8585
J. Phys. Chem. 1991,95, 8585-8597
Sensitivity of Molecular Structure to Intramolecular Potentials Roberta Susnow,+ Robert B. Nacbbar, Jr.,* Clarence
and Herschel Rabitz*yt
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, and Merck Sharp & Dohme Research Laboratories, Rahway, New Jersey 07065 (Received: November 9, 1990)
This paper applies sensitivity analysis methodology to molecular mechanics calculations. Both potential parameter sensitivity coefficients and Green's function sensitivity coefficients are developed. These sensitivities, respectively, show the relationship between the potential and molecular structure as well as reveal how the structure in one region of a molecule is related to that in another area of the molecule. Examples of acetaldehyde and propanal, where the sensitivites are depicted graphically, illustrate the sensitivity methodology. The sensitivities reflect the symmetry and complexity of the molecule being studied. Even in these simple illustrations nonbonded sensitivity behavior is evident.
I. Introduction Molecular mechanics is a computational procedure designed to give structures and energies for molecules. This procedure involves the geometry optimization of a molecular aggregate, whose atoms interact through a chosen force field to yield a stationary structure. In principle, this method allows for the study of a wide variety of molecular structures without the need for experimental data for the particular structures. Thus, the desire to quantitatively understand the structures and energies of various molecules coupled with the advent of high-speed digital computers over the past 20 years has resulted in molecular mechanics becoming one of the standard tools of theoretical structural organic and biophysical chemistry.' The key assumption in this technique is the acceptance of a static classical model of the molecule. Perhaps, more critical is the adoption of a chosen functional form of the potential energy, which is often obtained empirically,(thus, molecular mechanics is sometimes called the empirical force field method), although the approach can be taken with a force field from any source. The intramolecular forces in a molecule are modeled in terms of potential energy functions of its internal coordinates. These potential energy functions contain adjustable parameters, which for a given class of molecules, can be fit to a body of computational or experimental dataa2 The fitting of these parameters by least-squares attempts to give the best possible agreement between the model and the available data, while providing estimates of the accuracy of the energy functions with the optimized parameters.3-S A serious question concerns the quality of the resultant molecular mechanics determined structure in relation to the evident approximate nature of the potential. Even as potentials become more accurate, basic issues will still remain concerning how features in the potential act together to produce a particular molecular structure. The tools of sensitivity analysis will be introduced in this paper for these purposes. The goal of sensitivity analysis in the present context is to quantitatively assess the relationships between the collection of molecular structural observables and the parameters in the potential. The three types of sensitivities discussed here are (i) Cartesian position, (ii) internal coordinate, and (iii) Green's function sensitivites. Cartesian position and internal coordinate sensitivities pertain to understanding how the potential (Le., its parameters) controls the molecular geometry. The Green's function describes how the molecular structure or atomic positions throughout the molecule will respond to a particular force exerted on an atom or several atoms anywhere in the molecule, while restraining center of mass motion. The latter Green's function matrix may ultimately be useful when considering functional modification of a molecule for purposes of structural alteration. All of these sensitivity questions are addressed here by the computation of sensitivity coefficients as described later in this paper. The sensitivity analysis technique 'Princeton University. 'Merck Sharp & Dohme.
0022-3654/91/2095-8585$02.50/0
has been applied to chemical dynamics and kineticsb* and has recently been utilized in biostructural parameter fitting;9 here we extend it to other applications. As stated above, molecular mechanics applications have led to questions concerning the quality of the output molecular structure from the calculations, and the magnitudes of the sensitivity coefficients could suggest which characteristics of the potential are most important and, therefore, aid in pinpointing the areas in which improvements in the potential surface would be most beneficial. In addition, sensitivity analysis can uncover the role of subtle atomic interactions in a molecule. The role of particular forces or potential terms will often be obscured by their acting in concert to produce the net equilibrium structure; sensitivity analysis provides a quantitative means to assess these matters. In this present work, conformers of acetaldehyde and propanal, relatively simple molecules, are used to illustrate applications of the sensitivity methodology to molecular mechanics. In later work, the tools developed will be applied to more complicated molecules of greater intrinsic interest. Even beyond this realm, the same generic issues and questions arise when performing Monte Carlo or molecular dynamics calculations a t finite temperatures, and analogous sensitivity analysis tools should be applicable in these contexts. Furthermore, first-order sensitivity analysis is based on linear response theory and, thus, should have a close connection to similar concepts in molecular modeling. In this paper, section I1 will describe the molecular mechanics and sensitivity methodology, while section I11 will provide some illustrations of these techniques. Section IV will present a brief summary. 11. Methodology There are a large number of force fields and related computer programs available for structural analysis, and sensitivity analysis could, in principle, be incorporated into any of these programs. We have chosen to use BIGSTRN-3, which contains four force fields and was developed a t Princeton and Merck between 1980 and 1986.1° For the analysis, we focus on Allinger's MM2 force field because this field is rather robust with respect to the variety of functional groups it can model." However, we emphasize that ( 1 ) Burkert, U.;Allinger, N. L. Molecular Mechanics;American Chemical Society: Washington, DC, 1982. (2) Engler, E. M.; Andose, J. D.; Schleyer, P. v. R. J . Am. Chem. Soc. 1973, 95, 8005. Ermer, 0.; Lifson, s. J . Am. Chem. Soc. 1973, 95, 4121. Maple, J. R.; Dinur, U.;Hagler, A. T. Proc. Narl. Acad. Sci. U.S.A. 1988, 85, 5350. (3) Lifson, S.; Warshel, A. J . Chem. Phys. 1968, 49, 51 16. (4) Warshel, A.; Levitt, M.; Lifson, S.J . Mol. Spectrosc. 1970, 33, 84. ( 5 ) Hagler, A. T.; Lifson, S. Acta Crystallogr. 1974. 830, 1336. (6) Rabitz, H. Chem. Rev. 1987, 87, 101. Rabitz, H. Science 1989, 246, 221. (7) Smith, M. J.; Shi, S.; Rabitz, H. J . Chem. Phys. 1989, 91 (2), 1051. (8) Judson, R. S.; Shi: S.;Rabitz, H. J . Chem. Phys. 1989, 90 (4), 2263. (9) Thacher, T.; Rabitz, H. J . Am. Chem. Soc. 1991, 113, 2020. (10) Nachbar, R. 8.; Mislow, K. QCPE Bulletin 1986, 6, 96 (program 514).
0 1991 American Chemical Society
8586 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 the methodology is general and could be applied to other potential forms. BIGSTRN-3 was modified to compute the necessary derivatives for the sensitivity analysis and the derivatives were then utilized in a separate sensitivity analysis program.12 This section will first review the general form of the MM2 force field and then describe the details of the sensitivity analysis. Allinger's MM2 force field contains seven energy functions, which include various parameters. The seven functions are energies of stretching, bending, torsion, out-of-plane, stretch-bend, nonbonded, and dipole-dipole interactions. The sensitivities to the dipole-dipole parameters were not studied here because acetaldehyde and propanal do not contain any dipole-dipole interactions in Allinger's force field model. A. MM2 Energy Functions. The equations shown below are those used in the BIGSTRN-3 program. Although they appear to be different from those published by Allinger", they are equivalent and BIGSTRN-3 produces results identical, within machine precision, to those from MM2. 1. Stretching Energy.
E, = !hks(rij- r ~ +)f/zk,sc(rij ~ - r0)3
(1)
The parameters involved in the sensitivity analysis are s, = 2.0 The reference bond length is ro, and k, is the potential constant. The distance between the ith and jth atoms is rij. Each force field bond type has a characteristic ro and k,, while s, applies to all bonds. 2. Bending Energy.
A-', ro (A), and k, (kcal mol-' A-2).
+ I/zkbbc(6- 60)'
Eb = Yzkb(6 -
(2)
The parameters involved in the sensitivity analysis are b, = 0.75 rad4, Bo (rad), and kb (kcal mol-' rad-2). The reference angle is 60 and kb is the bending potential constant. The valence angle is 6, and each angle type has a characteristic Bo and kb. 3. Torsion Energy.
E, =
&(uO
+ UI
COS
(4)
+
02 COS
(24)
+ ~3 COS (34))
(3)
The coefficients in the Fourier expansion, uo, uI, u2, and u3 (kcal mol-'), are the parameters involved in the sensitivity analysis. The torsion angle 4 is formed from the sequentially bonded atoms I, J, K , and L. The torsion angle is between the plane defined by atoms I,J,K and the plane defined by atoms J,K,L. A positive torsion angle denotes a clockwise rotation of plane J,K,L with respect to the reference plane I,J,K when viewing the assembly from J to K." 4. Out-of-Plane Distortion Energy.
Eo = t/2ko(6)2 + f / 2 k o % ( V
(4)
The parameters involved in the sensitivity analysis are 0, (rad4) and k,, (kcal mol-' rad-2), the potential constant. The out-of-plane angle 6 applies to trigonal atoms only. Consider the group I,J,K,L, where L is the central atom and is bonded to atoms I, J, and K , which define the plane. When atom L distorts out of the plane three out-of-plane angles are then computed each involving the bond vectors I-L, J-L, and K-L, respectively, and the I,J,K plane. The angle is positive or negative depending on whether atom L is above or below from the I,J,K plane. 5. Stretch-Bend Energy.
Esb = kdrij - ro)(o - 60)
(5)
The arameter involved in the sensitivity analysis is ksb (kcal mol-' be-l rad-'), the stretching-bending potential constant. In the Allinger potential, the parameters ro and 60 are taken as the same as those arising in the corresponding stretching (eq 1) and bending (eq 2) potentials. In computing the sensitivities to ro and 60, we retain this constraint,
Susnow et al.
6. Nonbonded Interaction Energy. Evdw = AeBDiJ+ C / O $
The parameters involved in the sensitivity analysis are A (kcal mol-'), B (A-I), C (kcal mol-' A6), and F = 0.915, the hydrogen offset ratio. Dij is the nonbonded distance between the ith and jth atoms. A and B (B C 0) are parameters in the repulsive portion of the potential, and C (C C 0) is the parameter in the attractive portion of the potential. In general, A, B, and C for a given interatomic interaction are derived from the characteristic van der Waals radii (ri and rj) and "hardnessn (ti and t j ) of the involved atom types:
where r* = ri €*
#
A = c*(2.9 X IO5 kcal mol-I)
(6b)
B = -I2,5(r*)-' C = (-2.25 kcal mol-')t*(r*)'
(6c)
+ rj (A), e*
(64 For C and H interacting,
=
(titj)'/?
In the Allinger potential, where a hydrogen atom is bonded to a carbon atom, the offset ratio F displaces the hydrogen position used to compute Do closer to the carbon to account for hydrogeJ's anisotropic charge distribution. The new C-H bond vector Dkp is calculated as follows: Dkp = Fp - ik= F(Fi - ik) (7)
-
where Fp is the new hydrogen position vector, Fi is the original hydrogen position, and is, is the carbon position. 7. DipoleDipole Interaction Energy. kpijpk,(COS x - 3 COS CY COS j3) (8) Edd = rJc The parameters involved are the bond moments p,, and pkl (debye) and the dielectric constant t = 1SO. The conversion factor is k = 14.396 (kcal mol-l AS D-2) and the angle between the dipoles is x. The line between the midpoints of the bonds is rand the angles between dipole axes and line along which r is measured are a and 6. Each bond between dissimilar atom types has a characteristic bond moment p. Bond pairs sharing an atom do not contribute to this potential. 8. Total Potential Energy. The total potential energy is then + v = X E S C E b + X E , + C E O+ CEsb (9) B. M t i v i t y Analysis. The parametric sensitivites considered here are the Cartesian position sensitivities, S,,,;, = ari,,/ap, (i = a particular atom and a = x , y, or z Cartesian component), and internal-coordinate sensitivities, Ijn= acj/dpn (cj = bond length, valence angle, torsion angle, or out-of-plane angle), where p, is a particular parameter. In addition, the Green's function, Gi,,;is = 8ri,a/ajj8, will be calculated as the response of the ith atom in Cartesian direction a to an externally applied force, jj$, on the jth atom in Cartesian direction 8. is the Cartesian response of the ith atom and Iln is thejth internal coordinate response (e.g., bond length increase or decrease) resulting from a disturbance of the nth parameter value. The Green's function could also be converted to internal coordinates if desired. If p, occurs in several places in a molecule (e.g., r, for the C-H bonds of a methyl group), it is treated as a single parameter; this restriction could be relaxed for the probing of localized parameter contributions. In general these response relations are expressed as dri,a = CSi,a;ndpn
(10 4
dcj = XIj,, dp,
(lob)
dri,a = XGi+qj, djj,
(1h)
n
i.6
(1 1) Allinger, N. L. J . Am. Chem. SOC.1977, 99,8127. (12) The new version of BIGSTRN-3 will be submitted to QCPE. (13) For sign convention, see: Klyne, W.; F'relog, V . Experientiu 1960, 16. 521.
(6a)
The spatial component nature of eq 1Oc allows for directional input-output responses. In the context of the present paper we are not concerned with making particular coordinated parameter changes dp, or introducing special forces dfj,. Rather, the focus
Sensitivity of Molecular Structure to Potentials is on the fundamental sensitivites themselves. Molecular mechanics involves static equilibrium, whereby the goal of the structure optimization is to minimize the potential to yield a net force of zero on the molecule. Fi,, = -av/ar,,, = o (11) The Cartesian position sensitivity coefficient arra/ap,, is calculated by taking the derivative (Le., explicit and implicit) of eq 11 with respect to p,,. This yields the sensitivity equation
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 85%7
C. Mean-SquareSemitivities. Two types of root-mean-square averages are calculated for the Cartesian- and intemal-coordinate normalized sensitivities. These averages disregard the signs of the sensitivities, but allow for easier comparison among sensitivities. Type Ia averages the normalized Cartesian sensitivities over all the Np parameters of the potential and the 3 Cartesian components of each atom. This yields the average sensitivity, Sp, for each atom i in the molecule.
-4
SP =
(18)
Type Ib averages the normalized internal-coordinate sensitivities over the Np parameters, producing the average sensitivity, of each bond, valence angle, torsion angle, or out-of-plane angle.
4b.
or in matrix form JS+K=O (13) where J is the Jacobian (Hessian), S is the matrix of sensitivity coefficients, K is the inhomogeneity, and the desired matrix, S, may be determined from the Green’s function matrix G, which satisfies the equation JG = 1 (14) The right-hand side of eq 14 is actually only unity in a 3N - 6 dimensional subspace, corresponding to the removal of the rigid body center of mass rotational and translational motion. The calculation of the Green’s function matrix from the generalized inverse of the symmetric Jacobian matrix proceeds by eliminating the rigid body translation and rotation about the molecule’s center of mass.I4 Thus we may solve eq 13 as S = -J-’K = - C K (15) The Cartesian position sensitivities will be log normalized to allow for comparison among sensitivities with respect to different types and ranges of parameter values. Log normalization of Si,, for example, yields the atomic position change per unit fractional change in the parameter value
From this point on, unless otherwise stated, references to the Cartesian sensitivities imply log-normalized sensitivities. From the position sensitivities, Si,,;, = ari,,/ap,,, the internalcoordinate sensitivities are calculated by using the chain rule and the Wilson S vectors (ar/dri,,, ae/ari,,, ai+,,, dC$/ariJl5 These sensitivities include bond length, dr/ap,,, valence angle, af?/ap,, out-of-plane angle, aS/ap,, and torsion angle, aC$/ap,,,sensitivities and will be log normalized in a manner analogous to that of the Cartesian sensitivities. The Green’s function elements, G,,,., = ar,,,/af,,,e, give direct information about structural relationshlps within the molecule. The elements illustrate how an atom’s position will shift if a force is exerted in a particular direction on that atom or on other atoms in the molecule, as depicted by eq 1Oc. The Green’s function matrix is square and of dimension 3N, where N is the number of atoms in the molecule. A root-mean-square reduced Green’s function matrix, Gf,may be calculated from the Green’s function matrix to produce a less detailed but nevertheless valuable measure of the significance of the Green’s function sensitivities. The root-mean-square reduced Green’s function matrix has elements that are an average over the sum of the squares of each 3 X 3 Cartesian block consisting of the Green’s function elements for a pair of atoms ( i j ) . This averaging is represented by a summation over Cartesian components (Y and 8.
The resulting matrix of dimension N is square. (14) Strang, G. Linear Algebra and its Applications; Academic Press: New York, 1976; pp 130-138. (15) Wilson, E. B., Jr.; Decius. J. C.; Cross,P. C. Molecular Vlbrorions; Dover Publications: New York, 1955; pp 54-61.
Gb= 4i-
(19)
Type IIa averages the normalized Cartesian sensitivities over all the 3N atomic coordinates, yielding the average molecular sensitivity, $,’”, to a particular parameter, p,,.
Type IIb averages the internal coordinate sensitivites over the N , number of bonds, valence angles, torsion angles, or out-of-plane angles, producing the average sensitivities of these coordinates to the particular parameter.
-4
=
(21)
Each of the reduced or mean-square sensitivities in eqs 18-21 gives a limited but valuable overall measure of molecular structure parameter response. The next section will make use of these measures. 111. Illustrations Two molecules were studied by using the methodology described above. Figure la-c shows the lower (a minimum) and higher (a transition state) energy conformers of acetaldehyde and one conformer of propanal, with the atoms numbered for easy reference (Le., C(2), C(3)). The acetaldehyde conformers have C, symmetry, with an x-y symmetry plane chosen for convenience, while the propanal conformer has CIsymmetry. All three structures were rigorously optimized to an rms gradient of less than lod kcal mol-’ ,&-I. The higher energy acetaldehyde conformation possessed a single imaginary frequency. The various bond lengths, valence angles, torsion angles, and out-of-plane angles are listed in Tables 1-111, where I,J,K,L index the internal coordinates. A. Cartesian Position Sensitivities. The MM2 energy expressions and their parameters were described in section 11. The meaning of the Cartesian sensitivity coefficients, ar,,/a In p,,, is evident from an examination of eq loa. A positive (negative) coefficient indicates that a fractional increase in a particular parameter, p,,, results in an increase (decrease) in ri,,. Although here we are not concerned with actually altering the potential parameter, this interpretation is nonetheless useful and will be helpful in the following discussions. The structural effects to which a parameter contributes can be classified as local or distributed. Local effects are defined as being associated with an internal coordinate whose directly involved parameters are being studied. Less local or distributed effects are sensitivities associated with other internal coordinates that are indirectly influenced by parameters operating elsewhere in the molecule. The internal coordinates involved are the bond lengths, valence angles, torsion angles, and out-of-plane angles. Because nonbonded potential terms do not refer to any particular internal coordinate, consequently, their geometry changes are described exclusively in terms of distributed effects. The stretch-bend potential involves predominantly local effects. The dipole-dipole potential is not relevant to the cases of acetaldehyde and propanal, but would be a distributed interaction in other relevant molecules (e.g., 1,2dichloroethane). The distributed effects are best discussed by referring to pictures of the geometry shifts. In the sensitivity pictures to follow, the
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
8588
Susnow et al.
TABLE I: Internal Coordinates for the Lower Energy Acetaldehyde ConformeP
I 1 2 2 3 3 3 1 1 3 2 2 2 5 5 6 1 1 1 4 4 4 1 3 4
index J K L 2 3 4 5 6 7
bond length, A 1.209 1.514 1.1 I4 1.1 14 1.1 14 1.114
valence angle, rad
torsion angle, rad
out-of-plane angle, rad
2.157 2.084 2.04 1 1.937 1.920 1.920 1.891 1.891 1.902
2 2 2 3 3 3 3 3 3
3 4 4 5 6 7 6 7 7
2 2 2 2 2 2
3 3 3 3 3 3
5 6 7 5 6 7
3 4 1
4 1 3
2 2 2
0 2.093 -2.093 3.141 -1.048 1.048
0 0 0
"The indices I,J,K,L refer to atom numbers in Figure la. TABLE 11: Internal Coordinates for the Higher Energy Acetaldehyde ConformerO
I 1 2 2 3 3 3
index J K 2 3 4 5 6 7
1 1 3 2 2 2 5 5 6 1 1 1 4 4 4
2 2 2 3 3 3 3 3 3 2 2 2 2 2 2
L
3 4 4 5 6 7 6 7 7 3 5 3 6 3 7 3 5 3 6 3 7
bond length, 1.209 1.514 1.1 I4 1.1 14 1.1 14 1.1 14
valence angle, rad
torsion angle, rad
out-of-plane angle, rad
2.148 2.083 2.050 1.937 1.920 1.920 1.894 1.894 1.897
(C)
Figure 1. Lower (a) and higher (b) energy acetaldehyde structures. These structures have C, symmetry with an x-y symmetry plane. Atoms 1-5 lie in the x-y plane. Part (c) shows the Propanal structure. This structure has C,symmetry. -3.140 -1.045 1.045 0 2.096 -2.096
1 3 4 2 3 4 1 2 4 1 3 2
0 0 0
"The indices I,J,K,L refer to atom numbers in Figure Ib.
darker structure shows the original geometry, while the lighter structure, which is drawn from transformed coordinates, shows how the geometry would shift if the particular parameter were increased. The pictures are meant to give a qualitative understanding of the role of the parameters expressed in terms of geometric shifts that would occur upon reoptimization with a new parameter value. Thus, as suggested by eq 10, the following transformation is made
for a plot with a particular parameter pn. The magnification scale factor y > 0 is arbitrarily chosen to make the geometrical response pictorially evident. In cases where the shifts are difficult to discern, the geometric shifts are deliberately magnified by appropriately increasing y. Therefore, the figures should not be quantitatively interpreted. 1. Acetaldehyde. The geometry of acetaldehyde is controlled by the stretching, bending, torsion, out-of-plane, nonbonded, and stretching-bending potentials. There are no out-of-plane distortions, owing to symmetry, and there are no sensitivities that would cause a geometrical response orthogonal to the x-y plane if the atom being distorted is in the x-y plane. The sensitivities will be studied for each of the terms of the respective six contributions to the energy potential. The parameters and the interactions they describe in each of the potential functions are listed in Table IV. The subscript in parentheses indicates the internal coordinate or interaction to which the parameter refers. The figure captions for each case also explain the role of the probed parameters. a. Stretching Effects: t3Q,a/aIn ro, ari,a/aIn k,,ar,,/a In s,. The first step in studying the sensitivities to the stretching parameters is to evaluate the deviations of the bond lengths from
The Journal of Physical Chemistry, Vol. 95. No. 22, 1991 8589
Sensitivity of Molecular Structure to Potentials
TABLE 111: Internal Coordinates for Propanal" index I J K L 1 2 2 3 2 4 3 5 3 6 3 7 7 8 7 9 7 10 1 2 3 1 2 4 3 2 4 2 3 5 2 3 6 2 3 7 5 3 6 5 3 7 6 3 7 3 7 8 3 7 9 3 7 10 8 7 9 8 7 10 9 7 IO 1 1 1 4 4 4 2 2 2 5 5 5 6 6 6
2 2 2 2 2 2 3 3 3 3 3 3 3 3 3
3 5 3 6 3 7 3 5 3 6 3 7 7 8 7 9 7 1 0 7 8 7 9 7 1 0 7 8 7 9 7 1 0
1 3 4
3 4 1
4 1 3
2 2 2
bond valence torsion out-of-plane length, A angle, rad angle, rad angle, rad 1.209 1.517 1.1 14 1.1 16 1.1 16 1.533 1.115 1.115 1.1 15 2.1 58 2.082 2.042 1.908 1.884 1.954 1.877 1.918 1.919 1.919 1.943 1.936 1.878 1.880 1.885 -0.142 1.896 -2.27 1 3.024 -1.220 0.894 3.133 -1.057 1.042 1.009 3.101 -1.08 1 -1.054 1.037 3.137
(b)
Figure 2. (a) Sensitivity of lower energy acetaldehyde conformer to roc) = 1 SO9 A, the C(2)-C(3) stretching parameter. No magnification. (b) Sensitivity of higher energy acetaldehyde conformer to ro13)= 1.1 13 A, the aldehyde C-H stretching parameter. No magnification.
-0.006 -0.008 -0.007
"The indices I,J,K,L refer to atom numbers in Figure IC.
their reference (ro)values. In both optimized conformations of acetaldehyde all the bond lengths have positive deviations from their ro values. The C(3)-C(2) bond has the greatest deviation, probably due to repulsion between the methyl and the HC=O groups. These deviations will be referred to in discussing the local and distributed effects. The sensitivities to the ro parameter are discussed in reference to the stretching potential, although there is some contribution to the sensitivities from the stretching-bending potential. In general, this contribution is in accordance with the local and distributed effects outlined below. As might be expected, sensitivity analysis reveals that some local effects relate to increases or decreases in bond length. In the present case, we find that ari,,/a In ro > 0 and ari,,/a In k, < 0 have a simple intuitive interpretation. As ro is increased, the bond length will increase, and as k, is increased, the bond length tends toward ro. The bonds react in this way so as to reduce increases in stretching energy that would result from increases in ro or k,. The bonds react in a similar way, ari,,/a In s, < 0, to increases in s, (i.e., s, is made less negative): all the bond lengths decrease, and C(3)-C(2) has the greatest decrease because it has the largest deviation from its ro value. The methyl group H-C-H angles and the HC=O angle compress for an increase in s,, apparently as a result of all the bond lengths decreasing. In general, for the stretching parameters, the sensitivities of both conformers of acetaldehyde are very similar. Table V shows the range of sensitivities, local and distributed for each of the parameters. The ordering of the stretching parameters is as
expected from the form of the potential. Figure 2a depicts the sensitivity to the ro(,)parameter (C-C), for the lower energy conformer. Locally, as expected, increases in this parameter result in increases in the C(2)-C(3) bond for both conformers. The interesting observation concerns the distributed effects especially evident for the methyl group: all the other bond lengths decrease and the methyl H-C-H angles and H C 4 angle expand. These effects can be understood in terms of decreased steric repulsion between the methyl and the carboxaldehyde groups, thus allowing for their local geometries to relax toward their reference configurations. The magnitude of the distributed effect would, in general, be difficult to assess without sensitivity analysis. Clearly, local effects will be more predictable than distributed ones. Figure 2b illustrates the sensitivity to increases in the ro(3parameter (aldehyde C-H) for the higher energy conformer. in both conformers, locally, the C-H bond increases. Distributively, in the case of the higher energy conformer, all the other bonds decrease, while the methyl H-C-H angles and the H C = O angle expand. This expansion in turn causes the methyl group and the C=O bond to tilt away from each other. b. Bending Effects: ari,,/t3 In e,, ari,,/a In kb ari,,/a In b,. Analogous to stretching above, inspection of valence angle deviations from their do values is the first step in understanding the sensitivities to the bending parameters. The methyl HCH angles and the HC=O angle have negative deviations from eo, likely resulting from repulsion between the methyl and H C 4 groups. The remaining five angles have positive deviations from eo. As with the stretching potential, the sensitivities to the eo bending parameter are discussed in relation to the bending potential, although there is some contribution to the bending sensitivities from the stretching-bending potential. Again, this contribution is in agreement with the local and distributed effects outlined below for the bending potential. The sensitivities corresponding to local expansions and compressions of bond angles are somewhat more involved than local
8590 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 TABLE I V MM2 Parameters for Acetaldehyde function param stretching
bending
torsion
stretching-bending
value -2.000 1SO9 1.113 1.113 1.208 633.42 662.22 662.22 1554.77 0.7544 1.902 1.91I 2.032 2.138 2.100 46.07 53.27 53.27 66.22 53.27 0.455 0.180 0.000 0.275 -0.267 -0.167 0.000 -0.100 12.956 17.275 0.915 13630.0 16152.0 -4.16670 -3.85800 -77.0920 -144.970
nonbonded
TABLE V Rome of Sensitivities for Acetddehvde range intnl function param (log-norm) units coord stretching r, to 1 A bond length ks -IO-'to IF5 sc -10-5 to -IV bending e, -10-1 to I rad valence angle kb -Io-' to lo-3 b, -IO4 to IO4 torsion 00 0 rad torsion angle uI to 02 u3
stretching-bending nonbonded
F
-10-5 to 10-5 -IO-' to IO-' -IO-' to IO" to-10"
B
IO-'
A
10-5 to 10-3
ksb
c
A rad
A
bond length valence angle bond length
to
1 0 - to ~
stretches. Increases in 6, lead to angle expansion (Le., aei/a In 0, > 0), while increases in kb lead to expansion or compression, aOi/a In kb > or < 0, depending upon the initial deviation from 0,. A negative deviation favors expansion and a positive one favors compression, that is, the angles tend toward their reference values. This can also be rationalized by an examination of the role of k b under these circumstances. The H C 4 angle in both conformers expands with an increase in 0, or kb However, the response of the methyl HCH angles differs between the two conformers, indicated by expansion of all three HCH angles in the lower energy conformer, but the expansion of only two angles and the compression of the third in the higher energy conformer. The range of sensitivities for the bending parameters is shown in Table V.
Susnow et al.
units
A-' A
kcal mol-'
A-2
rad-' rad
descrpn coefficient cw--c(3) methyl CH aldehyde CH C 4
coefficient methyl HCH C( 2)-C(3)-H(methyl) C( 3)-C( 2)-H(ald) C( 3)-C(2)=O HC=O
kcal mol-' rad-2
kcal mol-!
H - C ( 3)-C(2)-H
H(methy1)-C( 3)-C(2)=O
H C 4 H-C( 2)
