UNIVERSITY OF LJUBLJANA
FACULTY OF MATEMATICS AND PHYSICS
SEMINAR II
Electron-electron correlation in He
Author: Jure Kokalj
Mentor: doc. dr. Matjaž Žitnik
May 19, 2005
Abstract
In this seminar we deal with electron-electron correlation in atoms
and how it is taken into account by some computational methods
(Hartree-Fock and Configuration Interaction). For that purpose we
study excited states of helium, which is the simplest correlated atomic
system. At the beginning good quantum numbers are presented and
approximate wave function with adequate symmetries is constructed.
Firstly, Hartree-Fock method is briefly discussed and its agreement
with experimental observations. To achieve better description of reality Configuration Interaction method is presented. To get more insight
into the nature of electron-electron correlations reduced probability
densities are constructed. It is shown that some properties of doubly excited states, like stability against autoionization is reflected in
reduced probability density maps.
1
Contents
1 Introduction
3
2 Characterization of He states
3
3 Calculation of states in He
4
3.1 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.1 Experiments and Hartree-Fock model . . . . . . . . . . 6
3.2 Configuration Interaction method . . . . . . . . . . . . . . . . 10
4 Reduced probability density
11
4.1 Examples of reduced probability density . . . . . . . . . . . . 14
5 Autoionization
16
6 Conclusion
17
2
1
Introduction
Helium atom is the simplest atom for which we cannot find the exact solution
of Schrödinger equation for stationary states
Ĥψ = (−
h̄2 2
h̄2 2
2e2
2e2
e2
∇1 −
∇2 −
−
+
)ψ = Eψ
2m
2m
4π0 r1 4π0 r2 4π0 r12
(1)
This 6-dimensional problem (3 coordinates for each electron) includes noncentral term - Coulomb repulsion of electrons. Due to that, we are forced
to search for the approximate solutions. These can be found by several
mathematical methods (Hartree and Hartree-Fock [1, 2], Multiconfiguration
Hartree-Fock (MCHF)[2], configurational interaction (CI) method, quasiseparabilty of wave functions in hyperspherical coordinates [3, 4], etc.). Some
methods give better results than the others, but in this seminar we are particularly interested in how they approach to the problem electron-electron
correlation. For example, Hartree method describes electrons moving independently in screened central-symmetric potential of the nucleus and other
electron. Thus, the movement of electrons is statistically independent and
there is no correlation among them [5]. So Hartree and Hartree-Fock methods are bad choice for describing correlations. We will rather use large-scale
conventional configuration interaction (CI) method, which is able to accurately describe atomic wave functions.
To achieve sufficient agreement with experiments the models need fully correlated wave functions. As we will see later, the correlation is especially
high when both electrons are excited (doubly excited state) and have to be
considered in zero order approximation. On the other hand, methods used
for solving (1) are mostly mathematical manipulations with hidden physical
meaning. The study and systematization of correlation offers an insight into
physical phenomena which can lead to the solution of the above problem.
2
Characterization of He states
First we look for the quantum numbers that are appropriate for describing
states of He. Since Hamilton operator (Ĥ) doesn’t prefer any particular direction in space, the square of total orbital angular momentum of electrons
L̂2 = (ˆl1 + ˆl2 )2 commutes with it. Therefore the total orbital angular momentum (L) and it’s projection on z-axis (ML ) are good quantum numbers.
There are no spin operators in Hamiltonian and therefore the square of total
spin Ŝ 2 = (ŝ1 + ŝ2 )2 commutes with Ĥ. So, S and MS are good quantum
numbers too.
3
The states with different ML or different MS are degenerate. We will use spectroscopic symbols 2S+1 L to describe states in He, but for better descriptions
and illustration we will use some additional approximate quantum numbers.
3
Calculation of states in He
Hydrogen atom is the only atom for which we can analytically calculate
state functions. In search for the state functions we can help ourselves with
hydrogen functions. The first and very crude approximation would be to put
each electron in one electron orbital of hydrogen atom with Z = 2 (charge of
nucleus). The wave function would be
ψ(r1 , r2 , ms1 , ms2 ) = Ψ1 (r1 )Ψ2 (r2 )
ψ(r1 , r2 , ms1 , ms2 ) = Rn1 l1 (r1 )Yl1 ml1 (Ω1 )χms1 Rn2 l2 (r2 )Yl2 ml2 (Ω2 )χms2
(2)
(3)
where Rnl (r) are radial function of hydrogen state for Z = 2, Ylml are spheric
harmonics and χms define the spin orientation. We can write such state in
more compact way
ψ(r1 , r2 , ms1 , ms2 ) = hr1 |n1 l1 ml1 ms1 i hr2 |n2 l2 ml2 ms2 i
(4)
In general this state in not an eigenfunction of total orbital angular momentum L̂ and total spin Ŝ, but we can form such function by choosing appropriate linear combination of functions (4) with the same set of (n1 l1 n2 l2 )
but different ml1 , ml2 , ms1 and ms2 . Such function is an eigenstate of operators L̂ and Ŝ, and is so a better approximation to the eigenstate or Ĥ.
hr1 r2 |n1 l1 n2 l2 LML SMS i =
X
X
hl1 l2 ml1 ml2 |LML ihs1 s2 ms1 ms2 |SMS i
ml1 ml2 ms1 ms2
hr1 |n1 l1 ml1 ms1 ihr2 |n2 l2 ml2 ms2 i
(5)
Above hl1 l2 ml1 ml2 |LML i and hs1 s2 ms1 ms2 |SMS i are the Clebsch-Gordan
coefficients.
We know that electrons are fermions and that two electrons with the same
spin cannot be at the same place. To account of the fermion nature of electrons we must make wave function antisymmetric. Antisymmetric function
can be easily made with particular liner combination of (5).
|n1 l1 n2 l2 LML SMS ias =
√1 (hr1 r2 |n1 l1 n2 l2 LML SMS i − hr2 r1 |n1 l1 n2 l2 LML SMS i)
2
4
(6)
Antisymmetric function is a better approximation of the eigenstate of
He. Spin part of the function is very simple and well known. It can be either
singlet state S = 0 (1 L) which is antisymmetric 1 or triplet state S = 1 (3 L)
which is symmetric2 . Energy of the state does not depend on quantum number MS and we will omit it. Now we can write the wave functions with good
total orbital angular momentum, good spin and which are antisymmetric.
We remember that our one electron functions are hydrogen like function
hr|nlml i = Rnl (r)Ylml (Ω)
(7)
Singlet state (S = 0):
1 X
hl1 l2 ml1 ml2 |LML i(hr1 |n1 l1 ml1 ihr2 |n2 l2 ml2 i
|n1 l1 n2 l2 1 LML i = √
2 ml1 ml2
+hr2 |n1 l1 ml1 ihr1 |n2 l2 ml2 i) (8)
Triplet state (S = 1):
1 X
|n1 l1 n2 l2 3 LML i = √
hl1 l2 ml1 ml2 |LML i(hr1 |n1 l1 ml1 ihr2 |n2 l2 ml2 i
2 ml1 ml2
−hr2 |n1 l1 ml1 ihr1 |n2 l2 ml2 i) (9)
This functions include symmetric properties of Hamiltonian, but are still
far away from being a true eigenstates because they totally ignore the Coulomb
repulsion between electrons. To find better eigenstates we will briefly discuss
two possibilities: Hartree-Fock and Configuration Interaction method (CI).
3.1
Hartree-Fock
Hartree-Fock (HF) method is a variation method by which we can calculate a better radial functions than Rnl (r) (hydrogen like radial functions).
It approximates Coulomb repulsion between electrons by central-symmetric
potential. Each electron moves in the Coulomb potential of a nucleus which
is screened by approximate central-symmetric but non-Coulomb potential of
the other electron.
HF
To calculate Rnl
(r) the model uses variation principle, requiring that variation of energy for small variation of radial function must be stationary
(δE = 0)3 . We also demand that one electron functions stay orthogonal
1
For antisymmetric spin part of the function, the spatial part must be symmetric.
For symmetric spin part of the function, the spatial part must be antisymmetric.
3
E = hψ|Ĥ|ψi
2
5
2
Figure 1: Two radial functions R1s of configuration 1s2 1 S are drawn. Red
one is hydrogen like function (Z = 2) and is used as a first approximation for
HF
R1s
(blue), which is calculated with HF program of computational package
ATSP [2].
and normalized. This brings Lagrange multiplicators into the variational
function:
δ(hn1 l1 n2 l2 2S+1 LML |Ĥ|n1 l1 n2 l2 2S+1 LML ) −
X
λij (hΨi |Ψj i − δij ) = 0 (10)
i,j
This lead to the set of two differential equations for each electron radial
function. Since we need to know one electron radial function (it gives a
screening potential) to calculate the other, the method is iterative starting
from initial approximation of radial function (these can be Rnl ). An example
is given in Figure 1, showing that because of screening, electrons are on
average further away from nucleus.
3.1.1
Experiments and Hartree-Fock model
To investigate He states we usually supply energy to the ground state in
order to make excited states. The absorbed energy matches the energy of
the excited state. In He we have two types of discrete excited states; singly
excited states and doubly excited states.
Singly excited states consist of one electron in ground state (1s orbit) and
another in higher orbit (for example 2s, 2p, 3s, 3p, etc.). As absorbed energy raises the other electron occupies higher and higher shell, until it gets
enough energy to leave the He+ ion and becomes free. The energy level at
which we get ionized He+ is called the first (N = 1) ionization threshold and
6
N=1 -53.4 eV
1
S
1sns
1s3s
1
P
1
3
D
1snp
1snd
1s3p
1s3d
S
1sns
1s3s
1s2p
-59 eV
-78 eV
1s2s
3
P
3
D
1snp
1snd
1s3p
1s3d
1s2p
1s2s
1s1s
Figure 2: Energy levels of HF 1sns 1 S, 1snp 1 P , 1snd 1 D and triplet states
with N = 1 ionization threshold are schematically shown.
is4 24, 6eV above the ground state. Some levels of HF states are shown in
Figure 2.
Doubly excited states are those with both electrons excited. We will focus
on states with one electron in second shell (2s or 2p), and the other in same
or higher shell (2s, 2p, 3s, 3p, 3d, 4s, . . . ). Some amount of energy is needed
to excite the first electron and another amount for the second electron. If
absorbed energy gets larger the second electron will occupy higher shell until
it will be torn from the atom. In that case, helium ion with one electron
in second shell (2s or 2p) and one free electron are produced. The energy
level at which this can happen is called second (N = 2) ionization threshold
and occurs in experiments 65.5eV above the ground state. In this seminar
our interest will mostly be doubly excited states with L = 1 and S = 0,
because they can be made efficiently from ground state by photoabsorption.
Some energy levels of doubly excited states in the HF model are shown in
Figure 3.
Let see now how the results from HF model meet experimental observations. Considering a photoabsorption spectrum in the region of doubly
excited states, we should expect to see three series of lines:
1. Series called 2snp 1 P , with states such as (2s2p 1 P , 2s3p 1 P , 2s4p 1 P ,
2s5p 1 P , . . . )
2. Series called 2pns 1 P , with states such as (2p2s 1 P , 2p3s 1 P , 2p4s 1 P ,
4
This energy is obtained with measurements and more sophisticated models.
7
1
2snp P
N=2 -12.6 eV
2s3p
2pns1P
1
2pnd P
2p3s
2p3d
2s2p
-18 eV
N=1 -53.4 eV
1
S
1sns
1s3s
1
P
1snp
1s3p
1s2p
-59 eV
-78 eV
1s2s
1s1s
Figure 3: Energy levels of HF (1sns 1 S, 1snp 1 P , 2snp 1 P , 2pns 1 P and
2pnd 1 P ) states with N = 1 and N = 2 ionization threshold are schematically
shown.
2p5s 1 P , . . . )
3. Series called 2pnd 1 P , with states such as (2p3d 1 P , 2p4d 1 P , 2p5d 1 P ,
2p6d 1 P , . . . )
The first two series are very similar and we expect to see them in the photoabsorption spectrum having approximately the same intensity. This is easily
seen, because the ratio in single configuration approximation is given by
σ2s3p
|h1s|r|3pih1s|2si|2
=
≈1
σ2p3s
|h1s|r|2pih1s|3si|2
σ2p3d = 0
(11)
The third series can not be excited by one-photon absorption from the ground
state due to the selection rules.
The photoabsorption spectrum of doubly excited states in He was firstly
measured in 1965 by Madden and Codling [6] and is shown in Figure 4.
We can see from Figure 4, that there is only one strong series. Another
8
Figure 4: Photoabsorption spectrum with one dominant series [7].
series is about 40 times weaker and the third series is not even observed.5
Same spectrum with better resolution is shown in Figure 5.
Figure 5: Photoionization spectrum in range of doubly excited states is
shown. On (a) only n+ 1 P and n− 1 P resonances are seen. The higher
peaks belong to n+ series and much smaller to n− series. At the vicinity of
n− 1 P resonances are hardly observed nd 1 P resonances which are shown on
(b). The figure is taken from [11].
5
It was observed much later in 1992 [8].
9
There must be something wrong with HF model. Let’s see how good
eigenstates of Hamiltonian the HF functions really are. If we calculate matrix
elements of Ĥ for HF single configuration functions 2s3p, 2p3s and 2p3d with
symmetry 1 P , we get
2s3p
−15.38 eV

=  1.20 eV
0.23 eV

Hij
2p3s
1.20 eV
−15.45 eV
0.09 eV
2p3d

0.23 eV
0.09 eV 
.
−15.02 eV
(12)
In this basis we can find better eigenfunctions with diagonalization of Hij :
3+
−14.20 eV

0
= 
0

H̃ij
3−
0
−16.65 eV
0
3d

0

0

−15.00 eV
(13)
where the new eigenfunctions are given by
1
3+ 1 P = 0.72 2s3p + 0.68 2p3s − 0.13 2p3d ≈ √ (2s3p + 2p3s)
2
1
3− 1 P = 0.70 2s3p + 0.71 2p3s − 0.14 2p3d ≈ √ (−2s3p + 2p3s)
2
1
3d P = 0.003 2s3p + 0.19 2p3s − 0.99 2p3d ≈ 2p3d
(14)
From above equations we can see, that doubly-excited states cannot be
well described (not even in zero order approximation) with one configuration n1 l1 n2 l2 1 P . In doubly excited states the 2snp and 2pns configurations
are highly mixed. States with + mixing (for example: √12 (2snp + 2pns))
are called n+ and states with − mixing ( √12 (−2snp + 2pns)) are called n−
states. Such mixing occurs because electrons are much more correlated as it
is approximated in HF model.
3.2
Configuration Interaction method
From HF example (14) we have learned that single configurations
n1 l1 n2 l2 2S+1 LML are not good approximations for describing correlations between electrons. To obtain more accurate wave function we must consider
linear combination of single configurations. We can make a linear combination of HF functions, but better method is to optimize radial functions and
configuration weights for a given state in the frame of the Multiconfiguration
HF. However, this method is not suitable, if the density of levels is too high.
10
Simpler but more efficient method is to take a linear combination of
functions n1 l1 n2 l2 2S+1 LML , which have hydrogen-like functions (Z = 2)
for radial parts. When we excite ground state to doubly excited state,
the most probable transition is to states with 1 P symmetry (∆L = 0, ±1,
∆S = 0). The appropriate basis set for such excited states consists of configurations with 1 P symmetry. An example of basis is 393 single configurations [9]: 121 sp configurations ((2s2p − 12s12p)1 P ), 110 pd configurations
((2p2d−12p12d)1 P ), 90 df configurations ((3d4f −12d12f )1 P ) and 72 f g configurations ((4f 5g − 12f 12g)1 P ). To find eigenstates of Hamiltonian in such
basis we calculate all matrix elements hn1 l1 n2 l2 1 P |Ĥ|n01 l10 n02 l20 1 P i. From diagonalization of this matrix we can calculate eigenstates and eigen energies
of Ĥ. The i-th eigenstate has a form:
|i 1 P ML i =
X
cin1 n2 l1 |n1 l1 n2 (l1 + 1) 1 P ML i
(15)
n1 n2 l1
Just as we got tree series in HF example, we also have here three series.
States with major coefficients with same sign at 2snp1 P an 2pns1 P configurations belong to n+ series. States with major coefficients with opposite
sign at 2snp1 P an 2pns1 P configurations belong to n− series and the states
with major coefficient at 2pnd 1 P configuration belong to so called nd series.
Energies of this states are shown schematically in Figure 6.
Photoabsorption cross-section from the ground state to these doubly excited states (n+ , n− and nd) is now much different as it was for HF functions
[11]:
σ3+ : σ3− : σ3d = 4240b : 100b : 10b.
(16)
This explains well the photoabsorption spectrum in Figure 5. We calculate
photoabsorption spectrum with the help of matrix elements hi 1 P ML ||r1 +
r2 ||1s2 1 Si for different energies of final states. But we have to be careful due to processes described in Autoionization section; final states include
some admixture of non-localized functions and also in some measurements
we cannot distinguish decay of doubly excited state from direct ionization of
ground state.
4
Reduced probability density
Now when we have accurate wave functions we try to visualize electronelectron correlation. The reduced probability density equals the two-electron
probability density |hi|ii|2 integrated over thee Euler angles which determine
orientation of the electron-nucleus-electron system. The reduced probability
11
Singlet LS Allowed States
68
66
+
n+(1Po)
He N=2
n-(1Po)
nd (1Po)
64
Energy (eV)
62
60
+
1
He N=1 1sns (1S)
1snp ( P)
1
1snd ( D)
24
22
20
1
1s2s ( S)
Metastable
State
18
16
2
0
2 1
1s ( S) Ground State
Figure 6: Energy levels calculated with configuration interaction method.
(Figure is taken from [12].)
density depends on three ”internal” coordinates only and gives the probability to find three particles in constellation of a triangle defined by one particle
in each corner. The most convenient set of internal coordinates for our presentation consists of r1 , r2 , the distance of each electron from the nucleus
and θ12 , the angle at which the two electrons are seen from the nucleus (Figure 7). To obtain the reduced density for a given doubly excited state, we
combine wave functions (15) with different projections M of the total orbital
angular momentum in such way to get rid of the coordinates defining the
orientation of the triangle; the integration over Euler angles reduces then to
multiplication by 8π 2 . The reduced densities of states which differ only by M
are obviously equal. The sum of two-electron probability density over M also
does not depend on the orientation of the coordinate system. Combining the
two together we obtain the reduced probability density
ρi =
8π 2 X 2S+1
|hi
LM |i 2S+1 LM i|2 .
2L + 1 M
12
(17)
(1)
r12
r1
(2)
r2
q12
Z=2
Figure 7: Coordinates for reduced probability density
Inserting (15) into (17) we arrive to
ρi (r1 , r2 , θ12 ) =
FKi (r1 , r2 )PK (cos θ12 ),
X
(18)
K
where PK are Legendre polynomial of order K and FK are linear combination
of pairs of hydrogen-like helium radial orbitals [10]. To visualize the correlations we calculate the conditional probability density ρic for an electron to
be found at a given r2 , θ12 when the other electron is at distance r1 from the
nucleus.
ρi (r1 , r2 , θ12 ) 2 2
r1 r2 ,
(19)
ρic (r1 |r2 , θ12 ) =
ρiu (r1 )
where ρiu (r1 ) represents the unconditional probability density for one electron
to be at r1 in the i-th state,
ρiu (r1 ) = r12
∞
Z
0
r22 dr2
π
Z
0
sin θ12 dθ12 ρi (r1 , r2 , θ12 ).
(20)
The normalization is then
Z
0
∞
dr1 ρiu
= 1,
Z
0
∞
dr2
Z
π
0
sin θ12 dθ12 ρic (r1 |r2 , θ12 ) = 1.
(21)
The autoionization probability which we will discuss later is closely related
to the unconditional density for the interelectron distance r12 (Figure 7).
The new variable is easily introduced into (18) instead of θ12 because of one
to one correspondence with the old variable for each pair r1 , r2 :
θ12 = arccos
2
r12 + r22 − r12
.
2r1 r2
(22)
The new volume element is r1 r2 r12 dr1 dr2 dr12 , reflecting complete geometric
equivalence of the three edges in the triangle. Similarly to (20, 21) the
13
unconditional probability density for electron-electron distance is given by
ρiu (r12 )
and
= r12
∞
Z
0
Z
0
4.1
r1 dr1
∞
Z
0
∞
r2 dr2 ρi (r1 , r2 , r12 )
dr12 ρiu (r12 ) = 1.
(23)
(24)
Examples of reduced probability density
To demonstrate our tools we plot some conditional and unconditional densities. First we look at unconditional probability density ρu (r1 ). Some examples are shown in Figure 8, from which inner and outer shell of electrons are
well seen. Inner and outer shell are also observed in conditional densities,
Figure 8: Unconditional reduced densities for 3+ 1 P , 3− 1 P and 3d 1 P
states are shown. States are calculated by CI method using basis set of 393
single configurations [9] and Hamiltonian is diagonalized by the truncated
diagonalization [14]. For calculation of probability density only first ten
single configurations are used.
which give also better illustration of correlation. From unconditional density
ρu (r1 ) also the probability for some conditional density (the probability to
find first electron at r1 in conditional density) is revealed. Lets take a look at
the behavior of some conditional densities (ρic (r1 |r2 , θ12 )) for HF-model and
CI functions. In Figure 9 are four reduced densities.
Two (A, B) are calculated using HF radial functions, and second two (C, D)
are calculated with CI method. We see that distribution of second electron
in HF model is mainly spherically symmetric around the nucleus and that
first electron doesn’t influence on it much. The asymmetry is mostly the
14
1
A) 2s3p P HF
C)
1
3- P CI
r1=4rB
r1=4rB
1
B) 2p3s P HF
D)
1
3+ P CI
r1=4rB
r1=4rB
Figure 9: Conditional reduced densities for second electron (r1 , θ12 ), when
the fist electrons (r1 ) is 4rB away from the nucleus. Reduced probability
densities (C, D) are obtained in same way as for Figure 8.
consequence of Pauli exclusion principle taken into account in HF model6 .
Reduced densities C and D show the distribution of second electron for 3− 1 P
and 3+ 1 P states. It is seen that electrons are much more correlated in this
state then they are in HF states 2s3p 1 P (A) and 2p3s 1 P (B).
It is interesting to see also the unconditional density ρiu (r12 ), which tells us
the probability for certain interelectron distance r12 . As mentioned before
two electrons can’t be at the same place in triplet state, so we can expect
them to be on average further apart in triplet state than in singlet state.
Singlet and triplet reduced probability densities shown in Figure 10 meet
this expectations very well.
6
The exclusion principle is best seen in reduced densities of triplet state, where the
reduced densities for second electron are zero at the position of first electron and very
small in its vicinity.
15
Figure 10: Unconditional reduced probability for singlet(2+ 1 P ) and triplet
(2+ 3 P ) states. Reduced probability densities are obtained in same way as
for Figure 8.
5
Autoionization
Since the first observation of helium doubly excited states [6] it was clear
that states are really a resonances which decay very fast by autoionization.
Final products of this channel are free electron and He+ ion in the ground
state. So one electron is moving with energy , while another electron remain
bound to helium nucleus in the 1s state. Therefore we will denote such state
as 1sl 2S+1 L . The process in which the He+ is formed from doubly excited
states is called autoionization and in fact the photoabsorption spectrum on
Figure 4 was measured by detecting the yield of ions as a function of incoming
photon energy.
The autoionization decay rate7 Γa is closely related to the matrix element
−1 1
h1sp 1 P |r12
|i P i and therefore to ρiu (r12 ). If the ρiu (r12 ) is higher for small
r12 then the Γa is higher and doubly-excited state decay faster. In other
words, if probability for electrons being closely together is high then the
autoionization is more probable. We can interpret that also in classical way.
If two electrons are often close together the transfer of energy from one
electron to another (and therefore autoionization) is more probable. From
photoionization spectrum (Figure 5) it is seen that n+ 1 P states have much
bigger autoionization rates than n− 1 P states. Therefore we can expect
ρiu (r12 ) to be bigger at small r12 for n+ 1 P than for n− 1 P and smallest for
nd 1 P . An example in Figure 11 surely confirms such expectation.
7
Lifetime of a resonance is given by decay rate: τ =
16
h̄
Γ
.
Figure 11: Unconditional reduced probability for n = 3 states. Secondary
maximum at r12 ≈ 4 indicates high autoionization rate of 3+ 1 P resonance.
Reduced probability densities are obtained in same way as for Figure 8.
From more elaborate calculations [9] we find out that
Γa (3+ ) : Γa (3− ) : Γa (3d) = 10000ns−1 : 180ns−1 : 0.18ns−1 ,
(25)
which is in perfect agreement with experiments and our considerations.
6
Conclusion
In this seminar we have presented basic differences between HF and CI
method. Although HF give satisfying results in many cases (higher singly
excited states) it is not appropriate for states in which electron-electron correlations are important. For doubly excited states the correlations were presented by the conditional reduced probability densities. Investigations of
helium is vivid in last few years due to observation of new phenomena like
UV decay of doubly excited states [9], metastable atoms, time scales of decay
[13], etc. For well description of this processes good known wave functions
are crucial and therefore electron-electron correlation must be taken into account properly. Althought many phenomena are well described, some are
still waiting for appropriate teoretical description. Some measurements of
photoabsorption spectra in the range of doubly excited states and with addition of external field (electric and magnetic) where made. Those spectra
are quite complicated and are till now left without teoretical background.
17
References
[1] Franc Schwabl, Quantum Mechanics, Translated by Ronald Kates,
Berlin [etc.]; Springer, 1991
[2] Charlotte Froese Fischer, Tomas Brage, Per Jönsson, Computational
atomic structure: an MCHF approach, Bristol, Philadelphia : Institute
of Physics Publishing, cop. 1997
[3] C. D. Lin, Classification and supermultiplet structure of doubly excited
states, Phys. Rev. A29, 1019–1033 (1984)
[4] Ugo Fano, Dynamics of electron excitation, Physics Today, Volume 29,
Issue 9, September 1976, pp.32-43
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