Direct variation of the second-order reduced density matrix andpositive semidefinite programming
NAKATA Maho1
11

Advanced Center for Computing and Communication, RIKENIntroduction

2

The Reduced-Density Matrix Method

2.1
The basic equation for chemistry is known and it iscalled the Schrödinger equation, however, it is also known
that the Schödinger equation is too diﬃcult to solve [1].
This is a motivation for direct determination of the secondorder reduced density matrix (2-RDM) [2] instead of solving the Schödinger equation, since it is much simpler thanwavefunction by following reasons: (i) we can calculate
the total energy and all physical proprieties without approximation by the 2-RDM (ii) the number of variables
is always four regardless how many electrons exist in thesystem, and We hope we can construct wave mechanics
without waves, and much simpler.
One major obstacle for the RDM method that the 2RDM should satisfy the N -representability condition, thattrial 2-RDM should have its parent wavefunction, otherwise, it is non-physical. It is not understood very muchin two ways: approximate or complete solution to the N representability is not very well understood, and how to dopractical and systematic calculation had been unknown.
There are some early attempts, but faded away.
In 2001, interest returned to the RDM method when
Nakata et al. [3] formulated the problem as a primal
semidefinite programming problem. They performed a direct variational calculation of the 2-RDM employing the P ,Q and G conditions using the well-established semidefinite
programming solver known as SemiDefinite Programming
Algorithm (SDPA) [4]. This approach was applied to manyfew electron atoms and molecules. Their results using theP , Q, and G conditions were very encouraging; yielding
around 100 to 130% of correlation energy as well as being able to produce the dissociation curve of the nitrogen
dimer in good agreement with fullCI. Zhao et al. included
the T 1 and T 2 conditions in addition to the P , Q and G
conditions in this approach for their calculations on small
molecules [5]. The accuracy of their results are comparable to coupled-cluster singles and doubles with perturbative triples (CCSD(T)) at equilibrium geometries. The
calculated correlation energies range typically from 100 to101%. Mazziotti confirmed the findings of Nakata et al. [3]and Zhao et al. [5], by applying these conditions to larger
systems. He accomplished this by implementing a semidefinite programming solver [6]. Mazziotti et al.’s thoroughstudy of these N -representability conditions was important to establish the validity of the approach.

The Schrodinger equation in Chemistry

The ground state energy calculation of a non-relativistic
and time-independent, N -electron molecular system underthe Born-Oppenheimer approximation is the most important problem It can be obtained as the lowest eigenvalue
E of the electronic Schrödinger equation:HΨ(z) = EΨ(z).
The Hamiltonian in second quantized form is expressed asfollows:
X i †1 X
wi1 i2 a† a† aj aj ,H=vj ai aj +2 i i j j j1 j2 i1 i2 2 1ij
1 2 1 2†where a and a are creation and annihilation operators,
ij
and where vji and wk`
are one- and two-particle terms arecalculated as follows:!
ZM1 2 X ZAi∗
vj = ψi (z) − ∇ −ψj (z)dz,2rA
A=1«1ψ` (z 1 )ψk (z 2 )dz 1 dz 2 .
|z 1 − z 2 |in which ZA is the atomic number of the nucleus A, riA is
the distance between the electron i and nucleus A, and rij
is the distance between two distinct electrons and {ψi }∞i=1
is the one-particle complete orthnormal system. We usually choose finite subset from {ψi }∞
i=1 , and it is called “basis set”.
ijwk`=2.2Zψi∗ (z 1 )ψj∗ (z 2 )The reduced density matrix and variatioanl
methodThe definition of 1- and 2-RDM in the second-quantizedversions are following:Xγji =tp hΨp |a†i aj |Ψp i,
pΓijk`=1X
tp hΨp |a†i a†j a` ak |Ψp i,2 pP
P
where D = p tp |Ψp ihΨp |, tp ≥ 0, p tp = 1.
The ground state energy Emin can be calculated by minimizing the total energy over 1- and 2-RDMs.
XEmin = mintp hΨp |H|Ψp ip=In this paper, we present summary of our research.

„min{Xijvji γji +Xijk`

ij ijwk`Γk` }.

2.3

N -representability condition and the RDMmethod

N -representability condition on 1- and 2-RDMs is firstcoined by Coleman, that whether trial 1- or 2-RDM has
its N -particle ancestor wavefunction.
Γ → D.

Table 1. Ground state energies (in diﬀerences from that offull CI) calculated by the RDM method imposing the P ,Q, G, T 1, T 20 conditions from SDPARA 7.3.2, and thoseobtained by CCSD(T), and Hartree-Fock. The last column
shows the full CI energies. The correlation energies (0%for Hartree-Fock and 100% for full CI) in percentage arealso shown in the second row.
system state basis

If there are, then N -representable, and no, then non
N -representable. The variational determination of 2RDM can be formulated using E N , a complete set of N presentable RDMs like folows.
X i i X ij ij
Emin = min {v j γj +wk` Γk` }.
Γ∈E Nijijijk`

where Ẽ N is approximately N -representable 2-RDM which
satisfies necessary condition:
Ẽ N = {Γ|Γ satisfies P , Q, G, T 1 and T 20 conditions}.
We refer such varational method as the “the RDMmethod”.
2.4

Positive semidefinite programming

The definition of the standard type semidefinite programming is given by folloiwng:(P )min : Σm
k=1 ck xks.t. : X = Σm
k=1 F k xk − F 0 , X  O,

where the symbol Sn is the space of n × n symmetric matrices. The Lagrangian dual (D) of the problem (P ) canbe derived as:
(D)

1A
128

CH21A
128

8 (4)

NH31A
130

10 (5)

CH32 A00
230

9 (5)

C21 Σ+g36

12 (6) 1

10 (5)

PQGT1T20+0.000 63
100.4+0.000 59
100.4+0.000 49
100.4∆ECCSD(T)+0.008 74
99.55+0.005 80
99.42+0.007 46
99.62+0.141 98 −55.624 71
100
0+0.100 67 −38.962 24
100
0+0.128 75 −56.304 89
100
0+0.000 31
100.3+0.000 39
101.2+0.004 01
99.67+0.055 98
99.86+0.094 54 −39.644 14
100
0+0.285 66 −75.642 111000∆E∆EHF

EFCIijk`

Since practical complete N -representability condition on2-RDM is not accessible, we need practical condition on
1- and 2-RDMs. For 1-RDM, practical complete condition
is known: if eigenvalue of 1-RDM (occupation number bynatural orbital) are lie in [0, 1], then it is N -representable,
but for 2-RDM, such condition is not known, and we only
know that some necessary conditions like P , Q, G, T 1 andT 20 conditions. Explicit form of these conditions can befound in literature [5]. These necessary conditions statesthat P -matrix, Q-matrix, G-matrix, etc. are all positive
semidefinite; eigenvalues of these matrices are larger thanor equal to zero. Moreover, complete N -representability
condition of 1-RDM can be restated as: given 1-RDM isN -representable if and only if γ and I − γ are positive
semidifinite, where I is an identity matrix.
Finally, the ground state problme using the 2-RDM asa basic variable can be formulated as:
X i i X ij ijẼmin = min {v j γj +wk` Γk` }.
(1)Γ∈Ẽ NN−NH2max : F 0 • Ys.t. : F k • Y = ck (k = 1, . . . , m)
Y  O.

We can formulate the RDM method as the standard typesemidefinite programming of huge size [3, 5]. We can use
one of the implementations as a black box solver.

3Results and Conclusion

We showed the outline of the RDM method with applications to molecular systems. Some feature of this method
are: (i) it is an ab initio method, which is rigorously thesame as the Schrödinger equation for the ground state; (ii)the number of variables is always four, regardless of thesize of the system. The major obstacle for this method
is the fundamentally diﬃculty of obtaining the complete
N -representability conditions for the 2-RDM; therefore weused approximate ones: P , Q, G, T 1 and T 20 . The considered problem becomes a semidefinite programming problem and we presented results for NH−
2 , CH2 , NH3 , C2 ,CH3 , in Table 1 using a supercomputer with a highly eﬃcient semidefinite programming solver, SDPARA. We obtained fairly good energies comprable to CCSD(T).
Acknowledgement
The author was supported by the Special Postdoctoral Researchers’ Program of RIKEN, and is partially
supported by Grant-in-Aid for Scientific Research (B)21300017.
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Chem. Phys., 114, 8282, (2001)[4] K. Fujisawa, K. Nakata, M. Yamashita and M.
Fukuda, “SDPA project: solving large-scale semidefinite programs”, J. Oper. Res. Soc. of Jpn., 50, 278,
(2007).
[5] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, “The reduced density matrix method for electronic structure calculations andthe role of three-index representability conditions”, J.
Chem. Phys. 120, 2095, (2004)[6] D. A. Mazziotti, “Realization of Quantum Chemistry without Wave Functions through First-Order
Semidefinite Programming”, Phys. Rev. Lett., 93,213001 (2004).