Effective local potentials for quantum many particle systems in excited states Sourabh Singh Chauhan Many particle system Density Functional Theory HK theorem KS equations Two particles in one dimensional box Further works References Effective local potentials for quantum many particle systems in excited states Sourabh Singh Chauhan School of physical sciences Project guide-Dr. Prasanjit Samal NISER
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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
• 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
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)