correlated one particle states

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)