GEMINALS IN DIRAC COULOMB HAMILTONIAN EIGENVALUE PROBLEM

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GEMINALS IN DIRAC-COULOMB HAMILTONIAN EIGENVALUE PROBLEM Grzegorz Pestka, Miros?aw Bylicki, Jacek Karwowski Instytut Fizyki, Uniwersytet Miko?aja Kopernika, Torun Third International meeting: MATHEMATICAL METHODS FOR AB INITIO QUANTUM CHEMISTRY Laboratoire J. A. Dieudonneé CNRS et Université de Nice-Sophia-Antipolis 19-20 October 2007

  • positive-energy-state-projected approach

  • free electron

  • dirac

  • coulomb hamiltonian

  • variational approach

  • equation


Publié le : mardi 19 juin 2012
Lecture(s) : 42
Source : math.unice.fr
Nombre de pages : 56
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GEMINALS IN DIRAC-COULOMB
HAMILTONIAN EIGENVALUE PROBLEM
Grzegorz Pestka, Mirosław Bylicki, Jacek Karwowski
Instytut Fizyki, Uniwersytet Mikołaja Kopernika, Torun´
Third International meeting:
MATHEMATICAL METHODS FOR AB INITIO QUANTUM CHEMISTRY
Laboratoire J. A. Dieudonneé
CNRS et Université de Nice-Sophia-Antipolis
19-20 October 2007Workshop Nice 2007 1
CONTENTS
Introduction
Variational approach to the one-electron Dirac equation
The controversy about the two-electron Dirac-Coulomb equation
Specific features of the algebraic approach
Relativistic Hylleraas CI
Artifacts and limits of accuracy
What can we learn from the complex scaling?
Positive-energy-state-projected approach versus non-projected one
When the positive-energy-space projection is necessary?
Final remarksWorkshop Nice 2007 2
DIRAC, LÉVY-LEBLOND AND SCHRÖDINGER EQUATIONS
Dirac equation:
2 32 3
l
(V E)I ; c( p)
2
4 54 5
= 0;
2 s
c( p); (V E 2mc )I
2
l s
, - two-component spinors;
– 2 2 Pauli matrices;
I – 2 2 unit matrix
2
The non-relativistic limit (c!1) – Lévy-Leblond equation:
2 3 2 3
l
(V E)I ; ( p)
2
4 5 4 5
= 0:
s
( p); 2mI c
2Workshop Nice 2007 3
DIRAC, LÉVY-LEBLOND AND SCHRÖDINGER EQUATIONS
The second pair of LL equations:
1
s l
c = ( p) :
2m
s
Elimination of from the first pair gives:

2
( p)
l
+ (V E)I = 0;
2
2m
2
Since ( p) = p I , we get two identical Schrödinger equations.
2
Their solution correspond to two spinorbitals with spins and:
2 3 2 3
1 0
l l
4 5 4 5
= ; and = :

0 1Workshop Nice 2007 4
SPECTRUM OF ONE-ELECTRON DIRAC HAMILTONIAN
E
E
D
POSITIVE−ENERGY CONTINUUM
L
2
mc
0
IONIZATION THRESHOLD
DISCRETE BOUND−STATE ENERGIES,
FORBIDDEN ENERGY GAP
0
D
IN THE CASE OF A FREE ELECTRON
2
2
mc

2mc

NEGATIVE−ENERGY CONTINUUM
SWorkshop Nice 2007 5
ENERGY FUNCTIONAL
2 3
l
c
hjHji
a
4 5
Rayleigh quotient: K[ ] = ; =
s
hji
c
b
q
2
2
Dirac: K[ ] = W + W + 2mc T
D +
Lévy-Leblond: K[ ] = T + (W +W )
L +

l l s s
1 hjVji h jVj i
2
W = mc ;

l l s s
2 hji h j i
l s s l
1 hj pj ih j pji
T = :
l l s s
2m hjih j iWorkshop Nice 2007 6
ENERGY FUNCTIONAL
l l
hjVji
LL energy functional: K[ ] =T +
L
l l
hji
l s s l
1 hj pj ih j pji
T = :
l l s s
2m hjih j i
s l
Kinetic balance condition: ( p) )
l 2 l
1 hj( p) ji
T =
l l
2m hji
8
<
variational LL and Schrödinger
Kinetic balance condition )
:
eigenvalue problems are equivalent.Workshop Nice 2007 7
ALGEBRAIC REPRESENTATION: MODEL SPACE
Basis set expansion of the components of the trial function
N N
L S
X X
L L L S S S
= C ; = C
k k
k=1 k=1
leads to the algebraic approximation to the Dirac equation.
The kinetic balance condition implies
S L
Hf gHf( p) g
whereHfg – space in which is expanded.
This condition is necessary for the correct behaviour of the variational
procedure applied to the Dirac equation.

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