1013
Excited States and Positive Ions of SF6 P. Jeffrey Hay Contributionfrom the Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545. Received July 19, 1976
Abstract: The excited electronic states of SFband the lowest states of SFbf are investigated with a b initio calculations in an extended Gaussian basis. The ordering of the states of SFs+ as predicted by orbital energies of SFs or by SCF calculations on SFs+ differs from the ordering predicted by calculations which include electron correlation. The excited states of SFs are treated by improved virtual orbital (IVO) calculations where the excited state orbitals are determined in the field of the appropriate ion core. The lowest excited states correspond to excitations into the 6alg orbital, a valence-type orbital with considerable S -F antibonding character. Higher excited states involve excitations into diffuse Rydberg 4s, 4p, or 3d orbitals. The term values for states involving the 6alg orbital vary by as much as 13 000 cm-I depending upon which orbital is excited, while the term values for true Rydberg states are not affected greatly by the nature of the ion core. Oscillator strengths for the dipole-allowed states are reported, and some excitations from core orbitals in x-ray absorption are investigated.
The SF6 molecule has been studied by a variety of experimental techniques including UV and vacuum tJV,] electr~n-impact,~ p h o t ~ e l e c t r o n and , ~ ~ ~x-ray ~pectroscopy.~The assignment of the lowest states of SF6’ has been controversial since four peaks are observed in the photoelectron spectrum in the region 15--20eV but five electronic states are predicted to lie in this region. Theoretical interpretations have been based on orbital energies from S C F calculationslO~i I on SF6, from many-body perturbation theory,’ and from Xu-scattered wave calculations.I2 In the only ab initio treatment to date of the excited states O f SF6, GianturcoI3 employed a small Gaussian basis which would not have sufficient flexibility to treat any states involving diffuse Rydberg orbitals. Using theoretical estimates for the term values of Rydberg states, Robin14 assigned many of the peaks in the UV and electron-impact spectra of SF6. In this study an extended Gaussian basis with added diffuse functions is used to calculate the excited states of SF6. In the improved virtual orbital (IVO) methodI5 each state is represented as a single electron in the field of a “frozen” ion core of particular symmetry. Although this simple approach neglects such effects as relaxation and electron correlation, the calculated results appear to be in good agreement with the experimental information and to be at variance with some of the earlier theoretical interpretations. The agreement between theory and experiment is aided by the customary procedure of correcting the calculated excitation energies by the difference between the Hartree-Fock and experimental ionization potentials. The interpretation of the SF6+ states themselves is based on the results of CI calculations using the generalized valence bond (GVB) orbitals of SF6.I6
Details of the Calculation Positive Ions. Three different techniques were employed for the states of SF6’. The simplest method uses Koopmans’ theorem to assign the IP as -E,, where the orbital energies are taken from the Hartree-Fock calculation on SF6. The (1 ls7pld/9s5p) Gaussian basis contracted to [4s3pld/3s2p] discussed in the preceding paper was used.16 Self-consistent calculations were performed on the ion states in the same basis to assess the importance of orbital relaxation effects. For the degenerate states such as 2 T ~arising u from the (tI,Js configuration, the x, y , and z tl, components were required to be spatially equivalent by the use of an average open-shell Hamiltonian. Allowing the tl component to be nonequivalent in the configuration (tlux)2(tluy)2(tl,z)’ resulted in an energy lowering of approximately 0.01 au (0.3 eV). The open-shell calculations allowed complete mixing of the open- and closed-shell orbitals of a given symmetry.
The GVB-CI calculations for SF.5’ used the six-pair generalized valence bond (GVB) calculation of the neutral molecules as a starting point.16 The GVB natural orbitals were transformed to octahedral symmetry functions, and the resulting 41 orbitals formed the basis set for the CI calculations. I n terms of these orbitals the Hartree-Fock configuration is represented by ( 4 a 1 , ) ~( 5 a 1 , ) ~( 3 t 1 , ) ~(4t1,)~( 3 1 , ) ~(2e,)4 (3e,)4 (lt2# (lt1,)6 (It& (6a1,)O (6t1,)’ (4e,)O. The 11 core orbitals not listed in this configuration describe the sulfur Is, 2s, 2p and fluorine 1s electrons, and these orbitals were held doubly occupied in the CI calculations. All single and double excitations relative to the reference configurations in the above space of orbitals were generated. The reference configuration consisted of the Hartree-Fock configuration for SF6. For the SF6+ states the reference states comprised all configurations of a particular symmetry obtained by removing one electron from the Hartree-Fock configuration. Thus the three 2 T ~reference u states corresponded to the (3tl,), (4tlU),and (5t1,) “ho1e”states of SF6+.Theconfigurations were then screened by perturbation theory and ones au were rewith estimated energy contributions of 5 X tained in the final variational CI. The number of final spin eigenfunctions in the CI calculations for SF6 and SF6+ (and the total number of spin eigenfunctions before selection) are as follows: ‘AI, 408 (1591); 2E, 1018 (7864); 2T~,,2T2u2253 (14697); and *TI,, 2T2g 1883 (8913). More than one octahedral symmetry can occur in a particular calculation since the D2h point group, the highest symmetry group containing only one-dimensional representations, was employed. To assess the reliability of the selection procedure the SF.5 calculations were repeated using all 1591 spin functions without selecting. The resultant total energy was 0.12 eV lower (-994.1907 compared to -994.1862 au) than the energy obtained using 408 selected configurations. Based on other experience to date we would expect the relative energy differences for the IP’s themselves should be at least as accurate as the 0.1 eV error in absolute energies introduced by the selection procedure. Excited States. In order to describe excited states which involve diffuse orbitals an additional set of basis functions centered on the sulfur atom was added to the valence basis. This consisted of three s functions with Gaussian exponents of 0.024,0.0076, and 0.0025, two sets of p functions with exponents 0.020 and 0.007, and a set of d functions with exponent 0.016. These should be appropriate for describing the 4s, 5s, 6s, 4p, 5p, and 3d Rydberg orbitals of sulfur. Most of the exponents were determined by optimizing the energy for the quintet configuration (3~)~3p,3p,3p, C$ of the sulfur atom, where C$ represents the Rydberg orbital. The outermost s and Hay
/ Excited States and Positive Ions of SF.5
1014 Table 1. The Ionization Potentials for SFs from Previous
Table 11. The Ionization Potentials of SF6 from the Present Calculations
Calculations
von Niessen et aLC Gianturcou Orbital
-ti
lti, 14.75 5tlu 16.36 15.93 ltru 3e, 17.25 l t ~ ~20.88 23.22 4t1,
Roosb 18.2 19.0 19.4 19.4 22.2 24.7
19.18 19.95 20.41 20.41 23.36 25.67
16.71 17.68 17.91 18.47 20.81 23.41
ltl, 5t1u ltzu 3% 1 t2g 4t1,
15.88 16.76 16.84 17.52 18.74 21.84
GVB-CI
ExDtl‘
(I
Reference 1 1.
19.15 19.96 20.39 20.39 23.29 25.64
18.63 19.44 19.89 19.65 22.83
16.88 18.05 18.09 18.98 20.95
15.7 17.0 17.0 18.6 19.8 22.9
Total energy of SFs, au -994.0287 -994.0287 -994.1862
~~~
Reference lob.
-6
MBPT Xa-SWd
-el
-6
Total energy of SF6, au -990.1283 -992.93 13 -993.7867 Reference loa.
Ilartree-Fock SCF‘
Orbital
References 5 and 6.
Reference
1 L.
p exponents, however, were chosen arbitrarily. In the improved virtual orbital (IVO) procedure for molecules the energy of a state arising from the excitation & $1 is calculated by constructing the hamiltonian for orbital dl with the appropriate ion core obtained by removing orbital $&.Thus for a configuration @ I 2 . . . Ij.Jn24kf#Jl
-
n
Hlvo = t 4- VN 4I=
(2J; - Ki) I
+ J&f K&
d
v~ + i= I 2Ji - Ki + k = 1 2aJk - bKk n
where d is the degeneracy of the ion core. For triply degenerate T states of the ion one has a = % and b = $$ for singlet coupling with the virtual orbital and a = 5/6 and b = 1 for triplet coupling. For doubly degenerate E states a = 314 and b = 0 for singlet states and a = 3/4 and b = 1 for triplet states. The excitation energy Ac(k I) is then given by
-At(k
1) =
* O = ld’l?’l
In the final state if excitation one has @f
where the plus and minus signs refer to singlet or triplet states, respectively. The coulomb and exchange operators are denoted by J and K. The excitation energies are then given by A E = € 1 - €&, where the el’s are the eigenvalues of Hlvo obtained in the space orthogonal to the occupied orbitals. Since all the ion states of interest in SF6 are degenerate, the hamiltonian is averaged over all components. The IVO hamiltonian then has the form
H1vo = t
many-electron wave functions the quantity P k / is multiplied by an appropriate factor related to the number of terms in the final state ]TI,,wave function. (Excited orbitals from the 5t1, excitation were used to calculate transition moments.) The initial state is the single-determinant ’AI, wave function
€1
- ck
where t/ is the orbital energy from the IVO calculation and €& is the orbital energy of the occupied orbital in the HartreeFock calculations. This procedure yields excitation energies which are average of configurations for the singlet or triplet manifolds. Since most of the excited orbitals are quite diffuse the splitting within these multiplets is expected to be quite small (-0.05 eV). The only valence excited orbital, as will be shown, is a nondegenerate alg orbital with no multiplet structure. In general one expects the IVO excitation energies to be in error by about the same error in the computed ionization potential for the ion core. For this reason the excitation energies are corrected by the difference between the Hartree-Fock and the experimental IP’s. As will be discussed this can lead to difficulties when the assignment of the ion states is uncertain. Transition Moments. Transition moments for the dipoleallowed transitions were computed rising the relation where k and I refer to the orbitals involved in the k I excitation. To obtain the overall transition moment between the +
Journal of the American Chemical Society / 99:4
@k
. . . 4 k q k . . . #n?nl
and
@/
correspond to a tlu(x)
= 2-”21@lTl . . . ($k?’/
+ $ / ? ‘ k ) . . . $n?’nl
-
(1) alg
(2)
and
-
vofx =
For a tl, becomes
t2g(XY)tl,b)
(2/2l/’)(tlU(x)lxlalg) = 2 1 ’ 2 P k /
(3)
t~ transition the quantity in parentheses in (2)
+Gb);& ;fY)
+ tZg(XZ)G(Z) + tlu
