Hellmann-Feynman forces

From VASP Wiki
Revision as of 14:16, 19 March 2019 by Karsai (talk | contribs) (Created page with "Within the finite temperature LDA forces are defined as the derivative of the generalized ''free energy''. This quantity can be evaluated easily. The functional <math>F</math>...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Within the finite temperature LDA forces are defined as the derivative of the generalized free energy. This quantity can be evaluated easily. The functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): F depends on the wavefunctions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \phi , the partial occupancies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f , and the positions of the ions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): R . In this section we will shortly discuss the variational properties of the free energy and we will explain why we calculate the forces as a derivative of the free energy. The formulas given are very symbolic and we do not take into account any constraints on the occupation numbers or the wavefunctions. We denote the whole set of wavefunctions as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \phi and the set of partial occupancies as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f .

The electronic groundstate is determined by the variational property of the free energy i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 0=\delta F(\phi ,f,R)

for arbitrary variations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \phi and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f . We can rewrite the right hand side of this equation as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {\partial F}{\partial \phi }}\delta \phi +{\frac {\partial F}{\partial f}}\delta f.

For arbitrary variations this quantity is zero only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {\partial F}{\partial \phi }}=0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\frac {\partial F}{\partial f}}=0 , leading to a system of equations which determines $\phi$ and $f$ at the electronic groundstate. We define the forces as derivatives of the free energy with respect to the ionic positions i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mbox{force}}={\frac {dF(\phi ,f,R)}{dR}}={\frac {\partial F}{\partial \phi }}{\frac {\partial \phi }{\partial R}}+{\frac {\partial F}{\partial f}}{\frac {\partial f}{\partial R}}+{\frac {\partial F}{\partial R}}.

At the groundstate the first two terms are zero and we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mbox{force}}={\frac {dF(\phi ,f,R)}{dR}}={\frac {\partial F}{\partial R}}

i.e. we can keep $\phi$ and $f$ fixed at their respective groundstate values and we have to calculate the partial derivative of the free energy with respect to the ionic positions only. This is relatively easy task.

Previously we have mentioned that the only physical quantity is the energy for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \sigma \to 0 . It is in principle possible to evaluate the derivatives of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): E(\sigma \to 0) with respect to the ionic coordinates but this is not easy and requires additional computer time.

Retrieved from "https://www.vasp.at/wiki/index.php?title=Hellmann-Feynman_forces&oldid=7332"