In this case, Darwin Smith's theorem applies, thus problem can be very simplified.
This is very important theorem that we do not obtain 1-RDM from fake wave function.
However, in doublet state, we don't have such symmetry, thus we have degenerated ground state. The total energy can be calculated very accurately, but fails to calculate the dipole moment which should be non-zero but zero.
I have a presentation in 7/25, so I'll be Canada on 7/27 16:00. I'll miss some important
presentations. If possible I'll have a talk.
I cannot attend this conference... though ... what a pity... Japan is located
too far from Canada. I'll go to US in this August. Toooooo hard schedule.
There will be a session by Erdahl....
(At CSC2009, Garnet and Ayers kindly told me about this conference, thanks!)
Non-commutative polynomial optimization and the varianional RDM method by Stefano Pironio
A standard problem in optimization theory is to find the minimum of a polynomial function subject to polynomial inequality constraints. We introduce a generalization of this problem where the optimization variables are not real numbers, but non-commutative variables, i.e., operators acting on Hilbert spaces of arbitrary dimension. We show how semidefinite programming (SDP) can be used to solve this problem. Specifically, we introduce a sequence of SDP relaxations of the original problem, whose optima converge monotically to the global optimum.
Our method can find applications to compute the ground state energy of quantum many-body systems. In particular, it gives a new interpretation to and should strengthens the RDM method used in quantum chemistry to compute electronic energies. Our method provides a computation technique for many-body systems that is not based on states (and thus directly linked to entanglement) but that is rather based on the algebraic structure of quantum operators.---------------------------------------------------------------------------------------------
I asked him to send me a preprint!
Maybe I cannot sleep until their preprint is sent to me.
What they are doing? Some magic? Yes I'm sure!
SDP now become very popular indeed!!!
only for SDPA-GMP. It is not a constructive. So I decided to implement whole part of BLAS
and possible part of LAPACK. It took very long and still ongoing.
It's very portable and my motivation is providing a reference library, and not optimized one.
(of course this is in my TODO list)
SDPA-QD, DD version were done in a day or so.
I was invited to have a presentation at Shinjyuku, in 2009/2. I don't know which society
is interested in this, so your advice is really appreciated.
just in some case 2-RDM method fails to converge.
The gap of the Primal objective function and Dual objective function can be larger than 1e-4.
In this case we discard the result and re-run SDPA. For some molecules, it takes 2 weeks,
then discard? I was quite frustrated.
Maybe late of 2005 or 6, Yamashita-san came to the Univ. of Tokyo and we had some discussion,
and lunch. He told me that Nakata Kazuhide-san's student did some preliminary work. Their
result is somewhat incomplete as implement multiple precision arithmetic in Java and it is
not an implementation of floating point number!
So I decided to implement SDPA-GMP based on SDPA 6.
Using GMP is quite easy. just replace double to mpf_class. One large obstacle was it uses
BLAS/LAPACK routines. I implemented all routines used by SDPA 6.
This result was applied to the Hubbard model of large correlation limit. and published as JCP 2008. Also SDPA-GMP has been released in 2008, too.
It's well known that Cho and Cohen and Frishberg have already found it.
However, Nakatsuji's result is the most important as he showed necessary and sufficient
condition. So the density equation is equivalent to Schroedinger equation. Others didn't
prove the sufficient part.
It seems Erdahl has already noticed about it. Unfortunately he never published it.
It's written in his dispersion relation paper, but he knew a bit before. I don't remember
exactly but RDM news or something like that mentioning dispersion operator.
Nakatsuji sensei and Yasuda sensei sometimes told me he is a great professor. But I didn't
know who he is.
In Japanese, his name is 伏見康治. In old style Roma spelling, his name was Husimi Kodi.
(I believe) we now use Hepburn spelling, then Fushimi Koji. That's why I cannot find his name
while searching papers.
Of course he is a really great physicist.
Unfortunately, he passed away last year.
I's a pity I couldn't find any applicable new N-representability conditions.
After I finished two papers, I was not sure where I go. I applied more N-representability condition like: Weinhold-Wilson inequality (this is a subset of Davidson's inequality) but energy gain were only order of 1e-4 hartree. Totally exhausted and want to take some rest. I moved to Tokyo Univ. as post doc (and there, I was really exhausted).
But I was very lucky - otherwise I cannot win my Ph.D - because Braams and Percus are also trying to do variational calculation on 2-RDM for realistic molecules via semi-definite programming around 2000. Especially when I saw Braams' NFS fund proposal, what he's trying to do was almost the same as what I did. I think he was also very very surprised :)
Fukuda-san (specialist of optimization) were also at there of course after JCP 2001 paper was out. Note that Erdahl and Jin had already done by their SDP in 2000 or so, this was included in Cioslowski's density matrix book.
In 2004, Fukuda-san send me a preprint. Incorporating T1/T2 conditions in the variational space and results were surprisingly good. With PQG, we cannot do chemistry due to lack of accuracy but now it is comparable to CCSD(T). I thought that it was a great breakthrough in this field...
Then I visited New York University for one month. and started collaboration.
I met Braams there. He is really smart person I ever met. I asked why he did notice 2-RDM
method, and he replied as just he saw on some books. Oh how smart he is.
Also, I never thought that someone will find a really effective and applicable N-representability
condition in 10 years. And Braams did find it just three years.
Nakatuji-sensei said "oh you are doing chemistry" when I copying Herzberg, for geometries
of diatomic molecules.
When the system becomes larger, PQ conditions become worse.
Nakatuji-sensei told me that "why it's so good?" "I'm not sure, but I'm very sure that C2 are the first result that this method fails. because with PQ, I got 800% of correlation energy error"
However, fortunately or unfortunately, it was false. I got the correlation energy 108% or so...
Also staffs and colleagues told me about "double dissociation" or "triple dissociation". Surprisingly, PQG rules! The result was published as JCP 2002.
What I did in the first year - very simple extension - just describe openshell system, and nothing has been changed in the theory. I could determine some systems like H2O, Be, etc after three years, nothing more. Density equation is very unstable. I was really disappointing about it and something I should change the way I do. At first Nakatsuji-sensei was not happy with my change, of course, he found the density equation (or contracted schroedinger equation).
I read many older papers. Garrod, Fusco, Mihailovic, Rosina, Kijiwski are the early pioneers to trying to variational calculation, and surprisingly, their results for Be are very promising. So I decided to do variational calculation. All papers are quite pessimistic about P, Q, and G conditions. Yes, I know it. However, only two papers at the three years makes me crazy. I thought that I can reproduce Be result, and at least I know by Kummer's paper or Erdahl's paper that P Q G are compact (in what sense?) and Hamiltonian is linear functional, so we have at least well-defined minima (yes we can have results! if there are no numerical issues). also G-condition is somewhat related to BCS type wavefunction, I hoped that it can apply to some systems.
At that time I thought that I can reproduce Be result, but may fail for larger systems like H2O
but _AT LEAST_ I can get 2-RDMs (compactness of P,Q,Gdomain and if not numerical issues) and publish a paper! Oh what a joy!
In 1999/3/30, I asked at fj.sci.math about optimization of linear functional over semidefinite constraint. Ono-san kindly replied as there is an established field in mathematical programming and some implementations are available via internet. Kojima-lab is the one of the active lab in Japan.
So I started. I read many Prof.Kojima's resumes but I just understand semi-definite programming can be used for variational 2-RDM.
I e-mailed to Mituhiro Fukuda, now he's my friend, about it. I proposed him a very simplified problem and he kindly show me how to translate it to "primal" standard type problem.
It took only one month to understand how I "play" with SDPA. Implementation was a breeze. With P and Q condition, Be energy was -17 au or something, and it is expected :)
Incorporating G-condition was really tough. Really exhausted. I tried, tried and tried. all are my stupid bugs, and misunderstanding of SDP or something like that.
Finally I got some results until fall. I fixed the final stupid bug. Then I got many results. Lab seminar was held at 99/12/7, I presented some results. I was very happy about it. and
Nakatsuji-sensei told me "this is a very important result". it was my great pleasure!!
This result was published as JCP 2001.
After I submitted a manuscript, referee comment was very affirmative. However, due to
my laziness, I just left for a while. Nakatsuji-sensei scold me. If I were bit more smart, it was
published in 2000.
The first significant appearance of reduced density matrix was due to P.A.M. Dirac for the Hartree-Fock theory. After, Husimi Kodi (modern Japanese may write as Fushimi Koji) defined p-th order reduced density matrices. Unfortunately his paper had been ignored for long time as his paper has been submitted just before the World War II. Per-Olov Löwdin and Mayer's paper are the first ones that using second-order reduced density matrix as basic variables. Note that Lowdin was a student of Dirac.
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Nakata Maho is a scientist and interested in reduced density matrix related theories, optimization and multiple precision arithmetics.