Molecular Orbital Theory of Sulfur and Selenium RadicalsInan Chen and T. P. Das Citation: The Journal of Chemical Physics 45, 3526 (1966); doi: 10.1063/1.1727368 View online: http://dx.doi.org/10.1063/1.1727368 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/45/10?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 45, NUMBER 10 15 NOVEMBER 1966

Molecular Orbital Theory of Sulfur and Selenium Radicals*

INAN CHEN

Fundamental Research Laboratory, Xerox Corporation, Rochester, New York

AND

T. P. DAS

University of California, Ri'Derside, California (Received 27 June 1966)

The electronic structures of various sulfur and selenium radicals were studied by semi empirical molecular orbital theory in the form of a parameter theory. The ESR data (g values) were used to determine the actual value of the parameter ex, which is the coefficient of the s orbital i~ the ~ondin.g h:;,brid of the SeSe) atom in the radical. The physical significance of a so determined was studIed by mvesttgatmg some quantum chemical quantities such as valence state energies E. and bond strengths B as functions ?f a. As a result, a criterion for the determination of the parameter value was deduced, namely, the selectlOn of that value of a which maximizes B21 E.I. ., . .

From theoretical considerations and experimental data in the literature, the relative SIgnS of the prmcipal values of the hyperfine-structure tensor of selenium were determined, and the magnitudes of the isotropic and anisotropic components were evaluated.

I. INTRODUCTION

MOLECULAR orbital theory combined with optical and electron spin resonance (ESR) spectra has

been very successful in understanding the electronic structure of impuritiesl and defects2 in solids.

Much work has been published on the ESR of sulfur and selenium radicals (see Table I). Gardner and FraenkeP reported on the ESR of liquid sulfur which consists of long-chain biradicals. ESR in solid sulfur (CS2-insoluble, supersublimation sulfur) was observed by Pinkus and Piette.4 They also attributed the signal to the biradical structure of sulfur although the g value was quite different from that of liquid sulfur. In amor­phous and hexagonal crystalline selenium, which con­sists of helical chains of atoms, a resonanCe line with a g value very close to that of free electrons has been observed.' In addition to this "sharp" line, in the preceding paper Sampath6 also discussed a "broad" line which extended over several kilogauss, and had a larger g value. The magnitude of g varied from 2.3 to 2.8 depending on the sample preparation. He has in­terpreted6 a set of experiments on the broad and sharp resonances by assuming that the signals stem from the unpaired but interacting electrons at the ends of chains,

• Part of this work was initiated when T. P. Das was a SU=er consultant of the Fundamental Research Laboratory, Xerox Corporation, Rochester, N.Y.

1 S. Sugano and R. G. Shulman, Phys. Rev. 130, 517 (1963); I. Chen, C. Kikuchi, and H. Watanabe, J. Chern. Phys. 42, 186 (1965).

2 G. D. Watkins and J. W. Corbett, Phys. Rev. 134, A1359 (1964).

aD. M. Gardner and G. K. Fraenkel, J. Am. Chern. Soc. 78, 3279 (1956).

4 A. G. Pinkus and L. H. Piette, J. Phys. Chern. 63,2086 (1959). 6 P. I. Sampath and R. C. Keezer, Bull. Am. Phys. Soc. 10,

613 (1965); G. B. Abdullaev, N. 1. Ibragimov, Sh. V. Mamedov, T. Ch. Dzhuvarly, and G. M. Aliev, Dokl. Akad. Nauk Azerb. SSSR. 20, No. 10, 13 (1964).

• P. 1. Sampath, J. Chern. Phys. 45, 3519 (1966), preceding paper.

and concluded that the existing experimental evidence is not inconsistent with this model.

The radicals produced by uv irradiation of organic sulfur and selenium compounds were investigated by Windle et al.7 in both liquid and solid states. Clark et al. and others8 studied the radicals produced in sodium dithionite and thiosulphate crystals by x and gamma irradiations and identified them as S02-' Cook et al.9 irradiated a single crystal of sodium hydrogen selenite and observed an ESR signal which they at­tributed to the Se02- radicals.

Theoretical studies of the electronic structures of sulfur and selenium chains have been published by Gaspar 10 Olechna and Knox,l1 and Sandrock and Treusch.12 None of these authors has considered the structure of the chain ends which constitutes a deviation from the periodic structure.

In this paper we present a molecular orbital theory of sulfur and selenium chain ends and radicals. The calculation is carried out in the form of a parameter theory. The coefficient a of the s orbital in the bondi~g hybrid is chosen as the parameter. The spectroscop~c splitting factor g is computed from the molecular orbI­tals, and, by comparison with ~e obser:red value, the actual value of the parameter IS deterrnmed. In many cases, the observed g values can be reproduced by a parameter value which can be justified by considering the bond strength and the valence state energy as functions of the same parameter.

7 J. J. Windle, A. K. Wiersema, and A. L. Tappel, J. Chern. Phys. 41,1996 (1964).

8 H. C. Clark, A. Horsfield, and M. C. R. Symons, J. Chern. Soc. 1961, 7; R. L. Eager and D. S. Mahadevappa, Can. J. Chern. 41,2106 (1963).

8 R. J. Cook, J. R. Rowlands, and D. H. Whiffen, Mol. Phys. 8, 195 (1964).

10 R. Gaspar, Acta Phys. Acad. Sci. Hung. 7, 289 (1957). 11 D. Olechna and R. S. Knox, Phys. Rev. 140, A986 (1965). 12 R. Sandrock and J. Treusch, Solid State Commun. 3, 361

(1965) . 3526

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SULFUR AND SELENIUM RADICALS 3527

TABLE I. Observed and calculated g values of sulfur and selenium radicals. [The values of the parameter (a) used to obtain the calculated g values are also included.]

Group Radical Ref. No.

I S, liquid 3 S, solid 4

Se, solid 5

Se, solid 6

SClL 7 S(CH2)ClL 7 S(CH2)CsH6 7 S (CH2) CH (NH2) COOH 7

Se (CH2) CGlL 7 Se (CH2) 17CHa 7 Se (CH2) nClL 7

II S02- 8 Se02- 9

We shall divide the radicals in Table I into two groups: singly bonded radicals and doubly bonded radi­cals. In the first group, the atom containing the un­paired electron (Atom A) is bonded to one neighbor (Atom B) and, in the second group, Atom A is bonded to two neighbors (B-A-B).

The sand p atomic orbitals of Atom A are hybridized into bonding and lone-pair hybrids as shown in Fig. 1.

In Secs. II and III, respectively, the molecular orbi­tals and the g values of singly bonded and doubly bonded radicals are calculated. The results are dis-

FIG. 1. Hybrid orbit­als and coordinate sys­temsof(a) singly bonded radicals A-B; (b) dou­bly bonded radicals B­A-B.

z

\-____ x

(a)

z

y

\-___ .. x

(b)

gcalc(a) gob.

sp-Hybrid model ..--Electron model

2.024 2.028(0.42) 2.014(0.45) 2.0044

2.0039 (sharp) 2.0998(0.42) 2.044(0.45) 2.3"-'2.8 (broad)

2m61 2.024 2.029 (0 .42) 2.014(0.45) 2.024 2.028

2.102} 2.094 2.100(0.42) 2.045 (0 .45) 2.099

2.005 2.0078(0.38) 2.0053(0.42) 2.0132 2.0121(0.36) 2.0031 (0.41)

cussed in Sec. IV. Some numerical results are given in the Appendix. Using the principle formulated in this paper, we present calculations of g values for some possible paramagnetic centers in amorphous selenium in the following paper.

II. SINGLY BONDED RADICALS

In the case of singly bonded radicals, A-B, the hybridization is such that there are two bonding and two lone-pair hybrids. One of the bonding hybrids, ,PI, is occupied by the unpaired electron; another one, 1/12, forms a bond with Atom B. Each one of the lone-pair hybrids contains a pair of electrons.

In order to keep the number of adjustable parameters at a minimum so that we can evaluate the parameters without requiring too much experimental data, we make an assumption on the hybridization. In general, the bond angles are different, but not strikingly so, from those of the tetrahedral hybrids in which all four bonds have exactly the same s character; thus we assume the s characters for the two bonding hybrids equal to each other but different from the s characters for the lone pairs. The S characters for the latter are also assumed equal.

The parameter 0: can be defined through the expres­sions of the bonding hybrids. By choosing a coordinate system as shown in Fig. l(a), we can write these hybrids as

unpaired: 1/11 = O:SA+ (1-0:2)iZA,

bonding: 1/12= 0: SA +[ (1- 20:2) j (1-0:2) ]lX A

-0:2j(1-0:2)lZA,

1/131 0: 1/1. =(!-0:2)!SA-[2(1_0:2)]!XA

lone pair:

lone pair:

1 0:(1-20:2)! ±-YA- ZA

V2 [2(1-0:2)]i' (1)

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352R 1. CHEN AND T. P. DAS

where the coefficients are related by the orthonormali­zation condition and symmetry. The bonding hybrid of Atom B is denoted by

(2)

where {3 is determined by the bonding scheme of Atom B, and PB is a linear combination of XB, YB, ZB. For example, in the case of an analogous organic radical, B is a carbon atom bonded tetrahedrally to four neigh­bors, and so (3=O.5, corresponding to that of four equivalent Sp8 hybrids. In the case of chain ends, B is the atom adjacent to the end of the chain, and is

bonded to two neighbors with an angle between the bonds. Assuming the angle is the same as the regular chain bond angle, 8=108(S) and 8=105.5(Se),1O we obtain {3=0.485 for sulfur, and (3=0.459 for selenium.

The secular equation for the bonding between 1/;2 and 1/;B,

=0, (3)

was solved to obtain the antibonding and bonding orbital energies,

Ell = EA+EB-2SEAB±[(EA+EB-2SEAB)2-4(1- S2) (EAEB- EAB2) Jt E2 2(1- S2) ,

(4)

where EA =a2E sA+ (1-a2

) EpA,

EB = {32EsB+ (1-{32) EpB

(5)

(6)

are the orbital energies of 1/;2 (and 1/;1) and 1/;B, respec­tively, and EAB is the exchange energy which can be approximated by the empirical formula18.14

(7)

S is the overlap integral between 1/;2 and 1/;B. The anti­bonding and bonding orbitals are given by

0/1 = CA11/;A +CB11/;B,

0/2= CA21/;A+CB21/;B, (8)

where 1/;A=1/;2 of Atom A, and the coefficients

CAn= (EAB - SEn)/[(En-EA)2+(EAB- SEn) 2

+2S(En- EA) (EAB - SEn) J! (9) and

CBn= (En- EA ) /[(En- EA)2+ (EAB- SEn) 2

+2S(En-EA) (EAB-SEn)J!

with n= 1 for antibonding and n= 2 for bonding orbi­tals, respectively.

A schematic diagram of the molecular orbital energy levels is shown in Fig. 2.

We now calculate the g shift for the unpaired electron in 1/;1 using the formula15

.1gij=2S-L[(O I L;\ n)(n I Lj 1 OJ/Ceo-En)], (10) n

where S- is the electronic spin-orbit coupling constant, and the indices 0 and n refer to the ground and excited states, respectively. In this formula, one of the angular momentum operators L comes from the spin-orbit

13 M. Wolfsberg and L. Helrnholz, J. Chern. Phys. 20, 837 (1952) .

14 C. J. Ballhausen and H. B. Gray, Inorg. Chern. 1, 111 (1962); H. Basch, A. Viste, and H. B. Gray, J. Chern. Phys. 44,10 (1966).

15 A. Abragarn and M. H. L. Pryce, Proc. Roy. Soc. (London) A20S, 135 (1951).

interaction and the other from the orbital Zeeman effect. The former is a short range effect, and hence the contribution from the overlap region can be ignored. This is not so for the second effect. In this problem, 10 )=1/;1, Eo=EA • Thus, we have effectively

(1/;11 Li 1 o/n)= CAn (1/;11 Li 11/;2) (11)

for the first L operator, and

(1/;11 Li 1 o/n)= (CAn+CBnS) (1/;1\ Li \1/;2) (12)

for the second L operator.2 We also note that the co-

-20

'VI EI

/ \ -40 / \

/ \ -80

>-

/ \ / \

(.!)

II: UJ / \ Z UJ

-100 / \

V...:!4 E \

" A ) "'a Ee

-+t- *{L / /

"'3 "'4 " /

" 'i'2 H I E2

-120

-140

A A-B B

FIG. 2. Molecular orbital energy levels of sulfur chain ends as an example of singly bonded radicals, a=0.42. Energy is in 10' em-I.

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SULFUR AND SELENIUM RADICALS 3.129

ordinate axes chosen in Fig. l(a) correspond to the principal axes of the g tensor. Thus we have

fl.gz = 2S-(1-a2) I(EA - EL ),

fl.gv=2S-{t[CAn(CAn+CBnS)](1_2a2)+ a2

},

n-l EA-En EA-EL

fl.gz=O,

fl.gij(i~j) =0, (13)

where EL = (!-a2 ) E sA+ (t+a2) EpA is the orbital en­ergy of the lone-pair hybrids. In order that the un­paired hybrid be higher in energy than the lone-pair hybrids, (EA> EL), the value of a is restricted to 0~a<0.5. The lower limit corresponds to the case where the bonding hybrid is a pure p orbital, and the upper limit is the value for a regular tetrahedral hybrid which has the largest bonding strength.l6 Although it is rather certain that the actual value of a will be closer to the upper limit than to the lower one, there is no a priori reasoning to fix it at this moment. Therefore the g values are calculated for various values of a in this range. The determination of the actual value of a is discussed in Sec. IV.

The average of g values in all directions

is plotted in Figs. 3 and 4 for sulfur and selenium radicals, respectively.

In this calculation, the overlap integrals were calcu­lated from Hartree-Fock radial functions17 and the interatomic distances determined by the x-ray analysis (S, Se chains) or the covalent radii (S-C, S-O, Se-C, Se-O).I6 The atomic orbital energies EsA, EpA, etc. were obtained from Ballhausen and Gray's book.IS The spin­orbit coupling constants S- were calculated from the atomic spectroscopic data compiled by MooreI9 as S- (S) = 0.384 X 10-3 cm-r, and S- (Se) = 1.58 X 10-3 cm-l •

III. DOUBLY BONDED RADICALS

In these radicals, there are three bonding hybrids and one lone pair. One of the three bonding hybrids is occupied by the unpaired electron as before. Again, the three bonding hybrids are assumed to be equivalent. Using the coordinate system shown in Fig. l(b), we can write the four hybrids as

unpaired: tfl =aSA+ (j)tXA - (t-a2)!ZA,

bonding: tf2=aSA- (i)iXA+ (!)tYA- (t-a2)~ZA,

bonding: tf3=aSA- (i)txA - (!)!YA - (!-a2)!ZA,

lone-pair: tf4= (1-3a2)!SA+3!aZA. (15)

Similar calculation as in the previous section leads to the molecular orbital levels shown schematically in Fig. 5, and the following g shifts:

Before we start the numerical calculation for S02-and Se02-, we have to determine the ionization states of the atoms in this radical ion. In the ionic extreme, one may visualize each oxygen atom as having a nega­tive charge and the sulfur (selenium) atom as having a positive charge. Calculations based on this ionization state give unreasonably large g shifts. This is not unexpected. The electronegativity difference between S (Se) and 0 is 1.0 (1.1) which is only half of the

16 L. I. Pauling, The Nature of Chemical Bond (Cornell Univer­sity Press, Ithaca, N.Y., 1960).

(16)

electronegativity difference between alkali metals and halogens. I6

For such systems, it is more reasonable to apply Pauling's electroneutrality principlel6 and assume that the more electropositive S (or Se) atom is neutral, and

17 R. E. Watson and A. J. Freeman, Phys. Rev. 123, 521; 124,1117 (1961); E. Clementi, C. C. J. Roothaan, and M. Yoshi­mine, ibid. 127, 1618 (1962).

18 C. J. Ballhausen and H. B. Gray, Molecular Orbital Theory (W. A. Benjamin Inc., New York, 1964).

19 C. E. Moore, "Atomic Energy Levels," Nat!. Bur. Std. (U.S.) eire. No. 467 (1949).

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3530 I. C H (N_" A N~D T. P. D A S

g

.5 a

FIG. 3. Calculated g values and B2 I E. I vs a for sulfur-carbon radicals (5-C), and sulfur chain ends (5-S). The observed value gob. (5-S) is that of liquid sulfur.

the two oxygen atoms share one unit electronic charge. (OO·5-S00·5-). The atomic orbital energies of 0 0.5- were obtained by interpolating the energies of neutral and singly charged atomsF·20 The overlap integrals were calculated from radial functions2o of 0- and neutral sulfur (selenium) wavefunctions.

There is no a priori reasoning for the determination of the hybridization coefficient {3 in this case. Therefore we calculated the g values for four {3 values ranging from 0.35 to 0.50. The results are plotted in Figs. 6 and 7. The choice of the value for {3 is discussed further in the following section.

IV. DISCUSSION

From Figs. 3, 4, 6, and 7 we see that in many cases the observed g values can be reproduced by the calcu­lation with a reasonable value of a (see also Table I). The only exceptions are the cases of solid sulfur and selenium, which are discussed later.

20 E. Clementi and A. D. McLean, Phys. Rev. 133, A419 (1964).

A. Maximum B2 I E. I Principle

In order to give a physical significance to the value for a which reproduces the observed g value, we con­sider the following quantum chemical quantities as functions of a.

The bonding strength of the bonding hybrid 1{;2 is defined by16

(17)

and the bonding energy is proportional to the square of this quantity. Therefore a strong bond requires a large B2. The latter increases from 3 to 4, its maximum value, as a increases from 0 to 0.5.

The second quantity we consider is the valence state energy of the atom, E., which is the sum of the energies of all electrons in their valence state hybrids,

g

gOBS (Se-C) 2.10F-____________ -H-__ _

2.05

.2 .3 a

.4 .5

FIG. 4. Calculated g values and B2 I E. I vs a for selenium­carbon radicals (Se-C) and selenium chain ends (Se-Se).

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SULFUR AND SELENIUM RADICALS 3531

for singly bonded radicals and

for doubly bonded radicals. This quantity measures the degree of "inconvenience" for the atom to prepare for bonding. Thus, bonding becomes easier as E» becomes increasingly more negative, which occurs as ex decreases. Therefore, we expect that the actual value of ex will be determined by a compromise between the above two conditions. The product B2 I E» I is plotted in Figs. 3, 4, 6, and 7 as a function of ex, together with the g values. It can be seen that the curve has a maximum near or at the ex value which reproduces the observed g values. Based on the above mentioned theoretical considerations and these empirical results, it seems the "maximum B2 I E» I" is a reasonable criterion for the determination of ex.

B. Klessinger's Optimization

Recently, Klessinger21 determined the optimum valence hybrid of H20 molecule by systematic variation of the hybridization parameter until the total electronic energy was a minimum. The purpose of this calculation was to use the result as a guide for calculations on larger molecules where full optimization of valence hybrids is too cumbersome. His result gives 17%"",18% s character for the optimum oxygen hybrid. By the method described in the previous paragraph, namely,

-30

'tt, 'tt,' E

,-. ----, '

-50 I \ I , , \

-80 I \ I \

~Epl , I

-1/12 ~3 E. \ ./, ./,'

\ 'fB 'l'B ;-- --EB

\ /

-100

I -120 \ /

\ 'tt2 ~~' /

'-+t- -Tt-' E2

-140

A B-A- B

FIG. 5. Molecular orbital energy levels of SeO.- as an example of doubly bonded radicals, a=0.36. Energy is in 103 em-I.

• 1 M. Klessinger, J. Chern. Phys. 43, S117 (1965).

TABLE II. Observed and calculated principal values of the g tensor.

gl g. gl

SClIa

Obs 1.997 2.025 2.057

Calc (a=0.42) 2.0023 2.024 2.061

SeO.-

Obs 1.9967 2.0268 2.0062

Calc (a=0.36) 2.0023 2.0267 2.0077 ({3=0.45)

maximum B2 I E. I, we determined the optimum valence hybrid for oxygen in H20 as ex=0.42, corresponding to 17.6% s character, in good agreement with Klessinger's result. Klessinger also pointed out that his result is at variance with conclusions based solely on maximum overlap considerations. The same has been found in the present work.

C. Pauling's Total Bond Energy

The total bond energy is given by Pauling16 as

Eb= bB2-ex2(Ep- E.), (20)

where b is a constant and the second term is the pro­motion energy which also appears in the expressions for E •. The maximum value of Eb, indicating the most stable bond, is another criterion that we could use for the determination of ex. However, the constant b for the radicals is not known. We have tried the value ob­tained from the known bond energy16 and bond anglell

of selenium chains. With this b (for the regular sites in the chain), Eb is maximum at a very small ex value which has no relation with the observed g value. Hence Eb cannot be used as a criterion unless we have the correct b value for the chain end.

D. Determination of {3 for Doubly Bonded Radicals

Encouraged by the success of the "maximum B2 I E. I" principle in singly bonded radicals, we have attempted to determine the {3 value for the oxygen bonding hybrids in S02- and Se02- radicals by the same principle. We plot the corresponding quantity for the oxygen atom as a function of (3, together with g values, in Figs. 6(b) and 7 (b). It can be seen from these figures that, for Se02-, the combination ex=0.37, which gives (B2 IE. \)A a maximum (A=Se), and {3=0.45, which corresponds to maximum (B21 E.lh, (B=O), reproduces the ob­served g values best. For S02- the agreement with the experimental value is not as good as that of Se02-' A modification for improving the situation is discussed below .

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3532

g

2.10

.1 .2 .3 .4 Q

1. CHEN AND T. P. DA5

2.0

2.02

g a =.40

.38

.36

2.01

2.00 (bl

1.99'----"''-----'-----'-----'-.35 .40 .45 .50

2.10

2.08

2.06

2.04 Q=.40 .38 .36

g .34 2.02

gOBS

2.00

1.98

(bl

1.96

.5 .35 .40 .45 .50 {3

FIG. 6. Calculatedg values and B21 Ev I vs a (a) and vs (3 (b) of 502- radicals.

FIG. 7. Calculatedgvalues and B21 E.I vs a (a) and vs (3 (b) of 5e02- radicals.

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SULFUR AND SELENIUM RADICALS 3533

E. 11" Unpaired Electron Model

In the above calculation, it was assumed that the unpaired electron orbital has the same hybridization as the bonding hybrids. Another possible model is that the unpaired electron is in a pure p orbital (1I"-electron model) and the bonding and lone-pair orbitals share the rest of Sp2 orbitals. The molecular orbital energy levels are shown in Fig. 8. Calculations based on this model result in g values smaller than the observed values in all cases except S02-, as shown in Table 1. For S02-(Fig. 9), the 1I"-electron model gives better agreement with the experimental value than the previous sp hybrid model. Clark et al. 8 have already proposed this 11" elec­tron model in their experimental paper. In their work, S02- radicals were produced from sodium dithionite, Na2S204, either by moistening with water or by x-ray irradiation. On the other hand, Se02- radicals were formed by loss of an OR fraction from the RSeOa- ion of sodium hydrogen selenite, NaRSeOa, single crystals. 9

Since the parent compounds have different structures in both cases, it is not unreasonable that the two seemingly very similar radicals have different electronic structures.

F. Anisotropy of the g Tensor

The anisotropy of the g tensor has been reported for some radicals discussed in this paper. Calculated prin­cipal values are in good agreement with the observed ones as shown by the two examples in Table II. The negative shift in gl may be due to the interaction with the higher-lying d orbitals.

-30

'1', '1',' ,--~E,

I \ I \

I \ / \

/ \ t3 ~ ~ \ ----\ EA

\ \ \ \ I/Ia I/I~

-50

-eo

-100

\ /'--E-a

-

-120 \ /

\-H--H-~~ '1'2 '1'2'

-140

A B-A-B B,B'

I-------f------;?"--- 90BS 9

1.990L---..l----i.2~--.3=----'::.4--~.5:--

a

FIG. 9. Calculated g values and B21 Ev I vs a for S02- radicals calculated from ... -electron model. Energy is in 1(}3 em-I.

G. Two-Parameter Theory

The one-parameter theory can easily be generalized to a two-parameter theory by introducing two different hybridization coefficients for the bonding and the un­paired hybrids, respectively. In case of SCRa radicals, the best fit to the observed g values was obtained for a (unpaired) =0.4, and a (bonding) =0.45. Similar re­sults were obtained for the other radicals. It seems, therefore, that the assumption of one-parameter theory is generally a satisfactory one.

H. Hyperfine Structure

The hyperfine structure (hfs) due to the 7.5%­abundance 77Se isotope was observed in Se02- radicals.9

Using the wavefunction 1/;1 of Sec. III, and the following formulas22 for the isotropic (con tact) A s and anisotropic (dipole) Ad components of the hfs tensor

As=f,rge!3eg,J3n 1 1/;1(0) 12,

Adii=~ge!3eg,J3n(r-a)4p(1/;lll(l+1)-31? 1 1/;1),

A dij =ige!3eg,,{3n(r-a)4P(1/;II-Hlilj+Mi)I,1/;I), (21)

the principal values of the hfs tensor can be expressed as

A 1=A.+2Ad,

(22)

FIG. 8. Molecular orbital energy levels of S02- radicals with 22 J. S. Griffith, The Theory of Transition Metal Ions (Cambridge ..--electron-model structure, a=0.42. University Press, London, 1961).

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3534 I. CHEN AND T. P. DAS

where A.=f,rgJ3.gJ3n I 1/;.(0) IV, Aa= !gJ3.gJ3n (,--3 )4p(1-a2). (23)

This result, I Ai I> I A2 I = I Aa I, is in qualitative agreement with the observed values: Ai=670, A 2= ±270, Aa=±240 Mc/sec. These values give A.=393, Aa= 138 Mc/sec, if the signs are positive, and A.=53, A d=308 Mc/sec if the signs are negative.

Using the Hartree-Fock radial function,17 we obtain (,--3 )4p= 9.23 a.u., hence for a=0.36 (Table I), we have Aa=327 Mc/sec, in agreement with the second .al­ternative. On the other hand, an accurate calculatIOn of A. is very difficult because of the spin-polarization effect. If we simply use the value of the Hartree-Fock 4s function at r=O, and a=0.36 (Table I), we obtain A.( calc) = 2.2X 104 Mc/sec which disagrees with the ob­served value even in order of magnitude. This disagree­ment is probably due to the lack of information on the difference petween the radial functions of different spins.2a ,24 Since we have determined the relative si.gns of the principal values to be the second alternative, we derive the empirical value of 11/;.(0) 12 as 0.48 a.u. for Se atoms from the observed A •. Using this result and a = 0.42 (Table I), we predict the hfs of the free chain-end radical in amorphous selenium, where the dipole components are smeared out, as A.(Se) = 72 Mc/sec (25.7 G).

I. Paramagnetic Centers in Solid Sulfur and Selenium

As mentioned in the beginning of Sec. IV, the ob­served g values of solid sulfur4 and selenium5•s do not

fall in the range of calculated values shown in Figs. 2 and 3. In liquid sulfur,3 the observed concentration of spins is in good agreement with the number of chains predicted by the thermodynamic theory which, in turn, has been checked with viscosity and viscoelasticity measurements.25 However, in solid sulfur and selenium there are no independent experiments nor theoretical work to definitely identify the paramagnetic species as free chain-end radicals. Using the maximum B21 E. I principle described in Sec. IV.A, we would expect the free chain ends to give g values gcale(S) = 2.028, and gcalo(Se) = 2.10. However, the observed values are gobs(S) =2.0044, and gobs(Se) =2.0039 for the sharp line and gob.(Se) =2.3,.......2.8 for the broad line. There­fore, it is unlikely that the signals stem from such free chain ends. In the following paper, we calculate the g values for several possible models of the paramagnetic centers in solid sulfur and selenium, and discuss the liklihood of the models from the viewpoint of the g value. In summary, the results are, most probably, that the sharp line with a g very close to that of free electrons stems from an oxygen impurity atom bonded to the chain-end atom; and the broad line observed in selenium arises from interacting chain ends.

ACKNOWLEDGMENTS

Helpful discussions with Professor C. Kikuchi (Uni­versity of Michigan), Professor R. S. Knox (University of Rochester), and many colleagues of this laboratory, particularly P. I. Sampath and J. H. Sharp, are grate­fully acknowledged. The author would also like to thank Dr. J. H. Becker for his continuous interest and encouragement during this work.

APPENDIX

The molecular orbitals and their energies (in 103 cm-1) determined by this calculation are summarized in the following.

Sulfur chain ends (Fig. 2)

E1 = - 28.19 'lt1 = 1.071 (0.42s+0.908b)A-1.043(0.485s+0.87Sp)B

EA = -106.88 1/;1= (0.42s+0.908p1) A

&=-117.621/;3,4= (O.S69s+0.822p3,4) A

E 2= -132.34 'lt2=0.S41 (O.42s+0.908P2h+0.593 (0.48Ss+0.875p ) B

Sulfur-carbon radicals (Fig. 2)

E1 = -14.42 'lt1 = 1.095 (0.42s+0.908p2) A -1.114(0.5s+0.866p)n

EA=-106.88 1/;1= (0.42s+0.908h)A

EL = -117.621/;3,4= (0.569s+0.822p3,4) A

E 2= -128.78 'lt2=0.S79(O.42s+0.908b)A+O.S42 (0.5s+0.866p )B

23 V. Heine, Phys. Rev. 107, 1002 (1956); D. A. Goodings, ibid. 123,. ~706 (1961). . . 24 R. Bersohn, J. Chern. Phys. 24, 1066 (1956); H. M. McConnell, ~bid. 24, 764 (1956) ; D. B. Chesnut and H. M. McConnell, zb~d.

28;,; ~.7 d!~;~;ans. Faraday Soc. 48, 515 (1952); F. Fairbrother, G. Gee, and G. T. Merrall, J. Polymer Sci. 16,459 (1955); A. V. Tobolsky and A. Eisenberg, J. Am. Chern. Soc. 81, 780 (1959).

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SULFUR AND SELENIUM RADICALS

Selenium chain ends (Fig. 2)

E1 = -31.65 'l'1 = 1.048(0.42s+0.908p2h-1.027 (0.459s+0.888p) B

EA = -101.29 1/;1= (0.42s+0.908p2) A

EL = -113.211/;3.4 = (0.569s+0.8223.4) A

&=-124.19 'l'2=0.552(0.42s+0.908p2)A+0.589(0.459s+0.888P)B

Selenium-carbon radicals (Fig. 2)

E1 = -17.08 'l'1 = 1.101 (0.42s+0.908p2) A -1.085(0.5s+0.866p h

EA = -101.29 1/;1 = (0.42s+0.908p1) A

EL=-103.75 1/;3.4= (0.569s+0.822p3,4) A

E2= -125.15 'l'2=0.547(0.42s+0.908p2) A+0.578(0.5s+0.866p h

S02- (sp hybrid model, Fig. 5)

E1 = -40.54 'l't,'li/ = 1.037(0,38s+0.925p2,3) A -0.981 (0.45s+0.893p) B.B'

EA = -104.54 1/;) = (0.38s+0.925b) A

E2 = -130.36 'l'2,'l'2' = 0.526(0.38s+0.925P2.3) A +0.623 (0.45s+0,893p) B.B'

E L= -135.38 1/;4= (0.753s+0.658p4) A

S02- (71" electron model, Fig, 8)

E1 = - 38.58 'l'l,'l'l' = 1.037 (0.42s+0.908p2,3) A -1.001 (0.45s+0.893p) B,B'

Ep=-94.00 1/;1= (b) A

E2= -131.92 'li2,'l'2' =0,541 (0.42s+0.908P2.3) A+0.606(0.45s+0,893p) B,B'

&= -141.25 1/;4= (0,804s+0.594p4) A

se02- (sp hybrid model, Fig. 5)

E1 = -44.64 'l'l,\[I"l' = 1.041 (0.36s+0.933P2.3) A -0.923 (0.45s+0.893p ) B.B'

EA = -97.50 1/;1 = (0.36s+0.933p1) A

E2= -126.02 'li2,'li2' = 0.476(0.36s+0.933p2,3)A+0.678(0.45s+0.893p) B,B'

EL = -136.51 1/;4 = O. 782s+0.623p4) A

Se02- (71" electron model, Fig. 8)

E1 = -42.99 'l'1,'li/ = 1.039(0.41s+0.912p2,3) A -0.95 (0.45s+0.89p) B,B'

Ep=-87.00 1/;l=(bh

E2 = -127 . 89'l'2, 'l'2' = 0.499(0.41s+0.912p2,3) A +0.653 (0.4Ss+0.89p) B,B'

E L = -140.77 1/;4 = (0.814s+ 0.58p4) A

3535

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