Dependence of Relativistic Effects on Electronic
Configuration in the Neutral Atoms of
d- and f-Block Elements
J. AUTSCHBACH,1, * S. SIEKIERSKI,2 M. SETH,3, * P. SCHWERDTFEGER,3 W. H. E. SCHWARZ1,4
1
Theoretical Chemistry, Department of Chemistry, University of Siegen,
D-57068 Siegen, Germany
2
Department of Radiochemistry, Institute of Nuclear Chemistry and Technology,
Dorodna 16, PL-03-195 Warszawa, Poland
3
Department of Chemistry, University of Auckland, Private Bag 92019, Auckland, New Zealand
4
Theoretical Chemistry Group, College of Chemistry, Shanghai Jiao Tong University,
200240 Shanghai, People’s Republic of China
Received 27 August 2001; Accepted 30 November 2001
DOI 10.1002/jcc.10060
Abstract: Although most neutral d- and f-block atoms have ndg⫺2(n⫹1)s2 and (n⫺1)fg-2(n⫹1)s2 ground configurations, respectively, where g is the group number (i.e., number of valence electrons), one-third of these 63 atoms prefer
a higher d-population, namely via (n⫹1)s3nd “outer” to “inner” electron shift (particularly atoms from the second
d-row), or via (n⫺1)f3nd “inner” to “outer” electron shift (particularly atoms from the second f-row). Although the
response to the modified self-consistent field is orbital destabilization and expansion for (n⫹1)s3nd, and stabilization
and contraction for (n⫺1)f3nd, the relativistic modification of the valence orbital responses is stabilization in both
cases. This is explained by double perturbation theory. Accordingly, electron configuration and relativity trigger the
orbital energies, the orbital populations and the chemical shell effects in different ways. The particularly pronounced
relativistic effects in groups 10 and 11, the so-called gold maximum, occur because of particularly efficient cooperative
nonrelativistic shell effects and relativistic stabilization effects (inverse indirect effect) at the end of the d-block.
© 2002 Wiley Periodicals, Inc.
J Comput Chem 23: 804 – 813, 2002
Key words: relativistic effects; d-block elements; lanthanides and actinides; electronic configurations
Introduction
The many unique properties of platinum, and even more so those
of gold, have attracted the attention of researchers for a long time.
The special behavior of gold and its compounds1– 6 are commonly
attributed to the relativistic energy stabilization and radial contraction of the atomic 6s shell, and to opposite effects of the 5d shell,
though somewhat different views are sometimes presented concerning the physical origins (see, e.g., refs. 1–12).
When compounds of heavy elements and their properties are
discussed, several mechanisms must simultaneously be taken into
account. First, the (so-called classical) overlap, charge transfer,
and electron-correlation (dispersion) effects, and the atomic shellstructure or configurational effects—and, second, the relativistic
modifications. The latter ones consist of two contributions. First
there is the direct (Dir) influence of relativistic kinematics, when
the valence electrons penetrate the atomic core shells and approach
a highly charged nucleus “at high speed.” The direct one-electron
contributions at lowest relativistic order are the mass-velocity
variation, the spin-orbit coupling, and the Darwin effect. Second,
there is the so-called indirect (Ind) response to the self-consistent
Hartee–Fock field of all the relativistically modified occupied
orbitals, i.e., the relativistically modified nuclear shielding.1
Since the relativistic calculations of the ground states of all
neutral atoms by Desclaux,13 it has been known that the relativistic
modifications (⌬rel) of properties (prop) of atomic orbitals (AO) nl
vary roughly in proportion to Z,2 where Z is the nuclear charge:
共⌬ relprop兲/prop ⫽ 关 prop ⫺ prop0 兴/prop
⫽ 1 ⫺ 1/关 prop/prop 0兴 ⫽ ␥ prop共nl 兲 䡠 Z 2/c 2.
*Present address: Theoretical Chemistry, Department of Chemistry,
University of Calgary, Calgary T2N-1N4, Alberta, Canada
Correspondence to: W. H. E. Schwarz; e-mail:
[email protected]
Contract/grant sponsors: Deutsche Forschungsgemeinschaft (Bonn),
Fonds der Chemischen Industrie (Frankfurt), and the Siegen and Jiao
Tong Universities (to W.H.E.S.), and the Marsden Fund (to P.S.)
© 2002 Wiley Periodicals, Inc.
(1)
Relativistic Effects in d- and f-Block Elements
Figure 1. Ratio of relativistic to nonrelativistic atomic 6s orbital
energies, ␧/␧0, vs. Z2, where Z is the nuclear charge, for Z ⫽ 55 (Cs)
to Z ⫽ 100 (Fm), after Desclaux.13 The curve at the bottom represents
the values of hydrogen-like ions (the nonlinearity of its increase is due
to the higher order relativistic contributions, compare with Fig. 4).
The superscript 0 indicates nonrelativistic values. c is the velocity
of light, c ⬃ 137 atomic units. Many [ prop/prop0] values were
tabulated by Desclaux.13 We call the proportionality factor ␥prop,
defined in eq. (1), the fractional relativistic correction factor
(RCF). The RCF has been introduced here to remove the smooth
overall (Z /c)2 dependence of relativistic effects, thereby serving as
a “magnifying glass” to observe the relativistic trends within the
whole periodic table more easily. The RCFs are all of the same
order of magnitude between about ⫹1 and ⫺1, but sizeable systematic and irregular variations can occur along the rows of the
periodic table. The RCFs also depend strongly on the angular
momentum quantum number l. Within the columns of the periodic
table, however, the RCFs generally vary only slightly with increasing quantum number n of the respective valence shell1 (see also
refs. 14 and 15).
Although these trends are rather smooth and general, some
atomic orbitals seem to behave quite exceptional, the most spectacular cases being the 6s orbitals of Pt and of Au (and of the
heavier actinides) (see Fig. 1). The respective ratios of orbital radii,
r/r,0 exhibit pronounced minima (relativistic contraction) for those
Z-values, for which the ␧/␧0 in Figure 1 take pronounced maxima
(relativistic stabilization). Pyykkö1 has coined the phrase gold
maximum for this phenomenon around nuclear numbers Z ⫽
78,79, and he had already pointed out that relativistic s-orbital
effects show extrema also for Cu and Ag.
Searching for the deeper physical reason of such seemingly
nonsystematic relativistic orbital effects, we note that the four
mentioned atoms Cu, Ag, Au, and Pt possess ndg⫺1(n⫹1)s1
ground configurations instead of the more common ndg⫺2(n⫹1)s2
ones. g is the group number in the periodic table, viz. the number
of valence electrons. Most elements of the periodic table have
filled s2 shells. The only further exceptions with open s1 shells are
the 3d atom Cr and several 4d-atoms between Nb and Ag (and,
trivially, the g ⫽ 1 elements H and the alkali metals). Finally, Pd
even has an empty 5s0 shell.
805
The objective of the present work is to investigate the smooth
trends as well as the irregularities of atomic orbitals over the
periodic system, and their relations to configurational differences.
The numerical data (previously determined for the ground configurations previously by Desclaux13 and supplemented here by additional data determined for excited configurations with the help of
the ab initio code of Dyall, Grant et al.22) are empirically analyzed
in detail in the next section. To understand those variations of
orbital parameters in physical terms, rather detailed theoretical
explanations are presented in the Theoretical Explanations section
on the basis of double perturbation theory. Then our conclusions
are summed up in a compact manner.
We have confined ourselves to the self-consistent field approximation. However, our results are not specific for the Dirac–Fock
model. Also, relativistically corrected density functional calculations,15 which account for post-SCF effects, have led to similar
explanations.
Trends and Irregularities
We discuss d- and f-block elements with different orbital occupations of their nd (n⫹1)s and (n⫺1)f nd configurations, respectively. We preferentially analyze the energy changes of the s,p,d,f
valence orbitals and of some core orbitals. The slightly different
trends of orbital radii and of the respective chemically relevant
overlaps25 will only be discussed briefly. From the computed data,
we empirically extract the periodic trends and their relativistic
modifications, which influence the lower part of the periodic table
so strongly, and which form (at least some of) the reasons for the
rather peculiar chemical properties of several of those elements.1
We just mention the color of gold or the low cohesion energy of
liquid Hg.
d/s Configurational Differences: s Valence Orbitals
In Figure 2, computed nonrelativistic and relativistic s-valence
orbital energies of the 5d elements are presented. The very special
Figure 2. Nonrelativistic (䊐) and relativistic (■) 6s orbital energies ␧
of 5d elements from groups g ⫽ 2 (Yb) to g ⫽ 12 (Hg), in eV. The
numbers ⌬rel␧ at the bottom in the figure are the relativistic stabilization energies of the respective atoms. Note the different valence
cofigurations 5dg⫺26s2 and 5dg⫺16s1.
806
Autschbach et al.
•
Vol. 23, No. 8
•
Journal of Computational Chemistry
behaviors of platinum and gold along the 5d row in the real
relativistic case (and also in the nonrelativistic model case) are
evident. This special behavior must be connected with the increase
of relativistic stabilization, when the AO population changes from
dg⫺2s2 to dg⫺1s1.
In Figure 3, the RCFs ␥␧ for the 4s, 5s, and 6s orbital energies
of the 3d, 4d, and 5d elements, respectively, are plotted vs. group
number g. In addition, the RCF for the (n⫹1)s orbital of hydrogen
(i.e., without any shielding electrons) is marked in the figures.
Atoms with the same type of configuration, i.e., with dg⫺2s,2 with
dg⫺1s,1 or with dgs,0 respectively, are connected by dashed lines,
while the values of the lowest energy ground configuration are
connected by bold lines. (Note that the Aufbau principle does not
strictly hold for near-degenerate orbitals at the ab initio selfconsistent field level.) The dashed lines are quite monotonous.
This holds for data points referring both to the electronic ground
configurations as well as to the energetically exited configurations.
Even the virtual 5s orbital of Pd in Figure 3b fits well into the
general pattern of occupied orbital energies.
Four empirical findings from these computed numbers can be
noted:
1. The partial electronic shielding of nuclear attraction by many
other electrons (i.e., by (Z ⫺ 1) electrons in a neutral atom)
increases the RCF of the s orbitals above the hydrogenic
values. It indicates a relativistically increased effective nuclear
attraction, i.e., relativistic deshielding.1 This holds not only for
the energies of the s valence orbitals (Fig. 3), but also for other
expectation values.
2. When going from group 2 to group 12, the nuclear charge and
the respective number of shielding nd electrons increase, and
simultaneously the RCFs of the (n⫹1)s orbitals increase. That
is, the (n⫹1)s shell is incompletely shielded by the nd electrons.
These trends are well known.1
3. Comparing different configurations, we see that the RCF of the
(n⫹1)s-orbitals is increased more by an additional inner shielding nd electron than by an additional shielding (n⫹1)s electron
in the same shell. For instance, going from dg⫺2s2 to dg⫺1s1, the
RCFs of ␧(n⫹1)s increase by amounts of 0.16 to 0.21 (that is,
by 40 –70%) in the 3d4s row, by 0.15 to 0.20 (30 to 60%) in the
4d5s row, and by 0.12 to 0.15 (25 to 30%) in the 5d6s row.
4. When one goes down from the 3d to the 4d to the 5d elements
(Fig. 3a– c), the RCF of the respective ␧(n⫹1)s increases. This
increase corresponds to the significant increase of the number
of partially shielding electrons in the lower rows. The increase
even overcompensates the decrease of the hydrogenic RCF
␥␧(ns) for increasing n, when going from the 4s to the 5s to the
6s AO, respectively (see Fig. 4) (note the hydrogenic order of
RCFs: 2s ⬎ 1s,3s ⬎ 4s ⬎ 5s etc.).
It is very satisfactory that the apparent irregularities of the
s-valence orbitals of d-block elements can be represented as regular trends of the configurations and orbital occupation schemes.
The four empirical rules given above for the atomic RCFs help to
systematize the chemical behaviors of the respective compounds.
Figure 3. Fractional relativistic correction factor RCF ␥␧(s) [eq. (1)]
of the s-valence atomic orbital energy of d-block elements vs. group
number g (⫽ number of valence electrons). (a) 4s level of 3d elements.
(b) 5s level of 4d elements. (c) 6s level of 5d elements. The respective
values for the hydrogen atom are also indicated. ■ ⫽ dg⫺2s2, 䊐 ⫽
dg⫺1s, Œ ⫽ dg configurations.
d/s Configurations: d Valence Orbitals
The RCF ␥␧(1s) of hydrogen-like systems is of the order of ⫹0.25
(Fig. 4); for neutral many-electron atoms ␥␧ of the 1s core orbital
Relativistic Effects in d- and f-Block Elements
807
d/s Configurations: Core Orbitals
It has to be noted that the transfer of an electron from the (n⫹1)s
shell into the nd shell does not only increase the RCFs of both the
(n⫹1)s and nd valence orbitals (points 3 above), but also of all the
other, innermore core shells, though to lesser extents. The respective configurational increases of ␥␧ are presented in Table 1. There,
⌬s3d␥␧ is defined as follows: because of the smooth dependence
of ␥␧ on the nuclear charge Z, one may set
␥ ␧共Z ⫺ 1兲 ⫹ ␥ ␧共Z ⫹ 1兲 ⬇ 2 䡠 ␥ ␧共Z兲
Figure 4. RCF ␥␧(ns) of hydrogenlike like ns orbitals versus nuclear
charge Z. Note the nonmonotonous variation with principal quantum
number n.
(2)
Then, for the s/d, i.e. dg⫺isi vs. dg⫺i⫹1si⫺1 configurational change
of atoms with nuclear charges Z/[Z⫾1] and valence electron numbers g/[g⫾1] (such as Cr/[V⫹Mn] or Pd/[Rh⫹Ag]), we set
⌬ s3d␥ ␧ ⫽ ␥ 共Z, d g⫺is i 兲 ⫺ ␥ 共Z, d g⫺i⫹1s i⫺1兲
⬇ 关 ␥ 共Z ⫺ 1, d g⫺1⫺is i 兲 ⫹ ␥ 共Z ⫹ 1, d g⫹1⫺is i 兲兴/ 2
is of similar magnitude.13 We have seen that the ␥␧ of the (n⫹1)s
valence orbitals in neutral many-electron atoms are significantly
larger than the respective ones in hydrogen-like ions. They are in
general even larger than the ones of the innermost 1s orbitals. It
comes about because the self-consistent potential in a manyelectron system differs considerably from the Coulomb potential
(see below). The statement, sometimes found in the literature (for
a recent example see ref. 19) that direct relativistic effects contribute significantly only to the inner core states, and that valence
orbitals are relativistically modified only because they are orthogonal to the core orbitals is misleading. It had explicitly been
disproved already in ref. 23.
The empirical trends in the computed data are:
1. The RCFs of the d valence (and the f) orbitals of neutral atoms
behave quite differently from that of the s orbitals.1,13–17 Although the ␥␧ of hydrogenic 3d or 4f orbitals is still positive
(spin-orbit averaged values are about ⫹0.05 and ⫹0.025, respectively), the relativistic correction of all the other, shielding
orbitals of the neutral atoms reduces the RCF of the outer d(and f-) orbitals to negative values, as is well known (relativistically increased shielding) (see Fig. 5).
2. The value of the RCF for the d-orbitals increases from larger
negative to smaller negative values along a period from group
3 to group 12. For the s-orbitals (Fig. 3) the RCF also increases,
however, from smaller positive to larger positive values.
3. Also, as for the ␥␧(s), an additional nd-electron increases the
␥␧(d) more than an additional (n⫹1)s-electron (see Fig. 5a). For
both ␧s and ␧d, ␥␧(dg⫺2s2) ⬍ ␥␧(dg⫺1s) ⬍ ␥␧(dg), however, as
mentioned before, with ␥␧(s) ⬎ 0 but with ␥␧(d) ⬍ 0.
4. There is no monotonous trend down the periodic table for the
␥␧(d) (see Fig. 5b). For group 3 elements: ␥␧(4d) ⬎ ␥␧(3d) ⬎
␥␧(5d); there is a change over at group 5, and for the higher
groups: ␥␧(4d) ⬎ ␥␧(5d) ⬎ ␥␧(3d). We note the similarity with
the (hydrogenic) s orbitals, where there also occurs a change of
trend for lower principal quantum numbers, as mentioned
above (Fig. 4).
⫺ ␥ 共Z, d g⫺i⫹1s i⫺1兲;
(3)
Figure 5. Fractional relativistic correction factor [␥␧, eq. (1)] of the
spin-averaged d-valence atomic orbital energies. (a) Second row
d-block elements with 4dg⫺25s2 (■), 4dg⫺15s (䊐) and 4dg (Œ) configurations versus group number g. (b) 3d (■), 4d (䊐), and 5d (Œ)
elements with ndg⫺2 (n⫹1)s2 atomic configurations.
808
Autschbach et al.
•
Vol. 23, No. 8
•
Journal of Computational Chemistry
Table 1. Difference ⌬ (n⫺1)s3nd ␥ ␧ of the RCF [␥␧ of eq. (1)] for nl-Orbitals of Neighboring d-Elements with
ndg⫺2 (n ⫹ 1)s 2 vs. ndg⫺1 (n ⫹ 1)s1 Configurations, or ndg⫺1 (n ⫹ 1)s1 vs. ndg Configurations,
Respectively (after Desclaux13).
40
(n ⫹ 1)s 3
nd:
(n ⫹
nd
np
ns
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
V, 25M n vs.
a
24Cr (3d4s)
23
1)s
1)f
1)d
1)p
1)s
2)d
2)p
2)s
Ni, 30Zn vs.
a
29Cu (3d4s)
28
Zr, 43Tc vs.
Nb, 42Mo
(4d5s)a
41
43
Rh, 47Ag vs.
b
46Pd (4d5s)
45
Tc, 48Cd vs.
Ru, 47Ag
(4d5s)a
44
0.179
0.083
0.080
0.069
0.157
0.061
0.061
0.056
0.179
0.146
0.068
0.055
0.34
0.232
0.068
0.055
0.151
0.086
0.053
0.047
0.009
0.010
0.006
0.008
0.014
0.010
0.009
0.010
0.008
0.008
0.009
0.008
0.008
0.001
0.001
0.001
0.002
0.001
0.002
0.001
0.001
77
Ir, 80Hg vs.
Pt, 79Au
(5d6s)a
78
0.147
0.112
0.046
0.035
0.022
0.008
0.006
0.006
0.001
0.001
0.001
g is the group number. j-weighted averages for the spin-orbit split p, d, and f shells.
ndg⫺2 (n ⫹ 1)s2 vs. ndg⫺1 (n ⫹ 1)s1.
b
ndg⫺1 (n ⫹ 1)s vs. ndg .
a
and similarly for cases such as [Mo⫹Ru]/Tc or [Pt⫹Au]/[Ir⫹Hg],
etc. The general trends are very similar for the 3d4s, for the 4d5s,
and for the 5d6s cases, both for the dg⫺2s23dg⫺1s and for the
dg⫺1s3dg configurational changes.
f/d Configurations: the s Valence Orbitals
The 4f and 5f elements (lanthanides and actinides) provide another
remarkable example of the drastic influence of configurational
changes on the RCFs of valence orbitals (Fig. 6). Although having
usually two electrons in the outer 6s and 7s shells, respectively, the
two series of the lanthanides and of the actinides each exhibit quite
varying values of ␥␧(s) for the valence (n⫹1)s orbital. The variation is governed by the distribution of the other valence electrons
on the two innermore nd and (n⫺1)f valence levels. The different
related ground configurations are (n⫺1)f g⫺2 or (n⫺1)f g⫺3 nd1 or
(n⫺1)f g⫺4 nd2, each with (n⫹1)s2, where g is the number of
valence electrons, as before. Again, the specific variations of the
orbital energies (and of other orbital properties13–17) are qualitatively evident already at the nonrelativistic level (Fig. 7). The
relativistic modifications are large, systematical, and monotonous.
They do not change the nonrelativistic trends of orbital energies
within a row, but enhance them.
According to Figure 6, the f elements show lower values of
␥␧(s) when a more outer nd electron is transferred to a more inner
(n⫺1)f orbital. That is contrary to the d elements, which show
higher values of ␥␧ when the outer (n⫹1)s electron is transferred
to the inner nd orbital. Transfer of an outer electron to an inner
orbital improves the nuclear shielding and thereby is expected to
reduce the electron binding energies ␧. These trends are indeed
seen in Figures 2 and 7 for both the d- and the f-elements for
s3d and d3f, respectively, for both the relativistic case and the
nonrelativistic approximation. However, what is unexpectedly different, is that the configurational variations of ␧(s) are reduced by
relativity for the d-elements (Fig. 2), but enhanced by relativity for
the f-elements (Fig. 7).
Figure 6. Fractional relativistic correction factors ␥␧(s) [eq. (1)] of the
s-valence atomic orbital energy of f-block elements, (a) Lanthanides,
(b) Actinides, vs. valence electron (group) numbers g ⫽ 2 to 17 and 14
(nuclear charges Z ⫽ 56 to 71, and 88 to 100), respectively. ■ ⫽
f g⫺2s2, 䊐 ⫽ f g⫺3d s2, Œ ⫽ f g⫺4d2 s2 configurations.
Relativistic Effects in d- and f-Block Elements
Figure 7. Nonrelativistic (䊐) and relativistic (■) 7s orbital energies ␧
of the actinides, in eV, vs. number of valence electrons g. The numbers
at the bottom in the figure are the relativistic stabilization energies.
809
Figure 8. Energetic and radial RCF ␥␧ and ␥r of the 3s, 3p, . . . , 6s
orbitals of Yb with excited configurations f g⫺3d1s2 (“s”), f g⫺3d2s1
(“d”), f g⫺2d1s1 (“f ”).
f/d Configurations: the Other Orbitals
The variation of ␥␧ for different atomic orbitals upon nd3(n⫺1)f
electron transfer is displayed in Table 2. The definition of ⌬d3f␥␧
corresponds to the one for ⌬s3d␥␧ given above [eq. (3)]. Although
the data for the core orbitals are similar in Table 1 for (n⫹1)s3nd
of the d-elements and in Table 2 for nd3(n⫺1)f of the f-elements,
the most loosely bound (n⫹1)s, nd and (n⫺1)f valence orbitals of
the f-elements behave in an “anomalous” manner.
In addition, we have carried out calculations for three excited configurations of the Yb atom: f g⫺3d1s2, f g⫺3d2s1 and
f g⫺2d1s1. The changes of ␥␧ (and ␥r2) are displayed in Figure 8.
The trends for all the orbitals are as described before, except
concerning the 6s valence orbital in d/f configurations. For the
nd3(n⫺1)f configurational changes with (n⫹1)s1 outer shell,
␥␧(6s) [and ⫺␥r2(6s)] still increase (although only a little),
while for the nd3(n⫺1)f configurational changes with (n⫹1)s2
outer shell ␥((nd)s) decreases as ␥(nd) and ␥((n⫺1)f) do. We
note the great similarity of ␥␧ and ⫺␥r2, i.e., the trends of
energetic stabilization and of orbital contraction correspond to
each other.
The trends emerging from the computed data are summarized
in Table 3.
Table 2. Difference ⌬ nd3(n⫺1)f␥ ␧ of the RCF for nl Orbitals of Neighboring f Elements with
(n ⫺ 1)fg⫺3 nd vs. (n ⫺ 1)fg⫺2 Configurations, or with (n ⫺ 1)fg⫺4 nd2 vs. (n ⫺ 1)fg⫺3 nd
Configurations, Respectively (after Desclaux13).
nd 3
(n ⫺ 1)f:
(n ⫹
nd
np
ns
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
(n ⫺
1)s
1)f
1)d
1)p
1)s
2)f
2)d
2)p
2)s
57
La, 58Ce vs.
56Ba, 59Pr
(4f5d)a
93
64
Gd vs. 63Eu,
(4f5d)a
⫺0.077
⫺0.085
⫺0.016
0.001
0.003
0.012
0.015
⫺0.001
0.014
⫺0.56
0.011
0.014
0.015
0.004
0.004
0.005
0.003
0.004
0.004
65
Tb
90
Th vs. 89Ac,
(5f6d)b
⫺0.051
⫺0.123
0.003
0.012
0.004
0.010
0.011
0.002
0.003
0.003
0.003
g is the group number. j-weighted averages for the spin-orbit split p, d, and f shells.
(n ⫺ 1)f g⫺3 nd(n ⫹ 1)s2 vs. (n ⫺ 1)f g⫺2 (n ⫹ 1)s2.
b
(n ⫺ 1)f g⫺4 nd2(n ⫹ 1)s2 vs. (n ⫺ 1)f g⫺3 nd(n ⫹ 1)s2.
a
91
Pa
Np, 96Cm vs.
94Pu, 95Am
(5f6d)b
⫺0.080
0.014
0.021
⫺0.605
0.011
0.011
0.011
0.004
0.003
0.003
0.003
810
Autschbach et al.
•
Vol. 23, No. 8
•
Journal of Computational Chemistry
Table 3. Sign and Magnitude of ␥␧ and Its Changes upon “Outer” to “Inner” Valence Electron
Transfer in Neutral Atoms.
Change of ␥␧ upon
(n ⫹ 1)s 3 nd
Orbitala
nl
␥␧
(n ⴙ 1)s
large positive
strong increase
strong decreaseb/
small increasec
nd
np
ns
large negative
positive
large positive
strong increase
increase
increase
strong decrease
small changes
small increase
(n ⴚ 1)f
(n ⫺ 1)d
(n ⫺ 1)p
(n ⫺ 1)s
large negative
small negative
positive
large positive
increase
small increase
small increase
small increase
strong decrease
small increase
small increase
small increase
nd 3 (n ⫺ 1)f
a
Valence orbitals in bold face italics.
For fg⫺3 d1s2 3 fg⫺2 d0s2.
c
For fg⫺3 d2s1 3 fg⫺2 d1s1.
b
Theoretical Explanations
Our aim is now to understand the regular trends and the irregularities. To this end we investigate the interplay of relativistic
effects and of configurational changes in detail with the help of
double perturbation theory.18 We restrict our analysis here to the
orbital energies.
Perturbation Theoretical Approach
We define Hamiltonians with the following conventions. The first
superscript refers to relativity: 0 means the nonrelativistic approximation, and 1 the relativistic first order correction (1Dir ⫽ direct,
1Ind
⫽ self-consistent indirect). The second superscript indicates
the reference configuration ( 0 ) or, respectively, the change due to
transfer of an electron from an outer to an inner valence orbital
( 1 ). Numerical values for the Yb atom (Z ⫽ 70 with (c/ Z)2 ⫽
3.83) are presented in Table 4.
1. F 00 is the Hartree–Fock operator of the nonrelativistic atom for
the reference configuration, for example, dg⫺2s2. For bound
orbitals ␾00, the nonrelativistic energies ␧00 are negative, of the
order of ⫺101 eV for valence orbitals,
具 ␾ 00兩F 00兩 ␾ 00典 ⫽ ␧ 00 ⬍ 0.
(4)
2. F 01 represents the coupled Hartree–Fock change of the nonrelativistic self-consistent field (due to changes of all the occupied orbitals) upon the change of the valence electron configuration. If a valence electron is transferred from an “outer” to
an “inner” valence orbital, i.e., (n⫹1)s 3 nd 3 (n⫺1)f, the
nuclear screening is improved. Most orbitals respond with a
slight expansion (antiscreening response in the sense of Le
Chatelier’s principle), especially the outer valence orbitals
(concerning 2具r典 ⬇ 具r2典 by about 10 to 20%). That is, F 01 acts
dominantly repulsive (i.e., F 01 ⬎ 0) in those inner atomic
Table 4. One Electron Energy Contributions (in eV) for Orbitals of Yb (Z ⫽ 70) with (Excited)
Configurations [Xe] 4f135d6s2 (“s”), 4f135d26s (“d”), and 4f145d6s (“f”), Orbitals 4s to 6s, Spin Orbit
Averages for p, d, f, Average Configuration Values.
AO
具r典/Å
␧00
(“s”)
␧00
(“d”)
␧00
(“f”)
␧10
(“s”)
␧10
(“d”)
␧10
(“f”)
␧01
(s 3 d)
␧01
(d 3 f)
␧11
(s 3 d)
␧11
(d 3 f)
6s
5d៮
5p៮
5s
4f៮
4d៮
4p៮
4s
2.2
1.8
0.76
0.65
0.4
0.3
0.3
0.26
⫺5.5
⫺6.7
⫺36.9
⫺61.9
⫺28.8
⫺217
⫺365
⫺447
⫺4.9
⫺4.8
⫺34.9
⫺59.6
⫺26.6
⫺215
⫺363
⫺445
⫺4.8
⫺3.2
⫺31.4
⫺55.5
⫺18.4
⫺207
⫺355
⫺436
⫺0.5
⫹1.5
⫺1.8
⫺9.8
⫹6.8
⫹3.3
⫺24.5
⫺69.6
⫺0.9
⫹0.8
⫺2.6
⫺10.6
⫹5.9
⫹2.3
⫺25.5
⫺70.6
⫺1.0
⫹0.7
⫺2.8
⫺10.7
⫹4.8
⫹1.2
⫺26.5
⫺71.5
0.4
2.0
2.0
2.3
2.3
2.2
2.2
2.3
0.1
1.6
3.4
4.1
8.2
8.1
8.6
8.8
⫺0.3
⫺0.7
⫺0.8
⫺0.8
⫺0.9
⫺0.9
⫺0.9
⫺0.9
⫺0.1
⫺0.1
⫺0.3
⫺0.1
⫺1.1
⫺1.1
⫺1.1
⫺1.0
␧00 is the nonrelativistic energy, ␧10 the relativistic correction, ␧01 the change due to s 3 d or d 3 f “outer” 3 “inner”
electron transfer, and ␧11 the double perturbation correction.
Relativistic Effects in d- and f-Block Elements
Figure 9. Coulomb repulsion Jnl,n’s (in a.u., logarithmic scale) between an electron in the nl subshell (K ⫽ 1s; L ⫽ 2s,2p; M ⫽ 3s,3p,3d;
N ⫽ 4s,4p,4d,4f; O ⫽ 5s,5p,5d; P ⫽ 6s,6p) and an electron in the n’s
shell (■ 4s, 䊐 5s, Œ 6s; the respective curves are only dawn to guide
the eye). The abscissa values are the logarithms of the radial expectation values, ln具r典, of the nl subshells. (Note the change of order of
s,p,d,f at the N shell.) The dashed horizontal lines indicate the point
charge Coulomb repulsion 具n’s|1/r|n’s典, i.e., the maximum screening of
the nuclear charge by an electron localized at the nucleus. All orbitals
refer to the nonrelativistic radon atom.
regions, which are enclosed by the respective valence orbitals,
with a repulsion maximum in the region of the additionally
occupied valence shell. If the s population is reduced (as for
s3d), F 01 exhibits a pronounced minimum at the nucleus. As
shown by Figure 9, the nuclear screening, felt by an n⬘s orbital,
corresponds to nearly one charge unit if the screening nl orbital
has a smaller n quantum number, particularly for n ⬍ n⬘ ⫺ 1
(inner screening); for n ⫽ n⬘ ⫺1, the screening is only slightly
reduced. For n ⫽ n⬘, the screening is significantly reduced, note
the logarithmic scale. An outer electron (n ⬎ n⬘) still has a
small, though nonnegligible screening power (outer screening).1,17 The nonrelativistic orbital energies of the modified
configuration are (to lowest order)
具␾00 ⫹ ␾01 兩F00 ⫹ F01 兩␾00 ⫹ ␾01 典 ⫽ ␧00 ⫹ ␧01 ,
(5)
␧01 ⫽ 具␾00 兩F01 兩␾00 典 ⬎ 0
(6)
with
for outer 3 inner electron transfer (see, e.g., Table 4).
3. F10 is the relativistic correction to F00. It contains two types of
terms, as mentioned before. The first contribution is the “direct”
relativistic one-electron correction F1Dir,0. The sum of the
mass-velocity and Darwin terms is attractive, i.e., energy lowering and stabilizing, ␧1Dir,0 ⬍ 0. (As long as we discuss only
811
s orbitals, or the spin-orbit averages of p, d, and f orbitals, we
may neglect the spin-orbit coupling term in our qualitative
discussions.) The influence of the direct term is only significant
in the immediate vicinity of the nucleus,16 i.e., it is especially
large for s and p1/2 orbitals. The second contribution F1Ind,0 is
the “self-consistent” change of the electronic shielding of the
nuclear attraction due to the relativistic modification of all the
other occupied orbitals. The relativistic modifications of these
screening effects [due to relativistic contraction of the occupied
s and p orbitals; to additional spin-orbit splitting of the p
orbitals; and to (indirect) expansion and spin-orbit splitting of
the d and f orbitals] contribute in the whole volume of the
atom.16 F1Ind,0 acts repulsive (ordinary indirect effect due to the
relativistically contracted s and p1/2 orbitals) or attractive (inverse indirect effect16 due to the relativistically expanded p3/2,
d and f orbitals), i.e., ␧1Ind,0 ⬎ 0 and ␧1Ind,0 ⬍ 0 are both
possible. The relativistic Hamiltonian for the atom with the
reference configuration 0 is (F00 ⫹ F10), with orbital energy
(␧00 ⫹ ␧10). At the level of first order perturbation theory, the
relativistic correction of the orbital energies is
具␾00 兩F10 兩␾00 典 ⫽ ␧1Dir,0 ⫹ ␧1 Ind, 0 ⫽ ␧10 ⬍ 0, or ⬎ 0,
if ␾00 is an s or p orbital, or if ␾00 is a d or f orbital.
(7)
This is known empirically1,13–16 (compare also Table 4). (Concerning p orbitals: ␧1,0(p) is slightly positive in a few cases in
the upper left region of the periodic table.)
4. F11 is the relativistic correction to F01, i.e., it is the configurational
difference of the relativistic changes of screening from all the
slightly modified occupied orbitals in the two different configurations. The relativistic Hamiltonian of the atom with the modified
valence electron configuration and its eigenvalues are, respectively
(F00⫹F10⫹F01⫹F11) and (␧00⫹␧10⫹␧01⫹␧11). According to
double perturbation theory, ␧11 can be expressed in many different
forms,18 for instance as
␧ 11 ⫽ 具 ␾ 00兩F 11兩 ␾ 00典 ⫹ 2 䡠 具 ␾ 00兩F 10兩 ␾ 01典
or ⫹ 2 䡠 具␾10 兩F01 兩␾00 典.
(8)
Concerning the first contribution, we remember that “outer” 3
“inner” electron transfer is partially compensated by self-consistent relaxation of the charge cloud. Therefore, the respective
relativistic corrections, which are also of different signs depending
on the l values of the occupied orbitals, are expected to cancel each
other partially. Therefore, we may at first skip the further discussion of 具F11典.
In the first representation of ␧11 given by eq. (8), the second term
is the average of F10 ⫽ F1Dir,0 ⫹ F1Ind,0 over the configurational
change of the density of the valence orbital under discussion,
␳01 ⫽ 2 䡠 ␾00 䡠 ␾01. We note that ␳01 is oscillatory, and zero on the
average, while F10 heavily weighs the contributions from near
the nucleus. In the alternative representation, the second term is the
average of the configurational change of the self-consistent field,
F01 ⬎ 0, over the relativistic density change of the valence orbital
␾00, ␳10 ⫽ 2 䡠 ␾00 䡠 ␾10. ␳10 is also oscillatory, with zero average.
It is positive more inside and negative more outside for s and p
812
Autschbach et al.
•
Vol. 23, No. 8
Table 5. Qualitative Behavior Expected for the RCF ␥␧ of Valence
Orbitals in Different Configurations of d- and f-Block Atoms
Orbital
Configurational Changes
(n ⫹ 1)s 3 nD
nd 3 (n ⫺ 1)f
(n ⫹ 1)s
the positive ␥␧ increases
to more positive
values
the positive ␥␧ varies a
little, or decreases to
less positive values
nd
the negative ␥␧ increases
to less negative values
the negative ␥␧ decreases
to more negative values
the negative ␥␧ decreases
to more negative values
(n ⫺ 1)f
orbitals, and it is vice versa for valence d and f orbitals. Therefore,
it sounds reasonable that the numerical integrations yield ␧11 as
significantly negative in the (n⫹1)s3nd cases (␧11 stronger than
⫺1 䡠 ␧01 䡠 Z2/c2), but comparatively less negative or even positive
in the nd 3 (n⫺1)f cases (␧11 weaker than ⫺1/2 䡠 ␧01 䡠 Z2/c2).
The Relativistic Correction Factor
The RCFs of orbital nl in the two different configurations o, and i
(the one with more occupation of the inner valence orbital), are
␥ 0 ⫽ ␧ 10/共␧ 00 ⫹ ␧ 10兲 䡠 c 2/Z 2
and ␥ i ⫽ 共␧10 ⫹ ␧11 兲/共␧00 ⫹ ␧10 ⫹ ␧01 ⫹ ␧11 兲 䡠 c2 /Z 2 .
(9)
We now ask, when does the RCF of some AO in a d-s or in an f-d
configuration increase or decrease upon an “outer” to “inner”
electron transfer. From eq. (7) follows:
␥ i ⬎ ␥ 0 if ␧11 䡠 ␧00 ⬎ ␧10 䡠 ␧01 ,
i.e. if 共⫺␧11 兲/␧01 ⬎ ␧10 /共⫺␧00 兲,
(10)
and vice versa, respectively, because ␧01 ⬎ 0. From the qualitative
discussions and from the numerical data mentioned above, we can
then conclude: If (⫺␧11)/␧01 is strongly positive, as for the s3d
cases, the ␥␧ (and the ⫺␥r2) increase for all orbitals. But if
(⫺␧11)/␧01 is not significantly positive (i.e., small or even negative), as for the d3f cases, the ␥␧ increase, as usual, for sure only
for the inner core orbitals with large |␧00|. However, ␥i ⬍ ␧o may
hold for the s, d, and f valence AOs in the d3f cases, particularly
in cases where ␧01, the orbital energy dependence on the type of
configuration, is large. These theoretical expectations are summarized in Table 5. They reproduce the empirical findings shown in
the Trends and Irregularities section.
Summary and Conclusions
We have analyzed the configurational and relativistic changes of
atomic valence and core orbitals and their interplay. The unex-
•
Journal of Computational Chemistry
pected trends found empirically in the computed data can be
understood perturbation-theoretically.
The properties of atomic orbitals such as orbital energies or
orbital radii, which determine the chemistry of the elements, are
strongly dependent on the electronic configurations, i.e., on the
distribution of the given number of valence electrons among the
near degenerate valence AOs. Note the nd and (n⫹1)s orbital
energies of the transition metal atoms, which still form a subject of
analysis and educational discussion, also do not follow the Aufbau
rule.24 The interplay of one-electron kinetic and nuclear attraction
energies, of two-electron repulsion energies, and of relativistic
correction energies in combination with orbital relaxation effects
as a response to the configuration-dependent self-consistent field,
results in lowest energies for dg⫺2s2 configurations of most d-bock
atoms, and for fg⫺2s2 configurations of most f-bock atoms in their
neutral states.
However, for about one-third of these 63 elements (La,Ac with
g ⫽ 3; Ce,Th with g ⫽ 4; Nb,Pa with g ⫽ 5; Cr,Mo,U with
g ⫽ 6; Pd,Pt,Gd,Cm with g ⫽ 10; Cu,Ag,Au with g ⫽ 11, etc.), a
lower energy is achieved for more electron density in the d shell.
This comes about through “outer” to “inner” s3d transfer in
d-bock atoms, but through “inner” to “outer” f3d electron transfer
in f-bock atoms. Thereby the s and d AOs become expanded and
more weakly bound in the d-block cases (Fig. 2), and the s, d, and
f AOs become contracted and more strongly bound in the f-block
cases (Fig. 7). Consequently, the weight of the different valence
AOs in molecular orbitals is also influenced differently.25
These opposite electron transfers (outer s to d, or inner f to d)
and the connected AO changes are accompanied by increased
relativistic stabilization of the valence orbitals in both cases. Note
the same sNd order in Figures 2, 3, and 5a for ␧ and ␥; but
different dNf orders in Figures 6 and 7. In the case of d-block
atoms, the expansion and destabilization of the valence orbitals
upon s3d transfer is partially compensated by relativity. Namely,
the configurationally destabilized, softened (n⫹1)s AO becomes
particularly strongly stabilized by relativity (Fig. 3), and the configurationally destabilized nd AO becomes comparatively little
destabilized by relativity (Fig. 5a), after the s population has been
reduced. As a consequence, the largest relativistic effects in the
(n⫹1)s valence shell and the smallest relativistic effects in the nd
valence shell occur for group 12 elements in the case of (n⫹1)s2
configurations, for group 11 elements in the case of ns1 configurations, and for group 10 elements in the case of vanishing (n⫹1)s
population (compare also Fig. 2 of ref. 20).
In the case of f-block atoms, the contraction and stabilization of
the valence orbitals (at least the d and f ones) upon f3d transfer
is also enhanced by relativity. The configurationally stabilized and
contracted nd and (n⫺1)f AOs become comparatively weakly
destabilized by relativity, and the configurationally contracted
(n⫹1)s AO becomes comparatively strongly and additionally stabilized by relativity in the case of an outer s2 configuration (Fig. 6).
The astonishing difference comes about, because the orbitals basically respond to the intra-atomic charge redistribution of “outer”
to “inner” s3d or of “inner” to “outer” f3d electron transfer,
whereas the self-consistent relativistic effects strongly depend on
the s population. Relativistic corrections contribute significantly to
the increased fine tuning in heavy element chemistry, but with
different tendencies than the basic nonrelativistic mechanisms.
Relativistic Effects in d- and f-Block Elements
813
Acknowledgments
We thank Professor Wang for the hospitality at Shanghai JTU.
References
Figure 10. Change of experimental and theoretical properties of the
atoms down the column 11 (Cu, Ag, Au). Reff (M⫹,2) is the effective
crystal radius of the singly charged cations with digonal coordination,21 and IP is the first atomic ionization potential. 具r典s is the radial
expectation value, and ␧(s) the Dirac–Fock eigenvalue of the (n⫹1)s
valence AO.
Chemistry is, in its essence, a qualitative science. To a large
extent, it can be explained at the level of the single electronmolecular orbital approximation, where the electron repulsion
effects are accounted for only on the average. The wealth of
general chemistry is well ordered with the help of the periodic
table. The characteristic variations along the periods are dominated
by configurational shell effects, i.e., by the properties and behaviors of the different (n⫺1)f, nd, and (n⫹1)sp near-degenerate
valence subshells and their dependencies on the population distributions among them.
The relativistic effects, which increase down a column in
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involved relation with the nonrelativistic shell effects. One example for the cooperation of nonrelativistic and relativistic effects had
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Cu, Ag, Au (see Fig. 10). Other interesting examples are the g ⫽
4 elements Ti, Zr, Ce, Th, or the g ⫽ 10 ones Gd, Cm.
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