Question

(a) Based on a typical bond energy of \(150 \mathrm{k}_{\mathrm{B}} T\) and a typical bond length of \(0.15 \mathrm{nm}\), use dimensional analysis to estimate the frequency of vibration of covalent bonds. (b) Assume that the Lennard-Jones potential given by \\[ V(r)=\frac{a}{r^{12}}-\frac{b}{r^{6}} \\] describes a covalent bond (though real covalent bonds are more appropriately described by alternatives such as the Morse potential that are not as convenient analytically). Using the typical bond energy as the depth of the potential and the typical bond length as its equilibrium position, find the parameters \(a\) and \(b\), Do a Taylor expansion around this equilibrium position to determine the effective spring constant and the resulting typical frequency of vibration. (c) Based on your results from (a) and (b), estimate the time step required to do a classical mechanical simulation of protein dynamics.

Short answer

The frequency of vibration derived from dimensional analysis is \(\frac{150 k_B T}{h}\). The parameters \(a\) and \(b\) for the Lennard-Jones potential can be found by setting the derivative equal to zero and using the given bond length and energy. From the Taylor expansion around equilibrium, the effective spring constant is found as \(k=V''(r_0)=72a/r_0^{14}-42b/r_0^{8}\) and the frequency of vibration as \(f=\frac{1}{2\pi}\sqrt{k/m}\). A suitable time step for classical mechanical simulations of protein dynamics can be derived from this frequency.

Step-by-step solution

Dimensional Analysis

Assuming the typical energy associated with a covalent bond to be \(150 k_B T\), where \(k_B\) is the Boltzmann constant and \(T\) is the temperature, and the typical bond length to be \(0.15 nm\). The frequency \(\nu\) of vibration can be estimated as the energy divided by Planck's constant (\(h\)), which gives \(\nu =\frac{150 k_B T}{h}\). This is a very basic estimation and does not take into account factors such as the specific type of bond or environmental conditions.

Lennard-Jones Potential

Assuming the Lennard-Jones (L-J) potential describes a covalent bond (although more appropriate descriptions exist), which is represented by \(V(r)=\frac{a}{r^{12}}-\frac{b}{r^{6}}\). Here, \(a\) and \(b\) are the L-J parameters. By setting the first derivative of \(V(r)\) with respect to \(r\) equal to zero, we find the equilibrium position of the potential \(r_0\), which is given by \(r_0=\left(\frac{2a}{b}\right)^{1/6}\). The depth \(V_{min}\) of the potential is the value of \(V(r)\) at this minimum, which is \(-\frac{b^2}{4a}\). From these two equations, and given the typical bond energy as the depth of the potential and the typical bond length as its equilibrium position, we can solve for \(a\) and \(b\).

Taylor Expansion

Carrying out a Taylor series expansion of the L-J potential around the equilibrium position, we find the effective spring constant \(k=V''(r_0)=72a/r_0^{14}-42b/r_0^{8}\). The frequency of vibration can be approximated as \(f=\frac{1}{2\pi}\sqrt{k/m}\), where \(m\) is the reduced mass of the two particles.

Time Step Estimation

The time step required to accurately simulate the dynamics of a protein can be determined from the highest frequency of vibration in the system. This is typically given by the smallest value of \(\Delta t\), such that \(f \Delta t << 1\). This ensures that the simulation does not miss any significant dynamics.

Key concepts

Dimensional Analysis

Dimensional analysis is a fundamental tool in physics and engineering that involves the study of dimensions of physical quantities. It's used to check the plausibility of derived equations and computations by considering the units involved. When it comes to estimating the frequency of vibration of covalent bonds, dimensional analysis comes in handy. Let's take the example from the exercise where the typical bond energy is given as \(150 k_B T\). The frequency (\(u\)) can be estimated using the relationship that involves energy (E) and frequency (\(u = E/h\)), where \(h\) is the Planck constant. Since energy has dimensions of mass \times length²/time² and Planck's constant has dimensions of mass \times length²/time, their ratio simplifies to 1/time, which coincides with the dimensions of frequency. Thus, from the viewpoint of dimensional analysis, the formula \(u = \frac{150 k_B T}{h}\) is dimensionally consistent for computing the frequency of vibration.

Using dimensional analysis helps to ensure that the calculations you're making are based on sound principles, even if other factors such as environmental conditions or bond types are not accounted for. For students who are new to physical concepts, this can be a very valuable way to check their work and gain a better intuitive understanding of the relationships between different physical quantities.

Lennard-Jones Potential

The Lennard-Jones potential is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules depending on their distance. This potential function is key to understanding the behavior of atoms in condensed matter physics. When applied to the covalent bond, which the exercise suggests despite its limitations, the potential \(V(r)\) is given by the equation \(V(r) = \frac{a}{r^{12}} - \frac{b}{r^{6}}\). The \(r^{12}\) term describes the repulsion at short ranges due to overlapping electron orbitals, and the \(r^{6}\) term represents the attraction at longer ranges as a result of dipole-dipole interactions.

Even though the Lennard-Jones potential is an approximation and real covalent bonds are better represented by more complicated potentials, it is still helpful for students to understand the typical behaviors of molecular interactions. It simplifies the computational process and allows for the analysis of atomic and molecular systems with reasonable accuracy, especially when exploring fundamental concepts and performing classical mechanical simulations.

Taylor Expansion

Taylor expansion is a powerful mathematical tool used to approximate functions that are not easily solvable. In the context of the Lennard-Jones potential, the Taylor expansion is used to describe how the potential energy changes when atoms are displaced slightly from their equilibrium position. By expanding the L-J potential around this point, scientists can extract useful information, like the effective spring constant in the system which is a measure of the bond's stiffness. Mathematically, the effective spring constant \(k\) is calculated by differentiating the potential twice with respect to the distance \(r\) and evaluating it at the equilibrium position \(r_0\), denoted as \(V''(r_0)\).

The result from a Taylor expansion gives a quadratic term that can be used to approximate the potential near the equilibrium, leading to harmonic motion. For students, understanding this application of Taylor expansion is crucial, as it translates a complex potential energy function into a simple form that describes how the bond responds to small displacements—a cornerstone in molecular dynamics simulations.

Classical Mechanical Simulation

Classical mechanical simulation is a computational method used to predict the motion and interaction of particles using classical mechanics principles. In the context of molecular dynamics, it helps scientists and engineers understand and predict the behavior of proteins and other complex molecular systems. The key is to choose an appropriate time step, which is informed by the highest vibration frequency present in the system. This time step must be small enough to capture all relevant motions within the protein structure without being so small as to make the simulation computationally unfeasible.

Students should grasp that the proper time step ensures accuracy in the simulation: it allows for capturing the natural movement of atoms and molecules without 'skipping' over periods of significant dynamic change. The approximation \(abla t << 1/f\), where \(abla t\) is the time step and \(f\) the frequency of vibration, is a rule of thumb to ensure that the simulation will reflect the real-world physical properties as closely as possible. When engaging in classical mechanical simulations, this aspect of computational modeling is as critical as the physical model itself.