Potential energy curves for doubly positive diatomic ions

JOURNAL

OF MOLECULAR

SPECTROSCOPY

9,

18-29

(1962)

Potential

Energy Curves for Doubly Part II. Predicted States and Transitions

Positive of Nif,

Diatomic Oi’,

Ions

and NOi+

.4. C. HURLEY Division of Chemical

Physics, CommorLwealth Organization, Fishermen’s Rend,

Scientijic und Industrial Melhorrrne, dustralia

Kesearrh

The theory developed in Part I is used to predict potential curves and spectroscopic constants for a number of states of the ions N?, O:‘, and N02+. The calculations provide an accurate description of a band which has recently been assigned to the transition Nf’, d ‘2;: + a Gi and suggest that a number of other band systems may be observable for these ions under suitable experimental conditions. The variations in bond length of X2, 02, and NO under single and double ionization are consistent with simple molecular orbital t,heory. I. INTRODUCTION

A t#heory has recently been developed for determining potential energy curves for a doubly positive diatomic ion AB2+ in terms of those for a related neutral molecule CD (1). This theory can be used to obtain curves for excited st&atesof ABL+ as well as the ground state curve and has been used to assist. in the ident’ification of a transition in Ni+ (2, 3). This ident,ification has two important, (‘onsequences. It gives a much more precise and t,horough test of the accuracy of t(he theory for the states involved than does the electron impact dat,a used previously (I), and it suggests that other transitions in AR?+ ions may he observable under suitable experimental condit.ions, The most favourable systems from the experiment,al point, of view are probably t,he ions Nz+, Oi’, and N02+ formed from at#mospheric gases. In this paper the t,heory of (1) is used to predict a number of states and transitions in these systems from the wealth of spectroscopic data on t,he corresponding neutral molecules CZ , X2 , and CI\‘. The spectroscopic notat,ion of Herzberg (./t)is used t.hroughout.; lengths are expressed in rm and fanergies in cm-l. II. (‘HOICE

OF W.4LE

FACTOR

With t,he above choke of units the rrlat,ionship est,ablished in (1) btltwrrn the binding energy B,, of a state of AR’+ and the corresponding quantity R,, for CD becomes 18

PREDICTED

STATES

OF N:+, 0;’

B&R) = -(K/R)

AND

N02+

+ t2B,(tR)

19 (I)

with K = ef/hc. Two choices of the scale factor t were considered in (I), viz., t = (Td.v(00)/T,,,,(00))1’2,

(3

where Td.%( m> and T, ,J W> are the mean kinetic energies of the valence shell electrons in the separat’ed ions A+ + B+ and the separated atoms C + D, respectively, and 1 = (T,I( w)/Tn( ,)}I’?

= (EJ_IY,)“~.

(3)

Here Td( 00) and !Pn( co ) are the mean kinetic energies of all electrons in A+ + B+, C + D, respectively, and Ed , E, are t’he corresponding total energies. In Ref. 2 it was shown that, Eqs. (1) and (3) give better agreement with Carroll’s (S) experimental results for t’he transition Ni+, d I&+ + a I&+ than do Eqs. (1) and (2). In t,hese calculations rather crude approximations (Morse functions) were used for the d l&f and a l&+ states of C? . In order to distinguish between errors inherent in Eqs. (l)-(3) and those arising from the use of Morse functions, the pot.ential curves for t.he d l&+ and a l&,+ st,ates of Ni+ have been recalculated using, for the corresponding C, states, accurate potent’ial curves derived by the Rydberg-Klein-Rees (RBR) method’ (5), The equilibrium nuclear separat,ions R, , the binding energies Bd(R,) and the derivatives Bd’(R,)( =O), Bd”(R,), Bd”‘(R,) and Bi”(R,) were t’hen evaluated analytically. The relations of Dunham (6) were used to derive the constants B e , we , a, , WJe , D, , and be for the two st,ates of N”,’ and hence the constants yoo, &‘, B,“, Do’, and Do” were obt,ained for comparison with Carroll’s (3) experimental values. The result,s are shor;vn in Table I (calculations A and B). Equation (3) clearly leads to significantly better agreement with experiment than does Eq. (2). For this reason all subsequent’ calculations are based on Eq. (3). In Table I the results of two simpler calculations based on Eq. (3) are also shown. In t,hese calculations t#wotypes of Morse curve B,,(r) = D, -

D{l

-

exp[-_P(r

-

P-,)])~

(4)

were employed for the Cf states. These curves differ in the choice of the paramet.ers D and p. For calculation C we have D = u,~/~~w.P, while for calculation

/3= (8~?~c/h)“2(WrR.~)“~,

(5)

D,” D = D,,

fi = (2ir2pc/Deh)1’2we.

(6)

1 Since the important regions of the Cz curves for this calculation are near the minima (Ref. 5, Eqs. (4) (in fact below the third VI‘b ra t’Ional level) Rees’ quadratic approximation and (5)) is adequate. 2 This type of curve was used in t.he calculatious of Ref. 2.

20

HURLEY TABLE

CALCULATED

AND

OBSERVED

PARAMETERS

d %+ + Parameter A R c Db

Calculationsa Experiment,

(S)

I

67,870 64,350 64,430 64,080 62,903.18

I

FOR

O-O RAND

THE

OF THE

TRANSITION N%*,

a ‘Za’

BO’

Bo"

2.264 1.805 1.809 1.749 1.8644

2,“B”., 1.817 1.813 1.796 1.8801

DO”

4’ 10.5 8.8 8.7 11.6 7.1

X X x X x

10-c lo-” 10-e 1O-6 10-G

9.7 8.1 8.3 9.0 Ii.9

x x x x X

10-G 10-G lo-” lWG LO-6

a B scale factor t from Eq. (2) ; RKR method for G curves. H scale factor t from Eq. (3) RKR method for C? curves. C scale factor t from Eq. (3); Eqs. (4) and (5) for C, curves. D scale factor t from Eq. (3); Eqs. (4) and (6) for CZ curves. Spectroscopic data for Cn states from Herzlwrg (4). Jksociation limits as in Fig. 3. b Assuming Do0 (CA) = 6.50 ev (7).

;

We see from Table I that the result’s obtained using Eqs (4) and (.5) are almost identical with t,hose obtained using t,he accurate RIXR curves. This is because t#heimportant region of a C, curve for these calculations is that near the minimum.’ It is well known that Eqs. (4) and (5) provide an accurate approximation to the RBR curve in this region (5, 8). However, for larger nuclear separations the curve given by Eqs. (4) and (.i) may diverge rapidly from the accurate curve and it may be in error by several electSron volt,s at the dissociat#ion limit. Alt,hough these larger nuclear separations are unimportant for determining t,he spectroscopic c0nstant.s of AB*+ ions they are important in determining the stabilities and lifetimes of t,he stat#es (see Sect,ion IV). III.

SPECTROSCOPIC

COKSTANTS

FOR

AB2+ STATES

In view of the results obt,ained in the previous section the calculation of these spectroscopic constant’s is based on Eqs. (l), (3), (4), and (5). There are t,wo propert,ies of the t values given by Eq. (3) which simplify the calculations. Firstly, since the tot,al energies of the atoms and ions are very much larger than the excitat,ion energies of the lower excit,ed stat,es, the same value of t may bc used for all states of a given ion AB9+, irrespect,ive of the state of excitation of the dissociation products. Secondly, to within the accuracy to be expected from F:q. (I), the atomic cxcitat,ion energy AE,, of the dissociation product,s of a state of CLDis related to the corresponding excitation cncrgy for AB?+ by t,he equation A&

=

t”AB,,

.

ii)

We see from Table II t,hat Eq. (7) gives values of AEd correct t,o within 0.2.i ev for the low-lying st#ates of S+ and O+. These two propert#ies of the t values, and the use of Eqs. (4) and (5) for the pot,ential curves of the neutral molecule CJD, ensure that t,he spectroscopic con-

PREDICTED

STATES

OF N:+, O:+ AND

TABLE

N02+

21

II

IONIC EXCITATION ENERGIES (ev)a Interval

12E,

Ed

N+ from C, t2 = 1.4285 [Eq. (3)] 3P -

‘L)

3P -

‘S

1.804 3.833

1.898 4.051 0+ from N, t2 = 1.3662 [Eq. (3)] 3.324 5.016

4s _ 20 Vi - 2P a From the table of Moore

3.256 4.883

(9)

&ants obtained for the states of AB2+ are independent of the dissociation limits assumed for the various states a.nd of the value assumed for Doo(CD). Thus if U,(T) is the potential curve of a state of CD relative to the ground state dissociation products and Ud(R) is a similar curve for the corresponding state of AB2+ t#hen (see Fig. 1) I&,(r) = T, + D, -

D,o -

U,,(r) = AE,

-

U,(T).

(8)

We then have, using Eq. (l), Ud(R) = AEd -

B@)

= AEd + (K/R)

-

t2B,(tR)

= (K/R)

+ (AEd -

PAE,)

+ PUn(tR),

so t’hat, using Eqs. (4) and (7), we obtain U,(R)

= (K/R)

+ P( T, -

D,O + D(1 -

e-x)2),

(9)

where 2 = ptR -

pr, .

The calculation of the spectroscopic constants of the states of ?Ji+, Oi’, and n’02+ was based on Eqs. (5) and (9). These equations involve only the constants T a.2 , r, , and W,Z, which are known accurately for all stat,es of the correspynding neutral molecules Cz , X2 , and CN which we consider. The dissociation energy D,O, which is somewhat uncertain for Cz and CN, appears only in the additive const,ant - PD,O and so does not affect the spectroscopic constants calc*ulated from Eq. (9). However, the D,O values do affect the calculated appea.rante potentials of the states. In order to determine t,he spectroscopic const’ants from Eq. (9) we first require the equilibrium nuclear separat,ion R, , that is the nuclear separat,ion (if any) for which T’n’(R,) = 0

(10)

HURLEY

C+D ,

Nuclear

FIG. 1. Relationship RI1

of energies for molecule

separatlan

CD (see Eq. 63))

d

Equ&ion

(10) may be expressed in the form y1C.r) = !/2(a),

where

&(Z) = c -

2 log(.r + cl),

ill)

with c = log (iY/.3/2D),

cl = 131“

Equations (11) are in a form which is particularly suitable for graphical solut,ion. Figure 2 shows the funct,ions yl and y2 for typical values of t,he constant,s c and ti (t’hose obtained for the ground state of 0:‘). The two intersections give the values of x at the minimum and maximum in the potential curve. The values for any other stat,e are obtained simply by shifting the curve y:! parallel to the .r and ‘y ases as indicated by the values of c and d appropriate t.o t,he stat,e in cluestion. It is clear that for values of c and d outside a cert,ain region t#hecurves .yl and ~2 will not inhersect. This corresponds to an unst#able st,ate of P1R*+, whose potent,ial curve is repulsive for all nuclear separations. The approximate values of R, obt.ained from Fig. 2 lvere refined to sewn significant figures by Newt’on’s method and the quant,ities Ud(R,), CT,,"(K,), I;,,"'(R,), and lii'(R,) determined from Eq. (9). Dunham’s relat,ions (6) wrc

PREDICTED

STATES

FIG. 2. Determination

OF N;+, O;+ AND

of equilibrium

nuclear

N02+

23

separation

then used to derive the constants T, , B, , r, ,3 we , 01, , and wexe for the stable states of Ni+, 0:’ and N02+. The results of these calculations are shown in Table III. The calculated appearance potential I of each state of AB*+, i.e., the vertical ionization potential from t’he ground state of AB, and some observed values of this quantity are also shown. The labelling of the states is taken direct,ly from that for t’he corresponding neutral molecule (4). IV. POTENTIAL

CURVES

FOR

LARGE

NUCLEAR

SEPAR.ATIOXS

Although, as we have seen, Eqs. (4) and (5) provide accurate curves for the states of CD near the minima they may be very inaccurate for the larger nuclear separations which correspond to the maxima in the potential curves for AB’+ ions. For certain of the states which we consider, viz., N? : X I&+, A 3ZU+,B 3Ho, a ‘II, , the experimental data (,&lo) are sufficiently extensive for accurate RKR curves to be plotted out to a nuclear separation which corresponds t)o t’he region of the maximum in t,he 05’ CLI~WS. These RKR curves have been used together with Eq. (1) and (3) to calculate the curves for these 0:’ states shown in Fig. 4. For the other states of O2‘+ and the st,ates of Ni+ and N02+ (shown in Figs. 3 and 5) there is insufficient data t,o provide information on the form of the curves in the segion of the maxima. For these st.ates Morse curves derived from Eqs. 3Since we need no longer distinguish fz and K, for AB2+ from T and T, for CD, ae revert to the usual spectroscopic notat,ion r, for AB’+.

HURLEY

24

TABLE PREDICTED P(ev)

SPECTROSCOPIC CONSTANTS

Id(ohs)

_

Deidev)

III FOR STATES OF

T,.

Wr

N?, O?,

OP &

&

AND

NO*+

a,

r,(lO 8Cm)

N%’ from CP, P = 1.4285, p-4 = 7.00349 50.72 48.04 43.99 $2.73

43.8 42.7

unstable 1.0 1.0 1.i “. 3

69,480 51,370 10,290 .5,020

1655 1596 l-125 17’24

21.3 25.2 19.0 “1.2

1.823 1.733 1.579 1.8’27

0.0279 0.0306 0.0250 0.0272

1.149 1.179 1.235 1.148

1518 1253 1505

“7.2 20.3 17.7

1 ,729 1.434 1.611

0.0336 0.0’241 0.02.32

1.180 1.295 1 .2”” “_

1.808 1.358 1.558 1.976

0.0299 0.0261 0.0287 0.0358

1.080 1.246 1.163 I.285

1.434 1.528 1.373 2.066

0.0149 0.0279 0.0252 0.0219

1 .“l” I

1.982 1.695 1.929

0.035i 1 .OG

unstable

46.14 43.77 4’2.63

43.8 42.7

0.4 1.7 2.4 O:+frorn

48.44 $2.49 42.15 39.64

0.6 1.0 0.7 0.5 unst,able 1.2

43.6Y2 43.65 42.85 35.48

1.ci 1.i 5.1

31,350 3 )370 0 (339.97O)b.C

N.‘*, t2 = 1.3662, ~1.4= 7.99972 1777 26.1 115,530 1170 30.6 il,250 68,330 1405 23.5 47 )fi40 95’2 38.4 80,620 80,660

74,300 0

1453 1367 1804 2363

13.6 ?2.5 19.8 19.4

1 ,175 1 ,239 1.010

(27%) lOO)“.C NO*+ from CN*, t* = 1.3917, g_, = 7.46857 unstable unstable 42.8i 38.72 38.10

39.8

2.2 2 1 3.0

38,110 (i, 700 0 (30tj,630)h,c

1908 lliA’2 “03’2

31 .5 19.0 18.1

0.0236 0.0224

1.154 1.082

n Spectroscopic data for Cs, X2. and CN from Herzberg 14), supplemented by the mow recent review of Wilkinson (IO) and references therein. b Relat,ive to the ground states of the neutral molecules 52, OS, and NO. c Assuming Db(C?) = 6.50 ev (7), D$(N~) = 9.756 cv (71, II,? = 5.14-l ev (7), D#(CN) = 7.54 ev (II), DS(NO) = (i.187 ev (7). rl Reference I. The assignment of t,he two ohserved appearance pokntials of N:’ diff~~rs from that of Ref. 1 (compare Fig. 3 and Ref. I, Fig. 2). (1) and ((i) have been used. Although, as Table I shows, these curves are less accurate near the minima than those derived from Eq. (5), they yield correct values for infinite nuclear separat’ion and probably provide a het,ter approximation in t,he region of the maxima of the AB’+ curves. In order to construct Morse curves from Eqs. (4) and (6) it is ncwssary to assign dissociation limit#s for the states concerned. For some of the statw.

PREDICTED

STATES OF N;‘, 0;’ AND N02+

25

34

32

FIG. 3. Potential curves predicted for N?. The zeroth vibrational levels of the observed transition d l&---t a ‘2: (2, 5) and the two observed appearance potentials of N? ions (1) are shown. a, these dissociation limits are uncertain; b, if this curve is taken to the limit 3P + 3P an unstable state results in disagreement with Table III.

notably Ce: A “H, , c ITI, , and h-,: B Q, , C 3~~ the limits shown in F’igs. 3-5 are uncertain; other equally plausible assignments can be made. In these cases the dissociation limits have been chosen to give approximate agreement wit.h the calculations of Section III. From t,hese curves we may determine bhe effective dissociation energy Drff for each of the states of Ni’, O”,‘, and NO*+, i.e., the height of the maximum of the potential energy curve above the minimum. Values of Deff obtained in &is way are shown in Table III. The lifetimes of the various states for dissociation by tunnelling through the potential hill may also be calculated from the curves of Figs. 3-5 using the WKB approximation (13). This leads to values which range from the order of 1 set for O”,‘, A %,,-+ to 10500set for O:‘, X ZQ+. These lifetimes are so long that tunnelling may be neglected entirely except perhaps for the highest vibrational levels.

HURLEY

26

35

28

FIG. 4. Potential Rydberg-Klein-Rees

curves predicted for 0;‘. a, for these states N:! curves obtained by the method were used (1%‘); b, these dissociation limits are unrertian. V. DISCUSSIOS

The equilibrium nuclear separations r, predicted for the low-lying st,at,cs of n_;*, O?, and K’CP+ show some intereskg t.rends when compared with the values for the ground states of the neutral molecules and singly charged ions, viz. (in unit,s of 1OF cm), y,(S2) r,,(?;‘)

= 1.097 (IO), r,(02) = 1.116

(d), r,(Of+)

= 1.207 (4), r&NO)

= 1.151 (,/t);

= 1.123 (/t), r,(NO+)

(IS)

= 1.062 (10).

The variations in r, values may be interpreted in t.erms of simple molecular orbital theory. The ground-stat’e configurations of t,he parent molecules are as

PREDICTED

FIG. 5. Potential curves N02+ ions (1) is shown.

STATES

predicted

OF N;+, O;+ AND

for X02+.

The

observed

27

NO*

appearance

potential

of

follows:4

x0

KK(a,2s)“(o,2s)2(a,2p)2(a,2p)4(a,2p)

(13)

02 : KK(a,2s)?(a,2s)‘(u,2p)2(~,2p)4(?rg2p)2 From the molecular orbital correlation diagram (Ref. 4, p. 329) we expect the outermost orbitals of (13) t#o have the following characterist.ics: a,2p weakly bonding, *,;2p &rongly bonding, 7r,2p strongly antibonding. The changes of bond lengths on ionization predictBed in Table III and Figs. 3-5 provide a faithful reflection of these characterist’ics. We consider first, the a I&f state of xi’. For this state the agreement between the calculated and observed B,” values shown in Table I provides dire& experimental confirmation of the bond length given in Table III and Fig. 3. The state Nz+, a Q,+ is obtained by removing two weakly bonding a,2p electrons from K2 , X l&,+. From Eq. (12) we see that the removal of the first a,2p elect,ron

4 The symmetries implied by the subscripts NO which lacks a center of symmetry.

g and u are, of course, only approximate

for

HURLEY

28

(S, , S l&+ --f Nz+, X Q,+) leads t,o a small increase (0.019) in r, ; Table III predict,s a similar small increase (0.032) in rc on removal of the stcond a& electron (Nz+, X Z&f ---f Ni+, a I&+). The states Ki’ , X 3~, , and b III, are obtained from I%?+, X ?&+ by removal (Jf a st,rongly bonding ?r,2p electron. Figure 3 shows that these two st’atcs have upproximat,ely the same value of T? and that this value is considerably greater than that for xi’, a lZ,+. Again the %,,- stat,e of Ni+, obtained by removing two st,rongly bonding *,2p electrons from r\72, X l&,+ has t)he largest, rr valrw of nil the KS’ states. The ground stat,es of OS , 02+, and Oi+ (Fig. 4) show a t#rend which is t,ht> reverse of that for N1 , N?+ and Kf’, a I&,+. For the oxygen systems it is the strongly antibonding rQ2p electrons which are being removed and this is rcfleeted in the decrease of the successive r, values. Finally t,he X Zy+ and 1-1Q, states4 of N02+ represent an intermediate caPe (Fig. 5). Here the first ionization SO, X %, * K-O+, X I&+ removes a strongly antihonding 7,2p ele&ron and leads to a reduction of 0.089 in T, (Eq. (12)). The states ,I’O”+, X ?&+, and A %,, arc t,hen obtained by removing a weakly bonding au2p elect,ron and a strongly bonding ~,2p electron, respectively. The predict)ed increases in re (0.020 for 9 7&,+ and 0.092 for A “II,) from Table III again r&e& t’hese properties of t,he orhitals. The removal of the bonding ant,ibonding pair of electrons (7r,2p)(7rU2p) in the transition NO, X ?HI, j ?;O”+, Ll Q,, leaves r, almost unahered. _Uernatively, these changes of bond length on ionization may be interpreted in terms of t,he binding and antibinding regions of space occupied by the orbit,als (I ,5). This int,erpretation, which is based on the elect(rostatic theorem of Hellmann and Feynman (1.5, 16), is in some ways more direct and intuitively satisfying than t,hr above interpretation in terms of standard molecular orbital theory. In either case the conclusions are identical and these qualitatively reasonable fcaturcs of Icigs. 3-5 add to our confidence in t’he accuracy of the spectroscopic constantas predicted in Table III. It is felt that t,he accuracy of these predictions may be comparable wit’h the agreement’ shown in Tahle I (calculation C) with C’arroll’s observations, and it is hoped that these calculations may assist, in the identification of ot,her t,ransitions in xi+, OS’, and NOZ+ ions.

The author is indebted to l)r. H:un for reading t,he manuscript. KE(~EIVEI)

January

V. IV. Maslen

for helpful

discussions

22, 1962 REFERENCES

I. A. C. HI.RLEY AND V. W. MASLEN, .I. Chem. Fhys. 34, 1919 (1961). 2. P. K. CARROLL AND A. C. HURLEY, .I. (‘hem. Ph,ys. 36, 2247 (1961) 9. P. Ii. (:ARROLL, Can. J. Phys. 36, 1585 (1958).

and t,o I)r. S. S.

PREDICTED 4. 5. 6. 7. 8. 9. IO. Il. 12. 13.

STATES

OF N:+, 0;’

AND

N02+

29

G. A. J. T.

HERZBERG, “Spectra of Diatomic Molecules.” Van Nostrand, Princeton, 1950. L. G. REES, Proc. Phys. Sot. 69, 998 (1947). L. DUNHAM, Phys. Rev. 41, 721 (1932). L. COTTRELL, “The Strengths of Chemical Bonds.” Butterworths Publications, Ltd., London, 1958. J. T. VANDERSLICE, E. A. MASON, W. G. MAISCH, AND E. R. LIPPINCOTT, J. Mol. Spectroscopy 3, 17 (1959); 6, 83 (1960) (Errata). C. E. MOORE, “Atomic Energy Levels,” Natl. Bur. Standards Circular No. 467 (1949). P. G. WILKINSON, J. Mol. Spectroscopy 6, 1 (1961). H. T. KNIGHT AND J. P. RINK, J. ChenL. Phys. 36, 199 (1961). J. T. VANDERSLICE AND E. A. MASON, J. Chem. Phys. 30, 129 (1959); 33, 614 (1960) (Errata). L. D. LANDAU AND E. M. LIFSHITZ, “Quantum Mechanics.” Pergamon Press, London, 1958.

BERLIN, J. Chem. Phys. 19, 208 (1951). 16. H. HELLMANN, “Einftihring in die Quantenchemie.” 16. R. P. FEYNMAN, Phys. Rev. 66. 340 (1939).

14. T.

Deuticlce,

Leipzig,

1937.