K. BALASUBRAMANIAN*
2. Relativistic effects and methods
There are several review articles (Pitzer 1979, Pyykk6 and Desclaux 1979, Krauss and Stevens 1984, Balasubramanian and Pitzer 1987, Malli 1982a, b, Christiansen et al. 1985, Pyykk6 1986, 1988, Balasubramanian 1989a, b, 1990a, b) which deal with the importance of relativistic effects on the chemical and spectroscopic properties of molecules containing very heavy atoms. For a detailed description of these effects and a survey of the literature, the reader is referred to these reviews. However, in this chapter
we will elucidate the basic nature of relativistic effects and in particular their role in lanthanide and actinide chemistry. We will also describe the basic relativistic quantum chemical methods briefly. F o r more details on these methods, the reader is referred to the above-mentioned reviews.
2.1. The nature of relativistic effects
Relativistic effects can be defined as differences in chemical and spectroscopic properties arising from the true velocity of light as opposed to the assumed infinite velocity of light in non-relativistic models. Relativistic effects are far more important for molecules containing very heavy atoms since the inner electrons in such atoms are subjected to the electric field generated by a large number of protons. Consequently, to maintain balance with the large electrostatic force the inner electrons of very heavy atoms move with speeds that are comparable to the speed of light. For example, it is estimated that the ls electrons of post-lanthanide atoms such as gold attain 60~o of the speed of light.
There are several effects of relativity on chemical bonding and spectroscopic properties. We first focus on a relativistic effect called the "mass-velocity" effect. As the electron approaches the speed of light, its mass increases with velocity since in special theory of relativity mass is no longer a constant and varies as
M - M°
where M o is the rest mass (mass when v = 0) and M is the actual mass. It is clear that when v approaches 60~o of c, M becomes
M = 1.25 M o.
That is, even at 60~o of the speed of light the mass of an inner ls electron of the gold atom increases by 2 5 ~ . The 2 5 ~ increased mass means an increase in its kinetic energy.
Since the total energy is conserved, the electrostatic energy has to become more negative. The electrostatic energy of an electron is inversely proportional to its averaged orbital radius and this evidently leads to the contraction of the inner ls orbitals of atoms such as Au, Pt, T1, etc. To keep balance with the core orbitals, which by far experience most of the relativistic effects, the valence orbitals also contract. F o r example, the 6s orbital of the gold atom contracts. Bagus et al. (1975) showed that nearly half of the lanthanide contraction is actually attributed to relativity. We discuss this in detail in the next section.
Another relativistic effect of considerable importance in the chemical properties of molecules containing very heavy atoms is the spin orbit interaction. The s p i n - o r b i t interaction is a relativistic effect in the sense that it goes to zero in the limit c --. oe due to the presence of~ 2 (e = l/c) term in the numerator of the s p i n - o r b i t operator. Physically it arises from the coupling of the orbital angular m o m e n t u m (1) of the electron with its spin (s). As Z (the atomic number) increases, the coupling of l and s increases significantly. Thus a new quantum number for the atoms, j = 1 + s, becomes a good
34 K. BALASUBRAMANIAN
A
ILl
I'
1.5"
1.0"
P b
G•e
;n Fig. 1. The spin-orbit coupling in the fourth group atoms C - P b . The 3Po-3P 2 atomic energy separations are shown. Reproduced from Balasubramanian (1989a, b).quantum number and 1 and s are no longer good quantum numbers. This, in turn, introduces significant mixing of electronic states of different spin multiplicities and different I values for an atom.
Figure 1 shows the periodic trend in the 3Po-3P 2 splitting of the group IV elements where the suffixes are the J quantum numbers. Note the dramatic increase in the 3P0-3P z splitting of Pb (1.3 eV) compared to Sn (0.4 eV). The ground state of the Pb atom is thus 88% 3P o and 12~o 1S o compared to Sn which is 97?/0 3P o and only 3% 1S 0.
Non-relativistically, electronic states of different spin multiplicities do not mix and thus the mixing of 3p with 1S is purely a relativistic effect. Likewise, the J = 2 state of Pb is 70% 3P 2 and 30% 1D 2.
The spin-orbit coupling makes significant impact on the electronic structure and spectroscopy of lanthanide and actinide molecules. First electronic states of different spin multiplicities and spatial symmetries are mixed by spin-orbit coupling. Conse- quently, the conventional AS = 0, AA = + 1 selection rules, where S and A are total spin and angular momentum along the molecular axis, do not hold for diatomics containing lanthanides and especially actinides. This is because the intensity of such electronic transitions depend on the magnitude of an integral called the transition moment. The spin-orbit coupling increases the transition moments and therefore the transition probabilities of electronic transitions which are rigorously zero non-relativ- istically. This is brought about primarily through mixing of different electronic states which do not mix non-relativistically.
The spin-orbit effect can destabilize chemical bonds and enhance chemical reactiv- ity through coupling of different electronic states. For example, the Do of Pb 2 (Bala- subramanian and Pitzer 1983) is reduced to half of its value primarily through the spin-orbit effect. For example, the D e of Pb2 is ~,, 1.8 eV in the absence of spin-orbit coupling, but becomes 0.88 eV when spin-orbit effects are included. This is primarily because the spin-orbit stabilizations of atomic states and molecular states are not always the same. In the case of Pb, the ground state of the lead atom is considerably more stabilized than the Pb 2 dimer and thus the D e is decreased significantly compared to the D e of S n 2 which is 1.94eV. The ground state of P b 2 is a mixture o f 3~---~ g (0:),
+ + 2 1 + + 2
l e g (0g)(rc) as well as Zg (0g)(Tzu).
The enhancement of reactivity through spin-orbit coupling occurs when potential energy surfaces of electronic states of different spin multiplicity cross. For molecules containing very heavy atoms, if this happens, the spin-orbit coupling at the points of intersection could be large leading to a significant number of non-adiabatic surface hoppings from one surface to the other. Typically high spin electronic states of heavy atoms do not insert into H 2 but the low spin states do. Hence if the high spin surfaces cross the low spin surfaces there could be considerable interaction between the two surfaces primarily brought about by spin-orbit coupling in the heavy species. In this event, there could be hopping of electrons from high spin to low spin states non- adiabatically and thus the high spin atoms, which are intrinsically non-reactive, become more .reactive. This is exemplified by several studies from our laboratory on the reactivity of third-row transition-metal atoms. For example, spin-orbit coupling enhances the reactivity of high spin state Ir with H 2 (Balasubramanian and Dai 1990).
In this chapter, we show how the reactivity of Hfwith H 2 is contrasted compared to Zr and H 2 due to lanthanides and relativistic effects.
2.2. Relativity and tanthanide contraction
Bagus et al. (1975) have studied the effects of relativity and the lanthanide contrac- tion (gradual decrease of atomic radii from La to Lu) on Hf to Bi atoms which follow the lanthanides. It is important to know how much of the contraction on the 6s valence orbital is caused by relativity as opposed to incomplete screening of the 4f shells.
To accomplish this goal, Bagus et al. made Hartree-Fock calculations on "pseudo- atoms" corresponding to Hf, Re, Au, Hg, T1, Pb and Bi without 4f electrons, but with the atomic number reduced by 14. The orbital energies of the pseudo-atoms were compared with Dirac-Fock orbital energies of the real atoms by Bagus et al. Of course, this technique would thus separate the contraction effect due to 4f shells and relativity.
Typically the seven projections of the 4f shells occupy a small volume and thus fail to screen the nucleus. This results in an incomplete screening of the 4f-sheU electrons.
Consequently, e.g., the 6s orbital of the Hf atom is not fully shielded by the 4f shell of electrons. This, in turn, leads to a contraction of the outer 6s orbital. The contraction of the radii of the outer electrons is more commonly referred to as the "lanthanide con- traction". However, not all of the orbital-radii contraction arises from the incomplete screening of the 4f shells. As a matter of fact, nearly half of the contraction is due to relativity.
Table la reproduces the results of Bagus et al. (1975) which shows a critical com- parison of the orbital energies in Hartree atomic units for both the HF pseudo- atom (without 4f shells) and the actual atom for which relativistic Dirac-Fock energies are listed. Note that the actual orbital energies are negative and for convenience the magnitudes of these energies are shown in table la. The difference in the orbital energies of the pseudo-atom and HF atom measures the effect of the 4f shell. The difference in the D H F orbital energy and HF orbital energy measures the effect of relativity.
As seen from table la, the difference in the 6s orbital energy of the pseudo-Hfand Hf (HF) is 0.0299 Hartrees while the difference between Hf (HF) and Hf (DHF) energy is
TABLE la
Orbital binding energies in a.u. of atoms Hf-Bi. Both pseudo, relativistic and non-relativistic atoms are compared. Reproduced from Bagus et al. (1975).
5d 6s 6p
(5d3/2) (5d5/2) (6Sl/2) (6Pl/2) (6P3/2)
pseudo-Hf 0.3192 0.1805
H~HF) a 0.2992 0.2104
H~DHF) b 0.2473 0.2355 0.2397
pseudo-Re 0.4660 0.2031
Re(HF) 0.4538 0.2347
Re(DHF) 0.3972 0,3661 0.2783
pseudo-Au 0.5372 0.1905
Au(HF) 0.5210 0.2208
Au(DHF) 0.4935 0.4287 0.2917
pseudo-Hg 0.7191 0.2288
Hg(H F) 0.7142 0.2610
Hg(DHF) 0.6501 0.5746 0.3280
pseudo-Tl 0.9472 0.3162
TI(HF) 0.9683 0.3611
TI(DHF) 0.8945 0.8062 0.4492
pseudo-Pb 1.1772 0.4025
Pb(HF) 1.2245 0.4589
Pb(DHF) 1.1388 1.0360 0.5665
pseudo-Bi 1.4131 0.4906
Bi(HF) 1.4874 0.5582
Bi(DHF) 1.1389 1.2710 0.6862
0.1836 0.1924
0.2114 0.1765
0.2268 0.2398
0.2751 0.2199
0.2693 0.2862
0.3385 0.2612
a Reference [1] of Bagus et al. (1975).
b Reference [3] of Bagus et al. (1975).
TABLE lb
Comparison of radial expectation values (R) for Hf, Re, Au, Hg, TI, Pb and Bi. Reproduced from Bagus et al.
(1975).
5d 6s 6p
d3/2
d5/2 Pl/2 P3/2pseudo-Hf 2.5048 4.6934
Hf(HF) 2.2277 4.0684
Hf(DHF) 2.3376 2.4198 3.6939
pseudo-Re 2.0326 4.2162
Re(HF) 1.7999 3.6942
Re(DHF) 1.8301 1.9047 3.2770
pseudo-Au 1.7228 4.2230
Au(HF) 1.5433 3.7006
Au(DHF) 1.5359 1.6185 3.0609
pseudo-Hg 1.6040 3.7500
Hg(HF) 1.4327 3.3284
Hg(DHF) 1.4312 1.4987 2.8434
pseudo-Tl 1.5042 3.3294
TI(HF) 1.3412 2.9669
TI(DHF) 1.3387 1.3940 2.5792
pseudo-Pb 1.4214 3.0475
Pb(HF) 1.2671 2.7242
Pb(DHF) 1.2641 1.3119 2.3916
pseudo-Bi 1.3506 2.8336
Bi(HF) 1.2046 2.5939
Bi(DHF) 1.2012 1.2439 2.2429
4.2434 3.9262
3.5166 4.0123
3.7532 3.4569
3.0739 3.5162
3.4116 3.1366
2.7802 3.1862
O.U.
0.3(
0.21
0.20
0.15
1 I I
Cu ...
o . " • . . . • . . . •
" ' • ' " - e "
\ \
\
1 I
4 5
n
/ Au e /
~ ' A u
\ \
% p S I AI.J
I I 0
..,"Hg
Z n .."
o. ,"
"""... .."'
""o"
C ~ H g
N \ N
N N
N
. . . DHF HF
I I I
6 4 5
n
ps Hg
I
6
4.5
4.C
O, II ,
3.5
3 £
2.5
I
. . . . DHF
~ H F
I I
p ps. Au I I / / / / / /
Ag
..'"'ct". AU
Cu ""...
"'"o
I 1 I
4 5 6
n
Fig. 2• Illustration of lanthanide contraction• Magnitude of orbital energies of valence s electrons of radii Cu, Ag, Au as well as Zn, Cd and Hg. Also shown are the expected values. The label ps is the pseudo-atom without 4f 14 shells. Reproduced from Bagus et al. (1975).
38 K. BALASUBRAMANIAN
0.0293. The former energy of stabilization is due to the incomplete screening of the 4f a4 shell while the latter energy of stabilization is due to relativity. This means that the effect of the lanthanide contraction is comparable to the effect of relativity.
Figure 2 shows systematically the magnitude of orbital energies of Cu, Ag and Au as well as the Zn, Cd and Hg triad. The plot joining the pseudo-atom (without 4f 14 shells and without relativity) is almost collinear (expected trend). The dotted curve is the actual D i r a c - F o c k trend including relativistic effects and 4f 14 shells while the full curve is the H F line without relativistic effects. It is clear that the dramatic change in trend for Au and Hg is due to both incomplete screening of the 4f 14 shell and the effect of relativity.
Table lb shows the orbital radii expectation values for the pseudo-, H F and D H F atoms. Figure 2 shows the orbital radii expectation values obtained by Bagus et al. As seen from both table lb and fig. 2, the 6s~/2 orbitals of H f - B i are contracted both due to lanthanide contraction and relativity. F o r the gold atom (fig. 2), the contraction due to lanthanide contraction is 0.523 a.u. The relativistic contraction is 0.64 a.u. Consequent- ly, the relativistic contraction effect is larger than the lanthanide contraction effect for the gold atom. F o r the Hf atom, the contraction is 0.625 a.u. while the relativistic contraction is only 0.3745 a.u. Hence, for Hf, the lanthanide contraction is larger than the relativistic contraction.
The above effect of relativistic and lanthanide contraction has dramatic impact in the chemical and spectroscopic compounds of molecules containing Hf to Bi. For example, the color of gold is due to this effect. Figure 3 compares the ionization potentials of Cu, Ag and Au atoms. As seen from fig. 3, the IP of gold is substantially larger than Cu and Ag, which is an anomaly since generally the IP decreases as one goes down the group.
The trend in fig. 3 is akin to the D H F orbital energies of Cu, Ag and Au computed by Bagus et al. (fig. 2). Hence, it is clear that nearly half of the anomalous behavior of the gold atom is due to relativity while the other half is due to the lanthanide contraction.
The lanthanide and relativistic contractions have other important consequences on molecular and chemical properties. Figure 4 compares the equilibrium bond distances of NiH, P d H and PtH. Note that the P t H diatomic has a significantly shorter bond distance compared to P d H which is attributed to the contractions of the outer 6s orbital of the Pt atom arising from both lanthanide contraction and relativity.
Figure 5 compares the Des of Cuz, Ag 2 and Au 2 triad. Once again, the trend from Cu 2 to Ag 2 is reversed in going from Ag 2 to Au 2. That is, instead of the expected decrease in the binding energy of Au z it actually increases. This is also due to the contraction of the 6s valence orbital of the gold atom which leads to a stronger and shorter bond for Au 2 compared to Ag z.
Atoms such as Hg, T1, Pb and Bi form considerably weaker bonds compared to the lighter members in the same group. This too is a consequence of the contraction of the 6s orbital of these elements. Note that for H g - B i the 6s 2 shell is complete. The unusual stability of the 6s shell leaves the filled 6s 2 shell inert in that it does not actively participate in bonding. Hence, two Hg atoms form only a van der Waals dimer Hg2, while Cd 2 and Zn 2 are strongly bound. For the same reason Hg 2 + is unusually stable (Nessler and Pitzer 1987).
1"
o,.9 . 5
9 . 0
8 . 5
8 . 0
7 . 5
Au
Fig. 3. The IPs of Cu, Ag and Au. Reproduced from Balasubramanian (1989a, b).
1.65
1.60
£
== ~.55
1.50
1 4 5 i i
NiH PdH PtH
Fig. 4. The Re values of the 2A 3/2 states of NiH, PdH and PtH diatomics.
2.4- 2.2.
t 2.0
1.8-
1.6"
Au2
Cu~2
Fig. 5. The Des of schematic Cu2, Ag 2 and A u 2. Reproduced from Balasubra- manian (1989a, b).
The formation of H g - H g bonds would require p r o m o t i o n of an electron from the 6s orbital to the 6p orbital. Figure 6 compares the n s - n p p r o m o t i o n energy by comparing the
(ns2)lS-(nsnp)lP
energy separations of Zn, Cd and Hg. As evidenced from fig. 6, mercury has got an usually larger p r o m o t i o n energy resulting from stabilization of the 6s orbital due to the lanthanide and relativistic contractions. Consequently, these effects are extremely significant and often dramatically alter the spectroscopic proper-40 K. B A L A S U B R A M A N I A N
, 1 ~ 6.0
W
5.5
7 . 0 -
H9
6.5 Zn~
5.0
Cd
Fig. 6. The rts 2 (1S)- nsnp(1P) energy separations for Zn, Cd and Hg. Reproduced from Balasubramanian (1989a, b).
ties and the reactivities of the third-row transition-metal compounds as well as compounds of H g - B i .
2.3. Basic relativistic quantum techniques
Since the difference between non-relativistic and relativistic mechanics is in the treatment of the speed of light, the natural origin for all relativistic methods of treating energy levels is the Einstein energy expression,
E 2 = m 2 c 4 + p2c2.
If one uses the quantum postulates to introduce operators for E and p as it is done to derive the SchrBdinger equation from energy conservation laws, one obtains an equation called the K l e i n - G o r d o n equation which involves second derivatives both in spatial coordinates and time. -This leads to the possibility that the probability density p = ~ , * could be negative which makes it difficult to interpret p.
In an attempt to overcome the above-mentioned difficulty, Dirac discovered an equation now well known as the Dirac equation which is essentially a relativistic analog of the Schr6dinger equation. The resulting equation for a single electron in a central Coulombic field is
HDO'= Et), where
where the ap are the 2 x 2 Pauli matrices and I is the 2 x 2 identity matrix.
The Dirac Hamiltonian for a many-electron atom can also be written as H D = Z h o ( i ) + Z - - , 1
i i < j rij
where h D (i) is the one-electron Dirac Hamiltonian, h o ( i ) = (oti.Pi + fli c2 - Z / r i ) .
Note that the above Hamiltonian ignores the two-electron relativistic Breit interaction.
Introduction of the Breit interaction as a perturbation shows that it is very small in the
valence region and more important for the properties of core electrons, for which this makes a significant contribution.
Since the one-particle Dirac Hamiltonian involves 4 x 4 matrices instead of scalar functions and differential operators, the solution of the Dirac equation is a vector of four components. This is referred to as a four-component spinor which takes the form
~9 1 V P.k(r) Z,,.(O, 4)
nkm =
rLiQ.k(r )
• _ k m ( O ' ~ ) J ,where
Zkm ( 0 ' ~)) = Z
C(l~j,m-a,a)Yz
1 ". m - a ( 0 , ¢ ) ¢ 1 / 2 , a a = + 1 / 2Y]' - " is a spherical harmonic,
1 / 2 = = , 4 V / U = = ,
are the Pauli spinors,
C(1½j; m - a,a)
are the Clebsch-Gordon coefficients, k is the relativistic quantum number, defined ask = j'+3,1 if j = / - ½ ,
= - ( j - ½ ) , if j = 1+½, and 2 is defined as
2 = k, if j = / - ½ ,
= - ( k + 1), if j = / + ½ .
The
Q.k
are known as the small components andP.k
are the large components. They satisfy the following coupled differential equations for a central force field,dP"k kP"*( 2 )
d--r- + - - r - \ ~ + aEv(r) -
e.k ] Q.k = O, dQ,k kQ.k
dr r
~[v(r) -e.k]P.k = O.
Desclaux's (1975) numerical D i r a c - F o c k implementation in a computer is widely used to generate relativistic numerical all-electron wavefunctions for almost any atom in the periodic table. This appears to be one starting point for
ab initio
relativistic quantum computations of molecules containing very heavy atoms.The small components
Q,k s
in relativistic four-component spinor solutions make small contributions especially for valence properties. Their contributions are far more important for core electrons. This can be best illustrated by a comparison of the large and small components of the 6s valence orbital of the lead atom. Figure 7 reproduces this comparison made by Lee et al. (1977). As seen from fig. 7, theQ6s~/2
spinor makes an42 K. BALASUBRAMANIAN 1.0
0.5
o
X
"= ( 3
-0,I
I i
- - - 6s (HF) 6s=/z (DHF)
-I,( I I
1.0 4.0 9.0
Radius la.u.)
Fig. 7. A comparison of small and large components of the 6s1/2 spinor of the Pb atom. Note that the Q6sl/2 components make negligible contribution in the valence region. Reproduced from Lee et al. (1977).
appreciable contribution only near the nucleus and is far less important in the valence region.
The above discussion leads to a natural approximation called the Pauli approxi- mation to the Dirac equation which is tantamount to ignoring the small Q components.
This is the method used in practice for most of the chemical problems.
Another starting point for relativistic effects is the perturbational Breit-Pauli Hamiltonian,
H n e = H o + HD + HMV + H s o ,
where
H o -- non-relativistic Hamiltonian,
~2
H D-- +~-(VzV) (Darwin),
0~ 2
H M v = _ __ y ' p4 (mass-velocity), 8
= y ,j × e , ) . ( s ,
r i i~j rij
where ~ is the fine-structure constant, and
(spin-orbit),
I + E 1 "
The Pauli Hamiltonian is ideally suited for carrying out relativistic corrections as a first-order perturbation to a non-relativistic Hamiltonian. However, the Pauli terms have been used with considerable success in variational self-consistent field (SCF) calcula- tions. Wadt and Hay (1985) used the above Hamiltonian within the Cowan-Griffin approximation to include relativistic effects for heavy atoms except that they typically do not include the spin-orbit effect variationally in the molecular calculations.
There are several other semi-empirical methods such as the relativistic extended Hfickel method formulated by Pyykk6 and co-workers. The Dirac-Slater multiple X~
method (Case 1982) has also been employed to include relativistic effects approximate- ly. Zerner and co-workers use the INDO/C1 method to study lathanide compounds.
We will discuss these in section 3.2.
3. Theoretical methods for lanthanide and actinide molecules