Sampling phonon spectra from molecular-dynamics simulations - Revision history

Liebetreu: Reverted edits by Liebetreu (talk) to last revision by Jona

2026-04-15T12:27:04Z

Reverted edits by Liebetreu (talk) to last revision by Jona

Liebetreu: /* Step 4: Compute the power spectrum for each normalized velocity autocorrelation function */

2026-04-14T14:50:50Z

Step 4: Compute the power spectrum for each normalized velocity autocorrelation function

← Older revision		Revision as of 14:50, 14 April 2026
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	The phonon spectral function is the power spectrum of <math>f_{s}(t)</math> and is obtained by performing the following Fourier transformation:		The phonon spectral function is the power spectrum of <math>f_{s}(t)</math> and is obtained by performing the following Fourier transformation:

	<math>		<math>
	g(\omega)=\sum_{s=1}^{types}g_{s}(\omega)=\left\| \sum_{s=1}^{types}\int_{-\infty}^{\infty}f_{s}(t)e^{-i\omega t}\right\|^{2}.		g(\omega)=\sum_{s=1}^{types}g_{s}(\omega)=\left\| \sum_{s=1}^{types}\int_{-\infty}^{\infty}f_{s}(t)e^{-i\omega t}\right\|^{2}.

Liebetreu: /* Step 3: Compute normalized velocity autocorrelation function for each NVE simulation */

2026-04-14T14:50:25Z

Step 3: Compute normalized velocity autocorrelation function for each NVE simulation

← Older revision		Revision as of 14:50, 14 April 2026
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	The '''normalized velocity autocorrelation function''' for an <math>N</math>-particle system is given by		The '''normalized velocity autocorrelation function''' for an <math>N</math>-particle system is given by

	<math>		<math>
	f(t)=\sum_{s=1}^{types}f_{s}(t)=\frac{\langle \sum_{s=1}^{types}\sum_{i=1}^{N_{s}}\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T+t) \rangle}{\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T)}.		f(t)=\sum_{s=1}^{types}f_{s}(t)=\frac{\langle \sum_{s=1}^{types}\sum_{i=1}^{N_{s}}\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T+t) \rangle}{\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T)}.
	</math>		</math>

	The brackets <math>\langle ,\rangle</math> denote a thermal average which has to be computed over different MD trajectories and starting times <math>\Delta T</math> within each trajectory. The sum over <math>i</math> runs over the atoms within each species, and the sum <math>s</math> is over all atomic species contained in the simulated system.		The brackets <math>\langle ,\rangle</math> denote a thermal average which has to be computed over different MD trajectories and starting times <math>\Delta T</math> within each trajectory. The sum over <math>i</math> runs over the atoms within each species, and the sum <math>s</math> is over all atomic species contained in the simulated system.

Liebetreu at 14:49, 14 April 2026

2026-04-14T14:49:41Z

← Older revision		Revision as of 14:49, 14 April 2026
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	[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]		[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]
	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of <~~math~~>~~CsPbBr_3~~</~~math~~> which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr<sub>3</sub> which is discussed in more detail below.

	In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.		In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.
	<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->		<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->

	== Phonon spectra step-by-~~setp~~ ==		== Phonon spectra step-by-step ==

	For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:		For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:

Liebetreu at 14:49, 14 April 2026

2026-04-14T14:49:16Z

Jona at 10:12, 8 November 2025

2025-11-08T10:12:20Z

← Older revision		Revision as of 10:12, 8 November 2025
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	[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon ~~spectral function~~.]]		[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]
	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.

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	For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.		For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.

	[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr$_{3}$ at 500K. '''Right:''' Atom-resolved phonon ~~spectra~~ for CsPbBr$_{3}$ at 500K.]]		[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr$_{3}$ at 500K. '''Right:''' Atom-resolved phonon DOS for CsPbBr$_{3}$ at 500K.]]

Csheldon: /* Anharmonic ratteling in CsPbBr$_{3}$ */

2025-11-06T14:27:41Z

Anharmonic ratteling in CsPbBr$_{3}$

← Older revision		Revision as of 14:27, 6 November 2025
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	Fig. 3 shows the atom-resolved normalized autocorrelations and phonon spectra.		Fig. 3 shows the atom-resolved normalized autocorrelations and phonon spectra.
	A peak in the Cs$^{+}$ cation phonon ~~stectral~~ function is visible around 1THz. This peak can be assigned to Cs$^{+}$ rattling frequencies coupling to optical phonon modes formed by the oscillations of the lead bromide framework.		A peak in the Cs$^{+}$ cation phonon spectral function is visible around 1THz. This peak can be assigned to Cs$^{+}$ rattling frequencies coupling to optical phonon modes formed by the oscillations of the lead bromide framework.

	For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.		For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.

Csheldon at 14:06, 6 November 2025

2025-11-06T14:06:37Z

← Older revision		Revision as of 14:06, 6 November 2025
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	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.

	In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on ~~maping~~ to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.		In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.
	<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->		<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->

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	<references/>		<references/>
	<noinclude>		<noinclude>

	==Related tags and articles==		==Related tags and articles==
	[[Molecular dynamics calculations\|Molecular-dynamics calculations]],		[[Molecular dynamics calculations\|Molecular-dynamics calculations]],

Huebsch: /* Step 5: Compute averages and check for convergence */

2025-10-22T07:23:09Z

Step 5: Compute averages and check for convergence

← Older revision		Revision as of 07:23, 22 October 2025
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	To check for convergence, $f(t)$ and $g(\omega)$ obtained for each [[NVE ensemble\|NVE trajectory]] can be successively averaged. To this end, plot a single trajectory, compared to an average over 2 trajectories, and so on. If needed, the above steps can be repeated to generate additional data to reach the desired accuracy.		To check for convergence, $f(t)$ and $g(\omega)$ obtained for each [[NVE ensemble\|NVE trajectory]] can be successively averaged. To this end, plot a single trajectory, compared to an average over 2 trajectories, and so on. If needed, the above steps can be repeated to generate additional data to reach the desired accuracy.
	{{NB\|tip\|For further information on phonon signal analysis Ref{{cite\|lahnsteiner:prb:2022}} might be a helpful source.}}		{{NB\|tip\|For further information on phonon signal analysis Ref{{cite\|lahnsteiner:prb:2022}} might be a helpful source.}}
	~~{{NB\|mind\|Rotational modes as well as drifts show up as spectral weight at $\omega${{=}}$0$.}}~~

	== Example ==		== Example ==

Huebsch at 06:18, 22 October 2025

2025-10-22T06:18:55Z

← Older revision		Revision as of 06:18, 22 October 2025
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	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.

	In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on maping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence. It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.		In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on maping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.
			<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->

	== Phonon spectra step-by-setp ==		== Phonon spectra step-by-setp ==

← Older revision		Revision as of 14:49, 14 April 2026
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	[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]		[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]
	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of <~~math~~>~~CsPbBr_3~~</~~math~~> which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr<sub>3</sub> which is discussed in more detail below.

	In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.		In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.
	<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->		<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->

	== Phonon spectra step-by-~~setp~~ ==		== Phonon spectra step-by-step ==

	For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:		For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:

← Older revision		Revision as of 10:12, 8 November 2025
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	[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon ~~spectral function~~.]]		[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]
	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.

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	For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.		For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.

	[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr$_{3}$ at 500K. '''Right:''' Atom-resolved phonon ~~spectra~~ for CsPbBr$_{3}$ at 500K.]]		[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr$_{3}$ at 500K. '''Right:''' Atom-resolved phonon DOS for CsPbBr$_{3}$ at 500K.]]

← Older revision		Revision as of 12:27, 15 April 2026
Line 1:		Line 1:
	[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]		[[File:PhononDOS.png\|500px\|thumb\|Fig. 1: '''Left:''' Shows convergence analysis of normalized velocity-autocorrelation function. '''Right:''' Convergence analysis of phonon DOS.]]
	[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr~~<sub>3</sub>~~ which is discussed in more detail below.		[[:Category:Phonons\|Phonon]] spectra can be obtained as the power spectrum of the normalized velocity-autocorrelation function {{cite\|reissland:book:1973}}{{cite\|lahnsteiner:prb:2022}}. The velocities of the ions and hence the velocity-autocorrelation function are recorded during a [[:Category:Molecular dynamics\|molecular dynamics (MD) simulation]]. Fig. 1 shows the example of CsPbBr$_3$ which is discussed in more detail below.

	In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.		In contrast to [[Computing the phonon dispersion and DOS\|the phonon DOS computed by Fourier interpolation of the force-constant matrix]], analyzing the power spectrum does not rely on mapping to a model Hamiltonian. It naturally accounts for anharmonic contributions, as well as temperature dependence.
	<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->		<!---It is also possible to obtain a [[Phonons from fitting the force-constant matrix to molecular dynamics\|force-constant matrix based on MD trajectories]] via fitting the effective potential energy surface which has the advantage of providing direct access to the phonon dispersion.--->

	== Phonon spectra step-by-~~step~~ ==		== Phonon spectra step-by-setp ==

	For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:		For the setup of the [[Molecular dynamics calculations\|MD simulation]] and choice of [[:Category:Ensembles\|ensemble]], two aspects need to be taken into account:
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	=== Step 3: Compute normalized velocity autocorrelation function for each NVE simulation ===		=== Step 3: Compute normalized velocity autocorrelation function for each NVE simulation ===

	The '''normalized velocity autocorrelation function''' for an ~~<math>~~N~~</math>~~-particle system is given by		The '''normalized velocity autocorrelation function''' for an $N$-particle system is given by
			\begin{equation}
	~~<math>~~
	f(t)=\sum_{s=1}^{types}f_{s}(t)=\frac{\langle \sum_{s=1}^{types}\sum_{i=1}^{N_{s}}\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T+t) \rangle}{\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T)}.		f(t)=\sum_{s=1}^{types}f_{s}(t)=\frac{\langle \sum_{s=1}^{types}\sum_{i=1}^{N_{s}}\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T+t) \rangle}{\mathbf{v}_{i}(\Delta T)\mathbf{v}_{i}(\Delta T)}.
	~~</math>~~		\end{equation}
			The brackets $\langle ,\rangle$ denote a thermal average which has to be computed over different MD trajectories and starting times $\Delta T$ within each trajectory. The sum over $i$ runs over the atoms within each species, and the sum $s$ is over all atomic species contained in the simulated system.
	The brackets ~~<math>~~\langle ,\rangle~~</math>~~ denote a thermal average which has to be computed over different MD trajectories and starting times ~~<math>~~\Delta T~~</math>~~ within each trajectory. The sum over ~~<math>~~i~~</math>~~ runs over the atoms within each species, and the sum ~~<math>~~s~~</math>~~ is over all atomic species contained in the simulated system.

	=== Step 4: Compute the power spectrum for each normalized velocity autocorrelation function ===		=== Step 4: Compute the power spectrum for each normalized velocity autocorrelation function ===

	The phonon spectral function is the power spectrum of ~~<math>~~f_{s}(t)~~</math>~~ and is obtained by performing the following Fourier transformation:		The phonon spectral function is the power spectrum of $f_{s}(t)$ and is obtained by performing the following Fourier transformation:
			\begin{equation}
	~~<math>~~
	g(\omega)=\sum_{s=1}^{types}g_{s}(\omega)=\left\| \sum_{s=1}^{types}\int_{-\infty}^{\infty}f_{s}(t)e^{-i\omega t}\right\|^{2}.		g(\omega)=\sum_{s=1}^{types}g_{s}(\omega)=\left\| \sum_{s=1}^{types}\int_{-\infty}^{\infty}f_{s}(t)e^{-i\omega t}\right\|^{2}.
	~~</math>~~		\end{equation}

	=== Step 5: Compute averages and check for convergence ===		=== Step 5: Compute averages and check for convergence ===

	To check for convergence, ~~<math>~~f(t)~~</math>~~ and ~~<math>~~g(\omega)~~</math>~~ obtained for each [[NVE ensemble\|NVE trajectory]] can be successively averaged. To this end, plot a single trajectory, compared to an average over 2 trajectories, and so on. If needed, the above steps can be repeated to generate additional data to reach the desired accuracy.		To check for convergence, $f(t)$ and $g(\omega)$ obtained for each [[NVE ensemble\|NVE trajectory]] can be successively averaged. To this end, plot a single trajectory, compared to an average over 2 trajectories, and so on. If needed, the above steps can be repeated to generate additional data to reach the desired accuracy.
	{{NB\|tip\|For further information on phonon signal analysis Ref{{cite\|lahnsteiner:prb:2022}} might be a helpful source.}}		{{NB\|tip\|For further information on phonon signal analysis Ref{{cite\|lahnsteiner:prb:2022}} might be a helpful source.}}

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	cp CONTCAR POSCAR		cp CONTCAR POSCAR
	done		done
	This bash script will create {{FILE\|POSCAR}}_i where ~~<math>~~i~~</math>~~ runs from 1 to 10. These serve as initial structures including inital velocities for the [[NVE ensemble\|NVE simulations]].		This bash script will create {{FILE\|POSCAR}}_i where $i$ runs from 1 to 10. These serve as initial structures including inital velocities for the [[NVE ensemble\|NVE simulations]].

	Secondly, sample velocities from [[NVE ensemble\|NVE simulations]] for each initial structure. An {{FILE\|INCAR}} file can look as follows:		Secondly, sample velocities from [[NVE ensemble\|NVE simulations]] for each initial structure. An {{FILE\|INCAR}} file can look as follows:
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	The '''PhononDOS.py''' script can be used to compute the phonon spectral function for a given [[NVE ensemble\|NVE simulation]] folder containing an {{FILE\|vaspout.h5}} file created with the aforementioned {{FILE\|INCAR}} file. The script will create a file called <code>total_ps.dat</code> containing the total phonon spectral function. The partial phonon spectra of the atomic species are written to files <code>ElementKey_ps.dat</code>. As input, the script needs a folder name containing a {{FILE\|vaspout.h5}} file, and the second input argument has to be the simulation time step of your simulation in fs. The written files will contain the frequency in <code>THz</code> as the first column. The second column will contain the phonon spectra computed as the power spectrum of the velocity autocorrelation function.		The '''PhononDOS.py''' script can be used to compute the phonon spectral function for a given [[NVE ensemble\|NVE simulation]] folder containing an {{FILE\|vaspout.h5}} file created with the aforementioned {{FILE\|INCAR}} file. The script will create a file called <code>total_ps.dat</code> containing the total phonon spectral function. The partial phonon spectra of the atomic species are written to files <code>ElementKey_ps.dat</code>. As input, the script needs a folder name containing a {{FILE\|vaspout.h5}} file, and the second input argument has to be the simulation time step of your simulation in fs. The written files will contain the frequency in <code>THz</code> as the first column. The second column will contain the phonon spectra computed as the power spectrum of the velocity autocorrelation function.

	=== Anharmonic ratteling in CsPbBr~~<sub>~~3~~</sub>~~ ===		=== Anharmonic ratteling in CsPbBr$_{3}$ ===
	[[File:CsPbBr3PhononDOS.png\|140px\|thumb\|Fig. 2: Snapshot of a ~~<math>~~2 \times 2 \times 2~~</math>~~ CsPbBr~~<sub>~~3~~</sub>~~ simulation box at 500K as used in the simulations for the convergence analysis.]]		[[File:CsPbBr3PhononDOS.png\|140px\|thumb\|Fig. 2: Snapshot of a $2 \times 2 \times 2$ CsPbBr$_{3}$ simulation box at 500K as used in the simulations for the convergence analysis.]]
	In the following the convergence of the phonon DOS will be exemplified on the CsPbBr~~<sub>~~3~~</sub>~~ in the cubic phase at 500K. A snapshot of the used simulation box is shown in Fig. 2. The CsPbBr~~<sub>~~3~~</sub>~~ consists of a cubic lead bromide framework which is covalently bonded. The cavities formed by the cubic lead-bromide framework are filled with weakly bonded Cs~~<sup>~~+~~</sup>~~ cations. This makes the CsPbBr~~<sub>~~3~~</sub>~~ a good example to test methods for anharmonic phonons.		In the following the convergence of the phonon DOS will be exemplified on the CsPbBr$_{3}$ in the cubic phase at 500K. A snapshot of the used simulation box is shown in Fig. 2. The CsPbBr$_{3}$ consists of a cubic lead bromide framework which is covalently bonded. The cavities formed by the cubic lead-bromide framework are filled with weakly bonded Cs$^{+}$ cations. This makes the CsPbBr$_{3}$ a good example to test methods for anharmonic phonons.

	The convergence of the phonon spectral function of ~~CsPbBr<sub>~~3~~</sub>~~ is visualized in a single plot shown in Fig. 1. The yellow line shows an average over a single trajectory. The more red the lines are, the more trajectories have been used for computing the average. The dark red line shows the average computed over all 10 trajectories. Based on the plot in Fig. 1, it is possible to conclude that enough data was obtained to properly converge the phonon spectral function.		The convergence of the phonon spectral function of CsPbrBr$_{3}$ is visualized in a single plot shown in Fig. 1. The yellow line shows an average over a single trajectory. The more red the lines are, the more trajectories have been used for computing the average. The dark red line shows the average computed over all 10 trajectories. Based on the plot in Fig. 1, it is possible to conclude that enough data was obtained to properly converge the phonon spectral function.

	Fig. 3 shows the atom-resolved normalized autocorrelations and phonon spectra.		Fig. 3 shows the atom-resolved normalized autocorrelations and phonon spectra.
	A peak in the Cs~~<sup>~~+~~</sup>~~ cation phonon spectral function is visible around 1THz. This peak can be assigned to Cs~~<sup>~~+~~</sup>~~ rattling frequencies coupling to optical phonon modes formed by the oscillations of the lead bromide framework.		A peak in the Cs$^{+}$ cation phonon spectral function is visible around 1THz. This peak can be assigned to Cs$^{+}$ rattling frequencies coupling to optical phonon modes formed by the oscillations of the lead bromide framework.

	For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs~~<sup>~~+~~</sup>~~ rattling modes were tuned to adjust the thermal conductivity of the material.		For further information it is advised to take a look at Ref{{cite\|lahnsteiner:prb:2022}} or Ref{{cite\|lahnsteiner:jpcc:2024}} in which Cs$^{+}$ rattling modes were tuned to adjust the thermal conductivity of the material.

	[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr~~<sub>~~3~~</sub>~~ at 500K. '''Right:''' Atom-resolved phonon DOS for CsPbBr~~<sub>~~3~~</sub>~~ at 500K.]]		[[File:AtomReslovedPhononDOS.png\|500px\|thumb\|center\|Fig. 3: '''Left:''' Shows atom-resolved normalized velocity autocorrelation function for CsPbBr$_{3}$ at 500K. '''Right:''' Atom-resolved phonon DOS for CsPbBr$_{3}$ at 500K.]]