Water molecule bond length: Difference between revisions
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=== Newton Equations === | === Newton Equations === | ||
<math>F = ma</math> where <math>F = - \nabla V</math>. | |||
=== Integration === | === Integration === | ||
The finite differences (Euler method) are | |||
<math> | |||
\begin{align*} | |||
v(t) &= \frac{x(t+\Delta t) - x(t)}{\Delta t} \\ | |||
a(t) &= \frac{v(t+\Delta t) - v(t)}{\Delta t} | |||
\end{align*} | |||
</math> | |||
and | |||
<math> | |||
a(t) = \frac{F(x(t))}{m} | |||
</math> | |||
==== Velocity Verlet Algorithm ==== | |||
A very good and easy to implement integration method is velocity Verlet: | |||
<math> | |||
\begin{align*} | |||
x(t + \Delta t) &= x(t) + v(t) \Delta t + \frac12 a \Delta t^2 \\ | |||
v(t + \Delta t) &= v(t) + \frac12\left( a(t) + a(t+\Delta t) \right) \Delta t | |||
\end{align*} | |||
</math> | |||
=== Potential Function === | === Potential Function === | ||
Revision as of 21:34, 12 October 2020
Introduction
Classical Mechanics
Newton Equations
where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=-\nabla V} .
Integration
The finite differences (Euler method) are
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align*} v(t) &= \frac{x(t+\Delta t) - x(t)}{\Delta t} \\ a(t) &= \frac{v(t+\Delta t) - v(t)}{\Delta t} \end{align*} } and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t) = \frac{F(x(t))}{m} }
Velocity Verlet Algorithm
A very good and easy to implement integration method is velocity Verlet:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align*} x(t + \Delta t) &= x(t) + v(t) \Delta t + \frac12 a \Delta t^2 \\ v(t + \Delta t) &= v(t) + \frac12\left( a(t) + a(t+\Delta t) \right) \Delta t \end{align*} }
Potential Function
Lennard--Jones potential with parameters for TIPS model:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align*} A &= 580.0 \times 10^3 kcal A^{12}/mol \\ B &= 525.0 kcal A^6/mol \end{align*} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is Ångströms.
Temperature/ Initial distribution
The initial velocity of the hydrogen atom is chosen randomly from the Maxwell-Boltzmann distribution at given temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} \exp\left[- \frac12 \frac{mv^2}{k_B T} \right] }
Results
Issues
1D statement
