Full Potential Electronic Structure Calculations and the Concept of Stress Fields and Energy Densities for Total Energy
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FULL POTENTIAL ELECTRONIC STRUCTURE CALCULATIONS AND THE CONCEPT OF STRESS FIELDS AND ENERGY DENSITIES FOR TOTAL ENERGY CALCULATIONS
P. Ziesche Technische Universitit Dresden, Institut ffir Theoretische Physik, Mommsenstr. 13, 0-8027 Dresden, Germany
ABSTRACT With the availability of reliable full atomic cell orbitals the possibility arises to calculate pressure or stress, restoring or relaxation driving Hellmann -Feynman forces, and total energies (especially of defects) alternatively and directly via stress tensor fields and energy densities, two local quantities. Although quantum mechanical stress field and energy density can not be defined uniquely, there is a recent interest in these quantities, because integrals with physical meaning are gauge invariant. The mentioned fields can be defined (i) for the full many-body description with the exact one-particle density matrix and pair distribution function as well as (ii) for the Kohn-Sham one-particle description with LDA or beyond (gradient expansion approximation). If the local stress field for a special system once is constructed, then the global stress tensor and /or forces on nuclei can be calculated via the stress theorem and the force theorem by means of unit cell surface integrals. The energy density can be derived from the terms of the stress field by taking the trace and can be used to calculate defect energies without bothering about the thermodynamic limit. SYSTEM, CONCEPT, AND THEOREMS Quantum mechanical many-body systems with Coulomb interaction and within BornOppenheimer approximation (nuclear charges Z, at nuclear sites RB) in the ground state are considered. The concept of quantum mechanical
"* momentum flux
density or local stress (tensor) or stress (tensor) field, denoted by
Q), [1 - 4] and (g
"* energy density, denoted by
e(_r), [4,5]
is used. The stress field a(_) and energy density e(_r) are defined in such a way that stress theorem, force theorem, and energy theorem hold [3,4]. The stress theorem is a generalized virial theorem and relates for finite clusters (cluster volume f9, cluster energy E)
E R
o ._._E =
d3r
or extended crystals (lattice translation vectors ai, unit cell volume Qo, bulk energy cb) 1
-1
0 ab
d3r
S0o-(
Mat. Res. Soc. Symp. Proc. Vol. 253. Ic)1992 Materials Research Society
(2
86
the global stress a (l.h.s.) to the averaged local stress (r.h.s.) [3,4]. An identical rewriting of Eq. (2) is [6]
a
dS a()o (E&).
(3)
Here we assume that there are several nuclei in the unit cell with positions R_. Thus we can think the unit cell Po to consist of atomic cells 01 with surfaces S1. The force theorem is a momentum balance [3,4] in local version Oa(_)=_(r_),
•F_16(1: - _R__)
(4)
Li = Zi[E(r) - fl(r)]•.
(5)
_fj_) =
or integral version
# dS g:()
=
l,
S,
f(r)
is the Hellmann-Feynman force density, F is the Hellmann-Feynman force on the
l-th nucleus, E(r) is the electric field (times lej) caused by the nuclei and the electrons, El(zr) is the field of the l-th nucleus. S, is the surface of a volume c
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