Hellmann-Feynman forces - Revision history

Huebsch: Huebsch moved page Forces to Hellmann-Feynman forces without leaving a redirect

2022-04-08T13:46:26Z

Huebsch moved page Forces to Hellmann-Feynman forces without leaving a redirect

← Older revision	Revision as of 13:46, 8 April 2022
(No difference)

Huebsch at 08:02, 7 April 2022

2022-04-07T08:02:29Z

← Older revision		Revision as of 08:02, 7 April 2022
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	[[Category:~~Structural Optimization~~]][[Category:Forces]][[Category:Theory]]		[[Category:Ionic minimization]][[Category:Forces]][[Category:Theory]]

Miranda.henrique at 14:39, 19 February 2021

2021-02-19T14:39:37Z

Karsai: 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 $F$ ..."

2019-03-19T14:16:46Z

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>..."

New page

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> depends on the
wavefunctions <math>\phi</math>, the partial occupancies <math>f</math>, and the positions of the
ions <math>R</math>. 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 <math>\phi</math> and the set of partial occupancies as <math>f</math>.

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

<math>
0 = \delta F(\phi,f,R)
</math>

for arbitrary variations of <math>\phi</math> and <math>f</math>. We can rewrite the right hand side of
this equation as

<math>
\frac{\partial F}{\partial \phi} \delta \phi +
\frac{\partial F}{\partial f} \delta f.
</math>

For arbitrary variations this quantity is zero only if <math>\frac{\partial F}{\partial \phi}=0</math>
and <math>\frac{\partial F}{\partial f}=0</math>, 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.

<math>
\mbox{force} = \frac{d F(\phi,f,R)} {d R} =
\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} .
</math>

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

<math>
\mbox{force}= \frac{d F(\phi,f,R)} {d R} = \frac{\partial F}{\partial R}
</math>

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 <math>\sigma \to 0</math>. It is in principle possible to evaluate
the derivatives of <math>E(\sigma \to 0)</math> with respect to the ionic
coordinates but this is not easy and requires additional computer time.

----
[[Category:Structural Optimization]][[Category:Forces]][[Category:Theory]]

← Older revision		Revision as of 14:39, 19 February 2021
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	Within the finite temperature LDA forces are defined as the derivative		Within the finite temperature, LDA forces are defined as the derivative
	of the generalized ''free energy''.		of the generalized ''free energy''.
	This quantity can be evaluated easily. The functional <math>F</math> depends on the		This quantity can be evaluated easily. The functional <math>F</math> depends on the
	wavefunctions <math>\phi</math>, the partial occupancies <math>f</math>, and the positions of the		wavefunctions <math>\phi</math>, the partial occupancies <math>f</math>, and the positions of the
	ions <math>R</math>. In this section we will shortly discuss the variational		ions <math>R</math>. In this section, we will shortly discuss the variational
	properties of the free energy and we will explain why we calculate the		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		forces as a derivative of the free energy. The formulas given are
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	For arbitrary variations this quantity is zero only if <math>\frac{\partial F}{\partial \phi}=0</math>		For arbitrary variations this quantity is zero only if <math>\frac{\partial F}{\partial \phi}=0</math>
	and <math>\frac{\partial F}{\partial f}=0</math>, leading to a system of		and <math>\frac{\partial F}{\partial f}=0</math>, leading to a system of
	equations which determines $\phi$ and $f$ at the electronic groundstate.		equations which determines <math>\phi</math> and <math>f</math> at the electronic groundstate.
	We define the forces as derivatives of the free energy		We define the forces as derivatives of the free energy
	with respect to the ionic positions i.e.		with respect to the ionic positions i.e.
Line 44:		Line 44:
	</math>		</math>

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

Hellmann-Feynman forces - Revision history

Huebsch: Huebsch moved page Forces to Hellmann-Feynman forces without leaving a redirect

Huebsch at 08:02, 7 April 2022

Miranda.henrique at 14:39, 19 February 2021

Karsai: 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 F..."

Karsai: 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 $F$ ..."