Thermodynamic integration with harmonic reference: Difference between revisions

Revision as of 07:57, 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 system harmonic in Cartesian coordinates (0,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 \mathbf{x}} ) 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,<math>\mathbf{x}} } + \Delta A_{0,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 \mathbf{x}} \rightarrow 1}

</math> 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,<math>\mathbf{x}} \rightarrow 1}</math> 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,<math>\mathbf{x}} \rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_{0,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 \mathbf{x}} } \rangle_\lambda

</math> 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,<math>\mathbf{x}} }

</math>

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,\mathbf{x}} = A_\mathrm{el}(\mathbf{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}(\mathbf{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 \mathbf{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 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 \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 . The

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_{0,\mathbf{x}}(\mathbf{x}) = V_{0,\mathbf{x}}(\mathbf{x}_0) + \frac{1}{2} (\mathbf{x} - \mathbf{x}_0)^T \underline{\mathbf{H}}^\mathbf{x} (\mathbf{x} - \mathbf{x}_0) }

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 The Helmholtz free energy (<math>A</math>) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,<math>\mathbf{x}</math>) as follows
 :<math>
-A_{1} = A_{0} + \Delta A_{0\rightarrow 1}
+A_{1} = A_{0,<math>\mathbf{x}</math>} + \Delta A_{0,<math>\mathbf{x}</math>\rightarrow 1}
 </math>
-where <math>\Delta A_{0\rightarrow 1}</math> is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)
+where <math>\Delta A_{0,<math>\mathbf{x}</math>\rightarrow 1}</math> is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)
 :<math>
-\Delta A_{0\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_0  \rangle_\lambda
+\Delta A_{0,<math>\mathbf{x}</math>\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_{0,<math>\mathbf{x}</math>}  \rangle_\lambda
 </math>
 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>
-\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>\mathbf{x}</math>}
 </math>