Electron Correlation Dynamics in Atomic CollisionsElectron Correlation Dynamics in Atomic Collisions
Chapter 1 Introduction
1.1 Probability of a transition
1 Single particle probability
2 Two particle probability
3 N particle binomial probability
4 Multinomial probabilities
1.2 Cross sections and reaction rates
1 Differential cross sections and transition probabilities
2 Total cross section
1.3 Quantum features
1 The Hamiltonian
2 The electronic wave amplitude
3 The transition probability
1.4 Correlation
1 Definition of correlation
2 Correlated probabilities
3 Other applications
Chapter 2 Single electron transition probabilities
2.1 Formulation
1 Complete atomic system
2 Atomic electrons
3 Evolution of an electron
4 Partial wave expansion
2.2 Excitation probabilities
1 1s - 2s,2p transitions
2 n,l,m -> n,l',m' transitions
2.3 Mass transfer probabilities
1 n,l,m -> n',l ',m' transitions
2 1s - 1s transitions
3 Special features of electron capture
2.4 Ionization probabilities
1 Continuum of the target
2 Continuum of the projectile
2.5 Long range Coulomb effects
1 Continuum of target and projectile
2.6 Other methods
1 Strong electric fields
2 Strong projectile fields
3 Over the barrier model
2.7 Observations
2.8 Appendix
1 Classical cross sections
2 Tables of ionization cross sections and probabilities
Chapter 3 Formulation of multi-electron transition probabilities
3.1 Terms in the Hamiltonian
3.2 The many electron wavefunction
1 Separation of the electronic from the nuclear wavefunction
2 Interaction representation
3 The evolution operator
3.3 Transition probability and cross section
1 Multi-electron effects and electron correlation
2 Scattering, relaxation and asymptotic regions
3 Waves and particles
4 Long range Coulomb terms
5 Exclusive and inclusive cross sections
6 The active electron approximation
7 Electron exchange
3.4 Methods of computation
3.5 Classification of multiple electron transitions
3.6 Appendix
1 Separation of electronic terms
Chapter 4 Independent electron approximation
4.1 Single atoms
1 Full atomic Hamiltonian
2 Uncorrelated Hamiltonian
3 Binomial distributions
4 Comparison to experiment
4.2 Systems of atoms
1 Full Hamiltonian for systems of atoms
2 Independent subsystems
4.3 Time dependent Dirac Fock
Chapter 5 Statistical methods
5.1 Statistical energy deposition model
5.2 Fokker Planck method
1 Fokker Planck equation
2 Application to atomic collisions
3 Simple model
5.3 Maximum entropy method
5.4 Intermediate systems
Chapter 6 Correlated multi-electron electron transition probabilities
6.1 Correlation
1 Asymptotic states
2 Scattering operator
3 Analytic expressions
4 Multi-center correlation
5 Dielectronic processes
6.3 Shake probabilities
1 Simple shake factors
2 Generalized shake factor
3 High energy ratios in helium
6.4 The Liouvillian
1 The Hamiltonian as a generator of dynamics for little systems
2 The Liouvillian as a generator of dynamics for big systems
Chapter 7 Perturbation expansions
7.1 Formulation
1 Binomial distributions with shake
2 Time ordering
3 Spatial correlation
7.2 Method of expansion in the scattering potential V
1 Scattering amplitude
2 Pseudostates
3 One electron amplitudes in pseudostates
7.3 Many body perturbation theory
1 Diagrams and matrix elements
7.4 Observations
1 Z^3 terms
2 Cross sections and ratios
Chapter 8 Projectiles carrying electrons
8.1 Formulation
1 Basic ideas
8.2 The first Born approximation
1 The effective projectile charge
2 Virtual impact parameter method
3 Application to the independent electron approximation
8.3 Examples
1 Screening terms
2 Antiscreening terms
8.4 Impulse approximation
8.5 Observed results
Chapter 9 Reactions with photons
9.1 Introduction
9.2 The Hamiltonian
1 Dipole approximation
2 L,V and A forms of the dipole operator
9.3 Photoionization (Photo effect)
1 Gauge dependence
9.4 Compton and Raman scattering
1 Compton scattering
2 Compton scattering and photoionization
3 Raman scattering
9.5 Observations
9.6 Appendix
1 Gauge transformations
2 Two electrons in a magnetic field
3 Two electrons in an electromagnetic field
4 Mathematical scattering factors
5 Expansion parameters
6 Names of processes
Chapter 10 Relations between charged particle and photon reactions
10.1 Multipoles
1 Ratios of cross sections
2 Compton and charged particle scattering
10.2 Dipole terms
10.3 Dipole and non-dipole terms
1 Ratios of cross sections differential in energy transfer
2 Ratios of total cross sections
10.4 Observations
Appendices
A.1 Hydrogenic wavefunctions
1 Spherical harmonics
2 Coordinate space
3 Momentum space
A.2 A special integral
A.3 The Green function
A.4 Equivalence of observables in the particle and wave pictures
A.5 Binomial and multinomial moments
A.6 N body systems
One of the central questions of science is: how are complex things made from simple things? In many cases larger systems are more complicated than their smaller subsystems. In biology and chemistry the issue is how to understand large molecules in terms of atoms. In atomic physics one may strive to understand properties of many electron systems in terms of single electron properties. The general theme is interdependency of subsystems, or `correlation'.
Correlation may be regarded as a conceptual bridge from properties of individuals to properties of groups or families. In atoms and molecules correlation occurs because electrons interact with one another -- the electrons are interdependent. This electron correlation determines much of the structure and dynamics of many electron systems, i.e., how complex electronic systems are made from single electrons. Complexity is the more significant idea, but complexity may be seldom, if ever, understood. Correlation is the key to complexity.
Understandably, much has been done on the correlation of static systems. There are many excellent methods and computer codes to evaluate energies and wavefunctions for complex atomic and molecular systems. However, the dynamics of these many electron systems is less well understood. Hence, the dynamics of electron correlation is a central theme in this book.
The dynamics of electron correlation may affect single electron transitions. However, this effect is sometimes difficult to separate from other effects. Correlation is usually dominant in multiple electron transitions for fast collisions since there is not enough time for the collision partners to interact more than once. This means that multiple electron transitions in fast collisions provide an unobstructed view of the dynamics of electron correlation.
Since there are numerous processes that depend on the transition of more than a single electron, multiple electron transitions are of interest in their own right. Also recent advances in the production of highly charged ions and in synchrotron radiation provide new experimental tools for studying multiple electron transitions.
So the motivation for this book is to look at an emerging topic that is of interest in its own right and that impacts on a question of broad scientific interest, namely, the question of correlation (or the many body problem, or the construction of complex atomic systems ...).
The audience for whom this book is intended includes specialists in the field, graduate students and engineers. The first chapter includes a review of basic concepts in simple probability, classical scattering theory and a brief introduction to quantum mechanics. Then a background in one electron transitions in atomic and molecular transitions is developed. The third chapter addresses the formulation of interactions in many electron atomic and molecular systems. Later chapters deal with interactions of many electron targets with charged particles, with projectiles carrying electrons, and with photons.
An effort has been made to include practical methods for analysis and interpretation of data which can be used without extensive theoretical background or large computer codes. While much of this book focuses on relatively simple fast interactions, examples for intermediate and slow interactions related to chemistry, biology and condensed matter are also included. The emphasis in this book is on ideas and techniques that are simple enough for a beginning graduate student to pick up quickly and interesting enough to attract the attention of a person with a little curiosity.
The following is the LaTeX file for the introduction to
Chapter 6 on Correlation.
\section{Correlation}
There is a significant difference between complex\index{complexity!meaning of}
and merely large. This difference is related to the the notion of
correlation\index{correlation}\index{correlation!definition}
which defines the rules of interdependency\index{interdependency}
in large systems\index{correlation!dynamic}.
The relevant question is: how may one make complicated things from simple ones?
Biological systems are complex because the atomic and molecular subsystems
are correlated. From the point of view of atomic physics correlation in
condensed matter\index{condensed matter}, chemistry\index{chemistry} and
biology\index{biology} is determined at least in part by electron
correlation\index{correlation!electron}\index{multiple electron!N body}
in chemical bonds\index{bonds} and the complex interdependent structures of electronic
densities\index{density, electron}.
Understanding correlation in this broad sense is a major challenge
common to most of science and much of technology. This is
sometimes referred to as the many body problem\index{many body problem}.
In a general sense correlation is a conceptual bridge
from properties of individuals to properties of groups or
families\index{correlation!families and individuals}.
The concept of correlation\index{correlation!common examples}
arises in many different contexts.
`Individual' may mean an individual electron, an individual
molecule or in principle an individual person, musical note
or ingredient in a recipe. In this book individual refers to
electron for the most part. In this case the interaction
between individuals is well known, namely $1/r_{12}$. However, that does not mean
that electron correlation is well understood in general. Although much has been done
to investigate correlation in various areas of physics, chemistry,
statistics\index{statistics}, biology and materials science, in many cases
little is well understood except in the limit of weak correlation.
In this weak limit one may begin to address the question of
what correlation is and how it works. A more fruitful
question for either more complex\index{complexity!meaning of} systems or exact calculations
of strongly correlated simple systems may be `what are the patterns
(or rules) of correlation?'. Perhaps both the challenge
and the reward in understanding correlation lies in finding
better ways to ask the question.
\begin{sloppypar}
Correlation may be defined and studied in both
space\index{correlation!spatial} and time\index{correlation!time}.
Examples of time correlation include time ordering
effects\index{time ordering}
[section 7.1.1] and memory\index{memory}. Most studies have been
confined to two particle or pair correlation since experimental studies of
higher order correlation are often difficult.
The emphasis in this text is on spatial correlation.
\end{sloppypar}
Correlation\index{correlation!definition}
may be defined mathematically\index{correlation!mathematical}
as the deviation from a product of single particle terms,
e.g., $\psi(r_1,r_2,...r_n) - \psi_1(r_1)\psi_2(r_2)...\-....\psi_n(r_N)$.
So correlated means non-separable or interdependent.
Physically, correlation\index{correlation!physical}
between $N$ electrons occurs because the
electrons influence one another, i.e. they interact with one another.
This interaction\index{correlation!interaction}
\index{potential!correlation}\index{correlation!potential}
is called the correlation interaction $v$ and is the
difference between the Coulomb interaction between the electrons
and the mean field\index{mean field potential} of the electrons.
Thus, correlation in a wavefunction\index{correlation!wavefunction}
is the difference between the given wavefunction and a
Hartree Fock\index{Hartree Fock} wavefunction
(a product wavefunction found using a mean field\index{mean field potential}
for the electrons symmetrized to
include the identical nature of the electrons).
The electron symmetry is neglected in the simpler Hartree approximation\index{Hartree approximation}.
In practice one defines correlation by defining the
uncorrelated\index{correlation!uncorrelated limit}\index{uncorrelated} limit.
For time dependent systems considered in this book the uncorrelated
limit is defined by the time dependent Dirac Fock
(TDDF)\index{time dependent Dirac Fock (TDDF)} approximation,
which is the relativistic generalization of the Hartree Fock approximation.
Any quantity that is uncorrelated in one representation (e.g. $b$ space)
is also uncorrelated in the conjugate space of the Fourier transform
(e.g. $q$ space)\index{correlation!waves and particles}\index{waves and
particles!correlation}\index{particles and waves!correlation}.
Mathematical correlation has been studied
in a broad variety of systems.
There are, for example, statistical studies
of correlation\index{correlation!statistical}
\index{statistics}
in the spatial patterns of ${\rm H}_2{\rm O}$ molecules
in water\index{correlation!water} [Stanley, 1996],
sequencing of codons in DNA\index{correlation!DNA}\index{DNA}
[Peng {\it et al.}, 1995a],
heartbeat intervals in healthy and diseased
hearts\index{correlation!heartbeats}
[Peng, {\it et al.}, 1995b]
feeding patterns of birds\index{correlation!bird
feeding patterns} [Viswanathan {\it et al.}, 1996],
corporate growth\index{correlation!corporate growth}
[Stanley {\it et al.}, 1996],
and even a possible description of free will
\index{correlation!free will}\index{free will}
[Stanley, 1996].
In these studies correlation has been used
to bring `order out of randomness'.
The surprising strength of these long range
correlations and their effect on scale invariance
\index{scale invariance}\index{correlation!scale invariance}
has been ascribed to a multiplicity of pathways
between individuals. Without the multiplicity
of pathways the correlation would decrease exponentially
with distance rather than as a more slowly decaying
finite power law.
Memory\index{memory}\index{correlation!memory},
a key element of living consciousness\index{consciousness}
\index{correlation!consciousness}, is also non-localized
\index{non-locality}\index{correlation!non-locality}.
Memory is not contained in just one single nerve cell,
but the information\index{correlation!information}
\index{information} is is stored in a number of cells
which interact with one another [van Hemmen, 1996].
Thus correlation is used to understand complex
systems\index{complexity!meaning of}\index{correlation!complexity}.
Like correlation complexity is defined by its opposite.
`Complex' means `not simple'.
The connection of these more general examples of
statistical correlation to electron correlation
has not yet been carefully studied.
The term `electron correlation'\index{correlation!first use of}
seems to have been first used in
physics in 1933 by Wigner and Seitz\footnote{Wigner [Rhoades, 1986, p.35] commented that,
`Physics does not even try to give us
complete information about the events around us --
it gives information about the correlation
between those events.'}\footnote{Somewhat earlier Mozart elegantly noted, \\
\hspace{3em} `Nor so I hear in my imagination the parts successively, \\
\hspace{3em} but I hear them, as it were, all at once.' }.
The modern definition of correlation in
static\index{correlation!static} atomic
systems was established by L\"{o}wdin [1959, 1995].
Many of the methods now used to determine correlation in electronic wavefunctions
were introduced by Kutzelnigg, Del Re and Berthier [1968],
who point out some earlier uses of correlation in mathematical statistics\index{statistics}.
Significant contributions have been made by Davidson [1976] and
by Froese Fischer [1996, 1977].
Correlation in single electron transitions\index{correlation!single
electron transitions} has been reviewed by Webster {\it et al}. [1978] and by Crowe [1987].
Electron correlation in molecules\index{correlation!molecules} and
solids\index{correlation!solids}\index{condensed matter} is discussed in
the book of Fulde [1995].
Ziesche [1995] has suggested a connection between correlation,
which may be used to describe the degree of mixing in a dynamic system,
and entropy\index{entropy}\index{correlation!entropy}.
Since understanding properties of macroscopic materials in terms of microscopic atomic
properties is interesting, it is sensible to note
stochastic\index{correlation!stochastic uses}
uses of correlation [Balescu, 1975; Huang, 1987] since statistics are
ultimately useful in dealing with large numbers of atoms.
Correlation is often introduced in stochastic and statistical mechanics\index{statistics}
[Balescu, 1975] as a generalization of the notion of
statistical\index{correlation!statistical mechanics} standard
deviation of a distribution\index{correlation!standard deviation},
\begin{eqnarray}
\sigma^2 = < (x-)^2> \; .
\label{6.1.1}
\end{eqnarray}
\noindent If $\sigma^2$ is not zero then the distribution is not localized
at a single value $$, but rather the distribution is spread out. This
leads to the idea that correlated functions are also spread out, i.e., not
confined to a single $x_i$ (e.g. denoting a single particle), but connect $x_i$
and $x_j$.
Thus, if $\cal{P}$ is some physical property, then $\cal{P} - <\cal{P}>$
is a measure of the correlation of the property $\cal{P}$.\footnote{In a
general sense theoretical physics
has moved from strict causality (perfect correlation) in the nineteenth century
to quantum mechanics (variable correlation) in the twentieth century by expanding
the idea and use of correlation. The notion of
chaos\index{correlation!chaos}, which usually implies
weak correlation, is a further expansion of the concept of correlation. Thus,
correlation ranges from fully causal $\delta$-like distributions to totally random
distributions. It has been suggested that the notion of a single exact result
is an ideal limit of perfect correlation between variables,
and further that this notion of unique truth may be unnecessarily
restrictive [McGuire, J.H., 1995; Fish, 1982; Derrida, 1988, 1976].}
\vskip 1em
{\it BBGKY hierarchy.}
Consider the one particle\index{BBGKY hierarchy}\index{density, electron}
\index{correlation!function!BBGKY}
distribution function\index{probability distribution!BBGKY hierarchy}
given by,
\begin{eqnarray}
f_1(r_i) = \rho(r_i) =\psi^*\psi \; .
\label{6.1.3}
\end{eqnarray}
\noindent If $f_2(r_1,r_2) = f_1(r_1)f_1(r_2)$,
the two particle distribution\index{probability distribution!two particle}
function $f_2$ is uncorrelated.
In general, however, $f_2$ is correlated and may be written as,
\begin{eqnarray}
f_2(r_1,r_2) = f_1(r_1)f_1(r_2) + g_2(r_1,r_2) \; ,
\label{6.1.4}
\end{eqnarray}
\noindent where $g_2$ is called the two particle
correlation\index{correlation!two particle correlation function} function.
For three particles one may generally write,
\begin{eqnarray}
f_3(r_1,r_2,r_3) &=& f_1(r_1)f_1(r_2)f_1(r_2) \EB
&+& f_1(r_1) g_2(r_2,r_3) + f_1(r_2)g_2(r_1,r_3) + f_1(r_3)g_2(r_1,r_2) \EB
&+& g_3(r_1,r_2,r_3) \; ,
\label{6.1.5}
\end{eqnarray}
\noindent and so forth for $f_4,f_5...f_N$,... This is called a cluster
expansion\index{cluster expansion}
of the distribution function $f_N$ in terms of the $j^{th}$ order
correlation\index{correlation!jth order correlation function}
functions $g_j$.
The $N+1$ particle distribution function can be generated from the $N$ particle
distribution function via the two particle correlation
potential\index{correlation!potential} $v(r_i-r_j)$.
This is referred to as the BBGKY (Bogoliubov-Born-Green-Krikwood-Yvon)
hierarchy\index{BBGKY hierarchy}\index{correlation!function!BBGKY}
[Balescu, p86].
\vskip 1em
{\it Generalized correlation coefficient.}
If $\cal{P}$ is a one particle operator,
then for a two electron system characterized by a distribution
function $f_2$ the generalized correlation coefficient
$\tau$\index{correlation!generalized correlation coefficient}
is defined as follows,
\begin{eqnarray}
\tau_f = {\int\int g_2(x_1,x_2) {\cal P}(x_1) {\cal P}(x_2) dx_1dx_2\over (\int f_1(x)
{\cal P}^2(x)dx) - (\int f_1(x) {\cal P}(x)dx)^2} \; .
\label{6.1.2}
\end{eqnarray}
\noindent Here, (i) $\tau=0$ if the electrons are uncorrelated, and
(ii) $\vert\tau\vert<1$. Most importantly, $\tau$ is independent of choice
of basis functions. This generalized correlation coefficient has been
applied by Christensen-Dalsgaard [1988] to atomic structure, but
application has not yet been made to dynamic correlation in atoms and
molecules.
\vskip 1em
{\it Degree of correlation K} [Grobe {\it et al}., 1994].
A canonical representation\index{correlation!degree of correlation}
of\index{degree of correlation K}
an exact wavefunction $\Psi(r_1,r_2)$ for a two electron
system is to expand $\Psi$ in terms of single particle Slater determinants $\psi(r_1,r_2)$
[Cowan, 1981], namely,
\begin{eqnarray}
\Psi(r_1,r_1) = \sum_i c_i \ \psi_i(r_1,r_2) \; .
\label{6.g.1}
\end{eqnarray}
\noindent
Here $\Psi$ may be regarded as an entangled
state\index{entangled states}\index{correlation!relation to
entangled states} (i.e., sum) in the set of pure states $\{\psi\}$.
The Slater determinant above describes orthonormal single particle orbitals
for fermions and a product of identical orbitals for bosons.
The normalization condition on $\Psi$ requires that $\sum_i |c_i|^2 = 1$.
The average probability $P_i^{av}$ is given by $\sum_i |c_i|^4$.
Note that $P_i^{av} \leq 1$ and that $P_i^{av} = 1$ corresponds to a
wavefunction represented by a single orbital, which is uncorrelated.
The inverse of this is an average `number' of effectively non-zero mixing probabilities,
which is a way to measure correlation (or extent of entanglement).
Thus, the degree of correlation is defined as,
\begin{eqnarray}
K = (P_i^{av})^{-1} = \left[ \sum_i |c_i|^4 \right]^{-1} \; .
\label{6.g.2}
\end{eqnarray}
Correlation mixes otherwise independent particle orbitals.
$K$ is a measure of the degree of mixing which corresponds to information
loss\index{correlation!entropy} from a `pure' uncorrelated state.
Consequently, the logarithm of $K$ is sometimes called the
St\"{u}ckelberg entropy\index{entropy}.
Grobe {\it et al}. [1994] have used this degree of correlation $K$ to analyze both electron
hydrogen scattering and photo detachment of electrons from atoms.
Ziesche [1995] has developed a quantum kinematic measure of correlation
strength related to entropy and information
theory\index{correlation!information theory} as a distributive property
of an $N$ body system. Gersdorf {\it et al}. [1996] have used this method to
describe the correlation entropy\index{correlation!entropy}
of the ${\rm H}_2$ molecule.
Correlation is important in chemistry\index{correlation!chemistry}
and biology\index{correlation!biology}.
For example neither life nor death nor most of the
processes in between may be described in terms of the
transitions of a single isolated electron.
Correlation is necessary.