On the interpretation of the NEXAFS spectrum of molecular oxygen

30 August 1996

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 259 (1996) 21-27

On the interpretation of the NEXAFS spectrum of molecular oxygen Hans/~gren a, Li Yang a, Vincenzo Carravetta b, Lars G.M. Pettersson c a Institute of Physics and Measurement Technology, LinkOping University, S-58183 Link6ping, Sweden b Istituto di Chimica Quantistica ed Energetica Molecolare del C.N.R., Via Risorgimento 35, 56100 Pisa, Italy c FYSIKUM, University of Stockholm, Box 6730, S-113 85 Stockholm, Sweden

Received 10 May 1996; in final form 7 June 1996

Abstract

Calculations with extended basis sets and full intrachannel coupling of discrete and continuum states suggest an altemative interpretation of the NEXAFS spectrum of molecular oxygen. In this interpretation the roles of valence-Rydberg mixing and intrachannel discrete-continuum interaction are stressed at the expense of the exchange splitting mechanism. Calculations indicate that only the doublet-parent channel holds discrete strong transitions with o-* character, while the quartet-parent channel contains preferentially Rydberg-like transitions.

1. Introduction Like its infrared and UV spectra, the X-ray absorption spectrum of molecular oxygen contains special features not encountered in common molecules. The particular electronic structure of molecular oxygen implies a significant distortion of the common four-fold pattern of NEXAFS spectra; a strong resonance plus a weak Rydberg series in the discrete part and multielectron excitations superimposed on a shape resonance structure in the near-continuum part. In the 02 spectrum it is only the first feature, the strong discrete resonance, that is retained, while the other three features apparently are replaced by two bands of complex discrete structures just below the first core ionization threshold. A common interpretation is that, due to the comparatively long intramolecular bond, the "shape resonant" or* excitation is pulled below the ionization threshold and appears there, together with the Rydberg states, in two paramagnetically (exchange)

split series of transitions [ 1 ]. Several factors have been proposed that complicate this picture, like the coexistence of dissociative and bound potentials with avoided crossings [2,3], special configuration interaction and Rydberg-valence mixing [ 3 ], and no consensus has been reached whether the exchange-induced splitting of the o'* resonance really contributes to both groups of transitions or only to the first of them. Besides, despite the effort to characterize the involved states, little attention has been devoted to the actual intensity distribution. In the present work we take advantage of the recently developed direct atomic orbital static exchange technique (STEX) [4] and reinvestigate the 02 NEXAFS spectrum. By means of this technique we can reach two limits important for the interpretation of NEXAFS spectra, namely the basis set limit and the full intra-channel coupling limit including the continuum. These two limits allow a close-to-correct "independent particle" interpretation of the spectrum. We

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H. ~gren et al./Chemical Physics Letters 259 (1996) 21-27

22

complement the calculations by atomic population-, quantum defect- and so-called r 2 analysis in order to further characterize the states.

core ionized s t a t e ( 4 ~ - , Sz = 1/2) = (c#x~y~, + c , x ~ y ~ + c~x~,y~) l / x / 3 ,

(4)

core ionized state ( 2 ~ - , Sz = 1/2) = (2ct~x,~y,~ - c,~xt3y,~ - c,~x,~yt~) 1/v/-6,

(5)

2. Method and calculations

In the static exchange, or improved virtual orbital, approach the photoabsorption spectrum is obtained from single excited states where the virtual orbitals are eigenvectors of a one-particle Hamiltonian that describes the motion of the excited electron in the field of the remaining molecular ion. The electronic relaxation is accomplished by a ASCF procedure and is assumed independent of the excited electron, with the excitation levels converging to a common ionization limit. In molecular oxygen there are two such limits to consider, represented by the quartet ( 4 ~ - ) and doublet ( 2 ~ - ) states of the core hole molecular ion. The static exchange Hamiltonian corresponding to ionization of the core (c) shell of 02 is expressed as 7-[ = F -

Z (cjjJj 4-CKjKj ) , j=c.x,y

(1)

where spin-restricted orbitals are assumed, F is the standard Fock operator for double occupancy of the orbitals and J and K are the usual Coulomb and exchange operators. Cg and cx are coefficients depending from the spin-coupling among the open shell orbitals (including the core c, valence x, y and excited orbitals). Each ordered spin-coupling sequence defines a configuration state function (CSF) with a specific energy and a specific static-exchange Hamiltonian, see Table 1. For photoabsorption the CSFs are also defined by imposing a dipole selection rule and, in a one-particle picture, by that the photo-excitation does not change the internal spin-coupling. The openshell static exchange Hamiltonians are then obtained by spin-couplings in the following order; the openshell valence orbitals - the core orbital - the excited orbital. For photoabsorption in molecular oxygen the following CSFs need to be considered:

core excited state ( Sz = 1) = (3v~c,~x,~y,~ - v,~c~x,~y,~ - v,~c,~x~y,~ (6)

-Vc, C,~x,~y~)I/v/12,

core excited state (Sz = 1 ) = (2v~cl~x~y,~ - v ~ c ~ x ~ y ~ - v,~c~x~y~ ) 1 I x ~ 6 , (7) where x and y are the ~gx and 7rgy orbitals, c the core orbital, v the excited orbital and the core excited states have 3y~- or 3I] symmetry, depending on the symmetry of the excited orbital. The CSFs in Eqs. (6) and (7) correspond to excitations with the parent ion in the quartet and in the doublet state, respectively; they define two excitation-ionization channels, here called the quartet and the doublet channels. States belonging to different channels do interact, but in this particular case it has been shown by Kosugi and co-workers [5] that this interaction is negligible both for valencelike and Rydberg excited states. We have further tested the independent particle approximation by performing multi-configuration SCF calculations for the cr* transition (with a paramerization tested in Ref. [6] ), and obtained comparable oscillator strengths; 0.105 (MCSCF) and 0.102 (STEX). In the AO basis the STEX Hamiltonian matrix is constructed directly as "~t'~ab = hab 4- Z [ Z ( a b l c d ) D ~ c d

- (aclbd)D~]

(8)

cd by modifying the density D corresponding to double occupancy according to

DcdJ = Dcd -- ~ z_.., z j~,x,y Dr

=

Dcd 4- Z

v~ v~j

'

cx.iVcjVdj '

(9) (10)

j=c,x,y

ground state ( 3 E - , Sz = 1) = x,~y,~ ,

(2)

core ionized state (4E-, Sz = 3/2) = c,~x,~y,~ ,

(3)

where Vcj is component c in MO vector j, D is the AO density matrix, and cg~ and CKj are the same coefficients as in Eq. ( 1 ) and in Table I, and can easily

H. ,~gren et al./Chemical Physics Letters 259 (1996) 21-27

23

Table 1 Coefficients of the STEX Hamiltonian for the two core ionization channels of 02 Coupling coefficient

Channel doublet

core orbital ~gx orbital ,'rrgy orbital

quartet

CJ

CK

CJ

CK

1.0 1.0 1.0

-2/3 - I/6 - 1/6

1.0 1.0 1.0

-4/3 -4/3 -4/3

be derived from expectation values of the Hamiltonian over the open-shell CSFs in Eqs. (6) and (7). The construction of 7-[ is determined directly from oneand two-electron integrals computed in the atomic orbital basis and obtained as a last step in the iterative calculation of the Fock matrix in AO basis. The bound molecular orbitals from SCF and ASCF optimizations are expanded in a relatively small basis set, while the construction of the STEX matrix is obtained in an augmented, large but non-redundant basis set. The occupied orbitals are then projected out from the basis set, and the projected STEX Hamiltonian is diagonalized completely to yield eigenvalues and eigenvectors. Intensities are obtained from dipole integrals between initial (Eq. (2)) and final (Eqs. (6) and (7)) CSFs expanded on non-orthogonal MO sets. The eigenpairs of excitation energies, obtained by adding the ionization potential to the eigenvalues of the STEX matrix, and the corresponding oscillator strengths provide a primitive spectrum for a Stieltjes imaging procedure in the continuum [7]. We have performed a series of basis set investigations and present results obtained with a basis of TZPD (triple zeta plus polarizing and diffuse) type with (5s, 4p, 3d) functions [8] for the bound orbitals, This basis set has been augmented in the calculation of the STEX Hamiltonian, with a large diffuse basis with (20s, 18p, 20d) functions which we consider being close to the basis set limit. All calculations are performed at the equilibrium geometry (R = 2.281 au) in the broken C2v symmetry enforcing core orbital localization [9], using the DISCO program modified for STEX calculations [4].

3. Results and discussion The independent channel spectra for core excitations corresponding to a parent-ion in a quartet or in a doublet state and their summed spectrum are shown in Fig. 1 and in Fig. 2, respectively. Note the different energy range for the total spectrum in Fig. 2 that includes also the strong ls-zr* excitation. The spectra are generated with a common gaussian broadening function of FWHM = 0.2 eV, without further attempt to simulate vibrational (or dissociative) broadenings. The different characters of the two spectra are remarkable; most intensity is gathered in the doublet spectrum and appearing there as states of o- symmetry, while the weaker quartet spectrum shows two progressions of 7r and o- symmetry. Another important difference is the ratio between discrete and continuum intensities. This ratio is substantially larger in the doublet than in the quartet spectrum. Opposite to the discrete parts the intensity between the quartet and doublet channels considerably exceeds the statistical value of 2 in the near continuum. This deviation close to the ionization threshold thus makes up for the anomalous discrete intensity distribution between the quartet and doublet spectra. In the far continuum (actually already at 40 eV above the ionization threshold) the expected statistical ratio is retained. We thus interpret that much of the salient differences between the doublet and quartet NEXAFS spectraofO2can be referred to the differences in the staticexchange potential of the two channels. These differences are due to the different spin couplings of the core and valence 7rg orbitals which are reflected by the c x coefficients in Table 1. From this point of view we can say that the different intensity distributions of the two spectra are induced by an exchange effect due to

24

11. ,~gren et al./Chemical Physics Letters 259 (1996) 21-27

02 NEXAFS m

IP=542.1 eV FWHM=O.2 eV A 2.0 t~

1.6

quartet

"5 1.2

0 0.4 0.0

,

IP=543.2 eV F-WHM=O.2 eV 2.0 vc - 1.6

doublet

.0

(~

~ 0.8 o 0.4 0.0 538

540

5~,2 Photon energy (eV)

5,~4

Fig. 1. STEX independent channel spectra for core excitation of 02; the two channels correspond to a parent-ion in a quartet or in a doublet state.

02 NEXAFS IP=542.1 eV FWHM=O.2 eV 2.0

max=18

1.o

0.5

0

F/ 529.0

532.0

535.0 538.0 Photon energy (eV)

,;' !~ ~1O I~ I,/,/' li :'

541.0

Fig. 2. Total O-Is NEXAFS spectrum of 02 by STEX.

544.0

H. ]tgren et al./Chemical Physics Letters 259 (1996) 21-27

the particular open-shell structure of 02. In the quartet spectrum a large part the o-* resonance intensity still resides in the continuum, as in a "normal" spectrum, while in the doublet spectrum the lowering of the o-* resonance into the discrete part is more complete. The irregular and strong discrete o- type peaks in the latter spectrum are consistent with this interpretation. In the quartet system we have both o- and 7r transitions with more intensity for the latter as expected in Rydberg like progressions, In our calculation the dominant o- peaks thus derive from the strong doublet spectrum. Considering population and r 2 values (see Table 2) the first two sigma peaks at 539.8 and 540.2 eV look like a split Is-o-* with O-2p contributions. The third peak at 541.5 eV corresponds also to an antibonding O'u level, but with 3s character close to the oxygens and Rydberg p character in the distantregion. This strong transition, which also reflects a shift of intensity from the continuum, can be considered as a mixed tr*-Rydberg state. Even the 4th o- peak at 542.5 eV has some reminiscent o-* character. Being above the ionization threshold of the quartet channel it is however considered to be broadened by the interaction with the continuum in the total spectrum. Our results in Table 2 and in Fig. 1 show that indeed those states with most valence character are gathered at lower energies in the spectra. Their r 2 numbers, although being the lowest for the excited states in Table 2, are still quite large. The o-* character has apparently been distributed over the spectrum, as also indicated by the dominance of transitions with o- symmetry type and by the irregularity of the energy and intensity distributions. Furthermore, there is no sign of a statistical intensity relation between the discrete states of the two channels as expected for pure Rydberg tran.sitions; this is consistent with the different discretecontinuum intrachannel interaction discussed above, The weaker w-series seems to have a more pure Rydberg character. Considering the quantum defect and the r 2 numbers, there is an increase in Rydberg character as one proceeds over the spectrum in energy. In particular, there is more Rydberg character in the second group of states than in the first one. This fits with observations in two recent experiments, by resonant Auger spectra, which show more non-resonant character for excitation energies closer to threshold [ 10], and by NEXAFS spectra of 02 physisorbed on differ-

25

ent surfaces in which the second group of transitions is depleted by the surface in a way that is consistent with a "volume" effect. Also the solid 02 NEXAFS spectrum seems to have low intensity in the appropriate energy region [ 2]. Kosugi and co-workers [5,3] performed calculations of the 02 NEXAFS spectra with techniques that bear resemblance to the presently adopted method. However, in both their papers they separate the treatment of valence o-* states and of Rydberg states, the latter are obtained by a procedure in which they are orthogonalized to the o'* orbital. In the present work we allow all states belonging to a given channel to mix, by constructing and diagonalizing the channel Hamiltonians without any constraint, except for orthogonalization to the occupied orbitals. The static exchange approach uses a frozen core potential that is usually optimized for the core ionized state. Using such a potential a good description of Rydberg and continuum orbitals is, in general, obtained, while for the lowest excited valence-like levels in "normal" molecules, (i.e. to a 7r* or a o'* orbital which penetrates into the core) the excitation energy is shifted upwards typically in the order of one eV [4,6]. As shown in a recent work [ 6] the oscillator strengths given by the static exchange approach still simulate the full ASCF values quite well even for pure valence like levels, typically within 10%. The effect of using a potential optimized for the bare core hole state is less clearcut when there is strong valence-Rydberg mixing which distributes the valence character into several states. Kosugi and co-workers [ 5,3 ] optimized, as pointed out above, the "o-*" state of 02 independently, and used orbitals from this state for the potentials. They obtain the o-* state at lower energies than here predicted; thus in the doublet channel it is placed lower than the Rydberg levels (we get strong Rydberg mixing), and in the quartet channel it was placed among the Rydberg levels (we get most of the o-* character in the continuum). In order to further elucidate the difference of these two approaches, we tested static exchange Hamiltonians for the "polarized" ion that were constructed from orbitals optimized for the Is-o-* or the ls-cr* states and which then were diagonalized for the full spectra. We obtain (in both cases) a doublet spectrum with a strong low-lying state at 537 eV followed by a weak Rydberg progression and a quartet spectrum with considerable o-* intensity in the

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H. ~gren et al./Chemical Physics Letters 259 (1996) 21-27

Table 2 O-Is NEXAFS spectrum of 02 by STEX No. of

Energy

Oscillator strengths ( × 10-2)

Population

level

(eV)

length

velocity

char.

0.2602 0.0904 0.0953 0.4030 0.0000 0.0000 0.0175 0.0519 0.2364 0.0145 0.0000 0.0242

0.3269 0.1061 0.0708 0.4327 0.0000 0.0000 0.0114 0.0392 0.2270 0.0242 0.0000 0.0157

po" po"1 p~" ( s+p ) o" (d+p)tr d6 (d+p) 7r ( p+d ) ~" ( p+d ) tr ( p+d ) tr dt~ ( d+p ) ~"

1.99 2.14 2.26 2.79 3.05 3.06 3.08 3.49 4.22 5.61 5.76 6.00

32.56 31.41 47.48 45.08 81.98 70.93 61.91 78.21 55.84 63.99 68.45 60.65

0.0623 0.0071 0.1524 0.0217 0.0000 0.0329 0.0623 0.0887 0.0687 0.0000 0.0424

0.0908 0.0162 0.1089 0.0124 0.0000 0.0212 0.0886 0.0644 0.0501 0.0000 0.0270

po" po"1 plr (d+p)o" dB (d+p)~ (p+s)7"r ( p+d ) ~r (p+d)odB (d+p) 7r

2.10 2.29 2.30 3.05 3.07 3.09 3.43 3.56 5.31 5.82 6.19

32.38 43.12 50.54 82.13 71.07 62.17 63.88 76.41 63.66 68.70 60.65

doublet spectrum b 1 539.77 2 540.24 3 540.55 4 541.46 5 541.74 6 541.76 7 541.77 8 542.09 9 542.45 10 542.78 11 542.80 12 542.83 quartet spectrum b 1 539.02 2 539.52 3 539.53 4 540.65 5 540.66 6 540.68 7 540.96 8 541.04 9 541.63 10

541.71

11

541.76

~a

r2 (au)

a • is the effective principal quantum number. b The doublet ionization threshold (543.2 eV) has been obtained by adding the experimental value of the exchange splitting (1.1 eV) 115] to the ionization threshold for the quartet channel (542.1 eV) obtained by a ASCF calculation. discrete 538-541 eV region. In this approach we thus obtain results more in line with those of Kosugi and co-workers [5,3], but still different since we include the valence state(s) in the diagonalization as before. We believe that the original approach using the potential from core hole optimization is more physical since if the valence character is distributed into several states there is no single state with strong core screening. We can also motivate the choice by experience from other molecules [ 6 ] and by the fact that it seemingly gives better experimental agreement also for 02. Nevertheless, the considerations described above and the differences in results, reflect some o f the difficulties of interpreting the N E X A F S spectrum o f 02. The experimental investigators seem to converge on an interpretation where the exchange induced split

peaks reside in the first band and that this splitting is small, below 1 eV [ 2 , 1 0 - 1 2 ] . Estimates coming from calculations only, have been quite scattered [ 13,5,13,14], with both small and large values o f the splitting. The present calculations might shed light on this controversy as we obtain several peaks with o'* character in the doublet spectrum. The first strong peak at 539.8 eV would then be consistent with the small exchange split (0.7 eV) o f the o-* excited state, while the second strong peak at 541.5 eV would be consistent with the large split (2.6 eV). If one on the basis o f the present calculations is to choose side in this controversy, it would be for the small exchange splitting. However, because of the strong valence-Rydberg mixing and because of the distribution o f intensity between discrete and continuum parts o f the spectra

H. ]tgren et al./Chemical Physics Letters 259 (1996) 21-27 with strong intensity only in one of the two channels, the most consistent conclusion is that the notion of an exchange induced splitting of the o-* excitation is inadequate for the analysis of the NEXAFS spectrum of molecular oxygen. We can rather safely make this statement, because the notion of an exchange induced splitting is an independent particle (independentchannel) notion, and because we believe to have obtained the NEXAFS spectrum of 02 in the true independent particle limit. The exchange interaction does still remain as a relevant, although indirect, factor for the interpretation of the spectrum, because the anomalous intensity distributions in the doublet and quartet spectra origin in the difference in their respective potentials, which in turn refers to an exchange induced effect.

Acknowledgement This work was supported by a contract from the Italian and Swedish Science Research Councils, CNR and NFR.

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