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A note on the electronic structure of UF6
15 April 1978
VoIume 55, number 2
A NOTE ON TEE ELECTRONIC
STRUCfWRE
OF IJF,
Ame ROSÉN Deswtmetzt Of~hysicr. C3iaImers Universïty of Technology, Göteborg, Sweden Received 7 JuIy 1977 Revised mamwxipt received 12 December 1977
A non-reiativïstic Hartrce-Fock-Slater method has been uscd to calculate oneelectron energy levels, optical excïtations and ionization energïes for DF,. The spacíng and the leve1 ordering for some of the valence band levels are somewhat different compared witb earlier results obtained with the NS Xo method uGng the muffim-tin epproximation. The effect of relativity bas been investigated by comparing witb earlier calculated relativistic Dirac-Slater values. It is found that the reJativïstic effects are important and give quïte different structure compared wïth a non-relativistic treatment. Experimental absorption data and ionization energies are much better re-produced using the relativistic vahres compared witb the non-
relativïsticones.
f. Iutroduction The electronic structure of UF, bas been the subject of experimental as well as theoretical investigations for a considerable period *. However, in recent years the interest has increased considerably because UFg is a possrble candidate for the laser-induced isotape enrichment procedure. Gewis et al. [l] have studied the absorption spectra and observed a number of electronic transitions with a complex vibronic structure. The analysis of the absorption data was based on relativistic Dirac-Slater molecular calculations [2] accordiug to the procedure that bas been described iu earlier papers [3 1. Ee multiple scattering Xo (MS ti) method has also been applîed by two dìfferent groups of werkers to calculate ene-electron eigenvalues and optical transitions for UFg [4,5]. The leve1 ordering and the spacing between the valence levels in those calculatious are however somewhat different probably due to the muffi-tin approxïmation and the overlap procedure used by Boring er al. [4] _ It was therefore found to be interesting to perform a new non-relativistic calculation using the self-consistentcharge HFS method in which no “muffm-tin” ap-
’ A review of absorption
spectroscopy up to date and a lot of new data bas been given in ref. [ 11.
proxiruation is implemented [6] _ Relativistic effects as spin-orbit splitting and energy leve1 shifts can then also be investigated by compariug with the DiracSlater calculations [2]. fn the non-relativistic Hartree-Fock-Slater mode! the one-electron hamiltonian is given as (in Hartree atomic units) h=-$v2+
v(r),
where V(r) is the sum of the Coulomb aud exchauge potentials. This last part is obtained from the molecular charge density using the Slater Xcr method with the exchange parameter OL= 0.70. A vsriational method is used to fmd the molecular wavefunctions which are approximated by a linear combination of symmetry orbitals. The atomic orbitals which are used for the construction of the symmetry orbitals are generated with an atomic SCF program. Application of a variational procedure then gives the secular equation. _ The Hartree-FockSlater and overlap matrix elements are evaluated by using the discrete variational method as described in earlier calculations [7]. After each iteration the molecular orbitals are analyzed according to the Mulliken population scheme [8] in terms of the input atomic basis fuuctions ïu order to determine orbital charges (occupation numbers). These orbital charges are used to construct a new mo311
Volurne55, number 2
lecular charge density required to generate the molecular potential. The basis functions for uranium have been generated in the 5f3 6d 7s2 atomic confïguration. Koelling et al. [2] used the atomic confîguration 5fz12 5f!,n 6din 6d$2 7s& in the generation of the basrs functions. A new relativïstic molecular calculation wasperfom~ed for UFg in connection wïth thïs work using basis functions generated in the 5f$ 5en 6d$2 6d$ 7sf12 atomic configuration. The molecular eigenvalues were found to be less than 0.15 eV uniformly shifted compared with those of Koelling et al. [2] _ The differente between the nonrelativistic values given in this work and the relativistic ones calculated by Koelling is therefore within the uucertainty attributable to relativistic effects.
2. Ground state properties The calculations were performed using the experimental equilibrium bond length R(U-F) = 3.768 au as in the DS calculations. The point group of the UFg molecule is Oh_ Different theoretical eigenvalues in the valence region are presented in _fîg_1_ To the left in the fïgure are the MS XCYresults as calculated by Boring et al. [4] and Maylotte et al. [S] , using bond lengths of 3.742 au and 3.761 au, respectively. These Ievels are then compared with the HFS values calculated in this work and the correspondïng relativistic DS eigenvalues [2]. The splitting of the non-relativistic levels ïnto the relativistic ones has been indïcated by using the compatability relations between the single aud double group of 0,. The width of the valente band is about the same Ïn all calculations. The non-relativistic calculations give the same leve1 ordering for the last occupied valence levels tlu, ak, tk and t2, while the ordering is somewhat different for the áeeper levels tZgr Llu and eg_ The spacing between the Ievels varies a great deal between the two MS Xor calculotions which is probably due to the muffïm-tin approxìmation wïth different type of overlap aud slïghtiy different bond lengths. The HFS values on the other hand are calculated without using the “muffm-tin” approximation and should therefore give a more realïstic description of the electronic structure. In the relativistic calculations the Ievels are brought together in certaiu groups with comparatively large splittings as for example for 312
15 April 1978
CHEMICAL PHYSICS LEITERS
Non-Relativistic
Rebtivktïc :
-m
42
-14 1
-I
Fig. 1. Comparïson of different non-relatïvïstic and relativïstic eigenvalues for IJFg- Col-ws A and B represent the values calculated by Boring et al. [4] and Maylotte et al. [S] using the MS Xorscheme. The columns denoted HFS 2nd DS represent the non-relativistic values calculated in this werk and the corresponding relativistic ones obtained by KoeUing et al. [2]_
the last occupied tlu level. We notice alsc that the valence levels are more tightly bound while the optical levels (of mainly Sf-character) have somewhat hïgher energïes in the relativistic treatment. Thïs is due to the mixïng in the valence band of the uranium 6~96~1~2 and 6~3 ~2 levels which are more bound in an atomic relativîstic calcuiation compared with a nonrelativistic one. The atomic 5f5j2 and 5f7/2 levels on the other hand are expanded due to the increased shielding of the nuclear charge from the more tightly bound s and p electrons. These effects are also seen from a Mulliken population analysis which gives in the non-relativïstic case for the uranium 6p and Sf electrons the occupation nurnbers 5.71 and 3.20 compared with 5.90 and 2.75 in the relativistic case. The r,et charges on the uranium and fluorine atoms are +1.554, -0.259 in the nonrelativístic calculations compared with +1.705 and -0.284 for the relativistic ones. This shows that the commonly used crystal field model with a 6+ metal ion and l- ligand ions is not in agreement with mo-
VoIume55, number 2
15 April 1978
Ch’EMICALPHYSICS LETTERS
lecular calculations as has also been found for Sdmetal hexafluorides [9].
” Fs
3. Excited state properties Using the ground state calculation in fig_ 1 optical transitions can be evaluated as the differente in oneelectron eigenvalues. In order to take into account relaxation effects and the non-validity of Koopmans’ theorem in the HFS method, better values are obtained by using the transition state procedure [lO] _ A glance at fig. 1 shows that the energies for nonrelativistic electronic transitions from the valence band to the excited levels should be of the order of 1 eV smaller than the corresponding relativistic values. The energy for the transition tl,, + a2,, is 1.46 eV using the ground state eigenvalues and 1.42 eV in a transition state calculation. This should be compared with the corresponding relativïstic values of 2.30 eV and 2.78 eV, respectively. The experimental value is 3.04 ev. Generally the relativistic calculations [2] give rather good agreement with the experimectal values [ll.
I....I....l....l,...i....l.... 18 lï Bi”d,“cJ
19
15
15
14
13
energy
2. Comparison of experimental spectra of KaAson et al. [ 121 wïth leve1 structures as given in the ground state calculations in fíg. 1 and transition state caiculations for ionïzation fiom the last occupied valence IeveL The position of the last occupied leve1 in all the calculations was chosen so that it coincides wïth the vertical ionization energy for the fust peak in the experimental spectra. Fig.
4. Ionization energies tained between
The ionization energies for UF, have been measured earlier by Potts et al. [l 1J and have more recently been measured Dy Karlsson et al. 1121 with a resolution of the vibrational structure. A recordïng of theïr vaIence band spectra is presented in Eg. 2. NO assignments of the different peaks were made in thís last work. A non-relativistic transition state calculation for the last occupied tlu leve1 gave a value of 12.1 eV w.hïch should be compared with the relativistic value of 12.9 eV [2] _This should be compared with the first experimental ionïzation energy of 14.14 eV 112,131. The relativistic and non-relativistic groundstate molecular levels are compared with the experimental ionïzation energïes in fig. 2. The position of the last occupied leve1 was chosen so that it coincides wïth the experimental vertical ionization energy for the first peak in the valence spectra. The reïativistic calculations reproduce the positïons of the experïmental peaks very well for the ground state as well as transition state calculations. A shift of 0.2 eV is ob-
these calculations.
AU tbe non-relativ-
istic calculations on the other hand do not reproduce the spectra so well. A more crïtical test on the calculations would be to calculate a theoretical ionïzation spectrum. Thìs involves calculations of dipole-matrix elements between the valence hand wavefunctions and the out-going electron in the experiments wlüch is a rather complicated procedure. A better and easïer way to check the assignments can be obtaïned from X-ray emission data from the valence band to tightly bound molecular levels (or atomic levels) located at some of the atoms in +be molecule. The allowed transitions are in that case @ven by the electric dipole radiation selection rules. Such data now seems to be available for some light molecules [ 131.
Acknowkdgement The calculations were performed at Gothenburg 313
Volume 55, number 2 University
Cornputing
W. Germany. Fricke pitality
Thanks
for arranging during
S. Larsson
CHEMICAL Centre and at GSL Darrnstadt,
@ven to Professo; B. my vïsit at GSI and for the hosshouId be
that time. The author
for vaIuable
is indebted
to Dr.
comments.
References [ 11 W. B. Lewìs, L. B. Asprey, L. H. Jones, R_ S. McDowell, E. W. Rabideau and R. T. Paine, J. Chem. Phys. 65 (1976) 2íO7. [2) D. D. Koelling, D. E. EUis and R. J. Bartlett, J. Chem. Phys. 65 (1976) 3331. [3 ] A. Rosén and D. E. Eilis. Chem. Phys. Letters 27 (1974) 595; J. Chem. Phyr. 62 (1975) 3039. [4] M. Boring and J. W. Moskowitz, Chem. Phys. Letters 38 (1976) 185.
314
PHYSKCS LETTERS
15 April 1978
[S] D. H- Maybtte. R. L. St.Petersand R. P- Messmer, Chem. Phys_ Letters 38 (1976) 181. 16] A. Rosén, D. E. EU&, H. Adachì and F. W. Averill, J. Chem. Phys_ 65 (1976) 3629. [7] D. E. Ellis and G. S. Païnter, Phys. Rev. B2 (1970) 2887. [S] R. S. MuBiken, J. Chem. Phys. 23 (1955) 1833,184l. [9] A. Rosén and D. E. Ellis, Z. Phy$k A283 (1977) 3. [ 101 J. C. Slater, The selfconsïstent field for molecules and solids (McGraw-Hïll, New York, 1974). [ll] A- W- Potts, H. J. Lempka, D. G. Sheets and W. C. Price, Phil. Trans. Roy. Sec. 268 (1970) 59. [12] L. Karlsson. L. Mattsson, R. Jadrny, T. Bergmarkand K. Siegbahn, Physica Scripta 14 (1976) 230. [ 131 K. Siegbahn, L. Werme, 8. Crennborg, J. Nordgren and C. Nordlïng, Phys. Letters 41A (1972) 111; L. 0. Wenne, J. Nordgren, H. Agren, C. Nordling and K. Siegbahn, Z. Physïk A272 (1975) 131.
VoIume 55, number 2
A NOTE ON TEE ELECTRONIC
STRUCfWRE
OF IJF,
Ame ROSÉN Deswtmetzt Of~hysicr. C3iaImers Universïty of Technology, Göteborg, Sweden Received 7 JuIy 1977 Revised mamwxipt received 12 December 1977
A non-reiativïstic Hartrce-Fock-Slater method has been uscd to calculate oneelectron energy levels, optical excïtations and ionization energïes for DF,. The spacíng and the leve1 ordering for some of the valence band levels are somewhat different compared witb earlier results obtained with the NS Xo method uGng the muffim-tin epproximation. The effect of relativity bas been investigated by comparing witb earlier calculated relativistic Dirac-Slater values. It is found that the reJativïstic effects are important and give quïte different structure compared wïth a non-relativistic treatment. Experimental absorption data and ionization energies are much better re-produced using the relativistic vahres compared witb the non-
relativïsticones.
f. Iutroduction The electronic structure of UF, bas been the subject of experimental as well as theoretical investigations for a considerable period *. However, in recent years the interest has increased considerably because UFg is a possrble candidate for the laser-induced isotape enrichment procedure. Gewis et al. [l] have studied the absorption spectra and observed a number of electronic transitions with a complex vibronic structure. The analysis of the absorption data was based on relativistic Dirac-Slater molecular calculations [2] accordiug to the procedure that bas been described iu earlier papers [3 1. Ee multiple scattering Xo (MS ti) method has also been applîed by two dìfferent groups of werkers to calculate ene-electron eigenvalues and optical transitions for UFg [4,5]. The leve1 ordering and the spacing between the valence levels in those calculatious are however somewhat different probably due to the muffi-tin approxïmation and the overlap procedure used by Boring er al. [4] _ It was therefore found to be interesting to perform a new non-relativistic calculation using the self-consistentcharge HFS method in which no “muffm-tin” ap-
’ A review of absorption
spectroscopy up to date and a lot of new data bas been given in ref. [ 11.
proxiruation is implemented [6] _ Relativistic effects as spin-orbit splitting and energy leve1 shifts can then also be investigated by compariug with the DiracSlater calculations [2]. fn the non-relativistic Hartree-Fock-Slater mode! the one-electron hamiltonian is given as (in Hartree atomic units) h=-$v2+
v(r),
where V(r) is the sum of the Coulomb aud exchauge potentials. This last part is obtained from the molecular charge density using the Slater Xcr method with the exchange parameter OL= 0.70. A vsriational method is used to fmd the molecular wavefunctions which are approximated by a linear combination of symmetry orbitals. The atomic orbitals which are used for the construction of the symmetry orbitals are generated with an atomic SCF program. Application of a variational procedure then gives the secular equation. _ The Hartree-FockSlater and overlap matrix elements are evaluated by using the discrete variational method as described in earlier calculations [7]. After each iteration the molecular orbitals are analyzed according to the Mulliken population scheme [8] in terms of the input atomic basis fuuctions ïu order to determine orbital charges (occupation numbers). These orbital charges are used to construct a new mo311
Volurne55, number 2
lecular charge density required to generate the molecular potential. The basis functions for uranium have been generated in the 5f3 6d 7s2 atomic confïguration. Koelling et al. [2] used the atomic confîguration 5fz12 5f!,n 6din 6d$2 7s& in the generation of the basrs functions. A new relativïstic molecular calculation wasperfom~ed for UFg in connection wïth thïs work using basis functions generated in the 5f$ 5en 6d$2 6d$ 7sf12 atomic configuration. The molecular eigenvalues were found to be less than 0.15 eV uniformly shifted compared with those of Koelling et al. [2] _ The differente between the nonrelativistic values given in this work and the relativistic ones calculated by Koelling is therefore within the uucertainty attributable to relativistic effects.
2. Ground state properties The calculations were performed using the experimental equilibrium bond length R(U-F) = 3.768 au as in the DS calculations. The point group of the UFg molecule is Oh_ Different theoretical eigenvalues in the valence region are presented in _fîg_1_ To the left in the fïgure are the MS XCYresults as calculated by Boring et al. [4] and Maylotte et al. [S] , using bond lengths of 3.742 au and 3.761 au, respectively. These Ievels are then compared with the HFS values calculated in this work and the correspondïng relativistic DS eigenvalues [2]. The splitting of the non-relativistic levels ïnto the relativistic ones has been indïcated by using the compatability relations between the single aud double group of 0,. The width of the valente band is about the same Ïn all calculations. The non-relativistic calculations give the same leve1 ordering for the last occupied valence levels tlu, ak, tk and t2, while the ordering is somewhat different for the áeeper levels tZgr Llu and eg_ The spacing between the Ievels varies a great deal between the two MS Xor calculotions which is probably due to the muffïm-tin approxìmation wïth different type of overlap aud slïghtiy different bond lengths. The HFS values on the other hand are calculated without using the “muffm-tin” approximation and should therefore give a more realïstic description of the electronic structure. In the relativistic calculations the Ievels are brought together in certaiu groups with comparatively large splittings as for example for 312
15 April 1978
CHEMICAL PHYSICS LEITERS
Non-Relativistic
Rebtivktïc :
-m
42
-14 1
-I
Fig. 1. Comparïson of different non-relatïvïstic and relativïstic eigenvalues for IJFg- Col-ws A and B represent the values calculated by Boring et al. [4] and Maylotte et al. [S] using the MS Xorscheme. The columns denoted HFS 2nd DS represent the non-relativistic values calculated in this werk and the corresponding relativistic ones obtained by KoeUing et al. [2]_
the last occupied tlu level. We notice alsc that the valence levels are more tightly bound while the optical levels (of mainly Sf-character) have somewhat hïgher energïes in the relativistic treatment. Thïs is due to the mixïng in the valence band of the uranium 6~96~1~2 and 6~3 ~2 levels which are more bound in an atomic relativîstic calcuiation compared with a nonrelativistic one. The atomic 5f5j2 and 5f7/2 levels on the other hand are expanded due to the increased shielding of the nuclear charge from the more tightly bound s and p electrons. These effects are also seen from a Mulliken population analysis which gives in the non-relativïstic case for the uranium 6p and Sf electrons the occupation nurnbers 5.71 and 3.20 compared with 5.90 and 2.75 in the relativistic case. The r,et charges on the uranium and fluorine atoms are +1.554, -0.259 in the nonrelativístic calculations compared with +1.705 and -0.284 for the relativistic ones. This shows that the commonly used crystal field model with a 6+ metal ion and l- ligand ions is not in agreement with mo-
VoIume55, number 2
15 April 1978
Ch’EMICALPHYSICS LETTERS
lecular calculations as has also been found for Sdmetal hexafluorides [9].
” Fs
3. Excited state properties Using the ground state calculation in fig_ 1 optical transitions can be evaluated as the differente in oneelectron eigenvalues. In order to take into account relaxation effects and the non-validity of Koopmans’ theorem in the HFS method, better values are obtained by using the transition state procedure [lO] _ A glance at fig. 1 shows that the energies for nonrelativistic electronic transitions from the valence band to the excited levels should be of the order of 1 eV smaller than the corresponding relativistic values. The energy for the transition tl,, + a2,, is 1.46 eV using the ground state eigenvalues and 1.42 eV in a transition state calculation. This should be compared with the corresponding relativïstic values of 2.30 eV and 2.78 eV, respectively. The experimental value is 3.04 ev. Generally the relativistic calculations [2] give rather good agreement with the experimectal values [ll.
I....I....l....l,...i....l.... 18 lï Bi”d,“cJ
19
15
15
14
13
energy
2. Comparison of experimental spectra of KaAson et al. [ 121 wïth leve1 structures as given in the ground state calculations in fíg. 1 and transition state caiculations for ionïzation fiom the last occupied valence IeveL The position of the last occupied leve1 in all the calculations was chosen so that it coincides wïth the vertical ionization energy for the fust peak in the experimental spectra. Fig.
4. Ionization energies tained between
The ionization energies for UF, have been measured earlier by Potts et al. [l 1J and have more recently been measured Dy Karlsson et al. 1121 with a resolution of the vibrational structure. A recordïng of theïr vaIence band spectra is presented in Eg. 2. NO assignments of the different peaks were made in thís last work. A non-relativistic transition state calculation for the last occupied tlu leve1 gave a value of 12.1 eV w.hïch should be compared with the relativistic value of 12.9 eV [2] _This should be compared with the first experimental ionïzation energy of 14.14 eV 112,131. The relativistic and non-relativistic groundstate molecular levels are compared with the experimental ionïzation energïes in fig. 2. The position of the last occupied leve1 was chosen so that it coincides wïth the experimental vertical ionization energy for the first peak in the valence spectra. The reïativistic calculations reproduce the positïons of the experïmental peaks very well for the ground state as well as transition state calculations. A shift of 0.2 eV is ob-
these calculations.
AU tbe non-relativ-
istic calculations on the other hand do not reproduce the spectra so well. A more crïtical test on the calculations would be to calculate a theoretical ionïzation spectrum. Thìs involves calculations of dipole-matrix elements between the valence hand wavefunctions and the out-going electron in the experiments wlüch is a rather complicated procedure. A better and easïer way to check the assignments can be obtaïned from X-ray emission data from the valence band to tightly bound molecular levels (or atomic levels) located at some of the atoms in +be molecule. The allowed transitions are in that case @ven by the electric dipole radiation selection rules. Such data now seems to be available for some light molecules [ 131.
Acknowkdgement The calculations were performed at Gothenburg 313
Volume 55, number 2 University
Cornputing
W. Germany. Fricke pitality
Thanks
for arranging during
S. Larsson
CHEMICAL Centre and at GSL Darrnstadt,
@ven to Professo; B. my vïsit at GSI and for the hosshouId be
that time. The author
for vaIuable
is indebted
to Dr.
comments.
References [ 11 W. B. Lewìs, L. B. Asprey, L. H. Jones, R_ S. McDowell, E. W. Rabideau and R. T. Paine, J. Chem. Phys. 65 (1976) 2íO7. [2) D. D. Koelling, D. E. EUis and R. J. Bartlett, J. Chem. Phys. 65 (1976) 3331. [3 ] A. Rosén and D. E. Eilis. Chem. Phys. Letters 27 (1974) 595; J. Chem. Phyr. 62 (1975) 3039. [4] M. Boring and J. W. Moskowitz, Chem. Phys. Letters 38 (1976) 185.
314
PHYSKCS LETTERS
15 April 1978
[S] D. H- Maybtte. R. L. St.Petersand R. P- Messmer, Chem. Phys_ Letters 38 (1976) 181. 16] A. Rosén, D. E. EU&, H. Adachì and F. W. Averill, J. Chem. Phys_ 65 (1976) 3629. [7] D. E. Ellis and G. S. Païnter, Phys. Rev. B2 (1970) 2887. [S] R. S. MuBiken, J. Chem. Phys. 23 (1955) 1833,184l. [9] A. Rosén and D. E. Ellis, Z. Phy$k A283 (1977) 3. [ 101 J. C. Slater, The selfconsïstent field for molecules and solids (McGraw-Hïll, New York, 1974). [ll] A- W- Potts, H. J. Lempka, D. G. Sheets and W. C. Price, Phil. Trans. Roy. Sec. 268 (1970) 59. [12] L. Karlsson. L. Mattsson, R. Jadrny, T. Bergmarkand K. Siegbahn, Physica Scripta 14 (1976) 230. [ 131 K. Siegbahn, L. Werme, 8. Crennborg, J. Nordgren and C. Nordlïng, Phys. Letters 41A (1972) 111; L. 0. Wenne, J. Nordgren, H. Agren, C. Nordling and K. Siegbahn, Z. Physïk A272 (1975) 131.