Adiabatic TDDFT + discussion
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- Oswin Gordon Hunter
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1 Adiabatic TDDFT + discussion University of Warsaw & University of Jyväskylä INT Program INT-13-3 Quantitative Large Amplitude Shape Dynamics: fission and heavy ion fusion Seattle, September 23 - November 15, 2013
2 Outline 1. DFT 2. TDDFT 3. Instantons (this is what I will not talk about) 4. Adiabatic expansion 5. Adiabatic equations 6. Adiabatic solutions 7. Discussion
3 What is DFT? Density Functional Theory: A variational method that uses observables as variational parameters. J. D., J. Phys.: Conf. Ser. 312, (2011)
4 Which DFT? J. D., J. Phys.: Conf. Ser. 312, (2011)
5 How the nuclear EDF is built? LDA Gogny, M3Y, Local energy density is a function of local density Non-local energy density is a function of non-local density
6 Configuration representation of the DFT and TDDFT W. Kohn and L.J. Sham Phys. Rev. 140, A1133 (1965) cited times DFT TDDFT
7 TDDFT Cited 2600 times
8 Instantons
9 We present a description of nuclear spontaneous fission, and generally of quantum tunneling, in terms of instantons, that is, periodic imaginarytime solutions to time-dependent mean-field equations. This description allows comparisons to be made with the more familiar generator coordinate (GCM) and adiabatic time-dependent Hartree-Fock (ATDHF) methods. It is shown that the action functional whose value for the instanton is the quasiclassical estimate of the decay exponent fulfills the minimum principle when additional constraints are imposed on trial fission paths.
10 Instantons In the adiabatic limit, the instanton method reduces to the time-odd ATDHF equation, with collective mass including the time-odd Thouless-Valatin term, while the GCM mass completely ignores velocity-momentum relations. This implies that GCM inertia generally overestimates the instanton-related decay rate.
11 Instantons
12 Instantons
13 TDDFT quantized TDDFT corresponds to the time-evolution equation of a quantum state moving along the manifold parameterized by ρ mn (t) Explicit quantum picture is recovered by: Path integrals Quantization Instantons Collective Hamiltonian
14 Collective coordinates
15
16 How to solve the ATDDFT equation J. D. and J. Skalski, Nuc. Phys. A369 (1981) 123
17 How to solve the ATDDFT equation J. D. and J. Skalski, Nuc. Phys. A369 (1981) 123
18 CDC Cyber 170 series 60-bit 40MHz CPU 256 kwords memory of 12-bit words
19 How to solve the ATDDFT equation J. D. and J. Skalski, Nuc. Phys. A369 (1981) 123
20 Perturbation expansion of ρ
21 Perturbation expansion of energy
22 Perturbation expansion of energy
23 Adiabatic expansion of ρ
24 Adiabatic TDDFT equation (ATDDFT)
25 ATDDFT equation ATDDFT mass parameter
26 How to solve the ATDDFT equation 1) For several collective variables use the chain rule: 2) Work in the particle-hole basis: 3) Use the fixed-point method and iterate:
27 How to solve the ATDDFT equation Cranking mass parameter (1), (2), (3), infinity = ATDDFT(B) mass parameter
28 How to solve the ATDDFT(B) equation Cranking mass parameter (1), (2), (3), infinity = ATDDFT(B) mass parameter
29 Quantization B. Podolsky, Phys. Rev. 32, 812 (1928)
30 Can TDDFT penetrate the barrier? Homework exercise for the proud owners and users of the TDDFT codes: Start 224 Ra axial
31 Can TDDFT penetrate the barrier? Homework exercise for the proud owners and users of the TDDFT codes: Next 224 Ra axial
32 Can TDDFT penetrate the barrier? Homework exercise for the proud owners and users of the TDDFT codes: Later? 224 Ra axial
33 Is TDDFT classical or quantal?
34 Can TDDFT penetrate the barrier?
35 Can TDDFT penetrate the barrier?
36 Thank you
37 Can TDDFT penetrate the barrier? Homework exercise for the proud owners and users of the TDDFT codes: Next 224 Ra axial
38 Can TDDFT penetrate the barrier? Homework exercise for the proud owners and users of the TDDFT codes: Later 224 Ra axial
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