Computational Solid State Physics ??????? ?8?

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Computational Solid State Physics ??????? ?8?

• 8. Many-body effect II
• Quantum Monte Carlo method

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Quantum diffusion Monte Carlo method

• Diffusion Monte Carlo method to calculate the
  ground state
• Importance sampling method
• How to treat the Pauli principle
• fixed node approximation

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Schrödinger equation in atomic unit
How to solve the Schrödinger equation for many
electrons?
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Time-dependent Schrödinger equation
Imaginary Time
The ground state wave function can be obtained in
the limit of infinite time.
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Diffusion equation with branching process for the
ground state wave function
diffusion
branching
Diffusion equation holds for
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Diffusion equation for particles
Flux
diffusion flux drift flux
D diffusion constant, vd drift velocity
Conservation of number of particles
Diffusion equation
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Rate equation
Rgt0 growth rate Rlt0 decay rate
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Time-dependent Greens function
Boundary Condition
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Time evolution of wave function
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Short time approximation
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Greens function of the classical diffusion
equation
The transition probability from x to y can be
simulated by random walk in 3N dimensions for N
electron system.
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Greens function of the rate equation
The branching process can be simulated by the
creation or destruction of walkers with
probability GB
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Importance sampling
analytical trial fn.
Diffusion equation with branching process
Diffusion
Branching
Drift
Local energy
Quantum force
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Biased diffusion Greens function
Kinetic energy operator
Drift term
The transition probability from x to y can be
simulated by biased random walk with quantum
force F in 3N dimensions for N electron system.
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Detailed balance condition
To guarantee equilibrium
Acceptance ratio of move of the walker from x to
y
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DMC and Importance-sampled DMC for the hydrogen
atom
Branching process
suppression of branching process
DMC
Importance-sampled DMC
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Schematic of the Greens function Monte Carlo
calculation with importance sampling for 3
electrons
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Evaluation of the ground state energy
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How to remove the condition ?

• Fixed node approximation
• to treat wave functions with nodes
• Fixed phase approximation
• to treat complex wave functions

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Fixed node approximation
Importance sampling on condition
Wave function f is assumed to have the same nodes
with ?D.
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Pauli principle for n like-spin electrons
Slater determinant
Slater determinant has nodes.
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Fixed phase approximation
Importance sampling on condition
Wave function f is assumed to have the same phase
with
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Ground states of free electrons
D.M.Ceperley, B.J.Alder PRL 45(1980)566
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Transition of the ground state of free electrons

• Unpolarized Fermi fluid
• Polarized Fermi fluid
• Wigner crystal

n concentration of free electrons
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Problems 8

• Calculate the ground state wave function of a
  hydrogen atom, using the diffusion Monte Carlo
  method.
• Consider how to calculate the ground state
  energy.
• Calculate the ground state of a hydrogen atom,
  using the diffusion Monte Carlo method with
  importance sampling method. Assume the trial
  function as follows.
• Derive the diffusion equation for
  in importance sampling method.