Thermodynamic integration with harmonic reference: Difference between revisions
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A_{1} = A_{0} + \Delta A_{0\rightarrow 1} | A_{1} = A_{0} + \Delta A_{0\rightarrow 1} | ||
</math> | </math> | ||
where <math>\Delta A_{0\rightarrow 1}</math> is anharmonic free energy. The latter term can be determined by means of thermodynamic integration | where <math>\Delta A_{0\rightarrow 1}</math> is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI) | ||
:<math> | :<math> | ||
\Delta A_{0\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_0 \rangle_\lambda | \Delta A_{0\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_0 \rangle_\lambda | ||
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with <math>V_i</math> being the potential energy of system <math>i</math>, <math>\lambda</math> is a coupling constant and <math>\langle\cdots\rangle_\lambda</math> is the NVT ensemble average of the system driven by the Hamiltonian | with <math>V_i</math> being the potential energy of system <math>i</math>, <math>\lambda</math> is a coupling constant and <math>\langle\cdots\rangle_\lambda</math> is the NVT ensemble average of the system driven by the Hamiltonian | ||
:<math> | :<math> | ||
\mathcal{H}_\lambda = \lambda \mathcal{H}_1 + (1-\lambda)\mathcal{H}_0 | \mathcal{H}_\lambda = \lambda \mathcal{H}_1 + (1-\lambda)\mathcal{H}_0 | ||
</math> | </math> | ||
Free energy of harmonic reference system within the quasi-classical theory writes | |||
:<math> | |||
A_{0,\vct{x}} = A_\mathrm{el}(\vct{x}_0) - k_\mathrm{B} T \sum_{i = 1}^{N_\mathrm{vib}} \ln \frac{k_\mathrm{B} T}{\hbar \omega_i} | |||
</math> | |||
with the electronic free energy <math>A_\mathrm{el}(\vct{x}_0)</math> for the | |||
configuration corresponding to the potential energy minimum with the | |||
atomic position vector <math>\vct{x}_0</math>, | |||
the number of vibrational degrees of freedom <math>N_\mathrm{vib}</math>, and the angular frequency $\omega_i$ of vibrational mode <math>i</math>. | |||
Revision as of 07:51, 1 November 2023
The Helmholtz free energy (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/":): A ) of a fully interacting system (1) can be expressed in terms of that of harmonic system (0) as follows
- 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/":): {\displaystyle A_{1} = A_{0} + \Delta A_{0\rightarrow 1} }
where 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/":): {\displaystyle \Delta A_{0\rightarrow 1}} is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)
- 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/":): {\displaystyle \Delta A_{0\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_0 \rangle_\lambda }
with 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/":): {\displaystyle V_i} being the potential energy of system 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/":): i , 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/":): {\displaystyle \lambda } is a coupling constant 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/":): {\displaystyle \langle\cdots\rangle_\lambda} is the NVT ensemble average of the system driven by the Hamiltonian
- 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/":): {\displaystyle \mathcal{H}_\lambda = \lambda \mathcal{H}_1 + (1-\lambda)\mathcal{H}_0 }
Free energy of harmonic reference system within the quasi-classical theory writes
- 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/":): {\displaystyle A_{0,\vct{x}} = A_\mathrm{el}(\vct{x}_0) - k_\mathrm{B} T \sum_{i = 1}^{N_\mathrm{vib}} \ln \frac{k_\mathrm{B} T}{\hbar \omega_i} }
with the electronic free energy 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/":): {\displaystyle A_\mathrm{el}(\vct{x}_0)} for the configuration corresponding to the potential energy minimum with the atomic position vector 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/":): {\displaystyle \vct{x}_0} , the number of vibrational degrees of freedom 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/":): {\displaystyle N_\mathrm{vib}} , and the angular frequency $\omega_i$ of vibrational mode 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/":): i .