Polarizability gradients from frost model wavefunctions

Vutun~c 97. nunlber 6

CHEMICAL

PHYSICS

_A first ~ppro~im.rIioii to the mean molecular pularI/.Mrt~ Q is readily ckuldted from Frost model waveitiw00sand.furs31u131cd hycfrocxbons, the results 11 ] arc in guud .qreernerll with cxperiinent. The cal~~I~J~I~IIS .rrr‘ also very economical: for a molecule \\ lth 31 electrons the wavefunction [2] is 3pprosimated hy Jll .iiitis> iniiietrired product of II floating spherx.11 ~JUSSIJII orbds (FSGOs). This econo~ny permits

Is

to

~~~%ihk

f’kli

ill

~nihxwles cuntJiiiing

cOllV~llIiOllJ~

t IJItlL’L‘

i:OCk

more atoms than

apprOXhS

[;]

such

aS

cou-

SCil~llll%.

11 is iiitcrNi~lg thercforc thdt results presented here shin\ 111~1these .Aiiittrdly crude wavefunctions also >~i~?l~i good csrim.ues of gradients of the incan polariLatxlir! . Tlirrc is considerable current interest [3,5] in 111~s~nwiemlJr prupcrtics as they describe li,o. the intc~ul~>

of

or perhaps a HCH ansle, are approximately equal for all hydrocarbons and may therefore be transferred among chemically similar molecules. However in practice it is often necessary to assume that one or more of these derivatives are in fact transferable in order to deduce the remaining derivatives [4.6-S 1. Ab initio estimates may provide a useful guide to such ap-

length

I _ Introduction

~pphc~tIl~l1

3 June 1983

LETTERS

111~

isotropic

vibrational

Kanlan

spectrum.

an even inoder.rtely accurate method ofc.kul.rting these grJdicnts is useful for two reasons. 1-rrst it 11.1s been suggsted [-F,6--S] that the esperi-

prosimations. difficulties

Second,

encountered

apart from the experimental when measuring absolute

Raman intensities. considerable difficulties are encountered in estracting the a@R from the experimental data. The most annoying and persistent of these arises [4.9, IO] from the nature of the non-linear relationship between I,,, and the a&R. At best this leads to an uncertainty in the sign of &&3R but more often it results in a multiplicity ofvalues for the ar&R all of which represent the data equally well. The results presented here provide information on both the sign and the transferability of aajaR_

in this context

rmwtally d~spi~crt~ne~~t

d~rernlined

qumrities

coordinate

where R is a such .ts a C-H or C-C bond

2. Theory

&x/W,

The Frost model [2] approximates

0 009-2614/S3/0000-0000/S

the wavefunc-

03.00 0 1983 North-Holland

Volume 97, number6

CHEMICALPHYSICS LETTERS

tion for a 2n electron system by the antisymmetrized product of n doubly occupied floating spherical gausSian orbitals (FSGOs). The total energy is minimized by optimizing the position and size (exponent) of each gaussian orbital. Given such a wavefunction, it has been shown [l] that a first approximation to the mean polarizability LYis given by

where & is the exponent of gaussian i. The spherically symmetrical character of FSGOs permits only the mean polarizability to be determined in this manner [I] _Estimates of the polarizability derivatives ih-&lR are obtained by calculating (Yat the equilibrium geometry and at a number of geometries corresponding to finite distortions along R. The results are used to calculate the coefficients of a polynomial expansion of Q in R; the linear coefficient is taken as a&R_ Experimental estimates of &x/aR were taken from the !iterature. They have been determined as follows [4-6] _The experimental polarized (IVV) and depolarized (IVR) intensities are combined to yield the isotropic (Ii,,) and anisotropic (l~niso) Raman intensities:

&so

= &,H = 3(ar/aQ)‘K

_

As is indicated in these equations, for the vibration corresponding to the normal coordinate Q these intensities are proportional respectively to (aa/aQ)3 and (&y/dQ)‘, where (Yis the trace and y the anisotropy of the polar&ability tensora. The constant of proportionality K depends, among other things, on the frequency and temperature_ The derivative &r/i3Q, with respect to the ith normal coordinate is related to the derivative wi?h respect to the jth displacement coordinate &@Rj by means of the L matrix,

aOjaQi = CLji aajaRj_ i The L matrix [ 1 l] relates the internal and normal coordinates R = LQ, and is obtained from a normalcoordinate analysis_ If the &x/laR and L are known, the do/dQ and hence Ii,, are readily calculated_ How-

3 Juue 1963

ever, the inverse calculation is made ambiguous by the non-linear relationship Of Iiso and &r/dQ. For a molecule with II totally symmetric modes there will be 2” solutions for the aalaR_ Suffciently precise intensity data obtained from the spectra of isotopomers can be used to reduce the number of acceptable solutions and the relative signs for the various aafaR can often be established in this way [9.10]. At best the mnnber of solutions is then reduced to 2 which are related by a simultaneous change of sign for all the acr/aR. However, the absolute sign of these derivatives will remain ambiguous_

3_ calculations Calculations were performed by miniiizing the energy using a generalized reduced gradient program GRG2 [ 12]_ Derivatives were evaluated by calculating LYat several different geometries for each internal coordinate_ As only the totally symmetric part of any distortion gives rise to a non-zero first-derivative of cr. it was often convenient to use geometries in which all symmetry equivalent internal coordinates were displaced equally. Generally, angle distortions of O-5” arc, and bond distortions of 0.001-0.050 au enabled separation of first derivatives from higher-order terms. The values of the derivatives are sensitive to the energy convergence criterion. A value of lU?/EI < 10m6 was found to give results which could not be significantly improved upon. The choice of geometry is not unambiguous_ It would be consistent to use the enemy-minimized geometry resulting from the FSGO wavefunction and then to distort it to find the derivatives. However. apart from consistency, there is no good reason for doing so, since the Emin geometry determined from FSGO wavefunctions may depart significantly from the experimental geomerry. Also, since it is polarizability rather than energy which is of interest, there is no reason to believe that rhe FSGO geometry will give results closer to experiment. For those cases where both geometries were tried, the experimental geometry gave better agreement for polarizability derivatives, and it is therefore used throughout this work. Where available the equilibrium geometry was preferred over the ground vibrational state average geometry; data were obtained from the following 495

references: cyclo-e~li6

3 June 1983

CHEMICAL PHYSICS LETTERS

Volume 97. nurnbcr 6 CH, [IS], C2H6 [Id], [ 161.

C3H8 1151 and

4. Results and discussion Calculated values of &x/%2 are displayed in table l_ The number in parentheses after each entry indicates the uncerrdinty in iWaR due to the combined influcnce of higher-order derivatives and the uncertainty in the equilibrium geometry. The latter is particularly important ior the symmetric deformation mode of eth.me: for 0(HCH) = 1073 lo (the assumed equilibrium vdue) acu/ao,,,, = -0.2-l A3/radian while for 0 (1 ICIi) = 107.7V (the ground-state average value) this derivative becomes -0.1 !I AS/radian. Table 1 also presents a representative set of esperimental polxiL:ability derivatives_ These are rarely more precise than ~10% due to uncertainties

in intensity

mexurcments but most authors will not attach an unsertsinty cstirllzlte to the aalaRbecause there are sensirivr functions of the L matris whose accuracy is dif-

ficult to assess. Exceptions to this are the results of Montero and co-workers (last column) and for these both sources of uncertainty have been considered. For a given molecule the relative signs of the derivatives

have been detemlined experimentally; the absolute signs have been established by assuming that in all cases &x/arCH is positive [Z]. For methane this sign has been confirmed by ab initio calculation [24] _ The agreement between calculated and experimental derivatives must be considered very satisfactory, both as regards sign and magnitude, in view of the spread in the experimental determinations. For derivatives involving stretching coordinates the calculated derivatives usually agree with the results of Montero and coworkers within one or two times the estimated uncertainty_ The situation is not quite so good for derivatives involving angle deformations but these derivatives are usually small and therefore not well determined esperimentally_ Furthermore we calculate significant second derivatives for these deformations which suggests that a harmonic interpretation of the experimental data may be inadequate_

1‘JblC I

CJILUIJIC~ md --

\lOkCUk

c\permtertt.d

--R -I)

pol.mzAlity

----_..--

derivatives aalaR

_-_---_____

dc.

1’)

C\F.

b,

Cl 14

‘Cl I

1.17 (1)

1.0s c,

C‘Zllb

‘Cl1 ‘CC %I!,

1.18 (2) 0.91 (2) -0.2-l (5)

1.08 d, 0.92 d) <0.3 d, >-0.3

c‘,iLz

‘CIIlC112)

1.21 (2)

1.20 II-)

‘CC UCII.(C112)

023s (4)

0.80 9

LycI+C31&j -_--_-.--_.-~~_-__-

-0.033 (5)

OCCC

0.066

ret 1 ‘CC ~c’Il>

1.13 (2) 1.11 (2) 0.21 (2)

_ _

(5)

1.135 1.70 e) 1.31 c) -0.12 e)

::;;

::

-0.0-U g) -0.09 ‘1) 0.078

g)

0.1s h)

1.09 f) 1.0s t-) -0.19 f)

(55) i,

1.092 (55) j) 1.096 (55) j) -0.14 j)

1.10 (15)j) 1.10 (14)

~0.2 j) >-0.2 co.2 j) >-0.2

J, “Cl1 1~A GuuIt~ne0us iucrass of .dl HCH .mgles of one CH 3 group: oC~I, is an increase of the HCH angle of one CH2 group. t’)Un,t:- _A~/_%= 1.1 1161 x lO*O C’ rn’ J-‘/m or X3/rddi.m I 1 -11261 X 104’ C2 m2 J-t/rddian. 2) Ref. [171. d)Kef. (181. ‘)Ref. [7]. “Ref. [191_ ‘I) For ;1 C-Cf12 -C unit in cyclohexme [6]. P) I-clr A C-CfIz-C unit in polyethgtrne [20]. i, RCI-_ pi]. j) Ref. 1221. -196

Volume 97. number 6

CHEMICAL PHYSICS LElTERS

It should be noted that in all cases our calculations confinn

the accepted

sign of the polarizability

deriva-

tives_ Our derivatives may also be compared with the results of other calculations. For methane John et al. [24] obtain &+rCH = 1.154 &&3/A and van Hemert and Blom [25] report 1.050 a3/A. Both calculations use a finite-field perturbation SCF technique_ Svendsen and Tangaa [26] employ a variation-perturbation CNDO approach for their calculations on small hydrocarbons. They report as follows: for methane &//ar,, = 0.96, for ethane &@rCH = 1.09, aa/ar,, = 0.69; all in a3/& It is clear from this comparison and the results in table 1 that our technique for calculating aa/aR is as good as, and usually superior to, those previously reported. The calculated polarizability derivatives for stretching C-H bonds show good transferability. There is a slight, but real, trend in the series propane > ethane > methane > cyclopropane which parallels the equibrium C-H bond lengths used in the calculations: 1.0960, 1.0877, 1.0858 and 1.0780 a, respectively_ This suggests the variation in a&/arCH is due to a variation in the equilibrium C-H bond length rather than the variation in chemical environment. This is confirmed by calculations which show that &@rCH for the CH, group of propane is l-18(2) as/A if the equilibrium C-H bond length is taken to be 1.088 A. The values of aa/arcc for ethane and propane are in good agreement but differ from that calculated for cyclopropane which is intermediate between that found experimentally for C-C and C=C bonds [IS] _This is in agreement with the intermediate nature of the C-C bond of cyclopropane, as evidenced for instance by the bond length. The polarizability derivatives for deformation coordinates are very dependent on the choice of equilibrium geometry and as such may be expected to be much more variable from one molecule to another_ However. a comparison of &/aBc+ (X = H, C) with the equilibrium XCX angle permits the following generalization: for saturated hydrocarbons i3a/aOcx, is negative (positive) if B,(XCX) is less (greater) than the tetrahedral angle (109.47”). Although this generalization must fail for a (hypothetical) molecule containing a C-CH,-C group with exact tetrahedral geometry, it may prove to be a useful guide. It is in ac-

cord with the available experimental

3 June 1983

evidence:

for

cyclohexane eHCH = 107.50 1271, amjaecH, is negative [~];SCCC = 11 l-40”, positive; for ethane 6~~11 = 107.31° 1141, negative (table 1); and for 2,2-dimethylpropane ~HCH = 109_O” [28], negative [ 29]_

If p-type gaussians are used for the n electrons, Frost model wavefunctions also yield good estimates of the polarizability of unsatured molecules [30] _ This suggests that the approach to polarizability derivatives presented here may be extended to these molecules and such calculations are in progress_

Acknowledgement MPB gratefully acknowledges receipt of financial support from the Australian Research Grants Scheme; RSW thanks SAENET for the use ofcomputing facilities

References A-T. Amos and J-A. Yoffe, Chtm. Ph>s. Letters 31 (1975) 57.

-4-A. Frost, J. Chem. Phys. -$7 (1967) 3707.37 14. J-E. Greedy, G-B. B.~rkay and N.S. Hush. Chcm. Phys.

[41

161 171 181 191 IlO1 1111 1121 1131

23 (1977) 9; P-W_ Lzmghoff, Xl. Karphts and R.P. Hurst. 1. Chem. Phys. 4-l (1966) 505; H-D. Cohen and C.C_J. Roothrtan. J. Chem. Plt)s. 43 (1965) S34_ 3%.Gussoni, in: Vibrtrtiotxtl intensities in infrared md Raman spectroscopy. eds. W.P. Person and G. Zrrbi (Elsevier. Amstemm. 1962) ch. 11; 31. Gussoni, in: Adwnces in infrared and Ramsn spcctroscopy, Vol. 6. eds. R.J.H. Cldrk and R.E. Hester (ileyder London, 1979) ch. 2. H.W. SchrGtter and H.\V. Kl?khner, in: Ratnan spectroscopy of gases and liquids. cd. A. Wcbcr (Springer, Berlin 1979) dl_ 4_ R-G_ Snyder. J_ Mol. Sprctry_ 36 (1970) 104. R.G. Snyder. Vij.ms Parishad Anuundha~ P3tril.J I-l (1971) 139. S. Montero and G. de1 Rio. Jlof. Ph>s. 31 (1976) 357. V.T. Alekunyan and S.Kh. Samvekyan. J. Mol. Spectry. 45 (1973) 79. Y.P. Bogtard snd R. Haines, J. Mol. Spcctry. 90 (1951) 267. S. Califano. VtbrationAl st.aes (Wiley. Nrxx York. 1976). B.A. Jfurtagh and X.4. Stunders. Math. Progr. 14 (1976) 41. D-L. Gmy .tnd A-G. Robiette, \Iol. Ph>s. 37 (1979) 1901.

497

Volume

97. number

6

CHEMICAL

J.L. Duncan, D.C. McLean and -4-1. Bruce. J. Mol. Struct. 71 (1979) 361. D.R. Lide. J. Chem. Phys. 33 (1960) 1514. R.J. Butcher and W-J. Jones, J. Mol. Spectry. 17 (1973) 64. H.W. Schrcjtter and H-J. Bernstein. J. Mol. Spectry. 12 (196-l) 1; 11.J. Ber ‘stein. 1. Mol. Spectry. 22 (1967) 122. 1‘. Yoshino and HJ. Bernstein. Spectrochim. Acta 14 (1959) 117. N.I. Prokofev.~ and LX. Sverdlov. Opt. Spectry. 16 (1961) 20-I. S. Abbdte. 11. Gussoni .md C. Zerbi, J. Chem. Phls. 73 (I 9SW -1680. 1). Bcrmejo, R. Escnbdno and J.M. 0rz.1. J. Mol. Spectry. 65 (1977) 3-15. S Montero .md co-\\urhcrs. priv.ite communiwtion.

PHYSICS LETTERS

3 June 1983

1331 R-P_ Betl, Trans. Faraday Sot. 38 (1942) 422; T_ Yoshino 3nd H_J_ Bernstein, J. Mol. Specq. 2 (1958) 213. [24] LG. John, G.B. Bacskay and N.S. Hush, Chem. Phys. 38 (1979) 319. [25] XC. van Hemert and C.E. Blom. Mol. Phys. 43 (1981) 229. [ 261 E-N. Svendsen and J. Targaa, J. Raman Spectry. 7 (1978) 268. [27] N.S. Chiu, J.D. Embank and L. Schifer. 1. Mol. Stiuct. 86 (1982) 397. [ZS] B. Beqley, D.P. Brown and J-J. blonashan, J. Mol. Struct. 4 (1969) 233; W. Zeil, J. Haase and 51. Dakkouri. 2. Naturforsch. 22a (1967) 1641. 1291 K.A. Taylor and L.A. Woodward. Proc. Roy. Sot. A264 (1961) 558. [30] J-A_ Yoffe. Chem. Phys. Letters 54 (1978) 562.