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Orientation-translation coupling for an encaged diatomic molecule in β-quinol clathrate
CHEMICAL PH”SlCS
Volume 85. number 3
ORIENTATION-TRANSLATION IN fl-QUINOL
COUPLING
LEITERS
15
FOR AN ENCAGED
DIATOMIC
J~~nuxy 198,
MOLECULE
CLATHRATE
ShoJl HIROKAWA Departmorr
of Ckmrsrry
Recctved 6 October
Faculty of Scrence. A’>OIO Unrr ersry
h’yoro 606. fapan
1981
1981. III final iorm 19 October
Ortenl3tton-translation couphng IS studred quanrum-mechamcatly m the case of rhc mormn of N2 enaged in C)qurnot clathratc. whcrc rotstlonal motion IS regzndcd as hbratron Two cmpmcal guest-host potcntuls arc examtned to cuplam fwtnfrared c~prrtmcnts The coupbng manly tnfluences the translauon irequency to cause the shalt and modulate the wdth of ths spbttmg.
There couphng grees of tsolated however,
has been much mtcrest m the effect of the between the rotattonal and translatronal defreedom upon the infrared spectra of m3trLxmolecules [ I] Almost 3ll theorettcal studtes, have been concerned wtth CDSCSwhere the
Nz IS -1 1 X 10mz6 csu cm2 [5]. Summmg up the guest-host tnteracttonspves the Internal fteld U(0.c) seen by guest We dtvtde rt tnto .uR(w), UT(r) and the orrentatton-tr3nsl3tton coupltng URT(w,r) to express the hamtltomsn 3s
rotattonal rnotton can be regarded as free rotatton This note treats the c3.se where the rotatronal matron
H(w.r)=TR+TT+CJ(w,.)=HR+@-+ffRT,
IS regarded 3s hbratton, and we shah examme the molecular motton of Nz encaged m the p-phase of qutnol clathrate. A cage m j3qutnol IS formed by SIX qutnol molecules and has Ss symmetry [2,3]. Roughly speaking, the gcomettxrl shape of the cage IS spherztl We 3s sume a space-fLued coordmate system OXYZ tts ongtn IS 3t the center of the c3ge and the Z 3~s IS along the symmetry aus The ortentatron of the tntrrnucle3r 3x1s of the guest molecule and the posttlon of tts
wtth
center of mass (cm.)
are wntten
0,
f, respecttvely.
The guest-host tnteractton IS, 3s tn a prevtous paper [4], assumed to be made up from the 3tomatom (aa) potentud
(1) where R,, denotes the dtst3nce between an 3tom I of the guest and an atom] of a host molecule, and from the electrostatic multlpole-mulhpole utteraction. It IS assumed that the host molecule has a quadrupole moment Qh of the carbon nng and a dipole moment “rh for each OH group The quadrupole moment of 0 009~2614/82/0000-0000/S
0175
0 1982 North-Holland
HR = TR +
UR(w) , H= = TT + U=(r),
(2)
(3)
where TR, T* 3re rotattonl and kmettc energy opcr3tors. rcspecttvely. We .tssumc F-(f)
= (r$RIURT(w,r)l~R)
,
(4
where rLR IS the ground state of hbration The rot3tronal motion of N2 IS typtcf of the ltbmtton [6,7]. In 3ctmd computattons the ground st3te tn 3 hmdermg potrnttal h Ylo(w) IS adopted instead of th3t of HR, because the hmdermg potenttal of thrs form IS 3 good approvmatton to UR If the guest molecule has no moleculsr structure, hke 3n atom, the couphng URT between the rotatton3l and translatton3l degrees of freedom vantsltes. The aver3ge effect of the Itbratton upon the constramed trllnslation of the c m. IS taken mto account by URT It wrll become clerr thst the constratned translatton of the guest molecule ut II175
Volun~
S5
CHEMICAL
number 3
The unu~ of a,, b,, xc IO8 at mot-’ c3t mot-’ A6 and the unlils of et, Irth 9rr‘ IO-” cs” cm3 And to-” csu cm B
113 8 WC aN0 aNtI kc bNo bNtt @h “‘11
3971 ‘818 0 1968 3 6’9 3 61-t 1 237 -5 6 096
IO 6 0 4 4
189 127 9019 502 558 I 378 -2 8 0 8256 ---
00
cy
labels R. T ar2 asslgncd when cxh level IS regardzd as the excltcd stat2 of hbratlon or tmnslatlon, and allowed transtnons from the ground state m Lr-mfrared adsorprlon are denoted as IR Only p-N, IS consld2r2d JS befor (41 Eupenmentally the fibration fr2qucncy IS 50 7 cm-l 18 j and the translation frcquen-
-
00
70 5 60 7 49 7
.tl 8 IR 319 IR 23 2 00
603
ill 476 451 IR310 IR223 00
T R T T T T gr
Table 3
Energy Ievcts for potrttUJt
HR
HT
103 0 5-t 3
1173 LOS I IR 6-12 IR 520
00
376
00
set B (in umts of em-‘)
1090 107 7 IR 518 IR 537 00
1106 108 7 IR 558 IR 51 2 00
1037 545 tR 54 3 IR 530 00
R R T T gr
106 7
105 5 7-t 0 682 365 00
IR IR
I#=. Results calculated quantum-mcchamcally on the basts of the potcntlal parameter sets A and B of table I are presented m tables 1 and 3. In tables Z-4, the
66 3 65 -1 61 9 60 7 1R31 5 IR 29 8 00
tR 1126
59 0
bratIon should b2 dcscrlbed by J? (or the Ike). Then the Iibratron-translatton couphng wlti occw through
98 7 51 6
Janu3r) 1982
Energy Iesels ior potenrrd set B on the basts 01 the StrucIure of host crystat oi SO2 clathrafe (in units of cm-’ f
paramcrers
x
15
Ptt\ SICS LLTTERS
-
91 8 tR IR
77 1 52.i 381 00
906
T
765
T
601 IR516 IR379 00
R T T gr
(sometimes called the rattlmg frrqucncy) IS 53 5 em-l [9] In the case of set A, rhe vaiues of the aa potentxd pammcters appro~~ateIy correspond to thos2 fitted to the Lennard-Jones rntermoI2cuIar potential [IO] _ However, HT,fl and H al1 @ve two levels of translatton whose enrr@es ar2 much Iowzr than the rkpcrlmental one This mconslstency w1t.h rkpenment IS cluefly due to madequacizs m the aa potentlal “fh2 values of Set B of table 1 ar2 adJusted to @vc the expcnmental frzqucnclzs. For this set, allthough the constrruncd translation due to HT IS amsotroplc, oRT makes the translatron ahnos~ :sotroplc The eficct of the tugher-ord-zr couplmg HRT IS small Replacmg flT by UFT = URT(o,,r) m H1, we obtam Hz, wherco, IS the cIass1cal2qud~br~u~~ ortentatlon whrch IS along rhe Z axis We r2~hze from tabfe 3 that, for the lowest exctted states. the couphng bctwiten the rotatronal and transiattonal degrees of freedom is mostly mvolvcd m the smut efiect I/CRTof molecular onentatlon and almost rvclus~vely aficcts tile trnnslatlonal degrees of freedom. The hbratlon IS only slightly mflucnced by the couphng smce the fluctuation m the posltlon of the c m IS small Srmdar effects would also be evpectcd m some molccular hqulds, as thr clathratc 1s constdered to be a good example of the ceIi model of hqutds. The suggestton by van der WAS [l l] that the guest molecule tends to orient itself psraliel to th2 wall of the cage whan rts c m approaches the wail s not suitable for the present case The crystal srructure of the N2 clathrate has so far been assumed to br the same as that of th2 H,S clathrate as obtamcd by X-ray m2asurements [3]. The SO, clathrate has a somewhat dlffcrent structure [7], for example, with respect fo the mchnatlon of
CtlChlICAL
Volume 85, number 3
l’lI\SlCS
the plane of a host molecule. The results of table -I have been computed on the basrs of the host crystal structure of the SO, clathrate and the potentral set B We see that the firstc\crted level of translatron m table 4 doffers from that m table 3 A prcvrous result [ 131 also had an analogous dlsagrccment wtth ckpcrtment
The matn reason for these dtfficulttcs
IS tltat
the constratned
translatton
IS largely dependent
the short-range
rcpulstve part of the aa potenttal
on and
LLTTCRS
I5 J.mu~r) 1982
T Tcrao and Mr. S MJtsur ior useiul drscussrons on Pqumol clathratc He also wrshes to thank Professor H Matsuda ior valuable drscusstons on thus work He IS also mdebted to Professor H Chlhara for a sugcsttvc comment
on the clJthratc
m the early stage of
thts scrrcs ofstudres He also .&nowfedgcs the Computer Center, instrtute for hlolccular Sctencc, for the use of the HITAC hl-ZOOH, and the Data Proccssmg Center of Kyoto Umvcrsrty
sensttwe to the cage structure.
If we assume a relation u,, = b, v(d, +d,)16/7. where 4, d, are van der Waals ra d II [ 131 ,‘then we hare/= 12033 for set B It IS to be noted that the XI potentral wrth/= 1 [ 141 somctrmcs employed dots not lead to success in the present problem. To undcrstand rhe guest-host rntcractron thoroughly we shall need calculations clatltrate
on the clectronrc
state of qutnol
We summartze our drscussrons to conclude The cage structure ot the N, clathratc may be r~ssumcd to be the same as that of the HIS clathrate We h.rve obtamed the aa potentral parameters fitted to the eupertments. Finally, the trend that the guest molecule becomes parallel to the cqudrbrurm dtrcLtron IS esscntral to the onentatron-transkrtion couplmg The statrc clicct of molecular orrenwtron IS dommant The cuuphnggrcatly mfluences the tr.mslatron rather thdn the IIbratron for the lowest energy levels wtth respect to the shrit and the wrdth of the sphttmg
The autlror would like to thank Professor T Yamamoto
for constant cnrouragcmcnt,
for helpiul
conwrsattons
Dr K Oh.rda
on machmc calculattons,
Dr
References [ I J K D hlbller anJ I\! C Ro~l~s~l~~ldl-.ir-tnlr.rrcd sP’r~rn,stop, (wllcy-lnIerwcnce. NC\\ 1 orb, 197 I) 131 D I: P;r\rIm Jnd tl bl Poacll, J Cbcm Sot (1947) 108 [3 1 T C N’ Mrk J S Tsc, C Tsc. K Lee and Y cbon> J Chcm Sot Perhm II (1976) 1 I69 [-II S tlrrohJ\vzr. Blol Pb)s 36 (1978) ‘-9 151 AD Duckmgbam. R L Drscb and DA Dunmur J Am Cbcm Sot 90 (1968) 3105 161 tl Mc~cr. $1 C bl O’Brren snd J II ran Vlcch l’roc Ru> Sot Al-13 (1958)lIJ
181 A A + Crbson, R Cocand T A Swtf. J W,gn Rcwn 2-t (1976) 103 [9J J C Rur~wl. II \lr‘y~r .md P L Rrib.rrds J Cb~m Pb)s 53 (1963)-1391 [ 101 N C Par~on~gc.md R C Psmbcrton Tr.mc I‘.trad.r~ SOL 63(1967)3ll [III J H \.I” dcr \V.rA J Pbys Clwm Sob& 18 (1961) 82 [ 131 S tIrro1raw.r. Intern J Quantum Cbcm I8 (1980) 533 [ 13) A I hlr3lgorodsh~, ~lolLcular LQw3lS Jfld nlolrculss (A\wdLmlc Press I&\r 1 .,rh, 1973) [l-1] R A Scorn .md tl A Sclwr.r:.r J CIILIII Pb>s 45 (1966) 2091
Volume 85. number 3
ORIENTATION-TRANSLATION IN fl-QUINOL
COUPLING
LEITERS
15
FOR AN ENCAGED
DIATOMIC
J~~nuxy 198,
MOLECULE
CLATHRATE
ShoJl HIROKAWA Departmorr
of Ckmrsrry
Recctved 6 October
Faculty of Scrence. A’>OIO Unrr ersry
h’yoro 606. fapan
1981
1981. III final iorm 19 October
Ortenl3tton-translation couphng IS studred quanrum-mechamcatly m the case of rhc mormn of N2 enaged in C)qurnot clathratc. whcrc rotstlonal motion IS regzndcd as hbratron Two cmpmcal guest-host potcntuls arc examtned to cuplam fwtnfrared c~prrtmcnts The coupbng manly tnfluences the translauon irequency to cause the shalt and modulate the wdth of ths spbttmg.
There couphng grees of tsolated however,
has been much mtcrest m the effect of the between the rotattonal and translatronal defreedom upon the infrared spectra of m3trLxmolecules [ I] Almost 3ll theorettcal studtes, have been concerned wtth CDSCSwhere the
Nz IS -1 1 X 10mz6 csu cm2 [5]. Summmg up the guest-host tnteracttonspves the Internal fteld U(0.c) seen by guest We dtvtde rt tnto .uR(w), UT(r) and the orrentatton-tr3nsl3tton coupltng URT(w,r) to express the hamtltomsn 3s
rotattonal rnotton can be regarded as free rotatton This note treats the c3.se where the rotatronal matron
H(w.r)=TR+TT+CJ(w,.)=HR+@-+ffRT,
IS regarded 3s hbratton, and we shah examme the molecular motton of Nz encaged m the p-phase of qutnol clathrate. A cage m j3qutnol IS formed by SIX qutnol molecules and has Ss symmetry [2,3]. Roughly speaking, the gcomettxrl shape of the cage IS spherztl We 3s sume a space-fLued coordmate system OXYZ tts ongtn IS 3t the center of the c3ge and the Z 3~s IS along the symmetry aus The ortentatron of the tntrrnucle3r 3x1s of the guest molecule and the posttlon of tts
wtth
center of mass (cm.)
are wntten
0,
f, respecttvely.
The guest-host tnteractton IS, 3s tn a prevtous paper [4], assumed to be made up from the 3tomatom (aa) potentud
(1) where R,, denotes the dtst3nce between an 3tom I of the guest and an atom] of a host molecule, and from the electrostatic multlpole-mulhpole utteraction. It IS assumed that the host molecule has a quadrupole moment Qh of the carbon nng and a dipole moment “rh for each OH group The quadrupole moment of 0 009~2614/82/0000-0000/S
0175
0 1982 North-Holland
HR = TR +
UR(w) , H= = TT + U=(r),
(2)
(3)
where TR, T* 3re rotattonl and kmettc energy opcr3tors. rcspecttvely. We .tssumc F-(f)
= (r$RIURT(w,r)l~R)
,
(4
where rLR IS the ground state of hbration The rot3tronal motion of N2 IS typtcf of the ltbmtton [6,7]. In 3ctmd computattons the ground st3te tn 3 hmdermg potrnttal h Ylo(w) IS adopted instead of th3t of HR, because the hmdermg potenttal of thrs form IS 3 good approvmatton to UR If the guest molecule has no moleculsr structure, hke 3n atom, the couphng URT between the rotatton3l and translatton3l degrees of freedom vantsltes. The aver3ge effect of the Itbratton upon the constramed trllnslation of the c m. IS taken mto account by URT It wrll become clerr thst the constratned translatton of the guest molecule ut II175
Volun~
S5
CHEMICAL
number 3
The unu~ of a,, b,, xc IO8 at mot-’ c3t mot-’ A6 and the unlils of et, Irth 9rr‘ IO-” cs” cm3 And to-” csu cm B
113 8 WC aN0 aNtI kc bNo bNtt @h “‘11
3971 ‘818 0 1968 3 6’9 3 61-t 1 237 -5 6 096
IO 6 0 4 4
189 127 9019 502 558 I 378 -2 8 0 8256 ---
00
cy
labels R. T ar2 asslgncd when cxh level IS regardzd as the excltcd stat2 of hbratlon or tmnslatlon, and allowed transtnons from the ground state m Lr-mfrared adsorprlon are denoted as IR Only p-N, IS consld2r2d JS befor (41 Eupenmentally the fibration fr2qucncy IS 50 7 cm-l 18 j and the translation frcquen-
-
00
70 5 60 7 49 7
.tl 8 IR 319 IR 23 2 00
603
ill 476 451 IR310 IR223 00
T R T T T T gr
Table 3
Energy Ievcts for potrttUJt
HR
HT
103 0 5-t 3
1173 LOS I IR 6-12 IR 520
00
376
00
set B (in umts of em-‘)
1090 107 7 IR 518 IR 537 00
1106 108 7 IR 558 IR 51 2 00
1037 545 tR 54 3 IR 530 00
R R T T gr
106 7
105 5 7-t 0 682 365 00
IR IR
I#=. Results calculated quantum-mcchamcally on the basts of the potcntlal parameter sets A and B of table I are presented m tables 1 and 3. In tables Z-4, the
66 3 65 -1 61 9 60 7 1R31 5 IR 29 8 00
tR 1126
59 0
bratIon should b2 dcscrlbed by J? (or the Ike). Then the Iibratron-translatton couphng wlti occw through
98 7 51 6
Janu3r) 1982
Energy Iesels ior potenrrd set B on the basts 01 the StrucIure of host crystat oi SO2 clathrafe (in units of cm-’ f
paramcrers
x
15
Ptt\ SICS LLTTERS
-
91 8 tR IR
77 1 52.i 381 00
906
T
765
T
601 IR516 IR379 00
R T T gr
(sometimes called the rattlmg frrqucncy) IS 53 5 em-l [9] In the case of set A, rhe vaiues of the aa potentxd pammcters appro~~ateIy correspond to thos2 fitted to the Lennard-Jones rntermoI2cuIar potential [IO] _ However, HT,fl and H al1 @ve two levels of translatton whose enrr@es ar2 much Iowzr than the rkpcrlmental one This mconslstency w1t.h rkpenment IS cluefly due to madequacizs m the aa potentlal “fh2 values of Set B of table 1 ar2 adJusted to @vc the expcnmental frzqucnclzs. For this set, allthough the constrruncd translation due to HT IS amsotroplc, oRT makes the translatron ahnos~ :sotroplc The eficct of the tugher-ord-zr couplmg HRT IS small Replacmg flT by UFT = URT(o,,r) m H1, we obtam Hz, wherco, IS the cIass1cal2qud~br~u~~ ortentatlon whrch IS along rhe Z axis We r2~hze from tabfe 3 that, for the lowest exctted states. the couphng bctwiten the rotatronal and transiattonal degrees of freedom is mostly mvolvcd m the smut efiect I/CRTof molecular onentatlon and almost rvclus~vely aficcts tile trnnslatlonal degrees of freedom. The hbratlon IS only slightly mflucnced by the couphng smce the fluctuation m the posltlon of the c m IS small Srmdar effects would also be evpectcd m some molccular hqulds, as thr clathratc 1s constdered to be a good example of the ceIi model of hqutds. The suggestton by van der WAS [l l] that the guest molecule tends to orient itself psraliel to th2 wall of the cage whan rts c m approaches the wail s not suitable for the present case The crystal srructure of the N2 clathrate has so far been assumed to br the same as that of th2 H,S clathrate as obtamcd by X-ray m2asurements [3]. The SO, clathrate has a somewhat dlffcrent structure [7], for example, with respect fo the mchnatlon of
CtlChlICAL
Volume 85, number 3
l’lI\SlCS
the plane of a host molecule. The results of table -I have been computed on the basrs of the host crystal structure of the SO, clathrate and the potentral set B We see that the firstc\crted level of translatron m table 4 doffers from that m table 3 A prcvrous result [ 131 also had an analogous dlsagrccment wtth ckpcrtment
The matn reason for these dtfficulttcs
IS tltat
the constratned
translatton
IS largely dependent
the short-range
rcpulstve part of the aa potenttal
on and
LLTTCRS
I5 J.mu~r) 1982
T Tcrao and Mr. S MJtsur ior useiul drscussrons on Pqumol clathratc He also wrshes to thank Professor H Matsuda ior valuable drscusstons on thus work He IS also mdebted to Professor H Chlhara for a sugcsttvc comment
on the clJthratc
m the early stage of
thts scrrcs ofstudres He also .&nowfedgcs the Computer Center, instrtute for hlolccular Sctencc, for the use of the HITAC hl-ZOOH, and the Data Proccssmg Center of Kyoto Umvcrsrty
sensttwe to the cage structure.
If we assume a relation u,, = b, v(d, +d,)16/7. where 4, d, are van der Waals ra d II [ 131 ,‘then we hare/= 12033 for set B It IS to be noted that the XI potentral wrth/= 1 [ 141 somctrmcs employed dots not lead to success in the present problem. To undcrstand rhe guest-host rntcractron thoroughly we shall need calculations clatltrate
on the clectronrc
state of qutnol
We summartze our drscussrons to conclude The cage structure ot the N, clathratc may be r~ssumcd to be the same as that of the HIS clathrate We h.rve obtamed the aa potentral parameters fitted to the eupertments. Finally, the trend that the guest molecule becomes parallel to the cqudrbrurm dtrcLtron IS esscntral to the onentatron-transkrtion couplmg The statrc clicct of molecular orrenwtron IS dommant The cuuphnggrcatly mfluences the tr.mslatron rather thdn the IIbratron for the lowest energy levels wtth respect to the shrit and the wrdth of the sphttmg
The autlror would like to thank Professor T Yamamoto
for constant cnrouragcmcnt,
for helpiul
conwrsattons
Dr K Oh.rda
on machmc calculattons,
Dr
References [ I J K D hlbller anJ I\! C Ro~l~s~l~~ldl-.ir-tnlr.rrcd sP’r~rn,stop, (wllcy-lnIerwcnce. NC\\ 1 orb, 197 I) 131 D I: P;r\rIm Jnd tl bl Poacll, J Cbcm Sot (1947) 108 [3 1 T C N’ Mrk J S Tsc, C Tsc. K Lee and Y cbon> J Chcm Sot Perhm II (1976) 1 I69 [-II S tlrrohJ\vzr. Blol Pb)s 36 (1978) ‘-9 151 AD Duckmgbam. R L Drscb and DA Dunmur J Am Cbcm Sot 90 (1968) 3105 161 tl Mc~cr. $1 C bl O’Brren snd J II ran Vlcch l’roc Ru> Sot Al-13 (1958)lIJ
181 A A + Crbson, R Cocand T A Swtf. J W,gn Rcwn 2-t (1976) 103 [9J J C Rur~wl. II \lr‘y~r .md P L Rrib.rrds J Cb~m Pb)s 53 (1963)-1391 [ 101 N C Par~on~gc.md R C Psmbcrton Tr.mc I‘.trad.r~ SOL 63(1967)3ll [III J H \.I” dcr \V.rA J Pbys Clwm Sob& 18 (1961) 82 [ 131 S tIrro1raw.r. Intern J Quantum Cbcm I8 (1980) 533 [ 13) A I hlr3lgorodsh~, ~lolLcular LQw3lS Jfld nlolrculss (A\wdLmlc Press I&\r 1 .,rh, 1973) [l-1] R A Scorn .md tl A Sclwr.r:.r J CIILIII Pb>s 45 (1966) 2091