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K-shell photoionization in molecular nitrogen
CHEMICAL
Volume 51, number 1
K-SHELL PHOTOIONIZATION
PHYSICS
IN MOLECULAR
1 October
LETTERS
1977
NITROGEN
T.N. RESCIGNO Theoretical Atomic and Molecular Physics Group, Livermore. chlifomiiz 94550. USA
University
of Chlifornia. Lawrence
Livermore
Laboratory.
and P.W. LANGHOFF Department
of Chemistry,
Indiana (Iniversity,
Bloomington,
Indiana 47401,
USA
Received 27 June 1977
Stieltjes-Tchebycheff calculations in the static-exchange approximation cross section in molecular nitrogen.
1. Introduction Photoabsorption [l ,2] and electron impactexcitation [3--61 studies of the K-shell photoionization cross section in molecular nitrogen reveal the presence of a weak Rydberg series, and resonance-like structures at energies above and below the 1s ionization potential [7]. The origin of the series has been clarified in terms of an isomorphism between valence excitations in NO and K-shell excitations in N2 [ 11, and that of the resonance features by recent XLYcalculations [8,9], which attribute the high- and lowenergy structures to f- and d-wave resonances, respectiveiy, in the final-state orbitals. Although the X~Ycalculations are in qualitative accord with the measured cross section, the highenergy resonance structure is apparently quite sensitive to the potential employed, and there are quantitative discrepancies between the theoretical [8, V] and experimental results [6]. A similar observation is made in connection with recent theoretical studies of valenceshell spectra in molecular nitrogen [lo, 111, suggesting that nonlocal Fock potentials are perhaps required for quantitatively reliable predictions of shape resonances in molecular photoionization cross sections. Moreover, although the description of the two prominent nitrogen Kedge structures in terms off- and d-wave resonances is helpful, further clarification along the lines of
are reported of the K-shell photoionization
Mulliken’s early analysis of intense transitions into n* and u* valence-like orbitals [12] is also desirable. An alternative theoretical investigation of the K-edge photoionization cross section in molecular nitrogen based on orbitals of good symmetry type and the nonlocal Fock potential would clearly be of considerable interest. In the present letter, Stieltjes-Tchebycheff calculations [ 13,141 in the static-exchange approximation are reported of the K-shell photoionization cross section in molecular nitrogen, complementing previously reported related studies of the valence-shell cross sections [I 11. Conventional Roothaan-Hartree-Fock . calculations and normalizable gaussian basis sets are employed in the construction of occupied and improved virtual molecular orbitals of good symmetry type. A satisfactory description is provided of the observed weak Rydberg series [l] , and the StieltjesTchebycheff profile obtained from an appropriate moment analysis [ 15,161 of the calculated pseudo-spectrum is in good quantitative accord with the recent electron impactexcitation measurements of the photoionization cross section [6], particularly in the vicinity of the high-energy shape resonance_ The origins of the low- and highenergy structures are discussed in terms of excitations into IT* and u* valence-like orbitals, respectively, and their partial Rydbergization 65
Volume 5 1, number 1
CHEMICAL PHYSICS LETTERS
. 112, 17-191 into 3dn and 4fu atomic orbit&, iespectively, thereby providing further clarification of the d- and f-wave natures of the resonances. The small discrepancies remaining between theory and experiment are attributed to core reiaxation, two-electron excitation, and final-state lineshape effects, which are treated in a subsequent more detailed report. The theoretical development is given in section 2, computational results are described and discussed in section 3, and concluding remarks are made in section 4.
2. Theoretical
development
Nonlocal potentiaIs of appropriate symmetry for the study of 10~ + ko,, knu and 10~ + ko kng excitations in molecular nitrogen are obtainid from conventional ground-state Hartree-Fock calculations employing gaussian basis sets [20] in the forms
1 October 1977
where = T + Vn + V$N-i) (4) HY is the one-electron hamiltonian for the 7th channel, are employed in conjunction with the StieltjesTchebycheff technique 113-161 to circumvent the difficulties that arise in constructing continuum solutions of eq. (2) In this way, photoionization cross sections corresponding to the staticexchange approximation are obtained without explicit construction of the scattering functions @_ The necessary physical information is contained in the pseudo-spectrum of transition frequencies and oscillator strengths {Gr, xr, i = 1, M)
where 6, and Ed are the lug or 1 uu orbital and energy, respectively, and P is the dipole operator. These provide approximate invariant spectral moments r251
v(N-l) =]. r
(21j - Kj) + Jr + Kr ,
(1)
O-+-Q where Jj and Kj are the customary Coulomb and exchange operators, respectively. The so-called improvedvirtuai-orbital potential of eq. (1) is appropriate for describing the motion of an electron excited from the @h occupied orbital in the frozen (Coulomb and exchange) field provided by the remaining N-l electrons [21-231. Consequently, scattering solutions of the one-electron problem, (T+
v,+
~.$v-1’-&5;=0,
(2)
where T and V, are the kinetic ener,T operator and nuclear-frame potential, respectively, corr*spond to the so-called static-exchange approximation 1243. The nonlocal and anisotropic nature of the potential ~~~-1) has apparently discouraged determinations of the continuum orbitals @z appearing in eq. (2) employing conventional techniques, motivating the introduction of approximate local-exchange potentials and partial-wave expansions IS-1 01. h the present studies, variationally determined pseudo-spectra CT?, y, i = 1, M) of good symmetry type satisfying
(3)
2.. (--k) = ig
(zT)-k
xy
, k = 0,1, . . . .
(6)
from which the corresponding oscillator-strength profile is obtained following the previousIy described Stieltjes-Tchebycheff technique (13-l 61.
3. Computational
results
A FJartree-Fock function of good quality is constructed near the equilibrium internuclear separation [26] and used in forming the appropriate V:Nw" potentials for the excitation of loa and 1 u,, electrons in molecular nitrogen. Previously described basis sets of gaussian orbitafs of appropriate symmetry [l l] _ are employed in the construction of M w 25-term pseudo-spectra [eqs. (3) to (6)J in each case. In table 1 are shown the first five terms in the calculated pseudo-spectra for the four dipole allowed symmetries. These discrete-like transitions in the tail of the vaIence-shell spectra are made to converge on the correct nitrogen 1s ionization potential (409.9 eV) by empIoying the experimental value [7] for er in eq. (5). Evidently, the spectrum is dominated by an &tense la, + lrrg transition at x 397 eV having an
-
Volume 5 1, number 1
1 October 1977
CHEMICAL PHYSICS LETTERS
Table 1 Discrete transition frequencies (ev) and oscillator strengths near the K-edge in molecular nitrogen (ZT/Fy) lug + nsu
lug” “Uu
lou+
407.1/1.46(-02) 408.6/4.06(-03) 409.2/1.73(-03) 409.5/9.08(-04) 409.6/5.95 (-04)
407.4/l-37(-04) 408.7/1.80(-05) 409.214.39 (-06) 409.5/l .57 (-06) 409.6/8.50(-07)
406.2/2.32(-03) 408.3/1.16(-04) 409.0/1.28(-0.5) 409.4/1.27(-04) 409.5/2.36(-04)
nog
a)
luu -, n7rg 396.9/2.57(-01) 408.5/1.42(-03) 409.1/7.20(-04) 409.4/3.94(-04) 409.6/2.35 (-04)
a) Values corresponding to eqs. (5) obtained from improved-virtual-orbital calculations as discussed in text. Numbers in parentheses give appropriate powers of ten.
f number = 0.26. High intensity
transitions into valence-Iike ‘II* orbitals are discussed by Mulliken [12], who notes that the participation of compact 2p atomic orbitals can give rise to large transition moments in these cases. Indeed, the 1cru + lrrg resonance is an extreme example of an n -+ rr* transition [27], and similarly intense In, --f lrrg (rr --f rr*) and 20, + lrrg (n + rr*) transitions contribute to the valence excitation spectrum in molecular nitrogen [ 111. In contrast to the intense lo, + lng transition, the other members of the I uu -+ nrrg series, and of the other three dipole spectra, are wek and correspond to excitations into more diffuse Rydberg-like molecular orbitals comprised of diffuse atomic orbitals. The calculated discrete spectrum is compared with experimental values and the spectral assignments of Nakamura et al. [I] in table 2. The latter assignments are made on basis of an approximate isomorphism between the energy ordering of unoccupied orbitals
in NO with the expected ordering of unoccupied orbitals in (Is-‘) Nz [I]. That is, since one of the Kshell electrons in N, is missing in the potential of eq. (l), the screening of the nitrogen nuclei is incomplete, resulting in a potential similar to that found in NO. The first six calculated excitations in table 2 are evidently in one-to-one correspondence with the spectral assignments made on this basis. Except for the first intense la, + lng transition, the calculated excitation energies are within = 0.5 eV of the measured values, which latter are in excellent quantitative accord with the observed valence-shell excitations in NO [28]. Moreover, the measured intensity profile [1] is in general accord with the calculated f numbers. The discrepancies between the calculated transition frequency and osciliator strength for the intense lo, + lrrg low-energy resonance and the recent electron impact-excitation measurements [6] can presumably be attributed to core relaxation. By contrast,
Table 2 Discrete transitions near the K-edge in molecular nitrogen Experimental a)
Theoretical b)
-
assignment
frequency
f-number
assignment
frequency
f-number
luu - 2png la, + 3sug lug-+ 3pn,
400.8 405.6 406.5 406.7
0.195 F 0.02
1UQ--, la,4ug 17rg lug-f 2a, lug-* 30,
396.9 406.2 407.1
2.57 (-01) 2,32(-03) l-46(-02)
lug + 3pau
lo, lo, lo, lug
+ -, + +
4sug 3drrg
407.7 407.9
3dcrg
408.3
4pu,
408.5
lug-+ 4plru
408.5
luu -+ 50 g luu 4 2rrg lug’3a” lugluu~
40”
6~s
407.4 408.3 408.5 408.6
1.37 (-04) 1.16(-04) l-42(-03)
408.7
4.06(-03) l-80(--051
409.0
1.28(-05)
a) Transition frequencies (ev) and spectral assignments taken from the photoabsorption studies of Nakamura et al. [ 11. Resonance f number taken from van der Wiel et al. [ 61; the earlier measurement of Wuilleumier and Krause [ 21. based on Auger line intensitiesgives 0.12 f 0.05. b) Valuestakenfromtable 1.
67
Volume 51, number 1
CHEMICAL PHYSICS LETTERS
1 October 1977
the Rydberg-like orbitals are generally less sensitive to relaxation effects. Part of the discrepancy can also be attributed
to shift and width
effects
1291,
since
the resonance line presumably contributes to I$ production by autoionization, to N’ production by predissociation, and to Np formation by Auger effect [3]. The comparison between theory and experiment in the case of the 1 uu + Ing resonance is further complicated by the fact rhat the generalized oscillator strength for this transition is evidently a rapidly varying ftinction of momentum transfer, and the extraction of the dipole limit may be contaminated by a quadrupole component [30]. Consequently, a meaningful detailed comparison between theory and experiment in the case of the low-energy resonance will require further investigation, and the present results can be tentatively regarded as satisfactory. The recent Xol calculations 18,9] give a lo, + 1ng resonance with a transition frequency = 6 eV above the present calculation and 3 eV above experiment, and an f number (0.22) in good accord with the measured value [6]. In these calculations the intensity of the transition is attributed to a resonance contribution in the d-wave component of a partial-wave expansion of the 1~~ orbital in a body-centered coordinate system. This description provides further insight from an alternative perspective into the presence of valence-like IT* orbitals in first-row diatomits and other reIated species familiar to molecular spectroscopists [27]. The large d-wave contribution to the 1~ orbital can also be understood on basis of its correIation with the 3dn atomic orbital in the united-atom limit, and by noting that the orbital is partially Rydbergized [17-191 at the equilibrium internuclear separation t. to the (3~;~ 1~;) 3~g state but are clearly re!evant to the present discussion. Moreover, the large d-wave contribution to the l~rg orbital involved in valence-shell rr -* T* and n + TT*excitations in molecuIar nitrogen [ 10,l l] is also clarified on this basis. Perhaps it is heIpfu1 to note in this connection that Mull&en’s assignment of 17% 3d7r character to the lng orbital at Re refers only to the rg3d contriiution; the ~g2P component aIso contributes substantially to the d-wave in a partial-wave expansion at the molecular center. Consequently, the 19 is valence-like in size at R,, but of high d-wave character. By contrast, at Re/2 the 1~g apparently swells into a nonpenetrating atomic 3dn orbital, and thus is fuIIy Ryclbergized 117-191.
t Mulliken’s calculations refer [17-191,
68
405
410
415
Excitation
420
425
Energy E (eV)
Fig. 1. Oscillator-strength distributions for excitation of the lag orbital
in molecular
nitrogen
constructed
from
the
Stieltjes-Tchebycheff procedure in the static exchange approximation as discussed in the text; (a) lug + knu excitation; (b) lug - kou excitation.
In figs. 1 and 2 are shown the photoionization continua for 1og and 1 uu ionization, respectively. Also shown in the figures are the discrete spectra, presented in the form of the so-called Stieltjes derivative 113, IS]. The continua are constructed from the Stieltjes-Tchebycheff technique employing between 16 to 24 spectral moments [eq. (6)], which are found to provide particularly smooth profdes in each case. The results of figs. 1 and 2 are in general qualitative accord with the previously reported Xar calculations [S, 91, although there are some important quantitative differences. Specifically, the Xar calculations place the high-energy shape resonance in the 1 og + ka, channel approximately 6 eV above the peak of the present calculations for this channel (= 416 ev), and give a maximum which is a factor two larger than the present result. In addition, the present calculations give a monotonically decreasing cross section in the 1 ag + klr, channel, whereas the ?Gx reiufts give a maximum. As in the case of the lo, + 14 low-energy resonance, the XLZstudies attribute the high-energy iog --t ko, resonance to a particular partial wave, in this case the I = 3, or f-wave, component. The ?GY
1 October 1977
CHEMICAL PHYSICS LETTERS
Volume 51, number 1 I
I
tion. Of course, the 0,2p component also contributes substantially to the f-wave in a partial-wave expansion of the 30, orbital at the molecular center. Consequently, the 3u, is valence-like in size at the equilib-
I
(cl)
rium internuclear i (b)
-I
-I
410
405
Excitation
415 Energy
420
425
E (eV1
excitation of the 10~ orbital in molecular nitrogen: (a) lau - kug excitation; (b) lo, -, kngexcitation.
Fig. 2. AS in fig. 1, for
study provides clarification from an alternative perspective of low-lying valence-like u* orbitals familiar in first row diatomics. In the case of Nz, however, the effective potential apparently does not support an additional unoccupied bound valence orbital other than the lng, and the photoabsorption spectrum in this case, in addition to exhibiting lug --t no, Rydberg transitions associated with diffuse atomic excitations, contains a resonance-like structure in the associated continuum. The corresponding u* resonance contribution to the valence photoionization spectrum in N2 has been discussed previously [IO, 111, and, in accord with the isomorphism [l] described above, the valence photoionization spectrum in NO is expected to exhibit a similar feature. High intensity charge-transfer spectra involving excitations into valence-like u* orbitals are discussed in the early work of Mulliken 121, who has also noted recently that the (3~;‘3u,) f lZ= or V, state in N2 should appear in the photoionization continuum Cl 81. His analysis is also applicable in case of the (1~~~30~) 1 2: core state of interest here. Moreover, the large f-wave character of the 30, orbitals participating in both the core and valence N -+ V,, transitions can be understood on basis of their correlations with 4fu atomic orbitals in the united atom limit, and their partial Rydbergization at the equilibrium internuclear separa-
acter. By contrast,
separation,
but of high f-wave charat smaller internuclear separation
the 30, is expected to swell into a nonpenetrating atomic 4fu Rydberg orbital. The oscillator strengths for absorption in the four channels of figs. 1 and 2 are combined, and compared with the recent electron impactexcitation measurements [6] in fig. 3. Evidently, the theoretical and experimental values are in excellent accord, with the threshold behavior at = 405 eV accounted for by the discrete portions of the spectra, and the rise at x 410 eV attributable to the continua. The doublepeaked structure in the z 415 to 420 eV region has been attributed to simultaneous K- and L-shell excitation [3,5], which is not treated explicitly in the present calculations_
4. Concluding
remarks
Theoretical
investigations
of the recently
measured
25I
01 r’ 405
I
410 Emtatton
Fig. 3.
I
I
475
420
Energy
I 425
E (eV)
Oscillator-strength distribution near the K-edge in
molecular nitrogen; Stieltjes-Tchebycheff results from f%s. 1 and 2; 0 electron-impact excitation measurements [6] ; the indicated error bars correspond to nominal uncertainties of 2 10%.
69
_
Volume 51, number 1
M-shell photoionization cross section in molecular nitrogen are reported employing the StieltjesTchebycheff procedure in the staticexchange approximation and square-integrable moIecular orbitals of good symmetry type. The intense resonance below the 1s ionization potentiF in the measured cross section is attributed to la, + 1~~ excitatidn into a valence-like ‘II* orbital, in accordance with Mulliken’s early analysis of n + Z* transitions. The observed higher-lying excitations correspond to weaker Rydberg transitions associated with diffuse molecular orbitals. The predicted transitions are in one-to-one correspondence wi_th assignments made on basis of an expected isomorphism between valence-shell excitations in NO and core excitation in N2_ Quantitative comparisons between theory and experiment indicate that the low-lying valence-like 15 orbital is somewhat more sensitive to core relaxation
and width and shift
effects than are the more diffuse Rydberg orbitals. In accordance with Mulliken’s early analysis of o + u* charge-transfer spectra, the high-energy structure in the measured cross section is attributed to a resonance contribution in the 10~ + ko, channel from a valenceIike u* orbital. The theoretical and experimental results are in excellent quantitative agreement, although the measured shape rescnance is split into a doublepeaked structure apparently due to simultaneous Kand Lshell excitation. The large d- and f-wave natures of the low- and high-energy resonances, respectively, predicted by recent m calculations are further clarified by the observations that the 1~~ and 30, orbit& are partially Rydbergized at the equilibrium internuclear separation, and correlate with 3dn and 4fo atomic orbitals, respectively, in the united atom limit.
Acknowledgement We thank M.J. van der Wiel and R-E. Kennerly prep-tits of their respective work prior to publication. Helpful conversations with R.S. MulIiken and B.I. Schneider are also gratefully acknowledged_ and W.hl. St. John for providing
References [l]
70
1 October 1977
CHEMICAL PHYSICS LETTERS
hi. Nakamura, M. Sasanuma, S. Sate, M. Watanabe, H. Yamashita. Y. Iguchi, A. Ejiri, S. Nakai,
S. Yamaguchi, T. Sagawa, Y. Nakai and T. Oshio, Phys. Rev. 178 (1969) 80. VI F. Wuilleumier and M-0. Krause, International Conference on Inner-shell Ionization Phenomena, Georgia Institute of Technology, p. 773. 131 M-J. van der Wiel, Th.M. El-Sherbini and CF. Brion, Chem. Phys. Letters 7 (1970) 161. [41 M.J. van der Wiel and Th.M. El-Sherbini, Physica 59 (1972) 453. I51 G.R. Wight, C.E. Brion and M.J. van der Wiel, I. Electron Spectry. 1 (1972/73) 457. t61 R.B. Kay, Ph.E. van der Leeuw and M.J. van der Wiel, to be published. [71 K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. HedBn, K. Hamrin, U. Gelius, T. Bergmark, L-0. Werme, R. Manne and Y. Baer, ESCA applied to free molecules (North-Holland, Amsterdam, 1969) p_ 63. [81 J-L. Dehmer and D. Dill, Phys. Rev. Letters 35 (1975) 213. !9] 3.L. Dehmer and D. Dill, J. Chem. Phys. 65 (1976) 5327. [lOI J.W. Davenport, Phys. Rev. Letters 36 (1976) 945. “ r111 T.N. Rescigno, CF. Bender, B-V. McKay and P-W. Langhoff, Phys. Rev. A (1977), to be published. iI21 R.S. Mulliken, J. Chem. Phys. 7 (1939) 14,20; R.S. Mull&en and C.A. Rieke. Rept. Progr. Phys. 8 (1941) 231. 1131 P-W. Langhoff, Chem. Phys. Letters 22 (1973) 60. r141 P.W. Langhoff and C.T. Corcoran, Chem. Phys. Letters 40 (1976) 367. 1151 P.W. Langhoff and CT. Corcoran, J. Chem. Phys. 61 (1974) 146. WI CT_ Corcoran and P-W. Langhoff, J. Math. Phys. 18 (1977) 651. 1171 R.S. Mulliken. Chem. Phys. Letters 13 (1972) 137, 141; Intern. J. Quantum Chem. 8 (1974) 817. WI R.S. Mulliken, Chem. Phys. Letters 25 (1974) 305; 46 (1977) 197 (191 R.S. Mull&en, Accounts Chem. Res. 9 (1976) 7. r203 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [211 H.P. Kelly, Phys. Rev. 136 (1964) B896. r221 P.W. Langhoff, M. Karplus and R.P. Hurst, J. Chem. Phys. 44 (1966) 505. I231 W.J. Hunt and W.A. Goddard, Chem. Phys. Letters 3 (1969) 414; 24 (1974) 464. [24] N.F. Mott and H.S.W. Massey, The theory of atomic collisions (Oxford 412,522,578.
Univ. Press, London,
1965) pp_
(251 P.W. Langhoff, J. Chem. Phys. 57 (1974) 2604. [26] P.E. Cade, K.D. Sales and AC. Wahl. J. Chem. Phys. 44 (1966) 1973. [27] J.G. Calvert and J.N. Pitts. Photochemistry (Wiiey, New York, 1966). (281 F.R. Gilmore, J. Quant. Spectry. Radiative Transfer 5 (1965)
369.
[29] U. Fano, Phys. Rev. 124 (1961) 1866. 1301 J.S. Lee, T.C. Wong, R.A. Bonham and R.E. Kennerly, to be published.
Volume 51, number 1
K-SHELL PHOTOIONIZATION
PHYSICS
IN MOLECULAR
1 October
LETTERS
1977
NITROGEN
T.N. RESCIGNO Theoretical Atomic and Molecular Physics Group, Livermore. chlifomiiz 94550. USA
University
of Chlifornia. Lawrence
Livermore
Laboratory.
and P.W. LANGHOFF Department
of Chemistry,
Indiana (Iniversity,
Bloomington,
Indiana 47401,
USA
Received 27 June 1977
Stieltjes-Tchebycheff calculations in the static-exchange approximation cross section in molecular nitrogen.
1. Introduction Photoabsorption [l ,2] and electron impactexcitation [3--61 studies of the K-shell photoionization cross section in molecular nitrogen reveal the presence of a weak Rydberg series, and resonance-like structures at energies above and below the 1s ionization potential [7]. The origin of the series has been clarified in terms of an isomorphism between valence excitations in NO and K-shell excitations in N2 [ 11, and that of the resonance features by recent XLYcalculations [8,9], which attribute the high- and lowenergy structures to f- and d-wave resonances, respectiveiy, in the final-state orbitals. Although the X~Ycalculations are in qualitative accord with the measured cross section, the highenergy resonance structure is apparently quite sensitive to the potential employed, and there are quantitative discrepancies between the theoretical [8, V] and experimental results [6]. A similar observation is made in connection with recent theoretical studies of valenceshell spectra in molecular nitrogen [lo, 111, suggesting that nonlocal Fock potentials are perhaps required for quantitatively reliable predictions of shape resonances in molecular photoionization cross sections. Moreover, although the description of the two prominent nitrogen Kedge structures in terms off- and d-wave resonances is helpful, further clarification along the lines of
are reported of the K-shell photoionization
Mulliken’s early analysis of intense transitions into n* and u* valence-like orbitals [12] is also desirable. An alternative theoretical investigation of the K-edge photoionization cross section in molecular nitrogen based on orbitals of good symmetry type and the nonlocal Fock potential would clearly be of considerable interest. In the present letter, Stieltjes-Tchebycheff calculations [ 13,141 in the static-exchange approximation are reported of the K-shell photoionization cross section in molecular nitrogen, complementing previously reported related studies of the valence-shell cross sections [I 11. Conventional Roothaan-Hartree-Fock . calculations and normalizable gaussian basis sets are employed in the construction of occupied and improved virtual molecular orbitals of good symmetry type. A satisfactory description is provided of the observed weak Rydberg series [l] , and the StieltjesTchebycheff profile obtained from an appropriate moment analysis [ 15,161 of the calculated pseudo-spectrum is in good quantitative accord with the recent electron impactexcitation measurements of the photoionization cross section [6], particularly in the vicinity of the high-energy shape resonance_ The origins of the low- and highenergy structures are discussed in terms of excitations into IT* and u* valence-like orbitals, respectively, and their partial Rydbergization 65
Volume 5 1, number 1
CHEMICAL PHYSICS LETTERS
. 112, 17-191 into 3dn and 4fu atomic orbit&, iespectively, thereby providing further clarification of the d- and f-wave natures of the resonances. The small discrepancies remaining between theory and experiment are attributed to core reiaxation, two-electron excitation, and final-state lineshape effects, which are treated in a subsequent more detailed report. The theoretical development is given in section 2, computational results are described and discussed in section 3, and concluding remarks are made in section 4.
2. Theoretical
development
Nonlocal potentiaIs of appropriate symmetry for the study of 10~ + ko,, knu and 10~ + ko kng excitations in molecular nitrogen are obtainid from conventional ground-state Hartree-Fock calculations employing gaussian basis sets [20] in the forms
1 October 1977
where = T + Vn + V$N-i) (4) HY is the one-electron hamiltonian for the 7th channel, are employed in conjunction with the StieltjesTchebycheff technique 113-161 to circumvent the difficulties that arise in constructing continuum solutions of eq. (2) In this way, photoionization cross sections corresponding to the staticexchange approximation are obtained without explicit construction of the scattering functions @_ The necessary physical information is contained in the pseudo-spectrum of transition frequencies and oscillator strengths {Gr, xr, i = 1, M)
where 6, and Ed are the lug or 1 uu orbital and energy, respectively, and P is the dipole operator. These provide approximate invariant spectral moments r251
v(N-l) =]. r
(21j - Kj) + Jr + Kr ,
(1)
O-+-Q where Jj and Kj are the customary Coulomb and exchange operators, respectively. The so-called improvedvirtuai-orbital potential of eq. (1) is appropriate for describing the motion of an electron excited from the @h occupied orbital in the frozen (Coulomb and exchange) field provided by the remaining N-l electrons [21-231. Consequently, scattering solutions of the one-electron problem, (T+
v,+
~.$v-1’-&5;=0,
(2)
where T and V, are the kinetic ener,T operator and nuclear-frame potential, respectively, corr*spond to the so-called static-exchange approximation 1243. The nonlocal and anisotropic nature of the potential ~~~-1) has apparently discouraged determinations of the continuum orbitals @z appearing in eq. (2) employing conventional techniques, motivating the introduction of approximate local-exchange potentials and partial-wave expansions IS-1 01. h the present studies, variationally determined pseudo-spectra CT?, y, i = 1, M) of good symmetry type satisfying
(3)
2.. (--k) = ig
(zT)-k
xy
, k = 0,1, . . . .
(6)
from which the corresponding oscillator-strength profile is obtained following the previousIy described Stieltjes-Tchebycheff technique (13-l 61.
3. Computational
results
A FJartree-Fock function of good quality is constructed near the equilibrium internuclear separation [26] and used in forming the appropriate V:Nw" potentials for the excitation of loa and 1 u,, electrons in molecular nitrogen. Previously described basis sets of gaussian orbitafs of appropriate symmetry [l l] _ are employed in the construction of M w 25-term pseudo-spectra [eqs. (3) to (6)J in each case. In table 1 are shown the first five terms in the calculated pseudo-spectra for the four dipole allowed symmetries. These discrete-like transitions in the tail of the vaIence-shell spectra are made to converge on the correct nitrogen 1s ionization potential (409.9 eV) by empIoying the experimental value [7] for er in eq. (5). Evidently, the spectrum is dominated by an &tense la, + lrrg transition at x 397 eV having an
-
Volume 5 1, number 1
1 October 1977
CHEMICAL PHYSICS LETTERS
Table 1 Discrete transition frequencies (ev) and oscillator strengths near the K-edge in molecular nitrogen (ZT/Fy) lug + nsu
lug” “Uu
lou+
407.1/1.46(-02) 408.6/4.06(-03) 409.2/1.73(-03) 409.5/9.08(-04) 409.6/5.95 (-04)
407.4/l-37(-04) 408.7/1.80(-05) 409.214.39 (-06) 409.5/l .57 (-06) 409.6/8.50(-07)
406.2/2.32(-03) 408.3/1.16(-04) 409.0/1.28(-0.5) 409.4/1.27(-04) 409.5/2.36(-04)
nog
a)
luu -, n7rg 396.9/2.57(-01) 408.5/1.42(-03) 409.1/7.20(-04) 409.4/3.94(-04) 409.6/2.35 (-04)
a) Values corresponding to eqs. (5) obtained from improved-virtual-orbital calculations as discussed in text. Numbers in parentheses give appropriate powers of ten.
f number = 0.26. High intensity
transitions into valence-Iike ‘II* orbitals are discussed by Mulliken [12], who notes that the participation of compact 2p atomic orbitals can give rise to large transition moments in these cases. Indeed, the 1cru + lrrg resonance is an extreme example of an n -+ rr* transition [27], and similarly intense In, --f lrrg (rr --f rr*) and 20, + lrrg (n + rr*) transitions contribute to the valence excitation spectrum in molecular nitrogen [ 111. In contrast to the intense lo, + lng transition, the other members of the I uu -+ nrrg series, and of the other three dipole spectra, are wek and correspond to excitations into more diffuse Rydberg-like molecular orbitals comprised of diffuse atomic orbitals. The calculated discrete spectrum is compared with experimental values and the spectral assignments of Nakamura et al. [I] in table 2. The latter assignments are made on basis of an approximate isomorphism between the energy ordering of unoccupied orbitals
in NO with the expected ordering of unoccupied orbitals in (Is-‘) Nz [I]. That is, since one of the Kshell electrons in N, is missing in the potential of eq. (l), the screening of the nitrogen nuclei is incomplete, resulting in a potential similar to that found in NO. The first six calculated excitations in table 2 are evidently in one-to-one correspondence with the spectral assignments made on this basis. Except for the first intense la, + lng transition, the calculated excitation energies are within = 0.5 eV of the measured values, which latter are in excellent quantitative accord with the observed valence-shell excitations in NO [28]. Moreover, the measured intensity profile [1] is in general accord with the calculated f numbers. The discrepancies between the calculated transition frequency and osciliator strength for the intense lo, + lrrg low-energy resonance and the recent electron impact-excitation measurements [6] can presumably be attributed to core relaxation. By contrast,
Table 2 Discrete transitions near the K-edge in molecular nitrogen Experimental a)
Theoretical b)
-
assignment
frequency
f-number
assignment
frequency
f-number
luu - 2png la, + 3sug lug-+ 3pn,
400.8 405.6 406.5 406.7
0.195 F 0.02
1UQ--, la,4ug 17rg lug-f 2a, lug-* 30,
396.9 406.2 407.1
2.57 (-01) 2,32(-03) l-46(-02)
lug + 3pau
lo, lo, lo, lug
+ -, + +
4sug 3drrg
407.7 407.9
3dcrg
408.3
4pu,
408.5
lug-+ 4plru
408.5
luu -+ 50 g luu 4 2rrg lug’3a” lugluu~
40”
6~s
407.4 408.3 408.5 408.6
1.37 (-04) 1.16(-04) l-42(-03)
408.7
4.06(-03) l-80(--051
409.0
1.28(-05)
a) Transition frequencies (ev) and spectral assignments taken from the photoabsorption studies of Nakamura et al. [ 11. Resonance f number taken from van der Wiel et al. [ 61; the earlier measurement of Wuilleumier and Krause [ 21. based on Auger line intensitiesgives 0.12 f 0.05. b) Valuestakenfromtable 1.
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CHEMICAL PHYSICS LETTERS
1 October 1977
the Rydberg-like orbitals are generally less sensitive to relaxation effects. Part of the discrepancy can also be attributed
to shift and width
effects
1291,
since
the resonance line presumably contributes to I$ production by autoionization, to N’ production by predissociation, and to Np formation by Auger effect [3]. The comparison between theory and experiment in the case of the 1 uu + Ing resonance is further complicated by the fact rhat the generalized oscillator strength for this transition is evidently a rapidly varying ftinction of momentum transfer, and the extraction of the dipole limit may be contaminated by a quadrupole component [30]. Consequently, a meaningful detailed comparison between theory and experiment in the case of the low-energy resonance will require further investigation, and the present results can be tentatively regarded as satisfactory. The recent Xol calculations 18,9] give a lo, + 1ng resonance with a transition frequency = 6 eV above the present calculation and 3 eV above experiment, and an f number (0.22) in good accord with the measured value [6]. In these calculations the intensity of the transition is attributed to a resonance contribution in the d-wave component of a partial-wave expansion of the 1~~ orbital in a body-centered coordinate system. This description provides further insight from an alternative perspective into the presence of valence-like IT* orbitals in first-row diatomits and other reIated species familiar to molecular spectroscopists [27]. The large d-wave contribution to the 1~ orbital can also be understood on basis of its correIation with the 3dn atomic orbital in the united-atom limit, and by noting that the orbital is partially Rydbergized [17-191 at the equilibrium internuclear separation t. to the (3~;~ 1~;) 3~g state but are clearly re!evant to the present discussion. Moreover, the large d-wave contribution to the l~rg orbital involved in valence-shell rr -* T* and n + TT*excitations in molecuIar nitrogen [ 10,l l] is also clarified on this basis. Perhaps it is heIpfu1 to note in this connection that Mull&en’s assignment of 17% 3d7r character to the lng orbital at Re refers only to the rg3d contriiution; the ~g2P component aIso contributes substantially to the d-wave in a partial-wave expansion at the molecular center. Consequently, the 19 is valence-like in size at R,, but of high d-wave character. By contrast, at Re/2 the 1~g apparently swells into a nonpenetrating atomic 3dn orbital, and thus is fuIIy Ryclbergized 117-191.
t Mulliken’s calculations refer [17-191,
68
405
410
415
Excitation
420
425
Energy E (eV)
Fig. 1. Oscillator-strength distributions for excitation of the lag orbital
in molecular
nitrogen
constructed
from
the
Stieltjes-Tchebycheff procedure in the static exchange approximation as discussed in the text; (a) lug + knu excitation; (b) lug - kou excitation.
In figs. 1 and 2 are shown the photoionization continua for 1og and 1 uu ionization, respectively. Also shown in the figures are the discrete spectra, presented in the form of the so-called Stieltjes derivative 113, IS]. The continua are constructed from the Stieltjes-Tchebycheff technique employing between 16 to 24 spectral moments [eq. (6)], which are found to provide particularly smooth profdes in each case. The results of figs. 1 and 2 are in general qualitative accord with the previously reported Xar calculations [S, 91, although there are some important quantitative differences. Specifically, the Xar calculations place the high-energy shape resonance in the 1 og + ka, channel approximately 6 eV above the peak of the present calculations for this channel (= 416 ev), and give a maximum which is a factor two larger than the present result. In addition, the present calculations give a monotonically decreasing cross section in the 1 ag + klr, channel, whereas the ?Gx reiufts give a maximum. As in the case of the lo, + 14 low-energy resonance, the XLZstudies attribute the high-energy iog --t ko, resonance to a particular partial wave, in this case the I = 3, or f-wave, component. The ?GY
1 October 1977
CHEMICAL PHYSICS LETTERS
Volume 51, number 1 I
I
tion. Of course, the 0,2p component also contributes substantially to the f-wave in a partial-wave expansion of the 30, orbital at the molecular center. Consequently, the 3u, is valence-like in size at the equilib-
I
(cl)
rium internuclear i (b)
-I
-I
410
405
Excitation
415 Energy
420
425
E (eV1
excitation of the 10~ orbital in molecular nitrogen: (a) lau - kug excitation; (b) lo, -, kngexcitation.
Fig. 2. AS in fig. 1, for
study provides clarification from an alternative perspective of low-lying valence-like u* orbitals familiar in first row diatomics. In the case of Nz, however, the effective potential apparently does not support an additional unoccupied bound valence orbital other than the lng, and the photoabsorption spectrum in this case, in addition to exhibiting lug --t no, Rydberg transitions associated with diffuse atomic excitations, contains a resonance-like structure in the associated continuum. The corresponding u* resonance contribution to the valence photoionization spectrum in N2 has been discussed previously [IO, 111, and, in accord with the isomorphism [l] described above, the valence photoionization spectrum in NO is expected to exhibit a similar feature. High intensity charge-transfer spectra involving excitations into valence-like u* orbitals are discussed in the early work of Mulliken 121, who has also noted recently that the (3~;‘3u,) f lZ= or V, state in N2 should appear in the photoionization continuum Cl 81. His analysis is also applicable in case of the (1~~~30~) 1 2: core state of interest here. Moreover, the large f-wave character of the 30, orbitals participating in both the core and valence N -+ V,, transitions can be understood on basis of their correlations with 4fu atomic orbitals in the united atom limit, and their partial Rydbergization at the equilibrium internuclear separa-
acter. By contrast,
separation,
but of high f-wave charat smaller internuclear separation
the 30, is expected to swell into a nonpenetrating atomic 4fu Rydberg orbital. The oscillator strengths for absorption in the four channels of figs. 1 and 2 are combined, and compared with the recent electron impactexcitation measurements [6] in fig. 3. Evidently, the theoretical and experimental values are in excellent accord, with the threshold behavior at = 405 eV accounted for by the discrete portions of the spectra, and the rise at x 410 eV attributable to the continua. The doublepeaked structure in the z 415 to 420 eV region has been attributed to simultaneous K- and L-shell excitation [3,5], which is not treated explicitly in the present calculations_
4. Concluding
remarks
Theoretical
investigations
of the recently
measured
25I
01 r’ 405
I
410 Emtatton
Fig. 3.
I
I
475
420
Energy
I 425
E (eV)
Oscillator-strength distribution near the K-edge in
molecular nitrogen; Stieltjes-Tchebycheff results from f%s. 1 and 2; 0 electron-impact excitation measurements [6] ; the indicated error bars correspond to nominal uncertainties of 2 10%.
69
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Volume 51, number 1
M-shell photoionization cross section in molecular nitrogen are reported employing the StieltjesTchebycheff procedure in the staticexchange approximation and square-integrable moIecular orbitals of good symmetry type. The intense resonance below the 1s ionization potentiF in the measured cross section is attributed to la, + 1~~ excitatidn into a valence-like ‘II* orbital, in accordance with Mulliken’s early analysis of n + Z* transitions. The observed higher-lying excitations correspond to weaker Rydberg transitions associated with diffuse molecular orbitals. The predicted transitions are in one-to-one correspondence wi_th assignments made on basis of an expected isomorphism between valence-shell excitations in NO and core excitation in N2_ Quantitative comparisons between theory and experiment indicate that the low-lying valence-like 15 orbital is somewhat more sensitive to core relaxation
and width and shift
effects than are the more diffuse Rydberg orbitals. In accordance with Mulliken’s early analysis of o + u* charge-transfer spectra, the high-energy structure in the measured cross section is attributed to a resonance contribution in the 10~ + ko, channel from a valenceIike u* orbital. The theoretical and experimental results are in excellent quantitative agreement, although the measured shape rescnance is split into a doublepeaked structure apparently due to simultaneous Kand Lshell excitation. The large d- and f-wave natures of the low- and high-energy resonances, respectively, predicted by recent m calculations are further clarified by the observations that the 1~~ and 30, orbit& are partially Rydbergized at the equilibrium internuclear separation, and correlate with 3dn and 4fo atomic orbitals, respectively, in the united atom limit.
Acknowledgement We thank M.J. van der Wiel and R-E. Kennerly prep-tits of their respective work prior to publication. Helpful conversations with R.S. MulIiken and B.I. Schneider are also gratefully acknowledged_ and W.hl. St. John for providing
References [l]
70
1 October 1977
CHEMICAL PHYSICS LETTERS
hi. Nakamura, M. Sasanuma, S. Sate, M. Watanabe, H. Yamashita. Y. Iguchi, A. Ejiri, S. Nakai,
S. Yamaguchi, T. Sagawa, Y. Nakai and T. Oshio, Phys. Rev. 178 (1969) 80. VI F. Wuilleumier and M-0. Krause, International Conference on Inner-shell Ionization Phenomena, Georgia Institute of Technology, p. 773. 131 M-J. van der Wiel, Th.M. El-Sherbini and CF. Brion, Chem. Phys. Letters 7 (1970) 161. [41 M.J. van der Wiel and Th.M. El-Sherbini, Physica 59 (1972) 453. I51 G.R. Wight, C.E. Brion and M.J. van der Wiel, I. Electron Spectry. 1 (1972/73) 457. t61 R.B. Kay, Ph.E. van der Leeuw and M.J. van der Wiel, to be published. [71 K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. HedBn, K. Hamrin, U. Gelius, T. Bergmark, L-0. Werme, R. Manne and Y. Baer, ESCA applied to free molecules (North-Holland, Amsterdam, 1969) p_ 63. [81 J-L. Dehmer and D. Dill, Phys. Rev. Letters 35 (1975) 213. !9] 3.L. Dehmer and D. Dill, J. Chem. Phys. 65 (1976) 5327. [lOI J.W. Davenport, Phys. Rev. Letters 36 (1976) 945. “ r111 T.N. Rescigno, CF. Bender, B-V. McKay and P-W. Langhoff, Phys. Rev. A (1977), to be published. iI21 R.S. Mulliken, J. Chem. Phys. 7 (1939) 14,20; R.S. Mull&en and C.A. Rieke. Rept. Progr. Phys. 8 (1941) 231. 1131 P-W. Langhoff, Chem. Phys. Letters 22 (1973) 60. r141 P.W. Langhoff and C.T. Corcoran, Chem. Phys. Letters 40 (1976) 367. 1151 P.W. Langhoff and CT. Corcoran, J. Chem. Phys. 61 (1974) 146. WI CT_ Corcoran and P-W. Langhoff, J. Math. Phys. 18 (1977) 651. 1171 R.S. Mulliken. Chem. Phys. Letters 13 (1972) 137, 141; Intern. J. Quantum Chem. 8 (1974) 817. WI R.S. Mulliken, Chem. Phys. Letters 25 (1974) 305; 46 (1977) 197 (191 R.S. Mull&en, Accounts Chem. Res. 9 (1976) 7. r203 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [211 H.P. Kelly, Phys. Rev. 136 (1964) B896. r221 P.W. Langhoff, M. Karplus and R.P. Hurst, J. Chem. Phys. 44 (1966) 505. I231 W.J. Hunt and W.A. Goddard, Chem. Phys. Letters 3 (1969) 414; 24 (1974) 464. [24] N.F. Mott and H.S.W. Massey, The theory of atomic collisions (Oxford 412,522,578.
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[29] U. Fano, Phys. Rev. 124 (1961) 1866. 1301 J.S. Lee, T.C. Wong, R.A. Bonham and R.E. Kennerly, to be published.