10002
J. Phys. Chem. 1993,97, 10002-10010
Rotational-Electronic Splitting of Matrix-Isolated NH/ND B ‘A in Argon Cages of Ob and Symmetry. Spectroscopic Analysis and Theoretical Interpretation Carsten Bhdauer, Martin Winter, Olev Sid,+ Georg Jansen, Bernd A. Hess, and Ulrich Schurath’ Institut fur Physikalische und Theoretische Chemie der Universitiit Bonn, Wegelerstrusse 12, 0-5300 Bonn 1. Germany Received: May 14, 1993.
Electronic state-dependent Ar-NH(X,a,b) potential surfaces from ab initio calculations have been combined with a classical model of a doped rare-gas matrix to calculate electronic matrix shifts and local-mode frequencies which compare favorably with spectroscopic observations. It is shown that N H / N D substitutes one Ar atom in sites of oh and D3h symmetry which occur with equal probability over a wide range of experimental conditions, while interstitial trapping and impurity effects could be excluded. The previously reported site-dependent rotational4ectronic (rotronic) splitting in the a l A state of N H is shown to be insensitive to deuteration. The partial lifting of the 10-fold degeneracy in the lowest rotronic level of the a l A state could be modeled for both site symmetries by evaluating the effect of the rare-gase cage in first-order perturbation theory. The NH(a)-Ar,, interaction is constructed from the average of a recently published pair of ab initio NH(alA’)-Ar and NH(ulA”)-Ar potential surfaces. Electronic splitting is introduced via a “difference” potential surface of NH(a)*Arlz which was obtained by quantum mechanical methods, as described in detail in an accompanying paper. The perturbation gives rise to rotronic splittings which differ in cages of oh and 0 3 1 symmetry. The selection rules governing transitions from the electronic ground state to the sublevels of N H a l A in cages of oh and 4 s symmetry are consistent with the observed spectra.
1. Introduction The electronic spectra of NH/ND isolated in various rare gas matrices have been studied extensively.l-l0 In solid argon the electronicsingletand triplet states of NH have been characterized in absorptionand/or laser-induced fluorescence studies up to the c 1 I I state.” The properties of the radical in the X %- and b lZ+ states are those of a nearly free rotors2 Rotational levels N 1 1 in these states exhibit large isotope-dependent splittings, which are in excellent agreement with the predictions of the rotationtranslation coupling (RTC) model of Friedmann and Kimel,12J3, while being incompatible with crystal-field models of hindered rotors in trapping sites of octahedral ~ y m m e t r y . ~Extensions ~J~ of the crystal-field model to sites of lower symmetry16 are equally incompatiblewith the hindered rotor structure of NH/ND in its 1 states. An apparently different type of narrow splitting has been observed in the emission and fluorescence excitation spectra of matrix-isolatedNH and ND u 1A. This multiple splitting, which is superimposed on each rotational line of the u X spectra, shows very little isotopic dependence, and-unlike the RTC splitting in the X 32-and b 1Z+ states-varies by less than 30% in Ar, Kr,and Xe matrices.? It could be shown that the complex splitting of the rotational lines represents a superposition of two simpler sublevel schemes, which are associated with radicals in different trapping sites. While it was obvious that the splitting in the rare gas cage must be due to a partial lifting of the 10-fold rotronic degeneracy of the a ‘A, 51 = J = 2 state, the interaction mechanism and the nature of the two trapping sites, which were formed in nearly equal proportions over a wide range of experimentalconditions, remained speculative. It was suggested that NH is trapped in substitutional sites of oh and 0 3 1 symmetry. Supporting evidence for the coexistence of substitutional oh and metastable D3h sites comes from infrared and inelastic neutron scattering investigations of matrix-isolated methane.”-19 However, the nearly equal abundance of NH in both trapping sites
-
~~~
t Permanent address: Institute of Physics, Estonian Academy of Sciences,
Riia 142, Tam, Estonia. *Abstract publiihcd in Advance ACS Abstracts, September 1, 1993.
0022-3654/93/2097-10002$04.00/0
contrasts with crystal structure determinationsof pure and doped matrices by X-ray methods: only low percentages of the thermodynamically unstable hcp structure were detected by these bulk methods.20.21 It is the main purpose of this work to correlate the observed electronic matrix shifts and the rotronic splittings of matrixisolated NH/ND in the a l A state with the nature (substitutional; interstitial) and symmetry (oh,&A) of the trapping sites. This was achieved by combining laboratory experimentswith matrix modelling calculations, using ab initio interaction potentials between argon and NH radicals in the X %-, a ‘A, and b 1Z+ states.22 The electronic level splitting of the a lA state in sites of oh and D3h symmetry was derived on the basis of quantummechanical calculations, which yielded the difference between the interaction energies of the two electronic components of NH u ‘A as a function of the orientation in a rigid cage of 12 argon atoms. The quantum mechanical calculations are described in detail in an accompanying paper.23 2. Spectroscopic Observations 2.1. Experiment. The experimental techniquesandthe heliumtransfer cryostat have been described in detail elsewhere.’” NH (ND) was generated by photolyzing matrix-isolated NH3 (ND3) with vacuum UV radiation from a Lyman a source (microwave discharge through 2% Hz in helium, LiF window) or from a microwave discharge in argon seeded with 5% Br2, which emits several strongBr linea in the 145-1 63-nm range through a sapphire window.24 Molecular hydrogen from the photodissociation of NH3 was removed by annealing. The matrix-isolated radicals were excited to the a lA, b ‘Z+, and A states with a pulsed tunable dye laser (Lambda Physik FL3002, pumped with 308nm radiation from an LPX 100 excimer laser). The system was calibrated with Ar lines of a hollow cathode lamp by means of the optogalvanic effect. Emission was dispersed with a Spex 1402 double monochromator with suitable gratings, and detected with photomultipliers (RCA3 1034,Hamamatsu R33 10). Signals were recorded by standard techniques (continuous-wave photocurrent measurements, boxcar gated detection, or photon counting). Owing to the long lifetime of the a IA state (740 ms in 0 1993 American Chemical Society
Argon Cages of
oh and D3h Symmetry
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 10003
, u.
1007
0 - 4 - 2
0
ND 'R(O>
2
4
R e l a t i v e Wavenumber
6
8
10
S c a l e Ccn-'1
-
Figure 1. sR(0) line splitting in broad-band-detected NH/ND a 'A X 'E- fluorescence excitation spectra at 6 K, showing smallness of the isotope effect. The wavenumber scale gives relative displacements from the D-line positions of both isotopomers.
-
argon), the (0,O)band of the a 'A X 32-transition could be simultaneouslyexcited, and its fluorescencedetected in the photoncounting mode, by synchronizing the pulsed laser with a mechanical chopper behind the entrance slit of the monochromator. The c I l l state was accessed from the long-living a 1A reservoir state, which was populated uia the A 311 state with a second homemade tunable dye laser. The bandwidth of the tunable dye laser (typically 0.2 cm-I) was sufficient for most purposes. High-resolution spectra of the b 12+ X 3Z- and a IA X 3 2 - transitions were obtained with an intracavity etalon. 2.2. Isotopic Study of the a 1A X 3 ETransition in Argon. We have previously published high-resolution fluorescence excitation spectra of the a l A X 3 2 - transition of N H in Ne, Ar, Kr, and Xe matrices (cf. Figure 3 in ref 2). The fine structure of the SR(0) line, which can be most clearly resolved in argon, consists of two superimposed SR(0) lines with different fine structures. The lines are attributed to N H radicals trapped in physicallydifferent but energeticallynearly degenerate sites. Both sites are formed in nearly equal proportions over a wide range of experimental conditions. N H radicals in either site can be selectively excited and/or selectively detected in emission, by tuning the laser and/or the monochromator to a site-specific component line (i.e., a line that does not overlap with the component lines of the other site). Here we report on a complementary study of the ND radical. Broad-band-detected (i.e., site-unselective) fluorescence excitation spectra of both isotopomers (SR(0) lines only) are plotted on relatiuewavelength scales in Figure 1. The origins of the relative scales are chosen to coincide with the intense D components of the sR(0) lines (to obtain absolute wavenumbers of N H and ND in uacuo, add 12 627.0 and 12 607.6 cm-l, respectively, to the relative scales). Note that the fine structures of N H and N D are nearly identical, in contrast to the RTC-induced rotational level splitting in the electronicground state, Figure 2 in ref 2, which exhibitsa dramatic isotope effect. Because the rotational parameter B(ND)is nearly a factor of 2 smaller than B(NH), a 'hotn RR,Q(1) line can be readily identified by its temperature dependencence in the corresponding fluorescence excitation spectra of ND (Figure 2). Note that another structureless "hot" line appears at approximately 12 650 cm-l. While this is the expected position of the sR(l) line, an unambiguous assignment is not possible because the intensity is much weaker than expected on the basis of gas-phase HBnlLondon factors.25 A blowup of the better resolved low-temperature spectrum (Figure 3) shows that the fine structure and the
-
+
ND 'R(1)
12580
12600
12620
12640
W Q v e nunb e r
12660
C c n-I1
-
Figure 2. Temperature dependence of the broad-band-detected a *A X Q- fluorescence excitation spectra, showing a 'hot" RR,Q(1) line of ND, as well as a tentatively identified 'hot" sR( 1) line. The sR(0) line of NH is also present, but no corresponding hot lines are observed.
ND/Ar
6K
-
D3h
n
+
12583
12593
12603
12613
Wavenumber [cm-'1
Figure 3. Blowup of Figure 2, showing the sR(0) and RR.Q(1) lines of ND at 6 K in absorption. The 'hot" line, which is rather weak at 6 K,
has been enhanced 10-fold. The partially overlapping spectral featurea cage) and 2 (Dah cage) are marked. associated with trapping sites 1 (4
intensity distribution of the "hotn RR,Q(1) line differ from the sR(0) line. This can be seen more clearly in the site-resolved excitation spectra (Figure 4ab), despite the much poorer signalto-noise ratio. The difference in site 1, (Figure 4a) is attributed to the g u selection rule in oh symmetry, uide infra, and to the spin-rotational splitting of about 0.8 cm-1 between the J = 1 and J = 2 sublevels of N = 1 in the electronic ground state. Our assignments of the line components, also indicated in the spectra, are summarized in a level scheme (Figure 5 ) . The study of the "hot" line structure thus corrects and completes the results of our N H study in ref 2. While all component lines of the site 1, including those of the "hot" RR,Q(1) line, could be unambiguously assigned, some ambiguity remains in the interpretation of the six 'hot line" components of site 2 (Figure 4b). Only five out of six a 'A state component levels in Figure 5 are accurately known
-
Blindauer et al.
10064 The Journal of Physical Chemistry, Vol. 97, No. 39, I993
-6K-
12585
-
12595 12605 Wavenumber [cm-' ]
12615
Figure 4. ND in argon, 6 K unless otherwise indicated: site-selectively detected (I 'A X 32- fluorescence excitation spectra. (a) ND on site 1 (4); (b) ND on site 2 (&); the "hot" RR,Q(l) line was recorded at two temperatures, to show the doublet character of the three noisy lines. Site
Site Oh X=
NH
ND
35.2
16.2
Eg 33.5
NH
T2UX=(33.3)
14.2
32.5
D,,
21 146.5 21147.0 cm-' Figure 6. N H in argon. (a) High-resolution fluorescence excitation spectra (laser line width = 0.04 cm-') showing absorption profiles of the b '2+ X 32-, QP(0) line, for broad-band and site-selectivedetection, as indicated, on the RR,Q(1) line of the (I 'A X Q- transition. (b) Fit of broad-band-detected absorption profile in (a), obtained by superimposing two Lorentzians. The line widths, relative positions, and intensity ratio were treated as fitting parameters.
-
ND (14.1) 13.3
A'; E"
11.6 10.8
E'
8.7
E' A;
+
30.8
alA 27.0
7.6
25.3
5.8
30.0
Tzg
7.9
ZPL
PSB
E, 1 2 6 0 0 t x Em-'
12600+x ,cm-' 29.0 28.2
28.1 27.2
E"
A;+ E" A;+ E' E" A;
16.3 15.6
+
x3c-
A';+ E '
Figure 5. Level scheme for the two trapping sites of N H and ND in argon, rotational levels in the X 32-state, and component levels of the matrix-split a 'A, J = 2 state. Levels which are known with h0.2 cm-' relative accuracy are shown as solid lines. Dashed level positions are approximate.
from theSR(0) line structure. We will show in section 4 that the transition to a sixth component level drawn in Figure 5 is electric dipole forbidden from the ground state, N"= 0. Its position can only be estimated from rather noisy siteresolved spectra of the "hot" RR,Q( 1) line (Figure 4b) or from an extremely weak sixth component of the SR(0) line which can be surmised in the most intense site-resolved excitation spectra of site 2. 2.3. R d u t i o n of Trapping Sites in Other Electronic Transitim. With an intracavity etalon the bandwidth of the dye laser could be sufficiently reduced to determine the line shape of the b IC+ X 3C-, QP(0) line of NH in argon at 5 K. Owing to efficient radiative b )E+ a )Arelaxation," emission from the a * Astate could be either broad-band or narrow-band detected, to yield excitation spectra for the simultaneousexcitation of both sites or for either site separately. The results are shown in Figure 6a. Figure 6b demonstrates that the superposition of two purely Lorentzian lines of 0.10- and 0.1 1-cm-1 width, shifted only 0.076 cm-l relative to each other, yields a perfect fit of the broadband-detected excitation line with a dip in the center. The intensity ratio of the least-squares fitted Lorentzian components is 5 1:49, in excellent agreement with an earlier determination based on site-selective excitation spectra of the a 1A state.2
.-
-
7
I
L-i
29570
29580 29590 29600 Wavenumber [cm-l ]
29610
-
Figure 7. N H in argon. Site-resolved fluorescence excitation spectra in the zero phonon line-region of the A 311 X ?S- (0,O) band. Emission from the a IA state, which is populated by matrix-induced intersystem crossingfollowed by radiationleas/radiative cascadingvia the b l2+state? wasdetectedonsitesptcificmponentlinesof thealA+X%transition.
Because a small percentage of NH matrix-isolated in argon cascades from the A 3 I I to the a )Astate,' it was possible to obtain site-resolvedexcitation spectra of the A 3 I I state (Figure 7). These consist of a structured zero-phonon line (zpl) adjacent to a phonon sideband, only part of which is shown in Figure 7 (note that the sidebandzpl intensity ratio is enhanced by saturation effects in the laser excitation of the electric dipole allowed A 3 I I X 3Ctransition). The sites are practically degenerate (relative matrix shift less than the width of the narrowest absorption feature) but differ by the splittingsand intensity ratios of their zpl components. The splittings, which are obviously the II state analogue of the
-
Argon Cages of
oh and D3h Symmetry
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 loooS
TABLE I: Parametrization of ab Mtio NH-Ar Potential
Surfaces from Ref 22: List of Best-Fit Parameters for JZq 1 paramete? NH(X"-).Ar NH(a 'A).& NH (b lZ+).Ar
n
i
.
21 680
~
~
'
I
~
"
~
I
'
21720 21 760 Wavrnumbrr [cm - 1 ]
'
~
~
I
~
"
~
Figure-%. Two-step excitation of the c I I I state via the a IA reservoir (0,O) state, v,, = 30 314.4 cm-I. Rotationally rcsolved c I l I b band showing effect of temperature on rotational line intensities. As shown by the insert, this is indirect evidence for matrix-inducedsplitting of the degenerate In state (into g, u component levels in oh cage). +
level splitting in the a l A state, amount of 4 (site 1) and 6 cm-' (site 2), in good agreement with an indirect determination by Bondybey.Io Matrix-induced electronic splitting is also present in the c In state, as evidenced by the temperature dependence and spacing of the R(0) and Q(l) lines in the c l I I b I2+transition; cf. spectra and inserted level scheme in Figure 8. Owing to the complexity of the c a excitation spectrum, which consisted of several unassigned component lines, no site dependence has been resolved. Combining the line separation of 38 cm-* in Figure 8 with the previously measured spacing of 30 cm-I between the lowest rotational levels of NH b 'E+in solid argon3yields 8 cm-I for the matrix-induced c In state splitting.
-
I
"
~
~
21 800
-
I
~
~
15 216 853 -3 990 304 2741 169 2 835 715 2.877 03 0.025 19 4 . 2 6 4 74 0.080 83 0.075 43 0.004 14 0.057 47 4 . 0 1 8 31 -121 421 -123 533 346 746 -132 644 -5 123 494 ~ ' I ~ 492 484 18 768 3.5397 0.716 14
20 238 083 -8 533 996 -964 068 9 349 147 3.073 06 4 , 6 9 0 72 4 . 4 6 7 01 0.175 04 0.040 25 0.116 35 0.112 47 4 , 0 4 0 24 -175 805 -151 980 -558 133 -347 717 -4 465 799 ~ ~ ~ l 1490 690 -439 861 2.6686 1.081 41
22 259 360 -861 720 7 126 439 12 878 574 3.182 29 0.500 17 4 . 3 3 4 74 0.281 91 0.048 53 4 . 0 6 4 91 0.054 34 4.055 46 -310 742 69 814 723 406 -12 976 -1 335 643 -3 541 288 -38 124 2.3555 0.508 52
* Note that the fit is heavily weighted for near and intermediate range interaction distances. Therefore the fitting parameters c6('), c6('), etc., may deviate from common long-range dispersion coefficients. dipole moments of NH in the electronic states of interest ( ~ c ( x , = 1.474 D; p ( # )= 1.462 D; p(b) = 1.386 D30) differ so little that the inductivecontributioncausesunrealistically smallmatrix shifts if combined with state-independent atom-pair potentials. Ab initio potential surfaces for one argon atom interacting with NH in the X,a, and b electronic states have recently become available.22 A 21-parameter function of the form31
3. Model Calculations 3.1. Matrix Shifts. It has been shown in previous publications26-2*that electronic matrix shifts and other photophysical properties of matrix-isolated diatomic guests can be calculated under favorable conditions, if electronic state-dependent interaction potentials of the guest with the rare-gas atoms of the host crystal are known. Most importantly, it is possible to discriminate between different sites (single, double substitution) by comparing the model-calculated properties with site-resolved spectra of the guest. The model, which is described in detail elsewhere,26 considers a rare-gas crystal of prescribed structure, usually fcc. A spherical cluster of typically 225 mobile rare-gas atoms is marked out in the center of the rigid crystal. The guest molecule is nested in an interstitial or substitutional site in the center of the cluster. The energy of the system, which is a function of the coordinates i of all mobile atoms including the guest, is defined as a sum over pairwise interactions of each mobile atom with its (fixed or mobile) neighbors, taken over all neighbors contained in an interaction sphere of prescribed size, plus an inductive contribution arising from the guest's dipole and/or higher multipole moments which can be treated self-consistently, if necessary. The relaxed equilibrium structure of the cluster is found by minimizing the energy of the system. While matrix effects in the spectra of NBr could be reasonably well reproduced by the cluster model using electronic stateindependent Lennard Jones-type atom pair potentials V(k.k), V(,), and V(hnr), but electronicstatedependent dipole moments ofthe gues1,26-29 this strategy failed for matrix-isolated NH: the
D(,.)
e-(&/r-W
if r
< Do,
otherwise D(r) = 1
was fitted to the ab initio data: r is the distance from a specified origin on the internuclear axis (see below) to the argon atom; the angle 8 is spanned by the hydrogen atom, the specifiedorigin, and the argon atom; El represents the short-range repulsion; thelongrange interaction E2 is quenchedat short distances by thedamping function D; the anisotropy is modelled by the Legendre polynomials P,(cos e). The origin of the coordinate r was treated as an additional free parameter: variation of a parameter c,,, between 0 and 1 shifts the origin from the hydrogen atom along the internuclear axis to the nitrogen atom. The 2 1 parameters, which fit the ab initio data within the numerical accuracy, are listed in Table I. Summation over all pairpotentials of the doped cluster yields the 'solvation energy" W(R)of the guest. It is normalized by subtracting the reference energy WO of a pure argon cluster from the energy of the doped cluster, to remove the unphysical
‘I
lo006 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
?-
TABLE II: Olwewed dComputed Electronic Matrix Shift8 A- of NH in Argon (cm-l). matrix shift in substitutional site of transition
o hSym
D 3 h SYm
Blindauer et al.
in intentitial site of oh s y m
X
NH(X3Z-)-Ar224
? 2 lib. mode
:
114,7 [ Ilcm]
NH o IA + X 3 c exptl theory
-58.1 -50.7
-58.1 -50.2
xx
q,
+13
C 1 trans. mode
64.5 [Ilcm]
2 trans. mode 60.9 [ Ilcm]
NH b Iz+ X 3 z -91.3 exptl -91.4 theory -7 1.7 -12.3 -23 Since sufficiently accurate od, ode,' are unavailable for matrixisolated NH,the following experimental shifts are listed: Amc(ooX) = ‘R(0)(matrix) -‘R(O) (gas); Amc(b-X) = QP(0) (matrix) -QP(O) (gas). +
dependence of the solvation energy on the size of the cluster: 100
[ Ilcm] Ni
W(& depends on the position vectors inof all atoms which constitute the system under consideration. N ( ~ 2 2 5for a pure argon cluster) is the number of mobile atoms, N, (=135) runs over all atoms inside each interaction sphere, i,,- i,j0 denotes the change of the relative position vector between argon atoms i and j d u e to relaxation, VRG is the potential between two argon atoms interacting over a distance i,, (Lennard-Jones parameters: t = 98cm-l,u= 3.35A39,and P H i s theanisotropicNH._Arpotential given by eq 1in analytical form. The_differencesW(R)’W(&” between the solvation energies W(R)of N H in electronically excited states and in the electronic ground state X ’2- give rise to electronic matrix shifts. A single argon atom in an off-axis position splits the (NH a 1A)-Ar interaction potential into A’ and A” components by lifting the S2 = f2 degeneracy of the linear geometry.22 This splitting, which is largest near the T-shaped configuration,increases rapidly at shorter distances but does not exceed a few cm-l for realistic NH-Ar separations in the matrix. For the purpose of calculating electronic matrix shifts, the splitting of the guest-host interaction potential F H in the a 1A state was neglected, and the average of the a lA‘and a ‘A” components was used in eq 2. The electronic splitting will be reintroduced in section 3.3: Minimizationof thesolvation energies W(R) of substitutionally doped fcc clusters yielded N H X 2-oriented along a Ck axis, while the orientation along a C, axis was energetically preferred in the a l A and b 1Z+states. However, the structure of the relaxed clusters hosting N H on a substitutional site differed very little from the crystal structur: of pure argon. Solvation energies W(R)determined for fixed orientations of the guest do not yield the correct matrix shifts, because the potential surface for rotation of N H in the substitutional cage is very shallow, with barriers 0.6) of Matrix-Isolated NH/ND in fcc Argon high-frequency local modw [cm-'1 electronicstate, site Calcd obsd NH X 3rsubstitutional site 114.7 (2 librational modes, see text) rotation 64.5; 60.9 (2 degenerate) 68.5 (ref 2) NH a l A substitutionalsite 72.5; 69.7; 6 2 2 5 9 . 7 72 (ref 3) interstitial site 168.8 (2 librational); 126.3; not obsd
-
99.6 (2 degenerate)
modes were also well reproduced by the calculations.] There are, however, five normal vibrations at higher frequencies, some of them degenerate, which involve large-amplitude vibrations of N and H around their equilibrium positions. The two librational modes at 114.7 cm-1 are an unphysical artifact of the harmonic approximation since, as already stated, the barrier against free rotation of N H in the cage is in the order of 20 cm-l only. The other normal vibrations at 64.5 and 60.9 cm-l represent translational modes of the radical in the rare-gas cage, one parallel and two perpendicular to the molecular axis. Table I1 shows that the translational mode frequencies are in good agreement with a local mode which has been identified in the a l A X 3Zspectrum of N H in argonSz We have also calculated the normal modes of an argon cluster doped with N H a 1A. Table I11 lists four prominent local modes with M(q)> 0.6. Because the excited radical is aligned along a Cb axis of the matrix, the resulting local modes are nondegenerate. The highest frequency in Table I11 coincides with a prominent feature in the phonon sideband of the local-mode-induced b 1Z+ a 1A transition of matrixisolated NH/ND in solid argon.z3 Note that an earlier calculation of the translational modes of NH in rare-gas solids, which was based on Lennard Jones-type N-Ar and H-Ar pair potentials, yielded much poorer results.* The considerable improvement in the present work is clearly due to the introduction of electronic state-dependent ab initio NH-Ar potentials.22Also listed in Table 111 are local-mode frequencies which were calculated for the hypothetical case of interstitially trapped N H a IA in argon (oh sites). The results are incompatible with the highest observed local mode of 72 cm-1 in the a 'A state, vide supra. This supports our conclusion that interstitially trapped N H is not involved in the observed spectra, which are fully compatible with substitutionally trapped NH. 3.3. Rotatio~land Electronic Splitting of the JI 'A State in ObmdDjbSites. Evidence has been presented in sections 3.1 and 3.2 that the two equally abundant classes of matrix-isolated N H radicals, which are nearly degenerate in their electronictransitions to the lowest singlet states but can be easily distinguished spectroscopicallyby virtue of their different level splittings in the a 1A state? are trapped in single substitutional sites of oh and D3h symmetry. We will now show that the expected matrixinduced level splitting of N H a 'A, 52 = J = 2 in an argon matrix is in very good agreement with this hypothesis. For this purpose the crystal field model, which to our knowledge has only been applied to diatomic rotors in nondegenerate electronic Z states, must be modified to include the degenerate a 'A state of NH. The rotational eigenfunctionsof a diatomic molecule in a state with electronic angular momentum quantum number 52 and total angular momentum quantum number J is given by34,35
-
-
The orientation of the molecular axis with respect to a coordinate system which is attached to the center of mass of the molecule and parallel to a space-fixed coordinate system' is specified as
usual by angles 8 and 4. [The following conventions are used: the coordinate system in oh symmetry is spanned by the C, axes of the cage, while in D3h symmetry the z, x, and y axes are directed along the C3 axis, a Cz axis, and a Cl axis (cf. Figures 6 and 7 of the accompanying paperz3).] The d&,are the Wigner functions, and the &, are the reduced rotational matrix elements.36 For a 'A, J = 52 = 2 state, which is of interest here, the electronic wave function may be written as follows:
x is the total azimuthal angle of the electrons, r denotes the remaining 3n - 1 electron coordinates, and E1 is a singlet spin function. A 'A, J = 2 state gives rise to 10 wave functions:
where only x is considered explicitly as electronic coordinate. In the gas phase they are degenerate to a very good appr~ximation.~~ However, when N H a l A is trapped in an argon matrix, the interaction with the matrix imposes an external perturbation, which causes a partial lifting of the degeneracy. If the simplifying assumption is made that the center of mass rests in the center of the cage, the perturbation operator H', which describes the interaction of the electrons and nuclei of N H with those of the matrix, is a function of only three coordinates: 8 and (6, which describe the orientation of the molecular axis, and the azimuthal angle x of the electrons. The effect of k'on the rotational levels may then be calculated in degenerate first order perturbation theory by diagonalizing a 10 X 10 matrix with the following elements:
Because only rare-gas cages of oh and Dph symmetry are considered, it is possible to block-diagonalize the perturbation matrix by introducing symmetry-adapted eigenfunctions,38 which are suitable linear combinations of the eigenfunctions In oh symmetry the @& span the irreducible representations E, @ E,, @ Tz, @ Tzu,in D3h the irreducible representations are AI' e AI" 2E' @ 2E". One therefore expects that the N H a *A, 52 = J = 2 state in oh symmetry splits into two levels which are 2-fold degenerate and two levels which are 3-fold degenerate, while a perturbing cage of D3h symmetry should give rise to two nondegenerate and four 2-fold degenerate levels. The 10symmetry-adaptedfunctionsfor the consideredtrapping sites of oh and D3h symmetry may be uniformly written in the following form:
$;ziz.
~ ( x , M J=)N k ( A k v , 4 ) u / & ) sin 2~
+
Bk(44)( 1
cos 2x) (8a)
k = 1-5; Ak, B k = real functions of 8, 4. In Oh symmetry the first-order perturbation energies can be computed directly, because in the decomposition each irreducible representation is obtained only once:
Because in D3h symmetry the irreducible representations E' and E'' appear twice, the diagonalization of small 4 X 4 matrices cannot be avoided. To evaluate the first-order perturbation
Blindauer et al.
loo08 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
theor matrix shifts of NH a l A in crystal fields of
WM(0,4)= ( l / r ) c d x (sin 2x)fir(x,0,4) sin 2x
(1 1)
oh SYm
-47.3 E, (2) -47.6 Tzu(3)
They may be constructed from two contributions: (a) The "averaged" potential Wav(O,Q),which is summed over all pairwise interactions between NH a 'A and the argon atoms of the host crystal; the A' and A'r components of the NH.Ar pair potential are averaged, as outlined in section 3.1. (b) A "difference" potential AW(O,Q), which is the analogue of the splitting of the interaction energy for the NH(a lA)-Ar complex into Ar and A" components. AW(O,Q) was obtained for orientations compatible with C, or higher symmetries by quantum chemical methods as the difference between approximate interaction energies of the a 'A" and a 'A' components with the surrounding matrix (see below). The contribution W, is zero except for orientations of lower than C, symmetry and is expected to make only a negligible contributionto the matrix elementswhich are needed to calculate the first order perturbed energiesE:')* (cf. eq 9). Therefore W, and W, may be approximated as follows:
This can be justified because the a 'A' and a lArr components behave approximately like cos(2x) and sin(2;y), respectively. Wav(O,g) alone may be regarded as giving rise to purely rotational splitting, similar to the splitting of a diatomic rotor with J = 2 in a nondegenerate electronic state, while the +l/*AW(O,t$) and -1/2AW(6,Q) terms cause an additional electronic splitting. Wav(0,4) can be evaluated in either of the following approximations: (a) The soft cluster approximation: NH is assumed to rotate in a cluster of 224 mobile argon atoms. While the orientation 8, Q of the molecular axis changes, the cluster relaxes continuously with respect to all particle coordinates and with respect to the center of mass coordinates of the guest. (b) The hard cluster approximation: N H is assumed to rotate around its fixed center of mass, which coincides with the center of a rigid cage of argon atoms held at their ideal crystal positions. Approximation (a) is attractive from a classical point of view, because it couples the classical degrees of freedom of the system. Quantum mechanically, however, such strong coupling is unlikely in the absenceof accidentalres0nances.~3In view of the fact that the quantum mechanical evaluation of AWhad to be carried out for a hard cluster,~,Wav(e,t$) was also evaluated in approximation (b). Owing to the small size of the N H radical the potential surfaces Wav(0,4) differ very little in both approximations, the main difference being a shift of the surface in approximation (b) to higher energies by less than 10 cm-I. Since similar shifts were obtained for the potential surfaces Wav(0,4)of the X 3 2 - and a 'A states, the computed electronic matrix shifts are not very sensitive to the approximation used. The ab initio determination of the electronic energy splitting AW(O,t$) is the subject matter of an accompanyingpaperS23Briefly, AW(O,+) was calculated for the electronically excited radical in the center of an Ar12 superm molecule^. The argon atoms were kept frozen at their ideal lattice positions in fcc or hcp crystals, forming cages of oh or D3h symmetry. AW(8,4) was calculated for 11 orientations of N H a lA in a cage of Oh symmetry, and for 19 orientations in a cage of D3h symmetry. By transferring these sets of computed Wav(Or,Q,) and AW(Of,Q,) to all symmet-
-53.0 Tu (3) -55.3 & (2)
036 S Y m
-47.3 AI'' (1) -48.0 E" (2) -49.9 E' (2) -50.4 E" (2) -52.3 AI' (1) -52.7 E' (2)
obsd sublevels of NH a IA in different trapping sites
site 1 12 635.2 A 12 633.5 A*
12 627.0 D 12 625.3 D*
site 2 =12 633.3 A* 12 632.5 B 12 630.8 B* 12 630.0 C 12 628.1 C* 12 627.2 D
weighted averagesb 12 630.3
12 630.3
Comparison of theoretical matrix shifts with observed term values. Thedegcneracica of calculated sublevelsare given in parentheses, following the symmetry classification of the level. Observed term values are identified by upper case letters following the nomenclature introduced in ref. 2. The statistical weights of the experimental levels in &h are based on the assignment shown in Figure 5. a
rically equivalent orientations, a dense network of orientations was generated. Complete surfaces Wav(O,Q)and AW(O,#) were obtained by splining. The matrix-perturbedsublevelsof thea 'Asfatein ohsymmetry were finally obtained by integrating W,(O,Q) and W,(O,g) over O and 4, using the Ak(O,4) and Ek(O,4) of the symmetry-adapted eigenfunctions, and in D3h symmetry by diagonalizingthe Er and E" blocks of the perturbation matrix. The results are listed in Table IV. They arereported as matrix shiftsLwhichwere obtained by subtracting the solvation energies ( W(R)) of the electronic ground state, as outlined in section 3.1. The matrix shifts of -50.7 cm-l for NH a IA in sites of oh symmetry, and -50.2 cm-l in sites of Djh symmetry,which are listed in Table 11, are averages of the sublevels in Table IV, weighted with the degeneracies of the component levels. 4. Discussion
As outlined in the Introduction, it is the main purpose of the present work to correlate the observed electronic matrix shifts and splittings of matrix-isolated NH/ND in the a 1A state with the nature (substitutional versus interstitial) and symmetry (oh, D3h) of the trapping sites. We have proven that NH resides in two trapping sites of nearly 1:1 abundance,which are energeticallydegenerate(relativematrix shifts 0.076 cm-I and
