Quantum chemical Green’s function approach to
correlation in solids
Christian Buth, Viktor Bezugly, Uwe Birkenheuer
One-particle excitations, which describe the energetics of an infinite system after
removal or adding of a single electron, are the most fundamental excitations in
solids. They provide the band structure of periodic compounds and also the
fundamental band gap of crystalline materials.
The one-particle excitations are given by the many-body wave functions of the
excited (N –1)- and (N +1)-particle eigenstates of the system under consideration. Usually there is a close relation between these correlated wave functions
and the single-determinant one-particle configurations
Φa (−k) = ĉψak ΦHF
and Φr (k) = ĉ†ψrk ΦHF
(1)
where ψak and ψrk are occupied and unoccupied Bloch orbitals, respectively,
and ΦHF is the Hartree-Fock reference determinant of the neutral N -particle
system. This way it is possible to assign band (and spin) indices a and r to the
correlated (N –1)- and (N +1)-particle states of a periodic system and to arrive
at a ’correlated’ band structure.
The local Hamiltonian approach is an efficient way to calculated such a band
structure by means of quantum chemical post-Hartree-Fock correlation methods
(for details see separate report). Yet, often, there also exist (N –1)- and (N +1)particle eigenstates in condensed matter which are dominated by three-particle
(or more) configurations such as the (2h,1p)-configurations ĉ†ψrk ĉψak0 ĉψbk00 ΦHF .
They give rise to so-called satellite peaks in photoemission spectra. Preferentially, they occur in small band gap systems, where the energy to create an additional electron-hole pair is relatively small. They also show up quite frequently
in conjunction with inner-valence hole states or highly excited (N +1)-particle
states. No band indices can be assigned to such states, and as soon as these satellites states start to interfere heavily with the band structure generating
’one-particle’ states describes above, the one-particle (or band structure) picture
of a solid breaks down.
The local Hamiltonian approach is not able to describe satellite states. Therefore we have developed, as an alternative, a quantum chemical ab initio Green’s
function formalism for infinite periodic systems. It is based on the algebraic
diagrammatic construction (ADC) which is well-established for molecular systems [1–3]. It sets out from the following analytical ansatz
Σ(ω)
= Σ∞ + M + (ω) + M − (ω)
with
†
M ± (ω) = U ± (ω11 − C ± )−1 U ±
(2)
(3)
for the self-energy Σ(ω) and determines the unknown quantities Σ∞ , U ± , and
C ± by expanding them in powers of the residual interaction V from Ĥ = Ĥ0 +V
1
and comparing the various contributions with the corresponding Feynman diagrams for the self-energy. The resulting scheme is termed ADC(n), where n is
the order in V up to which the ADC ansatz recovers the diagrammatic series
for Σ(ω) completely. Yet, in fact, the above ansatzes for M ± (ω) go much further because they effectively establish an infinite summation of proper Feynman
diagrams in addition to the infinite summation of Feynman diagrams already
provided by the Dyson equation. Another important point about the ADC formalism is, that it allows to replace the numerically cumbersome pole search in
−1
the Green’s function G(ω) = G−1
(which gives the excitation
0 (ω) − Σ(ω)
energies) by an eigenvalue problem Bx = ωx with

†
† 
H0 + Σ ∞ U + U −

B =  U+
(4)
C+ 0
−
−
U
0
C
which can be solved very robustly by block-Lanczos diagonalization [4, 5].
We have developed a new variant of the ADC formalism which allows to perform such ADC calculations for infinite periodic systems. This cannot be done
by simply switching from canonical Hartree-Fock orbitals to Bloch orbitals, because the resulting matrices are simply too large. Localized orbitals had to be
introduced together with suitable configuration selection schemes which take
fully into account the translational symmetry of the compounds, the compactness of the localized orbitals and the pre-dominantly short-range character of
electron correlation [6]. We have implemented this scheme starting from the
WANNIER code (which has been developed at the mpipks some years ago [7])
in a strictly quantum chemical way, using flexible Gaussian-type basis functions
and being ab initio in the sense that we use the full non-relativistic many-body
Hamiltonian of our system.
correlation included
local Hamiltonian
-16
CO-ADC(2)
E (eV)
-18
Hartree-Fock
HF chain
-20
-22
H
F
-24 L
k
[
F
H
H
F
]
F
∞
X
Abbildung 1: Band structure of infinite HF zig-zag chains.
The program is operating, and one of the first applications was a comparison of the valence band structure of infinite hydrogen fluoride (HF) chains
2
H
in zig-zag arrangement (like in crystalline HF) with the results obtained by
the local Hamiltonian approach [8]. The outer-most three F 2p-like valence
bands have been chosen for that purpose and a standard cc-pVDZ basis set
is employed. The local Hamiltonian matrix elements are extracted from MRCI
(multi-reference configuration interaction) calculations with all (1h)-, (2h,1p)and (3h,2p)-configurations being included in the CI space (like in previous studies [9]). The result of these comparison is shown in Fig. 1. The correlated
energy bands coincide within 0.0-0.2 eV which impressively demonstrates the
ability of the new crystal-orbital-ADC(2) formalism. In particular it shows that
the perturbative treatment of the off-diagonal matrix elements of Ĥ0 which is
inherent to our new CO-ADC technique is well justified.
30
25
bulk LiF
20
E (eV)
15
10
correlation included
5
CO-ADC(2)
0
Hartree-Fock
-5
L
L
X
W
K
L
Abbildung 2: Band structure of bulk LiF using 19 unit cells as support for the
local three-particle configurations.
Bulk systems can be describes by our CO-ADC program as well, as can be seen
from our study on LiF, a rock-salt-like ionic crystal. It mainly served to check the
required extent of the local three-particle configurations entering the auxiliary
matrices U ± and C ± . Inclusion of all configurations with localized orbitals from
up to the 2nd nearest-neighbor unit cell (19 cells all together) were necessary to
produce the sound upward shift of about 3 eV of the valence bands discernible
in the correlated band structure of LiF (see Fig. 2). Of course, conduction bands
can be handled by our Green’s function approach as well, as is corroborated by
the unoccupied band occurring at 15-30 eV.
The new CO-ADC formalism is a very promising scheme, and we are currently
working on improving both, the formalism itself and its implementation. In
particular, we plan to link the CO-ADC code to a more advanced periodic
quantum chemical program package such as the CRYSTAL code [10].
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Literatur
[1] J. Schirmer, Phys. Rev. A 26 (1982) 2395.
[2] J. Schirmer, L. S. Cederbaum, O. Walter, Phys. Rev. A 28 (1983) 1237.
[3] L. S. Cederbaum, in Encyclopedia of Computational Chemistry, Vol. 2, ed.
P. V. R. Schleyer, (John Wiley & Sons, Chichester, New York, 1998), p. 1202.
[4] C. Lanczos, J. Res. Nat. Bur. Stand. 45 (1950) 255.
[5] H.-D. Meyer, S. Pal, J. Chem. Phys. 91 (1989) 6195.
[6] C. Buth, U. Birkenheuer, M. Albrecht, P. Fulde, Phys. Rev. B, submitted –
arXiv:cond-mat/0409078.
[7] A. Shukla, M. Dolg, P. Fulde, H. Stoll, Phys. Rev. B 57 (1998) 1471.
[8] C. Buth, V. Bezugly, M. Albrecht, U. Birkenheuer, in preparation.
[9] V. Bezugly, U. Birkenheuer, Chem. Phys. Lett. 399 (2004) 57-61.
[10] V. R. Saunders, R. Dovesi, C. Roetti, R. Orlando, C. M. Zicovich-Wilson,
N. M. Harrison, K. Doll, B. Civalleri, I. J. Bush, P. D’Arco, M. Llunell, CRYSTAL
2003 User’s Manual (Universitá di Torino, Torino, 2003).
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