function V,rr(cp) is usually approximated by a Fourier series truncated after several terms. Specifically, if at least one of the rotating fragments has a symmetry plane containing the rotation axis, the commonly used representation is
In the absence of symmetry elements. similar expansions in sine and cosine functions are used. Fourier approximations are also used to represent
= F. +
5 Fk cos ky
183-194
calculations for vibrational (IR and Raman) spectra require the model dipole moment and polarizability operators. The procedure given above makes it possible to obtain the theoretical torsional spectrum provided that the potential function V&y) is known. The inverse problem of constructing (or refining) the potential function of internal rotation consists in optimization of the coefficients vk in the expanSiOn (Eq. (5)) in accordance with the criterion of minimizing the functional R(V) =
&Yi’ 1
(6)
k=l
where the coefficients F, are determined by treatment of a set of the F(q) values as calculated for different points along the coordinate axis of the internal rotation angle ip. A variational technique is best suited to the calculation of the energy spectrum and wavefunctions which are relevant to the model Hamiltonian (Eq. (3)) using the approximations in Eqs. (5) and (6). The basis set used here is composed of eigenfunctions of free rotation, (27r-“‘e=‘kq [19]. However, an equivalent basis set made up of real functions [8,9]: & = (27)-l’* t+!~;= 7i-‘12 cos kp
k=
1,2,...
@i = rTT’12sin kv
k=
1,2,...
(7)
is usually utilized. The eigenvalues Em (torsional energy levels) are found by diagonalization of the Hamiltonian matrix. The eigenvectors contain the coefficients of expansion of wavefunctions with respect to the basis set of Eq. (7), i.e.
The wavefunctions relative probabilities
376 (1996)
where b, is the weighted deviation of the calculated value of some property h from the corresponding experimental (or preassigned) one:
F(cp), e.g. F(y)
Structure
can be used to calculate the of torsional transitions. Such
The cri value characterizes the degree of accuracy of the ith experimental value or the admissible level of discrepancies between the calculated and preset values. Usually, thef, are torsional transition frequencies, although additional information can also be included, e.g. the energy difference between conformers (AH,), the geometry of the conformers (the positions of the potential function minima), and barrier height estimates AH’. The coefficients L/k are refined, starting from some initial approximation, by means of the iterative Gauss Newton procedure [21]. Corrections AI’,< to the current values of the potential function variable parameters are determined from the system of linear equations J’JAV
= -J’b
(11)
where S is the vector of the weighted deviations, J is elements Jacobian matrix with the the Jii = a&,/aV, = (l/~~)~d$~‘~/dV~ and AV is the sought vector of corrections to the potential function coefficients. If the property J; comprises the torsional as a frequency v,, or AH, (i.e. it is represented difference of eigenvalues AE,,, = E,, - E,,), then the relevant Jacobian elements are readily calculated on the basis of the Hellmann-Feynman
A. V. Abramenkov
(‘I al.lJournal
qf MolceuiarStructure 376
theorem:
= z
((@PII c0s.M I Qn) ~ wn, I cos_bP I*,))
(12)
(The lait expression is obtained with regard to a specific form of the function (Eq. (5) for the approximating potential.) Calculation of J,, for other propertiesji is discussed in detail in Ref. [22]. The peculiarity of the problem under consideration lies in the fact that the system of Eqs. (11) for determination of the corrections Av, is illconditioned every so often. This results in slow convergence or, sometimes, the impossibility of obtaining a final solution. The reason for these difficulties is the strong correlation between the parameters of the model. An approach allowing one to surmount these difficulties lies in application of the singular value decomposition (SVD) [23] to the Jacobi matrix J: J = UAW’
(13)
where U and W are orthogonal matrices and A is a rectangular matrix with non-negative diagonal elements Xii, which are called singular values, and zero off-diagonal elements. Substitution of small singular values by zeros is tantamount to modification of the initial problem where the near linear dependence between unknowns is replaced by the exact one. Coefficients of this dependence are contained in the corresponding column of the matrix W. The modified problem possesses a better conditionality but gives a larger residual sum of squared deviations. By choosing a proper number of singular values assumed to be zero, one can reach the point where there is a tolerable compromise between the conditionality of the problem and accuracy of reproducing the given data. The technique based on the SVD makes it possible to avoid any explicit exclusion of dependent unknowns: instead, from all the possible solutions, a least norm solution is adopted. The SVD method outlined above is a standard method for the solution of linear problems 1241. However, its application to the nonlinear problem of optimization of the potential parameters offers an additional advantage. Applying the singular
11996) 183-194
187
analysis to the linearized form (Eqs. (11)) of nonlinear inverse problems reveals linear dependences between corrections to the current values of the parameters involved in the optimization. Eqs. (11) are redefined at each iteration step and, concerning the parameters themselves, we can only speak of local dependences. Finding corrections with a minimum norm leads to the solution closest to the starting approximation. Clearly, such a solution should be given preference if the initial potential has some physical meaning (e.g. if it is obtained in quantum mechanical calculations). A more detailed discussion of questions connected with ill-conditionality in the problem of refining the potential function of internal rotation with instructive examples of real calculations can be found in Ref. [25] and references cited therein.
4. Practical use of the nonrigid model On general theoretical grounds, one would expect that the nonrigid model, being a higher order one, would allow reproduction of the real molecular behavior more accurately than the semirigid model. What actually happens is that the advantages of the nonrigid model do not always make themselves evident and depend on the formulation of the problem. The point is (see above) that the distinction between the two models consists only in the form of F(p). As an illustration of the character and order of magnitude of such differences, Fig. 1 shows curves of F(p) for semi- and nonrigid models of the acryloyl fluoride molecule. However, as was shown in Ref. [26], the form of F(p) may be varied over a wide range by nonlinear transformations of the coordinate cp. In particular, there is a transformation converting F(p) into a constant independent of cp. As this takes place, the spectrum of the model Hamiltonian in Eq. (3) is preserved. The transformation of ‘p, of course, modifies the potential function, but the contraction or extension along the p axis does not change the extremal values of V(p), i.e. the barrier height and energy difference between conformers are preserved. Jn an analogous way, it is shown in Ref. [27] that the effect of geometry relaxation during internal
1.6 1 0
I I 30 Interna6P
I
T
rotatiogn0
angle
‘~~egrees)
I 150
I 160
Fig. I. Cumparison of the functions F(q) for semi- and nonrigid mod& of internal rotation in acryloyl fluoride as calculated using quantum mechanical data obtained at the MP2/6-3lG* lcvcl [II].
rotation is radically indistinguishable from modifcation of the form of the potential function. The latter is expressed. in general, by changes in the values of the higher harmonic coefficients V, in Eq. (5). From this, it is inferred, in particular, that it is impossible to refine simultaneously both the functions F(p) and V,,(cp) by means of fitting parameters using the experimental vibrational frequencies. Therefore, if the problem is merely to find some model potential providing the maximum accurate representation of the observed spectrum of the torsional vibrations (without any additional requirements on the physical meaning of this potential), then both semi- and nonrigid models, and an even simpler model with F = constant, would provide equally good results. It is selfevident that for the different models different potential functions will be obtained. However, substitution of any one of these functions into the Hamiltonian (Eq. (3)) together with the relevant F(p) will yield the same spectrum. Nevertheless, in actual practice there is no constancy of the barrier height obtained using different models for the same molecule [26]. As mentioned in Ref. [27], the reason for this discrepancy is most likely
neglect of the pseudopotential component that is transferred into Vrff from the kinetic energy operator. i.e. the result mentioned above is obtained under the assumption that -(d/dp)F(d/dpj is the exact expression for the kinetic energy operator, and the rest of the Hamiltonian represents the properties of the real potential function, If definition of 9 (reference atoms selected) and the kinetic energy expression are dictated by physical reasons and cannot be arbitrarily changed, then unavoidable dif‘ferences may arise between the results obtained with semirigid and nonrigid models. In particular, when it is necessary to obtain comparable results for different molecules one must USC compatible definitions and expressions for the kinetic energy to determine the potential energy functions. An analogous situation also occurs when it is necessary to conform with a model at another physical level (for example, with the potential energy surface as calculated from solution of the electron Schrodinger equation in the Born-Oppenheimer approximation). Therefore. a nonrigid model of internal rotation should be used when physically correlated functions of the potential and kinetic energies are desired.
A. V. .khmenko~~
rt al./Journal
~/^Molrcubr
The principal problem in using the nonrigid model is obtaining data on the changes in the geometrical parameters during rotation of the molecular moieties as the angle cp is varied. There is no way of obtaining such information from experiment. At best, the experimental data may provide the geometrical parameters of the stable conformers corresponding to the minima on the potential curve. However, it should be expected (and this is confirmed by quantum mechanical calculations) that the largest changes in the geometry of the top and frame take place in configurations corresponding to the potential energy maximum, i.e. near the top of the barrier. In early work it was suggested that simple empirical functions describing the dependence of internuclear distances and the valence and dihedral angles (which characterize the molecular geometry) on the internal rotation angle p be used. Parametrization of these expressions was carried out, taking into account the experimentally determined geometry of conformers (i.e. as a rule, at two points on the potential curve). As an example, the expression Aa = a( 1 ~ cos 2~) was used in Ref. [6] for the construction of a nonrigid model of the glyoxal molecule. Here Acr is the change in the CC=0 bond angle with respect to its value for the trans conformer (‘p = 0). The disadvantages of such empirical accounting for relaxation of geometrical parameters are obvious. First, the available experimental information is inadequate to judge the dependence of geometrical parameters on the angle p. Because of this, the form of the functions employed is based on some a priori arguments that may result in serious errors, especially when p is far from its equilibrium value(s). Second, such functions must necessarily be extraordinarily simple because many parameters cannot be determined from the available data. Nevertheless, even with such a crude approach, the results obtained in Ref. [6] unambiguously bear witness to the significance of using a nonrigid model. Information about relaxation of geometrical parameters during internal rotation has improved considerably over the last 10 years. Now ab initio quantum mechanical data on organic molecules containing 10 or more atoms have become available even at fairly high theoretical levels (i.e. from
Structure 376 (1996) 183-194
189
calculations using extended basis sets and accounting for electron correlation). A number of the limitations to computational structural chemistry in its present stage of development and the conditions under which computational determination of molecular structures can be nearly comparable to high-quality experimental determinations were considered by Boggs [28]. When internal rotation is concerned, complete optimization of all the geometrical parameters (except QG)must be performed at a sufficient number of fixed values cp along the entire potential curve. In doing so, we concurrently obtain the theoretical potential function of internal rotation and a comprehensive pattern of changes in the molecular geometry during rotation of the molecular fragments with respect to each other. Furthermore, the results obtained may be introduced into standard equations such as Eqs. (5) and (6). In principle, this is sufficient to permit the calculation of the theoretical spectrum of the torsional vibrations when the energy levels and wavefunctions are obtained with the model Hamiltonian (Eq. (3)). Note, however, that the use of the ab initio potential function for Veff is not completely correct. As noted above, V,, is a pseudopotential and contains some (in general, small) terms depending only on the coordinates rather than on the impulses. These terms, however, are conceptually identical with the kinetic energy and depend on the nuclear masses. Thus, Veff should depend on the masses, in contrast to the quantum mechanical potential function, in the Born-Oppenheimer approximation. This feature is particularly important when constructing a potential function for several isotopomers [36,37]. A detailed discussion of this problem can be found in Ref. [25]. It is known that the distortions of the potential surfaces give rise to systematic errors in the calculation of the torsional frequencies (see, for example, Ref. [29]). It is reasonable to mention here that the definition of the torsional coordinates for the majority of molecules in vibrational problems is usually incorrect owing to the impossibility of partitioning the rotational and vibrational motions exactly. Thus, as a rule, ab initio potential functions are not sufficiently accurate to describe quantitatively the torsional spectrum. However,
190
A.V. Abramenkov
et ai.,%wnal
qf’M&cular
they offer a physically meaningful initial approximation for construction of the potential function by solution of the inverse problem using the set of experimentally measured frequencies as outlined in Section 2. Even in the case of ill-conditionality, using the SVD technique makes it possible to find a potential which is closest to the initial theoretical potential and allows reproduction of the vibrational frequencies at the level of accuracy of their experimental determination. In seminal papers which appeared in 1979 [30] and 1983 [31], the torsional potential function for 1,3-butadiene was constructed using quantum mechanical torsional energy data. Torsional frequencies were calculated for comparison with the experimental values. In these pioneering studies, the refinement of the potential (see above) using the experimental torsional frequencies was not carried out. However, some nonrigid models were constructed on the basis of ab initio geometrical results of Boggs and co-workers [32,33] and molecular mechanics calculations. The approach based on using optimized quantum mechanical molecular geometries (and energies) to construct the model kinetic energy operator and torsional function has gained acceptance as a promising one: after optimized geometries, obtained in highlevel calculations [34,3.5], became available, the torsional potential functions of 1,3-butadiene [36] and isoprene [29] were constructed using the ab initio data for the initial functions and then these were refined using the experimental torsional frequencies. Concurrently and independently, an analogous approach was used by Gomez et al. [37] for the construction of the potential functions of internal rotation of the methanol isotopomers. This approach has also been used successfully to construct refined potential functions for internal rotation of acrolein [38], glyoxal [39], acryloyl fluoride [40.41], fluoroacryloyl Buoride [42] and nitrobenzene [43]. Based on the body of knowledge obtained from these investigations, we can state with assurance that the nonrigid model makes it possible to arrive at more trustworthy and physically meaningful results than with a semirigid model. This is reflected, in particular, in the fact that the estimates of NY0 and the barrier heights have more realistic
Strucrure
376 (1996 i 1X3-194
values than those obtained by treatment of the same experimental data in the framework of a semirigid model. The explicit trend has been toward decreasing the gap between the estimates obtained and available experimental data as the level of quantum mechanical calculations is increased, i.e. when more correct values of geometrical parameters are involved. Let us consider some examples to depict the various magnitudes of effects which are caused by nonrigidity of molecular moieties during internal rotation. 4.1. h’itrobenzene For this molecule, semi- and nonrigid models lead to practically identical results [43]. This is because of the symmetry of the top and the small absolute changes in the geometrical parameters. In fact, it is possible to say that the moieties of the nitrobenzene molecule are nearly rigid. Therefore, for some molecules a semirigid model is a reasonable approximation. However, this can be determined reliably only after comparison of both models. From this point of view, it is always desirable to use the nonrigid model as a more general one. 4.2. Fluoroacryloyljuoride Although the top and the frame of this molecule are not symmetrical, they are nearly balanced (their centers of mass are posed near the axis of internal rotation). Hence, in the semirigid approximation, the value of the function F depends weakly on p, being nearly constant. Balance decreases the effect of changes in the geometrical parameters on the F function. Moreover, as was pointed out in Ref. [42], these changes occur as if they are in antiphase. This nearly results in the total mutual compensation of their effects. Thus, the functions F(p) for semi- and nonrigid models of fluoroacryloyl fluoride practically coincide, the differences being only a few units in the third significant figure, i.e. by less than 1% [42]. 4.3. 1,3-Butadiene As a third example,
the comparison
of the results
A.V. Abramenko~
0
et al.lJournnl
quantum
mechanical
Table 1 Expermental
rotatic
Transition
of the functions F(p) for senw and nonrigid data obtained at the MP2/6-31G* level.
and calculated
frequencies Exptl. [44]
of torsional
Structure
376 (1996)
transltions
I
angle l[cYegrees)
models
of internal
of 1.3-butadiene
191
183-194
I
I
I
I
30 InternaeP
Pig.2. Comparison
of Molecular
rotation
I
150
180
in 1.3-hutadiene
as calculated
using
(cm-‘)”
Nonrigid
Semirigid
Cak.
Err01
Calc.
Error
tln,lS 2-4 3-5 4-6 s-7 6- 8 l-9
321 60 316 60 311.10 305.70 299.60 292.70 285.20 274.30
320.60 316.72 311.99 306.42 299.97 292.57 284.1 I 214.42
0.12 0.89 0.72 0.37 -0.13 -1.09 0.12
320.65 316.RO 312.11 306.56 300.12 292.73 284.27 274.57
-0.95 0 20 1 01 0 86 0 52 0.03 -0.93 0.27
gauche o--20--2+ l--31--3+
269.90 262.40 255.90 214.90
270.8 I 261.89 255.43 214.99
0.91 -0.51 -0.47 0.09
269.23 261.13 252.53 213.53
-0.67 -1.27 -3.37 -1.37
o-2 I-3
-1.00
* ’ Calculations were performed for semirigid and nonrigid I’> = 942.2, V, = -92.6, V, = -60.3, Va = -13.4cm-‘.
0.65 models
with
the same potential
1.27 function:
VI = 550.5,
V, = 1061.6,
192 Table 2 Experimental Transition
A. V. Abramenkov
and calculated
frequencies Exptl. [45]
rl ul./Joumul
of torsional
yf‘Moleculur
transitions
of acryloyl
Strucrurr
376 (1996)
fluoride
183-194
(~rn~‘)~
Nonrigid Calc.
Semirigid Error
Calc
Error
tram o-1 l-2 2-3 3-4 4-5 5-6 6-7 778 8%‘)
116.72 114.86 113.06 111.26 109.48 107.68 105.85 103.98 102.08
116.72 114.87 113.05 111.26 109.48 107.68 105.86 103.99 102 07
0.00 0.01 -0.01 0.00 0.00 0.00 0.01 0.01 -0 01
116.84 115.14 113.46 111.77 110.07 108.34 106.56 104.72 102 81
0.12 0.28 0.40 0.51 0.59 0.66 0.71 0.74 0.73
ci.7 O-1 1 2 2-3 3-4 4-5 S-6 6-l
101.31 100.15 98.85 97.59 96.31 95.16 94.14
101.44 100.07 98.79 97.58 96.40 95.23 94.05
0.07 PO.08 PO.06 Po.o1 0.09 0.07 -0.09
103.46 102.08 100.80 99.57 98.36 97.15 95.92
2.09 1.93 1.95 1.98 2.05 1.99 1.78
cl a Calculations were performed for semirigid and nonrigid models C; = 113.0, V(4) = -122.8, V(5) = -8.7 and V(6) = 12.5 cm.-‘.
of calculations using semi- and nonrigid models for 1.3-butadiene are given in Fig. 2 and Table 1. They are based on ab initio quantum mechanical calculations at the MP2/6-3 lG* level (detailed results of these calculations including construction of the refined potential function of internal rotation will be published later). Although the degree of “nonrigidity” of the 1,3-butadiene molecule, i.e. the amplitude of changes in the geometrical parameters during internal rotation, is important, the semi- and nonrigid models give nearly identical results over a large portion of the curve (in the region from 0” up to 135”). This is connected with the phenomenon of mutual compensation of the effects caused by changing each of the gcomctrical parameters (see acryloyl fluoride). The major differences between the functions F(v) for the two models arise only in the region up = 150-180“. Hence it is expected that accounting for geometry relaxation will have an appreciable effect on the shape of the potential curve in the region near the gauche minimum and especially on the estimation
0.05 with the same potential
1.37 function
[41]: V, = 71.7, Vz = 1944.8,
of the barrier height separating the potential wells of the gauche conformers. The results of torsional frequency calculations using the same potential function optimized for a nonrigid model are compared with the experimental data in Table 1. The difference in the discrepancies is not as marked as in the case of acryloyl fluoride (see below). For the trans conformer the average magnitude of the errors does not change. However, the discrepancies between the calculated and experimental torsional frequencies for the gauche conformer are several times larger with the semirigid model. 4.4. Acryloyljhoride The geometrical parameters for both the semiand nonrigid models are taken from the results of ab initio quantum mechanical calculations at the MP2/6-31G* level [41]. The functions F(p) (Eq. 6) for internal rotation in acryloyl fluoride are shown in Fig. 1. The differences observed here are the most
A.V. Abramenkov
et al.iJournal
of MolecularStructure
marked ones in comparison with other molecules studied [38-431. It is of interest that the difference between the values F(p) calculated with the semiand nonrigid models is greatest at q = lSO”, whereas the dependence of the individual geometrical parameters on (o usually has the form of a curve with an extremum near 90” (see, for example, Fig. 2 in Ref. [41]). Probably, there is partial compensation of effects caused by changing the different geometrical parameters. Note also that this situation is probably a typical one, since the same pattern is observed to a greater or lesser extent for the majority of molecules studied by us. The influence of the choice of the molecular model on the calculated vibrational frequencies can also be seen in Table 2. Here the potential function was optimized for a nonrigid model so that the deviations of the calculated frequencies from the experimental ones are less than 0.1 cm-’ (see Table 3 in Ref. [41], columns denoted A). As is easily seen, transition to the semirigid model (provided that the shape of the potential function is preserved) results in a marked increase in frequency errors, especially on going from the trans to the cis conformer. This correlates with the character of discrepancies in the functions F(p). It is selfevident that after optimization of the potential function for the semirigid model it is possible to attain the same good reproduction of the experimental frequencies as in the nonrigid model, but the shape of the potential function will be changed in this case. (Table 4 in Ref. [41] gives an idea about the magnitude of possible variations of the potential function.)
5. Conclusions The above consideration of semi- and nonrigid models of internal rotation demonstrates the generality of the approach based on the latter model. Application of a nonrigid model means following the ideology that a known error is preferable to a chance possibility of obtaining a nearly correct answer. However, it is always necessary to keep in mind that the results obtained from the nonrigid model are also significantly dependent on the level of the quantum mechanical calculation
376 (1996)
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providing the character relaxation.
of the molecular
193
geometry
Acknowledgements Yu.N.P. thanks the ULB for an international scientific collaboration grant and the Russian Foundation for Fundamental Investigations (Project 93-3-18386) for financial support.
References [l] [Z] [3] [4]
H.H. Nielsen, Phys Rev., 40 (1932) 445 E.R. Wilson. Chem. Rev., 27 (1940) 17. B.L. Crawford, J Chem Phys., 8 (1940) 273. R. Meyer and Hs.H Giinthard, J Chem. Phys., 4Y (1968) 1510. [S] R. Meyer and E.B. Wilson, I. Chem. Phys., 53 (1970) 3969. [6] R. Meyer, J. Mol. Spectrosc., 76 (lY79) 266. [7] Hs.H. Giinthard, Studies in Physical and Theoretical Chemistry, Vol. 23, Elsevier, Amsterdam, 1983, pp. 133147. [8] J.D Lewis, T.B. Malloy. T.H. Chao and J. Laane, J. Mol. Struct., 12 (1972) 427. [9] J.D. Leuis and J. Laarle, J. Mol. Spectrosc., 65 (1977) 147. [lo] J. Laane, MA Harthcock, P.M. Killough, L.E. Bauman and J.M. Cooke, J. Mol. Spectrosc., 91 (1982) 286. [ll] M.A. Harthcock and J. Laane. J. Mol. Spectrosc., 91 (1982) 300. [12] Y.G. Smeyers, M.L. Senent. F.Y.J. Pefialver and D.C. Moule, J. Mol. Struct. (Theochem), 287 (1993) 117. [13] A. Nifio, C. MuAoz-Care and D.C. Moule. J. Mol. Struct., 318 (1994) 237. [14] R.A. Weersink. D.T. Cramb, SC. Wallace and R.D. Gordon, J. Chem. Phys.. 102 (1995) 623. [15] Y.G. Smeyers, A. Niiio and D.C. Moule, J. Chem. Phys.. 93 (1990) 5786. [16] B. Kirtman. J. Chem. Phys., 37 (1962) 2516. [17] C.R. Quade and CC. Lin, J. Chem. Phys., 38 (1963) 540. [18] J.V. Knopp and CR. Quade, J. Chem. Phys., 48 (1968) 3317. [19] P. Meakin. D.O. Harris and E. Hirota. J. Chem. Phys., 51 (1969) 3775. [20] KS. Pitzer, J. Chem. Phys., 14 (1946) 239. [21] Y. Bard. Nonlinear Parameter Estimation, Academic Press, New York, 1974. 1221 A.V. Abramenkov, Russ. J. Phys. Chcm., 69 (1995) 94X. (231 J.H. Wilkinson and C. Remsch, Linear Algebra: Handbook for Automatic Computation, Vol. 2, Springer, Bcrhn, 1971. [24] CL. Lawson and R.J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englcwood Cliffs, NJ, 1974.
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1251 A.V. Abramenkov and Yu.N. Panchenko, Russ. J. Phys. Chem., 69 (1995) 944. [26] C.S. Ewig and D.O. Harris, J. Chem. Phys., 52 (1970) 6268. [27] R.M. Lees, J. Chem. Phys., 59 (1973) 2690. [ZS] J.E. Boggs, in A. Domcnicano and I. Hargittai (Eds.), Accurate Molecular Structures, Oxford University Prcqs, New York, 1992, Chapter 13. [29] Yu.N. Panchenko, V.I. Pupyshev, A.V. Abramenkov, M. Traetteberg and S.J. Cyan, J. Mol. Struct., 130 (1985) 355. [30] Ch.W. Bock, Ph. George, M. Trachtman and M. Zanger, J. Chem. Sot., Perkin Trans. 2, (1979) 26. (311 Yu.N. Panchenko, A.V. Abramenkov, V.I. Mochalov, A.A. Zenkin, G. Keresztury and G. Jalsovszky, J. Mol. Spectrosc., 99 (1983) 288. [32] P.N. Skancke and I.E. Boggs, J. Mol. Struct., 16 (1973) 179. [33] S. Skaarup, J.E. Boggs and P.N. Skancke, Tetrahedron, 32 (1976) 1179. [34] G R. De Mar& in I.G. Csizmadia and R. Daudel (Ed%), Computational Theoretical Organic Chemistry, NATO ASI Ser. C, 67 (1981) 371; G.R. De MarC and D. Ncisius, J. Mol. Struct. (Theochem), 109 (1984) 103. [35] Ch.W. Bock, P. George and M. Trachtman, Theor. Chim. Acta, 64 (1984) 293. [36] Yu.N. Panchenko, A.V. Abramenkov and Ch.W. Bock, J. Mol. Struct., 140 (lY86) 87.
Structure
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[37] P. GOmez, M. Fcrnindcz and V. Botclla, J Mol. Struct. (Theochem), 122 (1985) 317. [38] G.R. De Mar& Y&N. Panchenko and A.V. Abramcnkov, J. Mol. Struct., 160 (1987) 327. [39] G.R. De Mar& Yu.N. Panchenko and A.V. Abramenkov, J. Phys Chem., 96 (1992) 2111. [40] Ch.W. Bock, A.V. Abramenkov,G.R. De Mare andYu.N. Panchenko, J. Comput. Chem., 13 (1992) 718. [41] G.R. De Mart-, Yu.N. Panchenko, A.V. Abramenkov and Ch.W. Bock, Can J. Chem., 71 (1993) 656. (Note that in Tablr 2 of this paper, the coefficient F4 in the Fourier expansion of the F(q) function was inadvertently reported as 0.2306cm-‘; the correct value is F4 = 0.0231 cm-‘. This error is restricted only to the published value of F4; all the calculations reported in the paper were performed with the correct value of F+) [42] G.R. De Mar& Yu.N. Panchckno and A.V. Abramenkov, Russ. J. Phys. Chcm., 69 t 1995) Ii 17. [43] L.S. Khaikin, Ch.W. Bock, A.V. Abramenkov, 1.P. Fabrichnyi, 0-E. Grikina and L.V. Vilkov, XV Austin Sympossum on Molecular Structure, March 21-23, lYY4, IJniverslty of Texas, Austin, TX, S 24, 1994, p. 125. [44] R. Engeln, D. Consalvo and J. Reuss, Chem. Phys., 160 (1992) 427. [45] J.R. Durig, R.J. Berry and P. Groner, J. Chem. Phys., 87 (1987) 6303.
MOLECULAR STRUCTURE ELSEVIER
Journal of Molecular Structure 376 (1996) 183-194
Effective potential function of internal rotation nonrigid approximation’ A.V. Abramenkov”,
Ch.W. Bockb, G.R. De MarF*,
in the
Yu.N. Panchenkod
‘Laboratory of Molecular Structure and Quantum Mechanics, Choir of Physicul Chrmistry. Department of Chrmistry, M.V. Lomonosov Moscow State lhriversity, Moscow 119899, Rmsiaw Federution bChen~istry Department. Philadelphia College of Textiles and Science, Philadelphiu, PA19144. WA ‘Lnborotoire de Chimw Physique Moldculaire, FacultC des Sciences, CP 160/09. UkiversitC Libre de Brrr.relles, Av F.-D. RomewIt 50, B-1050 Brussels. Belgium
Received 21 July 1995; accepted 4 September 1995
Abstract The feasibility of the construction of potential functions of internal rotation in the framework of semi- and nonrigid one-dimensional models is considered. The nonrigid approach under discussion is based on using ab initio quantum mechanical calculations to obtain informalion on molecular geometry relaxation during internal rotation and to find a physically meaningful starting approximation of the potential. Furthermore. the theoretical potential obtained is refined according to the criterion of the best agreement between the calculated and experimental frequencies of the torsional vibrations. During this latter optimization, some additional conditions may be imposed which support the maximal closeness of the results to the starting approximation. The refined potential is sensitive to the accuracy of determination of molecular geometrical parameters. For this reason, the quantum mechanical calculation should be performed at a rather high level of theory (accounting for electron correlation) to obtain reliable results. The comparison of semi- and
nonrigid models of internal rotation is performed using nitrobenzene, fluoroacryloyl fluoride, 1Jbutadiene fluoride as examples.
1. Introduction When studying intramolecular motions with large amplitudes such as hindered internal rotation. pseudorotation of cyclic molecules or inversion and ring puckering, the molecular models used involve, as dynamic variables, only the internal * Corresponding author ’Dedicated to Professor James E. Boggs on the occasion of his 75th birthday.
and acryloyl
coordinates corresponding to these motions and the angular variables characterizing the overall rotation. The simplest case in point is the so called one-dimensional model (one internal degree of freedom plus overall rotation). Many practically important problems can be successfully solved in the framework of the one-dimensional approximation. This is true, in particular, for the description of torsional vibratidns and internal rotatidn of molecules with one top or several weakly interacting
0022-2860/96/$15.00 IQ 1996 Elsevier Science B.V. All rights reserved SSDI 0022-2860(95)09098-3
tops.
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A. V. Abrammkm
er ul.~/ounml
of Moleculm
Sixty years ago, the semirigid model of internal rotation [l-3] was developed, which considers a molecule as consisting of two rigid moieties (referred to as top and frame) rotating around their connecting axis. In spite of its simplicity, such a model provides a means for the correct description of qualitative peculiarities of the motions under consideration. Moreover, in the framework of this approximation, it is often possible to achieve quantitative agreement between the calculated and experimental torsional vibrational spectra. A more general approach was developed at the end of the 1960s with major contributions being reported in papers by Meyer and co-workers [446], Gtinthard [7] and Laane and co-workers [8-l I]. Some general expressions were obtained for the model Hamiltonians for any arbitrary kind of motion with large amplitude, with due regard to the interaction of internal degrees of freedom with the overall rotation. In particular, a model for the torsional vibrations and internal rotation was suggested, which was called “flexible” model by the authors [5,6]. We prefer the term “nonrigid model” used by Giinthard [7]. It is assumed that the top and frame may be distorted during internal rotation, i.e. all the geometrical parameters of such a molecular model depend on the angle of internal rotation p. If was also pointed out [6] that such an approach, remaining within the framework of the one-dimensional approximation, is a sort of alternative for more complicated (or “combined” [6]) multi-dimensional models accounting for other vibrational degrees of freedom at the lcvcl of small vibrations. Therefore, the nonrigid model was introduced as a means of accounting (at least partially) for interactions between the considered motion with large amplitude and other molecular vibrations. If it is necessary to take into account the interactions of several motions with large amplitude (for example, interaction of rotations of two tops or rotation of top and inversion), the nonrigid onedimensional model cannot provide a means for solution of this problem. A multi-dimensional model should be used here. constructed within the framework of the same formalism as a one-dimensional model [lO,ll]. For example, a
Srrrrcturc 376 (1996 i 183-194
two-dimensional model was used in Ref. [12] for describing the rotation of the methyl moieties in acetone and trans-dimethylglyoxal. An analogous approach was applied to analysis of the interaction between internal rotation of the methyl top and inversion of the aldehyde moiety in the ground and lower excited electronic states of acetaldehyde [13.14] and thioacetaldehyde [15]. Note that multidimensional models of motions with large amplitude can be constructed using either rigid or nonrigid approximations. In the latter case, the gcomctrical parameters of the moving moieties are assumed to depend on several variables. In the present paper. we shall restrict our consideration to a one-dimensional nonrigid model of hindered internal rotation and discuss some problems related to the practical construction of potential functions of internal rotation in this approximation.
2. The model The general forms of the Hamiltonians for rigid and nonrigid models of internal rotation are essentially the same. The relevant equations are well known [4-6,8,16-191. They are given here without any further comments, viz. 2
where cp is the internal rotation F = (t1/2)G,, and g is the determinant matrix G:
G=
-I21
122
-I23
x2,
41
x3q
angle, of the
(2) -41
-42
qd
Kp2
xp3
y.pp
Here IV are the elements of the usual inertia tensor; XI,- = X,+,;= cj mj[rl x (8rj/jla(p)]l are the elements corresponding to interaction between internal and overall molecular rotations; YVP = c, m,(iAj/c3p) (i3rj/dp); m, and ri are the mass and radius vector,
A. C’. .4hramenkov et al.lJournai of Molecular Structure 376 (1996,
respectively, of the,jth atom in the molecular frame of reference. The g and F values are functions of the atomic coordinates and masses and do not depend on momenta. The second and third terms in the kinetic energy operator, which are primarily responsible for the interaction of internal and overall rotation of a molecule, are therefore usually combined with the potential function. This gives the effective potential VefF:
The distinction between rigid and nonrigid models manifests itself in the character of the dependence of the G matrix elements on the internal rotation angle ‘p. In the case of the semirigid model, such a dependence is due to the change in the position of the molecular center of mass during rotation of the top with respect to the frame. This results in changes of radius vectors of all the atoms in the center of mass coordinate system. If the top is a symmetric one, i.e. its center of mass is situated on the axis of internal rotation, then, in the framework of the semirigid model, the G matrix does not depend on ‘p, and F in the kinetic energy operator becomes merely a constant (internal rotation constant). For the nonrigid model, along with the change of position of the center of mass, there is a parametric dependence on 9 for all the internal coordinates (internuclear distances. valence and dihedral angles). Because of this, in general, the G matrix in the nonrigid approximation will be a function of p even in the case of a symmetric top. There is a further feature distinguishing the nonrigid from a semirigid model. For the internal coordinate which is relevant to the angle ‘p, the latter is usually determined by the axis of internal rotation and a pair of atoms which are not aligned with this axis. One of these two atoms must belong to the top and the other to the frame. In the semirigid model, the different choices of dihedral angles differ from one another only by an additive constant (phase), i.c. the determination of the angle p is essentially unambiguous.
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The value of the angle cp in nonrigid models depends on the reference atoms selected in the top and frame: selection of different reference atoms corresponds to nonequivalent definitions of cp which are related by some nonlinear transformation. This should be taken into consideration, e.g. in comparing potential functions obtained in different ways (for example, by different choice of coordinates). According to the results of numerous quantum mechanical calculations, internal rotations usually have less effect on the structural parameters of heavy (skeletal) atoms. It is therefore reasonable to use such reference atoms for the definition of the internal rotation angle. to minimize possible discrepancies related to angle definition. For the practical determination of the function F(p) there are at least two approaches. In the first, F is calculated as an ordinary rotational constant using the reduced inertia moment of internal rotation 1, [3], viz. F=-
h hC1,
Z, being explicitly expressed in terms of instant atomic coordinates and masses. This is achieved by using an implicit coordinate transformation depending on y and diagonalizing the G matrix
POI. The second approach is based on the direct calculation of the G matrix elements, the necessary derivatives ar,/&c being calculated by means of numerical differentiation [6,10,11]. Both methods are equally applicable to semiand nonrigid models, because they make it possible to introduce easily a parametric dependence of top and frame geometries on the internal rotation angle cp. In principle, the second technique can be used to analyze interactions between internal and overall molecular rotations by calculating cross terms X,, and estimating their contribution to F(p) and u(y). If the first technique is used, these terms are implicitly redistributed between the principal momenta of inertia and the reduced moment of inertia of internal rotation. However, in practice, the first technique is applied more frequently, probably because it is simpler computationally. Because of its conventional character, this is the technique used in our calculations.
186
A. V. Abramenkov
3. Computational
et al.lJournnl
of Molecular
details
The model potential function V,rr(cp) is usually approximated by a Fourier series truncated after several terms. Specifically, if at least one of the rotating fragments has a symmetry plane containing the rotation axis, the commonly used representation is
In the absence of symmetry elements. similar expansions in sine and cosine functions are used. Fourier approximations are also used to represent
= F. +
5 Fk cos ky
183-194
calculations for vibrational (IR and Raman) spectra require the model dipole moment and polarizability operators. The procedure given above makes it possible to obtain the theoretical torsional spectrum provided that the potential function V&y) is known. The inverse problem of constructing (or refining) the potential function of internal rotation consists in optimization of the coefficients vk in the expanSiOn (Eq. (5)) in accordance with the criterion of minimizing the functional R(V) =
&Yi’ 1
(6)
k=l
where the coefficients F, are determined by treatment of a set of the F(q) values as calculated for different points along the coordinate axis of the internal rotation angle ip. A variational technique is best suited to the calculation of the energy spectrum and wavefunctions which are relevant to the model Hamiltonian (Eq. (3)) using the approximations in Eqs. (5) and (6). The basis set used here is composed of eigenfunctions of free rotation, (27r-“‘e=‘kq [19]. However, an equivalent basis set made up of real functions [8,9]: & = (27)-l’* t+!~;= 7i-‘12 cos kp
k=
1,2,...
@i = rTT’12sin kv
k=
1,2,...
(7)
is usually utilized. The eigenvalues Em (torsional energy levels) are found by diagonalization of the Hamiltonian matrix. The eigenvectors contain the coefficients of expansion of wavefunctions with respect to the basis set of Eq. (7), i.e.
The wavefunctions relative probabilities
376 (1996)
where b, is the weighted deviation of the calculated value of some property h from the corresponding experimental (or preassigned) one:
F(cp), e.g. F(y)
Structure
can be used to calculate the of torsional transitions. Such
The cri value characterizes the degree of accuracy of the ith experimental value or the admissible level of discrepancies between the calculated and preset values. Usually, thef, are torsional transition frequencies, although additional information can also be included, e.g. the energy difference between conformers (AH,), the geometry of the conformers (the positions of the potential function minima), and barrier height estimates AH’. The coefficients L/k are refined, starting from some initial approximation, by means of the iterative Gauss Newton procedure [21]. Corrections AI’,< to the current values of the potential function variable parameters are determined from the system of linear equations J’JAV
= -J’b
(11)
where S is the vector of the weighted deviations, J is elements Jacobian matrix with the the Jii = a&,/aV, = (l/~~)~d$~‘~/dV~ and AV is the sought vector of corrections to the potential function coefficients. If the property J; comprises the torsional as a frequency v,, or AH, (i.e. it is represented difference of eigenvalues AE,,, = E,, - E,,), then the relevant Jacobian elements are readily calculated on the basis of the Hellmann-Feynman
A. V. Abramenkov
(‘I al.lJournal
qf MolceuiarStructure 376
theorem:
= z
((@PII c0s.M I Qn) ~ wn, I cos_bP I*,))
(12)
(The lait expression is obtained with regard to a specific form of the function (Eq. (5) for the approximating potential.) Calculation of J,, for other propertiesji is discussed in detail in Ref. [22]. The peculiarity of the problem under consideration lies in the fact that the system of Eqs. (11) for determination of the corrections Av, is illconditioned every so often. This results in slow convergence or, sometimes, the impossibility of obtaining a final solution. The reason for these difficulties is the strong correlation between the parameters of the model. An approach allowing one to surmount these difficulties lies in application of the singular value decomposition (SVD) [23] to the Jacobi matrix J: J = UAW’
(13)
where U and W are orthogonal matrices and A is a rectangular matrix with non-negative diagonal elements Xii, which are called singular values, and zero off-diagonal elements. Substitution of small singular values by zeros is tantamount to modification of the initial problem where the near linear dependence between unknowns is replaced by the exact one. Coefficients of this dependence are contained in the corresponding column of the matrix W. The modified problem possesses a better conditionality but gives a larger residual sum of squared deviations. By choosing a proper number of singular values assumed to be zero, one can reach the point where there is a tolerable compromise between the conditionality of the problem and accuracy of reproducing the given data. The technique based on the SVD makes it possible to avoid any explicit exclusion of dependent unknowns: instead, from all the possible solutions, a least norm solution is adopted. The SVD method outlined above is a standard method for the solution of linear problems 1241. However, its application to the nonlinear problem of optimization of the potential parameters offers an additional advantage. Applying the singular
11996) 183-194
187
analysis to the linearized form (Eqs. (11)) of nonlinear inverse problems reveals linear dependences between corrections to the current values of the parameters involved in the optimization. Eqs. (11) are redefined at each iteration step and, concerning the parameters themselves, we can only speak of local dependences. Finding corrections with a minimum norm leads to the solution closest to the starting approximation. Clearly, such a solution should be given preference if the initial potential has some physical meaning (e.g. if it is obtained in quantum mechanical calculations). A more detailed discussion of questions connected with ill-conditionality in the problem of refining the potential function of internal rotation with instructive examples of real calculations can be found in Ref. [25] and references cited therein.
4. Practical use of the nonrigid model On general theoretical grounds, one would expect that the nonrigid model, being a higher order one, would allow reproduction of the real molecular behavior more accurately than the semirigid model. What actually happens is that the advantages of the nonrigid model do not always make themselves evident and depend on the formulation of the problem. The point is (see above) that the distinction between the two models consists only in the form of F(p). As an illustration of the character and order of magnitude of such differences, Fig. 1 shows curves of F(p) for semi- and nonrigid models of the acryloyl fluoride molecule. However, as was shown in Ref. [26], the form of F(p) may be varied over a wide range by nonlinear transformations of the coordinate cp. In particular, there is a transformation converting F(p) into a constant independent of cp. As this takes place, the spectrum of the model Hamiltonian in Eq. (3) is preserved. The transformation of ‘p, of course, modifies the potential function, but the contraction or extension along the p axis does not change the extremal values of V(p), i.e. the barrier height and energy difference between conformers are preserved. Jn an analogous way, it is shown in Ref. [27] that the effect of geometry relaxation during internal
1.6 1 0
I I 30 Interna6P
I
T
rotatiogn0
angle
‘~~egrees)
I 150
I 160
Fig. I. Cumparison of the functions F(q) for semi- and nonrigid mod& of internal rotation in acryloyl fluoride as calculated using quantum mechanical data obtained at the MP2/6-3lG* lcvcl [II].
rotation is radically indistinguishable from modifcation of the form of the potential function. The latter is expressed. in general, by changes in the values of the higher harmonic coefficients V, in Eq. (5). From this, it is inferred, in particular, that it is impossible to refine simultaneously both the functions F(p) and V,,(cp) by means of fitting parameters using the experimental vibrational frequencies. Therefore, if the problem is merely to find some model potential providing the maximum accurate representation of the observed spectrum of the torsional vibrations (without any additional requirements on the physical meaning of this potential), then both semi- and nonrigid models, and an even simpler model with F = constant, would provide equally good results. It is selfevident that for the different models different potential functions will be obtained. However, substitution of any one of these functions into the Hamiltonian (Eq. (3)) together with the relevant F(p) will yield the same spectrum. Nevertheless, in actual practice there is no constancy of the barrier height obtained using different models for the same molecule [26]. As mentioned in Ref. [27], the reason for this discrepancy is most likely
neglect of the pseudopotential component that is transferred into Vrff from the kinetic energy operator. i.e. the result mentioned above is obtained under the assumption that -(d/dp)F(d/dpj is the exact expression for the kinetic energy operator, and the rest of the Hamiltonian represents the properties of the real potential function, If definition of 9 (reference atoms selected) and the kinetic energy expression are dictated by physical reasons and cannot be arbitrarily changed, then unavoidable dif‘ferences may arise between the results obtained with semirigid and nonrigid models. In particular, when it is necessary to obtain comparable results for different molecules one must USC compatible definitions and expressions for the kinetic energy to determine the potential energy functions. An analogous situation also occurs when it is necessary to conform with a model at another physical level (for example, with the potential energy surface as calculated from solution of the electron Schrodinger equation in the Born-Oppenheimer approximation). Therefore. a nonrigid model of internal rotation should be used when physically correlated functions of the potential and kinetic energies are desired.
A. V. .khmenko~~
rt al./Journal
~/^Molrcubr
The principal problem in using the nonrigid model is obtaining data on the changes in the geometrical parameters during rotation of the molecular moieties as the angle cp is varied. There is no way of obtaining such information from experiment. At best, the experimental data may provide the geometrical parameters of the stable conformers corresponding to the minima on the potential curve. However, it should be expected (and this is confirmed by quantum mechanical calculations) that the largest changes in the geometry of the top and frame take place in configurations corresponding to the potential energy maximum, i.e. near the top of the barrier. In early work it was suggested that simple empirical functions describing the dependence of internuclear distances and the valence and dihedral angles (which characterize the molecular geometry) on the internal rotation angle p be used. Parametrization of these expressions was carried out, taking into account the experimentally determined geometry of conformers (i.e. as a rule, at two points on the potential curve). As an example, the expression Aa = a( 1 ~ cos 2~) was used in Ref. [6] for the construction of a nonrigid model of the glyoxal molecule. Here Acr is the change in the CC=0 bond angle with respect to its value for the trans conformer (‘p = 0). The disadvantages of such empirical accounting for relaxation of geometrical parameters are obvious. First, the available experimental information is inadequate to judge the dependence of geometrical parameters on the angle p. Because of this, the form of the functions employed is based on some a priori arguments that may result in serious errors, especially when p is far from its equilibrium value(s). Second, such functions must necessarily be extraordinarily simple because many parameters cannot be determined from the available data. Nevertheless, even with such a crude approach, the results obtained in Ref. [6] unambiguously bear witness to the significance of using a nonrigid model. Information about relaxation of geometrical parameters during internal rotation has improved considerably over the last 10 years. Now ab initio quantum mechanical data on organic molecules containing 10 or more atoms have become available even at fairly high theoretical levels (i.e. from
Structure 376 (1996) 183-194
189
calculations using extended basis sets and accounting for electron correlation). A number of the limitations to computational structural chemistry in its present stage of development and the conditions under which computational determination of molecular structures can be nearly comparable to high-quality experimental determinations were considered by Boggs [28]. When internal rotation is concerned, complete optimization of all the geometrical parameters (except QG)must be performed at a sufficient number of fixed values cp along the entire potential curve. In doing so, we concurrently obtain the theoretical potential function of internal rotation and a comprehensive pattern of changes in the molecular geometry during rotation of the molecular fragments with respect to each other. Furthermore, the results obtained may be introduced into standard equations such as Eqs. (5) and (6). In principle, this is sufficient to permit the calculation of the theoretical spectrum of the torsional vibrations when the energy levels and wavefunctions are obtained with the model Hamiltonian (Eq. (3)). Note, however, that the use of the ab initio potential function for Veff is not completely correct. As noted above, V,, is a pseudopotential and contains some (in general, small) terms depending only on the coordinates rather than on the impulses. These terms, however, are conceptually identical with the kinetic energy and depend on the nuclear masses. Thus, Veff should depend on the masses, in contrast to the quantum mechanical potential function, in the Born-Oppenheimer approximation. This feature is particularly important when constructing a potential function for several isotopomers [36,37]. A detailed discussion of this problem can be found in Ref. [25]. It is known that the distortions of the potential surfaces give rise to systematic errors in the calculation of the torsional frequencies (see, for example, Ref. [29]). It is reasonable to mention here that the definition of the torsional coordinates for the majority of molecules in vibrational problems is usually incorrect owing to the impossibility of partitioning the rotational and vibrational motions exactly. Thus, as a rule, ab initio potential functions are not sufficiently accurate to describe quantitatively the torsional spectrum. However,
190
A.V. Abramenkov
et ai.,%wnal
qf’M&cular
they offer a physically meaningful initial approximation for construction of the potential function by solution of the inverse problem using the set of experimentally measured frequencies as outlined in Section 2. Even in the case of ill-conditionality, using the SVD technique makes it possible to find a potential which is closest to the initial theoretical potential and allows reproduction of the vibrational frequencies at the level of accuracy of their experimental determination. In seminal papers which appeared in 1979 [30] and 1983 [31], the torsional potential function for 1,3-butadiene was constructed using quantum mechanical torsional energy data. Torsional frequencies were calculated for comparison with the experimental values. In these pioneering studies, the refinement of the potential (see above) using the experimental torsional frequencies was not carried out. However, some nonrigid models were constructed on the basis of ab initio geometrical results of Boggs and co-workers [32,33] and molecular mechanics calculations. The approach based on using optimized quantum mechanical molecular geometries (and energies) to construct the model kinetic energy operator and torsional function has gained acceptance as a promising one: after optimized geometries, obtained in highlevel calculations [34,3.5], became available, the torsional potential functions of 1,3-butadiene [36] and isoprene [29] were constructed using the ab initio data for the initial functions and then these were refined using the experimental torsional frequencies. Concurrently and independently, an analogous approach was used by Gomez et al. [37] for the construction of the potential functions of internal rotation of the methanol isotopomers. This approach has also been used successfully to construct refined potential functions for internal rotation of acrolein [38], glyoxal [39], acryloyl fluoride [40.41], fluoroacryloyl Buoride [42] and nitrobenzene [43]. Based on the body of knowledge obtained from these investigations, we can state with assurance that the nonrigid model makes it possible to arrive at more trustworthy and physically meaningful results than with a semirigid model. This is reflected, in particular, in the fact that the estimates of NY0 and the barrier heights have more realistic
Strucrure
376 (1996 i 1X3-194
values than those obtained by treatment of the same experimental data in the framework of a semirigid model. The explicit trend has been toward decreasing the gap between the estimates obtained and available experimental data as the level of quantum mechanical calculations is increased, i.e. when more correct values of geometrical parameters are involved. Let us consider some examples to depict the various magnitudes of effects which are caused by nonrigidity of molecular moieties during internal rotation. 4.1. h’itrobenzene For this molecule, semi- and nonrigid models lead to practically identical results [43]. This is because of the symmetry of the top and the small absolute changes in the geometrical parameters. In fact, it is possible to say that the moieties of the nitrobenzene molecule are nearly rigid. Therefore, for some molecules a semirigid model is a reasonable approximation. However, this can be determined reliably only after comparison of both models. From this point of view, it is always desirable to use the nonrigid model as a more general one. 4.2. Fluoroacryloyljuoride Although the top and the frame of this molecule are not symmetrical, they are nearly balanced (their centers of mass are posed near the axis of internal rotation). Hence, in the semirigid approximation, the value of the function F depends weakly on p, being nearly constant. Balance decreases the effect of changes in the geometrical parameters on the F function. Moreover, as was pointed out in Ref. [42], these changes occur as if they are in antiphase. This nearly results in the total mutual compensation of their effects. Thus, the functions F(p) for semi- and nonrigid models of fluoroacryloyl fluoride practically coincide, the differences being only a few units in the third significant figure, i.e. by less than 1% [42]. 4.3. 1,3-Butadiene As a third example,
the comparison
of the results
A.V. Abramenko~
0
et al.lJournnl
quantum
mechanical
Table 1 Expermental
rotatic
Transition
of the functions F(p) for senw and nonrigid data obtained at the MP2/6-31G* level.
and calculated
frequencies Exptl. [44]
of torsional
Structure
376 (1996)
transltions
I
angle l[cYegrees)
models
of internal
of 1.3-butadiene
191
183-194
I
I
I
I
30 InternaeP
Pig.2. Comparison
of Molecular
rotation
I
150
180
in 1.3-hutadiene
as calculated
using
(cm-‘)”
Nonrigid
Semirigid
Cak.
Err01
Calc.
Error
tln,lS 2-4 3-5 4-6 s-7 6- 8 l-9
321 60 316 60 311.10 305.70 299.60 292.70 285.20 274.30
320.60 316.72 311.99 306.42 299.97 292.57 284.1 I 214.42
0.12 0.89 0.72 0.37 -0.13 -1.09 0.12
320.65 316.RO 312.11 306.56 300.12 292.73 284.27 274.57
-0.95 0 20 1 01 0 86 0 52 0.03 -0.93 0.27
gauche o--20--2+ l--31--3+
269.90 262.40 255.90 214.90
270.8 I 261.89 255.43 214.99
0.91 -0.51 -0.47 0.09
269.23 261.13 252.53 213.53
-0.67 -1.27 -3.37 -1.37
o-2 I-3
-1.00
* ’ Calculations were performed for semirigid and nonrigid I’> = 942.2, V, = -92.6, V, = -60.3, Va = -13.4cm-‘.
0.65 models
with
the same potential
1.27 function:
VI = 550.5,
V, = 1061.6,
192 Table 2 Experimental Transition
A. V. Abramenkov
and calculated
frequencies Exptl. [45]
rl ul./Joumul
of torsional
yf‘Moleculur
transitions
of acryloyl
Strucrurr
376 (1996)
fluoride
183-194
(~rn~‘)~
Nonrigid Calc.
Semirigid Error
Calc
Error
tram o-1 l-2 2-3 3-4 4-5 5-6 6-7 778 8%‘)
116.72 114.86 113.06 111.26 109.48 107.68 105.85 103.98 102.08
116.72 114.87 113.05 111.26 109.48 107.68 105.86 103.99 102 07
0.00 0.01 -0.01 0.00 0.00 0.00 0.01 0.01 -0 01
116.84 115.14 113.46 111.77 110.07 108.34 106.56 104.72 102 81
0.12 0.28 0.40 0.51 0.59 0.66 0.71 0.74 0.73
ci.7 O-1 1 2 2-3 3-4 4-5 S-6 6-l
101.31 100.15 98.85 97.59 96.31 95.16 94.14
101.44 100.07 98.79 97.58 96.40 95.23 94.05
0.07 PO.08 PO.06 Po.o1 0.09 0.07 -0.09
103.46 102.08 100.80 99.57 98.36 97.15 95.92
2.09 1.93 1.95 1.98 2.05 1.99 1.78
cl a Calculations were performed for semirigid and nonrigid models C; = 113.0, V(4) = -122.8, V(5) = -8.7 and V(6) = 12.5 cm.-‘.
of calculations using semi- and nonrigid models for 1.3-butadiene are given in Fig. 2 and Table 1. They are based on ab initio quantum mechanical calculations at the MP2/6-3 lG* level (detailed results of these calculations including construction of the refined potential function of internal rotation will be published later). Although the degree of “nonrigidity” of the 1,3-butadiene molecule, i.e. the amplitude of changes in the geometrical parameters during internal rotation, is important, the semi- and nonrigid models give nearly identical results over a large portion of the curve (in the region from 0” up to 135”). This is connected with the phenomenon of mutual compensation of the effects caused by changing each of the gcomctrical parameters (see acryloyl fluoride). The major differences between the functions F(v) for the two models arise only in the region up = 150-180“. Hence it is expected that accounting for geometry relaxation will have an appreciable effect on the shape of the potential curve in the region near the gauche minimum and especially on the estimation
0.05 with the same potential
1.37 function
[41]: V, = 71.7, Vz = 1944.8,
of the barrier height separating the potential wells of the gauche conformers. The results of torsional frequency calculations using the same potential function optimized for a nonrigid model are compared with the experimental data in Table 1. The difference in the discrepancies is not as marked as in the case of acryloyl fluoride (see below). For the trans conformer the average magnitude of the errors does not change. However, the discrepancies between the calculated and experimental torsional frequencies for the gauche conformer are several times larger with the semirigid model. 4.4. Acryloyljhoride The geometrical parameters for both the semiand nonrigid models are taken from the results of ab initio quantum mechanical calculations at the MP2/6-31G* level [41]. The functions F(p) (Eq. 6) for internal rotation in acryloyl fluoride are shown in Fig. 1. The differences observed here are the most
A.V. Abramenkov
et al.iJournal
of MolecularStructure
marked ones in comparison with other molecules studied [38-431. It is of interest that the difference between the values F(p) calculated with the semiand nonrigid models is greatest at q = lSO”, whereas the dependence of the individual geometrical parameters on (o usually has the form of a curve with an extremum near 90” (see, for example, Fig. 2 in Ref. [41]). Probably, there is partial compensation of effects caused by changing the different geometrical parameters. Note also that this situation is probably a typical one, since the same pattern is observed to a greater or lesser extent for the majority of molecules studied by us. The influence of the choice of the molecular model on the calculated vibrational frequencies can also be seen in Table 2. Here the potential function was optimized for a nonrigid model so that the deviations of the calculated frequencies from the experimental ones are less than 0.1 cm-’ (see Table 3 in Ref. [41], columns denoted A). As is easily seen, transition to the semirigid model (provided that the shape of the potential function is preserved) results in a marked increase in frequency errors, especially on going from the trans to the cis conformer. This correlates with the character of discrepancies in the functions F(p). It is selfevident that after optimization of the potential function for the semirigid model it is possible to attain the same good reproduction of the experimental frequencies as in the nonrigid model, but the shape of the potential function will be changed in this case. (Table 4 in Ref. [41] gives an idea about the magnitude of possible variations of the potential function.)
5. Conclusions The above consideration of semi- and nonrigid models of internal rotation demonstrates the generality of the approach based on the latter model. Application of a nonrigid model means following the ideology that a known error is preferable to a chance possibility of obtaining a nearly correct answer. However, it is always necessary to keep in mind that the results obtained from the nonrigid model are also significantly dependent on the level of the quantum mechanical calculation
376 (1996)
183-194
providing the character relaxation.
of the molecular
193
geometry
Acknowledgements Yu.N.P. thanks the ULB for an international scientific collaboration grant and the Russian Foundation for Fundamental Investigations (Project 93-3-18386) for financial support.
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