effective local potentials for quantum many particle...

Effective localpotentials for

quantummany particle

systems inexcited states

Sourabh SinghChauhan

Many particlesystem

DensityFunctionalTheory

HK theorem

KS equations

Two particlesin onedimensionalbox

Further works

References

Effective local potentials for quantum manyparticle systems in excited states

Sourabh Singh Chauhan

School of physical sciencesProject guide-Dr. Prasanjit Samal

NISER





Many particlesystem


HK theorem

KS equations


Further works

References

Overview

1 Many particle system

2 Density Functional Theory

3 HK theorem

4 KS equations

5 Two particles in one dimensional box

6 Further works

7 References





Many particlesystem


HK theorem

KS equations


Further works

References

Many Particle system I

Consider a molecule having N nuclei and Ne electrons. TotalHamiltonian:-

Hmol = He + TN + ˆVNN

By Born-Oppenheimer appoximation we can seperate nuclearand electronic motion. i.e. for electrons we have to solveHeψ = Eeeψ

Figure: Different approaches for solving many particle system





Many particlesystem


HK theorem

KS equations


Further works

References

Density Functional theory I

• Density ρ(r) is the probability of finding an electron involume confined between ~r and ~r + d~r. It is

ρ(~r) = Ne

∑s

∫dr2dr2....drN |ψ0(~r, s, ~r2, ~r3.... ~rN )|2

Figure: Functional meaning





Many particlesystem


HK theorem

KS equations


Further works

References

HK theorem 1 I

• There exists one to one correspondence between externalpotential and ground state density.

Second theorem says–the total ground state density functionalE0[ρ] has its minimum value at the density equal to the groundstate density of system. Here

E0[ρ] = F [ρ] +

∫v(~r)ρ(~r)d3r

subject to constraint∫ρ(~r)d3r = N . Here

F [ρ] =< ψ|T + Vee|ψ >





Many particlesystem


HK theorem

KS equations


Further works

References

Determination of energy Functional I

According to KS ansatz:-If we take a system of N non interacting electrons subjected topotential VKS , it is possible to choose this potential such thatground state density of this system is same as ground statedensity of an interacting system subject to some externalpotential Uext. Now variational minimization leads to

((−1/2)∇2 + VKS)ui(~r) = Eiui(~r).........1

where

VKS = U(~r) +

∫ρ(~r

′)

|r − r′ |+δExc

δρ

Hence energy functional can be written as

E0[ρ] = T1[ρ] +1

2

∫drdr

′ ρ(~r)ρ(~r)′

|~r − ~r′ |+ U [ρ] + Exc[ρ]





Many particlesystem


HK theorem

KS equations


Further works

References

Determination of energy Functional II

Therefore once we know Exc we can solve equation (1) alongwith the constraint of number of particles self consistently toget final solution to the system.





Many particlesystem


HK theorem

KS equations


Further works

References

Two particles in one dimensional box I

Aim– To study the analogy of HK theorem for excited states.System– Two noninteracting particles in 1D infinite squarewell. Considering Two systems with potentials v(x) and v(x)

′

having same density. We can write:-[−1

2

d2

dx2+ v(x)

]φi(x) = εiφi(x) (1)[

−1

2

d2

dx2+ v(x)

′]φ

′i(x) = ε

′iφ

′i(x) (2)

where

ρ(x) = φ21(x) + φ21(x) = φ′21 (x) + φ

′21 (x) (3)

Taking rotation by an angle θ(x) as the transformation We get

φ = R(θ(x))φ′





Many particlesystem


HK theorem

KS equations


Further works

References

Two particles in one dimensional box II

where R is given by

(cos(θ(x)) sin(θ(x))−sin(θ(x)) cos(θ(x))

)For new modified potential we get:-

v′(x) = ε

′i +

φ′i(x)

2φ′i(x)

(4)

Energy difference in terms of wave functions and defining

∆ = ε1 − ε2

∆′

= ε′1 − ε

′2

Now in the expression of ∆′

substituting φ′i in terms of φi we

get

¨θ(x)ρ(x) + ˙θ(x) ˙ρ(x) + f(φ1, φ2,∆,∆′, θ) = 0 (5)

wheref = 2∆φ1φ2 −∆′[2φ1φ2cos(2θ(x)) + (φ22 − φ21)sin(2θ)] (6)





Many particlesystem


HK theorem

KS equations


Further works

References

Two particles in one dimensional box III

• Solve for θ(x).

• Get the wavefunctions.

• Get the potential v′(x) having same state density

Now the aim is to look for multiple potentials for differentparameter values.





Many particlesystem


HK theorem

KS equations


Further works

References

Lowest Excited state I

Corresponding density:-

ρ(x) = 2[sin2(πx) + sin2(2πx)]

Putting that in equation differential equation:-

f = 6π2sin(πx)sin(2πx)−∆′[4sin(πx)sin(2πx)cos(θ(x))

+2sin2(2πx)− sin2πxcos(θ(x))]

By symmetry of φ1, antisymmetry of φ2 and symmetry of ρ(x)about x = 1/2 we find θ to be antisymmetric about x = 1/2 soθ(1/2) = 0.As x→ 0 we take large theta limit and assuming sin and cos tobe rapidly oscillating in that limit we drop those term from thesecond order diff. equation. Finally we get

¨θ(x)ρ(x) + ˙θ(x) ˙ρ(x) + 2∆φ1(x)φ2(x) = 0 (7)





Many particlesystem


HK theorem

KS equations


Further works

References

Lowest Excited state II

Solution of this equation in the limit x→ 0 takes the form:-

Figure: eqn

This reduces to :-

θ(x) ∼ a

x+ b+ cx+O(x2) (8)

So for a physical solution we need new wave functions to benot only normalized but also to have a→ 0 as x goes to zero.For normalization of φ

′1∫ 1

0φ

′1(x)2dx− 1 = R = 0 (9)





Many particlesystem


HK theorem

KS equations


Further works

References

Figure: 1-2 configuration ∆′=10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20

Ren

orm

aliz

atio

n-1

initial value of derivative of theta at x=1/2(a.u.)

Deltaprime=10


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20

Ren

orm

aliz

atio

n-1

initial value of derivative of theta at x=1/2

1-2 configuration

Delta prime=15





Many particlesystem


HK theorem

KS equations


Further works

References

Table: Table for the value of a for Normalized wavefunctions only forpositive values of dθ/dx| at x = 1/2

∆′

A B C

10 -0.005834 -0.06096 -0.0795915 -0.000074 -0.06442 -0.0775520 0.004940 -0.06812 -0.0766025 0.009001 -0.07168 -0.0761530 0.012501 -0.07444 -0.07594





Many particlesystem


HK theorem

KS equations


Further works

References

Figure: Values of a for small values of x as function of parameter ∆′

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

10 15 20 25 30

vale

of a

delta prime

’avalfinal’ u 1:2’avalfinal’ u 1:3’avalfinal’ u 1:4

f(x)





Many particlesystem


HK theorem

KS equations


Further works

References

Figure: Physical solutions fot∆

′= 15

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

phi1

(x)

and

phi2

(x)

x

Physical solutions for delaprime =15

’phi1’ u 1: 201’phi2’ u 1:201


Potential wth a = 0

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6 0.8 1

pote

ntia

l

x

infinite square well potential for 1-2 configuration delta prime =15





Many particlesystem


HK theorem

KS equations


Further works

References

Second excited state I

For this state we get density as :-

ρ(x) = 2[sin2(πx) + sin2(3πx)]

An approach similar to lowest excited state is used for thisstate also. For various parameter values R is plotted versusinitial conditions of θ(1/2)For ∆ = ∆

′= 40 → physically acceptable solution having

infinite square well potential.But for higher values of ∆

′we get some physically acceptable

solutions (∆′

= 160) with various potentials. For that value ofparameter corresponding θ(x) , wave function and potentialsare plotted as a function of x.





Many particlesystem


HK theorem

KS equations


Further works

References

Second excited state II


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5

Ren

orm

-1

theta(x)at x=1/2

delta prime =40 for 1-3 configuration

delta prime =40





Many particlesystem


HK theorem

KS equations


Further works

References

Second excited state III

Table: Table for the value of ‘a’ for Normalized wave functionsvarious values of dθ(1/2) (for 1-3 configuration)

∆′

A B C D E

20 -0.00379147 - - - -40 -0.00002159 -0.00303702 -0.00676243 -0.00683759 -0.0032940380 0.0131049 -0.0150336 -0.0136453 -0.00817969 -0.00580624

120 -0.00131377 -0.015926 -0.0122711 -0.0158277 -0.00410315160 -0.00434915 -0.0174066 -0.0145585 -0.00435088 -0.00237201





Many particlesystem


HK theorem

KS equations


Further works

References


physical solution

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

wav

efun

ctio

ns

x

normalized Wave functions for delta prime=40 and a=0

’phi1’’phi2’

1.414*sin(pi*x)1.414*sin(3*pi*x)


potential

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6 0.8 1

pote

ntia

l

x

infinite square well potential for 1-3 configuration delta prime =40





Many particlesystem


HK theorem

KS equations


Further works

References


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5

Ren

orm

-1

theta(x)at x=1/2


delta prime =160

Figure: Value of rotation for physically acceptable solution





Many particlesystem


HK theorem

KS equations


Further works

References

Figure: Normalized Wavefunctions

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

wav

efun

ctio

ns

x


delta prime =160wavefunction 2

Figure: Alternate effectivepotential

-100

-50

0

50

100

150

200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v(x)

x

aternate potential

’vx_160’





Many particlesystem


HK theorem

KS equations


Further works

References

For ∆ 6= ∆′

I

Figure: Renormalisation-1 for∆

′= 50

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20

Ren

orm

-1

Initial condition theta(1/2)

Renorm-1 for 1-2 configuration Deltaprime=50

DElta prime=50

Figure: Renormalisation-1 for∆

′= 100

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-40 -30 -20 -10 0 10 20

reno

rm-1

initial conditions theta(1/2)

renorm-1 v/s initial condition delta prime=100

’100renorm’





Many particlesystem


HK theorem

KS equations


Further works

References

For ∆ 6= ∆′

I

Figure: Wave functions for∆

′= 100

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

wav

e fu

nctio

ns

x

wave functions for delta prime =100

’100phi1’ u 1:143’100phi2’ u 1:143

Figure: Corresponding φ′

1

-1

-0.5

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

phi1

prim

e(x)

x

wave functions phi1 for delta prime =150,100,50

delta prime150100

50





Many particlesystem


HK theorem

KS equations


Further works

References

Figure: Multiple effective potentials

-1000

-500

0

500

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pote

ntia

ls

x

Different effective local potentials for a non zero

’pot150_1’’pot50_3’

’pot100_1’’pot201’





Many particlesystem


HK theorem

KS equations


Further works

References

Conclusions

• ‘There exists one to one correspondance between theexternal potential and lowest excited state of a givensymmetry’.

• To investigate this statement we need to:- Get alternatev′(x) → wave functions → check symmetry of φ

′i.

• If symmetry is same and we have different potentialstatement is wrong.

• Even for ∆′ 6= ∆ one may get same potential having same

symmetry.

• This approach is similar to constraint search approach.Here we are trying to find out φ

′i subject to constraint that

density remians same.

• For one to one correspondance in any general excited statewe must check for ground state density first.





Many particlesystem


HK theorem

KS equations


Further works

References

Further works

• Continuous values of ∆′.

• Harmonic oscillator potential.

• Putting time dependence in the potential.

• Including interacion in system.

• Including strong correlation amongst electrons.





Many particlesystem


HK theorem

KS equations


Further works

References

References I

• Physics Review A 85, 032517(2012)

• O. Gunarason and B. Lundqist Phys. Rev. B.

13 4274(1976)

• Introduction to quantum mechanics, Griffiths

• Physics of Atoms and Molecules, Bransden and

Joachain

• A Primer in Density Functional Theory by C.

Fiolhais, F. Nogueira, M. Marques

• http://www.nyu.edu





Many particlesystem


HK theorem

KS equations


Further works

References

Acknowledgments

I am did this semester project under the proper guidance of Dr.Prasanjit Samal who helped me in solving my queries.





Many particlesystem


HK theorem

KS equations


Further works

References

THANK YOU

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