Electronic effect on the rotational energy of a diatomic molecule

Volume 41, number 3

CHEMICAL PHYSICS LEITERS

1 August 1976

ELECTRONK EFFECT ON THE ROTATIONAL ENERGY OF A DIA’FOMIC MOLECULE A. ASGHARIAN of Fhysìcs, National Unìversìry of Iran. Tehran. Iran

Department

Received 15 April 1976

The rotational hamiltonian for a diatomlc molecule has been rederived frorn the total classical hamiltonian. This procedure directly ïntroduces the effect of electronic motion which is ordinardy neglected i? zero-order approximation. Kronig’s rotational hamiItonian is discussed and shown to be an approximation of our fïndings. Our general result is then specialized to 1x states, and the theory tested by calculating the observed fractional discrcpancy between the experimentally determined H35C1 energy leve1 constant Yoz and its predicted value from Dunham’s theory. When all corxections are summed, the results are in good agreement with experiment.

1. Introduction In prevíous papers [ 1,2] , the diatomic molecule bas been treated in some detail in an attempt to remove discrepancies between experimentally determined values of the rotational energy leve1 constant Y& and values as predícted by Dunham’s theory [3S] _ Nevertheless, smal1 discrepancies stil.l remain. In thïs paper, the rotational hamiltonian has been reexamined from a somewhat different point of view to determine the effect of electronïc motion which is normally neglected in zero-order approximations. In the usual approach [6,7], it is assumed that in zero order, the nuclear motion can be separated from the electronic motion. Thus, if L ís the total electronic angular momentum, i.e. the internal electronïc angular momentum plus electronic angular momentum due to the rotation of the molecule, and if J is the total anguIar momentum of the molecule, one normally takes the angular momentum of the nuclei as 1 - L and writes the rotational hamiltonian of the nuclei as

(1) where A, is the moment of inertia of the nuclei. Here, HRo is the leading term in the rotational hamiltonian HR for the entire molecule; one then attempts to correct for the effect of thë electrons via perturbation theory by introducing hígher-order terms.

We show that it is feasible to introduce the effect of the electronic motion in the initial stage. In our procedure, we wrïte the total classical hamiltonian for all partícles in the molecule relative to a fmed coordínate system, and transform to a rotating system, thus introducíngli, the intemal angular momentum of the electrons (i.e. Li would be the total rotational angular momentum of the molecule if it were not rotating). From this, we obtain po and p+,, the momenta conjugated to B and cp, respectiveljj, whích in turn yields our gene& rOtatiOti hamiltonían Operator HR. The relationship between OUYHR and Kronig’s hamiltonian is then investigated, and Kronig’s result is shown to be an approximatíon. We then specialize our general HR to ‘Z states which yietds nR = (9 - @PA

(2a)

A =A,+A,,

(2b)

where 4 is the moment of inertia of the electrons and A ís the total moment of inertia. The externat rotational angular momenturn, J -Li, is íntroduced, and ít is shown that HR of eqs. (2) contains part of the previously derived corrections plus an additional correcting term due to the electronic motion. As a test of the theory, the HR of eqs. (2) was used to compute the fractiond deviation between the experimentally determined [8] H35Cl energy leve1 con513

1 August

CHEMICAL PHYSICS LETTERS

Volume 41, number 3

stant_Yoz and its theoretical vake from Dunham’s theory. The value-of 4 was calculated with Nesbet’s [9] electronic wavefunction for H?sCl. When all previous corrections [ 1,2] are summed with the new correction for electronic motion, the results are in excellent agreement with theory. The significante of these fìnding is briefIy discussed.

1976

where

(W Iz

=CMk[(-qkc0sefSi:su1e)2+Sk]

=ro sin2e +rtcos2e,

(6b)

and 2. Theory

Li, = clc$(Qkkk

The total classical hamiltonian Îor aII particles of a diatomic molecule relative to a fiued coordinate system is

L,

H= f CM-&

+_vk’f 2;) + V&,),

(3)

where Mk is the mass of the kth particle with coordinates Xk,Yk, sk, Rkj is the distance between the kth and ith particles, and V(R,j) is the total potential energy of the system. Let US define the molecular axis as the mean position joining the two nuclei, averagld over al1 effects caused by the rapid orbital motion of the elecfrons about the nuclei; thïs is aïso an axis of symmetry for the system. Consider a new cocrdinate system f, n, 3; rotating with the molecule and related to the fied x,y, z system by the following transformation equations: x=--~sir.p--7jc0s~c0se y = .$cosy2 = nsina

qsi?qcc0se i- ~COSO.

+~COSC~S~, -i-~sincpsin0,

- Sktih-)>

= I.u -3s e + Liv sh 9.

(7a) Ub)

In deriving eqs. (5) and (6), we have used the diagonality of the moment of inertia tensor. In eqs. (7), Lig, Lis, and Lir are the components of the electronic internal angular momentum relative to the 4, q, and 5 axes respectively. The angular momenta pe and p, conjugate to 0 and cp may be readily derived from eq. (5): p* =r$) fL$,

(W

=I,$ö+L,. @bl % The rot+ionai hamiltonian is that part of eq. (5) containing 8 and $. By sohing eqs. (8) for 6 and uj by substitution in eq. (S), we obtain as the exact expression for the rotational hamiltonian HR of the entire molecu!e,

(44

HR =@,2-L~>12~~~@~-LiL~J~=.

(4b)

which we adopt as our quantum mechanical operator. Before proceeding to further reduction of eq. (9), it is instructive to compare these results with those of Kronìg f7]. The rotational hamiltonian of the nuclei, eq. (l), may be written

(4c)

Nere, we take the {-axis along the molecular axis, and its spherical polar angles in the x,,., z system are

~~~ = 4 ~~~~~~ f (p2 sin2e),

(9)

(10)

where J,Lis the reduced mass of the nuclei and p is the internuclear distance. We may defme the total electronie angular moinentum as L, = I&ë f Lig) L, =r,,tj

* L,,

UW

(1 lb)

where 1; and &.. are *Grecomponents oî Le moment of inertia of the electrons. In terms of these, the an-

CHEMICALPHYSICSLETI’ERS

Volume4 1, number 3 gdar momenta p* =pp2&L p, = w2+

of eqs. (8) become

$=A. Wa)

E’

sin20 + L,.

Wb)

8 and (I, from eqs. (10) and (12), and on neglectìng the dependence of LE and L, on 8 and (p, one obtaïns On eliminating

HRo = (po - L,)2/2t(p2 f (pv - L,j2j2pp2

sin2e;

(13)

this is Kronig’s result for the rotational hamiltonian of the nuclei. To obtain the rotational energy of the whole molecule, one must use eq. (9). if on the other hand, one uses eq. (13), the following errors are introduced: (1) By comparing eq. (10) to eq. (S), one notes that the electronic rotational energy (j&ë2 i- ~Ie,$2), has been neglected, which introduces an error of the order of M,&+z, (i.e. the ratio of the electron mass to the nuclear mass). (2) Neglecting the dependence of LE and L, on 6 and (j in deriving eq. (13) from eqs. (12) introduces another error of the same order m,/m,. (3) By comparing eq. (10) to eq. (S), ene fu:ther notes that the Coriolis interaction energy, (Li@ + Lu+), has been neglected, which introduces an error of the order of (?r~,jrn,)~. (4) Disregarding the presence of the electrons in defìníng the reduced mass ~_rof the two nuclei in eq. (10) introduces another error of the order @~,/rn,)~, the so-caìled reduced mass effect. We now return to eq. (9) which by neglecting second order effects may be further simplified to obtain rotational energy levels. Here, 1, in the denominator, is given by eq. (6b). Since 1, refers to rotation about the molecular axis, we may neglect it in comparison to I& Also, due to the cylindrical symmetry of the molecule aöout the < axis, I,., = 4. Finaiiy, by restricting ourselves to moleccles in ‘Z states (i.e., Lis = 0),

eq. (9) reduces to (2a’)

where J2 = pz i pz/sin28, Lf

=L$+L$

and on comparison

with eq. (2),

Gd’)

Here, we may take JE =&+&,

(141

where the use of the reduced mass lu of the nuclei introduces an error of second order which can be neglected. In eq. (2a’), J, the total angular momentum of the molecule, is given by J=K+Li,

(15)

where K is the extemd rotational angular mornentum (i.e. that due to the rotation of the axis of the molecule), and Li is the previously defïned internal angular momentum of the electrons. Substitution of eqs. (14) and (15) into eq. (2a’) yields HR = (K2 c 2K-Li)/2(jq.32 + IC,>_

(16)

Eq. (16) determines the rotational energy of a nonvibrating dîatomic molecule in the ‘Z state. The electronie-rotational term, 2K-Li, and another ten& electron-vibrational. have previously been considered and theír effects on Yo determined [1,2] . in Dunham% simple theory, the equivalent expression for eq. (16) is K2/2w2_ Thus, we need only consider the effect on Yo of the new term let in the denominator of eq. (16). According to Dunham’s theory, the rotational energy leve1 constant is given by the wel1 known expression D= Yo = 4B$Ue_

(17)

The electronic moment of inertia, I&, depends both dïrectly and indirectly on the internuclear distance p_ The direct dependence results from the tendency of the electrons to foliow the nuclei; we may approximate this wïth suffìcient accuracy by the linear relation I& =I()(I f6c),

HR = (J2 - L$$,

1 August 1975

(W

where 6 is a constant and E is the relative change in nuclear distance, defined as

W’)

E = (P - P&J.

h j2b,

Here pn is the equiiibrium dïstance, and’lo the electronie moment of inertia at equilibrium distance. The indirect dependence of le on p fo!lows from the fact 515

Volun?c 41, number

3

CHEMICAL

PHYSICS

that when the molecule is rotating, the electrons are subjected to a centrifugal force. Therefore, tbe electronk moment of inertia increases as the rotational quantum number increnses; one can show that this effect is given by =&(l f e)% (19) ‘e Then, from cqs. (18) and (IS), the total dependence of l& on e to this order is f& =$(l

+Se)(í

+ #.

(20)

T!le substitution of eq. (20) ìhto eq. 426) and the calculation of the resultíint fractional change in Yu from Dunham’s value, eq. (17), yields AD/D = 2BeI*S )

(211

where 10 and 6 are to be obtained

3. Application

theoretically.

to Hz5C1 molecule

Tlie values of10 and 6 in eq. (21) are computed from Nesbet’s wavefunction [9] . Our .$, q,< axes correspond to his x,y, z axes; on noting that (_$) = , we can write

(22j Since (rf> and are given by Nesbet relative to the chlorine nucleus, the value of Iti computed from eq. (22) must be transformed to the C.m. of the system. These transformed values of I& at p equal to 2.3085, 2.4085 (= pO) and 2.5085 au are equal to 23.958, 24.327 (= 10) and 24.701 au, respectively. From these results, via eq. (18), we find 6 = 0.37. Finally, from eq. (21), we obtain AD/D = 0.87 x 10-3,

(23)

as the fractional shift due to the. term I& in eq. (16). When the previcusly mentioned corrections (electronic-vibrat‘anal, electronic-rotational) amounttig to -0.29 X iO- 3 [1,2] , are added to eq. (23), one obtains m1.D = 0.58 X 10-3, in very close agreement

516

(24) with the observed fractional

LETTERS

1 August

1976

differente of (0.649 f 0.065) X 90e3 [8) between the experimental and Dunham’s theoretical value.

4. Conclusions It has been shown that the effect of electronic motion may be directly introduced into the rotational hamiltonian in a simple marmer. At first glance, the contribution to m/D for H35Cl from the electronic moment of inertia may seem surprisingly large. lt is seen, however, that the relatively large proportionality constant 6 enters Iinearly in eq. (21). This sterns from eq. (18) which reflects the direct dependence of IC on the internuclear distance p, and may be interpreted as being due to the electrons redistributing themselves to mininüze tbe electronic energy of the molecule. On the other hand, the contribution to AD/D from the centrifugal force, eq. (lg), vanishes because the moment of inertia of the nuclei varies in exactly the same maner with nuclear distance, thus cancelling out this effect. Additional calculations and data are required lor a further test of the tlreory.

Acknowledgement The author is grateful to dr. Jeremiah N. Silverman for several helpful discussions and for reading the manuscript.

References [i J R.M. Herman and A. Asgharian, J. Mol. Spectry. 19

(1966) 305. 12) R.M. Herman and A. Asghanan, [3] [4] [S]

[6] [ 71 [SI [9]

J. Chcm. Phys. 45 (1966) 2433.. J.L. Dunhnm, Phys. Rev. 41 (1932) 721. J.H. van Vlack, J. Chem. Phys. 4 (1936) 327. 1. Sandeman, Proc. Roy- Sec. Edinburgh 60 (1940) 210. B. Rosenblum et al., Phys. Rev. 109 (1958) 400. R. de L. Kranig, Band spectra and molecular structure (Macmillan, New York, 1930). D.H. Rank et al., J. Opt. Sec. Am. 52 (1962) 1. R.K. Nesbet, J. Chem. Phys. 41 (1964) 100.