496
J . Phys. Chem. 1984, 88, 496-500
Molecular Orbital Approaches to the Calculation of Vibrational Circular Dichroism Teresa B. Freedman and Laurence A. Nafie* Department of Chemistry, Syracuse University, Syracuse, New York 13210 (Received: May 3, 1983)
A new molecular orbital description of vibrational circular dichroism (VCD), the nonlocalized molecular orbital model (NMO), is developed. In the CNDO approximation, the NMO-VCD intensity is expressed in terms of one-center charge and rehybridization contributions and two-center current contributions. The NMO model, which is the molecular orbital analogue of previous charge-flow models, is compared to the other molecular orbital approaches to VCD, the localized molecular orbital model (LMO), and the atomic polar tensor model (APT). The APT expressions are shown to arise directly as an approximate form of the NMO expressions. Calculations of the CH-stretching VCD in L-alanine are presented for the three molecular orbital approaches.
Introduction The detailed understanding of vibrational circular dichroism (VCD) spectra, leading ultimately to stereochemical information about molecules in solution, depends on theoretical models with which VCD intensities can be both accurately and conveniently calculated.' Models thus far proposed differ in their treatment of the contribution of the electrons to the electric dipole transition moment, and, more particularly, to the magnetic dipole transition moment. The earliest general model, the fixed partial charge (FPC) model,2 has been modified to include the effects of charge flow along bond^.^,^ The latter models have been recently compared5 to the bond dipole theory! A dynamic polarization model' has also been proposed. The molecular orbital approaches provide a more complete treatment of the electronic contributions to both infrared and VCD intensities, and avoid the necessity of choosing and adjusting empirical charge and charge-flow parameters. The localized molecular orbital (LMO) model8 has been used to calculate VCD intensities in very good agreement with e ~ p e r i m e n t . ~ More recently, we proposed the atomic polar tensor (APT) modello as a more efficient, but somewhat less accurate molecular orbital approach, which does not require localized orbitals. Finally, we have used vibronic coupling theory to develop exact expressions for VCD intensity which use momentum and angular momentum operators directly." Numerical calculations using the latter method have not yet been carried out. In this paper we present a more complete nonlocalized molecular orbital (NMO) approach which is based on atomic contributions to the dipole moment derivative. The model is shown to be the molecular orbital analogue of the charge-flow model^,^^^ and the APT expressions10are shown to result directly as an approximate form of the N M O model. Theoretical Background The infrared and VCD intensities of a transition between levels (1) (a) L. A. Nafie and M. Diem, Acc. Chem. Res., 12, 296 (1979); (b) T. A. Keiderling, Appl. Spectrosc. Reu., 17, 189 (1981); (c) L. A. Nafie in "Vibrational Spectra and Structure", Vol. 10, J. R. Durig, Ed., Elsevier, Amsterdam, 1981. (2) (a) C. W. Deutsche and A. Moscowitz, J . Chem. Phys., 49, 3257 (1968); 53, 2530 (1970); (b) J. A. Schellman, ibid., 58, 2882 (1973). (3) S. Abbate, L. Laux, .I. Overend, and A. Moscowitz, J. Chem. Phys.,
75, 3161 (1981). (4) M. Moskovits and A. Gohin, J . Phys. Chem., 86, 3947 (1982). (5) P. L. Polavarapu, Mol. Phys., 49, 645 (1983). (6) L. D. Barron in "Optical Activity and Chiral Discrimination", S. F. Mason, Ed., Reidel, Dordrecht, 1979. (7) C. J. Barrett, A. F. Drake, R. Kuroda, and S. F. Mason, Mol: Phys., 41, 455 (1980). (8) (a) L. A. Nafie and T. H. Walnut, Chem. Phys. Lett., 49,441 (1977); (b) T. H. Walnut and L. A. Nafie, J . Chem. Phys., 67,1491, 1501 (1977); (c) L. A. Nafie and P. L. Polavarapu, ibid., 75, 2935 (1981). (9) T. B. Freedman, M. Diem, P. L. Polavarapu, and L. A. Nafie, J . Am. Chem. SOC.,104, 3343 (1982). (10) T. B. Freedman and L. A. Nafie, J . Chem. Phys., 78, 27 (1983). (11) L. A. Nafieand T. B. Freedman, J . Chem. Phys., 78, 7108 (1983).
0022-3654/84/2088-0496$01.50/0
v and u' of normal mode Q, are proportional to the dipole and rotational strengths, respectively, given by
DE".= lPL:"42
(1)
R;,"= Im (p5t.m;a)
(2)
where pi", and m,","are the electric dipole and magnetic dipole transition moments, respectively. For a fundamental transition at frequency w,
+
RBo = Im (OICZneRn C-erkll,)n
k
for n nuclei with position R,, momentum P,, charge Zne,and mass M,,, and k electrons with position rk, momentum P k , charge -e, and mass m. The electric and magnetic dipole moments and dipole transition moments of a molecule are written as sums of nuclear and electronic contributions, for example, p = pE + pN. The nuclear contributions to the transition moments are readily evaluated as
(mN)lo=
CR,,o X Zne n
where zeroes denote equilibrium values. The electronic contributions are normally evaluated by expanding the electronic integrals as Taylor series in Q,. The electronic contribution (mE)10 cannot be obtained from a similar expansion in Q,." Instead, calculational models for VCD employ approximate position-dependent expressions for (mE)lO. In the fixed partial charge model, the electrons serve merely to screen the nucleus, resulting in a fixed partial nuclear charge En which follows the nuclear displacement (aRn/dQ,)o= s,. The FPC expression for the rotational strength is2
Since charge redistribution is known to occur during the course of a vibration, the FPC expressions for both the dipole and rotational strengths can be improved by including charge-flow (CF) term^.^,^ Following the treatment of Moskovits and Gohin: the change in charge at nucleus n during normal mode Qa, Le., the charge flux (aEn/aQa),, is considered to result from currents In, along the bonds connecting atom n and atoms 1 C(aIn//aPa)o = - ( a t n / d Q a ) o /
(7)
where P, is the time derivative of Q,. With our notation, the rotational strength including these charge flow contributions is
0 1984 American Chemical Society
Calculation of Vibrational Circular Dichroism
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 497 tronic and nuclear terms are based on atomic contributions. The molecular orbital expressions for pE can be written as a sum of atomic one-center and two-center contributions. To a first approximation the two-center terms can be ignored, as is assumed in the CNDO and other semiempirical methods. We thus define CL = C n p nand
The molecular orbital models retain the full nuclear charges Z,, and calculate the electronic contribution to the dipole moment and dipole moment derivative by using the ground-state LCAOM O wave function. The approaches differ in the approximate form used for the electronic contribution (mE)lO.In the localized molecular orbital model (LMO),s the electronic contribution to the dipole moment is considered to be a sum of contributions pE = &pkE from localized molecular orbitals at equilibrium centroid giving position rkO,O
( a ~ / a Q a ) o= X(aKn/aQa)o n
(1 3)
and consider an approximate, atomic expression for the rotational strength, eq 14, having the same general form as the FPC and L M O expressions, which are also based on local contributions.
In the following discussion we consider the specific form of the N M O model in the CNDO approximation, which we have used for all our previous MO calculations. The CNDO molecular dipole moment is written asi6
where l i s the diagonal unit tensor. Intrinsic terms due to orbital rocking have been ignored. In calculations using the LMO model, (apk/dQ,)o is obtained as a finite nuclear displacement derivative (FND) requiring a separate localized M O calculation for each normal mode. The LMO method can, in principal, be made more efficient by using finite electric field perturbation (FEF) methods to calculate the equilibrium nuclear displacement derivatives of pkI2-l5 However, whereas for most S C F wave functions, including CNDO, the molecular property ( a p / a R,) is readily obtained by this method, a localized property (a pk/BR,J0 cannot be accurately calculated by an F E F procedure, especially for large molecules.10 The more approximate atomic polar tensor model has proved to be an efficient molecular orbital method for calculating dipole and rotation strengths. In the APT approximation, the rotational strength is expressed as
where
( a ~ / a R , ) o = Z,eI + (apE/aRn)o (1 1) The components of (dp/aR,), are the elements of the atomic polar tensor for atom n. The electronic contribution to this molecular property can be calculated by finite electric field perturbation methods at the equilibrium geometry using the relationship
(
=
-[$(&)I0
/
Nonlocalized Molecular Orbital Model We now develop a second molecular orbital approach not requiring localization which we will refer to as the nonlocalized molecular orbital model (NMO). In this method both the elecA. Komornicki and J. W. McIver, Jr., J . Chem. Phys., 70, 2014
~
L. A. Nafie and T. B. Freedman, J . Chem. Phys., 75, 4847 (1981). P. L. Polavarapu and J. Chandrasekhar, Chem. Phys. Lett., 84, 587 86, 326 (1982).
P. L. Polavarapu, J . Chem. Phys., 72, 2273 (1982).
The terms in the first summation correspond to a charge ( e 2 , eP,,) following the nuclear motion s,, the second term is a charge-flux term involving a change in the electron population of atom n, and the third term gives the contribution of rehybridization at atom n. When we consider the approximate rotational strength expression, eq 14, the first and third of these contributions are properly considered as originating at the nuclear position Rn,o. However, in analogy to previous developments of the charge-flow model^,^,^ the charge-flux term (dP,,/aQ,) should be considered to arise from currents between atom n and all the other atoms in the molecule C(aZn//aPa)o = e(dPnn/dQa)o
(12)
where u, p are Cartesian directions, F is a perturbing electric field, and t is the electronic potential energy.I2 This method does not require localization and, coupled to force field calculations which yield the nuclear trajectories, the atomic polar tensor elements generated from a single molecular orbital calculation can be used to calculate infrared and VCD intensities for all the normal modes of a molecule.
(12) (1979). (13) (14) (1981); (15)
.
where 2, is the core charge and P,, the Mulliken electron population on atom n. The second term is an s-p hybridization term for second row elements involving the off-diagonal density matrix element for atom n, Pg!2p,,where ln is the Slater exponent and zi, is a unit vector in the a (=x,y,z) direction. For third row elements different hybridization terms, including p-d hybridization, are required, but for simplicity will not be included here. Taking the derivative of eq 15 with respect to normal mode Q, we find
(17)
for current Zfl/ defined as positive for positive charge flowing from atom n to atom 1 and where Pa is the mass-weighted momentum associated with the mass-weighted normal coordinate Q,, Pa = iuaQa.
We now express the charge flux term in eq 16 as the equivalent summations
The last term clearly expresses the charge flux in terms of the vector directed between two atoms times the current flowing between the atoms. We can now write the contribution (ap,/aQ,)o as (16) J. A. Pople and D. L. Beveridge, “Approximate Molecular Orbital Theory”, McGraw Hill, New York, 1970.
498
The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
From eq 14, the CNDO expression for the rotational strength becomes RTo(NMO) =
This expression is the CNDO-MO analogue of the charge-flow expression, eq 8, and can be shown to exhibit the proper origin independen~e.~-~,~' The total two-center current contribution for atoms n and I can be combined to give
that is, it can be viewed in terms of current flowing between the atoms and as contributing from the midpoint of the two atomic centers. In contrast to the charge-flow models the N M O expression incorporates the one-center electronic contributions more completely, and is not necessarily restricted to charge flow along bonds. In practice, however, the current derivatives are estimated by using eq 17 for bonded atoms only and from the requirement of overall charge neutrality. For a cyclic molecule this procedure results in some indeterminancy in the values of (aInI/dPa)o.The N M O model calculations require a finite nuclear displacement method in which only the one-center terms (dP,,,/dQ,), are calculated directly. We emphasize that the atomic populations and density derivatives result from the molecular orbital calculation and are not empirical, adjustable parameters as in the FPC and charge-flow models. Comparison of Models When the APT model was initially proposed we developed the expressions as an approximate form of the LMO model.1° In fact, the APT model is not a localized model and the APT rotational strength expression involves only nuclear summation. The APT model can best be understood as an approximate form of the NMO expressions. In terms of atomic displacements, eq 14, which leads to the N M O model, can be written as
Freedman and Nafie Comparing eq 23 with eq 10 for the APT model, we see that the APT expression results from ignoring the contribution of the second term in the magnetic dipole transition moment above and summing over n in the remaining terms. Whereas in practice the atomic polar tensor elements are obtained by the finite electric field perturbation technique, it is revealing to consider the CNDO expressions for the magnetic dipole transition moment in the APT model in terms of atomic contributions defined in eq 19. For example
The first summation in this expression is the perfect following term, identical with that in eq 20. The second term can be interpreted as taking the part of the current between any two atoms due to the vibrational momentum of atom m to be a contribution from the moving nucleus rn. Similarly the last term takes the contribution to the rehybridization of atom n due to the motion of atom m to be located at atom m. The derivatives in eq 24 with m = n should have the largest contributions and, since then Rm,fl = 0, these contributions are the same in the APT and NMO models. However, in the APT model currents are not restricted to bonded atoms. For m # n there are errors introduced by the APT approximation which will be largest for local nuclear motion which induces significant electronic currents in a distant part of the molecule. It is also of interest to compare the localized and nonlocalized molecular orbital models. In the LMO model, electronic charge motion is included as displacement of the LMO centroids during vibration. Since the L M O s are normalized units always occupied by two electrons (in closed shell molecules considered here), no charge flow between LMO's enters the LMO expression, eq 9. In the CNDO approximation, both rko,oand (dpk/dQa)o = -2e (ar,o/aQ)o are sums over one-center atomic orbital contributionsls
where Clrkis the LCAO coefficient for the ,uth orbital of the nth atomic center in the localized molecular orbital k . For a given bonding LMO between atoms A and B with centroid along the line connecting A and B, the magnetic dipole contribution in the LMO model is approximately
R?, =
Substituting R , , = Rn.O- Rm.Oand s,
= (dRm/aQ,)o yields
(17) A. E. Hansen and T. D.Bouman in "Advances in Chemical Physics", Vol. 44, I. Prigogine and S . A. Rice, Ed., Wiley, New York, 1980.
(18) P. L. Polavarapu and L. A. Nafie, J . C h e w Phys., 75, 2945 (1981).
Calculation of Vibrational Circular Dichroism
The Journal of Physical Chemistry, Vol. 88. No. 3, 1984 499
TABLE I: Rotational and Dipole Strength Calculations for the CH-Stretching Vibrations of L-Alanine Using the Localized Molecular Orbital, Nonlocalized Molecular Orbital, Atomic Polar Tensor, and Fixed Partial Charge Models
L- Alanine-N-d,
3009 2993 2970 2930
2.21 5.13 4.80 2.87
1.77 -9.06 18.6 -7.35
0.018 -7.53 15.0 -6.50
2.22 5.16 4.74 2.82
3007 2993 2932
1.45 4.72 3.25
0.944 -0.372 -1.23
-0.250 0.550 -0.490
1.44 4.70 3.12
2970
5.74
4.58
1.11
5.80
1.40 -9.23 12.2 -5.58
1.42 1.34 0.45 0.41
CH,=yrn
0.545 -1.71 1.59 -0.365
0.57 0.58 0.45 0.30
0.54 -2.43 5.58 -1.15
C*H CH,SYm
-0.065 0.073 0.0016
0.53 0.67 0.52
0.70 -0.53 -0.39
CH, aSyrn CHSasym CH,SYm
0.0345
0.37
2.11
CH,aSYm
L-Alin~nc-,l.~-d,-C*-d, 0.980 -0,812 0.139
1.25 1.22 0.43
L- AI nnin e-N-d 3- C-d, a
See ref 9 and 10.
Dipole strength (esu' cm')
X 10".
.-1.57
Rotational strength (esu' cm')
since the contributions from other atomic centers is small. For a lone pair L M O on A, only the terms for atom A are included. Since the density matrix element is PIrv= 2zkCIrkCvkand the we see that the electronic electronic population is P,, = E,,"PIrfi, contribution to the perfect following term for atom A in the N M O model can be obtained by moving rk,o,oto RA,O for the terms involving (dRA/dQ,)o above and summing over the LMO's involving atom A. The rehybridization N M O terms can be recovered in a similar fashion. The remaining terms for the A-B bonding L M O in eq 27 are
(28) The L M O is normalized,
0.73
C,,Cck12 = 1, so we have approximately (29)
If we define
eq 28 becomes
which can be compared with the N M O current contribution, eq 21. These two equations become equal to one another under the = l/Z(RA,O + RB,o)and that ( N A B / ~ P ~as) o , restriction that rktO,O. defined above, describes all of the current flowing between atoms A and B. Neither of these restrictions is unreasonable. The above discussion considered the case where the LMO is completely localized between atoms A and B. In fact, some "tailing" is always observed, and these terms for atoms C # A, B enter the complete LMO expression as current-type contributions for nonbonded atoms.
Calculations and Discussion In order to compare the three molecular orbital approaches described here, we have carried out dipole and rotational strength calculations for the CH-stretching vibrations of L-alanine-N-d, (CH,CH(ND,+)COO-), ~ - a l a n i n e - N - d ~ - C * - (CH3CDd~ (ND3+)COO-), and ~-alanine-N-d,-C-d,(CD3CH(ND3+)COO-) using C N D O wave functions. The nuclear trajectories, (dR,/ aQa)o,were determined previously from an in-depth vibrational a n a l y s i ~ .Results ~ ~ ~ ~for ~ the LMO model were obtained with a (19) M. Diem, P. L. Polavarapu, M. Oboodi, and L. A. Nafie, J . Am. Chem. SOC.,104, 3329 (1982).
X
C*H
lo4".
finite nuclear displacement method for each normal mode separately, while the APT results, using the finite electric field perturbation technique to determine the dipole moment derivatives at equilibrium, were obtained from a single MO calculation for all the normal modes. The APT and LMO methods and results for alanine have been published p r e v i o u ~ l y . ~ J ~ The N M O model also requires a finite nuclear displacement calculation to obtain the derivatives (dPm/dQ)oand (dPz,,2,/dQ),,. However, the standard CNDO-MO program16is readily adapted to the N M O model, and, since no localization is required, results for each mode are calculated with considerable savings in time compared to the LMO model. In these calculations the currents have been restricted to bonded atoms only. The bond current derivatives (dZ,,/dP,), were obtained from eq 17 and the requirement that z,(dP,,,/dQa)o = 0 due to charge n e ~ t r a l i t y . ~ Since, excluding nonbonded interaction, alanine is noncyclic, unique values are found for the n - 1 bond currents in terms of the n atomic charge fluxes. Our results for the three molecular orbital approaches and the best FPC calculationm are compared in Table I. The dipole strengths for the LMO and N M O methods are identical since the determination of (dp/dQ,), is independent of localization. The APT (FEF) procedure yields very similar (to within 1%) dipole strengths.1° We previously showed that the CNDO results closely reproduce all the features in the experimental infrared absorption spectra of all three isotopomers within a single scale factor? Close agreement between the signs and relative magnitudes of the rotational strengths calculated with the LMO model and the experimental VCD spectrum were also demonstrated. As we have discussed previously? the observation of diminished CH-stretching dipole strengths in aqueous relative to nonaqueous environments may account for a large part of the discrepancy in magnitude between the C N D O results and experiment. In the FPC calculation the fixed partial charges were scaled up to give overall dipole strengths comparable to experiment. Unlike the CNDO results, however, the relative FPC dipole strengths do not agree with the observed pattern. The FPC rotational strength calculation underestimates the contributions of the methine and symmetric methyl stretch in alanine-N-d, and fall one to three orders of magnitude below the experimental values for the other two isotopomers. Compared to the LMO results and the observed VCD, the APT model is in better agreement for the antisymmetric methyl stretches, whereas the NMO model is in better agreement for the methine and symmetric methyl C H stretches. This contrast is especially apparent for the C-d3and C*-dl isotopomers in which the nuclear vibrational motion is more localized. As can be seen from the stereoprojections of the nuclear and LMO centroid m ~ t i o n the , ~ methine stretch in ~-alanine-N-d,-C-d, and the symmetric methyl stretch in ~-alanine-N-d~-C*-d~ were two cases in which electronic motion (reflected in LMO centroid dis-
-
(20) B. B. Lal, M. Diem, P. L. Polavarapu, M. Oboodi, T. B. Freedman, and L. A. Nafie, J . Am. Chem. SOC.,104, 3336 (1982).
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The Journal of Physical Chemistry, Vol. 88, No. 3, 1984
placement) is observed in portions of the molecule in which the nuclei are not moving. As noted above, it is precisely in these cases that the APT approximation introduces errors compared to the N M O equations, and indeed an incorrect sign of the rotational strength of these modes is calculated with the APT model but not the N M O model. In L-alanine-N-d3 where the nuclear motions are more coupled and spread out, both the APT and N M O methods yield satisfactory results. In the antisymmetric methyl stretching modes, the N M O results are poor. For these modes most of the LMO centroid motion is observed in bonds within and adjacent to the methyl group. It is possible that these vibrations induce some “current” contributions between nonbonded methyl hydrogens, which are implicitly included in the LMO and APT models but not in the N M O methods as applied here. Such currents may also be responsible for the generally lower rotational strength magnitudes obtained by using the NMO model compared with the LMO model. The discrepancy for the antisymmetric methyl couplet may also arise from the fact that the methyl force field was adjusted such that both the degeneracy splitting and LMO sign pattern agreed with e ~ p e r i m e n t . ~ We should note that both the electric and magnetic dipole transition moments arise as a small difference between large quantities, the nuclear and electronic contributions. For the NMO model, the electronic bond current contributions were required to obtain the large magnitude of the methine VCD and both the sign and magnitude of the symmetric methyl VCD. Including nonbonded interactions in the NMO model in an empirical manner to simulate induced ring-type current contributions may improve the results of the N M O calculations. The neglect of current effects and charge flux are also responsible for the poor agreement in both the dipole and rotational strength values from the FPC model. In addition, contributions of lone pair electrons, which may be significant for some types of vibrations, are included in the N M O model in the rehybridization contribution, but are neglected in the FPC and empirical charge-flow models.
Conclusions The usefulness of a theory or model in understanding and analyzing experimental phenomena depends on several factors. For the molecular orbital models presented here, these include the accuracy with which the experimental intensities are reproduced, the ease of implementation of the model, and the cost and speed of the calculation. We should also consider the degree to which the model provides descriptive insight into sources of VCD intensity which can be applied to the analysis of spectra of similar molecules without carrying out detailed calculations. We must emphasize that in comparing any model of VCD with experiment a crucial variable is the force field used to obtain the nuclear trajectories. Of the MO approaches, the LMO model yields calculated intensity patterns which agree best with experiment, and appears to provide the most accurate description of the electronic contributions to the magnetic dipole transition moment. Only internal LMO current effects, which can be described in part by orbital rocking terms,8b are neglected. The method is also the most expensive and time consuming to use. The cost of carrying out localized finite nuclear displacement MO calculations for all the normal modes, or even all the normal modes in a specific spectral region of a large molecule can be prohibitive. The LMO model is most useful for final calculations once spectra have been assigned and the force field carefully refined. From stereoprojections of the motions of nuclei and LMO centroids during normal modes, sources of intensity, in particular patterns of induced electronic currents extending over several atoms or coupled oscillator type motions, can be detected which are of use in understanding the spectra of other molecules.21
Freedman and Nafie With the N M O model, localization is not required and, since no significant modifications to the CNDO-MO program are needed, it is the most readily employed. The rotational strength expression (eq 20) for each normal mode requires only nuclear trajectories, equilibrium nuclear positions, and the electronic density matrix at both equilibrium and displaced nuclear positions. The N M O model appears to be quite satisfactory in accounting for the VCD intensity for distinctive readily assigned features of the spectrum (for example, the lone methine stretch) which may be useful in determining absolute configurations. There are, however, weaknesses in the model which may affect the accuracy of the calculations. Discrepancies between the N M O and LMO calculations may arise from two sources: (a) positioning the perfect following and bond current contribution at the nuclei or half way between nuclei, respectively,rather than at the centroids of electron density, and (b) describing currents only in terms of bonded atoms. As noted earlier, simulated effects of ring-type currents could be included in a modified N M O calculation. The APT model involves the most approximate form for the magnetic dipole moment, while at the same time it is the most efficient. When APT’S are used, the descriptive picture of local electronic contributions is lost, and as noted above the model fails when there are electronic currents in the absence of nuclear displacement. The modifications of the MO programs to include both the electric field perturbations and the electronic potential energy gradient calculation are quite extensive and the programs are not yet widely available. The initial execution of the M O program to obtain the APT parameters is equivalent in cost to performing several NMO-FND calculations, but these parameters, once calculated, can be used to obtain the intensities for all the normal modes of any isotopic species of the molecules, for any force field. Therefore, as with the FPC model, there is almost instant feedback on intensities for the whole spectrum as the force field is modified. For a large molecule such as 3-methylcyclohexanone in which the normal modes involve considerable coupling among internal coordinates, we find that the APT results show very good agreement with both experiment and the LMO results for all the C H stretching vibrations except the symmetric methyl motion. The APT model has proved to be of prime utility in obtaining the required accurate nuclear trajectories. The N M O model can be qualitatively compared to the LMO model in the following way. The LMO model describes the motions of localized orbital centroids in response to nuclear displacements. The centroids are associated with either bonding or atomic nonbonding orbitals in the valence shell. In the N M O model the valence electronic contributions are organized into perfect following, charge flow between atomic centers and atomic rehybridization (angular redistribution) terms. The first two come largely from bonding centroid motion whereas the latter is quite important in describing the motion of lone pairs; however, the overall difference is simply a different kind of partitioning of the electronic contribution for the two models. Qualitative comparison of the AFT model to the LMO model has been provided previously and will not be discussed here.1° The development of the NMO formalism presented in this paper has been carried out within the CNDO approximation. While this restriction is not necessary it is consistent with the overall approximate aim of the NMO model as a simple approach to VCD intensities. Acknowledgment. The authors acknowledge financial support from the National Science Foundation (CHE 80-05227) and the National Institutes of Health (GM-23567). Registry No. L-Alanine, 56-41-7. (21) L. A. Nafie, M. R. Oboodi, and T.B. Freedman, J Am Chern. SOC., in press.
