MODELING MATTER AT NANOSCALES 6.The theory of molecular orbitals for the description of nanosystems (part II) 6.04. The density matrix

MODELING MATTER AT

NANOSCALES

6. The theory of molecular orbitals for the description of nanosystems (part II)6.04. The density matrix.

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Electron density from multielectronic wave functions

The case of N identical antisymmetrized particles (fermions, as electrons are), characterized in the configuration space by t1,t 2,.. tN spatial and spin coordinates each, moving under the influence of a framework of fixed potentials (as their mutual interactions and those with nuclei) is the typical quantum system where nanoscopic phenomena and structures occur…

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The case of N identical antisymmetrized particles (fermions, as electrons are), characterized in the configuration space by t1,t 2,.. tN spatial and spin coordinates each, moving under the influence of a framework of fixed potentials (as their mutual interactions and those with nuclei) is the typical quantum system where nanoscopic phenomena and structures occur…

where: is a permutation operator working on subindexes of each of the N coordinatesp is the “parity” of each permutation (1 or 2)

NI

p

NIP ,...,,1,...,,ˆ2121

P̂

The wave function of a certain I state of such system can be given by:

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Let us consider that:

is a volume differential of the configuration space regarding spatial and spin coordinates and allowing integration on all coordinates of all and each one of the electrons

Ndddd ...21


A simple planar representation of the configuration space, where all electrons are placed at their respective coordinates, gives the idea that isolating a portion corresponding to a certain coordinate permits to evaluate how the whole system behaves in such portion:

t2, t3, …,tNt1


To integrate the electron population in the configuration space we adopt a convention regarding particle coordinate differentials, and will agree that:

is a notation for volume differentials of the remaining space after pointing to certain reference particle coordinate volume elements in positions t1, t2, ... of interest.

Ndddd ...321

Ndddd ...432,1


To integrate the electron population in the configuration space we adopt a convention regarding particle coordinate differentials, and will agree that:

is a notation for volume differentials of the remaining space after pointing to certain reference particle coordinate volume elements in positions t1, t2, ... of interest.

Ndddd ...321

Ndddd ...432,1

It means that refers to the volume differential of the entire electron system, except those oftl, tm,… point coordinates.

..lmd

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From quantum mechanical principles, it is well known that if YI is a normalized wave function of a multielectronic system in the I state, then:

means the probability that electron 1 has to be inside the volume and spatial element dt1 with coordinates t1, while electron 2 is at the same time in volume dt2 with coordinates t2, and so on for the entire system of N particles.

NNINI ddd ...,...,,,...,, 2121*

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From quantum mechanical principles, it is well known that if YI is a normalized wave function of a multielectronic system in the I state, then:

means the probability that electron 1 has to be inside the volume and spatial element dt1 with coordinates t1, while electron 2 is at the same time in volume dt2 with coordinates t2, and so on for the entire system of N particles.

NNINI ddd ...,...,,,...,, 2121*

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The integral of this product in the whole space is normalized to 1.

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Density matricesIf we are only interested on the electron population at the t1

coordinate then:

12121*

1 ,...,,,...,, dN NINI

is the probability to find a portion of all N electrons at the t1 spatial and spin coordinates of the configuration space corresponding to any I state of the system.

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coordinate then:

12121*

1 ,...,,,...,, dN NINI


r(tl) is the reduced density function at the tl coordinate of the configuration space in an N – electron system, and is expressed as a non – dimensional number per unit of the spatial – spin “volume”.

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coordinate then:

12121*

1 ,...,,,...,, dN NINI


ll dis the non – dimensional number giving the electron population in the dtl volume differential at tl coordinate of the Hilbert space.

Therefore:

r(tl) is the reduced density function at the tl coordinate of the configuration space in an N – electron system, and is expressed as a non – dimensional number per unit of the spatial – spin “volume”.

12121

*11 ,...,,,...,,'|' dN NINI

Density matricesThe generalized density matrix of one particle is then given by:

where t1 and t1´ are different coordinates in the configuration space corresponding to the subset of one particle. It is also known as the reduced density matrix of first order.

12121

*11 ,...,,,...,,'|' dN NINI



It means a mapping of density originated by the one particle portion of the Hilbert space of the whole system.

12121

*11 ,...,,,...,,'|' dN NINI



In this conditions the previously defined probability term r(tl) is the diagonal element of this matrix for each tl coordinate.

lll |

It means a mapping of density originated by the one particle portion of the Hilbert space of the whole system.

Density matricesIn a similar way, the reduced density function of two interacting particles can be defined as:

2,1321321*

21 ,...,,,,...,,,2

, dN

NINI

It is the probability to find a portion of all N electrons at two t1 and t2 spatial and spin coordinates anywhere in the configuration space corresponding to a certain I state of the system.


2,1321321*

21 ,...,,,,...,,,2

, dN

NINI

mlml dd ,expresses the pair population of particles in the dtldtm differential volume of such space at coordinates tl andtm.

Therefore:



2,1321321*

21 ,...,,,,...,,,2

, dN

NINI

mlml dd ,expresses the pair population of particles in the dtldtm differential volume of such space at coordinates tl andtm.

Therefore:

is the number of order M sets in the total of N elements.

!!

!MNM

NM

N


Density matrices

The generalized density matrix of two particles is given by:

where t1, t1´, t2, t2´ are different coordinates in the configuration space corresponding to the subset of particles 1 and 2. It is also known as the reduced density matrix of second order.

2,1321321*

2121 ,...,,,,...,,','2

|'' dN

NINI

Density matrices



2,1321321*

2121 ,...,,,,...,,','2

|'' dN

NINI

It means a mapping of the given pair of particles in the considered portion of the Hilbert space of the system.

Density matrices



In this conditions the previously defined probability term r(tl,tm) is the diagonal element of this matrix for a given pair of particles.

2,1321321*

2121 ,...,,,,...,,','2

|'' dN

NINI

mlmlml |,

It means a mapping of the given pair of particles in the considered portion of the Hilbert space of the system.

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Density matrices

It can be proof that if two or more indices are equal, the matrix collapses to 0, because the particle antisymmetrization.

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Density matrices


0, ll It is the case of diagonal elements of two particle density matrices:

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Density matrices


Two electrons can not hold the same coordinates and it corresponds to the physical evidence that two Fermions with the same parallel spins can not coexist at small distances. It is the so called “Fermi’s hole”.

0, ll It is the case of diagonal elements of two particle density matrices:

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Density matrices

All these expressions of any order p:

are two dimensional matrices where one function give columns and the conjugate the rows, or vice versa.

pNpINpI

ppp

dp

N,...2,1321321

*

2121

......'...'...''

...|''...'

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Density matricesAs a consequence:

N

d

ddNd NNNI

111

112121*

11

|

,...,,,...,,

is the number of electrons because Y is normalized to 1.

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Density matricesAs a consequence:

is the number of electron pairs.

N

d

ddNd NNNI

111

112121*

11

|

,...,,,...,,

is the number of electrons because Y is normalized to 1.

21

|, 2121212121

NN

dddd Similarly:

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Quantum properties depending on density matrices

Any kind of quantum operator, being symmetrical with respect to particle indexes can be developed as:

...ˆ'ˆ'ˆˆˆ)(!3

1)(!2

1)()0(

lmnlmn

lmlm

ll AAAAA

corresponding (0), (l), (lm), (lmn),… to operate on zero, one, two, three,… particles, respectively.

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Any kind of quantum operator, being symmetrical with respect to particle indexes can be developed as:

...ˆ'ˆ'ˆˆˆ)(!3

1)(!2

1)()0(

lmnlmn

lmlm

ll AAAAA

corresponding (0), (l), (lm), (lmn),… to operate on zero, one, two, three,… particles, respectively.

Primes indicate that terms with two or more identical subindex are excluded.

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In the case of two particles, taken as an example, the expectation value ã1,2 corresponding to the value of property A related to their mutual interaction in the given I state of the system is:

2121212,1

2,1213212,1321*

*

21*

2,1

|ˆ

,...,,,ˆ,...,,,2

ˆ2

1

ˆ'~

ddA

dddAN

dANN

dAa

NINI

IlmI

Ilm

lmI

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In the case of two particles, taken as an example, the expectation value ã1,2 corresponding to the value of property A related to their mutual interaction in the given I state of the system is:

2121212,1

2,1213212,1321*

*

21*

2,1

|ˆ

,...,,,ˆ,...,,,2

ˆ2

1

ˆ'~

ddA

dddAN

dANN

dAa

NINI

IlmI

Ilm

lmI

It must be observed that this transformation can only be made for cases where t1 = t1’ and t2 = t2‘.

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The physical expectation value of the property A at the I state of the system can be obtained from, and de facto depends on, the diagonal of the corresponding generalized density matrix.

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In general, the expectation value ã corresponding to the value of property A in the given I state of the system in terms of the generalized density matrices is:

...|ˆ

|ˆ

|ˆ~

ˆ~

3213213213,2,1

2121212,1

11110

*

dddA

ddA

dAa

dAa II

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In the case of a transition from state I to any state J of the system, the matrix element related with the change of property A in terms of the generalized density matrices is:

...|''ˆ

|'ˆ

ˆˆ

2121212,1

1111

*

ddA

dA

dAA

IJ

IJ

JIJI

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Therefore, for evaluating a transition between two quantum states YI and YJ of a multi – particle system we can define the transition matrix densities as:

12121

*11 ,...,,,...,,'|' dN NJNIIJ

2,1321321

*

2121

,...,,,,...,,','2

|''

dN

NJNI

IJ

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Density matrices in terms of molecular orbitals

In the special case of a ground state Y0 that is a solution of a Hartree – Fock problem, the generalized density matrix of one l particle can be expressed in terms of molecular spin – orbitals of the Slater determinants and it remains as:

'

'|*

lii

lii

llHF

lHF

n

where the sum runs over all spin orbitals expressed in Y0 and ni is their corresponding electron occupation.

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In the special case of a ground state Y0 that is a solution of a Hartree – Fock problem, the generalized density matrix of one l particle can be expressed in terms of molecular spin – orbitals of the Slater determinants and it remains as:

'

'|*

lii

lii

llHF

lHF

n

As the molecular orbitals are orthonormal, it holds again that:

Nndndi

ii

lliillHF 2

)()(

where the sum runs over all spin orbitals expressed in Y0 and ni is their corresponding electron occupation.


Expressing the spatial component of MO’s in terms of a complete orthonormal atomic basis set:

)()( lili c


Expressing the spatial component of MO’s in terms of a complete orthonormal atomic basis set:

and performing the appropriate substitutions:

)()( lili c

iii

lli

iii

ililii

iliil

cn

ccn

ccn

n

2

*

,

*

2

)()(

)()(

)()(

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Energy and density functions

Being the p density matrix element:

i

iii ccnp *

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Being the p density matrix element:

i

iii ccnp *

The expression of energy of the system remains as:

,

,

*

Fp

FccnEi

iii

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Then the density function depends on the HF density matrix p as:

)(

)()()( *

,

pTr

p

p lll

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The density function is the trace of the Hartree – Fock density matrix in terms of an orthonormal basis of atomic orbitals.

)(

)()()( *

,

pTr

p

p lll

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The density function is the trace of the Hartree – Fock density matrix in terms of an orthonormal basis of atomic orbitals.

)(

)()()( *

,

pTr

p

p lll

This expression is very useful in cases when a given quantum state can not be expressed in terms of a wave function and it is in terms of a basis to define densities.

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Consequently:

If the density matrix of a system is available it can be obtained the expectation value of the energy, as well as it was developed in terms of a complete orthonormal basis set.

,

)()( FpTrTrE pFFp

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An application to dipole moments

The operator for an electrical moment in a multielectronic system could be expressed as:

i

ireD )(ˆ

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The operator for an electrical moment in a multielectronic system could be expressed as:

remains that the expectation value of the electric or dipole moment is only depending from the generalized density matrix of first order:

i

ireD )(ˆ

...|ˆ

|ˆ~~

2121212,1

11110

ddA

dAaaWhen substituted in the previous expectation value formula:

111

1111 |~

dre

dreD

because only one particle terms have sense.

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Consequently, the electric or dipole moment of transition between two states I and J can be written as:

1111

*

|

ˆˆ

dre

dDD

IJ

JIJI

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Consequently, the electric or dipole moment of transition between two states I and J can be written as:

1111

*

|

ˆˆ

dre

dDD

IJ

JIJI

In the case of dealing with electronic states of nanoscopic systems becomes evident that a change in the distribution of the multielectronic cloud means changing the quantum state of the system.

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MODELING MATTER AT NANOSCALES 6.The theory of molecular orbitals for the description of nanosystems (part II) 6.04. The density matrix