correlated one particle states
DESCRIPTION
Correlated One Particle States B. Weiner Department of Physics, Pennsylvania State University DuBois PA 15801 J. V. Ortiz Department of Chemistry, Kansas State University, Manhattan, KS 66506-3701. One Particle Theory. N-particle State totally determined by - PowerPoint PPT PresentationTRANSCRIPT
Correlated One Particle States
B. WeinerDepartment of Physics,
Pennsylvania State UniversityDuBois PA 15801
J. V. OrtizDepartment of Chemistry, Kansas State University,
Manhattan, KS 66506-3701
One Particle Theory
N-particle State totally determined by
• A set of Generalized Spin Orbitals (GSO’s) spanning one particle space
• A set of occupation numbers of these GSO’s
rjj 1;, r
rjN j 1;
Generalized Spin Orbitals
k
kjkkjkj cc rrr,
kk
jj
rr
rr ,,
SSS
S
SS
,,Operator
Spin theofcomponent
of seigenvalue ofset Spec
ZY,X,OperatorVector
OperatorPosition
of seigenvalue ofset Spec
0
0
00
S
Q
QQr
Occupation Number
is the probability that an electron belonging to a group of N-electrons in
a specific N-electron state is somewhere in the region of space/spin
described by
jN
j,r
First Order Reduced Density Operator
FORDO
jjrj
jND
1
1
rjj 1;, r rjN j 1;&
jjrj
j ψψND
1
1
rje ji j 1;
Produce the same FORDO
Antisymmetrized Geminal Power State (AGP)
sjjsjjsjj
jj NN
N
N
N
cc
g
22
11
21
21
2
1
Geminal
sjjsjjcg
1
spaceelectron one of basis lorthonormaan formthat
s)(CGSO' OrbitalsSpin General Canonical
are 21;
tsCoefficien Canonical Real0
srj
c
j
j
equal becan 1; theof some
i.e.greater becan degeneracy
,degeneratedoubly least at of sEigenvalue
of FORDO
2
1
1
sjn
gg
cn
nggD
g
j
jj
sjsjjjsjj
equal becan 1; theof some
i.e.greater becan degeneracy
,degeneratedoubly least at of sEigenvalue
AGP of FORDO
22
22
1
1
1
sjN
ggD
NggD
j
sjsjjjsj
j
NN
NN
A. J. Coleman has proved (Reduced Density Matrices pp 142-
144), that
fashion 1-1 ain
1;1; sjnsjN jj
1
2
12
1
12
1
12
2
21
12
2
2
,,
11
1
1
ˆ
ˆ
N
N
N
N
N
N
N
N
N
j
jjj
sjjj
jsjj
j
j
j
nnjS
nnS
S
jSnN
2
1; and 1;
1; and 1;
1; and 1;
N
g
g
rjsjc
rjsjn
rjsjN
jj
jj
jj
sj
,ψψV
SUSU
g
sjjj
N
1
Span Linear
subspaces on theact that
22
group the tobelonging
tionstransforma
toinvariant always is
times-s
2
If the geminal is more than two fold degenerate then the invariance group
is bigger
FORDO same thehave all
2,,0;,,
sAGP' ofset The
real is ,,
,,
1
1
r
jj
sji
ji
sjj
g
eecg sjj
c
rr
c
kjkjk
sjkrkjj
skksjji
kskjj
sjjsjjsjj
ψψψψkjSnn
ψψψψkjSecc
ψψψψjSnS
ggD
N
Nkj
N
N
NN
ˆˆ
ˆˆ
ˆ1
21
11
11
2
2
2
2
2
22
Second Order Reduced Density Operators (SORDO’s)