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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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 >
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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]
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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′
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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)
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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)
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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)
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Figure: 1-2 configuration ∆′=15
-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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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)
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Figure: 1-2 configuration ∆′=15
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
Further works
References
Second excited state II
Figure: 1-3 configuration ∆′=40
-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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
Further works
References
Figure: 1-3 configuration ∆′=40
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)
Figure: 1-3 configuration ∆′=40
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
Further works
References
Figure: 1-3 configuration ∆′=160
-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 for 1-3 configuration
delta prime =160
Figure: Value of rotation for physically acceptable solution
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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 =160 for 1-3 configuration
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’
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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’
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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’
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
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.
Effective localpotentials for
quantummany particle
systems inexcited states
Sourabh SinghChauhan
Many particlesystem
DensityFunctionalTheory
HK theorem
KS equations
Two particlesin onedimensionalbox
Further works
References
THANK YOU