Pseudohamiltonians and Quantum Monte Carlo
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PSEUDOHAMILTONIANS AND QUANTUM MONTE CARLO A. Bosin * , V. Fiorentini * ** , A. Lastri t and G. B. Bachelet t *Dipartimento di Scienze Fisiche, Universit6 di Cagliari, 09124 Cagliari, ITALY "•Fritz-Haber-Institut der Max-Planck-Gesellschaft, W-1 Berlin 33, GERMANY tDipartimento di Fisica, Universiti di Trento, 38050 Povo (TN), ITALY t Dipartimento di Fisica, Universitk di Roma "La Sapienza", 00185 Roma, ITALY
ABSTRACT A new method is presented for the generation of valence-only local hamiltonians, or pseudohamiltonians, within the DFT-LDA framework. At the moment these promise to be very useful tools to overcome the problem of eliminating the core electrons from many QMC calculations. INTRODUCTION The need for the introduction of a local valence-only hamiltonian operator comes from the practical limitations that face one when using Quantum Monte Carlo (QMC) techniques[1, 31. The strong dependence of the computational time on the atomic number Z of the atoms involved, which has been estimated to be proportional to Z` 5 - Z6_5 for the Green's Function QMC[4], sets an upper limit on the number of the electrons that can be accounted for, making all-electron QMC simulations feasible only for few-electron systems. Many techniques have been developed to get rid of core electrons in QMC simulations[l, 2] but we will concentrate here on the method of Bachelet et al.[2] (BCC). The elimination of core electrons via effective or pseudo-potentials was successfully accomplished long ago[5] in connection with solid state calculations and in recent times the most relevant developement appears to be the "ab-initio", non-local norm-conserving pseudopotential (NCPP) by Hamann, Schliter and Chiang[6] (HSC). While a straightforward use of such a non-local hamiltonian is not possible in connection with many QMC techniques[1], a generalized local electron-ion hamiltonian can sometimes replace the non-local NCPP while keeping most of the beauty of the HSC. BCC have shown that the most general form for a valenceonly local hermitian operator, bounded from below, good for fixed-nodes QMC[7] and of spherical symmetry (like the atomic full-core hamiltonian it replaces), is the following
pseudohamiltonian[2] (PH): fips
=
-1V2+ Vrps(R) + U(R)
Vps (R)
=
-~Via(ri)Vi +
2
U(R) = Zv,o.(r, +
(1)
~I~2
(2)
1
(3)
where i spans the valence electrons, a(r), b(r) and vo, 0 (r) are radial functions that satisfy[2]:
Sa(r) + 1 > 0 I.a(r) + b(r) + 1 > 0
Vr >0
Mat. Res. Soc. Symp. Proc. Vol. 291. 01993 Materials Research Society
(4)
22
b(r) =0 v, 0.(r)
w > ro>(5) Z
-
a(r) = 0
r
where r, is the core radius, Z,, = Z - N, and N, is the number of core electrons. Unfortunately a(r), b(r) and v, 0,(r) must be determined simultaneously and undergo the additional conditions of Eq. 4, which have no equivalent in the HSC theory, so the simple procedure of HSC must be replaced by a complex non linear optimization scheme[2][8, 10]. Up to now PH have been generated in different ways[2][8, 10] the most satisfactory being those of references [9, 10].
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