San Jose State University

From the SelectedWorks of Juana Vivó Acrivos

December, 1961

HYPERFINE SPLITTING IN AROMATICFREE RADICALSJuana Vivó Acrivos, San José State University

Available at: https://works.bepress.com/juana_acrivos/79/

..

UNIVERSITY OF CALIFORNIA

Lawrence Radiation Laboratory Berkeley, California

Contract No. W-7405-eng-48

HYPERFINE SPLITTING IN ARO·lv'IATIC FREE RADICALS

J. V. Acrivos

December 1961

5].·

:J .V. ACRIVOS\Department of Chemistry, Lawrence Radiation Laboratory

University of California, Berkeley, California .I

The h;me1"finG st:ructurc of ·the E�"=! spectra of f'.cee :radicals iG a

prol)c:r·ty of th0 system of electrons and nuclei which const,i tute the molecule.

It arises from.the energy 9.f interaction of the unpaired elect�ons end nuclear

S!)inn ·o·hich is of the .form

11hcre T is a. tenso1 .. of second ranlt, I is th� nuclear spin o.nd J the to·tal elec-� I � �

tron a..rigular mo1:wntum.. II?, other ·wordn, a nucleus o� spip. ;£ t. se1-..reo a.c a probe

to the elect,ronic struct,ure of a conrple:i-: molccttlar free radicel. The m,-..;asure-

rent of this· coupling energy is in general of :9:_,,,il:10 im.portencc, because it

pcrrili ts th:e direct dcterr;,-hia.tion of th:::: elccfa .. cn ·densities, end its theorc·G:i.ce.l

tu:.r"C.

In particular; it ;:is not unco!!ImOn i'or the ESR spectra of :rrcc :i-:e.d.icalo in

solutions to· possess a hypdr:f'ine struct;�'e, cs nhown in Fig. l for the p­

benzosenicquinone ion an.d $' o:r which the J:·c.dio:t.ion induced ·crm1sitiono :!.� the

Paschen-:eack. high f'ield ap}?rmd.mation o.re coverned by the Bol:n.-> relo:i;ionship:

-1- .uCRL· 9998

hw O ia. the eneiy;i · difference between the electronic Zeeman levels in the ab­

sence �f the hyperfine interaction a.ndl\. io a conotant chara�teristic .of the . . n . ·. . . . .

. . . z .

free re.d.ical. and the nucleus n, of gyro�gnetic ratio rn�·. H is the applied

external rr.sgnetic f'ield. In· the aroma.tic r,-elcctron free re.dicalz, the sif;!I!,a 1 bonded protons ·w:tth apin I = .2

are rcoponaible fo:r the obs.erved structtn'e.

Th:!.s ex:ge:rimental evidence· baa led therefore to the concluf3ion that the spin

polarization is trunomi ttccl through a valence bond bet1-;een the a- and 'fr-

electrc�l� "'.:ll.ich Ut) to that time were cons�dered o.s non interacting groups.

Weissman1 and McCon.nell2 have brought fo:r.1-che the idea tllc.t us originally

proposed by Abragam. llorowi tz and. Pryce3 in their study of the l2Yr>e:-L"f'i1?,e

OIJli tting of the iono of the transition 111.etal c;c11ics, this 'bonding can be ex­

plaihe.d in terms of .a mn{3Uetic �nt dencity mat,rix. in t-1hich all the e):citcd

coni'igura.tions cm1.tri'buting to the g1·0,.i..-rid state through. ·the correl.a.tion inter­

action betm�en the tt-ro grOlll?S would. have to be included. T'nis a.p:proo.ch, which ·

.has been success�y developed and applied to niur.�rous cases in the literature,

·will now be given in some det�il. . :trs�? will be :me.de o:f' the dons:lty matrix for­

malism developed by Lowdin4 end._McWeeney.�

. i

-2- UCRL 9998·

The D�nsity J:.'in:triX Form.ule.tion

The electronic system described by· 1�.:.tomri in ?t�lecules 11 ·

is given, within . i .

.. · . . . 6 th.e Born-Oppenheimer approximtion, in terms of e complete set of atomic . ' \ .

orbitals under the restriction that the Pauli principle be obeyed� A direct-

ly mJasui"�ble pbysico9I1emical proper<ty of this system n has then e.n ex-gecta-

·i:iion value,

. : .·' / . (2)

whe::re _n _(�) is a function associated with the obneT'roble, aD:d ·o/ (�)J�·:_··�l�e

:ra.Y� ·1\.,,�1.c·tion.· uh:lcl1 c1..cscribes ..,Ghe elcc·t:conic sto.te vhen x. spans t:he · con11)lete . ' I'\.,

s:9ace U�:.d. spin com:·dir .. a:tes (1·,s) oz the !f electx·ons,_(::: = =="\ , x .... , �9'1' ,::. .. _.)o ·v (::)

. . . ' .. I\, "'. . . . I\., ·. 'v.J.. I\,� • . l\:.,J . 'J\,

.is e:q;:t\1nsed,· in g�neral, as a SUl)erposition of configurations.·,- Thus,

(3). :, .

uhei'e ·:ea.ch-�(�) is. given by the· e.ntic�uoetriz:�d product _Cir" the s-t;,gle ·part;icle

molecular _sp�-or�itals 'O'k(:£1),. �rbich ro.·-...� ·in tu1"n J :i..,es.r combinations of the

complete se� .of �tomic orbitals ¢(xi

):

or ( h,.)

where Tis

orbitals ¢ into ·that oi ·m.o1ecu1.e.r· orbitals: ��"u · The atomic' orbitals c1re: character-·"' 4'\.,

ized by the· fourfold· sy.awol v = (n,. P,, m," :ni· ), in such a manner tne.t: ' ., fi

�v· t�:: )= R { r) Y n .� ( 0, �) f .. ( m ). - "" n[ "'n:. . b s (5)

UCRL 9998

-3-

�1here (r� e, �) are the spheri�al polar coordinates of the electron

witl1 respeCt \to the nucieus y. 8n (�) deperids onlir on the distanc�r1•om v and Y

.em(e,_q,)' ia · a spherical harmo�i�� {se� for instance:

Ref. (7 ).

p .. · 57). .Also J. · t; ·the azin11:1t.al quant�: �umber ., indicates. . . . . .· .. . . . .·. . . the irreducibl� representation or the ·spatial-synunet;y (a, p, d, .�.}�· . . . , .

. . . • , ., • • , I . . '

.. · and m the _particular- component of. th�· 2.e. +. 1·.· fo,J.d degenerate state·s · :' I

o:r· orp1ta1 ariguiar· ·momentum., wh11e n,· .. the pr1n�:t:pa1 quantum number

di'stinguishes· ·th�.- �;ri�us : states· 1,·1:tth . the same:· ·cransformation . ' '

propei?ty _but with di��ei�ent: �nergy. Finally the spin fun�tion f�,

i�. equa.1 to a· (?r (3. :a�cording to wheth�r �a·=,+ � o� - �. . .. • • • i • '

The l!K: ar�. ·given.: approxin1ate.1y by .. e1�her one

. �r th� ro11ow1ng

two· conventional •e:itpre�.sionrl; . , ( see .for · inst arice Ref. , ( 7)., p ,o; 232): .

·.: •.• . A; , .By th� s1ai'e;�P�ui�g 't\l'�le�ce' Bo�d-1 :(VB) ���1� (8) (9): ·, ...

... · ... '.' .

. ,· ·, �- ', .. ' ...... · . . ... . . . . .. .·.... ... ' .. , . . . . . . . :

. .

: .: �K- = � -� '[tk1��1 )v°k2<�2)· • ·if1�(�) l, =,hr� Ht � DK. {Ga)·-

.. :/."·· ·: ·: · B, "i;JY. the nH9_lect.ilar ·ofbita.1'n '(MO) method developed· by

HUck;e·1. (lo)· arid Milliken (1i) : ' : : : . . .,. ;. .. . . .

. ·. : .. :··· ; . . . . . .. '. ' . . . . ., ... ' . ··1 ..

. . . . .. · �ic'= 1 [llrk1<::s1>tk2<!2>·. ·t1,N(�)J C DK, (j = m: ( -"l)P fl

. . . . ; \ . . . .

. . . . ' . . - ;( ?b)

w�.e�-� _-P�,}�. �h-�. Sl�ter d�te�inant..

·: .... · ; . . : ,ft (l.). ,fr . {l) :�:..:, 11 (1).. · ..... kl .k2 .· kn

. D�.<� •. ·. ��1 <_2,) t�c2�2>, - ___ ;,,1�1('2,>.- -

The eigenvalue c;>f:_ the spin is indicated by ·a superscript, a bar , •, " ' 'o ,, I• I • • • • ' I

for. Ills .= -.- _l/2 a:nd not�;ng f�r m6 = +. �/2 •. . .. . ' ' ' ... . .. '. . -. ' . .. . -����-����������������--���---������-��----��.�-����-���---��������--�--�-�

t Jg·\ interchanges t� �pin in the · n :pairs of orbits -w�ch are .bonded together �no. E permutes- the electrons among the spin-·oz,bituls, with �he· correszx>nding pari:l;ies

r and p. The transformation % in Eqll { 4) in the valence bond, VB method is the unit matrix when hybrid orbitals have been taken into account. · ·�

-4-UCRL 9998

The function !2. (x.)is in general expanded in tenris of· zer�.; one, two;"" '\,

. . •

. f'

I ' 1

· etc. J.': article operatqrs:

. (7)

�nd according to Eq. (2).°its e:h.1:)ectation value is: : .· ·.

+ -2

•.·.., , .

. (8)

·where · the· operation inside the b);'a.cket is curried . out before the prin1e sign• ' .

• 1

is eli-r.•·inatcd •. .Her� p�, p2 •••. Pi� � defined .e.s,-.the o�, ·two, etc. electron

c1cnsity matrices.

P1 (�;�{) = Ni ,:fe(:f.]_, *'2• • • • �) / (:£J., �' ••• �j )( f::._)

Pi:f.i•�;tCI,��>.. f�-kJJ «:£i,�• � .. :IN) *)!,<:£J.,�• • • ·�><�>

fl o •. D·• o • o o o o o • o 11 o e 1> 0 0 0 Q O O O O O & Q O O •

and

(9)

The inr9ortance·of-the density matrix is due to_ the physical significance

of the diagonal elements. Thus:

�1(�1f �1> .. �l:.� -�umbe� �f particles in �he .. system mul�i!)lied by the

pl"obability of finding one with:L'l'l :the· volume element d�, cen-. . . . . . : . . - .

.

-5- ·uCRL ·9998

p2(!1,�2� .�l;�2) d!:1 d.£2 = 2 x number_ of p?,irs in the system. , .. . . .' . � . . .

· multiplied by.the probability of simultaneously· finding· . . .

one particle in each of the volume elements_d�1 and d£2

c:��t�red about. i1

anq �-2,P • respeot�vely o

.Adcord:tng t� .Eq� ··(3.) then: .· .. . . .. : ..... · · ...

: where

Prh;2;,:;,nll 1 ,2•; .. ·�n 1 )� � PIO, Pn(I0\!1,2,, .. n; 1',2 1 J"·n') .. ' . . . . .. . . .

(9}, *

PKA = CK CA

Pn (1., 2J1. �- • • n I l .•., 2', • • • n' ) =

(N��)! J IJ!K(i,2�.�•n,···N·) 'i'*(1<2 1 �;·;�n 1 ,: •• N)(�)

with the relationship between successive density matri�es:

(N�n)�n(la2,·� 0 nf1 1 �2·�··gn i ) =

J Pn+l (1, 2, • · •n,n+l I 1', 2.1, •; �n 1 ,n+l) �+l

For brev�ty th� coordinates are indicated by the particle index.

Also the space and spin coordinates may now be-separated according

to McWeeny; .

· a a . * a(3 * .

. p1 (KA)l;l') = p1 (K.,AI 1;1') a(l) a {1') +p1 (K.,AJl;l') a(l) f3 (1') . . . .

.· . � a · * ·.· . (3 (3·· . .'4 . .

, ·+·p1(K.,AJ1;1_f) �(1) .a ,(l� )� p1(K.7AJl2l') t3(1) f3n(i') .

. ·. · · (10)

· where the ·.p1 are spiniess density.matrices obtained by integrating

p1 over the spin.coo�dinates •. The spin product to -which the

spatial function belongs is indicated by a raised symbo+. In

particular, if there exists an axis of quantization�' the spin

. component in that direction is :

. · . -

-6-

(Sz> = f [��(l)pi(l; 1')]1, = i d�l

l\ a. a � � ·= 2 f (p1 (1; 1) - p1{1·, 1) ]d£l

. .

""· ½ i: ql_(l; 1) d£1 '·

. UCRL: 9998

(11)

Foi� states of definite spin· the s·pinless and spin density matrices

·. Pi and q1 .. are then;

P1 (1; � 1) = :}: p - [p . (KA j 1;_ l') + p (K� I lJ l')] . : KA .'}p\ ? , . . : : : ,l · . ·

ql {lj l'). = .� P� [pl (�i\ J l;:. �') · - Pi (KA fl; 1')] K

(12)

with. the following physical signtficance of· the·diafl;onal elements:

p1(1J 1) df1. = probability of finding any· eleot�on within

.. the volume e�ement d£l centered at �1• · . .

q1 (lJ ·1) d�l .= P::r9babi.lity of f.ifl:d.i.ng ·a.n U..l'1pa�red electron

-� . with spin + 1/2 within the volume element d:_:1· at !:i·

... minus that of finding' it' with spin _. 1/2 0 ·

-7- UCRL 9998

THE HYPERFINE INTERACTION HAMILTONIAN

The magnetic int�raotion between particles is a relativistic

.effect. Fermi i2 d�rived the expression for the energy of inter-. . '

action between the electron and nuclear spin from Di�ac 1 s theory

of _relativity.. A d�r1vat1on consist�nt w�:th these prC>p�rti��. �as.: develop�d b_y Casimir13 and· Ferreil 14 as follows.. The hyperfine

energy is obtained by. considering the· motion or,i an electron in

the field,

�n = curl �n

i�duced by the·nucleus1 of moment �n" where the vector potential·

�n at a_ distance� from the hucleus·n is defined by:

<iiv. �n = o

�n �n x �.'A = curl - = ---n r r3

The energy of a particle in t�is magneti� field is given by

the classical Hamiltonian of electrodynamics (see fqr instance (7)

p. 108) [atomic units are use_d throughout this development - see

Appendix ( ) ] ;

i ( H = Z 2 pk

.....

l )' 2 c�n k + potential Energy

Where the transition to quantum mechanics is achieved by replacing

the momentum p·by the operator - 1 V if the particles do not �· 11V ' •

possess spin. F�r t;tn electron with �pins=,� and intrinsic mag-

netic moment· !::e = 13· � • g., an additional contribution to the energy,' .'.:." ; • .• : · • • ,.

. : • .

arises from the' s·ca1ar product !:e • �n· The electrostatic inter-

action between the nucleus and the · elec-tron charge clouds also

contributes to the energy (quadrupole t�rms)�but is non zero oniy

. -8-UCRL. 9998

£or nuclei of apin I =·1 and will not be considered here. ·Thus

the hyperfine energy of the system�.·in th� absence· of quadrupole

interactions is:

Jlen - (l2CP - .!. A )2 � 1

2 P2]' - .µ • B hf - . -· a -n - -e -n

=�·{f3[(pol;:n�r

+·�nX£ �P).--..1..A2]·+µ •VxVx.�} . - 3 3 · - 2 n . ..._e - - r .. .. r r · · c . · ., · l . r X p [ ( ·µ . . µ' ] 1

- - 2f3 µ � - - + µ 0 'v V " .::n.) .. v2- ::!!. . . n r3 . -e - � - _ r . - r �

_ _ { 213 �n} + µ • [ v (µ • 'v) 1 µ v2 J 1 rv -e -. -n - 3 -n - r .

- -µ •µ V -2 2 1} 5-e -n- r. (13)

Use has been made of the. definition of the orbital angular

momentum t = r x p and of the vector identity curl curl= grad. - "" -

div - (grad) 2 _keeping only the linear terms in �n· The two terms

arising from the electron spin magnetic moment have been separated

according to their symmetry transformation property. The first

is the anisotropic term

[ . . : . 1 2 ] 1 [ 3r r 1 ]

µ 0 V (µ • V) - - µ V - = µ, 11 � - - • µ -e - . -n - 3 -n - r -n . u 3 -e

. . ·

r r

It transforms under rotation as a sph_erical harmonic of order

t = 211

In magnet1c fields or presently attainable magnitudes the

�lectron and nuclear spin magnet1o··moments are

µ = f3 a. • g . -e -

(14)

where� and �o are respectively the Bohr and ·nuclear magnetons and

-where th� g-f�c�or for the electron with gn�i�otropic� is a tensor

. �9-UCRL 9998

of second rank p · The an�sotropic. energy may·therefo�e be written

as !n _• �n ° ! where �n is a tensor_· or second :rank. \'/.hen g is iso­

. tropi� ·�:;he term ·1n. !� is ·

· l3z (xsx + ysY + z� z -·r2sz ·

1 ... ( a .., a 1) z = 5· - -5 - . - . � ( 6n =. g gn I f3 f f3o) n - . n_ . r

=) ((p cos 2

.e • 1) �z + � s�n e cos e_ (1�

1cp�-+ e"'"1<P�+))

6n 4?T. 1/ 2 z 5 r . . . - . / +

= --s <s> £2 Y2o<e,cp) � +.J2 CY21<�,�) � +Y2-1Ce,<V) ! llo r- . .

(15a) . .,71,.Y

·By rotating the axis ., x �z� the other two components are

obtained and the· general expression is Pake's15 vector identity: .. 6 .

!n'�n·� = 7[!+�+£+�+!+!'.l (15b)

·where

1 (. · 2 .

).( z z )B = - 3 cos 0 - 1 I a - I ' s - 2 -n - -n -

* D=C (complex conjugate of C).

* F = E (complex conjugate of �E). -

Here (r�e# �) are the polar spherical coordinates of the electron

with respect to.the nucleus n and the ·z axis is that of an

, applied external field. + - + !n' !nJ) � and�

UCRL-99�8

· are the spin raising and lowering operators (ff:: = ;x

+ i eY) which connectI

states �rith �:: ,.::::,.1, or �s: = ::!: �o an_d vanish for statrs of definite spine '. .Vlhen the:r"e exists � axis o! �uant\zation _z, · the �Sotr?pic �arm in a state:

,Qf definite SJ:?in is then. \

6 . 2 . , . (3 cos 0 - i) � t2 ; (15c)

Now from this expr.ession �e reason fo� the nomenclature is clear, since

th8 te;::m is ang�ar· dependent, · wit..11 principal values: 1 . 6

-1/2_ -j-

a = ·n

6 1/2 · n .-:,

That is, a is a tensor of zero trace which will vanish: when averaged over all n orientations as is the case for s-electl·ons (.t =O). · The angular dependence on

Y 20 { 0, 4>) in Eq. · (!Sc) implies then that the term co11.tribution will be non zero £or � :!(

the states· (1 l �-1 )'/ (1 2

m2) of the _electron (s·ee Eq. (5) ) whe·n (2 1122) satisfy the

· .propertie� of the .sides of �- triangle of even pe:rimeter (t�iangular rule for the

addition of angular momenta):

! I '•

··111..1 + 121? ·2 ? 1·11-ll.2 f• • 0 • •

and for

Th� iaotr.opic term is obtained from the claseica� theory of Coulon'lb

. :fields [s�e for �tance J?a.n:o!sky and Phillips (14) page 3) J,

-11- : UCRL-9996

(16)

' . : .

where the delt� fu.nc;ti_on is defined by:

{ 6(;._- r ) ·= O, fo,r � F In,.

·.J

(� :(�

1

- �) = f(�). , . for f(t) continuous � tJie field regi�n·: i. . ,·.-. . . . . :

Tli.J..o term is spherically symmetric and is therefore �on-zero only for a-electrons. §

. . ·. ,. '. . . . . . ': .... ,· . . .. ', : ! ;,,

· The_ hyperfine interaction Hamiltonian for the N-elee;tron aystc::�1. of a com�lex

mo_lcc�:!lar free radical can be e,cpanded �n ter!X"t..S of one electron operators:

(17) . ; � . )

where :: : :·

)-( • I

L { J,, • - AiIf: . • ::; ,. .....,1

,t; , . ....... ,. .. ...

. lif(�Y - 6n � 3 · n r. · ,. · ; · · .·.. 1 .n

·3 r. (;t. , a.)+ "'"'1. n 1n -i

r. 1_ n

+ � .� (� � !nf!�J:I!� \ ' •; ., •• ··.!.··'•

... : '

. . · :. In ·02:d�t to;det�rmine :the ·expec.tatio_n value cf. )-(hf ,it must b� .noted,

that 'the· operator· contains the :nuclear. spin coordinates, which.must therefore be. ' . . : . . . . .

· included· in· th� ·wave £1lll

ction used in Eq. (2). I£ IV, q,k are the ground and excited

state eigenfunctions of the_ ele·ctroni'c Hamiltonian 11 then to fb:st order, and as

. i •• • •.

pointe� out by 'Pryce (15)

.. �: {iv-k'��K-!�'>�} e

(18)

is an eig.enf�ction of the c�mplete. Hamiltonian, where E, E

K . are the energies

associated With the corresponding electronic states and (:, is the nuclear spin

· eigenfunction. ' l:

. ( ·-12- ·UCRL-999�,\:

. -�·polyatomic molecules the orbital degeneracy ,, when it e:dsts in

the gr�"UL"1.d �t�te, _i.s quench�d as a consequence of the Jalili.-Tellez- theoram< :,:;°' I :{\. ··< .... -)

{16, '17). This mean!) that L� /\ll)it and/ LJ'" are zero.. The expectation

value for· the ·hyperfine interaction is then:

,hf l {A I .. : S + I.. • A. •_:_ s}. · h n�. f"'I • • �-: .n K, . _.(19)

where, as·;�uming that there e,dsta an e:1:ternal a.Jda· of. qua.V1.tization, Z,. ·;· ., · ..

(a)

or

(b)

, · .

. . z ' '?. A r- s= =n.�

8 Sz = _;. 6 ·iz /2 :) n· n S 1 · qln'

A = � . 6 q1 n ::, . n n

(q - q (l·l). -�).. ln "'."' 1 ' r,1 =tn, · ·

(19a)

(19b)

! «· !,.·' � ·v· � -

The molecular axis :2� as well �.$ the eA'"teTnal a.tis of quantization Z

must be �a.ken into accou11t in the a.."llisoti-opic terin. . The general case arises · .

when the el�ctr�n is in an atomic 01·bit_al not centereid about n [see Fig. 1]. •

· Then. the electronic and molecular coordinates ai-e separated by malting use of the

addition .theor�m of spherical ha!'monics (see for instance (7) p.· 370)

-13-

.. 2

. UC:RL-999?. (

and

2(3 cos 01 . n 1) = (3 cos e -· 1). P2 (cos 151). vn

r1

� CO/'il °/11, �,l\ n • r1 COS 81

r.l sin e { ; >· 1 .n '

sin 1\' .

(20)_

if elcctz:on .<l) la �� an atomic �rbit�l centered a�out V : The signi�cant.

contributions to (19b) will then :axise from. atomic orbitals centered about n

and v. I£ n is a hydrogen atom with an electron in a is orbit, then the

significant non-zero. contribution.a will arise .fFom one-center inte.grale about. .

,'•

v and two-center integrals about v - and n. Wnen Eq. (20) is substituted into . .

(19b) the terms containing the molecular coordinates �vill come outside of the

integral sign: { 2·. ·· =1·P

2{costJ:).

( 3 cos 9 vn - 1) :; l

Q l ( 1 ; l) d t l rl .. . . n

Ql (l;l) dJ'l }

• .(l9b} . J

Thus,· i! the .molecule uridergoe.s rapi� t�?Ill;,ling, as in s_�lutione, the average

valu� of the _anisotropic co�pling .vanishes due to the proporties. of the spherical

harmonic of ?rder. 2. . The physiqal ir.npli�ation he�e ie that. �n solutions the

hyperfine structure of the ESR spectra 'Will arise_ only from the isotropic

contact term. · This 'property ,was derived by Weissman (18) •

APPLICATION TO THE HYPER FINE INTF.R ACTION IN AROMA'i'IC .b""'REE ·RADICALS

UCRL-.9998

....

T'he raacti vity of the aror.ciatic: iilolecules lie� in their �bility to add

or lose: an elect°:i:"on from the �-el�ctron systen'l,.- thu� forming, a faee radical. ' I • • .: • � • � •

'

These are ·stabilized in a.· propei-: ·sol�e�t il'l order to :record their ESR spectra,

where tl�e obaerved hyperfine .structure arises from the contact interaction

ter1�1. of the'. unpaired election. at ·the.1'luclei of the a:rom.atic and· o. · protons ill

1. - . b t. . . ul d t· ., 1 t r 13 NI 4 a 1phatlc au s 1tuents, �J.; � ·he S.1.'\'..e e on - or r .• . . · . . ' .

This r..1.1.eans that the

--elect:rons of an aromatic frt:e ra0:ical posses a certain degree of unpaired

cha:�acter,. ,i,hich is due to a a. ·- iI' t?orrelation h,�teraction. ... : ·:

·.,:The .. best wave functions for an· a1·omatic ·free ra.dical are thµs obtained

by .firs.t assu:rning sig1na·-pi separability v1here the o- - : and 1T-electronf? are

. cm"lsidered as individual groups in. the electrostatic field of each 9th.er see

:for instance (6), p. 249 . The configurations which. take into account the

e,dstence of the O' - "fl' valence bonds, indicated by the ESR eJ:periments, are

then introduced by 1neans of a se:r:iea pel·tu:rbation. The p:rocedure a which is

based on the theorems developed by Lykoe· and Parr (19) 0 is as follows:

A. The electronic system to a first appro1dmation ia described by:. r:

ttso = 1(j7r

[<ia i<rr)J-= \8 1 I_) on·\\ .. (21) ..

where _( 1

l_) and {.ll) are independollt antisymmetric £unctions of the <1- and

"ll'-electrons respectively. They are given in terms of a compl�te orthonormal

set of Slater determinants which describe the system:

l l l

.. < Lo> .= Al < [1 > + Az < Lz> + • · •

(JI� = Bl {Ill) + B2 (Ilz) + •• • (22}

-15- . UCRL 9998

where· * 1 · l * . . . · *' .

.f 'Y.0'¥.0 � = J ( �0) ( �o) d�0 = J (II) (rr) d�

1r = 1

(1�0) is a singlet and 'Y.0·and (n) have t�e same spin multiplicity.:

Also: ( 1z0 and '(JI) are ortho{?;o�l since the CI• and w-electrons

move in orbitals. w�ich ar� respectively ay.mmetrip and antisym­

metric with respect �o the plane of the aromatic ring.

B. The cr-� bonding properti is introduced.by the interactio�

of �0

·with the configurations !K� where the a-electron wave function

is in a� electronic triplet state or in a state of.higher multi-. .

.plicltyJ (�K), leading to the ground state wave function:

(23)

In the absence·of an external magnetic field., the electronic

energy of the syetem is obtained by expanding the spin-free

Hamiltonian up·to two particle operators:

= z . --i;-··.

. 2

}

· · {Ek Zn k=;-electrons . 2 .· n rkn .

(24)

� The first term,in dl) represents the kinetic energy of the elecr

trona and Zn is the-charge of the bare nuclei .. The subindices

indicate the system or electrons on which the operators act. By .. .

sigma-pi separability it is·meant that (1z0) is an eigenCUnction

of the cr-electron.operators alone$ while (n) is the eigenfunction

of the effective Hamiltonian

H = R(l) (n) + � H(2) -� -core, 2• -1rn (25)

where

-16- UCRL 9998

11<1) (7r) = !:!il) + 2� ('11'), [�� ('11'} = ��. ('II') - !� ('11')] .•.... core

GO (1r) _arises from the last tewn in Eqo (24)., which is the two--cr

parti�le correltationfunction between the a- and n-electronso In:

_the first approximation the 'lf-elec:trons move in the electrostatic

:,otential .field set up by the· nuclei and the a-�lectrons., that is;;

0 � l ·1 1 1 · icr < '11'1) t'll'(l) = 2 {J rl,1' P1[ { zo)( Zo) I 1 'H' ) la.fa' ) 'if/ .,r

<1)

;�<'11'1) t '11'(1) = � JC r/ l' !1. l' P1 [ {

lzoH lzi/1 l •; 1 11 Jt 'll'(1) \"=li�' •.. J

when w1r

(1) is a one-electron n··mo ..

The cr-n correlation int·eraction:

l 1 G =-.�­-cnr 2. cr1r ran

will now mix the excited configurations into the ground state

! _ � 2S+�& :A [ (�K) (rr)]K - -9.!r . \..� .. i .

where �S+le is a projection operator defin�d by L<5wdin which . .

-

selects the states of ( 2S+l) spin multiplicity. For a single

_(26)

{27)

Slater determinant n0

in Eq. (27), the eigenfunction of s2 is then:

n2S+l9 Do = � ckDk- k=O

(28)

where Dk_ia the aum of the Sl�ter determinants which-result by

applying a�l the. possible k-spin permutations to n0 and _n is the

total number of pairs of orbitals· with o·pposite spin. · T'ne

coefficients ck are then determined from recurrence symmetry

-� · relationships ts;e Lowdin ( 4)]. For a doublet state these are

given by:

.

:!f. = (-l)k; cn�l)-1 CO

-17- UCRL 9998

(29)

where c0 is obtained by direct application of 2�;+-l�,. Thus., to

'first order, the ground atate·wave function is:

<H>o�{'1-'l! - �

- .1

'1! .- O K=l <�>KK <H>oo K

H:·�re all the {i:1) and (If1) satisfy the sigma-pi.· separability

theorams and therefore the·expectation value of H between the � . �

and

where AeK is the energ� of excitation of the virtual processes

which give rise to the hyperfine interaction.

The expectation val�e of the hyperfine interaction energy

is now obtained from the wave function

Y=te

where 1' is given by Eq .. · (30) and e is the nuclear spin eigen-

(30)

(31)

! .f'Unction4 The excited spin states in Eq. (18) have been neglected ·

here �ince they a�e of the order of 6-nlLSE, where for protons

· 6r/ AE � 8 · x 10-7• The hyperfine i�teraction for the aromatic

free radicals in solution is then (see Eq. 19a):

-18- UCRL 9998.

= §2! � 6 Iz s�·q ··�\ · 3, n n -� ln/ _. n . ·. . .s. . , · .

(

\ ..

(32)

where _·q1n is 'the·:on�_-elect�on spin_ density at the nucleus n. ·-Th�-

one electron deneity matri�:-' · . l

•: ...

can be derived to first order from Eq. (so);

That an unpaired w-el�c·tr�n does not, contribut� _t_o the a-electron

-��n�.i�i�s. in !o :t� �oted ���m#

since the'one ... electron·densitymatr:tces ·or ·tlie separate groups

are additive'. : The . effective one electron density :matrix will

the·refore only contain the cr·oss· terms· .pl (KO J l;l 1 ) • · .. .

The coupling constante for an-aromatic ·free radical with

total ·spin S = 1/2 can readily be calculated now from Eq. (32) ..

The complete set o� Slater determinants which describe the n­

electron. system are assumed.to be eigenfunctions of s2 • This is

true in general since the doublet state project-ion operator can

--19- UCRL· 9998

be applied successively to then-electron antisymetrized products

(ll) and to: the· ··f: J�(�) (IT) f lo 'l'he ·matrix elements h,etw�en !0 and 'IKare to be evaluated betwe�n antisymmetrized products ··of the form:

and·.

O�ly three electrons need to be considered here since al� the . .

others are assumed to be paired. Although the excited state 1a

doubly degenerate� only the E1 (c�V) linear 9ombinat;o� given. . .

above contributes to the hyperfine couplingo ; ! • ' • • •

I

._. Since_ 2cr7r

1.s a . single_ particle o�er�to� :1�. bo_tl?, �he a- and

�-electrons _non zero terms will arise ortty from antisymmetrized

·products which differ by not more than one set .. _i� each· system,i . .

* . .

\ for instance Acr0

., aK) and (1r1

, nj ). Thus, in terms of the compl�te

set of Slater.determinants which describe the system, see Eq. {22): . ! K 1 1 .

<Gcm>oK = fj :S1 (oJBJ) J (2 r12 (l-£12>P2(0ICj1. 2; l'. 2 I) }�t1

dxldx2

' ' • I . ,•

where in order to separate the space and spin coord�nates the .. .. .

Derac ·�dentity. [�ee f�r· _inatan.ce., Dirac (22) p�. 221] .,,

X . 1 (0 1) 8

. =. 2 l . 0 .

�12 .;_ £12 �(l+4!!_1· !!.2) • aY = � (� -t}; .. z -= i (1 o,s = ·t 9 -1)

is used� Here the� operate on the electron spin functions

-20- UCRI:, 9998

a = {�) an� 13 = (�)., while �12 interchanges the_ space .....

coordinates of electrons land 2o Thus if Z is the external axis

· of . quantization-:

'_<0crro>o

K= - ½ �- B1<0'j B�)/ f4 �12 :!i !:.� r�2 P2(0Kjl,2;1t,2')i1:,1dz.t�

. . . ' . .· 1J .. . . . . . ·. ,, . . ·. . . . . . 2'=2

ar.:.d

= - � {j B1 (c jB�)J.r�

2 _qI(n11rj I 1, 2) q1 ( cr0cr;I 1,2) d�1 d£2

Here it is assumed that ·(z0

) and (�K ) can be expressed in

tei-ms of � single · Slater detemninant while (II) must be .. approxiimted'r,.

. '

by a series expansion where B1 and Bt are the coefficients of (rr1 f

in the sta�es -�0 and �Kand ci is the coefficient of the spin

pe1�utation (see· Eq. 28) . which makes the components of.� 0 and YK identical except for one set of 9fbital�-3� ·p

1 a� two for p2.

T'ne a0

3 aK

' and �i and rij are molecular orbitals given by

Eq.. (4):

...

cro = T6(2} = � To1(b)£ . p,

. ; . * + ·.

:; crK = TK(�) = -� TK.e(a).e•

[(b)£]

(a ).t

1

c£1)1 l+S12 = -../2-

-1l'-S12<?>.e2

(34)

-21-

. .

UCRL 9998

• . where the linea� transf'.ormat1ons !o and !K and !i ·�nd !j are

column vectors operati� respectively �n �he space of the a- and .-:. ..

1r-orbitals j 1n:auch a manner, that the·ene:rgy of the two systems

is independently minimized;.. ·.,

5<(1zq} 1110'

1 (1zo» = 0 .

a< (TI.) I J11r ,1 (rr)> =:= o .

(pil and 4>_12 are d·;treo1ied orbita.1st cent.�red �t .ineilYl;boring atoms.

l and 2 ., see Fig., 3., used in ··forming the sigma bonding and anti­

bonding or·bitala;; (b)1 and (a)1. S is the overlap matrix for

the c:(2p�) orbitals, Tilus in Eq. (32)-the effective one electron.;..a

density matrix is:·

· ( * 1, o l t ) KOK ) , . . p1 a0a-r -a iJ 1,. K

p ( 1 J 1 t ) = . Z · K · B ( C · B .�) B. ( C--B: ) . l · . ·. · . Kijk �eK i j .j K _K-l{ (36)

f;.�;--di;;;t;d-h;;;id-;t;?rl�-�;bit;i;-c;;;-;�;-i;;t;;;;-v;�-vi;�k---<2� > are; a) for the carbori atoms in a planar aromatic ring:

in X .. - cos x· 0 C: (2s)

0

<P3

l . · 1 . i. 1 �2 cos X ,..;2 a 1?- JC ,./2

1 . ·l .1 :;j2 cos x ,.J2 sin X -"1 2 O

j I• i C; (2px)

C; (2py)/2 cos f?!

, pos X = -Vcos �-1 I

C: (2Pz 0 0 0 1 / \C: ( 2pz ), ·: A . ,

where f3 is the bond:·angle $2�3 ( = 120° for sp2 � hybrids. as in . benzene� and b) for the tet�anedral orbitals in the CH3 group:

2pa\ /sin a - _co� a . o · O \ ;c;(2S). 1 l 1 0 .'/�"".

·. (35a}

=• Ts oos a ,JS -a n a "Y·:·�t;;,

l cos "' l a in rv . l · -l ) \c • t 2 ) I J sin ct = -y2cof ···u

2h2

2h3

.Js . '-" :Js � � :J2 .JG . �-.\ Py 1 . ' l . 1 -1 ·. .

·:JS cos a Ts sin ct -�2. Ts _C: (2pz) (35b) . . - . /."..

where 0 is the bond angle (2pcr)(2h1), i = 1,2J 3 (e = 2 arc sin�2/3 > .· 1r/2, for true tetrahedral hybrids as in CH4). . .. ..................................................... _ ................. -......... -� .............................. � ................... �-..................... ._ ............ ...,,,, ... -..., .. ..

-22- UCRL 9,998

If an ave�age energy 8e = �€K ·ma� be assumed for a_aingle type·. . *

or a-orbital excitation (a �.o) the sum.over the comple�e set

K above yields:

Now, ·since the a-orbitals are local�zed.; t:P,e .significant . *' ....

. . contributions �o Kf j a.re the one .and two center . integrals bet wee¥

bonded atoms. Conseqdentl:i· the ,/ontEl:a't t·errif w'i11· be non zero oi-ily­

for a-orbitals cente�ed about the particular nucleuso · This·�eans

that for a;,:,omat�a protons H\ the C-H b0nd orbitals ne�d only be . .• •·

cona-idered in Eqq 36 while, fo1., the ring skel�tbn nuclei c13 and

N�4 all the directed ·:··a-orbitals and .the· 1s· orbital about. the : .. ;·7 ,·, ·• ,.. . . \

center· mus�. be taken· into a_cc��t. · :.Thus ., for an aromatic proton

bonded .to the nth oarbon __ ,atom{ : �; i

.;,. · ' :" . 1cl-1 i j� 0

• • .

P1(l;l.' )H #= � on2 x [-pi(H:1S H:1sl1;�_1·)+p1(Cn:4>1 Cn:<t>1flzl')]. . .· 2b.e(l-s ) 2 · ·

·

(37) ···. �nd fo:r 'c13 on (�14 ) 1n' the ring skeleton at �n= · .. ··

. ·"_ �s

+ �- . 'It!}' 2 .x. P1 (Cn;ls · Cn; 2S.�l;l 1 )}

. 2Lie ' ( 1-s ) 2 . . , · . • . .

where

•25- UCRL 9998

exchange integrals correspond to the virtual excitation of the-

three a-bonds at en, see Fig. 2. Thus,

· · 1, l

1 .• -.·-· . . _,,__ K11 = (sin2x K88 + cos 2x KXl�-12: ·i ) (1-a 2 ) �- == K. 0 (1-s 2 ) 2

mr ·. zz �z zz • . l . . · ·

T22. �. (cos 2x IC:B ·+ (1 + sin2x) IC'.':� -K2 1 2').(l-s2)-½ = K •. (i-s2)-½

J,_7MT' - 2 zz . 2 . zz zz . 2

ir3::s = (co�2x ic:s + (1+ sin2xl �.:. �· :3' )(l�s2) -l= I{ • (1-s2) -½� · . 2 zz 2 · zz zz 3

(38)

1·1l).01"e ·�he para.mete� x varies t'!i th the a-bond angle., a ee Eq. (35)

(for cp2 hybrids ain2x = l/3}., oos2

x = 2/3). The carbon non-zero :• "'

cn0 cente1-i. integrals. are· K:: and I{�� =,)Ir� w�e�e �he atomic ·

orbit�ls indicated by (s-.11x.,y,z) ai��.: th�_ C (2s1 2px' �py.,_2pz). If

·the. t;m center int�grala a.re neglected then f'or sp2 hybrids:

where

In this approximation, the hyperfine split.ting constants ., are . .

given in the form derived by McConnell and Chesnut (23) and

MacLa.c'hlan, Dea..-rin,.an and Lefebrre ( 24):

Acis � Q1cPin ·'. Q2o<Pfn-1 +··vfn+1>

where

(39)

(40)

-23- UCRL-9998

is the probability of finding an unpaired ,r-electron in the-neighborhood of . th

. . .then atom in the ring, and the st�\tes (II.e) and (1\_) differ at mst by one

mo. ('rr nk,7T:k,,).· The directed orbitals 4> and 4> 1 are respectively centered

� /fl ni n i about the neighboring atoms C and C ,, see Fig. 3. Also,. the contribution

1 n n from the exchange integrals correspond to the. virtual excitation of the three

a-bonds at C, see Fig. 2. Thus., n

11 2 SS . 2 w...XX l I 1 I 2 -¼ 2 -½ K . = ( sin X K · + cos X K - K )(1-s ) 2 = K1 • (1-s ) 2 .mr · zz zz zz . . . ·

K22 = (cos2:x. Kss + {l+sin2x) � _ i<:2'2 1,)(l-s

2 f½ = K ·(lr.s2)-½ mr 2 zz 2 zz zz . 2 ·

K33 = (cos2

x Kss +· (1+sin2:x.) � - K3 1 3.

1

)(1-s2 r½ = K •(1-s2)-½ 7itr 2 zz .:2 zz zz . 3

i3S = Kss = K mr zz s (38) . 2 where the parameter x. varies with the a-bond angle, see Eq. (35).(for sp hy-

brids sin2x. = 1/3, co�?x. = 2/3). The carbon non-zero one C';=nter integrals ·are

t'6 = 'ifY' where the atomic orbitals indicated by (S, s,x, y, z) are the C (18,zz zz . .

2S, 2px' 2py' 2.Pz). hybrids :

where

2 If the two center integrals are neglected then for sp ·

K = ±: Kss + g �3 zz 3 zz

.(39)

-24- UCRL 9998

In this approximation, the hyf;erfine splitting constants, are ' · .

given in the form derived by McCo�ell an�. Chesnut (2· ) and

MacLachlan;·Dearman·and Lefebvre (24); and Karplus.and'Fra�nkel (25):

. 1r

A 1 = % Ptn ·. H. . .

S = �1r· _qs_c_K_s_2_· - # [qao- = <l�J?.2)s �!�1)al>]. v • .6€ t (1-S )

{40)

. · · 471' q11c<K> · 2 1 Qlc = T 6 ci5 Ae(l-

s2

) • (quc ::: <.I c:

4128 (o

) I )], (<R) = 5CK1+2K2)]

where the parameter·cos2x = 2/3 for exact sp2 hybrids as in

b Th it b id th t f. . 2

0

h b • d . th QI enzene� :us, may e sa a or s,p,., y ri s . .- e. s ·-are ..

-:,� universal constants ..

The experimental' evidence is that then-electron coupling

to· an aliphatic ,side-. chain ,proceeds significantly only to the

a-protons� This can be explained on the basis of hyperconjugation

as Bersohn ·(26) and M�cLachlan (27) have done s�ccessfully for the

tolu-_and xylosemiquinones and for Wlirster's blue· ions. The -cH3 (or -NH

2) fr�gme�t is attached to the aromatic system through

the C-C cr-bond but the excited triplet states which contribute

-25- UCRL 9998

to the H� splitting are those of the three C-H a-bonds. This

means that there exists a correlation interaction, leading to

a valence bond ., between the· w- and the r.·CH3 a-ele�trons.. .

. . . . .

Since the three protons in -cH3, see Fig. 3, are exactly

equivalent, due to the rapid rotations about the c-c axis which / ' ..,

• I ' "• •

oceti.:r. in solution, the· tm:ae·e directed C-H bonds may be· reduced

according· to c5v symmetry:_: ..

(1/-./3 : 1/�· 1/�)

�.-:;.;·-�!. (41)

where the (b1.) .and ·(a1) (1 =·1., 2 ., 3) refer to the c·H1 bonds, see

·Eq.· (34)�

The effective density matrix for H3 is then,

. . Kii p� *

p (1.;l'}l = . L . mr on P1(cricri _l;lt) .

l · H3 11,2�3 e

The highest contribution, ho�ever, arises from the virtual . *

excitations o-2 -+ .a2 since the. exchange integral is between

orbitals with maximum overlap:

22 . z ' z 1 : Hl H! · Hl H2 K'11'7T � Kzz • (Kzz - Kzz ) = K

. (42)

(43)

-26- UCRL 9998

where z, z' are respeotively.carbon 2�z·orb1tals at Cn_and en,

(see E'ig. 3) and (Hi., H2, H3) are the H:1S orbitals for the

CI\� protons, ·. '

· The c13 splitting in -CH3 arises from the vil"tual procea·ses .

(a00, � a�0

, ). (a1 + a�) and (c:1S -+ a�); (i=O�l,2,3 in th� c-cH3

group).

p1(l;l' )0, = P�n[� p1(a a*j1;1 1 )+ �', p1 (1s a*/1;1 1 )] (44)

l'ihere

CC· C I C' · = K - K · ·

mr . mr

.

' .

C and C' are respectively the hybrid.orbitals. centered at en and

en, in Fig.' 3. The highest ··contr.ibutions are ., hmtever, from the

one center integrals K:, , · which is the same pr�cess a� for

aromatic protons, and Ks· ·_'I'he hyperfine coupling constants for

the -CH3 group are then

.(45)

· where

-27- UCRL 9998

sin2a = i/4 tor tetrahedral hybrids� aee Eq. 35 and r12 is the

distance between two H atoms in the �H3 groups,

T'.a1s appr.oxima.t1on, where the spin density qln 1n the a-

electron plane·- is pJ;toportional to the p�obability of finding an

unpaired n-eleotron 1n that neighborhood Pon' ia_verified by

evaluating the Q's by two different methods� . *

A) The excitation energy of the virtual processes A€(a -+O) * ·

·�nd the exohange integrals K: . are obtained semiempirically.

The fi�st ia evaluated in the Heitler-London (28) approximation,_

whe�� for a diatomic molecule A-B the bond energies are;

and therefore

K - s2

J Ae(a -:?=- a*) = 2 :AB A: AB

· 1 - S ·. AB

(46)

E0

is the kinetio energy of the electrons and JAB and KABare the coulomb and exchange energies respectively. For Slater

type o�bitals it can· be.shown that the contributions to the total

bond energy are 10 to· 15% from JAB and the rest from KAB' and

since SAB < l,

(47)

where Dis the dissociation energy or the bond. The experimental

valu�s of inteJ?eat are then (se·e. for instance Cottrell (28),

chapter 9)

D(C6H5 - H) = _.162 .'­

D( c6�5c�2-H) = �_133 �-

-28-

\

(ac:H � .138)

(AeH_ �- . -113) ;:,

(Lle(c-c) = .11s )'

UCRL 9998

(48)

T'ae carbon atomic ex�hange integrals have peen ·evaluated.- by

Ufford (50) from the spectroscopic term values [see for instance• • I • 1' t •

Condon and Short ;ey ( 31 ) . p. _ l 7 8 ] , th�s ;

K�: =·<G1> \� � <G1(2p,2S)>c = o·.1062

• ..xx 1 +l+l 5 3 . 2 ( . ) . . Azz = 2 Kzz = 2 <F2> = 50 <F 2P,2P >c = o •. 9i3o.

where <G1> and .. <F2

> are the term. �alue �verages for. the ground

and excited E1tatea or C evaluated by Ufford. For sp2 .hybrids:

K°

�= c; <a1>c + <F2>

0l = o. 044os

(49)

.. · The two· -�ent�r int�graia have· b�en· evalu�ted ·ror sp2 . hybrids by

Altman ·(s1')°

�1th S�ate:r atomic orbitals. For interelectronic . ' ...

distances found.in aromatic compoundS J [see for instance (32)

M234] these· are·

z•zr .

. . Kzz (R0

= 2.53) = -0.1003

z•z•c Kzz · 2.91) = .. 0�04557

z•z• ) . Kzz (3.49 � -0.015

2 1 2•c ) · Kzz 2.53 = 0.03557

K!: 'ls_ 1

(2. �2) = O. 02737

(49)

Altman also evaluated the carbon one center integrals which were

higher 1n magnitude than those given by Ufford� however, this

discrepancy 1a unlikely to occur for the two center integrals

-29- UCRL 9998

since the Slater o�bitals are known to be a better approximation

far from the nucleus than close to it. T'ne exchange integrals in

Eq.. ( 40) and ( 45) are then: .

K1. = 0. 01669

K2 = K3

=.0.0085

K = •0.030

Kc,� K1'

lne values.of K6

and Ae' have been.estimated by Karplus and

Fraenkel:

K6

= • 023

Ae' = 10

The proportionality constants thus. obtained Q0

, are given in

Table I •.

(50)

B) If the probab.ility of finding an unpaired -rr-electron in the

neighborhood or the nth ring atom, p0

� is known then the Q's may

. be evaluated directly from the ESR spectra. The two conventional

methods or ap.proach to the problem., the VB and MO have bee� used ·

successfully, and the comparison with the constants obtained above

is given in Table I.

Hence, the agreement between theory and experiment is evident

from Table I. It is also important to note that, as Das et al

·: (50) have pointed out in the theory of the hyper-fine interaction· · 14 of the N · atom, the oontribution to the contact term from all

the electrons 1n the system must be taken into account. Indeed.,

for !7!0lecular free radicals Karplus and Fraenkel .{ 25) and

Pai.-ticle-.

C

., 1r

··

. .2 li.

_1/2::: · ' .. 1/2

t.

2.0023.

.... ·• ' .· · ·o�ar.·1· 4; :} · .

. . /.: . ' . \ ) ·-.·-1,2 · ·-._lo40=:i-3:' · .. ·· ..

. . ··.·

.··. 5 "2· ' .. · o· 71::· ... , .� /4 . ., ;J � .. ' ·.

. . • ..... :

I��3.' ... : ·3/2··::. ·_1.4rr4

:_3/2._. o.4285_

:..29a-

.. . q · 6 :iclO ..._.n .... (2Ra.J)

12.·�-0.

. 20.31.

.10.95

·76.00

21.37

6.198

Brr :J

0 ·(.1 ·(b)_.n -m

(oe);

.5o6.2 {IS)

' ) "."I

1 'D• • c.;. 9. --5 J •• ,. Q

(W�(2S). 569 (2s)2

. 2 17.15 R(O)

'.._1390 ( 2S )2 .

119.0 · R(O) 2·.1!�, oS6 . ( 28 )2 · .

.. "'"'( ) 2 33.93 .!.t O · c�eo )2

.)tJ

� co..lculG,ted frou ·the .

·:· . 0.. ,Q .· "2 .

-14;

. r

. i

o(c) j �--

:bt:)J:>�Ctiob.·

.. �)·o· �--

or

·. :-:�a,,:

-4

( C) ·-1:�. 7' . .: , .-:t.8 ·

.-8·

••· ·( ,· t . ..

. l

,,·:· .

::,·· UCRL-9998

:.ci?J.ct!.l3ted. :t��om.. �hhc .·unr,ai:t\�d electron· dc��!;:t:t:,j :�1�l ];SR . · -�pactr;;.,

:�.

. ;:.1 h (q) ..:... .. ":' .

..

. •. >··

· .. _'.. -"\(n) :·+39 • .) .• · .·._..... . .. -

·- .

Q,,�·::-·: . J:l'.:l

·. . -� ·.

...

-29h- UCRL-9998

{ e.) T'ne g-fa.cto;-s. ( gn = i1 n ) are taken from the table of �rimentaJ. value�

. n given by Rams�y., N. F. > "?luclear Moment:3 U, .John Wiley and Sons., Inc, N. Y � ·, ; · � ·

(1953)..:

{:b) Except· :for the: lI:IS atomic orbital the value of·. R{o)_ 2 �s bee� evaluated·. . . . .

from IIa.rtree.-Fock "trrave functions, see Hartree, D; :a; ''The·. Calculation: of I

· ·A:romic Structures", John Wiley and Sons, Inc_., N�Y.· (1957)(.Append.ix 2),

· Tioothaen, C. C !' J., L. M. Sachs and A .W. Wei�s., ·Rev�. Mod. Phys. �' J.86' .

( 1960 �; Allen; : L. c. 1 J. Cb.em. Phys .J!!:.1, ll56 ( 1961).

(c) 'l:1 J i's the contribution from neighborning atoms in the 1f.-electron system..::. .

. ' .

(d)

(e)

othe1.. than C.

From the work in references· (2), (24), (26), and (34).

�1--om the work in reference ( 2:'{) �

(f) From the ESR spectrum of CH( COOH)2 _

( 42)( 43 ).' • I •

{g) · ;From the EMR _shifts of V (C�02)3 (49) .... .

(h) . From �he work of references (2), (26), and (45).

(i) From.('the -work of :references (26)(27) and {52).

(j) From· the work in reference {36).

(k) From :the l·TOrk in reference�(25) and (47).,

(1) Contribut:i.on from the viri;ual procesfl (aCO """"'7 aC0*) determined from. the <f-3 ..

apli tting of the dihydr�ysemiquino_ne ion ( 52) �-. ·

(m) , From the work in reference (52) for the methyl derivatives of the p­

benzosemiqu.inone ion.

(n) From. the work of references {25), (37) and (41) for the c13 ·splitting in the·.

CH3

radical. . .. .. . ,4 ( o) From the work. in reference ( 50) for atomic r ..

(p)

{q)

(r)

(a)

-29c- UCRL-9998

From the -work in reference ( 4o).

From the �mrk 1n reference (39) :f'or the tetracyanoethylene· negative ion.

From the work in ref�1·ence { ) for the Wurs�::t''s flue ions. i

.

From the work- 1n reference ( 46) for the tetrai'luono p-benzos� ::quinone ion.

'

-30-

UCRL 9998

Ben Jemia and Lefebvre (51) estimate that for each c.-H bond . . *

excitation the virtual processes (c:ls � e1 ) account for 15%

(-4.12 oe) of the total· contributions, to the c13 'splitting in

the CH3

and CH radicals.

The a .priori knowledge or Pon is therefore or utmost

importance since it enables the proper assignment of a chemical

structure to a free radical from itt�.· .. ESR structurer�··,:;:;.,

Applications

The importance of the two particle correlation 1n a chemical

bond is the most significant contribution to be derived from . .

this type of calculation. In general, it is found that the

hyperfine coupling constants at the end of a bond are of opposite

. sign, and:therefore arise from antiparallel spin polarization.

This is a consequence of the combined effects of the Pauli exclusion

principle and the correlation interaction, which demand that the,

probability or finding two electrons or unlike as well as like

spin in the same neighborhood be zero.

Using the VB method to ;nterpret the hyperfine splitting

in triphenyl methyl1 Brovetto and Ferroni (53) found that the I � ' o "

1r-electron densities alternate in sign for neighb?ring. aromatic

carbon a·toms, whereas the s_ingle determinant Ruckel MO approxi-

. mation without overlap would predict no change in sign (26). Thia

however is a consequence of the mathematical approximations. Since

in the VB method the dominating terms are the bond exchange

energies., the electrons remain far apart., in the resulting wave

function., slightly more than the actual repulsion energy would

demand (5). Conversely in the MO approximation� only the electrons

-31- UCRL 9998

of.like spin remain apart due to the antisymmetry condition

(P.ermi correlation) but if the correlation property is introduced

by means or a series expansion where the.perturbation is the two . '

el.ectron ±ntera·ction,. the two methods converge to the same

answer,

An example of academic interest, in which both approximations

have bee� used1 1s the allyl radical discussed by McConnell and

Chesnut 3 The VB wave function is discribed in terms of four_--.-

4' particles, three electrons and one hole, with 2,;, = 2 cannonical'..:>.

structu��es in the Pauling-Rummer diagram, (see for instance (7)

chapte:r 13):

(52)

·where

(Ill) = :k Cl� C2 cl -c2 �cs 0 -C3

�l ·c cl C2

1· . 2

(II2) =;� b JS b . C3

Here the Slater determinant wave functions are expressed

symbolically. The atomic orbitals _which are bonded together are

connected by an arrow pointing towards the spin function a. and

the unpaired orbital is bonded to·the hol�.

The probability of finding an unpaired electron 1n the

neighborhood· of Cn is then obtained directly from:the super­

position diagram:·

.,.32-.

n1 �+l( 1) J 1· �

P · = � b-c . P1J·' ((n) = � B1(n1)l. n� · ij 2 ij . . . · · i · .

UCRL 9998

whe�e ·bis the number �f �-eleotron bonds, c1j the number or

closed contours� the auperp�sition diagram �f (ni) and (llj) and

n13 the number of bonds which separate n from the hole �n a

closed contour. The term� are zero if the hole and C� are in • • • 1 �

different closed contours •. Thus, for the·allyl radical the

VB approximation �elda;

2 Pio

=

P30 = 3·

1 P20·= - 3

In the MO approximation .the· transformation which.minimizes the

on� 1l'-electron e�ergy has been obtained by Chalvet and Daudel

(54)

. ! ,= :(�: :::: o. 7�17· 0.4537 -0.7217

with energies

.El � l. 4934 f3

·e2 = -1. 540 f3

e5 = -1.339 f3

o. 6022) -0.8994

0.6022

� is.the benzene resonance integral and the corresponding lcao

mo•a are uaed to generate the Slater determinants which form the

basis set for the expansion of the w�electron system, see Eqa (22):

(53)

where

and

0.9249

.2049

B= ,

, �133 . �·. 0660

2 ( Ill ) = ) · ' � 171" l ,r 31 f ·

2 C rr2 > · = J I 1r irr 37! 3 J I

:\ rr3 h = 2� I '7ri 71" 21T s 11

.. .,; -,A c I I '11"1 ¾'ll"s I l

2<rrs>2 = l c I I i1 'lT2n3 I l

�

and (2) 1T . 1T

Poi=

Pos = o.s1

1r

p02 = -0.11

-35-

{lll.) (II )

(II)::':-, . 2 ' - . - : . ( l!3 ) 1

{Il.3)2

UCRL 9998

� lh'll"2=ir3!1 - � 1171"11T2'll"31!} .·

ll1r11r21r

3II r

Here the sign of the coupling constant is also found to alternate

, along the.bonds. This is a general characteristic or the type

of compound defined as an nAlternant Hydrocarbon. *1

The problems or actual interest are more complicated than

the allyl ��d1cal, and as the number of electrons increase the.

MO approximation with the two electron correlation taken into

account is found to be most suitable. For some aromatic free

radicals the method of ''alternant molecular orbitals" developed

by Lowdin (4) can.be applied with some degree of success. An

-34- UCRL 9998

alternant hydrocarbon is a compound where'the aromatic system

can be divided into two subsets of atoms, I and II, in such a

manner, that.no two atoms of a subset are bonded �ogether. It is

then a �haracteriatio �f th� aiternant hydrocarbons that the . . '

single electron ·MO's occur in pairs., with equidistant energy

.separ·.:i. .. i;ion from a zero level,· in the same manner as electrons

and holes are situated with respect to the conduction band:

. . . .

-For convenience the 'i are orthonormal atomic orbitals which are

obtained from the real atomic orbitals$ by means of the following

transformation:

or

[s - f ,h{l) �(1) d . 6 '], 'ft ij - •1 �j �l - ij

{55)

. The. _ele�tronic. �ystem may then .be described _in .terms . of _the lcao

mo •_s:

. ' .

which are used to construct the single Slater determinant D used . '

to generate the doublet state:

(II) = 20 D-

where

(56)

-35-

D = I lt1I tlII • •' tnr fnI! to l I

UCRL 9998

_ � k1! (oos e )2n-.1 <s1n e >j f1 I·· ·v .v .. ·v< > .tc . J .. ·t 11j=l ·.1� \ · � k k .. k+j k+j o

11 • • ·tktic, • • ·t (k+j >""'lk+J) • :"' o 11} when the p�ameter e is independent of k., and th_e unpaired

electron orbital �s denoted by t0

• Here the .significance of the

angle e is as follows:

e = 90° , t iI = -1/1 iII = ,; 1, = upper half.,· antibonding MO' s

The ground state wave function will correspond to a value of

tan e < 1 since fore= 45 ° the exchange integrals tend to zero.

This means t.hat in Eq. (5b) tan e is a perturbation which intro­

duces configuration interaction into the �-elect�on system.

For an odd alternant hydrocarbon with (2n+l) aromatic carbon

atoma and ( 2n+l) electrons the wave function is then (55)

2(n) = l:.. 1r cos2e {I It t · · ·,; v; ,ft J f �N i=l., ••• n 1 . 1 1 n n o

+ c ( 3) Z tan 8 [ J J • • • t ,fr 1 i/1 • .. 11-1 J J • • · t t t · · · II0

1 i 1 i O • 2 i 1 1 0

- � 11 • .. ,,, 1 w 1, t O

• . . I 11 + · .. J c 55)

where the coefficient for the doublet operator for three eleqtrons

is c0

(3) = 2/3 [see for instance, Lowdin (4)] and N is the normal­

ization constant.

-56- UCRL 9998

· 4 [ 2 n 2 N = n �os 01 l + 3 z tan e1

+ 1==1 .,

0•n · . 1=1 . . ·]

The n-eleotron density m�trix correct to second order is

then:

�·Jhere p0 is the density matrix for the orthonormal set ! and for

the sake of simplicity the superscr�pt n has been eliminated.

Thus� in the neighborhood ot Cv:

· 1- · S1 2 · 2 n 2

Pov = . lTvo (1-9 � tan 01)· - l+� z tan2(\+··· -- i=l

1

+ ! .z Tv1 tan 01 [Tvi, + � Tvi tan 01] + · · ·}

i - ·".SI ---

(57)

Now sine� the symmetry requires that for an odd alternant · hydro­

carbon T0µ_Ct.t· in II) = 0 the 1r-electron density in the neighborhood

or C� w1�l be negative. ·The general result is thus applicable to

the allyl radical which is the smallest odd altern?,nt free radicai.

The even alternant hydrocarbons .,_ with 2n aromatic atoms form

two types of' free radicals according to whether an electron is ·, ,·:

added or abstracted from the system. The �-electron wave function

to� the poa�tive ion with (2n-l) electrons is then:

:-37- UCRL--9998

2(n}+) = . 1T . {cos2 ei1=1, ••• n-1

fi+ .

. cos ·e 11 t , · · · t ?Jr ?Jr. f I n 11 n-1 n-1 n

where

+ sin en I l?Jrl?/rl •• ·'l/rn-·l'n-11/J'n 11

+ i � t� ei [:as en 01+½II • • ·t/i ,i/J'n .. '. I 1)

• • • ?Jr . 1/r. 'W' • � • I ·1- �2 I I • . • t°. 7/>�i I 1/r • • • II:i. 1 n 1 n .

+ sin 0 ( 11 • .. ?Jr. ?Jr. ,t , j • • II . + -�-f

n . 1 1 n

'Tr 4. n-J.N = · · I I cos e. [1 + J .Z tan 2

e 1 + • • ·]+ i=l • • ·n-i ·· 1 i=l -- . :) ·---- .. ·-

For the. neg�tive �on �he extra electron is to be placed in an orbital ortho-

Thus,

7/r . III. = - sin .enfr + cos eJ/J' , •n, rtrn n n

(58a)

'""58-. UCRL 9998

· 2(rr}- = 1T · coa 0;1.

(l.·+tan�fJ�){coa enllt1fi .. ;i/ti"i'n ' 11 .. . i=l,·•·n

itJN_

+ sin . e I It 1,r, 1 • · ·tn i �, ,t, J 11 + � :!� tan e1 [aos en {JI·• ·ifit�{i_,···. 11

- - 1 II , ·· · w t , t , ···I I-; I I · · ·,; w , t , · · · 11 } ·2 . '.

1 :� n . . 2 1 i n .

+ sin en{l l.,•'·t;1.t1 ,Wn· • • 11-½l 1 • • ·_tii/ti,tn'; · · l J

.,

- � ll···�t:1.. 1:t�,.;.11JJ ... J

The �-electron'density matrices for the two free radicals will '

' ' . ' ,, .. . . j

' • • • • '

ha�e _similar .f'unct�onal_ �-epe�dence �n e1: . · ..

(58b)

1 · t 2 n-� 2 · · 2 2 p0 (+ ion). . n':'"l . _[l- 9 � tan e1][pnnc?s 0n +: -�n'n'sin anl

1+.?.. z tan2e , . :1:-l 1 ·

31=1 1 ..

4 · l +.5 �� tan 01 [Pus + _s .tan. e1(p11 + .p1,1_, )_]

· 0 · ·: 2 ·2 .'n-l :2· · · ·. · + P

nn , a1n2en c1 + tan en - 9 1!1 t_an _01 ]_ + ...

Now,since Coulson and Rushbrooke (56), Pople (57) and �icLacklan (58) have shown that P

n'�' = P

nn the physical significance of

Eq. · (59) ia that for a constant a-� parameter �·the·correct

(59)

-39- UCRL 9998

electron densities £or the even alternan� hydrocarbons with C2v

,

or higher, symmetry may be appro:r'"1mat�d from the Huckel MO

approximation;

P (e p) · ·n-1(

T2 + T.2,ov . x. = 1 + §. .z tan2e vi vi

Po. 'V (Ruckel) 9 i=l 1 . . 2T2 ....

(60) vn

�hus it the electron density at the 2n carbon atoms could be

·measured, the ta�2e1

correlation paramete�s wouid be obtained by

inverti�g the ma.triX in Eq. (60). This in gen�ral is not possi9le

sir..ce some of the substituent in the aromatic system do not have

a magnetic moment or the· abundance of the paramagnetic nuclei

is beneath:the detection limit. The validity or the Ituckel·

approximation is indeed eviden� f�om the ESR spectra.of the ions

of anthracene,, tetrace:ne., perylene and naphthacene observed by . .

Weissman de Boer and Conradi (59) and Carringtop, Dravnicks and

Symons (60), Here the splitting for the unipositive and negative

ions is different indicating that there exists a different ' ,.

· correlation interaction for two different ions where:

p0v(exp for the +·ion)N 8 n-1

p0y(exp tor the ion)

= l+� 1!1

i:1.1�:-�:ta The C?mplexity of the pi,..oblem increases further as heteroatoms

are introduced into the aromat1c·system.· However, if the Coulomb

and resonance integrals for the heteroatom can be estimated with

some degree of accuracy (61), then the single determinant Huckel

MO'a predict electron densities in fair agreement with the . ,

experimental values (26)(45). Hence, these compounds. are only

-40- UCRL.9998

pseudoalternant hydrocarbons since the bonding and antibonding

s�etry orbitals are not equidistant from a zero level. However,

. the tw� particle cQrr�1�t16n can still be evaluated .from Eq. (60), .

' .•. . .

Also, great caution should be \practiced in using the universal

value o:r the a-,r parameter· _since � is· 'a .function of .the a-bond

angle.and the a�bond distances.

SUl'-'.!MARY

In conclusion, it is important to note that

(a) For '?Very type of cr-bond in an aroma_tic free .radical,

there exists .a sem:tempirical constan�, known as the a-� parameter,

which, together with the probability of finding an unpaired�-. \ . . ,.

electron in that neighborhood, may be used to predict the hyperfine • j

' •

splitting constant at the nucleus of an atom forming the bond.

(b) The sign or the hyperfine coupling constant.is found

to alternate along a chain of atoms as a consequence of strong,

two electron correlation interaction which demands opposite spin

polarization at ·t;he ends of a chemical bond,.

(o) The' success of the Ruckel MO approY..imation� with a

single Slater determinant, in predicting the correct �-electron

densities in alternant as well as pseudoalternant free �adicals

is due to the fact that the correction for two particle correlation·

int�raotions·can be introduced by pairing the·orbitals which lie

eymmet�ically above and below a··zero level. The correct electron

density ia then given by the Huckel approximation with a small

co�rection which is a ·.function of a �-electron cor�elation

.. par�eter e1 .•

-41-

REF.EREHCES

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, . i

la. .. , Weissman., S.I.; J_:- Chem. Phys. 25, 890 (1956).

2. McConnell, ll,M • ., J. Chem ... Phys. 24, 632., 764 {1956).

2a.. McConnell, H.M. j J. Chem. Phys. 23, l.188 ( 1958 ) •

UCRL-9998

2b, McConnell, 11.M. and ;D.B. Chesnut; J. Chem. �hys. 28, 107 (1958). \

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-42- UCRL-9998

17. P,:yce,. M.H. L., Proc. Phys. Soc. A63, 25 ( J.950).

18. Jahn, Ii.A. and E, Teller, Proc. Roy; Soc. (London) Al61, 220(1937), H.A., -

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Jarrett, H.s • ., J. Chem. Phys. 25, 1289 (1956).,

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carbon anq. �cH3 group 0:: .(

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corresponding. nuc1ei ·has. been indicated by an arrow in the:. .

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