9512.net
泫襞恅踱
絞ヶ弇离ㄩ忑珜 >> >>

Developments for Reference--State One--Particle Density--Matrix Theory


APS/123-QED

Developments for Reference每State One每Particle Density每Matrix Theory

arXiv:physics/0402059v1 [physics.chem-ph] 12 Feb 2004

James P. Finley Department of Physical Sciences, Eastern New Mexico University, Station #33, Portales, NM 88130? and Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan 113-8656
(Dated: February 2, 2008)

Abstract
Brueckner orbitals, and the density of the Brueckner reference-state, are shown to satify the same cusp condition 每 involving the nuclear charges 每 as natural- and Hartree每Fock-orbitals. Using the cusp condition, the density of a determinantal state can be used to determine the external potential, if the determinantal state is from either Hartee每Fock or Brueckner-orbital theory, as well as, determinant states obtained by many other formalisms that are de?ned by a one-body operator, if a portion of the one-body operator 每 the portion not associated with the kinetic energy or external potential 每 generates a well behaved function when acting on an occupied orbital. Using this relationship involving a determinant and its external potential, a variation of Reference每 State One每Particle Density每Matrix Theory [arXiv:physics/0308056] is formulated, where the trial wavefunctions are universal, in the Kohn-Sham sense, since they do not depend on the external potential. The resulting correlation-energy functionals, are also, universal, except for a relatively small term involving the portion of the expectation value of the external potential with the trial wavefunctions that appears beyond the ?rst order. The same approximate energy functionals that were shown to be valid for the previous v -dependent, Reference每State One每Particle Density每Matrix Theory [arXiv:physics/0308084], are shown to be valid for the current approach, except that the use of the LYP and Colle每Salvetti functional appear more natural within the current approach, since these functionals are universal ones. And since the BLYP and B3LYP functionals contain the LYP functional, these approaches are also better suited with the current approach.

?

Electronic address: james.?nley@enmu.edu

1

I.

EXTERNAL POTENTIAL DETERMINED BY THE ONE-PARTICLE DEN-

SITY MATRIX AND THE PARTICLE DENSITY

There is a one-to-one correspondence between determinant states and their one-particle density-matrices [1, 2]. Because of this correspondence, it is convenient to denote a determinantal state that is determined by a one-particle density-matrix, say 污 , simply by |污 . In addition, any function, say G, that depends on 污 , can be written as G(污 ); this same notation also indicates that G is determined by the corresponding determinant, |污 . Consider the following noninteracting Hamiltonian:
n 污 Hs = i 1 2 ?2 ?r i + v ( r i ) + w ? 污 (xi ) ,

(1)

where the external potential v is given by a ?xed set of point charges v (r ) = ?
m﹋{Rnuc }

Zm , |R m ? r|

(2)

and the summation is over the coordinates of the nuclear point charges, denoted by {Rnuc }; furthermore, w ? 污 may be non-local and this operator can depend on the spin-coordinate 肋 , where the spatial and spin coordinates are denoted collectively by x; in addition, the 污 superscript appended to w indicates that this operator may also depend on 污 (or equivalently |污 ). Consider a determinantal state, say |污 , that satis?es the Schr“ odinger Eq:
污 Hs |污 = E污 |污 ,

(3)

where the Hamiltonian Hs is given by Eq. (1); furthermore, the noninteracting eigenstate,
? |污 , can be expressed by a unique set of occupied orbitals, denoted by {肉o ↘ 污, f 污 }; each of

these orbitals satisfy the following one-particle Schr“ odinger Eq:
污 污 污 污 ? ? f 污 肉x考 (x) = ?i考 肉x考 (x), 考 = 汐, 汕, 肉x考 ﹋ {肉o ↘ 污, f污 },

(4)

where the one-body operator is given by
污 1 2 ? f 污 = ? 2 ?r + v (r) + w (x).

(5)

In addition, we require the operator w 污 to be Hermitian and satisfy
|R?r|↙0
污 lim |R ? r|w 污 (r, 肋 )肉x考 (r, 肋 ) = 0, for all R.

(6)

2

In order to emphasize an exclusive dependence upon R, we modify the limit in this Eq., giving
r↙R
污 lim |r ? R|w 污 (r, 肋 )肉x考 (r, 肋 ) = 0. for all R.

(7)

? Let us also mention that the set of unoccupied orbitals 每 orthogonal to {肉o ↘ 污, f 污 } 每 is ? denoted by {肉u ↘ 污, f 污 } and, in addition, all of our spin-orbitals 肉i考 (x) have the following

form: 肉i考 (x) = 聿i考 (r)考 (肋 ), 考 = 汐 or 汕, (8)

where the spatial and spin portions are given by 聿i考 (r) and 考 (肋 ), respectively, and the spatial functions 聿i考 (r) are permitted to be unrestricted 每 two spin orbitals do not, in general, share the same spatial function, i.e., (聿i汐 = 聿i汕 ) is permitted. Multiplying Eq. (4) by |r ? R| followed by taking a limit of this term vanishing, gives ? ? Zm ? 污 2 肉x考 (x) = 0, (9) lim |R ? r| ?? 1 ?r ? 2 |r?R|↙0 |R m ? r|
m﹋{Rnuc }

where we have used Eqs. (2), (5), and (6). (Note that this Eq. is also the cusp condition
污 [3, 4], however, in that case, 肉x考 is a natural orbital and 污 is the one-particle density-matrix

of an interacting target-state, say 朵.) In order to obtain an exclusive dependence upon R, we, again, modify the limit in this Eq, giving
1 2 lim |r ? R| ?? 2 ?r ?

r↙R

?

m﹋{Rnuc }

Since the second term vanishes unless (R ﹋ {Rnuc }), we have ? ?
r↙R

Zm ? 污 肉x考 (x) = 0. |R m ? r|

?

(10)

which can be written as
r↙R

lim |r ? R| ? ? 1 ?2 ? 2 r

m﹋{Rnuc }

污 汛RR Zm? 肉x考 (x) = 0,
m

(11)

污 污 ?2 肉x考 (x) = lim |r ? R|肉x考 (x)?1 ? 1 2 r

汛RR Zm.
m

(12)

m﹋{Rnuc }

De?ning the left side by
污 污 污 ?2 肉x考 (x), T (肉x考 , R) = lim |r ? R|肉x考 (x)?1 ? 1 2 r

r↙R

(13)

3

we can write
污 T (肉x考 , R) =

汛RR Zm .
m

(14)

m﹋{Rnuc }

For a set of spatially restricted orbitals:
污 肉x考 (x) = 聿污 x (r )考 (肋 ),

(15)

it is readily proven that we have T (聿污 x , R) =
m﹋{Rnuc }
污 Multiplying Eq. (11) by (肉x考 (x∩ ))? , and summing over all occupied orbitals from the set

汛RR Zm.
m

(16)

? {肉o ↘ 污, f 污 }, gives

r↙R

where the one-particle density matrix is given by 污 (x, x∩ ) =
? x考﹋{肉o ↘污,f 污}

lim |r ? R| ? ? 1 ?2 ? 2 r

?

m﹋{Rnuc }

汛RR Zm? 污 (x, x∩ ) = 0,
m

?

(17)

污 污 肉x考 (x) (肉x考 (x∩ ))

?

(18)

and it is readily proven that we have T (污, R) =
m﹋{Rnuc }

汛RR Zm ,
m

(19)

where T (污, R) = lim |r ? R|污 (x, x∩ )?1 ? 1 ?2 污 (x, x∩ ). 2 r
r↙R

(20)

Since this expression is invariant to the variable x∩ , we can choose (x∩ = x), yielding T (污, R) = lim |r ? R|污 (x, x)?1
r↙R

?2 污 (x, x∩ ) ?1 2 r

x∩ = x

.

(21)

污 Since Eq. (11) is also satis?ed by the complex conjugate orbital, 肉x考 (x)? , it is readily shown

that we have T (污 ? , R ) =
m﹋{Rnuc }

汛RR Zm,
m

(22)

4

where T (污 ? , R) = lim |r ? R|污 (x, x)?1
r↙R

?1 ?2 污 (x∩ , x) 2 r

x∩ = x

.

(23)

Adding together Eq. (19) and (22), using (21) and (23), and using the following identity:
1 2 ?r 污 (x, x) = ?2 1 2 ?r 污 (x, x∩ ) ?2 x∩ = x

+

?2 污 (x∩ , x) ?1 2 r

x∩ = x

,

(24)

we get T (老s 污 , R) = 2
m﹋{Rnuc }

汛RR Zm,
m

(25)

where
s ?1 1 2 ?r 老s T (老s ?2 污 (x), 污 , R) = lim |r ? R|老污 (x) r↙R

(26)

and 老s 污 (x) is the spin density, i.e., 老s 污 (x) = 污 (x, x). (27)

Eq. (17) is also valid for 污 (r, 肋 ; r∩, 肋 ) replacing 污 (x, x∩ ); making this substitution and summing over the spin-variable 肋 we obtain the same expression, as Eq. (17), except that it involves the spinless density matrix 老1 , given by 老1 (r, r∩) =


污 (r , 肋 ; r ∩ , 肋 ),

(28)

and it is readily proven that we have 1 T (老1 , R) = T (老, R) = 2

汛RR Zm,
m

(29)

m﹋{Rnuc }

where T (老1 , R) and T (老污 , R) are de?ned by Eqs. (21) and (26), respectively; 老 is the electron density, i.e., 老(r) = 老1 (r, r). (30)

Consider the set of (ground and excited) determinantal states, denoted {|污 v }, that are eigenfunctions of Hs , given by Eq. (1), where the states from the set, {|污 v }, are obtained from all w 污 that satisfy Eq. (7), and from all Coulombic external-potentials v , given by Eq. (2). From the density of any one of theses states, say 老, we can determine its Coulombic external-potential v by using Eqs. (29) and (2). Hence, v is a unique function of the density. In other words, we have v (老), and this function is de?ned for all densities that are from this set of determinantal states, {|污 v }. 5

II.

INVARIANCE OF OCCUPIED-ORBITAL TRANSFORMATION

We now partition the operator w 污 into the following four components:
污 污 污 污 w 污 = wex + wde + woc + wun ,

(31)

where the excitation (ex), de-excitation (de), occupied (oc), and unoccupied (un) parts are given by the following expressions:
污 wex =

r考 ? ww考 ∩ ar考 aw考 ∩ , w考r考∩ w考 ? wr考 ∩ aw考 ar考 ∩ , r考w考∩ w考 ? wx考 ∩ aw考 ax考 ∩ , w考x考∩ r考 ? ws考 ∩ ar考 as考 ∩ , r考s考∩

(32) (33) (34) (35)

污 = wde

污 woc =

污 wun =

and the occupied- and unoccupied-orbitals are, respectively, given by
污 污 ? 肉w考 , 肉x考 ﹋ {肉o ↘ 污, f 污 }, 污 污 ? ﹋ {肉u ↘ 污, f , 肉s考 肉r考 污 }.

(36) (37)

The results from the previous Sec. indicate that v is a function of 老 for any 老 determined
污 from Hs , given by Eq. (1) 每 or, equivalently, any 老 determined from the one-body operator 污 ? f 污 given by (5) 每 when the operator w satis?es Eq. (7). Using the partitioning method

given above, Eq. (7) becomes
污 污 污 lim |r ? R| [woc (r, 肋 ) + wex (r, 肋 )] 肉w考 (r , 肋 ) = 0 ,

|r?R|↙0

for all R,

(38)

污 污 and wun . indicating that Eq. (7) can be satis?ed with any choice of wde

The above relation is satis?ed when we have
污 污 lim |r ? R|woc (r, 肋 )肉w考 (r , 肋 ) = 0 , 污 污 lim |r ? R|wex (r, 肋 )肉w考 (r , 肋 ) = 0 .

r↙R

(39a) (39b)

r↙R

It is easily proven that a determinantal state |污 that satis?es Eq. (3) 每 and the corre污 sponding density 老 from |污 每 does not depend on woc ; so, when considering the statements

appearing in the last paragraph of the previous section, we can relax the requirement that 6

Eq. (7) be satis?ed, and only require Eq. (39b) to be satis?ed. In other words, the density of a determinantal state, that satis?es Eq. (3), can be used to determine its external potential, given by Eq. (2), by using Eq. (29), if (39b) is satis?ed. An equivalent statement refers to ? the one-body operator f 污 : The density of a determinantal state can be used to determine its external potential, given by Eq. (2), by using Eq. (29), if (39b) is satis?ed, where the ? orbitals de?ning the determinantal state |污 are the occupied eigenfunctions of f 污 , de?ned
污 污 by Eq. (5). Note that the woc and wun portions of the operator w 污 are at our disposal, since 污 污 is determined by wex , the determinantal state does not depend on these components; wde

since w 污 is required to be Hermitian. (The Hermitian requirement can be dropped by using a biorthogonal basis set.)

III.

HARTREE每FOCK DETERMINANTAL STATES

We now show that the set of Hartree每Fock determinantal states, say {|而 ?}, are members of {|污 v }, indicating that their Coulombic external-potentials v can be uniquely determined by their electron density, i.e., v (老), by using Eq. (29). The occupied, canonical Hartree每Fock orbitals satisfy the following single particle Eq:
而 ? 而 ? 而 ? ?而 F ? 肉x考 (x) = ?x考 肉x考 (x), 而 ? ?而 考 = 汐, 汕, 肉x考 ﹋ {肉o ↘ 而 ?, F ?}

(40)

where the Fock operator is given by
1 2 ?而 F ? = ? 2 ?r + v (r) + ?1 而 ? r12 而 ?(x2 , x2 ) dx2 + v ?x (x),

(41)

and the one-particle density-matrix for the Hartree每Fock reference-state has the following form: 而 ?(x, x∩ ) =
?而 x考﹋{肉o ↘而 ?,F ?}
污 furthermore, the exchange operator, v ?x , is a non-local operator that is de?ned by its kernel, 而 ? 而 ? 肉x考 (x) (肉x考 (x∩ )) ;

?

(42)

?1 ?r12 污 . Therefore, for an arbitrary function, say 肉 , we have
而 ? v ?x (x1 )肉 (x1 ) = ?

?1 dx2 r12 而 ?(x1 , x2 )肉 (x2 ).

(43)

Equating Eqs. (40) and (41) with (4) and (5), for (污 = 而 ?), we have
? w而 (x) =

?1 而 ? r12 而 ?(x2 , x2 ) dx2 + v ?x (x),

(44)

7

and it is easily seen that Eq. (7) is satis?ed; so, the Hartree每Fock states are members of {|污 v }, and we have v (? ?) where ? ? is the Hartree每Fock density: ? ?(r) =


而 ?(r, 肋 ; r, 肋 ).

(45)

The Hartree每Fock Eqs. are usually solved using a iterative, self consistent ?eld (SCF) approach, where the (m ? 1)th iteration is given by
?m 而 ?m 而 而 ?m ?而 F ?m?1 肉x考 (x) = ?x考 肉x考 (x), 而 ?m ?而 ﹋ {肉o ↘ 而 ?, F 考 = 汐, 汕, 肉x考 ?m ?1 }

(46)

?m?1 and its easily seen that Eq. (7) is satis?ed for w 而 , so all determinantal states determined

during the SCF approach are also members of {|污 v }. ? ∩ , that is given by Consider another Hermitian Fock-type operator, say F 污
污 ?∩ = F ?污 + v F ?x 污 +

(47)

污 where the excitation (ex) portion of the additional exchange-operator v ?x is zero: +

污 v ?x +

ex

= 0.

(48)

Hence, according to Sec. II, Eq. (39b) remains satis?ed and, in addition, the determinantal
∩ ?污 , is the as same the determinantal state from state de?ned by the occupied orbitals, from F ?污 ; it is a member of {|污 v }; so, again, the density of this determinantal state can be used F

to determine the external potential v , Eq. (2), by using Eq. (29). Furthermore, since the
∩ ?而 occupied eigenfunctions from F ? , given by ∩ ∩而 ? 而 ? ∩而 ? ?而 F ? 肉x考 (x) = ?x考 肉x考 (x),
∩而 ? ?∩ } 考 = 汐, 汕, 肉x考 ﹋ {肉o ↘ 而 ?, F 而 ?

(49)

?而 ?而 di?er only by a unitary transformation from the F ?, F ? }. There? occupied orbitals, {肉o ↘ 而 fore, and of course, the one-particle density-matrix obtained from these occupied orbitals are equivalent: 而 ?(x, x∩ ) =
?∩ } x考﹋{肉o ↘而 ?,F 而 ?
∩而 ? ∩而 ? 肉x考 (x) (肉x考 (x∩ )) ,

?

(50)

where this one-particle density-matrix 而 ? is the same one appearing in Eq. (42).

8

IV.

GENERALIZED FOCK OPERATOR

?污 , where its occupied orbitals satisfy Consider a generalized Fock-operator F ?污 肉 污 (x) = 汍污 肉 污 (x), 考 = 汐, 汕, 肉 污 ﹋ {肉o ↘ 污, F ?污 } F x考 x考 x考 x考 ?污 is given by and F ?污 = ? 1 ?2 + v (r) + F 2 r
?1 污 污 r12 污 (x2 , x2 ) dx2 + v ?x (x) + v ?co (x).

(51)

(52)

Comparing the above two Eqs. with (4) and (5), we obtain a new de?nition for w 污 : w 污 (x) =
?1 污 污 r12 污 (x2 , x2 ) dx2 + v ?x (x), +? vco (x).

(53)

and substituting this expression into Eq. (39b), gives
r↙R
污 污 lim |r ? R| [? vco (r, 肋 )]ex 肉w考 (r, 肋 ) = 0, for all R.

(54)

Hence, if this relation is satis?ed, the determinantal state de?ned by the occupied orbitals, ?污 , is a member of {|污 v }; so, again, the density of this determinantal state can be from F used to determine the external potential v , Eq. (2), by using Eq. (29).

V.

BRUECKNER DETERMINANTAL STATES

We seek solutions of the time-independent Schr“ odinger equation, Hv |朵 = E|朵 , (55)

where Hv denotes the Hamiltonian operator de?ned by the external potential v , where the Hamiltonian is independent of the number of electrons when it is expressed in second quantization: Hv =
i考j考

? |j考 )a? aj考 + (i考 |h i考

1 2 i考j考

? (i考j考 |k考 ∩ l考 ∩ )a? i考 ak考∩ al考∩ aj考 k考∩ l考∩

(56)

where our Hamiltonian is spin-free; the spin-free integrals are written using chemist*s notation [5]: ? |j考 ) = (i考 |h (i考j考 |k考 ∩ l考 ∩ ) =
1 2 聿? i考 (r) ? 2 ?r + v (r) 聿j考 (r)dr, ?1 ? 聿? i考 (r1 )聿j考 (r1 )r12 聿k考∩ (r2 )聿l考∩ (r2 ) dr1 dr2 ,

(57) (58)

9

and the creation and annihilation operators, a? i考 and ai考 , correspond to the unrestricted spin-orbitals, 肉i考 , de?ned by Eq. (8). The wavefunction of interest |朵 can be generated by a wave operator ?污 : ?污 |污 = (1 + 聿污 )|污 = |朵 , (59)

and the second relation de?nes the correlation operator, 聿污 ; furthermore, |污 is any determinantal reference-state that overlaps with the target state: ( 污 |朵 = 0). Brueckner orbital theory [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] is a generalization of Hartree每Fock theory that utilizes a single-determinantal state that has the maximum overlap with the target state [22, 23]. By de?nition, if |而 is a Brueckner reference-state, then the target state, |朵 , contains no singly-excited states from |而 :
r考 而w考 | 朵 = 0,


(60)

and the singly-excited states are given by
r考 |而w考 = a? r考∩ aw考 |而 ,


(61)

where the Brueckner-state occupied- and unoccupied-orbitals are, respectively, given by
而 而 ﹋ {肉o ↙ 而 }, , 肉x考 肉w考 而 而 肉r考 ∩ , 肉s考 ∩ ﹋ {肉u ↙ 而 },

(62) (63)

and this notation indicates that the occupied orbitals determine 而 ; furthermore, the unoccupied orbitals also determine 而 since the union of the two orthogonal sets (of orbitals) is a complete set. Note that, unlike the orbitals that are de?ned by Eq. (36), the occupied orbitals that satisfy Eq. (62) are not completely de?ned; they are invariant to a unitary transformation; similarly, the unoccupied orbitals that satisfy Eq. (63) are also invariant to a unitary transformation. Using a set of these orbitals, the Brueckner one-particle density-matrix is given by 而 (x, x∩ ) =
x考﹋{肉o ↙而 }
而 而 肉x考 (x) (肉x考 (x∩ )) ,

?

(64)

10

and, for future use, we mention that the virtual orbitals de?ne the following two-body function: 百而 (x, x∩ ) =
r考﹋{肉u ↙而 }
而 而 肉r考 (x) (肉r考 (x∩ )) ,

?

(65)

where, for a complete set of one-particle functions, the sum of the two gives the Dirac delta function: 汛 (x, x∩ ) = 百而 (x, x∩ ) + 而 (x, x∩ ). (66)

Since our Hamiltonian, given by Eq. (56), is spin-free, it is easily demonstrated that we have [24]
r考 污w考 |朵 = 0, for 考 = 考 ∩ and 污 |朵 = 0 ;


(67)

hence, we can modify the de?nition for a Brueckner reference-state, given by Eq. (60), and only consider the spin-conserving matrix-elements:
r考 而w考 | 朵 = 0.

(68)

Because of spin symmetry, Eq. (67) certainly holds when |污 is a determinantal state that ?2 , e.g., a closed-shell is an eigenfunction of the total spin angular-momentum operator, S ground-states with spatially restricted spin orbitals. However, this identity should also hold in more general cases, since, diagrammatically speaking, the spin state 每 either 汐 or 汕 每 must be conserved along an oriented path [24], and w考 and r考 ∩ are on the same oriented path. In order to simplify our discussions, henceforth, we only consider cases where Eq. (67) holds; however, the result are easily generalized to the more general case, e.g., when the Hamiltonian is spin-dependent. Substituting Eqs. (55) and (59) into (68), sequentially, we obtain
r考 r考 r考 r考 r考 0 = 而w考 |朵 = 而w考 |Hv |朵 = 而w考 |Hv ?而 |而 = 而w考 |Hv |而 + 而w考 |Hv 聿而 |而 ,

(69)

and the vanishing of the above matrix elements involving Hv is know as the Brillouin每 Brueckner condition [7, 9, 9, 22, 25, 26]. Writing the operator-product Hv 聿而 in normalordered form [24, 27, 28, 29] with respect to the reference state |而 , the last matrix element of the above Eq. becomes
r考 r考 而w考 |Hv 聿而 |而 = 而w考 | (Hv 聿而 )1 |而 ,

(70)

11

where the the one-body portion, (Hv 聿而 )1 , can be partitioned in the following manner: (Hv 聿而 )1 = [(Hv 聿而 )1 ]op + [(Hv 聿而 )1 ]re , (71)

and where the open (op) portion and remaining (re) portions have the following explicit forms [30, 31]: [(Hv 聿而 )1 ]op = [(Hv 聿而 )1 ]re =
而 w考r考 而 r考w考 r考 ? Uw考 ar考 aw考 , w考 ? Ur考 aw考 ar考 + 而 r考s考 s考 ? Ur考 as考 ar考 ? 而 w考x考 w考 Ux考 ax考 a? w考 ;

(72) (73)

furthermore, the one-body matrix-elements are de?ned by
j考 而 而 Ui考 = 肉j考 | (Hv 聿而 )1 |肉i考 ,

(74)

and the orbital indices are given by the right side of Eqs. (62) and (63); this choice is indicated by the 而 superscripts appended to the summations, i.e.,


. (Note that the

de?nition of an open operator given above di?ers from the de?nition used by other authors [29, 32, 33].)
j考 for In the above matrix elements, the ones that do not preserve the spin, i.e., (Ui考


考 = 考 ∩ ), are omitted, since they can easily be shown to vanish for a spin-free Hamiltonian. (The vanishing of these matrix elements occurs, diagrammatically speaking, since the spin state 每 either 汐 or 汕 每 must be conserved along an oriented path [24], and i考 and j考 ∩ are on the same oriented path.) Substituting Eq. (71) into (70) and using (72) and (73), gives
r考 r考 | [(Hv 聿而 )1 ]op |而 . 而w考 | (Hv 聿而 )1 |而 = 而w考

(75)

Since the one-body operator-product [(Hv 聿而 )1 ]op can also act within the one-body sector of the Hilbert space, we have the following identity:
r考 而 而 而w考 | [(Hv 聿而 )1 ]op |而 = 肉r考 | [(Hv 聿而 )1 ]op |肉w考 .

(76)

Substituting Eq. (70) into the Brillouin每Brueckner condition, Eq. (69), and using Eq. (75) and (76), and also the following identity:
而 ?而 )ex |肉 而 = 而 r考 |Hv |朵 , 肉r考 | (F w考 w考

(77)

12

involving the Fock operator, Eq. (41), yields
而 而 而 ?而 )ex |肉 而 + 肉 而 | (? 肉r考 | (F w考 r考 vco )ex |肉w考 = 0,

(78)

而 where the introduced correlation potential v ?co , by de?nition, satis?es

而 (? vco )ex = [(Hv 聿而 )1 ]op ,

(79)

而 污 ?而 )ex and (? and the operators, (F vco )ex , are de?ned in an analogous way as wex , as indicated

by Eqs. (31) through (35). Since the above form of the Brillouin每Brueckner condition, given by Eq. (78), is satis?ed by all pairs of orbitals involving one unoccupied-orbital and one occupied-orbital, we have ?而 F = 0, (80)

ex

where the generalized, or exact, Fock operator is de?ned by
而 ?而 = F ?而 + v F ?co .

(81)

?而 with the one given by Sec. IV, Eq. (52), and using Eq. (41), Comparing this de?nition of F we see that, for (污 = 而 ), the two de?nitions are equivalent, except that in this Sec. we require
而 the excitation (ex) portion of the correlation potential v ?co to satisfy Eq. (79); by arbitrarily ?而 , and this eigenvalue Eq. is given by de?ning the other portions of v ?而 we can diagonalize F

co

Eq. (51) for (污 = 而 ): ?而 肉 而 (x) = 汍而 肉 而 (x), F x考 x考 x考 and, furthermore, Eq. (54) becomes
而 而 lim |r ? R| [? vco (r, 肋 )]ex 肉w考 (r, 肋 ) = 0, for all R, 而 ?而 }, 考 = 汐, 汕, 肉x考 ﹋ {肉o ↘ 而, F

(82)

r↙R

(83)

而 where (? vco )ex is given by Eq. (79). Hence, if this relation is satis?ed, Bruckner determinantal

states {|而 } are member of {|污 v }; so, again, the density of a Brueckner determinantal state can be used to determine its external potential v , Eq. (2), by using Eq. (29). Using the results from appendix (B), we have
而 而 x考 ? 而 r考 ? 而 lim |r1 ? R| [? vco (x1 )]ex 肉w考 (x1 ) = lim |r1 ? R| Cw考 hv1 肉x考 (x1 ) + Dw考 hv1 肉r考 (x1 ) , (84)

r1 ↙R

r1 ↙R

13

where ? v 1 = ? 1 ?2 + v ( r 1 ) , h 2 r1 (85)

and there are summations over the repeated indices x考 and r考 for the orbital sets {肉o ↘ ?而 } and {肉u ↘ 而, F ?而 }. (The coe?cients C x考 and D r考 are de?ned by Eqs. (B38) and 而, F w考 w考 (B39).) Unfortunately we have been unable to prove that Eq. (83) is an identity by using Eq. (84). So, as an alternative approach, consider the case where the above identity, given by Eq. (83), is not necessarily satis?ed. As in the derivation Eq. (9), by multiplying Eq. (82) by |r1 ? R| followed by taking a limit of this term vanishing, gives the following identity that must be satis?ed:
而 而 ? v1 肉 而 (x1 ) + lim |r1 ? R| [? lim |R ? r1 |h vco (r1 , 肋 )]ex 肉w考 (r 1 , 肋 ) = 0 , w考

r1 ↙R

r1 ↙R

(86)

where we have used Eqs. (81), (85), and (41) and, also, omitted the Coulomb and exchange
而 as terms, since these terms vanish; furthermore, we have used the decomposition of v ?co 而 de?ned by Eq. (31), and have chosen (? vco )oc to be zero, since, according to the discussion

within Sec. II, this portion is at our disposal; the one-particle density-matrix 而 is invariant to this choice. Substituting Eq. (84) into (86), gives
r考 ? 而 ? 而 ? x考 h lim |r1 ? R| C w考 v1 肉x考 (x) + Dw考 hv1 肉r考 (x) = 0,

r1 ↙R

(87)

where ? x考 = 汛w考,x考 + C x考 . C w考 w考 Since both terms from the above identity are independent, apparently, we must have ? v1 肉 而 (x) = 0, lim |R ? r|h x考 ? v1 肉 而 (x) = 0. lim |R ? r|h r考 (89) (90) (88)

|r?R|↙0

|r?R|↙0

Substituting these relations into Eq. (84) proves that Eq. (83) is an identity. Hence, the density of the Brueckner-determinantal state, |而 , can be used to determine the external potential v , Eq. (2), by using Eq. (29). Note that Eq. (89) is identical to Eq. (9), the cusp 14

condition, except that the orbitals are now Brueckner; Eq. (89) can also be used to prove all relations within Sec.I that appear after Eq. (9), including the one above that states that the density of the Brueckner-determinantal state can be used to determine the external potential.

VI. A.

ONE-PARTICLE DENSITY-MATRIX THEORY Variational Brueckner orbital theory

Reference-state one-particle density-matrix theory [30, 31, 34] is based on Brueckner orbital theory [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Unlike many other density functional formalisms based on variants of the Konh每Sham method, the correlation operator for this approach is non-local. The approach also uses an energy functional that depends on the one-particle density-matrix of a reference determinantal-state, and not the exact one from the target state, where the energy functional is partitioned into the exact exchange-energy and a correlation-energy functional that is non-universal, since this functional depends on the external potential v . We now modify this formalism to remove 每 for the most part 每 the dependence of the correlation energy-functional upon the external potential. However, an additional term that describes a portion of the potential energy is also included that does not have an analog in Kohn每Sham approaches. However, this term can be easily treated once the kinetic energy functional is known or, in many cases, this term can be neglected, since it is probably quite small. For convenience, we refer to the previous works [30, 31, 34] as being v -dependent, even though we still retain some v -dependence in the correlation-energy functionals for the current approach under consideration. In this previous work [30, 31, 34], we introduced four v -dependent trial wavefunctions
灰) 每 say |朵( 污v , where (灰 = I, II, III, and IV) 每 that are de?ned with respect to an exter-

nal potential v and a one-particle density-matrix, where 污 is from a single-determinantal reference-state, |污 . The ?rst trial-wavefunction |朵(I) 污v is simply the target state of interest, say |朵N v , with the single excitations removed:
污 |朵(I) 污v = (1 ? P11 ) |朵N v ,

(91)

where |朵N v is a ground-state determined by the external potential v and the number of 15

electrons N and, furthermore, the spin-conserved projector for the singly-excited states is
污 P11 = w考﹋{肉o ↙污 } r考﹋{肉u ↙污 } r考 r考 |污w考 污w考 |,

(92)

where we assume that Eq. (67) holds, but this requirement can easily be dropped by append污 r考 ing the states |污w考 the the right side of Eq. (92). Note that the P11 subspace is completely


污 is also invariant to a unitary transformation of occupied, or virtual, determined by |污 ; P11

orbitals [13]. We now require the state |污 to be a member of {|污 v }, so Eq. (91) becomes
污 |朵(I) 污v∩ = (1 ? P11 ) |朵N v∩ ,

污 ﹋ {|污 v }, 污 ↙ v, 污 ↙ N,

(93)

where the prime superscripts appended to the external potentials, i.e., v ∩ , emphasizes that the potential de?ning the states, |朵(I) 污v∩ and |朵N v∩ , may di?er from the one determined by 污 (indicated by 污 ↙ v ). However, we now restrict these potentials to be equivalent, i.e., (v ∩ = v ), and generate the target state using a wave operator:
污 污 |朵(I) 污v = (1 ? P11 ) |朵N v = (1 ? P11 ) ?污v |污 , 污 ﹋ {|污 v }, 污 ↙ v,

(94)

and since the external potential is a unique function of 污 (or the density 老 of 污 ), we can use the function v (污 ) to express the wave operator as a unique function of 污 : ?污v = ?污 污 ﹋ {|污 v }, 污 ↙ v, 污 ↙ N, and for are target state, we have |朵污 = |朵N v , 污 ↙ v, 污 ↙ N. (96) (95)

Therefore, we are assuming that the target state is completely determined by 污 . This is a reasonable assumption, since the Hamiltonian is completely determined by 污 , since 污 gives the number of electrons, N , and the external potential v . In addition, however, we must also make the assumption that the wave operator is, or 每 at least, in principle 每 can be uniquely de?ned so that it generates only one exact eigenstate 每 the ground state 每 from all 污 that have (污 ↙ v ). This implies, however, two wavefunctions can di?er only by a constant, say c, if they are obtained from density-matrices that determine the same external potential: | 朵 污 = c| 朵 污 ∩ , if 污 ↙ v , 污 ∩ ↙ v ∩ , and v = v ∩ , 16 (97)

where the 污 subscript also indicates the normalization of the target state: 污 | 朵污 = 1 . (98)

By substituting Eqs. (95) and (96) into (94), we obtain a trial wavefunction that can be assumed to be determined by 污 :
污 污 ) |朵污 , 污 ﹋ {|污 v }, ) ?污 |污 = (1 ? P11 |朵(I) = (1 ? P11 污

(99)

and we have |朵(I) = c|朵(I) 污 污∩ , 污 |朵(I) = 1. 污 if 污 ↙ v , 污 ∩ ↙ v ∩ , and v = v ∩ , (100) (101)

The second trial-wavefunction |朵(II) is de?ned with respect to the target state expressed 污 by an exponential ansatz: (|朵 = eS污 |污 ), where |朵(II) is generated by removing the single污
污 from the cluster-operator S : excitation amplitudes S1

|朵(II) = e(S污 ?S1 ) |污 , 污 where, as in Eq. (95), we have S污v = S污 污 ﹋ {|污 v }, 污 ↙ v, 污 ↙ N. The third trial-wavefunction |朵(III) can be generated by its wave-operator: 污 ? 污 |污 = |朵(III) ? , 污



(102)

(103)

(104)

?污 ?污 can be written as a ? 污 = eS |污 ), where S that can be expressed in an exponential form: (?

sum n-body excitations, with the exclusion of a one-body operator:
污 污 ?污 = S ?2 ?3 S +S +﹞﹞﹞ .

(105)

? 污 is a solution to the coupled cluster equations [27, 28, 29, 35, 36, 37, The wave operator ? 38, 39, 40, 41] with the single excitation portion removed:
污 ?污 (1 ? P11 ) Hv ? op,cn

= 0, 污 ↙ v, 污 ↙ N,

(106)

where only the open (op) and connected (cn) portions enter into the relation. This expression ?污 de?nes the trial functional |朵(III) using Eq. (104) and, again, Eq. (95) is satis?ed with ? 污 replacing ?污 . 17

The fourth trial wavefunctions |朵(IV) is not considered here, except to mention that it 污 is obtained by solving the con?guration-interaction equations [5, 41, 42] in an approximate way, i.e., by neglecting the single-excitation portion.
灰) All of the trial states |朵( share the property that they contain no single excitations, 污

污 灰) i.e., (P11 | 朵( = 0), and they generate the target state |朵而 when their reference state 污

satis?es (|污 = |而 ), where |而 is the determinantal state constructed from occupied Bruckner orbitals. In other words, we have
灰) | 朵( = | 朵而 . 而

(107)

Since the target state is a solution of the Schr“ odinger equation (55) with an exact energy, say EN v , we have EN v = 朵污 | H v | 朵污 = E1 [污, v ] + Eco [污, v ] = Hv 污 , 朵污 | 朵污 (108)

where the ?rst-order energy E1 is given by the expectation value of the Hamiltonian involving the reference state, 污 |Hv |污 , and the correlation energy Eco is de?ned above as (EN v ? E1 ); furthermore, the introduced notation Hv the Hamiltonian involving the target state |朵污 . Using the trial wavefunctions, we can de?ne variational energy-functionals that depend on the one-particle density-matrix: ?灰 [污, v ] = Hv E
污灰 污

indicates the expectation value of

? (灰) [污, v ], = E1 [污, v ] + E co

(109)

?, given by where we use the notation for the expectation value of an operator, say A ? 污灰 = A
灰) ? (灰 ) 朵( 污 |A|朵污 灰) (灰 ) 朵( 污 | 朵污

(110)

? (灰) as and the last relation within Eq. (109) de?nes the correlation-energy functionals E co ?灰 ? E1 ); furthermore, the ?rst order energy is given by (E E1 [污, v ] =
1 2 ?r 老1 (r, r∩) ?2
r∩=r

dr +

v (r)老(r) dr + EJ [老] + Ex [老考 1 ],

(111)

where the Coulomb and exchange energies have their usual forms: 1 2 1 ?Ex [老考 1] = 2 EJ [老] =
?1 dr1 dr2 r12 老(r1 )老(r2 ), 汕 汕 汐 ?1 老汐 dr1 dr2 r12 1 (r1 , r2 )老1 (r2 , r1 ) + 老1 (r1 , r2 )老1 (r2 , r1 ) ,

(112) (113)

18

and the spin-components of the one particle density matrix are given by 老汐 1 (r 1 , r 2 ) =
x汐﹋{肉o ↙污 } ? 污 聿污 x汐 (r1 ) (聿x汐 (r2 )) = 污 (r1 , 1, r2 , 1), ?

(114) (115)

老汕 1 (r 1 , r 2 )

=
x汕 ﹋{肉o ↙污 }

污 聿污 x汕 (r1 ) 聿x汕 (r2 )

= 污 (r1 , ?1, r2 , ?1).

Substituting Eq. (111) into (109) gives the following: ?灰 [污, v ] = E
1 2 ?r 污 (x, x∩ ) ?2
x∩=x

dx +

? (灰) [污, v ], v (r)老(r) dr + EJ [老] + E xc

(116)

where the exchange-correlation energy-functionals are de?ned by ? (灰) [污, v ] = Ex [污 ] + E ? (灰) [污, v ]. E xc co (117)

Returning to our energy functionals, Eq. (109), let the functional derivative of these functionals yield two-body functions that serve as the kernels of exact Fock operators:
(灰 ) 汎污v (x1 , x2 ) =

?灰 [污, v ] E 汛污 (x2 , x1 )
?1 r23 污 (x3 , x3 ) dx3 + 糸污灰v xc (x1 , x2 ),

(118)

2 = 汛 (x2 ? x1 ) ? 1 2 ?2 + v ( r 2 ) +

where the kernels of the exchange-correlation operators, 糸污灰v xc (x1 , x2 ), are obtained from the exchange-correlation energy-functionals: 糸污灰v xc (x1 , x2 ) = ? (灰) [污, v ] ? (灰) [污, v ] 汛E 汛E co xc ?1 = ? r12 污 (x1 , x2 ), 汛污 (x2 , x1 ) 汛污 (x2 , x1 ) (119)

where the last relation uses Eqs. (117) and the identity: 汛Ex [污 ] ?1 污 = ?r12 污 (x1 , x2 ) = vx (x1 , x2 ), 汛污 (x2 , x1 )
污 污 and the function vx (x1 , x2 ) is the kernel of the exchange operator, denoted by v ?x .

(120)

Using the variation theorem, and by noting the identity given by Eq. (107), it becomes obvious 每 as in our previous v -dependent approach [30, 31] 每 that the minimizing of the ?灰 [污, v ], subject to the constraint that the one-particle density-matrix comes functionals E from a single-determinantal state, yields ?灰 [而, v ], EN v = E ? (灰) [而, v ]. Eco [而, v ] = E co 19 (121) (122)

where EN v and Eco the electronic-energy and correlation energy arising from the target state, de?ned by Eqs. (108); furthermore, 而 is the one-particle density-matrix of the Brueckner reference-state |而 that determines v : 而 (x, x∩ ) =
w ﹋{肉o ↙而 } ? 肉w (x)肉w (x∩ ), 而 ↙ v,

(123)

and the Brueckner orbitals satisfy the following equivalent conditions: ?(灰) |肉w = 0; 肉w ﹋ {肉o ↙ 而 }, 肉r ﹋ {肉u ↙ 而 }, 而 ↙ v, 肉r |汎 而 ?(灰) 而 = 0, ? 1?而 汎 而 (124a) (124b)

where these orbitals do not depend of 灰 每 any trial wavefunction gives the same results. A unique set of occupied and unoccupied orbitals is obtained by requiring the occupied ?(灰) to be diagonal: and unoccupied blocks of 汎


?(灰) 肉 而 (x) = 缶 而 肉 而 (x), 肉 而 ﹋ {肉o ↙ 而 }, 汎 而 w w w w ?(灰) 肉 而 (x) = 缶 而 肉 而 (x), 肉 而 ﹋ {肉u ↙ 而 }. 汎 而 r r r r

(125a) (125b)

(灰 ) 而 ?而 ↘ 而, 汎 } Henceforth, the orbitals sets that satisfy Eqs. (124) and (125) are denoted by {肉o (灰 ) 而 ?(灰) . Since theses orbitals, ?而 and {肉u ↘ 而, 汎 }, indicating that they are determined by 而 and 汎 而

而灰 and their energies, can, perhaps, depend on 灰 , it is more precise to denote then by 肉i and

缶i而 灰 , but we suppress the 灰 superscripts to keep the notation less cluttered. Substituting Eq. (118) into Eqs. (125) gives generalized, canonical Hartree每Fock Eqs:
1 2 ?1 + v ( r 1 ) + ?2 ?1 而 灰v 而 而 r12 而 (x2 , x2 ) dx2 + 糸 ?xc (x1 ) 肉i (x1 ) = 缶i而 肉i (x1 ).

(126)

VII.

(灰 ) ?co TREATMENT OF THE CORRELATION-ENERGY FUNCTIONALS E

The Hamiltonian operator Hv can be written in normal-ordered form [24, 27, 28, 29] with respect to the reference state |污 :
?1 污 2 , Hv = E1 [污, v ] + {v }污 + {? 1 1 ,2 2 ? }污 + {r12 }

(127)

20

?1 污 2 where {v }污 , {? 1 1 ,2 are terms from the external potential, kinetic energy and 2 ? }污 , and {r12 }

electron-electron interactions: {v }污 =
i考j考
1 2 {? 2 ? }污 =

(i考 |v |j考 ){a? i考 aj考 }污 ,
? 1 2 (i考 |? 2 ? |j考 ){ai考 aj考 }污 , i考j考

(128) (129)
老 污 (i考 | (? vJ +v ?x ) |j考 ){a? i考 aj考 }污 ,

?1 污 {r12 }1,2 =

1 2 i考j考

? (i考j考 |k考 ∩ l考 ∩ ){a? i考 ak考∩ al考∩ aj考 }污 + k考∩ l考∩ i考j考

(130)
老 Furthermore, the Coulomb v ?J operator satis?es:

老 v ?J 耳 (r 1 ) =

?1 r12 老(r2 ) 耳(r1 ) dr2,

(131)

污 and the exchange operator v ?x is given by Eq. (43).

Substituting Eqs. (127) into (109) gives
(灰 ) (灰 ) (灰 ) (灰 ) ?co E [污, v ] = Vco [污, v ] + Tco [污 ] + Uco [污 ]

(132)

where these terms are identi?ed as the potential, kinetic, and electron-electron-interaction contributions to the correlation-energy functionals:
(灰 ) Vco [污, v ] = {v }污

污灰

= {v } 污灰
污灰
1 2 = {? 2 ? } 污灰 ,

(133) (134) (135)

(灰 ) 2 Tco [污 ] = {? 1 2 ? }污 (灰 ) ?1 污 Uco [污 ] = {r12 }1,2

污灰

?1 = {r12 }1,2

污灰 ,

and where we have also introduced a more condensed notation where the vacuum state is understood to agree with the trial wavefunction. Similarly, substituting Eqs. (127) into (108) gives Eco [污, v ] = Vco [污, v ] + Tco [污 ] + Uco [污 ] where Vco [污, v ] = {v } 污 ,
2 Tco [污 ] = {? 1 2? } 污

(136)

(137) (138) (139)

?1 Uco [污 ] = {r12 }1,2 污 .

21

VIII.

APPROXIMATIONS

If we know the exact correlation energy Eco for some Brueckner one-particle densitymatrix, say 而 ∩ with some external potential, say v ∩ , where 而 ∩ ↙ v ∩ , then using Eq. (122), we obtain the following reasonable approximation: ? (III) [污, v ] = Eco [而 ∩ , v ∩ ](而 ∩ =污,v∩ =v) , 而 ∩ ↙ v ∩ , E co (140)

where a similar approximation has been used previously in the v -dependent approach [30, 31, 34], and we are are assuming, as we have previously, that this approximation is most appropriate for (灰 = III). Similarly, a reasonable approximation for the components of the correlation-energy functionals are given by the following prescriptions:
(III) Vco [污, v ] = Vco [而 ∩ , v ∩ ](而 ∩ =污,v∩ =v) , 而 ∩ ↙ v ∩ , (III) Tco [污 ] = Tco [而 ∩ ](而 ∩ =污 ) (III) Uco [污 ] = Uco [而 ∩ ](而 ∩ =污 ) ,

(141) (142) (143)

and the three functionals: Vco , Tco and Uco , are known for a Brueckner one-particle densitymatrix 而 ∩ that determines the external potential v ∩ , i.e., 而 ∩ ↙ v ∩ . Of course, we have many Brueckner one-particle density-matrices 而 coming from the same external potential v ; presumable, we have one 而 from each N 每electron sector of the Hilbert space, for external potentials v with nondegenerate ground states.
(灰 ) ?co dependence on the external potential v comes The correlation-energy functionals E (III) exclusively form the correlated potential-energy-functional Vco . In many cases it is reason(III) able to assume that the this functionals, and the kinetic energy one Tco , are small, since

the potential- and kinetic-energy contributions are treated well in ?rst order. Therefore, the following approximation seems reasonable: ? (III) [污 ] > Uco [而 ∩ ](而 ∩ =污 ) , E co (144)

and this approximation yields a universal functional in the Kohn每Sham sense 每 the functional does not depend on v . However, even the potential energy contribution to the correlation(灰 ) energy functional, Vco , can be viewed 每 in a more general sense 每 as being universal, since if

this functional is known for an arbitrary external potential, than it is known for other cases, since the manner in which this functional depends on the external potential is the same for 22

all systems. On the other hand, if we use a model system to approximate the functionals, this will certainly not generate exact functionals for real systems, only approximation ones. Using the helium atom as a model system where Uco is presumable known, the previous approximation becomes ? (III) [污 ] > Uco [而he ](而he =污 ) , E co (145)

where 而he is the Brueckner one-particle density matrix from the helium atom. Assuming the Hartree每Fock one-particle density-matrix, say 而 ?he , is approximately equal to the Brueckner one, 而he , we have ? (III) [污 ] > Uco [? E 而he ](? 而he =污 ) , co (146)

and for a closed-shell systems that use spatially-restricted spin-orbitals, given by Eq. (A1), we have ? (III) [老1 ] > Uco [? E ?1he ](? ?1he =老1 ) , co (147)

where 老1 is the spinless one-particle density matrix [2, 43], given by Eq. (A2). As demoncs strated in Appendix A, we can use the well known approximation for Uco [? ?1he ] given by the

Colle and Salvetti functional [44, 45], giving ? (III) [老1 ] > E cs [老1 ] = U cs [? E ?1he =老1 ) , co co co ?1he ](? (148)

cs , uses four empirical parameters that are determined where this functional, denoted by Eco

using data from the helium atom. In addition, it is reaily veri?ed that this approximate correlation-energy functional neglects the potential- and kinetic-energy components, Vco and Tco , and these terms vanish when the exact one-particle density matrix, say 忙1 , from the target state |朵污 , is equal to the reference state one, 污 ; these two terms are considered to be small, or small enough to neglect, when (忙1 > 污1 ), as in the approach used when deriving the Colle每Salvetti functional. (Note that Vco vanishes if the density from the reference-state is the same as the density from the target state, as is the case for the Kohn每Sham method.) Using an identical derivation as in the v -dependent approach [34], it is readily veri?ed that the well known density-dependent approximation for the Colle每Salvetti functional [45], given by the LYP functional 每 at least for closed shell ground states 每 remains valid for the current approach:
cs lyp Eco [老] > Eco [老],

(149)

23

where the density 老 dependence is associated with the reference state, and not the target state. An electron gas de?ned by an constant external potential is not a member of {|污 v }, as in the case for an electron gas with periodic boundary conditions. However, let us assume that we can generalize the functional v (老) so that it yields the appropriate constant value, say vg , from the constant-density of an electron gas, say 老g ; so, we have (v (老g ) = vg ). And if we denote the one-particle density-matrix of the Brueckner reference state for an electron gas by 而g , we get ? (III) [污, v ] > Eco [而g , vg ](而g =污,vg =v) . E co (150)

However, since the correlation energy Eco does not depend on on the constant external potential vg , we cannot make the substitution (vg = v ), and so we obtain an approximation that yields a universal functional:
(III) (gas) ?co E [污 ] > Eco [而g ](而g =污 ) ,

(151)

(gas) where Eco is the correlation energy of an electron gas, and this approximation is also

identical to the approximation used in the v -dependent approach [30, 31], but with a slightly di?erent interpretation and derivation. Furthermore, in order to include 而g in the set {|污 v }, we only need to require v (老) to vanishes for any constant density: (v (老g ) = 0), where we consider two external-potentials that di?er by a constant to be equivalent. Starting with Eq. (151), except using a uniform electron gas, the same correlation energyfunction used in the local density approximation (LDA) [46] was shown to be valid with the v -dependent approach, for closed shell ground states [34]; however, in contrast to the Kohn每 Sham approach, the density dependence of the correlation-energy function is only associated with the reference state, and not, in addition, the target state. Furthermore, using an identical derivation as in the v -dependent approach, it is easily veri?ed that the correlation-energy ? (III) within the curfunctional from LDA can also be used, as well, as an approximation for E
co

rent approach under consideration. Furthermore, since the exchange-energy functional in the current approach is identical with the one from the v -dependent method, the exchangeenergy functionals that are valid in the v -dependent approach are also valid in the current approach, including the Dirac exchange-functional, and the augmentation of this functional with the Becke exchange correction [47]. Hence, as in the v -dependent approach, the LDA 24

and the method known as BLYP are also valid in the current method. (At least for closed shell ground states.) Furthermore, it is readily veri?ed that the B3LYP approach [48, 49] 每 that was demonstrated to be a reasonable approximation within the v -dependent approach for closed-shell ground-states [34] 每 remains valid for current approach under consideration. While all functionals that have been shown, so far, to be valid approximations for the v -dependent approach, are also valid in the current approach, the use of the LYP and Colle每 Salvetti functional appear more natural within the current approach under consideration, since these functionals are universal ones that do not have a dependence on the external potential. And since the BLYP and B3LYP functionals contain the LYP functional, these approaches are also better suited with the current approach.

APPENDIX A: CONNECTION WITH COLLE每SALVETTI FUNCTIONAL

In order to keep the discussion simple, we only consider closed-shell singlet states that are well described by a single determinantal-state, where we use spatially-restricted spinorbitals, given by 肉j考 (x) = 聿j (r )考 (肋 ); 考 = 汐, 汕. (A1)

By using these orbitals, it is easily demonstrated that the one-particle density-matrix 污 is determined by the spinless one, as indicated by the following relation: 1 污 (x1 , x2 ) = 老1 (r1 , r2 )汛肋1 肋2 . 2 (A2)

Hence, any functional of 污 now becomes a functional of 老1 ; So, if we use the Hartree每Fock spin-less one-particle density matrix, say ? ?1 , Eq. (108), becomes EN v = E1 [? ?1 , v ] + Eco [? ?1 , v ]. (A3)

Substituting Eq. (136) into this expression for (污1 = ?1 ), and neglected the terms Vco and Tco , we have EN v > E1 [? ?1 , v ] + Uco [?1 ], and using Eq. (111), we have EN v >
2 ?1 (r, r∩) ?1 2 ?r ?
r∩ =r

(A4)

dr +

v (r)? ?(r) dr + EJ [? ?] + Ex [? ?考 1 ] + Uco [?1 ]. 25

(A5)

Consider the total electron-electron potential energy, given as the expectation value in?1 volving the target state |朵老1 and the electron-electron repulsion energy operator {r12 }0 : ?1 {r12 }0



=

?1 朵老1 |{r12 }0 |朵老1 朵老1 |朵老1

(A6)

where the operator is given by
?1 {r12 }0 =

1 ? (i考j考 |k考 ∩ l考 ∩ )a? i考 ak考∩ al考∩ aj考 , 2 i考j考 k考∩ l考∩

(A7)

?1 and the 0 subscript appended to {r12 }0 indicates normal-ordering with respect to the true

vacuum state, | . Using the Hartree每Fock closed-shell reference-state |? ?1 , instead, as the vacuum-state, it is readily demonstrated that we have
?1 {r12 }0 ? ?1 ?1 = {r12 }1,2 ? ?1

+ EJ [? ?] + Ex [? ?1 ] = Uco [? ?1 ] + EJ [? ?] + Ex [? ?1 ],

(A8)

?1 where we have use Eq. (139); in addition, {r12 }1,2 is given by Eq. (130), where the suppressed

superscript, 污 每 the vacuum state 每 is set to ? ?1 . It is well known that the total electron-electron potential energy can also be expressed using the (diagonal portion of) the two-particle, spinless density-matrix from the target state [2, 43]:
?1 {r12 }0 ? ?1

=

?1 ?1 ? r12 忙2 (r 1 , r 2 ) d r 1 d r 2 ,

(A9)

and we can use the approximate expression, involving a two-body function ?, for the twoparticle spinless-density-matrix 每 valid for closed-shell systems 每 that was derived by Colle and Salvetti [44]:
?1 忙? ?2 (r1 , r2 ) 1 + ?2 (r1 , r2 ) ? 2?(r1 , r2 ) , 2 (r 1 , r 2 ) = ?

(A10)

and the two-particle density-matrix, from the Hartree每Fock reference-state, is given by 1 1 ?(r1 )? ? (r 2 ) ? ? ?1 (r1 , r2 )? ?1 (r2 , r1 ), ? ?2 (r1 , r2 ) = ? 2 4 where we also have
?1 r12 ? ?2 (r1 , r2 ) dr1dr2 = EJ [? ?] + Ex [? ?1 ].

(A11)

(A12)

Substituting Eq. (A10) into (A9) and using (A12), we have
?1 {r12 }0 ? ?1 cs = EJ [? ?] + Ex [? ?1 ] + Eco [? ?1 ],

(A13)

26

cs where Eco is the Colle每Salvetti correlation-energy functional [44, 45]: cs Eco [? ?1 ] = ?1 r12 ? ?2 (r1 , r2 ) ?2 (r1 , r2 ) ? 2?(r1 , r2 ) ,

(A14)

and this functional, after a series of approximations, is developed into one that does depends explicitly on ? ?1 [45]. (Note that ? ?2 is determined by ? ?1 , as indicated by Eq. (A11).) Comparing this Eqs. (A8) and (A13), gives the desired result:
cs Eco [? ?1 ] = Uco [? ?1 ].

(A15)

and from Eq. (A4), we have
cs EN v = E1 [? ?1 , v ] + Eco [? ?1 ],

(A16)

in agreement with the Colle每Salvetti electronic energy expresion used in their derivation of
cs Eco [44], where E1 [? ?1 , v ] is the Hartree每Fock energy.

APPENDIX B: DERIVATION OF Eq. (84)

Using the occupied and unoccupied Brueckner orbitals, given by Eqs. (62) and (63),
而 respectively, we can express the one-body operator (? vco )ex in the following manner:

而 )ex = (? vco

而 )ex |w考 a? r考 | (? vco r考 aw考 ,

(B1)

w考

r考

where, as in (H聿而 )1 , the vanishing terms involving the matrix elements that do not preserve
而 the spin state, , i.e., r考 ∩| (? vco )ex |w考 for 考 = 考 ∩ , are omitted. The matrix elements in the

above expression can be computed using the kernel from the operator:
而 r考 | (? vco )ex |w考 = 而 而 ex 而 肉r考 (x)vco (x, x∩ )肉w考 (x∩ ) dx dx∩;

(B2)

so, if we have the an expression for the matrix element on the left-hand side, that has the general form given by the integral on the right-hand side, we should by able to obtain the
而 ex 而 而 kernel vco (x, x∩ ) and, therefore, (? vco )ex 肉w考 , from the following de?nition:

而 而 (x) = (x))ex 肉w考 (? vco

而 而 ex (x∩ ) dx∩ . (x, x∩ )肉w考 vco

(B3)

In order to obtain an expression for the term on the left side of Eq. (86), below we obtain the diagrammatic expansion of the open portion of (H聿而 )1 , where this operator gives the 27

而 matrix elements r考 ∩ |v ?co |w考 using Eqs. (79) and (B2). We then use this matrix element to 而 而 determine v ?co 肉w考 and obtain the identity given by Eq. (84).

It is well known that the correlation operator 聿而 is given by a linked-diagram expansion, where all disconnected pieces are open [29, 38, 50]. Since 聿而 does not contain a one-body portion, it is easily demonstrated that the open portion of (H聿而 )1 is connected 每 all disconnected pieces from 聿而 are connected by the Hamiltonian H . Using the diagrammatic formalism presented in Appendix C and elsewhere [30], consider the following example of a diagram that contributes to the open portion of (H聿而 )1 :

?而 1 百而 (x1 , x2 )g (x2 , x1∩ )肉 而 (x1∩ )肉 而 ? (x1 )a? aw考 , = F r考 r考 w考 (B4) where the repeated indices, w考 and r考 , are summed over and where the two body function is given by g (x2 , x1∩ ) = 1 ?4 ∩ ?而 5 百而 (x5 , x6 )百而 (x6 , x4 )r ?1r ?1 ? 汍 而 (x4 , x5 )F 62 41∩ 百而 (x2 , x3 )F而 3 百而 (x3 , x1 ); 16 而 (B5)

furthermore, it is understood that there are no integrations over x2 and x1∩ on the right side of Eq. (B5), since these variables are not repeated indices according to the following convention: When determining which dummy indices are repeated indices, indices appearing within operators are not counted. So, for example, the indices x2 and x∩1 appear only once in the above Eq, and not two times, since the dummy indices from the Coulombic operator,
?1 ?1 i.e., x2 (and x6 ) from r62 and x∩1 (and x4 ) from r41 ∩ , are not counted.
而 而 The orbitals 肉w考 and 肉r考 , presented in Eq. (B4), can be any Brueckner orbitals, as de?ned

by Eqs. (62) and (63). However, for convenience we choose the canonical orbitals that are ?而 , as de?ned by Eq. (82). Using these orbitals the Brueckner one-particle eigenfunctions of F density-matrix, 而 , and the orthogonal function, 百而 , are given by 而 (x, x∩ ) =
?而 } x考﹋{肉o ↘而,F
而 而 肉x考 (x) (肉x考 (x∩ )) ,

?

(B6) (B7)

百而 (x, x∩ ) =
?而 } r考﹋{肉u ↘而,F

而 而 肉r考 (x) (肉r考 (x∩ )) ,

?

where these functions are also given by Eqs. (64) and (65).

28

而 According to Eq. (79), the diagram from Eq. (B4) also contributes to (? vco )ex . Comparing

Eqs. (B1) and (B4) we see that the following term: ?而 1 百而 (x1 , x2 )g (x2 , x1∩ )肉 而 (x1∩ )肉 而 ? (x1 ), F r考 w考
而 contributes to the matrix element r | (? vco )ex |w ; furthermore, and diagrammatically speak-

ing, removing the incoming and outgoing free-lines from the operator given by Eq. (B4), yields
1

?而 1 百而 (x1 , x2 )g (x2 , x1∩ ), = F
1∩

(B8)

而 而 ex vco )ex , as de?ned by Eq. (B2); and this diagram contributes to vco (x1 , x1∩ ), the kernel of (?

furthermore, we use the following diagrammatic representation for this two-body function:
而 ex vco (x1 , x1∩ ) =

vco
1∩ 1

(B9)

where, in addition, the non-dummy indices x1 and x1∩ in the above two diagrams, by our convention, correspond to the vertices of the omitted outgoing and incoming lines, respectively. Note that we have labeled these indices in the diagrams above; however, we will often omit these labels in similar (kernel) diagrams below. As a slight variation of the diagram given within Eq. (B4), consider the following diagram: 1? 而? 而 (x1 )a? (x1∩ )肉r F而 1 百而 (x1 , x2 )g (x2, x1∩ )肉w r aw , 2

=

(B10)

where the additional factor of

1 2

comes from the diagonal term, given by Eq. (C20a). As in

而 the diagram within Eq. (B4), this diagram contributes to (? vco )ex ; the corresponding diagram 而 ex that contributes to the kernel vco (x1 , x1∩ ) can be expressed in two alternative forms:

1

1

=
1∩ 1∩

=

1? F而 1 百而 (x1 , x2 )g (x2 , x1∩ ), 2

(B11)

where the ?rst diagram on the left side replaces the incoming and outgoing omitted-lines with dotted lines; this form gives a visual aid in determining the excitations involved and a psudo hole-line for the diagonal term to reside on. 29

The diagrams appearing in Eqs. (B8) and (B11) are examples of one-body kernel-diagrams ?而 1 vertex and the omitted incomingwhere the omitted outgoing free-line is attached at a F
?1 line is attached at the x1∩ vertex of a rj 1∩ operator. (In this particular case we have (j = 4),

according to Eq. (B5)). Furthermore, note that the Fock operator is acting upon excited ?而 1 百而 (x1 , x2 ) term. Summing over all diagrams of this type, we have orbitals, giving the F ?而 1 百而 (x1 , x2 )G(x2 , x1∩ ) F = + + + (B12) +﹞﹞﹞,

where the two-body function G(x2 , x1∩ ) is obtained from the in?nite-order expansion, and, for brevity, we have only displayed the ?rst four diagrams of the series. Introducing a diagrammatic symbol for G(x2 , x1∩ ), we can represent the above expansion in the following manner: ?而 1 百而 (x1 , x2 )G(x2 , x1∩ ) = F
1∩ 1

G
2

(B13)

As a slight variation of the diagrams from the series appearing in Eq. (B12), we also have diagrams that have the Fock operator acting upon occupied orbitals, e.g., 1? F而 1 而 (x1 , x2 )g (x2 , x1∩ ), 3

=

(B14)

where g (x2, x1∩ ) is given by Eq. (B5). Summing over all diagrams of this type, as in Eqs. (B12) and (B13), we have
I

?而 1 而 (x1 , x2 )I (x2 , x1∩ ) = F

2

1∩ 1

(B15)

The diagram sums represented by Eqs. (B13) and (B15) include all diagram where the ?而 1 vertex and the omitted incoming-line is atomitted outgoing free-line is attached at a F
?1 tached at the x1∩ vertex of a rj 1∩ operator. Two examples where both incoming and outgoing

omitted-lines are connected to Fock operators are given by the following two diagrams:

?而 1∩ , ?而 1 百而 (x1 , x2 )l(x2 , x1∩ )F = F

(B16)

30

?而 1∩ , ?而 1 而 (x1 , x2 )m(x2 , x1∩ )F = F where, in these examples, we have

(B17)

1 ?4 ?而 5 百而 (x5 , x6 )百而 (x6 , x4 )r ?1 r ?1 而 (x2 , x3 )百而 (x3 , x1∩ ), (B18) 汍而 而 (x4 , x5 )F 42 63 96 1 ?4 ?而 5 百而 (x5 , x6 )百而 (x6 , x4 )r ?1 r ?1 百而 (x2 , x3 )而 (x3 , x1∩ ), (B19) 汍而 而 (x4 , x5 )F m(x2 , x1∩ ) = 43 62 96 l(x2 , x1∩ ) = and again, we can sum over all diagrams of these types: ?而 1∩ = ?而 1 百而 (x1 , x2 )L(x2 , x1∩ )F F
1∩ 1

L
2

(B20)
2

M

?而 1∩ = ?而 1 而 (x1 , x2 )M (x2 , x1∩ )F F

1∩ 1

(B21)

The other two cases of interest involve diagrams where both incoming and outgoing
?1 omitted lines are attached to the two-body part of the Hamiltonian, e.g., rij , and diagrams ?而 1∩ vertex and the omitted outgoingwhere the omitted incoming free-line is attached at a F ?1 line is attached at the x1 vertex of a rj 1 operator. Examples of these two case are given by

the following two diagrams:

=

p(x1 , x1∩ ),

(B22)

= where, for these examples, we have p(x1 , x1∩ ) =

?而 1∩ , n(x1 , x1∩ )F

(B23)

?1 1 ?4 ?而 5 百而 (x5 , x6 )百而 (x6 , x4 )r ?1 ? ? 汍 而 (x4 , x5 )F 61∩ r41 而 (x1 , x2 )F而 2 而 (x2 , x3 )F而 3 而 (x3 , x1∩ ). 128 而

(B24) n(x1 , x1∩ ) =
1 ?4 ?而 5 百而 (x5 , x6 )百而 (x6 , x4 )r ?1 r ?1 而 (x1 , x2 )F ?而 2 而 (x2 , x3 )而 (x3 , x1∩ ) 汍 而 (x4 , x5 )F 63 41 16 而

(B25) Summing over all diagrams of these types, gives P (x1 , x1∩ ) ?而 1∩ N (x1 , x1∩ )F = = 31
P
1∩ 1

(B26)
1

N
1∩

(B27)

Since the diagrams represented in Eq. (B13), (B15), (B20), (B21), (B26), and (B27)
而 ex include all possible diagrams that can contribute to vco (x1 , x1∩ ), using these expression and

Eq. (B9), we have the following diagrammatic and algebraic relations:
vco
1∩ 1 1

I

2

1

=
1∩

G
2

+
M

1∩ 1 2

+
1∩

L
2

(B28)
N
1∩ 1

+

1∩ 1

+

P
1∩ 1

+

,

而 ex ?而 1∩ ?而 1 百而 (x1 , x2 )L(x2 , x1∩ )F ?而 1 而 (x1 , x2 )I (x2 , x1∩ ) + F ?而 1 百而 (x1 , x2 )G(x2 , x1∩ ) + F vco (x1 , x1∩ ) = F

?而 1∩ . (B29) ?而 1∩ + P (x1 , x1∩ ) + N (x1 , x1∩ )F ?而 1 而 (x1 , x2 )M (x2 , x1∩ )F +F Substituting this expression into Eq. (B3), and reordering terms, we have
而 而 而 而 ?而 1∩ 肉w考 (x1∩ ) (x1∩ ) + N (x1 , x1∩ )F (? vco (x1 ))ex 肉w考 (x1 ) = P (x1, x1∩ )肉w考

(B30)

?而 1 而 (x1 , x2 )I (x2 , x1∩ )肉 而 (x1∩ ) ?而 1 百而 (x1 , x2 )G(x2 , x1∩ )肉 而 (x1∩ ) + F +F w考 w考 ?而 1∩ 肉 而 (x1∩ ) ?而 1∩ 肉 而 (x1∩ ) + F ?而 1 而 (x1 , x2 )M (x2 , x1∩ )F ?而 1 百而 (x1 , x2 )L(x2 , x1∩ )F +F w考 w考 If we substitute the above expression into Eq. (83), it is easily demonstrated that the ?rst two terms from the above expression vanish, e.g., for the ?rst term, we have
r1 ↙R
而 (x1∩ ) = 0, lim |r1 ? R|P (x1 , x1∩ )肉w考

for all R,

(B31)

where the variable x1 within P (x1 , x2 ) is the independent variable for functions of the general
?1 ?1 form rj 1 而 (x1 , xi ) or rj 1 百而 (x1 , xi ), and the dummy indices in these function: xi and xj , are

integrated over; furthermore, and in general, all other variable that appear in diagrams that contribute to P (x1 , x1∩ ) 每 e.g., x1 , x2 , x3 , x4 , x4 , and x6 for p(x1 , x1∩ ) as presented in
?1 Eq. (B24) 每 are also dummy integration variables. Hence, since the functions rj 1 而 (x1 xi ) and ?1 1 2 1 2 rj 1 百而 (x1 , xi ) do not and contain a laplacian term 每 i.e., ? 2 ?r1 而 (x1 , xi ) or ? 2 ?r1 百而 (x1 , xi )

每 or a singularity 每 i.e., |r1 ? Rm |?1 每 the above identity holds. Using a similar analysis, we also obtain the following identity:
r1 ↙R

?而 1∩ 肉 而 (x1∩ ) = 0. lim |r1 ? R|N (x1 , x1∩ )F w考

(B32)

Substituting Eq. (B30) into (83), and using the above two identities, we get
r1 ↙R
而 而 lim |r1 ? R| [? vco (x1 )]ex 肉w考 (x1 ) =

(B33)

r1 ↙R

?而 1 而 (x1 , x2 )I (x2 , x1∩ )肉 而 (x1∩ ) ?而 1 百而 (x1 , x2 )G(x2 , x1∩ )肉 而 (x1∩ ) + F lim |r1 ? R| F w考 w考 ?而 1∩ 肉 而 (x1∩ ) , ?而 1∩ 肉 而 (x1∩ ) + F ?而 1 而 (x1 , x2 )M (x2 , x1∩ )F ?而 1 百而 (x1 , x2 )L(x2 , x1∩ )F +F w考 w考 32

and this expression can be written as
而 而 lim |r1 ? R| [? vco (x1 )]ex 肉w考 (x1 ) =

r1 ↙R

(B34)

r1 ↙R

?而 1 而 (x1 , x2 )Aw考 (x2 )+ F ?而 1 百而 (x1 , x2 )Bw考 (x2 ) , lim |r1 ? R| F

where
而 ?而 1∩ 肉 而 (x1∩ ), (x1∩ ) + M (x2 , x1∩ )F Aw考 (x2 ) = I (x2 , x1∩ )肉w考 w考 而 ?而 1∩ 肉 而 (x1∩ ). (x1∩ ) + L(x2 , x1∩ )F Bw考 (x2 ) = G(x2 , x1∩ )肉w考 w考

(B35) (B36)

Substituting into Eq. (B34) the Fock operator, Eq. (41), it is easily seen that the terms involving the Coulomb and exchange operator vanish, so we have
而 而 lim |r1 ? R| [? vco (x1 )]ex 肉w考 (x1 ) =

r1 ↙R

(B37)

r1 ↙R

? v1 而 (x1 , x2 )Aw考 (x2 ) + h ? v1 百而 (x1 , x2 )Bw考 (x2 ) , lim |r1 ? R| h

? v1 is given by Eq. (85). Using Eqs. (64) and (65), we obtain Eq. (84), where where h
而 x考 (x2 ))? = Aw考 (x2 ) (肉x考 Cw考

(B38) (B39)


r考 而 Dw考 = Bw考 (x2 ) (肉r考 (x2 ))?

x考 and the terms that do not preserve the spin state, e.g., Dw考 , ore omitted, since these terms

vanish; furthermore, there are summations over the repeated indices x考 and r考 for the ?而 } and {肉u ↘ 而, F ?而 }, respectively. orbital sets {肉o ↘ 而, F

APPENDIX C: DIAGRAMMATIC FORMALISM FOR THE CORRELATION ENERGY Eco

In order to keep the notation less cluttered, for this section we use a combined spin-spatial
而 而 notation for spin-orbital indices; for example, 肉w考 is now denoted by 肉w .

A diagrammatic expansion for correlation-energy Eco , or for 聿而 using Lindgren*s formalism [29, 38, 50]), is easier to obtain when all operators involved are written in normal-ordered form [24, 28, 29, 51]. For example, the Hamiltonian, given by Eq. (56), can be written as ?而 } + {r ?1 }而 , H = E1 [而 ] + {F 12 33 (C1)

where E1 [而 ] = 而 |H |而 =
w

? |w ] + 1 [w |h 2 ? |j ] + [i|h

([ww |xx] ? [wx|xw ]) ,
wx

(C2) (C3) (C4)

?而 } = {F
ij

([ww |ij ] ? [wi|jw ]) {a? i aj }而 ,
w

1 ?1 {r12 }而 = 2

? [ij |kl]{a? i ak al aj }而 , ijkl

and the integrals are now spin-dependent as indicated by the square brackets [﹞ ﹞ ﹞ ] [5]. ?而 }, is appropriate, since this term is the FockDenoting the one-body portion of H by {F operator, except that the second quantized operators are normal-ordered with respect to ?而 . The two body portion of H the |而 vacuum state, instead of the true vacuum | , as in F
?1 ?1 }而 emphasizing that this operator is determined by r12 and the vacuum is denoted by {r12 ?1 state |而 ; furthermore, except for the shifted vacuum, the two-body portion of H is r12 ,

when this operator is expressed in second quantization. For a perturbative treatment, we partition the Hamiltonian into a zeroth-order Hamiltonian H0 and a perturbation V : H = H0 + V, (C5)

where we require the reference state |而 to be an eigenfunction of H0 , a one-body operator: H0 | 而 = E0 |而 , ?ij a? i aj ,
ij

(C6) (C7)

H0 =

and the zeroth-order Hamiltonian is de?ned by its matrix elements; we choose them by requiring the following relation to be satis?ed: ?ij = ?ji = ?而 ij , where ?而 wr = 0,
而 ? 而 而 ?而 wx = 肉w |fo |肉x , 而 ? 而 而 ?而 rs = 肉r |fu |肉s ,

(C8a)

(C8b) (C8c) (C8d)

34

?而 and f ?而 , are determined by the reference state |而 , but the and the one-body operators, f o u ?而 and f ?而 upon |而 is at our disposal; the orbital subspaces are, again, de?ned dependence of f
o u

by Eqs. (62) and (63). Using the above choice, our zeroth-order Hamiltonian becomes
而 H0 = w,x﹋{肉o ↙而 } ? ?而 wx aw ax + r,s﹋{肉u ↙而 } ? ?而 rs ar as ,

(C9)

而 where the appended 而 superscript indicates that H0 now depends on the reference state |而 .

A linked diagram expansion for 聿而 and Eco [而 ] is known to exist for a zeroth-order Hamiltonian that is a diagonal, one-body, operator [24, 29, 41, 50, 52, 53, 54, 55, 56]. A diagonal
而 form for our one-body operator, H0 , is obtained when we choose its orbital sets 每 {肉o ↙ 而 }

and {肉u ↙ 而 } 每 to satisfy the following conditions:
而 ? 而 而 |肉x = 汛wx ?而 |fo 肉w w, 而 ? 而 而 肉r |fu |肉s = 汛rs ?而 r,

(C10a) (C10b)

?而 } and {肉u ↘ 而, f ?而 }, indicating where we denote these particular sets of orbitals by {肉o ↘ 而, f o u 而 ? or f ?而 . that they are uniquely determined by |而 and their one-particle operator, f
o u 而 Using these orbitals, H0 can be written as 而 ?而 + u ?而, H0 =o

(C11)

而 而 ? 而 and u ? 而 每 are the occupied and unoccupied portions of H0 )oc 每 (H 0 where these terms 每 o 而 and (H0 )un 每 and are given by the following:

?而 o

=

? ?而 w aw aw ,

(C12a) (C12b)

?而 } w ﹋{肉o ↘而,f o

?而 = u where our partitioning can be written as

? ?而 r ar ar ,

?而 } r ﹋{肉u ↘而,f u

而 + V而 . H = H0

(C13)

Using the above notation, our zeroth-order Hamiltonian in normal-ordered form can be written as
而 ?而} + u ?而 , H0 = E0 [而 ] + {o

(C14)

35

? 而 is already normal-ordered; the constant term E0 [而 ] is the zeroth-order energy of where u |而 :
而 H0 |而 = E0 [而 ] |而 ,

(C15)

and is given by E0 [而 ] = ?而 w.
?而 } w ﹋{肉o ↘而,f o

(C16)

Note that the ?rst-order and the correlation energies, E1 [而 ] and Eco , do not depend the zeroth-order energy E0 [而 ], The perturbation V而 , de?ned by Eqs. (C13), can also be written in normal-ordered form: V而 = Vc而 + V1而 + V2而 , (C17)

where, from Eqs. (C1), and (C14), the individual terms are given by the following expressions: Vc而 = E1 [而 ] ? E0 [而 ], ?而 } ? {o ?而} ? u ?而, V1而 = {F
?1 V2而 = {r12 }而 .

(C18a) (C18b) (C18c)

?而 } and {r ?1 }而 , are given by Eqs. (C3) and (C4), The one- and two-body parts of H , {F 12 respectively. The Goldstone diagrammatic representation of these operators can be written in the following manner [24, 29, 50, 52, 53, 54, 55, 56]: ?而 } = {F , (C19a)
?1 {r12 }而 =

. (C19b)

The one-body part of the perturbation V1而 is usually represented by a single diagrammatic operator. However, for our purposes, it is convenient to use separate diagrammatic operators ?而 } is presented by Eq. (C19a). for the three terms on the right side of Eq. (C18b), where {F Since the other two terms are diagonal, it is appropriate to simply represent them as un?lled

36

arrows: ?而} = ?{o , (C20a) ?而 = ?u . (C20b) In contrast, hole- and particle-lines, by themselves, are represented by ?lled arrows: and . As a slight alternative to the usual approach to evaluate the diagrams of the correlation energy Eco and the correlation operator 聿而 [5, 24, 29, 50, 52, 53, 54, 55, 56], we associate
? an internal hole-line corresponding to a w -occupied orbital with a 肉w (x1 )肉w (x2 ) factor; ? we associate a particle line corresponding to an r -unoccupied orbital with a 肉r (x2 )肉r (x1 )

factor, where x1 and x2 denote the dummy integration variables that arise from the vertices. ?而 from second-order Using this convention, the sole diagram involving the Fock operator F perturbation theory can be evaluated in the following manner: ?而 1 而w (x1 , x2 ) ﹞ F ?而 2 而r (x2 , x1 ), d x1 d x2 F

?1 = (汍而 rw )

(C21)

where
而 而 汍而 rw = ?w ? ?r ,

(C22)

?而 i denotes the Fock operator F ?而 每 and the repeated indices 每 r and w 每 are summed over; F ?而 i ﹞ ﹞ ﹞ )﹞ indicates that F ?而 i exclusively acts given by Eq. (41) 每 acting upon (xi ); the term (F within the brackets; furthermore, the w th component of the (one-particle) density-matrix 而 is denoted by
? 而w (x1 , x2 ) = 肉w (x1 )肉w (x2 );

(C23a)

the r th orthogonal-component of 而 is denoted by
? (x2 ), 而r (x1 , x2 ) = 肉r (x1 )肉r

(C23b)

where, for a complete set of orbital states, we have [55] 汛 (x1 ? x2 ) =
w

而w (x1 , x2 ) +
r

而r (x1 , x2 ),

(C24)

37

which is a shorthand notations for 汛 (x1 ? x2 ) = 汛 (r1 ? r2 )汛肋1 肋2 . (C25)

If we remove the top interaction from the diagram given by Eq. (C21), we see that this is a ?rst-order diagram that contributes to the one-body portion of the correlation operator 聿而 [29, 38, 50]. Since the in?nite-order sum of all one-body diagrams for 聿而 must vanish for a Bruckner orbital description, this diagram can be omitted from the expansion for the correlation energy Eco However, we will still consider it as a simple example to illustrate our approach and notation. In order to further compress our notation, we use the convention that all repeated dummy ?而 i to exclusively act upon the indices are integrated over and restrict the Fock operator F ?而 i 汐∩ (xj , xi )汐(xi , xj ) = 汐∩ (xj , xi )F ?而 i 汐(xi , xj )); ?rst variable of any two-body function, i.e., (F Eq. (C21) can then be written as
?1 ? ? = (汍而 rw ) F而 1 而w (x1 , x2 )F而 2 而r (x2 , x1 ),

(C26a)

and the other two diagrams from second-order perturbation theory have the following forms:

1 ?1 ?1 )?1 r12 r34 而w (x1 , x3 )而r (x3 , x1 )而x (x2 , x4 )而s (x4 , x2 ), = (汍 而 2 rwsx

(C26b)

1 ?1 ?1 )?1 r12 r34 而w (x1 , x3 )而r (x3 , x2 )而x (x2 , x4 )而s (x4 , x1 ), = ? (汍 而 2 rwsx where
而 而 汍而 rwsx = 汍rw + 汍sx .

(C26c)

(C27)

Let us also mention that when determining which dummy indices are repeated indices, it is not necessary to count indices appearing within operators. So, for example, the indices x1 and x2 appear twice in Eq. (C26a), and not three times, since the dummy indices from the ?而 1 and F ?而 2 , are not counted. Fock operators, i.e., F ? 而 } and ?u ?而, The diagonal terms arising from the zeroth-order Hamiltonian, given by ?{o and represented by Eqs. (C20), ?rst appear in third order. For example, the following two 38

? 而 } and ?u ? 而 into the diagram on the left side of diagrams can be obtained by inserting ?{o Eq. (C26a): = ? (??w ) ? ?而 2 而r (x2 , x1 ), F 而 (x1 , x2 )F 2 而1 w (汍 而 rw ) (C28a) = (??r ) ? ?而 2 而r (x2 , x1 ). F 而 (x1 , x2 )F 2 而1 w (汍 而 ) rw (C28b) ? 而 } generates an additional hole line when inserted into a diagram The hole-line operator {o and, therefore, a factor of ?1 is included when diagram (C28a) is evaluated, where this factor cancels the ?1 factor from ??w . Since this type of cancellation always occurs, as an ? 而 } insertions, and treat {o ? 而 } vertices as ones alternative, we associate a factor of ?w for {o ? 而 is associated with a ??r factor. Keep in mind, that do not generate additional hole lines; u also, that these operators generate an additional energy-denominator factor, e.g., 汍而 rw , when inserted into a diagram. The individual diagrams depend, in part, on each of the 而w components, given by Eq. (C23a), and the orthogonal components 而r , given by Eq. (C23b). In addition, each diagram depends on the set of orbital energies {?而 }, which are at our disposal. In order to make each diagram an explicit functional of the one-particle density-matrix 而 , given by 而 (x1 , x2 ) =
w

而w (x1 , x2 ),

(C29)

and its orthogonal component, 百而 , given by 百而 (x1 , x2 ) =
r

而r (x1 , x2 ),

(C30)

where 百而 depends, explicitly, on 而 : 汛 (x1 ? x2 ) = 而 (x1 , x2 ) + 百而 (x1 , x2 ), (C31)

we choose all occupied orbitals to be degenerate, with energy ?而 o ; also, we choose all unoccupied orbitals to be degenerate, with energy ?而 u . With these choices, the zeroth-order Hamiltonian, given by Eqs. (C11) and (C12), becomes
而 H0 = ?而 o ?而 } w ﹋{肉o ↘而,f o 而 a? w aw + ?u ?而 } r ﹋{肉u ↘而,f u

a? r ar ,

(C32)

39

and since this operator is invariant to a unitary transformation of occupied or unoccupied ?而 and f ?而 每 any set of orbitals de?ning 而 is appropriate 每 orbitals, it no longer depends on f o u so we can write
而 H0 = ?而 o w ﹋{肉o ↙而 } 而 a? w aw + ?u r ﹋{肉u ↙而 }

a? r ar .

(C33)

It is easily proven that all perturbative orders, except for the zeroth-order, depend on the orbital-energy di?erence 汍而 , given by
而 汍而 = ?而 o ? ?u ,

(C34)

而 而 and not on the individual orbital-energies, ?而 o and ?u . Therefore, we can choose (?u = 0),

and so our only parameter is 汍而 . With this choice we have
而 ?而 , H0 = 汍而 N

(C35)

?而 is the number operator for the occupied orbitals, where N ?而 = N
w ﹋{肉o ↙而 }

a? w aw ,

(C36)

and it gives the total number of occupied orbitals when acting on a single determinant. In the one-particle Hilbert space, this operator is the projector for the occupied subspace 每 spanned by {肉o ↙ 而 } 每 or, the one-particle density-matrix operator: ?而 = N
w ﹋{肉o ↙而 }

|肉w 肉w | = 而 ?.

(C37)

Using the above two expressions, let us generalize the de?nition of 而 ?: 而 ?=
w ﹋{肉o ↙而 }

a? w aw ,

(C38)

and write the zeroth-order Hamiltonian in a simpli?ed form, given by
而 H0 = 汍而 而 ?.

(C39)

By normal-ordering this expression, we have
而 H0 = 汍而 N而 + 汍而 {而 ?},

(C40)

40

where N而 is the number of particles within |而 , and from Eq. (C14), we get the following identities: E0 [而 ] = 汍而 N而 , ? 而 } = 汍 而 {而 {o ?}, ? 而 = 0; u (C41) (C42) (C43)

furthermore, our zero- and one-body portion of the perturbation, Eqs. (C18a) and (C18b), have the following modi?ed forms: Vc而 = E1 [而 ] ? 汍而 N而 , ?而 } ? 汍而 {而 V1而 = {F ?}. (C44a) (C44b)

? 而 , represented by Eq. (C20b), does Eq. (C43) indicates that the unoccupied operator, u ? 而 }, represented by Eq. (C20a) not appear in the expansion of the correlation-energy Eco ; {o and given by 汍而 {而 ?}, is associated with a factor of 汍而 . Each diagram now becomes an explicit functional of 而 and 百而 . For example, the second-order diagrams can be written in the following manner:

1? ? = 汍? 而 F而 1 而 (x1 , x2 )F而 2 百而 (x2 , x1 ),

(C45)

1 1 ?1 ?1 = 汍? r r 而 (x1 , x3 )百而 (x3 , x1 )而 (x2 , x4 )百而 (x4 , x2 ), 4 而 12 34

(C46)

1 1 ?1 ?1 = ? 汍? r r 而 (x1 , x3 )百而 (x3 , x2 )而 (x2 , x4 )百而 (x4 , x1 ), 4 而 12 34

(C47)

where 百而 is given by Eq. (C31). Higher order examples are presented elsewhere [30] and, in addition, a method that yields diagrams for the correlation-energy Eco that explicitly depend on the one particle density-matrix, 而 . It is well known that the correlation operator 聿而 is given by a linked-diagram expansion, where all disconnected pieces are open [29, 38, 50]. Using our approach here, these diagrams 41

can be evaluated in an identical manner as the diagrams for the correlation energy Eco ; Eq. (C48) gives an example of a ?fth-order one-body 聿而 diagram:

而? ? 而 1? = 汍? 而 F而 1 百而 (x1 , x2 )g (x2 , x1∩ )肉w (x1∩ )肉r (x1 )ar aw ,

(C48)

where the repeated indices, w and r , are summed over and where the two body function is given by Eq. (B5). (This diagram is identical to the (H聿而 )1 diagram appearing in Eq. (B4), but that diagram is evaluated slighly di?erent, since there is no energy denominator associated with H .)

[1] J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cambridge, Mass., 1986). [2] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). [3] S. M. Poling, E. R. Davidson, and G. Vincow, J. Chem. Phys 54, 3005 (1971). [4] E. R. Davidson, Reduced Density Matrices in Quantum Chemistry (Academic Press, New York, 1976). [5] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Macmillian, New York, 1982). [6] K. A. Brueckner, Phys. Rev. 96, 508 (96), (See Ref. 4 for more citations from Brueckner and coworkers.). [7] R. K. Nesbet, Phys. Rev. 109, 1632 (1958). [8] W. Brenig, Nucl. Phys. 22, 177 (1961). [9] P. O. L“ owdin, J. Math. Phys. 3, 1171 (1962). [10] W. Kutzelnigg and V. H. Smith, J. Chem. Phys. 41, 896 (1964). ’? i’ zek, Phys. Scripta 21, 251 (1980). [11] J. Paldus and J. C [12] R. A. Chiles and C. E. Dykstra, J. Chem. Phys. 74, 4544 (1981). [13] L. Z. Stolarczyk and H. J. Monkhorst, Int. J. Quantum Chem. 18, 267 (1984). [14] N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavachari, and G. W. Trucks, Chem. Phys. Lett. 164, 185 (1985).

42

[15] N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavachari, and G. W. Trucks, Chem. Phys. Lett. 164, 185 (1989). [16] K. K. Raghavachari, J. A. Pople, E. S. Replogle, M. Head-Gordon, and N. C. Handy, Chem. Phys. Lett. 167, 115 (1990). [17] K. Hirao, in Self Consistent Field Theory, Studies in Physical and Theoretical Chemistry (Elsevier, Amsterdam, 1990), vol. 90. [18] J. F. Stanton, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 97, 5554 (1992). [19] C. Hampel, K. A. Peterson, and H. -J. Werner, Chem. Phys. Lett. 190, 1 (1992). [20] G. E. Scuseria, Chem. Phys. Lett. 226, 251 (1994). [21] I. Lindgren and S. Solomonson, Int. J. Quantum Chem. 90, 294 (2002). [22] D. H. Kobe, Nucl. Phys. 3, 417 (1971). [23] L. Sh“ afer and H. A. Weidenm“ uller, Nucl. Phys. A174, 1 (1971). ’? [24] J. Paldus and J. C i’ zek, Adv. Quantum Chem. 9, 105 (1975). [25] W. Brenig, Nucl. Phys. 4, 363 (1957). [26] L. Sch“ afer and H. A. Weidenm“ uller, Nucl. Phys. A 174, 1 (1971). ’? [27] J. C i’ zek, J. Chem. Phys. 45, 4256 (1966). ’? [28] J. C i’ zek, Adv. Chem. Phys. 14, 35 (1969). [29] I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer Series on Atoms and Plasmas (Springer-Verlag, New York, Berlin, Heidelberg, 1986), 2nd ed. [30] J. P. Finley, arXiv:physics/0308056. [31] J. P. Finley, Phys. Rev. A (2004), (Approved for publication). [32] I. Lindgren, Phys. Scripta 32, 291 (1985), (See also ibid. 32, 611 (1985). [33] I. Lindgren and D. Mukherjee, Phys. Rep. 151, 93 (1987). [34] J. P. Finley, arXiv:physics/0308084. [35] J. Hubard, Proc. Roy. Soc. London A 240, 539 (1957). [36] F. Coester, Nucl. Phys. 7, 421 (1958). ’? i’ zek and J. Paldus, Int. J. Quantum Chem. 5, 359 (1971). [37] J. C [38] I. Lindgren, Int. J. Quantum Chem. S12, 33 (1978). [39] R. J. Bartlett and G. D. Purvis, Int. J. Quantum Chem. 14, 561 (1978). [40] J. A. Pople, R. Krishnan, H. B. Schlegel, and J. S. Binkley, Int. J. Quantum Chem. S14, 545 (1978).

43

[41] F. E. Harris, H. J. Monkhorst, and D. L. Freeman, Algebraic and Diagrammatic Methods in Many-Fermion Theory (Oxford University Press, New York, 1992). [42] H.f Schaefer III, Electronic Structures of Atoms and Molecules每A Survey of Rigorous Quantum Mechanical Methods (Addison-Wesley, Reading, 1972). [43] R. McWeeny, Rev. Mod. Phys. 32, 335 (1960). [44] R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1975). [45] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). [46] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [47] A. D. Becke, Phys. Rev. A 38, 3098 (1988). [48] A. D. Becke, J. Chem. Phys. 98, 5648 (1993). [49] P. J. Stephens, J. F. Devlin, and C. F. Chabalowski, J. Phys. Chem. 98, 11623 (1994), (see also internet address http://www.gaussian.com/q3.htm). [50] I. Lindgren, J. Phys. B 7, 2441 (1974). [51] N. N. Bogoliubov and D. V. Shirkov, Introduction to the Field of Quantized Fields (Wiley, New York, 1959), (English translation). [52] J. Goldstone, Proc. R. Soc. London A 239, 267 (1957). [53] N. M. Hugenholtz, Physica 27, 281 (1957). [54] P. G. H. Sanders, Adv. Chem. Phys. 14, 365 (1969). [55] S. Raimes, Many-Electron Theory (North-Holland, Amsterdam, 1972). [56] S. Wilson, Comp. Phys. Rep. 2, 389 (1985).

44


婝翑妀蟈諉

載嗣眈壽恅梒ㄩ
Unity3d-Building-ScenesParticle-Systems笢荎恅楹祒_芞恅
Please view the Particle Scripting Reference here....Particle Systems work by using one or two ...density of smoke, and high Energy to simulate ...
荎---價衾燠禎怓煦票腔薜赽薦疏呾楊腔弝け醴梓躲趿
become one of the main research direction of video...state transition density, especially when the ...Distributed particle filters for wirelesssensor net ...
蕉旃荎逄棵颯硌鰍-棵砱恁寁睿竘扠-▽轎煤狟婥▼
瞰:Albert Einstein, who developed the theory of ...some respected ideas in elementary particle physics...one of the most sensational developments in recent...
載嗣眈壽梓ワㄩ

All rights reserved Powered by 泫襞恅踱 9512.net

copyright ©right 2010-2021﹝
泫襞恅踱囀暌棚奜讕蝤畏觬陎硊裔蹅肢翕芛﹝zhit325@126.com|厙桴華芞