Thermodynamic integration with harmonic reference: Difference between revisions
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# integrate <math>\langle V_1 -V_{0,\mathbf{x}} \rangle</math> over the <math>\lambda </math> grid and compute <math>\Delta A_{0,\mathbf{x}\rightarrow 1}</math> | # integrate <math>\langle V_1 -V_{0,\mathbf{x}} \rangle</math> over the <math>\lambda </math> grid and compute <math>\Delta A_{0,\mathbf{x}\rightarrow 1}</math> | ||
Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve position of the center of mass and is therefore unsuitable for the use in conventional TI). Furthermore, the simulations with <math>\lambda \rightarrow /math> 0 | Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because <math>V_{0,\mathbf{x}}(\mathbf{x})</math> is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve position of the center of mass and is therefore unsuitable for the use in conventional TI). Furthermore, the simulations with <math>\lambda \rightarrow </math> 0 | ||
Revision as of 07:36, 2 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[1] (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,\mathbf{x}\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_{0,\mathbf{x}} \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,\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
- compute 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} in vibrational analysis
- use the data obtained in the point 2 to 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 \underline{\mathbf{H}}^\mathbf{x}} that defines the harmonic forcefield
- perform NVT MD simulations for several values of 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 \in \langle0,1\rangle} and 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 \langle V_1 -V_{0,\mathbf{x}} \rangle}
- integrate 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 V_1 -V_{0,\mathbf{x}} \rangle} over 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 \lambda } grid and compute 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}}
Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because 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})} is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve position of the center of mass and is therefore unsuitable for the use in conventional TI). Furthermore, the simulations 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 \lambda \rightarrow } 0