Dipole Derivative Distribution: A Useful Adjunct to ... - ACS Publications

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J. Phys. Chem. 1993,97, 11578-1 1579

11578

Dipole Derivative Distribution: A Useful Adjunct to the Potential Energy Distribution Weili Qian and Samuel Krimm' Biophysics Research Division and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 Received: August 17, 1993; In Final Form: September 27, 1993'

The infrared intensity of a normal mode is shown to be composed of projected contributions from dipole derivatives in displacement coordinates onto the dipole derivative in the normal coordinate. Such a dipole derivative distribution provides insights into the origin of the band intensity not generally obtainable from the potential energy distribution.

Thedescriptionof a normal modeis often given by the potential energy distribution (PED), which is defined by the relative contribution of each displacement coordinates{to the total change in potential energy during the normal vibration Q,. The fractional contribution of each coordinate is given by'

TABLE I: Potential Energy Distribution and Dipole Derivative Distribution for Three Normal Modes of Isolated Glycine. 1384 cm-1 1096 cm-l 873 cm-I (24.2)d

Sb

NC s where Ffiis the diagonal force constant matrix element, L,,is the eigenvector matrix element

cc s c=o s c-0 s NH2 b

a

and &is theeigenvalue. (Contributionsof off-diagonalelements, which can be negative as well as positive, are typically small and are usually neglected in this representation.) The infrared intensity associated with a normal mode, A,, is given by2

A , = 4 2 . 2 5 ( a ; / a ~ , ) ~km mol-'

(3)

where a;/aQa is the normal-coordinate derivative of the dipole moment in D A-1 amu-llt. From (2) we see that

(4) where a;/&!?, is the dipole derivative in the displacement coordinate. Modern quanJum-mechanical programs permit the ready calculation of the ap/aSi and therefore of A,. However, it is not always easy to see from the PED which Si are primarily responsible for the observed intensity. We suggest that the dipole derivativedistribution (DDD) is a useful way to gain this insight. From (3) and ( 4 ) we see that

a;

-r

A , = 42.25-

aQa

a; --7

aQa

Therefore, the quantity

representsthecontribution that eachSimakes to the final infrared intensity. Since this contribution-depends on the projection of each eigenvector component on ap/aQa, it provides a different measure of the relative importance of the St than does the PED, .Abstract published in Aduance ACS Absrracfs, November 1, 1993.

0022-365419312097-1 1578$04.00/0

NH2 w NH2 r NCC d CH2 b CH2 w

(2 16.9)d

(7.8)d

laji/a@ PEW DDDf PEW DDW PED. DDDf 1.66 0.70 6.42 5.39 1.45 2.28 0.13 1.68 0.35 0.09 0.35 0.50 0.92 3.72 1.56 0.94 1.63 0.67 1.64

7 12 22

-6.5 -5.4 1.6 25.2 -5.7 -1.7

54

41.4

27

95.6 -21.3 54

-1.1 6 59

-1.6

15.7

1.1

CH2 r 24 -2.3 CH2 tw 9 2.3 CCO d -1.5 C4ib 9 29.6 22.0 COH b -10.4 18 64.0 C=Oob 19 5.6 NCt 1.7 CCt -2.1 C-0 t 4.2 a Reference 3. Local symmetry coordinate;s = stretch, b = bend, w = wag, r = rock, d = deformation, tw = twist, ib = in-plane bend, ob = out-of-planebend, t = torsion. e Magnitude of dipole derivative, in unitsof D A-' amu-l/Z. Infrared intensity, in unitsof kmmol-1. Potential energy distribution, components 1 5 . /Dipole derivative distribution, components 2 5 % of intensity, in units of km mol-'. which depends on ILial. Thk projection could be small becauseof the values-of L,, and/_orap/aSi, but also because of the angle between ap/aSi and ap/aQa. The collection of terms (6) for all Si represents the DDD for the mode. These points are illustrated by several modes from a scaled ab initio force field analysis of isolated glycine.' For most modes, the PED components are generally representative of the relative contributions to the DDD; in such cases, either one or two Si predominate, or the ap/aS, are generally proportional to the Fii, But for others, some of which are shown in Table I, this is not the case. The 1384-cm-1 mode is mainly CH2 wag (w) according to the PED,but its intensity derives primarily from C - 0 inplane bend and C-0 stretch (s), with opposing contributions from a large n_umber of other coordinates, undoubtedly because of the small &/asfor CHI w. For the strong band calculated at 1096 cm-I, the main contribution to the PED is from NC s, but the intensity derkes in greater part from C-0 s, primarily because oJits large ap/aS. Although NC s and COH bend have similar aCrlaS, their contributions to the intensity are opposite to their ratio in the PED, undoubtedly a reflection of their orientation relative to ap/aQ and therefore their projection on this vector. The weak 873-cm-1 mode is mainly NH2 rock in the 0 1993 American Chemical Society

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PED,but this coordinate makes the smallest contribution to the intensity and in fact opposes thecontributions that determine the final intensity. These examples illustrate the additional physical insights into the origins of observed infrared bands that can be provided by the DDD. The DDD is also preferable to the PED as a guide in assigning observed bands during the refinement of a scaled ab initio force field.

The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11579

Acknowledgment, This research was supported by NSFGrants DMR-9110353 and MCB-9115906. References and Notes (1) Califano, S.VibmtionalSrarss;John Wiley & Sons: London, 1976. (2) Person, W.; Zerbi, G. VibrationalIntensities in Infrared and Roman Spectroscopy;Elsevier: Amsterdam, 1982. ( 3 ) Qian, W.; Krimm, S.To be submitted for publication.