计算凝聚态物理研究

*11月15日～21日在美国菲尼克斯举行的SC2003超级计算机会议上，第22届国际高性 能计算机性能TOP500排行榜如期揭晓。从榜单上看，虽然第一名仍被基于矢量技术的 日本地球模拟器以35.86Tflop/s的Linpack性能值稳占，第二名也依然是Linpack值为 13.88Tflop/s、安装于美国能源部洛斯阿拉莫斯国家实验室的ASCI Q，但是从第三名 开始便引出了本届排行榜的第一个亮点——机群（Cluster）系统的提升，包括最高名 次的提升和总数量的提升。
Top 10 in 2010
A Chinese system called Nebulae, build from a Dawning TC3600 Blade system with Intel X5650 processors and NVidia Tesla C2050 GPUs is now the fastest in theoretical peak performance at 2.98 PFlop/s and No. 2 with a Linpack performance of 1.271 PFlop/s. This is the highest rank a Chinese system ever achieved. There are now 2 Chinese systems in the TOP10 and 24 in the TOP500 overall.
The basic lemma of Hohenberg-Kohn
The ground state density n(r) of a bound system of interacting electrons in some external potential v(r) determines this potential uniquely. or v(r) = F[n(r)] or to say: n(r) completely characterizes the system
About the VASP code
1. Written by Georg Kresse, Martijn Marsman, and Jurgen Furthmuller Computational Physics, Faculty of Physics Universitat Wien Sensengasse 8, A-1130 Wien, Austria
*simulations in other scales: molecular dynamics finite element method
First Principles Calculations Based on Density Functional Theory ( DFT )
What does “first principles” mean? -Also called ab initio calculations
History of Density Functional Theory
WALTER KOHN
1998 Nobel Laureate in Chemistry for his development of the densityfunctional theory
1998 Nobel Laureate in Chemistry for his development of computational methods in quantum chemistry
计算凝聚态物理研究
HPC first principles calculations my work in computational physics
1946年2月15日，第一台电子数字 计算机ENIAC问世，这标志着计算 机时代的到来。
时钟频率100 KHz，能在1秒钟的时 间内完成5000次加法运算。
System with more electrons (e.g. N=100) M~3300 ~10150
---Density functional theory provides just a solution to such large systems
History of Density Functional Theory
2. http://cms.mpi.univie.ac.at/VASP/
3. Characteristics The code is easy to use: just prepare four files for each calculation The system that modeled can be as large as having several hundred atoms The results are reliable It is ever developing to include more functions The code is well parallel programmed so it runs fast enough It is used worldwide
1 2
n(r )n(r ' )drdr ' | r r'|
Approximation for Exc[n(r)]
The local density approximation (LDA)
ExLcDA[n(r)] exc[n(r)]n(r)dr
For a uniform electron gas:
(
1 2
2
veff
(r)
j
)
j
(r)
0
Where
veff (r) v(r)
|
n(r r
') r'
|dr'vxc
(r
)
and
vxc
(r)
E x c[n(r )] n(r)
The total energy of the system:
E j j Exc[n(r)]
vxc
(r
)n(r
)dr
• US DOE ASCI systems are claiming the first four positions of the TOP500. The new IBM ASCI White system at Lawrence Livermore National Laboratory is the new number one with 4.9 TFlop/s Linpack performance. This system is built with 512 nodes, each of which contains 16 IBM Power3 processors using a shared memory. This type of hierarchical architecture is more and more common for systems used in HPC .(Year 2000)
应用是最欠缺的 意味机会也最多
My cluster at School of Physics
系统总的双精度浮点峰值理论性能达~24万亿次每秒（2.4T）(峰值计 算能力公式=主频*每个时钟周期执行指令数*CPU核数*系统CPU总数; 即,2.66*4*4*56=2384Gflop)
One of the applications in scientific research---ab initio calculations of condensed matter
the system is completely specified by பைடு நூலகம்(r)
The Hohenberg-Kohn Variational Principle
E[n(r)] T[n(r)]U[n(r)] V (r)n(r)d 3r
or E[n(r)] T0[n(r)]VH [n(r)] Exc[n(r)] V (r)n(r)d 3r
Projects in my group
Doping of graphene hydrogen fuel cell Pt catalyst carbon alloy to replace Pt which kind of structure is the key how to realize it
i / t H
Or the static one:
H E
For hydrogen atom: H 22 e2
2m r
En 13.6 / n2 (eV )
{
nlm Rn (r)Ylm ( ,)
History of Density Functional Theory
H2 (or He) with two electrons early approaches Heitler and London (1927) Mullikan (1928) James and Coolidge (1933, the most successful ) Ψ depends on M parameters M~33N , N is the number of electrons M can be significantly reduced
• Jaguar, which is located at the Department of Energy’s Oak Ridge Leadership Computing Facility and was upgraded earlier this year, posted a 1.75 petaflop/s performance speed running the Linpack benchmark. Jaguar roared ahead with new processors bringing the theoretical peak capability to 2.3 petaflop/s and nearly a quarter of a million cores. One petaflop/s refers to one quadrillion calculations per second. (Year 2009, being No. 1 till now)
where
VH [n(r)]
n(r)n(r') d 3rd 3r' | r r'|
History of Density Functional Theory
Variational principle: E = min{ E[n(r)] }
We have single electron equation (Kohn-Sham):
JOHN A. POPLE
History of Density Functional Theory
Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria.
The Schrodinger Equation (in 1926):
-Means ‘from the beginning’ -No parameters from experiments used
What can we calculate?
-Electronic structure of condensed matter -magnetic properties -optical properties -structural optimization of defects -molecular dynamics simulations
exc (n)
0.458 rs
rs
0.44 0.78
Simple but the results are surprisingly useful
Beyond the local density approximation
GGA (generalized gradient approximation) meta-GGA Hybrid functionals that includes the HF exact exchange