Recent progresses in the variational reduced-density-matrix method

.
Recent progresses in the variational
.
reduced-density-matrix method

中田真秀 (NAKATA, Maho)
maho@riken.jp
http://accc.riken.jp/maho/

理化学研究所 (RIKEN), Advanced Center for Computing and Communication

The 50th Sanibel Symposium (February 24 - March 2, 2010)

NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method
Sanibel symposium 2010/2/25 1 / 34

Collaborators current and past

福田光浩 (Fukuda Mituhiro) Michael Overton
安田耕二 (Yasuda Koji) Zhengji Zhao
Bastiaan J. Braams
中田和秀 (Nakata Kazuhide)
Jerome K. Percus
藤澤克樹 (Fujisawa Katsuki) 江原正博 (Ehara Masahiro)

山下真 (Yamashita Makoto) 中辻博 (Nakatsuji Hiroshi)


Overview

Introduction of the RDM method.
Recent results.
Some open problems.


Part 1

Introduction of the RDM method.


What is the RDM method in short?



The RDM method: 2-RDM as basic variable




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Equivalent to the Schrodinger equation




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Ground state energy: Minimize directly!




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Ground state energy: Minimize directly!
N-representability condition; the only one approximation


Our goal: doing chemistry from the ﬁrst
principle, faster calculation and deeper
understanding

§ ¤
Our target ¥
¦

ab initio...theoretically and practically good
approximation
faster method ...mathematically simpler
deeper understanding...electronic structure


understanding

§ ¤
Our target ¥
¦

approximation


The ground state and energy calculation
¨
[Husimi 1940], [Lowdin 1954], [Mayer 1955], [Coulson 1960], [Rosina 1968]


¨
∑ 1 ∑ i1 i2 † †
H= vij a† a j +
i
w a a a j2 a j1
ij
2 i1 i2 j1 j2 j1 j2 i1 i2


¨
∑ 1 ∑ i1 i2 † †
H= vij a† a j +
i
w a a a j2 a j1
ij
2 i1 i2 j1 j2 j1 j2 i1 i2
The ground state energy becomes...
E g = min Ψ|H|Ψ
∑ 1 ∑ i1 i2
= min vij Ψ|a† a j |Ψ +
i
w j j Ψ|a† a† a j2 a j1 |Ψ
i1 i2
ij
2 i1 i2 j1 j2 1 2
∑ ∑
= min{ vij γij + wij1 ij2 Γij1 ij2 }
1 2 1 2
ij i1 i2 j1 j2


¨
∑ 1 ∑ i1 i2 † †
H= vij a† a j +
i
w a a a j2 a j1
ij
2 i1 i2 j1 j2 j1 j2 i1 i2
The ground state energy becomes...
E g = min Ψ|H|Ψ
∑ 1 ∑ i1 i2
= min vij Ψ|a† a j |Ψ +
i
w j j Ψ|a† a† a j2 a j1 |Ψ
i1 i2
ij
2 i1 i2 j1 j2 1 2
∑ ∑
= min{ vij γij + wij1 ij2 Γij1 ij2 }
1 2 1 2
ij i1 i2 j1 j2

Deﬁnition of 1, 2-RDMs
1
Γij1 ij2 = Ψ|a† a† a j2 a j1 |Ψ , γij = Ψ|a† a j |Ψ .
i1 i2 i
1 2 2

N-representability condition



[Mayers 1955], [Tredgold 1957]: Far lower than the exact one



[Mayers 1955], [Tredgold 1957]: Far lower than the exact one
N-representability condition [Coleman 1963]
∑ ∑
E g = min{ v jγ j +
i i
wij1 ij2 Γij1 ij2 }
P 1 2 1 2
ij i1 i2 j1 j2

γ, Γ ∈ P should satisfy N-representability condition:

Γ(12|1 2 ) → Ψ(123 · · · N)

γ(1|1 ) → Ψ(123 · · · N).
§ ¤
Encodes two-body effects completely. Very compact. ¥
¦


Approximate N-representability condition
Approximation (necessary) condition : where Physics and
Chemistry are

NAKATA, Maho (RIKEN, ACCC) Recent progresses in the variational reduced-density-matrix method 2010/2/25
Sanibel symposium 10 / 34

Chemistry are

P, Q-condition, ensemble 1-RDM condition [Coleman 1963]


Chemistry are

G-condition [Garrod and Percus 1964]


Chemistry are

k-th order approximation [Erdahl, Jin 2000] (aka k-positivity
[Mazziotti Erdahl 2001])


Chemistry are

¯
T1, T2, T2 , (T2)-condition [Zhao et al. 2004], [Erdahl 1978]
[Braams et al 2007] [Mazziotti 2006, 2007]


Chemistry are

¯

Davidson’s inequality [Davidson 1969][Ayers et al. 2006]


Chemistry are

¯

Davidson’s inequality [Davidson 1969][Ayers et al. 2006]

Construction of 2-particle density [Pistol 2004, 2006]


Summary 1: the RDM method is an ab initio
method


method

Can evaluate total energy exactly via 1 and 2-RDM


method

Can evaluate total energy exactly via 1 and 2-RDM
only one approximation is N-representability
condition (aka theory of everything)


understanding

§ ¤
Our target ¥
¦

approximation


Mathematically simpler:
number of variables are always four



Method # of variable (discritized) Exact?
Ψ N, (r!) Yes
Γ(12|1 2 ) 4
4, (r ) Yes



Ψ N, (r!) Yes
Γ(12|1 2 ) 4
4, (r ) Yes

Do not depend on the size of the system



Ψ N, (r!) Yes
Γ(12|1 2 ) 4
4, (r ) Yes

Do not depend on the size of the system
¨
Equivalent to Schrodinger eq. (ground state)


minimization of linear functional


minimization of linear functional

Eg = Min TrHΓ
Γ∈P
P = {Γ : Approx. N-rep.condition}


PSD type N-representability conditions


PSD type N-representability conditions

P,Q,G,T1,T2-matrix are all positive semideﬁnite ↔
eigenvalues λi ≥ 0
 
 λ1


0 



 λ2 


† 

 
 0

U ΓU =   ... 



 


 
0 λn

First application to Be atom
[Garrod et al 1975, 1976]
Calculation methods are not very well studied...


Realization of the RDM method for atoms
and molecules


and molecules

Eg = Min TrHΓ
Γ∈P


and molecules

Eg = Min TrHΓ
Γ∈P

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]
[Nakata-Nakatsuji-Ehara 2002]

Semidiﬁnite programming
We solved exactly for the ﬁrst time!


and molecules

Eg = Min TrHΓ
Γ∈P

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]

Semidiﬁnite programming
We solved exactly for the ﬁrst time!
Small enough “primal dual gap, feasibility” values show that total energies etc are MATHEMATICALLY correct


polynomial algorithm



Semideﬁnite programming: prima-dual
interior-point method polynomial algorithm



N-representability conditions: P, Q, G, T1, T2
polynomial



polynomial
Hartree-Fock: NP-hard (not O( N4 )! )



polynomial
HF ref. MP2, Coupled cluster: NP-hard, post
Hartree-Fock part is ponlynomial



polynomial
HF ref. MP2, Coupled cluster: NP-hard, post
Hartree-Fock part is ponlynomial
HF ref. Trancated CI: NP-hard, post Hartree-Fock
part is ponlynomial


Summary 2: the RDM method is a simpler
(and possibly faster) method



Number of variables are always four.



Minimization of linear functional.



Semideﬁnite programming solved exactly for the
ﬁrst time M.N.’s major contribution



Semideﬁnite programming solved exactly for the
ﬁrst time M.N.’s major contribution
polynomial algorithm (cf. Hartree-Fock is NP-hard).


understanding

§ ¤
Our target ¥
¦

ab initio...with theoretically and practically good
approximation


Physical and Chemical meaning of approx.
Theoretical


Theoretical
P, Q condition: electron and hole exist [Coleman].


Theoretical
G condition: exact for the AGP type Hamiltonian: BCS wave
function / superconductivity. [Coleman].


Theoretical
G condition: exact for high correlation of limit of Hubbard
model [submitted].
Practical


Theoretical
model [submitted].
Practical
P, Q and G condition: 100 ∼ 130% corr. [Nakata et al], [Mazziotti et al] [Eric et al]


Theoretical
model [submitted].
Practical

P, Q, G, T1, T2 condition: 100 ∼ 101% corr. [Zhao et al], [Nakata et al]


Theoretical
model [submitted].
Practical

P, Q, G, T1, T2 condition: 100 ∼ 101% corr. [Zhao et al], [Nakata et al]

P, Q and G condition: dissociation limit (sometimes fails).
[Nakata et al], [Mazziotti], [H. Aggelen et al]


The ground state energy of atoms and molecules [Nakata et al 2008]

System State N r ∆ EGT1T2 ∆ EGT1T2 ∆ ECCSD(T) ∆ E HF E FCI
C 3
P 6 20 −0.0004 −0.0001 +0.00016 +0.05202 −37.73653
O 1
D 8 20 −0.0013 −0.0012 +0.00279 +0.10878 −74.78733
Ne 1
S 10 20 −0.0002 −0.0001 −0.00005 +0.11645 −128.63881
O+ 2
Π g 15 20 −0.0022 −0.0020 +0.00325 +0.17074 −148.79339
2
1 +
BH Σ 6 24 −0.0001 −0.0001 +0.00030 +0.07398 −25.18766
CH 2
Πr 7 24 −0.0008 −0.0003 +0.00031 +0.07895 −38.33735
NH 1
∆ 8 24 −0.0005 −0.0004 +0.00437 +0.11495 −54.96440
1 +
HF Σ 14 24 −0.0003 −0.0003 +0.00032 +0.13834 −100.16031
SiH4 1
A1 18 26 −0.0002 −0.0002 +0.00018 +0.07311 −290.28490
F− 1
S 10 26 −0.0003 −0.0003 +0.00067 +0.15427 −99.59712
P 4
S 15 26 −0.0001 −0.0000 +0.00003 +0.01908 −340.70802
H2 O 1
A1 10 28 −0.0004 −0.0004 +0.00055 +0.14645 −76.15576

GT1T2 : The RDM method ( P, Q, G, T1 and T2 conditions)
GT1T2 : The RDM method ( P, Q, G, T1 and T2 conditions)
CCSD(T) : Coupled cluster singles and doubles with perturbation treatment of triples
HF : Hartree-Fock
FCI : FullCI


Application to potential energy curve
Dissociation curve of N2 (triple bond) the world ﬁrst result.
Potential curve for N2 (STO-6G)
Hartree-Fock
-108.5 PQG
FullCI
MP2
Total energy(atomic unit)

CCSD(T)

-108.55

-108.6

-108.65

-108.7

-108.75
1 1.5 2 2.5 3
distance(Angstrom)

Part 2

Recent results: non-size extensivity


Size-extensivity and consistency

Size extensivity or consistency is very important
property for a calculation theory.

E(A − −inﬁnity − − A) = E( A) + E(A)?



Not size consistnt: [Nakata-Nakatsuji-Ehara 2002]
(small deviation),
[Aggelen-Bultinck-Verstichel-VanNeck-Ayers 2009]
(fractional charge!)



Not size consistnt: [Nakata-Nakatsuji-Ehara 2002]
(small deviation),
[Aggelen-Bultinck-Verstichel-VanNeck-Ayers 2009]
(fractional charge!)
Not size extensive: [Nakata-Yasuda 2009]
PRA80,042109(2009).
CH4 , N2 non interacting polymers: slightly deviated
primal-dual interior point method is mandatory;
Monteiro-Bruner [Mazziotti 04] is inaccurate.


Size-extensivity: N2 polymer

N2 N2 N2 · · · N2 non interacting, N-rep.: PQG

0.0 Upper bound

Lower bound
Energy (10 a.u.)

Linear Fit
-3

-2.8

-2.0

-3.2
0.0 0.1

0.0 0.5 1.0
-2
(Number of molecules)

E(M) = −108.71553 + 0.00302M−2 . 3 × 10−4 au

Size-extensivity: CH4 polymer

CH4 CH4 CH4 · · · CH4 non interacting, N-rep.: PQG
0.0
Lower bound

Upper bound
Energy (10 a.u.)

-3.0
-4

-2.0

-3.1
0.0 0.1

0.0 0.5 1.0
-2
(Number of molecules)

Nither PQG nor PQGT1T2 are size-extensive

Size-extensivity: Inaccurate result by
Monteiro-Bruner method
H2 O: solved by Monteiro-Bruner method [Mazziotti 2004]: # of iteration req’ed
scale like exponential. Not converged with CO (double-ζ ).


Summary: the RDM method in short




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Ground state: minimize directly via semidef. prog.! [Nakata et al 2001]




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
N-rep: PQGT1T2 100 ∼ 101% [Zhao et al 2004]




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Polynomial method but takes very long time: H2O double-ζ 1 day




Γ 1 2 = 1 Ψ|a† a† a j2 a j1 |Ψ
i i
j1 j2 2 i1 i2

¨
Polynomial method but takes very long time: H2O double-ζ 1 day
§ ¤
¦
Hopeful and still lot of unknowns! ¥

How many iterations are needed?

How many iterations are required by

primal-dual interior-point method (PDIPM) or

Monteiro-Bruner method (RRSDP) [Mazziotti 2004]

P, Q, and G P, Q, G, T1, T2
algorithm ﬂops # iterations memory ﬂops # iterations memory
PDIPM r12 r ln ε−1 r8 r12 r3/2 ln ε−1 r8
RRSDP r6 none r4 r9 none r6
Note: when we stop the iteration is a big problem


How large these SDP are?

# of constraints
r constraints block
24 15018 2520x2, 792x4, 288x1,220x2
26 20709 3211x2, 1014x4, 338x1, 286x2

Elapsed time using Itanium 2 (1.3GHz) 1 node 4 processors.
System, State, Basis N-rep. r Time # of nodes
1
SiH4 , A1 , STO-6G PQGT1T2 26 5.1 days 16
H2 O, 1 A1 , double-ζ PQG 28 2.2 hours 8
H2 O, 1 A1 , double-ζ PQGT1T2 28 20 days 8
H2 O, 1 A1 , double-ζ PQGT1T2 28 24 days 8


Necessity of highly accurate solver
SDP results are usually not accurate; typically 8 digits or so.
When the ground state is degenerated, the SDP becomes
more difﬁcult when approaching to the exact optimal.


WE NEED MORE DIGITS, FOR EXAMPLE 60 DIGITS!



double (16 digits) 1 + 0.00000000000000001 1



double (16 digits) 1 + 0.00000000000000001 1
GMP (60 digits; can be arbitrary)
1 + 0.000000000000000000000000000000000000000000000000000000000001 1



double (16 digits) 1 + 0.00000000000000001 1
1 + 0.000000000000000000000000000000000000000000000000000000000001 1

GMP (GNU multiple precision)



double (16 digits) 1 + 0.00000000000000001 1
1 + 0.000000000000000000000000000000000000000000000000000000000001 1

GMP (GNU multiple precision) ⇒ necessity of highly
accurate solver, using multiple precision arithmetic
(SDPA-GMP) http://sdpa.indsys.chuo-u.ac.jp/sdpa/
GNU Public License

SDPA-GMP and Hubbard model
The 1D Hubbard model with high correlation limit
|U/t| → ∞: All states are almost degenerated.
The ground state energies of 1D Hubbard model
PBC, # of sites:4, # of electrons: 4, spin 0
U/t SDPA (16 digits) SDPA-GMP (60 digits) fullCI
10000.0 0 −1.1999998800000251 × 10−3 −1.199999880 × 10−3
1000.0 −1.2 × 10−2 −1.1999880002507934 × 10−2 −1.1999880002 × 10−2
100.0 −1.1991 × 10 −1 −1.1988025013717993 × 10 −1 −1.19880248946 × 10−1
10.0 −1.1000 −1.0999400441222934 −1.099877772750
1.0 −3.3417 −3.3416748070259956 −3.340847617248
PBC, # of sites:6, # of electrons: 6, spin 0
U/t SDPA (16 digits) SDPA-GMP (60 digits) fullCI
10000.0 0 −1.7249951195749525 × 10−3 −1.721110121 × 10−3
1000.0 −1 × 10−2 −1.7255360310431304 × 10−2 −1.7211034713 × 10−2
100.0 −1.730 × 10 −1 −1.7302157140594339 × 10 −1 −1.72043338097 × 10−1
10.0 −1.6954 −1.6953843276854447 −1.664362733287
1.0 −6.6012 −6.6012042217806286 −6.601158293375


Recent progresses in the variational reduced-density-matrix method

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