J. Phys. Chem. 1988, 92, 979-982
979
Photoionization and Photoabsorption Cross Section Calculations in Molecules: CH,, NH,, H,O, and HF I. Cacelli25 V. Carravetta,t R. Moccia,*+and A. Rizzot Dipartimento di Chimica e Chimica Industriale dell' Universitii and Istituto di Chimica Quantistica ed Energetica Molecolare del C. N.R., via Risorgimento 35, 56100 Pisa, Italy (Received: June 17, 1987)
Photoabsorption and photoionization cross sections of CH,, NH3, H20, and HF have been calculated by using extended L2 basis sets. The continuous spectrum results were obtained via a Stieltjes imaging procedure applied to the 'discretized" excitation spectrum yielded by the random phase approximation, RPA. The basis sets included functions all centered upon the heavy atom and the RPA calculations were performed both in the independent channels and in the interacting channels schemes.
Introduction In two previous papers'g2 the photoabsorption and the photoionization cross-section profiles were calculated for H 2 0 , NH3, CH,, and HF. The continuum orbitals were obtained in the static-exchange approximation (SEA) by using extended L2 basis sets which included suitable special oscillating functions. The resulting "discretized" oscillator strength distributions were converted to continuum oscillator strength densities via a Stieltjes imaging p r o c e d ~ r e . The ~ ~ ~onecenter expansion ( W E ) technique was employed to describe both the bound and continuum orbitals. The results were generally in good agreement with the experimental data and the neglect of the interaction among the excitation channels was considered the most probable cause of the discrepancies. More recent investigations on HC15 and H2Sasuggest that (i) the inclusion of the multichannel interaction actually improves the agreement with the experiment of the calculated photoionization cross section profiles for these molecules; (ii) the use of the random phase approximation (RPA) for the calculation of the one-electron transition density matrix elements, despite the fact that it is partially implemented and that the basis set although large is certainly not complete, noticeably improves the agreement between the length gauge (L) and the velocity gauge (V) results. It should be emphasized that the degree of gauge invariance is particularly important when, in order to assess the accomplishment of the approximation method employed, the calculated values are compared to the experimental data. For instance, the differences between L and Vresults of the SEA multichannel calculations5 might be so large as to make quite unclear the role played by the channel interaction. To estimate the extent to which these effects apply to second-row hydrides, similar multichannel RPA calculations have been performed for CH,, NH3, H 2 0 , and HF. This paper reports the results obtained for both photoionization and photoabsorption cross sections due to the valence shell excitation. Method and Computational Details Since a full description of the theoretical background and of the computational method employed is extensively reported in the l i t e r a t ~ r e l -here ~ only the main points are briefly outlined. Each excitation channel A is identified by the set of indexes A = (j,r,, r2,r),which refer to the promotion of an electron from the occupied orbital cpj, belonging to the symmetry rl,to an excited orbital, associated with the symmetry index rZ.thus giving rise to excited states of syemetry r. We can define the one-electron excitation operator OA: as '*A,)
bA21dIJ)
where '9, is the singlet excited state of the channel A, dois the Dipartimento di Chimica e Chimica Industriale. 'C.N.R. Researcher of Scuola Normale Superiore.
SCF closed-shell ground state, and p indicates the excited orbital of symmetry r2yielded by a SEA calculatiqn referred to the proper generates from bo hole parent ion. The excitation operator OAClt a proper linear combination of singly excited Slater determinants with the correct spin-spatial symmetry form. Although the cross sections here reported were calculated only with the RPA method, it was expedient, as was done p r e ~ i o u s l y ~ ~ ~ and as will be explained in the next section, to go through the SEA calculation in order to estimate the photoionization cross-section pro fi1e. The excited states were thus obtained by the following four methods: (a) single channel static-exchange approximation (SC-SEA); (b) multichannel static-exchange approximation (MC-SEA); (c) single-channel random phase approximation (SC-RPA); (d) multichannel random phase approxjmation (MC-RPA). The corresponding excitation operators fi are (a) SC-SEA: (b) MC-SEA:
F J = C C , 16t\rt ACl
(C) SC-RPA:
FA: = C[Xbvdbt- YA,,"~,] P
(d) MC-RPA:
F: = C[XA,,'d,t - Y,'d,+] All
While the C matrix is variationally determined, X and Y are defined by the RPA In the SC-SEA the excited orbitals are determined by imposing the additional constraint of orthogonality to the occupied orbitals. The calculations b, c, and d are performed by projecting the involved matrix elements on the SC-SEA states. A drastic reduction of the computer time, which, as verified in some instance, does not affect the final results,' is achieved by employing three different basis sets, all centered on the heavy nucleus: (i) The ground state and the bound orbitals of the singly excited final states are projected upon medium size basis sets of S T O s (typically 60 functions); (ii) the parent ion occupied orbitals that define the static-exchange potential are projected upon limited basis sets of STO's (30 basis functions on the average); (iii) the excited orbitals are projected upon large basis sets including, besides STOs, an adequate number of hydrogenic functions as well as special oscillating functions's2 obtained as a Cacelli, I.; Moccia, R.; Carravetta, V. Chem. Phys. 1984, 90, 313. Cacelli, I.; Carravetta, V.; Moccia, R. J. Phys. B 1985, 18, 1375. Langhoff, P. W.; In Electron-Molecule and Photon-Molecule CalliRescigno, T. N., McKoy, B. V., Schneider, B., Eds.; Plenum: New York, 1979; pp 183-224. (4) Langhoff, P. W. In Theory and Application of Moment Methods in Many-FermionsSystems; Dalton, B. J., Grimes, S . M., Vary, J. P., Williams, S . A., Plenum: New York, 1980; pp 191-212. (5) Cacelli, I.; Carravetta, V.; Moccia, R. Mol. Phys. 1986, 59, 385. (6) Cacelli, I.; Carravetta, V.; Moccia, R. Chem. Phys., in press. (7) Rowe, D. J. Reu. Mod. Phys. 1968,40, 153.
0022-3654/88/2092-0979$01.50/00 1988 American Chemical Society
980 The Journal of Physical Chemistry, Vol. 92, No. 4 , 1988 product of diffuse S T O s and trigonometric (cos kr) factors (more or less 80 functions for each channel). An adequate OCE description of the ground-state wave function close to the hydrogen nuclei requires the use of STO’s with high values of n and 1. In the present work values of n up to 9 and 1 up to 7 are employed. The MC-RPA calculations typically involved 250 to 300 excited states for each final-state symmetry when only the excitation from the valence shells were taken into account. The “discretized” oscillator strength distribution obtained by the previously listed methods is then processed by the Stieltjes imaging technique3v4to obtain the continuum oscillator strength density dflde, which is related to the photoabsorption/ionization cross section (in megabarns (Mb)) according to a(e)
TABLE I: Vertical Ionization Potentials (ev); Calculated Koopmans’ (SCF-WE) and Experimental Values”J2
IP, eV
N
CJ(u,)-k
k = 0, 1, 2, ..., 2n - 1
I=]
Here the wI)s are the excitation energies and thef;.’s are the oscillator strengths. The Stieltjes imaged spectra give principal representations of the oscillator strength distribution, with remarkable proper tie^,^^^ by which a continuous oscillator strength density dfldt may be obtained in a convergent way. For smoothly varying cross sections the dependence upon the chosen order n is often negligible. In our calculations n ranges typically from 17 to 22 so that several sets of Stieltjes derivative points are obtained. The cross section is then evaluated through a leastsquares fitting. Some care is required however in the M C calculations when strong discrete excitations are embedded in the electronic continuum of other channels and their electronic interaction is weak. In this case the behavior of the cross section close to the resonance is strongly dependent upon the order n.8 Actually the Stieltjes derivative prescription near the resonance peak may introduce an unphysical width which decreases as the order n increases. Hence to have a more reasonable account of these “quasi-stationary” states on the absorption spectrum, their contribution has been cancelled from the discretized spectrum and then added to the final Stieltjes cross-section profile by a Gaussian function centered at the resonant energy and subtending an area proportional to the original oscillator strength. Our fixed nuclei calculations cannot evaluate the width of these bands due to the vibronic coupling. Thus the full width at half-maximum (fwhm) was estimated from the photoelectron spectra (PES) available. We suppose that the Franck-Condon envelope of the resonant band is comparable with that appearing in PES and corresponding to emptying the occupied orbital involved in the resonant excitation. Hence the estimated fwhm of this FC envelope has been employed as width of the Gaussian function representing the effect of the resonant state. A measure of the coupling between the electronic discrete states and the underlying electronic continuum of another channel can be obtained with sufficient accuracy by taking the first derivative, at the resonant energy E,, of the least-squares-fitted histogram
Here fi is the electronic Hamiltonian, 4, is the resonant SC-SEA wave function, and the sum runs over the discretized states & lying in the continuum and considered as narrow wavepackets of mean energy &. This procedure corresponds to the application of the Fermi golden rule for the evaluation of the line broadening r,
r, = 2 ~ 1 ( 4 ~ (-f E)4,)12 i
E = E,
(8) Nesbet, R. K. Phys. Rev. A 1976, 24, 1065.
calcd
exptl
H20 13.81 15.78 19.25 36.63
12.61 14.73 18.55 32.20
CHI 14.40 25.35
14.20 23.05
NH,
11.36 16.72 30.74
= 2~~aa~~lO~~(df/de)
where e is the photon energy in hartrees, a is the Bohr radius in centimeters, and a! is the fine structure constant. The Stieltjes imaging algorithm converts a discretized spectrum of order N to a smaller spectrum of order n ( n
