Theor Chem Acc (1999) 102:97±104
DOI 10.1007/s002149800m94

Regular article
Equation for the direct determination of the density matrix:
Time-dependent density equation and perturbation theory*
Hiroshi Nakatsuji
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University,
Kyoto 606-01, Japan and Institute for Fundamental Chemistry, 34-4 Takano Nishi-hiraki-cho, Sakyo-ku, Kyoto 606, Japan
Received: 16 June 1998 / Accepted: 2 September 1998 / Published online: 8 February 1999

Abstract. The density equation proposed previously forthe direct determination of the density matrix, i.e. for thewave mechanics without wave, is extended to a timedependent case. The time-dependent density equation
has been shown to be equivalent to the time-dependent
SchroÈdinger equation so long as the density matrix,
included as a self-contained variable, is N-representable.
Formally, it is obtainable from the previous timeindependent equation by replacing the energy E withi
h @t@ . The perturbation theory formulas for the density
equation have also been given for both the timedependent and time-independent cases.
Key words: Density equation ± Time-dependent density
equation ± Perturbation theory

1 Introductioni

The non-relativistic electronic state is determined by the
SchroÈdinger equation constrained by the Pauli exclusion
principle. We restrict ourselves, for a while, to fermions
but the argument is also valid for bosons. The
SchroÈdinger equation is an equation of motion of the
wave function which depends on all the N-electron
coordinates, where N is the number of the electrons of
the system, and the Pauli exclusion principle is anantisymmetric condition for the exchange of any twoelectrons of the system. On the other hand, all theelementary observables in quantum mechanics arecomposed of one- and/or two-electron operators, sothat all such quantities are calculated only with theknowledge of the second-order density matrix (2-DM)
de®ned by [1]C 2 10 20 j12Z
 N C 2 W 10 20 3    N W 123    N dx3    dxN :* Contribution to the Kenichi Fukui Memorial Issue

The 2-DM depends only on the four coordinates, while
the wave function depends on the N-coordinates.
Therefore, it would be a great simpli®cation if we could®nd the equation-of-motion of the 2-DM itself.
Such an equation-of-motion was derived by thepresent author in 1976 as the density equation [2], whichis connected with the SchroÈdinger equation by thenecessary and sucient conditions. The Pauli conditionin the wave function space becomes the N -representability condition [3] for the density matrix. Figure 1
shows the relation between these two methods. A
problem in the density equation method is that the
N -representability condition is not yet completely
known, despite the eorts of many researchers [4]. The
Pauli (exclusion) principle is trivial in the wave function
space so that we still use the wave function approach
despite the simplicity of the 2-DM.
Let us de®ne the Hamiltonian of the system byXXv i 
w i; j ;2
H 1    N 

1

i>j

where v i and w i; j are one- and two-particle operators, respectively. The SchroÈdinger equation for thestationary state is written as
H 1    N W 1    N   EW 1    N  :3

The Pauli condition for the wave function is
P  W 1    N   ÿp W 1    N  ;4where P is a permutation operator and ÿp is plus orminus unity depending on the parity of the operator P.
The second-order density equation is written as [2]EC 2  v 1  v 2  w 1; 2C 2
Z
 3 v 3  w 1; 3  w 2; 3C 3 dx3
Z6

w 3; 4C 4 dx3 dx4 ;which is equivalently written as

5a

98

H n de®ned by Eq. (7a) is symmetric for the interchangeof the coordinates, n+1 and n+2, but that by Eq. (7b) isnot. The nth-order density matrix n-DMC n is de®nedbyC n 10    n0 j1    nZ
 N Cn W 10    n0 n  1    N 

Fig. 1 Wave function method vs density equation methodZ W 1    n n  1    N dxn1    dxN :H 2 12; 34 ÿ EC 4 10 20 30 40 j1234dx3 dx4  0 :5b

In general, the nth-order density equation is written as
[2]
"
#
n
n
X
X
n
v i 
w i; j C n
EC 
i

i>j

#
Z "
n
X
 n  1v n  1 
w i; n  1i1
 C n1 dxn1  n  1 n  22Z

w n  1; n  2C n2 dxn1 dxn2 ;6a

or equivalently asZ
H n 1    n; n  1 n  2 ÿ E
 C n2 10    n  20 j1    n  2dxn1 dxn2  0 ;6b
where Hn

is a reduced Hamiltonian de®ned by either

H n 1    n; n  1 n  2nnXXv i 
w i; j

i

i>j

"
#
n
X
1
w i; n  1
 N ÿ n v n  1 
2
i
"
#
n
X
1
w i; n  2
 N ÿ n v n  2 
2i

1N ÿ n N ÿ n ÿ 1w n  1; n  2 ;
2

7a

or
H n 1    n; n  1 n  2nnXXv i 
w i; j

i

i>j

"

 N ÿ n v n  1 nX#
w i; n  1i1
N ÿ n N ÿ n ÿ 1w n  1; n  2 :
2

7b

8

Using the energy density matrix G n , the density
equation is also written as [2]EC n  N Cn G n :9The de®nition of G n is self-evident by comparing Eq.
(9) with Eq. (6a). It was shown in the previous paper [2]that each density equation with n  2 for the set of the
N -representable density matrices is equivalent (connected by the necessary and sucient conditions) with the
SchroÈdinger equation constrained by the Pauli principle(see Fig. 1).
Here we note that the factor N Cn can be omitted if
C n is normalized to unity. However, for consistencewith the previous paper [2], we use the normalization
given by Eq. (8).
The density equation was left unsolved, despite its
potential utility, for two decades. Davidson and Harriman [5] pointed out that the number of unknowns included in the density equation is larger than the numberof equations, so that the equation is insoluble. From
Fig. 1, it is clear that such insolubility originates fromthe unknown nature of the N -representability condition:
if we have the complete N -representability condition, we
can solve the density equation, just as we can solve the
SchroÈdinger equation under the Pauli condition.
Recently, a breakthrough concerning this approach
was presented by Valdemoro's group [6, 7] and ourgroup [8]. Valdemoro's group tried to ®nd good approximate formulas representing C 3 and C 4 in termsof C 2 and C 1 ; putting these formulas into Eq. (5a) thedensity equation becomes an equation involving only
C 2 as a basic variable. Valdemoro and her coworkers[6] generalized the fermion anticommutation relation
written for the 1-DM and the ®rst-order hole density
matrix (1-HDM) to the second, third, and fourth-ordercases, and from these equations they derived approximate decoupling equations (which are referred to as IPH
approximation in Ref. [8]). For the 2-DM, this approximation was equivalent to the independent particle approximation. Nakatsuji and Yasuda [8], on the other
hand, utilized the hierarchy equations in the Green's
function method [9] and introduced the decoupling
approximations which were valid essentially up tosecond-order in electron correlations. As clearly
understood from the argument based on Fig. 1,
these decoupling formulas constitute the (approximate)
N -representability conditions. In this formalism, Valdemoro's approximation is shown to be valid up to ®rstorder in the correlations. They could solve the density
equation for more than ten atoms and molecules and theresultant energies and properties were in good agreement

99

with the full con®guration interaction (CI) (exact) values. Thus, we have been able to solve the 2-DM directlyfrom the density equation without any use of the wave
function. Wave mechanics without wave has thus beenrealized in our laboratory. This was certainly a big
breakthrough in the density approach in chemistry andphysics.
It is therefore time to publish the time-dependent
density equation. In this paper we deal with a generalization of the density equation to the time-dependentcase. Our basic variable is the time-dependent density
matrix which is connected with the time-dependent wave
function as
C n  C n 10    n0 ; t0 j1    n; tZ N Cn W 10    n0 n  1    N ; t0 
 W 1    n n  1    N ; tdxn1    dxN

10

and we want to derive an equation which is related to thetime-dependent SchroÈdinger equation
@
W
11
@t
by a necessary and sucient theorem. In Sect. 2, wederive such a necessary and sucient theorem. In Sect. 3,
we give the time-dependent density equation in a form
which includes only the density matrix as a variable andconsider a method to use it for the direct determinationof the density matrix. In Sect. 4, the perturbation theoryof the denstiy equation is studied brie¯y for both thetime-dependent and time-independent cases. Some concluding remarks will be given in Sect. 5.

H W  ih

2 Basic theorem
We consider a system composed of N fermions or bosons
speci®ed by the time-dependent Hamiltonian
H

N
X
i

v i; t NX

w i; j; t ;where v and w are respectively the one- and two-particle
operators which may depend explicitly on time t. We
assume that our density matrix C n 10    n0 ; t0 j1    n; t is
N -representable so that we can suppose the existence ofa wave function W 1    N ; t which satis®es Eq. (10).
Note that our time-dependent denstiy matrix is completely general, i.e. it depends not only on the two
independent sets of n-coordinates, 10    n0 and 1    n,but also on the two independent times, t and t0 . The
number of particles, N is supposed to be a constant of
motion, so thatZ
13C n 1    n; tj1    n; tdx1    dxn  N Cn :where S t0 ; t  hW t0 ; W ti.

G n  G n 10    n0 ; t0 j1    n; tZ W 10    n0 n  10    N 0 ; t0 
 H 1    n n  1    N ; t
 W 1    n n  1    N ; t n10  n1;...;N 0 N dxn1    dxN

 hW0 ; HWin :

14

15

Note that in Eq. (15) the Hamiltonian is de®ned with theunprimed coordinates and time, and operates only on
the right. In the last equality, the integral notation h inwhich denotesZ16
h in    n10  n1;...;N 0 N dxn1    dxN ;is used and the prime on W within h in means thatit depends only on the primed coordinates and time, i.e.
W0  W 10    N 0 ; t0 . Using this notation, Eq. (10) is
written as
C n  N Cn hW0 ; Win :100 

2.1 Theorem
The time-dependent density equation,ih@ n 0C 1    n0 ; t0 j1    n; t  N Cn G n 10    n0 ; t0 j1    n; t@t17with any single n which is larger than or equal to twon  2 constitutes a necessary and sucient condition
for the W 1    N ; t included in Eqs. (10) and (15) tosatisfy the SchroÈdinger equation
H W  i
h

12

i>j

Note thatZ
C n 1    n; t0 j1    n; tdx1    dxn  N Cn S t0 ; t ;Using the wave function W introduced in Eq. (10), wede®ne the energy density matrix for the time-dependent
system analogously to Eq. (10) as,

@
W :
@t

110 

2.2 Proof
We note that in Eq. (17) the operator @t@ written by theunprimed t operates only on the unprimed time t in C n .
The necessity is as follows. From Eqs. (10) and (15), we
can rewrite Eq. (17) as


 
@
W 0 :h18
W0 ; H ÿ i@tnThis is the density equation in the W-representation. It istrivial that when W satis®es Eq. (11), Eq. (18) follows
automatically. Next, we prove the converse, i.e. thesuciency. We ®rst consider the lowest-order density
equation with n  2. Then, Eq. (17) reads@ 2 0 0 0
C 1 2 ; t j12; t  N C2 G 2 10 20 ; t0 j12; t :19@tEquating 20 to 2 and integrating, we obtain from Eq. (19)the identity
ih

100

i
h

@ 1 0 0
C 1 ; t j1; t  N C1 G 1 10 ; t0 j1; t ;
@t


20

i
h

@ 0
W ; HW@t0

and furtherih@hW0 ; Wi  hW0 ; H Wi :@t

21The right-hand side of Eq. (21) can be rewritten as
hW0 ; H WiZZ v 1; tC 1 dx1  w 1; 2; tC 2 dx1 dx2 ;22

where we have used Eq. (12) and the permutation
symmetry for the coordinates of W 1    N ; t. We havealso used the abbreviations
ZZ
23
v 1; tC 1 dx1  v 1; tC 1 10 ; t0 j1; t10 1 dx1 :Hereafter such abbreviations will be used frequently.
Similarly, the integral hH 0 W0 ; Wi is written as
hH 0 W0 ; Wi
Z
Z
 0 0
1
 v 1 ; t C dx1  w 10 ; 20 ; t0 C 2 dx1 dx2 ; 24
where v and w are actually equal to v and w, because
they are real. We now introduce an N -particle function
U 1    N ; t by@
W 1    N ; t25
U 1    N ; t  H 1    N ; t ÿ ih@t0

and consider the integral hU ; Ui:
@
Wi
@t
@
@
@
h 0 W0 ; ih Wi :
ÿ hih 0 W0 ; H Wi  hi@t@t@t
The ®rst term of Eq. (26) can be transformed tohhU0 ; Ui  hH 0 W0 ; H Wi ÿ hH 0 W0 ; i26

hH 0 W0 ; H WiZ
 N C1 v 10 ; t0 G 1 10 ; t0 j1; tdx1
Z
 N C2 w 10 ; 20 ; t0 G 2 10 20 ; t0 j12; t dx1 dx2
Z

Z
@
 ih
v 10 ; t0 C 1 dx1  w 10 ; 20 ; t0 C 2 dx1 dx2@t@ ih hH 0 W0 ; Wi@t@27 hH 0 W0 ; ih Wi :@t
The ®rst equality was obtained similarly to Eq. (24),since W and HW have the same symmetry for theinterchange of coordinates. In the second equality wehave used Eqs. (19) and (20), and in the third equality
we have used Eq. (24). Next, we transform the third termof Eq. (26).

@hW0 ; H Wi
@t0


@
@
0
i
h hW ; Wi
 ÿi
h
@t
@t


@
@
h W :
 i
h W0 ; i
@t
@t
 ÿi
h

28

In the second equality we have used Eq. (21). Inserting
Eqs. (27) and (28) into Eq. (26), we obtain
hU0 ; Ui  0 :
0

29
0

for any t and t. When t  t, this is
hU; Ui  030which gives U  0, i.e. the SchroÈdinger equation (Eq.
11). Thus, we have shown that the second-order density
equation (Eq. 19) is a necessary and sucient conditionof the SchroÈdinger equation. In general, when we havethe density equation (Eq. 17) with n  2, we can always
obtain the second-order density equation (Eq. 19) by adirect integration over the last n-2 coordinates. Therefore, the above proof also holds for the general case.
Thus, we have proved that the time-dependent density
equation (Eq. 17) with any single n n  2 is a necessary
and sucient condition for the W to satisfy the timedependent SchroÈdinger equation (Eq. 11). (QED)
2.3 Implication of the theorem
Because the theorem is a necessary and sucient
theorem, it shows that the density equation (Eq. 17)
for any single n n  2 is just equivalent to the
SchroÈdinger equation (Eq. 11) so long as the density
matrix is N -representable. Among others, the simplest
equation is the second-order density equation (Eq. 19)
with n  2. Here, the density equation is de®ned using
W explicitly, but our purpose is to solve the density
equation not in the W-space but for the set of the density
matrices. For this purpose, we have to transform theenergy density matrix G n such that it depends only on
the density matrix, since G n was de®ned in Eq. (15)using the wave function W. Such a transformation will
give an equation-of-motion of the N -representabledensity matrix itself, as will be shown in the next section.
The following properties of the density equation areimportant:
1. As in the time-independent case, the equation for
n  1 (®rst-order equation) is not a sucient condition, though it is evidently a necessary condition. Thisis due to the fact that our Hamiltonian (Eq. 12)
includes up to the two-particle interaction terms. For
the case in which the Hamiltonian is composed ofonly the one-particle term (e.g. the model Hamiltonian in the time-dependent Hartree-Fock theory), the®rst-order density equation also constitutes a necessary and sucient condition.
2. In general, when the Hamiltonian of the system
includes up to p-particle interaction operators, it is101easy to show that the necessary and sucient theorem
holds for the density equation higher than the p-thorder n  p.
3. The permutation symmetry of the wave function W
required in the proof of the theorem was only forthe three types of exchange operators, P12 ; P1i andP2i 3  i  N , in the pairwise exchanges in bothsides of bra and ket, i.e.
P W P W  W W;

P  P12 ;

P1i ;

P2i

Although this requirement alone looks at ®rst sight tobe much weaker than requiring W to be symmetric orantisymmetric for any exchange operators (i.e. requiring C to be N -representable), it can be shown thatthese two requirements are the same1 [10]. Therefore,
the theorem is valid only for the N -representable setof the density matrices.
4. When the system is time-independent, the present
time-dependent density equation reduces to the previous time-independent density equation. For thestationary state, the time variable would appear as;32 C n 10    n0 j1    neÿiE tÿt =h

0

33

0

34

W 1    N ; t  U 1    N e
or
C
G

n

0

0

i1



3  i  N :31ÿiEt=hthe method to use the density equation for the direct
determination of the density matrix.
We rewrite the Hamiltonian of the system given by
Eq. (12) as
"
#
n
n
N
n
X
X
X
X
H
v i; t 
w i; j; t v i; t 
w i; j; t

0

Z 

n

0

0

n

0

0

9
EC 1    n j1    n  N Cn G 1    n j1    n
which is just the density equation for the timeindependent system, and the constant E is the energyof the system given byZ
35
E  G n 1    nj1    ndx1    dxn :

36


@
C n2 dxn1 dxn2  0 ;@t37

where Hnis the reduced Hamiltonian given by

H n 1    n; n  1 n  2 : tnXv i; t inXw i; j; t

i>j

"
#
n
X
1
w i; n  1; t N ÿ n v n  1; t 
2
i
"
#
n
X
1
w i; n  2; t N ÿ n v n  2; t 2i

1    n ; t j1    n; twhere E is an appropriate constant. Inserting theseequations into Eq. (17), we obtainw i; j; t :hH n 1    n; n  1 n  2 : t ÿ i0

 G n 10    n0 j1    neÿiE tÿt =hj1

Inserting Eq. (36) into the density equation (Eq. 17) or(Eq. 18) and performing the integration over the last N ÿ ncoordinates, we obtain the density equation in the form

0

0

in1

i>jn1

1    n ; t j1    n; t

n

i>j

n
X



1
N ÿ n N ÿ n ÿ 1w n  1; n  2; t ;
2

38a

or
H n 1    n; n  1 n  2 : t


n
X
i

v i; t "nXw i; j; t

i>j N ÿ n v n  1; t nX#w i; n  1; t

i

3 Time-dependent density equation
In this section, we transform the time-dependent density
equation (Eq. 17) into the form which includes only thedensity matrix as a self-contained variable, and consider
1

The requirement given by Eq. (31) means that the W should besymmetric or antisymmetric for the exchanges P12 , P1i and P2i
3  i  N . However, this requirement is completely general: any
exchange operator Pjk can be written as Pjk  P1j P2k P12 P2k P1j , sothat such W should actually be an eigenfunction of any exchange
operators, all of the eigenvalues being either +1 or ÿ1. (See, e.g.
Ref. [10]. Since any permutation operator can be written as a
product of exchange operators, the requirement given by Eq. (31) iscompletely general. (In a previous paper (Ref. 2), the author has
tried to expand the space of the density matrix from the N representable one in the last two paragraphs of Sect. IIIA, but theresultant space turned out to be not wider than the N -representable
space for the reason given above.)1
N ÿ n N ÿ n ÿ 1w n  1; n  2; t :(38b)
2
In Eq. (37), we have used the abbreviation as in Eq. (23).
As the reduced Hamiltonian, we may use either Eq.
(38a) or (38b). The former is symmetric for thecoordinates n  1 and n  2, but the latter is not. The
two particles n  1 and n  2 are the representatives ofthe last N ÿ n particles.
The density equation (Eq. 37) is written in a following
equivalent form. Using the recurrence formula for thedensity matrix
Z
C p 10    p ÿ 10 p; t0 j1    p ÿ 1p; tdxp
N ÿ p  1 pÿ1 0
1    p ÿ 10 ; tj1    p ÿ 1; t ;
C
p
39

102

we can rewrite Eq. (37) as
i
h

@ n
C
@t "
#
n
n
X
X

v i; t 
w i; j; t C niZ "

 n  1i>jv n  1; t nX#w i; n  1; t

i1
 C n1 dxn1  n  1 n  22
Z
 w n  1; n  2; tC n2 dxn1 dxn2 :40

From the previous theorem, the density equation(Eq. 17), (Eq. 37) or (Eq. 40) with any single n n  2 isequivalent to the SchroÈdinger equation for the set of the
N -representable density matrices. That is, to solve the
SchroÈdinger equation (Eq. 11) imposing W to be symmetric (for bosons) or antisymmetric (for fermions) is
equivalent to solving the density equation (Eq. 17), (Eq.
37) or (Eq. 40) imposing the density matrix to be N representable. This equivalence is just like the one shownin Fig. 1 for the time-independent case. The density
equation gives the exact density matrix without any
ambiguity when it is solved for the N -representable ones.
Thus, the time-dependent density equation (Eq. 37) or(Eq. 40) is self-contained equation for the time-dependent density matrix, and so may be considered as giving
an equation-of-motion of the N -representable density
matrix. There, every solution of the density equation
should be identical with the one obtained indirectly fromthe SchroÈdinger equation.
The simplest and practically most important density
equation is the second-order density equation given by

Z 
@
2
h
H 12; 34 : t ÿ i@t
 C 4 10 20 30 40 ; t0 j1234; t dx3 dx4  0 ;

41

or
i
h

@ 2
C  v 1; t  v 2; t  w 1; 2; tC 2
@tZ
 3 v 3; t  w 1; 3; t  w 2; 3; tC 3 dx3
Z6

w 3; 4; tC 4 dx3 dx4 :42

Here, the variable is essentially the 2-DM C 2 irrespective of the number of the particles involved in thesystem. This would be a quite remarkable simpli®cation
over the SchroÈdinger equation in which the variable W
depends on all the N coordinates. In the density
equation, N is included in the reduced Hamiltonian
only as a constant factor, so that an increase in N would
not cause a diculty in solution. In the SchroÈdinger
equation, however, it is well known that an increasein N causes a tremendous diculty in solution.

A note is necessary here that the density equation(Eq. 41) or (Eq. 42) involve not only C 2 but also C 3and C 4 so they are indeterministic by themselves.
However, the necessary and sucient relation as givenin the proof and as depicted in Fig. 2 means that they
are deterministic when the complete N -representability
condition is given as a subsidiary condition. This implies
that the N -reprersentability condition describes the
explicit and exact relations of C 3 and C 4 in termsof C 2 and C 1 .
The density equation given by Eq. (40) or (42) is similar
to a member of the coupled chain of hierarchy equations
often appearing in the many-body theory (e.g. Green's
function) [9, 11]. We also refer to the BBKGY (BornBogoliubov-Kirkwood-(H. S.) Green-Yvon) equation(see, for example, Ref. [12]). In particular, in the timeindependent limit, Eq. (40) becomes identical with amember of the coupled hierarchy equations derived forthe density matrix by Cho [13] and by Cohen and Frishberg [14], who showed only the necessary condition.
However, the implication of our density equation is essentially dierent from that of these hierarchy equations.
As discussed in the previous paragraph, every density
equation higher than second order is deterministic in that it
is equivalent to (a necessary and sucient condition of)
the SchroÈdinger equation. The higher-order equations are
not required for the exact solution in contrast to the usualinterpretation of the hierarchy equations.
So far, we have supposed that our density equation is
to be solved for the N -representable density matrix. In
such a case, all the solutions would be the exact physicalsolutions. However, since the complete (necessary andsucient) N -representability conditions are not yet
known at present, such a presupposition would not berealistic. So, we may relax the constraint, i.e. we solve
the density equation imposing only the known andtractable N -representability conditions. Such a procedure is possible for the density equation since it is a nonvariational equation. Even in such a case, the density
equation would give correct exact solutions within the
N -representable set, though it may also give some nonphysical solutions in the space outside of the N -representable set. The properties and the number of these
non-physical solutions would depend strongly on the
N -representability (necessary) conditions which havebeen imposed simultaneously on the density equation.
These non-physical solutions could be eliminated, inprinciple, using the rest of the N -representability conditions, although we are not sure whether the alreadyknown N -representability conditions are enough forcompletely eliminating these non-physical solutions. In
some cases, the experimental values for some propertiesof the system may well be used, since our knowledge on
the complete N -representability condition is still limited.
In the above procedure, the choice of the N -representability conditions to be imposed simultaneously on
the density equation are very important, in order to effectively reduce the non-physical solutions. If the choice
is improper, the number of the non-physical solutions
would be too many to select out the correct ones. Nevertheless, the above procedure would be more realistic
than the variational solution for which the complete N -103representability conditions are required before solution
in order to obtain a correct upper bound of the energy2 .
In the density equation, some of the conditions may be
used after solution for selecting the correct solutions.
Such a procedure seems furthermore to be more convenient since some of the known N -representability
conditions are too complex to be used as simultaneous
constraints. They are more easily used as the criteria of
the N -representability for the given density matrix obtained as a solution.
In the previous solution of the time-independent
density equation [8], the 3- and 4-DM, C 3 and C 4 ,were decoupled with the use of C 2 and C 1 , using theGreen's function technique for the correlation expansion
correct up to the second-order. Though our formulas
might be far from the correct N -representability condition, they certainly worked for many molecules in calculating the 2-DM without any use of wave functions.
The calculated C 1 and C 2 were completely N -representable and almost N -representable, respectively. A
similar method of solution is expected for the timedependent density equation.

Z 

n

H0 ÿ i
h


@
n2
C
dxn1 dxn2  0 ;
@t 0

46

which is the density equation for the unperturbed
system, and
Z 

@
n
h
H0 ÿ i

@t



n2
n n2
 H1 Ckÿ1 dxn1 dxn2  0 ;Ckk  1; 2; . . .47which is the density equation for the kth-order correcn2
. The normalization condition for the densitytion, Ckmatrix (Eq. 13) gives
Z

n2

C0

1    n  2; tj1    n  2; t dx1    dxn2  N Cn2 ;
48

and
Z
n2
1    n  2; tj1    n  2; t dx1    dxn2  0;
Ckk  1; 2; . . . :49n

When the unperturbed Hamiltonian H0 is time independent, the solution of Eq. (46) may be written in theform given by Eq. (33).

4 Perturbation theory for the density equation
We next consider the perturbation theory for both thetime-dependent and time-independent density equations.
We refer to the earlier perturbation theory formulation
for the dendity matrix [16, 17]. When the Hamiltonian of
the system is divided into the unperturbed part and theperturbation partH  H0  kH1

43

the reduced Hamiltonian given by Eq. (38) is divided
similarly asnn

H n  H0  kH1

44

n
n
where H0 and H1 have the same form as Eq. (38)except that they are composed of v0 , w0 and v1 , w1 ,respectively. When the perturbation is the so-called onen
particle perturbation, H1 is composed only of theoperator v1 , and the application would be simpli®ed. As
in the conventional perturbation theory in the W-space,
we assume that the density matrix Cn2 is expanded in
power series in the perturbation parameter kXn2
C n2 
k k Ck45
k0where k denotes the order of the perturbation.
4.1 Time-dependent caseInserting Eqs. (44) and (45) into the density equation(Eq. 37), and equating the coecient of each power in k
to be zero, we obtain
2

Garrod and Percus have proposed a variational method in which
only the selected N -representability conditions are used to get thelower bound to the ground state energy. (see Ref [15]).

4.2 Time-independent caseHere, we further assume, in addition to Eq. (45), that theenergy E of the perturbed system is expanded asX
k k Ek :
50
E
k0

Inserting Eqs. (45) and (50) into the time-independent
density equation (Eq. 6b), we obtain a set of equations
Z
n
n2
dxn1 dxn2  0 ;51
H0 ÿ E0 C0
Z

n

n2

 H0 ÿ E0 C1

n

n2

 H1 ÿ E1 C0 dxn1 dxn2  0 ;
and
Z "

52

n
n2
n
n2
 H1 ÿ E1 Ckÿ1 ÿ
H0 ÿ E0 Ck dxn1 dxn2  0;

k  2; 3; . . .

k
X
j2

#
n2
Ej Ckÿj

53

where Eq. (51) is the equation for the unperturbed
system, Eq. (52) the equation for the ®rst-order correction to the density matrix, and Eq. (53) the equation for
the kth-order correction. The normalization condition
gives the same equations as Eqs. (48) and (49).
The expression of the kth-order correction to theenergy, Ek is obtained by integrating Eqs. (52) and (53).
It includes both of the kth- and (k ) 1)th-order correcn2n2
and Ckÿ1 . However, we show below thattions, Ckn2
we need only Ckÿ1 for the calculation of Ek . The energyof the perturbed system is written as

104

Z

55b

density [2] is also easily extended to the time-dependentcase.
The perturbation theory for the density equation
would be useful when the density equation for the unperturbed system is solved exactly. When we take electron correlation as a perturbation, the Hartree-Fockequation which is the unperturbed density equation [2],and the correlated density equation [2], which includes
the correlation correction to all orders, may be divided
into each order using the present perturbation theory.

where k is a dummy perturbation parameter de®ned in
Eq. (43). Dierentiating Eq. (50) with respect to k,equating the result with Eq. (55a), and making thecoecient of each power of k zero, we obtainZn n2
Ek  k N Cn2 ÿ1 H1 Ckÿ1 dx1    dxn2 :Acknowledgements. This paper is dedicated to the memory of thelate Professor Kenichi Fukui who was a leader of theoreticalchemistry in Japan and who was very interested in this approachand encouraged the author throughout this series of studies. The
author would also like to thank Dr. K. Hirao for useful discussions.
Part of this study has been supported by the Scienti®c Research
Grant from the Ministry of Education, Science and Culture, Japan(Nos. 155310 and 175444).

E  N Cn2 ÿ1H n C n2 dx1    dxn2 :54

We use the Hellmann-Feynman theorem in the formZ
@E
@H n n2C
 N Cn2 ÿ1
dx1    dxn2 ;
55a
@k
@k
or
Z

H n

@C n2dx1    dxn2  0 ;@kk  1; 2; . . .

56
n2

This equation requires only Ckÿ1 for the calculation of
Ek which somewhat reduces the labour in calculating the
perturbation energy. Further, the Hellmann-Feynman
theorem expressed by Eq. (55b) gives the equationsZn n2dx1    dxn2  0 ;
57a
H0 C1
and
Z
n n2dx1    dxn2
k H0 CkZ k ÿ 1nn2H1 Ckÿ1 dx1    dxn2  0 ;57b

which shows an interrelation between the k th- andk ÿ 1th-order corrections. The expression corresponding to Eq. (56) was derived in the W-representation by
Carr [18] and LoÈwdin [19].
5 Concluding remarks
In this paper, the author has generalized the density
equation to the time-dependent case and obtained an
equation-of-motion for the N -representable density
matrix. The equation has the form which is formallyobtainable from the previous time-independent equation
by replacing E with ih @=@t. Its properties are alsosimilar to the previous ones. Therefore, some of theprevious results are easily extended to the time-dependent case. An example is the time-dependent HartreeFock equation [20] which is easily derived from thepresent time-dependent density equation by using theindependent particle approximation, just like the Hartree-Fock equation derived from the density equation ina previous paper [2]. The equation for the correlated

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