Download 1. The Dirac Equation

1
Contents
1 The Dirac Equation
1.1 Klein-Gordon Equation
1.2 Dirac’s Free Particle Equation
1.3 Hydrogenic Solutions of Dirac’s Equation
1.4 Electron Spin Angular Momentum and Magnetic Moment
2 Dirac Fock Method
2.1 A Many Electron Hamiltonian
2.2 Atomic Solutions
3 Relativistic Effective Core Potentials
3.1 Effective Core Potentials
3.2 Relatavistic Effective Core Potentials
3.3 Defining the Core
4 Relativistic Effects in Atomic Systems
4.1 Direct vrs. Indirect effects
4.2 Chemical Consequences of Relativistic Effects
5 References
2
1. The Dirac Equation
One of the first attempts at a relativistic quantum mechanical wave equation was the Klein-Gordon
equation. Though it is theoretically sound from the perspective that it is consistent with both
classical quantum mechanics and the special theory of relativity, it has several unsavory features
which keep it from being a very powerful tool in relativistic quantum mechanics. Dirac later
developed his own relativistic wave equation which did not have some of the shortcomings of the
Klein-Gordon equation and that some of the supposed ‘‘errors’’ that the KG equation gave rise to
were actually illustrating some new physics. The Dirac equation is only rigorous for a one particle
system, but has been used as a starting point for a number of approximate many electron methods.
1.1 Klein-Gordon Equation
The special-relativistic expression for the kinetic energy may be used to form the classical free
particle Hamiltonian
[
E = c m2 c2 + p2
]
12
(1.1)
The analogous quantum mechanical expression may be constructed by by replacing the classical
)
momentum, p, with its quantum mechanical operator, p which yields the free particle wave
equation
 ∂

)
 ih ψ =  c m 2c 2 + p2
 ∂t 
(
1 2
)
ψ
(1.2)
This equation, however, does not satisfy some of the conditions required by special relativity. The
wave equation is not invariant to a Lorentz transform, and the square root term introduces
ambiguity. The Klein-Gordon equation, first proposed in 1927 rectifies both of these problems
simply by taking the square of the original energy expression and extending the result to a quantum
mechanical wave
equation:
 ∂ 2
)
E ψ =  ih  ψ = m2c 4 + p2 c2 ψ
 ∂t 
2
(
)
(1.3)
The resulting wave equation is Lorentz invariant and well defined, but suffers from other problems.
Negative energy solutions to this equation are possible, which do not have a readily obvious
explanation, and the probability density, ψ∗ψ, fluctuates with time as does its integral over all
space.
The foibles of the Klein-Gordon equation may make it a poor equation for the electron, but those
weaknesses helped to point Dirac in the right direction and to develop a single particle equation
which successfully surmounted all these problems.
1.2 Dirac’s Free Particle Equation
In order for a wave equation to satisfy the special relativistic requirement of Lorentz invariance,
derivatives in space and time must all appear in the same order. The K-G equation illustrates that
an expression which satisfies this condition but is non-linear in the space and time derivatives gives
3
rise to anomalous results. Dirac set out to find an equation which was first order in space and time
derivatives. The result of his efforts, the Dirac equation, is difficult to motivate, impossible to
prove, and far more complicated that the non-relativistic analog. However, Dirac’s wave equation
for a single particle satisfies all the requirements of special relativity and quantum mechanics, and
is able to predict the properties of one particle systems with remarkable accuracy.
Dirac’s equation for an electron in field-free space is given by
(p) 0 − α⋅ p) − βmc)ψ = 0
(1.4
h ∂
)
p0 = i
c ∂t
(1.5)
where
)
and p is simply the three component momentum. In order to determine the nature of the three
components of a and b, it is useful to compare the modified equation
(p) 0 − α⋅ p) − βmc)* ( p) 0 −
)
α p⋅ − βmc) ψ = 0
(1.6)
to the Klein-Gordon equation. Equations (1.3) and (1.6) are equivalent if we enforce the conditions
[ αi ,α j ]+ = αiα j + α jα i = δij
(1.7)
where ai represents b for i=0 and ax, ay, and az for i = 1, 2 and 3, respectively. In order for this set of
four objects to fulfill these anti-commutation relations, each a must be, minimally, fourdimensional.
One set of matrices which obey similar anti-commutation relations are the Pauli spin matrices: sx,y,z.
These are 2x2 matrices, however, and there are only three of them, so they are not useful in their
usual form. If the a’s are defined as
 12x2
α0 = 
 0
0
αi = 
 σi
0 

−12x2 
σi

0 
(1.8)
(1.9)
Where 12x2 is a two by two identity matrix and si are the Pauli spin matrices. Though these four by
four
matrices do not represent the only set of matrices which satisfy the anti-commutation relations, they
may only differ by a similarity transform from this set.
Because the Dirac equation contains operators represented by four dimensional matrices, the
solutions {y} must be represented by a four component vector
4
 ψ1 
 
 ψ2 
ψ= 
 ψ3 
ψ 
 4
(1.10)
The interpretation of the components of this four-vector is not readily evident. Expressing the Dirac
equation in full matrix form provides some elucidation
 )
 (p 0 − mc )

0

)

−p z

 − p) + i p)
x
y

(
)
0
(p) 0 − mc)
(
)
)
− p x − ip y
)
pz
)
(
)
−p z
)
)
)
− p x + ip y
)
(p0 + mc)
0
(
)
)
− p x − ip y
)
pz
)  ψ1 

  ψ 2  = 0
 ψ
0
  3 
)
(p0 + mc )   ψ 4
(1.11)
The non-relativistic electronic wave-function has two components corresponding to the a and b
components of spin angular momentum. In the non-relativistic limit, p0 approaches mc, and the
terms which couple y1 with y2 drop out. What remains are four eigenvector equations, with
approximate eigenvalues of +m0c2 for y1 and y2, and -m0c2 for y3 and y4. y1 and y2, then, may be
interpreted as the a and b components of positive energy, electron-like solutions, but the solutions
which are dominated by y3 and y4 do not possess a readily evident interpretation. Dirac deduced
that these solutions correspond to a particle with the same mass as the electron but an opposite
charge, and dubbed these particles positrons. Far from being figments of a theorist’s imagination,
only three years after Dirac First suggested them in 1930, positrons were observed experimentally.
1.3 Hydrogenic Solutions of Dirac’s Equation
In its present form, the Dirac equation cannot even address the hydrogen atom, since rit has no
provision for an external potential. If we introduce a scalar and vector potential, φ+ A , to the
momenta of the field free Dirac equation
p0 →
r
p→
p0 −
φ
c
r
r A r
p− = π
c
(1.12)
(1.13)
then the Dirac equation becomes


φ
p0 − − α⋅π − βmc ψ = 0
c
(1.14)
If the potential terms are not explicitly time dependent, we may make the rearrangement
[βmc + cα
⋅ π +]ψφ= Eψ
(1.15)
The left hand side defines the Dirac Hamiltonian for a single particle in a time-independent
electro)
magnetic field. In the absence of a magnetic field the Dirac Hamiltonian, H p , may be expressed in
full matrix form as
5
(
)
 φ +mc2


0
)

Hp = 
)
cpz

)
 )
 c p x + i py
(
)
)
cp z
)
)
px + i pyl
0
(φ + mc2 ) (
)
)
)
c( px − ipy ) (φ − mc2 )
)
−cpz
0
(p) x − ip) y )



0

2 
φ − mc 
)
pz
(
(1.16)
)
This Hamiltonian is now ready to take on the task of an electron in the field of a nucleus.
Because of the spherical symmetry of the nuclear potential, the Schrödinger equation for the
hydrogenic atom may be simplified via separation of variables into a radial equation and two
angular equations. The angular solutions are given by the spherical harmonics, Ylm l ( θ, φ) , and the
radial solutions by
−
R nl ( r) = N nl e
r

na 0 
l
2r  2l +1  2r 

 Ln +l  na 
 na 0 
 0
(1.17)
where L2ln ++1l (x) are the associated Laguerre polynomials, and n, l, and ml are the principal quantum
number, orbital angular momentum quantum number, and the z-projection of l, respectively.
The hydrogenic solutions to the Dirac equations may also be achieved analytically, though the
) )
resulting eigenfunctions cannot be so simply expressed. Because of the presence of the α ⋅ p term,
)
)
) ) )
)
the s and l operators do not commute with H . The components of the j = l + s operator,
)
however, does commute with H , and so the eigenfunctions of the Hamiltonian may be
)
) )
)
eigenfunctions of j 2 and jz , l 2 and s 2 also commute with the Hamiltonian, and so we may still
associate a particular l and s value with each solution, y.
It is useful, at this point, to express the four-component wave-function as a pair of two component
spinors, f and q:
 ψ1 
 
 ψ 2   φ
ψ = = 
 ψ3   θ
 
ψ4
(1.18)
Such a separation is dictated by the nature of the 4x4 Pauli spin matrices, and makes it possible to
work with the familiar two-spinors of non-relativistic theory. Capitalizing on the commutation of
)
)
)
H with j , each spinor may be expressed as the product of a two-component eigenfunction of j
which is dependent upon the spin and spatial angular coordinates and a radial function:
j,m j
φ = g( r)χ l
j,m j
; θ = f ( r )χ l
(1.19)
) j,m
The eigenfunctions of j , χl j , may be expressed as sums of direct products of the more familiar
)
)
eigenfunctions of l and s and the appropriate Clebsh-Gordon coefficients:
j= ( l +1 2),m j
χl
+)
= χ (j,m
=
j
l + m j +1 2
2l +1
( 22) +
Yl,m j −1 2 φ11
l − mj +1 2
2l +1
(
Yl,m j +1 2 φ1−122
)
(1.20)
6
j= ( l −1 2),m j
χl
−)
= χ (j,m
=−
j
l − m j +1 2
2l +1
Yl,m j −1
( )+
1 2
2 φ1 2
l + mj + 1 2
2l +1
(
Yl,m j +1 2 φ1−122
)
(1.21)
where Yl,m j ±1 2 are the spherical harmonics and
 1
φ11 22 = α =  
 0
 0
; φ1−12 2 = β =  
 1
(1.22)
Next, it is useful to define the operator
)
)
K = σ ⋅ l +1
(
)
(1.23)
)
)
which commutes with J . The eigenvalue equation associated with K is given by
) ±)
±)
K χ(j,m
=± kχ(j,m
j
j
(1.24)
if j = l −1 2
 j +1 2
k = 
−( j −1 2 ) if j = l + 1 2
(1.25)
where
Because of the spherical symmetry of the nuclear potential, the eigensolutions may be further
separated on the basis of their response to spatial coordinate inversion. The parity operator
commutes with the full Hamiltonian, and so the final eigenfunctions, ψ j,m j , must also obey the
eigenvalue equation
βψ (− r) = ±ψ (r )
(1.26)
This necessarily dictates one of the following forms for the four-component eigensolutions:
or

(+ ) 
+)
 g(r )χ 
ψ (j,m
=
j
 f( r )χ(− )
(1.27)
 −g( r)χ( −) 
−)


ψ (j,m
=
j
 f ( r)χ ( +) 
(1.28)
It is possible, through the use of the identity
)
)
)  ) ) σ ⋅ l 
σ ⋅p ≡ (σ⋅ r ) r ⋅ p +
r 

and the property
(1.29)
7
±)
m)
(σ ⋅)r )ψ (j,m
= −ψ (j,m
j
j
(1.30)
to completely separate out the angular functions from the eigenvalue equation, and thereby achieve
two coupled, complex differential equations for the radial functions f(r) and g(r). The resultant
equations are greatly simplified by the substitutions
f(r) = F(r)/r and
to yield
g(r) = iG(r)/r
(mc2 − E − Z r )F(r ) − c dGfr(r ) − kcG( r) r = 0
(−mc 2 − E − Z r )F(r ) + c dGfr(r ) − kcG(r ) r = 0
(1.31)
(1.32)
(1.33)
The solutions to these differential equations may be determined by first solving the asymptotic
equations in the limit as r → inf . These solutions, which are of the form


mc


F( r) = 1+ E mc exp −
2 
 h 1− E mc r 


mc
2


G(r ) = 1 − E mc exp  −
2 
 h 1 − E mc r 
2
(1.34)
(1.35)
are then multiplied by an undetermined power series for both components and the bound state
solutions are sought. The final form of each wave-function is uniquely determined by the recursive
relations of the power series and the boundary conditions, but may not be expressed in a simple
general form. The resulting electronic energies are given by
En = m 1 +

n − j − 1 2 +

(Zα )2
( j +1 2 )
2
2
+ (Zα) 

2
(1.36)
where a is the fine structure constant, and is given by in atomic units.
The four-component nature of the Dirac eigenfucntions gives rise to many interesting differences
when compared to the non relativistic solutions. The contributions from the first two components of
the wave-function tend to be much larger than the contributions from the final two components for
electron-like solutions. For this reason the upper two-component spinor is known as the large
component of the wave-function, while the lower spinor is known as the small component. By
orthogonality the opposite is true for the positron-like solutions. Since the angular and spin portions
of the wave-functions are normalized, the magnitude of the large and small contributions is
determined entirely by the radial functions, f(r) and g(r). Given the asymptotic form of the radial
equations and the orbital energy , the ratio of their contributions may be expressed as
g( r) Zα
≅
f (r ) 2n
(1.37)
In general, the ratio will probably be larger than this approximate value in regions close to the
8
nucleus, where the contribution of g(r) is the greatest. The contribution of the small component,
then, may be significant for sufficiently heavy nuclei. Though hydrogenic ions with very heavy
nuclei may not be of great practical interest, this relationship will have implications for heavy
many-electron atoms where the innermost electrons experience a large portion of the full nuclear
charge and hence are closely related to the analogous hydrogenic systems.
One interesting consequence of the four component wave-function may be observed in the
associated probability density. Scalar wave-functions, y, which possess radial and angular nodes
will have the same radial and angular nodes in their probability density, y*y. The four-component
wave-function, however, gives rise to a probability density which sis the sum of the probability
densities associated with the large and small wave-functions, f and c, respectively. The radial
component of f and c, possess different numbers of nodes and, in general, none of these nodes will
coincide. Similarly, the angular contributions of f differ from those of c by a single unit of angular
momentum. Therefore, although f*f and c*c may possess either radial of angular nodes individually,
their sum, their sum will be node-less.
The energy eigenvalues of the hydrogenic solutions to the Schrödinger equation are only dependent
upon n, the principal quantum number, while the Dirac hydrogenic eigenvalues are dependent on
both n and j. The spectral dependence on j is upheld by experimental observation, however the
degeneracy of eigenvalues for solutions with the same j values but differing l values is not
observed in nature. The breaking of this degeneracy is known as the Lamb shift and its origin has
been attributed to the difference in the interaction of the different j eigenfunctions with the vacuum
fluctuations predicted by quantum electrodynamics. The magnitude of the Lamb shift is much
smaller than the splitting introduced by spin orbit coupling and is largest for the n = 2, j = 1/2
shells of the hydrogenic atom. In this case, the splitting introduced by the Lamb shift is
approximately 10% of the magnitude of the energy separation of the p1/2 and p3/2 eigenfunctions. It
should not be too surprising that the predictions of the Dirac equation cannot entirely reproduce
experimental observation, since it has its roots in special relativity. In order for any physical theory
to definitively represent reality, it must at least be consistent with the general theory of relativity.
1.4 Electron Spin Angular Momentum and Magnetic Moment
One of the triumphs of the Dirac equation was its explicit connection to electron spin. The Dirac
equations does not predict electron spin as a relativistic property, observable only under relativistic
conditions. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic
limit yields the Schrödinger equation with a correction term which takes account of the interaction
of the electron’s intrinsic
magnetic moment with the magnetic field Using the definition of the
)
Dirac Hamiltonian, H D , given in ( 1.15), we may arrive at the equation
)
(H D + φ)2
c 2 = (βmc + α ⋅ )π
2
(1.38)
Due to the anti-commutation properties of the components of a, we may expand the right hand side
to reveal that
)
(H D + φ)2 c −2 = m2 c2 ( α ⋅ π)(α ⋅ π)
If we define our ai matrices as products of a matrix r and the si4 matrices where
(1.39)
9
 0
ρ = 
12x2
σ
12x2 
 and σi4 =  i
0 
0
0

σ i
(1.40)
so that ai = rsi4, then we may express (α ⋅ π)(α ⋅ π) as
(α ⋅ π)(α ⋅ π) = (σ ⋅ )π( σ ⋅ )π
(1.41)
where we have used the identity r2 = 1. This may be rearranged to the form
(σ ⋅ )π(σ ⋅ )π= π ⋅ π+ iσ(π × π)
(1.42)
via sundry vector identities. The cross-product of p with itself does not drop out. This is because p
is the sum of two vectors, p and A, so the cross terms of the cross products remaining, making the
cross-product of with p itself
(π × π) = −ih∇ ×A = −ihB
(1.43)
Returning to our original equation (1.39), we may make this substitution to arrive at
)
(H D + φ)2 c −2 = m2 c2 + π2 + hσ4 B
(1.44)
In the non-relativistic limit this gives us an Hamiltonian of the form
)
π2
h
H D = −φ +
+
σB
2m 2m
(1.45)
The final term indicates that a term which corresponds to a body with a magnetic moment of
h
−
σ in a magnetic field B must be added to give the proper energy. This prediction of electron
2m
spin as a property which should exist in the non-relativistic limit is a clear indication that the Dirac
equation could get the right answers for the right reasons.
10
2. Dirac Fock Method
The Dirac-Coulomb Hamiltonian is one of the most widely used special-relativistic, many-electron
Hamiltonians. This Hamiltonian may be utilized in conjunction with a Hartree-Fock-like wavefunction in what is known as the Dirac-Hartree-Fock (DHF) method. The DHF method has a
special status in quantum chemistry as it often us utilized to benchmark relativistic effects in the
absence of electron correlation. Such benchmarks can provide a gauge of the accuracy of more
approximate methods that attempt to include relativistic effects as a perturbation of the nonrelativistic Hartree-Fock case as well as methods which employ transformed, simplified versions of
the Dirac-Coulomb Hamiltonian.
2.1 A Many Electron Hamiltonian
The Dirac equation has been shown to successfully treat the interactions of electrons with nuclei in
a special-relativistic manner, and so represents a good starting point for a special-relativistic
manyelectron Hamiltonian. What is needed next is a description of the electron-electron
)
interaction, and an associated quantum mechanical operator, g ij . If the electron-electron interaction
is not altered significantly by the introduction of special relativity, then the standard, non1
)
relativistic coulomb operator, g ij = , might not be a bad guess. With the coulomb interaction, the
rij
Hamiltonian for an n electron system would be given by
n )
n 1
)
H = ∑ hD (i ) + ∑
i=1
i> jrij
[
]
n 1
)
)
= ∑ φ ( ri ) + cα( i) ⋅ p(i ) + mc2 + ∑
i=1
i > j rij
n
(2.46)
)
This is known as the Dirac-Coulomb Hamiltonian, H DC . The associated wave equation,
)
)
H DCψ = Eψ , is not Lorentz invariant, and so H DC does not represent a proper special relativistic
Hamiltonian. In order to obtain a two-electron interaction term which is consistent with special
relativity, it is necessary to turn to Quantum Electrodynamics. In order to cast the QED electronelectron interaction term into a reasonable form, it is necessary to expand it in perturbative series in
orders of the fine structure constant, a. Retaining only the terms which contribute up to order a2,
gives the coulomb operator plus the Breit interaction term
(
)(

α i ⋅ rij α j ⋅ rij
1
)B
g (i , )j = −  αi ⋅α j +
rij 
rij2
)


(2.47)
In practice, however, the full Breit magnetic interaction term is often cumbersome to implement,
and so an approximation of the Breit operator known as the Gaunt operator, may be used
α i ⋅ αj
)
g G ( i, j) = −
rij
(2.48)
The Gaunt operator is not gauge invariant, but it avoids having to solve integrals over operators
11
more complicated than
1
, and includes the largest contributions of the Breit interaction.
rij
More rigorous forms of the molecular Hamiltonian have been suggested, but they are, at best, only
approximately Lorentz covariant to some order in a. Because the two-electron magnetic interaction
terms are typically small, the contributions beyond second order in a are typically chemically
unimportant. Because of this the more rigorous Hamiltonians, which typically involve more
complicated operators than even the Breit interaction, have received far less attention from
quantum chemical investigators. The Dirac-Coulomb, Dirac-Coulomb-Gaunt, and Dirac-CoulombBreit Hamiltonians have become the most widely accepted four-component special relativistic
molecular Hamiltonians.
2.2 Atomic Solutions
Atomic eigenfunctions of the D-C Hamiltonian may not be achieved analytically as were the
hydrogenic solutions of the single particle Dirac equation. Instead, it is necessary, as it was in the
non-relativistic theory, to appeal to approximate models such as the Dirac-Hartree-Fock (DHF)
method. In a manner analogous to Hartree-Fock theory, DHF begins with the assumption that an nelectron atomic wavefunciton, y, can be represented by an antisymmetrized product of n single
particle functions, { fi }. In contrast to the HF method and in accordance with the 4-component
)
nature of the single particle operators, h D (i ), the single particle functions fi are 4-component
spinors. If we make the approximation that each electron experiences a central potential only, then
the single particle functions may be separated into radial, angular, and spin parts analagous to the
hydrogenic eigenfunctions. Anticipating the existence of electronic and positronic solutions, we
may propose two separate forms for fi, analogous to the hydrogenic solutions
j,m
φ i j ( +)
1  pi (r )χ(i +) 
= 
r  iq i (r )χi(− )
(2.49)
j,m
φ i j ( −)
1  −p i (r )χi(− )
= 
r  iq i (r )χi(+ ) 
(2.50)
and
where χi( ±) are two component spinors dependant upon the spin and angular coordinates and the
)
j,m
quantum numbers l, j and mj, and are given by j the eigenfunctions ψ i j presented for the
hydrogenic case in Section 2.1.3. The form of the radial functions has, again, been chosen such that
the resultant wave equations and electronic energy expression are simplified. The total
wavefunction formed by these one particle functions may be expressed as
n
ˆ  ∏ φ ( r )
ψ =A
 i =1 i i 
φ1( r1 ) φ 2 ( r1 ) ... φn ( r1)
1 φ1 ( r2 ) φ2 (r2 ) ... φ n (r2 )
=
n! :
:
:
φ1( rn ) φ 2 ( rn ) ... φ n ( rn )
(2.51)
12
where Aˆ is the antisymetrizer operator. If this form of the atomic wavefunction is assumed, and the
Dirac-Coulomb Hamiltonian is utilized, then the electronic energy is given by
)
E = ψ H DC ψ
1 n 
1
1 
ˆ
= ∑ φ i φ i h d + ∑  φiφ j φiφ j
− φi φ j φ jφi
2 i.j 
r12
r12 
i= 1
n
n
(2.52)
n
1
= ∑ i i hˆ d + ∑ ij ij
2 i.j
i= 1
Once again, use of equations 1.29 and 1.30 allows for complete separation of the angular and radial
contributions to the single electron terms. Since the angular portions of the wavefunction are
normalized, the one-electron terms reduce to
∞ 
 d k
 d k 
i i hˆ d = ∫ dr mc 2p *i p i − mc 2q *i q i + φ( r ) p*i pi + q *i q i q i* +  pi + pi*  +  
 dr r 
 dr r  
0 
(
)
)
where k is the eigenvalue associated with operator K , given by 1.24.
(2.53)
13
3 Relativistic Effective Core Potentials
One of the most fundamental assumptions in chemistry is that low lying core electrons are
relatively inert, and are not perturbed by a molecular environment. This assumption is supported by
the chemical similarity of elements in the same column of the periodic table. Most of the important
chemical properties of atoms and molecules are determined by the interaction of their valence
electrons with the valence electrons of other atomic or molecular species. For molecules containing
heavy atoms, large computational savings may be realized if a particular mathematical form for the
atomic fore orbitals is assumed, a priori, so that the total number of parameters which must be
optimized in, say, the computation of a Hartree-Fock wavefunction may be reduced. This
approximation on its own is known as the frozen core approximation and will be explored in some
detail in chapter 3.
A more extensive savings may be achieved if, instead of simply assuming a particular form for the
core orbitals, all terms describing the interaction of the electrons in these orbitals with each other
and those outside the core region are simply replaced by a scalar "effective potential". This
provides the additional benefit of reducing the basis set size requirements, since no basis functions
are now needed to explicitly describe the core orbitals. This has the effect of drastically reducing
the number of two-electron quantities, such as electron repulsion integrals, which must be
considered.
In correlated electronic structure calculations, the molecular orbitals which correspond to atomic
core orbitals are often excluded from the active orbital space. Even when such orbitals are included
in the active space, the resultant contributions are typically small for most chemical properties. The
use of effective core orbitals in these methods ideally should not significantly affect the quality of
the resulting molecular property predictions. Because correlated methods scale higher than HartreeFock with respect to basis set size, the decreased basis set size resulting from the use of effective
potentials will introduce in an even more dramatic computational savings over all-electron
methods.
3.1 Effective Core Potentials
The objective of the Effective fore potential method is to construct and are solely dependant upon
the coordinates of the valence electrons, but take into account the influence of the inert, core
electrons. The all electron Hartree-Fock model for an n-electron system begins with an
antisymmetrized Hartree product wavefunctions of the form
ˆ (φ φ ...φ )
ψ HF = A
1 2
n
(3.54)
where Aˆ is the antisymetrizer operator and { φi } are the single particel eigenfunctions of the Fock
operator
)
)
∇2 Z l(l + 1) n )
F =− r − +
2 + ∑ Jj − K j
2
r
2r
j =1
(
)
)
where the action of the operators J j and K j is defined by
)
(3.55)
14
 r
) r
r 1
r  r
J jφ i( r1) = ∫ d r2φ *j ( r2 ) φ j( r2 ) φ i( r1)
r12


 r
)
r
r 1
r  r
K jφ i( r1) = ∫ d r2 φ *j ( r2)
φi ( r2 ) φ j( r1)
r12


(3.56)
(3.57)
If we now divide the orbitals into a group of Nc core orbitals and Nv valence orbitals, we may reexpress the fock operator as
)
∇2r Z l(l + 1) ) core ) val
F =−
− +
+V
+V
2
r
2r 2
(3.58)
where
Nc )
)
)
V core = ∑ Jα − K α
α=1
(
)
Nv )
)
)
and V val = ∑ Ji − Ki
(
i =1
)
(3.59)
In order to reduce the computational intensity of solving the Fock equations for the molecular case,
it would be useful to develop an equation analogous to the Fock equation which does not contain
non-local terms which depend upon the core orbitals. Such an equation would take the form
 1 2 Z eff l(l +1) ) val ) eff 
− ∇r −
+
+ V + V φ i = ε iφi
 2

r
2r 2
(3.60)
)
Where the full coulomb and exchange terms of V core have been replaced by an effective core
)
potential (ECP), V eff , a local potential. The form of this effective potential is typically derived
from numerical Hartree-Fock atomic solutions, in accordance with the frozen core approximation.
)
Several methods have been suggested for obtaining V eff . If we consider 3.60 for a valence atomic
)
orbital of angular momentum l, φ li , it is possible to obtain an analytic form for Vleff , the local
potential which will exactly reproduce
)
Vleff
1 2 ) f 
 ∇ − Vval φ
Z eff l( l +1)  2 r

= εl +
−
2 +
l
r
2r
φi
l
i
(3.61)
)
where the l subscript acknowledges the unique form of V eff for each value of valence obital
)
angular momentum l. Unfortunately, this expression for Vleff is only valid for φi, since the last term
in 3.61 is singular when φi = 0. To circumvent this problem, the full atomic valence orbitals, { φi,
}, may be replaced by approximate pseudo-orbitals, { χi, }, which are nodeless. Ideally we would
like to obtain a which will produce atomic valence pseudo-orbitals which are as close to the
original orbitals, { φi }, as possible. The use of approximate valence orbitals brings a new definition
)
of V eff , since the valence orbitals now satisfy the equation
 1 2 Z eff l(l +1) ) f
) eff  l
l l
− ∇r −
+
+
V
+
V
val
l χ i = ε i χi
2
 2

r
2r
(3.62)
15
)f
)
Because Vval
will differ from Vval , the ECP must represent not only the core-core and corevalence interactions, but also the parts of the valence-valence interaction which were lost in the
)f
)
switch from φi to χi. If the difference Vval
between and Vval is large then the ECP, which will
contain valence-valence interaction terms for the atomic case, will bias the potential in the
molecular valence region towards the atomic case.
One popular method of forming the nodeless χi takes a linear combination of the appropriate full
electron valence orbital, φv, with all of the full electron core orbitals, χc, to give
χ v = a v φv + ∑ a cφ c
(3.63)
c
where the parameters av and ac are selected by minimizing the kinetic energy of χv. Because of the
impostition of the normalization condition upon χv, av will always be less than one, which means
that χv will be less than φv outside of the core region, where the φc’s will have little to no
)
)f
contribution. This could potentially introduce a substantial difference in Vval and Vval
and thereby
introduce large valence-valence terms to the ECP.
One χi selection scheme which gets around the problem of valence-valence contributions to the
ECP is the technique of selecting shape consistent pseudo-orbitals. This method requires that χi
exactly match the original valence atomic orbital, φi, for r ≥ Rmax, where R max is the radius at which
φi experiences its outermost maximum. Inside this radius, a function of the form
4
χi = ∑ c mr m + N ,
m =0
r < Rmax
(3.64)
where N = l+2, in most cases. The coefficients, cm, are restricted by the requirement that the zeroth
through third derivatives of χi and φi are equal, and χi that is normalized. The resultant χi are
generally known as shape-consistent pseudo-orbitals.
)
By solving (3.61) for Vleff we obtain
)
Z
Vleff = εil + eff
r
1 2 ) f  l
 ∇ − Vval  χi
l( l +1)  2 r

−
+
2
l
2r
χi
(3.65)
)
The total Vleff for an atom, then, is written
l
)
)
Veff = ∑ Vleff ∑ lm lm
l
m =− l
(3.66)
so that the orthonormality of the spherical harmonics lm pairs each atomic wavefunction with the
)
proper Vleff term. The radial wavefunction of the valence orbitals is required to be orthoganal to
the core orbitals of the same angular momentum as that valence orbital. If an atomic core contains
orbitals with angular momenta up to a certain value lmax then valence orbitals of a greater orbital
angular momentum than lmax do not have any such orthogonality constraints on their radial
)
wavefunctions. Vleff for χl ( l > lmax ), then, will be largely independent of l, since the only l
dependant terms which should be represented are the core-valence exchange terms, which should
not be very large. It is convenient, then, to approximate (3.66) as
16
lmax −1 )
l
)
v
)
)
Veff = Vlmax + ∑ Vleff − Vlmax ∑ Vleff ∑ lm lm
l= 0
l
(3.67)
m=− l
Once the nodeless pseudo-orbitals have been selected, it is possible to simply solve 3.65
numerically. Once this has been done, a parameterized analytic form may be fit to the numerical
data. One common
form is given by
e −α ir
Vl (r ) = ∑ A i N i
ri
i
2
(3.68)
Once the ECP has been modeled by some set of functions, then (3.60) is complete and may be
implemented for a chosen set of valence orbitals.
3.2 Relatavistic Effective Core Potentials
Relativistic corrections to the Hartree-Fock model tend to be largest in the region immediately
surrounding the nucleus. This is because of the fact that the kinetic energy of the electrons is great
in this region. Therefore, it is reasonable to assume that the electrons in the innermost core orbitals
will experience the effects of relativity much more directly than will the higher lying valence
orbitals, which tend to have much less probability of existing close to the nucleus. The interaction
of the valence electrons, then, withthe nuclei and other electrons should be relatively well described
by the non-relativistic Hamiltonian. Although the valence orbitals do not experience large direct
relativistic effects, the changes in the core orbitals will have direct consequences on the nuclear
shielding and orthogonality constraints which the valence orbitals experience. These effects are
generally known as secondary effects. Since the behaviour of valence electrons is the primary
concern for quantum chemists, and the deeper core electrons of very heavy atoms should not
change significantly in a molecular environment, an ECP method which incorporates the direct
relativistic effects experienced by the core orbitals is of great practical interest.
Relativistic effective core potentials (RECP) may be obtained in a manner directly analagous to the
non-relativistic ECP’s. The starting point for RECP’s is the atomic Dirac-Coulomb-Fock equations.
The valence solutions, { φi }, are four-component spinors, but the large radial component generally
accounts for over 99% of the electronic density. Therefore, the re-normalized large components are
used as the starting points for the relativistic pseudo-orbitals, χi. The methods used to derive the
relativistic pseudo-orbitals are exactly the same as the methods used in the non-relativistic case.
Once { χi } have been determined, the RECP’s are obtained via a relation similar to 3.65. The DHF
) ) )
eigenstates, however, are now eigenstates of the total angular momentum operator, j = l + s , and
so the resultant RECP’s are no longer uniquely defined for a particular l value. Instead, the RECP’s
are dependant on l and j and are based on the relation
 1 2 Z l (l +1) ) f
) 
− ∇r − +
+ Vval + Vl,jeff  χ li , j = εil,j χ li , j
2
 2

r
2r
(3.69)
Since the RECP’s are typically used within the framework of non-relativistic quantum chemistry,
)
)
)
where single particle states are eigenfunctions of l and s rather than j , it is necessary to construct
)
a V eff which is only l dependant. This may be accomplished by statistically averaging over all of
the appropriate j-dependant RECP’s which are associated with a particular l value to give what are
17
known as l averaged RECP’s.

) AREP r
) REP r 
r
−1  ) REP
Vl
( r ) = (2l +1)  lV
1 (r ) + (l +1) V
1 (r)
l,j = l+
 l,j = l− 2

2
(3.70)
The total VAREP may be given by
)
)
r l
V AREP = ∑ VlAREP ( r ) ∑ lm lm
(3.71)
m =− l
l
completely analogous to the equation for Veff.
Alternatively independent RECP’s may be derived by starting with l-averaged pseudo orbitals.
Such orbitals may be obtained by statistically averaging the large component of the valence spin
orbitals, thereby producing l averaged atomic orbitals which are then utilized to construct nodeless
pseudo-orbitals



φ l = (2l + 1) lφ
1 + (l + 1) φ
1
l,j = l + 
 l,j = l− 2
2
−1 
(3.72)
The l averageing may also take place in an analogous fashion after the j dependant pseudo-orbitals,
φ
1 , have been formed.
l,j = l ±
2
When RECP’s are utilized to form valence LS eigenstates, the nuclear shielding and core shrinking
effects are taken into account, but the third major affect of relativity on the valence orbitals, spinorbit splitting, is not. The portion of the original RECP’s which was lost in the process of statistical
averaging over l-states, to give the spin-orbit interaction term
)
)
)
H SO = VREP − V AREP
(3.73)
)
This spin-orbit interaction operator, H SO , may be expressed as
)
) SO
) REP 

REP
H = ∑ V
1 −V
1
l,j =l − 
l  l , =j l +
2
2
1
1

(
−l +
−l −
 1

2
2
1
1
1
1
1

l , =j l + , mj l , j= l + , mj −
l , j= l − ,m j l , =j l − ,m j 
∑
∑
2
2
2l +1 m =− l + 1
2
2

 2l + 1 m j =− l − 1
j


2
2
3.74)
Since this correction is based solely upon the RECP’s, only one-electron integrals are required for
its evaluation. This correction term was first presented in the context of CI calculations, but may be
employed to estimate the extent of spin-orbit coupling of two HF wavefunctions.
3.3 Defining the Core
18
The definition of the core-valence division in the construction of ECP’s is usually done on the basis
of chemical intuition. For the first and second row atoms, this typically means that the ns and np
orbitals make up the valence space. For transiton metals, the valence space will include the (n-1)d
orbitals.
For group IIIA and higher, however, the (n-1)d -orbitals are usually assigned to the core space,
since the shell is full and they are considered chemically inert. These standard definintions of the
core-valence division are appealing from the point of view that they offer a simple, chemically
appealing model of the valence space, and they lead to a drastic reduction in the total number of
variables required to describe a full molecular wavefunction, especially for molecules containing
very large, main group atoms.
There are many instances, however, where the standard definitions of the core-valence division are
inadequate. One problem arises in the case of transition metals, where the (n-1)d-orbitals are
considered valence but the (n-1)s and (n-1)p-orbitals are considered part of the core. Because the d,
s, and p orbitals in the same principal quantum shell typcally have a significant radial overlap, the
exchange terms between them can be significant, and play an important role in the description of
the valence d orbitals. The local interaction terms of the effective potential terms, therefore, may be
inadequate. To rectify this, it is necessary to include the entire shell in the valence space. This has
the added benefit of producing ns and np orbitals with at least one radial node. This altered
definition of the core space has proven to give more accurate results for a variety of molecular
systems.
A related problem arises for the group IIIA and higher main-group elements. The radial extent of
the full (n-1)d shell tends to be more diffuse than that of the (n-1)s and (n-1)p shells. As a result,
the d orbitals share a significant spatial overlap with the valence orbitals, and the local
approximation of the full coulomb and exchange integrals again become a poor estimate. This
relationship is accentuated in the heavier elements where the valence s and p orbitals experience
relativistic contraction, while the d orbitals experience relativistic expansion (see Chap. 3 for a
discussion of the effects of relativity on atomic electronic structure). In these cases, the valence
space may be augmented by only the (n-1)d functions, or the total (n-1) shell, depending on the
importance of s-d and s-p exchange for a proper description of the molecular system in question.
A priori definition of the core space, then, is seen to be a difficult task. Some clues may be taken
from the atomic, all-electron solutions. The spatial extent of supposed core electrons may be
compared to the orbitals on the atomic state may both be used as core selection criteria. Ultimately,
however, the results of several levels of approximation must be compared to investigate the
stability of the predictions to an increase in the core space for each system that is investigated.
19
4. Relativistic Effects in Atomic Systems
The importance of ‘‘relativistic effects’’ in properly describing the electronic structure of
molecules containing heavy elements is often stressed in the literature. A logical definition of the
term might be ‘‘The differences in molecular property predictions made by the relativistic
Hamiltonian and the non-relativistic Hamiltonian’’. There are some problems with this definition.
The first of these problems is that it requires the existence of an exact relativistic Hamiltonian. This
may be resolved by choosing a single, well accepted approximate hamiltoninan such as the DiracCoulomb or Dirac-Breit Hamiltonian as the relativistic Hamiltonian. Another problem stems from
the extensive discussion of ‘‘correlation effects’’ in the quantum chemical literature. This term is
used to describe the difference between properties predicted by the Hartree-Fock wavefunctions
and the properties predicted by higher level, correlated wavefunctions. In order to study relativistic
effects independently of correlation effects, it is useful to define relativistic effects as the
differences between the predictions of the HF and DHF methods. In this way, both relativistic
effects and correlation effects may be viewed as corrections to the non-relativistic HF model.
Unfortunately, there are other approximations inherent in most molecular electronic structure
methods which stand in the way of isolating those differences which are purely due to the
introduction of relativity. The properties predicted by a particular molecular wavefunction are often
highly dependant on the quality of the finite basis set in which the wavefunction is expanded. This
especially true in the case of the DHF method, where the reliability of the predictions are not only
dependant upon the size and flexibility of the basis set as is the case for HF, but also upon the
proper balance of large and small component basis sets. The extraordinarily different basis set
requirements of the two methods make it difficult to determine what would constitute an equal
treatment of the two.
For atomic systems, however, the issue of basis set effects may be avoided by solving the one
dimensional DHF and HF radial equations numerically. This makes atomic systems a good starting
point for the investigation of isolated relativistic effects. Furthermore, characterization of the
differences in such properties as orbital size and extent, and orbital energies for isolated atomic
systems can help to identify periodic trends in relativistic effects without the perturbation of a
molecular environment. Models established in the high symmetry of the central force field of the
atom may then lend structure to descriptions of the valence orbital effects in molecules, where
heavy orbital mixing may be encouraged by the lower symmetry molecular environment.
In practice, DHF based four-component methods are often impractical to employ for systems of
real chemical interest. In these cases, more approximate two-component methods such as
relativistic perturbation theory are utilized in an attempt to include the most important relativistic
effects. In these cases, it is essential to have an estimate of the magnitude of the difference between
the relativistic and non-relativistic pictures in order to evaluate the validity of perturbative schemes.
4.1 Direct vrs. Indirect effects
A crude estimate of the importance of relativity in characterizing the atomic orbital energies and
spatial
extent may be acheived by invoking the Bohr model of the hydrogenic atom of charge Z. In this
classical picture, an electron in an orbit designated by principal quantum number n would travel at
a radius rn a velocity vn with a total energy En given (in A.U.) by
n2
rn =
mZ
vn = Z n
(4.75)
(4.76)
20
En = −
mZ 2
2n 2
(4.77)
where m has been retained intentionally. Since the electron is participating in accelerated motion,
the equations of special relativity cannot apply rigorously. However, it is possible to achieve an
estimate of the changes which a relativistic Hamiltonian would introduce for this system by
replacing m with the veclocity dependant mass
m=
m0
2
v
1− 2
c
=
m0
(4.78)
Z2
1− 2 2
n c
This relationship is illustrated in Figure 1.
1
0.9
n
r el/r rel
0.8
r
0.7
0.6
0.5
0.4
0
20
40
60
Z/n
80
100
120
Figure 1: Ratio of Relativistic Bohr Orbit to Non-Relativistic Bohr Orbit
These sorts of effects are termed direct effects, and usually affect the s and p type orbitals much
more dramatically because these orbitals tend to have a greater density near the nucleus, and hence
experience a greater fraction of the full nuclear charge.
The orbital shrinking experienced by s and p orbitals will, in turn, have an affect on the higher
angular momentum d and f orbitals. The greater density of the s and p orbitals in the nuclear
vicinity effectively crowds out the d and f orbitals through orbital orthoginality and sheilding
effects. The resultant expansion of the d and f orbitals and the associated shift of the orbital
energies is generally classified as an indirect relativistic effect. In atomic systems where this orbital
shrinking and expansion is highly pronounced, non-relativistic methods will obviously be
21
inadequate.
Another litmus of the necessity of a relativistic treatment is the contribution of the small component
to the total electron density predicted by the DHF method for a particular atomic orbital. The
pattern of electron density and orthogonality relationships associated with a four-component
wavefunction with appreciable large and small components are difficult to reproduce with a twocomponent wavefunction. The small component of the wavefunction tends to be largest very close
to the nucleus and therefore plays a substantial role in the screening of nuclear charge for the higher
energy orbitals. The contribution of the small component is usually greatest for orbitals with low
principal quantum numbers. This is illustrated by these plots of the small component of the
wavefunction for the s-like orbitals (j = 1/2, k = -1, l = 0) of Rn with principal quantum numbers n
= 1,2,3.
Figs.
Similarly, the small component to the 1s orbital of increasingly lighter elements tends to be smaller,
and the profile of the small components tends to more closely match that of the large component.
Therefore, the renormalized large component should give a closer estimate of the radial distribution
of the electron density for the lighter elements. This may be observed in the following plot of the
large and small components of the 1s DHF orbitals of Rn, Xe, and Kr.
22
Figs.
23
4.2 Chemical Consequences of Relativistic Effects
Some of the general trends which have been observed by quantum chemical investigators:
• Au vrs. Ag
- shift of the spectrum to include portions of the visible region
- AuH shorter than AgH
- "Halogenlike" behavior
• Tl vrs. In
- potential for 6s2 occupancy
- S-O splitting places p2/3 orbitals out of hybridization range
• General Trends
- shorter σ-bonds
- greater participation of d orbitals
- potential change in valancy
- L-S eigenstates may be poor approximation
24
25
26
27
28
5. References
R. E. Moss, Advanced Molecular Quantum Mechanics, (John Wiley and Sons, New York, 1973).
Ed. Stephen Wilson, Methods in Computational Chemistry, Vol. 2, (Plenum Press, New York,
1988).
P. Pyykkö, Relativistic Theory of Atoms and Molecules, (Springer-Verlag, Berlin, 1986).
Ed. S. Wilson, I. P. Grant, and B. L. Gyorffy, The Effects of Relativity in Atoms, Molecules, and
the Solid State
Balusubramanian, K. and Pitzer, K. in Ab Initio Methods in Quantum Chemistry--Part I, edited by
K. P. Lawley (Wiley and Sons, Ltd., 1987), pp. 287--319.
Ann. Rev. Phys. Chem., (84) 35, 357--385.