Biased molecular dynamics

The Hamiltonian of the physical 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/":): H(q,p)=T(p)+V(q),\;

with T(p), and V(q) being kinetic, and potential energies, respectively, can be extended by adding a bias potential ${\tilde {V}}(\xi )$ acting only on one or more selected internal coordinates of the system ξ=ξ(q):

${\tilde {H}}(q,p)=H(q,p)+{\tilde {V}}(\xi ).$

Presently, the following types of ${\tilde {V}}(\xi )$ are supported:

• Gauss function

${\tilde {V}}(\xi )=h\,{\text{exp}}\left[-{\frac {(\xi (q)-\xi _{0})^{2}}{2w^{2}}}\right],\;$

• harmonic potential

${\tilde {V}}(\xi )={\frac {1}{2}}\kappa (\xi (q)-\xi _{0})^{2}\;$

• Fermi function

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 \tilde{V}(\xi)= \frac{A}{1+\text{exp}\left [-D\frac{\xi(q)}{\xi_0} -1 \right ]} \; }

The probability density for a geometric parameter ξ of the system driven by a 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/":): H(q,p)=T(p)+V(q),\;

with T(p), and V(q) being kinetic, and potential energies, respectively, can be written 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/":): P(\xi _{i})={\frac {\int \delta {\Big (}\xi (q)-\xi _{i}{\Big )}\exp \left\{-H(q,p)/k_{B}\,T\right\}dq\,dp}{\int \exp \left\{-H(q,p)/k_{B}\,T\right\}dq\,dp}}=\langle \delta {\Big (}\xi (q)-\xi _{i}{\Big )}\rangle _{{H}}.

The term 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/":): \langle X\rangle _{H} stands for a thermal average of quantity X evaluated for the system driven by the Hamiltonian H.

If the system is modified by adding a bias potential 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/":): {\tilde {V}}(\xi ) acting only on a selected internal parameter of the system ξ=ξ(q), the Hamiltonian takes a form:

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/":): {\tilde {H}}(q,p)=H(q,p)+{\tilde {V}}(\xi ),

and the probability density of ξ in the biased ensemble is:

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/":): {\tilde {P}}(\xi _{i})={\frac {\int \delta {\Big (}\xi (q)-\xi _{i}{\Big )}\exp \left\{-{\tilde {H}}(q,p)/k_{B}\,T\right\}dq\,dp}{\int \exp \left\{-{\tilde {H}}(q,p)/k_{B}\,T\right\}dq\,dp}}=\langle \delta {\Big (}\xi (q)-\xi _{i}{\Big )}\rangle _{{{\tilde {H}}}}

It can be shown that the biased and unbiased averages are related via a simple formula:

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/":): P(\xi _{i})={\tilde {P}}(\xi _{i}){\frac {\exp \left\{{\tilde {V}}(\xi )/k_{B}\,T\right\}}{\langle \exp \left\{{\tilde {V}}(\xi )/k_{B}\,T\right\}\rangle _{{{\tilde {H}}}}}}.

More generally, an observable 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/":): \langle A\rangle _{{H}} :

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/":): \langle A\rangle _{{H}}={\frac {\int A(q)\exp \left\{-H(q,p)/k_{B}\,T\right\}dq\,dp}{\int \exp \left\{-H(q,p)/k_{B}\,T\right\}dq\,dp}}

can be expressed in terms of thermal averages within the biased ensemble:

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/":): \langle A\rangle _{{H}}={\frac {\langle A(q)\,\exp \left\{{\tilde {V}}(\xi )/k_{B}\,T\right\}\rangle _{{{\tilde {H}}}}}{\langle \exp \left\{{\tilde {V}}(\xi )/k_{B}\,T\right\}\rangle _{{{\tilde {H}}}}}}.

Simulation methods such as umbrella sampling use a bias potential to enhance sampling of ξ in regions with low P(ξ_i) such as transition regions of chemical reactions. The correct distributions are recovered afterwards using the equation for 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/":): \langle A\rangle _{{H}} above.

A more detailed description of the method can be found in Ref.. Biased molecular dynamics can be used also to introduce soft geometric constraints in which the controlled geometric parameter is not strictly constant, instead it oscillates in a narrow interval of values.

Supported types of bias potentials

Presently, the following types of bias potential are supported:

• Gauss function

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_{bias}(\xi) = h\,\text{exp}\left [-\frac{(\xi(q)-\xi_0)^2}{2w^2} \right ], \; }

• harmonic potential

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_{bias}(\xi) = \frac{1}{2}\kappa (\xi(q)-\xi_0)^2 \; }

• Fermi function

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_{bias}(\xi) = \frac{A}{1+\text{exp}\left [-D\frac{\xi(q)}{\xi_0} -1 \right ]} \; }

Andersen thermostat

• For a biased molecular dynamics run with Andersen thermostat, one has to:

1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=1 (MDALGO=11 in VASP 5.x), and choose an appropriate setting for ANDERSEN_PROB
3. In order to avoid updating of the bias potential, set HILLS_BIN=NSW
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. Define the bias potential in the PENALTYPOT-file

Nose-Hoover thermostat

• For a biased molecular dynamics run with Nose-Hoover thermostat, one has to:

1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=2 (MDALGO=21 in VASP 5.x), and choose an appropriate setting for SMASS
3. In order to avoid updating of the bias potential, set HILLS_BIN=NSW
4. Define collective variables in the ICONST-file, and set the STATUS parameter for the collective variables to 5
5. Define the bias potential in the PENALTYPOT-file

The values of all collective variables for each MD step are listed in the REPORT-file, check the lines after the string Metadynamics.

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