J. Phys. Chem. 1993,97, 6167-6174
6167
Application of Structure-Structure Sensitivity Analysis to Small Molecules Paul C. Jasien' Department of Chemistry, California State University San Marcos, San Marcos, California 92096
Thomas S. Thacher Biosym Technologies, 9685 Scranton Road, San Diego, California 92121 -2777 Received: February 5, 1993
Sensitivity theory is applied to determine trends in the structurestructure sensitivities in a number of small molecular systems. The analysis is based upon ab initio second-derivative matrices calculated at the HartreeFock self-consistent-field level. The effect of basis set on the calculated sensitivities was checked through a comparison of 3-21Gand 6-31G* calculations. It was found that although the absolutevaluesfor the sensitivities differed, the relative trends were unaffected by the difference in the basis sets. The use of this methodology provides a straightforward way of interpreting the structural couplings that arise in molecular systems without assuming specific forms and parametrizations of molecular potentials. The complementary use of constrained and unconstrained sensitivities is shown to be helpful in elucidating complex indirect couplings of coordinates in molecules.
Introductioo Chemists always seek to gain an understanding of molecular structure and reactivity in terms of various structural features. These relationshipscan be as simple as determining how sensitive a particular bond length is to deformations in an adjacant bond angle,or they may be more complex, such as trying to understand how bond lengths affect the energy of a transition state in a chemical reaction. As molecular systems of interest to chemists get larger and larger, the number of possible relationshipsof this type grows immensely. It is therefore advantageous to develop a systematic procedure for examining structural relationshipsin molecules. One such methodology is sensitivity analysis.192 Although sensitivity analysis has been used in the engineering community for many years,lJit has only recentlymade an impact in analyzing chemical system^.^.^ A number of papers on the application of sensitivity analysis to chemical kineticsM and molecular dynamics4+sv7 have appeared. More recently, there have been a number of publications which have dealt with the application of this technique to molecular structure and energetics. Thacher et a1.8 used sensitivity analysis to investigate the relationship between the parameters found in molecular mechanical force fields and the forces which determine crystal structure. Susnow et al.9 have used sensitivity analysis to investigatethe internal coordinate and barrier height parametric sensitivities of a number of amides. Lastly, Wong and Rabitz have applied sensitivity analysis to investigatethe uncertainties associated with free energy calculations.1° In this work, we apply sensitivity analysis to determine the structurestructure sensitivities in small molecules, as a prelude to its application to larger systems. The previously cited work on the application of sensitivity analysis to molecular mechanics and dynamics focused on elucidating the relationship between the potential energy expression and calculated observables. Structure-structure sensitivities have also been calculated, but again in the context of molecular mechanics where the structural dependencies are mediated by the potential energy expres~ion.~ The sensitivities calculated in this work use ab initio second derivatives. Because of this, the dependencies that are observed are not influenced by the choice of potential energy expression.
* To whom correspondenceshould be addressed.
It is hoped that, by first undertaking a rigorous examination of small systems, we will lay the groundwork for understanding larger systems which are of more immediate interest to chemists. In this way, we can provide a basis for elucidating structure structure relationships that are of interest to chemists in their quest to understand chemical bonding and structure-function relationships. The specific systems chosen for study in this work are water (HzO) hydrogen sulfide (H2S), hypochlorous acid (HOCl), hydrogen peroxide (H202), formaldehyde (HzCO), formoyl chloride (HCICO), glyoxylic acid ( C ~ H Z O ~and ) , three Lewis acid-base complexes of AlCl3. In addition to the basic structurestructure analysis, a few of these systems were investigatedmore fully to probe the effectsof basis set and molecular conformation on the calculated sensitivities, as well as the effect of the independent variable set on the sensitivity (vide infra).
Methodology The basis of sensitivity analysis as applied in chemistry is covered in detail in a number of other reports,ss here we will only review the principles important for the determinationof structurestructure sensitivities applicable to this work. In the following discussion, Vwill represent the potential energy function for the molecule in terms of the 3N- 6 internal coordinates (r,). F,will represent the force on internal coordinate i of a molecule, aV/ar,, and Hi,will be used to represent the second derivativeof Vwith respect to the internal coordinates i and j , @V/(ar, This quantity will also be referred to as the i j element of the Hessian. Using basic calculus,the variationsin the forces can be related to infinitesimalvariations in the internal coordinates as follows:
The first quantity in the summation is simply the i j element of the Hessian matrix (Hi,).Using the usual matrix notation to write this equation for all 3 N - 6 elements i, we get d P = H-di (2) Therefore, the Hessian, H,is, simply a transformation matrix which connects the vectors dF and di. In order to obtain the
0022-3654/93/2097-6167$04.00/0 0 1993 American Chemical Society
6168 The Journal of Physical Chemistry, Vol. 97, No. 23, 1993
structure-structure sensitivitiesof interesthere, we need to obtain the inverse quantity, (art/8F')Ft# . This differs from the Hessian by an interchange of the dependent and independent variables. Thesenew elementsaresimplythoseof theinverseHessian matrix. The inverse Hessian matrix is often referred to as the Green's function matrix.3 d i = H-'& (3) At this point a number of different structure-structure sensitivities (drt/arj)xkd, can be obtained, depending upon the set of independent variables (xk) held constant. In this work we will concentrate on only two types of structure-structure sensitivities: (44
Since all evaluations will be done at stationary points on the multidimensional potential energy surface (PES),all Fk = 0. Under these conditions, the first sensitivity (4a) corresponds to the variation of rt with respect to rj, allowing all other internal coordinatesto relax to their minimum-energy value. The second sensitivity (4b) corresponds to the variation of ri with respect to ri holding all f k constant and also holding Ft = 0. (Here, only FkZO). Comparison of the values obtained via sensitivities (4a) and (4b) yields information on the influence of indirect effects for a particular sensitivity. By holding the r k constant (expression4b), changes in rt not due directly to rj can be probed. In other words, changes in rj that affect rk which subsequently affects rt can be removed. These sensitivities will often be referred to as constrained, sinceallinternalcoordinatesnot involvedin thederivative are held constant in value. Analogously, the sensitivities from eq 4a will often be referred to as unconstrained, since the internal coordinates not involved in the derivatives are not held constant. The sensitivity coefficients from eq 4a can be obtained by performing the appropriate interchange of independent and dependent variablesin eq 3 (see ref 4, eq 13-16). Interchanging the first dependent and independent variables leads to
... ...
... ... It should be noted that the i,l elements, i=2,3N- 6, of the matrix are the derived structure-structure sensitivities with respect to the first internal coordinate. The sensitivities for the other coordinates may be obtained in a similar manner, until all (3N - 6)(3N - 7) sensitivities have been obtained. An analogous procedure to that outlined above may be used to obtain the sensitivities from eq 4b and will not be specifically discussed here. Since the units associated with the various derivatives above will differ depending on whether bonds or angles are involved, it is advantageousto normalize all the derivatives. This is usually done by evaluating the log-log normalized sensitivities (eq 6), where log is the natural logarithm. This results in dimensionless
Jasien and Thacher derivatives. Performing this normalization provides
a good starting point so that all of the calculated sensitivities can be directly compared to one another. It is these quantities which will be explored in this work. The use of log-log normalization to compare sensitivities pertaining to different classes of internal coordinates (e.g., bonds vs angles) is an arbitrary choice. The log-log derivatives indicate the fractionalchange in a given internal as a result of the fractional change in another internal. However, the structural effect normally associated with the fractional change of an internal is dependent on the type of internal. For example, the change of a bond length by 1 bohr is quite different from the change in a bond angle by 1 rad. One way to address this problem is to define a set of physically reasonable normalizationconstants. However, we feel that such a change would not affect the interpretation of the current results since it is qualitative in nature and is focused on relative trends rather than on absolute values. In order to simplify the discussion and notation in this paper, an abbreviation will be used to represent the derivatives for a particular sensitivity. The sensitivity (a log(rt)/a log(rj)),J as given in eq 6 will be written as r,/rk Since all sensitivities in this paper will be normalized as discussed above,there is no ambiguity as to which derivativeis being discussed, oncethe label constrained or unconstrained is used. At this point it should be noted that the sensitivities rl/rl and rj/rtare not merely inverses of each other. Although qualitatively related, these sensitivities are unique since they differ in their sets of independent variables. Therefore in reporting the results in this work, both sensitivities will be reported, but in general only one member of the pair will be discussed in detail.
Calculational Details In order to perform the sensitivity analysis as outlined in the previous section,it is necessary to start with the second derivative or Hessian matrix for the system of interest. In contrast to other works of this type where the Hessian matrix was obtained from parametrized model potentials,8s9the Hessians here were obtained in a purely ab initio manner. Specifically,these Hessians were obtained from ab initio Hartree-Fcck self-consistent-field(SCF) calculations. In some ways this is an improvement over previous analyses of this type, since no particular form of an empirical potential is assumed. However, it must be noted that for very large systems this method is presently intractable. All Hessians werecalculated at theoptimized minimum-energy structures of the molecule at the same theoretical level as was used for the Hessian evaluation. Unless otherwise noted, all reported data utilized the 6-3 1G* basis set as implemented in the Gaussian 90 electronicstructure program.lI Some studiesof the sensitivity coefficient dependence on the one-electron basis set were also done. The comparative basis sets used in these cases reported here are the 3-21G and 6-31G* bash sets.11 In the case of the H202 molecule, three structures were investigated: a C2 symmetry minimum-energy structurg, as well as the cis (Ca) and the trans (Cu) conformations. Since the cis and trans conformations are saddle points on the PES, Ft = 0 for all internal coordinates at these points as well. Comparison of the derived sensitivities for these three structures provides information on any changes occurring as a function of torsion angle. Results and Discussion H20, HpS, and HOCI. Given in Figure 1 are bar graphs of the normalized unconstrained structure-structure sensitivities for HzO, H2S, and HOCI. In looking at these data, the most striking
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6169
Structure Sensitivity Analysis of Small Molecules
".l
,
I
0.00 0.0
-0.10
-9 .g 'f rn
-0.1 -0.20
.e
.g
HSH H HOH H HOCl
3
-0.30
-0.2
-0.3
-0.40 I
1
2
3
4
5
.
-0.4!
6
1
. 2
. 3
Coordinate Set 1. XHlXY 4. LHXY/XY
2. XYlXH 5. XHILHXY
.
. 4
5
. 6
-
.
. 7
9
8
I
10
Coordinate Set 3. LHXYlXH 6. XYILHXY
Figure 1. Unconstrainedstructure-structuresensitivitiesfor HXY systems (X = 0, S and Y = H, Cl).
1. 4. 7. 10.
CHlCO CHz/CHl COlLHCO LH,CO I L H ~ C O
2. LHCOlCO 5. LHiCOICHi 8. CHl/LHICO
3. C O l C H 6. LHZCOlCHi 9. CHZlLHiCO
Figure 2. Unconstrained structure-structure sensitivities for H 2 C O .
TABLE I: Constrained and Unconstrained Sensitivitiesafor H2X
n.4. " . l
0.2
XH/XH LHXH/XH XH/LHXH
0.018 -0.147 -0.048
0.002 -0.061 -0.018
0.011 -0.144 -0.048
* Mk = 0 indicates unconstrained, b
k
0.001 -0.06 1 -0.018
= 0 indicates constrained.
feature is the large value of the LHOCl/OCl sensitivity in HOCl. This quantity is a factor of 3 larger than the analogous value in H20. The negative sign associated with this sensitivity indicates that, as the OC1 bond is stretched, the HOCl angle will decrease. The comparable sensitivities for H2O and H2S are smaller in magnitude but have the same sign. The sign of this derivative is consistent with that expected if the 1-3 interaction of the terminal atoms is important (i.e., as one bond length increases, the repulsion between terminal atoms decreases and the angle can then decrease). If one examines the XH/XY bond sensitivities, however, it is found that the sign of these sensitivities is positive. This indicates that a lengthening of one of the bonds will lead to a lengthening of the other bond. This is the opposite of what one would expect if the 1-3 interaction was the determining factor. It must be mentioned, however, that the absolute magnitude of the bond/ bond sensitivity is many times smaller than that for the angle/ bond. Comparingonly the sensitiviesfor H2O and H2S, it can be seen that the values for H2S are consistently smaller than for H2O. This difference in magnitude is expected since it is indicativeof the softer vibrational modes and weaker XH bond strength in H2S (DO298 = 90 kcal/mol) as compqred to H2O (DO298 = 119 kcal/mol).12 One interesting effect of C1 substitution on the sensitivities is the largedifference in the angle/bond sensitivities. As mentioned before, the LHOCl/OCl sensitivity is quite large, about 3 times larger than the corresponding LHOH/OH value in H20. However, the LHOCl/OH sensitivity is about 15 times smaller than the LHOH/OH value in H20. Such a large change in the sensitivitiescan be indicative of a decreasein the OH bond strength in HOCl as compared to H20. To further probe the sensitivities in this set of triatomics, the constrainedstructurestructure sensitivities (expression 4b) were calculated. Table I comparesthe constrained and unconstrained values for H2O and H2S. It can be seen that the two sets of sensitivitiesare really quite close in value, indicatingthat indirect coupling effects are not important for these systems. Thus, the large angle/bond sensitivity is not influenced significantly by the
*
I
-
0.0-
.?;
j
-0.2-
-0.4
-
-0.6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Coordinate Set 1. C H l C O LClCO/CO 7. LHCOKH 10. CHlCCl 13. COlLHCO 16. LClCO I LHCO 19. CCllLClCO
e
2. 5. 8. 11. 14. 17. 20.
CCllCO COlCH LClCOlCH LHCOlCCl CCllLHCO CO I LClCO LHCO I LClCO
3. 6. 9. 12. 15. 18.
LHCOlCO CCllCH COlCCl LclCO/Ccl CCllLHCO CHlLClCO
Figure 3. Unconstrained structure-structure sensitivities for HClCO.
bond-bond interaction. It should be kept in mind however that, due to the small number of total internal coordinates in these systems, the number of possible indirect couplingsis quite small. (More will be said about similar couplings later for the case of H202.1 HtCO and HCICO. Given in Figures 2 and 3 are bar graphs of the unconstrained structure-structure sensitivities for H2CO and HClCO. There is a great deal of information represented in these graphs, but we will concentrate only on a few of the larger magnitude sensitivities. Starting with H2C0, a cursory analysis shows that the most important coupling in H2CO is between the two HCO angles (LH,CO/LH2CO, no. 10) while the next largest are LHCO/CO (no. 2) and CH/CO (no. 1). The negative value for LHICO/ L H ~ C Oindicates that as one of the angles increases, the other decreases. Such a change is consistent with arguments based on 1-3 interactions as introduced before. The negative values for CH/CO and LHCO/CO are similarly consistent with 1-3 arguments. In addition, these are also consistent with the following argument based upon hybridization around the C atom. As the CO bond lengthens, the amount of p character in the bond increases. As more of the C p orbital is used for bonding to the 0, less p character (Le., more s character) is used in the CH bonds. Such an increase in the contribution of s character into these bonds would lead to a shorter CH bond as well as a smaller HCO angle (Le., LHCH becomes larger).
Jasien and Thacher
6170 The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 0.000
0.00
-0.10
." ~
-0.100
.g
H H2CO H HClCO
.g VJ
mi? 2
'8 rn
-0.20
cis
&a min
eS trans -0.30
-0.200 -0.40
-0.300
I
! 1
2
3
4
5
.
- 0 . 5 o - J . .
6
1
2
3
Coordinate Set 1. CHlCO
2. LHCOlCO
4. LHCOlCH
5. COlLHCO
. 5
. 6
' 7
-
. 8
9
I 10
CoordinateSet 3. COlCH 6. C H ~ L H C O
Figure 4. Comparison of common unconstrained structure-structure sensitivities for H 2 C O and HClCO.
Similar couplingsto those for H2CO are seen in HClCO (Figure 3). The angle/angle couplings (nos. 16 and 20) once again have the largestmagnitude. In addition,thevalues for CCl/CO, CH/ CO, and LHCO/CO (nos. 2, 1, and 3) are all reasonably large and negative as in H2CO. However, in assessing the degree of coupling, the CH/CO sensitivity in HClCO has been diminished somewhat relative to that in H2CO (vide infra). On the other hand, the angle/bond sensitivity,LHCO/CCl (no. 1l), is enhanced compared to the corresponding LH2COICHl value (Figure 2, no. 6) in H2CO. These observations are analogous to those for the HOX systems discussed previously. An interesting analysis of the two systems can be made by comparing the sensitivities which are common to both systems. In this way, we obtain a measure of how the electron-withdrawing C1 atom influences the structure-structure sensitivities. Figure 4 shows a direct comparison of these quantities. C1 substitutionleads to a decrease in CO/CH so that the value in HClCO is about one-half that in H2CO. In both cases, the quantity is negative, indicating that a lengthening of the CO bond will lead to a decrease in the CH bond. Chemically, an elongation and hence weakening of the CO bond should cause a shifting of electron density away from the 0 toward the rest of the molecule. If this density is shifted to the CH and CCl bonds, these bonds should be strengthened and hence shortened. This is illustrated by the negative value of CH/CO in both H2CO and HClCO. However, in the HClCO case, the electronegative C1 receives more of this shifted electron density and less goes to the CH bond. This is shown by the lower CH/CO coupling in HClCO relative to H2C0, but also by the relatively large CCl/CO, which is -3 times larger than the corresponding CH/CO coupling in H2CO and 6 times larger than the CH/CO coupling in HClCO. Referring next to the HCO angle sensitivity with respect to the CO bond, we find a negative coefficient which is larger in the chlorinated species. The fact that LHCO/CO is about 2 times more sensitive in HClCO is consistent with the argument above about the relative bond strengtheningof the CH bond in the two species (Le., a stronger CH bond would lead to a stiffer HCO angle). It is also possible that there is a large indirect effect between the CO and CCl bonds. Another quantity which shows interesting behavior is the coupling of the HCO angle and the included CH bond (no. 4). On going from H2CO to HClCO, the sensitivity changes sign. In H2C0, an increase in the CH bond will lead to a small increase in the HCO angle, while in HClCO an increase in the CH bond will decrease the HCO angle. It should be noted, however, that the positive value for H2CO is quite small in magnitude and may mask a more general trend of angle/bond couplings. When one compares the angle/bond sensitivities for included bonds like those above, the general trend is that an increase in
-
. 4
1. OHlOO 4. OHZIOHi 7. O O ~ L O O H 10. LOOH, l LOOH,
2 LOOH~OO 5. LOOH1lOHi 8.
OH1ILOOH1
3. OOlOH 6. LOOH,/OH, 9. OH,ILOOH~
Figure 5. Selected unconstrained structure+structure sensitivities for conformations of H 2 0 2 .
bond length leads to a decrease in bond angle. This can be seen for the HOX systems (Figure 1, nos. 3 and 4), H2CO (Figure 2, no. 2), and HClCO (Figure 3, nos. 3, 4, 7, and 12). The only exception seems to be for LHCO/CH in formaldehyde as mentioned above. Once again, such changes are consistent with 1-3 interaction arguments. A second general trend in bond/angle sensitivities is seen for bonds which are external to the angle. In general, these sensitivities seem to have positive signs, indicating that as the bond lengthens, the angle tends to increase. This can be seen for H2CO (Figure 2, no. 6) and HClCO (Figure 3, nos. 8 and 11). Similar trends have been noted in systems such as AlC13 and glyoxylic acid (vide infra). A great deal of interpretation can be made from the data in Figures 2-4. However, the points made above suffice to demonstrate the utility of the sensitivity analysis methodology to structurestructure couplings and how they are influenced by atom substitutions. Care must be taken, however, in interpreting these coefficients, since some may be quite small and it is not clear when the differences become significant. Keeping this in mind, the remainder of the paper will concentrate on those coefficients which are relatively large in magnitude. H202. Another fruitful area of application is monitoring structurestructure sensitivities during conformational changes in molecules. A similar study was performed in the work of Susnow et al.9 for a series of amide molecules. In this study, we chose the smaller H202 system. Given in the bar graph in Figure 5 are the unconstrained structure-structure sensitivities for three conformations of H202. The three types of bars for each coordinate set represent the results for the cis, C2 symmetry minimum and trans structures. The cis and trans conformations are first-order saddle points on the PES, while the C 2 structure lies at the minimum with a torsional angle of 116O (6-31G* SCF result). The sensitivities presented in Figure 5 are only those involving the OH and 00 bonds and the OOH angles, since the HOOH torsional angle is decoupled from these coordinates in the planar cis and trans structures (i.e., LHOOH/cn.plane = 0). The sensitivities LHOOH/r for the nonplanar minimum-energy structure will be discussed later. The most prominent feature of Figure 5 is the dominance of LOOH/00 (no. 2). For all conformations, an increase in the 00bond will lead to a decreasein the OOH angle. This sensitivity is slightly larger for the C2 structure than the cis and trans. The interdependence of these two parameters has been studied in great detail in the work of Getino et al.13 through ab initio calculations of the PES of H202. The next most important coupling in this molecule is 00/LOOH (no. 7). This sensitivity
-
Structure Sensitivity Analysis of Small Molecules
1
2
3
4
5
6
7
Coordinate Set
I.
Lnoonioo
OOlLHOOH 7. LOOHI00
4.
2. 5.
Lnoonion onimooH
3. LnooH I LOOH 6. LOOH I LHOOH
Figure 6. Unconstrained structure-structure sensitivities involving LHOOH for the C 2 minimum-energy conformation of H202.
is related to the previous sensitivity and will not be discussed further. Both of these sensitivities appear to peak in magnitude for the nonplanar structure C2 conformation. The only other significantdependency of structural parameters is between the two OOH angles (no. 10). Unlike the previously discussed coupling which were all relatively equivalent in the three conformations, the value of LOOHz/LOOH, is significantly larger in the trans conformation than for the either the cis or Cz conformation. For the trans conformation, as one OOH angle increases, the other decreases, while for the cis conformation,the opposite is true. This relationship is somewhat surprising since in the cis conformation the H's are much closer to each other and would be expected to have large 1-4 nonbonded interactions. This would lead one to expect larger couplings for the OOH angles. Since, as mentioned previously for the cis and trans conformations, the out-of-planetorsionalmode is of different symmetry than all the in-plane coordinates, to first order all sensitivities with resped to the torsion are zero. However, in the case of the C2 structure, the coupling of the torsional mode with the other coordinates is nonzero. Figure 6 graphically displays the set of unconstrained structurestructure sensitivities for the torsional angle with thevarious other internalcoordinates for the minimumenergy structure of H202. For comparison, the largest sensitivity from Figure 5 (no. 2, LOOH/OO) is listed alongside the others in Figure 6 (no. 7). The structurbstructuresensitivitiesfor this conformation show the presence of three pronounced couplings with the torsional angle. The first and largest is a positive coupling LHOOH/OO, indicating that as the 00 bond increases, the torsional angle should increase as well. The next largest coupling is LHOOH/ LOOH, which indicates that as the OOH angle increases, the torsional angle should decrease. This dependence may be explained by the fact that as the OOH angle increases, the 1-4 interaction between hydrogen atoms decreases, thereby allowing the HOOH angleto decrease slightly toward the cis conformation. Although this explanation is consistent with the LHOOH/LOOH sensitivity, it does not explain the previous magnitude and sign for the value of LHOOH/00. There remains one last sensitivity of reasonably large magnitude, LHOOH/OH. The negative sign for this sensitivity indicates that an increase in the OH bond length will lead to a decrease in the HOOH angle, which is consistent with the 1-4 interaction argument. By comparing the values for coordinate set no. 1 and no. 7 in Figure 6,it can be seen how the magnitude of the sensitivities for the couplings in Figure 5 compare with those in Figure 6. Figure 5 isdominatedby thevalue of LOOH/OO,yet when themagnitude
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6171 of this coupling is compared on the same scale with those of the torsional dependencies in Figure 6,it is dwarfed by the LHOOH/ 00 and LHOOH/OOH sensitivities. A further analysis of the H202 system was undertaken to determine the influence of indirect effects on the sensitivities for the three conformations of H202. Given in Table I1 are the calculated values for the constrained and unconstrained sensitivities for the cis, C2 minimum, and trans structures of H 2 0 ~ . Figure 7 presents graphically the results for the C2 minimumenergy conformation. Both Table I1 and Figure 7 indicate that the amount of indirect coupling for most of the coordinates studied is relatively insignificant. The noticeable exception is for the LHOOH/OO values, which differ dramatically for the constrained and unconstrained cases. This is indicative of a significant indirect influence in this quantity. The most likely candidate for indirect coupling is the HOO angle. This can be qualitatively reasoned out from the data in Table 11. By using the chain rule and the data from Table 11, a more quantitative estimate can be made utilizing eq 7. In this equation,the subscripts unc = unconstrained and c = constrained.
d(LHO0H) d(00) )unc=2(
(
d(LHO0H)
d(L0OH)
)l
~(LHOOH)
d(LO0H) d(O0) )c
d(00)
Given the relatively large values for LHOOH/LOOH(-0.91 1) andLOOH/00 (-0.437), it can be estimatedthat the first indirect coupling term in eq 7 accounts for a great deal of the difference for the unconstrained and constrained values of LHOOH/OO. In fact, the numerical value of the first term in eq 7 is 0.796, which in fact does account for almost all of the difference of 0.8 15 between the constrained and uncontrained values of LHOOH/OO. In this case, the use of constrained and uncontrained structure-structure sensitivities gives a very clear picture of the large LHOOH/OO sensitivity. The only other reasonably large percentage difference in the unconstrained and constrained values of the sensitivities is for the LOOHZ/LOOH~ sensitivity in the trans conformation. Here the differenceof 0.044 can be almost totally accounted for by an indirect couplingterm involvingthe 00 bond length (Le., LOOH2/ 00 and OO/LOOH~).However, in this case, the constrained sensitivity is larger than the unconstrained. This indicates that the relaxation of the 00 bond in the unconstrained case acts to mitigate the coupling of the two OOH angles. From a chemical perspective this certainly makes a great deal of sense, since it would be expected that whether or not the 00 bond relaxes should indirectly influence the sensitivity of the two angles. Glyoxylic Acid. Glyoxylicacid was chosen for this investigation for a number of reasons; one of these was due to the presence of an internal H bond (Figure 8) in the particular conformer studied. This molecule was chosen to study the effect of basis set on the calculated sensitivities. Since the description of this nonbonded interaction should be sensitive to the type of basis set used, it is a good test of the influence of basis set on the calculated sensitivity coefficients. Given in Table 111are the constrained structure structure sensitivities for glyoxylic acid derived from the Hessian calculated at the SCF level using a 6-31G* basis set. Due to the large number of possible sensitivities in this molecule, only sensitivities having an absolute value greater than 0.10 are tabulated. Given in Figure 9 is a bar graph of the calculated unconstrained structure-structure sensitivities derived from the 6-31G* SCF calculated Hessian compared with those from a 3-21G SCF calculated Hessian.
6172 The Journal of Physical Chemistry, Vol. 97, No. 23, 1993
Jasien and Thacher
TABLE Ik Constrained and Unconstrained Sensitivitiesafor H202
OH/OO LOOH/00 LHOOH/OO OO/OH OH2/0Hl LOOH1/OH1 LOOH2/OH1 LHOOH/OH 00/LOOH OH1/LOOH1 OH1/LOOH2 LOOH2/LOOH1 LHOOH/LOOHl OO/LHOOH OH/LHOOH LOOHl/LHOOH
0.016 -0.362 0.000 0.011 -0.009 0.027 0.021 0.000 -0.118 0.013 0.0 10 0.03 1 0.000 0.000 0.000 0.000
h F k = 0 indicates unconstrained; b
k
0.027 -0.367 0.000 0.017 -0.010 0.032 0.025 0.000 -0.1 15 0.016 0.013 -0.013 0.000 0.000 0.000 0.000
g
4
0.00
cn 4.50
. . . . . . . . . . . . . . . . 1 2 3 4 5 6 7 8 910111213141516 Coordinate Set
1. 4. 7. 10. 13. 16.
0.019 -0.367 0.000 0.013 0.004 -0.023 -0.012 0.000 -0.101 -0.009 -0.005 4.136 0.000 0.000 0.000 0.000
0.0 14 -0.433 0.000 0.009 0.003 -0.0 19 -0.010‘ 0.000 -0.1 17 -0.008 -0.004 -0.180 0.000 0.000 0.000
0.000
TABLE IIk Selected Sensitivities for Glyoxylic Acid (C2H204) with a 6-31G* Basis Set
0.50
-1.00
0.009 -0.437 0.419 0.005 0.001 0.009 -0.004 -0.161 -0.122 0.004 -0.002 -0.090 -0.91 1 0.005 -0.003 -0.04 1
= 0 indicates constrained.
1.00
‘g
0.003 -0.447 1.234 0.002 0.002 0.015 0.001 -0.175 -0.128 0.006 0.000 0.006 -0.97 1 0.0 16 -0.003 -0.044
OHlOO OOlOH LOOH,/OH, OH1 I LOOH, LHOOHI LOOH, LOOHI LHOOH
2. 5. 8. 11. 14.
LOOHIOO OH,/OH, LH~OH;OO OH, I LOOH, 00 I LHOOH
3. 6. 9. 12. 15.
LHOOH~OO LOOH,/OH, OOlLbOH * LOOH, I LOOH, OHlLHOOH
Figure 7. Comparison of constrained and unconstrained structurestructuresensitivities for the C2 minimum-energyconformationof H 2 0 2 .
C
0’2
H
Figure 8. Structure and atom numbering for glyoxylic acid.
The first thing that is noticeable from this plot is that these data are relatively insensitiveto the basis set used in the calculation. Despite the fact that there is a large difference in the accuracy of an unpolarized 3-21G and a polarized 6-31G* basis set for energetics, these results are essentially equivalent with regard to the sensitivities. This is not too surprising, however, since for many years quantum chemists have been calculatingvibrational frequencies of large molecules with small one-electron basis sets at the SCF level with relatively good success.14 This result indicates that much sensitivity information may be gleaned from less rigorous and less computationally expensive calculations. As always, a caveat exists, since if there are subtle effects that
set no.
coordinate set
sensitivity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
LHCC/LOICH LOICH/LHCC
-0.82 -0.57 -0.56 -0.55 0.35 -0.24 -0.2 1 0.21 -0.21 0.21 0.20 -0.17 -0.17 -0.15 -0.14 0.14 -0.14 -0.12 -0.1 1 0.11 -0.1 1 0.10 -0.10
Lo~cc/Lo2cc Lo2cc/Lo~cc
LHCC/LCOI LHOsC/LCOj L03CC/CC LHCC/L02CC LHCC/LO&C L03CC/CO2 L02CC/CO3 LOICH/COI
co3/co2
L02CC/CO2 CC/L03CC LOICH/CC LH03C/OH LO3CC/LHCC CC/COI LO2CC/LHCC
cc/co2
C03/LO2CC CH/COI
determine the structure of a molecule, it may be necessary to use a better basis set to obtain accurate structure-structure sensitivit ies. In glancing over the plot of the sensitivities for this system, the largestvalues (nos. 1,2,3, and 4) correspond to couplings between HCC and OCH angles as well as the OzCC and 03CC angles, respectively. That these angles should be highly negatively coupled is of no surprise, since they are adjacentand both influence the geometry around a carbonyl-type carbon. The next largest coupling is for LHCC/CO~(no. 3, which indicates that a lengtheningof the CO bond will lead to an increase in the opposite angle. This value is consistent with the trend outlined earlier for bonds that are externalto angles. This positive sensitivity is also seen in a number of other couplings of this type (nos. 5,10,11, and 16). The other correlationthat was presented previously involved the angle/included bond sensitivity. Here, too, the trend holds as shown by the negative sensitivity for the angle/bond couplings of this type (nos. 6, 7, 12, 14, and 17). Once again, the large number of possible sensitivities here lends itself to a great deal of analysis. However, we will be satisfied stopping at this point, having brought out the salient features of the insensitivity of the structure-structure sensitivities in this system to the one-electron basis set and the continued adherence of the angle/bond correlations previously mentioned.
Structure Sensitivity Analysis of Small Molecules
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6173 TABLE Iv: Selected Sensitivities for Lewis Acid-Base Complexes of AlCh
_..
0.2
complex AlC13- -H2C0
0.0
2.
.LI
-0.2
.* ."
2
AICI3- -HClCO (C1 bound)
-0.4
fn -0.6
AICl3- -H3CCI
-0.8
,."
. . . . . . _ _ _ _ _ _ _ _ _ _ _ _ . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Coordinate Set
Figure 9. Comparison of unconstrained structure-structure sensitivities for glyoxylic acid derived from 3-21G SCF and 6-31G* SCF Hessians.
b)
Figure 10. Structure of AlCl3 complexes with (a) H2C0, (b) HClCO, and (c) H3CC1.
AlCl3- -HzCO, AlCl3- -HClCO, and AlCl3- -H3CCl. These last examples are included as an illustration of the application of sensitivityanalysisto intermolecularinteractions. In these cases, the intermolecular interaction is a Lewis acid-base interaction between AlC13and a a-donor molecule with either C1or 0 acting as the donor atom. The energetics of these systems have been studied in detail by Jasien.15 Due to size of these systems, there are a tremendous number of possible structure-structure sensitivities that may be analyzed. Here we will only give a cursory analysisof these structure-structure sensitivities. The structures of these complexesare schematically indicated in Figure 10, and some of the unconstrained sensitivities are listed in Table IV. The first structurestructure sensitivityof note is that for the intermolecular distance coupling to the bond adjacent to the u donor (CX, X = 0,Cl). In Table IV this correspondsto the first entry for each complex. The large magnitude of the AlCl/CCl sensitivityin HClCO indicates a very large sensitivityto the CCl bond length. This is probably indicative of the overall weakness of the CCl bond and its susceptibility to external perturbations. Stretching this bond causes a rearrangement in electron density around the donor atom, making its electrondensitymore available for donation to the aluminum. The fact that this sensitivity is negative in all three complexes is consistent with the argument
.
.
.
.
coordinate set
sensitivity
AIO/CO LAIOC/CO LAIOC/AIO AIO/LAlOC AlCI/CCI LAlClC/CCI LAlClC/AlCl AICl/LAICIC AICl/CCl LAICIC/CCI LAICIC/AICI AICI/LAICIC
4.129 -0.202 -0.130 -0.239 -0.822 -0.006 0.049 0.159 -0.195 -0.1 12 -0.030 -0.066
that the AlC13 complexation perturbs the bonding in the donor molecule and subsequently weakens the CX bond in the base. The large effect in the HClCO case is not necessarily directly relatable to the strength of the intermolecular interaction, since at the 6-31G* SCF level the binding energies of the complexes are predicted to be 28.4, 3.4, and 9.5 kcal/mol for the H2C0, HClCO, and H3CCl complexes, respectively. Therefore, the complex showing the smallest binding energy actually shows the largest AlX/CX sensitivity. The magnitude of this sensitivityis probably more indicativeof the CX bond strength than the binding energy of the complex. The sensitivity of the intermolecular angle and the CX bond of the donor shows pronounceddifferencesin the threecomplexes. The H2C0 and H3CCl complexes show reasonably large sensitivities of -0.202 and -0.1 12, respectively, while the value for the HClCO complex is an anemic -0.006. The negative couplings indicate that a lengthening of the CX bond results in a decrease in the intermolecular angle. One possible explanation for the sign and magnitudeof these sensitivitiesin the complexes is related to repulsive interactions between the acid and the base. In all cases, lengthening of the CX distance effectively moves the two ineractingmonomersfurther apart, lowering repulsive interactions and allowing the intermolecular angle to relax slightly. The relatively large value of LAlOC/CO in the H2CO complex may be due to the close proximity of the two monomers, since the A10 distance is only 1.95 A.15 The relatively low value of LAlClC/ CCl in the HClCO complex may be due to the fact that the monomers are already quite far apart and there are few significant repulsive interactions, since here the A1-Cl distance is 2.61 A.15 The remaining structure-structure sensitivities in Table IV seem to show no other discernible trends. This once again may be due to the individual complexitiesof the various complexesas related to repulsive interactions as discussed above. In addition, as was mentioned in ref 15, the complexesshow varying degrees of weak hydrogen-bonding interactions between the hydrogen(s) on the donor molecule and the chlorine(s) on the acceptor. All of the subtleties above, as well as the fact that the magnitude of some of these sensitivities may be due to indirect effects, may lead to the lack of a readily discernible trend. The finer points of these sensitivitiesas they relate to reactivity of the Lewis bases will be more fully discussed in a subsequent report. Conclusion
The calculation of the structurestructure sensitivitiesdetermined with the current methodology provides insight into the interrelationship between structural features in molecules. The use of this methodology provides a straightforward way of interpreting the structural coupling that arises in molecular systems without assuming specific forms and parametrizations of molecular potentials. The effect of basis set on the calculated sensitivities was checked through a comparison of 3-21G and 6-31G* calculations. It was found that although the absolute values for the sensitivities differed, the relative trends were
6174 The Journal of Physical Chemistry, Vol. 97, No. 23, 1993
unaffected by thedifferencein the basis sets. Theuseof structurestructure sensitivities can provide a simple method by which structural correlations in molecules can be gleaned by simply scanning graphical representations of the sensitivities. Lastly, the complementary use of constrained and unconstrained sensitivities can elucidate complex indirect structure-structure relationships of coordinates. Acknowledgment. P.G.J. thanks the ACS-PRF (Grant 26276GB6) for supporting this work and the San Diego Supercomputer Center for an allocationof computer time to perform some of the calculations. References and Notes (1) Tomovic,R.; Vukobratovic,M. GeneralSensitiuityTheory;American Elsevier: New York, 1982. (2) Martens, H. R.; Allen, D. R. Introduction io Sysrems Theory; Menill: Columbus, 1969.
Jasien and Thacher (3) Rabitz, H.Science 1989,246, 221. (4) Rabitz, H. Comput. Chem. 1981, 5, 167. (5) Rabitz, H. Chem. Rev. 1987.87, 101. (6) Rabitz, H.;Kramer, M.;D a d , D. Annu. Rev. Phys. Chem. 1983, 34,419. (7) Chang, J.; Brown, N. J.; Rabitz, H. J. Phys. Chem. 1992,96,6890. (8) Thacher, T. S.;Hagler, A. T.; Rabitz, H. J . Am. Chem.Soc. 1991, 123, 2020. (9) Susnow, R.; Nachbar, R. B., Jr.; Schutt, C.; Rabitz, H. J. Phys. Chem. 1991,95, 10662. (10) Wong, C. F.;Rabitz, H. J . Phys. Chem. 1991,95,9628. (1 1) Frisch, M. J.; Head-Gordon,M.; Schlegel, H. B.; Raghavachari,K.; Binkley, J. S.;Gonzalez, C.; Defrem, D. J.; Fox, D. J.; Whiteaide, R. A,; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Ruder, E. M.; Topiol, S.;Pople, J. A. Gaussian Inc., Pittsburgh, PA. (12) Weast, R. C.,Ed. CRC Handbook of Chemistry und Physics, 63rd ed.: CRC Press: Bcca Raton. FL. 1982. (13) Gettino, C.; Sumpter, B.’G.; Santamaria, J.; Ezra, G. S.J. Phys. Chem. 1990.94.3995. (14) Has, B.A., Jr.;Schaad, L. J.; Carsky, P.; Zahradnik. R. Chem.Reu. 1986,86,709. (15) Jasien, P . G. J . Phys. Chem. 1992,96,9273.
