Constraints and redundancies for vibrations of linear and planar molecules

NOTES

463

REFERENCES 1. J. F. SCOTT AND K. NARAHARI RAO, J. Mol. Spectry. 16, 15 (1965). 2. J. F. SCOTT AND K. NARAHARI RAO, this issue [J. Mol. Spectry. 20, 438 (1966)]. 3. F. W. LOOMIS AND R. W. WOOD, Phys. Rev. 32, 223 (1928). 4. G. HERZBERG, “Infrared and Raman Spectra of Polyatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1945. Laboratory of Molecular Spectroscopy Infrared Studies Department of Physics The Ohio State C’niversity Columbus, Ohio Received May 1, 1966 t NASA

and

J. F. Scott

AND K.

NARAHARI RAO

Trainee.

Constraints and Redundancies for Vibrations Linear and Planar Molecules*

of

For linear or planar molecules, the internal displacement coordinates associated with certain types of intramolecular force fields are vibrationally incomplete. A linear constraint inequality involving such internal coordinates is automatically satisfied if the nonlinear redundancy appropriate to an augmented complete coordinate set is taken into account. An example is discussed. For molecules with linear or planar equilibrium configurations, Brown and St,einer (1) have suggested that the neglect of linear constraint inequalities involving the internal displacement coordinates associated with certain types of force fields is responsible for the false conclusion that these model force fields predict zero frequencies for out-of-line or out-of-plane bending vibrations. The object of the present note is to indicate that, in such cases, the coordinate systems associated with central, Urey-Bradley, or generalized valence force fields are incomplete, and that, in terms of an augmented complete set of coordinates, the linear constraint inequality is automatically implied by the nonlinear redundancy appropriate to the complete coordinate set. This realization, together with the fact that, redundancies have been extensively discussed in the literature, eliminates the necessit,y of considering constraints separately. A single example, such as the BF, molecule, illustrates t,he point. The generalized valence force field model specifies force constants associated with changes in the three bond lengths (ar,) and the three interbond angles (Sol,). These six coordinates are sufficient to define all planar vibrationally distorted configurations available to the five degrees of planar vibrational freedom, for which

* This work, performed under an Air Force contract with Wentworth Institute, was done at Air Force Cambridge Research Laboratories, L. G. Hanscom Field, Bedford, Massachusetts.

564

NOTES

Ht)\vever, the single degree of nonplanar vibrat,ional freedom requires for its specification a coordinate which is antisymmetric with respect to the equilibrium plane. Ko combination of the 6ri and &Y, satisfies this requirement, and these coordinates, which are subject to the linear constraint inequality 6cu, + 6az + 6cxa < 0

I”)

for nonplanar motions, do nclt define a tmique nonplanar configuration, since they can not distinguish between the pair of nonplanar configurations related by reflection in the equilil)rium plane. An additional coordinate is needed which is antisymmetric. (Consider an arbitrarily distorted vibrational configuration specified by the bond lengths which is the cc~mplement of angles toll , QZ, as)’ and the angle+, (1.2 , i’:: , ra), the interbond the angle between the bond r, and the normal through the central atom to the equilihrirtm plane (which may be taken without loss of generality as containing I’~and ra/. If the angles made by the projection of rl in the equilibrium plane with rz and 1’8, respectively, are denoted hy pa and py , the exact redundancy condition pertaining to the arbitrarily distorted mol~ct~le is given b) 6crl + sp, + b& = 0. From

simple

vector

geometry

it is easily

shown

j3.’ = cos-

that

(cos (Y?/CflR@)

and /33 = cos-’ (cos oLB/COY#I). ICspressions for 60, and condition

which are Taylor series expansions through second-order terms && from Eys. (4). Suhstitutitrn of these into Eq. (3) yields2 R = 6a, + 6a! + Ga:j -

+

Cl‘&,

+ S&S + ‘;rC

F&,2

+ $

(5)

it is evident that. the sect*nd-order (2) which applies to the inc~~mtrlete a function the Taylor

of the seven depetttlettt series expansiott

F&Y,2 + F&?

1

i

+ 2x F,,&,&Y, t#i + cross _~_ 101,is the interhond 2 The form

can he ~~htainetl the redrtndanc~

’ 2 (cot ale” + cot cyo”)&$* = 0.

Since both equilibrium angles CY?”and olg” exceed r/2, redundancy (5) implies the linear constraint inequality coordinate set (Er, , 6rn , 0:~ , &xl , 601~ , Gas). The vibrational potent,ial energy (P.E.), regarded as by variables (Srt , fSl’2,8r3 ,&YL ,601~~ 6~ , 64) is generated b17 = C”fJVJ

l-11))

terms

+ 2x

Fr,br$a,

in &a+, Sol@] + higher

terms ._

angle

for r? and

r.2 ,

is tpriginally obtained because the derivatives are double-valued. The equilibrium planarity loner sign combination.

(Y:!for r’a and rI , and w for r1 and I’:

of the inverse circular functions in Eqs. (4) condititm. Eq. (l), requires rejection of the

465

XOTEG

The vibrational P.E. is not completely defined by Eq. (6), it being required also that the P.E. be subject to the redundancy (5) and have a minimum at equilibrium. The necessary condition for this restricted minimum (2) is that) linear terms vanish in the expression SP = 6V + KR in which K is a Lagrangian

multiplier.

Therefore

f* = -K, The P.E.

consist.ent

jr = 0,

with the redundancy

f+ = 0.

(5) and which has a minimum

+ 2C

i#i

at) equilibrium

Fw, 6r,6rj + 2x Faa, 6a,bj i#i

+ 2x F,,dr&j i#j

+ 2x

+ cross terms in

6r&$,

z

is

(7)

F,aSrL&al

&x6+]

+

higher

terms

The harmonic P.E. is found by ignoring other than quadratic contributions in the expression which results from Eq. (7) when ST is re-expressed in terms of a complete nonredundant set of independent coordinates. The purely planar coordinates 6sr , 6s2, 6.~ , up, , & and the purely nonplanar coordinate 84 constitute such a set. The required coordinate transformation to which Eq. (7) is to be subjected is

and the harmonic SF

(harmonic,

P.E.

is thereby

planar)

= ?a!x

1

obtained

as the sum of purely planar

F,6s,2 + c

Sv (harmonic,

z

and nonplanar

parts:

F&32 + 2x F,,,6Ss6Sj i#j

nonplanar)

= ?$ (F, + 3-r/2~)642.

(9b)

The application of the transformation (8) to Eq. (7) adds no new types of quadratic contributions since the transformed contributions from S42 affect only higher order parts of the P.E. Quadratic cross-terms involving 84 and any planar coordinate are absent from Eqs. (9) since they vanish by symmetry. It is noteworthy that (1) The potential constants F, , F, , etc. which determine the purely planar frequencies are the generalized valence force const,ants originally defined in Eq. (6) with respect to arbitrary vibrational deformations which were not necessarily planar.

NOTES

4X

(2~ The generalized valence force field model requires F1 to he zero. Therefore, ljrovided K is positive, this model predicts a harmonic nonplanar frequency which is entirely determined by the magnitude of the intramolecular tensilm K; whereas t,he planar harmonic frequencies are unaffected by the intramolecular tension. (3) The constraint inequality (2) is automatically satisfied if the redundancy (5) appr~~ priate to a complete coordinate set is taken into account). (4) Analogous considerations obtain for the vibrations of any planar moleculr described by any force field (e.g. generalized valence, central, or Urey-Bradley) defined by force constants that, do not refer directly to a coordinate which is antisymmetric with respect to reflection in the equilibrium plane. A similar situat.itm is encountered in the applicatif>n oi f hr central force field to the out-of-line vibrations of linear molecules. REFERENCES 1. W. B. BROWN 2. B. CRAWFORD Wentworth

AND E. STEINER, J. Mol. Spectry. 10, 348 (1963). AND J. OVEREND, J. Mol. Spectry. 12,307 (1964).

Institute,

Hoslon, Massachusetts Received June 10, 1966

II. F;. FREEMAN