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Property surfaces and molecular symmetry
Volume
85. number
3
CHEMICAL
PROPER-I-Y SURFACES
AND MOLECULAR
PHYSICS LC-I-I-CRS
SYMMETRY
P-W. FOWLER Unwenuy Chemrcal Labomrov. Gwnbndge CB21EW. UK Rccewed
24 September
198 1.
III fmzd iorm 13 October
198 1
The number of mdependcnt mefficlents dcscribtng a property surfax IS obtatncd derwatwes are pven for the gcnenl lotear molecule and for a sclenlon of polyatomtu
Results to fourth
by group theory
1. Introduction
In the study of the effects of vtbratton on molecular propertres the use of properry surfaces IS well-estabhshed. These represent the varratron of a molecular property wrth the geometry of the molecule The most famrhar property surface ts the potentral energy surface but many other propertres of dntonucs and polyatomKS have been represented m thrs way. In thrs paper we use group theory to deterrmne the number of cocfficrents needed to descnbe such a surface As the property (P) rs usually wntten as a Taylor scrres in drsplacement coordrnates X, thrs task amounts to countmg the mdependent derrvatrves @P/ax,), ($P/aX,i3X,), _ for equrhbnum geometry_ Relatrons between the non-vamshmg denvatrves
have been consIdered for the potential energy [ I,?] and drpole moment surfaces [3] In what follows WC assume a non-degenerate electronic state. Ropertres of degenerate states wffl be consrdered in a later paper 2. Method The geometry of an N-atormc molecule IS descrrbed by 3N - 6 (3fV - 5 for a hnear molecule) rndependent intramolecular coordmates. These may be the curvrhnear rnternal drsplacement coordrnates 32, [4], the reduced nomral coordmates 9i or any surtable set X, The property surface 1s then
0 009~2614/87/0000-0000/S
02.75 0 1982 North-Holland
+
,z_ :p,,pY,x,x~+ _..
When the molecule has C, symmetry coeffrcrents of order III IS n(n2) = (M - I + nr)‘/nr’(M
the number
- 1)’ ,
of
(3
where 61 IS the number of mdependent coordmatcs. Watson [S] has described a procedure for obtamtng the number of mdependent force constants for a molccule of a grven symmetry and rt may be generahsed as follows If rn the pomt group of the molecule the rcprescntatron oi the M mtramolccular coordmates IS f,,, then the number of Independent coeffictents of order m for a property surface P IS the number of times r(P) occurs m the symmctrtc rrlth power of FVID. t f~,,,l under a symmetry operaThe character of [F&J tron R, y,,,(R), IS defined [5]
xW)l ,
x3(R) = ; [x(R)’
+
x3(@
+ 3x(WxW)
= f; rx@W
x..r(R) =A ]x(# + 3@)2
+ 6x(Wzx(R2)
+ 6~(fZ~)]
.
(3) + 2x(R3)1
9
(4)
+ 8x(Wx(R3) (9
The zeroth power of a representatton ts, trtvrally, Fo, the totally symmetnc representatron. For a nondegenerate specres F,, [yl IS ro for even IPI, r, for 313
Voluma
CHEMICAL
S5_ number 3
PHYSICS
odd HZ.Boyle f6] ~vcs powers (8)~= 3,4,5) of degenerate spcc~es ior a number of pomt groups The representation rvrs IS, except ior rhe diatomlc, reducrblc and the symmctrtc powers of a reductble reprcscnt3tlon f are
= C (I
PI
= x I
[r;j +a$brarb, rl21
+aFb
3. Results and discussion
(71
If+;] rb +,
f&f,, (81
D _b 26-atomrc 0,
Itnear matccule
D -b
For a given pomt group K’viB can be obtatned as a function of the number of atok [7]. As an example the symmetric nith powers (nz = 1,~, 3,4) of rViB for D,, molecules are gven m table 1_ The results for C,, molecules are obtamed by combmtng the coeffictents for g and u representatrons Rather than present the cumbersome general formulas for the nonlmcar pomt groups we gtve particular results for a number of small polyatomlcs in tables 2 and 3 Some esamples of apphcations of tables l-3 foilow
(762 + I)-3tormc
linear molecule
=x
a;! =ns’“6’&.
1
b, =X(2k -
I) = (k-
b2 =bS =b,
bJ=(k-1)(X--2) /A$=&$j=(k-
&=A&-
b, =ktl,k+
l)(ZR- 1)
1)
c3 = (k - 1)(X2 - -lx- + I) CJ =I(rc-I)3 E5=Cg={k1)(1x--1)* c7=fkl)Qk’-Z1[.+i) es = 1x-Q - 11’ d, = ;(lok=
-
IOX- +
& = flk/3)(& - l)(W d3 = (3k//L?MA da = 3k(X.- tp
1)(X”
1) &*=bS=bg=Xf2k-i)
b, = (k - I)” b,=bs=X(X.-I) b,=k2
1)”
Cl iii$2x_ - i)(.G -4ki.3) cz = 4cg = 4c,o = (4A/3)(2x.- l)(lk - 1)
3)(k’ - x f 1) - 51 t 2) - 44 + 1)
ds =& =&k- l)c_k-- l)@k’- 8ki.3) d, = (k/3)& - l)(lOk” - 12x-+ 5) da = (k/3)(k - 1)(10x12- 1ZR+ 2) dg = dro = &/3)(X - 1)(2/c- l)Z dl, = (L/6)& - 1)(X? + 1) dx2 = &/x-)(x-- 1)(I&’- 1) 314
15 Januztry I982
For reductbie representattons of large dtmenston the powers are more easdy calculated vta eqs. (3)-(S).
(6) [P]
LC-ITERS
C‘ = c* =
(k/3)(81’ + I) es =cq =X(x-- 1)(2/C- 1) cg = X(4X’- tk - 1) c6 = k(4k2 - 2x:+ 1) t7=Cs=X=CZkI) c9 = (x/3)(&’ - 1) C,(#= @/3)(2X’+ 1)
d, = C_X2/3)(5k’ f 4) d2 = (tr’/3)(lOk” - I) ds = (3&/2)Ui- 1)0X’ - 1) dJ =3X3(x-- 1) dS =de = (k/3)Qk - 1)(&k + 1) d7 = (~/3)(10~~ - 2h’ - k + 2) d8 = (k/3)(1OkJ - 2L’ - 4R - I) d9 =dsc = @/3)(4X-’ - 1) d, 1 = (k/6)(k + 1)(2k’ + 1) d12 = (k/6)(k t i)(k-’ - 1)
Volume
85. numbcr
Table 2 Fust and second some
polyzdomlc
htolrcule --e___--YYZ x11 XY,Z
CHCMICAL
3
powers of the wbrarmlwl molecules
Porn,,
of
%I czv
3X’ ZA, + B2 3A, +B, +7B2
C
?A,
XY3
Gb
Ai+A:+2E’
XY2Z2
c2v
3A,
xs32
C3v
+ 2B2 3A, + 3I3
XY.,
Td
A, +E+‘T=
\YaZW
CBV
-%A, +4c
AZ Y.,
J&d
+ 1C
+
A2 + ZB,
3Af + A” + 2Blg + BI” + Bq + 2B2” + ‘83” Alg+E,+X,” + Tzg + T2u
Pam,
6A’ 1A, + ‘B2 IOA, + 2A2 + 38, + 60, 6A, +A2+7E
CS C2\ CZ\ C3v
5Ai + Az+A; +5E +ZE’ J7A, + 8A2 + IOU, + IOU, IZA, +3A2 + l5E SA, +X+3T, + n-z ZOA, + 6Az + 26C I 8Ag T 7A” + lJB,g+7B,” + 7Bzg + 1 JB2” +6BJg+ llB3”
J’31,
C,v C3, -J-d
C3\ &d
7A1c + A,, + 2ii* +‘A2” 4 + 9Eg + 3C, + 4Tlg + 8T,” + 9T,, + 7Tz”
I&!
ir
*33TIg+SIT,” + -1Zl-~g + 36TZ”
Allpowers of rVIR gvc Ct , only. Thus a propert) of ttis symmetry (e g Q=_(E),5(E), . _) has precisely one COeffiClent at eaCh order of the property and a property surface of any other symmetry (.$)), .) ISldenucally zero. r-?,(-9. =x’x)
IOA 6A, +4U2 27AI + 6A2 + IOB, + IOBZ I-IA, + 6A, + ISC 9A, + Ai + 5A, +5A; + l3L + SC’ 524, + 33A, + 4OB, + 4ORz 38A* + l9Az + Hr. I3A, + -IA2 + IJE + IST, + 25-r. 80A, ; MA, + IlOC 60AL: * 38A” + 53B,g + 3811,” + 3snzg + 53112” + 3 I ll3g + S3B3”
‘2A,g + 9A,” + I 3Azg + I-l,&” + 33Lg + aoc,
3 I Diaronr~rnolccdcs
senes
ISA’ 9A, + 68, 48A, + l8A2 + 12B, + 38B2 28A, + I&~, + 42c l9A, + 9Az + SAi + 91~2’ +29c* + 13c IJ7A, + lOBA, + IlOB, + 120112 I OZA, + 63A, + 1651: 33A, + I?& +45L+SIT, + 69-l=, 270 \, + I85.\, + 455L 116Ag + I-18,\” + I88U,g + I-l8U,, + I48U2b * J88B2”
9’,\,g + jOA,” + 63A2& + 59A2” + l55Lg + 109C” + ,7tT,g + 199-r,” + 196Tzg + 190T2,
(e g surface has the full complement +3PI +6P2 + IOP, + 15Pa
of coefticlents
PO
Water rirolecule
The entry for XY2 of Czv symmetry m tables 2 and 3 shows that for totally symmetnc propertres the surface is PO +3qI +4P2 + 6P3 + 9P4(usmg hldl's for potentral surfaces), for B2 proper&pole moment along the II E&art axis the surface IS 1P1 + 2P2 + 4P3 + 6Pa PropertIes of A2 or B, symnzetry are unaffected by \nbratlon and their surfaces are Identically zero. For a property such as the nuclear tieldmg of one of the protons the sne symmetry ISlower than qv, m thus case C,, and the [S] notatron
t,es
oi
+ 11IU3g + 188B3” Oh
3 2
1981
E’oW
XY3
Oh
IS January
Table 3 Thud and fourth powrs oi rhc ~rbnuoml rcpr~s~mmon some polyaIom,c molcculcs -~_______ Xlolcc”lc
I&B!
rVJB
LCI-JIRS
Fo”P CS
XY6
reprcscn,a,Ion
PHYSICS
hke the
Tables
l-3
also give the number
of the L tensor whxh curvdmear coordmates For each coordmate
of components
relates the geometncally defmed 92, to the normal coordmatrs W,
00) 315
Volumr‘ 85 number 3
CllfMlC4L
PHYSICS LETTERS
wluch has the form of a property surface (of a peculiar hmd since, considered as a property surface, eq. (10) 1s teilmg us that the geometry of the molecule Is 3 function of the geometry of the molecule). The L tensor takes Its sunplest form for the symmetry coordmates S, @near combmations of the c;Pi) [4]. Thus for HZ0 the total number of components o L at each order IS Zrr(At) + “(Bz) where II(~) 1s the number of coefficl:nts for 3 surface of symmetry r, I e. L(H20)=
SL, + IO& + 16L, + 24L, + ._
(11)
The dcscrlptlon of these as numbers of mfcper~de~~t cocfficlents requires some clanficatlon. The group theory gives the number of L components at each order wh~Al cannot be related to each other by symmetry arguments alone In fact, for the L tensor we hnow that all elements can bc calculated from the harniomc force field, the structural parameters and the masses [A] Thus for Hz0 S parameters specify L(4& constants + rT + 0, + m. + mH) and for HDO 1 more piece of mformatlon (I+,) IS needed.
15 January 1982
For vector and tensor properties this transferabdlty IS mamtamed If we use an axis system defied by geometnc considerations alone For example the axes for the nor&near tnatomlc could be defined by umt vectors along the bonds ?, , r^z and the cross product (r^t X &). This non-orthonormal a_us system was used to specify the dipole moment surface of water [9]. The ;Lyes used to relate the &pole moment surface to experimental observations, the E&art zxeues,are defied by mass-dependent condlhons Thus m the example of Hz0 and HDO 1IO] the number of non-zero cocfticlents descnbmg the components along the a and b axes Increases from Hz0 to HDO (where It IS not possslble to defiie the positlo? of the Eckart axes by symmetry arguments alone).
Acknowledgement The author thanks the Science Research Council for tinanclal support
3 4 Isotoprc substrtutior~ References Wltlun the Born-Oppenheuner appro_xunatlon and after allowmg for posslblc changes m nuclear moments, etc., the property surface is unchanged by isotopic substltutlon. The number of Independent coefticients needed to describe a property should be deterrnmed usmg the group theory with the pomt group of the eqbtibnum framework of charges, not of )Irasses It IS useful to express the surface in the purely geometncally defined %? coordmates, because then the values of the coefficients are transferable from one Isotopic form to another. Use of mass-dependent reduced normal coordmates brings in successively more non-zero (though not mdependent) coefficrents as we descend m symmetry.
316
[I] L H~NY andG Amar. J hi01 SDCC~I~ 5 (1960) 319 121 L Henry and C Amat, J AloL S&r; 1; (1965) 168 131 D Papou%k and K Sanka. J BloL Spcctry 28 (I 968) 125 i-11 A R Hay. 1 M Wlls and G Strcy, MOI Ph>s 24 (1972) 1265. [S] J KC Watson. J hlol Spcctry 41 (1972) 229 (61 L L Boyle, Intern. J Quantum Chem 6 (1972) 725 [7] G Herzbeg, Blolccular spectra and S~IUC~IJ~~, Vol. 2 Infrared and Raman spectra (Van Nostrand, Prmceton. 1945) tables 35 and 36 [8] I hf MiUss.m Spectahst Pcrtodlal Reports - Thcoreual Chematry. Vol. 1. ed. R N Dl?ton (The Chemlal Socrcty, London, 1972) ch 4 [9] P W rowlcr, G Rdey and W-T Raynes. hlol Phys 42 (1981) 1463 [lOI P W. rowler and WT Raynes. hloL Phys 43 (1981) 65.
85. number
3
CHEMICAL
PROPER-I-Y SURFACES
AND MOLECULAR
PHYSICS LC-I-I-CRS
SYMMETRY
P-W. FOWLER Unwenuy Chemrcal Labomrov. Gwnbndge CB21EW. UK Rccewed
24 September
198 1.
III fmzd iorm 13 October
198 1
The number of mdependcnt mefficlents dcscribtng a property surfax IS obtatncd derwatwes are pven for the gcnenl lotear molecule and for a sclenlon of polyatomtu
Results to fourth
by group theory
1. Introduction
In the study of the effects of vtbratton on molecular propertres the use of properry surfaces IS well-estabhshed. These represent the varratron of a molecular property wrth the geometry of the molecule The most famrhar property surface ts the potentral energy surface but many other propertres of dntonucs and polyatomKS have been represented m thrs way. In thrs paper we use group theory to deterrmne the number of cocfficrents needed to descnbe such a surface As the property (P) rs usually wntten as a Taylor scrres in drsplacement coordrnates X, thrs task amounts to countmg the mdependent derrvatrves @P/ax,), ($P/aX,i3X,), _ for equrhbnum geometry_ Relatrons between the non-vamshmg denvatrves
have been consIdered for the potential energy [ I,?] and drpole moment surfaces [3] In what follows WC assume a non-degenerate electronic state. Ropertres of degenerate states wffl be consrdered in a later paper 2. Method The geometry of an N-atormc molecule IS descrrbed by 3N - 6 (3fV - 5 for a hnear molecule) rndependent intramolecular coordmates. These may be the curvrhnear rnternal drsplacement coordrnates 32, [4], the reduced nomral coordmates 9i or any surtable set X, The property surface 1s then
0 009~2614/87/0000-0000/S
02.75 0 1982 North-Holland
+
,z_ :p,,pY,x,x~+ _..
When the molecule has C, symmetry coeffrcrents of order III IS n(n2) = (M - I + nr)‘/nr’(M
the number
- 1)’ ,
of
(3
where 61 IS the number of mdependent coordmatcs. Watson [S] has described a procedure for obtamtng the number of mdependent force constants for a molccule of a grven symmetry and rt may be generahsed as follows If rn the pomt group of the molecule the rcprescntatron oi the M mtramolccular coordmates IS f,,, then the number of Independent coeffictents of order m for a property surface P IS the number of times r(P) occurs m the symmctrtc rrlth power of FVID. t f~,,,l under a symmetry operaThe character of [F&J tron R, y,,,(R), IS defined [5]
xW)l ,
x3(R) = ; [x(R)’
+
x3(@
+ 3x(WxW)
= f; rx@W
x..r(R) =A ]x(# + 3@)2
+ 6x(Wzx(R2)
+ 6~(fZ~)]
.
(3) + 2x(R3)1
9
(4)
+ 8x(Wx(R3) (9
The zeroth power of a representatton ts, trtvrally, Fo, the totally symmetnc representatron. For a nondegenerate specres F,, [yl IS ro for even IPI, r, for 313
Voluma
CHEMICAL
S5_ number 3
PHYSICS
odd HZ.Boyle f6] ~vcs powers (8)~= 3,4,5) of degenerate spcc~es ior a number of pomt groups The representation rvrs IS, except ior rhe diatomlc, reducrblc and the symmctrtc powers of a reductble reprcscnt3tlon f are
= C (I
PI
= x I
[r;j +a$brarb, rl21
+aFb
3. Results and discussion
(71
If+;] rb +,
f&f,, (81
D _b 26-atomrc 0,
Itnear matccule
D -b
For a given pomt group K’viB can be obtatned as a function of the number of atok [7]. As an example the symmetric nith powers (nz = 1,~, 3,4) of rViB for D,, molecules are gven m table 1_ The results for C,, molecules are obtamed by combmtng the coeffictents for g and u representatrons Rather than present the cumbersome general formulas for the nonlmcar pomt groups we gtve particular results for a number of small polyatomlcs in tables 2 and 3 Some esamples of apphcations of tables l-3 foilow
(762 + I)-3tormc
linear molecule
=x
a;! =ns’“6’&.
1
b, =X(2k -
I) = (k-
b2 =bS =b,
bJ=(k-1)(X--2) /A$=&$j=(k-
&=A&-
b, =ktl,k+
l)(ZR- 1)
1)
c3 = (k - 1)(X2 - -lx- + I) CJ =I(rc-I)3 E5=Cg={k1)(1x--1)* c7=fkl)Qk’-Z1[.+i) es = 1x-Q - 11’ d, = ;(lok=
-
IOX- +
& = flk/3)(& - l)(W d3 = (3k//L?MA da = 3k(X.- tp
1)(X”
1) &*=bS=bg=Xf2k-i)
b, = (k - I)” b,=bs=X(X.-I) b,=k2
1)”
Cl iii$2x_ - i)(.G -4ki.3) cz = 4cg = 4c,o = (4A/3)(2x.- l)(lk - 1)
3)(k’ - x f 1) - 51 t 2) - 44 + 1)
ds =& =&k- l)c_k-- l)@k’- 8ki.3) d, = (k/3)& - l)(lOk” - 12x-+ 5) da = (k/3)(k - 1)(10x12- 1ZR+ 2) dg = dro = &/3)(X - 1)(2/c- l)Z dl, = (L/6)& - 1)(X? + 1) dx2 = &/x-)(x-- 1)(I&’- 1) 314
15 Januztry I982
For reductbie representattons of large dtmenston the powers are more easdy calculated vta eqs. (3)-(S).
(6) [P]
LC-ITERS
C‘ = c* =
(k/3)(81’ + I) es =cq =X(x-- 1)(2/C- 1) cg = X(4X’- tk - 1) c6 = k(4k2 - 2x:+ 1) t7=Cs=X=CZkI) c9 = (x/3)(&’ - 1) C,(#= @/3)(2X’+ 1)
d, = C_X2/3)(5k’ f 4) d2 = (tr’/3)(lOk” - I) ds = (3&/2)Ui- 1)0X’ - 1) dJ =3X3(x-- 1) dS =de = (k/3)Qk - 1)(&k + 1) d7 = (~/3)(10~~ - 2h’ - k + 2) d8 = (k/3)(1OkJ - 2L’ - 4R - I) d9 =dsc = @/3)(4X-’ - 1) d, 1 = (k/6)(k + 1)(2k’ + 1) d12 = (k/6)(k t i)(k-’ - 1)
Volume
85. numbcr
Table 2 Fust and second some
polyzdomlc
htolrcule --e___--YYZ x11 XY,Z
CHCMICAL
3
powers of the wbrarmlwl molecules
Porn,,
of
%I czv
3X’ ZA, + B2 3A, +B, +7B2
C
?A,
XY3
Gb
Ai+A:+2E’
XY2Z2
c2v
3A,
xs32
C3v
+ 2B2 3A, + 3I3
XY.,
Td
A, +E+‘T=
\YaZW
CBV
-%A, +4c
AZ Y.,
J&d
+ 1C
+
A2 + ZB,
3Af + A” + 2Blg + BI” + Bq + 2B2” + ‘83” Alg+E,+X,” + Tzg + T2u
Pam,
6A’ 1A, + ‘B2 IOA, + 2A2 + 38, + 60, 6A, +A2+7E
CS C2\ CZ\ C3v
5Ai + Az+A; +5E +ZE’ J7A, + 8A2 + IOU, + IOU, IZA, +3A2 + l5E SA, +X+3T, + n-z ZOA, + 6Az + 26C I 8Ag T 7A” + lJB,g+7B,” + 7Bzg + 1 JB2” +6BJg+ llB3”
J’31,
C,v C3, -J-d
C3\ &d
7A1c + A,, + 2ii* +‘A2” 4 + 9Eg + 3C, + 4Tlg + 8T,” + 9T,, + 7Tz”
I&!
ir
*33TIg+SIT,” + -1Zl-~g + 36TZ”
Allpowers of rVIR gvc Ct , only. Thus a propert) of ttis symmetry (e g Q=_(E),5(E), . _) has precisely one COeffiClent at eaCh order of the property and a property surface of any other symmetry (.$)), .) ISldenucally zero. r-?,(-9. =x’x)
IOA 6A, +4U2 27AI + 6A2 + IOB, + IOBZ I-IA, + 6A, + ISC 9A, + Ai + 5A, +5A; + l3L + SC’ 524, + 33A, + 4OB, + 4ORz 38A* + l9Az + Hr. I3A, + -IA2 + IJE + IST, + 25-r. 80A, ; MA, + IlOC 60AL: * 38A” + 53B,g + 3811,” + 3snzg + 53112” + 3 I ll3g + S3B3”
‘2A,g + 9A,” + I 3Azg + I-l,&” + 33Lg + aoc,
3 I Diaronr~rnolccdcs
senes
ISA’ 9A, + 68, 48A, + l8A2 + 12B, + 38B2 28A, + I&~, + 42c l9A, + 9Az + SAi + 91~2’ +29c* + 13c IJ7A, + lOBA, + IlOB, + 120112 I OZA, + 63A, + 1651: 33A, + I?& +45L+SIT, + 69-l=, 270 \, + I85.\, + 455L 116Ag + I-18,\” + I88U,g + I-l8U,, + I48U2b * J88B2”
9’,\,g + jOA,” + 63A2& + 59A2” + l55Lg + 109C” + ,7tT,g + 199-r,” + 196Tzg + 190T2,
(e g surface has the full complement +3PI +6P2 + IOP, + 15Pa
of coefticlents
PO
Water rirolecule
The entry for XY2 of Czv symmetry m tables 2 and 3 shows that for totally symmetnc propertres the surface is PO +3qI +4P2 + 6P3 + 9P4(usmg hldl's for potentral surfaces), for B2 proper&pole moment along the II E&art axis the surface IS 1P1 + 2P2 + 4P3 + 6Pa PropertIes of A2 or B, symnzetry are unaffected by \nbratlon and their surfaces are Identically zero. For a property such as the nuclear tieldmg of one of the protons the sne symmetry ISlower than qv, m thus case C,, and the [S] notatron
t,es
oi
+ 11IU3g + 188B3” Oh
3 2
1981
E’oW
XY3
Oh
IS January
Table 3 Thud and fourth powrs oi rhc ~rbnuoml rcpr~s~mmon some polyaIom,c molcculcs -~_______ Xlolcc”lc
I&B!
rVJB
LCI-JIRS
Fo”P CS
XY6
reprcscn,a,Ion
PHYSICS
hke the
Tables
l-3
also give the number
of the L tensor whxh curvdmear coordmates For each coordmate
of components
relates the geometncally defmed 92, to the normal coordmatrs W,
00) 315
Volumr‘ 85 number 3
CllfMlC4L
PHYSICS LETTERS
wluch has the form of a property surface (of a peculiar hmd since, considered as a property surface, eq. (10) 1s teilmg us that the geometry of the molecule Is 3 function of the geometry of the molecule). The L tensor takes Its sunplest form for the symmetry coordmates S, @near combmations of the c;Pi) [4]. Thus for HZ0 the total number of components o L at each order IS Zrr(At) + “(Bz) where II(~) 1s the number of coefficl:nts for 3 surface of symmetry r, I e. L(H20)=
SL, + IO& + 16L, + 24L, + ._
(11)
The dcscrlptlon of these as numbers of mfcper~de~~t cocfficlents requires some clanficatlon. The group theory gives the number of L components at each order wh~Al cannot be related to each other by symmetry arguments alone In fact, for the L tensor we hnow that all elements can bc calculated from the harniomc force field, the structural parameters and the masses [A] Thus for Hz0 S parameters specify L(4& constants + rT + 0, + m. + mH) and for HDO 1 more piece of mformatlon (I+,) IS needed.
15 January 1982
For vector and tensor properties this transferabdlty IS mamtamed If we use an axis system defied by geometnc considerations alone For example the axes for the nor&near tnatomlc could be defined by umt vectors along the bonds ?, , r^z and the cross product (r^t X &). This non-orthonormal a_us system was used to specify the dipole moment surface of water [9]. The ;Lyes used to relate the &pole moment surface to experimental observations, the E&art zxeues,are defied by mass-dependent condlhons Thus m the example of Hz0 and HDO 1IO] the number of non-zero cocfticlents descnbmg the components along the a and b axes Increases from Hz0 to HDO (where It IS not possslble to defiie the positlo? of the Eckart axes by symmetry arguments alone).
Acknowledgement The author thanks the Science Research Council for tinanclal support
3 4 Isotoprc substrtutior~ References Wltlun the Born-Oppenheuner appro_xunatlon and after allowmg for posslblc changes m nuclear moments, etc., the property surface is unchanged by isotopic substltutlon. The number of Independent coefticients needed to describe a property should be deterrnmed usmg the group theory with the pomt group of the eqbtibnum framework of charges, not of )Irasses It IS useful to express the surface in the purely geometncally defined %? coordmates, because then the values of the coefficients are transferable from one Isotopic form to another. Use of mass-dependent reduced normal coordmates brings in successively more non-zero (though not mdependent) coefficrents as we descend m symmetry.
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[I] L H~NY andG Amar. J hi01 SDCC~I~ 5 (1960) 319 121 L Henry and C Amat, J AloL S&r; 1; (1965) 168 131 D Papou%k and K Sanka. J BloL Spcctry 28 (I 968) 125 i-11 A R Hay. 1 M Wlls and G Strcy, MOI Ph>s 24 (1972) 1265. [S] J KC Watson. J hlol Spcctry 41 (1972) 229 (61 L L Boyle, Intern. J Quantum Chem 6 (1972) 725 [7] G Herzbeg, Blolccular spectra and S~IUC~IJ~~, Vol. 2 Infrared and Raman spectra (Van Nostrand, Prmceton. 1945) tables 35 and 36 [8] I hf MiUss.m Spectahst Pcrtodlal Reports - Thcoreual Chematry. Vol. 1. ed. R N Dl?ton (The Chemlal Socrcty, London, 1972) ch 4 [9] P W rowlcr, G Rdey and W-T Raynes. hlol Phys 42 (1981) 1463 [lOI P W. rowler and WT Raynes. hloL Phys 43 (1981) 65.