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A theoretical study of hyperpolarizability effects in the measurement of molecular quadrupole moments
Volume
85. number
A
CHEhlICAL
1
PilYSlCS
LETTERS
THEORETICAL STUDY OF HYPERPOLARIZABILITY EFFECTS
IN THE MEASUREMENT OF MOLECULAR QUADRUPOLE hfOhIENTS Roger D AhlOS Unrversrry Chemrcal Lobomrory, Recewed
17 Scptcmber
Cambrrdge CBZ IEW
(IA’
1981. m timal form 25 September
1981
Calcuht~onsarc made of the hypcrpolaruabdlty conwburlon to the clcctrlc-f?rld~r~drrnt-InduL~d brrcfrmbcncc which ISused IO mcxure the quadrupolc moments of nondmolar molecules It ISconcluded 11~11h) pcrpol.wzabdlry A%ccs arc small for most, but not A. molecules
For centrosymmetnc molecules such as carbon dooxide or mtrogen the first non-vamshmg electnc multipole moment IS the quadrupole 0. The practicable techmque for measunng the quadrupole moment of a nondlpolar molecule utdises the electnc-field-gradlent-mduced birefnngence [l--5] This evpenment mvolves measunng the ellipnclty mduced m an mlrially linearly polarized beam of hght passed through a gas, or hquld. subjected to a hnown external field grahent. The blrcfnngence LSproportlonal to
IS
(%)w=o
the polanzabthty,
whim has to be obtamed
separately, and b IS a hyperpolanzsbtity term The fwo contnbutions can be separated by a study of the temperature dependence of the effect, however It has normally been assumed that the hyperpolanzabdlty IS neghgble Ln the only case where a temperature dependence study was made, for carbon dioxide [5], the hyperpolartzabthty term was Indeed found to be small The purpose of the present theoretlcal study LSto mvestigate whether this term IS neghgble for all molecules, and If not, to ldentlfy those for which a temperature dependence study would be desirable The hyperpolanzabdltyconslsts of three terms which anse from the distortton of the molecule by the external field grarhent [3-l,
,
=(-5~-tea~~J,,~,,)~=~
ahere eoflY Is the antlsymmetnc tensor The slmplcst way of evaluatmg wmLJ IS by a sum-over-states method usmg the perturbation-theory
cxprewon
[?I.
(~-~JcQ7.&.J=o= F [ Nl/(L, - Eg)3j x ((/IP,I/) -
$ b + O~scrc&l-,
where a
frequency of the molecule, Ihe first two of these terms and the hyperpokwbdlty slmphfies to Its lov.frequency hrmt,
cancel
+,
x
c*,
~OI~,lO~)~Ol~,I/~~llr~~plO~
[w%
(tol/+>(kl~,llX/
-
J%>cq
-
&J)‘l
Ir,rP IO)
+ ~OI~,Il~~fln~plk~~kl~~lO~) +, z,
[lh/(Ek
- &,)(E, - E,)‘l
x (~0l~,Ik~~kl~~ll~~/l~~~pl0~ + ~Ol~,I~~~ll~,Ik~~kl~rrplO~)
+ cc ,
2 b = B,@ . a!p - 93, 3p sap - S’C.J-~~~~~J~ 1B 17 .
where c and m are rhe electnc and mzgnetlc dipole operators. Thus expresslon has been evaluatc.l usmg SCF molecular orbrtals to generate the exated states The SCF wavefunctlons were obtamed m p:evlous stud:es of the multIpole moments and polanzabrhtles of N, [6],
At frequenaes
CH4 [71, CZH2. C2H4 and CzH,j [81,C3H,j
well below the first electromc
0 009-16 1482/0000-0000/S
excltatlon
07 75 0 1982 North-Holland
PI, Cl2 123
\‘olumc
85
nrrmbir
I
CffLSllCAL
PHjSICS
T.sblr 1 C.~fcuIxrd and c\prrrmentaf j3.51 xtlues oi the hypcrpofartztbtfttl krm b tn tmtts 01 10w6’ C’ rn’ J-’ Xtcthod 1 -0 0’9 -0 036 -0 034 -0 119 -0 13-i -0 092 -0 I42 -0 119 -0 553 -~
Xlrthod
2
CHf
-0O.U -0 06.f -0071 -0 182 -0 204 -0 139 -0212 -0 080 -0516
-0
L\P
-
058 -0’203 -04rOf
p &o
In thts case w-‘G approach using. &J-lt&rL=o
h-0
[Ifi&
- C&f
but wrth the molecular orbttals perturbed by an esternal electnc field F7 obt.uned by usmg the fmite-field method I 1 _j _This has th* advantage that one of the perturba..oErs IS treated m an rtcratrve manner, and as can be seen from table 1 (the column headed “method 2”). the resulttng values of v‘ are up to 50% dtfferent from those obtatned by the prevtous method A better method stall would be to treat ah three of the tields mvolved m u- ‘J usmg an tterattve techruque along the irnes of coupled Hartsee-Fock theory [ 13]_ If the wavefunctton m the presence of a static electnc field ts wrttten \Ir(Fcll) = @to) + F,d’)(F,) then
f
= 21n~?Irt’)(F*)I\I’~‘~(Bp)),
where B IS a magnetic field The coupled Hartree-Fock method funds the optrmum first-order changes to the molecular orbttals due to an external perturbatron,
_,
-3) + - -
the coupled Hartree-Fock
(o-‘c,$),=,
__evaluated by a sum-over-states
= c
(W-tG,,p)w=o
Consequentty for w_tG rs
= (W~~,)(~-~G,,p)o=Q
1981
tt follows that,
@h(f,) = @?’ + Fo#)(F
[ICI] CO, [5]. and CS, ]I I] and should be reasonably close to the Hartree-Foch hnuts for these molecules The resultmg estrmares of the hyperpolanzabthry term b aregrien m table 1 m the column headed “method I *‘_ A sum-over-states method usmg the vtrtua.l molecular orbttals IS not a very rehable means of calcuiattng any polanzab&ty as these orbrtals descrtbe the evcrted states rather poorly_ An altemauve way of obtauung w-tJ ts to use the de~nttton [2],
(w-‘Jd,
1 January
LFXTCRS
elpressron
= 4th hcocc (#)(F,)M~“(BP)) 1 Thus obtarrung a value of w-‘G by tfus method m~ofves separate calculations to get the orbttals perturbed by an electnc field, and a magnettc field, and combtmng the results. To obtam w_tJ from o~-l G the firute-field method IS used As thts is quite trme consummg, it has only been done for Nz. and the result is gtven rn table 1. This shows that the CHF method results m a further change of ==30% tn the calculated value of b for rutrogen The caiculattons could also be Improved by the consrderatton of correlatton and drsperston effects. Dcsprte &IS, the caiculated values shotrId be accurate enough to show whether or not the hyperpolanzabthty ts tmportartt for the molecules considered There IS comparattvely httle expenmentai evtdence regardtng the hy~rpolanzab~ty cont~button to the mduced btrefrtngenee There IS a measurement of the mduced btrefrtngence m methane [3], which, as thus molecule has no quadrupoie moment, must be due solely to the hyprpolanzabrltty term, gtving a value of b = -(0.4 + 0.1) X 10e60 C3 rnq Je2_ The tem~mture dependence of the btrefrmgence m CO2 has been measured 151, yreldmg a value for the hy~r~I~~b~ty of-(0 2 + 0.3) X 10v60 C3 mJ JW2 The calculated values of b are all negative, as are the values for the hydrogen atom (6 = -0.090 X 10W60 C3 rn’l Jm2 [2]) and the mert-gas atoms where thrs term IS deterred by the Verdet constant j14] _The calculated values generally increase in magmtude as the method used to obtam them rs Improved, and also as the basis set size IS mcreased, and are probably underesttmates. Collected m table 2 are the experimental vaiues of the quadrupole moments, and recent ab tmtio values. Also shown are values of the term $b, obtamed from the calculated values of b, and values of the expressron q b + @~,c~~~/x-T,
Volume 85. number I Table
CIIChllCAL
PIIYSICS
1982
1
Ab mmo from the of IO-@ wlcuhted IS the CC
[S-l I] and c~pcrm~cnt~l values [3 5.151 oi the quadrupolc moment. m umts oi lO-_4oC mZ The ralues oi $b XC t~hcn calcuhtcd values orb in table 1 and arc m umts )f 10mw C3 ma J-’ The values of $6 + O,pcir#T 31 20°C .llso I” U”lIS C3 ma J-’ are constructed from the c~pcruncnr~l values oi 0 .md Aa [ 161. urth the c\crpr,on of- acct>lc”c whcrc the value of 0 MS used Eth) lene has two lndcpcndcnt components of 0 - the VAIC quoted IS rhc o,, component u hers z ;L\I.S
-5_s -15 2 0 2-l 5
-49=03 -150=05 0
67 -.I
Imental
mcasurcments
exprnmcntal
values
the elceptlon
of 0 and (1
of CO,,
chid not separate
the exper-
the tcmperature-
and -Independent contnbutlons. the expenmental values of the quadrupolc moment are really valucs of
dependent
Aa IS the polanzab~ltty
-0 13 -048 -0 53 - I 37 -I 53 -I 04 - I 59 -@ 59 -387
66205 -33zO’ 53-_07 106-05 1’0~06
z
84 II 9 II I
calculated usmg the Note that since wth
yb nlculatcd __ --__-___
0 c\pcrlmenral
0
ab “IIIIO
wliere
I Jawx)
LLI-TIlRS
antsotropy
However from table 1 It can be seen that the hyperpolanzablhty contnbutes only 2-X to the total blrcfnngence for most molecules, wluch ts less than the typ~cal cupenmental error. and conscqucntly the neglect of tlus term ~s~ustlfied The two e\ceptlons to tlus are ethane and cyclopropane, where the hyperpolanzabthty accounts for over 10% of the total effect due to the smaller size of the O,,u,, contnbutlon. As the calculated valuesof b are probably underestimates, the hyperpolanzablllty may also make a s~grufkant contnbutlon to the blrefnngence III nitrogen Allowmg for the hyperpolanzabtity does not Improve the agreement between the ab uutlo md elpenmental values of the quadrupole moment for CzH6 and C,H,; mdeed It makes it worse, and the discrepancies probably reflect the fact that these are the molecules for wluch the expenmental and theoretical determinations of @are the most rhfficult i\s the temperature-independent term tn the btrefringence has been measured tn CO,, where tt makes a smaller percentage contnbuhon than m any of the other molecules constdered, tt should be posstble to observe tlus term tn nearly any molecule However a temperature-dependence study IS only necessary in an accurate determmatlon of
-9-l -87 3 -33 I-t0 3 45 2 -65 -110 75 5 3115
the quadrupole moment oi molcculcs such as cthanc cycloprop.me for wl~ich the overall cficctis sm.111 The author thanhs financial support
the Sclrncc
Research
Councd
or
ior
References
[ 11 A D Buchm~hx~: J Chc”~ 1%)~ 30 (1959) I580 131 A D Buchmch~m Jnd II C Lon~uc~-ll~gy~s 1101 I’hrs 1-t (196s) 63 131 A D Uuchm~h~m R L Dlsch .md D ,\ Dunmur J .\m Chcm Sot 90 (1968) 3104 I-t] M R 8~11~~113 A D Uuchl”&m~ .md J II \~dh~m\ Chcm Ph~s Lctrrrs 78 (1981) 421 151 \I R Ba~t.~gha A D Ituch~n~h~m D Nculnxh R h PwrcnsandJ II Wdhamr \lol 1’11)s -t3(1981) 1015 [6] R D Amos 1101 Phjs 39 (19SO) I 171 R D Amos Mel I’hvs 38 (1979) 33 [Sl R D Alnos and J II \11lh_1”1s Chc”~ Phjs Lc~rcrr 66 (1979) 171 191 R D An~osand J II \Vrllwms Che”~ 1%~~ LcItcrs8-t (1981) IO-I [IO]
J II Wdktms and R D Amos Chcm Phyc Lertcrs 70 (1980) I62 [ll] J tl Wdhamsand R D Amos.ChLm l’h~s Lcttcrs.66 (1979) 370 [I?] II D CohcnandCCJ Ro01h~n.J ChL”l Ph)s 43 (1965) S34 [ 131 R Bl Sw\cns. R Pnzcr and U N Lipccomb J Chem Phys 38 (1963) 550 [ 131 A D Buchmghamand hl J Jdnucson Mel Phys 22 (1971) I17 [15] A D Buckmgham C Grahamand J II Wdlu~ns to be pubbshcd [ 161 ht P Bogxwd. A D Bucl.mgham. R K PIerens and A H Whltc. I amday Trims I 74 (1978) 3008
85. number
A
CHEhlICAL
1
PilYSlCS
LETTERS
THEORETICAL STUDY OF HYPERPOLARIZABILITY EFFECTS
IN THE MEASUREMENT OF MOLECULAR QUADRUPOLE hfOhIENTS Roger D AhlOS Unrversrry Chemrcal Lobomrory, Recewed
17 Scptcmber
Cambrrdge CBZ IEW
(IA’
1981. m timal form 25 September
1981
Calcuht~onsarc made of the hypcrpolaruabdlty conwburlon to the clcctrlc-f?rld~r~drrnt-InduL~d brrcfrmbcncc which ISused IO mcxure the quadrupolc moments of nondmolar molecules It ISconcluded 11~11h) pcrpol.wzabdlry A%ccs arc small for most, but not A. molecules
For centrosymmetnc molecules such as carbon dooxide or mtrogen the first non-vamshmg electnc multipole moment IS the quadrupole 0. The practicable techmque for measunng the quadrupole moment of a nondlpolar molecule utdises the electnc-field-gradlent-mduced birefnngence [l--5] This evpenment mvolves measunng the ellipnclty mduced m an mlrially linearly polarized beam of hght passed through a gas, or hquld. subjected to a hnown external field grahent. The blrcfnngence LSproportlonal to
IS
(%)w=o
the polanzabthty,
whim has to be obtamed
separately, and b IS a hyperpolanzsbtity term The fwo contnbutions can be separated by a study of the temperature dependence of the effect, however It has normally been assumed that the hyperpolanzabdlty IS neghgble Ln the only case where a temperature dependence study was made, for carbon dioxide [5], the hyperpolartzabthty term was Indeed found to be small The purpose of the present theoretlcal study LSto mvestigate whether this term IS neghgble for all molecules, and If not, to ldentlfy those for which a temperature dependence study would be desirable The hyperpolanzabdltyconslsts of three terms which anse from the distortton of the molecule by the external field grarhent [3-l,
,
=(-5~-tea~~J,,~,,)~=~
ahere eoflY Is the antlsymmetnc tensor The slmplcst way of evaluatmg wmLJ IS by a sum-over-states method usmg the perturbation-theory
cxprewon
[?I.
(~-~JcQ7.&.J=o= F [ Nl/(L, - Eg)3j x ((/IP,I/) -
$ b + O~scrc&l-,
where a
frequency of the molecule, Ihe first two of these terms and the hyperpokwbdlty slmphfies to Its lov.frequency hrmt,
cancel
+,
x
c*,
~OI~,lO~)~Ol~,I/~~llr~~plO~
[w%
(tol/+>(kl~,llX/
-
J%>cq
-
&J)‘l
Ir,rP IO)
+ ~OI~,Il~~fln~plk~~kl~~lO~) +, z,
[lh/(Ek
- &,)(E, - E,)‘l
x (~0l~,Ik~~kl~~ll~~/l~~~pl0~ + ~Ol~,I~~~ll~,Ik~~kl~rrplO~)
+ cc ,
2 b = B,@ . a!p - 93, 3p sap - S’C.J-~~~~~J~ 1B 17 .
where c and m are rhe electnc and mzgnetlc dipole operators. Thus expresslon has been evaluatc.l usmg SCF molecular orbrtals to generate the exated states The SCF wavefunctlons were obtamed m p:evlous stud:es of the multIpole moments and polanzabrhtles of N, [6],
At frequenaes
CH4 [71, CZH2. C2H4 and CzH,j [81,C3H,j
well below the first electromc
0 009-16 1482/0000-0000/S
excltatlon
07 75 0 1982 North-Holland
PI, Cl2 123
\‘olumc
85
nrrmbir
I
CffLSllCAL
PHjSICS
T.sblr 1 C.~fcuIxrd and c\prrrmentaf j3.51 xtlues oi the hypcrpofartztbtfttl krm b tn tmtts 01 10w6’ C’ rn’ J-’ Xtcthod 1 -0 0’9 -0 036 -0 034 -0 119 -0 13-i -0 092 -0 I42 -0 119 -0 553 -~
Xlrthod
2
CHf
-0O.U -0 06.f -0071 -0 182 -0 204 -0 139 -0212 -0 080 -0516
-0
L\P
-
058 -0’203 -04rOf
p &o
In thts case w-‘G approach using. &J-lt&rL=o
h-0
[Ifi&
- C&f
but wrth the molecular orbttals perturbed by an esternal electnc field F7 obt.uned by usmg the fmite-field method I 1 _j _This has th* advantage that one of the perturba..oErs IS treated m an rtcratrve manner, and as can be seen from table 1 (the column headed “method 2”). the resulttng values of v‘ are up to 50% dtfferent from those obtatned by the prevtous method A better method stall would be to treat ah three of the tields mvolved m u- ‘J usmg an tterattve techruque along the irnes of coupled Hartsee-Fock theory [ 13]_ If the wavefunctton m the presence of a static electnc field ts wrttten \Ir(Fcll) = @to) + F,d’)(F,) then
f
= 21n~?Irt’)(F*)I\I’~‘~(Bp)),
where B IS a magnetic field The coupled Hartree-Fock method funds the optrmum first-order changes to the molecular orbttals due to an external perturbatron,
_,
-3) + - -
the coupled Hartree-Fock
(o-‘c,$),=,
__evaluated by a sum-over-states
= c
(W-tG,,p)w=o
Consequentty for w_tG rs
= (W~~,)(~-~G,,p)o=Q
1981
tt follows that,
@h(f,) = @?’ + Fo#)(F
[ICI] CO, [5]. and CS, ]I I] and should be reasonably close to the Hartree-Foch hnuts for these molecules The resultmg estrmares of the hyperpolanzabthry term b aregrien m table 1 m the column headed “method I *‘_ A sum-over-states method usmg the vtrtua.l molecular orbttals IS not a very rehable means of calcuiattng any polanzab&ty as these orbrtals descrtbe the evcrted states rather poorly_ An altemauve way of obtauung w-tJ ts to use the de~nttton [2],
(w-‘Jd,
1 January
LFXTCRS
elpressron
= 4th hcocc (#)(F,)M~“(BP)) 1 Thus obtarrung a value of w-‘G by tfus method m~ofves separate calculations to get the orbttals perturbed by an electnc field, and a magnettc field, and combtmng the results. To obtam w_tJ from o~-l G the firute-field method IS used As thts is quite trme consummg, it has only been done for Nz. and the result is gtven rn table 1. This shows that the CHF method results m a further change of ==30% tn the calculated value of b for rutrogen The caiculattons could also be Improved by the consrderatton of correlatton and drsperston effects. Dcsprte &IS, the caiculated values shotrId be accurate enough to show whether or not the hyperpolanzabthty ts tmportartt for the molecules considered There IS comparattvely httle expenmentai evtdence regardtng the hy~rpolanzab~ty cont~button to the mduced btrefrtngenee There IS a measurement of the mduced btrefrtngence m methane [3], which, as thus molecule has no quadrupoie moment, must be due solely to the hyprpolanzabrltty term, gtving a value of b = -(0.4 + 0.1) X 10e60 C3 rnq Je2_ The tem~mture dependence of the btrefrmgence m CO2 has been measured 151, yreldmg a value for the hy~r~I~~b~ty of-(0 2 + 0.3) X 10v60 C3 mJ JW2 The calculated values of b are all negative, as are the values for the hydrogen atom (6 = -0.090 X 10W60 C3 rn’l Jm2 [2]) and the mert-gas atoms where thrs term IS deterred by the Verdet constant j14] _The calculated values generally increase in magmtude as the method used to obtam them rs Improved, and also as the basis set size IS mcreased, and are probably underesttmates. Collected m table 2 are the experimental vaiues of the quadrupole moments, and recent ab tmtio values. Also shown are values of the term $b, obtamed from the calculated values of b, and values of the expressron q b + @~,c~~~/x-T,
Volume 85. number I Table
CIIChllCAL
PIIYSICS
1982
1
Ab mmo from the of IO-@ wlcuhted IS the CC
[S-l I] and c~pcrm~cnt~l values [3 5.151 oi the quadrupolc moment. m umts oi lO-_4oC mZ The ralues oi $b XC t~hcn calcuhtcd values orb in table 1 and arc m umts )f 10mw C3 ma J-’ The values of $6 + O,pcir#T 31 20°C .llso I” U”lIS C3 ma J-’ are constructed from the c~pcruncnr~l values oi 0 .md Aa [ 161. urth the c\crpr,on of- acct>lc”c whcrc the value of 0 MS used Eth) lene has two lndcpcndcnt components of 0 - the VAIC quoted IS rhc o,, component u hers z ;L\I.S
-5_s -15 2 0 2-l 5
-49=03 -150=05 0
67 -.I
Imental
mcasurcments
exprnmcntal
values
the elceptlon
of 0 and (1
of CO,,
chid not separate
the exper-
the tcmperature-
and -Independent contnbutlons. the expenmental values of the quadrupolc moment are really valucs of
dependent
Aa IS the polanzab~ltty
-0 13 -048 -0 53 - I 37 -I 53 -I 04 - I 59 -@ 59 -387
66205 -33zO’ 53-_07 106-05 1’0~06
z
84 II 9 II I
calculated usmg the Note that since wth
yb nlculatcd __ --__-___
0 c\pcrlmenral
0
ab “IIIIO
wliere
I Jawx)
LLI-TIlRS
antsotropy
However from table 1 It can be seen that the hyperpolanzablhty contnbutes only 2-X to the total blrcfnngence for most molecules, wluch ts less than the typ~cal cupenmental error. and conscqucntly the neglect of tlus term ~s~ustlfied The two e\ceptlons to tlus are ethane and cyclopropane, where the hyperpolanzabthty accounts for over 10% of the total effect due to the smaller size of the O,,u,, contnbutlon. As the calculated valuesof b are probably underestimates, the hyperpolanzablllty may also make a s~grufkant contnbutlon to the blrefnngence III nitrogen Allowmg for the hyperpolanzabtity does not Improve the agreement between the ab uutlo md elpenmental values of the quadrupole moment for CzH6 and C,H,; mdeed It makes it worse, and the discrepancies probably reflect the fact that these are the molecules for wluch the expenmental and theoretical determinations of @are the most rhfficult i\s the temperature-independent term tn the btrefringence has been measured tn CO,, where tt makes a smaller percentage contnbuhon than m any of the other molecules constdered, tt should be posstble to observe tlus term tn nearly any molecule However a temperature-dependence study IS only necessary in an accurate determmatlon of
-9-l -87 3 -33 I-t0 3 45 2 -65 -110 75 5 3115
the quadrupole moment oi molcculcs such as cthanc cycloprop.me for wl~ich the overall cficctis sm.111 The author thanhs financial support
the Sclrncc
Research
Councd
or
ior
References
[ 11 A D Buchm~hx~: J Chc”~ 1%)~ 30 (1959) I580 131 A D Buchmch~m Jnd II C Lon~uc~-ll~gy~s 1101 I’hrs 1-t (196s) 63 131 A D Uuchm~h~m R L Dlsch .md D ,\ Dunmur J .\m Chcm Sot 90 (1968) 3104 I-t] M R 8~11~~113 A D Uuchl”&m~ .md J II \~dh~m\ Chcm Ph~s Lctrrrs 78 (1981) 421 151 \I R Ba~t.~gha A D Ituch~n~h~m D Nculnxh R h PwrcnsandJ II Wdhamr \lol 1’11)s -t3(1981) 1015 [6] R D Amos 1101 Phjs 39 (19SO) I 171 R D Amos Mel I’hvs 38 (1979) 33 [Sl R D Alnos and J II \11lh_1”1s Chc”~ Phjs Lc~rcrr 66 (1979) 171 191 R D An~osand J II \Vrllwms Che”~ 1%~~ LcItcrs8-t (1981) IO-I [IO]
J II Wdktms and R D Amos Chcm Phyc Lertcrs 70 (1980) I62 [ll] J tl Wdhamsand R D Amos.ChLm l’h~s Lcttcrs.66 (1979) 370 [I?] II D CohcnandCCJ Ro01h~n.J ChL”l Ph)s 43 (1965) S34 [ 131 R Bl Sw\cns. R Pnzcr and U N Lipccomb J Chem Phys 38 (1963) 550 [ 131 A D Buchmghamand hl J Jdnucson Mel Phys 22 (1971) I17 [15] A D Buckmgham C Grahamand J II Wdlu~ns to be pubbshcd [ 161 ht P Bogxwd. A D Bucl.mgham. R K PIerens and A H Whltc. I amday Trims I 74 (1978) 3008