- Email: [email protected]
A dynamic algebra for rotation-vibration spectra of complex molecules
Volume 85, number 1
CHEMICAL PHYSICS LET-JFRS
A DYNAMIC ALGEBRA
Rccwcd
FOR ROTATION-VIBRATION
1 I August 198 I m final form 74 September
1 January 1982
SPECTRA OF COMPLEX MOLECULES
1981
An algcbralc approach IO try- and poly-atomic molcculcs IS prcscnhzd This approach IS appbcd 10 the stud) of Imear ITIJtomlc molcculcs It IS sueestcd that these trchnlques ma) be useiul III the desnlptlon of complc\ molecular rotatlonvibrarlon specKI
I Introduction Recently, one of us [I ] has suggested that algebratc methods may be useful m describmg molecular rotattonvlbratlon spectra In ref [I] the stmple case of a dtatomtc molecule was dtscussed Here, both descrlpttons, that tn terms of Schrodtngcr equatton [?I and that tn terms of an algebraic approach are equslly simple. However, for nlorc complex molecules, the Schrbdtngsr approach leads to several coupled dlfierentlal equations whose solution IS by no means simple Conversely, the algebrslc approach leads to a system of coupled algebraic equations whose solution IS stra&tiorward Furthermore, by exploltmg the group structure of the problem. tt IS possible to find simple onalyttc solutions to the complex case of several coupled vtbrattons and rotattons The purpose of thts letter IS to describe how the algcbratc approach can be applted to tnatomtc molecules, and to mdtcate a posstble generahzatton to more complex mo!ecules In ref. [I ] tt was shown that tt IJ posstble to construct the rotatton-vibration spectrum of dlatomtc molecules by mahtng use of the dynamtc algfbra U(4) Thts algebra was reahzed tn terms of four creation (annthtlatton) operators, dtvtded tnto a scalar, u (u) and a vector n: (rr,) operator It was also shown that, when the hamdtontan descrlbmg the molecule had a dynamic symmetry, O(4), the elgenvalue problem could be solved analyttcally and leads to the etgenvalues E(R:v,J,Pf)=Eo+A’(Rr+2)(“+f)-/!‘(”+~)~ where E 0, rl and B are appropriate
E&J)
=
c
Y,,
(u++)’ [J(J+
constants
+fM(J+1), This evpresston
was compared
(1) wtth the usual Dunham
evpanston
I)]’
LJ
A study of the connectton between classtcal (Schriidmger) approach and algebratc approach along the hnes dtscussed tn ref [3] for appltcattons tn nuclear phystcs, suggests that an algebratc descnptton of rotatton-vtbrauon spectra m tnntomuz molecules may be provided by two coupled algebratc structures, each with dynamtc algebra, U(-i) Thts can be Interpreted as couphng the rotattonal-viirattonal degrees of freedom of the two bonds It 1s convement to rcahze ths algebra agam tn terms cf creation and anruhdatton operators We thus Introduce two sets 32
0 GO9-2614/82/0000-0000/S
02.75 0 1982 North-Holland
Volume 8.5. number I ofboson
OpZF&OFS
H==H, +H,+
CflIlMICAL FIfYSKS LrnCRS Ql . 77; p and
0:.
&,
1
I 3anusr> 1982
and a ~o~r~span~r~~ hamrttontan
VQ
Each prece H I, HI! m eq (3) has the same form as eqs (-1) and (5) of rei [ I ] The mtcractton pounded into products of apcrators of system 1 and 2 rcspcctrvely, I c
(31 term I’t2 CWIbc c’r-
2 Dynamical symmetries One of thz adimrages of the dgebratc approach ts that $t 1s posstbfe, by studying the group structure of the problem, to cop=rruct z~naty~tc solur~ons of the ergcnvalue problem for the hamrltoman,H Thcsc .malytrc sofutlons ate known as dynamtc~ symmetnes [4&and ttre prtrucularly tmportant here beccrtuscOF the dtf~cult$~~ connected wtlt constructrng exa_ct sfrfutrons for systems of coupted non-hncar osedlrttors [Sj in ttr~sfetter, we focus our sttcnt~on on one solution, which appears to yteid spectra descrsbmg rotsttons and v~bratxons of hnear trtatomtc ntuiecules Tars IS obtamed by constdertng the case m whtch both hamtlronrans.ffl and 0, have the 0(4)syrnmetry dcscrtbrd rn rcf [I] and furthermore ihe tntcrnctron Vlt ts also budt wnh opewrors belongmg to the Ot-1) symmetry, I C,
Volume SS, number
I
U’“(4) @U”‘(4)
CHEMICAL
3
O”‘(1) 8 O’“‘(4) 3
PHYSICS
1 January 1982
LEl-KRS
O(4) > O(3) 3 O(2)
(7)
vibratton-rotation states of a certam electronic state of a trratomtc molecule are assumed to be contamed m a given trreductble representatton CL the group U(t)(4) o @(4) These are charactertzed by two numbers, Nt and A’?. the total number of quanta of species I and 2 respecttvely. The complete labelhng of states IS obtained by app1y111gstandard group thcoretrcal rechmques to the group cham eq. (7). The labels are IN,, N2, 9,. p2, (it, rl_). .I”, Jl) The values of nt and 172 contamed m each representation Nt . N2 are stmply gtven by ~7, =N,, N,--2, _ , IorO(‘\It=evenorodd)at1d~~=N~,N~-2, ., 1 or 0 (N3 = even or odd) The values of (rt , s-2) and .I are mstead more dlfticult to obtam A complete hstmg will be given m a longer pubhcatton. We have also Included a parrty quantum number R Thus appears because m the reductton from the group O(4) to O(3) (the rotation group) both pantws occur (for example one can have states wtth Jx = I’) The energy etgenvalues are given by All
E(~~‘,‘,.N~.0,.9~,(7,.T~),J’,nl)=~~-f(A,-C)17~(9t+7-)-$(A~-C)r1,(~~+7-) -+C[it(rt+1)+7$]
+BJ(J+l),
(8)
where E. IS a constant
wluch depends on _4 , , PI z C and A’,, IV, In order to show thdt tlus formula describes mdeed (although approummatcly) rotatton-vlbratron spectra of linear trtatomrc molecules, tt IS convement to Introduct rhc quantum numbers uI _u2, uj nnd I, through nr = A’, --2u, _q2 = A’-,-_lu,. T-, - = I,- T, = ,V, +N, - 3_“, - u, - 3uj Then, cq (8) can be rewritten as
-1C(“,+~)(“j+f)-c(u~+~)(“,+l)-c(”l+~)(”Z+l)+BJ(~+I) TIII~ expression
ca. be dnectly
compared
WIIII IIIC usual cxpansron
(9)
[6]
-
+X~~(“,+~)~“~+l)+X,~(“t+f)(“~+~)+Y~~(”,+I)(”j+~)+BJ(J+I)+
(10)
of the coupled algebraic cquatlons wth O(4) symmctr): thus produces a spectrum smtrlar to that gwn by eq (IO) s lth special values of the coefficrents w1,w~.w3, x~~,~~~,x~~,~~~.x~,x,~, xl3 and B Furthermore, drt mspcctton of the values of the quantum number 1, = ‘2 allowed by the group cham (7) shows that the solutton cif the coupled algebratc equations also produces auiorrlofrcal& the spectes Z, fl, A, approprtate to the double degenerate ndture of the u3 vtbratton In fact, tt provides a group theorettcal mterprctatron of the quantum number /2 mtroduccd errprrtcally As .m e-,ample the vtbratronal spectrum obtained by usmg eq (S) wtth Nt = 5O,N, = 50,A, = IS cm-t, C= I3 cm-‘.A. = 33 cm-t IS shown u-r fig I. where tt IS compared with the low-lymgvtbratlonsl spectrum of the CO2 molecules On top of each vlbrattonal st.tte there are butlt rotattonal bands Thts figure IS shown not so much to emphasrzc the agrcemcnt with experiment. but rarhcr m order to dtsplay the fact that the coupled algebratc system produces automatrcally all quahtatlve features of the observed spectra For a more quantttattve descrtptton, a breahmg of the O(t)(4) @ 0f2)(-l) symmetry must be consldered We also mention bnefl! that tt IS posstblc IO construct other analyttc solutrons to the Edgenvalue problem correspondmg to group chains which are dtfferent from
The solutron
(7). Although
we lntve not yet explored
m detad all posstble analytic
soluttons,
preltmmary
studtes show that tt
May be powble to descrlbc, with the U(l)(4) SJ U(‘)(4) group structure, both ngtd and normgld molecules. The latter point IS particularly Important in connection with the study of van der Waals molecules [7]. Fmally we note th.tt the approach described here IS related to that of Kclhnan et al [8], who drscussed tnatomtc molecules m terms of the group structure U(t)(3) 0 U(“)(3) The relatton between the approaches will be dacussed In detad m a longer pubhcatton. 31
I Jiu~~ry
CHIIMICAL PIIYSICS LEI-KRS
Volume 85. number 1
1982
3 OfU*
I)
21D-
2-
p=a-
-
-
-
_
I=J-
-
-
-
-
6-
ooo3,,
iI-
500(
L
Eap Th
Erp Th
I-
”
_
Exp Th
lOOI2; 02Ol"l
-
)2c
200(
lOO(
I
1IC I!.
otoii_ Y
(I1
Fig. 1. A compaaon between the cxpenmcntal vibratlorwl spectrum oi the CO2 molecule [6l and that calculated usmg q (8) ~lth A’, =50, h12= 5O.A I = 15 cm-‘. C= I3 cm-‘..t 2 = 33 cm-’ On top of each vibratIona lcrel there arc rotst~orul bandsas shown m the msert
3. Conclusions We have presented here an algebrruc approach to rotation-wbratlon spectra of trlatomlc molecules. and apphed tt to the sample case of hnear molecules The advantage of this approach ISthat It products automattcally the complex vtbrattonal and rotatlonal pattern of these molecules. Furthermore, it IS possible, usmg the same approach, to calculate all other properties of the molecule, such as mtenslues oiemlsslon and absorption hncs, excltatron probabdltles from one rotation--Vlbratlon level ut,y,u3,12,J to another ui,&,u;,fi,J’,etc. Fmally, the same ap 35
Volume 85 number 1
CHEAIICAI
PHYSICS LCI’TERS
1 January 1982
proxh can be generaked to even more complex molecules. For retra-atonuc molecules, one needs three coupled U(1) problems. and. m general, for N-atomic molecules. N-l coupled U(4) problems Although the complewty of the solutions mcreascs wth IV, the algcbratc structure of each U(4) problem IS sample enough that detailed calculatlons oi comple\ molecules mdy be powble m the near future.
We wsh to thanh R D Lcvmc, who stmwlated our Interest m an algebraic treatment of complex molecules, and A hndc and 1. Talm~ for useful dIscussIons This work was supported m part by the Stlchtmg voor Fundamcntecl Onderroeh der Matertc (F 0 M ) and In part under USDOE Contract EY-76-C-W-3074
References 11 j 121 131 [4) [S ] 161 171 181
36
I‘ l~chellc. Chim Phys Letters 73 (198 1) 181 T C Wwch and R B Bernhtcm. J Chem Phys 46 (1967) 4905 0 S ban Roosmalcn and A L: L D~epermk, Phps Lcrters 1OOB (!98 I) 299. I- Ixh~Uo,Comm Nucl Part Phls 8 (1978) 59 S A Rice. m Quantum d?. namlcs of molecules, cd R C Woollcy (Plenum Press. New York, 1980) p 257 G tlcrzbeF;. Moleculx spectra and molecular S1ructur t. Vol 2 (Van Nostrand. Prmccton, 1945) p 211 B Blancy and G E\\mg.Ann Rev Phjs Chcm 37 (1976) 553 %IE GAlman, r +marsnd R S Dcrry. 1 Chem Phys. 73 (1980) 2387, r Amar hl C LcUmanand R S Berry, J Chum Phys 70 (1979) 1973, hl C lieUman 3rd R S Berry. Chcm Phys. Letters 42 (1976) 327
CHEMICAL PHYSICS LET-JFRS
A DYNAMIC ALGEBRA
Rccwcd
FOR ROTATION-VIBRATION
1 I August 198 I m final form 74 September
1 January 1982
SPECTRA OF COMPLEX MOLECULES
1981
An algcbralc approach IO try- and poly-atomic molcculcs IS prcscnhzd This approach IS appbcd 10 the stud) of Imear ITIJtomlc molcculcs It IS sueestcd that these trchnlques ma) be useiul III the desnlptlon of complc\ molecular rotatlonvibrarlon specKI
I Introduction Recently, one of us [I ] has suggested that algebratc methods may be useful m describmg molecular rotattonvlbratlon spectra In ref [I] the stmple case of a dtatomtc molecule was dtscussed Here, both descrlpttons, that tn terms of Schrodtngcr equatton [?I and that tn terms of an algebraic approach are equslly simple. However, for nlorc complex molecules, the Schrbdtngsr approach leads to several coupled dlfierentlal equations whose solution IS by no means simple Conversely, the algebrslc approach leads to a system of coupled algebraic equations whose solution IS stra&tiorward Furthermore, by exploltmg the group structure of the problem. tt IS possible to find simple onalyttc solutions to the complex case of several coupled vtbrattons and rotattons The purpose of thts letter IS to describe how the algcbratc approach can be applted to tnatomtc molecules, and to mdtcate a posstble generahzatton to more complex mo!ecules In ref. [I ] tt was shown that tt IJ posstble to construct the rotatton-vibration spectrum of dlatomtc molecules by mahtng use of the dynamtc algfbra U(4) Thts algebra was reahzed tn terms of four creation (annthtlatton) operators, dtvtded tnto a scalar, u (u) and a vector n: (rr,) operator It was also shown that, when the hamdtontan descrlbmg the molecule had a dynamic symmetry, O(4), the elgenvalue problem could be solved analyttcally and leads to the etgenvalues E(R:v,J,Pf)=Eo+A’(Rr+2)(“+f)-/!‘(”+~)~ where E 0, rl and B are appropriate
E&J)
=
c
Y,,
(u++)’ [J(J+
constants
+fM(J+1), This evpresston
was compared
(1) wtth the usual Dunham
evpanston
I)]’
LJ
A study of the connectton between classtcal (Schriidmger) approach and algebratc approach along the hnes dtscussed tn ref [3] for appltcattons tn nuclear phystcs, suggests that an algebratc descnptton of rotatton-vtbrauon spectra m tnntomuz molecules may be provided by two coupled algebratc structures, each with dynamtc algebra, U(-i) Thts can be Interpreted as couphng the rotattonal-viirattonal degrees of freedom of the two bonds It 1s convement to rcahze ths algebra agam tn terms cf creation and anruhdatton operators We thus Introduce two sets 32
0 GO9-2614/82/0000-0000/S
02.75 0 1982 North-Holland
Volume 8.5. number I ofboson
OpZF&OFS
H==H, +H,+
CflIlMICAL FIfYSKS LrnCRS Ql . 77; p and
0:.
&,
1
I 3anusr> 1982
and a ~o~r~span~r~~ hamrttontan
VQ
Each prece H I, HI! m eq (3) has the same form as eqs (-1) and (5) of rei [ I ] The mtcractton pounded into products of apcrators of system 1 and 2 rcspcctrvely, I c
(31 term I’t2 CWIbc c’r-
2 Dynamical symmetries One of thz adimrages of the dgebratc approach ts that $t 1s posstbfe, by studying the group structure of the problem, to cop=rruct z~naty~tc solur~ons of the ergcnvalue problem for the hamrltoman,H Thcsc .malytrc sofutlons ate known as dynamtc~ symmetnes [4&and ttre prtrucularly tmportant here beccrtuscOF the dtf~cult$~~ connected wtlt constructrng exa_ct sfrfutrons for systems of coupted non-hncar osedlrttors [Sj in ttr~sfetter, we focus our sttcnt~on on one solution, which appears to yteid spectra descrsbmg rotsttons and v~bratxons of hnear trtatomtc ntuiecules Tars IS obtamed by constdertng the case m whtch both hamtlronrans.ffl and 0, have the 0(4)syrnmetry dcscrtbrd rn rcf [I] and furthermore ihe tntcrnctron Vlt ts also budt wnh opewrors belongmg to the Ot-1) symmetry, I C,
Volume SS, number
I
U’“(4) @U”‘(4)
CHEMICAL
3
O”‘(1) 8 O’“‘(4) 3
PHYSICS
1 January 1982
LEl-KRS
O(4) > O(3) 3 O(2)
(7)
vibratton-rotation states of a certam electronic state of a trratomtc molecule are assumed to be contamed m a given trreductble representatton CL the group U(t)(4) o @(4) These are charactertzed by two numbers, Nt and A’?. the total number of quanta of species I and 2 respecttvely. The complete labelhng of states IS obtained by app1y111gstandard group thcoretrcal rechmques to the group cham eq. (7). The labels are IN,, N2, 9,. p2, (it, rl_). .I”, Jl) The values of nt and 172 contamed m each representation Nt . N2 are stmply gtven by ~7, =N,, N,--2, _ , IorO(‘\It=evenorodd)at1d~~=N~,N~-2, ., 1 or 0 (N3 = even or odd) The values of (rt , s-2) and .I are mstead more dlfticult to obtam A complete hstmg will be given m a longer pubhcatton. We have also Included a parrty quantum number R Thus appears because m the reductton from the group O(4) to O(3) (the rotation group) both pantws occur (for example one can have states wtth Jx = I’) The energy etgenvalues are given by All
E(~~‘,‘,.N~.0,.9~,(7,.T~),J’,nl)=~~-f(A,-C)17~(9t+7-)-$(A~-C)r1,(~~+7-) -+C[it(rt+1)+7$]
+BJ(J+l),
(8)
where E. IS a constant
wluch depends on _4 , , PI z C and A’,, IV, In order to show thdt tlus formula describes mdeed (although approummatcly) rotatton-vlbratron spectra of linear trtatomrc molecules, tt IS convement to Introduct rhc quantum numbers uI _u2, uj nnd I, through nr = A’, --2u, _q2 = A’-,-_lu,. T-, - = I,- T, = ,V, +N, - 3_“, - u, - 3uj Then, cq (8) can be rewritten as
-1C(“,+~)(“j+f)-c(u~+~)(“,+l)-c(”l+~)(”Z+l)+BJ(~+I) TIII~ expression
ca. be dnectly
compared
WIIII IIIC usual cxpansron
(9)
[6]
-
+X~~(“,+~)~“~+l)+X,~(“t+f)(“~+~)+Y~~(”,+I)(”j+~)+BJ(J+I)+
(10)
of the coupled algebraic cquatlons wth O(4) symmctr): thus produces a spectrum smtrlar to that gwn by eq (IO) s lth special values of the coefficrents w1,w~.w3, x~~,~~~,x~~,~~~.x~,x,~, xl3 and B Furthermore, drt mspcctton of the values of the quantum number 1, = ‘2 allowed by the group cham (7) shows that the solutton cif the coupled algebratc equations also produces auiorrlofrcal& the spectes Z, fl, A, approprtate to the double degenerate ndture of the u3 vtbratton In fact, tt provides a group theorettcal mterprctatron of the quantum number /2 mtroduccd errprrtcally As .m e-,ample the vtbratronal spectrum obtained by usmg eq (S) wtth Nt = 5O,N, = 50,A, = IS cm-t, C= I3 cm-‘.A. = 33 cm-t IS shown u-r fig I. where tt IS compared with the low-lymgvtbratlonsl spectrum of the CO2 molecules On top of each vlbrattonal st.tte there are butlt rotattonal bands Thts figure IS shown not so much to emphasrzc the agrcemcnt with experiment. but rarhcr m order to dtsplay the fact that the coupled algebratc system produces automatrcally all quahtatlve features of the observed spectra For a more quantttattve descrtptton, a breahmg of the O(t)(4) @ 0f2)(-l) symmetry must be consldered We also mention bnefl! that tt IS posstblc IO construct other analyttc solutrons to the Edgenvalue problem correspondmg to group chains which are dtfferent from
The solutron
(7). Although
we lntve not yet explored
m detad all posstble analytic
soluttons,
preltmmary
studtes show that tt
May be powble to descrlbc, with the U(l)(4) SJ U(‘)(4) group structure, both ngtd and normgld molecules. The latter point IS particularly Important in connection with the study of van der Waals molecules [7]. Fmally we note th.tt the approach described here IS related to that of Kclhnan et al [8], who drscussed tnatomtc molecules m terms of the group structure U(t)(3) 0 U(“)(3) The relatton between the approaches will be dacussed In detad m a longer pubhcatton. 31
I Jiu~~ry
CHIIMICAL PIIYSICS LEI-KRS
Volume 85. number 1
1982
3 OfU*
I)
21D-
2-
p=a-
-
-
-
_
I=J-
-
-
-
-
6-
ooo3,,
iI-
500(
L
Eap Th
Erp Th
I-
”
_
Exp Th
lOOI2; 02Ol"l
-
)2c
200(
lOO(
I
1IC I!.
otoii_ Y
(I1
Fig. 1. A compaaon between the cxpenmcntal vibratlorwl spectrum oi the CO2 molecule [6l and that calculated usmg q (8) ~lth A’, =50, h12= 5O.A I = 15 cm-‘. C= I3 cm-‘..t 2 = 33 cm-’ On top of each vibratIona lcrel there arc rotst~orul bandsas shown m the msert
3. Conclusions We have presented here an algebrruc approach to rotation-wbratlon spectra of trlatomlc molecules. and apphed tt to the sample case of hnear molecules The advantage of this approach ISthat It products automattcally the complex vtbrattonal and rotatlonal pattern of these molecules. Furthermore, it IS possible, usmg the same approach, to calculate all other properties of the molecule, such as mtenslues oiemlsslon and absorption hncs, excltatron probabdltles from one rotation--Vlbratlon level ut,y,u3,12,J to another ui,&,u;,fi,J’,etc. Fmally, the same ap 35
Volume 85 number 1
CHEAIICAI
PHYSICS LCI’TERS
1 January 1982
proxh can be generaked to even more complex molecules. For retra-atonuc molecules, one needs three coupled U(1) problems. and. m general, for N-atomic molecules. N-l coupled U(4) problems Although the complewty of the solutions mcreascs wth IV, the algcbratc structure of each U(4) problem IS sample enough that detailed calculatlons oi comple\ molecules mdy be powble m the near future.
We wsh to thanh R D Lcvmc, who stmwlated our Interest m an algebraic treatment of complex molecules, and A hndc and 1. Talm~ for useful dIscussIons This work was supported m part by the Stlchtmg voor Fundamcntecl Onderroeh der Matertc (F 0 M ) and In part under USDOE Contract EY-76-C-W-3074
References 11 j 121 131 [4) [S ] 161 171 181
36
I‘ l~chellc. Chim Phys Letters 73 (198 1) 181 T C Wwch and R B Bernhtcm. J Chem Phys 46 (1967) 4905 0 S ban Roosmalcn and A L: L D~epermk, Phps Lcrters 1OOB (!98 I) 299. I- Ixh~Uo,Comm Nucl Part Phls 8 (1978) 59 S A Rice. m Quantum d?. namlcs of molecules, cd R C Woollcy (Plenum Press. New York, 1980) p 257 G tlcrzbeF;. Moleculx spectra and molecular S1ructur t. Vol 2 (Van Nostrand. Prmccton, 1945) p 211 B Blancy and G E\\mg.Ann Rev Phjs Chcm 37 (1976) 553 %IE GAlman, r +marsnd R S Dcrry. 1 Chem Phys. 73 (1980) 2387, r Amar hl C LcUmanand R S Berry, J Chum Phys 70 (1979) 1973, hl C lieUman 3rd R S Berry. Chcm Phys. Letters 42 (1976) 327