Difficulties with VMC The many-electron wavefunction is unknown Has to be approximated Use a small model system with no more than a few thousand electrons May seem hopeless to have to actually guess the wavefunction But is surprisingly accurate when it work
• The many-electron wavefunction is unknown • Has to be approximated • Use a small model system with no more than a few thousand electrons • May seem hopeless to have to actually guess the wavefunction • But is surprisingly accurate when it work Difficulties with VMC
The limitation of vmo Nothing can really be done if the trial wavefunction isn't accurate enough Therefore,there are other methods Example:Diffusion QMC
The limitation of VMC • Nothing can really be done if the trial wavefunction isn’t accurate enough • Therefore, there are other methods • Example: Diffusion QMC
Diffusion Monte Carlo >DMC is a quantum Monte Carlo method that uses a Green's function to solve the Schrodinger equation. >DMC is potentially numerically exact,meaning that it can find the exact ground state energy within a given error for any quantum system. >For bosons,DMC scales as a polynomial with the system size,but for fermions,DMC scales exponentially with the system size.This makes exact large-scale DMC simulations for fermions impossible. >The fixed-node approximation can yield very accurate results for fermions
Diffusion Monte Carlo DMC is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. For bosons, DMC scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible. The fixed-node approximation can yield very accurate results for fermions
One Experiment Done Using QMC Total Energy Calculations -Can use Monte Carlo to calculate cohesive energies of different solids. -Table shows how much more accurate the QMC calculation can be. Table 1.The Cohesive Energy of Ge Obtained Using Three Different Methods Method Used Cohesive Energy (eV/atom) LDA Calculation 4.59 Diffusion QMC Calculation 3.85 Experiment 3.85
One Experiment Done Using QMC Total Energy Calculations -Can use Monte Carlo to calculate cohesive energies of different solids. -Table shows how much more accurate the QMC calculation can be
Application to Microscale Heat Transfer Boltzmann Transport Equation (BTE)for phonons best describes the heat flow in solid nonmetallic thin films Difficult to solve analytically or even numerically using deterministic approaches Alternative is to solve the BTE using stochastic or Monte Carlo techniques
Application to Microscale Heat Transfer • Boltzmann Transport Equation (BTE) for phonons best describes the heat flow in solid nonmetallic thin films • Difficult to solve analytically or even numerically using deterministic approaches • Alternative is to solve the BTE using stochastic or Monte Carlo techniques