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A NEW VARIATIONAL EXPRESSION OF THE EFFECTIVE EXCHANGE AND CORRELATION
ENERGY AND RELATED APPROXIMATIONS
L. Dagens
To cite this version:
L. Dagens. A NEW VARIATIONAL EXPRESSION OF THE EFFECTIVE EXCHANGE AND CORRELATION ENERGY AND RELATED APPROXIMATIONS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-83-C3-88. �10.1051/jphyscol:1972311�. �jpa-00215045�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-83
A NEW VARIATIONAL EXPRESSION OF THE EFFECTIVE
EXCHANGE AND CORRELATION ENERGY AND RELATED APPROXIMATIONS
L. DAGENS
Commissariat B 1'Energie Atomique. Centre d'Etudes de Limeil.
B. P. no 27, 94-Villeneuve-Saint-Georges, France
RBsumB. - Une expression nouvelle de l'energie effective d'kchange et de corrblation est Ctablie B partir d'une formule variationnelle de Luttinger et Ward. Elle est exprimke en fonction de la fonction de Green g du systkme Blectronique non uniforme et est stationnaire quand G est variC autour de sa valeur exacte. L'approximation fondamentale de ce travail s'obtient en remplacant S
par l'approximation d'Hartree ; I'interaction effective est alors 6valuBe en se limitant a diverssous- ensembles de diagrammes respectant la condition de @-dCrivabilit6 de Baym. On d6finit les appro- ximations maximales vis-&-vis d'un ensemble de Z-diagrammes irrkductibles et on construit celles d'ordre I et I1 par rapport au nombre d'interactions CcrantCes.
Abstract. - A new variational expression of the effective exchange and correlation energy functional E x c [n(v)] is derived, using the Luttinger and Ward variational energy functional. E x c is expressed in terms of the density-dependent exact Green function S of the inhomogeneous elec- tron gas and is stationary when G is varied about its exact value. The basic approximation studied in this paper is obtained when 9 is replaced by the Hartree approximation S H . The corresponding effective interaction satisfies the compressibility theorem. It involves the set of all irreducible polarization graphs without self-energy parts. Each partial subset defines a specific approximation.
According to Baym, only @-derivable approximations are to be considered. To each basic subset of irreducible Z-diagrams correspond one and only one maximal approximation. The first and second order maximal approximations with respect to the number of dynamically screened inter- action lines are explicitly given.
I. Introduction. - The self-consistent field method of Kohn and Sham [I] (SCKS) is an extension of the approximate HFS method [2]. It allows an exact calculation of the ground state energy E of a non- uniform electron gas and its derivatives with respect to the external potential Vo, and the number of elec- trons. Consequently the electronic density p(r), the static limits of the screening function and the various dielectric functions [3] may be calculated exactly in the framework of the SCKS method.
The exact expression of the KS effective exchange and correlation energy Exc[p] as a functional of p(r) is not known. The p4I3 approximation (GaspBr [4], Kohn and Sham [I]) is valid when p(r) is a slowly varying function of r. It can be applied to the heavy atoms and ions but not to the valence band of simple metals [5], [6], [7]. The same can be said about the inhomogeneity correction of Herman and al. [%I.
More precise expression are required for the a priori calculation of (say) cohesive energy (Hedin and Lundquist [6] p. 168). The core electrons (p,(r)) and the valence electrons (n(r) ; p = p, + n) are consi- dered separately and Exc[p] is written as a sum of Ex,[n] (which is the quantity considered in this work)
and a complementary term including the effects of the core electrons and their interaction with the valence electrons. A good approximation to this term may be given in the form
Unpublished numerical calculations on simple metals atoms (Dagens and Desclaux) have shown that the appropriate value of A is almost always very near the Gasp&-Kohn-Sham value.
We propose a new method for the calculation of Exc[n], which leads t o explicit results when the density fluctuation n(r) - no is small. It is a Green function method, which uses a variational expression of the energy due to Luttinger and Ward [9] (see also Klein [lo] and Baym [ll]).
We derive first (Section 11) a general expression Ex, (S[n]) in term of the exact one-electron Green function 9, appropriate to the inhomogeneous electron gas whose density is n(r), using the Luttinger-Ward formula. It is stationary with respect to variations of S around its exact value.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972311
C3-84 L. DAGENS The replacement of the exact Q[n] by the Hartree- Green function QH[n] leads to the basic approximation of this paper (Section 111). The error made is of second order with respect to the deviation 943,. A new general approximation for the effective exchange and corre- lation interaction function X(q) (= - 4 nG/q2) is then derived and its general properties are studied. The basic approximation satisfies the compressibility theorem. It can be expressed in terms of the complete set of the irreducible polarization graphs without self-energy parts. The complete static irreducible screening function calculated with the linearized SCKS equations generalises an interpolating formula due to Hubbard [12] (Geldart and Taylor 1131, Singwi et al.
[14], Shaw [15]).
We consider in Section IV some explicit approxi- mations related to specific choices of the E-diagrams implied in the Luttinger-Ward @ (or Z) functional.
These choices are submitted to the very restricting condition of @-derivability (Baym [Ill). The concept of maximality with respect to a chosen set of irredu- cible Z-diagrams is defined. It leads to a complete determination of the infinite set of E-diagrams to be used. The dynamically screened interaction which must be used is also a one-to-one function of the chosen set of irreducible Z-diagrams.
A hierarchy of approximations is obtained by extending this chosen set. Note that the X,, approxi- mation (relative to the unscreened H F Z-diagram), studied elsewhere by 'other methods (Geldart and Taylor [13] ; Dagens [7], [16]), is @-derivable, but not maximal. The maximal approximations of first and second order with respect to the number of screened interaction lines included in the chosen Z-set diagrams are given explicitly. The first order approximation leads to a screening function studied by Geldart and Vosko [5]. The second order one is new.
11. A variational green function expression of Ex,. -
An interacting electron gas is subjected to an external potential Vo(r). Hohenberg and Kohn [17] have shown that the ground state energy is given by the minimum value of the universal functional
with respect to n(r), with
5
6n dr = 0. The first term is, symbolically, the interaction energy with the exter- nal potential. F includes the kinetic energy of the electrons and their interaction energy.No accurate approximations of F[n] are known, even in the Hartree limit. Kohn and Sham write F as
where T[n] is the exact kinetic energy functional for the Hartree equations. Eq. (3) is a definition of the
effective exchange and correlation energy. Comparison of (2) and (3) gives
where the last term is the Hohenberg and Kohn Hartree energy functional. It follows from (4) that the value of Ex, is always strictly smaller than the true (conventional) exchange and correlation energy.
The T[n] functional is calculated by writing n(r> = C ( qi(r) l2 ;
OCC
( 5 ) a variation of the auxiliary effective spin-orbitals q i leads to the SCKS equations.
Ex, will be expressed in terms of the one-electron Green function of the T = 0° inhomogeneous electron gas. A variational expression of the ground state energy E due to Luttinger and Ward [9] (Nozikres [18]
p. 225) is used :
with
The Hartree terms have been written out explicitly.
Eo = E,[no], where no is the mean value of n(r) and Go the uniform free electron gas Green function. The
<< tr )) operation is in general a product of four-
dimensional integrations and spin summations such as
The functional @ is given by the sum of all irredu- cible E-diagrams (energy diagrams ; see Klein [lo]), excluding the Hartree diagrams (see (6)), and with the exact Green function 9 in place of the unperturbed one. @ is related to the indirect self-energy operator by the relation
where Z is given by the sum of the skeleton Z-diagrams.
Z { Q ) is said to be @-derivable (Baym [11]) ; this property must be preserved when approximations are made (Section IV).
As a consequence of (6), E is stationary with respect to Q variations around its true value [9, 10, 11, 181.
A new expression of Ex, results from (4) and (6).
We note first that Q is a universal functional of n(r) (Kohn and Sham [19]), which satisfies the identity
A NEW VARIATIONAL EXPRESSION OF THE EFFECTIVE EXCHANGE C3-85
When Q[n] is substituted into (6) an exact formula for E[n, V o ] is obtained. If an approximate 9 [ n ] is used, the corresponding E[n, V o ] deviates from the exact one by a second order term.
In the same way, the Hartree Hohenberg and Kohn functional EH[n, V o ] results from the Hartree Luttinger and Ward functional calculated with the Hartree Green function functional GH[n]. It is worth noting that the SCKS Green function QScK,[n] is identical to s H [ n ] as a functional of n(r)
Q H b I = 9,,K,[nl - (12)
The proof is straightforward : n(r) determines comple- tely the local SC potential which generates it ; conse- quently it determines the associated Green function.
The Hartree energy functional EH is then given by the formula
Thus the z functional gives, in the Hartree approxi- mation, that part of the kinetic energy which is not included in Eo(no).
The combination of eq. (4), (6) and (13) gives the wanted result
Exc[nl = @ { Q b l ) + { %I ) - { Q H b 1 ) . (14)
This important, although rather formal, result is the starting point of approximations studied in the next parts of this work.
This expression has the following important proper- ty : it is stationary around the exact functional Exc[n]
with respect to variations of G[n] (subjected to the condition (1 1)) about the exact 9 [ n ] .
111. A general approximation for Ex, and the related effective interaction. - I. A simple approximation for Exc[n] is obtained by replacing in the basic for- mula (14) the exact manybody Green function 9 [ n ] by the Hartree Green function QH[n]. The two z terms cancel and Ex, becomes simply
Excbl = Q, ( %ml> ; (1 5 ) due to the stationary property of (14), the error is only of second order with respect to the deviation between G[n] and GH[n].
An interesting limiting case is that of an homoge- neous electron gas. SH[no] reduces to the free electron gas Green function :
(16) where
Substituting (16) into (15) we obtain the exchange and correlation energy AE(no) of a homogeneous electron gas (in this exceptional case, the effective energy Exc(no) is identical to the true energy A E ( ~ , ) ) :
When @ is calculated with the RPA E-diagrams only, eq. (18) gives exactly the usual RPA approxima- tion for AE (obtained by introducing in the Hubbard formula the Lindhard dielectric function ; see, for instance, Hedin and Lundquist [6] p. 7 8 ) ; this shows that the basic approximation cr 9 -t 9, )) gives results which compare well with the best evaluations of AE.
2. We assume in the following that n(r) fluctuates moderatly (but not necessarily slowly) about its mean value no. Its Fourier coefficients n, ( q # 0 ) are much smaller than no. Ex, [n] is expanded to second order in n,. Owing to the translational invariance of the imperturbed system we have
where third and higher order terms have been neglect- ed. This equation is the definition of the effective exchange and correlation interaction X(q) (Harri- son [20] ; this quantity is frequently written as -
4 nG(q)/q2). X(q) is related to the exact static screening function n(q) by the (( Hubbard formula )> [12]
where nO(q) (q = q, 0 ) is the Lindhard screening function, and n(q) the exact irreducible polarization function.
3. The approximate X(q) corresponding to the basic approximation (15) is given by the second derivative of @ with respect to n,. n, is related to the Hartree S. C. potential V , by the relation
n, = ",(a> vq + O(2) ;
SO X(q) is given by the approximate formula
i. e., using formula (9)
where the self-energy function Z(p) is given by the sum of all the skeleton Z-diagrams, calculated for 9 = Go.
This approximate X(q) is related to a certain subset of the irreducible polarization graphs. The following expression results directly from formula (20) and the definition of @ in term of E-diagrams :
L. DAGENS
where the sum is over the set 3 of all irreducible polarization diagrams without self-energy parts.
The general approximation considered in this section satisfies the compressibility theorem when the exchange and correlation energy is given by the corresponding approximation (18). This theorem expresses that
u u
lim X(q) = @ { Go ) =
,
AE(no) . (23)q + o an0 an,
The important fact is that Q, is only a implicit functio- nal of n(r), through the quantity QH[n]. The relation (23) is still valid when Z, in (22), is replaced by a @-deri- vable approximation.
IV. Study of a class of @-derivable approximations.
- 1) @-DERIVABLE AND MAXIMAL APPROXIMATIONS. -
The basic formula (21) allows a calculation of the effective interaction X(q) with a second order error with respect to the first and second n, derivatives of 9 - 9,.
An explicit calculation can be made only when some convenient subset of the skeleton Z-diagrams is retain- ed. It is worth noting that the knowledge of the functional @ itself is not necessary when using for- mula (22). In fact, the summation of an infinite subset of graphs is easier for Z than for @.
We seek first an approximation for Z { Q ), corres- ponding to a definite choice of skeleton Z-diagrams.
This choice is subjected to an important constraint : the approximate Z { 9 ) must be @-derivable. That is, a @ { 9 ) must exist such that (9) is true. A non
@-derivable approximation would be meaningless, since no expression of the energy can then be unam- bigously defined (Baym [I I]).
The functional Z ( 9 ) is given by the sum of all skeleton Z-diagrams, with the Hartree diagrams excluded. It is also given by the sum of the irreducible Z-diagrams of figure 1, with the appropriate dynami- cally screened interaction W { 9 ) replacing the bare Coulomb interaction u(q). The corresponding serie is ordered according to the number of screened inter- actions W { Q ) :
Z { Q , w } = z, { Q , w ) + Z,, { Q , w ) + ... . (24) When bare interactions are used the diagram I is the H F diagram and I1 is the second order exchange graph.
It is known that a straightforward bare interaction expansion is meaningless. Hedin and Lundquist [6]
have argued that a screened-interaction-expansion may converge in the metallic density range. All the subse- quent considerations rest on this hypothesis.
A given level of approximation will be defined by a definite subset of irreducible Z-diagrams (to be refered
FIG. 1. - Diagramatic definition of the irreducible skeleton self-energy operator.
to as the basic Z-set) and by a definite dynamically screened interaction W { Q ).
The important fact is that the @-derivability condi- tion imposes drastic constraints upon the choice of W ( Q ), once the basic Z-set is selected. The cr mini-
mal )> choice W = v (the bare Coulomb interaction)
lead always to @-derivable approximations. The first order maximal approximation will be considered later.
The maximal approximation with respect to a given basic Z-set correspond to the greatest subset of skeleton Z-diagrams such that the corresponding approximate Z { Q ) is @-derivable. The summation of these diagrams leads to an approximate Z ( 9 , W { Q I),
where W { Q ) is completely defined by the basic Z-set. By increasing this Z-set (including new irredu- cible Z-diagrams with more and more interaction lines) a hierarchy of maximal approximations is obtained, of which the first two will now be considered.
2) THE @-DERIVABLE APPROXIMATIONS OF ORDER I. -
The basic Z-set includes only the (( Hartree-Fock,) diagram I of figure I. The first order minimal approxi- mation is then given by the Z-diagram of order e2, with a bare interaction v(q). Z, ( Go, u ) is substituted into (21) and the resulting approximate effective inter- action is found to be
in terms of the two first order unscreened polarization graphs A and B of figure 2. This approximation have been studied by Geldart and Taylor [13] and by the author [7], [I61 ; XAB is shown to be an upper bound of the exact Hartree-Fock effective interaction [16].
We construct now the first order maximal approxi- mation and the corresponding dynamically screened interaction W, { Q ).
We consider first the associated maximal set of E-diagrams. The opening of any one of the electron lines (the graphical representation of the operation 6/69 applied to this line [ll]) must give a Z-diagram which is reducible to Z,. This implies that every closed
A NEW VARIATIONAL EXPRESSION OF THE EFFECTIVE EXCHANGE
A B
FIG. 2. - The two first order bare polarization graphs. II II 11
electron line of this E-diagram (and of the derived Z-diagrams) possesses exactly two vertices. This property gives a complete characterisation of the wanted maximal E- and Z-sets.
A few graphs of the maximal Z-set are shown in figure 3. The sum of this set is Z1 ( S , W, ( 9 ) ). W, is the RPA screened interaction given by the integral equation represented in figure 3. WI is required only for Q = Go (see eq. (21)) and is then a function the transferred energy-momentum q = q, o :
FIG. 4. - The screened irreducible skeleton polarization graphs considered in this work.
3) THE SECOND ORDER @-DERIVABLE MAXIMAL APPROXIMATION. - This approximation is defined by the two irreducible Z-diagrams I and I1 of figure 1.
The associated maximal set of E-diagrams is first determined. By opening any electron line of any of these E-diagrams we obtain a Z-diagram which must be reducible to Z, or Z,, (Fig. 1). Every closed electron line appears then as an insertion into an interaction line of one of the two polarization graphs, no(q) and n*(q) (Fig. 5).
1 1 I1 11 I I
1 I
4Il
If = 1 + : 1' f I I II
11 I1 I II
1 I II I II
I + ; II 11 1 ! I1
II = I U
w, TI Ir 1 I I1 II
FIG. 3. - Diagramatic equations of the maximal first order
WE
dynamically screened interaction and the associated self-energy operator.
Z, .( Go ), the self-energy functional obtained, is substituted into eq. (21). The resulting effective interaction is
(27) FIG. 5. -The second order maximal screened interaction.
where the polarization graphs ( A, B, C ) shown in figure 4 must be calculated using the RPA dynamically screened interaction.
This approximation has been studied by Geldart and Vosko [5] and Geldart and Taylor [13]. Exact numerical values of the sum TC:~,-(~) are available only in the limit q -t 0 [5]. The same quantity is computed in [13] for the whole range of I q I values, using a static screened interaction. According to Geldart and Vosko [5] this is a very bad approximation (at least in the limit q -+ 0) and a more accurate computation is still needed for I q I > k,.
The appropriate screened interaction WI, ( ) is given by the integral equation represented in figure 5.
When the electron gas is homogeneous, this equation reduces to the S. C . equation
where T C ~ differs from n; of eq. (25) since the screened interaction is different.
The general formula (21) gives then
C3-88 L. DAGENS
where n is the sum of all the polarizations graphs of figure 4, calculated with the dynamically screened interaction (28).
This approximation is new. A quantity similar to nA(q, o) has been studied by Kleinman [21] and Langreth [22]. Using their approximation we found that WI, is at most 10 % smaller than WI. Thus
xiBC and are expected to be very similar. Preli- minary numerical results in the q = 0 limit shown that the difference is at most 5 % in the metallic density range. On the other hand, it is well known that X,(O), calculated without screening, is zero ; no numerical results are yet available for X,, even in the q = 0 limit.
an approximate one ; (b) a @-derivable subset of Z-diagrams is chosen ; a systematic way of doing this choice has been described, based on the concept of maximality with respect to a given subset of irreducible Z-graphs.
No new numerical results are yet available. The approximation XAB(q) is the only one accurately known for the whole range of q [7], [13]. xiBc can be
exactly calculated from the results of Geldart and Vosko [5] in the q # 0 limit and is 10 to 20 % lower than XAB(0) in the metallic density range. No accurate values for q = 0 are available : it results from the work of Taylor and Vosko that the use of a static screened interaction leads (when q = 0) to values of xi,, no Conclusion. - A systematic approach to the calcu-
better than the simple XAB approximation ; thus an lation of the effective exchange and correlation inter- accurate numerical calculation using frequency depen- action (giving the zero frequency density response
dant screened interaction is still needed to obtain a function) has been described. It involves a Green
definite improvement over XAB approximation for non function variational expression. Two steps are then
zero q values.
required : (a) the exact Green function is replaced by
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