- Email: [email protected]
Calculation of the potential energy curves for the low-lying doublet and quartet states of the CN radical
_ cXL&JIATIOi OF.THE PO& FOR THE LOW-LYING Dot&LET _.. Haruhiko
rro,
xi~~~hi
D~JZW&E of clianiry. F& Kazuo Diridon
ozmu
ENERGyCU&__.; AND QUARTET :
2, -rak~~~hi
NAGAT&
of Sckncr, 27~ iI&cdy
.: -.
I -.
.:
_
::-‘_.:-._:.:.
SA-i-ES~- OF TEIk CN RiDICiL ~. -&II~U’K~NDOW,
of Tokyo. H&o,
~o20
Bunkyo-ky ?bi&
TAICATSU~-Hiroki b+ICAMUR4 of ZtXcorelicnl Shrdia It&itufe f& ~G~cuLv Scienrc.M~o&z# Okazab d44, Jqti
KUCHIE~
ii3. &an
.-
;- -‘r I_ :
:
-~
-. .:
and Yoshihiro OSAMURA, Facuhy of Science +
Tedmology, Keio University. Hipshi, Kohoh-u-X-u. Yokohama 223. Japan
Received 20 Febntaty 1985
l-be potential energy--es of the low-lying X22+. A=&, B’Z+. =Z*. and 4ll states of CN are calculated by the MC $iCF (CA!5 SCF) method. l’ltcir \lbratiottaI 1-a and tbc mokcuIar constants obtained are in good agreement with those ductmined in our -t expaimemaI atzaIysis of the CN (B’Z*-X22+) emission spectrum. Se%etaIintensity anomaIies in the obsa-wd spectrum are asaii to perturbations between the B2E+ and +ll states with the foIIouing vibrational quantum nmnbefsz
1, InLntroduction EIectronicalIyexcited states of the CN radical have heen e..tensiveIy studied by various spectrascopic and theoreticalmethods [l-17]_ In particular, the Iow-lying doublet states (x *E+, A%,, and B2EC) have been studied in detail mostly from.anaIyses of the intense A*II,;X’2’ and B*Z+-X*Z? emission spectra [2]: A remarkable feature of the em&ion Spectrumof CN is the local perturbations [3-131, the best known being the strong perturbation between the _B?Z*, u = 0 &nd .A’&, D= 10 states observea in- the reaction of organic molecuIes with active nitrogen (e-g., ref_ [3D__ --. -~. -. Many intensity anomalies have been observed ~. ~-’ Resent ad&s: its. Yatima+i,
National titute for Envirom&tal Tsuktk, pa+ci 305. Japan _ -
Stud-.
..
in the CN (B2Zt-X ‘Z+) emission produced by reactions of a number of cyanides with metastable argon atoms, Ar(‘Pc& [S-13], and they have been assigned to the perturbations hetweenB2Zf and 4Z+ [5-lo], 411 [12,13], or higher vibrationai states of AzIIi [ll]- According to our recent analysis of these intensity anomalies by spectral simulation 111-131, the following rovibrational IcveIs‘of .the B2Xt and 411 state. interact with e&h other: (ua, Un; N) F (9, x; xi), (11, x + 2; 33), (IL x.+ 3; lo), (14, x i 6; 7,10), (17,-x t 11: 17,19), (18, x + 13; 30), where u and ,iV stana for the vibrational atid ~otati&ial quantum numbers, +pectively. However, the VibrationaI quantum number of the perturbing‘II. statc:x, h&-emained ktcertai& because only the viir&ional intervals of 411 have been determined by this simdati&andysis_- ~ -. _The presence-of a 411 s&e close t0 B 22 k was First suggestid by Schaefer and Hcil p4] from thc&
0301-01~/85/SO3~ 0 &vier Science PubEshers B-V_ (North-HoIIand Physics Publishing Division)
:
ab initio caIcuIation with conf@ration inter-a& tions (CI) with a rehtively small number of configurations_ Their predictions of the spectroscopic constants of ‘II were sufficient to stinudate our interest of the experimentalsearch for the an state [12,13], but they were not accurate enough to explain the observed spectmI anomaha Ahhough a Iarge scale muIticonfi~~tion SCF (MC SCF) cahxdation was made by Das et aL [lSb none of the quartet states was included_ In order to confirm our recent e_xpezxirnentaI fmdings en the ‘II state [lZJ3] and to estimate the vibrational quantum number, _r_ MC SCF calculations have been carried out on the relevant excited states of CN_ In an attempt to assign the vibrational IeveIsof the jll state causing the perturbation the v+bmtionaIenerggesand the mokcuIar cons*%rnts have been estimated from the theoret&I potential curves in the present study_ The cakulation method is descrii in section 2 The rest&s of the caIcuIation are presented in section 3_ Section 3 is aIso devoted to discussions of the perturbations: B’S+A’l& (se&on 3-Q B’S+ -jfI (section 3-2). and B’Z*-“SC (section 3.3)_
Z Method of c!ak&hu LL
Pofenriui meqy
curves
As stated in section 1, the primary purpose of the present work is to obtain the relative energies of the B’Y and 411 states instead of their absolute vaIues_Therefore, it is important to choose such a computationaI scheme that systematic errors originating from rehtxationand electron correIation in the derived energies of the& states are nearIy identical In this sense, the CK method based on the Hartree-FocIc orbitaIs which optimize the grormd state energy is- not neazzuiIy suitable, because it is diflkuh to choose the best configuration (functional) space in which et-rots due to rekxation and correlation can be *-ted in a well-baIanced manner_ As a rest& Ci calculations of a very large scale are inevitabk As an aiternative method, we have used the MC SCF method in the present study_ From the CI
caIcuIatkms of Schaefer and .HeiI [14]. the dominant configurations are reported to be {lo’2a’3a’40’5alc’) (0.9304) for X2x+, {lc?202302405a21r”) (O.SSS7) + (1a’2a23~*4&ul&!n) (03682) for -B ‘Se, and 10*2o’3a’4&aZlrr22n} (O-9624) for %, where the numbers in parentheses are the coefIkients of the normahzed wavefunctions AccordingIy, u-e have optimized the la, 20, 30, 4al 50, lr, and 2s orbitaIs so as to minimize the energy of each state- AII the coefficients of the ekctronic contigurations generated from this orbital space, except for those corresponding to the excitation from the la and 20 orbitaIs, are aIso optinked_ Thus our MC SCF calculation is one of the so-caIIed CAS SCF (complete active space SCF) methods [lS]. The inactive and active subspaces are {lo, 2a) and (3a. 4~. 5o,l-;r, 27r}. respectively- The dimension of the variational space in which the coefficients of each configuration are optirftized is =500_ The errors of this MC SCF calculation are txpectecl to be nearly equal for the B’Z* and ‘II states_ In order to confm this estimate. we have carried out much laker MC SCF calculations with two more active orbitals (resulting in a dimension of = 5000) for each of the B’Z+ and % states at one point near the potential minimum_ The energiesare 129 and 134 eV lower for B’Z+ and ‘II, respectively *, than those derived from the 500 dimensional calculation_ This result ensures that our compact MC SCF calculation provides estimates of the energy parameters suitable for our primary purpose_ The basis set of Dunning (9s5p)/[33p] is augmented by d-type gaussian pokrization functions of single-l quality placed at each nucleus, where 5 = 0.75 and 0.8 for the carbon and-nitrogenatoms, respectively_The energies are found to be insensitive to the choice of the < parameters. The program for the MC SCF is taken from the GAII’ESS package [19]_ One of the practical problems in dealing with the excited states is that one of the orbitals which has to be included in the ~LCI~~ofanill~scalc,ailha~onof ZSOOO.has also been made for B’X+_ The energy of this sate is Iowcrcd by 1.33 cl’ from that of the 500 dimadd MC .SCF -on
active space happens, to k_ re&&ed by +nothq orbital of the virtual space in an early:stage of the. iteration_ Such ,m interc&nge,of o;bit& may kin the convergence or Icad to ti false convergence to, an undesired state This iirbital intqc&m~e is preeluded in the .sqer-Ci aipro&h base$ on. the BrilIouin theorem [lS]_ In the present gcenemhzed study, however- this diffikulty @ been remokd by using t&e &crlap bctw?n +e orbitals obtained in consecutive iteration stcps~. If the-orbital ir$erchange is *countered, the correspon&ng qverlap .bccomes~vcry small, and this provides a warningSince no~_convergence has Fn at@ined for +e B’Z* state at bond distances snorter thq l-14 A, we have carried out a nearly full -CI _cakula~on which is based .on the MC SCF orbitak for the X*x* state and the two additional virtual orbitals (= 8800 configurations)_ The results have been scaled so as to bc joined smoothly to the MC SCF curve calculated at R > 1.14 i\_ 2.2. VIbrational energies and related molecular constants The vibrational energies, E,, and the rotational constants, J3,. for the X*Z*. A’& B*Z’, 4x* and 4ri states have been determined by numerical integgtion of the S&r&linger equation- In view of the limited accuracy of the present potential curves, especially near the dissociation Jimits,. the outer part of the potential curve, R > 1.65-1.75 A, is connected smoothly to [20] V( R),c=
0, - G/R”,
(1)
whet-c D, is the. observed dissociation energy for each state and n and q arc the constants which are determined by the procedure described in ref_ Ill]_ For each electronic state, u is found to b,” = 6_ On the potential curve for R -c 1.65-1.75 A which was obtained in the present calculation, numerical integration has been carried out using the Aitkin-Lagrange interpolatioti. The E, and B,. values are fitted to [l] E,=T,+o,(ai1/2)--c;,x,(u+1/2)*
(2).
and (3)
~Bv=B=-aa,(u+1/2) inthercgion~ofO<~,(1+24.
-_
In tabk1 and fig_ -1 are shown ihe moiecuhir constants and the potential curves, respectively_-
.
:.
Table I
cfxatants for the clsctronic states of CN. -ia
Moldar
Ohs b’
: z.
Cak d)
Cl
-e)
x*x+
T,
0-z
0.0 2068.7
%“c B, =c
&Ii T, UC *c-s
B, =c B’x+ T -c
(ic
*,-G BC =e
0.0
211.5.S
o-0
1939.2
o-0 207g_5
.-.
13.14 1.900 0.0173
IL05 1.864 O-0171
1454 1.610 0.0151
13.00 l-90 O-019
1_146 1814-4 1288 X.717 0.0175
1394 19120 11.90 1.705 0_0160
l-S83 1621.4 16.74 1.403 0_0149
~1013 17872 122s 1.69 O-018
3.193 2164.1 2025 I_970 0.0222
j-410 22152 2354 1.926 0*0251
3.765 17653 3253 1.636 0.0262
3.114 22755 2494 I-97 O-Q22
4-731 1433.4 13.03 I-413 OvO161
4_Oi6 1249.4 14-82 1174 O-0152
?z* T, us %-% & a, ‘n
Obr r, x=0
$1
GE *SC B, %
Gait x=1
x=2e
c)
_d)
~.
536 5326 5.64 550 5.909 217 1.561 2_45 231 2499 873.0 1102 1133 1164 11132 i415 15-6 15.6 15.6 S-44 1.188 O-927 1_200 x235 1217 0.0174 0.0174 O-0174 O-0148 0.0154
a) Thi mokcular anstants are defmed in cqs. (2) hd (3)- 1 Units arc T, in eV; s; oex,, Be and a, in c&-l_ _..~ ” Tak&i from refi[l]. c, Obtained & the present study: d’ Taken from ret 1141. ‘1 Taken fi-om rcf_ [lSj_ 9 Takenfrom ref.[13]_ m.. 1. * The B’X*. o=9 level is assumed to be p.&wbed by the 4l-I. t?=x heL ~.. ” Tc (dl-I)-T(B22+)_ e
fi. hoer aL / Pomuidma-gy
84 EfeV c-rr
8-O i-
1.0
l-5
cxmwfw A-
CNradicai
levels of the A*& state which perturb the B*Z?u = 0,5.10P 12, and 14 IeVek -are .compared in table 2 with those determined from analyses of the observed perturbatiotis [3,4,11]. -The rotationaI levels of these states come very close to each other at a certain vakof N be&se of the difference in their rotationaI constants (= 1.5 and = 1.9 cm-!. for the A211i and B*Z* states, respectiveIy~see tables 1 and 2)_ In the present caIcuIation, the rotational levek- of the AZTIi state which perturb the levels of B*B* listed above are prkdicted as in table 2 by using the calculated molecular constants for these states listed in table 1. By a comparikn of the present theoretical assignments of the perturbed v&rational levels of A*I& with the corresponding assignments in the observed spectrum [3.4,11], the uncertaintiesof the theoreticalassignments of the vibrational quantum numbers, u,, are estimatedtobe+lforu,~2Oand+2for20( Us 5 30_ These uncertainties, which arise from errors in the estimation of the molecxdarconstants, are small enough to predict the B*Z+-A211i perturbations at lower vibrational levels v, (5 20) with reasonable accuracy in the present calcdatiOQS.
The @culated mokcular constants for the doublet states are in good agreement with the observed constants. However. slight discrepancies exist between the observed and caLdated values for the molecular constants and the vibrational energg levels of the doublet electronic states: (1) The cakuhted T, values for the A’& and B’S+ states are, respectively, 0.25 and 0.22 eV higher than the observed vahxs. The difference in these deviations. O-03 eV or 240 cm-‘, is much smaller than the intervals of the vibrational IeveIs of the two states (= 1800-2000 cm-!) (see table 1). The axxsponding difference was O-17 eV in the cakxdations of !Scbaeferand Heil[14), and O-05 eV in those of Das et al. [15]. Consequently, the relative energgesof the A’& and B’Z’ states are better reproduced in the present caktdations. The caiculted W, vaIues are systematicaIIyIarger than the observed vahxesby 2.3,5_5 and 2.4’S* while the caIcuIated B, values are systematically sxnakr by 1.8.0.6. and 2.4% for the X’FP A’&, and B’Z* states. respectively. (2) The present assignments of the vibrational
Table 2
=B
=,
Ohs =A
0 5 10 12 14
10 17 24 27 30
-’ vibh& put-on b, Vibmional
cstnawd
LB
CaIc. iv=’
=A
4.7.11.15 12 0 16.X9.20.23
10 16 23 26 28
2x30.34
4
N”
60 61 45 55 24
quantumnumberof
&e B2Z* state &~ere &e caused by the A*l-I; state is obscmed [3.4.111. quantum number of the A*& state whichis fo pauub the listed vibraional 1cr.d of B’Z*
(3.4.1 ll=’ Rotational qntum Iulmbas a-here the puturbati~ between the B=T+ and A’l& states are rqx~nal[3.4,11]. The pcnurbacioas at several mLMionaI Ievck baause of rhc spin-doubkss of clu roxarkmal Icvdr e Vibr;ltional quantum number of Lhe A%, state which is
pt-dctatinthcprcscnt~Liontopcrmrb~lincd viirationaIIcvd of B’Z+ (seesection3.Q * Ftotariod quantumnumberw&z the B*Z* - A*& permixuionisprcdictaIiathcprcsaxtstudybyusingthc ukuIawd moIaxlIaramstamsIistcdin tabIe1.
3.2 Assignnxmt of the vibrational levek causing the . . _ B%‘-.fII perhybaion -.
TJk quantum number x of the vibrational level of 4H whidiperhirbs the v=9 Ievef of B*Z+ has remained uncertain [13] (see section i)_ The tiresent study shows. that x = 0 is the most probable assignment by’ the foliowing comparison of the observed and calculated energy levels and molecular constants for this state:. (1) As shown in fig 1, the calcufated viirational levels of u = 3; 6. and 11 of 4H are very close to the u= 12, 14, and 17 levefs of B’x+; respectively_ This leads to the assignment that x = o_ (2) The systematic errors in the ckiktdated molecular constants for these states are estimated as follows: The T, values for B*Z+ and A’& estimated from the present calculations are 0.22 and 025 eV higher than the experimental values, respectively (see section 3_1)_ Accordingly~ the excited state energies areoverestimated nearly equally in the present computational scheme Since ah the relevant states are tteated in the same active space, theT, valuefor4Hcanalsobcassumedtobe eV with a possible unoverestimated by 2025 certainty of = O-05 eV (see below). Consequently, the T, value of 4H is estimated to be 566 eV from the calculated T, value, 591 eV_ On the other hand, the T, value for ‘II is estimated from the deperturbation analysis of the B*X+-X22+ tail band spectrum 1131 to be 5.64 eV under the assumption that x = 0 (see table l), being in good agreement with the theoretical value estimated above (3) The relative energies of the 4H and B*Z* states are essentially unchanged even in the MC SCF calculations that have higher aceuracy with a dimension of = SOOO_As descriied in section 2.1, the energy of ‘II .relative to B*B* derived from the 5000 dimensional c+ufations is 0.05 eV (400 cm-‘) lower than that from the 500 dimensional cakufations. However, this energy difference is smakr than those between the Iower vibrational levels of the Tf state (i: 1100 cm-‘). Therefore. it is unlikely that the above estimates of the relative energies are incorrect by more than one vibrational quantum number_ .’
: (4)me
d&rturbation
hy&
@J~-&:&~~.-.
vides the 0:. and -B, yahes for the’*Hst&e k% that ‘X .G 0..or 1 : listed in iable 1; the as.&iptions result in we = 1102 or’ li33 cm-?, aird:TB; = ~200 or 1217 cm-‘, respectively_ The_ corres@nding : calcuiations‘are o, = 11132 cm-,? and B;= &l-188 cm-’ (see. table I)_ These -_values are in -~good agreement with those obtained from the simula- _ tion analysis for x = 0. No calculation has been made to examine how these molecular~con&ms are affected if electron correlation is further taken into account However, in view of the qluality of. ourcalc;l~tions_&erelative errors in the calculated values of o, and B, for the 41S state are expected to be roughly equal. to those for the doublet states. For this reason, these constants for 411 are estimated to be o, = 1055:109(j cm-’ and B, = 1_20-122 cm-’ by application of the ~sys: tematic corrections for the doublet states (see section 3.1). The corrected 0, vahte is -still in good agreement with the estimate for x= 0, but the corrected B,vaIue indicatestheremainingpossibihtythatx=l_ From the above discussion, the most probable assi~entisx=O_Thespec~simulationofthe B2Z*-4H perturbation observed eat the ~B’Z+, v = 9 (IV = 56) level [13] has revealed that the vibrational energy levels of the perturbing 4H state he close to the B*ZE’ ~ v = 10 level and that the rotational levels of the B’X+- v =_ 9 and 4H, v 7 x levels cross at such a high N level because of the difference in the rotational constants (= l-76 and =1_19cm-‘forB2~C,.u=9and4~L, v=xIevels, respectively). The present calculation predicts that the va = 10 and II, = 0 levels he dose together (see fig 1). Therefore, the B’Z~-4H perturbation. inthev,=9levelcanbe~~edasthatcaused~ by the v=O level of 411_ The B2Z*-411 per-turbations observed at the B’Z*, v = 17 and 18 levels cannot be explained unambiguously in the present cakulation, because the energies of the perturbing vibrational levels of the 4H state-lie nearthe dissociation region so that the_cakuIated energies have hmited accuracy (see ~section 2.2). Never&&s, the quantum numbers of ‘the viirational levels of de 4H -state ‘which perturb the. 38 = l-7 and 18 levels are estimated to be II and 13, respectively, using the assignment x=0 esti-.
, s6
mated in the present cakuktion ~.and &e vibrational i.nteFnls of ?II determined from the anaiysis of the observed B22+-%I pexp&ations[l3]_ Thnq the present theoretical caIcuIations confirm that the -our pl+vious -experimental -finding a.nomalies in the emission speetla (Baz+-x’z+; uu = 9,11,12,14,17, and 18) are caused by the
perturbation by the411 state. 3-R yibralional
_
iecelk of ‘X+
The perturbation between thenB’Z* and 4X+ states has been observed at the ur, = 11, N = 20 and DB = 13, N F 9 levels of B’Z+ [5-lo]_ The corresponding vibrationaI quantum numbers of the 42+ state are estimated as follows: -
4_Condndingremarks We have carried out MC SCF caIcuIations for the low-Iying doublet and quartet states of the CN radical The vibrationaI energies and the rotational constants for the B2Z* and ‘II states obtained in the caIcuIations support the assignments made in our previous spectroscopic study_ The viirational levels of B2Z+ with u= 9,11,12,14,17, and 18
are perturbed by the follow& viirkional kvels of 41Tr respectiveIyr x, x-k& *.-i-3,.*+6, X+,11; and- x + 13, where the most probable assignment of x is O_ _lIe gissignmen_ts.of_ the. Io_wer -Ieve% on 4 x + 6, are more-reliable than the, last two_ In addition to this perturhatiop, the: eations in the u = 11. and 13 levels of B!X* and the y and y +- 3 Ieveh of-42+. are also discussed, where y is estimatedtobes+l.~ After the present study. was cornpIe&, the vibrational numberings of. the. 411 state of CN. were derived from fan ab ~initio caIcuIation [21]. The rest& .of this &kuIation showed _tbat the vibrational levels of 41i which perturb the B’Z+, D = 12 level is expected to be un = 3 or 4, which is consistent with the present resu1t.s.
_4cclm&dgement me cakuIation has been carried out at the Computer Center of the Institute for MolecuIar Science sup_ported by the -Joint Studies Pr&I.ram (1982-1984) The authora are grateftd to DE T_A_ Miller, D-C_ Cartwright and H--J_ Werner for their fruitful advice and discussions.
R&[l] K_P_Huba
and G_ Herzberg in: Mokular _qccua and Ilzobxh suuccur=. YOL iv. Gxtstanrs of diawmic xuolccuk(V+nNosmmd,-on.1979). f2] M_R Gorhal and ML Savadat& Chax~ Rev_ 82 (1982) [3] g
Radford and HP_ Brokia. J. cZhcm_ Php
38 (1963)
[4] r&g anan Jr, I MoL Spxay_ 19 0974) 106. = [5] JH IColts and D-W. Sew, in: Rc+k hamediates
in
the gas phase, a.L D-w_ setsa (Acadanic Press, New York, 1979) p_ 151_ [6] J+ Coxon. DA Ramsay and D-W. Scucr, Can I Phys. 53 (l975) 1587_ m TA Milk. RS Framd ani RW.-FK14 I Cham Phys 65 0976) 3790_ [S] J_LUCool&R_Zq+i&dT.A_hfi!kr.J__chcrntiys. 68 0978) 4763.
[la]
1274 DC Cartarright and_PJ_ Hay. As-&bys. 383_
Phyr 62 0984) l508_ J- 257 (1982)
..
rro,
xi~~~hi
D~JZW&E of clianiry. F& Kazuo Diridon
ozmu
ENERGyCU&__.; AND QUARTET :
2, -rak~~~hi
NAGAT&
of Sckncr, 27~ iI&cdy
.: -.
I -.
.:
_
::-‘_.:-._:.:.
SA-i-ES~- OF TEIk CN RiDICiL ~. -&II~U’K~NDOW,
of Tokyo. H&o,
~o20
Bunkyo-ky ?bi&
TAICATSU~-Hiroki b+ICAMUR4 of ZtXcorelicnl Shrdia It&itufe f& ~G~cuLv Scienrc.M~o&z# Okazab d44, Jqti
KUCHIE~
ii3. &an
.-
;- -‘r I_ :
:
-~
-. .:
and Yoshihiro OSAMURA, Facuhy of Science +
Tedmology, Keio University. Hipshi, Kohoh-u-X-u. Yokohama 223. Japan
Received 20 Febntaty 1985
l-be potential energy--es of the low-lying X22+. A=&, B’Z+. =Z*. and 4ll states of CN are calculated by the MC $iCF (CA!5 SCF) method. l’ltcir \lbratiottaI 1-a and tbc mokcuIar constants obtained are in good agreement with those ductmined in our -t expaimemaI atzaIysis of the CN (B’Z*-X22+) emission spectrum. Se%etaIintensity anomaIies in the obsa-wd spectrum are asaii to perturbations between the B2E+ and +ll states with the foIIouing vibrational quantum nmnbefsz
1, InLntroduction EIectronicalIyexcited states of the CN radical have heen e..tensiveIy studied by various spectrascopic and theoreticalmethods [l-17]_ In particular, the Iow-lying doublet states (x *E+, A%,, and B2EC) have been studied in detail mostly from.anaIyses of the intense A*II,;X’2’ and B*Z+-X*Z? emission spectra [2]: A remarkable feature of the em&ion Spectrumof CN is the local perturbations [3-131, the best known being the strong perturbation between the _B?Z*, u = 0 &nd .A’&, D= 10 states observea in- the reaction of organic molecuIes with active nitrogen (e-g., ref_ [3D__ --. -~. -. Many intensity anomalies have been observed ~. ~-’ Resent ad&s: its. Yatima+i,
National titute for Envirom&tal Tsuktk, pa+ci 305. Japan _ -
Stud-.
..
in the CN (B2Zt-X ‘Z+) emission produced by reactions of a number of cyanides with metastable argon atoms, Ar(‘Pc& [S-13], and they have been assigned to the perturbations hetweenB2Zf and 4Z+ [5-lo], 411 [12,13], or higher vibrationai states of AzIIi [ll]- According to our recent analysis of these intensity anomalies by spectral simulation 111-131, the following rovibrational IcveIs‘of .the B2Xt and 411 state. interact with e&h other: (ua, Un; N) F (9, x; xi), (11, x + 2; 33), (IL x.+ 3; lo), (14, x i 6; 7,10), (17,-x t 11: 17,19), (18, x + 13; 30), where u and ,iV stana for the vibrational atid ~otati&ial quantum numbers, +pectively. However, the VibrationaI quantum number of the perturbing‘II. statc:x, h&-emained ktcertai& because only the viir&ional intervals of 411 have been determined by this simdati&andysis_- ~ -. _The presence-of a 411 s&e close t0 B 22 k was First suggestid by Schaefer and Hcil p4] from thc&
0301-01~/85/SO3~ 0 &vier Science PubEshers B-V_ (North-HoIIand Physics Publishing Division)
:
ab initio caIcuIation with conf@ration inter-a& tions (CI) with a rehtively small number of configurations_ Their predictions of the spectroscopic constants of ‘II were sufficient to stinudate our interest of the experimentalsearch for the an state [12,13], but they were not accurate enough to explain the observed spectmI anomaha Ahhough a Iarge scale muIticonfi~~tion SCF (MC SCF) cahxdation was made by Das et aL [lSb none of the quartet states was included_ In order to confirm our recent e_xpezxirnentaI fmdings en the ‘II state [lZJ3] and to estimate the vibrational quantum number, _r_ MC SCF calculations have been carried out on the relevant excited states of CN_ In an attempt to assign the vibrational IeveIsof the jll state causing the perturbation the v+bmtionaIenerggesand the mokcuIar cons*%rnts have been estimated from the theoret&I potential curves in the present study_ The cakulation method is descrii in section 2 The rest&s of the caIcuIation are presented in section 3_ Section 3 is aIso devoted to discussions of the perturbations: B’S+A’l& (se&on 3-Q B’S+ -jfI (section 3-2). and B’Z*-“SC (section 3.3)_
Z Method of c!ak&hu LL
Pofenriui meqy
curves
As stated in section 1, the primary purpose of the present work is to obtain the relative energies of the B’Y and 411 states instead of their absolute vaIues_Therefore, it is important to choose such a computationaI scheme that systematic errors originating from rehtxationand electron correIation in the derived energies of the& states are nearIy identical In this sense, the CK method based on the Hartree-FocIc orbitaIs which optimize the grormd state energy is- not neazzuiIy suitable, because it is diflkuh to choose the best configuration (functional) space in which et-rots due to rekxation and correlation can be *-ted in a well-baIanced manner_ As a rest& Ci calculations of a very large scale are inevitabk As an aiternative method, we have used the MC SCF method in the present study_ From the CI
caIcuIatkms of Schaefer and .HeiI [14]. the dominant configurations are reported to be {lo’2a’3a’40’5alc’) (0.9304) for X2x+, {lc?202302405a21r”) (O.SSS7) + (1a’2a23~*4&ul&!n) (03682) for -B ‘Se, and 10*2o’3a’4&aZlrr22n} (O-9624) for %, where the numbers in parentheses are the coefIkients of the normahzed wavefunctions AccordingIy, u-e have optimized the la, 20, 30, 4al 50, lr, and 2s orbitaIs so as to minimize the energy of each state- AII the coefficients of the ekctronic contigurations generated from this orbital space, except for those corresponding to the excitation from the la and 20 orbitaIs, are aIso optinked_ Thus our MC SCF calculation is one of the so-caIIed CAS SCF (complete active space SCF) methods [lS]. The inactive and active subspaces are {lo, 2a) and (3a. 4~. 5o,l-;r, 27r}. respectively- The dimension of the variational space in which the coefficients of each configuration are optirftized is =500_ The errors of this MC SCF calculation are txpectecl to be nearly equal for the B’Z* and ‘II states_ In order to confm this estimate. we have carried out much laker MC SCF calculations with two more active orbitals (resulting in a dimension of = 5000) for each of the B’Z+ and % states at one point near the potential minimum_ The energiesare 129 and 134 eV lower for B’Z+ and ‘II, respectively *, than those derived from the 500 dimensional calculation_ This result ensures that our compact MC SCF calculation provides estimates of the energy parameters suitable for our primary purpose_ The basis set of Dunning (9s5p)/[33p] is augmented by d-type gaussian pokrization functions of single-l quality placed at each nucleus, where 5 = 0.75 and 0.8 for the carbon and-nitrogenatoms, respectively_The energies are found to be insensitive to the choice of the < parameters. The program for the MC SCF is taken from the GAII’ESS package [19]_ One of the practical problems in dealing with the excited states is that one of the orbitals which has to be included in the ~LCI~~ofanill~scalc,ailha~onof ZSOOO.has also been made for B’X+_ The energy of this sate is Iowcrcd by 1.33 cl’ from that of the 500 dimadd MC .SCF -on
active space happens, to k_ re&&ed by +nothq orbital of the virtual space in an early:stage of the. iteration_ Such ,m interc&nge,of o;bit& may kin the convergence or Icad to ti false convergence to, an undesired state This iirbital intqc&m~e is preeluded in the .sqer-Ci aipro&h base$ on. the BrilIouin theorem [lS]_ In the present gcenemhzed study, however- this diffikulty @ been remokd by using t&e &crlap bctw?n +e orbitals obtained in consecutive iteration stcps~. If the-orbital ir$erchange is *countered, the correspon&ng qverlap .bccomes~vcry small, and this provides a warningSince no~_convergence has Fn at@ined for +e B’Z* state at bond distances snorter thq l-14 A, we have carried out a nearly full -CI _cakula~on which is based .on the MC SCF orbitak for the X*x* state and the two additional virtual orbitals (= 8800 configurations)_ The results have been scaled so as to bc joined smoothly to the MC SCF curve calculated at R > 1.14 i\_ 2.2. VIbrational energies and related molecular constants The vibrational energies, E,, and the rotational constants, J3,. for the X*Z*. A’& B*Z’, 4x* and 4ri states have been determined by numerical integgtion of the S&r&linger equation- In view of the limited accuracy of the present potential curves, especially near the dissociation Jimits,. the outer part of the potential curve, R > 1.65-1.75 A, is connected smoothly to [20] V( R),c=
0, - G/R”,
(1)
whet-c D, is the. observed dissociation energy for each state and n and q arc the constants which are determined by the procedure described in ref_ Ill]_ For each electronic state, u is found to b,” = 6_ On the potential curve for R -c 1.65-1.75 A which was obtained in the present calculation, numerical integration has been carried out using the Aitkin-Lagrange interpolatioti. The E, and B,. values are fitted to [l] E,=T,+o,(ai1/2)--c;,x,(u+1/2)*
(2).
and (3)
~Bv=B=-aa,(u+1/2) inthercgion~ofO<~,(1+24.
-_
In tabk1 and fig_ -1 are shown ihe moiecuhir constants and the potential curves, respectively_-
.
:.
Table I
cfxatants for the clsctronic states of CN. -ia
Moldar
Ohs b’
: z.
Cak d)
Cl
-e)
x*x+
T,
0-z
0.0 2068.7
%“c B, =c
&Ii T, UC *c-s
B, =c B’x+ T -c
(ic
*,-G BC =e
0.0
211.5.S
o-0
1939.2
o-0 207g_5
.-.
13.14 1.900 0.0173
IL05 1.864 O-0171
1454 1.610 0.0151
13.00 l-90 O-019
1_146 1814-4 1288 X.717 0.0175
1394 19120 11.90 1.705 0_0160
l-S83 1621.4 16.74 1.403 0_0149
~1013 17872 122s 1.69 O-018
3.193 2164.1 2025 I_970 0.0222
j-410 22152 2354 1.926 0*0251
3.765 17653 3253 1.636 0.0262
3.114 22755 2494 I-97 O-Q22
4-731 1433.4 13.03 I-413 OvO161
4_Oi6 1249.4 14-82 1174 O-0152
?z* T, us %-% & a, ‘n
Obr r, x=0
$1
GE *SC B, %
Gait x=1
x=2e
c)
_d)
~.
536 5326 5.64 550 5.909 217 1.561 2_45 231 2499 873.0 1102 1133 1164 11132 i415 15-6 15.6 15.6 S-44 1.188 O-927 1_200 x235 1217 0.0174 0.0174 O-0174 O-0148 0.0154
a) Thi mokcular anstants are defmed in cqs. (2) hd (3)- 1 Units arc T, in eV; s; oex,, Be and a, in c&-l_ _..~ ” Tak&i from refi[l]. c, Obtained & the present study: d’ Taken from ret 1141. ‘1 Taken fi-om rcf_ [lSj_ 9 Takenfrom ref.[13]_ m.. 1. * The B’X*. o=9 level is assumed to be p.&wbed by the 4l-I. t?=x heL ~.. ” Tc (dl-I)-T(B22+)_ e
fi. hoer aL / Pomuidma-gy
84 EfeV c-rr
8-O i-
1.0
l-5
cxmwfw A-
CNradicai
levels of the A*& state which perturb the B*Z?u = 0,5.10P 12, and 14 IeVek -are .compared in table 2 with those determined from analyses of the observed perturbatiotis [3,4,11]. -The rotationaI levels of these states come very close to each other at a certain vakof N be&se of the difference in their rotationaI constants (= 1.5 and = 1.9 cm-!. for the A211i and B*Z* states, respectiveIy~see tables 1 and 2)_ In the present caIcuIation, the rotational levek- of the AZTIi state which perturb the levels of B*B* listed above are prkdicted as in table 2 by using the calculated molecular constants for these states listed in table 1. By a comparikn of the present theoretical assignments of the perturbed v&rational levels of A*I& with the corresponding assignments in the observed spectrum [3.4,11], the uncertaintiesof the theoreticalassignments of the vibrational quantum numbers, u,, are estimatedtobe+lforu,~2Oand+2for20( Us 5 30_ These uncertainties, which arise from errors in the estimation of the molecxdarconstants, are small enough to predict the B*Z+-A211i perturbations at lower vibrational levels v, (5 20) with reasonable accuracy in the present calcdatiOQS.
The @culated mokcular constants for the doublet states are in good agreement with the observed constants. However. slight discrepancies exist between the observed and caLdated values for the molecular constants and the vibrational energg levels of the doublet electronic states: (1) The cakuhted T, values for the A’& and B’S+ states are, respectively, 0.25 and 0.22 eV higher than the observed vahxs. The difference in these deviations. O-03 eV or 240 cm-‘, is much smaller than the intervals of the vibrational IeveIs of the two states (= 1800-2000 cm-!) (see table 1). The axxsponding difference was O-17 eV in the cakxdations of !Scbaeferand Heil[14), and O-05 eV in those of Das et al. [15]. Consequently, the relative energgesof the A’& and B’Z’ states are better reproduced in the present caktdations. The caiculted W, vaIues are systematicaIIyIarger than the observed vahxesby 2.3,5_5 and 2.4’S* while the caIcuIated B, values are systematically sxnakr by 1.8.0.6. and 2.4% for the X’FP A’&, and B’Z* states. respectively. (2) The present assignments of the vibrational
Table 2
=B
=,
Ohs =A
0 5 10 12 14
10 17 24 27 30
-’ vibh& put-on b, Vibmional
cstnawd
LB
CaIc. iv=’
=A
4.7.11.15 12 0 16.X9.20.23
10 16 23 26 28
2x30.34
4
N”
60 61 45 55 24
quantumnumberof
&e B2Z* state &~ere &e caused by the A*l-I; state is obscmed [3.4.111. quantum number of the A*& state whichis fo pauub the listed vibraional 1cr.d of B’Z*
(3.4.1 ll=’ Rotational qntum Iulmbas a-here the puturbati~ between the B=T+ and A’l& states are rqx~nal[3.4,11]. The pcnurbacioas at several mLMionaI Ievck baause of rhc spin-doubkss of clu roxarkmal Icvdr e Vibr;ltional quantum number of Lhe A%, state which is
pt-dctatinthcprcscnt~Liontopcrmrb~lincd viirationaIIcvd of B’Z+ (seesection3.Q * Ftotariod quantumnumberw&z the B*Z* - A*& permixuionisprcdictaIiathcprcsaxtstudybyusingthc ukuIawd moIaxlIaramstamsIistcdin tabIe1.
3.2 Assignnxmt of the vibrational levek causing the . . _ B%‘-.fII perhybaion -.
TJk quantum number x of the vibrational level of 4H whidiperhirbs the v=9 Ievef of B*Z+ has remained uncertain [13] (see section i)_ The tiresent study shows. that x = 0 is the most probable assignment by’ the foliowing comparison of the observed and calculated energy levels and molecular constants for this state:. (1) As shown in fig 1, the calcufated viirational levels of u = 3; 6. and 11 of 4H are very close to the u= 12, 14, and 17 levefs of B’x+; respectively_ This leads to the assignment that x = o_ (2) The systematic errors in the ckiktdated molecular constants for these states are estimated as follows: The T, values for B*Z+ and A’& estimated from the present calculations are 0.22 and 025 eV higher than the experimental values, respectively (see section 3_1)_ Accordingly~ the excited state energies areoverestimated nearly equally in the present computational scheme Since ah the relevant states are tteated in the same active space, theT, valuefor4Hcanalsobcassumedtobe eV with a possible unoverestimated by 2025 certainty of = O-05 eV (see below). Consequently, the T, value of 4H is estimated to be 566 eV from the calculated T, value, 591 eV_ On the other hand, the T, value for ‘II is estimated from the deperturbation analysis of the B*X+-X22+ tail band spectrum 1131 to be 5.64 eV under the assumption that x = 0 (see table l), being in good agreement with the theoretical value estimated above (3) The relative energies of the 4H and B*Z* states are essentially unchanged even in the MC SCF calculations that have higher aceuracy with a dimension of = SOOO_As descriied in section 2.1, the energy of ‘II .relative to B*B* derived from the 5000 dimensional c+ufations is 0.05 eV (400 cm-‘) lower than that from the 500 dimensional cakufations. However, this energy difference is smakr than those between the Iower vibrational levels of the Tf state (i: 1100 cm-‘). Therefore. it is unlikely that the above estimates of the relative energies are incorrect by more than one vibrational quantum number_ .’
: (4)me
d&rturbation
hy&
@J~-&:&~~.-.
vides the 0:. and -B, yahes for the’*Hst&e k% that ‘X .G 0..or 1 : listed in iable 1; the as.&iptions result in we = 1102 or’ li33 cm-?, aird:TB; = ~200 or 1217 cm-‘, respectively_ The_ corres@nding : calcuiations‘are o, = 11132 cm-,? and B;= &l-188 cm-’ (see. table I)_ These -_values are in -~good agreement with those obtained from the simula- _ tion analysis for x = 0. No calculation has been made to examine how these molecular~con&ms are affected if electron correlation is further taken into account However, in view of the qluality of. ourcalc;l~tions_&erelative errors in the calculated values of o, and B, for the 41S state are expected to be roughly equal. to those for the doublet states. For this reason, these constants for 411 are estimated to be o, = 1055:109(j cm-’ and B, = 1_20-122 cm-’ by application of the ~sys: tematic corrections for the doublet states (see section 3.1). The corrected 0, vahte is -still in good agreement with the estimate for x= 0, but the corrected B,vaIue indicatestheremainingpossibihtythatx=l_ From the above discussion, the most probable assi~entisx=O_Thespec~simulationofthe B2Z*-4H perturbation observed eat the ~B’Z+, v = 9 (IV = 56) level [13] has revealed that the vibrational energy levels of the perturbing 4H state he close to the B*ZE’ ~ v = 10 level and that the rotational levels of the B’X+- v =_ 9 and 4H, v 7 x levels cross at such a high N level because of the difference in the rotational constants (= l-76 and =1_19cm-‘forB2~C,.u=9and4~L, v=xIevels, respectively). The present calculation predicts that the va = 10 and II, = 0 levels he dose together (see fig 1). Therefore, the B’Z~-4H perturbation. inthev,=9levelcanbe~~edasthatcaused~ by the v=O level of 411_ The B2Z*-411 per-turbations observed at the B’Z*, v = 17 and 18 levels cannot be explained unambiguously in the present cakulation, because the energies of the perturbing vibrational levels of the 4H state-lie nearthe dissociation region so that the_cakuIated energies have hmited accuracy (see ~section 2.2). Never&&s, the quantum numbers of ‘the viirational levels of de 4H -state ‘which perturb the. 38 = l-7 and 18 levels are estimated to be II and 13, respectively, using the assignment x=0 esti-.
, s6
mated in the present cakuktion ~.and &e vibrational i.nteFnls of ?II determined from the anaiysis of the observed B22+-%I pexp&ations[l3]_ Thnq the present theoretical caIcuIations confirm that the -our pl+vious -experimental -finding a.nomalies in the emission speetla (Baz+-x’z+; uu = 9,11,12,14,17, and 18) are caused by the
perturbation by the411 state. 3-R yibralional
_
iecelk of ‘X+
The perturbation between thenB’Z* and 4X+ states has been observed at the ur, = 11, N = 20 and DB = 13, N F 9 levels of B’Z+ [5-lo]_ The corresponding vibrationaI quantum numbers of the 42+ state are estimated as follows: -
4_Condndingremarks We have carried out MC SCF caIcuIations for the low-Iying doublet and quartet states of the CN radical The vibrationaI energies and the rotational constants for the B2Z* and ‘II states obtained in the caIcuIations support the assignments made in our previous spectroscopic study_ The viirational levels of B2Z+ with u= 9,11,12,14,17, and 18
are perturbed by the follow& viirkional kvels of 41Tr respectiveIyr x, x-k& *.-i-3,.*+6, X+,11; and- x + 13, where the most probable assignment of x is O_ _lIe gissignmen_ts.of_ the. Io_wer -Ieve% on 4 x + 6, are more-reliable than the, last two_ In addition to this perturhatiop, the: eations in the u = 11. and 13 levels of B!X* and the y and y +- 3 Ieveh of-42+. are also discussed, where y is estimatedtobes+l.~ After the present study. was cornpIe&, the vibrational numberings of. the. 411 state of CN. were derived from fan ab ~initio caIcuIation [21]. The rest& .of this &kuIation showed _tbat the vibrational levels of 41i which perturb the B’Z+, D = 12 level is expected to be un = 3 or 4, which is consistent with the present resu1t.s.
_4cclm&dgement me cakuIation has been carried out at the Computer Center of the Institute for MolecuIar Science sup_ported by the -Joint Studies Pr&I.ram (1982-1984) The authora are grateftd to DE T_A_ Miller, D-C_ Cartwright and H--J_ Werner for their fruitful advice and discussions.
R&[l] K_P_Huba
and G_ Herzberg in: Mokular _qccua and Ilzobxh suuccur=. YOL iv. Gxtstanrs of diawmic xuolccuk(V+nNosmmd,-on.1979). f2] M_R Gorhal and ML Savadat& Chax~ Rev_ 82 (1982) [3] g
Radford and HP_ Brokia. J. cZhcm_ Php
38 (1963)
[4] r&g anan Jr, I MoL Spxay_ 19 0974) 106. = [5] JH IColts and D-W. Sew, in: Rc+k hamediates
in
the gas phase, a.L D-w_ setsa (Acadanic Press, New York, 1979) p_ 151_ [6] J+ Coxon. DA Ramsay and D-W. Scucr, Can I Phys. 53 (l975) 1587_ m TA Milk. RS Framd ani RW.-FK14 I Cham Phys 65 0976) 3790_ [S] J_LUCool&R_Zq+i&dT.A_hfi!kr.J__chcrntiys. 68 0978) 4763.
[la]
1274 DC Cartarright and_PJ_ Hay. As-&bys. 383_
Phyr 62 0984) l508_ J- 257 (1982)
..