Download Relativistic Effects in Atomic Spectra

Relativistic Effects in
Atomic Spectra
Diploma Thesis
by
Robert Lang
Supervisor: Prof. Dr. Gero Friesecke
Submission Date: May 31, 2011
Technische Universität München
Fakultät für Mathematik
Contents
1 Introduction
5
2 Spectrum of the Hydrogen Atom
2.1 Non-relativistic Schrödinger Model . . . . . . . . . . . . . .
2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Weyl’s Criterion . . . . . . . . . . . . . . . . . . . . .
2.1.3 The Kato-Rellich Theorem . . . . . . . . . . . . . . .
2.1.4 Self-adjointness of the Hydrogen Hamiltonian . . . .
2.1.5 Non-relativistic Spectrum of the Hydrogen Atom . .
2.2 Relativistic Dirac Model . . . . . . . . . . . . . . . . . . . .
2.2.1 Self-adjointness of the Dirac Operator . . . . . . . . .
2.2.2 Dirac Operator with Coulomb Interaction . . . . . .
2.2.3 Non-relativistic Limit and its Relativistic Corrections
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3 Spectrum of Many-electron Atoms
3.1 Non-relativistic Perturbation-theory Model . . . . . . . . . . . .
3.1.1 Definition of the PT Model . . . . . . . . . . . . . . . .
3.1.2 Principal Results . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Energy Levels and Spectral Gaps . . . . . . . . . . . . .
3.2 Relativistic Effects in Asymptotic PT States . . . . . . . . . . .
3.2.1 P-contribution and Darwin Term . . . . . . . . . . . . .
3.2.2 Spin-orbit Coupling . . . . . . . . . . . . . . . . . . . . .
3.2.3 Relativistic Energy Levels, Spectral Gaps and Splitting .
3.3 Relativistic Corrections to Lithium . . . . . . . . . . . . . . . .
3.3.1 Shifted Energy Levels . . . . . . . . . . . . . . . . . . . .
3.3.2 Effective Nuclear Charge . . . . . . . . . . . . . . . . . .
3.4 Relativistic Corrections to Beryllium, Boron and Carbon . . . .
3.5 Relativistic Corrections to Nitrogen, Oxygen, Fluorine and Neon
3.6 Non-local Relativistic Corrections . . . . . . . . . . . . . . . . .
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4 Summary and Conclusion
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A Appendix
A.1 Sobolev Spaces . . . . . . . . . . . . . . . . . .
A.2 Spherical Harmonics and Laguerre Polynomials
A.2.1 Spherical Harmonics . . . . . . . . . . .
A.2.2 Laguerre Polynomials . . . . . . . . . . .
A.3 Notations in Relativistic Quantum Mechanics .
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3
1 Introduction
Die Physik ist für die Physiker eigentlich viel zu schwer.1
David Hilbert
The history of physics has shown an interesting progress of the comprehension of manyparticle systems: in classical mechanics, the three-body problem has been known not being
solvable generally. In electrodynamics, also the two-body case has become unsolvable.
When quantum mechanics was emerging, new phenomena and paradoxes have occurred
and one-particle systems needed to be discussed anew. Today, in quantum field theory or
string theory representing modern physics, we are not even sure about the vacuum.
In this thesis we investigate relativistic effects in atomic spectra, particularly the spectral gaps in many-electron atoms. We derive explicit perturbative results from first principles of quantum mechanics, the theory of special relativity and Dirac theory. The basis
we are working on is the non-relativistic perturbation-theory (PT) model, developed in
[FG09b] and [FG10]. It provides asymptotic solutions for the eigenstates and eigenvalues of the second-shell atoms Lithium to Neon. The key idea thereby is induced by the
observation that in many-electron atoms the electron-electron interaction is dominated
by the electron-nucleus interaction in the limit of large nuclear charges, Z → ∞, the
so-called asymptotic limit. Therefore, the interaction between different electrons can be
treated perturbatively for large Z. The asymptotic PT states are approximations of the
Schrödinger eigenstates, but even in the case of neutral atoms their ground-state quantum
numbers coincide with those found in experiments. Comparisons of the asymptotic energy
gaps to experiments with highly-charged ions confirm the PT model. We use the NIST
database, [Yu.10], for the experimental data. We emphasize that the results obtained
within the PT model are independent of any semi-empirical input like Hund’s rules or the
Hartree-Fock method using a single Slater determinant of hydrogen orbitals as a starting
point.
Of course, the experimental spectrum includes relativistic effects. These are not covered
by the PT model. The main goal of this thesis is the derivation and discussion of all
first-order relativistic corrections to the energy levels of Lithium to Neon. The three
contributing corrections are well-known in the one-particle case: firstly, the relativistic
energy-momentum relation adjusts the classical kinetic energy. Secondly, retardations
effects due to a finite speed of light are represented by the Darwin term. Finally, the
spin-orbit coupling incorporates the spin of the electrons. We show briefly how these
corrections terms arise from the non-relativistic limit of Dirac theory. For the spin-orbit
coupling we restrict the main discussion to a local coupling between the spin of one electron
and its angular momentum. Terms taking the spin and angular momentum of different
electrons into account are assumed to be dominated by the local coupling. Our findings
concerning the relativistically corrected energy gaps are compared to the experimental
data from NIST. We will discuss the effects for each chemical element in detail.
1
Actually, physics is too hard for physicists.
5
1 Introduction
This thesis is divided mainly into two parts: to begin with we need to understand the
spectrum of the hydrogen atom as one-electron system.2 We introduce some fundamental
terms of spectral theory and prove the self-adjointness of the hydrogen Hamiltonian. The
hydrogen orbitals are essential for the PT model and its relativistic corrections, hence
we show in the Appendix a detailed derivation of them. We also introduce the Dirac
operator and discuss its self-adjointness and its spectrum. From the Dirac operator all
relativistic correction terms can be derived by expanding its resolvent around the classical
limit point.
The second part of this thesis starts with the presentation of the PT model and its
principal results. Combining these with the correction terms obtained from Dirac theory,
we derive in a general discussion all claimed relativistic corrections. In the subsequent
detailed discussion we specify the general corrections for all ground and first excited states
of Lithium, Beryllium, Boron, Carbon, Nitrogen, Oxygen and Fluorine. For Neon the PT
model offers only the ground state which undergoes some relativistic shift but no splitting.
In particular we find theoretically that some spectral gaps for Lithium and Oxygen
are not expected to feature relativistic corrections, since the relativistic corrections to
the involved energy levels are degenerate. This sort of degeneracy is well-known in the
analytically solvable hydrogen atom: there, the states 2s and 2p1/2 are degenerate in all
orders of perturbation theory in the fine-structure constant α0 . Indeed, the NIST data
feature the predicted degeneracies in the many-electron atoms, too.
Even the simplified local treatment of the spin-orbit coupling features the abolishment
of the mathematically indistinguishableness of the ground states of Boron and Fluorine,
and Carbon and Oxygen. When taking only the quantum numbers L and S into account
the ground states of the two considered pairs cannot be distinguished. The additional
total angular momentum, J, helps to label these ground states uniquely.
In this thesis we used JaxoDraw 2.0-1 to visualize the energy levels. The plots for
the spectral gaps and splittings were made using Microcal Origin 6.0.
2
6
The electron-proton system features a small and large mass scale, m and M , respectively, hence it is
usually considered as one-particle system. Due to a small reduced mass, µ ≡ mM/(m + M ) → m for
large M , this approximation is feasible.
2 Spectrum of the Hydrogen Atom
2.1 Non-relativistic Schrödinger Model
A good definition should be the hypothesis of a theorem.
James Glimm
This section is directed to understanding the non-relativistic spectrum of the hydrogen
atom. The Schrödinger equation, Eψ = Hψ, describes the eigenvalues, E ∈ σp (H), of
the hydrogen Hamiltonian H. We will investigate on which domain H is a self-adjoint
operator and prepare the set of its eigenfunctions which are key ingredients for the PT
model later on. We start with introducing some elementary definitions and theorems of
operator theory and spectral theory.
2.1.1 Preliminaries
Definition 2.1. Let X and Y be two Hilbert spaces and D(A), called the domain of A,
be a dense linear subspace of X. If the map A : D(A) → Y is linear, we call A a linear
operator in X and denote it by A : X → Y . A is called linear operator on X, if
D(A) = X. We call A a bounded linear operator, if
||A|| := sup{|Ax| : x ∈ D(A), |x| = 1} < ∞ .
(2.1)
In this case the non-negative real number ||A|| is called the norm of the linear operator
A. We denote the set of all bounded linear operators on X by B(X, Y ); if X = Y we
write briefly B(X). If A and B are linear operators in X, then A is said to be an
extension of B, B ⊆ A, if D(B) ⊆ D(A) and Ax = Bx for all x ∈ D(B).
Definition 2.2. A linear operator A : X → Y is said to be closed if for all sequences
(xn )n∈N ⊆ D(A) with xn → x ∈ D(A) and Axn → y ∈ Y , one has Ax = y.
Remark 2.3. In general, boundedness does not imply closeness. Furthermore, closeness
does not imply boundedness. When restricting to operators A ∈ B(X, Y ), then, by the
closed graph theorem, A is bounded ⇔ A is closed.
Theorem 2.4. Let A : X → Y be a linear operator in X. Then one has the equivalence
A is bounded
⇔
A is continuous .
Remark 2.5. If not stated otherwise, the continuity of linear operators refers to the norm
topologies on X and Y , respectively. We consider only Hilbert spaces with infinitely many
dimensions, since all linear finite-rank1 operators are compact2 , hence bounded.
1
2
A linear operator A : X → Y on X is said to be a finite-rank operator, if dim A(X) < ∞.
A linear operator A : X → Y on X is called compact, if A({x ∈ X : |x| ≤ 1}) is compact in Y , which
means that A(BX ) is relatively compact in Y .
7
2 Spectrum of the Hydrogen Atom
Definition 2.6. A linear operator A : X → Y in X is called boundedly invertible
if there is a bounded linear operator B ∈ B(Y, X) with the properties AB = idY and
BA ⊆ idX . In this case B is unique and we call it the bounded inverse of A.
Lemma 2.7. Let A : X → Y be a linear operator in X. Then the following statements
are equivalent:
(a) A is boundedly invertible.
(b) A is closed and bijective.
Definition 2.8. Let A : X → X be a linear operator in X. The spectrum of A is
defined as
σ(A) := {λ ∈ C : λ id − A is not boundedly invertible} .
(2.2)
The resolvent set of A is defined as ρ(A) := C\σ(A). Additionally, we define the point
spectrum, the continuous spectrum and the residual spectrum of A as
σp (A) := {λ ∈ C : λ id − A is not injective} ,
σc (A) := {λ ∈ C : λ id − A is injective and ran (λ id − A) 6= ran (λ id − A) = X} ,
σr (A) := {λ ∈ C : λ id − A is injective and ran (λ id − A) 6= X} .
Furthermore we define the discrete spectrum and the essential spectrum of A as
σdisc (A) := {λ ∈ σp (A) : λ is isolated in σ(A) and its eigenspace dimension is finite} ,
σess (A) := σ(A)\σdisc (A) .
We call λ ∈ σ(A) a spectral value of A.
Corollary 2.9. Let A : X → X be a linear operator in X. Then
σ(A) = σdisc (A) ∪ σess (A) ⊇ {λ ∈ C : λ id − A is not invertible} = σp (A) ∪ σc (A) ∪ σr (A) .
If A is a closed operator, then additionally
σ(A) = {λ ∈ C : λ id − A is not invertible} .
(2.3)
Proof. The statements follow immediately from Definition 2.8 and Lemma 2.7.
Theorem 2.10. Let A : X → Y be a linear operator in X. Then, its spectrum σ(A) is
a closed subset of C. If A ∈ B(X), then σ(A) is additionally non-empty and bounded,
hence compact. In both cases the resolvent map, RA : ρ(A) → B(X), λ 7→ (λ id − A)−1 ,
is an analytic function on the resolvent set ρ(A).
Remark 2.11. The proofs of these statements can be found in standard textbooks. In
the literature there are two different ideas to prove the non-emptiness and compactness of
a bounded linear operator: usually, as done in [RS80], one uses Liouville’s theorem from
complex analysis in combination with some Hahn-Banach corollary, making the spectrum
to be a non-constructive set. However, in [Kan09] it is proven by contradiction that the
spectrum of any element in a Banach algebra3 is non-empty and compact. Although
one has avoided the Hahn-Banach theorem in the second case, the spectrum is still nonconstructive.
3
8
Note that the linear space B(X), together with the operator norm || · ||, forms a Banach algebra.
2.1 Non-relativistic Schrödinger Model
From a mathematical point of view the spectrum can be investigated for many classes of
operators: normal4 operators, compact operators, . . . In quantum physics the operators
model (in many cases) the energy of the system. The related spectrum is interpreted as
the set of energy values the system can have, hence the spectrum needs to be a subset of
the real axis. Therefore our discussion is restricted to self-adjoint operators5 .
Definition 2.12. For a linear operator A : X → Y define its adjoint operator A∗ by
D(A∗ ) := {y ∈ Y : hAx, yi = hx, y ∗ i for some y ∗ ∈ X and all x ∈ D(A)} ,
(2.4)
and A∗ y := y ∗ . Then A∗ : Y → X is a linear operator in Y .
Remark 2.13. Due to the fact that A is densely defined, its adjoint is well-defined.
Indeed, in this case, y ∗ in (2.4) is uniquely determined. Note that D(A∗ ) does not need
to be dense in Y . This property is equivalent to the existence of a closed extension of A.
However, the adjoint A∗ is closed without any further constraints. For a proof of these
statements and the following two theorems see, for instance, [Rud91].
Definition 2.14. Let A : X → X be a linear operator in X.
(i) A is called symmetric or hermitian, if A ⊆ A∗ .
(ii) A is called self-adjoint, if A = A∗ .
Remark 2.15. In the case of bounded operators, A ∈ B(X), both terms coincide. Of
course, every self-adjoint operator is hermitian and, by Remark 2.13, also closed.
We now state a criterion which characterizes the self-adjointness of a hermitian operator:
it is possible to check self-adjointness when checking injectivity instead of surjectivity,
which is, in many applications, much easier. In [RS80] this is called the basic criterion of
self-adjointness:
Theorem 2.16. Let A : X → X be a linear operator in X. If A is hermitian, then the
following statements are equivalent:
(i) A is self-adjoint.
(ii) (±i)id − A is surjective.
(iii) A is closed and (±i)id − A∗ is injective.
Even more, when restricting to closed hermitian operators, we have:
Theorem 2.17. Let A : X → X be a linear operator in X. If A is hermitian and closed,
then the following statements are equivalent:
(i) A is self-adjoint.
(ii) σ(A) ⊆ R.
4
5
A linear operator A ∈ B(X, Y ) is called normal, if AA∗ = A∗ A.
While discussing symmetry properties of physical systems, also unitary operators are used. The timereversal symmetry calls for an anti-unitary operator. Note that any self-adjoint operator A induces
by A 7→ exp(iA) a unitary operator.
9
2 Spectrum of the Hydrogen Atom
2.1.2 Weyl’s Criterion
In the literature there is a characterization of spectral values being in the essential spectrum using so-called Weyl sequences [HS96]. We are going to prove a version of Weyl’s
criterion which is adapted to our purposes: every spectral value in the spectrum of a
self-adjoint operator is already an approximate eigenvalue.
Definition 2.18. Let A : X → X be a linear operator in X. A complex number λ ∈ C is
called approximate eigenvalue of A if for all ε > 0 there exists x ∈ D(A) with ||x|| = 1
and ||λx − Ax|| < ε. The set of all approximate eigenvalues of A is denoted by σap (A).
Remark 2.19. The definition of an approximate eigenvalue is equivalent to the existence
of a sequence (xn )n∈N ⊆ D(A) with ||xn || = 1 for all n ∈ N such that ||(λ id − A)xn || → 0
for n → ∞ . We call such a sequence (xn )n∈N a Weyl sequence for λ and A and
emphasize that in the literature often the additional condition “xn converges weakly to
0” is related to a Weyl sequence. For our purposes we do not ask for that restriction!
Theorem 2.20. Let A : X → X be a closed linear operator in X. Then
σp (A) ∪ σc (A) ⊆ σap (A) ⊆ σ(A) .
Proof. We first show the inclusion σp (A) ∪ σc (A) ⊆ σap (A):
(i) Consider λ ∈ σp (A). Then there is 0 6= x ∈ D(A) such that (λ id − A)x = 0. Denote
x
e := x/||x||, then ||e
x|| = 1 and (λ id − A)e
x = 0, hence λ ∈ σap (A).
(ii) Consider λ ∈ σc (A). We claim that (λ id − A)−1 : ran (λ id − A) → D(A) is bijective
but unbounded. Bijectivity holds, since λ ∈ σc (A) ⇒ (λ id − A) : D(A) → X is
injective but not surjective. Therefore (λ id−A) : D(A) → ran (λ id−A) is bijective.
Assume that (λ id − A)−1 : ran (λ id − A) → D(A) is additionally bounded, then
there is a linear operator B : X = ran (λ id − A) → D(A) which is bounded, too.
This contradicts λ ∈ σ(A), hence (λ id − A) : D(A) → ran (λ id − A) is unbounded as
claimed above. From this we get a sequence (yn )n∈N ⊆ ran (λ id − A) with ||yn || = 1
for all n ∈ N such that ||xn || := ||(λ id − A)−1 yn || → ∞. Denote x
en := xn /||xn ||,
then x
en ∈ D(A) with ||e
xn || = 1 for all n ∈ N. Finally, we have
||(λ id − A)e
xn || =
||yn ||
→0,
||(λ id − A)−1 yn ||
which shows that (e
xn )n∈N is a Weyl sequence for λ and A.
For the inclusion σap (A) ⊆ σ(A) consider a Weyl sequence for λ and A. Only the case
(λ id−A)xn 6= 0 for all n ∈ N is non-trivial. Therefore, assume that (λ id−A) : D(A) → X
is injective and denote
(λ id − A)xn
yn :=
,
||(λ id − A)xn ||
then yn ∈ ran (λ id − A) and ||yn || = 1 for all n ∈ N. Furthermore
||(λ id − A)−1 yn || =
||xn ||
→∞,
||(λ id − A)xn ||
hence (λ id − A) : D(A) → X is not invertible by Lemma 2.7. Finally, using Corollary 2.9,
we can conclude λ ∈ σ(A).
10
2.1 Non-relativistic Schrödinger Model
Lemma 2.21. Let A : X → Y be a linear operator in X. Then
(a) (ran A)⊥ = ker A∗ .
(b) If A is additionally closed, then (ran A∗ )⊥ = ker A .
Lemma 2.22. For any self-adjoint linear operator A : X → X in X one has σr (A) = ∅.
Proof. Assume that there is λ ∈ σr (A), i.e. (λ id − A) : D(A) → X is injective and
ran (λ id − A) 6= X. Then one has 0 6= y ∈ (ran (λ id − A))⊥ 6= {0}. By Lemma 2.21
we have y ∈ ker (λ id − A)∗ = ker (λ id − A∗ ) = ker (λ id − A), a contradiction. The last
equality is true since A is self-adjoint, hence A = A∗ and σ(A) ⊆ R.
Combining Theorem 2.20, Corollary 2.9 and Lemma 2.22, one has the following
Theorem 2.23. Weyl’s criterion
Let A : X → X be a linear operator in X. If A is self-adjoint, then
λ ∈ σ(A) ⇔ λ ∈ σap (A) .
For completeness we state the version of Weyl’s criterion which is often used in the
literature and refer for the proof, for instance, to [HS96]:
Theorem 2.24. Weyl’s criterion (essential version)
Let A : X → X be a linear operator in X. If A is self-adjoint, then one has
λ ∈ σess (A) ⇔ there is a Weyl sequence (xn )n∈N ⊆ D(A) for λ and A
with hxn , xi → 0 for all x ∈ X .
Remark 2.25. In physics x ∈ D(A) ⊆ X is called wave function, the Hilbert space X
is called state space. The property ||xn || = 1 for all n ∈ N of a Weyl sequence is crucial
for the statistical interpretation of the wave function. In a physical environment, we are
going to write ψ ∈ X instead of x, since the latter one is then used to denote the position
x ∈ Rn , n ∈ N, the wave function ψ is considered at.
Example 2.26. In order to show how Weyl’s criterion justifies some sloppy physical way
of handling spectral values, we give the following example, following the lecture by [Fri07]:
consider the Hilbert space6 X = L2 (R) of all square-integrable functions ψ : R → C with
1/2
R
the L2 –norm ||ψ||2 := R |ψ(x)|2 dx
< ∞. Furthermore, consider the linear operator
2
2
2
A = − d / dx in L (R).
Physical picture: for any k ∈ R, the function ψ(x) := eikx fulfills for λ := |k|2 the
equation (λ id − A)ψ = 0. Therefore ψ is an eigenfunction of A with eigenvalue |k|2 .
However, this line of arguments is not correct, since ||ψ||2 = ∞, i.e. ψ 6= L2 (R).
6
As usual we are dealing with equivalence classes of integrable functions in Lp (Rn ), p ≥ 1, and identify
functions which are λ–a.e. equal. Thereby λ is the Lebesgue measure on the measurable space (Rn , A),
where A denotes the complete Lebesgue sigma algebra on Rn . Then, (Lp (Rn ), || · ||p ), p ≥ 1, is a
Banach space.
11
2 Spectrum of the Hydrogen Atom
Mathematical picture: for ψ(x) = eikx define for ε > 0 the function ψε : R → C,
2
ψε (x) := cε e−εx ψ(x) .
The constant cε is chosen in such a way that ||ψε ||2 = 1. Since ψε ∈ C 2 (R) ∩ L2 (R), we
are indeed allowed to evaluate
Aψε (x) = ψε (k 2 − 4ε2 x2 + 4ikε + 2ε) .
Therefore we find pointwise for all x ∈ R:
(k 2 id − A)ψε (x) = (4ε2 x2 − 4ikεx − 2ε)ψε (x) → 0 for ε → 0 .
Furthermore, we have in L2 –norm:
||(4ε2 x2 − 4ikεx − 2ε)ψε (x)||2 → 0 for ε → 0 ,
hence (ψε(n) )n∈N , ε(n) := 1/n, is a Weyl sequence for |k|2 ≥ 0 and A. Since k ∈ R is
arbitrary, we know by Weyl’s criterion7 : [0 ,∞) ⊆ σ(A).
However, none of the functions, neither ψ(x) nor ψε (x), are eigenfunctions of A, but
for all ε > 0 the function ψε (x) gives rise to an approximate eigenvalue λ = |k|2 of A. So
we can conclude, [Fri07]:
Physical picture
Hilbert space is approximated:
ψ∈
/ L2 (R)
Eigenvalue equation is “solved” exactly:
(λ id − A)ψ = 0
Mathematical picture
Hilbert space is kept exactly:
ψε ∈ L2 (R)
Eigenvalue equation is solved approximately:
||(λ id − A)ψε || < c ε
The mathematical shortcoming that the “solution” ψ of the eigenvalue equation is not
element of the underlying Hilbert space is physically interpreted in the following way:
a single de Broglie wave ψ(x) = eikx does have a sharp momentum k, in conflict with
Heisenberg’s uncertainty principle. Therefore only some superposition of these plane
waves is physical:
Z
e
ψ(x) := C(l) eilx dl ,
where C : R → [0 ,∞) is some smooth density function for the momenta. If C is Gaussian
with mean k and variance 1/ε, i.e. C = N (k, 1/ε), then ψe = ψε . In this case, the
boundary of the uncertainty principle becomes sharp.
2.1.3 The Kato-Rellich Theorem
In this short chapter we state the Kato-Rellich theorem which will be used to prove the
self-adjointness of the hydrogen Hamiltonian. We thereby ask for a sufficient condition
that the perturbation S = T + A of a self-adjoint operator T remains self-adjoint. We
need to introduce the following
7
If one considers the operator A = − d2 / dx2 on domain D(A) = H 2 (R), the Sobolev space of order 2,
A is indeed self-adjoint. The Sobolev spaces are discussed in Appendix A.1.
12
2.1 Non-relativistic Schrödinger Model
Definition 2.27. Let A, T : X → Y be two linear operators in X with D(T ) ⊆ D(A).
The operator A is called T -bounded if there are constants a, b ≥ 0 such that
||Ax|| ≤ a||x|| + b||T x||
for all x ∈ D(T ) .
(2.5)
In this case we define the T -bound of A by
b0 := inf{b ≥ 0 : there is a ≥ 0 such that (2.5) holds} .
Remark 2.28. If A ∈ B(X, Y ), then A is T -bounded for all linear operators T : X → Y
in X with b0 = 0 and a = ||A||. However, in general it is not possible to fulfill (2.5) when
setting b = b0 .
Theorem 2.29. Kato-Rellich
Let T : X → X be a self-adjoint linear operator in X. If A : X → X is hermitian
and T -bounded with T -bound b0 < 1, then the perturbed operator S := T + A is again a
self-adjoint operator in X with domain D(S) = D(T ).
Proof. We refer to the literature, for instance [RS75].
2.1.4 Self-adjointness of the Hydrogen Hamiltonian
In this section our main goal is to show that the hydrogen Hamiltonian,
1
1
H =− ∆−
,
2
|x|
(2.6)
is a self-adjoint operator in L2 (R3 ) with domain H 2 (R3 ). Here, H k denotes the Sobolev
space of order k which is described in Appendix A.1. The structure of this operator is
motivated by classical physics and the correspondence principle.
Consider a classical particle with mass m > 0, velocity v ∈ R3 and momentum p =
mv ∈ R3 . The kinetic energy of this particle reads T = mv 2 /2 = p2 /2m ∈ R. In an
attractive Coulomb potential where this particle with elementary charge e < 0 interacts
with some other particle with charge −e > 0, its potential energy reads V = −e2 /4π0 |x|.
In classical Hamilton theory, the total energy is just the sum H = T + V ∈ R. We are not
going to use SI units and switch to units such that m = 1, ~ = 1 and e2 = 4π0 . So far
the particle is considered classically. Mapping the energy H to an operator by identifying
the momentum p with an operator, p 7→ −i ∇, we find H to be the operator stated in
(2.6). This way of motivating the hydrogen Hamiltonian by mapping classical observables
to operators is called correspondence principle.
Remark 2.30. The hydrogen Hamiltonian (2.6) is a so-called Schrödinger operator,
since it is a linear operator in the Hilbert space L2 (Rn ) and obeys the form ∆ + V , where
V is some real-valued function.
Theorem 2.31. Let A : L2 (Rn ) → L2 (Rn ) be the linear operator A = −∆. On the
domain H 2 (Rn ), A = −∆ is self-adjoint and σ(−∆) = σess (−∆) = [0 , ∞).
Proof. For both, the proof of self-adjointness and the proof for the spectrum, we refer to
the literature, for instance to [HS96]. Crucial ingredients for the second part of the proof
are Fourier analysis and Weyl’s criterion 2.24.
13
2 Spectrum of the Hydrogen Atom
For showing that not only the unperturbed Laplacian is self-adjoint on H 2 (Rn ), but also
the hydrogen Hamiltonian (2.6), we need the following
Theorem 2.32. Hardy’s inequality
Let n ∈ N with n ≥ 3, then one has for all f ∈ H 1 (Rn ):
Z
Z
4
|f (x)|2
dx ≤
|∇f (x)|2 dx .
2
2
|x|
(n
−
2)
n
n
R
R
(2.7)
Proof. We assume without loss of generality that f ∈ H 1 (Rn ) is a real-valued function.
Then, we have for all i = 1, . . . , n and all α ∈ R:
2
Z Z α2 x2i
xi
xi
2
2
0≤
∂i f (x) − α 2 f (x) dx =
(∂i f (x)) − 2α 2 f (x)∂i f (x) +
f (x) dx .
|x|
|x |
|x|4
Rn
Rn
The second term reads
Z Z
Z xi
xi
1
x2i
2
2
−α 2 ∂i f (x) dx = α
∂i 2 f (x) dx = α
− 2 4 f (x)2 dx ,
2
|x|
|x|
|x|
|x|
Rn
Rn
Rn
where we used ∂i f 2 = 2f ∂i f and integrated by parts. Since f ∈ H 1 (Rn ), there is no
surface term. Summing now over all components, i = 1, . . . , n, yields
Z n
2
α2
2
2
2
0≤
|∇f (x)| + α
−
f (x) + 2 f (x) dx =
|x|2 |x|2
|x|
Rn
Z 2
f (x)
=
dx .
|∇f (x)|2 + α(n − 2) + α2
|x|2
Rn
Both terms, |∇f (x)|2 and f (x)2 /|x|2 are non-negative. Therefore, we find an optimal
bound when minimizing over α:
α(n − 2) + α2 = min
⇔
α=
2−n
.
n
At this minimizer, the inequality reads:
Z (n − 2)2 f (x)2
2
0≤
|∇f (x)| −
dx ,
4
|x|2
Rn
which implies the claim.
Remark 2.33. The presented proof of Hardy’s inequality is adapted from [Fri07]. For
the proof and, actually, also for the inequality R(2.7), the hypothesis n ≥ 3 is necessary,
since the left-hand side of Hardy’s inequality, Rn f (x)2 /|x|2 dx, might be divergent for
n = 1 or n = 2.
Remark 2.34. In [RS75] a special case of Hardy’s inequality is proven: there, the so-called
“uncertainty principle” states that (2.7) holds for all f ∈ C0∞ (R3 ) ⊆ H 1 (R3 ). However,
in the literature there are other versions of Hardy’s inequality, too. For instance, in
[GGM03], there is an Lp -version for f ∈ W01,p , p ≥ 1.
Now we are able to prove our claim that the hydrogen Hamiltonian in at least three
dimensions and on a suitable domain is a self-adjoint operator:
14
2.1 Non-relativistic Schrödinger Model
1
.
Theorem 2.35. Let H : L2 (Rn ) → L2 (Rn ) be the hydrogen Hamiltonian H = − 12 ∆ − |x|
2
n
If n ≥ 3, then H is a self-adjoint operator on D(H) = H (R ).
Proof. Consider in L2 (Rn ) the linear operator T = − 12 ∆ with domain D(T ) = H 2 (Rn ).
1
Furthermore consider on L2 (Rn ) the linear operator A with Aψ(x) = − |x|
ψ(x) . We show
2
n
that A is T -bounded with T -bound b0 < 1: for ψ ∈ L (R ) we have:
Z
Z
1
1
4
2
2
2
||Aψ||2 = || ψ||2 =
|ψ(x)| dx ≤
|∇ψ(x)|2 dx =
2
2
|x|
(n − 2) Rn
n |x|
Z R
Z
4
4
=
∇ψ(x) · ∇ψ(x) dx =
(−∆ψ(x))ψ(x) dx ≤
(n − 2)2 Rn
(n − 2)2 Rn
4
≤
||∆ψ||2 ||ψ||2 ,
(n − 2)2
where we used first Hardy’s inequality8 , subsequently the Cauchy-Schwarz inequality.
Consider now some ε > 0, then one has for all a, b ≥ 0:
2
b2
b
1 2 2 b2
2 2
= ε a − 2ab + 2 ,
εa + 2 .
0 ≤ εa −
hence ab ≤
ε
ε
2
ε
Therefore we find:
Using
√
||Aψ||22
2
≤
(n − 2)2
1
2
2
2
ε ||∆ψ||2 + 2 ||ψ||2 .
ε
√
a + b for a, b ≥ 0, we finally arrive at:
√ √ 2
2 2
1
1
||Aψ||2 ≤
ε||∆ψ||2 + ||ψ||2 =
ε||T ψ||2 + ||ψ||2 .
n−2
ε
n−2
2ε
a+b≤
√
√ , then A is indeed T -bounded with T -bound b0 < 1. By Theorem 2.31,
Choose 0 < ε < n−2
2 2
T is self-adjoint, hence applying the Kato-Rellich theorem 2.29 finishes the proof.
2.1.5 Non-relativistic Spectrum of the Hydrogen Atom
For later convenience we introduce an additional parameter, Z ∈ N, describing the nuclear charge of the hydrogen-type Hamiltonian9 . Later on, when discussing the PT
model, this parameter will be crucial for our approach to many-electron systems.
Theorem 2.36. For Z ∈ N, let H = − 12 ∆ −
domain D(H) = H 2 (R3 ). Then
Z
|x|
be a linear operator in L2 (R3 ) with
(a) H is a self-adjoint operator with σ(H) = {−Z 2 /2n2 : n ∈ N} ∪ [0,∞) .
(b) For all n ∈ N, En := −Z 2 /2n2 is an eigenvalue of H. Its corresponding eigenspace
is n2 –dimensional, En ∈ σdisc (H).
8
9
The hypothesis of Theorem 2.32 is fulfilled, since H 2 (Rn ) ⊆ H 1 (Rn ).
For X being some chemical element, we call an ion “X–like” if it has as many electrons as the neutral
one, N , but some arbitrary nuclear charge Z ≥ N , compare Definition 3.9. Note that for being a
stable ion, the condition Z ≥ N must be fulfilled, compare [Fri03].
15
2 Spectrum of the Hydrogen Atom
Proof. We show only some aspects of the theorem. For the complete proof we refer to the
literature, for instance [Fri07].
(a) First we show that [0 ,∞) ⊆ σ(H) by proving that
(x − an )2
1
+ ik(x − an )
exp −
ψn (x) =
(2εn )3/4
4εn
(2.8)
denotes for some εn > 0 and an ∈ R3 a Weyl sequence for H and λ = 21 |k|2 . One
finds immediately that ||ψn ||2 = 1 for all n ∈ N. Applying 12 ∆ to ψn , one gets:
2
1
x
ix · k x · an
a2n
ian · k
3
k2
∆ψn (x) = ψn (x)
−
−
+ 2 +
−
−
.
2
8ε2n
2εn
4ε2n
8εn
2εn
4εn
2
The last term is canceled by λψn (x) and we arrive at
2 2 2 2
2
2 2 λ id + 1 ∆ + Z ψn (x) ≤ x ψn + an ψn + x · an ψn + 3 ψn +
8ε2 8ε2 4ε2
4εn 2
|x|
n
n
n
2
2
2
22 2 22
x · k an · k Z +
2εn ψn + 2εn ψn + |x| ψn .
2
2
2
(2.9)
√
We consider now a sequence 0 < εn → ∞ for n → ∞ and choose an := a0 εn ,
with 0 6= a0 ∈ R3 . With this, we find that all terms on the right-hand side of (2.9)
vanish in the limit n → ∞, since for all of them there is some q < 0 such that
|| · || = εqn · const. As instructive example we derive the third term explicitly:
Z √
√ 2
2
x · an 2
ε
(x
·
a
)
(x
−
a
εn )
1
n
0
0
=
ψ
exp
−
dx =
n
4ε2
16ε4n R3 (2πεn )3/2
2εn
n
2
Z
(x̃ − a0 )2
1
1
1
2
(x̃ · a0 ) exp −
=
dx̃ = 2 · const.
3/2
2
16(2π) εn R3
2
εn
√
In the second line we have switched to x̃ := x/ εn . The remaining three-dimensional
integral is a finite constant, hence
x · an 2
ψn lim =0.
n→∞ 4ε2
n
2
Similar calculations lead to q = −2 for the expressions of the first line in (2.9) and
to q = −1 in the second line. Altogether we arrive at
1 2
=0.
k
id
−
H
ψ
(2.10)
lim n
n→∞ 2
2
Since k ∈ R3 is arbitrary, Weyl’s criterion 2.23 implies the claim [0 ,∞) ⊆ σ(H).
(b) In Appendix A.2 we show that En = −Z 2 /2n2 < 0 is an eigenvalue of H. The
corresponding eigenspace is spanned by the orthonormal basis
Vn = {ψnlm ∈ L(R3 ) : l = 0, 1, . . . , n−1 and m = −l, −l +1, . . . , l −1, l} . (2.11)
16
2.1 Non-relativistic Schrödinger Model
The ψnlm are eigenfunctions of H, given in Theorem A.16:
ψnml (r, θ, ϕ) = Z 3/2 Rnl (Zr)Ylm (θ, ϕ) ,
where Ylm denote the spherical harmonics and Rnl are principally governed by the
associated Laguerre polynomials. However, we can calculate quickly the number of
elements in Vn :
n−1
X
#Vn =
(2l + 1) = n2 .
(2.12)
l=0
Together we have En ∈ σ(H) and dim span Vn = n2 < ∞, hence En ∈ σdisc (H).
In the PT model and for the discussion of its relativistic corrections, the hydrogen orbitals,
ψnlm , are crucial. The following table states their explicit form for the lowest quantum
numbers n = 1 and n = 2:
n
l
m
1
0
0
2
0
0
1
0
1
−1
ψnlm (r, θ, ϕ)
Z√3/2
π
Z 3/2
√
8π
Z 5/2
√
32π
Z 5/2
√
32π
5/2
Z
√
32π
e−Zr
−Zr/2
1 − Zr
e
2
r cos θ e−Zr/2
r sin θ cos ϕ e−Zr/2
r sin θ sin ϕ e−Zr/2
ψnlm (x)
notation
Z√3/2 −Z|x|
e
π
3/2
Z|x|
Z
√
1
−
e−Z|x|/2
2
8π
Z 5/2
√
x e−Z|x|/2
32π 3
Z 5/2
√
x e−Z|x|/2
32π 1
Z 5/2
√
x e−Z|x|/2
32π 2
|1i
|2i
|3i
|4i
|5i
Table 2.1: Hydrogen orbitals for the lowest quantum numbers n = 1 and n = 2. For the state (n, l, m) =
(2, 1, ±1) we have chosen the real part, |4i, and imaginary part, |5i, of exp (±imϕ) instead
of their complex linear combination. This is just a change of the basis functions and more
convenient for later use.
17
2 Spectrum of the Hydrogen Atom
2.2 Relativistic Dirac Model
Es gibt keinen Gott und Dirac ist sein Prophet.10
Wolfgang Pauli
When quantum mechanics arose in the 20th of the last century, the special and general
theory of relativity was already known and largely accepted. One of the big shortcomings of early quantum mechanics was the fact that its fundamental equation of motion,
namely the Schrödinger equation, was violating relativistic symmetry aspects: the time
appears in a first-order derivative, whereas the spatial momentum appears in a secondorder derivative. In principle, one can remove this flaw by implementing the relativistic
energy-momentum relation, E 2 = p2 + m2 , and, by the correspondence principle between
observables and operators, one can therewith motivate the relativistic Klein-Gordon equation. In the first time it was not clear how to interpret its solutions. Today, within
quantum field theory, the Klein-Gordon equation itself and its physical conclusions are
understood. However, we are not going to discuss these aspects and refer to the literature,
for instance [PS95].
In order to describe relativistic particles with half-integer spin, so-called fermions, relativistic quantum mechanics11 tells us to use the Dirac equation instead of the Klein-Gordon
equation. The latter one describes relativistic particles with integer spin, so-called bosons.
Our main goal of this section is to introduce the free Dirac operator, H0 , and its coupling to the Coulomb potential. For later use in the PT model, we also prepare the
non-relativistic limit, H∞ , and its first-order relativistic corrections. We refer mostly to
[Tha92].
2.2.1 Self-adjointness of the Dirac Operator
Definition 2.37. Consider the Hilbert space X = L2 (R3 , C4 ) = L2 (R3 ) ⊗ C4 , i.e. the set
of all L2 –functions ψ : R3 → C4 . The scalar product of ψ, φ ∈ X is defined by
4
X
hψ, φiX :=
hψi , φi i ,
i=1
where h·, ·i is the usual scalar product on L2 (R3 ). We define the in L2 (R3 ) ⊗ C4 the linear
free Dirac operator12
H0 := −ic0 α · ∇ + βc20 ,
(2.13)
10
There is no God and Dirac is his prophet.
Actually relativistic quantum mechanics is already a quantum field theory, since there is no relativistic
theory of one-particle quantum systems. Having the possibility of converting energy and mass, as the
special theory of relativity states, the number of particles is no longer a well-defined quantity of the
system. The idea of fields accommodates this physical principle in a very successful way, as QED,
QCD and the whole Standard Model show. One of the paradoxes when holding on to the one-particle
picture is the Klein paradox which is discussed frequently in the physical literature.
12
Again, we use atomic units, hence our Dirac operator is adapted to a fermion with mass m = 1. Later
on, for the discussion of the non-relativistic limit, we need the explicit dependence of H0 on the
speed of light c0 . In atomic units one has c0 = 1/α0 ≈ 137, where α0 denotes the electromagnetic
fine-structure constant.
11
18
2.2 Relativistic Dirac Model
where c0 > 0 is some parameter and (β, α) is a four-component vector with complex 4 × 4
matrices:
12
0
0 σi
β :=
and αi :=
i = 1, 2, 3 ,
0 −12
σi 0
where the complex 2 × 2 matrices σi , i = 1, 2, 3, are the so-called Pauli matrices, listed
and briefly discussed in Appendix A.3.
Remark 2.38.
(i) It is beyond the scope of this thesis to justify the form of the matrices α and β
from a group-theoretic point of view. However, our choice is called the PauliDirac representation and is convenient for our purpose, since β, also called γ0 ,
is diagonal. For a deeper examination of this, we refer to the multifarious physical
literature, for instance [PS95].
(ii) From a mathematical point of view, the four-component complex wave functions
ψ ∈ L2 (R3 ) ⊗ C4 are vectors. However, in physics they are called Dirac spinors,
due to their transformation law under some Lorentz transformation (ω, ϕ), where
ω, ϕ ∈ R3 denote some boost and spatial rotation, respectively:


P
3
i
exp
σ
(ϕ
−
iω
)
0
j
j
j=1 j
2
P
ψ .
ψ0 = 
3
i
0
exp 2 j=1 σj (ϕj + iωj )
We emphasize that, although ψ being a four-component vector, ψ is not a (relativistic) four-vector in the common physical nomenclature.
Our next step is the investigation of the domain of the free Dirac operator and its spectrum. With the following, we will be able to define the Foldy-Wouthuysen transformation,
UF W , which transforms the free Dirac operator into a diagonal matrix differential operator.
For n ∈ N consider the well-known Fourier transformation F : L1 (Rn ) → C(Rn ), which
maps integrable functions, ψ ∈ L1 (Rn ), to continuous functions, Fψ ∈ C(Rn ), by
Z
1
ψ(x)e−ip·x dx .
(2.14)
(Fψ)(p) :=
(2π)n/2 Rn
The Fourier transformation is a linear map on the Banach space of integrable functions
onto some complicated and cumbersome subspace of the set of continuous functions on
Rn . In fact, it may happen that the Fourier transform of ψ ∈ L1 (Rn ) is not again integrable: Fψ ∈
/ L1 (Rn ). In order to define a more handy version of the Fourier transformation, firstly we restrict F to L1 (Rn ) ∩ L2 (Rn ), which is a dense subspace of the Hilbert
space L2 (Rn ). Second, by Plancherel’s theorem13 , this restricted Fourier transformation,
F|L1 ∩L2 , can be uniquely extended to some unitary operator14 P : L2 (Rn ) → L2 (Rn ),
called the Plancherel transformation. Note that P is a continuous linear operator on
L2 (Rn ) onto L2 (Rn ) and ||Pψ||2 = ||ψ||2 for all ψ ∈ L2 (Rn ).
13
14
Plancherel’s theorem is discussed frequently in the literature, for instance in [Eva98] or [RS75].
A linear operator A : X → X in a Hilbert space X is called unitary, if it is isometric on X, i.e.
||Ax|| = ||x|| for all x ∈ X.
19
2 Spectrum of the Hydrogen Atom
Remark 2.39. We want to emphasize that the Plancherel transformation, P, does not
extend the Fourier transformation, F, although it is sometimes stated in the literature.
Note that in the first place F is defined on L1 (Rn ), but the restriction of the Plancherel
transformation to this space, P|L1 : L1 (Rn ) ∩ L2 (Rn ) → L2 (Rn ), is not defined for all
integrable functions. However, it is possible that the Lp spaces are nested: if one considers
the counting measure on N, then Lp = lp and the inclusion l1 ⊆ l2 holds.
Now we concentrate on the free Dirac operator, H0 : X → X, which is a linear operator in the Hilbert space X = L2 (R3 ) ⊗ C4 . In the following we will find a suitable
domain, D(H0 ) ⊆ X, of H0 . For this we introduce the Plancherel transformation for
linear operators in X, H0 ∈ Lin(X), by b : Lin(X) → Lin(X) and the following claim:
c0 : X → X,
H
c0 Pψ =! P(H0 ψ) , for all ψ ∈ D(H0 ) .
H
(2.15)
Since the Plancherel transformation, P, is unique and invertible, we find for the Plancherel
transform of the free Dirac operator:
c0 Pψ =! PH0 ψ = PH0 P −1 Pψ
H
⇒
c0 = PH0 P −1 .
H
c0 is again a linear operator in the Hilbert space X = L2 (R3 ) ⊗ C4 with dense15
In fact, H
domain P(D(H0 )). Therefore, the transformation b is well-defined.
Lemma 2.40. Let X = L2 (R3 ) ⊗ C4 and H0 : X → X be the free Dirac operator in X.
(a) The explicit form of the derivative operator H0 reads
c20 12
−ic0 σ · ∇
H0 =
.
−ic0 σ · ∇
−c20 12
(2.16)
c0 : X → X is given by the multiplicative matrix operator
Its Plancherel transform, H
2
1
c
σ
·
p
c
2
0
−1
0
c0 Pψ)(p) = (FH0 F )(p) =
Pψ(p) for all p ∈ R3 .
(H
c0 σ · p −c20 12
(2.17)
p
c0 (p) are λ1,2 (p) = −λ3,4 (p) = c20 p2 + c40 =: λ(p), p ∈ R3 ,
(b) The eigenvalues of H
where the corresponding diagonalization matrix, U : X → X, is given by
U (p)±1 :=
[c20 + λ(p)]14 ± βc0 α · p
c0 U −1 = βλ .
p
, with U H
2
2λ(p)[c0 + λ(p)]
(2.18)
c0 coincide: D(λ) = D(H
c0 ) = P(H 1 (R3 )⊗C4 ).
In particular, the domains of λ and H
(c) The operator W : X → X, W := U ◦ P is unitary and diagonalizes the free Dirac
operator: WH0 W −1 = βλ.
15
The fact that P(D(H0 )) is dense in X follows since the unitary Plancherel transformation is particularly
continuous: P(D(H0 )) = P(D(H0 )) = P(X) = X. The last equal sign follows since P is surjective.
20
2.2 Relativistic Dirac Model
Proof.
(a) The explicit form of the free Dirac operator H0 in (2.17) follows directly from the
definition of the matrices β and α. For its Plancherel transform, (2.14) tells us to
c0 .
map16 ∇ 7→ ip, which implies directly H
p
(b) A straightforward calculation yields for p ∈ R3 the eigenvalues ±λ(p) = ± c20 p2 + c40
c0 , and also the unitary matrix U (p) which diagonalizes H
c0 :
of H
c0 U −1 )(p) = βλ(p) .
(U H
For the inverse matrix U (p)−1 one uses the anti-commutator relation {αi , β} = 0,
i = 1, 2, 3. Furthermore, the identities β 2 = 14 and (α · p)(α · p) = p2 hold.
Note that the eigenvalue operator λ : X → X is just a multiplication operator,
P(D(H0 )) 3 Pψ 7→ λ(p)14 Pψ(p), for all p ∈ R3 . Since U : X → X is unitary and
c0 and λ coincide: D(H
c0 ) = D(λ). The form
β a constant matrix, the domains of H
of the eigenvalues ±λ(p) implies17
n
o
p
D(λ) = Pψ ∈ X : 1 + p2 (Pψ)(p) ∈ X .
(2.19)
Therefore, using for k = 1 and n = 3 the equivalent definitions of the Sobolev spaces
H k (Rn ) in Theorem A.8 (i) ⇔ (iii), we find
p
1 + p2 (Pψ)(p) ∈ L2 (R3 ) ⊗ C4 ⇔ ψ ∈ H 1 (R3 ) ⊗ C4 ,
Pψ ∈ D(λ) ⇔
(2.20)
1
3
4
c
hence D(H0 ) = D(λ) = P(H (R ) ⊗ C ).
(c) As composition of two unitary operators, U and P on X, the operator W = U ◦ P
is unitary, too. Using the results of part (a) and (b), we find immediately:
c0 U −1 = βλ .
WH0 W −1 = U PH0 P −1 U −1 = U H
(2.21)
Theorem 2.41. The free Dirac operator H0 in X = L2 (R3 )⊗C4 is a self-adjoint operator
on D(H0 ) = H 1 (R3 ) ⊗ C4 . Its spectrum is given by σ(H0 ) = (−∞, −c20 ] ∪ [c20 , ∞).
c0 ) = P(D(H0 )). Since the Plancherel transformation is
Proof. From (2.15) we know D(H
c0 )). Combining this we Lemma 2.40(b), particularly
unitary we have D(H0 ) = P −1 (D(H
c0 ) = P(H 1 (R3 ) ⊗ C4 ), we find: D(H0 ) = H 1 (R3 ) ⊗ C4 . Furthermore, from
D(λ) = D(H
Lemma 2.40(c) we know that the operator H0 is unitarily equivalent to the multiplicative
diagonal operator βλ, hence
σ(H0 ) = σ(βλ) = ran λ1,2 (·) ∪ ran λ3,4 (·) = (−∞, −c20 ] ∪ [c20 , ∞) .
For the remaining proof of the self-adjointness we refer to the short proof in [Tha92].
16
17
Note that we have already met this mapping for motivating the hydrogen Hamiltonian (2.6): p 7→ −i∇.
At this point it is important that the particle is massive, i.e. m > 0, since we are scaling, i.e. dividing
by the constant m2 c40 . However, we work in atomic units, hence we are not affected by this restriction.
In the massless case the here made implication would be not correct.
21
2 Spectrum of the Hydrogen Atom
Definition 2.42. The Foldy-Wouthuysen transformation UF W : Lin(X) → Lin(X)
is defined by UF W := P −1 ◦ W, where P : X → X denotes the Plancherel transformation
and W is defined in Lemma 2.40(c).
Theorem 2.43. The Foldy-Wouthuysen transformation is unitary and diagonalizes the
free Dirac operator:
p 2
0
−c0 ∆ + c40 12
−1
−1
p
.
UF W H0 UF W = P βλP =
0
− −c20 ∆ + c40 12
Remark 2.44.
(i) The Foldy-Wouthuysen transformation UF W diagonalizes the free Dirac operator in
the original space, L2 (R3 ) ⊗ C4 , whereas W diagonalizes in Plancherel-transformed
space P(L2 (R3 ) ⊗ C4 ).
(ii) From (2.43) one can guess the spectrum and domain of the free Dirac operator
H0 : by Theorem 2.31 we know already that the operator A = −∆ is self-adjoint
on the domain H 2 (R3 ) and σ(−∆) = [0,∞). Since the square-root function is
bounded
and Borel-measurable, the functional calculus allows to define the operator
p
2
−c0 ∆ + c40 , c0 > 0. The spectral mapping theorem suggests the spectrum of H0
to be as stated in Theorem 2.41. Also the domain
on which H0 is self-adjoint arises
√
from its diagonalized version symbolically: H 2 = H 1 . Of course, these arguments
are not rigorous, but they provide some intuition about the free Dirac operator.
(iii) It is remarkable that even the free Dirac operator features a negative spectrum. For
the non-relativistic spectrum of the hydrogen atom, we know from Theorem 2.36,
that negative spectral values correspond to the attractive Coulomb potential and
bounded physical states. Its discrete spectrum is empty if and only if we switch
off the nuclear charge (Z = 0). In the case of the free Dirac operator the negative
values spectrum do not correspond to bounded states, but to the existence of antiparticles. It is not possible to restrict the free Dirac operator to its positive part
without loosing its fundamental relativistic Poincaré covariance.
Definition 2.45. A symmetric operator A : X → X in a Hilbert space X is said to be
semibounded from below, if there is γ ∈ R such that
hAx, xi = hx, Axi ≥ γ||x||2 , for all x ∈ D(A) .
We call γA := sup{γ ∈ R : hx, Axi ≥ γ for all x ∈ D(A)} the lower bound of A.
Theorem 2.46. Let A be a self-adjoint operator in the Hilbert space X. Then: A is
semibounded from below with lower bound γA ⇔ λ ≥ γA for all λ ∈ σ(A) ⊆ R.
Proof. We refer to the literature, for instance [Rud91].
Corollary 2.47. For the free Dirac operator H0 on L2 (R3 ) ⊗ C4 one has18 :
inf{hψ, H0 ψi : ψ ∈ H 1 (R3 ) ⊗ C4 } = −∞ .
18
We suppress the subscript of the scalar product h·, ·iX .
22
2.2 Relativistic Dirac Model
2.2.2 Dirac Operator with Coulomb Interaction
Now we investigate the perturbation H = H0 + V of the free Dirac operator H0 by
some external field, given by the potential V . In [Tha92] the discussion considers general
matrix-valued potentials, but we restrict ourselves to the diagonal Coulomb potential
VC := φC 14 with φC = −Z/|x|. We will derive that the essential spectrum of some
perturbed Dirac operator remains the spectrum of the free Dirac operator, in contrast to
the non-relativistic case in Theorem 2.36.
Theorem 2.48. Let H0 be the free Dirac operator in L2 (R3 )⊗C4 . Let V be a multiplicative
operator with a Hermitian 4 × 4 matrix such that each component satisfies
|Vij (x)| ≤ a + b
c0
, for all 0 6= x ∈ R3 , i, j = 1, . . . , 4
2|x|
(2.22)
for some constants a > 0 and 0 < b < 1. Then, the operator H = H0 + V is self-adjoint
on D(H) = D(H0 ) = H 1 (R3 ) ⊗ C4 .
Proof. First of all one should compare the estimate (2.22) with the definition of T boundedness (2.5). In principle, one can recycle the ideas of the proof of Theorem 2.35:
Hardy’s inequality, applied to ψ ∈ H 1 (R3 ) ⊗ C4 , and the Kato-Rellich theorem imply the
self-adjointness of H. A detailed proof is elaborated in [Tha92].
Corollary 2.49. Let H0 be the free Dirac operator and VC the matrix-valued Coulomb
potential. Then, the linear operator H := H0 + VC in L2 (R3 ) ⊗ C4 is for Z < c0 /2 a
self-adjoint operator with domain D(H) = H 1 (R3 ) ⊗ C4 .
Remark 2.50. It is wrong to state that the perturbation of the free Dirac operator by the
Coulomb potential leads generally to a self-adjoint operator. For being able to apply the
Kato-Rellich theorem in the proof of Theorem 2.48, it is necessary to meet the relativistic
restriction Z ≤ 68 < c0 /2 (in atomic units). In the non-relativistic limit, c0 → ∞, this
restriction is irrelevant. It is known that the existence of such a restriction to the Coulomb
potential is not only due to our
√ approach. Indeed, [Tha92] shows that H = H0 + VC is
self-adjoint ⇔ Z ≤ 118 < c0 3/2.
The nomenclature “essential” spectrum can be motivated by the fact that σess is quite
stable under perturbations, in particular under compact perturbations. For our purpose
it is sufficient to state just two corollaries of Weyl’s essential-spectrum theorem, which is
in many applications, not only of relativistic quantum mechanics, one of the crucial tools.
We need the following
Definition 2.51. Let A, T : X → Y be two linear operators in X with D(T ) ⊆ D(A).
Then, the operator A is called T -compact, if A(i id + T )−1 is compact.
Theorem 2.52. Weyl’s essential-spectrum theorem
(a) Let A, B be two self-adjoint operators in the Hilbert space X on the same domain.
If the operator (i id − A)−1 − (i id − B)−1 is compact, then σess (A) = σess (B).
(b) Let T be a self-adjoint operator in the Hilbert space X. If the operator A is T –
compact, then S := T + A is a closed operator with D(S) = D(T ). Furthermore,
σess (S) = σess (T ).
23
2 Spectrum of the Hydrogen Atom
Proof. As already mentioned, both statements are actually corollaries of Weyl’s essentialspectrum theorem, which is proved, for instance, in [RS78].
Let X be a Hilbert space and M ⊆ X some open or closed subset. In the following
χM : X → {0, 1} will denote the characteristic function of M , i.e. χM (x) = 1 if x ∈ M
and χM (x) = 0 if x 6= M . Instead of χ{y∈X:||y||<R} (x), R > 0, we will write more
conveniently χ(||x|| < R).
Lemma 2.53. Let A, B be two self-adjoint operators in the Hilbert space X on the same
domain. If one has for all λ ∈ C\R that
(2.23)
lim (λ id − A)−1 − (λ id − B)−1 χ(||x|| ≥ R) = 0 ,
R→∞
then the following two statements are equivalent:
(i) T := (λ id − A)−1 − (λ id − B)−1 is compact.
(ii) TR := [(λ id − A)−1 − (λ id − B)−1 ] χ(||x|| < R) is compact for all R > 0.
Proof. The implication (i) ⇒ (ii) follows directly by the fact that χ is just some multiplicative bounded function, hence its product with a compact operator is again a compact
operator. Note that for this implication, (2.23) is not needed.
For the other direction, (ii) ⇒ (i), consider
||TR − T || = || (λ id − A)−1 − (λ id − B)−1 χ(||x|| ≥ R)|| → 0 for R → ∞ .
This means TR converges to T in the norm topology. By hypothesis TR is a compact
operator for all R > 0. Using the fact that K(X, Y ), the space of compact operators in
X to Y , is a closed subspace of B(X, Y ), the limit T is also compact.
Definition 2.54. A self-adjoint operator A : X → X in the Hilbert space X is called
locally compact, if the operator (λ id − A)−k χ(||x|| < R) is compact for all R > 0, some
λ ∈ C\R and some k > 0.
Lemma 2.55.
(a) If A is a locally compact, then (λ id − A)−k χ(||x|| < R) is compact for all R > 0,
for all λ ∈ C\R and for all k > 0.
(b) Both, the free and the Coulomb-perturbed Dirac operator, H0 and H = H0 + VC , are
locally compact.
Proof. We refer to the literature: (a) is proven in [Per83], a proof of (b) can be found, for
instance, in [Tha92].
Finally, we are able to prove the main claim of this section: the essential spectrum of (all
relevant) Coulomb-perturbed Dirac operators is just the same as the (essential) spectrum
of the free Dirac operator:
Theorem 2.56. Let H0 be the free Dirac operator and H = H0 + VC its perturbation by
VC = φC 14 with φC = −Z/|x| and Z < c0 /2. One has σess (H) = σess (H0 ) = σ(H0 ).
24
2.2 Relativistic Dirac Model
Proof. The idea of the proof is to use Weyl’s essential-spectrum theorem 2.52(a). It is
sufficient to prove the compactness of (i id − H)−1 − (i id − H0 )−1 .
As the proof of Theorem 2.48 shows, VC is H0 -bounded19 , hence we are allowed to use
the second resolvent equation. With this, we get for all λ ∈ C\R:
lim (λ id − H)−1 − (λ id − H0 )−1 χ(||x|| ≥ R) =
R→∞
= lim (λ id − H)−1 VC (λ id − H0 )−1 χ(||x|| ≥ R) ≤
R→∞
(2.24)
1
1
Z
≤
=0,
lim kφC χ(||x|| ≥ R)k ≤
lim sup
|Im λ|2 R→∞
|Im λ|2 R→∞ |x|≥R |x|
hence the hypothesis (2.23) holds. For the estimate we have used the identity
||(λ id − A)−1 || =
1
,
dist(λ, σ(A))
which is true for self-adjoint operators A and λ ∈ ρ(A) = C\σ(A) ⊇ C\R, compare
[Rud91].
Applying Lemma 2.55 (k = 1) yields that statement (ii) in Lemma 2.53 holds for
all λ ∈ C\R. Therefore, in the special case λ = i, Lemma 2.53 (ii)⇒(i) implies the
compactness of (i id − H)−1 − (i id − H0 )−1 and completes the proof.
2.2.3 Non-relativistic Limit and its Relativistic Corrections
We now derive the non-relativistic limit, H∞ , of the Dirac operator H = H0 + V . For
the following it is sufficient the potential V to be H0 –bounded and symmetric, which
is fulfilled for the diagonal Coulomb potential V = VC . In the following, within the
PT model, we will use this non-relativistic limit for investigating many electron-systems.
Furthermore, we state the first-order corrections of the non-relativistic eigenvalues which
we will need for our investigations of atomic spectra.
Consider the c0 -dependent Dirac operator20 H(c0 ) = c0 Q+c20 β +VC in the Hilbert space
X = L2 (R3 ) ⊗ C4 with domain D(H) = H 1 (R3 ) ⊗ C4 , where Q = −iα · ∇. In order to
get the physically-correct non-relativistic limit of the Dirac operator we need to subtract
the rest energy c20 from H(c0 ), since the rest energy does not have any correspondent in
classical non-relativistic physics. This statementp
becomes evident when considering the
relativistic energy-momentum relation, E(p) = c20 + p2 c40 : when subtracting the rest
energy, c20 , we get
E(p) −
c20
p2
p4
=
− 2 + O(c−4
0 ) → T (p) for c0 → ∞ ,
2
8c0
(2.25)
which is the classical kinetic energy T (p) of a particle with mass m = 1. The nonsubtracted E(p) diverges for c0 → ∞. However, even the operator H(c0 ) − c20 14 diverges
for c0 → ∞, hence H∞ = limc0 →∞ (H(c0 ) − c20 14 ) is an ill-defined quantity and one has
to find another way to define H∞ .
19
This fact is crucial for being able to apply the Kato-Rellich theorem which proves the self-adjointness
of the perturbed operator H.
20
In [Tha92] the setup of Dirac operators is more general: β can be substituted by some unitary involution
τ , and Q is allowed to be some (self-adjoint) supercharge with respect to τ .
25
2 Spectrum of the Hydrogen Atom
We use resolvents as an intermediate step and follow the ideas of [Ves69], elaborated
in [Tha92]. The key ingredient is the fact that the resolvent of H0 (c0 ) − c20 is analytic in
1/c0 around c0 = ∞. We have the following
Theorem 2.57. Let H(c0 ) = c0 Q + c20 β + VC be the Dirac operator in X as described
above. Let A : X → X be the subtracted operator A := H(c0 ) − c20 14 . Then, the resolvent
RA (λ) is analytic in 1/c0 around c0 = ∞ for all λ ∈ C\R with the expansion
∞
X
1
R (λ) ,
RA (λ) =
n n
c
0
n=0
(2.26)
where the right-hand side converges in the operator norm. The first two terms read
R0 (λ) = R∞ (λ)P+ = (λ id − H∞ )−1 P+ ,
1
1
R1 (λ) = P+ R∞ (λ) Q + QR∞ (λ)P+ ,
2
2
(2.27)
where we defined H∞ := 21 Q2 + VC P+ and its resolvent R∞ (λ). Moreover, P+ denotes the
(positive) projection operator, defined by
1 12 ± 12
1
0
P± := (14 ± β) =
,
(2.28)
0
12 ∓ 12
2
2
with the properties21 P+ P− = 0 = P− P+ and P±2 = P± .
Proof. We refer to [Tha92].
Corollary 2.58. We call H∞ P+ the non-relativistic limit of the Dirac operator.
Its explicit form reads
Z
1
12 .
(2.29)
H∞ P+ = − ∆ −
2
|x|
Remark 2.59. Comparing H∞ P+ with the definition of the hydrogen Hamiltonian in
(2.6), we see that this is sort of doubling the former one. The physical reason for this
is given by the spin of the electron: as a particle with spin 1/2 there are two directions
of the spin (“up” and “down”), but the energy, described by the Hamiltonian, does not
change when changing the spin direction; this is true only in this non-relativistic limit.
Introducing relativistic corrections, the energy of the system is actually dependent on the
direction of the electron spin.
Knowing the non-relativistic limit of the Dirac operator with Coulomb interaction, we are
now interested in relativistic corrections to its eigenvalues. As we know from Theorem
2.36, the hydrogen Hamiltonian features, besides the non-negative essential spectrum,
countably many isolated eigenvalues, En = −Z 2 /2n2 , n ∈ N. The dimensions of all
eigenspaces are finite: dim span Vn = n2 . Taking the spin as additional degree of freedom
into account, the eigenvalues of the non-relativistic limit H∞ P+ , are still En , but this time
their multiplicity is given by 2n2 .
21
Note that both properties hold when replacing β by some unitary involution τ .
26
2.2 Relativistic Dirac Model
Theorem 2.60. Let H(c0 ) be as above. For all eigenvalues En of H∞ P+ one has: the
operator H(c0 ) − c20 has k ≤ 2n2 distinct eigenvalues Enj (c0 ), j = 1, . . . , k, whose multiplicities sum up to 2n2 . Each Ej is analytic in 1/c0 around c0 = ∞ with the expansion
Enj (c0 ) = En +
1 j
V + O(c−4
0 ) .
c20 n
(2.30)
The Vnj , j = 1, . . . , k are eigenvalues of the self-adjoint matrix
1
Vab := hψ0a , Q(VC − En )Qψ0b i ,
4
(2.31)
where ψ0a , a = 1, . . . , 2n2 , forms an orthonormal system of eigenvectors of H∞ P+ coresponding to En .
In particular, for a non-degenerate eigenvalue22 of H∞ P+ with eigenfunction |ψ0 i, there
is only one eigenvalue of the matrix V11 :23
4
p
Z L · S πZ (3) 1
V11 = V1 = ψ0 − +
δ (x) ψ0 ,
(2.32)
+
8
2 |x|3
2
where we have defined the spin operator, S := 21 σ and the angular-momentum operator L := −ix∧p, compare Definition 3.2. Moreover, δ (3) denotes the delta distribution
in three dimensions, compare also the next Remark.
Proof. Again, we refer to [Tha92].
Remark 2.61.
(i) For a non-degenerate eigenvalue the matrix Vab in (2.32) is just one real number. Its
explicit form is a straight-forward calculation, compare [Tha92]. The delta distribution arises from the highly singular term ∆1/|x|. By the notation δ (3) (x) we target
the following important property: hψ0 |δ (3) (x)|ψ0 i = ψ0 (0), which will be important
for the explicit evaluation in Lemma 3.21.
(ii) For the degenerate eigenvalue En , n > 1, the matrix Vab , a, b = 1, . . . , n, must
be diagonalized. In particular the LS-correction will force us to change the basis
labeled with the quantum numbers (l, s), angular momentum and spin, and we
need to introduce the total angular momentum J := L + S. When discussing
relativistic corrections to the PT model, we will come back to this in detail.
(iii) It is important that√the operators in (2.32) are evaluated by some well-behaving
function |1i = Z 3/2 / π e−Z|x| . The operators (distributions) themselves are highly
singular and can not be treated isolated. The correction terms in (2.32) (in order
of their appearance) are called P-contribution, LS-coupling and Darwin term.
The first one can be motivated by expanding the relativistic energy-momentum
expansion as done in (2.25). There, the P-contribution is just the c−4
0 -term in the
expansion. The LS-coupling is some magnetic effect, induced from the orbiting electron. To get the right prefactor one has to take the Thomas precision into account.
22
We call an eigenvalue λ non-degenerate, if its eigenspace is one-dimensional, otherwise we call λ a
degenerate eigenvalue.
23
We
the “bra-ket” notation which is frequently used in physics: hf |A|gi := hf, Agi =
R use
†
f
(x)Ag(x)
dx, where A denotes some linear operator and f, g ∈ D(A).
n
R
27
2 Spectrum of the Hydrogen Atom
The Darwin term can be considered as retardation effect of the electromagnetic
field caused by the finite speed of light c0 < ∞. For further reading we refer to the
literature [Sch07] and [PS95].
Definition 2.62. Motivated from (2.32) and using the notations in Table 2.1 we define
for i = 1, . . . , 5 the following notations:
4 πZ
p
δ (3) (x) i .
and IiD := i IiP := i − i
8
2
We conclude this section with the following
Remark 2.63. In the physical literature, for instance [Sch05], the corrections in (2.32) are
derived using iteratively the Foldy-Wouthuysen transformation as introduced in (2.42).
From a mathematical point of view, this approach is only formal and cannot be justified
using operator theory. The main problem thereby is the fact that doing perturbation
theory one needs, in some sense, a small quantity. Corrections like the P-contribution,
−p4 /8, destroy the applicability of perturbation theory on the operator level. Indeed, there
are examples where the Foldy-Wouthuysen transformation diverges when taking higherorder “corrections” into account, compare [Tha92]. The approach we have followed uses
resolvents of operators as an intermediate step. This idea guarantees analyticity in 1/c0
around c0 = ∞ and the convergence of the expansion (2.26) in the operator norm.
28
3 Spectrum of Many-electron Atoms
3.1 Non-relativistic Perturbation-theory Model
Physik ist die Kunst, die passende Näherung zu finden.
Exakt rechnen kann jeder.1
Harald Friedrich
In this section we investigate atoms with more than one electron as in the hydrogen case.
In these systems there are additional interactions we have to take into account. Even at
the non-relativistic level there is the Coulomb repulsion between the iso-charged electrons
which destroys the analytically solvability of the system.
In this first step we present the perturbation-theory (PT) model, developed in [FG09b]
and [FG10], in which the interaction between the electrons can be treated perturbatively
if Z, the nuclear charge, is large. In this so-called asymptotic limit, the eigenfunctions
and their energy levels can be derived analytically. Taking these solutions as starting
point, we are going to derive in a second step the relativistic corrections of these energy
levels.
3.1.1 Definition of the PT Model
Considering the non-relativistic spectrum of a N -electron system, N ∈ N, we have to take
several aspects, some from physics, some from mathematics, into account. In the previous
chapter of this thesis we have introduced and prepared all the ingredients we bring now
together. In the following we concentrate on the discrete spectrum, σdisc (H(N )), where
the Hamiltonian H(N ) is defined as
N X
Z
1
12 + Vee 12 .
− ∆i −
H(N ) :=
2
|x
|
i
i=1
(3.1)
H is just the sum of N copies of the non-relativistic limit of the Dirac equation with
Coulomb interaction as derived in (2.29). The interaction between electrons is modeled
by Vee and in the non-relativistic limit it restricts to a purely electromagnetic term
Vee :=
X
1≤i<j≤N
1
1X
1
=
.
|xi − xj |
2 i6=j |xi − xj |
(3.2)
The spin does not occur in the Hamiltonian H(N ), hence the energies of the system,
described by the eigenvalues, are independent of the spin. However, the underlying Hilbert
space knows about the spin:
X = L2a (R3 × Z2 )N ⊗ C2 ,
(3.3)
1
Physics is the art of finding a convenient approximation. Anybody is able to calculate accurately.
29
3 Spectrum of Many-electron Atoms
where L2a denotes the anti-symmetric2 subspace of L2 , i.e. for all i, j = 1, . . . , N , i 6= j:
ψ(. . . , xi , si , . . . , xj , sj , . . .) = −ψ(. . . , xj , sj , . . . , xi , si , . . .) .
(3.4)
Lemma 3.1. Denote H(N ) the Hamiltonian of an N -electron system as defined above.
H(N ) is a linear operator in the Hilbert space X = L2a (R3 × Z2 )N ⊗ C2 with scalar
product
2 X
N Z
X
X
hψ, φiX :=
ψi∗ (xj , sj )φi (xj , sj ) dxj .
(3.5)
R3
i=1 j=1
sj =± 12
Furthermore, H(N ) is self-adjoint on the domain D(H) = (L2a ∩ H 2 )((R3 × Z2 )N ) ⊗ C2 .
Proof. Mainly, this follows from Theorem 2.35: we find directly the self-adjointness of
H(N ) − Vee 12 . The restriction to the subspace of anti-symmetric wave functions does not
affect this statement. For the self-adjointness of H(N ) itself it is sufficient to show that
Vee 12 is (H(N ) − Vee 12 )-bounded. This can be done similarly as in the proof of Theorem
2.35.
For clarity, we compare the non-relativistic hydrogen Hamiltonian, H in (2.6), with the
one-particle Hamiltonian, H(N = 1) = H∞ L+ = H12 , in detail: since 12 is just a unity
matrix, both operators feature the same eigenvalues, En = −Z 2 /2n2 , but their eigenspaces
differ. For H we know the corresponding bases: Vn , as given (2.11). Due to the spin,
which occurs as additional degree of freedom, we define the Cartesian product
Vnσ := Vn × Z2 .
σ
∈ Vnσ we have
This implies: dim span Vnσ = 2n2 . For ψnlm
1
0
σ
ψnlm (x, s) = ψnlm (x) δ 1 ,s
+ δ− 1 ,s
.
2
2
0
1
(3.6)
(3.7)
Definition 3.2. We introduce the following operators (σ = (σ1 , σ2 , σ3 ) denotes the three
Pauli matrices, listed in Appendix A.3):
S :=
N
X
S(k) :=
k=1
L :=
N
X
k=1
L(k) :=
N
X
k=1
k=1
J := L + S =
N
X
1
N
X
2
σ(k) spin angular momentum ,
−ixk ∧ ∇k angular momentum ,
(L(k) + S(k)) total angular momentum .
k=1
The index k denotes that every term in the sum acts only on the k-th electron. All
operators3 are defined on a suitable dense domain in the Hilbert space X, for instance
D(L) = {ψ ∈ X : ||(x ∧ ∇)ψ||2X < ∞} .
2
This restriction is caused by the Spin-Statistic Theorem [Pau40]. It states that elements ψ of a Hilbert
space which describe bosons are necessarily symmetric, whereas elements describing fermions are
necessarily anti-symmetric.
3
To be more precise, one should write J := L12 + S.
30
3.1 Non-relativistic Perturbation-theory Model
Definition 3.3. Denote H some Hamiltonian defined in a Hilbert space X. A linear
operator A in X is called conserved quantity, if [H, A] := HA − AH = 0, on the dense
domain D(H) ∩ D(A). Its eigenvalues are called good quantum numbers.
Remark 3.4. In physics, the eigenvalues of conserved quantities are called good quantum
numbers due to the following fact: a linear operator A is a conserved quantity if and only
if the statement that ψ ∈ D(H)∩D(A) is an eigenfunction of H with eigenvalue E implies
that Aψ is also an eigenfunction of H with the same eigenvalue. In this case, H and A
have the same system of eigenfunctions.
The following lemma states one big difference between the non-relativistic Schrödinger
theory and the relativistic Dirac theory: in a relativistic setup, the operators for angular
momentum and spin are no longer conserved quantities.
Lemma 3.5. Let be α as in Definition 2.37.
(a) For H(1) the operators L = L(1) and S = S(1) are conserved quantities.
(b) For H(N ) the operators L and S are conserved quantities.
(c) For the Dirac operator with Coulomb interaction, H = H0 + φC 14 , one finds4
[H, S(k)] = α ∧ ∇ = −[H, L(k)] 6= 0 .
Instead, the total angular momentum is a conserved quantity: [H, J(k)] = 0.
Definition 3.6. We denote by H0 (N ) := H(N ) − Vee 12 the Hamiltonian which does not
include the electron-electron interaction.5 Its ground state6 is denoted by V0 (N ). Let
P be the orthogonal projection onto V0 (N ), then the PT Hamiltonian is defined by
P H(N )P . The eigenvalue equation
P H(N )P ψ = Eψ,
ψ ∈ V0 (N ) ,
(3.8)
is called the perturbation-theory (PT) model.
Remark 3.7. The nomenclature “perturbation theory” is justified by the following consideration: Let ψ be the solution of H(N )ψ = Eψ, then the scaled function
e i , si ) := Z −3N/2 ψ(xi /Z, si ),
ψ(x
i = 1, . . . , N,
solves the eigenvalue equation
E
1
H0 (N ) + Vee ψe = 2 ψe .
Z
Z
(3.9)
(3.10)
Therefore, the PT model represents just the leading-order term one would expect from
usual perturbation theory: for large Z the interaction term Vee can be neglected and the
leading terms is governed by H0 (N ) and its eigenvalues.
4
One has to interpret the operators suitably: S(k) 7→ S(k) ⊗ 12 , and L(k) 7→ (L(k)12 ) ⊗ 12 .
Note that despite using the subscript 0, H0 (N ) does not describe a free system, since H0 (N ) still
contains the Coulomb interaction between electrons and the nucleus.
6
It is known, [Fri03] that for neutral atoms and positive ions, N ≤ Z, there exist countably many
eigenvalues Ei ∈ σdisc (H(N )) with E1 < E2 < . . . The eigenspace corresponding to the energy value
E1 is called ground state.
5
31
3 Spectrum of Many-electron Atoms
3.1.2 Principal Results
In the following theorem we collect some of the constitutive properties of the PT model,
justifying the ansatz and its nomenclature rigorously:
Theorem 3.8. Let N = 1, . . . , 10 be the fixed number of electrons, characterizing the
chemical element we are dealing with. Let n(N ) denote the number of energy levels of the
PT model, then:
(a) for all sufficiently large Z, the lowest n(N ) energy levels E1 < . . . < En(N ) , with
Ei = Ei (N, Z), of the full Hamiltonian H(N ) have exactly the same dimension,
total spin quantum number and total angular-momentum quantum number as the
PT
corresponding PT energy levels E1P T (N, Z) < . . . < En(N
) (N, Z).
(b) the lowest n(N ) energy levels of the full Hamiltonian H(N ) have the asymptotic
expansion
EjP T
1
Ej (N, Z)
as Z → ∞ .
(3.11)
= 2 +O
2
Z
Z
Z2
(c) for all j = 1, . . . , n(N ), the projections Pj onto the lowest n(N ) eigenspaces of H(N )
satisfy
1
PT
||Pj − Pj || = O
as Z → ∞ ,
(3.12)
Z2
where PjP T are the corresponding projectors for the PT model.
Proof. We refer to the original paper [FG09b].
Definition 3.9. The iso-electronic limit of some neutral chemical element X is the
sequence of ions with fixed electron number, N , but increasing nuclear charge Z → ∞.
Ions in the iso-electronic sequence containing X are called X-like.
Remark 3.10. The PT Hamiltonian, P H(N )P = P (H0 (N ) + Vee 12 )P includes still the
electron-electron interaction. The key
simplification is given by the finite dimension of
8
the ground state: dim V0 (N ) = N −2 < ∞, for 3 ≤ N ≤ 10. This fact ensures an analytic
solution of the PT model in the iso-electronic limit.
Key ingredients of the PT eigenstates are Slater determinants of the hydrogen orbitals in
Table 2.1. Due to their construction, see Definition 3.13, Slater determinants feature the
necessary antisymmetry condition (3.4) for fermions. In physics, [Sch05], and chemistry,
[Jen06], Slater determinants are used particularly as ansatz functionals for many-body
systems such as molecules. For example, the frequently used Hartree-Fock method takes
a single Slater determinant and derives, using the Rayleigh-Ritz variational principle7 ,
the ground-state eigenfunction. This method can be applied to any self-adjoint operator
which is bounded from below, compare Definition 2.45. However, as Table 3.1 shows, the
asymptotic PT ground states are not always given by just a single Slater determinant.
This effect occurring for Beryllium, Boron, and Carbon is not described by the HartreeFock ansatz as a matter of principle.
For the hydrogen orbitals we recycle the notation |ii, i = 1, 2, 3, 4, 5, from Table 2.1:
from now on, |ii denotes the corresponding state with spin “up”, whereas |īi denotes the
state with spin “down”.
7
We are going to use this method to determine the effective nuclear charge for Lithium later on.
32
3.1 Non-relativistic Perturbation-theory Model
Iso-electronic
sequence
Symmetry
Ground state
Dimension
H
2
S
|1i, |1i
2
He
1
S
|11i
1
Li
2
S
2
Be
1
S
|112i, |112i
√ 1
√1 |1133i + |1144i + |1155i
|1122i
+
c
2
3
1+c
√
√
3
with c = − 59049
(2 1509308377 − 69821) = −0.2310995 . . .
√ 1
√1 |11ijji + |11ikki
+
c
|1122ii
2
2
1+c √ 1
√1 |11ijji + |11ikki
+
c
|1122ii
2
1+c2
B
C
2
Po
3
P
for (i, j, k) = (3, 4, 5), (4, 5, 3), (5, 3, 4)
√
√
2
with c = − 393660
( 733174301809 − 809747) = −0.1670823 . . .
√ 1
2 |1122iji + c|11kkiji
1+c
√ 1
√1 |1122iji + |1122iji + c √1 |11kkiji + |11kkiji
2
2
2
1+c
√ 1
|1122iji
+
c|11kkiji
1+c2
1
6
9
for (i, j, k) = (3, 4, 5), (4, 5, 3), (5, 3, 4)
√
1
( 221876564389 − 460642) = −0.1056317 . . .
with c = − 98415
N
4
So
√1 (|1122345i
3
√1 (|1122345i
3
|1122345i
4
+ |1122345i + |1122345i)
+ |1122345i + |1122345i)
|1122345i
O
3
P
|1122iijki
√1 (|1122iijki
2
9
+ |1122iijki)
|1122iijki
for (i, j, k) = (3, 4, 5), (4, 5, 3), (5, 3, 4)
F
2
P
o
|1122iijjki
6
|1122iijjki
for (i, j, k) = (3, 4, 5), (4, 5, 3), (5, 3, 4)
Ne
1
S
|1122334455i
1
Table 3.1: Adapted from [FG09b]. Ground states of the N -electron Hamiltonian H(N ) in the isoelectronic limit, Z → ∞. The indicated wave functions are an orthonormal basis of the
ground state. The symmetry agrees with experiment for each sequence and all Z. Since L3
and S3 are good quantum numbers in the non-relativistic setup, we choose for the ground-state
wavefunction L3 = 0 and S3 to be maximal.
Definition 3.11. Let be n ∈ N and φi (xi , si ) be some distinct hydrogen orbital. Their
Slater determinant is defined by
n
Y
1 X
|φ1 (x1 , s1 ) . . . φn (xn , sn )i := √
sgn(π)
|φπ(i) (xi , si )i ,
n! π∈Sn
i=1
(3.13)
where Sn denotes the permutation group and π ∈ Sn is one of the n! permutations.
We emphasize the important message from Table 3.1 that the ground states of Beryllium,
33
3 Spectrum of Many-electron Atoms
Boron, and Carbon are not just a single Slater determinant, but some linear combination
of them. We now state one of the main results of the PT model:
Theorem 3.12. For N = 1, . . . , 10 the ground state of H(N ) has the spin, angular
momentum, and dimension as given in Table 3.1. In the iso-electronic limit, Z → ∞,
its ground state is asymptotic to the indicated vector space in the perturbative sense of
Theorem 3.8.
Proof. This is proven in [FG09b]. Here, we just outline the steps which has to be performed in order to derive this result: first of all one has to determine the ground states
V0 (N ) explicitly. A suitable choice of their bases ensures that the (finite-dimensional) matrix P H(N )P is as simple as possible: it obeys block-diagonal form and only some 2 × 2
matrices must be diagonalized additionally.8 These steps result in an analytic expression
for the asymptotic eigenstates and energy values.
Remark 3.13. In Table 3.1 we have used the spectroscopic notation for describing the
symmetries of the ground state. In general, 2S+1 X ν has the following interpretation: S
denotes the total spin. The angular momentum, L, corresponds to X via 0 ↔ S, 1 ↔ P ,
3 ↔ D, . . . The superscript ν relates to the parity of the state, p = ±1:
ψ(xi , si ) = (−1)p ψ(−xi , si ), simultaneously for all i = 1, . . . , n .
(3.14)
If the parity of ψ is odd, i.e. p = −1, the superscript is set to ν = 0. Otherwise, for an
even ψ, p = 1, we suppress the superscript ν.
Example 3.14. The ground-state symmetry of N, N = 7, is labeled by 4 S 0 . This means
total spin S = 3/2, total angular momentum L = 0 and odd parity. The ground state
of Oxygen, N = 8, obeys the symmetry 3 P , meaning total spin S = 1, total angular
momentum L = 1 and even parity.
3.1.3 Energy Levels and Spectral Gaps
Knowing the energy levels of the asymptotic states within the PT model, one can consider
differences between them. In order to determine all energy levels, as done in [FG09b], we
need the following notation:
Definition 3.15. Denote a, b, c, d ∈ {1, . . . , 5} some hydrogen orbital as in Table 2.1. We
define
(a|b) := ha|H(N = 1)|bi ,
Z
(3.15)
1
|c(x2 )i|d(x2 )i dx1 dx2 .
(ab|cd) :=
ha(x1 )|hb(x1 )|
|x1 − x2 |
R6
In the literature (ab|cd) is called the Coulomb integral.
Lemma 3.16. With the introduced notation we have:
(a) (a|b) = δab En with n = 1 for |ai = |1i, |1̄i, and n = 2 in the other cases.
(b) For (ab|cd) the following table holds:
8
(11|11)
(11|22)
(12|21)
(22|22)
(11|33)
(13|31)
(22|33)
(23|32)
(33|33)
(33|44)
(34|43)
5
Z
8
17
Z
81
16
Z
729
77
Z
512
59
Z
243
112
Z
6561
83
Z
512
15
Z
512
501
Z
2560
447
Z
2560
27
Z
2560
This is the reason why the eye-catching square roots in Table 3.1 come into play.
34
3.1 Non-relativistic Perturbation-theory Model
Proof. This can be derived using Fourier analysis. One needs to know the Fourier transforms of the functions the hydrogen orbitals consist of. The detailed proof is elaborated
in [FG09b].
Example 3.17. To get used to the very compact notations we now show an explanatory
calculation: we derive the energy of the asymptotic ground state of Lithium, |ψi = |11̄2i,
given in Table 3.1:
hψ|H(N = 3)|ψi =
+
*
3 X
X
Z
1
1
12 1(x1 )1̄(x2 )2(x3 ) =
12 +
− ∆i −
= 1(x1 )1̄(x2 )2(x3 ) 2
|xi |
|xi − xj | i=1
1≤i<j≤3
= 3h1(x1 )1̄(x2 )2(x3 )|H(1)|1(x1 )1̄(x2 )2(x3 )i + h1(x1 )1̄(x2 )2(x3 )|Vee 12 |1(x1 )1̄(x2 )2(x3 )i =
9
= 2E1 + E2 + hVee i = − Z 2 + hVee i ,
8
where we have used (a|b) = δab En . Using the even more compact notation 11 1̄2 23 for both,
|1(x1 )1̄(x2 )2(x3 )i and h1(x1 )1̄(x2 )2(x3 )|, the interaction between the electrons reads:
*
+
X
12 hVee i = 1(x1 )1̄(x2 )2(x3 ) 1(x1 )1̄(x2 )2(x3 ) =
|xi − xj | 1≤i<j≤3
Z
3
12
=
(11 1̄2 23 − 12 1̄1 23 + 12 1̄3 21 − 13 1̄2 21 + 13 1̄1 22 − 11 1̄3 22 )
×
3! R9
|x1 − x2 |
× (11 1̄2 23 − 12 1̄1 23 + 12 1̄3 21 − 13 1̄2 21 + 13 1̄1 22 − 11 1̄3 22 ) dx1 dx2 dx3 =
Z 1
1
2
2
2
=
(11 1̄2 − 12 1̄1 ) + (12 21 − 12 22 ) + (1̄1 22 − 1̄2 21 )
dx1 dx2 =
2 R6
|x1 − x2 |
Z
1
((11 11 1̄2 1̄2 + 1̄1 1̄1 12 12 ) + (21 21 12 12 − 2 · 11 21 22 12 + 11 11 22 22 ) +
=
2 R6
1
+ (1̄1 1̄1 22 22 + 21 21 1̄1 1̄1 ))
dx1 dx2 = (11|11) + 2(12|21) − (12|21) .
|x1 − x2 |
Note that the Coulomb integral is independent of the spin in the following meaning:
(āā|bb) = (aa|bb) and (aa|b̄b̄) = (aa|bb). Terms like (aā|bb̄) vanish, since the states with
different spin are orthogonal. Using the explicit expressions of the Coulomb integral in
Lemma 3.16(b), the energy of the asymptotic ground state of Lithium reads
5965
9
h11̄2|H(N = 3)|11̄2i = − Z 2 +
Z.
8
5832
(3.16)
The contribution to the energy coming from the interaction between electrons and nucleus,
H0 (N ), can be derived in the general setup of Slater determinants, and, therefore, for all
asymptotic states of the PT model:
Lemma 3.18. Denote φi , i = 1, . . . , N , some distinct hydrogen orbitals. One has
hφ1 (x1 ) . . . φN (xN )|H0 (N )|φ1 (x1 ) . . . φN (xN )i =
N
X
En(i) ,
i=1
where we use again n(i) = 1 for |φi i = |1i and n(i) = 2 in the other cases.
35
3 Spectrum of Many-electron Atoms
Proof. We have to replicate the first part of Example 3.17. For all N ∈ N one gets:
hφ1 (x1 ) . . . φN (xN )|H0 (N )|φ1 (x1 ) . . . φN (xN )i =
*
+
N
N
N X
Y
Y
1 X
sgn(π)sgn(σ)
φπ(i) (xi ) H(1)k
=
φσ(j) (xj ) =
N ! π,σ∈S
i=1
j=1
k=1
N
N
*
+
N
N X
Y
Y
1 X
=
sgn(π)sgn(σ)
φπ(i) (xi ) En(σ(k)) φσ(j) (j) =
N!
i=1
π,σ∈SN
=
j=1
k=1
N
X
N
Y
1 X
hφπ(i) (xi )|φσ(j) (xj )i =
sgn(π)sgn(σ)
En(σ(k))
N ! π,σ∈S
i,j=1
k=1
N
{z
}
|
=δπ,σ
=
1 X
N ! π∈S
N
N
X
k=1
En(π(k)) =
N
X
En(k) .
k=1
In the last step we have used that SN contains N ! elements, which completes the proof.
The structure of the energy of the asymptotic ground state of Li, |11̄2i, in (3.16) holds
generally in the PT model. Combing Lemma 3.16 and Lemma 3.18, we have the general
form of an energy state in the PT model:
E = −|A|Z 2 + BZ ,
(3.17)
where A = A(N ) ∈ Q is some negative rational number describing the interaction between
electrons and nucleus. For a fixed N = 3, . . . , 10 it is a constant, characteristic for each
chemical element. Besides this, B ∈ R, containing the Coulomb integral, describes the
electron-electron interaction. For all elements of the second shell, B is some positive real
number, but it depends on the state: ground state (GS or E0), first excited state (E1),
second excited state (E2), . . . lead to different values for B. These statements should be
compared to the asymptotic states, listed in [FG09b] or in the summary chapter 4.
As a direct consequence of (3.17), the energy gaps within one chemical element, Ei (Z)−
Ej (Z), are linear in Z. Due to our perturbative ansatz, we expect our results to agree
with experimental results for large Z → ∞. Therefore, it is convenient to consider the
spectral gaps divided by Z 2 , since this form reads
Ei (Z) − Ej (Z)
= (Bi − Bj )x ,
Z2
(3.18)
which is a straight line in x := 1/Z. Typically, these scaled energies are some meV. We
expect a good agreement with experimental data for small x → 0 (⇔ Z → ∞). Compare
this to FIG.4(a)-(g) in [FG10].
We derive explicitly the gap between the ground state of Lithium, |11̄2i, and its first
excited state |11̄3i. As done for deriving (3.16) one finds similarly for the first excited
state: hVee i = (11|11) + 2(11|33) − (13|31). The energy term coming from H0 (N = 3)
is indeed the same as for the ground state. Hence, with the results of Lemma 3.16, the
energy of the first excited state of Lithium reads:
9
57397
h11̄3|H(N = 3)|11̄3i = − Z 2 +
Z.
8
52488
36
(3.19)
3.1 Non-relativistic Perturbation-theory Model
Figure 3.1: Spectral gap between the ground state (GS) and first excited state (E1) of Lithium, where we
have averaged, by multiplicity, over J. The experimental data points “converge” for x . 0.20
to the straight line of the PT model. For x . 0.05, the plot shows the relativistic deviation
from this straight line which is the reason and motivation for this diploma thesis.
Therefore, the scaled gap between ground state and first excited state of Li reads:
E1 (Z) − E0 (Z)
57397 5965 1
464 1
=
−
=
.
Z2
52488 5832 Z
6561 Z
The comparison between this theoretical PT result and experimental data9 is shown
in Figure 3.1: we learn that the straight line predicted by the PT model explains the
experimental values accurately for 0.05 . x . 0.10, hence for 10 . Z . 20. On the
one hand one expects that the PT model works well if Z is “large enough”, Z & 10,
but on the other hand a large Z strengthens the Coulomb potential of the nucleus. This
implies a tighter binding between electrons and nucleus, hence an increase of the typical
velocity of the electrons. Therefore, if Z it “too large”, Z & 20, relativistic effects come
significantly into play. Since, in the sense of Theorem 3.8, the PT model is convergent in
the asymptotic limit, Z → ∞, it is necessary to investigate these relativistic effects.
9
All experimental data are taken from the NIST atomic-spectra database [Yu.10].
37
3 Spectrum of Many-electron Atoms
3.2 Relativistic Effects in Asymptotic PT States
Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche
Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist.10
Immanuel Kant
Definition 3.19. The relativistic Hamiltonian describing an N -electron system is
defined by11
1
(3.20)
Hrel (N ) := H(N ) + 2 H P (N ) + H D (N ) + H LS (N ) ,
c0
with the operators in Theorem 3.23 and Theorem 3.29.
In the following we use the notation IiX , i = 1, . . . , 5 and X = P, D as introduced in
Definition 2.62. The corresponding state with opposite spin is denoted by |īi.
3.2.1 P-contribution and Darwin Term
D E
D E
1 Lemma 3.20. The integrals i |x|k i = ī |x|1k ī , for i = 1, . . . , 5 and k = 1, 2, 3,
evaluate to the following table:
H
HH
k
i
1
2
3, 4, 5
1
Z
1
Z
4
1
Z
4
2
2Z 2
1 2
Z
4
1
Z2
12
3
∞
∞
1
Z3
24
HH
H
Proof. It is clear that
the integrals are independent of the spin direction, since |īi = A|ii,
0 1
where A =
. We find immediately:
1 0
1 T 1 1 ī k ī = i A
A i = i k i ,
|x|
|x|k |x|
where we have used AT A = A2 = 12 . We show only the case k = 1 explicitly:
Z
3
3 Z ∞
1
Z
1
4πZ
−2Z|x|
1 1 =
e
dx =
r e−2Zr dr = Z ,
r
π
|x|
π
3
R
|0
{z
}
=1/4Z 2
Z
2
3
1
Z
Z|x|
1
2 2 =
1−
e−Z|x| dx =
r
2
R3 8π |x|
3 Z ∞
Z 2 r3 −Zr
4π · Z
Z
2
r − Zr +
=
e
dr = ,
8π
4
4
|0
{z
}
=1/2Z 2
10
In every department of physical science there is only so much science, properly so-called, as there is
mathematics.
11
The overall pulled-out prefactor c−2
0 is due to Theorem 2.60. It ensures the correct physical unit of the
correction terms.
38
3.2 Relativistic Effects in Asymptotic PT States
Z
Z
1
4π · Z 5 ∞ 3 −Zr
Z
1
Z5 1
2
2
2
−Z|x|
(x3 + x1 + x2 ) e
dx =
3 3 =
r e
dr = .
{z
}
r
3 R3 32π |x| |
3 · 32π 0
4
{z
}
|
=|x|2
=6/Z 4
From the last line it is clear that the results for i = 3, 4, 5 coincide.
Lemma 3.21. The integrals IiX , for i = 1, . . . , 5 and X = P, D, are independent of the
spin-direction: IīX = IiX . They evaluate to the following table:
i
1
2
3, 4, 5
IiP
− 58 Z 4
13
− 128
Z4
7
− 384
Z4
1 4
Z
2
IiD
1
Z4
16
0
Proof. We start with X = P: denoting H = − 12 ∆ − Z/|x| the hydrogen Hamiltonian
(2.6), we know from Theorem 2.36 and Table 2.1 that |ii are eigenfunctions of H with
eigenvalue En , where n = 1 for i = 1 and n = 2 for the other cases i = 2, . . . , 5. We get
for IiP :
* 4 2 +
p 1 1 1
Z
1
2
2
i − i = −
i H +
.
En + 2ZEn i i + Z i 2 i
i = −
8
2
|x| 2
|x|
|x|
With the results of Lemma 3.20, k = 1, 2, the claim for X = P follows.
For X = D we have to deal with the delta-distribution δ(·), evaluating the eigenfunction
|ii at the origin. Inspecting Table 2.1 implies the vanishing terms for i = 3, 4, 5. For
i = 1, 2 we get for IiD :

 Z4 , i = 1 ,
πZ 3 πZ
2
δ (x) i =
|ψi00 (x = 0)|2 =
i Z
 4, i = 2 .
2
2
16
Any asymptotic state in the PT model is some real linear combination of Slater determinants. Considering the iso-electronic sequence which contains the neutral atom with
N = 3, . . . , 10 electrons, the general asymptotic state has the form
∗
|ψi =
N
X
k=1
(k)
(k)
αk |φ1 (x1 , s1 ) . . . φN (xN , sN )i ,
(3.21)
where N ∗ ∈ N denotes the number of Slater determinants contributing to the state and
(k)
αi ∈ R are the coordinates12 of the state. The φi ∈ Vnσ are hydrogen orbitals with spin,
as introduced in (3.6).
Example 3.22. For clarity, we consider the following two ground states:
(i) Lithium: |ψi = |11̄2i. This means: N = 3, N ∗ = 1, α1 = 1 and for the hydrogen
(1)
(1)
(1)
orbitals: φ1 = |1i, φ2 = |1̄i, φ3 = |2i.
12
Note that due to normalization of the asymptotic states one has
PN ∗
k=1
αk2 = 1.
39
3 Spectrum of Many-electron Atoms
(ii) Beryllium: |ψi =
√ 1
1+c2
√c
3
∗
|11̄22̄i +
(|11̄33̄i + |11̄44̄i + |11̄55̄i) . This more com√ 1 ,
1+c2
plicated state implies: N = 4, N = 4, α1 =
(k)
√ √c
3 1+c2
(1)
|2i, φ4 = |2̄i,
α2 = α3 = α4 =
(k)
(1)
and φ1 = |1i, φ2 = |1̄i for all k = 1, . . . , 4. Furthermore φ3 =
(2)
(2)
(3)
(3)
(4)
(4)
φ3 = |3i, φ4 = |3̄i, φ3 = |4i, φ4 = |4̄i and φ3 = |5i, φ4 = |5̄i.
Theorem 3.23. Let be X = P, D with the corresponding operators13 :
P
H (N ) :=
N
X
HiP
i=1
N
X
p4
i
=−
i=1
8
D
and H (N ) :=
N
X
HiD
i=1
N
πZ X (3)
=
δ (xi ) ,
2 i=1
(3.22)
where the subscript i tells us that the operator HiX acts only on the i-th electron. Then,
the relativistic correction to the general asymptotic state, |ψi in (3.21), is given by
∗
X
hψ|H (N )|ψi =
N
X
αk2
N
X
IX(k) E .
i=1
k=1
(3.23)
φi
Proof. Using the orthogonality of the hydrogen orbitals, hi|ji = δij , we get:
∗
X
hψ|H (N )|ψi =
∗
N
N X
X
k=1 l=1
(k)
(k)
(l)
(l)
αk αl hφ1 (x1 ) . . . φN (xN )|H X |φ1 (x1 ) . . . φN (xN )i =
∗
=
N
X
k=1
(k)
(k)
(k)
(k)
αk2 hφ1 (x1 ) . . . φN (xN )|H X |φ1 (x1 ) . . . φN (xN )i =
*
+
N∗
N
N Y
Y
N X 2 X
(k)
(k)
α
sgn(π)sgn(σ)
ψπ(i) (xi ) H1X
=
φσ(j) (xj ) .
N ! k=1 k π,σ∈S
i=1
j=1
N
{z
}
|
=δπ,σ I X (k)
|φπ(1) i
P
X
Besides the orthogonality we have used that all terms of H X = N
i=1 Hi contribute in the
same way, hence the global prefactor N has arosen. Now, δπ,σ removes one of the sums
P
over the permutation group. Inserting a decomposition of the identity, 1 = N
i=1 δi,π(1) ,
restricts the π-sum to the subgroup SN −1 , hence we have
∗
N
X
X
1
hψ|H (N )|ψi =
αk2
IX(k) E =
(N − 1)! k=1 π∈S φπ(1)
X
N
1
=
(N − 1)!
N∗
X
k=1
αk2
N
X
∗
X
i=1 π∈SN
|
δi,π(1) IX(k) E
φi
{z
π∈SN −1
=
N
X
k=1
αk2
N
X
i=1
IX(k) E .
φi
}
In the last step we have used that IX (k) E is independent of the permutation π.
φi
13
These operators are direct N -electron generalizations of the correction terms in Definition 2.62. For
N = 1 they restore to hi|H X |ii = IiX .
40
3.2 Relativistic Effects in Asymptotic PT States
3.2.2 Spin-orbit Coupling
Now we investigate the relativistic correction induced by the spin: the LS-coupling. For
the beginning we restrict to the one-particle operators L(k) and S(k). We know already
that in a relativistic setup the operators L(k) and S(k) are no conserved quantities,
Lemma 3.5(b), hence they are no longer appropriate. Instead, we have introduced the
total angular momentum J(k) = L(k) + S(k) which is a good quantum number for the
Dirac operator. In the non-relativistic setup, additionally L3 (k) and S3 (k) commute with
the Hamiltonian. Since we are restricted to the second shell, and |1i and |2i feature
l = 0, we need to consider only hydrogen orbitals with l = 1, i.e. |ii for i = 3, 4, 5. The
σ
σ
σ
σ
.
eigenvalue equations read L2 ψnlm
= l(l + 1)ψnlm
and S 2 ψnlm
= 43 ψnlm
2
σ
In addition, due to [Si (k), S (k)] = 0 for all i = 1, 2, 3, ψnlm are also eigenfunctions
of one 14 component of S(k). This statement is also true for the angular-momentum
σ
operator: one component of L(k) is diagonal in the basis formed by ψnlm
. We follow the
usual convention and choose L3 and S3 to be the diagonal ones. Also for these operators,
σ
σ
, for m = −l, . . . , l, compare (A.20), and
= mψnlm
we know the eigenvalues: L3 ψnlm
1 σ
σ
S3 ψnlm = ± 2 ψnlm .
Since electrons have spin 1/2, the Clebsch-Gordan decomposition yields (for l ≥ 1) only
two components: j = l − 12 and j = l + 21 . The eigenfunction of J 2 (k) are denoted by |j±i.
Note that the multiplicities of j = l − 21 and j = l + 21 sum up to 2(l − 12 )+1+2(l + 21 )+1 =
4l + 2, as expected from the multiplicity of the product state of spin 12 and angular
momentum l: 2 · (2l + 1) = 4l + 2.
Lemma 3.24. For the two one-particle states |j±i the following eigenvalue equations
hold:
l+1
l
|j−i .
(3.24)
L · S|j+i = |j+i , and L · S|j−i = −
2
2
Proof. Knowing from the above discussion that |j±i are eigenstates of J 2 , L2 and S 2 , we
get directly:
1 2
J − L2 − S 2 |j±i =
2


1
1
3
1
l±
l ± + 1 − l(l + 1) −
|j±i =
=
 −
2
2
2
4
L · S|j±i =
l
|j+i,
2
l+1
|j−i,
2
j =l+
1
2
,
j =l−
1
2
.
From this we are able to derive the energy corrections for the two possible J-states when
coupling the spin of one electron to its angular momentum:
Lemma 3.25. Let be n = 2, l = 1 and i = 3, 4, 5. We define |i±i := |i, j = l ± 12 i. With
this, we derive


1
Z 4 , j = 32 ,
Z L · S 96
I± := i ± i± =
 − 1 Z 4, j = 1 .
2 |x|3 48
2
14
In fact, since the commutator of the Pauli matrices σi is non trivial, [σi , σj ] = 2iεijk σk , it is impossible
to find a basis in which all three components of the angular-momentum operator S(k) are diagonal.
Compare Appendix A.3, where we define the totally anti-symmetric ε tensor.
41
3 Spectrum of Many-electron Atoms
Proof. We only have to combine Lemma 3.21 and Lemma 3.24. We consider only the
state |i+i for i = 3, 4, 5:
1 1
l
lZ 3
i + 3 i+ =
.
I+ = i + 3 L · S i+ =
|x|
2
|x|
48
Multiplying by Z/2 yields for l = 1 our claim. Similarly the claim follows for |i−i.
When considering an N -electron state with angular momentum L and spin S, the ClebschGordan decomposition contains L + S − |L − S| + 1 terms. The total angular momentum
J can be J = |L−S|, |L−S|+1, . . . , L+S, which is known as the triangular condition.
Note that for a one-particle state, angular momentum l = 1 and spin 1/2, the two terms
j = 1/2 and j = 3/2, appear as a special case. We introduce the following notation:
Definition 3.26.
(i) N + denotes the number of electrons with spin “up”, N − those with spin “down”:
N ± = 0, . . . , N,
with N + + N − = N .
(ii) For a general asymptotic state, |ψi in (3.21), we denote by Nk∗ , k = 1, . . . , N ∗ , the
numbers of l 6= 0−orbitals in the k-th term:
n
o
(k)
(k)
∗
Nk := # |ii, |īi, i = 3, 4, 5, in |φ1 . . . φN i .
Example 3.27. For the two states in Example 3.22 we have for Li: N1∗ = 1. For Be we
have: N1∗ = 0 and N2∗ = N3∗ = N4∗ = 2.
Remark 3.28. For a given angular momentum, L = 0, 1, 2, . . ., and total angular momentum, J = 0, 12 , 1, 32 , 2, . . ., of an N -electron state, the spin configuration must obey:
1
1
N
!
J = |L + N + − N − | = |L +
− N − | , for N − = 1, . . . , N − 1 .
2
2
2
(3.25)
Due to the closed first shell, n = 1 with |1i and |1̄i, the number of electrons with “spin up”
and “spin down” is at least one in both cases. Since the PT model restricts to the second
shell, N = 3, . . . , 10, we have always N ± = 1, 2, . . . , N − 1. However, it will turn out
that there are for some combinations of L, J and N two possible values for N − satisfying
(3.25), particularly for Carbon, Nitrogen, Oxygen and Fluorine.
Corresponding to Theorem 3.23 for X = P, D, we are now able to state in Theorem 3.29
the relativistic corrections which are induced by the spin-orbit coupling. The operator
HiLS defined therein takes only the diagonal term of the whole LS-correction into account:
!
!
N
N
N
X
X
X
X
L·S =
L(k) ·
S(k) =
L(k) · S(k) +
L(k) · S(k 0 ) . (3.26)
k=1
k=1
k=1
1≤k<k0 ≤N
We justify the restriction to the diagonal contributions as follows: the ground states
the relativistic corrections are derived for are given in Table 3.1, the first excited states
can be found in [FG09b]. All these PT states are derived using perturbation theory
with respect to the Coulomb interaction between different electrons. The asymptotic
42
3.2 Relativistic Effects in Asymptotic PT States
ground states become exact solutions of the Schrödinger equation for Z → ∞, where the
electron-nucleus interaction becomes more and more dominant. Therefore it is consistent
to take off also the electromagnetic interaction between different electrons when discussing
relativistic effects of the asymptotic states. By the local model we denote the restriction
to the diagonal term of L·S and tracing back the spin-orbit coupling to each single electron
and its own j-state. Compare this to the discussion in section 3.6 and also Remark 3.32.
Theorem 3.29. Let H LS (N ) denote the N -particle spin-orbit operator, defined by15
H
LS
(N ) :=
N
X
i=1
HiLS
N
Z X L(i) · S(i)
,
=
2 i=1
|xi |3
(3.27)
where the subscript i tells us that the operator HiLS acts only on the i-th electron. Let
the general asymptotic state, |ψi in (3.21), have the total angular momentum J, denoted
by |ψiJ . The spin structure is described by the two parameters N ± and the coefficients
Nk∗ . Then, for the chemical elements in the second shell, N = 3, . . . , 10, the relativistic
correction induced by the local spin-orbit coupling reads:
∗
hψ|H
LS
N
Z 4 N + 1 − 3N − X 2 ∗
αk Nk .
(N )|ψiJ =
96
N −2
k=1
(3.28)
Proof. As in the proof of Theorem 3.23 we use the orthogonality of the hydrogen states,
hence the orthogonality of the J 2 -eigenstates |i±i:
∗
hψ|H
LS
(N )|ψiJ =
N
X
αk2
N
X
i=1
k=1
1
(k) Z L·S (k)
(k) Z L·S (k)
−
+
.
φ + +N φi − φ −
N φi + N −2
2 |x|3 i
2 |x|3 i
From Lemma 3.25 we know that only the l 6= 0−orbitals have a non-trivial contribution.
Additionally, this contribution is independent of the explicit state |i±i, i = 3, 4, 5, hence
we find Nk∗ identical terms:
∗
hψ|H
LS
(N )|ψiJ =
N
X
k=1
(N
αk2 Nk∗
+
∗
N
− 1)I+ + (N − − 1)I−
Z 4 N + 1 − 3N − X 2 ∗
=
αk Nk ,
N −2
96
N −2
k=1
where we have used the explicit expressions for I±LS and I−LS = −2I+LS . Note that the
terms (N ± − 1)/(N − 2) appear due to the closed first shell: for all N = 3, . . . , 10, there
are two electrons in the first shell, |1i and |1̄i, with opposite spin direction. The other
N − 2 electron spins together with their angular momenta have to sum up to the total
angular momentum J.
Corollary 3.30. The sign of the LS-correction (3.28) depends only on N ± :
+
N +1
LS
−
sgn hψ|H (N )|ψiJ = sgn N + 1 − 3N = sgn
−2 .
N−
(3.29)
Particularly, the spin does not affect the energy levels ⇔ 3N − = N + 1.
15
P
Note that the operator i L(i)S(i) commutes with both the non-relativistic N -particle Hamiltonian,
H(N ), and the PT-Hamiltonian, P H(N )P .
43
3 Spectrum of Many-electron Atoms
Remark 3.31. Since we are restricted to the second shell, N = 3, . . . , 10, only for energy
states of Boron, N = 5, or Oxygen, N = 8, it is possible that the spin-orbit coupling
implies finally no relativistic correction.
3.2.3 Relativistic Energy Levels, Spectral Gaps and Splitting
We conclude briefly the general form of an energy state in the relativistic PT
model: combining Theorem 3.23 and Theorem 3.29, we find the relativistic generalization
of (3.17):
C
(3.30)
E = −|A|Z 2 + BZ + 2 Z 4 ,
c0
with A and B are as above and C ∈ Q. The relativistic parameter C can be both, positive
or negative. (3.30) implies a deviation also of the Z 2 -scaled spectral gaps (3.18):
J
1
EiJi (Z) − Ej j (Z)
1 Ji
∆C 1
Jj
C
−
C
=
(B
−
B
)x
+
=: ∆Bx + 2 2 ,
i
j
i
j
2
2
2
Z
c0
x
c0 x
(3.31)
where we have defined x := 1/Z. The J-independent constant ∆B is known from the nonrelativistic PT model, whereas ∆C must be derived using the results we have prepared
in this section. If ∆C > 0, then the straight line predicted by the non-relativistic PT
model is shifted upwards for x → 0 (⇔ Z → ∞). In the other case, ∆C < 0, the shift is
downwards.
Additionally, the relativistic corrections feature a splitting of the asymptotic PT states:
in general, different values of J cause different shifts of the energy states. It is convenient
to consider the splittings divided by Z 5 :
∆C
E Ji (Z) − E Jj (Z)
= 2 x,
5
Z
c0
(3.32)
which is again a straight line in x = 1/Z. Typically, due to the high power of Z, these
scaled energies are only a few µeV.
Remark 3.32. The form of the energy E(Z) in (3.30) shows that the relativistic correction, ∼ Z 4 , becomes the dominant contribution to the energy in the asymptotic limit
Z → ∞. But this is the limit we need the asyptotic states to be close to the Schrödinger
states. For the comparison to experimental data this implies that one can expect to find
an agreement to the local model for large, but not too large values of Z. Since there is
only a finite number of ions in each iso-electronic sequence experimentally available, the
precedure we are doing is still reasonable. However, for Z → ∞ perturbation theory at
first order in the relativistic sector is physically not sufficient: higher order corrections
become more important since they feature an even higher power of Z.
44
3.3 Relativistic Corrections to Lithium
3.3 Relativistic Corrections to Lithium
Die Physik erklärt die Geheimnisse der Natur nicht,
sie führt sie auf tieferliegende Geheimnisse zurück.16
Carl Friedrich von Weizsäcker
In this section we derive in detail the relativistic corrections to the asymptotic states of
Lithium-like ions17 , N = 3, of the PT model. We will apply the general considerations
of the local model to the ground state, ψ0 = |11̄2i, and first excited state, ψ1 = |11̄3i,
derived in [FG09b].
For the energy corrections18 we use the following notation: ∆EiJ denotes the total
correction of the ground state, i = 0, or first excited state, i = 1, with total angular
momentum J. If there is only one possible J, i.e. L or S vanish, we suppress the
superscript J. The J-independent corrections, P-contribution and Darwin term, sum up
to ∆EiP+D := ∆EiP + ∆EiD . The J-dependent spin-orbit correction is denoted by ∆EiLS,J .
3.3.1 Shifted Energy Levels
We start with the asymptotic ground state of Li: ψ0 = |11̄2i with L = 0, S = 1/2, hence
J = 1/2. Theorem 3.23 tells us how its relativistic correction due to the P-contribution
and the Darwin term reads:
∆E0P+D = 2I1P + I2P + 2I1D + I2D = −
37 4
Z ,
128
(3.33)
where we have used the explicit expression of Lemma 3.21. Due to L = 0, particularly
N1∗ = 0, the ground state does not have any spin-correction, hence ∆E0 = ∆E0P+D .
The first excited state of Li, |11̄3i has L = 1 and S = 1/2, hence there are two possible
total angular momenta: J = 1/2, 3/2. The J-independent corrections reads:
∆E1P+D = 2I1P + I3P + 2I1D = −
103 4
Z .
384
(3.34)
For the spin-orbit corrections, we apply Theorem 3.29. The spin structure, described by
N ± , must obey (3.25): for J = 1/2 we get N + = 1 and N − = 2, for J = 3/2 we get
N + = 2 and N − = 1. This yields
LS,1/2
∆E1
=−
1 4
1
LS,3/2
Z and ∆E1
= Z4 .
48
96
(3.35)
Proposition 3.33. The relativistic corrections to the ground state, |11̄2i, and first excited
state, |11̄3i, of Lithium are given by:
∆E0 = −
37 4
37 4
33 4
1/2
3/2
Z , ∆E1 = −
Z , ∆E1 = −
Z .
128
128
128
1/2
In particular one has the degeneracy ∆E0 = ∆E1 . The shifted energy levels are sketched
in Figure 3.2, and the comparison to experimental data is shown in Figure 3.3.
16
Physics does not explain the secrets of nature, it traces nature back to more fundamental secrets.
We will drop this cumbersome notation and denote it just by the term “Lithium”.
18
Note that there is a global prefactor of c−2
0 coming from the definition of the relativistic Hamiltonian
(3.20). When evaluating to numbers (in atomic units), we need to divide our results by c20 ≈ 1372 .
17
45
3 Spectrum of Many-electron Atoms
Li-E1 : |11̄3
3/2
ΔE1
J = 3/2
1/2
J = 1/2
ΔE1
Li-GS : |11̄2
ΔE0
J = 1/2
Figure 3.2: Relativistic corrections to the ground state (GS) and first excited state (E1) of Lithium,
Proposition 3.33. The PT states are shifted downwards. The red arrows indicate the degen1/2
eracy of ∆E0 and ∆E1 . Therefore, the spectral gap between these two states is expected
to be described by the non-relativistic PT model: ∆C = 0 in (3.31), see also Figure 3.3. This
sort of degeneracy is also present in the hydrogen atom, compare Proposition 3.35.
1/2
Remark 3.34. The degeneracy ∆E0 = ∆E1 provides an excellent check for our results: if two states feature the same relativistic corrections, the general form of the spectral gap, (3.31), implies ∆C = 0, hence their spectral gap should be described by the
non-relativistic PT model. Indeed, the experimental data affirm this theoretical result,
compare Figure 3.3, where the slope of the straight line is the same as derived in (3.1.3):
∆B = 464/6561. However, such a degeneracy between relativistically shifted states is
well-known in the hydrogen atom. Our model includes also the hydrogen degeneracy:
Proposition 3.35. Consider the hydrogen atom, N = 1, within the relativistic PT model.
Its ground state, first excited and second excited state, |1i, |2i and |3i, respectively, feature
the following relativistic corrections:
5 4
1 4
5 4
1
1/2
3/2
Z , ∆E2 = −
Z , ∆E2 = −
Z .
∆E0 = − Z 4 , ∆E1 = −
8
128
128
128
1/2
Particulary, we find the well-known19 degeneracy ∆E1 = ∆E2 .
Proof. We only need the explicit results of Lemma 3.21 and Lemma 3.25:
1/2
∆E0 = I1P + I1D , ∆E1 = I2P + I2D , ∆E2
3/2
= I3P + I− , ∆E2
= I3P + I+ .
1/2
Note that the terms contributing to the degenerate corrections ∆E1 and ∆E2
differ.
Corollary 3.36. Only the first excited state of Li features a relativistic splitting. From
Proposition 3.33 one finds immediately ∆C = 1/32, hence
3/2
∆E1
1/2
− ∆E1
Z5
=
1 1
x,
c20 32
(3.36)
again with x = 1/Z. The comparison to experimental data is shown in Figure 3.4.
19
In physics it is well-known, for instance [Sch05], that within the theory of Dirac hydrogen, the here
considered two relativistic corrections are degenerate in all orders of perturbation theory in the finestructure constant α0 = c−1
0 .
46
3.3 Relativistic Corrections to Lithium
Figure 3.3: Spectral gap between the ground state and first excited states of Lithium including relativistic
effects. The experimental data points for J = 1/2 “converge” to the straight line of the
PT model. This represents the degeneracy of the corresponding relativistic corrections in
Proposition 3.33. For J = 3/2 the data points “converge” to the result of the relativistic PT
model. It is instructive to compare this plot to Figure 3.1.
Figure 3.4: Splitting of the first excited state of Lithium J = 1/2 and J = 3/2: for decreasing x, i.e.
increasing Z, the agreement between our theoretical result and the experimental data points
becomes better. Therefore, in an asymptotic sense, the relativistic PT model can explain
qualitatively the splitting, but quantitatively the actual theoretical depth is not satisfactory.
See also the result of the effective PT model in Figure 3.5.
47
3 Spectrum of Many-electron Atoms
3.3.2 Effective Nuclear Charge
The splitting of asymptotic PT states for different values of the total angular momentum
J is a purely relativistic effect coming from the spin-orbit coupling. The P-contribution
and the Darwin do not affect the splitting, since
E Ji (Z) − E Jj (Z) = ∆E LS,Ji (Z) − ∆E LS,Jj (Z) .
(3.37)
Let us assume for a moment that it is possible to make a sequence of snapshots of the
first excited state of Lithium, |11̄3i. In each single shot there is only one electron, |3i,
contributing to the spin-orbit coupling. We ask now, what nuclear charge does this
certain electron face when averaging over many snapshots? One might expect that the
two electrons of the inner closed first shell, |1i and |1̄i, shield the nuclear charge Z = 3
to Zeff = 1. Using Table 2.1, we redefine
1
5/2
Zeff r cos θ e−Zeff r/2 .
|3i := √
32π
(3.38)
Our classical snapshot can be substantiated using the Rayleigh-Ritz variational principle
and results of [FG09a]. Recall the PT energy of the first excited state of Li, (3.19), and
how it was determined:
h11̄3|H(N = 3)|11̄3i = 2(1|1) + (3|3) + (11|11) + 2(11|33) − (13|31) .
(3.39)
The notation is the same as introduced in Definition 3.15, but this time we assume for
the state |3i some, so far, unknown effective nuclear charge Zeff . Its deviation from
the usual nuclear charge is denoted by
δ(Z) := Z − Zeff > 0 .
(3.40)
Lemma 3.37. Using the effective nuclear charge Zeff instead of Z for the state |3i, the
following holds:20
1
1 2
− ZZeff ,
(3|3) = Zeff
8
4
2
3
4
ZZeff (8Z 4 + 20Z 3 Zeff + 20Z 2 Zeff
+ 10ZZeff
+ Zeff
)
(11|33) =
,
5
(2Z + Zeff )
5
112Z 3 Zeff
.
(13|31) =
3 (2Z + Zeff )7
Furthermore, the spin-orbit corrections, I± , read
I+ =
1
1
3
3
, and I− = − ZZeff
.
ZZeff
96
48
Proof. The integrals for (a|b) and (ab|cd) are elaborated in [FG09a]. From Lemma 3.25
and its proof one finds immediately the expressions for I± .
Remark 3.38. For δ(Z) = 0, the terms in the recent Lemma 3.37 coincide with the prior
results in Lemma 3.16 and Lemma 3.25, respectively.
20
Note that Lemma 3.37 states only results we need for the current discussion. There are other terms,
like (11|33) or (23|32), which need to be generalized.
48
3.3 Relativistic Corrections to Lithium
Figure 3.5: Splitting of the first excited state of Lithium J = 1/2 and J = 3/2: the effective PT model
which uses the effective nuclear charge Zeff , see Table 3.2, agrees quantitatively for x . 0.3
with the experimental data. Also its qualitative behavior can be explained acceptably.
With these expressions the energy (3.39) becomes dependent on Zeff . Applying the
Rayleigh-Ritz variational principle now determines the effective nuclear charge, since Zeff
is chosen in such a way that is minimizes the energy of the first excited state of Lithium.
Proposition 3.39. The energy h11̄3|H(N = 3)|11̄3i, dependent on Z and Zeff , is minimal
at Zeff (Z) as stated in the following table:
Z
3
8
13
18
23
28
33
Zeff
1.0458
6.2661
11.3144
16.3344
21.3454
26.3522
31.3570
δ(Z)
1.9542
1.7339
1.6856
1.6656
1.6546
1.6478
1.6430
Z
4
9
14
19
24
29
34
Zeff
2.1266
7.2805
12.3197
17.3371
22.3470
27.3533
32.3577
δ(Z)
1.8734
1.7195
1.6803
1.6629
1.6530
1.6467
1.6423
Z
5
10
15
20
25
30
35
Zeff
3.1833
8.2918
13.3241
18.3395
23.3485
28.3543
33.3585
δ(Z)
1.8167
1.7082
1.6759
1.6605
1.6515
1.6457
1.6415
Z
6
11
16
21
26
31
36
Zeff
4.2209
9.3008
14.3280
19.3417
24.3498
29.3553
34.3591
δ(Z)
1.7791
1.6992
1.6720
1.6583
1.6502
1.6447
1.6409
Z
7
12
17
22
27
32
37
Zeff
δ(Z)
5.2471 1.7529
10.3082 1.6918
15.3314 1.6686
20.3436 1.6564
25.3511 1.6489
30.3561 1.6439
35.3598 1.6402
Table 3.2: Effective nuclear charge for |11̄3i from to the Rayleigh-Ritz variational principle
Remark 3.40. From Table 3.2 we see that for increasing Z the shielding effect of the
two inner electrons, |1i and |1̄i, weakens. In the case of an neutral atom, Z = N = 3, we
find indeed our classical guess δ(3) ≈ 2.
Corollary 3.41. The splitting in the first excited state of Li, |11̄3i, between the states
J = 1/2 and J = 3/2 is given by
3/2
∆E1
1/2
− ∆E1
Z5
=
1 1
x (1 − δx)3 ,
c20 32
again with x = 1/Z. The comparison to experimental data is shown in Figure 3.5.
49
3 Spectrum of Many-electron Atoms
3.4 Relativistic Corrections to Beryllium, Boron and
Carbon
Wer nichts als Chemie versteht, versteht auch die nicht recht.21
Georg Christoph Lichtenberg
The ground state of Beryllium is the first PT state which is not just a single Slater
determinant, but a sum of four Slater determinants, see Table 3.1 for the asymptotic
ground states, L3 = 0 and S3 is maximal:
c
1
Be
(3.41)
|11̄22̄i + √ (|11̄33̄i + |11̄44̄i + |11̄55̄i) ,
ψ0 = √
1 + c2
3
with
√
3 √
2 1509308377 − 69821 ≈ −0.23 .
(3.42)
59049
The dominant term, |11̄22̄i, contributes with almost 95%. The quantum numbers of ψ0Be
read L = 0 and S = 0, hence only J = 0 is possible. For the P-contribution and the
Darwin term we find:
c=−
2
2c2
P
P
D
D
P
P
D
I
+
I
+
I
+
I
+
I
+
I
+
I
=
1
2
1
2
1
3
1
2
1 + c2 1
+
c
1
21
2 55
+c
Z4 .
=−
1 + c2 64
192
∆E0P+D =
(3.43)
The spin-structure reads N + = N− = 2, hence
∆E0LS = −
c2 1 4
Z .
1 + c2 94
(3.44)
The first excited state of Be with quantum numbers L = 1 and S = 1 reads much easier:
ψ1Be := |11̄23i .
(3.45)
For the P-contribution and the Darwin term we find we find immediately:
59 4
∆E1P+D = 2 I1P + I1D + I2P + I2D + I3P = −
Z .
192
(3.46)
There are three possible total angular momenta: J = 0 (N + = 1, N − = 3), J = 1
(N + = 2, N − = 2), or J = 2 (N + = 3, N − = 1).
Proposition 3.42. The relativistic corrections to the ground state, ψ0Be , and first excited
state, ψ1Be , of Beryllium are given by:
21
1
2 19
+c
Z4 ,
∆E0 = −
1 + c2 64
64
21
20
19
∆E10 = − Z 4 , ∆E11 = − Z 4 , ∆E12 = − Z 4 ,
64
64
64
where c is given in (3.42).
21
He who understands nothing but chemistry does not truly understand chemistry either.
50
3.4 Relativistic Corrections to Beryllium, Boron and Carbon
Figure 3.6: Spectral gap between the ground state and first excited state of Beryllium for J = 0, Proposition 3.42. The experimental data points follow up to a small negative relativistic correction
the straight line of the non-relativistic PT model, compare Remark 3.43. Note the high
resolution on both axes.
Figure 3.7: Spectral gap between the ground state and first excited states of Beryllium for J = 1 (circles) and J = 2 (squares), Proposition 3.42. The qualitative behavior of the experimental
data points is described satisfactory. As expected from the asymptotic limit, the qualitative
agreement between theoretical values and data points increases when going to higher charged
ions.
51
3 Spectrum of Many-electron Atoms
Figure 3.8: Spectral gap between the ground state, J = 3/2 and first excited state, J = 0 of Boron,
Proposition 3.44. The experimental data follow with a slightly negative deviation the classic
PT prediction, compare Corollary 3.45. Note the high resolution on both axes.
Figure 3.9: Spectral gap between the ground state, J = 1/2, and first excited states of, J = 1/2 (circles)
and J = 5/2 (squares), of Boron, Proposition 3.44. In both cases the relativist PT model and
the experimental data agrees qualitatively. The relativistic prediction is dramatically better
for the lower total angular momentum of the first excited state.
52
3.4 Relativistic Corrections to Beryllium, Boron and Carbon
Remark 3.43. If one neglects the minor three Slater determinants of ψ0Be , then one
finds again a degeneracy: ∆E0 (c = 0) = E10 . However, since c ≈ −0.23 is small, the
experimental data are expected to follow almost the non-relativistic PT model prediction.
The J-dependent constants in (3.31) read in the case of Beryllium:
√
1363969 2813231 − 5 1509308377
−
≈ 0.0649 ,
∆B =
839808
1679616
(3.47)
c2 1
−3
≈ −0.0016 · 10 < 0 .
∆C(J = 0) = −
1 + c2 32
This time, in contrast to Corollary 3.36, the relativistic contribution has in total a negative
sign: ∆(J = 0) < 0. We can conclude: for the spectral gap between ground state and first
excited state we expect in the case J = 0 the experimental data points to follow a slightly
negatively shifted straight line. Indeed, the experimental data feature this behavior as
shown in Figure 3.6. Since |∆C(J = 0)| > 1/64, the corresponding values for J = 1, 2
are positive and about a factor ten larger. Therefore we expect some apparent positively
shifted straight line, compare Figure 3.7.
The ground state of Boron is again a sum of Slater determinants, whereas its first excited
state is a single one:
c
1
B
|11̄22̄3i + √ (|11̄344̄i + |11̄355̄i) ,
ψ0 = √
(3.48)
1 + c2
2
B
ψ1 = |11̄245i ,
with
√
2 √
733174301809 − 809747 .
(3.49)
393660
The quantum number for ψ0B read L = 1 and S = 1/2, whereas for ψ1B one has L = 1 and
S = 3/2. This yields a splitting for both states. Similarly as done for Li and Be, we find
the following
c=−
Proposition 3.44. The relativistic corrections to the ground state, ψ0B , and first excited
state, ψ1B , of Boron read:
1
137
1
133
1/2
3/2
2 43
4
2 39
∆E0 = −
+c
Z , ∆E0 = −
+c
Z4 ,
1 + c2 384
128
1 + c2 384
128
133 4
125 4
117 4
3/2
5/2
1/2
∆E1 = −
Z , ∆E1 −
Z , ∆E1 = −
Z ,
384
384
384
where c is given in (3.49).
Corollary 3.45. For Boron, using Proposition 3.44, we find immediately the following
table of relativistic deviations, parameterized by ∆C(JGS , JE1 ):
∆C
JE1 = 1/2
JE1 = 3/2
JE1 = 5/2
JGS = 1/2
0.0099
0.0303
0.0515
JGS = 3/2
- 0.0011
0.0197
0.0405
53
3 Spectrum of Many-electron Atoms
Remark 3.46. As for one spectral gap in Beryllium, we find one combinations of JGS
and JE1 such that ∆C < 0, compare Remark 3.43. The comparisons of our theoretical
results to experimental data confirm this prediction, see Figures 3.8 and 3.9.
The investigation of the Carbon spectrum demonstrates the first time serious boundaries of the applicability of our perturbative ansatz in the relativistic sector. Similar to
Figure 3.1, where relativistic effects have appeared in the first place, this time higher
order corrections appear significantly in the experimental data. We emphasize that these
deviations do not contradict the convergence of the asymptotic limit or the relativistic
expansion of energy levels: a purely physical reason defines the boundaries of applicability
of our theory as we will discuss in Remark 3.48.
We consider the ground state, ψ0C , and first excited state, ψ1C , of Carbon:
1
|11̄22̄45i + c|11̄33̄45i ,
ψ0C = √
1 + c2
1
1
ψ1C = √ √
(2|11̄22̄33̄i − |11̄22̄44̄i − |11̄22̄55̄i) −
6 1 + c2
− c (2|11̄44̄55̄i − |11̄33̄44̄i − |11̄33̄55̄i) ,
(3.50)
with
1 √
221876564389 − 460642 ≈ −0.1056 .
(3.51)
98415
The ground states obeys the quantum numbers L = 1 and S = 1, hence there is a splitting
into three states. For the first excited state there is no splitting, since L = 2 and S = 0.
One finds even more:
c=−
Proposition 3.47. The relativistic corrections to the ground state, ψ0C , and first excited
state, ψ1C , of Carbon are given by:
1
25
23
0
+c
∆E0 = −
Z4 ,
1 + c2 64
8
1
3
1
2 11
∆E0 = −
+c
Z4 ,
1 + c2 8
32
1
23
2 5
2
∆E0 = −
+c
Z4 .
2
1 + c 64
16
where c is given in (3.49). The relativistic correction to the first excited state is the same
as for the ground state J = 1: ∆E1 = ∆E01 . The energy levels are sketched in Figure 3.10.
Remark 3.48. As Figure 3.11 indicates we need to discuss some deviations between
the results of our relativistic PT model and the experimental iso-electronic sequence of
highly-charged ions. We start with the predicted degeneracy ∆E1 = ∆E01 : one would
expect that the data points follow the straight line coming from the non-relativistic PT
model. For x . 0.4 the experimental data do not follow this line, indicating that there
are additional relativistic effects we did not take into account. Since the deviation takes
place for higher values of Z compared to Figure 3.2, we interpret these as a relativistic
corrections of higher order. So one might ask why higher-order effects are present in the
plot actually? Relativistic effects are even more important for higher nuclear charges,
54
3.4 Relativistic Corrections to Beryllium, Boron and Carbon
C-E1
ΔE1
J=2
C-GS
ΔE02, ΔC < 0
J =2
J=1
J=0
ΔE01, ΔC = 0
ΔE00, ΔC > 0
Figure 3.10: Relativistic corrections to the ground state (GS) and first excited state (E1) of Carbon,
Proposition 3.47. All PT states are shifted downwards. The red arrows indicate the degeneracy of ∆E1 and ∆E01 . Therefore, one expects the experimental data points to follow the
prediction of the non-relativistic PT model, but this is found not to be true. The two other
gaps feature a smaller, J = 2, and a larger spectral gap, J = 0, hence one has ∆C(J = 0) > 0
and C(J = 2) < 0. In the asymptotic limit the experimental data also contradict the second
prediction. Compare to the discussion in Remark 3.48.
Figure 3.11: Spectral gap between the ground state and first excited states of Carbon for J = 0 (circles),
J = 1 (squares) and J = 2 (triangles), Proposition 3.47. Compare to the discussion in
Remark 3.48.
55
3 Spectrum of Many-electron Atoms
hence in the asymptotic limit our first-order ansatz in the relativistic sector runs out
of physical applicability. The perturbative ansatz in large nuclear charges and a large
speed of light is fine from a mathematical point of view. Physically, we need to take the
full relativistic theory into account when combining both expansions. However, there is
always just a finite number of experimental data points available and the magnitude of
the higher-order corrections for different chemical elements differ strongly, hence there is
indeed a satisfactory agreement between our theory and experimental data as shown for
Lithium, Beryllium, and Boron.
We emphasize that the qualitative agreement for not too high Z does still hold in the
carbon case: for J = 0 we expect a positive, for J = 2 we expect a negative relativistic
correction. ∆C(J = 0) > 0 is confirmed by Figure 3.11, if thought not quantitatively.
Also ∆C(J = 2) < 0 is confirmed when restricting to the region of applicability of the
first-order ansatz: 0.04 . x . 0.10. In comparison to the experimental data for J = 1
their downward deviation holds. Due to the strong Z-dependence, higher-order corrections
become more important for x . 0.04.
56
3.5 Relativistic Corrections to Nitrogen, Oxygen, Fluorine and Neon
3.5 Relativistic Corrections to Nitrogen, Oxygen,
Fluorine and Neon
Das Leben ist nur ein physikalisches Phänomen.22
Ernst Haeckel
Before considering further elements of the second shell, we summarize the most important
messages from the discussion so far: the results of the relativistic PT model coincide with
the experimental spectral gaps between J-split ground and first excited states qualitatively
in each case. The lower the number of electrons and the lower the energy in the split
level, the better our results agree also quantitatively. Additionally, the ordering of the
fine-structure states coincides with the experimental ones: for Li, Be, B and C always the
state with the lowest J value has the lowest energy. This changes now for N, O, and F.
For these elements our focus lies now on the ordering of the fine-structure levels.
Inspecting the quantum numbers L and S of the ground states, one finds experimentally,
and, for the first and second shell, also in the PT model, only five combinations:
L
S
1st shell
2nd shell
3rd shell
classification/chemical family [CHD05]
0
0
He
Be, Ne
Mg, Ar
alkaline earth metals and noble gases
0
1/2
H
Li
Na
alkali metals
0
3/2
N
P
nitrogen family (pnictogens)
1
1/2
B, F
Al, Cl
1
3/2
C, O
Si, S
boron family and halogens
carbon family and oxygen group (chalcogens)
Table 3.3: Ground-state quantum number of the first 18 elements. The PT model predicts the values
for L and S for the first and second shell correctly, compare Table 2.1. As we will derive in
Corollary 3.57, the relativistic quantum number J helps to distinguish Boron and Fluorine,
and Carbon and Oxygen.
From a theoretical point of view this can be explained by the so-called particle-hole
dualism: the number of p-orbitals in B and F, but also in C and O, sum up to six, which
is the maximal number of p-orbitals for elements of the second shell. So one has that the
four p-orbitals of C imply the same angular momentum and spin quantum number as the
two p-orbitals of O. Four existing orbitals are equivalent to the absence of two orbitals,
and the other way around. Additionally, the parity of the dual chemical elements coincide:
if two integers sum up to an even number, six, they are either both odd or both even. For
details we refer to the discussion in [FG09b].
We now consider the relativistic corrections to Nitrogen with the asymptotic ground
state
ψ0N = |11̄22̄345i ,
(3.52)
with L = 0 and S = 3/2, hence only J = 3/2 is possible. The first excited state is again
a sum of Slater determinants,
1
ψ1N = √ 2|11̄22̄3̄45i − |11̄22̄345̄i − |11̄22̄34̄5i ,
6
22
(3.53)
Life is just a physical phenomenon.
57
3 Spectrum of Many-electron Atoms
with L = 2 and S = 1/2, hence there are two possible values for the total angular
momentum: J = 3/2, 5/2. Similar calculations yield the following
Proposition 3.49. The relativistic corrections to the ground state, ψ0N , and first excited
state, ψ1N , of Nitrogen read:
237 4
Z ,
640
347 4
359 4
5/2
Z , ∆E1 = −
Z ,
=−
896
896
∆E0 = −
3/2
∆E1
where a sketch of the shifted energy levels is shown in Figure 3.12.
Remark 3.50. We suggest that the reversal of the two J-states in the first excited state
of Nitrogen as shown in Figure 3.13 is not correct. The NIST data claim that for Z ≤ 21
(i.e. for neutral N, and for the N-like ions O-II, F-III, Ne-IV, . . ., Sc-XV) the state J = 3/2
is higher than the state J = 5/2, but for Z > 21 (i.e. for the N-like ions Ti-XVI, V-XVII,
Cr-XIX, . . .) the order is reversed. As seen, this leads to quite unconvincing data points
in the iso-electronic sequence for Nitrogen-like ions, since actually we expect the sequence
to be smooth.
In the asymptotic limit, Z → ∞ ⇔ x → 0, the experimentally ordering of the J-states
coincides with our results of Proposition 3.49, see also Figure 3.12. Therefore we suggest
that for Z ≤ 21 the data points of the two sequences have to be exchanged.
Note that in [FG10] there is a similar statement concerning the quantum numbers of
the fourth and fifth excited state of Boron, 2 S and 2 P , respectively. There, for Z ≤ 22
the NIST data claim 2 S <2 P , and reversed for Z > 22. Again, in order to get smooth
spectral gaps for both states, it is suggested that this reversal is not correct.
Switching to Oxygen, first of all, we state some error in a single NIST datum: in the isoelectronic sequence of Oxygen, for the ion Br-XXVIII the ground state with J = 0 needs
to be changed to something which guarantees the monotony of the sequence, compare
Figure 3.14. However, the ground state of Oxygen reads
ψ0O = |11̄22̄33̄45i ,
(3.54)
with quantum numbers L = 2 and S = 1, hence three states with J = 0, 1, 2 are possible.
The first excited state is given by
1
ψ1O = √ 2|11̄22̄44̄55̄i − |11̄22̄44̄55̄i − |11̄22̄33̄55̄i ,
6
(3.55)
with quantum numbers L = 2 and S = 0, hence only J = 2 is possible. We find the
following
Proposition 3.51. The relativistic corrections to the ground state, ψ0O , and first excited
state, ψ1O , of Oxygen read:
1
635 4
587 4
Z , ∆E01 = − Z 4 , ∆E02 = −
Z ,
1344
56
1344
1
∆E1 = − Z 4 = ∆E01 ,
56
∆E00 = −
where a sketch of the shifted energy levels is shown in Figure 3.16.
58
3.5 Relativistic Corrections to Nitrogen, Oxygen, Fluorine and Neon
N-E1
5/2
ΔE1
J = 5/2
3/2
J = 3/2
ΔE1
N-GS
ΔE0
J = 3/2
Figure 3.12: Relativistic corrections to the ground state and first excited state of Nitrogen, Proposition
3.49. Again, all PT states are shifted downwards and in the first excited states the lower
level has lower total angular momentum, J = 1/2.
Figure 3.13: Experimental results for the spectral gaps between ground states and first excited states,
J = 3/2 and J = 5/2, of Nitrogen by the NIST database [Yu.10]. See the discussion in
Remark 3.50.
59
3 Spectrum of Many-electron Atoms
Figure 3.14: Experimental data from NIST, [Yu.10], for the spectral gap between ground state with J = 0
and the first excited state of Oxygen. The data point for Br XXVIII is strongly suggested
to be not correct.
Figure 3.15: Experimental results for the spectral gaps between ground states J = 0, 1, 2 and first excited state of Oxygen, Proposition 3.51. The degeneracy ∆E1 = ∆E01 is featured by the
experimental data. Furthermore, there is indeed a crossing of the gaps for J = 0 and J = 1,
compare the discussion in Remark 3.52 and compare Figure 3.16.
60
3.5 Relativistic Corrections to Nitrogen, Oxygen, Fluorine and Neon
O-E1
ΔE1
J =2
O-GS
J =1
J =0
J =2
ΔE01
ΔE00
ΔE02
Figure 3.16: Relativistic corrections to the ground state and first excited state of Oxygen, Proposition
3.51. The lowest level of the split ground state has J = 2, which is confirmed by the
experimental data. The ordering E(J = 0) < E(J = 1) holds experimentally only in the
asymptotic limit, see the reversal in Figure 3.15. For the ground state of neutral Oxygen one
finds experimentally the reversed ordering. The red arrows indicate again the degeneracy
∆E1 = ∆E01 , also found in the experimental data.
Remark 3.52. For Oxygen we expect from Proposition 3.51 that the ground state features a splitting into three energy levels with the ordering E(J = 2) < E(J = 0) < E(J = 1).
This ordering is different to the cases we had before since the values of J do not just follow
the pattern “high to low” or “low to high”, but they are really mixed. The ground state of
neutral Oxygen, J = 2, is confirmed experimentally, but the predicted ordering for J = 0
and J = 1 is not found experimentally, but one finds: E(J = 2) < E(J = 1) < E(J = 0).
However, the iso-electronic sequence for Oxygen features an interesting crossing of the
energy levels for J = 1 and J = 0. In contrast to Nitrogen, compare Remark 3.50, this
reversal happens smoothly, so we consider the experimental data to be correct. Therefore,
the theoretically predicted ordering appears to be correct for large values of the nuclear
charge. This is consistent with our perturbative approach and the convergence in the
asymptotic limit Z → ∞. Note that for all data points the spectral gap between J = 2
and the first excited state is the largest one, hence this is also the ground state of neutral
Oxygen, in agreement with our results.
Furthermore, the predicted degeneracy ∆E1 = ∆E01 is found in the experimental spectral
gaps for Oxygen: the accordant data points (squares in Figure 3.15) seem to follow the
straight line from the, even non-relativistic, PT model. Additionally, for the other states
J = 2 and J = 1 we expect, besides their ordering, a positive relativistic corrections,
which is also featured by the experimental data:
∆C(J = 2) =
563
611
, and ∆C(J = 0) =
.
1344
1344
(3.56)
61
3 Spectrum of Many-electron Atoms
Now we start the investigation of Fluorine. Its ground state reads
ψ0F = |11̄22̄344̄55̄i ,
(3.57)
with L = 1 and S = 1/2, hence two states, J = 1/2, 3/2 are possible. The first excited
state is quite similar,
(3.58)
ψ1F = |11̄233̄44̄55̄i ,
with L = 0 and S = 1/2, hence only J = 1/2 is possible. Again, we skip the explicit
calculation and state the following
Proposition 3.53. The relativistic corrections to the ground state, ψ0F , and first excited
state, ψ1F , of Fluorine read:
449 4
409 4
3/2
Z , ∆E0
=−
Z ,
896
896
397 4
∆E1 = −
Z ,
896
1/2
∆E0
=−
where a sketch of the shifted energy levels is shown in Figure 3.17.
Remark 3.54. In Fluorine we do not expect any degeneracy but the relativistic correction
for the spectral gap between ground state J = 1/2 and first excited state is much smaller
than for J = 3/2 in the ground state:
∆C(J = 1/2) =
13
3
≈ 0.01 , and ∆C(J = 3/2) =
≈ 0.06 .
224
224
(3.59)
This qualitative behavior can be observed in the experimental data, but the positive
relativistic correction is not yet visible for the experimentally available ions in the isoelectronic sequence for Fluorine.
In Figure 3.18 we show again also the quantitative agreement between our theoretical
results and the experimental data. As already concluded at the beginning of this section
the lower the energy state the better the qualitative agreement. This statement does hold
also for chemical elements with more than six electrons.
The last chemical element we consider in this thesis is Neon. The PT model offers only
the ground state,
(3.60)
ψ0Ne = |11̄22̄33̄44̄55̄i ,
with L = 0 and S = 0, hence no splitting of the ground state takes place: only J = 0 is
possible. We find the following
Proposition 3.55. The relativistic correction to the ground state of Neon reads:
∆E0 = −
15 4
Z .
32
Remark 3.56. Since in Neon all ten available hydrogen orbitals, |ii, |īi for i = 1, . . . , 5,
are occupied, excited states can not be derived in the PT model.
62
3.5 Relativistic Corrections to Nitrogen, Oxygen, Fluorine and Neon
F-E1
ΔE1
J = 1/2
F-GS
1/2
J = 1/2
ΔE0
3/2
J = 3/2
ΔE0
Figure 3.17: Relativistic corrections to the ground state and first excited state of Fluorine, Proposition
3.53. The lowest level of the split ground state has J = 3/2, which is confirmed by the
experimental data.
Figure 3.18: Experimental results for the spectral gaps between ground states J = 1/2, 3/2 and first
excited state of Fluorine, Proposition 3.53. The small predicted relativistic correction for
J = 1/2 is featured by the experimental data, actually it seems to follow the non-relativistic
prediction.
63
3 Spectrum of Many-electron Atoms
As claimed at the beginning of this section we now conclude our findings in Corollary
3.57: the chemical elements of the second shell whose ground states feature the angularmomentum quantum number L = 1 are distinguishable when taking the relativistic quantum number J into account:
Corollary 3.57. For the two pairs of quantum numbers (L, S) = (1, 1/2) and (L, S) =
(1, 3/2) there is a splitting into two distinguishable energy levels with J = 1/2, 3/2 and
J = 3/2, 5/2, respectively.
(a) The ground state of Boron has the quantum number (L, S, J) = (1, 1/2, 1/2). Its
dual, Fluorine, features (L, S, J) = (1, 1/2, 3/2).
(b) The ground state of Carbon has the quantum number (L, S, J) = (1, 3/2, 1/2). Its
dual, Oxygen, features (L, S, J) = (1, 3/2, 5/2).
In the relativistic setup only ground states of two second-shell elements, Beryllium and
Neon, remain indistinguishable in the quantum numbers (L, S, J) = (0, 0, 0).
Proof. For Boron and Fluorine one only has to inspect Propositions 3.44 and 3.53. For
Carbon and Oxygen the relevant Propositions are 3.47 and 3.51.
Remark 3.58. The experimental data confirm our theoretical results: for Oxygen and
Fluorine the ground states feature the highest possible value for J, whereas for Boron and
Carbon the lowest possible J is realized in nature.
64
3.6 Non-local Relativistic Corrections
3.6 Non-local Relativistic Corrections
Wer einen Fehler gemacht hat und ihn nicht korrigiert, begeht einen zweiten.23
Konfuzius
So far we have derived the relativistic corrections in the local model, compare section 3.2.2.
The fact that the asymptotic states we are dealing with coincide with the Schrödinger
states in the limit Z → ∞ has motivated us to restrict to the diagonal terms of L · S.
The set of good quantum numbers in the non-relativistic setup is given by (L, S, L3 , S3 ),
where we can choose without loss of generality one component of L and S. This is also
how the wave functions of the ground state from Table 3.1 are chosen: L3 = 0 and
S3 = L + S is maximal. As already done in the proof of Lemma 3.25 for the one-particle
state, we now switch to bases of the ground state where the many-particle operators L2 ,
S 2 , J 2 and J3 are diagonal. Again, without loss of generality, we choose for a given
eigenvalue J the corresponding magnetic quantum number to be maximal: J3 = J. The
transformation between these two bases is known as Clebsch-Gordan decomposition,
and we denote the new states by |L, S, J, J3 i, or, if the corresponding chemical element is
clear, by |J, J3 i. Having an orthogonal basis in the non-relativistic setup, the new basis
is again orthogonal. A general discussion of this technique can be found, for instance, in
[FH91]. We only note that the dimensions of the ground-state in the classical and the
relativistic basis indeed coincide, since the degeneracy for a given J is 2J + 1:
L+S
X
J=|L−S|
(2J + 1) = (1 + L + S − |L − S|)(1 + L + S + |L − S|) =
(3.61)
= 1 + 2(L + S) + 2LS − |L − S|2 = (2L + 1)(2S + 1) .
In Table 3.4 we state the Clebsch-Gordan decomposition for all ground and first excited
states. Table 3.5 summarizes the degeneracies of the ground-state and first-excited-state
eigenspaces of the non-relativistic PT model. For the non-interacting ground state we
have, [FG09b]:
2
8
dim V0 (N ) =
, for N = 1, 2, dim V0 (N ) =
, for 3 ≤ N ≤ 10 .
N
N −2
(3.62)
As already discussed, the dimensions of the interacting ground and first excited states
depend on their symmetry, described by the eigenvalues L and S:
dim(·)(N ) = (2L + 1)(2S + 1) .
In the following we will discuss in detail the case of the first excited state of Lithium and
Beryllium in order to illustrate the similarities and differences to the local model. We
start with Lithium: the interacting first excited state is six-dimensional, spanned by the
orthogonal bases of non-relativistic eigenfunctions
E1(3) = span{|11ii, |11ii : i = 3, 4, 5} .
23
(3.63)
If you make a mistake and do not correct it, this is called a mistake.
65
3 Spectrum of Many-electron Atoms
In terms of non-relativistic eigenvalues |L3 , S3 i we find the following:
|114i = |1, 1/2i ,
|115i = | − 1, 1/2i ,
|113i = |0, 1/2i ,
|113i = |0, −1/2i , |114i = |1, −1/2i , |115i = | − 1, −1/2i .
(3.64)
For the calculation in the local model we have chosen the wave function with L3 = 0
and S3 = 1/2, namely |113i. From Table 3.4 we get the Clebsch-Gordan decomposition
to lead to a direct sum of a two-dimensional and a four-dimensional eigenspace of the
N
state
Li
GS
1⊗2=2
E1
3⊗2=2⊕4
GS
1⊗1=1
E1
3⊗3=1⊕3⊕5
GS
3⊗2=2⊕4
E1
4⊗3=2⊕4⊕6
GS
3⊗3=1⊕3⊕5
E1
5⊗1=5
GS
1⊗4=4
E1
5⊗2=4⊕6
GS
3⊗3=1⊕3⊕5
E1
5⊗1=5
GS
3⊗2=2⊕4
E1
1⊗2=2
GS
1⊗1=1
Be
B
C
N
O
F
Ne
Clebsch-Gordan decomposition
Table 3.4: Clebsch-Gordan decomposition of ground and first excited states of the PT model. The syntax
is as follows: the first excited state of Lithium features angular momentum L = 1 and spin
momentum S = 1/2. Therefore the decomposition of the product of a three-dimensional and
two-dimensional vectorspace, 3 ⊗ 2, needs to be determined: 2 ⊕ 4, a doublet and a quartett.
N
H
He
Li
Be
B
C
N
O
F
Ne
dim V0 (N )
2
1
8
28
56
70
56
28
8
1
dim GS(N )
2
1
2
1
6
9
4
9
6
1
dim E1(N )
-
-
6
9
12
5
10
5
2
-
Table 3.5: Degeneracies of the non-interacting ground state, V0 (N ), the interacting ground state, GS(N ),
and first excited state, E1(N ), in the PT model
66
3.6 Non-local Relativistic Corrections
operator J 2 . The new doublet states |J, J3 i read:
r
r
2
1
|114i −
|113i ,
|1/2, +1/2i =
3
3
r
r
1
2
|113i −
|115i ,
|1/2, −1/2i =
3
3
(3.65)
and for the quartett:
|3/2, +3/2i = |114i ,
r
r
1
2
|3/2, +1/2i =
|114i +
|113i ,
3
3
r
r
2
1
|3/2, −1/2i =
|113i +
|115i ,
3
3
|3/2, −3/2i = |115i .
(3.66)
The sign convention and the explicit choice of the prefactors we have used is the same
as in [N+ 10]. Again we can just choose the value for J3 and follow the prescription of
the non-relativistic case: J3 is choosen to be maximal. For the ground state this means
|1/2, 1/2i, and for the first excited state |3/2, 3/2i. In Lemma 3.24 we have denoted these
two states by |j±i. Their LS-corrections read:
1 2
J − L2 − S 2 |1/2, 1/2i =
2
1 3
1 1 3
· −1·2− ·
|1/2, 1/2i = −|1/2, 1/2i ,
=
2 2 2
2 2
1 3 5
3
1
L · S|3/2, 3/2i =
· −2−
|3/2, 3/2i = |3/2, 3/2i .
2 2 2
4
2
L · S|1/2, 1/2i =
(3.67)
Taking the prefactor Z/2 and the results from Lemma 3.20 into account, we rederive
exactly the same results for the spin-orbit corrections of the first excited state of Lithium
as in the local model in (3.35):
LS,1/2
∆E1
=−
1
1 4
LS,3/2
Z and ∆E1
= Z4 .
48
96
(3.68)
Since the new basis are orthogonal, too, the local-model results for the P-correction and
the Darwin term do not change and we can state again Proposition 3.33.
The ground state of Beryllium is just one-dimensional, hence we discuss its first excited
state: the PT model tells us that L = 1 and S = 1 with the nine-dimensional eigenspace
√
E1(4) = span{|112ii, |112ii, |112ii + |112ii / 2 : i = 3, 4, 5} .
(3.69)
The state we have chosen for the local model reads |1123i. Following again he convention
of [N+ 10] we find for the relativistic Clebsch-Gordan states with maximal J3 :
r
r
1 1
1
√ |1123i + |1123i −
|2, 2i = |1124i ,
|1, 1i =
|1123i ,
2 2
2
r
r
r
(3.70)
1
1 1
1
√ |1123i + |1123i +
|0, 0i =
|1124i −
|1125i .
3
3 2
3
67
3 Spectrum of Many-electron Atoms
From this we find for the eigenvalues of L · S:
L · S|2, 2i = |2, 2i ,
L · S|1, 1i = −|1, 1i ,
L · S|0, 0i = −2|0, 0i .
(3.71)
Since the P-contribution and Darwin term can be again recycled from our discussion in
the local model, we arrive at the following
Proposition 3.59. In the non-local PT model the relativistic corrections to the ground
and first excited state of Beryllium are given by:
21
1
2 55
+c
Z4 ,
∆E0 = −
1 + c2 64
192
67 4
63 4
55 4
∆E10 = −
Z , ∆E11 = −
Z , ∆E12 = −
Z ,
192
192
192
where c is given in (3.42).
Remark 3.60. The comparison of the last results in Proposition 3.59 to their local
versions in Proposition 3.42 yields deviations of only a few percent: in the ground state
only the coefficient of c2 /(1 + c2 ) differs: −55/192 + 19/64 ≈ 1.0%.
In the first excited state the deviation for J = 0 reads (−67 + 63)/192 ≈ −2.1%. For
J = 1 one finds (−63 + 60)/192 ≈ −1.6%, and for J = 2 one has (−55 + 57)/192 ≈ 1.0%.
Note that the deviations for J = 0, 1 lead to a further downwards shift of the fine
structure states, whereas for J = 2 the predicted state of the local model is shifted
slightly upwars. However, the deviations are not very large and Figures 3.6 and 3.7
still hold, particularly the discussion in Remark 3.43: for instance the new value for the
relativistically-corrected energy gap
∆C(J = 0) = −
c2 1
≈ −0.0031 · 10−3 < 0 ,
1 + c2 16
which is twice the value in the local model (3.47). The qualitative feature that this
correction is negative does still hold in the non-local model.
We can also derive the corrections for Boron: again, the P-contribution and the Darwin
term can be used from our discussion of the local model.
Proposition 3.61. In the non-local PT model the relativistic corrections to the ground
and first excited state of Boron are given by:
1
47
1
43
1/2
3/2
2 125
4
2 113
∆E0 = −
+c
Z , ∆E0 = −
+c
Z4 ,
1 + c2 128
384
1 + c2 128
384
145 4
133 4
113 4
1/2
3/2
5/2
∆E1 = −
Z , ∆E1 −
Z , ∆E1 = −
Z ,
384
384
384
where c is given in (3.49).
Remark 3.62. Again, the deviations between the results in Proposition 3.61 and their
1/2
local version in Proposition 3.44 are small. The largest deviations occurs for ∆E1 :
(−145 + 133)/384 ≈ 3.1%.
Remark 3.63. As discussed in section 3.5, the experimental ground state of chemical
elements with 7 ≤ N ≤ 10 feature the highest possible value for J to be the lowest fine
68
3.6 Non-local Relativistic Corrections
structure state. In particular, two pairs of elements, namely Boron and Fluorine, and
Carbon and Oxygen, have the same ground state symmetry on the non-relativistic level,
but the total angular momentum was found experimentally to distinguish the members
of each pair. In the local PT model we have used the parameter N − to describe the
electronic structure. This parameter has made it possible to describe this behavior.
Now, as Table 3.4 shows, for a given angular momentum and spin momentum, the
Clebsch-Gordan decomposition does not know about the number of electrons. Therefore,
the decompositions for Boron and Fluorine lead to the same |J, J3 i states, hence also to
the same eigenvalues of the operator L · S. In the end the split state with the lowest
possible value for J appears to be the ground state.
We interpret this as follows: the asymptotic vector spaces for the ground states are
derived using a perturbative ansatz in the Coulomb interaction between the electrons. In
the limit Z → ∞ the electron-nucleus interaction is the dominant interaction. As already
states in Remark 3.32, this limit is actually incompatible with perturbation theory in the
relativistic sector. However, we use the operator L · S, with some additional prefactor,
which occurs in the non-relativistic limit of Dirac theory for non-degenerate eigenvalues.
From experiments it is known that for heavy nuclei, i.e. for atoms or ions which carry a
large nuclear charge Z, the so-called JJ-coupling instead of the LS-coupling is apparent.
The physical condition for the LS-coupling is given by large effects of electromagnetic
interaction between different electrons, but small effects of the spin-orbit coupling. Since
the latter ones are proportional to Z 4 , whereas the electron-electron interaction are proportional to Z, compare (3.30), we actually have to implement the JJ-coupling in our
model.
69
4 Summary and Conclusion
Questo grandissimo libro [...] – io dico l’universo – [...] è scritto in lingua matematica.1
Galileo Galilei
We state from [FG09b] all asymptotic PT eigenfunctions for the ground states and first
excited states we have used for our calculations. We also summarize all energy levels and
their relativistic corrections we have derived in this thesis whithin the local model: Tables
4.1, 4.2 and 4.3. A summary which emphasizes the spin structure, described by N ± , and
the theoretical J-ordering of the energy levels is shown in Table 4.4.
We have seen that doing perturbation theory in the relativistic sector on the asymptotic
PT states asks for a careful treatment of the two limits, since, for a fixed finite value of
the speed of light, the relativistic effects become dominant for large values of the nuclear
charge. One might treat the two limits, Z → ∞ and c0 → ∞, in such a way that the
hierarchy Z c0 is still respected.
The second conceptual question is due to the form of the spin-orbit coupling for N electron systems. The one-particle LS term emerges in the non-relativistic limit of Dirac
theory in a rigorous way. However, there is no N -particle Dirac theory; one actually
has to implement concepts of quantum fields. In particular, the spatial part of the LS
term, |x|−3 , cannot be generalized ad hoc to N -electron systems. In this thesis we have
used a local model which restricts the LS term to the diagonal contributions. Due to the
asymptotic states we are dealing with, this seems to be a resonable treatment: we treat
both electric and magnetic interactions between different electrons to be small compared
to the interaction between one electron and the higly charged nucleus. However, the
relativistic setup suggests in the first place to have only electromagnetic interactions in
mind and not to distinguish between electric and magnetic ones. The local model takes
this aspect consistently into account.
Describing the spin structure by N ± , we have been able to model the experimental
fact that for Oxygen and Fluorine the highest possible value for J appears to be the
ground state, whereas for their dual elements, Carbon and Boron, respectively, the lowest
possible value of J imply the lowest energy. In the semi-empirical Hund’s rules this reveral
is described by inserting a minus sign in front of the LS term for elements N = 7, 8, 9, 10.
In conclusion, the local model implements in a consistent way relativistic effects into
the non-relativistic PT model. The treatment of the concurrent limits, Z → ∞ and
c0 → ∞, needs to guarantee the applicability of first-order perturbation theory in the
relativistic sector. In comparison to experimental data from the NIST database we find
good qualitative agreements for the Z 2 -scaled spectral gaps. In particular, the prediction
of relativistic degeneracies in Lithium or Oxygen are featured by nature.
1
This grand book [...] – I mean the universe – [...] is written in the language of mathematics.
71
4 Summary and Conclusion
N
Li
Be
E
GS: 2 S
|11̄2i
EPT (Z)
− 89 Z 2 +
∆E(Z)
J = 1/2
E1: 2 P 0
|11̄3i
EPT (Z)
− 89 Z 2 +
∆E(Z)
J = 1/2
GS: 1 S
EPT (Z)
∆E(Z)
E1: 3 P 0
5965
Z
5832
37
− 128
Z4
57397
Z
52488
33
J = 3/2 − 128
Z4
c
√ 1
√
|11̄22̄i + 3 (|11̄33̄i + |11̄44̄i + |11̄55̄i)
1+c2
√
√
3
2 1509308377 − 69821 ≈ −0.2311
with c = − 59049
√
1
2813231 − 5 1509308377 Z
− 45 Z 2 + 1679616
4
1
21
2 19
J =0
− 1+c
+
c
Z
2
64
64
|11̄23i
EPT (Z)
− 45 Z 2 +
∆E(Z)
J =0
1363969
Z
839808
GS: 2 P 0
√ 1
1+c2
− 21
Z4
64
Z4
− 19
64
|11̄22̄3i +
√
2
with c = − 393660
EPT (Z)
− 11
Z2 +
8
∆E(Z)
J = 1/2
J = 3/2
E1: 4 P
|11̄245i
EPT (Z)
Z2 +
− 11
8
∆E(Z)
J = 1/2
J = 3/2
J = 5/2
1
6718464
1
− 1+c
2
1
− 1+c
2
-6.8457
-6.8455
(-13.7629)
-13.7674
(-13.5034)
-13.5079
− 20
Z4
64
J =2
-7.0578
(-6.8444)
37
− 128
Z4
J =1
B
(-7.0566)
-13.5077
-13.5075
√c (|11̄344̄i + |11̄355̄i)
2
√
733174301809 − 809747 ≈ −0.1671
√
16493659 − 733174301809 Z
4
43
137
+ c2 128
Z
384
133
39
+ c2 128
Z4
384
2006759
Z
839808
− 133
Z4
384
− 125
Z4
384
− 117
Z4
384
(-22.7374)
-22.7492
-22.7489
(-22.4273)
-22.4388
-22.4381
-22.4374
Table 4.1: Asymptotic ground states (GS) and first excited states (E1) of the relativistic PT model for
Lithium, Beryllium and Boron. The numerical value of E = EPT (Z) + c−2
0 ∆E(Z) is evaluated
at Z = N . Energies in brackets refer to the non-relativistic PT model for comparison.
72
N
C
E
GS: 3 P
EPT (Z)
∆E(Z)
E1: 1 D
EPT (Z)
∆E(Z)
N
GS: 4 S 0
√ 1
|1
1̄2
2̄45i
+
c|1
1̄3
3̄45i
2
1+c
√
1
with c = − 98415
221876564389 − 460642 ≈ −0.1056
√
221876564389
3 2
3806107
− 2 Z + 1119744 −
Z
3359232
25
1
+ c2 83 Z 4
J =0
− 1+c
2
64
4
1
3
11
J =1
− 1+c
+ c2 32
Z
2
8
23
5
1
+ c2 16
Z4
J =2
− 1+c
2
64
√1 √ 1
(2|11̄22̄33̄i − |11̄22̄44̄i − |11̄22̄55̄i) −
2
6 1+c
− c (2|11̄44̄55̄i − |11̄33̄44̄i − |11̄33̄55̄i)
with the same c as in the ground state
√
221876564389
− 32 Z 2 + 19148633
Z
−
5598720
3359232
11
3
1
+ c2 32
Z4
J =2
− 1+c
2
8
|11̄22̄345i
EPT (Z)
Z2 +
− 13
8
∆E(Z)
J = 3/2
E1: 2 D 0
EPT (Z)
∆E(Z)
√1
6
− 13
Z2
8
2437421
Z
559872
(-34.4468)
-34.4738
-34.4727
-34.4716
(-34.3202)
-34.4727
(-49.1503)
237 4
− 640
Z
-49.1976
24551357
Z
5598720
(-48.9288)
359 4
Z
− 896
-48.9800
2|11̄22̄3̄44̄55̄i − |11̄22̄345̄i − |11̄22̄34̄5i
+
J = 3/2
J = 5/2
347 4
− 896
Z
-48.9783
Table 4.2: Asymptotic ground states (GS) and first excited states (E1) of the relativistic PT model for
Carbon and Nitrogen. The numerical value of E = EPT (Z)+c−2
0 ∆E(Z) is evaluated at Z = N .
Energies in brackets refer to the non-relativistic PT model.
73
4 Summary and Conclusion
N
O
E
GS: 3 P
|11̄22̄33̄45i
EPT (Z)
− 74 Z 2 +
∆E(Z)
J =2
4754911
Z
839808
635
Z4
− 1344
587
− 1344
Z4
J =0
1
− 56
Z4
J =1
E1: 1 D
EPT (Z)
∆E(Z)
F
GS: 2 P 0
√1
6
− 74 Z 2
+
47726257
Z
8398080
635
− 1344
Z4
J =2
-66.8000
-66.7087
(-66.5360)
-66.6391
|11̄22̄344̄55̄i
− 15
Z2 +
8
∆E(Z)
J = 3/2
11982943
Z
1679616
(-87.6660)
− 449
Z4
896
-87.8411
− 409
Z4
896
J = 1/2
Ne
-66.8078
2|11̄22̄44̄55̄i − |11̄22̄33̄44̄i − |11̄22̄33̄55̄i
EPT (Z)
E1: 2 S
(-66.7048)
-87.8255
|11̄233̄44̄55̄i
EPT (Z)
− 15
Z2 +
8
∆E(Z)
J =0
GS: 1 S
|11̄22̄33̄44̄55̄i
EPT (Z)
−2Z 2 +
∆E(Z)
J =0
4108267
Z
559872
− 397
Z4
896
2455271
Z
279936
Z4
− 15
32
(-85.8342)
-85.9890
(-112.2917)
-112.5413
Table 4.3: Asymptotic ground states (GS) and first excited states (E1) of the relativistic PT model for
Oxygen, Fluorine, and Neon. The numerical value of E = EPT (Z) + c−2
0 ∆E(Z) is evaluated
at Z = N . Energies in brackets refer to the non-relativistic PT model.
74
N
state
L
S
R
J
N−
Li
GS
0
1/2
1
1/2
2
E1
1
1/2 −1
1/2
2
3/2
1
Be
B
GS
0
0
1
0
2
E1
1
1
−1
0
3
1
2
2
1
1/2
3
3/2
2
1/2
3
3/2
2
5/2
1
0
4
1
3
2
2
2
3
GS
E1
C
N
O
F
Ne
GS
1
1
1
1/2 −1
3/2
1
1
E1
2
GS
0
3/2 −1
3/2
5
E1
2
1/2 −1
3/2
4
5/2
5/2
3
3/2
2
7
0
5
1
1
4
0
2
4
3/2
7
1/2
5
GS
1
E1
2
GS
1
0
1
1
0
1
J(NIST)
1
1
1/2 −1
E1
0
1/2
1
1/2
5
GS
0
0
1
0
5
see Remark 3.50
see Remark 3.52
Table 4.4: Summary of the quantum numbers for angular momentum, L, spin momentum, S, parity, R,
and total angular momentum, J. We also state the parameter N − for the electron configuration used for the results of the local model. The last column shows the comparision to the
experimental data from NIST in the cases where we have found a disagreement between the
neutral atom of the iso-electronic sequence and the theoretical results.
75
A Appendix
A.1 Sobolev Spaces
Definition A.1. Let Ω ⊆ Rn be an open subset. The set of bump functions, or test
functions, is defined as
C0∞ (Ω) := {f ∈ C ∞ (Ω) : supp (f ) is compact} .
(A.1)
Remark A.2. Note that there are analytic functions which are not in C0∞ (Ω). However,
there are also bump functions which are not analytical.
For any function f ∈ C 1 (Ω) and g ∈ C0∞ (Ω) the integration-by-parts formula yields
Z
Z
(∂i f (x))g(x) dx = − f (x)∂i g(x) dx for i = 1, . . . , n .
(A.2)
Ω
Ω
The crucial observation is that the right-hand side of (A.2) is defined even if f is not
differentiable. Hence we are motivated to state the
Definition A.3. A function f ∈ Lp (Rn ), p ≥ 1, is called weakly differentiable w.r.t.
xi , i = 1, . . . , n, if there exists a function ϕi ∈ Lp (Rn ) such that
Z
Z
f (x)∂i g(x) dx = −
ϕi (x)g(x) dx
Rn
Rn
for all bump functions g ∈ C0∞ (Rn ). We call ϕi weak derivative of f w.r.t. xi and
denote it also by ϕi = ∂f /∂xi . Higher-order weak derivatives are defined iteratively.
Furthermore, we define the p-th Sobolev space of order k by
W k,p (Rn ) := {f ∈ Lp (Rn ) : f possesses all weak derivatives up to order k} .
The fact that L2 (Rn ) is a Hilbert spaces motivates the definition of the Sobolev space
of order k:
H k (Rn ) := W k,2 (Rn ) .
Remark A.4. We state the following without proof:
(i) If a weak derivative of f ∈ Lp (Rn ) exists, then it is unique.
(ii) If f ∈ Lp (Rn ) ∩ C 1 (Rn ), then the weak and the classical derivative coincide.
(iii) The Sobolev spaces H k (Rn ) are needed in this thesis since they provide ”natural”
domains of differential operators of order k.
For us, the most important examples are the Laplacian, A = −∆ in X = L2 (R3 ),
which is self-adjoint on domain D(A) = H 2 (R3 ), and the Nabla operator, B = ∇
in X = L2 (R3 ) ⊗ C4 , which is self-adjoint on domain D(B) = H 1 (R3 ) ⊗ C4 .
77
A Appendix
Definition A.5. Let k, p ∈ N. For f ∈ C0∞ (Rn ) the Sobolev norm || · ||k,p is defined as
1/p
n σ i p
XX
∂ f σ 
:= 
,
∂x i i
p
i=1

||f ||k,p
(A.3)
|σ|≤k
where σ ∈ Nn0 is a multiindex and |σ| :=
Pn
i=1
σi = |σ|1 .
Theorem A.6. The completion of the space of bump functions, C0∞ (Rn ), w.r.t. the
Sobolev norm, || · ||k,p , is an alternative way to get the Sobolev spaces:
||·||k,p
W k,p (Rn ) = (C0∞ (Rn ), || · ||k,p )
.
Proof. We refer to the literature, for instance [Eva98].
Remark A.7. Sometimes, for instance in [HS96], the Sobolev spaces are defined in the
first place as completion of C0∞ (Rn ).
For investigating the spectrum of the Dirac operator in section 2.2 we need a characterization of the Sobolev space of order k, H k (Rn ), in terms of the Fourier transformation,
or, more precisely, the Plancherel transformation:
Theorem A.8. Let k ∈ N and let P : L2 (Rn ) → L2 (Rn ) denote the isometric Plancherel
transformation. For a function f ∈ L2 (Rn ) the following statements are equivalent:
(i) f ∈ H k (Rn ) ,
(ii) 1 + |x|k Pf (x) ∈ L2 (Rn ) ,
(iii) (1 + |x|2 )
k/2
Pf (x) ∈ L2 (Rn ) .
Proof. For the equivalence (i) ⇔ (ii) we refer to the literature, for instance [Eva98]. The
implication
(iii) ⇒ (ii) follows immediately. For the other direction (ii) ⇒ (iii) firstly we
R
2k
get Rn |x| |Pf (x)| dx < ∞. We derive
2 k/2
k 1 + |x|
Pf (x)k22
Z
2 k
=
Rn
Z
1 + |x|
2
|Pf (x)| dx =
Z
k X
k
Rn l=0
l
|x|2l |Pf (x)|2 dx ≤
Z
χ(|x| ≤ 1)|Pf (x)|2 dx + Ck
χ(|x| > 1)|x|2k |Pf (x)|2 dx ≤
n
n
R
R
Z
≤ Ck ||f ||22 +
|x|2k |Pf (x)|2 dx < ∞ ,
≤ Ck
Rn
with a k-dependent finite number Ck := (k + 1) maxl=0,...k
used ||Pf ||2 = ||f ||2 < ∞.
k
l
. In the last line we have
Remark A.9. The last Theorem A.8 offers the great possibility to extend the definition
of the Sobolev spaces H k (Rn ) also to non-integer values k ≥ 1. Therefore, one can define
for f ∈ L2 (Rn ): f ∈ H π (Rn ) :⇔ (1 + |x|π )Pf (x) ∈ L2 (Rn ). For further reading we refer
to the literature, for instance [Tar07].
78
A.2 Spherical Harmonics and Laguerre Polynomials
A.2 Spherical Harmonics and Laguerre Polynomials
We consider Schrödinger operators H = − 21 ∆ + V in L2 (Rn ) on a suitable domain D(H).
The potential V is some real-valued function on x ∈ R. In many physical application the
potential obeys some symmetry properties. They are, for instance, cylindrically symmetric, or, as the Coulomb potential V (x) = −Z/|x|, Z ∈ N, even spherically symmetric.
Let us restrict to the Coulomb potential and the physical case n = 3. The main ideas
of the following derivation can be found in physics textbooks, for instance in [Jac06] or
[Sch07]. For classical reading we recommend [AS65].
We switch to the usual spherical coordinates x 7→ (r, θ, ϕ) ∈ [0, ∞) × [0, π] × [0, 2π]
which are suitable coordinates for isotropic systems. Making the factorization ansatz
ψ=
U (r)
P (θ)Q(ϕ) ,
r
(A.4)
and transforming the Laplacian into spherical coordinates, we find for the eigenvalue
equation:
1
Z
U (r)
− (∆r + ∆θ,ϕ ) − − E(r)
P (θ)Q(ϕ) = 0 .
(A.5)
2
r
r
The fact that the eigenvalue E(r) is a function only of the radial coordinate comes from
the spherical symmetry of the Coulomb interaction. We have separated the Laplacian
into a radial, ∆r , and a spherical part, ∆θ,ϕ , with ∆ = ∆r + ∆θ,ϕ :
1
1
2
1
∆r := ∂r2 r = ∂r r ∂r + ∂r = ∂r2 + ∂r ,
r
r
r
r
1
1
∂θ (sin θ ∂θ ) + 2 2 ∂ϕ2 .
∆θ,ϕ := 2
r sin θ
r sin θ
(A.6)
A.2.1 Spherical Harmonics
First we concentrate on the angular part of this equation and consider the Laplace equation
U (r)
∆ψ = (∆r + ∆θ,ϕ )
P (θ)Q(ϕ) = 0 .
(A.7)
r
For ψ 6= 0 this is equivalent to
r2 sin2 θ 2
sin θ
1
∂r U (r) +
∂θ (sin θ ∂θ )P (θ) +
∂ 2 Q(ϕ) = 0 .
U (r)
P (θ)
Q(ϕ) ϕ
(A.8)
Only the third term depends only on ϕ, hence it must be a constant c2 ∈ R:
∂ϕ2 Q(ϕ) = c2 Q(ϕ)
⇒
Q(ϕ) = exp(cϕ) .
(A.9)
The periodic boundary condition ψ(·, ·, ϕ) = ψ(·, ·, ϕ + 2π) implies exp(2πc) = 1. This
means c ∈ iZ. Therefore we introduce a new constant1 m := −ic ∈ Z, thus
Qm (ϕ) = exp(imϕ) .
1
(A.10)
This constant m is called magnetic quantum number.
79
A Appendix
Plugging this into (A.8) and dividing by sin2 θ yields:
1
m2
r2 2
∂r U (r) +
∂θ (sin θ ∂θ )P (θ) −
=0.
U (r)
sin θ P (θ)
sin2 θ
(A.11)
Only the first term depends only on r. Furthermore, only the second and third term
depends only on θ. Therefore, the first term must be a constant, C ∈ R, and the second
and third term must be the same constant with opposite sign. We arrive at:
C
2
∂r − 2 U (r) = 0 ,
r
(A.12)
2
1
m
∂θ (sin θ ∂θ ) + C −
P (θ) = 0 .
sin θ
sin2 θ
The differential equation for P is well-known in mathematical physics. We switch to
x := cos θ ∈ [−1, 1] and meet the general Legendre equation:
d
d
m2
2
(1 − x )
+ C−
P (x) = 0 .
(A.13)
dx
dx
1 − x2
For m = 0 this equation is called Legendre equation. This case relates to a Q(ϕ) = 1
and describes systems with azimuthal symmetry. We address now this case and make for
−1 ≤ x ≤ 1 the following power-series ansatz:
P (x) = x
α
∞
X
aj x j ,
(A.14)
j=0
with aj ∈ R α ∈ R. We have the following
Lemma A.10. Assuming that (A.14) is a solution of the Legendre equation, (A.13) for
m = 0, one has:
(a) a0 6= 0 ⇒ α ∈ {0, 1} and a1 6= 0 ⇒ α ∈ {−1, 0} .
(b) For j ≥ 2, the coefficients are recursively given by
aj = aj−2 ·
(α + j − 2)(α + j − 1) − C
.
(α + j − 1)(α + j)
In particular: a0 = 0 ⇒ a2n = 0 ∀n ∈ N and a1 = 0 ⇒ a2n+1 = 0 ∀n ∈ N.
(c) P (x) is convergent for all −1 < x < 1.
(d) P (x) is convergent for |x| = 1 ⇔ C = l(l + 1) for some l ∈ N0 .
(e) In the case of (d): α = 0 ⇔ l is even, α = 1 ⇔ l is odd.
Proof. The ideas of the proof can be found in [Jac06] and the references therein.
Remark A.11. The decomposition of the constant C = l(l + 1) is crucial for the physical
interpretation2 . As the Lemma tells us, it is necessarily that l ∈ N0 in order to make
2
This constant l is called angular momentum quantum number.
80
A.2 Spherical Harmonics and Laguerre Polynomials
the power series convergent at x ∈ {−1, 1}. In this case, the power series reduces to a
polynomial function with only odd or even powers of x. They can be cataloged by l. We
follow the usual convention and normalize these Legendre polynomials by Pl (1) = 1.
Theorem A.12.
(a) The Legendre polynomials (l ∈ N0 ) are given by Rodrigues’ formula:
1 dl 2
(x − 1)l .
Pl (x) = l
l
2 l! dx
(A.15)
(b) The set
r
[
l∈N0
{Ul : [−1, 1] → R, Ul (x) :=
2l + 1
Pl (x)}
2
is a orthonormal basis of the Hilbert space (L2 ([−1, 1]), h· , ·i) with hUl , Uk i = δl,k .
For a given l ∈ N0 we restrict the constant m ∈ Z to m = 0, 1, . . . , l and define the
associated Legendre polynomials Plm : [−1, 1] → R by
Plm (x) := (−1)m (1 − x2 )m/2
dm
Pl (x) .
dxm
(A.16)
Theorem A.13.
(a) The associated Legendre polynomials (l ∈ N0 , m = 0, ±1, . . . , ±l) are given by
Rodrigues’ formula:
Plm (x) =
l+m
(−1)m
2 m/2 d
(1
−
x
)
(x2 − 1)l .
2l l!
dxl+m
(A.17)
(b) The polynomials with negative m are proportional to those with positive m:
−|m|
Pl
(x) = (−1)|m|
(l − |m|)! |m|
P .
(l + |m|)! l
(A.18)
(c) The associated Legendre polynomials Plm (l ∈ N0 , m = −l, −l + 1, . . . , l − 1, l) are
solutions of the general Legendre equation (A.13).
(d) The set
s
[
l∈N0
{Ulm : [−1, 1] → R, Ul (x) :=
2l + 1 (l − m)! m
P (x)}
2 (l + m)! l
is for all m = −l, −l + 1, . . . , l − 1, l an orthonormal basis of the Hilbert space
(L2 ([−1, 1]), h· , ·i) with hUlm , Ukm i = δl,k .
We have already found the solutions Qm (ϕ) = exp(imϕ), m ∈ Z, of the azimuthal angle
(A.10). As we know, the set
1
{Um : [0, 2π] → C, Um (ϕ) = √ exp(imϕ)}
2π
m∈Z
[
(A.19)
81
A Appendix
forms an orthonormal basis of the Hilbert space (L2 ([−1, 1]), h· , ·i) with hUm , Un i = δm,n .
Therefore, the product of the Legendre polynomials Plm (cos θ) and the functions Qm (ϕ)
(with the corresponding prefactors) form a basis of square-integrable functions on the
surface of a sphere: we define the spherical harmonics Ylm : R2 → C by
s
2l + 1 (l − m)! m
1
(A.20)
P (cos θ)eimϕ ,
Ylm (θ, ϕ) := √
2 (l + m)! l
2π
again for l ∈ N0 and m = −l, −l + 1, . . . , l − 1, l.
Remark A.14. The prefactors and the orthogonality of both, Plm and Qm , imply that
the spherical harmonics fulfill: hYlm , Ykn i = δlk δmn . Furthermore, Ylm (θ + π, ϕ + 2π) =
(−1)l+m Ylm (θ, ϕ). The periodicity of ϕ is obvious. The changing sign comes from the
identity Plm (−x) = (−1)l+m Plm (x), which is clear from (A.15) and (A.16).
A.2.2 Laguerre Polynomials
We concentrate now on the radial part of (A.5). Since the Coulomb potential is only
negative, V (r) = −Z/r < 0, we investigate this eigenvalue equation for E < 03 . Using
the result that P (θ)Q(ϕ) = Ylm (θ, ϕ) are eigenfunctions of the spherical Laplacian, ∆θ,ϕ ,
with eigenvalue −C/r2 = −l(l + 1)/r2 , (A.5) reads:
1 2 l(l + 1) Z
− ∂r +
− + |E| u(r) = 0 ,
(A.21)
2
2r2
r
for a function u : [0,∞) → R. We have used the first form of the radial Laplacian (A.6).
We follow the calculation in [Sch07] and switch to the following quantities:
ρ :=
p
2Z
2|E|r and ρ0 := p
2|E|
⇒
V
ρ0
=− .
|E|
ρ
Therefore, the eigenvalue equation (A.21) reads:
ρ
l(l + 1)
2
+
− 1 u(ρ) = 0 .
∂ρ −
ρ2
ρ0
(A.22)
(A.23)
As in the discussion of the spherical part of the Laplacian, we make a power-series ansatz:
u(ρ) = ρ
l+1 −ρ
e
∞
X
ak ρ k .
(A.24)
k=0
The condition, that the solution is not exponentially divergent for large ρ, i.e. for large r,
implies that there must be some N ∈ N0 such that4 ρ0 = 2(N + l + 1) =: 2n. Therefore
n ∈ N and l = 0, 1, . . . , n − 1. From this, we can catalog the eigenvalues by n and find
3
Indeed, for E ≥ 0 this eigenvalue equation has no solution. This follows from the fact that σess (H) =
[0 , ∞) (see Theorem 2.36), hence E ≥ 0 cannot be an eigenvalue.
4
The constant N is called radial quantum number and the constant n is called principal quantum
number.
82
A.2 Spherical Harmonics and Laguerre Polynomials
directly from (A.22):
p
Z 2|E|r
Vρ
|E| = −
=
ρ0
2nr
⇒
Z2
0>E=− 2 .
2n
(A.25)
For the eigenfunction we consider the general Laguerre equation for Lsr : [0,∞) → R:
2
(A.26)
x∂x + (s + 1 − x)∂x + (r − s) Lsr (x) = 0 .
For s = 0 this equation is called Laguerre equation. We have the following
Theorem A.15.
(a) The associated Laguerre polynomials, Lsr , given by Rodrigues’ formula5
Lsr (x) =
ex x−s dr−s x −x r e e x ,
(r − s)! dxr−s
with r, s ∈ N0 , r ≥ s, are solutions of the general Laguerre equation.
(b) Furthermore, the associated Laguerre polynomials are given explicitly by
Lsr (x)
=
r−s
X
k=0
(−1)k
(r!) xk
.
k!(k + s)!(r − k − s)!
(c) For the normalization one has
Z ∞
(2r − s + 1)(r!)
xs+1 e−x [Lsr (x)]2 dx =
.
(r − s)!
0
Finally we can state the solution for (A.21) and, what was the claim of this whole section,
for (A.5). Ylm denote the spherical harmonics and Lsr denote the associated Laguerre
polynomials.
Theorem A.16. The eigenvalue equation (A.5) with eigenvalue En = −Z 2 /2n2 < 0 is
solved by ψnlm : [0,∞) × [0, π] × [0, 2π] → C with
ψnml (r, θ, ϕ) = Z 3/2 Rnl (Zr)Ylm (θ, ϕ) ,
where n ∈ N0 , l = 0, 1, . . . , n − 1 and m = −l, −l + 1, . . . , l − 1, l are admissible. Thereby
we have defined
s
l
U (r)
2 (n − l − 1)! −r/n 2r
Rnl (r) :=
= 2
e
L2l+1
n+l (2r/n) .
r
n
(n + l)!
n
5
In the literature there are several conventions for the Laguerre polynomials. Quite frequently they are
denoted by L̃kn (x), with n, k ∈ N0 , k ≤ n. These functions coincide with those in our convention when
identifying s ↔ k and n ↔ r − s.
83
A Appendix
The l, m–dependent prefactor of Rnl (r) is due to the normalization of Lsr . We want to
emphasize at this point, that Rnl (r) is independent of the magnetic quantum number m.
Furthermore, expressing the spherical symmetry of the Coulomb potential, the energy
eigenvalues En < 0 depend only on the principal quantum number n. However, we
have defined the solutions ψnlm in spherical coordinates. Transforming back to Cartesian
coordinates, these eigenfunctions are functions on R3 and, indeed, square integrable.
A.3 Notations in Relativistic Quantum Mechanics
The Pauli matrices σi , for i = 1, 2, 3, are defined as
0 1
0 −i
1 0
σ1 =
, σ2 =
, σ3 =
.
1 0
i 0
0 −1
(A.27)
All Pauli matrices are traceless, trace σi = 0, and hermitian, σi† := σiT∗ = σi . Even more,
any traceless hermitian 2 × 2 matrix can be written as some unique linear combination of
the Pauli matrices.
The Pauli matrices obey the following commutator relation: [σi , σj ] = 2iεijk σk , where
εijk denotes the totally antisymmetric tensor

 1 (ijk) = (123) or any cyclic permutation,
−1 (ijk) = (213) or any cyclic permutation,
εijk =
(A.28)

0
else.
In physics they are closely related to spin- 12 -particles: the operator describing their spin
angular momentum is given by S = 12 σ, compare Definition 3.2. When describing rela4
2
tivistic Dirac spinors
in C instead of non-relativistic Weyl spinors in C , the spin operator
σ 0
reads S ⊗ 12 = 21
=: 21 Σ.
0 σ
From a group-theoretic point of view, the Pauli matrices are just the infinitesimal generators of the three-dimensional Lie group SU(2). For this enlightening branch of group
theory we refer to the literature, for instance [FH91] and many physics textbooks. Important results are Stone’s Theorem, discussed in [RS75], and Nelson’s Theorem, discussed
in [Tha92].
84
Bibliography
[AS65] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables. Dover, 1965.
[CHD05] N. G. Connelly, R. M. Hartshorn, and T. Damhus. Nomenclature of Inorganic
Chemistry: IUPAC Recommendations 2005. Royal Society of Chemistry, 2005.
[Eva98] L. C. Evans. Partial Differential Equations. American Mathematical Society,
1998.
[FG09a] G. Friesecke and B. D. Goddard. Asymptotics-Based CI Models for Atoms:
Properties, Exact Solution of a Minimal Model for Li to Ne, and Application to
Atomic Spectra. Multiscale Modeling & Simulation, 7(4):1876–1897, 2009.
[FG09b] G. Friesecke and B. D. Goddard. Explicit large nuclear charge limit of electronic
ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table.
SIAM Journal on Mathematical Analysis, 41:631–664, 2009.
[FG10] G. Friesecke and B. D Goddard. Atomic structure via highly charged ions and
their exact quantum states. Physical Review A, 81:032516, 2010.
[FH91] W. Fulton and J. Harris. Representation Theory. Springer, 1991.
[Fri03] G. Friesecke. The Multiconfiguration Equations for Atoms and Molecules: Charge
Quantization and Existence of Solutions. Archive for Rational Mechanics and
Analysis, 169(1):35–75, 2003.
[Fri07] G. Friesecke. Quantum Mechanics for Mathematicians. Lecture notes, 2007.
[GGM03] F. Gazzola, H. Ch. Grunau, and E. Mitidieri. Hardy Inequalities with Optimal
Constants and Remainder Terms. Transactions of the American Mathematical
Society, 356(6):2149–2168, 2003.
[HS96] P. D. Hislop and I. M. Sigal. Introduction to Spectral Theory with Applications
to Schrödinger Operators. Springer, 1996.
[Jac06] J. D. Jackson. Klassische Elektrodynamik. de Gruyter, 2006.
[Jen06] F. Jensen. Introduction to Computational Chemistry. Wiley, 2006.
[Kan09] E. Kaniuth. A Course in Commutative Banach Algebras. Springer, 2009.
[N+ 10] K. Nakamura et al. Review of Particle Physics. Journal of Physics G, 37:075021,
2010.
[Pau40] W. Pauli. The Connection between Spin and Statistics.
58(8):716–722, 1940.
Physical Review,
85
Bibliography
[Per83] P. A. Perry. Scattering Theory by the Enss Method. Mathematical Reports, 1983.
[PS95] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory.
Westview Press, 1995.
[RS75] M. Reed and B. Simon. II - Fourier Analysis, Self-adjointness. Elsevier, 1975.
[RS78] M. Reed and B. Simon. IV - Analysis of Operators. Elsevier, 1978.
[RS80] M. Reed and B. Simon. I - Functional Analysis. Elsevier, 1980.
[Rud91] W. Rudin. Functional Analysis. McGraw-Hill, 1991.
[Sch05] F. Schwabl. Quantenmechanik für Fortgeschrittene. Springer, 2005.
[Sch07] F. Schwabl. Quantenmechanik. Springer, 2007.
[Tar07] L. Tartar. An Introduction to Sobolev Spaces and Interpolation Spaces. Springer,
2007.
[Tha92] B. Thaller. The Dirac Equation. Springer, 1992.
[Ves69] K. Veselic. The nonrelativistic limit of the Dirac equation and the spectral concentration. Glasnik Mathematicki, 4:231–240, 1969.
[Yu.10] Yu. Ralchenkom, F.-C. Jou, D. E. Kelleher, A. E. Kramida, A. Musgrove, J.
Reader, W. L. Wiese, and K. Olsen. NIST Atomic Spectra Database (version
4.0.1) [Online], 2010. Available: http://physics.nist.gov/asd.
86
Acknowledgement
In these final lines I try to express my gratitude to all the people who have supported
me during my studies and particularly during the last six months when working on this
thesis.
During more than five years Prof. Dr. Gero Friesecke has taught me in many lectures
and discussions in both mathematics and physics: at the beginning in the elementary
calculus courses, later on in lectures which have combined abstract mathematical formalism and problems arising in physics. I have learnt from him that mathematics which is
applied to physical questions helps to learn more about both issues. I am very thankful
to Prof. Friesecke for his confidence in me and for asking me to work on the perturbation
theory model in this diploma thesis and in the prior project thesis. I also thank you for
the opportunity to attend the workshop at Oberwolfach about “Mathematical Methods
in Quantum Chemistry” in the upcoming month.
During my whole studies I have been accompanied by Stefan Kahler. At the beginning
I have known him as a fellow student, then it emerged that we are indeed neighbors, and
actually we have become friends. During many night sessions we have discussed problems
arising not only in mathematics or physics and have learnt from each other. Thank you
for all your support, especially for preparing and testing me for upcoming exams and for
proofreading my diploma thesis.
I know that I am repeating myself, but, really, it is impossible to write a diploma thesis
within isolation! Also during my official absence at T39 in the last six months I have
found here always an open door for discussions and meetings. Thank you all very much
for your support.
Auch wenn ich eine etwas andere Meinung zur Schule, insbesondere zur Mathematik,
vertrete, so danke ich Reinhard Mey für seine wunderbaren Texte, Melodien und Gedanken.
Beinahe täglich begleiten sie mich; real oder oft auch rein imaginär.
Last but not least I am indebted to my parents and my sister. Everything I have reached
so far in my life I owe to my family. They have provided a comfortable environment I was
always able to pursue intensely my interests.
87