1 modeling matter at nanoscales 4. introduction to quantum treatments 4.02. the variational method

1

MODELING MATTER AT NANOSCALES

4. Introduction to quantum treatments4.02. The variational method

Schrödinger Equation

The solution of the Schrödinger equation gives the exact wave function of a system by means of the Hamiltonian operator that comes from the total energy of a system in classical mechanics:

where the “spectrum” of the operator is the set of eigenvalues Ek of all the K states of the system.

kkk EH ˆ

Schrödinger Equation

However, exact solutions for a Hamiltonian are scarce, although very relevant, and were mostly obtained for ideal conditions.

The variational theorem

The minimal energy as a criterion of accuracy

A wave function is more accurate or near to the exact eigenfunction when the expectation values of energy are minimal:

where yarb is an arbitrary wave function and y0 would be the eigenfunction (exact wave function) of the system.

00*0* ˆˆ EdHdH arbarb

Variational Theorem

Any expectation value of energy corresponding to a test wave function is equal or the upper limit to the lowest exact value of the system

Variational Theorem

Optionally, the variational theorem expresses that the best approximate wave function is that minimizing the expression:

where the denominator is a normalizing factor.This expression must depend on parameters to minimize energy with respect to them.

d

dHE

arbarb

arbarb

*

* ˆ

Variational Theorem

To this purpose, the arbitrary function can be developed in terms of fm possible states of the system that must be linearly independent, although not necesarily eigenfunctions of the operator:

where cm are coefficients of participation in the linear combination and can be considered as parameters serving to optimize the function.

cccarb ...2211

A simple case

A typical simple case where the arbitrary function depends on two states f1 and f2:

2211 ccarb

A simple case

This example could be referred to the electronic component of hydrogen molecule. The wave function is to be represented as a linear combination of functions describing the behavior of each electron as f1 and f2 .

+

=

1 2 arb

H2

A simple case

The variational function of the system can be written as:

and the conditions to yield the minimal energy with respect to coefficients would be:

dcccc

dccHccE

2211*22

*11

2211*22

*11

ˆ

02

1

cc

E0

12

cc

E

A simple case

The energy expression in terms of the previous linear combination is then:

dHcHccHccHcE

dHcHccHccHc

2*2

221

*2122

*1211

*1

21

2*2

221

*2122

*1211

*1

21

ˆˆˆˆ

ˆˆˆˆ

A simple caseDefining the term or matrix element:

expressing the energy of interaction between fm and fn local functions, and:

meaning the overlap between both functions in the space defined by t coordinates, we can write:

dHH ˆ*




dHH ˆ*

dS *




dHH ˆ*

dS *

22222112122111

21

22222112122111

21

ScSccSccScE

HcHccHccHc

A simple caseReordering, and deriving with respect to c1 and c2, we are ready to obtain their values for a minimal energy:

where the unknown are cm´s that will approach the exact energy according the variational principle.It must be observed that (Hmn – SmnE) terms are the coefficients of the cm variables.

0

0

2222221211

1212211111

ESHcESHc

ESHcESHc

A simple caseIn matrix terms, it means:

or(H – SE)C = 0

HC – SEC = 0

HC = SEC

0

0

2

1

22222121

12121111

c

c

ESHESH

ESHESH

A simple caseThe only non – trivial solution of this system, that is, the necessary and sufficient condition for the solution, is that the determinant of coefficients Hmn – SmnE must vanish:

or|H – SE| = 0

It is an equation of nth degree (in this case second degree) with n solutions for E.

022222121

12121111

ESHESH

ESHESH

A simple caseDeveloping the determinant:

If we substitute some terms to H11= H22 = W, H21= H12=w and the wave functions are normalized (S11=S22=1), the equation simplifies to a simple second degree equation in E:

0

22112122121122112

22211112222112211

2121121222221111

ESSESHESHHH

ESSESHESHHH

ESHESHESHESH

A simple caseDeveloping the determinant:

If we substitute some terms to H11= H22 = W, H21= H12=w and the wave functions are normalized (S11=S22=1), the equation simplifies to a simple second degree equation in E:

0

22112122121122112

22211112222112211

2121121222221111

ESSESHESHHH

ESSESHESHHH

ESHESHESHESH

021 2212

2212 ESES

A simple caseSolutions must be:

and therefore the energy only depends on parameters W, w and S12. It must be remembered that these solutions were deduced under the condition that the energy was minimal with respect to coefficients cm, not being them explicitly there.

2

12

22212

21212

2

212

22212

21212

1

12

122

12

122

21

21

S

SSSE

S

SSSE

The variational wave function

Solving these equations, the spectrum of eigenvalues (energies) of the system’s Hamiltonian is obtained. They must correspond to the corresponding Ek different states.

The variational optimal wave function must also be deduced after finding cm’s.


If the values of S12, W and w were considered known since the initial steps and we found values for E1 and E2, then c m values can be also calculated:

However, it is easy to realize that this system is unable to provide solutions different from 0 for c1 and c2 if other condition is not applied.

0

0

222211

112211

EcESc

EScEc


The solution of this problem can only be achieved if the state wave functions are also mutually orthogonal, giving:

wheredmn = 1 when m = n

dmn = 0 when m ≠ n

1...22

*

cc

dS


Then:

And consequently, solving the system:

1

0

0

22

21

221

211

cc

Ecc

cEc


Then:

And consequently, solving the system:

1

0

0

22

21

221

211

cc

Ecc

cEc

21

2

2

2

1 1

E

c

21

2

1

2

2 1

E

c

The variational solutionIn our case of interest

the variational wave function is then:

+

=

1 2 arb

H2

22

1

2

12

2

2

2211

21

21

2

11

EE

ccarbH

Notes and Conclusions

To obtain a reliable variational wave function it is also necessary to select appropriate f1 and f2 basis functions.For the variational method to obtain a wave function of a system approaching the exact it is necessary to find an appropriate parameter dependence of the basis reference functionsAs much appropriate parameters can be considered, providing accountable details of the object system, the resulting variational wave functions will be more reliable and near to the exact solution.

Notes and Conclusions

To obtain a reliable variational wave function it is also necessary to select appropriate functional forms for the f1 and f2 basis.

For approaching the exact wave function of a system by the variational method holds that as much appropriate parameters can be considered, the resulting variational wave functions will be more reliable and near to the exact solution.It works if such parameters can account energy relevant details of the object system.

1 modeling matter at nanoscales 4. introduction to quantum treatments 4.02. the variational method

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