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Spin-orbit coupling constants of transition metal atoms and ions in density functional theory
THEO CHEM Journal of Molecular Structure (Theochem) 361 (1996) I5- 19
Spin-orbit
coupling constants of transition metal atoms and ions in density functional theory M. Vijayakumar’,
M.S. Gopinathan*
Department of Chemistry, Indian Institute of Technology, Madras-600 036. India
Received 3 April 1995; accepted 7 April 1995
Abstract Spin-orbit coupling constants for various orbitals of first-, second- and third-row transition metal atoms and ions have been calculated using a quasi-relativistic density functional method reported earlier by us. Our results are comparable with the fully relativistic Dirac-Hartree-Fock method and with available experimental values. Keywords: Density functional theory; Local density functional; Spin-orbit
1. Introduction Reliable calculations of atomic and molecular properties of d- and f-series elements and molecules containing these elements present a formidable problem in theoretical chemistry [l]. In such systems, relativistic and electron correlation effects are important. While the multi-configuration Dirac-Hartree-Fock (MCDHF) method [2] takes into account both relativistic and correlation effects, there are several simpler local-density functional (LDFT) methods [3-61 that deal with correlation. A quasi-relativistic density functional method incorporating exchange and correlation effects was reported earlier by us [7]. This method has already been applied with reasonable success to various atomic properties such as total energy, orbital ionisation potential, spin-orbit coupling * Corresponding author. ’ Present address: Department of Chemistry, University of Pondicherry, Pondicherry 605 014, India.
coupling constant; Transition metal atoms
for some selected atoms [7j, pair correlation energies [8] and Auger energies [9]. It has also been extended to molecules [lo]. Here we apply the method to a detailed study of spin-orbit coupling constants of various orbitals in the first-, secondand third-row transition metal atoms and ions. The calculated values of spin-orbit coupling constants and radial moments, (T”), can be used to compute spin-orbit splitting in molecules [ 111.
2. Method We briefly outline the theoretical method here. Details can be found in Ref. [7]. This is an orbitalbased DFT method of the Kohn-Sham type, with exchange and correlation functionals derived from ab initio considerations of the properties of Fermi and Coulomb holes. It is virtually parameter free except for a universal parameter representing the ratio of Fermi and Coulomb hole radii. Relativistic terms are introduced into the
0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-1280(95)04297-O
16
M. VQayakumar. MS.
Table 1 Spin-orbit
coupling
constants
of Molecular
GopinathanlJournal
HFR”
DHFb
DFT
Exptd
2P 3P 3d
222.30 51.90 5.90
209.02 48.92 5.80
208.79 48.29 5.82
4P 4d 4f
13.60 1.40 0.25
13.05 I .44 0.27
12.77 I .41 0.27
206.30 48.20 5.74 12.60 1.41 _
5P 5d
3.02 0.27
3.15 0.29
3.16 0.29
3.28 0.32
a Ref. [13]. b Values computed using Eq. (3) of text with DHF orbital eigenvalues from Ref. [I 51. ’ Computed using eigenvalues of Eq. (1) in Eq. (3) of text. d Quoted in Ref. [ 151.
non-relativistic Schrodinger Hamiltonian through the non-relativistic reduction of the Dirac Hamiltonian [ 121.The final orbital equations to be solved are (see Eq. (53) of Ref. [7]):
[-
X
1(1+ 1) -+
pdj(r)
r2 =
u(r) + &dr)
f HD(r)
+ &ok)
1
+ UC(r) + usi
3. Results The spin-orbit coupling constants calculated by the present DFT method for various orbitals of plutonium, as a typical example, are given in Table 2 Spin-orbit
+ vex(r) + zlcc
<“i =
&
(%tj+ - %tj-)
DHFa
metal atoms (in Ry)
0.4321 0.0537 0.0024
0.4446 0.0546 0.0036
0.5488 0.0734 0.0020
cu
2P 3P 3d
1.0189 0.1365 0.0109
I.041 0.1350 0.0109
0.9702 0.1469 0.0074
MO
2P 3P 3d
5.3280 0.8980 0.1010 0.1302 0.0060
5.3281 0.8852 0.1021 0.1331 0.0085
5.1449 0.8134 0.0941 0.1029 0.0050
8.7380 1.5731 0.1878 0.2627 0.0196
8.7385 1.5473 0.1883 0.2673 0.0202
8.4769
66.4175 14.5848 1.8839 3.4152 0.3846 0.0509 0.0193
66.7729 14.4849
1.8786
1.8500
3.3796 0.3739 0.0511 0.0284
3.2732 0.3646 0.0462 0.0190
90.0977 20.1543 2.5749 4.8939 0.5553 0.0842 0.8405 0.0518
90.4748 19.9507 2.5682 4.8251 0.5382 0.0836 0.8563 0.0548
88.9339 19.8448 2.4989 4.7236 0.5321 0.0777 0.8329 0.0447
4P 4d 2P 3P 3d 4P 4d W
2P 3P 3d 4P 4d 4f 5d 2P 3P 3d 4P 4d 4f 5P 5d
DFTb
Expt.C
2P 3P 3d
Au
where E is the spin-orbit coupling constant, and rand s’ are the orbital and spin angular momenta respectively. The & values are calculated in the present method as well as in the DHF methods using the expression
for transition
Cr
(2)
H,,= ri.s'
constants
Orbital
Ag
where ZI’, ?I’~, tiex and vcc are the local density expressions for electron-electron Coulomb, selfinteraction, exchange and correlation terms as defined in Ref. [7]. The relativistic potential H,, Ho and &, are reSpeCtiVe]y the IIUSS-Velocity, Darwin correction and spin-orbit interaction terms which involve u(r) and its derivative [7]. The operator for the one-electron spin-orbit interaction can be written as
coupling
Atom
(1)
%tjPdj(r)
Here, v(r) is the central field potential: 21(r) = -22/r
361 (1996) 15-19
where en/j+ and [n/j_ are the eigenvalues of Eq. (1) withj, = 1 + (l/2) andj- = I - (l/2) respectively.
(in Ry) for plutonium
Orbital
d2 dr2’
Structure (Theochem)
1.5092 0.1764 0.2646
65.5122 14.4058
a Calculated using orbital eigenvalues from Ref. [15] in Eq. (3). b Calculated using eigenvalues of Eq. (I) of text in Eq. (3). ’ Calculated using experimental orbital ionisation energies from Ref. [16] in Eq. (3).
M. Vijayakumar. MS.
1.20
a-
owat~ _ &*--A
Sc Y LU 0.25
GopinarhanlJournal of Molecular Struc!ure (Theochem)
1st row 2nd row(xl0) 3rd row(x100)
Ti Zr Hf
2000-
ODOW AbObA x I L-I-
17
361 (1996) IS-19
DFT DHF Expt 3c ion
V Cr Mn Fe Co Ni Cu Nb MO Tc Ru Rh Pd Ag Ta W Re OS Ir Pt Au
c-30+-0
1st row
oww 6aw-a
2nd row(xl0) 3rd row(x100)
O!
(b) _r H&”
I
I
38
d
40
I
I
42 Atomic
44 Number(Z)
6
46
Fig. 3. Spin-orbit coupling constants for 4d orbital of secondrow transition elements and their triply positive ions.
!
0.00
s
SC Y Lu
n
* I B ’ I ’ V Cr Mn Fe Co Ni Cu Nb MO Tc Ru Rh Pd Ag Ta W Re OS lr Pt Au
Ti Zr Hf
Fig. 1. Spin-orbit coupling orbitals of the first-, second-
’
constants for (a) 2p and (b) 3p and third-row transition elements.
Table 1. These are compared with results of the HFR and DHF methods and with experimental values. The DFT values are clearly superior to those of the HFR method. It is gratifying that the relatively simpler DFT method closely approximates the DHF as well as experimental values. Table 2 gives the spin-orbit coupling constants
for the orbitals in some first-, second- and thirdrow transition elements. The DHF values given in this Table were calculated using reported orbital eigenvalues in Eq. (2). “Experimental” values were similarly computed using experimental orbital binding energies (quoted in Ref. [14]) in the same equation. Generally, the present DFT values are again quite close to the DHF and experimental values. The well known trends of spin-orbit coupling increasing with 2 and decreasing with I for a given Z, which follows from Dirac’s theory [12], are evident from this Table. The calculated DFT values of <,,, for 2p and 3p orbitals of the transition elements are shown 6500 _~ -I o+w m* 5500 - AHL-H
OFT DHF Expt. 3+lon
/
/+
4500 ‘i $
3500-
C ._ ;
25OO1500-
0 20
I 22
24 Atomic
I 26 Number(Z)
Fig. 2. Spin-orbit coupling constants for 3d orbital transition elements and their triply positive ions.
I 28
1 30
J
500
7’0
712 Atomic
of first-row
74
16
78
I 1
number(Z)
Fig. 4. Spin-orbit coupling constants for 5d orbital transition elements and their triply positive ions.
of third-row
M. Vijayakumar. MS.
18
~o~inathanlJaurna1 of Molecular Structure (Theochem)
361 (1996) IS-19
I 0.0
(a)
ooooo l .-u
DFT
I _-_-----_-_--_
(d)
t
DHF
2 -0.01 .c G Q -0.4
-0.06
[
70
b
I I 22 2; Atomic Numbe?&
0.00
I
w 000~
-0.10 a c ;; w--o.20 Q
2
-0.30
-0.10
’ 3’e
I
38
I;‘0
I I 42 44 Atomic Number(Z)
I 4’0 Ator%
I 44 Number(Z)
I
I
74 76 Atomic Number (2)
I
78
I
80
Fig. 5. Deviation from “experimental” values of calculated spin-orbit coupling constants by the DHF and DPT methods for (a) 2p orbital of first-row, (b) 2p orbital of second-row, (c) 3p of second-row, and (d) 4p of third-row transition elements.
I 28
(b) __ _____ -_---__----s
75
DFT DHF
I 46
48
I 46
48
graphically in Figs. l(a) and l(b). The <,,)values for the 3d, 4d and 5d orbitals of the first-, second- and third-row transition atoms and their triply positive ions are shown in Figs. 2-4. These Figures also show the experimental and DHF results for the neutral atoms. The experimental variation of &, with atomic number is closely followed by both
DHF and DFT methods. However, DFT values are consistently somewhat higher than DHF and experimental results. The values for the ions are always larger than those of the neutral atoms. This is expected since the ion has a greater effective nuclear charge, resulting in a decreasing expectation value for the radial distance of the electron from the nucleus. The deviations of the DFT and DHF results from the experimental values for 2p orbitals of the first-row and second-row transition elements are displayed in Figs. 5(a) and 5(b). Similar plots of deviations of the two methods for 3p of the second row and 4p of the third row are shown in Figs. 5(c) and 5(d). These curves show that the DFT and DHF results are very close and have nearly the same trend in their deviation from experiment. Two further comments may be added based on some of our results not reported here [14]: (1) removal of electrons from the outer shell has little effect on the spin-orbit coupling of inner orbitals; (2) electron correlation has little effect on spinorbit coupling constants, as can be shown by removing the correlation potential term, 21~in Eq. (2).
References [l] C.W. Bauschhcher and S.R. Langhoff, Adv. Chem. Phys., 77 (1990) 103. [2] J.P. Desclaux, Comp. Phys. Commun., 9 (1975) 31.
M. Vijayakumar. MS.
GopinathanlJournal of Molecular Structure (Theochem)
[3] D.R. Salahub, Adv. Chem. Phys., 69 (1987) 447. [4] R.D. Cowan, Theory of Atomic Structure and Spectra, University of California, Berkeley, 1981. [S] E.J. Baerends, J.G. Snijders, C.A. de Lange and G. Jonkers, in J.P. Dahl and J. Avery (Eds.), Local Density Approximations in Quantum Chemistry and Solid State Physics, Plenum, New York, 1984, p. 415. [6] J.H. Wood andA.M. Boring, Phys. Rev. B, 18 (1978) 2701. [7] M. Vijayakumar, N. Vaidehi and M.S. Gopinathan, Phys. Rev. A, 40 (1989) 6834. [8] M. Vijayakumar and MS. Gopinathan, J. Chem. Phys., 91 (1992) 6639. [9] M. Vijayakumar and M.S. Gopinathan, Phys. Rev. A, 44 (1991) 2850. [IO] M. Vijayakumar and M.S. Gopinathan, J. Chem. Phys., 98 (2993) 4009.
361 (1996) 15-19
19
[l l] E.M. Spain and M.D. Morse, J. Chem. Phys., 97 (1992) 4641. [12] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Academic Press, New York, 1957. [ 131 R.D. Cowan and D.C. Griffin, J. Opt. Sot. Am., 66 (1976) 1010. [l4] M. Vijayakumar, Ph.D. Thesis, Indian Institute of Technology, Madras, 1990. [15] J.P. Desclaux, At. Data Nucl. Data Tables, 12 (1973) 311. [16a] CRC Handbook of Chemistry and Physics, 68th Edition, 198771988, CRC Press, Rota Baton, FL, 1987. [16b] T.A. Carlson, Photoelectron and Auger Spectroscopy, Plenum, New York, 1975. [16c] M. Gerloch, Orbitals, Terms and States, Wiley, Chichester, 1986.
Spin-orbit
coupling constants of transition metal atoms and ions in density functional theory M. Vijayakumar’,
M.S. Gopinathan*
Department of Chemistry, Indian Institute of Technology, Madras-600 036. India
Received 3 April 1995; accepted 7 April 1995
Abstract Spin-orbit coupling constants for various orbitals of first-, second- and third-row transition metal atoms and ions have been calculated using a quasi-relativistic density functional method reported earlier by us. Our results are comparable with the fully relativistic Dirac-Hartree-Fock method and with available experimental values. Keywords: Density functional theory; Local density functional; Spin-orbit
1. Introduction Reliable calculations of atomic and molecular properties of d- and f-series elements and molecules containing these elements present a formidable problem in theoretical chemistry [l]. In such systems, relativistic and electron correlation effects are important. While the multi-configuration Dirac-Hartree-Fock (MCDHF) method [2] takes into account both relativistic and correlation effects, there are several simpler local-density functional (LDFT) methods [3-61 that deal with correlation. A quasi-relativistic density functional method incorporating exchange and correlation effects was reported earlier by us [7]. This method has already been applied with reasonable success to various atomic properties such as total energy, orbital ionisation potential, spin-orbit coupling * Corresponding author. ’ Present address: Department of Chemistry, University of Pondicherry, Pondicherry 605 014, India.
coupling constant; Transition metal atoms
for some selected atoms [7j, pair correlation energies [8] and Auger energies [9]. It has also been extended to molecules [lo]. Here we apply the method to a detailed study of spin-orbit coupling constants of various orbitals in the first-, secondand third-row transition metal atoms and ions. The calculated values of spin-orbit coupling constants and radial moments, (T”), can be used to compute spin-orbit splitting in molecules [ 111.
2. Method We briefly outline the theoretical method here. Details can be found in Ref. [7]. This is an orbitalbased DFT method of the Kohn-Sham type, with exchange and correlation functionals derived from ab initio considerations of the properties of Fermi and Coulomb holes. It is virtually parameter free except for a universal parameter representing the ratio of Fermi and Coulomb hole radii. Relativistic terms are introduced into the
0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-1280(95)04297-O
16
M. VQayakumar. MS.
Table 1 Spin-orbit
coupling
constants
of Molecular
GopinathanlJournal
HFR”
DHFb
DFT
Exptd
2P 3P 3d
222.30 51.90 5.90
209.02 48.92 5.80
208.79 48.29 5.82
4P 4d 4f
13.60 1.40 0.25
13.05 I .44 0.27
12.77 I .41 0.27
206.30 48.20 5.74 12.60 1.41 _
5P 5d
3.02 0.27
3.15 0.29
3.16 0.29
3.28 0.32
a Ref. [13]. b Values computed using Eq. (3) of text with DHF orbital eigenvalues from Ref. [I 51. ’ Computed using eigenvalues of Eq. (1) in Eq. (3) of text. d Quoted in Ref. [ 151.
non-relativistic Schrodinger Hamiltonian through the non-relativistic reduction of the Dirac Hamiltonian [ 121.The final orbital equations to be solved are (see Eq. (53) of Ref. [7]):
[-
X
1(1+ 1) -+
pdj(r)
r2 =
u(r) + &dr)
f HD(r)
+ &ok)
1
+ UC(r) + usi
3. Results The spin-orbit coupling constants calculated by the present DFT method for various orbitals of plutonium, as a typical example, are given in Table 2 Spin-orbit
+ vex(r) + zlcc
<“i =
&
(%tj+ - %tj-)
DHFa
metal atoms (in Ry)
0.4321 0.0537 0.0024
0.4446 0.0546 0.0036
0.5488 0.0734 0.0020
cu
2P 3P 3d
1.0189 0.1365 0.0109
I.041 0.1350 0.0109
0.9702 0.1469 0.0074
MO
2P 3P 3d
5.3280 0.8980 0.1010 0.1302 0.0060
5.3281 0.8852 0.1021 0.1331 0.0085
5.1449 0.8134 0.0941 0.1029 0.0050
8.7380 1.5731 0.1878 0.2627 0.0196
8.7385 1.5473 0.1883 0.2673 0.0202
8.4769
66.4175 14.5848 1.8839 3.4152 0.3846 0.0509 0.0193
66.7729 14.4849
1.8786
1.8500
3.3796 0.3739 0.0511 0.0284
3.2732 0.3646 0.0462 0.0190
90.0977 20.1543 2.5749 4.8939 0.5553 0.0842 0.8405 0.0518
90.4748 19.9507 2.5682 4.8251 0.5382 0.0836 0.8563 0.0548
88.9339 19.8448 2.4989 4.7236 0.5321 0.0777 0.8329 0.0447
4P 4d 2P 3P 3d 4P 4d W
2P 3P 3d 4P 4d 4f 5d 2P 3P 3d 4P 4d 4f 5P 5d
DFTb
Expt.C
2P 3P 3d
Au
where E is the spin-orbit coupling constant, and rand s’ are the orbital and spin angular momenta respectively. The & values are calculated in the present method as well as in the DHF methods using the expression
for transition
Cr
(2)
H,,= ri.s'
constants
Orbital
Ag
where ZI’, ?I’~, tiex and vcc are the local density expressions for electron-electron Coulomb, selfinteraction, exchange and correlation terms as defined in Ref. [7]. The relativistic potential H,, Ho and &, are reSpeCtiVe]y the IIUSS-Velocity, Darwin correction and spin-orbit interaction terms which involve u(r) and its derivative [7]. The operator for the one-electron spin-orbit interaction can be written as
coupling
Atom
(1)
%tjPdj(r)
Here, v(r) is the central field potential: 21(r) = -22/r
361 (1996) 15-19
where en/j+ and [n/j_ are the eigenvalues of Eq. (1) withj, = 1 + (l/2) andj- = I - (l/2) respectively.
(in Ry) for plutonium
Orbital
d2 dr2’
Structure (Theochem)
1.5092 0.1764 0.2646
65.5122 14.4058
a Calculated using orbital eigenvalues from Ref. [15] in Eq. (3). b Calculated using eigenvalues of Eq. (I) of text in Eq. (3). ’ Calculated using experimental orbital ionisation energies from Ref. [16] in Eq. (3).
M. Vijayakumar. MS.
1.20
a-
owat~ _ &*--A
Sc Y LU 0.25
GopinarhanlJournal of Molecular Struc!ure (Theochem)
1st row 2nd row(xl0) 3rd row(x100)
Ti Zr Hf
2000-
ODOW AbObA x I L-I-
17
361 (1996) IS-19
DFT DHF Expt 3c ion
V Cr Mn Fe Co Ni Cu Nb MO Tc Ru Rh Pd Ag Ta W Re OS Ir Pt Au
c-30+-0
1st row
oww 6aw-a
2nd row(xl0) 3rd row(x100)
O!
(b) _r H&”
I
I
38
d
40
I
I
42 Atomic
44 Number(Z)
6
46
Fig. 3. Spin-orbit coupling constants for 4d orbital of secondrow transition elements and their triply positive ions.
!
0.00
s
SC Y Lu
n
* I B ’ I ’ V Cr Mn Fe Co Ni Cu Nb MO Tc Ru Rh Pd Ag Ta W Re OS lr Pt Au
Ti Zr Hf
Fig. 1. Spin-orbit coupling orbitals of the first-, second-
’
constants for (a) 2p and (b) 3p and third-row transition elements.
Table 1. These are compared with results of the HFR and DHF methods and with experimental values. The DFT values are clearly superior to those of the HFR method. It is gratifying that the relatively simpler DFT method closely approximates the DHF as well as experimental values. Table 2 gives the spin-orbit coupling constants
for the orbitals in some first-, second- and thirdrow transition elements. The DHF values given in this Table were calculated using reported orbital eigenvalues in Eq. (2). “Experimental” values were similarly computed using experimental orbital binding energies (quoted in Ref. [14]) in the same equation. Generally, the present DFT values are again quite close to the DHF and experimental values. The well known trends of spin-orbit coupling increasing with 2 and decreasing with I for a given Z, which follows from Dirac’s theory [12], are evident from this Table. The calculated DFT values of <,,, for 2p and 3p orbitals of the transition elements are shown 6500 _~ -I o+w m* 5500 - AHL-H
OFT DHF Expt. 3+lon
/
/+
4500 ‘i $
3500-
C ._ ;
25OO1500-
0 20
I 22
24 Atomic
I 26 Number(Z)
Fig. 2. Spin-orbit coupling constants for 3d orbital transition elements and their triply positive ions.
I 28
1 30
J
500
7’0
712 Atomic
of first-row
74
16
78
I 1
number(Z)
Fig. 4. Spin-orbit coupling constants for 5d orbital transition elements and their triply positive ions.
of third-row
M. Vijayakumar. MS.
18
~o~inathanlJaurna1 of Molecular Structure (Theochem)
361 (1996) IS-19
I 0.0
(a)
ooooo l .-u
DFT
I _-_-----_-_--_
(d)
t
DHF
2 -0.01 .c G Q -0.4
-0.06
[
70
b
I I 22 2; Atomic Numbe?&
0.00
I
w 000~
-0.10 a c ;; w--o.20 Q
2
-0.30
-0.10
’ 3’e
I
38
I;‘0
I I 42 44 Atomic Number(Z)
I 4’0 Ator%
I 44 Number(Z)
I
I
74 76 Atomic Number (2)
I
78
I
80
Fig. 5. Deviation from “experimental” values of calculated spin-orbit coupling constants by the DHF and DPT methods for (a) 2p orbital of first-row, (b) 2p orbital of second-row, (c) 3p of second-row, and (d) 4p of third-row transition elements.
I 28
(b) __ _____ -_---__----s
75
DFT DHF
I 46
48
I 46
48
graphically in Figs. l(a) and l(b). The <,,)values for the 3d, 4d and 5d orbitals of the first-, second- and third-row transition atoms and their triply positive ions are shown in Figs. 2-4. These Figures also show the experimental and DHF results for the neutral atoms. The experimental variation of &, with atomic number is closely followed by both
DHF and DFT methods. However, DFT values are consistently somewhat higher than DHF and experimental results. The values for the ions are always larger than those of the neutral atoms. This is expected since the ion has a greater effective nuclear charge, resulting in a decreasing expectation value for the radial distance of the electron from the nucleus. The deviations of the DFT and DHF results from the experimental values for 2p orbitals of the first-row and second-row transition elements are displayed in Figs. 5(a) and 5(b). Similar plots of deviations of the two methods for 3p of the second row and 4p of the third row are shown in Figs. 5(c) and 5(d). These curves show that the DFT and DHF results are very close and have nearly the same trend in their deviation from experiment. Two further comments may be added based on some of our results not reported here [14]: (1) removal of electrons from the outer shell has little effect on the spin-orbit coupling of inner orbitals; (2) electron correlation has little effect on spinorbit coupling constants, as can be shown by removing the correlation potential term, 21~in Eq. (2).
References [l] C.W. Bauschhcher and S.R. Langhoff, Adv. Chem. Phys., 77 (1990) 103. [2] J.P. Desclaux, Comp. Phys. Commun., 9 (1975) 31.
M. Vijayakumar. MS.
GopinathanlJournal of Molecular Structure (Theochem)
[3] D.R. Salahub, Adv. Chem. Phys., 69 (1987) 447. [4] R.D. Cowan, Theory of Atomic Structure and Spectra, University of California, Berkeley, 1981. [S] E.J. Baerends, J.G. Snijders, C.A. de Lange and G. Jonkers, in J.P. Dahl and J. Avery (Eds.), Local Density Approximations in Quantum Chemistry and Solid State Physics, Plenum, New York, 1984, p. 415. [6] J.H. Wood andA.M. Boring, Phys. Rev. B, 18 (1978) 2701. [7] M. Vijayakumar, N. Vaidehi and M.S. Gopinathan, Phys. Rev. A, 40 (1989) 6834. [8] M. Vijayakumar and MS. Gopinathan, J. Chem. Phys., 91 (1992) 6639. [9] M. Vijayakumar and M.S. Gopinathan, Phys. Rev. A, 44 (1991) 2850. [IO] M. Vijayakumar and M.S. Gopinathan, J. Chem. Phys., 98 (2993) 4009.
361 (1996) 15-19
19
[l l] E.M. Spain and M.D. Morse, J. Chem. Phys., 97 (1992) 4641. [12] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Academic Press, New York, 1957. [ 131 R.D. Cowan and D.C. Griffin, J. Opt. Sot. Am., 66 (1976) 1010. [l4] M. Vijayakumar, Ph.D. Thesis, Indian Institute of Technology, Madras, 1990. [15] J.P. Desclaux, At. Data Nucl. Data Tables, 12 (1973) 311. [16a] CRC Handbook of Chemistry and Physics, 68th Edition, 198771988, CRC Press, Rota Baton, FL, 1987. [16b] T.A. Carlson, Photoelectron and Auger Spectroscopy, Plenum, New York, 1975. [16c] M. Gerloch, Orbitals, Terms and States, Wiley, Chichester, 1986.