Water molecule bond length: Difference between revisions
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=== Potential Function === | === Potential Function === | ||
[https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard--Jones potential] with parameters for [https://en.wikipedia.org/wiki/Water_model TIPS] model: | [https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard--Jones potential] with dimensionless parameters for [https://en.wikipedia.org/wiki/Water_model TIPS] model: | ||
<math> | <math> | ||
V(x) = 4 | V(x) = 4 \left[ \left( \frac1x\right)^{12} - \left( \frac1 x\right)^6 \right] | ||
</math> | </math> | ||
where the distance <math>x</math> is given in Ångstroms. | |||
where <math> | |||
=== Integration === | === Integration === | ||
Revision as of 22:00, 12 October 2020
Introduction
Classical Mechanics
Newton Equations
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 F = - \nabla V} . The system is 1D, thus gradient will be differential, 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 F= - \frac{d V}{dx}} .
Potential Function
Lennard--Jones potential with dimensionless 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 V(x) = 4 \left[ \left( \frac1x\right)^{12} - \left( \frac1 x\right)^6 \right] }
where the distance 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 x} is given in Ångstroms.
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{\frac{d}{dx}(V(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} }
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
