Thermodynamic integration with harmonic reference: Difference between revisions
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atomic position vector <math>\mathbf{x}_0</math>, | atomic position vector <math>\mathbf{x}_0</math>, | ||
the number of vibrational degrees of freedom <math>N_\mathrm{vib}</math>, and the angular frequency <math>\omega_i</math> of vibrational mode <math>i</math> obtained using the Hesse matrix <math>\underline{\mathbf{H}}^\mathbf{x}</math>. | the number of vibrational degrees of freedom <math>N_\mathrm{vib}</math>, and the angular frequency <math>\omega_i</math> of vibrational mode <math>i</math> obtained using the Hesse matrix <math>\underline{\mathbf{H}}^\mathbf{x}</math>. | ||
Finally, the harmonic potential energy is expressed as | |||
:<math> | :<math> | ||
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) | 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) | ||
</math> | </math> | ||
Thus, a conventional TI calculation consists of the following steps: | |||
# determine <math>\mathbf{x}_0</math> and <math>V_{0,\mathbf{x}}(\mathbf{x}_0)</math> in structural relaxation |
Revision as of 08:04, 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,\mathbf{x}} + \Delta A_{0,\mathbf{x}\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,\mathbf{x}\rightarrow 1}} is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)
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,\mathbf{x}} }
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 obtained using the Hesse matrix 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 \underline{\mathbf{H}}^\mathbf{x}} . Finally, the harmonic potential energy is expressed 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/":): {\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) }
Thus, a conventional TI calculation consists of the following steps:
- determine 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} 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 V_{0,\mathbf{x}}(\mathbf{x}_0)} in structural relaxation