34; TETRAVALENT
3. Physical theory of bonding in metals
3.6. The relativistic volume effect
The standard procedure (Rose 196t), for heavy atoms, is to solve the wave equation in a spherical potential with relativistic kinematics. In open-shell systems the charge density is spherically symmetrized by averaging over all azimuthal quantum numbers.
The wave equations to be solved in practice are, therefore, the coupled first-order differential equations for the radial components of the Dirac equation (Rose 1961)
g'K- (x+l)O~+cf~
1+ , (5la)r
cf'~
(~c- 1)cf~- + ( v - E ) O ~ ,
(51b)
r
where gK and f~ are the major and minor components of
q~~~=( g~)~~~ ~ (52)
where x is the Dirac quantum number,
Z~u
is a two-component Pauli spinor and a m is the radial component of the Pauli spin matrices. The properties and solutions of the Dirac equation for spherical symmetry have been reviewed in detail by Rose (1961), Grant (1970) and Desclaux (1973).In the RLMTO or RKKR methods the solutions corresponding to a given l are replaced by the two solutionsj = l _+.21- of the Dirac equation. The potential is specified by a set of parameters twice as large as the LMTO set (Andersen 1975, Godreche 1982, Brooks 1983, 1984a, Christensen 1984a). Thus, the Hamitonian and overlap matrices have the same form as the LMTO matrices, but, like the structure constants, are doubled in size. It is then convenient to think of the sets of potential parameters as giving rise to pure j-bands rather than/-bands. The charge density is reconstructed by taking the spherical average
n(r)= ~
fN~,j(E)[f~,j(E,r)+ g~,j(E,r)]dE,
(53)where
N~,j(E)
is the partial1,j
resolved state density.The immediate effect of separate j-projected stare density is that this leads to distinct occupation numbers for the two j-bands with a given I. In the limit of vanishing spin-orbit splitting, A ... of the two j-bands, the ratio of their populations becomes equal to the ratio of their degeneracies. But in the other limit, when A .... is much
2.5
2.0
1.5
1.0
0.5
Th Pa U Np Pu A m
An
Fig. 8. The ratio of the n u m b e r of electrons in the
j=l 71
and j = 1+~1 bands(l/(l+
1)), for zero spin-orbit splitting, for both d and f electrons in the actinide metals.greater than the bandwidth, W, the two j-bands are split apart and the j = l - ½ band fills first. The ratio R 2 = n~=z-a/2/nj=z+ 1/2 changes from 21/(21 + 2) to infinity as the ratio R 1 = As.o./W ~ increases, as long as there is a total of less than 2l electrons of type I. Therefore R 2 is an excellent measure of the importance of spin orbit interaction in the ground state. It has been evaluated in self-consistent, fully relativistic, band calculations (Brooks 1983) for the light actinide metals, A c - A m . The ratio for d and 3 and f electrons is plotted in fig. 8 as a function of atomic number. R 2 approaches 2 in the limit R 1 ~ 0 for f and d electrons, respectively. It is clear from fig. 8 that
g
there are considerable departures from this ratio for f electrons in the metals N p Am, but not for d electrons. The reason for this is that the preferential occupation of the j = 1 - ½ band is determined not by the spin orbit splitting alone, but also by the ratio R1, which is inversely proportional to the bandwidth. Thus, spin-orbit interaction does not have any major effect upon the bulk ground-state properties, even for heavy metals, unless the corresponding energy bands are narrow. This is exactly what does happen to the f bands with increasing atomic number, since there is both an increase in A .... and a decrease in W, producing an increase in R1.
The effect of preferential occupation of the " 5 J = 3 f band is made explicit by the relativistic form of eq. (47),
2nj t (
3 P V = ' { [ C j , , - V ( S ) ] + ~ ~ c ( S ) - e x ¢ ( S ) } + n j , , ( ( E ) j . , - C j , t ) 2 j + l + 2 , ,
)
Bj,l ,Uj,I/
(54) where n o w nr, j is the n u m b e r of partial l,j electrons, C~,j is the centre of t h e / , j - b a n d ,
#t,j is the effective mass and (E)z, ~ is the first energy moment. (E)~,~ is zero for an empty or filled band and is minimal for a half-filled band. When As.o./W is large, the
C O H E S I O N IN RARE E A R T H S A N D A C T I N I D E S 177
first moment vanishes for 0, l and 2 ( 2 / + 1) electrons and has minima at the centre of each partial/,j-band.
Therefore the trend in atomic volume across a transition metal series, which for a constant compressibility follows the pressure, will tend to increase towards the centre of the series if As.o./W is large.
3.7. Spin polarization and the cohesive energy
In LSDA the spin polarization energy may also be expressed in terms of radial integrals (Gunnarsson 1976, 1977, Janak 1977, Brooks and Johansson 1983)
ELSDA __ 1 ~ Jtrmtmr, (55)
sP 4 l,l'
where the LSDA atomic exchange integral matrices are given by
J,r = ~ f {r2~p2(r)(p2(r)A[n(r)]/n(r)} dr (56)
and A[n(r)] is a well known (von Barth and Hedin 1972) function of the density.
The LSDA exchange integrals for the f and d stares of free lanthanide and actinide atoms are plotted in figs. 9 and 10. In the self-consistent free-atom calculations the d " - i s configurations for the transition metals and f"ds2 configurations for the rare earths and actinides were used. Since for free atoms the eigenstates are bound states, there are only solutions at discrete energy eigenvalues and only one possible exchange interaction for a given value of ll'. In the solid state, where the conduction-electron bands are scattering states, the radial wave functions are continuous functions of the energy and the exchange integrals are energy dependent.
The reason that the f - d exchange integrals decrease across each series is the contrac- tion of the f shell, which decreases the overlap with the d states. The overlap between
L, O0 300
2O0 IO0
Lc~ Ce 5d
3 0 _ 2 0 - - E
10 Yb
Fig. 9. Calculated exchange integrals across the lantha- nide series for free atoms. The full lines are the L S D A exchange integrals, eq. (56), and the dashed lines are the Hartree Fock exchange integrals.
«00
E
200
r-~
I
Th
- ~ ~ ~ ~ ~ f s ~
~ f d ~ ~ ~ .
I
80
E
~ 0 - -
I l l i l I I I A A
Cm Lr
Fig. 10. The same exchange integrals as in fig. 9, but for the actinide series.
4f
i
05 1.0 1.5 Z.'O RAOIUS (o.u.)
2; 3;
Fig. l I. The calculated 4f and 5d radial densities for a free Gd atom.
4f and 5d densitles occurs over a relatively small region of space, corresponding to the outer part of 4f density and the inner part of the 5d density (fig. 11). When the 4f shell contracts, the region of overlap decreases.
As an example of the usefulness of eq. (56) we consider the cohesive energies of the transition metals (Gunnarsson et al. 1974, Friedel and Sayers 1977, Johansson et al. 1980, Brooks and Johansson 1983), shown in fig. 12a, which are not regular across the series. However, the cohesive energy, E c, of an elemental metal is defined as the energy difference between the free atom in its ground state and the energy of the metal per atom at zero temperature, and therefore contains a free-atom energy contribution (see sect. 2.2), AEatom , which is the preparation energy required to take the atom from its ground stare to a stare similar to the non-magnetic ground-state configuration of the metal. It is this contribution which behaves irregularly across the series. The cohesive energy may therefore be written as
E c = E b - - AEatom , ( 5 7 )
~- -~ 3 d series - - - t , - - 6 - - 4 d series - - o - - o - - 5 d series
a)
B"~k\,o_...o.
10£
~' 5.0 ô
c )
% ,
/~,,.
ùo, ~,
b ) / \ x
i i i i f i i i I
CoScl] V Cr MnFeCoNi Cu
Sr Y Zr NbMoTc RuRhPdAg YbLu Hf To W Re Oslr Pt Au
% i
. . . . . . . . d~
ds d 2 3 t. 5 ds dsds d~ ds ds ds 7 a 9 CeScTi V Cr IVnFeCoNi Cu Sr Y Zr NbNoTc RuRhPd Ag YbLu Hf To W Re O s I r Pt Au
Fig. 12. (a) The measured cohesive energies of the d transition metal series. (b) The calculated valence b o n d energies by application of eqs. (57) and (58) to the cohesive energies shown in (a).
COHESION IN RARE EARTHS AND ACTINIDES 179 thus defining the valence bond energy, Eh, which is expected to vary more smoothly across the series. If the relatively small contributions to the total energy of the atom from Hund's second rule and spin-orbit interaction are neglected, AEatom may be separated according to
AEatom = Ep - ~sp~LSDA' (58)
where Ep is the preparation energy required to take the atom from its ground state configurätion to the ma9netic ground-state configuration of the prepared atom, which for d electrons may be taken to be the sd n +* configuration. Ep may be obtained from experimental data (Moore 1958). The spin polarization energy is the LSDA equivalent of Hund's first rule energy (Gunnarsson and Lundqvist 1976) and is lost when the free atom is prepared in the non-magnetic ground state, and in the form (55) contains the coupling between open shells.
When Ep and --FLSDA --SP are added to the measured cohesive energies shown in fig. 12a, the valence bond energies of the three d transition metal series shown in fig. 12b are obtained. The valence bond energies, relative to the d "+ is non-magnetic ground stares of transition metal atoms, a r e - in contrast to the cohesive e n e r g i e s - approximately parabolic functions of the d occupation number. Thus, the valence bond energy, defined by eq. (57), is that part of the cohesive energy which is essentially a solid-state property, changes smoothly across a series, and is therefore most useful for interpolation (sect. 2.2).
If contributions to cohesion other than the transition metal "d band" contribution (Friedel 1969) were negligible, E b and the "d band" contribution would be equivalent.
Then E b could be estimated independently by assuming that the number of states function is a straight line and fitting the second moment of the d band by a rectangular density of stares function (Friedel 1969, Cyrot and Cyrot-Lackmann 1976). In general, however, the repulsive interactions between atoms, the sp contribution and any additional correlation energies in the solid that also show a pärabolic trend, are included in E b without being distinguishable in fig. 12b.
The cohesive energies of the trivalent rare earths should, therefore, be about 100 kcal/mol relative to the trivalent atomic stare (fnds2). However, the cohesive energy of gadolinium is slightly less 95 k c a l / m o l - although the configuration of the free gadolinium atom is f75d 6s 2. The difference is due to the multiplet coupling between the open 4f and 5d shells in the free gadolinium atom. This coupling may easily be estimäted in the local spin density äpproximation. In this approximation the multiplet coupling energy is --gd4fSd#4fBSd 1- ~ s in terms of the 4f-5d exchange interaction and the spin components of the 4f and 5d moments. We calculate J4fsd to be 0.105 eV (or 2.41kcal) for a G d atom. Therefore, with a 4f spin moment of seven and a 5d spin moment of one the coupling energy is 8.4 kcal.
In Gd metal the 4f moment remains saturated but the 5d states are itinerant and have a small moment of 0.475#B with a splitting of the spin-up and spin-down states at the Fermi energy due to exchange between the 4f and conduction electrons. The reduction of both the 5d moment and the 4f-5d exchange interaction means, that much of the exchange energy of multiplet coupling is absent in the solid. The exchange interaction, J«sd, is reduced to 0.092 eV (2.11 kcal) in Gd metal, hence the coupling
is 3.5 kcal. The difference between the 4f-5d multiplet coupling in the free atom and the solid is therefore 8.4 minus 3.5 kcal or about 5 kcal. The same coupling is also present with about the same magnitude in other trivalent free rare earth atoms. The cohesive e n e r g y - relative to the trivalent free a t o m s - should therefore be about 95 kcal/mol rather than 100 kcal/mol for metals where there is no f shell. Similarly, the measured cohesive energy of the trivalent actinide element curium is about 90 kcal/mol (Ward et al. 1975), which may be taken to be representative of the cohesive energy of the trivalent actinide metals - again relative to the corresponding trivalent (5f" 6d 7s 2) free-atom configuration.