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A SIMPLIFIED METHOD FOR COLLECTIVE EXCITATIONS IN REALISTIC SYSTEMS
B. Johansson, G. Arbman
To cite this version:
B. Johansson, G. Arbman. A SIMPLIFIED METHOD FOR COLLECTIVE EXCITATIONS IN REALISTIC SYSTEMS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-95-C3-97.
�10.1051/jphyscol:1972313�. �jpa-00215047�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-95
A SIMPLIFIED METHOD FOR COLLECTIVE EXCITATIONS IN REALISTIC SYSTEMS
B. JOHANSSON and G. ARBMAN FOA 4, Stockholm 80, Sweden
R6sum6. - Une methode simplifik pour le calcul &energies d'excitation collectives est dkve- loppCe pour des systBmes rkalistes de fermions. Comme test prkliminaire de cette mbthode, on a effectuC des calculs sur le gaz dY6lectrons avec des rksultats encourageants.
Abstract. - A simplified method for calculations of collective excitation energies in realistic many-fermion systems is developed. As a preliminary test of the method calculations were per- formed on the electron gas with encouraging results.
We propose a simplification in the correct treatment of collective excitations within the RPA (I), that consi- derably reduces the computational efforts involved in calculations of realistic systems. The procedure should be regarded as a zeroth order approximation, but we expect it to be useful for preliminary investiga- tions. Good agreement with correct RPA-results is obtained where these can be found (i. e. for the elec- tron gas). For realistic systems we obtain rather simple integrands over limited regions, and we believe calculations with our method to be feasible with the present generation of computers.
We will in this paper specify our treatment to plas- mons, which we believe is the most immediate appli- cation of our method. There exists a considerable amount of experimental data on plasmons [I]-[3], whereas the cr first-principle )) theoretical treatment has so far mainly been limited to the free-electron-gas and applications thereof to various crystals. The main limitation of these expressions is, of course, the assump- tion of a plane waves instead of the true one-electron wave-functions of realistic systems in the matrix elements involved. Finally, although we shall not deal with it here, our expression is a tool to investi- gate phase transitions [4].
I. Outline of method. - We assume the ground state wave function $, to be known. From this wave function an excited state (of collective nature) is defined by
*ill = A: $0 (1)
where A: operates on the ground state wave function and is given by
A: = C x$ a; bl + yzi b, a,, .
m.l
(1) With the RPA we will mean the time-dependent Hartree- Fock approximation. It should be noticed that in the case of the electron gas the RPA usually refers to the timedependent Hartree approximation and the notion of generalized RPA 1,
to the time-dependent Hartree-Fock approximation, i. e. with exchange effects included.
Here a + and b+ are creation operators for particles (above the Fermi surface) and holes (below the F. S.), respectively. Hence, this operator gives a superposi- tion of particle-hole pairs with amplitudes ~ 4 , ~ and y:i. Our simplification now consists of replacing these
quantities by m- and i-independent x, and y,, giving a cr restrictive >> collective excitation operator (').
Furthermore normalization gives
The energy wZOn is now obtained by minimizing the expression
~ : 0 1 1 = < $, I [A,, [H,A:ll- ' (4)
with respect to x, (or y,) using (3), within the usual RPA linearization procedure.
In so doing, we obtain
where
d = (8, - zi) + C [ ( j m I V I ni) - Cjm I V I in)]
m,i m,i n,j
(6)
The states to be summed (integrated) over now simply consist of those within the non-overlapping regions of two Fermi surfaces displaced by a vector q, as is shown in figure 1. In the limit q + 0, the intraband summation runs over states at the Fermi surface
(2) For assuming state independent amplitudes x, and y, in a crystal we have in mind intraband contributions to the plas- mon energy. When interband contributions cannot be neglected, the theory can easily be modified, by choosing different cons- tant xg and y4)s for different bands.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972313
C3-96 B. JOHANSSON AND G. ARBMAN
FIG. 1. - The shaded region in k-space indicates contributing hole states (below the F. S.) and particle states (above the F. S.).
only. An interesting point to notice is that when
( e [ > ( a' (, the excitation energy becomes purely
imaginary indicating a phase transition at I e I = I d 1.
11. Computational aspect. - By comparing our eq. (5)-(8) to an exact generalized RPA-expression for the plasmon energy, one finds that we have essentially achieved a decoupling of this quite complex system of coupled eq. [5]. The main remaining computational difficulty is then the evaluation of the multicenter integrals (jm I V I ni), where V is the bare Coulomb interaction ( r, - r2 I -'. The one-particle energies ei as well as the wave functions and the Fermi surface are assumed to be known from a previous calculation.
In principle, these quantities could be taken as those calculated with a local potential (e. g. of Hartree- Fock-Slater type). This would still be cr better )>
than an ordinary RPA-calculation, as the exchange is taken into account both in the one-particle quantities (albeit in an approximate form) and in the plasmon energies. In practice, however, we believe that the calculations should be performed in connection with a true Hartree-Fock evaluation of the band electron eigenstates and wave functions. The reason for this is that the numerical techniques involved in calculat- ing the multicenter integrals [6]-[8] are very similar to those involved in calculating the H. F. exchange part correctly. In the small q-limit, which is the region of interest for plasmons, we anticipate additional simpli- fication due to the fact that the intraband states involved are in a narrow region in k-space and hence differ little in energy. In particular, one should be able to do the integration in configuration space only once and use the stored integrals for all states in this energy region. To illustrate this, we express the band elec- tron wave functions in a form used by Abrikosov [9]
and Harris and Monkhorst [6]
M
$k = 1 Ci(k) I ki >
i = 1 (8)
I ki > = exp(2 nia - k. r) Qi(r - ap) . (9)
I(
The Slater-type orbitals (STO) cDi (i = 1,2, ..., M) are centered at each lattice site. For the narrow energy range associated with small q's, one set of optimized STO's should certainly be sufficient to give a good representation of all Bloch states in this energy range.
Following Harris and Monkhorst [6] one would then only have to calculate and store integrals of the type Qij(Q) = < Qi(r) I exp(2 xis-' Q. r) I Qj(r - ap) >
P
(10) where Q is a reciprocal lattice vector and p denotes a sum over lattice sites. The remaining part then consists of summing products Qij(Qp) @,,(Qv) multiplied by the inverse product of reciprocal vectors of increasing length and appropriate coefficients Ci(k)'s. Details of such a calculation can easily be deduced from the formalism presented in reference [6], where also a technique to evaluate the aij(Q)'s can be found.
111. Test results. - In order to test the validity of our assumption of state independent amplitudes, we have calculated the plasmon dispersion relation for the electron gas within the ordinary RPA (i. e. no exchange effects included) in the small-q limit. The exact RPA-result in this case reads
whereas we obtain the result
with cop, given as
From this we see that in the limit q = 0, the two expres- sions agree exactly, whereas for q f 0 our expression gives somewhat lower energies. Including exchange effects considerably complicates the calculation. As far as we know, no (( exact )) explicit generalized RPA-expression in the small-q limit has been published for the electron gas although there exist approximative ones [IOJ. Within our scheme, however, it can be rigourously shown [4] that the inclusion of exchange has the effect of lowering the plasmon energies. We believe this result to carry over not only to the exact generalized RPA-treatment of the electron gas, but also to. realistic systems like simple metals. Other results, primarily connected with phase transitions, can be found in reference [4]. Of these we would like t o mention the fact that the ferromagnetic H. F.-state of the electron gas is more unstable against density oscillations than the paramagnetic one.
IV. Conclusions. - We have derived a first-prin- ciple explicit dispersion relation for plasmons, valid within the generalized RPA. The only additional
A SIMPLIFIED METHOD FOR COLLECTIVE EXCITATIONS C3-97 approximation involved is the assumption of state-
independent amplitudes when describing the plasmon as a superposition of particle-hole pairs. If the energy differences between particle states and hole states for the main contributing particle-hole pairs are about the same, our additional approximation should be quite good. If they are not, the theory could easily be modified to include different constant amplitudes for a few different energy regions, and would still be useful.
The computational efforts involved for a crystal seem to be of the order of magnitude of a true H. F, calculation. Since this has recently been completed for simple (realistic) systems like metallic hydrogen and
lithium [Ill, we feel that for these cases, at least, dispersion relations could be calculated using our method. More interesting <( simple D systems (like A1 and K) should likewise be accessible to similar calcu- lations in the near future. However, since at the moment we don't have access to programs that can evaluate multicenter integrals in crystals, we are limited to the electron gas in our test calculations. These tests show that our expressions give results that agree rather well with those of exact RPA calculations. Finally, we feel that our expressions could serve as a starting point for further simplifying approximations, thus extending the region of applicability to more compli- cated systems as well.
References
[I] STEINMANN, Phys. Stat. Sol., 1968, 28, 437 and refe- [6] HARRIS (F. E.) and MONKHORST (H. J.), << Computa- rences therein. tional Methods in Band Theory >>, Plenum Press, [2] BAGACHI, DUKE, FEIBELMAN and PORTENS, Phys. Rev. New York-London, 1971.
Lett., 1971, 27, 998. [7] ELLIS (D. E.), Int. J. Quantum Chem., 1968, 2, 35.
[8] CALAIS (J.-L.), Arkiv for fysik, 1964, 28, 539.
[3] NILSSON (P. 0.)' Physica Scripta, 1971, 3, 87. [9] ABRIKOSOV (A. A.), U. S. S. R. Astronomical J., 1954, [4] JOHANSSON (B.), to be published in Annals of Physics. 31,112.
[lo] HUBBARD (J.), Proc. Roy. Soc. 1957, A 243, 336.
[5] BROWN (G.), << Unzzed Theory of Nuclear Models >>, [11] HARRIS (F. E.) and MONKHORST (H. J.), (this Procee- North-Holland publishing Co, Amsterdam, 1964. dings).
