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# Mitas_QMC - Quantum Monte Carlo methods Lubos Mitas North...

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Quantum Monte Carlo methods Lubos Mitas North Carolina State University Urbana, August 2006

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H  r 1 , r 2 , ... = E  r 1 , r 2 , ... - ground states - excited states, optical properties - responses to external fields - T>0, etc . .. H =− 1 2 i i 2 i,I Z I r iI i j 1 r ij E ion ion Robert Laughlin: Nobel Prize Talk, viewgraph #2
Electronic structure and properties of materials Hamiltonian of interacting electrons and ions H  r 1 , r 2 , ... = E  r 1 , r 2 , ... - ground states - excited states, optical properties - responses to external fields - T>0, etc . .. H =− 1 2 i i 2 i,I Z I r iI i j 1 r ij E ion ion

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Density Functional Theory : -based on one-particle density - exact functional is unknown - various approximations for : LDA (local density approx.) GGA (generalized grad. approx.) Problem: efficient but inaccurate (need accuracy 0.1 eV or higher) Alternatives ? Quantum Monte Carlo (QMC) Hartree-Fock : wavefunction as Slater determinant (antisymmetry) of one-particle orbitals Post-Hartree-Fock : expansion in excitations More advanced: CC, MBPT etc Problem: accurate but inefficient E tot = F [ r ] d r HF r 1 , r 2 , ... = Det [{ i r j }] F corr r 1 , r 2 , ... = n d n Det n [{ i r j }] Traditional electronic structure methods F
- cohesion, optical excitations, barriers : ~ 0.1 - 0.01eV ~ 1000 -100K - magnetism, superconductivity, spintronics : ~ 0.001 eV ~ 10K - QED (important) : ~ 0.000001 eV - recent calculations of sixth order QED corrections for He atom: 12 digit accuracy Nature employs energy, length, etc scales as a composer employs various orchestral instruments Accuracies which we need

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Density Functional Theory : -based on one-particle density - exact functional is unknown - various approximations for : LDA (local density approx.) GGA (generalized grad. approx.) Problem: efficient but inaccurate (need accuracy 0.1 eV or higher) Alternatives ? Quantum Monte Carlo (QMC) Hartree-Fock : wavefunction as Slater determinant (antisymmetry) of one-particle orbitals Post-Hartree-Fock : expansion in excitations More advanced: CC, MBPT etc Problem: accurate but inefficient E tot = F [ r ] d r HF r 1 , r 2 , ... = Det [{ i r j }] F corr r 1 , r 2 , ... = n d n Det n [{ i r j }] Accuracies which we need F
Post-HF methods: - convergence in one-particle basis sets is slow, ineffficient description of many-body effects - need to explictly evaluate integrals restricts functional forms which can be used DFT approaches: - difficult systematical improvement (the fundamental proof is not constructive) QMC: Traditional electronic structure methods and quantum Monte Carlo - use stochastic methods to map the many-body problem onto

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Mitas_QMC - Quantum Monte Carlo methods Lubos Mitas North...

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