Anal y sis of Pure Ro ta tional and Vi bra tion-rotational Spec tra of NaCl X 1Σ+

and Quan tum-chemical Cal cu la tion of Re lated Mo lec u lar Prop erties

J. F. Ogilvie,a* H. J. Aa Jensenb and S. P. A. Sauerc

aEscuela de Quimica, Universidad de Costa Rica, Ciudad Universitaria Rodrigo Facio, San Pedro de Mon tes de Oca, San Jose 2060, Costa Rica

bKemisk Institut, Syddansk Universitet, Campusvej 55, DK-5230 Odense M, Den markcDe part ment of Chem is try, Uni ver sity of Co pen ha gen, Universitetsparken 5,

DK-2100 Co pen ha gen φ, Den mark

We fit ted pub lished fre quency and wave num ber data for pure ro ta tional and vi bra tion-rotational spec tra,re spec tively, of 23Na35Cl and 23Na37Cl to de rive pa ram e ters re lated to po ten tial en ergy and to the ro ta tional gfac tor. For com par i son with these ex per i men tal data we un der took quan tum-chemical com pu ta tion of adi a -batic cor rec tions, ro ta tional and vi bra tional g fac tors, elec tric dipolar mo ment and its de riv a tive as a func tionof internuclear dis tance in a range near Re as a test of an al ge braic ap proach to spec tral anal y sis; ex per i men tal,0.0287 ± 0.0014, and cal cu lated, 0.02149, val ues of gr at Re are in mod er ate agree ment. The com bined re sultsare dis cussed from a point of view of com pu ta tional spec trom e try.

Key words: Spec tral anal y sis; Vi bra tion-rotational spec tra; Quan tum-chemical cal cu la tions; Com pu ta -tional spec trom e try; Ro ta tional and vi bra tional g fac tors.

IN TRO DUC TION

So dium chlo ride has played an im por tant role in the de -

vel op ment of in fra red spec tros copy since the ear li est mea -sure ments in the mid in fra red re gion. For in stance, Ångstromused a prism of min eral rocksalt to dis perse in fra red ra di a tion in his pi o neer ex per i ments, 1 and prac ti tio ners of in fra redspec trom e try since that time cus tom arily em ploy op ti calcom po nents of pure crys tal line NaCl such as win dows incells to con tain gas eous and liq uid sam ples for spec tral ex am -

i na tion. The rea son is that this com pound is in ex pen sive andtrans mits light sat is fac to rily for wave lengths from the ul tra -vi o let re gion to 16 µm in the mid in fra red re gion. In the far -ther in fra red there is strong ab sorp tion of crys tal line NaClwith a max i mum about 32000 m -1 at trib uted to a lat tice vi -bra tion, called a trans verse op ti cal mode, in which sub -lattices of atomic cen tres of types Na and Cl os cil late with

op po site phase. Whether these atomic cen tres cor re spond es -sen tially to atomic ions or to al most neu tral at oms is con tro -ver sial,2 but in the gas eous phase at tem per a tures above 1100K highly po lar di atomic mol e cules with stoichiometric for -mula NaCl are a ma jor con stit u ent of vapour over a sam ple ofmol ten NaCl. Such a heated sam ple can emit ra di a tion re sult -

ing from dis crete vi bra tion-rotational tran si tions from many

states ther mally pop u lated un der those con di tions,3 but spec -tra in ab sorp tion in mi cro wave, milli metre-wave and in fra red re gions are also readily ob serv able with ap pro pri ate in stru -ments.4-7 For ro ta tional tran si tions, quan tum num ber J for ro -ta tional an gu lar mo men tum al ters by only one unit, where asfor all ob served vi bra tion-rotational tran si tions vi bra tionalquan tum num ber v con cur rently al ters by only one unit.3,6 Re -

ported spec tra in volve vi bra tional and ro ta tional states withmax i mum val ues v = 8 and J = 118, re spec tively.

In an anal y sis8 of then avail able spec tra of mo lec u larNaCl for com par i son of its pa ram e ters with those of otherchlo rides of el e ments in group 1 − HCl, LiCl, KCl and RbCl – we as sumed a value for the ro ta tional g fac tor, gr, of 23Na35Clbased on rea son able as sump tions ac cord ing to an anal y sis of

this prop erty known for other mol e cules in elec tronic state1Σ+ or 0 +.9 To en sure that there be no sys tem atic de vi a tion be -tween mea sured wave num bers of tran si tions of 23Na35Cl and23Na37Cl and those cal cu lated on the ba sis of pa ram e ters forpo ten tial en ergy, 10 such in for ma tion is help ful or even es sen -tial. Be fore the pres ent work, no value of gr was known forNaCl from ei ther ex per i ment or re li able cal cu la tion of mo lec -

u lar elec tronic struc ture. The ro ta tional g fac tor is a mul ti pli -

Jour nal of the Chi nese Chem i cal So ci ety, 2005, 52, 631-639 631

Ded i cated to Pro fes sor Ching-Erh Lin on the Oc ca sion of his 66th Birth day and his Re tire ment from Na tional Tai wan Uni ver sity

cand within a fac tor of pro por tion al ity be tween ro ta tional an -gu lar mo men tum and a net mag netic dipolar mo ment in duced as a re sult of mo lec u lar ro ta tion,9 but a non-rotating mol e culein elec tronic state 1Σ+ or 0+ pos sesses no per ma nent mag neticdipolar mo ment apart from ef fects of mag netic nu clei. In con -trast a mol e cule in such an elec tronic state can pos sess a netelec tric dipolar mo ment, rel a tive to axes fixed in the frame of

the mol e cule; in gen eral for a heteronuclear di atomic mol e -cule there is such an elec tric dipolar mo ment, which for NaClat its equi lib rium internuclear sep a ra tion for the elec tronicground state has a value11 about four fifths that ex pected fornonpolarizable, or not in ter act ing, Na + and Cl- ions at thesame dis tance. Such a large mag ni tude of elec tric mo mentgreatly fa cil i tates ob ser va tion of pure ro ta tional tran si tions,

be cause the in ten sity of the cor re spond ing spec tral lines inab sorp tion de pends on the square of this quan tity. In this waymea sure ments of such tran si tions have been made for val uesof ro ta tional quan tum num ber J up to 76 for mol e cules withinvi bra tional states up to v = 4, and with great rel a tive pre ci -sion.7 Adding those data to our pre vi ous data set for NaCl8 en -ables not only an im proved and ex tended eval u a tion of pa -

ram e ters for po ten tial en ergy but also an es ti mate of the ro ta -tional g fac tor. This anal y sis and re lated cal cu la tions of mo -lec u lar elec tronic struc ture are the sub ject of this ar ti cle.

To aid such an anal y sis and the in ter pre ta tion of re sultsthere from, we ap ply com pu ta tional spec trom e try.12 By thisterm we mean the ap pli ca tion of re sults from cal cu la tions ofmo lec u lar elec tronic struc ture that can yield in for ma tion not

only about dom i nant prop er ties per ti nent to vi bra tion- rota -tional spec tra, such as to tal en ergy and elec tric dipolar mo -ment that are the prin ci pal fac tors af fect ing the wave num berand in ten sity of those spec tral lines in ab sorp tion and emis -sion, but also and more im por tantly the fur ther mo lec u larprop er ties such as ro ta tional and vi bra tional g fac tors thatalso af fect slightly but sig nif i cantly the wave num bers of

tran si tions; all these prop er ties are readily cal cu lated as afunc tion of internuclear sep a ra tion R. Al though the ro ta tional g fac tor is a mea sur able quan tity, most di rectly as a re sult ofsplit ting, due to the Zeeman ef fect,9 of spec tral lines for purero ta tional tran si tions, there ex ists no known mag netic ef fectof low or der that yields com pa ra ble in for ma tion about the vi -bra tional g fac tor, gv, but in prin ci ple this is an ex per i men tally

mea sur able quan tity. Con founding the in flu ence of the lat tertwo quan ti ties are two fur ther terms in an ef fec tive hamil -tonian ,(R) for mo tion of atomic nu clei and their as so ci atedelec trons in a di atomic mol e cule ex press ible in this form:10

$p (1 + gv(R) me/mp) $p /(2µ) + (1 + gr(R) me/mp) £2 J (J + 1)/(2 µR2) + V(R) + V’(R) + V”(R) (1)

Here $p is a lin ear mo men tum con ju gate to a vec tor to rep re -sent in stan ta neous internuclear sep a ra tion R, and the nextterm in volves the square of an an gu lar mo men tum that takesdis crete val ues £[J(J + 1)]1/2, with J in gen eral a non-negativein te ger or half in te ger; V(R) rep re sents the internuclear po -ten tial en ergy that is in de pend ent of nu clear mass, me is theelec tronic rest mass and mp is the protonic mass; £ ≡ h/2π,with Planck’s con stant h. The vi bra tional and ro ta tional g fac -tors, with typ i cal mag ni tudes of both |gr| and |gv| ≤ 1, ap pearin terms as ra dial func tions gr(R) and gv(R) that serve as cor -rec tions to the ki netic en ergy of nu clear mo tion par al lel andper pen dic u lar to the internuclear axis, hence per tain ing to vi -bra tional and ro ta tional mo tions, re spec tively. Of two fur therad dends in this hamiltonian that have no di rect phys i cal sig -nif i cance V’(R) con sti tutes a ra dial func tion for adi a baticcor rec tions to take into ac count a small de pend ence on nu -clear mass that would oth er wise af fect V(R), whereas V”(R)con sti tutes a ra dial func tion for nonadiabatic ef fects fur therto those within gr(R) and gv(R); as V”(R) is ex pected to beneg li gi ble for NaCl we omit it from fur ther con sid er ation.Cal cu la tion of V’(R), gr (R) and gv(R) is readily prac ti ca blethrough al go rithms that have been im ple mented in suites ofcom puter pro ce dures. As in for ma tion about only two, atmost, of these three ra dial func tions might be gleaned fromspec tral anal y sis of data of fre quency type in the ab sence ofex per i ments in volv ing ex ter nal elec tric or mag netic field, the func tion of com pu ta tional spec trom e try is to pro vide in for -ma tion about one or more of these ra dial func tions so as to en -able a phys i cal mean ing to be at tached to spec tral pa ram e tersas so ci ated with the re main ing func tions. Be cause all threecon tri bu tions to wave num bers of spec tral lines have muchsmaller – but still sig nif i cant – mag ni tudes than that from po -ten tial en ergy, the cal cu la tion of V’(R), gr(R) and gv(R) re -quires much less ac cu racy than would the cal cu la tion of V(R)to yield an ac cu racy com pa ra ble with that of a mea sure mentof wave num ber of an ac tual tran si tion.

ANAL Y SIS OF SPEC TRA

We in clude in our anal y sis all re ported spec tra for purero ta tional and vi bra tion-rotational tran si tions. Early mea -sure ments4 of mi cro wave spec tra yielded a few fre quen cies

632 J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 Ogilvie et al.

of lines for pure ro ta tional spec tra, four lines for 23Na35Cl upto v = 3 and three lines for 23Na37Cl up to v = 2, all for J = 2 ←J = 1 and with max i mum rel a tive pre ci sion ~3 × 10-6 . Sub se -quent ex ten sion5 of mea sure ments in a range [7, 24] of J wasachieved for 13 lines of 23Na35Cl up to v = 2 and six lines of23Na37Cl up to v = 1 but with rel a tive pre ci sion ~3×10-7. Forvi bra tion-rotational tran si tions com pris ing 1184 in to talnum ber mea sured in emis sion,3 the max i mum value of J is118 and the max i mum value of v is 8; al though the max i mumrel a tive pre ci sion of these mea sure ments for 23Na35Cl is ~6 ×10-6, for 23Na37Cl the rel a tive pre ci sion was ~10 -5. To the lat -ter data we added 13 unduplicated tran si tions for 23Na37Clmea sured in ab sorp tion,6 with max i mum rel a tive pre ci sion 2× 10-6. A set of these data com pris ing 1225 unduplicated tran -si tions in to tal was the sub ject of our pre vi ous anal y sis.8 Agreat ex ten sion of pure ro ta tional tran si tions into the sub- millimetre range was achieved7 in mea sure ment of 133 linesof 23Na35Cl and 58 lines of 23Na37Cl, up to J = 76 and up to v =4 and max i mum rel a tive pre ci sion ~3 × 10-8. Al though the lat -ter range7 was much greater than for pre ced ing mea sure ments of pure ro ta tional tran si tions,4,5 the range of both v and J sam -pled in vi bra tion-rotational tran si tions was much greater inturn, as spec i fied above. The prin ci pal ben e fit of the lat terpure ro ta tional tran si tions was their su pe rior pre ci sion, which en ables a greatly in creased sen si tiv ity to iso to pic ef fects as -so ci ated with atomic cen tre Cl.

As a ba sis of anal y sis of these data we ap plied this for -mula10 for dis crete val ues of en ergy EvJ

i for a par tic u lar iso to -pic spe cies, con sis tent with the ef fec tive hamiltonian above:

h c lk =

∞

=

∞

∑∑00

(Ykl + Zklv,Na + Zkl

v,Cl + Zklr,Na + Zkl

r,Cl)

(v + 1/2)k [J (J + 1)] l (2)

Apart from ve loc ity of light c, in this for mula Dun ham co ef fi -cients Ykl take into ac count the ma jor ef fects cor re spond ing to me chan i cal mo tions of vi bra tion and ro ta tion of two atomicnu clei and their as so ci ated elec trons about the cen tre of mo -lec u lar mass, and aux il iary co ef fi cients Zkl, of two types, takeinto ac count the ex tra-mechanical ef fects whereby elec tronsfail to fol low per fectly one or other atomic nu cleus in its vi -bra tional and ro ta tional mo tion. Ex tra-mechanical ef fects10

are sep a ra ble into vi bra tion-rotational con tri bu tions, in Zklv,

and fur ther ro ta tional con tri bu tions, in Zklr; co ef fi cients of

both these types have a de pend ence on in verse mass of sep a -rate atomic cen tres, thus ei ther Na or Cl, whereas co ef fi cients Ykl de pend on the re duced mass of the mol e cule as a whole.

The re la tion10 be tween these term co ef fi cients of types Ykl

and Zkl and terms in an ef fec tive hamiltonian arises from theex pres sion of the lat ter terms as ra dial func tions, thus ofinternuclear sep a ra tion R or a se lected func tional of R; in thepres ent case we em ploy a re duced dis place ment vari able z de -fined as

z = 2 (R − Re)/(R + Re) (3)

We de fine four ra dial func tions in terms of this vari able, forpo ten tial en ergy,

V(z) = V0 + h c c0 z2 (1 + j=∑

1

cj zj) (4)

for the vi bra tional g fac tor,

gv(z) = mp (j=∑

0

s jNa zj / MNa +

j=∑

0

sjCl zj / MCl) (5)

for the ro ta tional g fac tor,

gr(z) = mp (j=∑

0

tjNa zj/MNa +

j=∑

0

tjCl zj/MCl) (6)

and for adi a batic cor rec tions.

V’(z) = h c me (j=∑

0

ujNa zj/MNa +

j=∑

0

ujCl zj / MCl) (7)

For NaCl in its elec tronic ground state X 1E+ that is our pres -ent con cern, V0 ≡ 0; both gv and gr are cal cu lated as sums ofelec tronic and nu clear con tri bu tions: the nu clear con tri bu tion de pends on only atomic num bers and atomic masses, whereas the elec tronic con tri bu tion is cal cu lated for mally as a sum ofsquares of ma trix el e ments of nu clear op er a tors over elec -tron i cally ex cited states, called nonadiabatic ef fects. Adi a -batic cor rec tions are cal cu lated as the cor re spond ing ex pec ta -tion val ues within the elec tronic ground state. No part of theex pres sion for po ten tial en ergy de pends on nu clear mass, butex pres sions for vi bra tional and ro ta tional g fac tors and foradi a batic cor rec tions de pend on masses of sep a rate atomiccen tres as in di cated, and in clude also a ra tio of elec tronic andatomic masses; for the adi a batic cor rec tions we ignore thedif fer ence be tween a nu clear and an atomic mass of a givennu clide. Co ef fi cients tj

Na,Cl can be re lated to co ef fi cients pj ina ra dial func tion for elec tric dipolar mo ment of this form:9

p(z) = j=∑

0

pj zj (8)

In volving the value of the ro ta tional g fac tor at Re and the cor -

Anal y sis of Spec tra of NaCl J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 633

re spond ing value of p(Re) for which z = 0, the re la tions per ti -nent to anal y sis of data of NaCl are

t0Na = µ [gr(Re)/mp − 2 p0/(e Re MCl)] (9)

andt0

Cl = µ [gr(Re)/mp + 2 p0/(e Re MNa)] (10)

In par tic u lar, Dun ham co ef fi cients 13 Ykl de pend on re -duced mass µ,

µ = MNa MCl/(MNa + MCl) (11)

equi lib rium internuclear sep a ra tion Re and co ef fi cients cj inV(z). Aux il iary co ef fi cients of type Zkl

v,Na de pend on the pre -ced ing pa ram e ters plus MNa and co ef fi cients sj

Na in gv(z) anduj

Na in V’(z), and anal o gously for Zklv,Cl, whereas co ef fi cients

of type Zklr,Na de pend like wise on pa ram e ters within Ykl plus

MNa and co ef fi cients sjNa in gv(z) and tj

Na in gr(z), and anal o -gously for Zkl

r,Cl. All these ex pres sions are re ported ex plic itlyelse where.10 An im por tant as pect of an ap pli ca tion of Dun -ham’s for mal ism13 is that Dun ham co ef fi cients Ykl and aux il -iary co ef fi cients Zkl of var i ous types, in con sis tent sets, num -ber min i mally to re pro duce sat is fac to rily the wave num bersin the full data set on con duct ing a si mul ta neous fit of bothpure ro ta tional and vi bra tion-rotational tran si tions of all iso -to pic spe cies; fur ther such co ef fi cients Ykl and Zkl are set to

zero. The eval u ated co ef fi cients im ply val ues of Re plus ra -dial co ef fi cients of types cj, sj, tj and uj in fi nite sets thatthereby be come eval u ated; fur ther ra dial co ef fi cients are en -tirely un de fined, but are un likely to have zero val ues. For ap -pli ca tion to spec tra of NaCl, be cause avail able spec tra in -volve only 23Na, there is no pos si bil ity of eval u at ing co ef fi -cients s j

N a, tjN a and uj

Na di rectly from ex per i men tal data ofonly wave-number type, but ei ther in di rect meth ods or ap pli -ca tion of re sults from quan tum-chemical cal cu la tions is fea -si ble.

In anal y sis of spec tra of NaCl we first fit ted all tran si -tions ex cept the pure ro ta tional data7 with tran si tions in volv -ing large J. Thus for 17 pure ro ta tional and 1206 vi bra tion- rotational tran si tions of 23Na35Cl and 23Na37Cl within rangesν ≤ 8 and J ≤ 120, six in de pend ent fit ted pa ram e ters, spe cif i -cally Re and c0-c4, with two con strained pa ram e ters – t0

Na,Cl

es ti mated, ac cord ing to for mu lae 9 and 10, from elec tricdipolar mo ment11 and gr at Re fac tor taken to be zero – suf ficeto re pro duce the wave num bers for which the best rel a tivepre ci sion8 is 3 × 10-7. These pa ram e ters ap pear in the first col -umn of Ta ble 1. The cor re spond ing value of re duced stan dardde vi a tion $σ of this fit is 0.921; that this value is less than unity in di cates that pre ci sions of mea sure ments are es ti mated con -ser va tively. Al though those ra dial co ef fi cients in Ta ble 1might be con sid ered to pos sess a phys i cal mean ing in re la tion to their re spec tive ra dial func tions, one must bear in mindthat the pur pose of these co ef fi cients is to en sure a con sis tent

634 J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 Ogilvie et al.

Table 1. Coefficients of radial functions and other molecular parameters of NaCl X 1∑+, depending on extent of data set and constrainedparameters

Parameter 1223 data, [ t0Na, t0

Cl] 1412 data, [t0Na, t0

Cl] 1412 data, [t0Na] 1428 data, [t0

Na, ujNa, uj

Cl]

c0/m-1 15247257 ± 18 15247401 ± 22 15247425 ± 24 15247678 ± 22

c1 -2.076785 ± 0.000037 -2.0768284 ± 0.0000095 -2.0768239 ± 0.0000095 -2.0767952 ± 0.0000102c2 2.54160 ± 0.00027 2.53836 ± 0.00048 2.53836 ± 0.00048 2.537935 ± 0.00048c3 -1.8567 ± 0.0020 -1.84459 ± 0.00162 -1.84459 ± 0.00162 -1.84411 ± 0.00169c4 -0.2888 ± 0.0176 0.0198 ± 0.042 0.0195 ± 0.042 0.0747 ± 0.044c5 ... 1.97 ± 0.29 1.975 ± 0.29 1.635 ± 0.29c6 ... -2.21 ± 0.41 -2.22 ± 0.41 -2.697 ± 0.47s0

Cl ... ... -0.238 ± 0.032 -2.718 ± 0.037t0

Na [0.81] [0.81] [0.727] [1.0225]t0

Cl [-0.77] [-0.77] -0.8527 ± 0.0082 -0.5598 ± 0.0096t1

Cl ... -1.685 ± 0.067 ... . ..u1

Cl/106 m-1 ... -1.240 ± 0.126 ... a

U0,1/m-1 u 302.465982 ± 0.000047 302.466455 ± 0.000039 302.467076 ± 0.000039 302.469618 ± 0.000045

U1,0/m-1 u½ 135820.124 ± 0.091 135820.873 ± 0.106 135821.120 ± 0.115 135822.819 ± 0.110

Re/10-10 m 2.36080425 ± 0.00000058 2.36080240 ± 0.00000057 2.36079998 ± 0.00000057 2.36079006 ± 0.00000058ke/N m-1 108.687209 ± 0.000147 108.688406 ± 0.000171 108.688801 ± 0.000184 108.691521 ± 0.000177σ̂ 0.921 0.880 0.881 1.032a Coefficients uj

Na and u jCl are constrained to values presented in Table 3.

and min i mal set of Dun ham co ef fi cients13 Ykl and aux il iaryco ef fi cients10 Zkl that they rep re sent, and that al ter na tivemeans14,15 to de rive these Dun ham co ef fi cients in volve nosuch ra dial func tions, which be come ac cord ingly artefacts ofa par tic u lar math e mat i cal model.

De spite the fact that an ad di tional 189 pure ro ta tionallines7 in volve in their as sign ments quan tum num bers, spe cif -i cally v ≤ 4 and J ≤ 76, that are well within the to tal ranges of vand J of the vi bra tion-rotational tran si tions,3,6 their wavenum bers are un sat is fac to rily re pro duced, rel a tive to their pre -ci sion of mea sure ment that is 3 × 10-8 at best, with only thoseeight pa ram e ters. We ac cord ingly made fur ther fits in clud ingthose 189 pure ro ta tional lines of both iso to pic spe cies;7 withre ten tion of two con strained pa ram e ters t0

Na,Cl , not only c5 and c6 for po ten tial en ergy, re flect ing en hanced sen si tiv ity of theen tire set of data to ef fects of cen trif u gal dis tor tion, but also apa ram e ter t1

Cl re lated to the ro ta tional g fac tor and pa ram e teru1

Cl as so ci ated with adi a batic cor rec tions, re flect ing en -hanced sen si tiv ity of ad di tional data to iso to pic ef fects as so -ci ated with Cl, be come thus eval u ated. These re sults ap pearin the sec ond col umn of Ta ble 1. The value of $σ for this fit dif -fers im ma te ri ally from that of the pre ced ing fit. Com par i sonof val ues of pa ram e ters be tween the first two col umns of Ta -ble 1 in di cates that, ex cept c4 , only small al ter ations oc cur;hence for cor re spond ing pa ram e ters these fits are rea son ablysta ble. There is no in di ca tion that fur ther nonadiabatic ef -fects, pres ent within V”(R) in the ef fec tive hamiltonian, arere quired to be taken into ac count in this fit, con sis tent with(me/µ)2 ~ 10-9 for NaCl be ing smaller than the best pre ci sion.

Be cause the ad di tional pure ro ta tional data7 are so pre -cise, in a fur ther fit we avoided con strain ing a value of t0

Cl,but dis cov ered a fur ther pa ram e ter, of which s0

Cl seemed best, to be also nec es sary; re sults ap pear in the third col umn of Ta -ble 1. With a known pre cise value11 of p0 of 23Na35Cl and ourfit ted value of t0

Cl in serted in for mu lae 9 and 10, we es ti mategr(Re) = 0.00741 ± 0.00060 and t0

Na; the lat ter value was in -cluded as a con straint in it er a tive fit ting un til con ver gencewas at tained. This value of gr for 23Na35Cl is a gen u ine pre dic -tion as no value was known from other ex per i ment or cal cu la -tion at the time of these fits, but its re li abil ity rests on the as -sump tion of neg li gi ble adi a batic cor rec tions.

QUAN TUM-CHEMICAL CAL CU LA TION OFMO LEC U LAR PROP ERTIES

For this rea son, we cal cu lated adi a batic cor rec tionswith a wave func tion of type multi-configuration, self- con sis -

tent field (MCSCF)16 and com plete ac tive space (CASSCF)17

with eight elec trons in thir teen orbitals – five σ, three π andone δ orbitals – in cluded in the ac tive space and us ing(17s12p5d4f|7s6p4d3f) con trac tions for Na and Cl with theANO ba sis sets.18 The re ported val ues are cal cu lated with nu -mer i cal dif fer en ti a tion, per formed with the Westa pro gram,19

of MCSCF wave func tions and a lo cal de vel op ment ver sionof Dal ton.20

We ex press di rect re sults of adi a batic cor rec tions ateach internuclear dis tance R in the fol low ing form:

V’(R)/h c = ∆VNa(R)/MNa + ∆VCl(R)/MCl (12)

Values of ∆VNa,Cl and of the to tal V’(R)/hc of 23Na35Cl for tense lected val ues of R ap pear in Ta ble 2. We fit ted those val uesof ∆VNa,Cl to a poly no mial in z to eval u ate co ef fi cients uj

Na,Cl

con sis tent with for mula 7; for a quartic and a cu bic poly no -mial suf fi cient to re pro duce sat is fac to rily the cal cu latedpoints, the re sult ing co ef fi cients ap pear in Ta ble 3. The inter -nuclear dis tances se lected for these cal cu la tions em pha sizethe range near Re and all lie within the range of a strongly po -lar mol e cule, be cause the point of an avoided cross ing of thecurve for the elec tronic ground state of NaCl with an othercurve for po ten tial en ergy in an elec tron i cally ex cited state

Anal y sis of Spec tra of NaCl J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 635

Table 2. Adiabatic corrections ∆V for atomic centres Na and Cland total adiabatic correction V’(R) for NaCl X 1Σ+ as afunction of internuclear separation R

R/10-10 m ∆VNa/102 u m-1 ∆VCl/102 u m-1 V’(R)/h c m-1

1.0000 17262.3 45717.5 112.9112.0000 16986.5 45651.1 112.1492.2200 16976.1 45645.8 112.1162.2800 16974.3 45644.9 112.1102.3608 16972.3 45644.0 112.1042.4400 16970.7 45643.3 112.0992.5200 16969.3 45642.7 112.0953.0000 16964.4 45641.6 112.0834.0000 16962.6 45642.2 112.0788.0000 16961.9 45643.4 112.078

Table 3. Coefficients uj for adiabatic corrections for atomiccentres Na and Cl in NaCl X 1Σ+

j u jNa/106 m-1 u j

Cl/106 m-1

0 3093.8508 ± 0.0062 8320.4066 ± 0.01581 -9.908 ± 0.109 -4.468 ± 0.1802 26.96 ± 0.26 10.260 ± 0.1133 -36.95 ± 0.21 -5.75 ± 0.234 17.54 ± 0.32 0

also of sym me try class 1Σ+ is ~11 × 10-10 m. The dif fer ence inen er gies be tween these elec tronic states for internuclear dis -tances within the range of this do main, for in stance near ~8 ×10-10 m, is small enough to cause prob lems in cal cu la tions ofadi a batic ef fects that hence re quire care ful han dling.

With pro gram pack age20 Dal ton 1.2 for com pu ta tion ofmo lec u lar elec tronic struc ture and mo lec u lar prop er ties inan other de vel op men tal ver sion, we cal cu lated also the ro ta -tional gr and vi bra tional gv fac tors, elec tric dipolar mo ment pand its de riv a tive dp/dR with re spect to internuclear dis tancewith wave func tions16,21 of type multi-configurational, self- consistent field and com plete ac tive space17 (CAS). The cal -cu la tion of ro ta tional9,22-33 and vi bra tional32,33 g fac tors us inglin ear-response the ory and de tails of an im ple men ta tion ofthe ro ta tional g fac tor us ing ro ta tional Lon don orbitals,34 thevi bra tional g fac tor32 and the de riv a tive of elec tric dipolarmo ment35 is de scribed in gen eral as pects else where. We ini ti -ated cal cu la tions with a com plete ac tive space and nat u ralorbitals ob tained on diagonalizing the one-electron den sityma trix eval u ated through sec ond or der in Møller-Plesset per -tur ba tion the ory36,37 (MP2). The com plete ac tive space com -prised MP2 nat u ral orbitals 7σ+-11σ+, 3π-5π and 1δ, whereasMP2 nat u ral orbitals 1σ+-6σ+ and 1 π-2π were kept dou bly oc -cu pied through out these CAS cal cu la tions. The ba sis set em -ployed is based on po lar ized ba sis sets of me dium size.38,39

Be cause these ba sis sets are known40 to be in suf fi ciently flex -i ble for ac cu rate cal cu la tion of mag netic prop er ties, we sup -pressed com pletely the con trac tion of ba sis func tions andadded ad di tional po lar iza tion func tions, of d-type, and sec -ond po lar iza tion func tions, of f-type, with ex po nents in thesame set as the p-func tions ac cord ing to a phi los o phy of me -dium-sized po lar ized ba sis sets; the ex po nents of the ad di -tional func tions are pre sented in Ta ble 4, and re sults for23Na35Cl at five internuclear dis tances are pre sented in Ta ble5.

At each dis tance the value of the to tal vi bra tional g fac -tor is no ta bly larger than that of the ro ta tional g fac tor. Thenu clear con tri bu tion to both g fac tors is pos i tive and con stant,

0.48502, whereas the elec tronic con tri bu tions are both neg a -tive. The ab so lute value of the elec tronic con tri bu tion ishence slightly larger for the ro ta tional g fac tor than for the vi -bra tional g fac tor, con trary to what is found for H2

32 andHeH+,33 for which near Re the elec tronic con tri bu tion to gv

was four to seven times as large as the elec tronic con tri bu tionto gr. More over, rel a tive to the nu clear con tri bu tion, the elec -tronic con tri bu tions to both gr and gv are larger than for H2

and HeH+; the to tal g fac tors are thus much smaller for NaClthan for those mo lec u lar spe cies, which im plies that the elec -trons fol low more closely the mo tion of one or other as so ci -ated atomic nu cleus.

DIS CUS SION

Po ten tial en ergyPa ram e ters c0-c6 in Ta ble 1 de fine a ra dial func tion for

internuclear po ten tial en ergy in de pend ent of mass; this func -tion is pre cisely what one ex pects to re sult from an ex actquan tum-mechanical cal cu la tion of to tal mo lec u lar elec -tronic en ergy, in clud ing internuclear re pul sion, at each fixedinternuclear dis tance in a do main/10-10 m [2.09, 2.79]. Endpoints of that do main are ap prox i mately the clas si cal turn ingpoints of nu clear os cil la tion for vi bra tional state v = 8 of23Na35Cl, which is the most en er getic vi bra tional state forwhich ex per i men tal data are in cluded in the pres ent data set.That quan tum-mechanical cal cu la tion must in clude ‘rel a tiv -is tic ef fects’ that are nat u rally em bod ied within ex per i men taldata, so hence be come an arte fact of ap prox i mate com pu ta -tional ap proaches. The equi lib rium internuclear dis tanceRe/10-10 m = 2.36079998 ± 0.00000057 pre sented in Ta ble 1has a rel a tive un cer tainty 0.24 parts per mil lion, most ofwhich re sults from un cer tainty in fun da men tal phys i cal con -stants h and NA that are in volved in a con ver sion of spec tralpa ram e ter U0,1 ≡ Y0,1µ into that dis tance. That un cer tainty inRe is also about 1 per cent of ei ther nu clear di am e ter. For com -

636 J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 Ogilvie et al.

Table 4. Exponents of additional first and second polarizationfunctions

Type Na Cl

d 0.0163 6.59950.0057 2.7141

f 0.0465 0.35800.0163 0.12500.0057 0.0436

Table 5. Calculated rotational g factor, gr, vibrational g factor, gv,electric dipolar moment, p, and its gradient, dp/dR, for23Na35Cl in electronic state X 1Σ+ as a function ofinternuclear separation R

R/10-10 m gr gv p/10-30 C m dp/dR/10-20 C

2.2200 0.02003 0.06080 -28.442 -13.1052.2800 0.02076 0.05974 -29.240 -13.4582.3608 0.02149 0.05845 -30.345 -13.8942.4400 0.02200 0.05731 -31.461 -14.2822.5200 0.02234 0.05628 -32.618 -14.638

par i son the atomic masses 41 of 23Na and 35Cl have rel a tiveprecisions 1.3 × 10-4 and 1.1 × 10-3 ppm, re spec tively. For aquan tum-mechanical cal cu la tion to yield val ues of dif fer -ences of vi bra tion-rotational en er gies com pa ra ble with ex -per i ment, that cal cu la tion must achieve a rel a tive pre ci sion~3 × 10-8 on a ba sis of the best mea sure ments of ro ta tionalfre quen cies or at least ~1.6 × 10-6 on a ba sis of δc0/c0 forwhich val ues ap pear in Ta ble 1.

To fa cil i tate ac quir ing an un der stand ing of mo lec u larprop er ties and spec tral phe nom ena from a spec tral anal y sis,an ad van tage of this data set of NaCl is that mea sure ments ofspec tral wave num bers en com pass max i mum ranges [0, 8] ofvi bra tional quan tum num ber v and [0, 118] of ro ta tionalquan tum num ber J. Al though that level J = 118 is at tained foronly one vi bra tional state, v = 1, for other vi bra tional statesup to v = 6 the ro ta tional ex ci ta tion is al most as great. For vi -bra tional states v = 7 and 8, the ex tents of ro ta tional ex ci ta tion are J = 82 and 75, re spec tively. Each ro ta tional state is for -

mally char ac ter ised with its own func tion of ef fec tive po ten -tial en ergy, de fined as a sum of rotationless po ten tial en ergyand a cen trif u gal term,

Veff (R, J) = V(R) + h c Be J (J + 1) Re2/R2 (13)

In Fig. 1 we pres ent a graph that ex hib its curves of ef fec tivepo ten tial Veff (R, J)/h c for J = 0, 75, 82 and 118. Hor i zon talstraight lines within these curves de note en er gies of pure vi -bra tional states in the lower man i fold per tain ing to J = 0 andof ap prox i mately max i mum ob served vi bra tion-rotationalstates for J = 75, 82 and 118; the en ergy of state v = 3, J = 118prac ti cally co in cides with that of state v = 8, J = 75. In clu sionof that cen trif u gal term within the ef fec tive po ten tial mightpro duce a max i mum in the curve for R > Re, but all suchcurves pos sess the same as ymp totic limit for large R; for J =118 this max i mum en ergy oc curs at ~7.2 × 10-10 m, at whichpoint, not shown in Fig. 1, the dif fer ence be tween curves for J= 118 and J = 0 is ~3 × 104 m-1.

An other re sult of this anal y sis of vi bra tion-rotationalspec tra of NaCl is the pro duc tion of a pre cise value of theequi lib rium force co ef fi cient, ke/N m-1 = 108.688801 ±0.000184, which is about one fifth that of HCl, both for elec -tronic ground states of sym me try class 1Σ+; the equi lib riumbind ing en er gies De/106 h c m-1 of NaCl and HCl are 3.43 and3.575, re spec tively. De spite the sim i lar ity in bind ing en er -gies, these force co ef fi cients ex hibit a dis par ity that might beas so ci ated with var ied char ac ters of bind ing, largely ionic for NaCl but largely co va lent for HCl. Even at v = 8 the vi bra -tional en ergy of NaCl within elec tronic state X is only 0.087De, and re mains less than the en ergy of the most en er getic ro -ta tional state ob served for v = 0, as dem on strated in Fig. 1.

In clu sion of adi a batic cor rec tions in fits of spec tral dataOur pre dic tion that gr(Re) = 0.00741 ± 0.00060 is pred i -

cated on a sup po si tion that adi a batic cor rec tions have smallmag ni tudes rel a tive to the ef fect of the ro ta tional g fac tor.With cal cu lated val ues of adi a batic cor rec tions in Ta ble 2 wecan test that sup po si tion. In pre cise terms, we must com parecon tri bu tions to Z0,1 for both atomic cen tres Na and Cl fromadi a batic ef fects and from gr. With val ues of pa ram e ters in the third col umn of Ta ble 1 and known ex pres sions for these aux -il iary term co ef fi cients, for 23Na35Cl we find these con tri bu -tions/m-1 to be -0.00034 and 0.00038 for Na and 0.000046and -0.00029 for Cl. Al though the for mer mag ni tudes for Naare sim i lar, they are ir rel e vant be cause lack of data for an iso -to pic spe cies other than 23Na pre cludes fit ting t0Na in any case. The mag ni tude of the con tri bu tion of the adi a batic cor rec tion

Anal y sis of Spec tra of NaCl J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 637

Fig. 1. Curves of ef fec tive po ten tial en ergy V(R)/h c +Be J (J + 1) Re

2/R2 for J = 0, 75, 82 and 118 in or -der of in creas ing en ergy for 23Na35Cl in elec -tronic ground state X 1Σ+; the or di nate axis in di -cates en ergy/h c 106 m-1 rel a tive to the min i -mum of po ten tial en ergy. The lower man i foldof hor i zon tal lines in di cates en er gies of pure vi -bra tional states for v = 0,1,...8; the up per man i -fold of hor i zon tal lines ap plies to ap prox i -mately the ro ta tional state of ex per i men tallysam pled great est en ergy within a par tic u lar vi -bra tional state: thus a curve for J = 118 sup ports vi bra tional states v = 0-6, whereas a curve for J= 82 sup ports v = 7 and a curve for J = 75 sup -ports v = 8. On the scale of the fig ure the en ergyof state v = 3, J = 118 co in cides with that of state v = 8, J = 75.

to Z0,1 is only one sixth that from gr, but that ex tent might besig nif i cant. For this rea son we made fur ther fits of 1428 spec -tral data, in clud ing all data from ab sorp tion mea sure ments6

and elim i nat ing du pli cated data from emis sion spec tra,3 andin cluded all uj

Na and ujCl from Ta ble 3 as fixed pa ram e ters;

with t0Na at a value con strained to re pro duce the elec tric di -

polar mo ment of 23Na35Cl at Re,11 the re sults of a fit ap pear inthe fourth col umn of Ta ble 1. On the ba sis of the fit ted valueof t0

Cl, we thus pro duce an es ti mate gr(Re) = 0.0287 ± 0.0014with hence a larger mag ni tude than be fore. In the case of ei -ther 0.00741 or 0.0287, the mag ni tude is smaller than typ i calmag ni tudes for al kali halides, but is com pa ra ble with that of23Na19F for which gr = -0.00536 ± 0.00006 9 with an op po sitesign. The value of $σ for the lat ter fit is larger than for the pre -ced ing col umns of Ta ble 1 be cause es ti mates of un cer tain tiesof mea sure ments of wave num ber are more re al is tic for theab sorp tion spec tra6 than for the emis sion spec tra. 3

The lat ter es ti mate gr (Re) = 0.0287 ± 0.0014, whichamounts still to a pre dic tion be cause it was de rived sep a rately from the re sults of quan tum-chemical cal cu la tions, is about athird, or five es ti mated stan dard er rors, larger than the ac cu -rate com pu ta tional value in Ta ble 5, gr(Re) = 0.02149, al -though the or der of mag ni tude and sign are cor rect. We at trib -ute this dis crep ancy to ac cu mu lated er ror in spec tral data andtrun ca tion of se ries. Al though the val ues of ∆VCl in Ta ble 2are likely ac cu rate to about one part in 1000, those val ues forinternuclear dis tance near Re vary only slightly with R; nev er -the less the slope, u1

Cl/106 m-1 = -4.468 ± 0.180, of the ra dialfunc tion in the con tri bu tion to V’(z), which is a crit i cal pa -ram e ter for the fit ting of t0

Cl, ap pears sat is fac to rily de fined.From val ues of co ef fi cients pj in for mula 8 de rived from re -sults in Ta ble 5 and gv therein, we ex pect s0

Na = 0.688 and s0Cl

= -1.047; as there are no data for an iso to pic vari ant of Naother than 23Na, the value of s0

N a is in ac ces si ble from ourspec tral anal y sis, but the de rived value of s0

Cl pres ent in thefourth col umn of Ta ble 1 dif fers mark edly from that ex pectedvalue. As for H2,32 es ti ma tion of the ro ta tional g fac tor ap -pears eas ier than for the vi bra tional g fac tor.

CON CLU SION

Our prac tice here of com pu ta tional spec trom e try,where by re sults from cal cu la tion of adi a batic cor rec tions areap plied as con straints to ob tain re li able val ues of ro ta tionaland vi bra tional g fac tors through their re spec tive ra dial co ef -fi cients sj and tj in for mu lae 6 and 5, re spec tively, has pro -duced an ex per i men tal value of gr(Re) near that of our cal cu la -

tions, con firm ing the va lid ity of our al ge braic ap proach tospec tral anal y sis. Our cal cu la tions of ro ta tional and vi bra -tional g fac tors and adi a batic cor rec tions as a func tion ofinternuclear dis tance near the equi lib rium sep a ra tion pro videa sig nif i cant ad di tion to our knowl edge of these prop er ties ofdi atomic mol e cules, within a con text of an ef fec tive hamil -tonian in for mula 1.

AC KNOWL EDGE MENTS

We thank Pro fes sor Jens Oddershede for help ful com -ments. SPAS ac knowl edges sup port from the CarlsbergFoun da tion, and HJAaJ and SPAS ac knowl edge sup port from the Dan ish Nat u ral Sci ence Re search Coun cil, grant 21-02- 0467, and the Dan ish Cen tre for Sci en tific Com puting.

Re ceived Feb ru ary 1, 2005.

REF ER ENCES

1. Ang strom, K. Ofvers. Kongl. Vet. Akad. Forh. Stock holm1889, 46, 203.

2. Ogilvie, J. F. As pects of the Chem i cal Bond 1996 , in Con cep -tual Per spec tives in Quan tum Chem is try; Calais, J.-L.,Kryachko, E., Ed.; Reidel: Utrecht, Neth er lands, 1997; p127.

3. Ram, R. S.; Dulick, M.; Guo, B.; Zhang, K. Q.; Bernath, P. F. J. Mol. Spectrosc. 1997, 183, 360.

4. Honig, A.; Mandel, M.; Stitch, M. L.; Townes, C. H. Phys.Rev. 1954, 96, 629.

5. Clouser, P. L.; Gordy, W. Phys. Rev. 1964 , 134A , 863. 6. Uehara, H.; Horiai, K.; Nakagawa, K.; Fujimoto, T. J. Mol.

Spectrosc. 1989, 134 , 98. 7. Caris, M.; Lewen, F.; Winnewisser, G. Z. Naturforsch. 2002,

57a, 663. 8. Ogilvie, J. F.; Uehara, H.; Horiai, K. Bull. Chem. Soc. Ja pan

2000, 73, 321. 9. Ogilvie, J. F.; Oddershede, J.; Sauer, S. P. A. Adv. Chem.

Phys. 2000, 111, 475.10. Ogilvie, J. F. The Vi bra tional and Ro ta tional Spec trom e try

of Di atomic Mol e cules ; Ac a demic: Lon don, UK, 1998.11. de Leeuw, F. H.; van Wachem, R.; Dymanus, A. J. Chem.

Phys. 1970, 53 , 981.12. Ogilvie, J. F.; Oddershede, J. Adv. Quan tum Chem . 2005 , 48,

in press.13. Dun ham, J. L. Phys. Rev. 1932, 41, 721.14. Lathouwers, L.; van Leuven, P. Adv. Chem. Phys. 1982, 49 ,

115.

638 J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 Ogilvie et al.

15. Lathouwers, L.; van Leuven, P.; Deumens, E.; Ohrn, Y. J.Chem. Phys . 1987, 86, 6352.

16. Helgaker, T.; Jørgensen, P.; Olsen, J. Mo lec u lar Elec tronicStruc ture The ory; Wiley: Chichester, UK, 2000.

17. Roos, B. O. Adv. Chem. Phys. 1987, 69 , 399.18. Widmark, P.-O.; Persson, B. J.; Roos, B. O. Theor. Chim.

Acta, 1991, 79, 419.19. Ågren, H.; Flores-Riveros, A.; Jensen, H. J. Aa Phys. Rev.

1986, A34, 4606.20. Helgaker, T.; Jensen, H. J. Aa; Jørgensen, P.; Olsen, J.; Ruud,

K.; Ågren, H.; Auer, A. A.; Bak, K. L.; Bakken, V.;Christiansen, O.; Coriani, S.; Dahle, P.; Dalskov, E. K.;Enevoldsen, T.; Fernandez, B.; Hättig, C.; Hald, A.; Halkier,K.; Heiberg, H.; Hettema, H.; Jonsson, D.; Kirpekar, S.;Kobayashi, R.; Koch, H.; Mikkelsen, K. V.; Nor man, P.;Packer, M. J.; Pedersen, T. B.; Ruden, T. A.; Sanchez, A.;Saue, T.; Sauer, S. P. A.; Schimmelpfennig, B.; Sylvester- Hvid, K. O.; Tay lor, P. R.; Vahtras, O. DAL TON, a pro gramfor cal cu la tion of mo lec u lar elec tronic struc ture, Re lease 1.2,2001; http://www.kjemi.uio.no/soft ware/dal ton/dal ton.html

21. Olsen, J.; Jørgensen, P. J. Chem. Phys. 1985, 82, 3235.22. Sauer, S. P. A.; Spirko, V.; Oddershede, J. Chem. Phys . 1991,

153, 189.23. Sauer, S. P. A.; Oddershede, J.; Geertsen, J. Mol. Phys . 1992,

76 , 445.24. Sauer, S. P. A.; Spirko, V.; Paidarová, I.; Oddershede, J.

Chem. Phys . 1994, 184, 1.25. Sauer, S. P. A.; Ogilvie, J. F. J. Phys. Chem. 1994, 98 , 8617.

26. Ogilvie, J. F.; Oddershede, J.; Sauer, S. P. A. Chem. Phys.Let ters 1994 , 228, 183.

27. Sauer, S. P. A. Chem. Phys. Let ters 1996, 260 , 271.28. Sauer, S. P. A. Chem. Phys. Let ters 1998, 297 , 475.29. Enevoldsen, T.; Ras mus sen, T.; Sauer, S. P. A. J. Chem.

Phys. 2001, 114, 84.30. Ogilvie, J. F.; Cheah, S.-L.; Lee, Y.-P.; Sauer, S. P. A. Theor.

Chem. Acc . 2002, 108, 85.31. Sauer, S. P. A. Adv. Quan tum Chem. 2005 , 48, in press.32. Bak, K. L.; Sauer, S. P. A.; Oddershede, J.; Ogilvie, J. F.

Phys. Chem. Chem. Phys . in press.33. Sauer, S. P. A.; Jensen, H. J. Aa; Ogilvie, J. F. Adv. Quan tum

Chem . 2005, 48, in press.34. Gauss, J.; Ruud, K.; Helgaker, T. J. Chem. Phys . 1996, 105 ,

2804.35. Helgaker, T. U.; Jensen, H. J. Å.; Jørgensen, P. J. Chem.

Phys. 1986, 84 , 6280.36. Jensen, H. J. Aa; Jørgensen, P.; Ågren, H.; Olsen, J. J. Chem.

Phys. 1988, 88 , 3834.37. Jensen, H. J. Aa; Jørgensen, P.; Ågren, H.; Olsen, J. J. Chem.

Phys. 1989, 89 , 5354.38. Sadlej, A. J. Theor. Chim. Acta 1991, 79 , 123.39. Sadlej, A. J.; Ur ban, M. J. Mol. Struct. 1991, 234, 147.40. Kelly, H. M.; Fowler, P. W. Chem. Phys. Let ters, 1993, 206 ,

568.41. Audi, G.; Wapstra, A. H.; Thibault, H. Nucl. Phys. 2003 ,

A729, 337.

Anal y sis of Spec tra of NaCl J. Chin. Chem. Soc., Vol. 52, No. 4, 2005 639