Singlet-triplet anticrossings between ungerade states of H2

JOURNAL OP MOLECULAR

SPECTROSCOPY

Singlet-Triplet

63, 193-204

Anticrossings TERRY

(1976)

between Ungerade

A. ~JIILLER~ AND ROBERT

States of H,

S. FREUND~

Bell Laboratories, Murray Hill, New Jersey 07974 Singlet-triplet anticrossings in the Hz molecule have been observed between two ungerade states, B’(3p)e,+ and f(4p)Q,+. This is the first time that an observed HP anticrossing has involved a state which can radiate directly to the ground state. Analysis yields accurate values for the zero field separations between two pairs of rotational and vibrational levels. It also yields a value for the Fermi contact interaction in the triplet state as well as the difference in the orbital angular momentum ,c factors for the two states. From linewidth measurements, we deduce a rigorous lower limit to the radiative lifetime of the B’rZ,+ state and a (nearly equal) most reasonable value for it. It is shown that the perturbation between the two states is quite weak leading to little singlet-triplet mixing in zero field. The experimental data establish that the oscillator strength for the forbidden transition from the ground state to the f(4$)3X’uf state is at least seven orders of magnitude smaller than that of the allowed transition to the B’r&+ state. INTKODUCTION

anticrossin g experiments have been recently performed A number of singlet-triplet on simple atoms and molecules (1--g). In these experiments excited singlet and triplet states are prepared with unequal populations by electron bombardment. A magnetic field is applied to tune a spin component of the triplet state to essential degeneracy (the anticrossing point) with the singlet level. Here only a very small perturbation is required to equilibrate the populations of the two states. By monitoring the intensity of optical emission as a function of magnetic field from either (or both) states, the position of the anticrossing field can be determined. Such experiments have yielded considerable new information about excited-state energy level structures. For the simplest neutral molecules, Hz, Dz, and Hez, no singlet-triplet optical intercombination lines have ever been assigned. Thus, the relative energies of the entire singlet-triplet manifolds in these molecules had only been determined by (a) experimental extrapolations to Rydberg limits (5, 6, 11, 12) (b) theoretical ab initio calculations of the lowest triplet’s energy (13-16). Such methods are inherently less precise than direct spectroscopic measurements and in some cases considerable controversy has existed over the correct manifold separation (5, 13). From anticrossing experiments very precise zero-field separations for single-triplet pairs of rotational and vibrational levels can be determined. Combination of these results with intramanifold energies from optical work has resulted in new and much more precise values for the singlet-triplet separations in Hz (4, 5, S), Dz (7, 9), and He2 (6). 1Guest scientists at the Francis by the National Science Foundation.

Bitter National Magnet Laboratory, which is supported at M.I.T.

193 Copyright All rights

0

1976 by Academic

of reproduction

Press,

in any form

Inc. reserved.

194

MILLER

AND

FREUND

ORTHO-Ii2

MS MN MJ -0

0

B’(3P)

0

‘z:

v=3,N=O

f (4pPx: /‘O.N=O

t

0

-1

0 -1,

, 20 MAGNETIC

FIELD

40 IkGl

FIG. 1. Energy level diagram of the N = 0 c-t N = 0 anticrossing.

Analyses of anticrossing states. The magnetic

spectra have also led to other information

g factors of the anticrossing

In some cases, especially so-called forbidden has been determined.

about the excited

states have usually been determined.

anticrossings,

fine and hyperfine

structure

Analysis of the linewidth has led to values for the perturbations

between singlet and triplet levels and in a few cases radiative lifetimes of the states. Interestingly, although a number of anticrossings have been observed between singlet and triplet ungerade

singlets

states in Hz and Dz, all the levels involved can optically

vation of an anticrossing

connect

with the ground state

are gerade. Since only of Hz(Dz),

the obser-

between singlet and triplet ungerade states becomes desirable

for several reasons. Optical absorption experiments can be performed directly to the singlet state involved in the anticrossing, eliminating the need for a chain of optical data to determine

the singlet state’s absolute energy. In addition, from the anticrossing

experiment, the precise position of an ungerade triplet state is known, so that a singlettriplet absorption experiment may be contemplated. Finally, an anticrossing experiment involving an ungerade singlet radiating primarily in the VW region of the spectrum establishes the possibility of controlling the intensity of optical radiation at wavelength < - 1000 A by small magnetic field changes. In this paper we report observation of two separate singlet-triplet anticrossings involving

ungerade states of Hz (Figs. 1 and 2). One anticrossing occurs between the v = 3, N = 0 and the f(4p)%?&,+, ZI= 0, N = 0 ortho-Hz states; the other involves the B’(3p)Q,+, v = 3, N = 1 and the f(4~)~&,+, 21= 0, N = 1

B’(3p)‘zz+, anticrossing

para-Hz states. EXPERIMENTAL

METHOD

The experiments were performed inside a Bitter coil located at the Francis Bitter National Magnet Laboratory at M.I.T. The Bitter coil, described previously (3),

ANTICROSSINGS

195

IN Hz

40 20 MAGNETIC FIELD lkG1

I’IG. 2. Energy

level diagram

of the N = 1 ++ iV = 1 anticrossing.

allowed fields in excess of 13 Teslas (130 kG). As previously described (3), the vacuum chamber consisted of a 2 m long tube, which passed through the Bitter solenoid. An electron source fired electrons along the magnetic field exciting Hz molecules at a pressure of -10 mTorr. The light emitted by Hz from the most homogeneous field region was collected by a light pipe, subsequently dispersed by a monochromator, and detected by a photomultiplier whose count rate was monitored as a function of field strength. A schematic diagram of the apparatus is shown in Fig. 3. The narrowness (less than 50 G wide) of the present anticrossing signals required more sophisticated treatment of the magnetic field measurement and sweeping than described previously. The Bitter coil has spatial homogeneity over the sample volume of better than about one part in 104, so it represented no real problem with respect to the PICKUP COIL

TO CAPACITANCE

FIG. 3. Schematic mity Bitter coil.

diagram

of the apparatus.

It is located

MANOMETER

within

the 5 cm diam bore of a high-unifor-

196

MILLER AND FREUND

‘A!

A375

,,bl!

G

FIG. 4. The two N = 0 anticrossings,observed on the 4886.4 i line of ortho-Hz. This spectrum was accumulatedby signal averagingfour magnetic field scans for a total time of -1.5 min.

observed anticrossing lines. However, the temporal homogeneity, due to power fluctuations, was about one part in lo3 and thus was comparable to, or greater than, the anticrossing linewidths. In our early work (3) we had noted that these temporal fluctuations had to be eliminated in order to make accurate measurements of the magnetic field with the NMR system. A stabilization circuit existed, which sensed magnetic flux changes (dB/dt) through the voltage induced in a pickup coil inside the magnet. This voltage controlled a power supply which drove an independent set of coils built into the Bitter magnet. The field with the stabilizer energized is stable to about one part in lo5 for several seconds and about one part in lo4 for several minutes. While this arrangement is perfectly adequate for NMR measurements at a fixed field, the stabilizer, by its very presence, prevents the magnetic field from being swept through an anticrossing. What is necessary is a mechanism whereby the stabilizer circuit can be made to ignore a constant, linear field sweep while still compensating for random field fluctuations. To accomplish this goal the amplified output of the pickup coil is fed to one side of a differential amplifier, while the other side is fed by the voltage ramp, properly biased and compensated, which drives the linear field sweep. Since the sweeping field introduces a linear voltage ramp in the pickup coil, only random field fluctuations produce a net voltage at the output of the differential amplifier. With this arrangement field sweeps of up to 2 kG are possible with field fluctuations at the one part in lo5 level. It was in this way that the anticrossing shown in Fig. 4 was recorded. To determine the absolute magnetic field, the field scans must also be calibrated. This can be done with an independent run, with the NMR tube (see Fig. 3) inserted into the sample region (to avoid field differentials) and the field swept. The effect of any slow field drift can be compensated for by correcting the zero channel of the field sweep to a constant field value (as measured by a Hall probe) for both data and calibration runs. It was also possible, for some data runs, to place blip marks, corresponding to NMR signal positions at known frequencies, directly on the data trace. THEORY The 2 energy levels as a function of magnetic field are given by

Et = poH(gsMs

+ gdMd)

+ ft[H2, (MdP]

+ aMrMs f

SS.

(1)

ANTICROSSINGS Table1

Measured field positions of observed anticrossinys. As noted in the text, the absolute errors may be as great as 15 G, but the relative errors for each pair of anticrossings should be somewhat smaller.

f14p13$, N

197

IN- Hz

MN

v =0 “S

.-Mf

B’f3p#G,

v =3

N

‘S

MN

MI

Magnetic Field IGI

The first term is the linear Zeeman interaction, composed of the spin and orbital interactions with the field. The second term describes the quadratic Zeeman effect as a function of Hz and (M,@. The third term gives the hyperfine interaction which, as we have shown previously (9), can be adequately approximated by the Fermi contact interaction alone, The last term presents the spin-spin interaction in the triplet state. The singlet energy is E” = hvo + g&oHMNs

+ f[H”,

(M,v5)2],

(2)

where hvo is the zero field separation of the singlet and triplet states (singlet-triplet), the second term is the orbital Zeeman term, and f” is another quadratic Zeeman function. RESULTS

A. Measureme&s The N = 0 anticrossings were observed on the O-O, P(l), ,f”&+ + a”z,+ optical emission line at 4886.4 A (air) as an increase in light intensity (Fig. 4) and on the 3-1, P(l), B”&+ ---f El&,+ optical emission line at 7114.2 A as a decrease in intensity. All accurate field measurements were made on the stronger anticrossing signals carried on the triplet emission. In the case of the N = 1 anticrossing the P(2) and R(O), O-0, _f3&L+-+ a%,+ optical emission lines at 4906.3 and 4858.8 A were monitored, with the accurate field measurements made on the former. The measured line positions are given in Table 1. Each value is the average of three independent measurements which agree to better than 5 G. We believe the absolute Tablen

Experimental full widths at half height for observed anticrossings and their one standard deviation uncertainties,

198

MILLER

AND FREUND

line positions quoted in Table I to be accurate to <-15 G, the final error resulting from a combination of several factors, including spatial inhomogeneity over the sample volume, incomplete compensation of slow field drifts, residual line asymmetries, and statistical fluctuations in the line centers due to relatively low signal/noise. The measured linewidths given in Table II are averages of at least four independent measurements. The two anticrossings in N = 0 have the same width, within experimental error, so they have been averaged together. Similarly, the two N = 1 anticrossings have the same width. B. N = 0 Analysis

Selection rules (5) for anticrossings can be deduced from the fact that the component of total angular momentum along the magnetic field axis MJ (or MF if the total nuclear spin is nonzero) remains unchanged for any perturbation except a perpendicular electric field. For the N = 0 anticrossing, (in which MN f = M$ = 0), we have AMp = 0, so AMs = - AMr = 1. The Fermi contact hyperfme interaction satisfies this selection rule. For the N = 0 level the expectation value of SS vanishes, and the fact that M,+ = 0 requires a null contribution from the orbital Zeeman term. Thus, at anticrossing, Es - ET = 0 so from Eqs. (1) and (2) hvo = gsMspoH + (f” - f”) + aMrtMs.

(3)

The magnitude of the Fermi contact interaction can be determined from the separation of the two lines. The value of the quadratic Zeeman term f” - f” cannot be obtained from the available data, but it is important to determine its value as the magnitude and accuracy of hvo rest upon it. The quadratic Zeeman interaction is caused by the coupling of rotational level N with N’ = N f 1. Usually the strongest interaction is with the rotational levels N’ in the same vibrational and electronic state. However, since the states involved are 2, alternate rotation levels have different parity and cannot couple. The only level that can be coupled to an N = 0 Z,+ level is the N = 1 level (of the same parity) of a II, state of the same multiplicity. Using Eq. (6) of Ref. (17) for the Z-II coupling matrix (and remembering that the II state is a normalized combination of A =f 1 states), we have for N = 0 I(“& 2, = OIXHI3II, V’)l2

f”[H”,

(Jf~4~1 = C ll,o’

Ex, - En.u~

%.mm2 =sc

Il.99

~(32,2)=O,A=O~T-~1(L)~3n,v’,A= Ez.. -

&I..~

1)1” ,

(4)

where the sum runs over all other states. The f”[H”, (MN’)~] has exactly the same form except v = 3 and only singlet states are included in the sum. In the exact form, the sums are impossible to calculate. However, they may be estimated in the following way. The closest known level to the singlet is the v = 1, N = 1 level of the D(39)%, state located 452 cm-’ lower. Assuming the remaining terms are nearly negligible and/or roughly cancel, the sum is reduced to a single term. The matrix element is approximately separable into an electronic part and vibrational part. In-

ANTICROSSINGS Tablem

199

IN HP

Results fromtheanticrossings B'f3&+, Y

=3

-

f fqJl3q v -0.

NS=Nt=,, huof= ES-ET1

,I,~i Nti 1

128201+50MHz

-148199+_370MHz

4.2763+0.0017 cm-l

-4.943+0.012 cm

-1

I

voking the hypothesis

of free precession

for the electronic

part yields

f”[H”, (LI~N~)~]= (fi,H)? 1(ZJZJ’)12/(678 cm-‘),

(5)

where the vibration dependent factor is the Franck-Condon factor between the two relevant levels. Using potentials recently described by himself (18), P. Julienne has kindly calculated that 1(V/ o’) / 2 = 0.030. Th is result allows the evaluation of f” at the resonance field as 6.0 MHz. For the triplet case the nearest perturbing level is the (1(3p)311,, v = 2, LV = 1 level which is 507 cm-’ higher. Assuming the Franck-Condon factor is no larger here (which seems reasonable), we get a Zeeman shift f” of -5.4 MHz. Thus, one expects the total quadratic Zeeman effect to be f” - f” = - 11.4 MHz. As there are several approximations in this calculation, its absolute accuracy is difficult to verify, yet it seems reasonable to state that the quadratic Zeeman term introduces an uncertainty into the analysis of <-20 MHz. Coupling this with the experimental error of < -15 G, we feel hvo can be determined from Eq. (3) to an accuracy of <-SO MHz. The value of a, being essentially independent of quadratic Zeeman effects or systematic errors in absolute field measurement, is probably accurate to < -20 MHz. Thus we obtain the results in Table III, where we have assumed that gs has its free-spin value, 2.00232, PLO/~= 1.3996 G/MHz, and the speed of light c = 2.99792 X lOl”cm/sec. C. N = 1 Analysis The K = 1 analysis follows the same lines as the N = 0 work; however, it is somewhat more complicated. First, we note that the ~1: = 1 level corresponds to para Hz so that no hyperfine structure or perturbation can exist. However, we now have different ~MN levels, so spin-orbit coupling can satisfy the selection rule AMJ = 0, =-I. AMs =-A&~N The dominant Zeeman term of course remains the electron spin interaction, but as MN is not restricted to zero the orbital Zeeman terms can no longer be neglected. Since we are dealing with Z states the electronic orbital angular momentum is quenched so there is no direct paramagnetic contribution to gNt. This leaves only the so-called rotational magnetic moment, typically measured in ground-state ‘z1 molecules. The nuclear contribution to the rotation moment, and hence to gN, is easily calculated (19) to be - 5.45 X lo4 for any Hz state.

200

MILLER AND FREUND

There is also an indirect paramagnetic contribution (electronic contribution) to gN due to the partial un quenching of the electronic orbital angular momentum through the mixing of IT states via rotational uncoupling. Unfortunately, this term is consider_ ably more diffitult to calculate and is also probably considerably larger. The pertur­ bation expression for the paramagnetic contribution to gN, called gN(e), is given (20-23) by an expression similar to Eq. (4),

gN 8 (e) = (-2!/2)

L

{(l~,V

= 3,A = 0IT_1l(L)llII,v',A = 1)

n,V'

X (III, v', A = 112BTll(L) Il~, v = 3, A = 0>

+ (l~,

V

= 3, A = 0 12BT_ll(L) IIII, v', A = lWIT, v', A = 11 Tll(L) Il~, v', A = OJ} X (E1-z. v

-

Elrr .• ,)-l.

If we make essentially the same assumptions as we did to approximate

l' we

(6)

obtain

(7) where B is the rotational constant of the l~ state. Numerically we find gN 8 (e) =- 5.2 X 10-3 and similarly for the triplet, where the nearest interacting level lies 459 cm-l higher, gNt(e) = 5.6 X 10-3• Combining these gN 8C t) (e) with the nuclear contribution gives an effective gN (t). If in Eqs. (1) and (2) we neglect pct) and 55 we find that the separation of the two ob­ served lines should be proportional to (gN 8 - gN t), which by our calculation is - 0.0108. This predicts a separation of the two observed lines of ",285 G compared to a measured value of 411 G. While this agreement is not exact, it is certainly within the factor of 2 accuracy we estimated for our calculation of the 1'ct) for N = 0 using the same approximation. Furthermore, since the calculated and observed (see below) difference in g factors is well within a factor of 2 we shall assume the individual, calculated g factors are accurate to that factor of 2 (i.e., 5 X 10-3), which at the anticrossing field is a total uncertainty of ",370 MHz. Since the 1'ct) should be comparable for N = 1 and N = 0, they are quite negligible compared to the uncertainty in gN 8 - gN t. Likewise, using the measured value of the spin-spin interaction in the k(4p)3II u state (24), one can estimate a maximum shift of any level of the 3~ state to be considerably less than 50 MHz due to the 55 term. Thus, neglecting p, 1', and 55, from Eqs. (1) and (2) we obtain the simple formula for the anticrossing fields -hvo (8) H= -------------------------, ,uo[gs - gN t + MN 8 (gN 8 - gNt)] 8

where MN 8 =-1 and O. Solving Eq. (8) for hvo and (gN 8 - gN t), assuming that gN t = 5.1 X 10-3 with an error of 5 X 10-3, yields the results in Table III. The negative interval, hvo, indicates that the N = 1, v = 3 l~ state is below the N = 1, v = 0 3~ state. Its error limit is determined primarily by the assumed uncertainty in the calculation of gN t • The differ­ ence in the singlet and triplet g factors is clearly consistent with their calculated values but obviously the experimental result is more precise. Since the quantity (gN 8 - gN t)

ANTICROSSINGS

IN Hz

201

is largely determined by the separation of the anticrossings and is nearly independent of JzY~,to estimate its error, the uncertainties in f”, j”, and SS have been included. D. Radiative Lifetimes and Singlet-Triplet We have previously

shown that

Perturbations

the observed

anticrossing

width

(1, 8) (FWHH)

is given by Av =

where V = perturbation

(2/h) (4 1T/ / 2j7 + T%-~):,

matrix element between

(9)

the two states

and 7,s and 7T are, respectively, the radiative lifetimes of the singlet and triplet states. The lifetimes ~8 and rr have apparently never been directly measured for the B’%,+ or the j3&+ state. Nonetheless several lifetime measurements exist from which the values of 7s and 7T can be rather directly inferred. Phase shift measurements (25) of the lifetimes of the B(2p)‘Z,+ and C(2p)‘H, states give lifetimes of 0.8 and 0.6 nsec, respectively. The B’ state differs from the B state only in the promotion of the 2p electron to the 3p orbital. Such a promotion would slightly lessen the transition moment (based on the atomic model) but the increased v3 factor (for the vacuum ultraviolet transition to the ground state) probably roughly balances this effect. Thus it seems reasonable that T_s(B’%) = 1 nsec (within a factor of 2 or 3). Likewise, although no value of rr(j%Z) has been reported, there exist measurements for very similar states. MOMRIE results (23, 26, 27) for the ~(~P)~II, and K(~P)~II, states give a lifetime for each state of 32 rrt 5 nsec. Thus, we are inclined to believe that 7r(j?Z) = 30 nsec. If we use these values of 7r and ~8 we find that ? = 27s. If as a first approximation we neglect the perturbation term in Eq. (9) we predict an anticrossing linewidth of Av = 160 MHz. The important point is that this predicted width is larger than the observed anticrossing widths (Table II). Thus, we are led to believe that lifetime broadening accounts for the dominant part of the observed width. Inverting the procedure and solving for the experimental 7.9 from the widths yields 7s = 1.3 f

0.2 nsec

(N = 0),

~8 = 1.6 f

0.3 nsec

(Y = l),

where the error limits are derived solely from the statistical uncertainties of the measured widths. These derived lifetimes are reasonably self-consistent and obviously consistent with our estimates. Clearly the value of 1.3 nsec represents a rigorous lower limit to the lifetime. If the lifetime were shorter, the observed anticrossings would have to be broader than observed. On the other hand, the perturbation term in Eq. (9) may make a contribution to Av and hence rs could be longer than calculated. The importance of the contribution of the V term can be assessed in the following way. For N = 0 and 1 the origin of I/ is completely different. In N = 0, V is essentially determined by the Fermi contact interaction; for N = 1, V is determined by the spinorbit coupling (including spin-rotation) operator. There is clearly no reason for I/ to

202

MILLER

AND FREUND

be identical for the two cases. Yet it is unlikely that 78 would vary significantly between N = 0 and 1. Thus if only the lifetime term contributes to Av, the observed linewidths should be the same for the N = 0 and 1 anticrossing. From Table II, we see that the error limits on the measured widths overlap, reaffirming the lifetime broadening dominance. However, it does appear that the N = 1 anticrossings are not quite as broad as the N = 0 ones. It is therefore likely that the V term is not completely negligible, especially for N = 0. Thus, we would clearly favor the longer, N = 1, lifetime. The smallness of the perturbation matrix elements may be somewhat surprising. Clearly V cannot be greater than 31 MHz. Making the most reasonable assumptions for f, and the contribution for lifetime broadening, the Fermi contact perturbation, V, between the AT = 0 levels is probably less than 10 MHz with the spin-orbit coupling perturbation between N = 1 states still smaller. Clearly I/ is small compared with the diagonal value of the Fermi contact interaction in the ,f(4p)% state of 451 MHz. Likewise the diagonal value of the spin-orbit coupling constant in the K(4p)% state (the most similar state in which spin-orbit coupling does not vanish by symmetry) is reported (23) as - 314 MHz. The answer to this seeming paradox is that each of the actual perturbations is two or three times “forbidden.” First it may be noted that if one describes the molecular orbitals by the united atom model, both the Fermi contact and spin-orbit interactions vanish between states which differ in principal quantum number of the valence electron. Second, the overlap of the vibrational wavefunction for the z, = 0 and n = 3 levels should be poor, making these perturbations essentially “Franck-Condon forbidden.” Finally, the spin-orbit coupling is forbidden by still another selection rule. In the limit of precise case b coupling, the spin-orbit interaction vanishes between a pair of Z+ states or a pair of 2- states. Thus the coupling of the f(4~)“&+ and the B’(3p)%+ states is forbidden in this limit. This selection rule can, however, be broken by the admixture of II (and higher A states) into the wavefunction by other perturbations, principally rotational uncoupling. Indeed, in the previous anticrossing experiments on Hz and Dz gerade states derived from d orbitals, the rotational perturbations were so strong that these states tended toward Hund’s case a! and little significance could be attached to this selection rule. However, the present states, derived from p orbitals, are known to conform quite closely to case b coupling so that this selection rule should retain considerable validity. E. Singlet-Triplet

Transitions

As no singlet-triplet optical transition has ever been observed in Hz, it is interesting to calculate one’s expected strength. The oscillator strength for absorption from the ground state to a triplet is determined by the amount of singlet mixed into that triplet and the absorption oscillator strength to the singlet. The present anticrossing results provide the information needed to calculate the amount of singlet mixed into the two triplet levels studied (the singlet-triplet separation and the perturbation matrix element between them). If the wavefunction for the triplet state with a small admixture of singlet is written as

Is,t>=

a/t>+b/s)

ANTICROSSINGS

IN Hz

203

then the mixing coefficient is defined as w =

b/a.

To first order, this mixing coefficient is w =

V/km.

An absolute upper limit to V can be obtained from Eq. (9) by dropping the lifetime term, with the result that V/k < Av/4 = 31 MHz for N = 0 or 25 MHz for -V = 1. Thus, w < 2.4 X lo_4 for N = 0 and w < 1.7 X lo-* for N = 1. With more realistic estimates of the lifetimes, w would be several times smaller. Since the singlet-triplet oscillator strength is proportional to J, these are very highly forbidden transitions, more than seven orders of magnitude weaker than the corresponding singlet-singlet transitions. ACKNOWLEDGMENTS We thank L. Rubin, F. Silva, and Y. Golahny for designing and constructing the magnetic field sweep circuit and for their help during our visits. We also thank P. Julienne for his calculations of FranckCondon factors.

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AND FREUND

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