Correlated One Particle States
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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
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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;
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Generalized Spin Orbitals
k
kjkkjkj cc rrr,
kk
jj
rr
rr ,,
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SSS
S
SS
,,Operator
Spin theofcomponent
of seigenvalue ofset Spec
ZY,X,OperatorVector
OperatorPosition
of seigenvalue ofset Spec
0
0
00
S
Q
QQr
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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
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First Order Reduced Density Operator
FORDO
jjrj
jND
1
1
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rjj 1;, r rjN j 1;&
jjrj
j ψψND
1
1
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rje ji j 1;
Produce the same FORDO
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Antisymmetrized Geminal Power State (AGP)
sjjsjjsjj
jj NN
N
N
N
cc
g
22
11
21
21
2
1
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Geminal
sjjsjjcg
1
spaceelectron one of basis lorthonormaan formthat
s)(CGSO' OrbitalsSpin General Canonical
are 21;
tsCoefficien Canonical Real0
srj
c
j
j
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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
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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
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A. J. Coleman has proved (Reduced Density Matrices pp 142-
144), that
fashion 1-1 ain
1;1; sjnsjN jj
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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
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2
1; and 1;
1; and 1;
1; and 1;
N
g
g
rjsjc
rjsjn
rjsjN
jj
jj
jj
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sj
,ψψV
SUSU
g
sjjj
N
1
Span Linear
subspaces on theact that
22
group the tobelonging
tionstransforma
toinvariant always is
times-s
2
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If the geminal is more than two fold degenerate then the invariance group
is bigger
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FORDO same thehave all
2,,0;,,
sAGP' ofset The
real is ,,
,,
1
1
r
jj
sji
ji
sjj
g
eecg sjj
c
rr
c
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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)