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A semiempirical method for the calculation of the Slater-Condon parameters
ANNALS
OF PHYSICS:
20, 234-239 (1962)
A Semiempirical
Method Slater-Condon
the Calculation Parameters*
of the
F. LANE AXD CHUN C. LIN
NEAL
Department
for
of Physics,
of Oklahoma,
University
Norman,
Oklahoma
A semiempirical method for calculating the Slater-Condon parameters for the atoms of the first transition group is presented. This method is based on the use of effective nuclear charges which are determined from the experimental data of ionization potential. The calculated Slater-Condon parameters agree well with the experimental values and show considerable improvement over those obtained from the Slater orbitals (with Slater’s screening constants). Applications to the complex ions of the transition elements are discussed.
It is well known that the spacings of the multiplet levels of a given electronic of an atom can be expressedin terms of the Slater-Condon parameters (1, 2). For example, the (,3d)’ configuration gives rise to five states (3F, ‘D, 3P, ‘(7, ‘S) with splittings related to the int’egrals F2 and F, . A great deal of effort has been made to fit the observed multiplets by choosing empirical values of the Slater-Condon parameters (2’). Generally, these parameters can be chosen t,o give a satisfactory over-all agreement with experiments and several sets of empirical values of the Slater-Condon parameters have been given in the literature (8-8). From the definition of these parameters it is possible to calculate them by using the Slater orbitals (9) as the atomic wave functions; however, such calculated values are found to be considerably smaller than the empirical ones (10). Although more precise calculations can be performed by using the self-consistent field atomic wave functions and indeed calculations of this kind have been made (2)) it is desirable to have a simple way to obt,ain accurate estimates of these integrals. In this paper we shall present a semiempirical method for calculating the Slater-Condon parameters. The parameters Fz and Fq are defined (in atomic units) as ( 1, 2)) configuration
(1) F4(3d, 3d) = & * Supported
by the
Office
of Naval
/- R:,j(l)R;d(2) Research.
234
$:~r:
d,rl drz,
CALCULATION
OF
SLSTEIZ-CONDON
PSRAMETERS
23.5
where Rxd is the radial part, of the 3d orbitals; r< and r, represent the lesser and the great,er of r1 and Q . If we use the hydrogenic ware functions with an ndjust,able nuclear charge Z for t’he 3nl orbit,als, Eqs. (1) become F., = (202.499)
Z cK1
Fq = (11.7423) Z cm-’ The empirical values of F, for Ti III and V IV are 1110.4 crL1 and 1156 cm-’ rrspectlively (3). In order to fit Eq. (2) we would have to require Z = 5.#3 for Ti III and Z = 7.190 for V IV. The corresponding effect,ive nuclear charges as determined by Slater’s rule (7) arc 3.G and A.ciS, respectively. Since the effective nuclear charges required ho fit F? arc substantially larger t.han those determined from Slater’s rules, it appears t,hat the shielding of the 3d electrons by the inner core is not complet,e. The same sort, of deviation from t,he Slater rules has also been observed by Hartree from the Self-Consistent, Field wave functions (11). Furthermore, the difference in the Z value, as determined by Fr , between Ti III and V IV is 1.70 rather t,han 1.00 which is the difference of the true nuclear charges of the t#wo atoms. This indicates that the amount of the shielding due to the inner core electrons may vary according to t*hr charge of the atomic nucleus. The observations in t,he preceding paragraph suggest, t,hat it) is possible to obhain accurat,e values of Fa and Fj by using hydrogenic wave functions provided t,hat we choose a suitable effective nuclear charge to take into consideration the incomplete shielding of the inner core. For atoms wit,h one valence cl&ron, a convenient criterion for selecting t,he effective nuclear charge Z is to fit the observed ionization potential (1.2’) of t,he outer electron to the expression Z2/2n2. According to this procedure, the effective nuclear charges of hhe 3d rlrctron in SC III, Ti IV, V V, Cr VI, Mn VII, and Fe VIII are 4.05, .5.:-G, 6.56, 7.74, 8X3, and 10.00, respectively. In this isoelectronic sequence the effective charge increases by more than one unit in going from one atom to the nest. This is in qualitative agreement wit,h our previous observation of varying degree of inner core shielding. For atoms with (ad)” outer shell, the effective nuclear charge of the outer electrons can be determined from the ionization potent’ials in a similar way, i.e., by setting the energy (experiment,al) required to remove all the m valence electrons equal t,o mZ2/2n2. The values of effective nuclear charge obtained by this method are list’ed in Table I. In t#he case of Mn, Fe, Co, and Si, the ionization potentials of the various valence electrons are not, all available for spectroscopic data, and the effective nuclear charges can be determined only for Mn VII and Fe VIII. By comparing the sequences of Ti I, Ti II, Ti III; V I, V II, V III, V IV; etc., we note t’hat) screening due to each 3d electron is ap-
TABLE EPFECTIVEK;UCLEARCHARGESOFTHE
Atoms SC I SC II SC III Ti I Ti II Ti III Ti IV VI v II v III
Charges 2.971 3.500 4.047 3.821 4.327 4.859 5.349 4.628 5.106 5.624
I
3d ELECTRONSOFSC,
Ti,V,cr,Mn,~w Atcms
v IV v v Cr Cr Cr Cr Cr Cr Mn Fe
I II III IV V VI VII VIII
Fr Charges 0.130 6.563 5.386 5.874 6.355 6.858 7.359 7.742 8.883 10.00
proximately equal to 0.0 units (instead of 0.35 from Slater’s rules). Because of this apparent constancy throughout all the 2 values in Table I, we shall take the screening constant of each 3d electron as 0.500. With this empirical screening rule, we can obtain the effective nuclear charges of the Mn and Fe in all other ionization stages. (To avoid any possible confusion, the 2 values determined by Slate& rules will be referred to as the Slater effective nuclear charge). From these effective nuclear charges, we have calculated the Slater-Condon parameters and the results are shown in Table II. For the purpose of comparison, the same parameters as calculated by using Slater’s effective nuclear charges are also included. The experimental values are given in the last two columns of Table II. One may notice that our results do show a considerable improvement, over those obtained by Slater’s effective nuclear charges. Also there is some discrepancy between the empirical parameters evaluated by different authors, since it is possible t)o fit t#he observed spectra equally well by different sets of constants. The best agreements between the theory and experimenk are found for the singly and doubly ionized atoms. The calculated values become somewhat too small for atoms in the more highly ionized states. This may be due t$o increased contract,ions of the 3d orhitals as compared to the 3s and 3p functions which result in a slight increase of the effective nuclear charge. In view of the approximate nature of t,he multiplet theory, our calculated values should be regarded as quite satisfact,ory. In order to obtain an estimate of the Slater-Condon paramet#ers for Co and Ki, we have determined the effective nuclear charges of Co IX and ?Ji X by extrapolation from the series SC III, Ti IV, V V . . . , Fe VIII, as 11.12 and 12.24 respectively. Then for any other stages of ionization of these atoms, Z can be obtained by assuming a screening constant of 0.500 for each 3d electron. The results of the calculated F? and F4 as compared to the experimental values are given in Table III.
CALCULATIOX
OF
SLSTER-CONDON
TABLE CALCVLATED
Electron configurations
OBSERVED
;\toms
II
VALVES OF THE SLATER-CONDON (ALL IN Cm-‘)
____
This work F?
___
Slater’s
~-
rules
I;,
F2
F4
d3 cl’
SC I SC II
G0l.G 708.8
43.80 51.60
465.75 536.62
33.91 39.07
d’ d3
Ti I Ti II
773.8 876.2
56.33 G3.79
597.37 668.25
43.49 48.65
d'
Ti
III
983.9
71.63
739.12
53.81
d” d” d3
VI v II v TII
937.2 1034.0 1138.9
(is.23 75.27 82.91
729.00 799.87 870.75
53.07 58.23 113.39
dz
v IV
1241.3
90.37
941.G2
G8.55
d” dj (14
cr I Cr II Cr III
1087.0 1189.4 128G.9
79.40 86.59 93.G9
860.62 931.50 1002.4
62.65 67.81 72.97
d”
Cr IV
1388.8
101.1
1073.2
78.13
tP
Cr v
1490.2
108.5
1141.1
83.29
tP d6 d5 d’ d3 d? d8 d7 (16
Mn Mn Mn Mn Mm Mn Fe Fe Fe
I II III
1190.7 1291.9 1393.2 1494.4 1595.7 1696.9 1316.2 1417.5 1518.7
86.68 94.06 101.4 108.8 116.2 123.5 95.82 103.2 110.6
992.3 1063.1 1134.0 1204.9 1275.7 1346.G 1123.9 1194.7 1265.6
72.24 77.40 82.56 87.72 92.88 98.04 81.82 80.98 92.14
d”
Fe Fe Fe Fe
IV V VI VII
1020.0 1721.2 1822.5 1923.7
117.9 125.3 132.7 140.0
1539.0 1812.4 1883.2 1954.1
112.0.4 131.94 137.10 142.29
d”
d3 d” a W. h M. c L. d P. e M.
AND
I II III IV V VI
237
PBRAMETERS
M. CADI., Phys. Rev. 43, 322 (1933). A. CATALAX AND M. T. ANTL-NES, Z. Physik 102,432 (1936). E. OBGEL, J. Chenl. Phys. 23, 1819 (1955). TANABE AND S. SUGANO, J. Phys. Sot. Japan 9, 766 (1954). OSTROFSKY, Phys. Rev. 46, GOA (1934).
PARAMETERS
Experimental F?
F4
736.8" 7316
51.1 51
824" 845' 111oa lOl6*
54 55 83.16 G6
88W 115Gh 1220.7d 1325” 1456"
82 93.06 95 119
1067" 1319.3d 1152" lG88h 1508.4d 1758" lG2Jh
72 101.9 99 101 118.1 149 126
1410" 1600.7d 1935* 2OGP
110 127.1 109 181
1291” 154oc 1494. Id 1700. id
104 120 115.4 137.14
24020
211
238 TAI31,E PLATER-C'OSUON Electron configurations
II I
PARAMETERS
OF Co Slater’s
.4toms
co I co co
II III
ASD Xi rules
F1’
FI
l-l-41.8 1543.0 1644.3
105.0 112.3 119.7
1255.6 132G.4 139i.2
91.40 96.56 101.7
(16 d5 d4 (23 d2 d cl? 8 da
co IV co v co VI co VII co VIII co IX Xi I Ni II Ni III
1745.5 1846.8 1948.0 2049.3 2150.5 2251.8 1567.3 1668.6 1769.8
127.1 134.5 141.8 149.2 156.6 163.9 114.1 121.5 128.9
1468.1 1539.0 1609.9 1680.7 1751.6 1822.5 1387.1 1458.0 1528.9
106.9 112.0 117.2 122.4 127.5 132.7 101.0 106.1 111.3
d’ d” d5 d4 (13 d2 d
Ki Xi Ni Ni ?;i Xi Xi
1871.1 1972.3 2073.6 2174.8 2276.1 2377.3 2478.6
136.2 143.6 151 .o 158.3 165.7 173.1 180.5
1599.7 1670.6 1741.5 1812.4 1883.2 1954.1 2025.0
116.5 121.6 126.8 131.9 137.1 142.3 147.4
IV V VI VII VIII IS S
a W. M. CADY, Phys. Rezl. 43, 322 (1933). * Estimated from the graph in p. 379 of ref. B. c L. E. ORGEL, J. C’hem. Phys. 23, 1819 (1955). d Y. TANABE AND S:. SUGANO,J. Phys.Soc. Japan
Experimental F2
F,
16m 1613.4” 1612” 1796.-ld
120 128.5
1926+ 2240h 2670”
1722.9d 1568* 1893.6d
14ti.3
233.5
138.6 155.7
2195* 2871Cz
243.9
9, 766 (1954).
The effective nuclear charges can he determined presumably from atomic quantities other than the ionization pot’entials. Thus Hartree (11) has introduced a screening constant by comparing (T).&~computed according to t’he selfconsistent field wave functions with those derived from hydrogenic orbit,als. Using Hartree’s screening constants we have F2 = 1652 cm-‘, F4 = 120 cm-’ for V III and Fz = 1290 cm-‘, Fd = 93.9 cm-’ for Ti II which differ considerably from the calculated values in Table II. The ionization potential is essentially proportional to (l/r) while the Mater-Condon parameters depend on T<~/&?‘. Although the radial dependence of the Slater-Condon parameters is quite complicated, it should have more resemblance to 8 than T. Hence the effective nuclear charges determined from the ionization potentials should be adequate for calculating the Slater-Condon parameters.
C:ALCVL?1TIOS
OF
SLATER--CONDOS
PARAMETERS
239
Aside from their prominent, roles in the t’heory of atomic spectra, the SlaterCondon paramet,ers are also found to have much application in the study of cryst,als of complex salts of the transit#ion elements. For cryst)als of cubic structure, t,he energy levels are determined by FZ and Fq as well as t,he cryst,al field parameters Dy (13). The optical spectra, magnetic susceptibilities, and the energy of the complex formation are all related t’o t(hese parameters. Orgel (6) has further pointed out Ohat the covalency of these complex ions is determined by t’he relative magnitude of the Slat’er-Condon parameters as compared to Dq. The present, method of calculating F, and Fq should be useful in estimat,ing the energy levels in cases where arcurat,e spect,roscopic data is not available. From the analysis of the optical spectra it, has been found that the SlaterCondon parameters for the complex ions are generally smaller than those of t,he free atoms (10, 13). This can be explained on the basis of distortion of the 3d electron clouds by the ligands causing a decreaseof the effective nuclear charge. If we consider t’he bonding between t,he central atom and the ligands, the stabilizat.ion resulting from the overlap of electron clouds might cause some “spreading” of the 3d orbitals and thus decrease the effective nuclear charge as well as the Slater--Condon parameters. Wit#h the same reasoning we can also understand why t’he spin-orbit coupling ronst’ants decrease upon formation of complex ions. I~ECEIVED:
Jlay 28, 1962
1. E. LT.CO~D~~AND G.H. SHORTLEY, “The Theoyv of Btomic Spectra.” Cambridge Univ. Press, Lundon and New York, 1951. ,O. J. C. SLATER, ‘Quantum Theory of Atomic Structure,” VIUI. I. Mc(;raw-Hill, New York, 1960. 3. IV. hf. CADI., i’hys. f&o. 43, 322 (1933). 4. M. A. C~TALAX A~YD M. T. ANTTNES, Z. Physik 102, 432 (1936). 5. M. A. CAT.~T,.IN, F. ROHRLICH, AND A. G. SHENSTONE, Proc. Roy. Sot. (London) A221, 421 (I!&%). 6. L. E. OBGEL, J. C’hem. Phys. 23, 1819 (1955). 7. M. OSTROFSKY, Ph,ys. Rev. 46, 004 (193-l). 8. 2'. TANABE .~ND S. ~UGANO, J. Ph.ys. Sot. Jtrpnn 9, 766 (195-1). 9. J. C. SIUTER, Phys. Rely. 36, 57 (1930). 10. D. A. I~ROWN, 7. (Them. Phys. 28, 67 (1958). 11. 1). R. HARTREE, “The Calculation of Atomic St,ruct,ure,” pp. 126, 167. Wiley, New York, 1957. Circular of the Kational Bureau elf Standards 18. C. E. MOORER “Atomic Energy Levels.” i(i7, Washington, D.C., 1918. 1.9. See, for example, D. S. Mdk~-RE, Solid Stale Phys. 9, 399 (1959); W. Low, “Paramagnetic Resonance in Solids.” Academic Press, New York, 1960.
OF PHYSICS:
20, 234-239 (1962)
A Semiempirical
Method Slater-Condon
the Calculation Parameters*
of the
F. LANE AXD CHUN C. LIN
NEAL
Department
for
of Physics,
of Oklahoma,
University
Norman,
Oklahoma
A semiempirical method for calculating the Slater-Condon parameters for the atoms of the first transition group is presented. This method is based on the use of effective nuclear charges which are determined from the experimental data of ionization potential. The calculated Slater-Condon parameters agree well with the experimental values and show considerable improvement over those obtained from the Slater orbitals (with Slater’s screening constants). Applications to the complex ions of the transition elements are discussed.
It is well known that the spacings of the multiplet levels of a given electronic of an atom can be expressedin terms of the Slater-Condon parameters (1, 2). For example, the (,3d)’ configuration gives rise to five states (3F, ‘D, 3P, ‘(7, ‘S) with splittings related to the int’egrals F2 and F, . A great deal of effort has been made to fit the observed multiplets by choosing empirical values of the Slater-Condon parameters (2’). Generally, these parameters can be chosen t,o give a satisfactory over-all agreement with experiments and several sets of empirical values of the Slater-Condon parameters have been given in the literature (8-8). From the definition of these parameters it is possible to calculate them by using the Slater orbitals (9) as the atomic wave functions; however, such calculated values are found to be considerably smaller than the empirical ones (10). Although more precise calculations can be performed by using the self-consistent field atomic wave functions and indeed calculations of this kind have been made (2)) it is desirable to have a simple way to obt,ain accurate estimates of these integrals. In this paper we shall present a semiempirical method for calculating the Slater-Condon parameters. The parameters Fz and Fq are defined (in atomic units) as ( 1, 2)) configuration
(1) F4(3d, 3d) = & * Supported
by the
Office
of Naval
/- R:,j(l)R;d(2) Research.
234
$:~r:
d,rl drz,
CALCULATION
OF
SLSTEIZ-CONDON
PSRAMETERS
23.5
where Rxd is the radial part, of the 3d orbitals; r< and r, represent the lesser and the great,er of r1 and Q . If we use the hydrogenic ware functions with an ndjust,able nuclear charge Z for t’he 3nl orbit,als, Eqs. (1) become F., = (202.499)
Z cK1
Fq = (11.7423) Z cm-’ The empirical values of F, for Ti III and V IV are 1110.4 crL1 and 1156 cm-’ rrspectlively (3). In order to fit Eq. (2) we would have to require Z = 5.#3 for Ti III and Z = 7.190 for V IV. The corresponding effect,ive nuclear charges as determined by Slater’s rule (7) arc 3.G and A.ciS, respectively. Since the effective nuclear charges required ho fit F? arc substantially larger t.han those determined from Slater’s rules, it appears t,hat the shielding of the 3d electrons by the inner core is not complet,e. The same sort, of deviation from t,he Slater rules has also been observed by Hartree from the Self-Consistent, Field wave functions (11). Furthermore, the difference in the Z value, as determined by Fr , between Ti III and V IV is 1.70 rather t,han 1.00 which is the difference of the true nuclear charges of the t#wo atoms. This indicates that the amount of the shielding due to the inner core electrons may vary according to t*hr charge of the atomic nucleus. The observations in t,he preceding paragraph suggest, t,hat it) is possible to obhain accurat,e values of Fa and Fj by using hydrogenic wave functions provided t,hat we choose a suitable effective nuclear charge to take into consideration the incomplete shielding of the inner core. For atoms wit,h one valence cl&ron, a convenient criterion for selecting t,he effective nuclear charge Z is to fit the observed ionization potential (1.2’) of t,he outer electron to the expression Z2/2n2. According to this procedure, the effective nuclear charges of hhe 3d rlrctron in SC III, Ti IV, V V, Cr VI, Mn VII, and Fe VIII are 4.05, .5.:-G, 6.56, 7.74, 8X3, and 10.00, respectively. In this isoelectronic sequence the effective charge increases by more than one unit in going from one atom to the nest. This is in qualitative agreement wit,h our previous observation of varying degree of inner core shielding. For atoms with (ad)” outer shell, the effective nuclear charge of the outer electrons can be determined from the ionization potent’ials in a similar way, i.e., by setting the energy (experiment,al) required to remove all the m valence electrons equal t,o mZ2/2n2. The values of effective nuclear charge obtained by this method are list’ed in Table I. In t#he case of Mn, Fe, Co, and Si, the ionization potentials of the various valence electrons are not, all available for spectroscopic data, and the effective nuclear charges can be determined only for Mn VII and Fe VIII. By comparing the sequences of Ti I, Ti II, Ti III; V I, V II, V III, V IV; etc., we note t’hat) screening due to each 3d electron is ap-
TABLE EPFECTIVEK;UCLEARCHARGESOFTHE
Atoms SC I SC II SC III Ti I Ti II Ti III Ti IV VI v II v III
Charges 2.971 3.500 4.047 3.821 4.327 4.859 5.349 4.628 5.106 5.624
I
3d ELECTRONSOFSC,
Ti,V,cr,Mn,~w Atcms
v IV v v Cr Cr Cr Cr Cr Cr Mn Fe
I II III IV V VI VII VIII
Fr Charges 0.130 6.563 5.386 5.874 6.355 6.858 7.359 7.742 8.883 10.00
proximately equal to 0.0 units (instead of 0.35 from Slater’s rules). Because of this apparent constancy throughout all the 2 values in Table I, we shall take the screening constant of each 3d electron as 0.500. With this empirical screening rule, we can obtain the effective nuclear charges of the Mn and Fe in all other ionization stages. (To avoid any possible confusion, the 2 values determined by Slate& rules will be referred to as the Slater effective nuclear charge). From these effective nuclear charges, we have calculated the Slater-Condon parameters and the results are shown in Table II. For the purpose of comparison, the same parameters as calculated by using Slater’s effective nuclear charges are also included. The experimental values are given in the last two columns of Table II. One may notice that our results do show a considerable improvement, over those obtained by Slater’s effective nuclear charges. Also there is some discrepancy between the empirical parameters evaluated by different authors, since it is possible t)o fit t#he observed spectra equally well by different sets of constants. The best agreements between the theory and experimenk are found for the singly and doubly ionized atoms. The calculated values become somewhat too small for atoms in the more highly ionized states. This may be due t$o increased contract,ions of the 3d orhitals as compared to the 3s and 3p functions which result in a slight increase of the effective nuclear charge. In view of the approximate nature of t,he multiplet theory, our calculated values should be regarded as quite satisfact,ory. In order to obtain an estimate of the Slater-Condon paramet#ers for Co and Ki, we have determined the effective nuclear charges of Co IX and ?Ji X by extrapolation from the series SC III, Ti IV, V V . . . , Fe VIII, as 11.12 and 12.24 respectively. Then for any other stages of ionization of these atoms, Z can be obtained by assuming a screening constant of 0.500 for each 3d electron. The results of the calculated F? and F4 as compared to the experimental values are given in Table III.
CALCULATIOX
OF
SLSTER-CONDON
TABLE CALCVLATED
Electron configurations
OBSERVED
;\toms
II
VALVES OF THE SLATER-CONDON (ALL IN Cm-‘)
____
This work F?
___
Slater’s
~-
rules
I;,
F2
F4
d3 cl’
SC I SC II
G0l.G 708.8
43.80 51.60
465.75 536.62
33.91 39.07
d’ d3
Ti I Ti II
773.8 876.2
56.33 G3.79
597.37 668.25
43.49 48.65
d'
Ti
III
983.9
71.63
739.12
53.81
d” d” d3
VI v II v TII
937.2 1034.0 1138.9
(is.23 75.27 82.91
729.00 799.87 870.75
53.07 58.23 113.39
dz
v IV
1241.3
90.37
941.G2
G8.55
d” dj (14
cr I Cr II Cr III
1087.0 1189.4 128G.9
79.40 86.59 93.G9
860.62 931.50 1002.4
62.65 67.81 72.97
d”
Cr IV
1388.8
101.1
1073.2
78.13
tP
Cr v
1490.2
108.5
1141.1
83.29
tP d6 d5 d’ d3 d? d8 d7 (16
Mn Mn Mn Mn Mm Mn Fe Fe Fe
I II III
1190.7 1291.9 1393.2 1494.4 1595.7 1696.9 1316.2 1417.5 1518.7
86.68 94.06 101.4 108.8 116.2 123.5 95.82 103.2 110.6
992.3 1063.1 1134.0 1204.9 1275.7 1346.G 1123.9 1194.7 1265.6
72.24 77.40 82.56 87.72 92.88 98.04 81.82 80.98 92.14
d”
Fe Fe Fe Fe
IV V VI VII
1020.0 1721.2 1822.5 1923.7
117.9 125.3 132.7 140.0
1539.0 1812.4 1883.2 1954.1
112.0.4 131.94 137.10 142.29
d”
d3 d” a W. h M. c L. d P. e M.
AND
I II III IV V VI
237
PBRAMETERS
M. CADI., Phys. Rev. 43, 322 (1933). A. CATALAX AND M. T. ANTL-NES, Z. Physik 102,432 (1936). E. OBGEL, J. Chenl. Phys. 23, 1819 (1955). TANABE AND S. SUGANO, J. Phys. Sot. Japan 9, 766 (1954). OSTROFSKY, Phys. Rev. 46, GOA (1934).
PARAMETERS
Experimental F?
F4
736.8" 7316
51.1 51
824" 845' 111oa lOl6*
54 55 83.16 G6
88W 115Gh 1220.7d 1325” 1456"
82 93.06 95 119
1067" 1319.3d 1152" lG88h 1508.4d 1758" lG2Jh
72 101.9 99 101 118.1 149 126
1410" 1600.7d 1935* 2OGP
110 127.1 109 181
1291” 154oc 1494. Id 1700. id
104 120 115.4 137.14
24020
211
238 TAI31,E PLATER-C'OSUON Electron configurations
II I
PARAMETERS
OF Co Slater’s
.4toms
co I co co
II III
ASD Xi rules
F1’
FI
l-l-41.8 1543.0 1644.3
105.0 112.3 119.7
1255.6 132G.4 139i.2
91.40 96.56 101.7
(16 d5 d4 (23 d2 d cl? 8 da
co IV co v co VI co VII co VIII co IX Xi I Ni II Ni III
1745.5 1846.8 1948.0 2049.3 2150.5 2251.8 1567.3 1668.6 1769.8
127.1 134.5 141.8 149.2 156.6 163.9 114.1 121.5 128.9
1468.1 1539.0 1609.9 1680.7 1751.6 1822.5 1387.1 1458.0 1528.9
106.9 112.0 117.2 122.4 127.5 132.7 101.0 106.1 111.3
d’ d” d5 d4 (13 d2 d
Ki Xi Ni Ni ?;i Xi Xi
1871.1 1972.3 2073.6 2174.8 2276.1 2377.3 2478.6
136.2 143.6 151 .o 158.3 165.7 173.1 180.5
1599.7 1670.6 1741.5 1812.4 1883.2 1954.1 2025.0
116.5 121.6 126.8 131.9 137.1 142.3 147.4
IV V VI VII VIII IS S
a W. M. CADY, Phys. Rezl. 43, 322 (1933). * Estimated from the graph in p. 379 of ref. B. c L. E. ORGEL, J. C’hem. Phys. 23, 1819 (1955). d Y. TANABE AND S:. SUGANO,J. Phys.Soc. Japan
Experimental F2
F,
16m 1613.4” 1612” 1796.-ld
120 128.5
1926+ 2240h 2670”
1722.9d 1568* 1893.6d
14ti.3
233.5
138.6 155.7
2195* 2871Cz
243.9
9, 766 (1954).
The effective nuclear charges can he determined presumably from atomic quantities other than the ionization pot’entials. Thus Hartree (11) has introduced a screening constant by comparing (T).&~computed according to t’he selfconsistent field wave functions with those derived from hydrogenic orbit,als. Using Hartree’s screening constants we have F2 = 1652 cm-‘, F4 = 120 cm-’ for V III and Fz = 1290 cm-‘, Fd = 93.9 cm-’ for Ti II which differ considerably from the calculated values in Table II. The ionization potential is essentially proportional to (l/r) while the Mater-Condon parameters depend on T<~/&?‘. Although the radial dependence of the Slater-Condon parameters is quite complicated, it should have more resemblance to 8 than T. Hence the effective nuclear charges determined from the ionization potentials should be adequate for calculating the Slater-Condon parameters.
C:ALCVL?1TIOS
OF
SLATER--CONDOS
PARAMETERS
239
Aside from their prominent, roles in the t’heory of atomic spectra, the SlaterCondon paramet,ers are also found to have much application in the study of cryst,als of complex salts of the transit#ion elements. For cryst)als of cubic structure, t,he energy levels are determined by FZ and Fq as well as t,he cryst,al field parameters Dy (13). The optical spectra, magnetic susceptibilities, and the energy of the complex formation are all related t’o t(hese parameters. Orgel (6) has further pointed out Ohat the covalency of these complex ions is determined by t’he relative magnitude of the Slat’er-Condon parameters as compared to Dq. The present, method of calculating F, and Fq should be useful in estimat,ing the energy levels in cases where arcurat,e spect,roscopic data is not available. From the analysis of the optical spectra it, has been found that the SlaterCondon parameters for the complex ions are generally smaller than those of t,he free atoms (10, 13). This can be explained on the basis of distortion of the 3d electron clouds by the ligands causing a decreaseof the effective nuclear charge. If we consider t’he bonding between t,he central atom and the ligands, the stabilizat.ion resulting from the overlap of electron clouds might cause some “spreading” of the 3d orbitals and thus decrease the effective nuclear charge as well as the Slater--Condon parameters. Wit#h the same reasoning we can also understand why t’he spin-orbit coupling ronst’ants decrease upon formation of complex ions. I~ECEIVED:
Jlay 28, 1962
1. E. LT.CO~D~~AND G.H. SHORTLEY, “The Theoyv of Btomic Spectra.” Cambridge Univ. Press, Lundon and New York, 1951. ,O. J. C. SLATER, ‘Quantum Theory of Atomic Structure,” VIUI. I. Mc(;raw-Hill, New York, 1960. 3. IV. hf. CADI., i’hys. f&o. 43, 322 (1933). 4. M. A. C~TALAX A~YD M. T. ANTTNES, Z. Physik 102, 432 (1936). 5. M. A. CAT.~T,.IN, F. ROHRLICH, AND A. G. SHENSTONE, Proc. Roy. Sot. (London) A221, 421 (I!&%). 6. L. E. OBGEL, J. C’hem. Phys. 23, 1819 (1955). 7. M. OSTROFSKY, Ph,ys. Rev. 46, 004 (193-l). 8. 2'. TANABE .~ND S. ~UGANO, J. Ph.ys. Sot. Jtrpnn 9, 766 (195-1). 9. J. C. SIUTER, Phys. Rely. 36, 57 (1930). 10. D. A. I~ROWN, 7. (Them. Phys. 28, 67 (1958). 11. 1). R. HARTREE, “The Calculation of Atomic St,ruct,ure,” pp. 126, 167. Wiley, New York, 1957. Circular of the Kational Bureau elf Standards 18. C. E. MOORER “Atomic Energy Levels.” i(i7, Washington, D.C., 1918. 1.9. See, for example, D. S. Mdk~-RE, Solid Stale Phys. 9, 399 (1959); W. Low, “Paramagnetic Resonance in Solids.” Academic Press, New York, 1960.